The Development of Year 3 Students’ Place-Value Understanding:

Representations and Concepts

Peter Stanley Price Dip.Teach., B.Ed., M.Ed., A.C.P.

Centre for Mathematics and Science Education School of Mathematics, Science and Technology Education

Faculty of Education Queensland University of Technology

A Thesis submitted in partial fulfilment of the requirements

for the award of the degree of Doctor of Philosophy

March, 2001

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Keywords

Place value, base-ten blocks, Year 3, mathematical understanding, place-value software, representations of number, conceptions of number, electronic base-ten blocks, conceptual structures for multidigit numbers, feedback, misconceptions of number, independent-place construct, face-value construct, mathematics teaching with technology, number models, Payne-Rathmell model for teaching number topics.

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Abstract

Understanding base-ten numbers is one of the most important mathematics

topics taught in the primary school, and yet also one of the most difficult to teach and

to learn. Research shows that many children have inaccurate or faulty number

conceptions, and use rote-learned procedures with little regard for quantities

represented by mathematical symbols. Base-ten blocks are widely used to teach

place-value concepts, but children often do not perceive the links between numbers,

symbols, and models. Software has also been suggested as a means of improving

children’s development of these links but there is little research on its efficacy.

Sixteen Queensland Year 3 students worked cooperatively with the researcher

for 10 daily sessions, in 4 groups of 4 students of either high or low mathematical

achievement level, on tasks introducing the hundreds place. Two groups used

physical base-ten blocks and two used place-value software incorporating electronic

base-ten blocks. Individual interviews assessed participants’ place-value

understanding before and after teaching sessions. Data sources were videotapes of

interviews and teaching sessions, field notes, workbooks, and software audit trails,

analysed using a grounded theory method.

There was little difference evident in learning by students using either

physical or electronic blocks. Many errors related to the “face-value” construct,

counting and handling errors, and a lack of knowledge of base-ten rules were

evident. Several students trusted the counting of blocks to reveal number

relationships. The study failed to confirm several reported schemes describing

children’s conceptual structures for multidigit numbers. Many participants

demonstrated a preference for grouping or counting approaches, but not stable

mental models characterising their thinking about numbers generally. The

independent-place construct is proposed to explain evidence in both the study and

the literature that shows students making single-dimensional associations between a

place, a set of number words, and a digit, rather than taking account of groups of 10.

Feedback received in the two conditions differed greatly. Electronic feedback was

more positive and accurate than feedback from blocks, and reduced the need for

human-based feedback. Primary teachers are urged to monitor students’ use of base-

ten blocks closely, and to challenge faulty number conceptions by asking appropriate

questions.

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Table of Contents Keywords .........................................................................................................................i Abstract......................................................................................................................... iii Table of Contents ...........................................................................................................v List of Tables .................................................................................................................ix List of Figures.................................................................................................................x Supplementary Material ...............................................................................................x Statement of Original Authorship ..............................................................................xi Acknowledgments ...................................................................................................... xiii

Chapter 1: The Problem........................................................................................ 1 1.1 Recommendations for Changes in Mathematics Education............................1 1.2 The Learning of Place-Value Concepts..............................................................3

1.2.1 Conceptual Structures and Difficulties With Place-Value Concepts ...................3 1.2.2 Use of Number Representations............................................................................3

1.3 The Research Question.........................................................................................5 1.4 Overview of Research Methodology...................................................................5 1.5 Significance of the study.......................................................................................6 1.6 Outline of the Thesis .............................................................................................7

Chapter 2: Review of Literature........................................................................... 9 2.1 Chapter Overview.................................................................................................9 2.2 Issues in Mathematics Education......................................................................10

2.2.1 Students’ Active Involvement in Mathematics Learning...................................10 2.2.2 Number Sense......................................................................................................13 2.2.3 Use of Technological Devices ............................................................................15

2.3 Place-value Understanding ................................................................................17 2.3.1 Place Value ..........................................................................................................18 2.3.2 Place-value Understanding..................................................................................20

2.4 The Contribution of Cognitive Science to Mathematics Education .............21 2.4.1 Understanding Mathematics................................................................................22 2.4.2 Mental Models.....................................................................................................23 2.4.3 Analogical Reasoning..........................................................................................37

2.5 Teaching Place-value Understanding ...............................................................43 2.5.1 Teaching Approaches ..........................................................................................43 2.5.2 Building Place-Value Connections .....................................................................45 2.5.3 Use of Concrete Materials...................................................................................51

2.6 Computers and Mathematics Education .........................................................55 2.6.1 Claimed Benefits of Computers ..........................................................................55 2.6.2 Cognitive Aspects of Computer Use...................................................................57

2.7 Chapter Summary; Statement of the Problem ...............................................59 Chapter 3: Methods ............................................................................................. 61

3.1 Chapter Overview...............................................................................................61 3.2 Aims of the Study................................................................................................61 3.3 Variables ..............................................................................................................62

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3.3.1 Mathematical Achievement Level...................................................................... 62 3.3.2 Number Representation Format ......................................................................... 63

3.4 Data collection and analysis. .............................................................................63 3.5 Design Issues........................................................................................................63

3.5.1 Assumptions........................................................................................................ 63 3.5.2 Theoretical and Methodological Stance............................................................. 64

3.6 Pilot Study ...........................................................................................................68 3.6.1 Purposes of the Pilot Study................................................................................. 68 3.6.2 Selection of Pilot Study Participants.................................................................. 69 3.6.3 Pilot Study Procedures........................................................................................ 70 3.6.4 Pilot Study Data Collection and Analysis.......................................................... 70 3.6.5 Changes Made to Study Design After Pilot Study............................................. 70

3.7 Main Study ..........................................................................................................75 3.7.1 Selection of Participants ..................................................................................... 76 3.7.2 Teaching Program............................................................................................... 77 3.7.3 Instruments - First and Second Interviews......................................................... 83 3.7.4 Administration Procedures ................................................................................. 85 3.7.5 Data Collection and Analysis ............................................................................. 87

3.8 Validity and Reliability ......................................................................................92 3.9 Limitations...........................................................................................................93 3.10 Chapter Summary ..............................................................................................94

Chapter 4: Results................................................................................................ 97 4.1 Chapter Overview...............................................................................................97

4.1.1 Restatement of the Research Question............................................................... 97 4.2 Transcript Conventions Used in this Thesis....................................................98 4.3 Place-Value Task Performance Revealed in Interview Results ....................99

4.3.1 Methods used to Analyse Interview Data .......................................................... 99 4.3.2 Overview of Interview Results......................................................................... 100

4.4 Students’ Conceptions of Numbers ................................................................107 4.4.1 Grouping Approaches....................................................................................... 107 4.4.2 Counting Approaches ....................................................................................... 115 4.4.3 Face-Value Interpretation of Symbols ............................................................. 123 4.4.4 Summary of Approaches to Interview Questions ............................................ 132 4.4.5 Changeability of Participants’ Number Conceptions ...................................... 134

4.5 Digit Correspondence Tasks: Four Categories of Response .......................136 4.5.1 Category I: Face-Value Interpretation of Digits .............................................. 137 4.5.2 Category II: No Referents For Individual Digits ............................................. 137 4.5.3 Category III: Correct Total Represented by Each Digit, but Tens not Explained ........................................................................................................................... 140 4.5.4 Category IV: Correct Number of Referents, Tens Place Mentioned............... 141 4.5.5 Summary of Responses to Digit Correspondence Tasks................................. 142

4.6 Errors, Misconceptions, and Limited Conceptions ......................................143 4.6.1 Counting Errors ................................................................................................ 143 4.6.2 Blocks Handling Errors .................................................................................... 145 4.6.3 Errors in Naming and Writing Symbols for Numbers..................................... 149 4.6.4 Errors in Applying Values to Blocks ............................................................... 152

4.7 Use of Materials to Represent Numbers ........................................................158

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4.7.1 Counting of Representational Materials ...........................................................158 4.7.2 Use of Trial-and-Error Methods........................................................................162 4.7.3 Handling Larger Numbers.................................................................................164 4.7.4 Interpreting Non-Canonical Block Arrangements............................................167 4.7.5 Face-value Interpretations of Symbols .............................................................169 4.7.6 Predictions About Trading ................................................................................172 4.7.7 Feedback............................................................................................................176 4.7.8 Using Blocks To Discover Number Relationships...........................................186

4.8 Chapter Summary ............................................................................................192 Chapter 5: Discussion ........................................................................................ 193

5.1 Chapter Overview.............................................................................................193 5.2 Participants’ Ideas About Multidigit Numbers ............................................193

5.2.1 Participants’ Preferences for Grouping or Counting Approaches....................195 5.2.2 Comparison of Grouping and Counting Approaches .......................................198 5.2.3 Difficulties With Existing Conceptual Structure Schemes ..............................204 5.2.4 Face-value Interpretations of Symbols .............................................................208

5.3 Independent-Place Construct ..........................................................................213 5.3.1 Description & Definition of the Independent-Place Construct ........................213 5.3.2 Comparison of the Independent-Place Construct and the Face-Value Construct ............................................................................................................................214 5.3.3 Evidence for the Independent-Place Construct in This Study..........................214 5.3.4 Evidence of the Independent-Place Construct in the Literature.......................217 5.3.5 Written Computation and the Independent-Place Construct ............................221 5.3.6 Place-Value Tasks and the Independent-Place Construct ................................222

5.4 Participants’ Construction of Meaning..........................................................223 5.4.1 ‘Organic’ Understanding...................................................................................224 5.4.2 Participants’ “Invented” Answers .....................................................................225 5.4.3 Teaching, Learning, and Constructing Meaning ..............................................227

5.5 Effects of Physical or Electronic Base-Ten Blocks on Place-Value Understanding............................................................................................................227

5.5.1 Differences in Learning of Participants Using Physical or Electronic Blocks 228 5.5.2 Sensory Impact of Physical or Electronic Blocks.............................................228 5.5.3 How Numbers Are Represented by Physical or Electronic Blocks .................230 5.5.4 Development of Links Among Blocks, Symbols, and Numbers .....................232 5.5.5 Support for the Development of Number Concepts .........................................234

5.6 Place-Value Understanding Demonstrated by High- and Low-Achievement-Level Participants ......................................................................................................235

5.6.1 Similarities in Place-Value Understanding of High- and Low-Achievement-Level Participants ............................................................................................................235 5.6.2 Differences in Place-Value Understanding of High- and Low-Achievement-Level Participants ............................................................................................................236

Chapter 6: Conclusions ..................................................................................... 239 6.1 Chapter Overview.............................................................................................239 6.2 Conclusions About Answers to Research Questions ....................................239

6.2.1 Conceptual Structures for Multidigit Numbers Evident in Participants’ Responses.........................................................................................................................239 6.2.2 Misconceptions, Errors, or Limited Conceptions Evident In Participants’ Responses.........................................................................................................................240

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6.2.3 Effects of the Two Materials on Students’ Learning of Place-Value Concepts ........................................................................................................................... 243 6.2.4 Differences Between Place-Value Understanding of High- and Low-Achievement-Level Participants..................................................................................... 248

6.3 Implications for Teaching................................................................................249 6.3.1 Implications of Using Physical Base-Ten Blocks to Teach Place-Value Concepts .......................................................................................................................... 249 6.3.2 Implications of Using Electronic Base-Ten Blocks to Teach Place-Value Concepts .......................................................................................................................... 253 6.3.3 Implications of the Independent-Place Construct for Teaching Mathematics 255 6.3.4 Implications of Construction of Meaning for Teaching Mathematics ............ 256

6.4 Recommendations for Further Research.......................................................258 Appendix A – Design of Software used in the Study.............................................261 Appendix B - Overview of Teaching Session Content for Interviews and Teaching Phase of Pilot Study..................................................................................277 Appendix C – Summary of Pilot Study Teaching Program .................................279 Appendix D - Excerpt of Teaching Script of Pilot Study: Session 1....................281 Appendix E – Audit Trail Example.........................................................................283 Appendix F – Results of The Year Two Diagnostic Net, Used to Select Participants for the Main Study ..............................................................................287 Appendix G – List of Participants ...........................................................................289 Appendix H - Main Study Teaching Program.......................................................291 Appendix I - Main Study Interview 1 Instrument.................................................299 Appendix J - Main Study Interview 2 Instrument ................................................301 Appendix K – Letter Requesting Consent by Parents or Guardians of Prospective Participants ...........................................................................................303 Appendix L – Coding Teaching Session Transcripts for Feedback ....................305 Appendix M – Descriptions of Numeration Skills Targeted by Interview Questions and Criteria for Their Assessment ........................................................311 Appendix N – Transcript of Interview 1 Question 6 (a) with Terry ...................315 Appendix O – Transcript of Interview 2 Question 6 (a) with Hayden................319 Appendix P – Transcript of Low/Blocks Group Answering Task 28 (a)............321 Appendix Q – Transcripts of Task 4 (a) from 4 groups........................................325 Appendix R – Transcript Excerpts Showing Participants Predicting Equivalence of Traded Blocks........................................................................................................355 Appendix S – Transcript of Task 4 (d) from Low/Blocks Group........................365 Appendix T – Comparison Between Ross’s (1989) Model and a Proposed Model for Categories of Responses to Digit Correspondence Tasks...............................369 Appendix U – Sample Coding of Transcript for Feedback..................................371 References...................................................................................................................373 Supplementary Material – Hi-Flyer Maths Installation Files [CD-ROM] .........385

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List of Tables TABLE 2.1. Aspects of Place-value Understanding Described in the Literature... 26 TABLE 2.2. Task Performance Illustrating Limited Conceptions in Place-value

Understanding..................................................................................... 31 TABLE 3.1. Phases of the Research Design ........................................................... 75 TABLE 3.2. Participant Groups for the Main Study............................................... 76 TABLE 4.1. Transcript Notations ........................................................................... 98 TABLE 4.2. Summary of Participants’ Numeration Skills Identified in two

Interviews ......................................................................................... 102 TABLE 4.3. Summary of Numeration Skills Demonstrated by Each Participant and

by Each Group.................................................................................. 103 TABLE 4.4. Summary of Place-value Understanding Criteria Achieved by High-

achievement-level and Low-Achievement-Level Participants......... 105 TABLE 4.5. Summary of Place-value Understanding Criteria Achieved by

Participants in Computer and Blocks Groups .................................. 106 TABLE 4.6. Use of Grouping Approaches for Selected Interview Questions...... 113 TABLE 4.7. Use of Grouping Approaches by Each Group.................................. 113 TABLE 4.8. Use of a Counting Approach for Selected Interview Questions....... 121 TABLE 4.9. Use of Counting Approaches by Each Group .................................. 122 TABLE 4.10. Incidence of Face-value Interpretations for Written Symbols after

Selected Interview Questions ........................................................... 130 TABLE 4.11. Use of Face-Value Interpretations of Symbols by Each Group ....... 131 TABLE 4.12. Incidence of Approaches Adopted for Selected Interview Questions....

......................................................................................................... 133 TABLE 4.13. Response Categories for Interview Digit Correspondence Questions ...

......................................................................................................... 142 TABLE 4.14. Summary of Digit Correspondence Response Categories................ 143 TABLE 4.15. Participants’ Written Responses to Task 27 (b) ............................... 171 TABLE 4.16. Incidents of Feedback of Each Source per Group ............................ 177 TABLE 4.17. Percentage of Feedback Compared With Answer Status ................. 180 TABLE 4.18. Quality of Feedback Provided for Correct or Incorrect Answers..... 181 TABLE 4.19. Percent of Feedback for Correct Answers from Each Source.......... 182 TABLE 4.20. Percent of Feedback for Incorrect Answers from Each Source ....... 183 TABLE 4.21. Feedback Providing Answers from Each Source for Each Group ... 187 TABLE 5.1. Comparison of Results of Digit Correspondence Tasks Between This

Study and Ross (1989)...................................................................... 209 TABLE H.1. Overview of Teaching Program Tasks ............................................. 284 TABLE L.1. Source of Feedback .......................................................................... 307 TABLE L.2. Effects of Feedback .......................................................................... 307 TABLE L.3. Responses to Feedback..................................................................... 308

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List of Figures Figure 2.1. The face value of each individual numerical symbol, together with its

position relative to the ones place, determines the value it represents. ............................................................................................................ 18

Figure 2.2. Relationships inherent in base-ten blocks. ......................................... 41 Figure 2.3. Relationships among numbers, written symbols, and concrete

materials.............................................................................................. 46 Figure 2.4. Conceptual gap between written symbols and concrete materials. .... 48 Figure 2.5. The use of transitional forms to bridge the gap between written

symbols and concrete materials. ......................................................... 49 Figure 3.1. Dimensions of research design. .......................................................... 65 Figure 3.2. Original graphic images used on regrouping buttons in software used

during pilot study................................................................................ 72 Figure 3.3. Replacement graphic images used on regrouping buttons in software

used during main study....................................................................... 72 Figure 3.4. Sample Representing numbers task. ................................................... 80 Figure 3.5. Sample Regrouping task. .................................................................... 81 Figure 3.6. Sample Use of numeral expander task. .............................................. 81 Figure 3.7. Sample Comparison task. ................................................................... 81 Figure 3.8. Sample Counting task. ........................................................................ 82 Figure 3.9. Sample Addition task, including regrouping. ..................................... 83 Figure 3.10. Diagram showing objects used in interviews for Digit Correspondence

Task with misleading perceptual cues. ............................................... 85 Figure 4.1. Interview scores compared to use of grouping approaches. ............. 115 Figure 4.2. Interview scores compared to use of counting approaches. ............. 122 Figure 4.3. Interview scores compared to use of face-value interpretations of

symbols. ............................................................................................ 132 Figure 4.4. Proportions of feedback from each source for each group. .............. 178 Figure 5.1. Column counters in software representation of 248. ........................ 232 Figure A.1. Screen view of on-screen tutorial question with block representations.

......................................................................................................... 262 Figure A.2. Partial screen image from Rutgers Math Construction Tools, showing

block and symbol representations of a number. ............................... 263 Figure A.3. Screen view of Blocks Microworld showing block representation of a

number, nominating a cube as one. .................................................. 264 Figure A.4. Main screen of Hi-Flyer Maths. ....................................................... 266 Figure A.5. “Show as tens” feature activated. ..................................................... 268 Figure A.6. Number name window and numeral expander displayed................. 269 Figure A.7. A block is “sawn” into 10 pieces...................................................... 271 Figure A.8. “Add blocks” requester..................................................................... 272 Figure L.1. Data entry screen for feedback database. ......................................... 306

Supplementary Material Hi-Flyer Maths Installation Files [CD-ROM] ......................................................... 385

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Statement of Original Authorship

The work contained in this thesis has not been previously submitted for a degree or diploma at any other higher education institution. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made.

Signed: ________________________________

Date: ________________________________

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Acknowledgments

The completion of a thesis is a drawn-out, sometimes painful task that cannot be done without much assistance, both professional and personal, from many others. I gratefully acknowledge my indebtedness to the following people for their support over the past six years:

To my principal supervisor, Professor Lyn English, I offer my heartfelt appreciation for her patience, wisdom and unfailing support since I started this journey. Your example to me, Lyn, as an academic and colleague has always been of the highest standard, and I greatly appreciate your patience in leading me to the completion of the thesis. Thank you for believing in me and for giving me the space to finish.

To my associate supervisor, Dr Bill Atweh, I thank you also for your patience, support, and wisdom. Your ability to see past the data to what they reveal has been invaluable in helping me frame the last few chapters and in structuring what was quite a mess and turn it into a coherent account.

To my dear wife and partner, Trish, I can only say that a lesser person would have given up long ago. I deeply appreciate your love and support over what has ended up as a longer time than we could have imagined when I started. This has truly been a partnership, in which you have sacrificed your desires and your time to give me space to study, since 1993. Thank you from the bottom of my heart.

To my lovely, wonderful children, Mary, Andrew and Hannah, I express my deep love and devotion. You too have had to give up time with me, and to put up with your Dad’s frequent absences over a substantial part of your lives. I am immensely proud of each of you, and I look forward to seeing you grow and develop into the adults God intends.

To my parents, Rev and Mrs Stanley and Eva Price, I express my love and heartfelt thanks for everything you put into raising me. Though we are separated by great distance, I am aware of your constant support and prayers that you have provided all my life. Thanks, Dad and Mum.

To my colleagues and friends at Christian Heritage College, I express my heartfelt thanks and love for accepting me and supporting me in this endeavour. In particular, Dr Robert Herschell has been a constant friend, mentor and source of support over many years. Thanks, Rob, for believing in me, for giving me the chance to follow God’s call to teach others.

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To many colleagues, mentors and friends at the School of Mathematics, Science and Technology Education, QUT, thank you. I have had a very rewarding time at QUT over many years, and appreciate your input into my life and career, including the writing of this thesis. In particular, a sincere “thank you” to Professor Tom Cooper, A/Prof Cam McRobbie and Drs Cal Irons, Ian Ginns, Rod Nason and Jackie Stokes for your wise advice and counsel. And to my fellow PhDers over the past several years—Drs Neil Taylor, Carmel Diezmann, Kathy Charles, Mary Hanrahan, David Anderson, Stephen Norton, Anne Williams and Gillian Kidman—thank you all for your friendship and support.

Finally, but by no means least, I express my love and appreciation to the Lord Jesus Christ, without whom I could do nothing. My abilities and talents are from Him alone; my prayer is that I walk worthy of the calling He has placed on my life, as a faithful witness to His love and power.

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Chapter 1: The Problem The development of a competent understanding of place-value concepts by

primary students is a prerequisite for the learning of much later content of the school

mathematics curriculum. Children need to learn from the early primary school

grades1 how numbers are written in the base-ten numeration system, and to construct

accurate mental models for numbers, in order to develop a proficiency with

mathematics that will equip them to solve problems in later life. However, several

authors have noted that place-value concepts are difficult both for teachers to teach

and for students to learn (G. A. Jones & Thornton, 1993a; S. H. Ross, 1990). The

study described in this thesis investigated the teaching and learning of place-value

concepts using number representations in two formats: conventional base-ten blocks

and a computer software application.

1.1 Recommendations for Changes in Mathematics Education Several documents published over the past 20 years have recommended

important changes in the way mathematics is taught in schools. These documents

include Mathematics counts (Cockcroft, 1982), Everybody Counts (National

Research Council [NRC], 1989), Curriculum and Evaluation Standards for School

Mathematics (National Council of Teachers of Mathematics [NCTM], 1989), A

National Statement on Mathematics for Australian Schools (Australian Education

Council, 1990), and Principles and Standards for School Mathematics (NCTM,

2000). Three prominent topics in these documents are relevant to this study: (a) the

development of mathematical understanding, (b) the development of number sense,

and (c) the use of technology in mathematics classes.

The first recommendation for mathematics education identified as relevant to

this study, that more emphasis be given to students’ development of mathematical

1 N.B. Queensland primary schools include Years 1-7; the term primary as used in this thesis refers to this range of school class levels, which may be considered to be roughly equivalent to primary and elementary schools in the U.S.

2

understanding, underlies the advice contained in the policy documents listed in the

previous paragraph. The view of the NCTM (2000) is clear: “Learning mathematics

with understanding is essential” (p. 20). The documents embody a view of learning

as a sense-making activity (Mayer, 1996; McIntosh, Reys & Reys, 1992), in which

learners develop their own personal understandings of concepts to which they are

exposed. Thus the act of teaching is seen not as transmitting ready-formed

knowledge from teacher to learner, but rather as encouraging the learner to construct

concepts so that they make sense to him or her (Cobb, Yackel & Wood, 1992; NRC,

1989). The view of learning as a sense-making activity has special relevance for the

teaching of mathematics, because of its focus on abstract entities that need to be

conceptualised by each learner (Davis, 1992). If learners do not form appropriate,

accurate mental models of numbers, they will be hindered in attempting to solve

mathematical problems in meaningful ways. The literature is replete with

observations of students who, though they can do some computation, do so without

understanding the meanings behind the symbols and procedures used (e.g., Kamii &

Lewis, 1991).

Meaningful understanding of numbers is linked to the second

recommendation relevant to this study, that the development of number sense be

made a priority for mathematics teaching (McIntosh et al., 1992; NCTM, 2000;

Sowder & Schappelle, 1994). Number sense is regarded by many as an important

goal of mathematics education, enabling students to answer flexibly non-routine

questions that require a mathematical solution. Traditionally, mathematics was taught

so that students could answer routine arithmetic questions accurately, for future

employment in retail or manufacturing jobs (NRC, 1989). Today there is a greater

need for adults who can think mathematically and who can devise methods of

solving numerical questions in novel ways (NCTM, 1989).

The third recommendation for change in the way that mathematics is taught is

for the use of technological devices—calculators and computers—to be a matter of

course at all school grade levels (Australian Education Council, 1990; NCTM, 2000;

NRC, 1989). The question of how computer technology (referred to in this thesis as

“technology”) can best be incorporated in mathematics education is the subject of

some debate. Research such as that described here is needed to help answer questions

about the effects of technology on students’ learning. In particular, the computational

power and the representational capabilities of computers have the potential to assist

3

students to develop more meaningful concepts for numbers (Clements & McMillen,

1996; NCTM, 2000; Price, 1996, 1997). This potential needs further investigation.

1.2 The Learning of Place-Value Concepts The development of understanding of the base-ten numeration system is

foundational to all further use of numerical symbols, both in school and outside the

classroom. Thus, understanding how children develop place-value concepts, and the

difficulties they face in doing so, is of great importance to mathematics educators.

1.2.1 Conceptual Structures and Difficulties With Place-Value Concepts

Children’s difficulties in making sense of the meanings represented by

multidigit symbols have been reported widely in the literature (e.g., G. A. Jones &

Thornton, 1993a; Resnick, 1983; S. H. Ross, 1990). In particular, several authors

reported students having difficulty linking the abstract realm of numbers and their

symbolic and physical referents (e.g., Baroody, 1989; Baturo, 1998; Fuson, 1992;

Hart, 1989; Hiebert & Carpenter, 1992). In describing and analysing these

difficulties, several researchers have postulated children’s conceptual structures for

numbers (e.g., Fuson, 1990a, 1990b, 1992; Fuson et al., 1997; Resnick, 1983). A

number of conceptual structures, and several limited conceptions for numbers, have

been reported as being common among primary-age students. Such conceptual

structures feature prominently in much writing about children’s learning of place-

value concepts, and are considered by many, including this author, to be of critical

importance in understanding how children develop place-value concepts.

This thesis includes an analysis of evidence for conceptual structures for

multidigit numbers in the present study, and a comparison between that evidence and

reported findings of other researchers. Finally there is a discussion of possible links

between conceptual structures and participants’ use of two types of representational

material: physical and electronic base-ten blocks.

1.2.2 Use of Number Representations

Physical base-ten blocks.

Physical base-ten blocks, generally known in Queensland schools as

multibase arithmetic blocks [MABs], are regarded by many teachers as particularly

useful for helping students to build meaningful conceptual structures for multidigit

4

numbers (English & Halford, 1995). Developed by Dienes (1960) 40 years ago, they

have become the concrete materials of choice for teaching the base-ten numeration

system in many countries, including the USA, the UK, and Australia. Physical base-

ten blocks can be thought of as physical analogues of numbers, and mirror the

internal structures and relative magnitudes represented by the digits that make up a

written symbol (English & Halford, 1995). Students must reason analogically to use

the blocks effectively; that is, they must map the relations inherent in the blocks onto

the relations in the target realm (Gentner, 1983), the domain of numbers. In order for

physical base-ten blocks to be effective in representing numbers, it is important that

students’ attention be drawn to the analogical relationships that exist between the

blocks and the numbers they represent (Fuson, 1992).

Electronic base-ten blocks.

In light of the difficulties students have making links between numbers and

their referents, a number of suggestions have been made of teaching methods that

may help students to perceive connections among various forms of number

representation. One such suggestion is the use of computer-generated representations

for numbers (Clements & McMillen, 1996, Hunting & Lamon, 1995; NCTM, 2000).

Several software programs have been designed to model base-ten blocks

electronically on screen (e.g., Champagne & Rogalska-Saz, 1984; Rutgers Math

Construction Tools, 1992; P. W. Thompson, 1992). All use the capabilities of the

computer to enhance the number representations available to the user beyond those

provided by conventional physical blocks. For example, many of these programs

include number representations such as written symbols and representations of

regrouping actions on blocks, and link these representations tightly together so that a

change in one representation is mirrored by an equivalent change in the other

representations (see Appendix A). At the time the study was conducted, apart from

Rutgers Math Construction Tools the author only had access to descriptions of these

programs, and not to the programs themselves. Furthermore, none of the programs

included all the features that were felt to be desirable for teaching place-value

concepts; specifically, the author wanted the software to model multidigit numbers

with pictures of base-ten blocks on a place-value chart, to model regrouping actions

on the blocks, to show various symbolic representations for the numbers represented

by the blocks, and to play audio recordings of the number names. Because of the lack

5

of these features in available software, the author developed a new software program

for teaching place-value concepts, named Hi-Flyer Maths (described in Appendix A;

installation files available on CD-ROM in Error! Reference source not found.).

Central to this study is the effect of base-ten blocks, both physical and

electronic, on Year 3 students’ place-value conceptions of multidigit numbers. The

Hi-Flyer Maths software was used in the exploratory teaching study to assess these

effects.

1.3 The Research Question Based on the issues outlined in the previous section, the question investigated

in this study is

How do base-ten blocks, both physical and electronic, influence Year 3

students’ conceptual structures for multidigit numbers?

Within the context of Year 3 students’ use of physical and electronic base-ten

blocks, the following specific issues were of concern:

1. What conceptual structures for multidigit numbers do Year 3 students

display in response to place-value questions after instruction with base-

ten blocks?

2. What misconceptions, errors, or limited conceptions of numbers do

Year 3 students display in response to place-value questions after

instruction with base-ten blocks?

3. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in instruction with physical and

electronic base-ten blocks?

4. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in students’ achievement in

numeration?

1.4 Overview of Research Methodology The research questions were investigated using qualitative case studies

involving Vygotskian teaching experiments and Piagetian clinical interviews

(Hunting, 1983; Hunting & Doig, 1992). The study involved 16 Year 3 students

selected from a single primary school, half of each gender, and half of either high or

low mathematical achievement level (Table 3.2). The students were assigned to 4

6

groups of 4 students, each group comprising 2 boys and 2 girls, all of the same

achievement level. One high-achievement-level and one low-achievement-level

group were assigned to use physical base-ten blocks, and the other 2 groups used

computer software (electronic base-ten blocks). The groups each took part in 10

teaching sessions, involving up to a total of 45 place-value activities designed to

develop two-digit and three-digit place-value concepts. Each student was interviewed

individually both before and after the teaching sessions, to assess their place-value

understanding.

Each teaching session and interview was videotaped and audiotaped for later

transcription and analysis. As well, the researcher took field notes and the students’

workbooks were collected. The raw data from the teaching sessions and interviews

were transcribed for coding, principally using the grounded theory method described

by Strauss and Corbin (1990). Categories for participants’ responses emerged from

the data as they were analysed. These categories were compared with a framework of

conceptual structures identified in the literature.

1.5 Significance of the study There have been a number of suggestions for teaching strategies to help

students develop good place-value understanding, including the use of some means

of “bridging the gap” between numbers and physical number representations (Hart,

1989). One suggestion for bridging this gap is to use computer-generated

representations of numbers (e.g., Clements & McMillen, 1996; P. W. Thompson,

1992). However, there are few reports of in-depth investigation of the use of such

software, or of research-informed guidelines for future software development. In

particular, there is no evidence of analysis of children’s conceptual structures for

multidigit numbers as they use electronic base-ten blocks to learn place-value

concepts. Considering both the recommendations to use suitable place-value

software and the money invested in its development and purchase, there is a pressing

need for such research.

This study investigates the ideas that students have of numbers, and how

those ideas may be affected by the use of either physical or electronic base-ten

blocks. The study provides important findings in this field with significance for both

the teaching of place-value concepts generally, and the design and use of place-value

software.

7

1.6 Outline of the Thesis The thesis has 6 chapters. The current chapter provides an overview of the

study. Chapter 2 is a review of literature relevant to the study. Issues addressed are

current issues in mathematics education, place-value understanding, cognitive

science contributions to understanding of learning of place-value, the use of number

representational materials, and the use of computer software for teaching

mathematics. Chapter 3 contains a description of the methodology used in the study,

including assumptions and issues underlying the design, a description of the pilot

study and the main study, and discussion of validity, reliability, and limitations of the

design. Chapter 4 reports results of the study from the teaching sessions and

interviews. Chapter 5 comprises a discussion of the results in the light of other

reported research, and includes a description of a previously-unreported category of

student response to place-value questions, the independent-place construct. Chapter 6

concludes the thesis with a summary of findings, implications for the teaching of

place-value, and suggestions for further research in the area.

9

Chapter 2: Review of Literature

2.1 Chapter Overview This chapter comprises a review of literature relevant to the study, divided

into 5 main sections. The broad background to the research questions is related to

several current issues in mathematics education. Three issues relevant to this study

are (a) the development of mathematical understanding, (b) the development of

number sense, and (c) the use of technology in mathematics classes. These three

issues are linked in section 2.2 to the teaching of place-value concepts in primary

schools. Section 2.3 defines place value and place-value understanding for the

purposes of this thesis. This section identifies the skills that children need to develop

and introduces the desired mental models of numbers that are an important focus of

the study.

The contribution that cognitive science has made to the study of children’s

understandings of mathematics, and in particular place value, is summarised in

section 2.4. Two areas of cognitive science study in particular are described: mental

models and analogical reasoning. First, based on previous research, a framework of

four conceptual structures considered necessary for children to learn place-value

ideas is proposed, and three common limited conceptions of numbers are listed.

Second, analogical reasoning is an important consideration in the teaching of many

mathematical topics, including place-value concepts. Base-ten blocks are analogues

of the base-ten numeration system, and mirror the relations among digit places. A

focus on understanding of analogical reasoning is therefore important in considering

their use as representations of numbers.

Section 2.5 describes the teaching of place-value understanding, including the

use of physical models of numbers in teaching place-value concepts. It is shown that

there is evidence of a “conceptual gap” in the minds of many children between

written symbols and base-ten blocks, which a number of researchers have attempted

10

to bridge. One solution introduced in this section is the use of computer-generated

manipulatives. Section 2.6 includes a description of capabilities of modern

computers which make them potentially valuable for helping students to make

connections within many domains, including mathematics. Specifically, the

capability to present different representations of a concept shows promise for

representing numbers in several formats, with the aim of helping students to see

connections among them.

2.2 Issues in Mathematics Education Several issues of current concern in mathematics education are particularly

relevant to this study. This section describes three of these issues: students’ active

involvement in mathematics learning, development of number sense, and the use of

technology.

2.2.1 Students’ Active Involvement in Mathematics Learning

The view that students should actively participate in the process of learning

mathematics is a comparatively new one. As the NCTM (1998) noted, “the notion of

mathematics as something to be deeply understood, so that it can be used effectively,

has not always been a valued outcome of school mathematics” (p. 33). A

“traditional” model of mathematics teaching, typical of the first half of the 20th

Century, has been widely criticised (NCTM, 1989, 1991; NRC, 1989). This model

viewed the teaching-learning process as the transmission of information, and thereby

knowledge, from teacher to student. In this model the teacher was perceived to be the

source of information, “the sole authority for right answers” (NCTM, 1991, p. 3),

and the student was merely a passive recipient of the information. This model owes

much to behaviourist views of learning: namely, that “learning is conceived of as a

process in which students passively absorb information, storing it in easily

retrievable fragments as a result of repeated practice and reinforcement” (NCTM,

1989, p. 10). In contrast, recent recommendations for mathematics teaching and

learning (NCTM, 1989, 1991, 2000; NRC, 1989) portray a very different picture.

First, the learning process is now widely seen as one of individual construction of

understanding, in which new experiences are integrated with prior knowledge to

form understandings that are meaningful to the student (Simon, 1995). Second,

students are seen as “autonomous learners . . . . [who should] take control of their

11

learning” (NCTM, 2000, p. 21), to make sense of it for themselves. Third, the

teacher’s role is to be a “guide for exploring academic tasks” (Mayer, 1996, p. 152;

see also Sowder, 1994, p. 146), or an “[orchestrator of] classroom discourse in ways

that promote the investigation and growth of mathematical ideas” (NCTM, 1991,

p. 1).

A critical component of the view of mathematics learning described here is

the necessity of students making sense of what they learn (Mayer, 1996). If teachers

want their students to develop meaningful understanding of mathematical concepts,

then there is a need to consider many aspects of the learning environment that exists

in the classroom. One aspect of the learning environment of major relevance to this

study is the question of various interactions that take place, described below; this is

an important item of interest in the research described in this thesis. As explained by

McNeal (1995), “[by] studying classroom interactions, the observer could . . . infer a

particular individual’s knowledge . . . from observations of his/her interactions with

the objects or with other individuals” (p. 3). The following subsection addresses

interactions of three kinds that are of relevance to this study.

Student-teacher interactions.

If the view of learning as a constructive meaning-making activity is accepted,

then the interactions between students and teachers are of obvious importance.

“More than any other single factor, teachers influence what mathematics students

learn and how well they learn it” (NCTM, 1998, p. 30). Part of a constructivist model

of learning is a view that students construct mathematical knowledge as a product of

“interaction in social contexts” (Putnam, Lampert & Peterson, 1990, p. 134). As

Cobb and Yackel (1996) stated, “we consider students’ mathematical activity to be

social through and through because it does not develop apart from their participation

in communities of practice” (p. 180). This idea of a community of practice is implicit

in much recent writing about teaching mathematics (NCTM, 1991, 2000; NRC,

1989), and learning in general (Brown, Collins, & Duguid, 1989; Harley, 1993). The

NCTM’s (1991) recommendations for what a teacher can do to encourage the

development of a community of practice included: helping students to work together,

to rely more on themselves, to reason mathematically, to solve problems, and to

connect mathematics and its applications. More recently, the NCTM (2000) stated

the view that “teachers’ actions are what encourage students to think, question, solve

12

problems, and discuss their ideas, strategies, and solutions. The teacher is responsible

for creating an intellectual environment where serious mathematical thinking is the

norm” (p. 18). Similar advice was given by the NRC (1989), who described teachers’

actions as denoting supportive interaction with students: “encourage,” “help students

verbalise,” “build confidence” (pp. 81-82). Clearly, these authors believed that

interactions between a teacher and students are an important aspect of teaching and

learning mathematics.

Student-student interactions.

The second type of interaction, between student and student, is closely linked

to the first and has also received attention in the mathematics education literature.

One aspect of learning theories that typifies the differences between constructivism

and transmission models of learning is the focus on the interactions occurring among

students. As mentioned earlier, under the transmission model the student was

expected to receive knowledge passively without questioning it; modern learning

theories assert that discussion and debate among students is an essential part of the

learning process. Various benefits have been claimed for students learning in a social

community, whether in pairs, a small group, or a whole class (Akpinar & Hartley,

1996; Brown et al., 1989; Fox, 1988). These benefits include opportunity for

collective problem solving, development of skills of collaboration, development of

flexible thinking, and exposure of misconceptions.

Student-materials interactions.

The third type of interaction of interest here is interaction between a student

and learning materials, such as blocks or a computer. Interaction with materials is

linked with the two previous types of interaction, as the materials are an integral part

of the learning environment, and there is assumed to be an “interplay between

students’ cognitive activity and physical and social situations” (Nitko & Lane, 1990,

p. 5). The connection between learning and the learning environment was mentioned

by Kozma (1991), who stated that the learning process involves “extracting

information from the environment and integrating it with information already stored

in memory” (pp. 179-180). Writing specifically about computer learning

environments, Kozma stated that the learning process was “sensitive to

characteristics of the external environment, such as the availability of specific

information at a given moment” (p. 180).

13

The idea of situated cognition (Brown et al., 1989) addresses the question of

learning and its relation to the learning environment. Brown et al.’s idea, that a

learning environment constrains the learning activity of the students in that

environment, is important to this research. A central assumption of the situated

cognition view is that students “reason with what a situation affords them” (Winn,

1993, p. 16). In other words, the particular capabilities, or affordances (Salomon,

1998), provided by the materials available to a student can have an important

influence on the student’s learning. This view is supported by Kozma’s (1994b)

statement that

knowledge and learning are neither solely a property of the individual or of the environment. Rather, they are the reciprocal interaction between the learner’s cognitive resources and aspects of the external environment . . . and this interaction is strongly influenced by the extent to which internal and external resources fit together. (p. 8)

The research described here involves the investigation of children’s learning

when using one of two types of materials; the author assumed prior to the study that

the materials’ different special characteristics would have different effects on the

students’ learning.

2.2.2 Number Sense

The idea of number sense is related closely to development of mathematical

understanding, as it “typifies the theme of learning mathematics as a sense-making

activity” (McIntosh et al., 1992, p. 3). Number sense refers to a student’s familiarity

with numbers, and the ability to use numbers in sensible ways to answer

mathematical questions. It lacks a precise definition, but has been likened to “road

sense” (familiarity with a particular geographical area; Trafton, 1992) or

“friendliness with numbers” (Howden, 1989). The NCTM (1989) listed five

understandings demonstrated by students with good number sense. They “(1) have

well-understood number meanings, (2) have developed multiple relationships among

numbers, (3) recognize the relative magnitudes of numbers, (4) know the relative

effect of operating on numbers, and (5) develop referents for measures of common

objects and situations in their environments” (p. 38).

The need for students to possess number sense is extensively argued in the

literature (Australian Education Council, 1990; K. Jones, Kershaw, & Sparrow,

1994; McIntosh et al., 1992; NCTM, 1989, 2000; Sowder, 1988, 1992; Sowder &

14

Schappelle, 1994). Good number sense is important for making sense of

mathematical questions and for working out sensible answers. Perhaps the

characteristic that most easily sums up good number sense is flexibility, in finding

solutions to mathematical problems and in being able to see connections among

numbers in different ways. Trafton (1992) described a person with number sense as

having “a well-integrated mental map of a portion of the world of numbers and

operations and [being] able to move flexibly and intuitively throughout the territory”

(p. 79). A similar idea was proposed by Greeno (1991), who likened knowing and

learning “as an activity in an environment” (p. 175). Greeno connected number sense

with situated cognition, stating his view that “knowing the domain [e.g., the

mathematical domain] is knowing your way around in the environment and knowing

how to use its resources” (p. 175). In this view, number sense relates closely to the

ability to use available resources to make sense of the domain. Though Greeno was

writing specifically of mental resources, this idea is assumed here to apply also to

physical resources, as described in an earlier paragraph. In other words, a student

with good number sense could be seen as having not only a good idea of the

cognitive domain, but also of the physical environment, including how to use

available materials to answer mathematical questions.

For teachers to help students to develop good number sense involves helping

the students to develop a range of prerequisite understandings of numbers and

operations. This is borne out by McIntosh et al.’s (1992) description of number

sense:

Number sense refers to a person’s general understanding of number and operations along with the ability and inclination to use this understanding in flexible ways to make mathematical judgements and to develop useful strategies for handling numbers and operations. It reflects an inclination and an ability to use numbers and quantitative methods as a means of communicating, processing and interpreting information. (p. 3)

Prerequisite mathematical skills and understanding needed for good number

sense include proficiency with written algorithms, mental computation skills,

problem-solving ability, and place-value understanding (see NCTM, 2000, p. 32).

Teaching techniques suggested for developing number sense include the use of

calculators to investigate number magnitudes (Bobis, 1991; Schielack, 1991),

emphasising and encouraging sense-making (Sowder & Schappelle, 1994), and the

use of estimation activities (K. Jones et al., 1994; Lobato, 1993; Sowder, 1988,

15

1992). The particular focus of this research, the development of place-value

understanding, is directly relevant to the development of number sense. As explained

by the NCTM (2000),

understanding number and operations, developing number sense, and gaining fluency in arithmetic computation form the core of mathematics education for the elementary grades. As they progress from prekindergarten through grade 12, students should attain a rich understanding of numbers—what they are; how they are represented with objects, numerals, or on number lines; how they are related to one another; how numbers are embedded in systems that have structures and properties; and how to use numbers and operations to solve problems. (p. 32)

2.2.3 Use of Technological Devices

The third issue of particular relevance to this study is the use of technology in

mathematics teaching. There is common agreement that technological advances in

the general society outside schools lead to the need for different objectives in

mathematics education (NRC, 1989). These objectives will be seen in (a) the use of

different means of doing mathematics, and (b) having different emphases in the

curriculum.

Much has been written about the need to bring the procedures used in school

mathematics into line with those expected of workers in the 21st Century. It has been

pointed out (NCTM, 1989) that in the industrial age the goal of public schools was to

educate future shop assistants and factory workers, and so schools taught their

students so-called “shop keeper arithmetic” (Cruikshank & Sheffield, 1992; NCTM,

1989; NRC, 1989). There is no longer the same need for adults to be highly

proficient in written computation procedures; in its place is a need for workers and

citizens who possess a broader range of mathematical “concepts and procedures they

must master if they are to be self-fulfilled, productive citizens in the next century”

(NCTM, 1989, p. 3). These skills include developing methods to solve a variety of

problems; working cooperatively in teams; and reading, interpreting, and critically

evaluating quantitative data. The development of these skills is linked to the issues

discussed in the previous two sections—meaningful understanding of mathematics

and number sense—as well as the use of technological devices in mathematics

teaching.

One feature of the mathematics used by adults in homes and workplaces is the

use of calculators and computers to assist with a range of mathematical tasks

(Sparrow, Kershaw, & Jones, 1994). These include computation, storage of data, and

16

presentation of results of mathematical processes such as in spreadsheets and graphs.

It is assumed by writers of mathematics education policy documents that students in

schools will similarly have access to a range of technological devices to assist them

in learning mathematics (Australian Education Council, 1990; Australian Association

of Mathematics Teachers [AAMT], 1996; Cockcroft, 1982; NCTM, 2000; NRC,

1989). As the AAMT (1996) put it, “mathematics education must reflect the

influence of technology upon both mathematics and society” (p. 2). The NCTM

(1998) similarly recommended that schools improve their level of use of technology

also to match what happens in schools with what employers and others expect of

workers in the workforce:

Today’s jobs demand the use of mathematically driven technological tools. If schools do not have a level of technology equivalent to the level found outside schools, and if they do not prepare students appropriately with it, then they are placing their students at a serious disadvantage. (p. 43)

Merely increasing the amount of technology available in classrooms is not

sufficient to bring about the desired improvements in mathematics education,

however, and technological devices should not merely be added to existing programs

of instruction. Changes are also required in the methods of mathematics instruction,

so that appropriate tasks are set to answer with technological devices. As the NCTM

(1989) pointed out, “access to [calculator and computer] technology is no guarantee

that any student will become mathematically literate. Calculators and computers for

users of mathematics . . . are tools that simplify, but do not accomplish, the work at

hand” (p. 8). There is a need for detailed knowledge of the effects that calculators

and computers have on students using them, especially as they represent considerable

investments of finance and time by education departments and teachers. This topic is

reviewed further in section 2.6.

Electronic technology has the potential for several important effects on

mathematics curriculum. Technology makes mathematical skills such as written

computation easier, but it also hides processes that students in earlier times had to

consider, such as regrouping required for operations. Technology can also make

available to students mathematics learning experiences that previously were not

possible. For example, calculators can allow a student to investigate operations on

large numbers that would be too time-consuming with other mechanical computation

procedures. Computers similarly provide students with considerable computing

power, which can be used to represent mathematical and other domains with which

17

students can “interact” in ways not possible with any other technology. This study

investigates one such interactive learning environment in which a computer is used

to represent the domain of numbers for the purpose of developing students’

understanding of place-value concepts.

2.3 Place-value Understanding One area of the mathematics curriculum where the issues described in the

previous section warrant attention is in place value. The teaching of place-value

concepts is foundational for understanding of the base-ten numeration system, and is

thus central to the primary mathematics curriculum. As the NCTM (2000) stated,

“foundational ideas like place value . . . should have a prominent place in the

mathematics curriculum because they enable students to understand other

mathematical ideas and connect ideas across different areas of mathematics” (p. 15).

Issues such as those described in the previous section all have a potential impact on

the teaching of place value. It is this author’s view that recommendations for

students’ active involvement in learning mathematics and the development of

number sense have direct relevance for how place value is taught. Advances in

technology have a more indirect influence, through the capabilities they offer in the

area of models of numbers for place-value teaching (section 2.6).

The importance of place value in the primary mathematics curriculum and the

difficulty teachers experience in teaching place-value concepts to their students are

well documented (G. A. Jones & Thornton, 1993a; S. H. Ross, 1990). As Resnick

(1983) stated,

the initial introduction of the decimal system and the positional notation system based on it is, by common agreement of educators, the most difficult and important instructional task in mathematics in the early school years. (p. 126)

Teachers have difficulty teaching place-value concepts, and their students

have difficulty learning them (S. H. Ross, 1990). One source of difficulty for

teachers is that, as Skemp (1982) pointed out, mathematics’ “[conceptual structures]

are purely mental objects: invisible, inaudible, and not easily accessible even to their

possessor” (p. 281). Thus, teachers are limited in what they can know of what their

students are thinking with regard to mathematical entities. Research investigating

place-value understanding must address students’ conceptual structures for numbers

and how they are developed. This topic is dealt with in more detail in section 2.4.2.

18

2.3.1 Place Value

Place value refers to the feature of the base-ten system of numeration

(sometimes called the “Hindu-Arabic” system) in which each digit in a number

represents a precise amount, dependent on both the face value of the digit and its

position (Baturo, 1998; Miura & Okamoto, 1989). This contrasts with other

numeration systems that do not exhibit place-value, such as that of the ancient

Egyptians, who wrote a different symbol for each power of 10 (Irons & Burnett,

1994). In the base-ten numeration system the value represented by each digit is equal

to the product of the face value of the digit and the value assigned to the digit’s

position, relative to the rightmost whole number digit, or ones place (Fuson, 1990a;

Figure 2.1). Thus, though pairs of numbers such as “25” and “52” look very similar

(and may be confused by young children), the position of each digit determines its

value, giving a unique value to each different written symbol. This idea is at once

both simple and very powerful, and can be extended an indefinite number of places

to the left (for whole numbers) or the right (for decimal fractions).

Figure 2.1. The face value of each individual numerical symbol, together with its position relative to the ones place, determines the value it represents.

The base-ten numeration system is principally a system of written symbols by

which users record physical quantities and use them in calculations. Its importance

was emphasised by the NCTM (1998), who commented that “mathematical

symbolism and representation is one of the most significant achievements of

humankind” (p. 94). Associated with the written symbols are number words that are

alternative representations of numerical quantities. Whereas the written symbols

19

follow an entirely consistent mathematical system in which a quantity of ten of each

place equals one of the place immediately to the left, number words include

inconsistencies relating to the history of the language used. In English, there are

inconsistent words for multiples of 10, and for numbers from 11 to 19. Once a child

reaches the study of three-digit and four-digit numbers number naming is much more

consistent, but until a child reaches that stage the learning of numbers and their

names is very difficult. Thus, young children have a very challenging task of

developing understanding of a system that in its earliest, numerically simplest,

examples contains numerous inconsistencies.

The base-ten numeration system has been named as an “unnamed-value

positional value system of written marks” (Fuson, 1990a, p. 343) and a “regular

relative positional system” (Fuson, 1992, p. 136). In these two phrases Fuson has

captured three essential features of the base-ten numeration system: (a) Place names

and values are implicit in written numeric symbols; (b) the system is completely

consistent across all symbol positions; and (c) value is assigned according to each

digit’s position, relative to the point of reference, the ones place. The structure and

rules by which the conventional base-ten numeration system operates are not evident

from merely observing the written symbols, even if the meaning of each individual

symbol (“1,” “2,” etc.) is known (Fuson, 1992, p. 138). However, to those who are

familiar with the scheme’s conventions, each numerical symbol uniquely represents a

number. Apart from minor variations of the symbol used for the decimal point

(usually “.” or “,”), and leading or trailing zeros, each rational number is represented

by a unique symbol, and each sequence of digits stands for a unique number.

In summary, children need to understand the features of the base-ten

numeration system (Baturo, 1998; English & Halford, 1995; Fuson, 1990a, 1992;

Hiebert, 1988; Miura & Okamoto, 1989; Sowder & Schappelle, 1994). A number of

key features of the base-ten numeration system, in common with all place-value

numeration systems, are expressed in the following statements:

1. A discrete set of individual number symbols (base 10: 0 to 9), and a

decimal point marker, used in combination can uniquely represent any

rational number quantity.

2. Each place represents a power of the base, derived from its position

relative to the rightmost whole-number place.

20

3. The system is completely consistent across all places, in that the value

of each place is equal to the base number times the value of the adjacent

place on the right, and the value of the adjacent place on the left divided

by the base number.

4. The value represented by a digit is the product of the face value of the

digit and the value associated with the place of the digit.

5. The system allows operations on numbers to be represented

symbolically, and these operations work consistently on numbers in all

places.

From the previous discussion, definitions can be given for place value and for

place-value understanding. This study is concerned with understanding of the base-

ten numeration system only; hereafter “place value” will be used to refer only to

place value in the base-ten numeration system. For the purposes of this thesis, place

value is based on the description given by Miura and Okamoto (1989, p. 109), and

defined thus:

Place value is the property of the base-ten numeration system, by which the

numerical value represented by each digit of a written multidigit symbol is equal to

the product of the digit’s face value and the power of 10 associated with the digit’s

position in the numeral.

2.3.2 Place-value Understanding

For the purposes of this thesis, place-value understanding is described in

terms of both actions and conceptual structures. This approach is supported by Sfard

(1991), who described historical progress made in mathematical understanding as

having to do with both “operational” (actions) and “structural” (objects) conceptions,

dynamically linked together as professional mathematicians have struggled to

advance knowledge in the field. Sfard stated that “the ability of seeing a function or a

number both as a process and as an object is indispensible for a deep understanding

of mathematics” (p. 5). Though school students are not involved in the same level of

mathematical thinking as professional mathematicians, nevertheless students need to

develop both structural and operational conceptions for each new mathematical

concept. These two types of conception of numbers link students’ internal conceptual

structures of numbers and external physical models of numbers. In other words,

students are assumed to possess internal representational structures for numbers and

21

processes that are influenced as the students access and manipulate available external

representations of numbers (Hiebert & Carpenter, 1992; Putnam et al., 1990).

Internal and external representations and manipulations are closely linked, and in this

thesis they are considered together to be involved in students’ place-value

understanding.

The definition of place-value understanding used here takes into account

advice given by several authors, relating to both actions and conceptual structures.

This includes statements that children need to “construct number representations that

reflect the Base 10 numeration system” (Miura & Okamoto, 1989, p. 109),

“coordinate and synthesize a variety of subordinate knowledge about our culture’s

notational system for numbers” (S. H. Ross, 1989, p. 47), “develop flexibility in

representing and understanding multidigit numbers” (G. A. Jones, Thornton, & Putt,

1994, p. 122) and “[develop understanding of] the interpretation of numbers as

compositions of other numbers” (Resnick, 1983, p. 126).

For this thesis place-value understanding is defined thus:

A student possessing place-value understanding is able to use the place-

value features of the base-ten numeration system to form accurate, flexible

conceptual structures for quantities represented by written numerical symbols. The

student is able to manipulate numerical quantities in meaningful ways to answer

mathematical questions.

A student’s place-value understanding must be assessed at a deep level, by

probing the student’s conceptual structures for numbers (Skemp, 1982). Research in

this area generally uses observation of participants’ behaviour to posit “various

cognitive structures and processes believed to produce the behavior” (Putnam et al.,

1990, p. 65). The following section describes research into children’s cognition,

including the investigation of children’s conceptual structures for numbers.

2.4 The Contribution of Cognitive Science to Mathematics Education

As shown in the previous section, place-value understanding is an internal

phenomenon that a teacher or researcher cannot access directly. Research on

conceptual structures and analogical reasoning has particular applicability for the

study of mathematical understanding, in two respects. Findings about mental

structures and processes can explain how abstract number concepts are represented in

22

the mind, and cognitive science methods of research may be applied to the

investigation of learning of mathematics. This section outlines two aspects of

cognitive science research relevant to this study. First, the study of mental models is

described, including a form of mental model of particular importance to mathematics

understanding, conceptual structures for numbers. Second, analogical reasoning is

defined, and its relevance to the teaching and study of place-value concepts

demonstrated.

2.4.1 Understanding Mathematics

Understanding of mathematics relies on having internal mental

representations of numbers and the ability to manipulate them in meaningful ways,

because of the abstract nature of its content (Hiebert & Carpenter, 1992; Presmeg,

1992; Sfard, 1991). Thus learning mathematics involves representation of abstract

concepts, which may include physical representation of numbers using concrete

materials, but ultimately involves internal, mental, representations held in the mind

of the student (Baggett & Ehrenfeucht, 1992; Resnick, 1988; Sfard, 1991). As Davis

(1992) noted, “after all, mathematics is about thinking; there is a sense in which

mathematics exists only within the human mind” (p. 225). All numbers and

associated processes are abstract ideas rather than physical entities. For example, the

number three may be represented physically, using representations such as the

written symbol “3,” the verbal or written word “three,” a set of three counters, or a

picture of three objects. However, the number itself can never be perceived directly

by the physical senses (Sfard, 1991).

Because numbers are abstract entities, users can perceive and manipulate

them only mentally. For this reason children need proficiency with certain mental

skills to be successful in learning mathematics. For young children, the apparently

simple act of counting a group of objects demands a cluster of skills that must all be

present in order to correctly count, name, and then understand the number of objects

(English & Halford, 1995). These skills include correctly recalling the sequence of

number names, applying exactly one number name to each object, and understanding

that the last number name counted is the number of objects in the group (Fuson,

1992). At a higher level, children in middle primary school must possess more

advanced skills, relating to place-value understanding, numeration, the concept of

subtraction, and the algorithm itself (Fuson, 1990a).

23

The emphasis in mathematics on mental representations and processes makes

it particularly suitable for psychological research. Several authors have pointed out

the special relevance that psychological theories have for the teaching of

mathematics (Beilin, 1984; English & Halford, 1995; Glaser, 1982; Hiebert &

Carpenter, 1992). Cognitive science has supported mathematics education research in

two distinct ways: Firstly, theory derived from research into thinking generally has

been used to explain how students learn mathematical concepts; secondly,

mathematics researchers have used methods from cognitive science in their study of

mathematical understanding. As noted by English and Halford (1995), cognitive

science research findings have great relevance for understanding of mathematical

concepts. Several authors have written about psychological theories and how they

may explain mechanisms underlying mathematical understanding. For example,

Hiebert and Carpenter (1992) reported that they “[drew] quite heavily from insights

provided by work in cognitive science to deal with questions of learning and teaching

mathematics” (p. 66). The assumptions made by Hiebert and Carpenter with regards

to mental representations are adopted also in this thesis (section 3.5.1). The second

connection between psychological theory and mathematics education has been the

application of methods developed for cognitive science research to the investigation

of questions in mathematics learning (Ohlsson, Ernst, & Rees, 1992; Schoenfeld,

1992; Silver, 1994). These methods include think-aloud protocols, computer

simulations of mental processing of information, and generally inferring mental

models and processes from observed actions. Such cognitive science methods have

been applied to research into mathematics learning on topics such as problem solving

(Schoenfeld, 1992), mental computation (Hope, 1987), mathematics as a situated

mental activity (Silver, 1994), the cognitive complexity of subtraction algorithms

(Ohlsson et al., 1992) and geometric problem solving (Chinnappan & English, 1995).

2.4.2 Mental Models

Cognitive scientists are principally concerned with understanding human

thinking and how it is affected by external events (e.g., Greeno, 1991; Halford,

1993a, 1993b; Presmeg, 1992; Shepard, 1978). In seeking to understand the

workings of the human mind, cognitive scientists posit the existence of mental

models. These models are deduced from observations of people, often made under

experimental conditions, and are used to explain the observed behaviour. Mental

24

models have been defined by Halford (1993a) as “representations that are active

while solving a particular problem and that provide the workspace for inference and

mental operations” (p. 23). Greeno (1991) defined a mental model as

a special kind of mental representation, in that the properties and behavior of symbolic objects in the model simulate the properties and behavior of the objects rather than stating facts about them. . . . A model is a mental version of a situation, and the person interacts within that situation by placing mental objects in the situation and manipulating those symbolic objects in ways that correspond to interacting with objects or people in a physical or social environment. (p. 177)

Several researchers in the mathematics education field have investigated

mental models used by students as they learn mathematical concepts (e.g.,

Chinnappan & English, 1995; English & Halford, 1995; Fischbein, Deri, Nello, &

Marino, 1985; Hunting & Lamon, 1995).

Conceptual structures for multidigit numbers.

It is common practice for researchers investigating place-value understanding

to make deductions about “children’s inaccessible mathematical realities” (Cobb &

Steffe, 1983, p. 93) based on their performances on mathematical tasks (Davis, 1992;

Putnam et al., 1990; Resnick, 1983, 1987). As seen in Greeno’s (1991) definition in

the previous section, mental model is a broad term encompassing a range of internal

representations of situations. The particular type of mental model of interest in this

study is the mental models that students form to internally represent multidigit

numbers. These have been referred to variously as internal representations (English

& Halford, 1995; Hiebert & Carpenter, 1992), mathematical constructs (Sfard, 1991)

and conceptual structures (Bell, 1990; Fuson, 1990a, 1992; Fuson et al., 1997;

Skemp, 1982). In this thesis the term conceptual structures is used in the same sense

as Bell (1990) and Fuson (1990a, 1990b, 1992), to refer to the mental models

children use “for the formal mathematical words and marks used in the school

mathematics classroom” (Fuson, 1992, p. 56). As Fuson (1992) explained, children’s

conceptual structures vary in quality and usefulness:

Some of the conceptual structures are accurate and some are not; some are efficient and some are not; some are advanced and some are simple. To help children function effectively in mathematics, teachers need to reflect on how the classroom experiences they are providing their children are supporting children’s construction of accurate, efficient, and advanced conceptual structures for the mathematical marks, procedures, and concepts addressed in the classroom. (pp. 56-57)

Conceptual structures deduced by researchers and reported in the literature

fall into two broad groups: structures considered by authors to be necessary for the

25

development of place-value understanding, and structures that are limited

conceptions of numbers that hinder children’s mathematical understanding and

performance. Descriptions of conceptual structures of the first group, that are

believed to be necessary for the learning of place-value concepts, are given first in

this section. Common limited conceptions of multidigit numbers are described later

in this section.

The place-value literature includes a number of papers in which authors

provided descriptions of children’s conceptual structures; Table 2.1 shows a

summary of several of these descriptions. Some authors (Cobb, 1995; Cobb &

Wheatley, 1988; Miura & Okamoto, 1989; Miura, Okamoto, Kim, Steere, & Fayol,

1993; Resnick, 1983; S. H. Ross, 1989, 1990; Steffe, Cobb, & von Glasersfeld, 1988)

proposed stages or levels of understanding through which children are purported to

pass as they develop place-value understanding. Aspects of the schemes are

integrated into the proposed framework of conceptual structures described in this

section. Generally, authors devised schemes post hoc, during the analysis of

experimental data (A. Sinclair & Scheuer, 1993, p. 200). Other authors (e.g., Fuson

et al., 1997; Janvier, 1987), however, have disputed the validity of defining stages in

place-value understanding at all.

26

TABLE 2.1. Aspects of Place-value Understanding Described in the Literature

Researcher(s)

Aspect of place-value understanding

No. of stages or

levels Summary of findings

S. H. Ross, 1989, 1990

Acquisition of knowledge about two-digit numbers

Five

The author identified five stages in children’s acquisition of place-value understanding.

Steffe, Cobb, & von Glasersfeld, 1988

Concepts of ten Five

The authors identified five conceptions of ten constructed by children.

Cobb, 1995 Cobb & Wheatley, 1988

Concepts of ten Five

The authors identified five conceptions of ten in children’s counting after textbook instruction.

Resnick, 1983

Number knowledge / Development of decimal knowledge

Three

The author posited a mental number line and a part-whole schema preceding three stages of place-value understanding.

Fuson, 1990a, 1990b Fuson & Briars, 1990 Fuson et al., 1997

Conceptual structures for multidigit numbers

Not ap-plicable

The authors identified several conceptions of numbers, many of them limited conceptions.

Miura & Okamoto, 1989 Miura et al., 1993

Mental representations of multidigit numbers

Three

The authors observed three types of representation used by students to represent two-digit numbers with concrete materials.

A general overview of common features of the various classification schemes

summarised in Table 2.1 can be given, despite the diversity among them. First, the

authors each listed a number of levels at which children may operate in the place-

value domain. Some authors’ levels, the number of which varied from three to five,

were stages through which most children pass (e.g., S. H. Ross, 1990); generally,

however, they represented levels of expertise or maturity of understanding observed

in children (e.g., Miura et al., 1993). In fact, few studies attempted to track individual

students’ understanding over time. Rather, researchers usually described behaviour

common to several students at a particular point in time, as indicative of a particular

level of understanding. Second, the authors’ schemes each presented a sequence,

starting with initial immature understandings, with each successive level representing

better understanding of place value. This is clearly relevant to the teaching of place-

27

value concepts, the implied goal of which is to assist students to move to higher

levels of number understanding. The levels were often used as a means of

comparison of individuals and groups of children, to describe differences in

performance and understanding (e.g., S. H. Ross, 1990).

It is important to note here that the levels and stages proposed by the various

authors do not agree completely. This may be due to factors such as the date of the

research, the aims of the research, the philosophical stance of the author(s), and the

tasks provided to participants. Nevertheless, there does exist substantive agreement

among the various authors on internal structures revealed by observations of

children’s task performance, and so it is useful to draw them together for the purpose

of summarising the current state of knowledge of this field.

A framework of conceptual structures for place value.

This subsection describes a framework of four conceptual structures believed

to be necessary for the development of mature place-value understanding, based on a

synthesis of work in the field of place value research described in the previous

section. This framework was used to inform initial data analysis in this study and

then was subsequently compared with the study’s findings. The following paragraphs

describe the four conceptual structures in the proposed framework, including support

for each structure from the place-value literature.

Conceptual structure 1: Unitary construct. Early in their school years, and

even before the start of formal schooling, children are introduced to the idea of

numerical symbols. They learn to recognise the symbols for numbers 1 to 9 and to

associate each one with a number: the concept that refers to the numerosity of a

group of objects (Fuson, 1990b; Resnick, 1983). Resnick likened this conceptual

structure to a “mental number line” (p. 110), on which cardinal numbers are placed

in sequence from zero or one to the limit of a child’s counting. By having a mental

image of the number sequence, when counting a group of objects a child can

associate each element on the number line with an object, and give the name of the

last-mapped element as the total number in the group. Fuson et al. (1997) explained

the importance of this conceptual structure for later learning about base-ten numbers:

28

Multidigit numbers build on and use the unitary single-digit triads of knowledge for single-digit numbers. Thus, before children can learn about two-digit numbers, they must have learned for one to nine how to read and say the number word corresponding to each number mark, write the numeral corresponding to each number word, and count or count out quantities for each mark and number word one to nine. Because the number words for single-digit numbers in most languages and the corresponding written marks are arbitrary, most children learn most of the unitary single-digit triads as rote associations. (p. 138)

The unitary construct, though an essential component of early number

teaching and learning, can lead to a limited conception, common in older children,

that multidigit numerals represent only collections of single objects (Fuson, 1990b;

Fuson et al., 1997). This conception is described in more detail later in this section.

Conceptual structure 2: Tens and ones structure. Resnick (1983) proposed

that this structure followed an earlier “part-whole” construct, by which students learn

that quantities may be partitioned in different ways, especially when learning single-

digit addition and subtraction operations. Partitioning a multidigit quantity into

whole tens and leftover ones is a “unique partitioning of multidigit numbers”

(Resnick, 1983). At this stage, students learn counting number names for numbers

beyond 9, learn that numbers greater than 9 are separated into “tens” and “ones,” and

learn to write symbols for numbers using two digits. With this level of knowledge a

child may be able to carry out addition and subtraction operations that do not involve

trading, and may also be successful on many typical classroom and textbook

questions such as “How many tens are there in 36?” or “Circle the tens digit in 82.”

However, as several authors have pointed out (e.g., Cobb & Wheatley, 1988), this

type of question does not involve true place-value understanding, as it does not

address the multiplicative idea of 1 ten being composed of 10 ones, and children with

only this level of knowledge are not able to handle demands of operations which

include trading.

Conceptual structure 3: Ten as a unit. The third construct develops from the

second, and focuses on the fact that the tens digit in a multidigit number stands for a

collection of 10 single items. There are many variations of this conceptual structure

(cf. Cobb & Wheatley, 1988; Fuson et al., 1997); the common element of the various

constructs of this type is that ten is a single entity, made up of 10 units. By

understanding this idea, a student thinking at this level can mentally or physically

decompose a ten into 10 ones, or regroup 10 ones into a single ten, as the situation

demands.

29

Several researchers have identified the ten-as-a-unit construct. S. H. Ross

(1989) named this construct the construction zone stage, and explained the

understanding involved in this way: “Students know that the left digit in a two-digit

numeral represents sets of ten objects and that the right digit represents the remaining

single objects” (p. 49). Miura and Okamoto (1989) identified students who chose to

represent two-digit numbers using a canonical base 10 form: that is, so that the

number of tens material and ones material equalled the number of tens and ones,

respectively, in the written numeral. Numbers represented canonically can have no

more than nine in any place, unlike under the following construct, where groupings

of more than nine in a place are allowed. Steffe, Cobb, and von Glasersfeld (1988)

identified a concept of ten that is congruent with the ten-as-a-unit construct in their

study of children’s counting that they named ten as an iterable unit. This concept of

ten was held by students who could count using ten as a composite unit and the

remainder as units of one, and was also identified by Cobb and Wheatley (1988).

Conceptual structure 4: Flexible representations. The flexible representations

conceptual structure develops the understanding in the previous ten-as-a-unit

construct. The base-ten numeration system is written using a strict protocol of having

no more than nine in any one place. When concrete materials represent a number, it

is said to be a canonical representation if it has no more than nine in any single

place, or non-canonical if there are more than nine in any single place. For example,

75 can be represented as 6 tens and 15 ones, or as 4 tens and 35 ones, and so on. The

ability to understand multidigit numbers in non-canonical terms and to represent

them non-canonically is essential for proficiency with mental or written computation

(Greeno, 1991). The flexible representations construct represents a high level of

place-value understanding, and may be exhibited in a variety of ways. These include

the abilities to represent a given number in non-canonical form, to write the

numerical symbol for a number represented non-canonically, and to carry out mental

computation by flexibly partitioning multidigit numbers.

The idea that students need to develop the flexible representations construct is

contradictory to advice contained in a chapter written over 20 years ago, by Merseth

(1978). In her explanation of how to use concrete materials to teach addition and

subtraction algorithms, Merseth advised teachers to institute a trading rule, “that no

player may have more than nine objects in any column at the end of the individual

turn” (p. 64). With further knowledge of necessary conceptual structures for

30

multidigit numbers, many authors today are in favour of encouraging students to

develop more flexible understandings of how a number may be represented (e.g., G.

A. Jones et al., 1994; Resnick, 1983). By providing students with the idea that it is

never permitted to have more than 9 in a place, this “canonical arrangement only”

rule may restrict students’ thinking to the level of this base-ten structure construct.

Several researchers identified the flexible representations construct. Miura

and Okamoto (1989) and Miura et al. (1993) identified it as the highest category of

place-value understanding of their participants. After students represented a two-digit

number using blocks, researchers asked them to “show the number another way

using the blocks” (Miura & Okamoto, 1989, p. 111). The researchers categorised

students who did so using non-canonical arrangements of tens and ones blocks as

using a non-canonical base 10 representation. S. H. Ross (1989) described the ability

to determine the number represented by an arrangement of materials under this

construct as the understanding stage:

Students know that the individual digits in a two-digit numeral represent a partitioning of the whole quantity into a tens part and a ones part. The quantity of objects corresponding to each digit can be determined even for collections that have been partitioned in nonstandard ways. (p. 49)

Children’s limited conceptions of multidigit numbers.

As well as accurate conceptual structures, researchers conducting research in

this field have identified a number of common limited conceptual structures held by

children for multidigit numbers. Though misconceptions of place-value concepts

held by children are “very diverse” (A. Sinclair & Scheuer, 1993, p. 200), the

research literature contains references to a cluster of observed behaviours, each

indicating a basic conceptual misunderstanding. Three limited conceptions for

multidigit numbers commonly observed in children are: a unitary concept of

multidigit numbers, a face value construct, and a counting sequence concept. The

conceptions are outlined in Table 2.2, which also shows task behaviour that

illustrates the presence of each misconception.

31

TABLE 2.2. Task Performance Illustrating Limited Conceptions in Place-value Understanding

Limited conception Task Performance Example Illustration

Unitary concept of multidigit numbers: Multidigit numbers seen as unitary collections only.

Student represents a multidigit number as a collection of ones only. For example, 21 is represented as 21 ones only.

21:

Face value construct (a): Tens digits represent single units, not multiples of 10 ones.

Student represents a two-digit number as two sets of units. For example, 43 is represented by 4 ones and 3 ones.

43:

Face value construct (b): Digits representing traded amounts in written algorithms represent their face value only.

Student believes that a carried “1” in a written algorithm represents just one.

67 15 - 4 7 2 8

Counting sequence concept: Each number is one element in the sequence of counting numbers.

Student represents multidigit numbers as elements in counting sequence only. For example: 25 is the number after 24 and before 26.

Limited Conception 1: Unitary concept of multidigit numbers. The unitary

construct is one of the first steps towards understanding the base-ten numeration

system, as described earlier in this section. Single digit numbers are linked both to

single symbols 0 to 9 and to groups of fewer than 10 objects. However, it appears

that many children also retain this conceptual structure for numbers greater than 10

(Fuson, 1990b).

The place-value literature is replete with reports of children who see

multidigit numbers as collections of ones, or single elements, only. In other words,

they see 34, for example, not as 3 tens and 4 ones, but as 34 ones. For example,

Hughes (1995) found that when asked to show $67 with “play money” in $1, $10,

and $100 denominations, some children counted out 67 $1 notes. Miura and

Okamoto (1989) made very similar observations in their study of U.S. and Japanese

students’ cognitive representations of number. The researchers found that certain

participants held unitary concepts for two-digit numbers. These conceptual structures

32

were deduced by researchers from the representations of two-digit numbers which

participants produced using base-ten blocks. If a student showed 28 ones for the

number 28, for example, then the researchers inferred that the student was using a

unitary conception of multidigit numbers. This construct is closely linked to S. H.

Ross’s (1989) whole numeral stage: “[Children’s] cognitive construction of the

whole comes first—the numeral 52 represents the whole amount” (p. 49). Fuson

(1990a, 1990b, 1992) referred to ideas of multidigit numbers held by some children

as collected multiunits. She explained the children’s concepts this way: “The

collected multiunits are collections of single units: A ten-unit item is a collection of

ten single unit items, a hundred-unit item is a collection of one hundred single unit

items, . . . , and so forth” (Fuson, 1992, p. 142). Explaining the considerable

difficulties faced by English-speaking children in linking understanding of multidigit

numbers, their written symbols, and their spoken names, Fuson (1990a) blamed the

common construction by these children of unitary conceptual structures on the

“obfuscation of the underlying tens structure in English number words” (p. 357).

Limited Conception 2: Face value construct. The face value construct is also

a very common conceptual structure among children, according to place-value

researchers. It is defined as the idea that each digit in a multidigit numeral represents

only that number of ones: its face value. S. H. Ross (1989) defined a stage of place-

value understanding at which children exhibited this construct as the face value

stage:

Students interpret each digit as representing the number indicated by its face value. . . . but these objects do not truly represent groups of ten units to students in [this stage]; students do not recognize that the number represented by the tens digit is a multiple of ten. (p. 49)

The presence of the face-value construct points to a critical misconception of

multidigit numbers, but one that may be difficult to detect (S. H. Ross, 1990).

Though it is efficient to compute answers to multidigit questions as if each digit was

a single unit, many children apparently believe that each digit actually represents

only its face value. Researchers who were investigating a variety of mathematical

abilities have reported this construct. These abilities included children’s counting

(Cobb & Wheatley, 1988), representations of two-digit numbers (Miura & Okamoto,

1989; Miura et al., 1993; S. H. Ross, 1989, 1990), comparison of pairs of two-digit

and three-digit numbers (A. Sinclair & Scheuer, 1993), handling two-digit and three-

digit numbers in novel problem-solving exercises (Bednarz & Janvier, 1982), and

33

completing written algorithms (Fuson & Briars, 1990; Fuson et al., 1997; Kamii &

Lewis, 1991).

A task where the presence of the face-value construct is particularly important

is that of carrying out written computation. Though they can correctly carry out the

procedure, many children do not have a good idea of the values they are symbolically

manipulating, and regard each digit as representing only its face value. Fuson and

Briars (1990) called this conceptual structure concatenated single digits: “Even many

children who carry out the algorithms correctly do so procedurally and . . . cannot

give the values of the trades they are writing down” (p. 181). Similarly, Cobb and

Wheatley (1988) found that some students had a different conception of addition

questions when written vertically, compared to a horizontal presentation. They

concluded that the students understood two numbers added horizontally as one

number incrementing the other, whereas in vertical format, the operations were seen

either as separate single-digit tasks, or as separate tens and ones addition tasks.

A paradox needs to be clarified here. Place-value understanding requires a

person to understand the value represented by each digit of a number; however,

people proficient with written algorithms treat operations in each place as single-digit

sums, differences, products, and quotients. In other words, it is quicker, and

cognitively less demanding, to operate on digits in each place as if they were all

units, rather than to keep in mind the value actually represented by each digit as each

step is carried out. By doing this, those proficient in written computation thus take

advantage of the efficiency inherent in the base-ten numeration system’s notation,

referred to by other authors as the “unreasonable power of mathematics” (Fuson,

1992, p. 56), and “the beauty and seeming simplicity of the base-ten number system”

(Sowder & Schappelle, 1994, p. 343). Nevertheless, appreciation of this efficiency

does not develop automatically in students (Sowder & Schappelle), and care is

needed to teach this to students without causing lack of understanding. As Resnick

and Omanson (1987) noted, automatic performance in written arithmetic is

incompatible with continuing reflection on principles underlying the written

algorithms. For example, consider the following addition question using a

conventional written algorithm:

274 + 318

The algorithm is completed by considering a series of three single-digit sums:

34

4 1 2 + 8 + 7 + 3 + 1

This use of single-digit sums to complete a multidigit addition question is an

example of the trade-off involved in use of conventional written algorithms: In order

to achieve efficiency, the quantities represented by the digits in the various places are

ignored. As pointed out by Carpenter, Franke, Jacobs, Fennema, and Empson (1997),

“standard algorithms have evolved over centuries for efficient, accurate calculation.

For the most part, these algorithms are quite far removed from their conceptual

underpinnings” (p. 5). For children who have not developed accurate conceptual

structures for multidigit numbers, ignoring the meaning behind the symbols in this

way has the potential to hide the connections between symbols and the numbers they

represent.

The above comments illustrate the importance in place-value research of

distinguishing between children who operate according to the face-value construct,

and those who understand multidigit numbers according to the base-ten structure or

flexible groupings constructs. This importance is supported by S. H. Ross’s (1989)

comment that “pupils [holding the face value construct] may appear to understand

more than they actually do” (p. 50). Kamii and Lewis (1991) made the same point,

pointing out the inadequacies of standard achievement tests, which “tap mainly

knowledge of symbols” (p. 50), rather than understanding of numbers.

Limited Conception 3: Counting sequence construct. This limited conception,

similar to the unitary concept, was identified by Fuson (1992), who described a

limited understanding of multidigit numbers that she called “sequence multiunits.”

Students having this understanding of multidigit numbers conceive of them as

“[entities] within the number-word sequence: ‘Five thousand six hundred eighty

nine’ is the word after five thousand six hundred eighty eight and the word before

five thousand six hundred ninety” (p. 143). Fuson based her idea on observations of

two methods that certain students used to carry out multidigit addition or subtraction,

using counting procedures based on the position of each number in the sequence of

number names. Students possessing this construct either counted on (or back) from

one number using the counting number sequence by hundreds, tens, and ones; or

counted on or back within each place separately before combining the partial sums or

35

differences. For example, to compute 596 + 132, a student could count either “596,

696, 706, 716, 726, 727, 728” or “500, 600; 90, 100, 110, 120; 6, 7, 8; 728.”

Sources of children’s limited conceptions of multidigit numbers.

Many factors have been cited as causes of students’ difficulties in reaching a

level of understanding of place value that is robust, flexible, and efficient.

Knowledge of students’ common difficulties, and possible underlying causes for

them, is important in the planning of either teaching for or research into

understanding of the base-ten numeration system. Four sources of difficulty are

discussed in this section: (a) cognitive complexity, (b) over-emphasis on rules and

routines, (c) English language number names, and (d) lack of connections.

Cognitive complexity. As noted in the previous subsection, efficient

arithmetic computation is highly routinised, and effectively unthinking: The person

carrying it out is generally not thinking of the values involved (Hiebert, 1988). This

may be part of the reason that much instruction in mathematics is based on

unthinking use of routines. However, in order to check the accuracy of procedures

and to self-correct errors, it is important that a person carrying out computation has

the ability to reconstruct the meanings of the procedures involved, when needed

(Fuson & Briars, 1990).

The difficulty for primary teachers is that for students to be able to carry out

computational procedures while maintaining a mental representation of the quantities

involved may impose a greater processing load than many primary students can

handle. Boulton-Lewis (1993) and Boulton-Lewis and Halford (1992) referred to

Halford’s (1993a) structure mapping theory to explain the cognitive demands placed

on a student in learning about the base-ten numeration system. Boulton-Lewis and

Halford (1992) pointed out that in order to understand place value a student needs to

be able to make mappings at the system mappings level, possible from about 5 years

of age. They cautioned that processing loads will be increased if unfamiliar or

inappropriate analogues are used, and advised that

in order to reduce the load it is necessary to ensure that the child is able to recall automatically the relations between quantity, place value, and any symbolic and concrete representations of the task and uses less rather than more demanding [computational] strategies. (p. 8)

The question of how different concrete representations may be used in the

teaching of place value is addressed in section 2.5.3.

36

Over-emphasis on rules and routines. Several authors have cited premature or

over-emphasis on procedures as having a detrimental effect on children’s number

learning (e.g., Hiebert, 1988). This style of teaching and learning is often called a

“textbook” approach (Cobb & Wheatley, 1988; Fuson, 1990b, 1992; Kamii & Lewis,

1991). The characteristics of this approach include a strict sequence of instructional

steps according to the perceived difficulty of question types; few pictures, often

poorly linked to symbolic representations; and a rule-based approach to computation

(Fuson, 1992, pp. 149-150). These observations underline the advice, summarised in

section 2.4.1, that mathematics teaching today should be based not on the teaching of

rules and procedures, but on teaching for mathematical understanding and the

development of number sense.

English language number names. The irregular system of number names in

the English language is another source of difficulty in learning place-value concepts,

mentioned by many authors (Bell, 1990; Boulton-Lewis & Halford, 1992; Carpenter,

Fennema, & Franke, 1993; Fuson, 1990a, 1992; Fuson & Briars, 1990; Hughes,

1995; G. A. Jones & Thornton, 1993a; Miura & Okamoto, 1989; Miura et al., 1993).

Whereas the base-ten numeration system is used consistently in most nations today,

the spoken number names naturally vary with the language used. The number names

in the English language are not a consistent system across the range of spoken

numbers, and include a number of irregularities, especially in tens place names and

the teen numbers (Fuson, 1992). Similar irregularities also exist in other European

languages (Miura et al. 1993), such as French, in which are found such irregular

number names as quatre-vingt-quinze (“four-twenty-fifteen”) for the number ninety-

five. Such number name systems obscure the grouped tens structure of the base-ten

number system. There is a growing belief that this is a major cause of the difficulties

which European language-speaking students face in learning place-value concepts,

compared to other language speakers, including speakers of Asian (Bell, 1990;

Fuson, 1992) or Maori (Hughes, 1995) languages. Miura and her colleagues (Miura

& Okamoto, 1989; Miura et al., 1993) have claimed that their research comparing

place-value understanding of students who speak Asian and European languages

demonstrates that the respective structures in these languages influence conceptual

structures held by students who speak them. Hughes (1995) suggested that teachers

of English-speaking students incorporate more “transparent” English number names

in their lessons, to aid the students’ place-value understanding. Whether or not this

37

strategy is adopted, any teaching program that aims to develop efficient conceptual

structures for multidigit numbers in English-speaking students must take into account

the particular difficulties introduced by the number names in the English language.

Lack of connections. It is widely reported in the literature that many children

do not connect number symbols to their real-world referents or to number

representations such as blocks (Baroody, 1989, 1990; Hart, 1989). As a result, there

is considerable support for helping children build strong links between numbers and

number representations (Fuson, 1992; Hiebert, 1988; Hiebert & Wearne, 1992;

Resnick, 1987), including support for the use of computer software for this purpose

(Clements & McMillen, 1996; Fuson, 1992; Hiebert, 1984; Hunting & Lamon,

1995). The topic of building connections in place-value teaching through the use of

concrete materials is discussed in greater detail in section 2.5.3.

2.4.3 Analogical Reasoning

Analogical reasoning is the second branch of cognitive research relevant to

the teaching of place value. Analogical reasoning has been claimed to have a

particular importance in the study of thinking and reasoning (Gentner & Toupin,

1986; Goswami, 1992), and to be “the most important of all our reasoning processes”

(Grandgenett, 1991, p. 30). This section discusses the importance of analogical

reasoning as the cognitive mechanism underlying the use of materials such as base-

ten blocks to represent numbers.

Definition of analogical reasoning.

Though there is no single generally accepted definition of analogical

reasoning, several authors (English, 1997; Simons, 1984; Vosniadou & Ortony,

1989) have offered definitions for the term that share a number of essential features.

Vosniadou and Ortony’s (1989) definition will be used here: “Analogical reasoning

involves the transfer of relational information from a domain that already exists in

memory (usually referred to as the source or base domain) to that domain to be

explained (referred to as the target domain)” (p. 6).

Research into children’s analogical reasoning.

Analogical reasoning by children has not received much attention until

comparatively recently. As Goswami (1992) explained,

38

the reasons for this neglect were partly historical. According to piagetian [sic] theory, the ability to reason by analogy was a late-developing skill, emerging at around 11-12 years of age during the “formal-operational” period of reasoning. Younger children were thought to be incapable of reasoning by analogy, and consequently few people investigated their analogical reasoning skills. (p. 3)

However, several more recent studies of children’s analogical reasoning have

put these Piagetian claims in some doubt. Goswami (1992), in particular, argued that

the findings of Piaget and others on this point should be challenged on the

assumptions underlying their research. Goswami argued that when experiments are

designed that ensure that participants fully understand the task and the relations that

exist between terms, even very young children are capable of reasoning analogically.

Successful training in analogical reasoning skills both to children and to adults has

been demonstrated in the work of several researchers (Alexander, White, et al., 1987;

Alexander et al., 1989; Alexander, Wilson, et al., 1987; Bisanz, Bisanz, & LeFevre,

1984; Newby, Ertmer, & Stepich, 1995). Participants included 4- and 5-year-old

children, students aged 9 to 19, college students, and teachers of 4th grade and pre-

school classes. Results showed that the training was effective in each case. In the

case of the teachers, the training effects were found to transfer to the teachers’

students also (Alexander, Wilson, et al., 1987).

Though the application of analogical reasoning to science education has

received much research attention, there has been little study of the use of analogies in

teaching mathematical concepts. As English (1997) stated, “this appears to be a

serious omission, given the important role of analogy in mathematics learning”

(p. 192). Analogies are used by mathematics teachers, to teach a range of

mathematical ideas, including number, place value, and fractions. Some of the few

studies of analogical reasoning and mathematics have been those by Wilson and

Shield (1993) and English (1993, 1997).

Structural mapping theory.

The structural mapping theory of Gentner (1983, 1988, 1989) provides a

useful explanation of the mechanisms involved in analogical reasoning and the use of

mental models.

Definition of structural mapping. The structural mapping theory (Gentner,

1983) explains how commonalties between target and base domains are perceived

when reasoning analogically: “The central idea is that an analogy is an assertion that

a relational structure that normally applies in one domain can be applied in another

39

domain” (p. 156). In other words, the user perceives parallels between target and

base domain, based upon a common relational structure. The two domains are then

perceived to be correspondent, to the extent that relations among members of the two

domains can be mapped from target to base.

Gentner (1983) used Rutherford’s theory that an atom is like a solar system to

demonstrate the idea of structural mapping. In this example, though some attributes

of the solar system components cannot be mapped to an atom (such as colour and

temperature of the sun), key relations between the sun and planets (such as the

central body being more massive than, and attracting, the orbiting body) are mapped

directly onto relations between nucleus and electrons in the atomic domain. Under

the structural mapping theory, this is a general principle: Analogies have few

attribute matches, but many relation matches. Thus analogies can be distinguished

from literal similarity, abstraction, or anomaly (Gentner, 1983). This distinction also

applies to Halford’s (1993a) definition for cognitive representations:

A cognitive representation is an internal structure that mirrors a segment of the environment. The representation must be in structural correspondence to the environment and be consistent. Resemblance between the representation and the environment is not required, and representations are not ‘pictures in the head.’ (p. 69)

Application of structural mapping theory to place-value instruction. The

following major section (section 2.5) describes the use of manipulative materials for

the teaching of place-value concepts; the remainder of this section describes the

theory behind their use based on the structural mapping theory. Concrete materials,

such as bundling sticks and base-ten blocks, used to represent numbers “are

technically analogues [of numbers], and can be analyzed using analogy theory”

(Boulton-Lewis & Halford, 1992, p. 2). As Boulton-Lewis and Halford pointed out,

concrete representations of numbers mirror the structure of the domain of numbers.

Thus, the structure of the visible sticks or blocks is mapped onto the domain of

invisible numbers. Boulton-Lewis and Halford pointed out that the use of concrete

representations of numbers in mathematics teaching requires attention to two

important points. First, the representation itself should accurately model the structure

of numbers; second, children should be familiar with the representation to reduce the

cognitive load entailed in the use of the representation.

40

Base-ten blocks as analogues of numbers.

Base-ten blocks were developed by Dienes (1960), and are “probably the

most commonly used analogues in the teaching of numeration and computation”

(English & Halford, 1995, p. 105). Base-ten blocks qualify as analogues of numbers,

based on Gentner’s (1983) definition of structural mapping given in section 2.4.3.

First, there are no physical attributes that could be mapped from base-ten blocks to

numbers, since numbers are abstract entities. Second, there are a number of

relational similarities that can be mapped from base-ten blocks to base-ten numbers;

three of these are described in the following paragraphs.

The first feature of base-ten blocks that makes them effective analogues of

numbers is the fact that relative sizes of the four blocks map onto the relative values

of the four places represented (English & Halford, 1995; Figure 2.2). Bednarz and

Janvier (1982) described this feature of materials as representing numbers “so that

the rule of grouping is apparent (visible or explicit)” (p. 36). Individual base-ten

blocks are available in only four standard sizes: the one-block, a 1 cm cube; the ten-

block, a rectangular prism 1 cm x 1 cm x 10 cm; the hundred-block, 1 cm x 10 cm x

10 cm; and the thousand block, a 10 cm cube. Each of the three larger blocks has

sawn grooves, at 1-cm intervals, that provide a visual indication of the relation

between each larger block and a number of one-blocks. Thus, the size of each block

in relation to the size of a one-block maps directly onto the value of each place in

relation to the ones place. For example, as 100 is one tenth of 1000, and 10 times 10;

so also a hundred-block is one tenth the size of a thousand-block, and 10 times the

size of a ten-block. Between any pair of the four block sizes, the same mapping can

be made from the relative size of blocks to the relative values of the represented

numbers (see Figure 2.2).

Other materials available for the teaching of mathematics can be

manufactured according to other groupings, or can be so grouped by children using

them. For example, Unifix™ cubes can be grouped arbitrarily in any sized group and

so do not model the base-ten system in particular. Such materials may be termed

unstructured or semistructured analogues of multidigit numbers (English & Halford,

1995), indicating their lack of a built-in structure that directly models the base-ten

numeration system.

41

Figure 2.2. Relationships inherent in base-ten blocks.

Note. Based on figure from Mathematics education: Models and processes (p. 105), by L. D. English and G. S. Halford, 1995, Mahwah, NJ: Erlbaum.

The second mapping from base-ten blocks to the domain of numbers maps

the numerosity of a group of blocks of one size, onto the number represented in the

associated place. A set of blocks of the same size is used to represent a single digit

from one to nine, with the number of blocks being equal to the face value of the digit

represented. For example, 6 hundred-blocks represent the number 600. A

combination of the first two mappings described in this subsection is available in any

representation of numbers with base-ten blocks. In a base-ten block representation of

a number, such as 752, not only is the block representation of the entire number

proportional to its value (compared to a single one-block), but the representation of

any portion of that number—the tens part of it (50), or the hundreds and tens

expressed as tens (750), for example—is also proportional to that portion. Thus when

base-ten blocks are used to represent the steps in a computational algorithm, they do

so in a manner that preserves at every step a valid mapping from the block

representation for each number to its value.

The third mapping that base-ten blocks exhibit is that of trading relations

(Fuson, 1990a, 1992). In carrying out written or mental computation with multidigit

numbers, it often necessary to regroup a portion of a number in one place to another,

generally adjacent, place. This process of trading one-for-ten is essential for the

42

operations of addition, subtraction, multiplication, and division, which are important

components of the primary mathematics curriculum. Base-ten blocks effectively

model the trading process when one block is swapped for 10 of the next smallest

place, or vice versa. In the process, the size of the representation is preserved and so

can be mapped from the materials to the number. This mapping of size relations does

not occur with materials such as coloured chips or an abacus, as each chip or abacus

bead is the same size, making them less useful as analogues of numbers, particularly

early in children’s learning of place-value concepts (English & Halford, 1995).

The above paragraphs demonstrate that base-ten blocks incorporate the

systematicity principle, a further development of the structural mapping theory

introduced by Gentner and Toupin (1986). They defined the term in this way:

The systematicity principle states that a base [source] predicate that belongs to a mappable system of mutually interconnecting relations is more likely to be imported into the target than is an isolated predicate. A system of relations refers to an interconnected predicate structure in which higher-order predicates enforce constraints among lower-order predicates. (p. 280)

As demonstrated, base-ten blocks are capable of at least three different

relational mappings: (a) mappings between the sizes of individual blocks and the

values assigned to places, (b) mappings between the numerosity of a group of similar

blocks and the value of an individual digit, and (c) mappings between traded actions

on blocks and the corresponding regrouping carried out on numbers. These three

mappings together form a system of interrelated relations and so satisfy the

conditions for the systematicity principle described above. Thus, according to

Gentner and Toupin (1986) relations among base-ten blocks are “likely to be

imported into the target” (p. 280), adding further support to their use in teaching of

place-value concepts.

Cognitive load theory.

The cognitive load theory is also relevant to the teaching of place-value

concepts with base-ten blocks, being concerned with the demands placed on

students’ thinking processes by various instructional designs. Sweller (1999) pointed

out that current theories suggest that “we can process no more than about two to four

elements at any given time with the actual number probably being at the lower rather

than the higher end of this scale” (p. 5). In order to simultaneously manage larger

numbers of elements in working memory, it is necessary for learners to develop

schemas that “provide the means of storing huge amounts of information in long-

43

term memory” (Sweller, 1999, p. 11). Sweller defined a schema as “a cognitive

construct that permits us to treat multiple elements of information as a single element

categorised according to the manner in which it will be used” (p. 10). According to

cognitive load theory, base-ten blocks assist children to understand the base-ten

numeration system by helping them form such schemas that relate numerical

quantities, written symbols, and the blocks.

2.5 Teaching Place-value Understanding This section includes three subsections. Broad approaches to the teaching of

place-value understanding recommended in the literature are described in section

2.5.1. The focus is narrowed in section 2.5.2, to concentrate on the widely-stated goal

of helping students to build connections between numbers and number

representations. Section 2.5.3 describes the reported use of a range of concrete

materials in the teaching of place-value understanding, with particular emphasis on

base-ten blocks.

2.5.1 Teaching Approaches

A number of writers have described different ideas of how to teach place-

value concepts. Each of these teaching methods aims to help students to develop

links among number concepts, their real-world referents, and their representations by

symbols or physical analogues. Four recurrent themes evident in the literature on

place-value teaching are described in this section: (a) use of structured materials to

model numbers, (b) use of real-world problems, (c) teaching place-value concepts in

the context of computation, and (d) adopting a constructivist view of learning. These

four themes are by no means mutually exclusive; several authors included more than

one of these themes in their work.

Some writers have advocated a structured approach to teaching place value,

using concrete materials, and especially base-ten blocks (Fuson, 1990a, 1990b, 1992;

Fuson & Briars, 1990). In this approach the teacher continually reinforces the links

among written symbols, number names, and concrete materials. Bednarz and Janvier

(1982, 1988) recommended another approach that focused children’s attention on the

structure of the base-ten numeration system. Their particular focus was on the

groupings inherent in the base-ten numeration system; G. A. Jones and Thornton

(1993a) called this an explicit grouping approach. In their research, Bednarz and

44

Janvier presented students with various explicit groups of objects, often including

multiple groupings, or groups of groups. The objects used included cereal boxes

grouped by six into cases, and baskets of three cases; peppermints in rolls of 10, and

bags of 10 rolls; and paper flowers, each made of 10 sheets of paper, and grouped

into bouquets of 10 flowers. The research showed that some students did not

understand the groupings inherent in the base-ten numeration system, shown by the

fact that they attempted to answer questions involving multiple groupings without

inquiring about the number of objects in each group.

A second theme in the literature on place-value instruction is the use of real-

world problems to help students make links between symbols and real-world

application of mathematics. This approach has been recommended by Bednarz and

Janvier (1982, 1988), Hiebert (1989), and Hiebert and Wearne (1992).

The third theme is shown in a teaching approach recommended by several

writers (e.g., Fuson, 1990a, 1990b, 1992; G. A. Jones & Thornton, 1993b; G. A.

Jones et al., 1994), to teach place value in the context of computation or problem-

solving exercises, and particularly multidigit addition and subtraction exercises.

Several researchers, including Carpenter et al. (1993), Fuson (1990b, 1992), Kamii,

Lewis and Livingston (1993), and S. H. Ross (1989, 1990), argued that rather than

attempting to teach place-value concepts first, as a prelude to the teaching of addition

and subtraction, place-value learning should take place within the context of

computation. Fuson (1992) summarised this idea in her comment that

multidigit addition and subtraction are problem situations that permit crucial attributes of the named-multiunit words and the positional written marks to become evident and thus are excellent contexts within which children can construct place-value understandings. (p. 173)

One aspect of the idea that place-value concepts should be taught in the

context of their use in computation is the advice from several authors that children

should be encouraged to invent their own computational procedures, as a means to

gaining proficiency in understanding and using multidigit numbers. This advice has

been given by Carpenter et al. (1993), Duffin (1991), Kamii and Lewis (1991),

Kamii et al. (1993), Resnick and Omanson (1987), S. H. Ross (1989), Sowder and

Schappelle (1994), and P. W. Thompson (1992).

The fourth theme that emerges from the place-value literature is the use of a

constructivist approach to teaching place value. Some authors (e.g., Cobb, 1995;

Kamii & Lewis, 1991; Kamii et al., 1993) specifically mentioned constructivism as

45

the basis of their work; others (e.g., R. Ross & Kurtz, 1993; S. H. Ross, 1990; P. W.

Thompson, 1992) mentioned the idea of children constructing understanding,

indicating their acceptance of constructivist ideas. G. A. Jones et al. (1994) favoured

what they termed an interactive-constructivist approach, recommending that teachers

engage their students in “negotiated learning” as they solve problems that “challenge

and stretch” their abilities.

It is relevant to point out one view of constructivist teaching of place-value

concepts that excludes the use of concrete materials. Constructivism was a part of

Kamii et al.’s (1993) developmental approach that focused on the internal

construction of meanings in children’s minds. However, unlike most authors in the

field, including others who used constructivist teaching methods, Kamii et al. did not

use concrete materials to support learning. In their view, concrete materials are a

hindrance to children’s development of place-value concepts, because these concepts

derive from mental actions and higher-order constructions rather than from objects in

the external world (p. 201). It appears that Kamii et al.’s view is a minority one at

odds with that of the majority of other writers in the field, who recommend the use of

concrete materials for the teaching of place-value concepts.

2.5.2 Building Place-Value Connections

The findings of cognitive scientists are commonly used to inform the teaching

of mathematics, as explained in section 2.4. Mathematics researchers are typically

interested in the conceptual structures for numbers posited by cognitive scientists and

the relations that these structures have to external representations of numbers,

including written symbols and concrete materials. For example, Hiebert and

Carpenter (1992) described two assumptions in mathematics education research: that

“some relationship exists between external and internal representations” and that

“internal representations can be related or connected to one another in useful ways”

(p. 66). There is widespread support in the mathematics education literature for the

view that teaching place-value concepts involves assisting students to build

connections among numbers, verbal names, and written representations of numbers

(Baroody, 1989, 1990; Clements & McMillen, 1996; Fuson, 1990b, 1992; Fuson &

Briars 1990; Gluck 1991; Hart 1989; Hiebert & Wearne 1992; Merseth 1978; Payne

& Rathmell, 1975; Peterson, Mercer, McLeod, & Hudson, 1989; Peterson, Mercer,

Tragash, & O’Shea, 1987). The idea of “drawing connections between a set of

46

understandings and an appropriate symbol system [is] a central feature of all

learning, regardless of content” (Hiebert, 1984, p. 499); in the case of mathematics

learning, connections have to be made between the student’s understanding of

numbers and the written numeration system.

The development of place-value understanding involves the formation of a

number of interrelated connections, or links, in the student’s understanding. Three of

these links are illustrated in Figure 2.3, which portrays one view of the relations

among numbers, written symbols, and physical models. As already mentioned,

numbers are abstract entities that do not exist in the physical world. However, they

are represented in physical form in two principal ways: (a) through the numeration

system of written symbols and associated procedures (Skemp, 1982) and (b) through

various physical models known collectively as concrete materials. Students need to

understand the links among numbers and these two different sets of referents. Figure

2.3 shows that whereas written symbols represent numbers in an abstract, socially-

constructed manner (Cobb, 1995; S. H. Ross, 1990), concrete materials model

numbers analogically in a physical form that is closer to young children’s experience

(Hiebert, 1988; Hiebert & Carpenter, 1992).

Figure 2.3. Relationships among numbers, written symbols, and concrete materials.

It is clear that, though written symbols and concrete materials can both be

used to represent numbers (Hiebert, 1988), the relationships between numbers and

the two representations are of a different character. As explained in section 2.4.3,

concrete materials such as base-ten blocks model numbers analogically (Boulton-

Lewis & Halford, 1992): There is a direct relationship between the size of a number

and the size, numerosity, or both, of its physical representation. In contrast, written

symbols are “cultural tools” (Cobb, 1995, p. 380), “the shared symbol systems of

47

mathematics” (Putnam et al., 1990, p. 70), and only represent numbers as a function

of their socially-agreed meanings (Kamii & Lewis, 1991). There is no sense, for

example, in which the written symbol “7” or the words “seven,” “sept” (French), or

“qi” (Chinese) represent an actual number seven apart from the convention that

people using each of them agree that it stands for that number. As J. H. Mason

(1987) pointed out, the “symbolizing process” by which symbols stand for numbers

is often forgotten by teachers, and is never understood by many students (p. 76).

Students need to be made aware of the parallel relationship that is meant to

exist between symbols and concrete materials, so that they can take advantage of the

implied connections which exist between the two systems of representations. Hiebert

and Wearne (1992) summarised the idea that students need to see the connections

between written and material representations of numbers with their statement that

from a cognitive science point of view, it can be argued that building connections between external representations supports more coherent and useful internal representations. . . . Different forms of representation for quantities, such as physical materials and written symbols, highlight different aspects of the grouping structure, and building connections between these yields a more coherent understanding of place value. (p. 99)

Mathematicians have been concerned with helping children make links

between symbols and the ideas they represent for many years (Hiebert, 1984, p. 500).

For example, over half a century ago Van Engen (1949) wrote of the need for

teachers to consider how to help their students develop meanings for arithmetic.

Hiebert (1984) stated the view that making links is central to mathematics learning

and that “although it is not surprising that students have trouble connecting form and

understanding, the effect of not making these connections may be one of the most

serious problems in mathematics learning” (p. 499). Elsewhere, Hiebert (1989) again

stressed the centrality of this process to mathematics education. He stated that “it is

impossible to overemphasize the importance of helping children establish

connections between quantities and numerals and between actions on quantities and

operation signs” (p. 40).

Several authors have referred to a gap that appears to exist in the minds of

many children between the two systems of number representations—written symbols

and concrete models—and have referred to the need for teachers to work at “bridging

the gap” (e.g., Gluck, 1991; Hart, 1989; Hiebert, 1988). Figure 2.4 portrays what

Gluck referred to as “the very large gap between manipulatives and paper-and-pencil

48

tasks” (p. 10). Hart reported the same gap in preliminary results of a research project

entitled “Children’s Mathematical Frameworks.” She concluded from the poor

performance of the project’s participants that they were not making the needed

connections between manipulatives and written procedures. She summed up her

belief that students see the use of manipulatives and written procedures as

disconnected processes in her suggested subtitle for the project report: “Sums Are

Sums and Bricks Are Bricks” (p. 139). In her final paragraph Hart summed up her

thoughts:

Many of us have believed that in order to teach formal mathematics one should build up to the formalization by using materials, and that the child will then better understand the process. I now believe that the gap between the two types of experience is too large, and that we should investigate ways of bridging that gap by providing a third transitional form. (p. 142)

Figure 2.4. Conceptual gap between written symbols and concrete materials.

For many years authors have recommended a bridging approach to teaching

place value, that aims to bridge the gap between written symbols and concrete

materials. However, it needs to be realised that solutions to the problem seem rather

elusive. Several authors have warned that bridging the gap between numbers and

symbols, or numbers and materials “will not occur automatically” (Merseth, 1978,

p. 61). Fuson (1992) found that, even in the presence of appropriate concrete

materials, without appropriate guidance from a teacher to link blocks and written

symbols, some children did not make the relevant connections, and errors resulted.

Teaching strategies for improving these links have been suggested by several

authors. One such strategy is to allow students more time to construct links (e.g.,

Carpenter et al., 1993; Gluck, 1991; R. Ross & Kurtz, 1993). As Hiebert (1989)

pointed out, “connecting symbols with understanding is a difficult intellectual task,

49

and does not occur quickly” (p. 40); “students need time and many opportunities to

construct the connections for themselves” (p. 42). A second strategy is to strongly

emphasise connections between symbolic and concrete representations (Fuson, 1992,

p. 165) so helping to draw students’ attention to errors in their written computation.

Fuson explained that “when experimenters forced children to connect … marks

procedures to the blocks, the multiunit quantities always could help the children self-

correct the incorrect marks procedure” (p. 165). In fact even teachers who were

“forced” to link closely written symbols and blocks improved their understanding of

the represented meanings. Elsewhere, Fuson referred to the need to make “constant

use of the three sets of words” for number names, block names, and digit names

(Fuson & Briars, 1990, p. 182) and stated that links must be made “very tightly and

clearly” (Fuson, 1990b, p. 277). A third strategy for bridging the conceptual gap

between materials and symbols is to use an intermediate representation: This idea is

portrayed in Figure 2.5.

Figure 2.5. The use of transitional forms to bridge the gap between written symbols and concrete materials.

Various intermediate representations have been suggested to strengthen the

link between written symbols and concrete materials. Three suggestions for items to

bridge the gap between concrete materials and written symbols are (a) using material

that links concrete materials and symbols (Gluck, 1991), (b) pictorial representations

(Baroody, 1990; Peterson et al., 1987, 1989), and (c) computer-generated

50

representations (Champagne & Rogalska-Saz, 1984; Clements & McMillen, 1996; P.

W. Thompson, 1992).

First, Gluck (1991) devised a teaching method that involved the use of a

“place-value board” incorporating flip-over number labels for each digit and base-ten

blocks. Gluck claimed that this material could be used “to take students step by step

from the concrete, through the semi-concrete, and on to the abstract stage of

development” (p. 12). It is not clear what Gluck meant by the “semi-concrete” stage;

it appears that she was referring to activities in which students used base-ten blocks

and flip-books at the same time. The key idea behind Gluck’s place-value board is to

mirror actions on the blocks with changes in the written symbols, as also

recommended by Fuson (1992).

Second, the use of pictorial representations of numbers as intermediaries

between concrete materials and written symbols has a long history, going back at

least to Bruner (1966). Baroody (1990) stated that some writers, including Bruner,

hypothesised that use of pictorial models was “a necessary bridge between concrete

and abstract embodiments” (p. 283). Peterson et al. (1987, 1989) designed two

teaching experiments according to what they claimed was a “generally accepted

hierarchy for presenting a new skill [that] follows a concrete to abstract continuum”

(Peterson et al., 1989, p. i). They found success in teaching place value to students

with learning disabilities using a concrete-semiconcrete-abstract sequence: using

one-inch cubes, pictures of place-value sticks and cubes, and worksheets without

pictures, respectively. However, Fuson (1990a) argued that such a sequence was

based on a faulty understanding of concrete and symbolic representations of number:

The use of Bruner’s concrete-pictorial-abstract continuum in this context ignores the fact that the blocks and the written marks are not endpoints on a single continuum: They are structurally different systems that must be connected. Pictures have the same properties as the blocks (and different properties from the marks). . . . It is not clear at this time what, if any, advantages are provided by pictures, and there are definite disadvantages. (pp. 390-391)

A third suggestion for bridging the gap is to use computer-generated

representations of numbers. The software used in this study, like several other place-

value software applications described in the literature (Ball, 1988; Champagne &

Rogalska-Saz, 1984; Clements & McMillen, 1996; P. W. Thompson, 1992),

incorporates pictures of base-ten blocks. The connections made between these

pictures and the on-screen number symbols are provided by the software, so that

51

changes in one representation are reflected quickly in the other representation. Three

software applications that modelled base-ten blocks are described in Appendix A,

and compared to the software application designed for this study.

2.5.3 Use of Concrete Materials

Many types of concrete materials have been mentioned in the literature as

being appropriate for use in teaching place-value concepts (e.g., Baroody, 1990;

Bednarz & Janvier, 1982, 1988; Clements & McMillen, 1996; English & Halford,

1995; Hiebert & Carpenter, 1992; Hiebert & Wearne, 1992; Howard, Perry, &

Conroy, 1995; Nevin, 1992; Peterson et al., 1987; S. H. Ross, 1989; C. Thompson,

1990). These materials include

1. a wide variety of objects (including pictures of objects) that may be

used singly as counters, including wheels, lollies, flowers, or beans;

2. materials capable of being grouped into groups of 10, 100, and so on,

including bundling sticks, Unifix™ cubes, Multilink™ material, cereal

boxes, and paper flowers;

3. materials that include proportionally sized representations for ones,

tens, hundreds, and so on, including bean sticks (wooden sticks each

with 10 beans glued onto it), string lengths, and base-ten blocks;

4. materials that include representations for various places that are

distinguished from each other by colour or some other arbitrary feature,

including play money and coloured chips;

5. Cuisenaire rods, that are rods of different lengths and different colours

representing numbers from 1 to 10; and

6. materials that illustrate the sequence of number symbols, including

hundreds boards.

Some indication of the extent of use of concrete materials in mathematics

teaching is given by the results of studies in which researchers surveyed teachers

about their use of a range of concrete materials. The first, by Gilbert and Bush (1988)

surveyed grade 1, 2, and 3 teachers in 11 states of the U.S.; the second, by Howard et

al. (1995), surveyed 249 primary teachers in a metropolitan education region in New

South Wales. Both studies showed frequent use of concrete materials by the surveyed

teachers. Gilbert and Bush found that 65% of respondents reported using concrete

materials at least once per week; in Howard et al.’s study 62% of respondents

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reported using concrete materials “often,” with less than 1% of teachers reporting

that they did not use them at all. International data providing indirect evidence of the

use of concrete materials was provided by a report of the primary school phase of the

Third International Mathematics and Science Study [TIMSS]. Over 90% of teachers

of Grade 4 students from every one of 24 nations surveyed agreed that “more than

one representation (picture, concrete material, symbol, etc.) should be used in

teaching a mathematics topic” (Mullis et al., 1997, p. 151). The most common reason

given by teachers in the Howard et al. (1995) study for using concrete materials was

that “they benefit children’s learning,” chosen by 96% of teachers surveyed (p. 6).

The results of these studies make it clear that primary teachers collectively spend a

lot of time using a variety of concrete materials in mathematics lessons, and that

teachers believe that concrete materials have a beneficial effect on the children’s

learning of mathematics. However, results of research into this belief have been

equivocal (Hunting & Lamon, 1995; P. W. Thompson, 1992, 1994). This point is

discussed further in the following subsection.

The material used most often by teachers in the Howard et al. (1995) study

was base 10 material, used by 84% of respondents. The term “base 10 material” was

not defined by Howard et al., but is assumed to refer to base-ten blocks. As noted by

Howard et al., the finding that “number material”—base 10, Multilink, and Unifix—

is used more than any other is “hardly surprising given that the syllabus and many

commercial mathematics programs encourage the use of such material” (p. 6). These

findings underline the importance of research into the use of base-ten blocks, such as

that reported in this thesis.

Research into learning with base-ten blocks.

Despite the very common use of base-ten blocks in primary classrooms

(Gilbert & Bush, 1988; Howard et al., 1995), several authors have pointed out the

lack of consensus in results of research into number learning using base-ten blocks

(e.g., Hunting & Lamon, 1995; P. W. Thompson, 1992, 1994). As Thompson (1994)

pointed out, some research (such as that by Resnick & Omanson, 1987) has shown

little benefit from use of base-ten blocks for students learning place-value concepts,

whereas other studies (e.g., Fuson & Briars, 1990) did show significant gains in

student learning. Hunting and Lamon (1995) noted that results from several studies,

including a meta-analysis by Sowell (1989), “suggest that there is a large host of

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variables influencing the use of didactic materials, among these, type of material,

length of time used, teacher training, age of the students, whether students or teacher

chose the manipulative” (p. 55). As this statement demonstrates, questions of why

base-ten blocks may be effective in some cases and not in others are not easily

answered. P. W. Thompson (1994) suggested that

these contradictions [among findings of different studies] are probably due to aspects of instruction and students’ engagement to which studies did not attend. Evidently, just using concrete materials is not enough to guarantee success. We must look at the total instructional environment to understand effective use of concrete materials—especially teachers’ images of what they intend to teach and students’ images of the activities in which they are asked to engage. (p. 556)

One aspect of the effective use of place-value materials that has been

mentioned by several authors (Baroody, 1989; Hunting & Lamon, 1995; P. W.

Thompson, 1994) is students’ engagement with learning activities, that Hunting and

Lamon referred to as “cognitive engagement in sense making.” As discussed in

section 2.2.1, a prominent issue in mathematics education at present is the

development of number sense. In the context of the teaching of place-value concepts,

it is widely agreed that students must actively and sensibly consider the quantities

represented by place-value materials as they use them (e.g., Resnick & Omanson,

1987). Otherwise there is a risk that “just as with symbols, pupils can learn to use

manipulatives mechanically to obtain answers” (Baroody, 1989, p. 4; see also

Clements & McMillen, 1996).

Comments such as those reported in this subsection demonstrate a need for

further information about how students learn place-value concepts, and in particular

how that learning is influenced by materials such as base-ten blocks. As mentioned

by several authors, the use of concrete materials to teach place-value concepts has

great theoretical and intuitive appeal (e.g., Howard et al., 1995; Hunting & Lamon,

1995; Perry & Howard, 1994; P. W. Thompson, 1992, 1994). In light of this

widespread belief among educators that concrete materials should be effective for

teaching place-value ideas, there is a need for research that endeavours to find

reasons why such effectiveness is not always demonstrated. One aspect of this

research is discussed in the following subsection: There are now available a number

of software titles that model numbers by pictures of base-ten blocks; one hope held

for such software is that it may help overcome some of the drawbacks of physical

blocks.

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Computer-generated models of numbers.

In recent years a number of software applications have been developed to

model numbers and numerical relations on a computer display (Ball, 1988;

Champagne & Rogalska-Saz, 1984; Clements & McMillen, 1996; Hunting, Davis, &

Pearn, 1996; Hunting & Lamon, 1995; Rutgers Math Construction Tools, 1992; P.

W. Thompson, 1992). Three arguments in favour of the use of computer software to

model numbers are discussed in this subsection. Writers have argued that software

can (a) model numbers just as effectively as physical blocks, (b) provide a “cleaner”

form of manipulative, and (c) provide features not available with physical materials.

The first argument in favour of computer software to model numbers is based

on a view that physical models are not effective purely as a result of their tactile

properties, and that therefore software representations can be just as effective as

numerical models. “Computers might supply representations that are just as

personally meaningful to students as are real objects” (Clements & McMillen, 1996,

p. 271). Clements and McMillen argued for a reappraisal of what constitutes a

concrete manipulative in the context of teaching numeration ideas to children. They

asked the question “What does concrete mean?” (p. 270), and concluded that the

tactile, sensory nature of physical manipulative materials is not what makes them

useful for teaching about number. They argued that physical manipulation of

materials by children does not guarantee that they will generate the mental images

that their teachers expect, and that mathematical meaning is not contained within

physical materials (see also Hiebert et al., 1997; Holt, 1964; Hunting & Lamon,

1995; Perry & Howard, 1994; P. W. Thompson, 1994). Clements and McMillen

argued further that

mathematics cannot be packaged into sensory-concrete materials, no matter how clever our attempts are, because ideas such as number are not “out there.” As Piaget has shown, they are constructions—reinventions—of each human mind. “Fourness” is no more “in” four blocks than it is “in” a picture of four blocks. The child creates “four” by building a representation of number and connecting it with either real or pictured blocks. (p. 271)

A second argument advanced for the use of computer representations of

numbers is that they may provide “cleaner” manipulatives (Clements & McMillen,

1996). The difficulties that children have with physical manipulatives are sometimes

due to their “messy” nature. Students may miscount blocks, be distracted by

extraneous features of the blocks, or otherwise use them inaccurately (Champagne &

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Rogalska-Saz, 1984). Champagne and Rogalska-Saz noted that conventional

physical materials could distract students from the task at hand and cause difficulties

for teachers with managing materials; Hunting and Lamon (1995) and Touger (1986)

noted similar student difficulties produced by features of particular materials used to

represent numbers. In each case, it appears that properties of the materials intruded

on the student’s understanding of the mathematical relations the materials were

meant to model. As Clements and McMillen (1996) noted in relation to base-ten

blocks,

actual base-ten blocks can be so clumsy and the manipulations so disconnected one from the other that students may see only the trees—manipulations of many pieces—and miss the forest—place-value ideas. The computer blocks can be more manageable and “clean.” (p. 272)

The third argument in favour of computer models of numbers is that software

can incorporate features not possible with physical models (NCTM, 1998, p. 112).

Clements and McMillen (1996) compared physical blocks with computer-generated

blocks, and noted several aspects of computer materials that either improved on

features, or added features not available with physical materials. They noted

advantages of computer materials including flexibility of presentations, the ability to

store and retrieve configurations, the provision of aural and visual feedback, and the

capability to record student actions (Clements & McMillen, 1996, pp. 272-273).

Appendix A includes a further discussion of features of computer materials, in

relation to the software designed for this study.

2.6 Computers and Mathematics Education The purpose of this section of the thesis is to raise a number of issues from

the literature of particular relevance to the study of learning effects of computer

software. Four issues are addressed in this section: claimed educational benefits of

modern computers, features of software design that have the potential to enhance

mathematical learning, design considerations, and research into the use of computers

in mathematics education.

2.6.1 Claimed Benefits of Computers

Recent technological advances in hardware and software design are claimed

to have a number of claimed educational benefits. Three benefits of particular

relevance to this study are discussed in the following paragraphs: (a) improvements

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in students’ learning, (b) the promotion of interaction among students, and (c)

representation of conceptual relations in a knowledge domain (section 2.6.2).

Learning benefits.

A number of benefits for children’s learning have been claimed for

educational use of computers. Some of these general benefits are summarised here;

benefits that specifically have to do with cognition are described in section 2.6.2.

Fletcher-Flinn and Gravatt (1995) listed several advantages for learners provided by

computers, including

the presentation of realistic problems requiring interactive hypothetical-deductive reasoning, immediate feedback and self-evaluation, opportunities for collaborative learning in small groups, and ease of teacher monitoring and control. . . . Well-designed programs delivered by a computer can provide all of these benefits and, in addition, seem to be enjoyed by learners as shown by their positive attitudes and higher expectations about CAI [Computer-Aided Instruction]. (p. 232)

One general benefit claimed for computers is that computers may “provide

learning experiences not available by more ordinary means” (Champagne &

Rogalska-Saz, 1984, p. 44; see also NCTM, 1998, p. 96; NCTM, 2000, p. 25).

Similarly, Clements and McMillen (1996) suggested that in selecting software to

teach mathematics, teachers choose “computer manipulatives that . . . go beyond

what can be done with physical manipulatives” (p. 277). One issue addressed in this

study is the effects that features available only in computer software have on

students’ place-value learning.

Promotion of student interaction.

One benefit claimed for computers in classrooms is the promotion of student

interaction. The first type of interaction that has been claimed as the result of the

educational use of software is interaction between the user and the software (Akpinar

& Hartley, 1996; Helms & Helms, 1992; Kozma, 1994b; Stedman, 1995; Ullmer,

1994). Ullmer believed that the nature of interactive software required users to

change the way they learn:

Users of such [interactive] systems cannot ignore the technology and focus only on the content; a new level of instrumentality that may affect learning has been imposed on them. Consequently, the manner in which they perceive their relationship to the medium is invariably changed. But in this highly responsive environment, they gain increased control over their own learning activities and enjoy a more constructive role in learning. The shift in the learner’s role makes interaction, rather than passive assimilation, the key learning process. (p. 28)

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Kozma (1994b) described learning with a computer as a “complementary

process” (p. 11), in which both the user and the software construct representations

and perform procedures. This embraces the idea of “distributed cognitions”

(Jonassen, Campbell, & Davidson, 1994), in which the computer and its user form a

partnership, with the computer “assuming a significant part of the intellectual burden

of information processing” (K. E. Sinclair, 1993, p. 21).

2.6.2 Cognitive Aspects of Computer Use

One of the most prominent arguments in the literature in favour of the

educational use of computers is that they directly aid learners’ thinking, in ways not

possible with other teaching methods. This section addresses issues of cognitive

effects of computers on learners.

Computers as cognitive tools.

A number of authors have written that computers should be thought of as

cognitive tools of one sort or another (Edwards, 1995; Jonassen, 1995; McArthur,

1987; Pea, 1985; Salomon, 1988; K. E. Sinclair, 1993). Pea noted that it had been

common for proponents of computers in learning to claim that computers amplified

human thinking. Pea, however, declared that though efficiency may be one result of

learning with computers, other benefits were the result of reorganisation of thinking,

fostered by the software. Clark (1994) similarly maintained that the principal use of

software should be to support cognitive processes.

As mentioned earlier, some computer software has been criticised for its

behaviourist foundations, and for continuing a transmission model of teaching and

learning (Jonassen, 1995; Pea, 1985; K. E. Sinclair, 1993). Several authors have

proposed that educational software be designed so that instead of being used to

transmit knowledge, it constitutes a cognitive tool that can extend and develop a

student’s cognitive abilities (Jonassen, 1995; McArthur, 1987; Salomon, 1988).

Salomon explained this point in this way:

It is often said the computer-based tools can extend not human muscle or sensory functions, but cognitive, symbolic ones. To be a bit more specific, computer-based tools extend cognition to accomplish functions the cognitive apparatus could never accomplish on its own. (p. 129)

Similarly, in place of software that attempts to “control all learner

interactions” (p. 61), Jonassen (1995) suggested that computers be used as

58

intellectual partners or cognitive tools that “support, guide, and extend the thinking

processes of their users” (p. 62). One aspect of support for students’ thinking that

computers may provide is in the provision of an environment that encourages “an

active, experimental style of learning” (Cohen, Chechile, Smith, Tsai, & Burns,

1994, p. 237). McArthur (1987) stated that computers enable students to test “a wide

range of hypotheses . . . [which is] . . . an important way to exercise misconceptions

and learn” (p. 192). McArthur commented that in this respect computers are far

preferable to “the traditional paper-and-pencil medium [that] tacitly encourages the

students to think of such changes as mistakes to be avoided. On the contrary, the

ability to try out hypotheses rapidly, especially incorrect ones, is central to learning”

(p. 194). This feature is incorporated in the software used in this study; with little

effort users can quickly try different number representations to test their ideas.

Representation of conceptual domains.

Many writers have claimed that computers can benefit learning by

representing relations inherent in a conceptual domain (Babbitt & Usnick, 1993;

Becker & Dwyer, 1994; Bottino, Chiappini, & Ferrari, 1994; Cohen et al., 1994; De

Laurentiis, 1993; Edwards, 1995; Marchionini, 1988; Parkes, 1994). Most school

subject areas, including mathematics, involve the understanding of abstract concepts

and relations that exist among elements of the domain. A number of software

designers have used computer software to represent conceptual domains, and to

illustrate important conceptual relations, in ways that are claimed to improve

students’ understanding of the domains. This claimed benefit is widely reported in

the educational software literature, and is an important aspect of this study. These

software applications present what Parkes (1994) described as “a manipulable

problem space representation” (p. 199).

In representing ideas with computer software, designers often include

multiple representations of an idea (NCTM, 1998, p. 112). In so doing, it is hoped

that connections among elements of a domain can become evident to students

(Babbitt & Usnick, 1993; Becker & Dwyer, 1994). As De Laurentiis (1993) asserted,

excellent educational software will make explicit the associations in the body of knowledge that is being taught. This simplifies the student’s task of integrating this new body of knowledge into his or her own mesh of concepts. The software should also make it possible for the student to explore the associations, therefore enhancing his or her own mesh of concepts, and building an individualized representation of the world. (p. 7)

59

Software embodying this principle have been developed to teach topics

including theorem-proving problem solving (Parkes, 1994), high school chemistry

(Kozma, 1994a), common fractions (Babbitt & Usnick, 1993), and place value (P.

W. Thompson, 1992). The software designed for use in this study includes pictorial,

symbolic, and verbal representations of numbers that are linked closely together, in

an attempt to help student users develop their own conceptual links in the way

described here by De Laurentiis (1993).

2.7 Chapter Summary; Statement of the Problem The literature review presented in this chapter gives the background to the

problem investigated, stated below. Specifically, there are five findings from this

literature review that undergird the investigation described in this thesis: (a) Various

authors continue to encourage mathematics educators to develop meaningful

understanding and number sense in their students, (b) teaching of place-value

concepts is an important foundation for later mathematical study, (c) the teaching and

learning of place-value concepts is difficult and incompletely understood, (d)

computer technology appears to offer the promise of more effective teaching of

abstract domains, through its capabilities of presenting information in connected

ways, and (e) there is a need for up-to-date information about learning with computer

technology.

These findings lead to the statement of the problem for this study: How do base-ten

blocks, both physical and electronic, influence Year 3 students’ conceptual structures

for multidigit numbers? The investigation design used to address this problem is

described in chapter 3.

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Chapter 3: Methods

3.1 Chapter Overview This study investigated the development of understanding of place-value

concepts as students used either physical or electronic base-ten blocks, each of which

modelled numbers using a structured base-ten representation. In particular, this study

was planned to identify the Year 3 participants’ conceptual understandings of two-

digit and three-digit numbers, before, during, and after a program of 10 teaching

sessions. The study had five phases, which were trialled in a small pilot study:

selection of students, first interview, software training session, teaching program, and

second interview.

As shown in chapter 2, problems in teaching and learning place-value

concepts are very common, though they have been the subjects of discussion and

research for several decades. The current study was designed to contribute to this

discussion, by evaluating an innovative method of teaching place value with

appropriate computer software and comparing this with a conventional method using

base-ten blocks. The study design centred on a detailed descriptive analysis of what

happened as students used computer software or base-ten blocks to answer place-

value questions. Analysis of data from videotapes, participants’ written work, and the

researcher’s written records enabled discussion of differences between the effects of

using the software and the effects of using conventional base-ten blocks.

3.2 Aims of the Study The research question of the study was stated in section 1.3, and is repeated

here:

How do base-ten blocks, both physical and electronic, influence Year 3

students’ conceptual structures for multidigit numbers? Within the context of Year

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3 students’ use of physical and electronic base-ten blocks, the following specific

issues were of concern:

1. What conceptual structures for multidigit numbers do Year 3 students

display in response to place-value questions after instruction with base-

ten blocks?

2. What misconceptions, errors, or limited conceptions of numbers do

Year 3 students display in response to place-value questions after

instruction with base-ten blocks?

3. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in instruction with physical and

electronic base-ten blocks?

4. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in students’ achievement in

numeration?

3.3 Variables Two variables were controlled in pursuit of the above aims: mathematical

achievement level and mode of number representation. They were operationally

defined as follows:

3.3.1 Mathematical Achievement Level

This is defined as the level of each student’s mathematical achievement as

determined from results of the previous year’s The Year 2 Diagnostic Net

(Queensland Department of Education, 1996; hereafter referred to as the Year 2 Net).

The Year 2 Net is a diagnostic instrument administered by Year 2 class teachers in

every Queensland state school, and involves the diagnosis of mathematical abilities

in a range of areas including place value. Level of achievement was defined as high,

medium, or low, by dividing the available group of Year 3 students into thirds.

Students from the top third and bottom third were selected for involvement in the

study, to investigate any differences in place-value understanding, including the use

of representational materials, that related to the participants’ level of mathematical

achievement.

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3.3.2 Number Representation Format

The number representation formats used by participants in teaching sessions

were: (a) physical base-ten blocks, materials in hundreds, tens, and ones sizes that

can be manipulated by hand; or (b) electronic base-ten blocks, computer software

that models numbers using written symbols, verbal names, and pictures of base-ten

blocks that can be manipulated on-screen.

3.4 Data collection and analysis. A working definition for place-value understanding, given in chapter 2, has

been used to guide data collection and analysis: A student possessing place-value

understanding is able to use the place-value features of the base-ten numeration

system to form accurate, flexible conceptual structures for quantities represented by

written numerical symbols; the student is able to manipulate numerical quantities in

meaningful ways to answer mathematical questions.

The methodology adopted for this study was a combination of clinical

interviews and teaching experiments (Hunting, 1983). Responses of participants were

analysed as they related to the research sub-questions listed previously. Specifically,

evidence was collected as it related to participants’ conceptual structures for

numbers; misconceptions, errors or limited conceptions regarding numbers; and

effects on participants’ thinking about numbers that were apparently influenced by

features of each number representational format. The evidence principally came from

transcripts of videotapes of interviews and teaching sessions, supplemented by

researcher’s notes, participants’ written working, and software audit trails.

3.5 Design Issues This section comprises descriptions of assumptions underlying the design and

the theoretical stance taken with regard to five dimensions of research design.

3.5.1 Assumptions

Three major assumptions underlie the research described here. These

assumptions relate to how numbers are understood, how mathematics is learned, and

how a person’s thinking may be studied.

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Understanding of numbers.

The first assumption relates to the question of how people understand

numbers. Although numbers are generally considered to be abstract notions with no

physical objective existence, it is assumed that most people by their actions treat

numbers as entities, indicating that, for them, numbers do exist in some form (Sfard,

1991; Sfard & Thompson, 1994). A person’s conceptions of numbers are believed to

form a system that has a structural form and that incorporates rules by which the

conceptions may be manipulated (Ohlsson et al., 1992; Resnick, 1983). These

conceptions are also influenced by the person’s interactions with other people, and so

form part of the shared understanding of numbers held by the person’s social group

(Cobb & Yackel, 1996; Cobb et al., 1992).

Learning of mathematics.

The second assumption is that a person learning any topic has to construct

personal (and therefore to some extent unique) models of that topic. In learning

mathematics in particular, students are assumed to develop internal models, or

conceptual structures, for numbers (Fuson, 1990a, 1992). It is assumed that these

conceptual structures are influenced by interaction with physical, external

representations for numbers. Furthermore, there is assumed to be some relationship

between internal and external numerical representations, meaning that manipulation

of one has an effect on the other (Hiebert & Carpenter, 1992; Putnam et al., 1990).

Study of thinking.

The third assumption is that the nature of internal representations of numbers

may be deduced from a person’s responses to particular mathematical tasks (Resnick,

1983, 1987). This assumption is basic to cognitive research; it is assumed that a

person’s actions and speech are partially the product of internal mental structures

possessed by that person. Thus is it assumed that by studying a person’s actions, the

internal structures they hold for the domain under consideration may be deduced.

3.5.2 Theoretical and Methodological Stance

There are a number of theoretical and methodological considerations which

underlie this study. The study uses a design that may be described according to its

relation to five aspects of research methodology, illustrated in Figure 3.1 as continua

between pairs of opposing terms. The five aspects of the study’s design are the (a)

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underlying paradigm, (b) aim with regards to theory, (c) methodology, (d) type of

data collected, and (e) researcher’s role. The following paragraphs relate to Figure

3.1 and describe the study with reference to each dimension.

Figure 3.1. Dimensions of research design.

Paradigm.

The first dimension on which this study may be described is the question of

underlying paradigm. Paradigms that underlie research may be placed on a

continuum that extends from positivism on one side to non-positivism on the other.

Other terms have been used in opposition to positivism by various authors, including

the constructivist paradigm (Guba & Lincoln, 1989), phenomenological inquiry

(Patton, 1990), and the qualitative paradigm (Creswell, 1994); in this discussion the

term non-positivism is used.

The question of underlying paradigm needs to be addressed because of its

bearing on the choice of data collection and analysis methods. Rossman and Wilson

(1985) described “three distinct perspectives about combining methods” that they

labelled “the purist, the situationalist, and the pragmatist” (p. 629). As Rossman and

Wilson explained, the question of paradigm is considered by purists to be of vital

importance. Purists (evidently including Guba & Lincoln, 1989, and Creswell, 1994)

believe that one’s paradigmatic view must necessarily determine the research

methods to be used, because quantitative and qualitative methods derive from

“different, mutually exclusive epistemologic and ontologic assumptions about the

66

nature of research and society” (Rossman & Wilson, 1985, p. 629). However, this

view was contradicted by Miles and Huberman (1984), who stated that the two

positions “constitute an epistemological continuum, not a dichotomy,” and that

“epistemological purity doesn’t get research done” (p. 21). As explained later,

choices of method used for the current study were based on such a pragmatic view of

research design (Patton, 1990; Rossman & Wilson, 1985). Therefore, questions of

underlying paradigm are not given further discussion here.

Aim.

The second dimension of description in Figure 3.1 indicates whether the aim

of the study is to test a theory or theories proposed in advance of the collection of

data, or to generate new theory as a result of data analysis. Some research, especially

when conducted from a positivist perspective, sets out to propose a theory or theories

based on a review of literature and then to test those theories so that they may be

confirmed or refuted. Conversely, a strictly qualitative study generally commences

without any pre-conceived ideas of the likely results of the planned investigation, the

researcher expecting theory to emerge as the study proceeds (Creswell, 1994). This

study uses an adaptation of this approach described by Creswell (1994), in which the

researcher “advances a tentative conceptual framework in a qualitative study early in

the discussion” (p. 97). Theoretical models of children’s conceptual structures for

numbers were identified in the review of literature (section 2.4.2). These conceptual

structures have been used as starting points in the data analysis phase of the study

and have been compared with conceptual structures emerging from the data. These

two sources of data, the literature review and the data collection phase of the study

itself, have been compared and analysed in relation to each other for the purposes of

cross-validation (Wiersma, 1995).

Methodology.

Research studies may be described according to their overall methodology,

and located on a continuum from naturalistic inquiry on one hand, to experimental

research on the other. As already mentioned, some researchers believe that research

methods should be chosen to match the paradigm view that the researcher holds. For

example, Guba and Lincoln’s (1989) work implies two strongly-held assumptions:

(a) Positivism is an inadequate theory of the world and how things happen and

(b) use of quantitative, experimental, research methods is antithetical to the non-

67

positivist paradigm. Therefore, they argued that qualitative research is the only viable

option for a researcher studying social phenomena. However, this view has been

disputed by others (e.g., Patton, 1990; Yin, 1994). Yin argued against distinguishing

between qualitative and quantitative research on the basis of opposing philosophical

beliefs and contended that “there is a strong and essential common ground between

the two” (p. 15). Similarly, Patton stated that he “preferred pragmatism to one-sided

paradigm allegiance” (p. 38) and maintained that a methodology should be chosen

that is appropriate (a) for meeting the study’s purpose, (b) for answering the

questions being asked, and (c) for the resources available. This study is based on

such pragmatic considerations, although it utilises primarily a naturalistic inquiry

approach.

Type of data.

The fourth descriptive dimension is that of data type, shown in Figure 3.1 as a

continuum from qualitative to quantitative. There is widespread support in the

research design literature for an approach that incorporates both quantitative and

qualitative methods (e.g., S. A. Mason, 1993; Patton, 1990; Rossman & Wilson,

1985; Strauss & Corbin, 1990). Wiersma (1995) described a continuum between

quantitative and qualitative research and stated that “from a practical standpoint of

conducting research, quantitative and qualitative procedures are often mixed” (p. 14).

Likewise Best and Kahn (1993), noting that quantitative research had traditionally

dominated educational research, stated that “some investigations could be

strengthened by supplementing one approach [quantitative or qualitative] with the

other” (p. 212).

The use of mixed methods in a single study was given more detailed support

by Rossman and Wilson (1985, 1991), and Greene, Caracelli and Graham (1989). In

the first of these papers, Rossman and Wilson (1985) described three different

purposes for mixed-methods research. This list was added to by Greene et al. (1989),

and then expanded by Rossman and Wilson (1991) into a typology of four purposes

at the stages of research design or data analysis. The four purposes listed by these

authors are (a) corroboration, (b) elaboration, (c) development, and (d) initiation.

Briefly, corroboration refers to “classical triangulation where different methods are

employed to test the consistency of findings from one method to another” (Rossman

& Wilson, 1991, p. 2). Elaboration, also called “complementarity” by Greene et al.,

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is used to “illuminate different facets of the phenomenon of interest” (Rossman &

Wilson, 1991, p. 2). Development uses the results gained from one method to inform

subsequent investigation by the other method. Initiation is used only at the analysis

stage of a study to uncover “the unexpected, the paradoxical, or the contradictory”

(Rossman & Wilson, 1991, p. 4); in other words, initiation may be used to lead to

further questions for investigation. Rossman and Wilson (1991) pointed out that the

above four purposes for using mixed methods either may be planned in advance, or

may be decided upon after initial analysis as a study’s findings begin to emerge.

Although the study is predominantly qualitative, it also collects data in a

quantitative form; however, this quantitative data is used descriptively not

inferentially.

Role of researcher.

The researcher’s role is the fifth dimension on which this study is described.

Gold (1969) proposed a continuum of researcher roles, from complete participant, in

which the researcher becomes one of the participants under investigation, to

complete observer, in which the researcher is completely separate from the

participants. Between these two extremes, Gold identified roles of participant-as-

observer and observer-as-participant. In this study, the author was a participant-as-

observer; by taking the role of teacher for each group of students, he was an integral

part of the interactions that took place in each group. The researcher also observed of

what took place, mostly after the event via videotapes of the sessions.

3.6 Pilot Study A small-scale pilot study was conducted prior to the main study. The

following sections describe the pilot study’s purposes, the procedures followed, and

the results.

3.6.1 Purposes of the Pilot Study

The pilot study was used to test the feasibility of four aspects of the study

design: software design, teaching program, procedures, and data collection and

analysis. As a result of the pilot study, the design of the main study was modified in a

number of aspects, as explained in section 3.6.5.

The software. The software design (Appendix A) was tested to determine (a)

if the interface was clear to the users, (b) if the program contained any programming

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bugs that needed correcting, and (c) if any improvements were necessary to make the

software more effective in teaching place-value ideas.

Teaching program. The teaching program (Appendix B) was examined to

check that (a) 10 teaching sessions were sufficient to show some development of

place-value understanding, (b) the instructions to participants were clearly

understood by them, (c) teaching procedures were effective, and (d) there were

sufficient activities for the time available.

Procedures. Procedures including the interviews and teaching sessions

investigated whether (a) duration of teaching sessions and interviews was sufficient

to show development of place-value understanding, but not too long for the

participants’ attention spans; (b) placement of participants, researcher, camera and

microphone was suitable for clear video recording; and (c) arrangements for taking

students to and from their classrooms were suitable.

Data collection and analysis. These methods were examined to check

whether (a) sources of data were sufficient for developing triangulated descriptions

of participants’ actions and speech, (b) interview questions were appropriate to

identify place-value understanding, and (c) analysis methods facilitated the

identification of conceptual understanding of participants. In the end, the longer time

spent on coding and analysing transcript data from the main study led to changes to

data analysis that were not foreseen after the pilot study; see chapters 4 and 5 for

description of results and how they were analysed.

3.6.2 Selection of Pilot Study Participants

The pilot study was conducted at a school similar to that planned for the main

study. Participants in the pilot study were drawn from Year 3 classes at a small

primary school in a rural area north of Brisbane, Australia. Both the pilot and the

main studies were conducted using students at the Year 3 level, as that is the age at

which three-digit numeration is generally introduced in Queensland schools. At the

time of the pilot study (1997), the school had two Year 3 classes with approximately

50 students in total. Participant selection was made based on the previous year’s Year

2 Net (Queensland Department of Education, 1996). Results from this test were used

to divide the population of Year 3 students into three approximately equal groups,

defined as being of high, medium, and low mathematical achievement respectively.

In order to manage the time needed for data collection and analysis, only two pairs of

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participants were selected for the pilot study. A pair of students was selected at

random from each of the high and low achievement groups. Each pair was of one

gender, on the author’s assumption, based on classroom teaching experience, that

children at this age would commonly prefer to work with a peer of the same gender.

Random selection was used to assign the pair of girls to use the physical base-ten

blocks and the pair of boys to use the computer software (the electronic blocks).

3.6.3 Pilot Study Procedures

Place value of three-digit numbers is generally taught in the second term of

Year 3 in Queensland schools. The pilot study was timed to occur towards the end of

the first term of the school year, leaving time for the main study in the second term.

The researcher interviewed selected students individually in a quiet room, before and

after a teaching program of 10 sessions, described in the following paragraph.

The teaching program for the pilot study comprised 10 sessions for which a

teacher’s script was written in advance. An overview of the pilot study’s teaching

program (Appendix B & Appendix C) and the script used for the first session

(Appendix D) are appended to the thesis. The researcher led the participants through

a series of tasks, progressing from revision of two-digit numeration through to three-

digit numeration and two-digit addition and subtraction. If participants were unable

to complete all the tasks planned for a session, as was generally the case with the

low-achievement girls, then tasks were held over for the following session.

3.6.4 Pilot Study Data Collection and Analysis

There were three main sources of data in the pilot study: interviews; teaching

sessions; and software-generated records of user actions with the computer software,

known as an audit trail (Misanchuk & Schwier, 1992; Schwier & Misanchuk, 1990;

Williams & Dodge, 1993). Videotapes of the interviews and teaching sessions were

transcribed, including actions and dialogue by the researcher and the participants.

The transcripts were studied to identify any aspects of the main study which should

be modified in the main study. Preliminary data analysis was also carried out to test

analysis procedures planned for the main study.

3.6.5 Changes Made to Study Design After Pilot Study

As indicated earlier, the purpose of the pilot study was to investigate whether

any changes were indicated for the study design, in the areas of software design,

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teaching program, procedures, and data collection and analysis. Changes made for

the main study are summarised below in four sections, describing changes in

software design, procedures of participant selection, administration, the teaching

program, and data collection and analysis.

Changes in software design.

Minor changes were made to the software design (see Appendix A) as a result

of the pilot study results. One change was needed due to a bug in the program that

caused a difficulty when children clicked rapidly on buttons to add new blocks on-

screen. The Windows operating system recognises a pair of rapid mouse clicks as a

“double click” rather than two single clicks; in response to a double click the

software added only a single block. Thus, for example, if a child rapidly clicked six

times only three blocks were added to those on screen. The software was modified to

produce two blocks if a double click was made. A second modification was made to

the graphic images applied to two of the buttons on-screen. In the pilot study, the

graphics for the buttons by which regrouping actions were accessed were not clear to

the students. The graphics represented symbolically the idea of changing a larger

block for 10 smaller blocks (partitioning), and 10 small blocks for a larger block

(grouping), respectively (Figure 3.2). It was obvious that students did not recognise

these graphics as representing the actions as intended. The metaphors underlying on-

screen tools used to achieve these actions are a saw and a net; the button graphics

were therefore changed to pictures to match these tools (Figure 3.3), making the links

between the buttons and the tools clearer.

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Original Partitioning button graphic

Original Grouping button graphic

Figure 3.2. Original graphic images used on regrouping buttons in software used during pilot study.

Replacement Partitioning button graphic

Replacement Grouping button graphic

Figure 3.3. Replacement graphic images used on regrouping buttons in software used during main study.

Changes to selection procedures.

A difficulty was encountered early in the pilot study, regarding the ability of

one of the low-achievement students to understand the tasks. Of the two girls,

selected at random from the population of low-achievement Year 3 girls at the

school, Jenny (a pseudonym) was much more able than the other, Nina. Early in the

program it was found that whereas Jenny was ready to progress to more difficult

questions, Nina did not understand two-digit numeration concepts needed to make

progress in the teaching program. Consequently, on the one hand Jenny became

frustrated and started to lose interest in the activities, and on the other it was evident

that Nina needed considerable help to understand each question. The decision was

made to continue the teaching program with the girls separately, to continue to trial

the full 10 teaching sessions, and to decide at the end of the pilot study if individual

instruction might be more effective with low-achievement students. The same

difficulties were not experienced with the pair of boys, who for most of the program

worked amicably and cooperatively. There were occasions where one or other of the

boys made mistakes in answering a question, but the other student was able to state

the correct answer without causing any difficulties.

The difficulties described in the previous paragraph underlined the need to

select students for the main study who were able to cooperate in the learning

situation, especially since this study was exploratory, and its aim was not to

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generalise to all students of Year 3 age. Because of this finding the design of the

main study was modified to exclude students who might find participation difficult,

because of either specific learning difficulties or behavioural problems such as

Attention Deficit Disorder (ADD). This is explained in more detail in section 3.7.1.

Changes to administration procedures.

The pilot study was conducted with pairs of students working with either

base-ten blocks or the computer software; this was changed to groups of 4 students

for the main study. The initial use of small groups was based partly on the need for

each student in the computer group to have ready access to a computer, and partly on

constraints regarding videotaping facilities. To have more than two students using a

single computer would make it difficult for each student to have sufficient access to

the software. However, there was an observed lack of collaborative learning, which

was believed to be due to the small number of students in the pilot study’s teaching

sessions. Students tended to follow the researcher’s directions and answer his

questions, but not to consult with each other. Of course, collaborative learning was

not possible with the girls once they were separated. It was therefore decided to alter

the general design of the administration procedures, to involve groups of 4 students

at a time. To achieve this, it was necessary to use two video cameras for every

session and two computers for the computer groups. The use of two cameras was

needed to capture adequately interactions that occurred among 4 students and the

researcher. The two computers were needed for the computer groups, to give each

student sufficient access to a machine.

Changes to teaching program.

Changes were also made to the teaching program, to take advantage of the

larger groups and to encourage collaborative learning. The tasks planned for the main

study were very similar to those used in the pilot study; however, in place of teacher

directions explaining each step required, tasks were written that required each group

of four participants to discuss and complete the tasks with little direction from the

researcher. In this way, students were expected to exhibit more cooperative learning

and interaction within each group of four than took place with pairs or single

students in the pilot study. Because of these changes to the organisation and content

of teaching sessions, no attempt was made to analyse results of the pilot study in

depth.

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Changes to data collection and analysis procedures.

Analysis of the pilot study video tapes showed that the position of the video

camera during teaching sessions was of particular importance, and so this was

carefully planned for the main study. For the blocks groups, at times the

manipulations made by participants were not visible to the camera because of

obstructions, including piles of unused blocks. It was thus decided in the main study

to ensure at all times a clear line of sight for the camera. For the computer groups,

there was a different problem. Because the students sat facing the screen, it was not

possible to video both the students’ faces and the screen simultaneously with one

camera. A compromise between videoing the screen and videoing the student was

achieved in the pilot study by placing the camera slightly in front of the computer,

giving a side-on view of the screen and the student that was usually adequate. In the

main study with groups of four, two cameras were used on opposite sides of the

group, to give the best view possible of participants and blocks or computer screens.

This method is unsatisfactory for recording every interaction between participants

and the computer: The author strongly recommends the use of “split screen” methods

of video recording, which record a view of the computer screen and a view of

participants simultaneously, if the requisite technology is available.

The audit trails generated by the software during the pilot study were found to

be of limited usefulness, and so this feature was extended for the main study (see

Appendix E for an example of an audit trail generated during the main study). The

text files generated by the software recorded each time a button was clicked,

including the time on the computer system clock. However, it was found nearly

impossible to match these recorded actions with actions viewed on the videotapes.

The audit trails were modified in two ways for use in the main study. First, the time

recorded for each action included seconds as well as hours and minutes, to provide a

more accurate measure of when each action was taken. Second, more detail of each

action was recorded, to enable more accurate knowledge of what the student(s) did:

Each line of the audit trail identified the button clicked, the time, the blocks present

on the screen, and the number represented by the blocks overall. Audit trail data were

used only where video data were unclear and further information was needed to

determine what a student did with the computer.

Data collection sources were supplemented for the main study. Researcher

field notes and student workbooks were used, in addition to the videotapes and

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software audit trails. Field notes were not used in the pilot study, but were considered

desirable in the main study to add another source of data to support categories of

responses found through analysis of video transcripts. In the pilot study students’

written work either was collected on loose sheets of paper, or was written in the

students’ regular mathematics exercise books. It was considered desirable to collect

each student’s written work for verification of observations made from the

videotapes. Therefore, each student in the main study was given a workbook in

which all written work was dated and collected for later analysis as required.

Data analysis procedures for the pilot study were limited to transcription of

videotapes and initial coding of students’ responses. Video transcripts of the main

study were subject to analysis that was considerably more detailed, as described in

section 3.7.5.

3.7 Main Study The main study comprised five phases, summarised in Table 3.1. Four groups

of four Year 3 students were taught by the researcher in a teaching program of 10

lessons. Interviews before and after the teaching sessions were used to identify

differences in participants’ conceptual models of numbers before and after the

teaching phase. Each of phases I, II, and V was identical for both computer and

blocks groups. In phase III the two computer groups had an extra training session

prior to the teaching program, to familiarise them with the software. A parallel

session was not considered to be necessary for blocks groups, as the children were

familiar with the use of base-ten blocks from their class lessons. Phase IV was the

teaching phase, involving different treatments for the two cohorts. The study took

place over a 3-week period immediately prior to the mid-year break, after the

participants had been in Year 3 for almost half a school year.

TABLE 3.1. Phases of the Research Design

Phase: I II III IV V Blocks Groups (physical)

Selection & assignment of students

First interview

Teaching program using blocks

Second interview

Computer Groups (electronic)

Selection & assignment of students

First interview

Software training session

Teaching program using computer

Second interview

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3.7.1 Selection of Participants

The main study was conducted at a school that, like the school for the pilot

study, was a small state primary school in a semi-rural area north of Brisbane,

Australia. At the time of the study, the chosen school had two classes of

approximately 22 Year 3 students each, and a composite Year 3/4 class, with

approximately 6 Year 3 and 22 Year 4 students. When the teachers of these classes

were approached, one of the two Year 3 classes had already commenced teaching

hundreds place concepts, and so participants were selected from the other Year 3

class and the Year 3/4 class.

The previous year’s Year 2 Net (Queensland Department of Education, 1996)

results were used to rank the population of Year 3 students at the chosen school (see

Appendix F). The class teachers were asked to exclude from the population of Year 3

students any students who had been diagnosed as having either a specific learning

disability or a behavioural disorder, such as ADD, in order to exclude students who

might have difficulty completing the tasks or who might find cooperation in

groupwork difficult. Following this process, the top 4 boys and 4 girls, and the

bottom 4 boys and 4 girls were selected from the ranked list to participate in the

study. The top 8 participants are referred to hereafter as “high achievement level”

participants, and the bottom 8 participants as “low achievement level” participants.

Participants were assigned to 4 groups, as indicated in Table 3.2, each composed of 2

girls and 2 boys.

TABLE 3.2. Participant Groups for the Main Study

High Mathematical Achievement

Low Mathematical Achievement

Computer groups 4 students (2 male, 2 female) 4 students (2 male, 2 female) Blocks groups 4 students (2 male, 2 female) 4 students (2 male, 2 female)

As in the pilot study, each group was of a single mathematical achievement

level. This approach was supported by Fox (1988), who stated that “learning in small

groups is most effective when gaps in understanding between individuals are neither

too ‘great’ nor too ‘small’” (p. 36). In this thesis the groups are referred to as the

high/computer group, low/computer group, high/blocks group, and low/blocks group.

Appendix G contains a full list of participants, including their dates of birth and the

groups to which they were assigned.

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3.7.2 Teaching Program

As in the pilot study, the main study included a teaching program of 10

sessions (Appendix H). The following two sections describe the teaching approach

adopted and the lesson content.

Teaching approach.

The teaching program was based on a view of teaching of mathematics

described by G. A. Jones et al. (1994) as having a “constructivist orientation with a

strong emphasis on social interaction” (p. 119). It also agrees with Cobb et al.’s

(1992) view of learning as “an active, constructive process in which students attempt

to resolve problems that arise as they participate in the mathematical practices of the

classroom” (p. 10). This view was operationalised to include cooperative group

work; a sequence of learning activities that build on previously-understood concepts;

and the provision of freedom for students, within reasonable bounds, to choose for

themselves how to answer the questions asked. The groups of participants were

encouraged to cooperate with each other and to negotiate answers to the questions so

that, if possible, each group reached a consensus about the answer to each one. In

each question the students were asked to represent the quantities involved in each

question in symbolic form, with materials (blocks or computer software), or both.

The students were free to use different representations of the numbers to support

their answers, in keeping with the constructivist model of teaching employed.

The researcher took the role of teacher for all teaching sessions and used

appropriate teaching strategies to support and encourage learning by the participants

(see Confrey & Lachance, 2000, for a discussion of having the researcher do the

teaching in a teaching experiment). He encouraged students to discuss and negotiate

meanings of each question, the quantities involved, and possible solutions. The

researcher made suggestions to the participants, such as using the available

representational materials (physical or electronic blocks) to represent the numbers

involved, if the students did not seem to be making progress in answering a question.

He neither confirmed nor denied the validity of any solution proposed by participants

until the group members had discussed it and expressed their individual views of the

problem and possible solution. This was done for the same reasons cited by Fuson,

Fraivillig, and Burghardt (1992), to simulate a situation that is believed to be

common practice in classrooms, in which a teacher does not supervise each group of

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students continuously but monitors them infrequently as time permits. As Fuson et

al. explained,

an experimenter-intervention strategy was adopted that attempted to let children follow wrong paths until it did not seem likely that any child would bring the group back onto a productive path; the experimenter then intervened with hints to help the group but giving as little direction as necessary. This was done to provide maximal opportunities for the children to resolve conflicts and solve problems creatively. . . . This criterion was intended to reflect the reality of a classroom where a teacher monitoring six or more groups might not get to a given group for a whole class session but would be able to give support by the end of that time. (p. 47)

The problems were written in a sequence of increasing difficulty. New

problems were presented once the previous question had been answered to the

group’s satisfaction; the researcher inserted supplementary questions similar to any

that caused a group to have difficulties if he felt it was necessary. As is to be

expected, the 2 low-achievement groups did not complete as many questions as the

high-achievement groups by the end of the study.

Lesson content.

The teaching program was written to take into account features of both

physical and electronic base-ten blocks. Where specific mention was made of

features available only in the software, equivalent activities were included for the

blocks groups, using activities that would typically be used in a classroom. For

example, when the number name window feature of the software was used, the

teacher provided written symbols to the blocks participants, either by writing them

on paper or by showing them printed on cards.

There were 45 tasks (Appendix H) in the teaching program. Many of the tasks

were written as non-routine problems, to challenge and motivate the students, and

thereby to promote maximal learning (Sowder & Schappelle, 1994). The tasks all

required understanding of place-value concepts to complete them, and are examples

of five types of task found in the place-value literature. These task types are (a)

number representation, (b) regrouping, (c) comparison and ordering numbers, (d)

counting on and back, and (e) addition and subtraction. These types of task are

described in the following paragraphs, including reference to other researchers who

have used similar tasks. Instructions for tasks are given in full in Appendix H; task

numbers referred to in this section, and elsewhere, refer to the numbers used in the

appendix.

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For each task type there were two sets of tasks, the first involving two-digit

numbers and the second involving three-digit numbers. The tasks were sequenced

according to their relative difficulty and the need for comparatively basic skills to be

practised before the advanced tasks were attempted. Specifically, number

representation tasks were the first in the program, as these tasks involve basic skills

of demonstrating knowledge of written symbols, concrete representations, and verbal

names for numbers. These tasks were followed by regrouping tasks that also use

block representations and extend the skills needed for the number representation

tasks. Following this were comparison and ordering tasks, in which students

compared and ordered numbers presented as written symbols. The final task type for

each set of two-digit and three-digit number tasks was addition and subtraction.

These computation tasks relied on a number of skills in combination, including

knowledge of symbols, regrouping, and number facts, and thus were the last ones for

each set.

Numbers used for a single task were also sequenced according to the reported

difficulty that children have with different numbers. In particular, teen numbers were

used after other two-digit numbers, in view of the previously-mentioned difficulties

that teen number names introduce. Numbers that include zero digits were introduced

after other two-digit numbers, since zeros also cause well-documented difficulties for

children learning place value.

Description of task types, with examples.

In this sub-section each type of task is described, an example of each type is

given, and decisions made about the sequence of questions in each task type are

described. The full list of tasks is provided in Appendix H.

Number representation (Tasks 1-3, 28-30; see example in Figure 3.4). As

discussed in chapter 2, the ability to make connections among various

representations of numbers is generally considered to be fundamental to place-value

understanding (Fuson, 1992; Janvier, 1987). The tasks followed the “symbol-verbal-

concrete” model (Payne & Rathmell, 1975), which has been adopted by many

curriculum writers up to the present. In each task the student was given a number

representation (in written, concrete, or verbal form) and was asked to represent the

same number in one of the other two forms. Tasks of this general type have been

used by several researchers, including Boulton-Lewis (1993), Hughes (1995), Miura

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and Okamoto (1989), and S. H. Ross (1990). In this study these tasks were written in

sets of three for both two-digit and three-digit numbers; starting with concrete

representations, then verbal, and then symbolic; in each case converting the

representation to the other two forms. For example, in Task 1 (a) students were asked

to look at a block representation for the number 25 and to respond with the verbal

name and the written symbol for 25.

Task 1 - Representing numbers Look at the blocks put out by Mr Price. Say what number the blocks show. Write the number in your workbook. [Numbers were not printed on task cards provided to participants.] 25 61 13 40

Figure 3.4. Sample Representing numbers task.

As mentioned earlier, the numbers included in these tasks were sequenced

according the difficulty they provide for children, based on reports in the place-value

literature. For example, in Tasks 1 to 3 four examples were provided, beginning with

a number between 20 and 99 with more than 1 one. Following this was a number

with a number of tens and 1 one, as some children reportedly confuse such numbers

with teen numbers. Thirdly there was a teen number, and finally a number with zero

ones, regarded as the most difficult types of two-digit numbers. Similar sequences

were used in Tasks 28-30, with three-digit numbers.

Regrouping (Tasks 4-7, 31-34; Figure 3.5, Figure 3.6). An important

component of understanding the values represented by symbols is being able to

group or partition quantities represented into different arrangements (G. A. Jones &

Thornton, 1993a; G. A. Jones et al., 1994). For example, to show a sound

understanding of the symbol 35, a student should be able to represent 35 as 3 tens

and 5 ones, as 2 tens and 15 ones, or as 35 ones. This process of regrouping numbers

in different ways is essential for proficiency in written and mental computation,

though many students do not demonstrate this skill (Miura & Okamoto, 1989).

Regrouping tasks in this study required participants to regroup numbers in various

ways, including regrouping a single ten for 10 ones, all available tens for ones, or a

single hundred for 10 tens. Tasks 7 and 34 involved the use of a numeral expander to

investigate regrouping based on the written symbols.

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Task 4 - Regrouping Show the number with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook. 77 23 91 58

Figure 3.5. Sample Regrouping task.

Task 7 - Use of numeral expander (Computer groups) Show the number with the blocks. Turn on the numeral expander. Use the expander to

show the number in different ways. Write the number in two ways in your workbook. 34 96 52

Figure 3.6. Sample Use of numeral expander task.

Numbers included in these tasks were chosen to provide a variety of

examples, including a number with only 1 one. Because regrouping tasks are only

introduced in the Queensland mathematics syllabus in Year 3, more difficult

examples of regrouping with teen numbers and numbers with zero ones were not

included in the two-digit examples of this type of task.

Comparison and ordering numbers (Tasks 8-12, 35-39; Figure 3.7). These

tasks developed the ability of students to use their understanding of quantities

represented by symbols to compare pairs of numbers, or to order three or more

numbers. Tasks of this type have previously been used by G. A. Jones and Thornton

(1993a), and A. Sinclair and Scheuer (1993). To compare or order numbers students

need to have a good understanding of values represented by symbols, in particular

the value represented by each digit. For example, in order to correctly compare 51

and 39, a student must know that the tens only have to be compared, and that 5

represents 5 tens, which is greater than either 3 tens or 9 ones.

Task 8 - Comparing 2 numbers Tommy and Billy were arguing about who had more marbles. Tommy had 48 marbles, and Billy had 62 marbles. Who had more marbles? Show the numbers with the blocks. Explain your answer

in your workbook.

Figure 3.7. Sample Comparison task.

Numbers chosen for comparison and ordering tasks included pairs of numbers

in which one number in each pair had more tens, and the other number had more

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ones; the number of ones in the latter number was also the largest digit of the digits

involved (e.g., 48 & 62, 51 & 39). Ordering tasks included sets of three numbers in

which two numbers had the same number of tens (or hundreds in three-digit

examples), and in which digits were repeated in different positions in the various

numbers (e.g., 82, 37, & 88; 75, 57, & 54).

Counting on and back (Tasks 13-17, 40-43; Figure 3.8). Another type of task

that requires good understanding of place value is counting forwards or backwards,

also used by G. A. Jones and Thornton (1993a) and Boulton-Lewis (1993). Counting

on or back by ones involves the standard counting sequence that children learn early

in school (Resnick, 1983). This set of tasks also included counting by tens or

hundreds, either forwards or backwards, which requires knowledge of the values

represented by the tens and hundreds digits.

Task 14 - Counting back by 1s The Sunny Surfboard Company has 75 boogie boards left. If one is sold, how many are left? Then how many if another is sold? Say all the numbers in order from 75 back to 60. Show the numbers with the blocks. Write

them in your workbook.

Figure 3.8. Sample Counting task.

Numbers chosen for these tasks included numbers that allowed the sequence

to proceed for several numbers before either a teen number or a change of decade or

number of hundreds was required. For example, Task 13 involved counting back by

ones from 28, not requiring a change of decade until the tenth number in the

sequence. This was done to allow the participants to recognise the regular pattern in

which only one digit changes before having to deal with two digits changing at once.

Addition and subtraction (Tasks 18-21, 44-45; Figure 3.9). These tasks

involved application of prerequisite skills used in other question types, such as

regrouping and knowledge of digit values. Several researchers have recommended

that students be given tasks that require them to invent strategies to solve problems

of this type (Kamii et al., 1993; S. H. Ross, 1989). For this study these tasks were

presented as word problems, with no particular algorithm mentioned. Each task

involved a single operation, which is appropriate for students at this Year level.

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Task 19 - Addition A Space Race video game costs 75 dollars, and a set of batteries costs 19 dollars. How much will the game and the batteries cost? Show the numbers with the blocks. Discuss how to work it out with your group. Show how you work it out in your workbook.

Figure 3.9. Sample Addition task, including regrouping.

Numbers chosen for addition and subtraction questions included examples in

which there is no regrouping, followed by harder examples that involved regrouping

in one place (e.g., 28 + 31, 75 + 19, 95 – 23, 83 – 48). The researcher ensured that

students were able to complete addition and subtraction without regrouping before

the more difficult tasks were introduced.

3.7.3 Instruments - First and Second Interviews

Interviews were conducted before and after the teaching program. They were

in the form of “standardised open-ended” interviews, in which “all interviewees are

asked the same basic questions in the same order” (Fraenkel & Wallen, 1993,

p. 387). The following comments about particular items apply to both interviews; the

same questions were asked in each interview, with only the quantities involved

differing. The questions for the first and second interviews are listed in Appendix I

and Appendix J, respectively. The question numbers mentioned in this section apply

to both interviews.

Design criteria. Criteria adopted in designing the interviews were as follows:

1. Tasks were used by researchers in at least two other published studies;

2. Each task was to target one or more key components of place-value

understanding, based on the literature review; and

3. The whole interview was to take no longer than 20 minutes to

administer, considered a suitable length for the age of the students.

Categories of task. There are five task categories included in the interviews:

(a) number representation, (b) counting, (c) number relationships, (d) digit

correspondence, and (e) novel tens grouping. Other researchers have used these types

of task to probe students’ conceptual models of multidigit numbers, as described in

the following paragraphs.

Number representation tasks (Questions 1-3) were previously used by Miura

and Okamoto (1989), and Miura et al. (1993). In each task the participant was asked

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to represent the number shown by a written symbol using place-value material

(generally base-ten blocks).

Counting tasks (Question 4) were a component in G. A. Jones et al.’s (1994)

“framework for nurturing and assessing multidigit number sense” (p. 121): The

framework included a series of tasks of increasing difficulty, starting with counting

on by ones and progressing through counting on and back by tens to mental addition

and subtraction.

In number relationships tasks (Questions 5-6) participants were asked for a

number a little larger, a little smaller, much larger, and much smaller than a given

two-digit number. This task is an extension of an item used previously (G. A. Jones,

Thornton, & Van Zoest, 1992; G. A. Jones et al., 1994) that required students to

write a number a little more and a lot more than 42. To be successful in such an item,

a student needs to have good number sense; in particular, a clear idea of the relative

magnitude of numbers is required (Sowder & Schappelle, 1994).

Digit correspondence tasks (Question 7) were previously used by S. H. Ross

(1989, 1990) and Miura and colleagues (Miura & Okamoto, 1989; Miura et al.,

1993). Participants were asked to count a number of items between 10 and 40, and to

write the symbol that showed that number. The researcher asked the participant to

explain which of the counted items were represented by each digit in turn. A

variation of the digit correspondence task (Question 8), also used by S. H. Ross

(1989, 1990) and Miura et al. (1993), involves providing misleading perceptual cues

to the child that suggest a face value interpretation of the written symbol. For

example, if 13 objects are shared among three containers with one remaining, some

children will say that the digit 3 represents the three containers and the 1 the

remaining object (see Figure 3.10). This item was included in this study (as in studies

by Ross and Miura et al.) to test the robustness of the child’s understanding of digit

value in the face of misleading evidence.

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Figure 3.10. Diagram showing objects used in interviews for Digit Correspondence Task with misleading perceptual cues.

Novel tens grouping tasks (Question 9) were used by Bednarz and Janvier

(1982, 1988), who presented students with problem tasks involving items not

commonly used in place-value lessons, such as peppermints or paper flowers,

grouped in tens and hundreds. Bednarz and Janvier (1988) based the attention they

paid to groupings on their observation that “few children give a true interpretation of

the digit position in terms of groupings” (p. 300). To complete the tasks students had

to identify the groupings involved, deduce the relation between the groupings, and

then operate on the groupings to answer the given questions.

3.7.4 Administration Procedures

As described earlier, the study comprised five phases: selection of

participants, first interview, software training session, teaching program, and second

interview (Table 3.1). Administration procedures for each of these phases are

described in the following paragraphs.

Selection of participants. Sixteen students were selected for participation in

the study, as described in section 3.7.1. The researcher sought the consent of parents

or guardians of selected students for them to take part in the study (see Appendix K).

In all cases the parents or guardians of first 16 students selected for participation

gave their consent, and so selection of alternative participants was not needed.

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Interviews. Participants were interviewed both before and after the teaching

sessions, as described in section 3.7.3. Details of data collection procedures are

described in section 3.7.5.

Software training session. Participants using physical base-ten blocks were

already familiar with them from regular classroom lessons, but participants using the

software were not familiar with use of a computer for mathematics lessons, and the

software was totally new to them. To make the treatments more similar in this

respect, the 2 computer groups were given an extra session, prior to the first teaching

session, to familiarise them with the software. During these introductory sessions the

students were given the opportunity to experiment with the software and discover its

features. The researcher demonstrated any features that they did not discover for

themselves, or that they did not seem to understand.

Teaching phase. Participants were involved in a teaching program as

described in Appendix H. In the Queensland mathematics curriculum, which was

followed by the school chosen for the study, two-digit numeration is taught in Years

1 and 2, and three-digit numeration is introduced in Year 3. The Year 3 teachers at

the school generally introduced three-digit numeration in the second of four terms in

the school year. The teaching phase of the study was conducted at the end of term 2,

to match the usual timing of the topic. The Year 3 teachers of the study participants

were asked not to teach the topic to their classes until the study had concluded, in

order not to contaminate any learning effects produced during the teaching program;

both teachers involved complied with this request.

Four groups of 4 participants separately took part in the teaching program in a

room separate from the classroom, with the researcher taking the role of teacher.

Two groups used conventional physical base-ten blocks (blocks groups) and two

groups used electronic base-ten blocks (computer groups). Materials used by both

groups included task cards, workbooks, and pencils. Blocks groups used physical

base-ten blocks, and the computer groups had access to two computers with the

software installed (i.e., one computer between each pair of participants). Each

session was recorded using an audio cassette recorder and two video cameras, each

with an external microphone placed near the participants.

The sessions were conducted as follows. The first session commenced with

several activities designed to familiarise the participants with cooperative working, in

case they were not used to that mode of learning mathematics. This approach was

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recommended by Fox (1988), who commented that “groups must be shown how to

work cooperatively to get best results” (p. 37). When it seemed clear that the students

were comfortable with each other and with the researcher, the first task was

commenced. Each new task was introduced when the researcher was satisfied that

the participants had successfully understood the previous task. If it appeared that

further practice with a given task type was needed, the researcher introduced a

supplementary task or tasks before giving the students a task of the next type. The

researcher planned each session to last 20 minutes; in the main this was followed,

though some sessions exceeded this time by up to 10 minutes. On occasions when

the researcher decided that the students needed further practice with the last task in a

session, the following session commenced with a supplementary task of the same

type. On other occasions, in the interests of time remaining for the study and in view

of the participants’ competence on tasks of one type, the researcher decided that

certain tasks could be omitted. Otherwise, the next task was the next one in the

sequence listed in Appendix H.

3.7.5 Data Collection and Analysis

Guba and Lincoln (1989) summarised the role of a qualitative researcher in

the following statement:

The major task of the constructivist investigator is to tease out the constructions that various actors in a setting hold and, so far as possible, to bring them into conjunction—a joining—with one another and with whatever other information can be brought to bear on the issues involved. (p. 142)

In this study data from five sources were used to progressively triangulate, or

cross-validate, observations and conclusions (Wiersma, 1995, p. 264).

Data collection procedures.

Data came from several sources: a researcher’s journal, comprising field

notes and a field diary; videotapes and audio tapes of interviews and teaching

sessions; software audit trails; and participants’ workbooks. The researcher and an

assistant transcribed data from each source onto a computer. Video recordings were

transcribed, recording both actions and dialogue by the participants and the

researcher. Software audit trails saved as plain text files were copied from the

computers used in the teaching sessions. Hand-written data in the researcher’s field

notes and field diary and participants’ workbooks were transcribed into text files.

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Data from these various sources were used in combination to triangulate observations

and conclusions. The following paragraphs describe procedures followed for each

data source.

Researcher’s journal. The researcher kept a journal in which he recorded

field notes (notes taken during the teaching sessions) and a field diary (notes written

up at the end of each day of the teaching program; see Fraenkel & Wallen, 1993).

The field notes were used to record the researcher’s impressions of what was

happening as students attempted to complete tasks in the program; in particular, the

researcher commented on the apparent use of conceptual structures for numbers. The

field diary included notes about each day’s teaching, written in more detail. It

included questions about what occurred in the daily sessions, to direct the

researcher’s attention to particular aspects of the following sessions. By recording

comments at a time when they were fresh in the researcher’s mind, it was hoped to

provide insights about the students’ learning that may not have been accessible from

video transcripts alone.

Interviews. All participants were interviewed before the teaching sessions, as

far as possible on the same day. The second interviews were conducted after the

teaching sessions, again mostly on the same day. One participant, Yvonne, had to be

interviewed later than the others, after school resumed from the following vacation

break, as she and her family left in the last two days of the term for a holiday. The

researcher interviewed each participant individually, in a room separate from the

classroom. The researcher explained before each interview began that some items

might be too difficult for the student. This was necessary particularly for the first

interview, as students had not been taught three-digit numeration concepts in class

prior to that time. Each interview consisted of 9 questions (see Appendix I &

Appendix J), and was planned to take approximately 20 minutes per participant. The

two sets of interviews were videotaped and written responses to certain tasks were

collected. The resulting videos were transcribed, as described below.

Teaching sessions. Each lesson was audiotaped and videotaped. Two video

cameras were used simultaneously on opposite sides of the group, to record as many

of the occurring interactions as possible. As described in section 3.6.5, the cameras

were positioned carefully to avoid obstructions hiding the students’ actions, and in

the computer groups to record both participants’ faces and the computer screens as

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far as possible. All videotapes were transcribed, recording dialogue spoken and

actions taken by the participants and the researcher.

Software audit trails. The computer software used in the study generated an

audit trail as a plain text file for each session, timing and recording each major action

taken by the users, such as dragging a block or clicking the mouse on a button (see

Misanchuk & Schwier, 1992; Schwier & Misanchuk, 1990; Williams & Dodge,

1993). These text files were used to support the video transcripts where necessary;

the audit trails were referred to if actions taken by participants in the computer

groups could not be clearly determined from the videotapes.

Participants’ workbooks. Participants record their work during the teaching

sessions in workbooks provided by the researcher. Each day the participants dated

the page, and the workbooks were collected at the conclusion of each session, and

the contents transcribed onto a computer. Like the audit trails, workbooks were used

to support video transcript data, to clarify any actions of writing responses that were

not visible on the videotapes.

Data analysis.

Analysis of what took place in teaching sessions was centred on several

readings of the transcripts of session videotapes. The transcripts themselves contain

records of actions taken and verbal interactions among participants and the

researcher. At the start of the transcription process virtually all speech and actions

were recorded. However, after about half of the transcripts were completed, it was

clear that little was being revealed in descriptions of speech and actions that did not

relate to the mathematical tasks themselves. Therefore, for the remaining videotapes

only interactions relating to the tasks were transcribed. Transcripts from videotapes

were supported by data from audiotapes, participants’ workbooks, the researcher’s

field notes, and software audit trail records of user actions with the software. Once

the transcripts were completed they were read several times to ascertain categories of

participant action and speech emerging from the data. There were many candidates

for possible categories to consider: Over the 10 sessions the 4 groups attempted

approximately 30 mathematical tasks each, leading to a wide range of responses

relating to numbers, written symbols, and block or software numerical

representations.

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Once the raw data were transcribed data analysis began that involved

progressively organising and reducing the data until “focused conclusions” could be

made (Wiersma, 1995). The study is primarily exploratory and one of its major aims

is the generation of theory to describe and explain students’ use of place-value

materials in light of inferred conceptual structures for numbers. As Wiersma pointed

out, in qualitative research hypothesis generation and modification proceeds

throughout the study. Analysis of the data was conducted according to the grounded

theory approach of Strauss and Corbin (1990). This approach involves four main

phases: (a) review of literature, (b) open coding of data, (c) axial coding of data, and

(d) final integration of categories to form theory. These phases are summarised in the

following paragraphs.

Review of literature. In the first phase of the grounded theory approach, the

“technical” literature is reviewed, to “stimulate theoretical sensitivity” (Strauss &

Corbin, 1990, p. 50). Strauss and Corbin explained that

though you do not want to enter the field with an entire list of concepts and relationships, some may turn up over and over again in the literature and thus appear to be significant. These you may want to bring to the field where you will look for evidence of whether or not the concepts and relationships apply to the situation that you are studying, and if so what form they take here. (pp. 50-51)

This is the situation with this study. A number of categories of conceptual

understanding of numbers were found in the place-value literature (section 2.4.2).

These categories were used as starting points for the data analysis, but did not restrict

the search for new categories “that neither we, nor anyone else, had thought about

previously” (Strauss & Corbin, 1990, p. 50). The literature was also used as

“supplementary validation” (p. 52) in the succeeding phases of the study, to check

findings against previous work in the field.

Open coding of data. The second phase of grounded theory research is what

Strauss and Corbin (1990) called open coding of the data, and is linked closely to the

third phase of axial coding of data. Strauss and Corbin described open coding as

“breaking down, examining, comparing, conceptualizing, and categorizing data”

(p. 61). It involves first discovering categories in the raw data and naming them.

Following the naming of categories, they are developed in terms of their properties

and dimensions. This refers to the process of locating properties of each category on

a continuum. Strauss and Corbin describe several further procedures that can be used

in open coding, including questioning, comparing, and “waving the red flag.” Each

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of these procedures is designed to examine categories in further detail and to “break

through assumptions” (p. 84) regarding what the data show. In this study the

procedure described in this paragraph was carried out initially using Q.S.R.

NUD*IST (1994) software to code the data. Later this method was changed to use a

database designed by the author to analyse one particular category in the data,

feedback (see Appendix L). Categories identified in the review of literature and

categories emerging from the data were compared and used to cross-validate each

other (Strauss & Corbin, 1990; Wiersma, 1995), as mentioned in the previous

paragraph.

Axial coding of data. The third phase involved further refinement of the

categories defined in the previous stage. As Strauss and Corbin (1990) explained,

whereas open coding “fractures the data,” axial coding is used to put the data back

together, “by making connections between a category and its sub-categories” (p. 97).

Sub-categories are specific features of a category that give further detail about the

category, by describing conditions giving rise to it, its context, strategies that apply

to it and the consequences of those strategies. Strauss and Corbin introduced a

paradigm model to guide the process of axial coding. The paradigm model links a

category, or phenomenon, to its sub-categories in a linear fashion, as indicated:

Causal conditions → Phenomenon → Context → Intervening conditions →

Action/Interaction strategies → Consequences

The same procedures used in open coding, comparing and questioning, are

used in axial coding, but in axial coding the procedures are more complex. This

phase in the analysis involves “performing four analytic steps almost

simultaneously” (p. 107): (a) hypothesising the nature of relationships between

categories and sub-categories, (b) verification of hypotheses against the data, (c)

further search for the properties of categories, and (d) initial investigation of

variation in phenomena. Strauss and Corbin explained that in the coding phases

deductive and inductive thinking are used in turn repeatedly as hypotheses are

alternately proposed and checked. The final justification needed for a proposition is

that the relationship has been “supported over and over in the data” (p. 112).

Integration of categories to form theory. Strauss and Corbin (1990) labelled

the fourth phase as selective coding. This involves finally integrating the categories

previously identified and selecting the core category, “the central phenomenon

around which all the other categories are integrated” (p. 116). This phase involves

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five steps: (a) explicating the core category, (b) relating subsidiary categories to the

core category, (c) relating categories according to dimensions, (d) validating

relationships against the data, and (e) filling in categories that need further

refinement.

3.8 Validity and Reliability Any report of research should address questions of validity and reliability of

the study being reported on. As Burns (1990) stated,

with all data we must ask: (a) was the assessment instrument/technique reliable and valid; (b) were the conditions under which the data was obtained such that as far as possible only the subject’s ability is reflected in the data and that other extraneous factors had as minimal an effect as possible? (p. 189)

In quantitative research, reliability and validity questions refer to the

consistency and accuracy of test instruments for measuring the variables being

studied. For qualitative research, such as in this study, different reliability and

validity questions are needed. Rather than asking if observations are consistent with

others made at different times, or in different places, the question asked of qualitative

methods is whether observations made faithfully record what actually occurred

(Burns, 1990). These issues are addressed here in relation to three aspects of the

research: accuracy of raw observations, use of triangulation, and rigour of methods

of analysis.

Accuracy of observations. First, the researcher is an experienced primary

teacher, and as such is used to working with students, observing their reactions to

instruction and judging their understanding of subject matter. Videotapes and

audiotapes of each session have enabled actions and dialogue to be examined at a

level of detail that would not be possible in unrecorded situations. Though there is

obvious subjectivity inherent in any qualitative research, it is claimed that this

drawback is compensated for by the depth of insight into participants’ actions and

understandings afforded by the method. Burns (1990) stated that reliability of

qualitative research was enhanced by “delineation of the physical, social and

interpersonal contexts within which data are gathered” and that what is needed is

“careful and systematic recording of phenomena” (p. 246). The present chapter of

this thesis includes detail of reasons for and assumptions behind the research that

thus help to improve its reliability.

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Use of triangulation. As described in section 3.7.5, several different sources

of data were used for the study, to triangulate observations and findings. This is a

primary method for improving internal validity of observations in qualitative

research (Burns, 1990; Wiersma, 1995).

Rigour of methods of analysis. As has been noted by many qualitative

researchers, qualitative research methods have been criticised for their apparent lack

of rigour. Researchers who favour experimental research have rejected qualitative

research as being unscientific and sloppy. Strauss and Corbin (1990) developed the

grounded theory approach to qualitative research partly to address such concerns.

Conducted according to Strauss and Corbin’s advice, the grounded theory approach

involves a number of internal checks for validity, and requires the researcher to

check and re-check data to confirm conclusions.

3.9 Limitations Limitations of this study relate to three particular aspects of the design: the

size and representative nature of the sample, possible observer bias, and the use of

qualitative research methods. First, the sample size is just 16 students at one primary

school. This sample is too small and not sufficiently representative to generalise

findings to primary students in general. However, this is not the main intention of

this study. Fraenkel and Wallen (1993) explained that

in qualitative studies . . . it is much more likely that any generalizing to be done will be by interested practitioners—by individuals who are in situations similar to the one(s) investigated by the researcher. It is the practitioner, rather than the researcher, who judges the applicability of the researcher’s findings and conclusions, who determines whether the researcher’s findings fit his or her situation. (p. 403)

Thus it is argued that conclusions drawn in this study, as in qualitative

research generally, are to be viewed “as ideas to be shared, discussed, and

investigated further” (Fraenkel & Wallen, 1993, p. 403). The study has been used to

generate hypotheses that are likely to be of interest and relevance to primary teachers

and that may potentially lead to further investigation.

The second limitation is that only one researcher carried out all data

collection and analysis, introducing a possible source of bias. This is typical in

studies of this size and nature that do not have external funding. This concern is

addressed using triangulation; well-documented, comprehensive descriptions

(Wiersma, 1995); and an iterative process of hypothesising and checking. As

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described in section 3.7.5, triangulation of data has been achieved using several

means of collecting data. These data comprise careful, comprehensive notes from

each relevant incident. Data collection and analysis have been through a number of

iterations of hypothesis proposing, checking, and modification. It is claimed that

through these techniques a “logical basis” has been established for the validity of the

study’s findings (Wiersma, 1995, p. 223).

Finally, this study is limited because of its small scale. As is often the case

with qualitative research, the sample size was small, and the data were collected over

a short period of time. The obvious implication of these aspects of the study is that it

is risky to attempt to apply the study’s findings to Year 3 children generally. This

concern is handled by pointing out the different purposes of qualitative research and

its alternative epistemology. Qualitative research such as that described here attempts

to demonstrate a set of findings that applied in one particular situation and then

presents hypotheses about those findings that may be used to foster further study.

The situations investigated are not perceived as obeying certain laws of nature, but

rather as being constructed and understood individually by the participants in those

situations. In this study, the conceptions of numbers held by a small number of

children have been studied in depth, via recordings of their actions and spoken

dialogue. The proposed categories of response are compared with the results of

previous research, which strengthens the conclusions made. Conclusions about these

findings are presented for evaluation by the reader of the research based on the logic

inherent in the report, rather than being presented as a version of “the truth.”

3.10 Chapter Summary This chapter outlines the methodology employed in the study. The overall

design is exploratory in nature and is used to generate theory regarding Year 3

students’ understanding of two-digit and three-digit numbers. This theory is

investigated with relation to the students’ prior mathematical achievement levels and

to the mode of number representation used, either computer software or conventional

base-ten blocks. Results of these two sets of conditions have been studied in relation

to differences among student interactions and students’ development of place-value

understanding.

The overall design, a teaching experiment, is widely used in research into

mathematical understanding. A pilot study was conducted to trial various aspects of

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the study and appropriate modifications to procedures were made to the design. The

main study comprised 5 phases: participant selection, first interview, software

training session, teaching program, and second interview. Sixteen Year 3 students

from a single school were selected to take part in the study. Half of the students were

of low mathematical achievement and half of high mathematical achievement, based

results gained in the previous year using the Year 2 Net (Queensland Department of

Education, 1996).

The researcher took the role of teacher in the teaching program, teaching 4

groups of 4 participants for 10 daily sessions of 20 minutes duration. Sessions

involved students being presented with a series of tasks on cards, to be solved

cooperatively by each group. Tasks were written in a sequence of increasing

difficulty and were presented in order. New tasks were given as students appeared to

be ready for them; supplementary tasks were inserted as necessary, for extra practice.

All sessions involved the collection of qualitative data from several sources,

including a researcher’s journal and video transcripts. The method of data collection

and analysis used is the grounded theory approach of Strauss and Corbin (1990). The

method they have described was followed to generate theory regarding how students

learn place-value concepts and how the use of two modes of number representation

influences that learning.

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Chapter 4: Results

4.1 Chapter Overview

4.1.1 Restatement of the Research Question

The research question for this study, stated in section 1.3, is repeated here:

How do base-ten blocks, both physical and electronic, influence Year 3

students’ conceptual structures for multidigit numbers? Within the context of Year

3 students’ use of base-ten blocks or place-value software, the following specific

issues were stated as being of concern:

1. What conceptual structures for multidigit numbers do Year 3 students

display in response to place-value questions after instruction with base-

ten blocks?

2. What misconceptions, errors, or limited conceptions of numbers do

Year 3 students display in response to place-value questions after

instruction with base-ten blocks?

3. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in instruction with physical and

electronic base-ten blocks?

4. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in students’ achievement in

numeration?

Data from the interviews and the teaching phase of the study are described in

this chapter, as they relate to the above four questions. Section 4.3 comprises an

overview of data from the interviews, summarising the performance of the

participants on place-value tasks before and after the teaching phase of the study.

Section 4.4 includes a discussion of participants’ apparent number conceptions

evident in their responses. Section 4.5 summarises participants’ performance on digit

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correspondence tasks. Section 4.6 contains descriptions of errors, misconceptions,

and limited conceptions evident in participants’ responses. Section 4.7 includes an

outline of data relating to participants’ use of either base-ten blocks or computer

software to represent numbers.

4.2 Transcript Conventions Used in this Thesis Table 4.1 comprises a list of notations used in transcript excerpts quoted in

this thesis.

TABLE 4.1. Transcript Notations

Indication Notation Example

Speech Normal text Tell me what this means.

Actions Square brackets [] [Points to blocks.]

Pause or unfinished statement Ellipsis (…) It’s, … um …

Part of transcript omitted for brevity, clarity, or both

Em dash (—) —

Emphasis of point of analysis Italic script It’s still the same number!

Text inserted to aid clarity Parentheses () Where does this (block) go?

Numbers as written symbols

Single quotation marks (‘’) What does the ‘2’ mean?

Cardinal numbers referring to members of a set

Number words or figures in normal text

These three blocks go here.

Base-ten blocks Figure and place name Puts out 4 tens and 10 ones.

Identifying information is appended to transcript excerpts to aid the reader.

For interview transcripts, the number of the interview and the question are

abbreviated in parentheses at the end of each excerpt. For example, (I1, Qu. 2c) refers

to Question 2 (c) in Interview 1. In excerpts from teaching session transcripts, the

session number, group, and task number are similarly indicated. High-achievement-

level and low-achievement-level groups are indicated with the letters “h” and “l,”

respectively, and computer and blocks groups by “c” and “b,” respectively. For

example, (S6 h/b, T 23b) refers to Session 6 of the high/blocks group, attempting

Task 23 (b). Similarly, where necessary in the main text, the group to which a

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participant belonged is indicated by an abbreviation placed after the participant’s

name: for example, “Hayden (l/c).”

The author of the thesis conducted all interviews and all teaching sessions. To

indicate the different roles of the researcher in interviews and in teaching sessions, in

transcripts of interviews he is referred to with the label “Interviewer,” and in

teaching sessions with the label “Teacher.”

4.3 Place-Value Task Performance Revealed in Interview Results Results from the two sets of interviews are summarised in this section, in

order to provide an overview of performance of the participants at the start and at the

conclusion of the study. The interviews were intended to show any differences in the

learning of participants resulting from their using the two representational formats.

The results reported in this section show that such differences in performance by

participants using the two types of representational material were minor.

4.3.1 Methods used to Analyse Interview Data

Initial analysis of the interview responses was conducted by listing eight

different skills assessed during the interviews, and deciding on a criterion by which

to judge whether each participant had demonstrated each skill. The eight skills,

divided into 21 sub-skills, and the criteria by which the participants’ responses were

judged are listed in Appendix M. The identified skills and sub-skills mirror the

questions and part-questions very closely, because most questions targeted a

particular numeration skill. This varies for interview Questions 5, 7, and 8 only.

Question 5 was asked in four parts, asking the participant to state numbers that were

a little smaller, much smaller, a little greater, and much greater than a particular two-

digit number. Participants’ performance on these questions indicated that they were

able to state numbers close to the given number, or far from that number, but not

always both. Therefore the four question parts relate to two sub-skills, numbered 5a

and 5b. In the case of Questions 7 and 8, each question has been collapsed to a single

sub-skill. Questions 7 and 8 both asked participants to count between 20 and 40

objects and then to identify the referents for that number. Question 8 differs in that

the objects were grouped in such a way as to provide misleading perceptual cues

regarding the referents for the digits. The two questions both targeted the same basic

skill, but within two contexts; the sub-skills have been numbered 7a and 7b.

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The researcher decided on the criterion for each sub-skill by comparing the

intention of each question with the actual responses of participants, and making a

judgement about what was considered acceptable. For example, for sub-skills 1a to

1c (representing numbers with blocks), it was decided to allow at most one error in

counting the blocks for achievement of each criterion. This was found to be

necessary because of several participants who were clearly able to state the number

represented by the blocks, but who made a mistake in their first attempt at counting

the blocks. By stating a criterion for each sub-skill, the possibility of researcher bias

in deciding who had demonstrated each skill was reduced, and the reliability of

reported performance scores is improved. Reliability of these data was further

strengthened by having the coding of responses cross-checked by a second

researcher, also an experienced primary teacher. A score was determined for each

participant at each interview, based on a count of the sub-skill criteria achieved;

these scores are listed in Table 4.3 and referred to elsewhere in this chapter.

4.3.2 Overview of Interview Results

The scores relating to participants’ achievement of place-value criteria at the

interviews are summarised in a series of four tables, based on analysis of the

interview transcripts.

Table 4.2 indicates the numeration skills demonstrated by participants at each

interview. Three symbols are used to indicate the questions where participants’

demonstration of place-value understanding altered between the two interviews. A

vertical line ( | ) indicates that a criterion was achieved in both interviews. An

upward arrow ( ⇑ ) indicates that a participant achieved a criterion at the second

interview, but not at the first; a downward arrow ( ⇓ ) shows that the participant

achieved the criterion at the first interview, but not at the second. Shading is used

with upward arrows to add visual cues to improvements indicated in the table. The

data in Table 4.2 are summarised in Table 4.3, showing the overall improvement or

deterioration in the number of place-value criteria achieved by each of the 16

participants, and the combined score for each group. The data are further

consolidated in Table 4.4 and Table 4.5. Table 4.4 shows the aggregated scores for

participants of high-achievement-level and low-achievement-level, and Table 4.5

shows the aggregated scores for participants using blocks and participants using a

computer.

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Patterns in the interview data.

A number of comments may be made about the performance of individual

participants and groups at the two interviews, revealed in Table 4.2 and Table 4.3. It

is to be expected that a series of planned teaching sessions would result in

improvement of students’ understanding of place-value concepts; the shaded arrows

in Table 4.2 indicate the specific skills where this appears to have taken place. An

overview of the scores attained by the 16 participants shows that improvement on a

range of criteria occurred between interviews in the case of many participants, and

there were few criteria on which participants did worse at the second interview.

Individual scores ranged from 3 to 19 at the first interview, and 6 to 20 at the second

interview, and changes in individual scores ranged from +7 to -4.

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TABLE 4.2. Summary of Participants’ Numeration Skills Identified in two Interviews

Numeration Skilla

Participantb 1a 1b 1c 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 6a 6b 7a 7b 8a 8b 8c High/Blocks

Amanda | | | | | | | | | | | | | ⇑ ⇑ ⇑ | ⇓

Craig | | | | | ⇓ | | | | | ⇑ | | | | ⇑ ⇑ | | |

John | ⇑ | | | | | | | | | | | | | | | |

Simone ⇑ ⇑ ⇑ | ⇑ ⇑ ⇑ | | ⇓ ⇑ | | | ⇑

High/Computer

Belinda | | | | | | | | ⇓ ⇑ | ⇑ | | | | | | | | ⇓

Daniel | | | | | | | | | ⇑ ⇑ | | | | | | ⇑ ⇓ ⇑ |

Rory | | ⇑ | | | | | ⇓ | ⇑ | | | | | | | | ⇓

Yvonne | | ⇓ | | | | | | ⇓ | | | ⇓ | | ⇓

Low/Blocks

Clive ⇑ ⇑ | ⇓ | ⇓ | | ⇓

Jeremy | ⇑ | | ⇓ | ⇓ ⇑

Michelle | | ⇑ | ⇑ ⇑

Nerida | ⇑ | | | ⇑ ⇑ | | | ⇑ ⇑ ⇑ ⇑ ⇓

Low/Computer

Amy | | | ⇓ ⇑ ⇑ | | ⇓ | ⇑ ⇑

Hayden | | ⇑ | | ⇓ ⇑ | ⇑ | | | ⇑

Kelly | ⇑ ⇑ ⇑ ⇑ | ⇓ ⇓

Terry | | ⇑ | | | | ⇑ ⇑ | ⇑ ⇑ ⇑ | ⇓

Note. | - Criterion achieved at both interviews; ⇑ − Criterion achieved at Interview 2 only; ⇓ - Criterion achieved at Interview 1 only; no mark – Criterion not achieved at either interview. Criteria for numeration skills are described in Appendix M. aNumeration skills: 1 – Read block representation; 2 – Show block representation; 3 – Recognising three-digit block representations; 4 – Skip counting; 5 – Number relationships; 6 – Comparing pairs of numbers; 7 – Digit correspondence; 8 – Mental computation. bParticipants’ names are sorted alphabetically within groups in this and later tables. All participants’ names mentioned in this thesis are pseudonyms (Appendix G).

The number of place-value criteria attained by individual participants in the

two interviews are summarised as scores in Table 4.3. The circumstances of

Yvonne’s (h/c) second interview were different from the other 15 participants:

because her family went on a holiday before the end of the term, her second

interview was delayed for over 3 weeks. For this reason, her interview scores have

been discarded when calculating average group scores, and the row in Table 4.3

referring to Yvonne’s scores is greyed out.

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TABLE 4.3. Summary of Numeration Skills Demonstrated by Each Participant and by Each Group

Participant Group Score at Interview 1

Score at Interview 2

Increase (Decrease)

Amanda High/Blocks 15 17 2

Craig 18 20 2

John 17 18 1

Simone 7 14 7

Group Average: 14.3 17.3 3.0

Belinda High/Computer 19 19 0

Daniel 17 20 3

Rory 18 18 0

Yvonne 17 13 (-4)

Group Averagea: 18.0 19.0 1.0

Clive Low/Blocks 7 6 (-1)

Jeremy 6 6 0

Michelle 3 6 3

Nerida 8 14 6

Group Average: 6.0 8.0 2.0

Amy Low/Computer 8 10 2

Hayden 9 12 3

Kelly 4 6 2

Terry 9 14 5

Group Average: 7.5 10.5 3.0 Note. Maximum possible score per participant per interview was 21. aIn calculating average scores for the high/computer group, Yvonne’s scores have been discarded, as her second interview was conducted more than 3 weeks after teaching sessions were concluded.

The figures in Table 4.3 show that the aggregate scores for the 4 groups, if

considered on their own, would hide the differences in interview scores within

groups, that in some cases are greater than the differences between groups. For

example, Simone’s achievement of 7 more criteria at the second interview than the

first interview makes up more than half of the improvement (12 points) in the score

of the entire high/blocks group. Similar differences are evident in the scores achieved

by Yvonne (h/c), Nerida (l/b), and Terry (l/c) compared to their respective groups. In

the case of Yvonne, it is likely that her score was influenced by the circumstances of

her second interview, as mentioned earlier.

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With a small sample such as this, strong claims about the relative benefits of

use of computer software or blocks for learning place-value concepts based on the

interview scores would not be justified. Thus any conclusions that may be drawn

from the data are necessarily tentative; hypotheses that are suggested to explain any

apparent trends in the data will require further large-scale studies for testing. With

these comments in mind, the following observations are made regarding patterns in

the performance of the 4 groups at the interviews shown in Table 4.2:

1. The most improvement in scores is evident in 2 groups: the high/blocks

group and the low/computer group.

2. Certain participants appear to have been especially helped by the

teaching program used in the study, particularly Simone (h/b), Nerida

(l/b), and Terry (l/c).

3. Questions relating to skip counting (Skills 4a to 4d inclusive) showed

greater improvement among participants who had used the computer

than among those who had used the blocks.

One further observation can be made regarding Skill 8c, which involved

subtracting fewer than 10 ones from a number of tens (e.g., I1: 5 tens - 8). Question

9 (c), relating to this skill, was successfully completed by 7 fewer participants at the

second interview than at the first interview, and there was no participant who

improved on that question. The specific numbers used at each interview may help

explain this result. In Interview 1, participants were asked to subtract 8 ones from 5

tens; in Interview 2, the task was to subtract 6 ones from 7 tens. The particular

combination of numbers chosen may have led to a greater chance of error for

participants when considering the second version of the question. The high-

achievement-level participants who had answered the parallel question correctly at

the first interview but were incorrect at the second interview all gave the answers 63

or 61. This implies that they lost count of the tens and ones parts of the question,

either subtracting 6 from 7 to get the ones part of the answer, or subtracting 7 ones

instead of 6 ones from 70.

Achievement level and interview performance.

Although the differences in group scores between the first and second

interviews can be explained in light of individual performances, there are still

differences worth noting in the results summarised in Table 4.2 and Table 4.3. These

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two tables show a clear distinction between high-achievement-level participants and

low-achievement-level participants in their ability to meet criteria on interview tasks.

The interview scores are further collapsed in Table 4.4, showing the results from the

interviews arranged according to the mathematical achievement level of the

participants.

TABLE 4.4. Summary of Place-value Understanding Criteria Achieved by High-achievement-level and Low-Achievement-Level Participants

Achievement Level n Average score at

Interview 1 Average score at

Interview 2 Average increase

High 7 15.9 18.0 2.1

Low 8 6.8 9.3 2.5 Note. Scores of Yvonne (high/computer group) have been discarded. Maximum possible score per participant per interview was 21.

There was a marked difference in performance between the high-

achievement-level participants and low-achievement-level participants, with the

high-achievement-level participants achieving an average score that was

approximately twice that of the low-achievement-level participants at each interview.

Both cohorts improved between interviews; the low-achievement-level participants

had more room for improvement, and showed a slightly greater improvement,

increasing their scores by an average of 2.5 points. The difference in scores at the

first interview provides broad justification for the initial identification and selection

of high-achievement-level and low-achievement-level participants to participate in

the study. Though there were some anomalies in the performance of individual

participants, noted in the previous paragraph, in general high-achievement-level

participants showed a much better understanding of place-value than their low-

achievement-level counterparts.

Number representation formats and interview score.

The researcher’s intention was to form two equivalent groups of high-

achievement-level participants and two of low-achievement-level participants, with

one of group of each achievement level to use blocks and one to use computers in the

teaching sessions. However, the interview results show that there were marked

differences when comparing the 2 high-achievement-level groups with each other,

and also when comparing the 2 low-achievement-level groups, that raise a question

of the equivalence or the comparability of the pairs of groups of similar achievement

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level. Table 4.3 shows that at the first interview the high-achievement-level

participants who were to be in the computer group achieved an average score of 18.0

place-value criteria, compared to 14.3 among the high-achievement-level participants

who were to use the blocks. A similar difference is evident in the scores of the 2 low-

achievement-level groups: The computer group achieved an average score of 7.5

points, and the blocks group an average of 6.0 points. These differences are repeated

in Table 4.5, which shows that participants who used the computer started the study

with a score on average 1.9 points higher than participants who used blocks. At first

glance, these figures are cause for some concern, as it appears that the 8 participants

who used the computers started with a higher level of place-value understanding than

those who used the blocks.

TABLE 4.5. Summary of Place-value Understanding Criteria Achieved by Participants in Computer and Blocks Groups

Groups n Average score at Interview 1

Average score at Interview 2

Average increase

Computer 7 12.0 14.1 2.1

Blocks 8 10.1 12.6 2.5 Note. Scores of Yvonne (high/computer group) have been discarded. Maximum possible score per participant per interview was 21.

Two factors may help explain the reasons for the apparent inequality in levels

of place-value understanding shown in Table 4.5. First, differences in initial scores

varied among individuals more than expected, considering the results from the

previous Year 2 Net (Queensland Department of Education, 1996). Appendix F

shows that the performance on the Year 2 Net by the high-achievement-level students

selected to participate in the study showed little variation compared to results from

the interviews conducted in the study. In particular, Amanda and Simone, both in the

high/blocks group, were expected to perform better in the first interview compared to

their peers, based on the Year 2 Net results.

The second fact that may help explain the anomalies in the initial scores of

groups of participants is the method used to place participants in groups once the 16

participants had been selected. Initially, the students’ two class teachers assisted the

researcher to select 8 high-achievement-level and 8 low-achievement-level students,

and then the students were placed in pairs of the same gender. Four of the high-

achievement-level students, 2 girls and 2 boys, came from one class, and the

remaining 12 students were from another class. The researcher made the decision to

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separate the four children from the first class, so that there was not a group of 4

children from one class and 3 groups from another class. The teacher of the 12

students advised the researcher about which students she felt would work well

together, based on her knowledge of their friendship groups.

The researcher formed groups of 4 participants from these friendship pairs,

and randomly assigned each group to use the computer or the blocks. Thus the fact

that both computer group had somewhat higher levels of place-value understanding

prior to the commencement of the study was the result of a number of decisions

made for various pragmatic and research-oriented reasons, and the random

assignment of groups to each treatment. The variation in place-value understanding

of the 4 groups did not become evident until after the teaching phase had

commenced, as time did not allow the interviews to be transcribed prior to

commencing the teaching sessions.

4.4 Students’ Conceptions of Numbers Participants’ number conceptions and other information regarding participant

thinking are revealed through detailed analysis of the transcripts themselves, looking

at descriptions of the words spoken and the actions taken by participants as they

answered the questions. This analysis is described in this section, divided into

subsections, describing two broad approaches to interview questions, grouping

approaches (4.4.1) and counting approaches (4.4.2); and a common faulty

conception, the face-value interpretation of symbols (4.4.3). These results are

summarised in section 4.4.4, and comments are made about the changeability of

participants’ conceptions (4.4.5).

4.4.1 Grouping Approaches

A number of participants gave answers to interview questions that referred to

groups of 10 when dealing with numbers in the tens place. This is termed here a

grouping approach, and is considered to imply a concept of multidigit numbers that

recognises the groups of 10 around which the base-ten numeration system is based.

Transcripts of responses to interview Questions 1, 3, 6, 7, 8, and 9 show instances of

participants using a grouping approach. The following paragraphs describe how

individual participants used a grouping approach in answering each of these

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questions. At the conclusion of this section, Table 4.6 and Figure 4.1 summarise the

use of grouping approaches by each participant.

Question 1 (b) and (c): Interpreting non-canonical block representations

(e.g., asking the participant to say the number represented by 4 tens & 12 ones). In

answering these questions some participants grouped either ones or tens to make a

group of 10 blocks, and then counted the new group as a ten or hundred,

respectively, before finishing the count. For example, to interpret a block

arrangement comprising 4 tens and 12 ones, some participants first grouped 10 of the

ones together, then counted the tens including the new group of 10 ones, and then

added the remaining 2 ones:

Craig (h/b): I just got all the tens together here and I said to myself there’s 40 there and I

counted these, and there was 10 [ones] there. And so I thought I put them with

the tens so I know that there is 10 here. Then I counted the last two. So it’s 52.

(I1, Qu. 1b)

A comment is needed at this point about the possible use of a grouping

approach when answering Question 2: Using blocks to represent a two-digit or

three-digit number. Base-ten blocks allow students to take advantage of the grouped

structure inherent in the blocks themselves to represent the groups-of-ten structure in

the base-ten numeration system, as described in section 2.5.3. However, base-ten

blocks may also be used to represent numbers using a face-value interpretation of

numbers, as discussed later. Thus if a participant used the blocks to represent a

number canonically it is not possible to tell if the student had in mind the groups of

10 in the number, or a face-value construct for multidigit numbers. Therefore,

whereas it is possible to identify a counting approach (section 4.4.2) in a participant’s

response to Question 2, it is not possible to clearly identify the use of either a

grouping approach or a face-value construct in an answer to this question.

Question 3 (a) and (b): Interpreting non-canonical block representations of

three-digit numbers, and comparing them with written symbols (e.g., comparing

1 hundred, 2 tens, & 16 ones with 136). For these questions, participants were asked

to read a written symbol for a three-digit number. They were then shown three

examples of block arrangements one at a time, and asked if each arrangement

represented the same number as the written symbol. The third example was incorrect,

and targeted the face-value construct for multidigit numbers, discussed in section

4.4.3. The first two arrangements were of non-canonical representations for the

109

number on the card—for example, 1 hundred, 2 tens, and 16 ones for 136—and

could be answered using a grouping approach. This approach is demonstrated in the

following transcript excerpt showing Daniel (h/c) interpreting a collection of 17 tens

and 2 ones and comparing it with the symbol ‘172’:

Daniel: Mmm … [counts out 10 tens and places them together, then counts remaining

7 tens] — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Yeah that’s 10 there. 1, 2, 3, 4, 5, 6, 7.

Yep (the blocks represent the same number). (I2, Qu. 3b)

Question 6 – Comparing pairs of two-digit and three-digit numbers (e.g.,

compare 27 & 42, 174 & 147). In Question 6 participants were shown a pair of

written symbols for two-digit numbers, followed by a pair of symbols for three-digit

numbers. In each case they were asked which number was larger, and to explain their

reasoning. It was considered that participants were using a grouping approach if they

referred to the names of the places concerned when justifying their answers. For

example, consider the following transcript showing Rory (h/c) explaining which of

the symbols ‘27’ and ‘42’ represents the bigger number:

Rory: That one: ‘42.’

Interviewer: And how do you know it’s bigger?

Rory: Because it has more tens.

Interviewer: Uh-huh. And how many tens does it have?

Rory: ‘4.’

Interviewer: — Can you explain why that one’s bigger? I mean this one [‘27’] has a ‘7’ and

[you say] this one is smaller …

Rory: Because this one has 4 tens and 2 ones and that one has 2 tens and 7 ones.

(I1, Qu. 6a)

Rory, like several other participants, clearly knew that the position of each

digit determines its place name, and that tens are worth more than ones are. What is

not revealed by this nor other transcript excerpts is whether or not these participants

were aware of the “tenness” of a number in the tens place—the fact that “a ten” is a

collection of 10 ones. As S. H. Ross (1990) commented,

children may sound very knowledgeable as they speak of so many “tens and ones.” Yet in reality a child may be using a face-value interpretation in which “tens and ones” are merely names for different objects and have no real connection to “tenness.” (p. 14)

110

Despite this observation, it appears that responses like Rory’s do show an

awareness that tens and ones are not interchangeable, as students with face-value

interpretations of digits sometimes indicate. Nor did Rory’s response rely on the

counting sequence to justify why one number is larger than the other is: He indicated

that it was sufficient to check individual digits, in particular the tens digit, to

determine the larger number. For this reason, it is decided to include responses to

Question 6 that include reference to place names in the category of using a grouping

approach. Nevertheless, the points raised here should be kept in mind when

considering summaries of grouping and counting approaches given later in this

chapter.

Questions 7 and 8: Explaining referents for the digits in two-digit written

symbols (e.g., count 24 sticks, write symbol, & explain symbol). Participants were

asked to count a number of objects and to write the symbol for the number. They

were then asked to show which objects were represented by each written digit. Many

participants answered with a face-value interpretation of the symbols, but others

correctly showed the objects remaining after the ones had been taken out as the

referents for the tens digit. As in the case of responses to Question 6, again this type

of response may not indicate a complete understanding of the groups represented by

the tens digit. Nevertheless, it does show an awareness that the digit represents more

than its face value, and that the referents for the two digits together make up the

entire collection of objects. To distinguish between (a) the basic understanding that

the sum of the objects corresponding to the digits in a number must equal the entire

collection, and (b) the more advanced concept that a tens digit represents the product

of the digit’s face value and its place value, participants who indicated the correct

number of objects for the tens digit were asked “How can that digit stand for so

many?” If a participant said that the digit was a number of tens, the response was

categorised as showing a grouping approach. For example:

Interviewer: Does this part [‘3’] of your ‘37’ have anything to do with how many sticks

you have? Can you show me?

Rory (h/c): Yeah. [Picks up remaining 30 sticks] There.

Interviewer: All of them? How does that ‘3’ stand for all of those?

Rory: Because it’s 3 tens.

Interviewer: All right, so how many have you got in your hands there?

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Rory: [Just glances at them] 30. (I2, Qu. 7c)

Instances of a grouping approach to digit correspondence questions were not

very common; Table 4.6 shows that only 5 participants, all of them high-

achievement-level participants, showed a grouping approach at either interview in

answering these two questions. Section 4.5 includes a description of four distinct

categories of response to Questions 7 and 8, ranging from the grouping approach

described here to a face-value interpretation of digits. The grouping approach is

classed as a Category IV response, the highest level of response noted in this study.

Question 9 – Mental addition and subtraction (e.g., How many pieces of gum

in 3 packets of 10 sticks + 17 sticks?). In Question 9 participants were asked to work

out the answers to three questions: adding a group of tens and fewer than 10 ones,

adding a group of tens and between 11 and 19 ones, and subtracting fewer than 10

ones from a number of tens. To assist their thinking, at the first interview participants

were provided with packets of 10 pieces of chewing gum, and at the second

interview participants were provided with plastic bags each containing 10 clothes

pegs. In each case participants could handle and count the packets or bags, but they

were not permitted to open the collections to manipulate single items. Bags of pegs

permitted participants to see the pegs, and packets of gum allowed individual pieces

to be felt under the wrapper. Participants adopting the grouping approach used the

groups of 10 in each question to help them answer the question. For example, note

how Belinda (h/c) added 3 groups of ten and 17 single objects:

Belinda: 47. There’s um, three of them and then there’s a one, which would make a 40,

and then you put a ‘7’ on the end and it equals 47. (I1, Qu. 9b)

The same use of groups of 10 is shown in the following transcript in which

Terry (l/c) calculated 5 tens minus 8 ones. Terry subtracted 8 from 10, and then

added the remaining 4 tens:

Terry: 8, and there’s 10. [Moves packets to the left, counting quietly in tens] 42.

Interviewer: That was quick. How did you work that out?

Terry: ‘Cos I already knew. ‘Cos it’s 10, there only had to be 2 more because 9, 10.

Interviewer: And you know how many are in those packets?

Terry: Yup. It’s how you tell. ‘Cos there’s only 1 ten, 2 tens, 3 tens, 4 tens. So it must

be 40. (I1, Qu. 9c)

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Summary of the use of grouping approaches.

The use of grouping approaches by each participant is indicated in Table 4.6,

and group totals are summarised in Table 4.7. It should be noted that many

participants used a variety of approaches to answer further questions from the

researcher; other approaches are indicated in later tables in this chapter. For each

participant the responses at each of the two interviews are indicated in two adjacent

rows of Table 4.6, and the number of questions for which the participant used a

grouping approach at each interview is indicated in the last column.

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TABLE 4.6. Use of Grouping Approaches for Selected Interview Questions

Question Participant Interview 1b 1c 3a 3b 6a 6b 7 8 9a 9b 9c Count

High/Blocks Amanda 1 x x x x x 5 2 x x x x x x x 7 Craig 1 x x x x x x x 7 2 x x x x x x x x x x 10 John 1 x x x x x x x x x 9 2 x x x x x x x x 8 Simone 1 x 1 2 x x x x x x 6

High/Computer Belinda 1 x x x x x x x x x x 10 2 x x x x x x x x x 9 Daniel 1 x x x x x x x 7 2 x x x x x x x x 8 Rory 1 x x x x x x x x x x 10 2 x x x x x x x x x x 10 Yvonne 1 x x x x x 5 2 x x x x x x 6

Low/Blocks Clive 1 0 2 x 1 Jeremy 1 0 2 0 Michelle 1 0 2 x 1 Nerida 1 x 1 2 x x 2

Low/Computer Amy 1 0 2 x 1 Hayden 1 x x x 3 2 x 1 Kelly 1 0 2 0 Terry 1 x x x 3 2 x x x x x 5

Note. x – indicates use of a grouping approach in responding to the question.

TABLE 4.7. Use of Grouping Approaches by Each Group

Blocks Computer Total High 53 65 118 Low 5 13 18 Total 58 78 136

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It is clear from Table 4.7 that, overall, high-achievement-level participants

used the grouping approach far more often than low-achievement-level participants

did. On average, high-achievement-level participants used grouping approaches to

answer over 7 questions per interview, whereas the low-achievement-level

participants used them for just more than 1 question per interview. The clear

difference in the patterns of response of high-achievement-level and low-

achievement-level participants implies a markedly different level of understanding of

place-value. Overall, the computer groups used grouping approaches more often than

did blocks groups; however, this is considered to be due to differences of individual

members of these groups, as described earlier.

Scores achieved by the 16 participants at each interview are compared to the

number of times that a grouping approach was used in achieving those scores in

Figure 4.1. This scatter-plot graph shows a clear pattern of higher numbers of place-

value criteria being achieved by those participants who used grouping approaches the

most. Apart from one participant who achieved 14 criteria while using grouping

approaches only twice, participants who achieved more than 10 criteria at interviews

also used grouping approaches at least 5 times in the same interview. It should be

noted that, in this and later scatter-plot graphs, certain data points overlap others, so

that not all 32 data points are visible. This graph may be compared with Figure 4.2,

which shows a similar comparison between interview scores and counting

approaches.

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Figure 4.1. Interview scores compared to use of grouping approaches.

4.4.2 Counting Approaches

A second common approach to interview questions, adopted by several

participants, was based on consideration of individual ones in a number, rather than

groups of 10 ones or 10 tens. Participants’ responses of this type involved either

counting single one-blocks without grouping them first, or reference to the counting

sequence of number names, and so this approach is called a counting approach.

Counting approaches were characterised by participants ignoring the grouped aspect

of base-ten numbers, and treating multidigit numbers as collections of ones.

Representative responses to certain interview questions are summarised in the

following paragraphs.

Question 1 (b) and (c): interpreting two-digit and three-digit non-canonical

block representations. The counting approach was clearly evident among some

participants when attempting to name a number represented by a non-canonical

arrangement of blocks. For example, the following excerpt demonstrates that Jeremy

(l/b) used a counting approach to work out the number represented by 3 tens and 16

ones:

Jeremy: [Touches tens] 10, 20, 30, [touches ones] 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,

41, 42, 43, 44, 49, 46. (I2, Qu. 1b)

15

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20

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7

14

1919

17

20

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13

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3

6

8

14

8

109

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4

6

9

14

0

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6

9

12

15

18

21

0 1 2 3 4 5 6 7 8 9 10

Incidence of Grouping Approaches

Plac

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Crit

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Ach

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Jeremy’s answer in this instance was correct, but the inefficiency of the

method caused him to take longer than he might have using a grouping approach, and

there was a greater chance that a counting error could cause him to arrive at an

incorrect answer. In fact, Jeremy at first gave the answer 36 for this question, perhaps

because of a mistake at the change of decade from 39 to 40. Other participants also

made counting errors when counting 3 tens and 16 ones: Michelle (l/b) and Simone

(h/b) also gave the answer as 36, Hayden (l/c) and Nerida (l/b) as 47, and John (h/b)

as 44. Discussion of the relative efficiency and usefulness of grouping and counting

approaches is continued in section 5.2.2.

Question 2: Using blocks to represent a two-digit or three-digit number.

When asked to represent a two-digit or three-digit number using the blocks, some

participants chose to use what Fuson (1990a) called collected multiunits: They

represented the tens and ones digits of a number using only ones material. For

example, when showing 261, Daniel (h/c) selected 2 hundred-blocks, and then

started to count out 61 ones. He stopped when he had 20 ones, and changed his mind,

putting out 2 hundreds, 6 tens, and 1 one. Amy (l/c) used a similar approach when

asked at her first interview to show the number 134. She started to count out one-

blocks, apparently meaning to count 134 ones. She stopped when she reached 59,

and changed her method to putting out 10 tens and 34 ones. This approach, of

choosing multiple ones or tens to represent a multidigit number, is an example of a

counting approach. Rather than making use of the groupings inherent in the base-ten

numeration system, participants using multiunits to represent a multidigit number

count out blocks one at a time in until the end number is reached.

There is evidence that participants who used a counting approach for

Question 2 did so because they had not thought of using the already-grouped base-

ten material. This is shown in both examples mentioned above. Daniel changed to a

canonical representation for 261 himself, without input from the researcher.

Similarly, after a while Amy decided on her own not to try to count 134 ones, though

nevertheless she still chose to use 10 tens rather than 1 hundred-block and 34 ones

rather than 3 tens and 4 ones. It is quite possible that in these incidents participants

did not use a hundred-block because of a lack of familiarity with both three-digit

numbers and the base-ten blocks used to represent them, as at the time of the first

interview participants’ class teachers had not taught about the hundreds place.

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Question 6 – Comparing pairs of two-digit and three-digit numbers. Some

participants demonstrated a counting approach when answering Question 6. When

asked to justify their answer stating which of two numbers was larger, some

participants referred to the position of one or both numbers in the counting sequence.

For example, Hayden (l/c) explained in this manner when comparing 138 and 183 at

the first interview:

Interviewer: Which number is larger?

Hayden: 183.

Interviewer: OK, and how do you know it’s bigger?

Hayden: Because it takes longer than 138.

Interviewer: How do you know it’s going to take longer?

Hayden: Because you have to count to a 100 and then keep … count to um 83, and [for

the other number] you just have to count to 138. (I1, Qu. 6b)

A counting approach was also evident in the way that some participants

appeared to be influenced by the verbal names of the numbers in a question. For

example, in the following excerpt Michelle (l/b) appeared to have no reason for

believing 42 to be larger than 27, other than their respective names:

Michelle: [Points to ‘27’ then changes mind and points to ‘42’] No, that one [‘42’] is

bigger.

Interviewer: — And how do you know it’s bigger?

Michelle: Because it’s … that’s 27, that’s forty-se … 42.

Interviewer: Uh-huh, so how do you know 42 is bigger?

Michelle: Because they’re [‘27’] little and they’re [‘42’] bigger. (I1, Qu. 6a)

It may be that Michelle was thinking of some other reason for believing that

42 is greater than 27 other than the counting sequence. However, other authors (e.g.,

Resnick, 1983) have suggested that many children without an understanding of the

tens and ones nature of two-digit numbers picture numbers only as a sequence of

counting numbers. This would be consistent with Michelle’s statement in the

previous excerpt that (a) one number was 27 and the other was 42, and that (b) 27 is

little and 42 is bigger. Certainly the next example supports this argument, as it shows

Amy starting by referring to the verbal names of two numbers and then referring to

their position in the counting sequence.

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The name of a number appeared at times to trigger a response in some

participants that focussed on their knowledge of the counting number sequence. For

example, Amy (l/c), in comparing 38 and 61, started to say that 38 was bigger, until

she read the names of the numbers represented by the symbols. She started to say

that ‘61’ was “sixteen,” but corrected herself and immediately said that “That’s

bigger because it’s 61. So and that’s [‘38’] smaller.” She followed this with a clear

example of a counting approach, explaining that 61 was a bigger number than 38

because of their relative positions in the counting sequence: “‘Cos then it goes 40,

50, then 60” (I2, Qu. 6a). Another apparent example of a counting approach was seen

in a transcript in which Clive (l/b) compared the symbols ‘259’ and ‘295.’ Clive had

initially chosen 295 as the larger number, but noted that the written symbols had the

same digits, in different positions. When pressed, Clive said that he knew that 295 is

larger “because um it sounds like it’s the biggest number” (I2, Qu. 6b).

Questions 7 and 8: Explaining referents for the digits in two-digit written

symbols. An interesting phenomenon occurred among some participants when

answering Questions 7 and 8, that again indicates thinking that included the idea of

counting. Certain participants correctly rejected the idea that the two digits each

represented only their face value, but failed to explain the meanings of the digits in

terms of the groups of 10 and single ones. Instead, they explained that the two digits

in the symbol together represented the entire collection of objects, but that each

individual digit did not have a referent. This response ignores the grouped tens aspect

of multidigit symbols, and instead focuses on the entire set as a collection of single

objects: a counting idea. For example, Amanda (h/b) explained the referents for each

digit in the number 13 using a counting approach that incorporated her understanding

that the digits ‘1’ and ‘3’ combined in the symbol ‘13’ somehow represented more

than the sum of their individual values:

Amanda: If there’s only this three [takes the beads out of one cup and puts three out] by

itself then it won’t be 13.

Interviewer: Yes. All right, not on its own, no. All right, okay. Put them back in the cup

again. Now look at this part [‘1’] of your ‘13’: Does it have anything to do

with how many beads you have?

Amanda: Because there’s a one and you need another three, but that, it’s not like that,

because it has to be 13.

Interviewer: Uh-huh.

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Amanda: One … and three, doesn’t make it.

Interviewer: — So can you tell me why that’s a one? Er … what the one is for?

Amanda: — You need it because that’s how you count it, that’s how much they are.

Interviewer: Right, but the one … are you saying that the one doesn’t really stand for

anything? It’s just how you write it down, is that right?

Amanda: It means something, but that’s how … It means it’s part of the number, and

it’s um … you need it because um if you can’t have, if you don’t use it, it will

only be 3, not 13.

Interviewer: Uh-huh … it sounds like it’s to do with how you write it down?

Amanda: And it needs both of the numbers to make it. (I1, Qu. 8b)

Several participants explained the referents for individual digits in two-digit

numbers using explanations similar to Amanda’s response. These responses to digit

correspondence tasks are defined as Category II responses, according to the

hierarchy of response categories proposed in section 4.5. Participants responding as

Amanda did in the previous transcript were often forced to deal with contradictions

in their beliefs, due to their not recognising any number of sticks as corresponding to

each individual digit. The place of such contradictions in children’s development of

place-value understanding is discussed further in section 5.4, looking at evidence of

participants’ construction of meaning and how children managed apparent

contradictions as they perceived them in the information available to them.

A further example of counting approaches used when responding to the

question “How can that digit refer to so many objects?” is provided in the following

transcript excerpt. Hayden (l/c) gave a counting explanation for the fact that the face

value of the tens digit did not match the number of objects it represented:

Hayden: [The ‘7’] is a part of 30 … it’s a part of like in 30 it’s a part like … you count

to 30 and then you count 7 more and it ends up 37.

Interviewer: And what does this ‘3’ here mean?

Hayden: It’s up … it’s up to 30 … like if you count up to 30. (I2, Qu. 7b)

Question 9 – Mental addition and subtraction. Several participants opted for

a counting approach to answering mental addition and subtraction questions. In some

cases, participants used their fingers as an aid to counting; in others, they nodded

their heads or pointed at the desk, as if at imaginary objects. Unlike those who used a

grouping approach to consider separately the tens and ones parts of an addition or

subtraction question, participants using a counting approach stepped forward or back

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in the cardinal number sequence to find the answer. For example, participants using a

counting approach to answer the question 3 tens plus 17 ones generally chose an

inefficient approach of counting on from 30 by 17 steps. The following transcript

excerpt shows Kelly (l/c) using this method, and demonstrates a difficulty that it

introduces for the student. In counting on by 17 from 30, Kelly made an error and

reached the answer 43:

Kelly: [Touches each packet of gum] 10, 20, 30. [Counts on fingers by touching them

one by one on table] 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43.

43 pieces of gum.

Interviewer: 43. How did you do that?

Kelly: I counted them in tens and then I counted 17 more on.

Interviewer: Uh-huh. How did you know when you got to 17?

Kelly: I um in my head I counted out the um numbers and I just um did that [touches

fingers one by one on table] and I knew when I got to 17 ‘cos you have 10 and

you add a couple more on …

Interviewer: How many do you add on to make 17?

Kelly: Well you add um seven more on. (I1, Qu. 9b)

The above transcript clearly shows that Kelly was not using a grouping

concept for two-digit numbers. She knew that 17 was made of 10 plus 7, shown by

the fact that she used her fingers to count on 10 and then started again to add another

7. However, she evidently did not perceive of the ten as a group that could be added

straight to the 3 tens to make 4 tens, but rather saw the ten as 10 ones that had to be

added to 30 one at a time. For the same question Nerida (l/b) used an even more

inefficient counting strategy: she started at 17, and then counted on the 3 tens as 30

ones, using her fingers:

Nerida: [Counts quietly, looking around, then counts on her fingers] 47.

Interviewer: 47, well done. How did you do that?

Nerida: I counted um these 17 first then I counted 10, counted on by 10. — I went

from 17 and I counted on three times out of tens out of my hands. (I1, Qu. 9b)

In this instance Nerida’s strategy was successful: evidence of the care she

must have taken in carrying out the counting. However, the likelihood of making a

mistake with this method is clearly quite pronounced. Difficulties for students using

a counting approach are discussed further in section 5.2.2.

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Summary of the use of counting strategies.

Table 4.8 shows use of counting strategies by each participant at each

interview. These data are summarised for each group in Table 4.9. Incidence of

counting strategies by individual participants is compared with their interview scores

in Figure 4.2.

TABLE 4.8. Use of a Counting Approach for Selected Interview Questions

Question

Participant Inter-view 1b 1c 2a 2b 6a 6b 7 8 9a 9b 9c Count

High/Blocks Amanda 1 x x 2 2 0 Craig 1 0 2 0 John 1 x x 2 2 x 1 Simone 1 x 1 2 x x 2

High/Computer

Belinda 1 0 2 0 Daniel 1 0 2 x 1 Rory 1 0 2 0 Yvonne 1 x 1 2 x 1

Low/Blocks

Clive 1 x x 2 2 x x x x x 5

Jeremy 1 x x x 3 2 x 1 Michelle 1 x x 2 2 x x 2 Nerida 1 x x x x 4 2 x x x x 4

Low/Computer

Amy 1 x x x x x 5 2 x x x x x 5

Hayden 1 x x x x x x x 7 2 x x x x 4 Kelly 1 x x x x x x x 7 2 x x x x x x 6 Terry 1 x 1 2 x 1

Note. x – indicates use of a counting approach in responding to the question.

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TABLE 4.9. Use of Counting Approaches by Each Group

Blocks Computer Total High 8 3 11 Low 23 36 59 Total 31 39 70

Table 4.9 shows that, as in the case of grouping approaches (Table 4.6), there

was a clear difference in the frequency with which the high-achievement-level and

low-achievement-level participants used the strategy. However, this trend is reversed

in the case of counting strategies: Whereas high-achievement-level participants were

much more likely to use a grouping approach, the low-achievement-level participants

used counting approaches more than 5 times as often as high-achievement-level

participants. Differences between total use of counting strategies by blocks and

computer groups are minor.

Figure 4.2. Interview scores compared to use of counting approaches.

Figure 4.2 shows how interview scores related to the use of counting

approaches. Clearly, participants who showed better place-value understanding used

counting approaches infrequently. On the other hand, participants with weak place-

value understanding included participants who used counting approaches frequently

and others who did not do so. Figure 4.2 shows that it is possible to answer questions

like those in the interviews successfully using the less efficient approach of counting.

For example, one particular data point on the graph represents Hayden’s (l/c)

15

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Incidence of Counting Approaches

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performance on the first interview, at which he used a counting approach 7 times,

and achieved a score of 9 out of 21. However, it is highly likely that consistent use of

counting approaches would lead to difficulties in the future if a student did not learn

to switch to using the groups of 10 inherent in the base-ten numeration system: This

point is discussed further in section 5.2.2.

4.4.3 Face-Value Interpretation of Symbols

As discussed in section 2.4.2, a face-value interpretation of multidigit

numerical symbols is very common among children who are learning about the base-

ten numeration system. Researchers investigating a variety of aspects of place-value

understanding have found children who believe that each digit in a multidigit number

represents only its face value, rather than groups of 10, 100, and so on. Data

collected in this study reveal such ideas among several of the participants. In

particular, Questions 3 (c), 6, 7, and 8 prompted certain participants to use a face-

value construct in answering the question. The ways that face-value ideas were used

in each question are described in the paragraphs following. Note the comments in

section 4.7, regarding the use of base-ten material to represent multidigit numbers:

Responses to other interview questions may have been influenced by face-value

interpretations of digits without this being obvious.

Question 3 (c): Interpreting block representations of three-digit numbers with

misleading perceptual cues, and comparing them with written symbols. The task set

in Question 3 (c) was similar to that in Question 8, in that it offered participants

misleading perceptual cues about how an arrangement of blocks represented a

number. The blocks were arranged so that the numbers of blocks of each size

matched the three digits in the printed numerical symbol in order from left to right,

but so that the values represented by the blocks were incorrect. For example, in the

first interview the participants were shown 1 ten, 3 hundreds, and 6 ones in order

from left to right and asked whether or not they represented the number 136.

Most participants did not initially accept the three-digit block representation

presented to them in Question 3 (c) as correct. Considering the blocks presented, an

in particular the large number of hundred-blocks, it is perhaps not surprising that

even a student who held a face-value interpretation for multidigit numbers would

agree that the blocks represented the number. However, when the researcher offered

the counter-suggestion that each digit could in fact represent the number of blocks

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presented, some participants did accept the idea, indicating some willingness to

accept a face-value interpretation:

Interviewer: Do these blocks [1 ten, 3 hundreds, 6 ones] show that number [136]?

Terry (l/c): Well I already know [that they do not], ‘cos there’s a thousand [sic] in this

[top 2 hundreds] and there’s a hundred in this [lowest hundred-block] …

Interviewer: Mmm. So is that [block arrangement] the same as that [symbol on card]?

Terry: No.

Interviewer: Right, OK. Well let me just ask you another question, then. Could that ‘1’ [on

card] be for that [ten-block] and that ‘3’ be for those three [hundred-blocks]

and that ‘6’ be for those six [one-blocks]?

Terry: Oh yes! It does add up to that, does it?

Interviewer: Oh, Right.

Terry: ‘Cos it’s a hundred [points to ten-block], thirty [3 hundreds], six [6 ones]. Yes.

(I1, Qu. 3c)

Michelle (l/b) also initially rejected the face-value interpretation of the block

arrangement, but then offered her own, equally incorrect, block arrangement. She

apparently was not content to agree that 3 hundreds could represent the ‘3’ in ‘136,’

and changed the hundred-blocks for 3 tens. However, she left the 1 ten to stand for

the ‘1’ digit. Later she accepted the researcher’s counter-suggestion that the initial

block arrangement did match the written symbol.

Question 6: Comparing pairs of two-digit and three-digit numbers. Questions

6 (a) and 6 (b) required participants to compare two pairs of printed numerical

symbols. The numbers in Question 6 (a) were two-digit numbers, such that the

smaller number had a ones digit that was larger than either digit in the larger number;

the pairs were 27 and 42 in Interview 1 and 38 and 61 in Interview 2. The numbers

for Question 6 (b) were three-digit numbers, that had the same digits, with the tens

and ones swapped; the pairs at Interview 1 were 183 and 138, followed at Interview

2 by 295 and 259. The intention of these questions was to target face-value

interpretations of symbols, as a face-value interpretation should lead a participant to

choose the smaller number in Question 6 (a), and to state that numbers in Question 6

(b) were equal. As in other questions, the researcher offered face-value counter-

suggestions to participants who gave the correct answer, to test the stability of their

beliefs. Accepted counter-suggestions are indicated in the summary of face-value

interpretations in Table 4.10 by parentheses in the relevant cells of the table.

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Several participants provided face-value interpretations when answering

Question 6, without any counter-suggestion being offered. For example, Jeremy (l/b)

stated that 38 was bigger than 61 immediately on being asked:

Jeremy: [Points to ‘38’]

Interviewer: What number is that?

Jeremy: 38.

Interviewer: And why is 38 bigger?

Jeremy: Because it’s got a … 3 tens and 8 ones.

Interviewer: All right, and what’s the other number?

Jeremy: 61

Interviewer: And which one’s bigger?

Jeremy: 38.

Interviewer: That’s got 3 tens and 8 ones. And what’s this one [‘61’] got?

Jeremy: 6 tens and 1 ones. (I2, Qu 6a)

It is interesting that although Jeremy could correctly state the name of each

digit’s place, he ignored these labels in favour of a face-value interpretation of each

individual digit. A second example shows Terry (l/c) explaining why he believed that

259 and 295 were equal:

Interviewer: Can you tell me which of these two [‘259’ & ‘295’] is bigger?

Terry: You’re trying to trick me, aren’t you?

Interviewer: Well, I might be able to Terry.

Terry: Well, they are both bigger.

Interviewer: They’re both bigger? They’re both the same?

Terry: Yep.

Interviewer: And why are they both the same?

Terry: 259, 295.

Interviewer: So why are they the same? That doesn’t sound the same.

Terry: If you just turn around the ‘5’ and put the ‘9’ there, it’d be 259. (I2, Qu 6b)

Question 7: Explaining referents for the digits in two-digit written symbols.

Questions 7 and 8 were written purposely to target participants’ understanding of

two-digit written symbols, and to identify participants who held either face-value

interpretations of written symbols or correct grouping interpretations. In each

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question participants were asked to count a set of between 10 and 40 objects, to write

the written symbol for that number, and then to say which objects were represented

by each digit.

The results of Question 7 initially showed a considerable number of

participants who apparently held a face-value interpretation for the two-digit written

symbols involved (see Table 4.10). At the start of the question all participants easily

counted the objects and wrote the correct symbol for the number counted. The

researcher asked each participant if the number the participant had written

represented the entire group of objects, and most participants agreed that it did.

When asked about the referents for each digit, many participants indicated objects

that corresponded with only the face value of each digit. If that was the case, the

researcher asked them about the remaining objects: In Interview 1 there were 18 out

of 24 sticks left over, and in Interview 2 there were 27 out of 37. The following

excerpt is typical of transcripts of participants holding the face-value construct:

Interviewer: Does this part [‘4’] of your 24 have anything to do with how many sticks you

have?

Clive (l/b): [Frowns, nods]

Interviewer: Can you show me?

Clive: [Separates four sticks to his left]

Interviewer: Does this part [‘2’] of your ‘24’ have anything to do with how many sticks

you have?

Clive: [Puts out two sticks]

Interviewer: [Moves two sticks so they are above the ‘2’] So this ‘2’ is for two and then we

have another four [puts four sticks above ‘4’]. What about those [remaining

sticks] there?

Clive: They are the leftovers.

Interviewer: You said that this number [‘24’] was for all of the sticks. Do you still agree

with that?

Clive: [Nods]

Interviewer: All right, but you are saying now that the ‘4’ here [points to symbol] is for

those four [points to sticks] and the ‘2’ [symbol] is for those two [sticks] …

Clive: And they’re left over … by themselves. (I1, Qu. 7b)

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Clive’s answer that the remaining 18 sticks were “leftovers” is typical of

responses of many participants who apparently held a face-value interpretation for

two-digit numbers. Considering that it is almost certain that the participants invented

the ideas themselves, the similarity between responses such as the following is quite

remarkable:

Craig: Um. Oh, they’re extras. (I1, Qu. 7b)

Michelle: They’re just extras. (I1, Qu. 7b)

Terry: They’ll be left out. (I1, Qu. 7b)

Nerida: They’re left over. (I1, Qu. 8b)

Jeremy: They stay up because they’re not in there. (I2, Qu. 7b)

Kelly: Um, they don’t stand for any of them. (I2, Qu. 7b)

Simone: Those don’t count. (I2, Qu. 7b)

Amanda: Well they’re nothing then if that’s how that is. (I1, Qu. 7b)

Amy: Um, well, they would but they’re not included in that, um, these things.

(I1, Qu. 7b)

This collection of responses is considered important, as it reveals an aspect of

the participants’ beliefs about how symbols represent numbers that is evidently

common, but has not been reported in the literature before. Discussion of these and

other responses are continued in section 4.5, in which four categories of response to

digit correspondence tasks are identified.

Question 8: Explaining referents for the digits in two-digit written symbols

with misleading perceptual cues. Question 8 added another layer of difficulty to the

tasks in Question 7. Participants were asked to share a set of objects evenly into a

certain number of groups, resulting in equal-sized groups and leftover objects that

matched the digits in the written symbol, except that the groups were not groups of

tens and ones. In Interview 1, there were 13 beads to share evenly among three cups,

resulting in three cups of beads and one left over (Figure 3.10). In Interview 2, there

were 26 counters to share evenly onto six circles, resulting in six groups with two

remaining. Reports in the research literature (e.g., S. H. Ross, 1989, 1990) describe

children choosing incorrect interpretations of written symbols in the face of such

misleading perceptual cues.

As with Question 7, there were several variations of face-value interpretation

of written symbols evident in responses to Question 8. Some participants nominated

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a face-value interpretation without prompting by the researcher, nominating the

groups and leftover objects as referents for the digits in the written symbol. Other

participants initially did not choose these referents by themselves, but accepted them

later when the researcher suggested them. Some participants were unsure about the

researcher’s suggestion, and indicated that the face-value interpretation might be

correct, and still others rejected a face-value interpretation and gave a correct

interpretation of the digits.

The incidence of participants choosing a face-value interpretation for written

digits when faced with misleading perceptual cues was quite low (see Table 4.10).

The research literature, however, indicated that this pattern of response was quite

common. For example, S. H. Ross (1989) found that “nearly half” of the third-grade

participants in her study incorrectly chose a face-value interpretation of 26 objects

grouped in six groups and two single objects. In this study, however, even

participants who associated “remaining” ungrouped objects with the tens digit often

did not also associate the grouped objects with the ones digit: At the first interview, 5

participants chose the remaining bead as the referent for the ‘1’ in ‘13’ without

prompting; at the second interview, 2 participants chose the two single counters for

the ‘2’ in ‘26’ without prompting. On the other hand, no participant chose the three

cups in Interview 1 as referents for the ‘3’ digit for themselves, and at Interview 2

only 1 participant (Simone; h/b) initially said that the six groups were represented by

the ‘6.’

With prompting by the researcher a few participants were willing to accept

the face-value interpretation for the written digits suggested by the grouped objects.

However, even those participants who did accept the incorrect suggestion were

generally still reluctant to agree completely with the idea. In the following excerpt,

Yvonne (h/c) was clearly not totally convinced that the suggested face-value

interpretation was correct:

Interviewer: Let me say something to you: Some people would say that the ‘3’ is the three

cups and the ‘1’ is that one [bead]. Now is that right?

Yvonne: [Nods slowly]

Interviewer: You look a bit doubtful. Do you think it might be, or you think it is, or you are

sure it is, or … what do you think?

Yvonne: I think it is. (I1, Qu. 8b)

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Interpretation of digits in multidigit numbers is an important component of

understanding the base-ten numeration system. Section 4.5 includes more detailed

analysis of participants’ explanations for the meanings of the digits in two-digit

numbers, and descriptions of four categories of response to digit correspondence

questions.

Summary of the occurrence of face-value interpretations of symbols.

Table 4.10 indicates the incidence of face-value thinking in participants’

responses to Questions 3 (c), 6, 7, and 8, as described in this section. Each “x” in the

table represents a response to a question in which the participant finished answering

the question with a face-value interpretation of numbers. The criterion of noting the

participant’s final answer is adopted here to take into account the fact that in many

cases participants gave several differing answers to a question in the course of the

researcher’s questioning. The table reflects the considered response of each

participant after being questioned, rather than the initial response, or a response that

the participant gave in passing that but later denied or contradicted. Note that

instances in which a participant accepted a face-value counter-suggestion from the

researcher are included in Table 4.10, and again in the summaries of incidences of

face-value interpretations in Table 4.11, as indicated by parentheses. However, these

instances are not counted in the overview of approaches in Table 4.12. Previously

published accounts of digit correspondence tests do not include the effects of

researchers’ counter-suggestions, and so to enable comparison between this and other

studies the same method is applied. By accepting counter-suggestions participants

indicated a certain level of uncertainty in their minds about numbers, however,

supporting conclusions of this study that much of the participants knowledge about

numbers was quite tentative, and still being constructed (section 5.4).

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TABLE 4.10. Incidence of Face-value Interpretations for Written Symbols after Selected Interview Questions

Question Participant Interview 3c 6a 6b 7 8 High/ Blocks

Amanda 1 2 Craig 1 x 2 John 1 2 Simone 1 x 2 (x) x

High/ Computer Belinda 1 2 Daniel 1 (x) 2 Rory 1 2 Yvonne 1 (x) 2 (x)

Low/ Blocks Clive 1 x (x) x x 2 (x) (x) (x) x (x) Jeremy 1 x x x x 2 x x x x Michelle 1 x (x) x (x) 2 x x (x) x (x) Nerida 1 x x 2

Low/ Computer Amy 1 x 2 (x) x x (x) Hayden 1 x 2 Kelly 1 (x) (x) 2 x x x Terry 1 (x) x x x 2 x

Note. “x” indicates the existence of a face-value interpretation at the conclusion of the participant’s response. Parentheses () indicate that the participant did not volunteer a face-value interpretation, but accepted a face-value suggestion made by the researcher.

More information about responses indicated in the last two columns of Table

4.10 are included in Table 4.13, which indicates a range of responses to Questions 7

and 8, including face-value interpretations for written symbols. There is evidence

that face-value thinking evident in the answers to Questions 7 and 8 is at one end of a

continuum of responses to digit correspondence questions; this is discussed further in

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section 4.5. The summary of face-value interpretations used by members of each

group in Table 4.10 shows that low-achievement-level participants initiated face-

value interpretations without the researcher’s suggestion 10 times as often as high-

achievement-level participants did. There are some differences between blocks and

computer groups, but these appear to be related to differences of individual members

of each group.

TABLE 4.11. Use of Face-Value Interpretations of Symbols by Each Group

Blocks Computer Total High 3 (1) 0 (3) 3 (4) Low 19 (9) 11 (5) 30 (14) Total 22 (10) 11 (8) 33 (18) Note. Values not in parentheses represent incidents of face-value interpretations initiated by participants. Values in parentheses represent face-value interpretations suggested by the researcher and accepted by participants.

Figure 4.3 shows that, in general, participants who adopted face-value

interpretations of symbols achieved fewer place-value criteria than participants who

did not do so. This is not surprising, given the fact that face-value interpretations are

incorrect. Nevertheless, there were incidents of participants achieving high scores at

interviews who used face-value interpretations during interviews, supporting reports

in the literature indicating that this particular erroneous idea about numbers is quite

prevalent of among students of this age.

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Figure 4.3. Interview scores compared to use of face-value interpretations of symbols.

4.4.4 Summary of Approaches to Interview Questions

Previous tables in this chapter (Table 4.6, Table 4.8, & Table 4.10)

summarise the incidence of grouping approaches, counting approaches, and face-

value interpretations, respectively. Each of these tables shows thinking about

numbers demonstrated by each participant at each interview. Table 4.12 shows a

summary of each of the three earlier tables, to assist in comparing approaches

revealed by the interview data.

15

17

18

20

17

18

7

14

1919

17

20

1818

17

13

7

666

3

6

8

14

8

10

9

12

4

6

9

14

0

3

6

9

12

15

18

21

0 1 2 3

Incidence of Face-Value Interpretations of Symbols

Plac

e-Va

lue

Crit

eria

Ach

ieve

d

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TABLE 4.12. Incidence of Approaches Adopted for Selected Interview Questions

Participant Interview Groupinga Countingb Face-valuec Scored Amanda High/Blocks I 5 2 0 15 II 7 0 0 17 Craig I 7 0 1 18 II 10 0 0 20 John I 9 2 0 17 II 8 1 0 18 Simone I 1 1 1 7 II 6 2 1 14 Belinda High/Computer I 10 0 0 19 II 9 0 0 19 Daniel I 7 0 0 17 II 8 2 0 20 Rory I 10 0 0 18 II 10 0 0 18 Yvonne I 5 1 0 17 II 6 1 0 13 Clive Low/Blocks I 0 2 3 7 II 1 5 1 6 Jeremy I 0 3 4 6 II 0 1 4 6 Michelle I 0 1 2 3 II 1 2 3 6 Nerida I 1 4 2 8 II 2 4 0 14 Amy Low/Computer I 0 5 1 8 II 1 5 2 10 Hayden I 3 7 1 9 II 1 4 0 12 Kelly I 0 7 0 4 II 0 6 3 6 Terry I 3 1 3 9 II 5 1 1 14

Note. aGrouping approaches were noted in responses to 11 questions (Table 4.6). bCounting approaches were noted in responses to 11 questions (Table 4.8). cFace-value interpretations were noted in responses to 5 questions. The count of face-value incidents does not include instances where suggestions by the researcher were accepted (Table 4.10). dScore represents the number of criteria achieved at each interview (Table 4.2); maximum possible score per cell in Score column is 21.

Table 4.12 shows a summary which illustrates remarks made earlier about

differences between high-achievement-level and low-achievement-level participants:

In general, high-achievement-level participants adopted grouping approaches more

often and counting approaches and face-value interpretations less often than low-

achievement-level participants. It also appears that high-achievement-level

participants’ understanding of the grouped aspect of multidigit numbers was related

to the fact that they rarely adopted either inefficient counting approaches or incorrect

face-value interpretations of symbols. The relative instability of number conceptions

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of low-achievement-level participants in particular is addressed in the following

section.

4.4.5 Changeability of Participants’ Number Conceptions

One prominent feature of the interview data is the observation that on many

occasions some participants repeatedly changed their answers to questions as the

researcher continued to probe the reasoning behind their answers. Participants who

were unsure about the meanings of numerical symbols and block representations of

numbers often demonstrated thinking that was characterised by a willingness to

consider a range of ideas, apparently in an attempt to make sense of numbers and

numerical symbols. Often the opinions of these participants appeared not to be

completely formed, and were readily influenced by the researcher’s questions and

suggestions, including successive suggestions that contradicted each other. The

processes used by these participants to make sense of numbers match constructivist

ideas of learning; they used new information presented to them to compare with their

existing ideas about numbers, rejecting ideas that did not fit, and accepting others.

In the following transcript, Jeremy (l/b) compared printed symbols for 27 and

42. His initial response was that 27 was larger, apparently based on a face-value

interpretation of the digits, the ‘7’ being the largest digit present in the two numbers.

Interviewer: Can you tell me which of these numbers [‘27’ & ‘42’] is larger?

Jeremy: That one. [‘27’]

Interviewer: All right, what is that number?

Jeremy: 27.

Interviewer: Okay, and how do you that number is bigger than the other one?

Jeremy: Because it’s only got a ‘4’ in front of it.

The researcher twice attempted unsuccessfully to appeal to Jeremy’s

knowledge of the counting sequence, firstly by mentioning the verbal names for 27

and 42, and then secondly by asking which number would be reached first when

counting:

Interviewer: Uh-huh. What’s that number there?

Jeremy: — 42.

Interviewer: 42. And that’s 27, and 27 is bigger because of the ‘7’?

Jeremy: [Nods]

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Interviewer: Uh-huh. If you were counting and you were going to count up to a hundred,

say. Which one of those numbers would you come to first, 27 or 42?

Jeremy: That one. [‘42’]

Interviewer: 42 because it’s … smaller is it?

Jeremy: [Nods]

Only when the researcher suggested that 27 is larger because it is in the 20s

and 42 is in the 40s did Jeremy change his answer:

Interviewer: Uh-huh. All right. Someone said to me that this comes first because it’s in the

20s and that one comes later because it’s in the 40s. What do you think?

Jeremy: That one comes first. [‘27’]

Interviewer: So 27 comes first? So you agree with them that the 20s are first and then the

40s?

Jeremy: Yes.

Interviewer: All right, so do you want to change your answer? You’re now saying this one

is smaller?

Jeremy: Yeah, and that one [‘42’] is bigger.

Interviewer: All right 27 is smaller and 42 is bigger. And how do you that 27 is smaller?

It is interesting that when the researcher asked Jeremy to explain how he

knew that 27 is smaller than 42, Jeremy did not merely repeat the researcher’s earlier

suggestion about the counting sequence, but instead referred to the first digit of each

symbol:

Jeremy: Because it’s got a ‘2’ in front of it.

Interviewer: All right and that one has got?

Jeremy: A ‘4’ in front.

Interviewer: A ‘4’ in front. All right, well what about this ‘7’? ‘Cos you said the ‘7’ was

bigger before. What do you think?

Jeremy: ‘7’s bigger.

Interviewer: Right. So does that make that one bigger? Or is it still smaller?

Jeremy: Still smaller.

Interviewer: Right, even though it’s got a ‘7’? Even though the ‘7’ is bigger than the ‘4’?

This is … 27 is still smaller?

Jeremy: [nods] (I1, Qu. 6a)

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Another example of a participant attempting to use different pieces of

information to answer a question is provided in the following excerpt, in which the

researcher had just suggested to Hayden (l/c) that 27 might be larger than 42 because

of the digit ‘7.’ In explaining why 42 was larger, Hayden appealed to evidence from

the respective sums of the digits:

Hayden: Because um … ‘cos if when that makes 6 [‘42’] and that [‘27’] makes 9.

Interviewer: And 9 is bigger than 6 isn’t it? So does that mean this [‘27’] is bigger?

Hayden: No.

Interviewer: It’s not bigger? Even though 9 is bigger than 6?

Finding that the sums of the respective face values did not confirm his

answer, Hayden switched to a counting approach, referring to the relative order of 27

and 42 in the counting sequence:

Hayden: No, because if you count to 40 it takes longer. And if you count to 20 it takes

…

Interviewer: … less time?

Hayden: Yeah. (I1, Qu. 6a)

In an extended series of questions Terry (l/c) was questioned about 27 and 42

(see Appendix N for a full transcript). In his response, Terry called on a range of

knowledge he had about numbers and attempted to apply it to the question. In a

series of answers that changed in response to the researcher’s questions, Terry stated

that

1. 42 was larger than 27, because 42 is even;

2. 42 was larger than 57, because 42 is even;

3. 26 was larger than 42, because the ‘6’ was the largest digit;

4. 42 was larger than 26, because 42 is in the 40s and 26 in the 20s;

5. 42 was larger than 57, because 42 is even; and

6. 57 was larger than 42, because it is in the 50s.

Evidence of participants changing their minds when answering interview

questions is discussed further in section 5.4.

4.5 Digit Correspondence Tasks: Four Categories of Response In questions 7 and 8 the interviewer asked participants specific questions

about values represented by digits in two-digit numbers. Because the quantities

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represented by the digits of multidigit numbers is at the heart of the place-value

system, this type of question is regarded by other authors as quite critical for

revealing place-value understanding (e.g., S. H. Ross, 1989, 1990). This section

addresses the range of thinking revealed by participants’ responses to these

questions. Interview transcripts show a hierarchy of participant responses, that varied

in accuracy in interpreting two-digit written symbols. Four categories of thinking are

proposed in this section, with examples of each one provided from interview

transcripts.

4.5.1 Category I: Face-Value Interpretation of Digits

The type of response to digit correspondence questions showing the lowest

level of thinking about two-digit numbers is a face-value interpretation of digits,

defined here as Category I. Category I thinking was evidenced by participants’

statements that each digit represented only its face value, and that remaining objects

in the set represented by the two-digit symbol as a whole were not represented by

either digit. Examples of Category I thinking have been provided earlier (section

4.4.3), including a number of statements indicating the belief that not all objects were

represented by the two digits. This idea may set up a paradox for the student to

resolve: The two-digit symbol represents the entire set of objects, but the sum of the

referents for the two digits does not equal the same amount, meaning that some

objects are somehow without representation in the symbol. This problem is

overcome if a participant adopts a Category II response.

4.5.2 Category II: No Referents For Individual Digits

Category II responses indicated that a participant accepted the two-digit

symbol as representing the entire set of objects, but rejected the idea that each digit

had separate referents, on the basis that some objects would be left out. The

following transcript excerpt clearly shows a Category II response from Hayden (l/c):

Interviewer: Does this part [‘7’] of your ‘37’ have anything to do with how many sticks

you have?

Hayden: No.

Interviewer: I doesn’t? OK, can you tell me what that ‘7’ means?

Hayden: It’s a part of 30 … it’s a part of like in 30 it’s a part like … you count to 30

and then you count seven more and it ends up 37.

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Interviewer: And what does this ‘3’ here mean?

Hayden: It’s up … it’s up to 30 … like if you count up to 30.

Interviewer: Uh-huh. Can I show you something? If we have seven sticks like that [puts out

seven sticks], could we say that ‘7’ is for seven like that?

Hayden: No, because that … that’s not like [picks up three sticks] … that’s only 11

[sic].

Interviewer: Why have you got those three? That’s for the ‘3’ is it?

Hayden: No, those aren’t for the ‘3.’

Interviewer: It’s not for ‘3’? So the seven is not for ‘7’ either?

Hayden: No, because it doesn’t make um 37. It only makes 11 [sic].

Interviewer: But the whole number written down like that is for all of them?

Hayden: Yep.

Interviewer: But if you take just the seven it’s not … you can’t take part of them and say

that part is for that?

Hayden: No. (I2, Qu.7b)

Similar ideas are evident in the following three responses to the question

“What about the remaining objects?” asked after a participant initially gave a face-

value interpretation for the written digits:

Kelly (l/c): Um, they [individual digit symbols] don’t stand for any of them … If they’re

[the two digits] joined together, both of the numbers are for all of them.

(I2, Qu. 7b)

Jeremy (l/b): Put them together and it makes the number.… You put them all in together,

then you know what number, so you write them down and you get the number

with the sticks. (I1, Qu. 7b)

Amy (l/c): Yeah but you can’t make it though, just out of like … three [separates three

sticks] … like out of that [seven sticks]. ‘Cos then it wouldn’t be 37 still

though. (I2, Qu. 7b).

It appears that responses such as those quoted here represented the

participants’ rejection of face-value interpretations of multidigit symbols. In giving

such a response, the participants apparently recognised that each digit could not

represent only its face value and still be consistent with the meaning given to the

entire two-digit symbol. In trying to come to terms with the apparent contradiction of

their view, participants exhibiting Category II responses sometimes provided quite

creative ideas about how to make sense of the symbols. For example, at his second

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interview, Jeremy (l/b) indicated that the “extra” objects must be somehow recorded

within the two-digit symbol ‘37,’ though no symbol for them could be seen:

Jeremy: It’s got all [ruffles all the sticks while talking] … the sevens in here and the

threes.… They’re [the remaining sticks] in there too.

Interviewer: They’re in there too?

Jeremy: Yeah.

Interviewer: Right. So you know they’re part of this number.

Jeremy: [Nods]

Interviewer: Where are they written down, though?

Jeremy: In here [points to space between the three and the seven sticks]. (I2, Qu. 7b)

Terry (l/c) gave a category I response that was similar to category II, in that

he tentatively offered a suggested explanation for no referent being visible for the

remaining sticks. Terry appeared to suggest that, after taking out a set of three and a

set of seven from a set of 37 sticks, all the sticks remaining could somehow be

represented by the two digits ‘3’ and ‘7’:

Terry: They got gave away to sevens and threes, I suppose. (I2, Qu. 7b)

Amy (l/c) made another suggestion, indicating that she still believed in a face-

value interpretation of the digits, but that there was another possible reason why the

extra sticks were apparently not recorded in the symbol. Her idea seemed to be that

the entire group was recorded by the two-digit symbol, but that if each digit was

considered in turn, the objects represented by that digit were temporarily isolated

from the rest of the group:

Interviewer: But what about this ‘3’?

Amy: It means three [picks up three sticks] but it still won’t make 37.

Interviewer: Won’t it?

Amy: No.

Interviewer: How does that work? Because you said that number is for all the sticks.

Amy: Yeah and it includes these ones.

Interviewer: Yes.

Amy: But … when they’re out of a group it means they’re not part of the group.

Interviewer: Sorry, which ones are not part of the group?

Amy: These ones right now [points to the groups of three and seven sticks].

Interviewer: They’re not part of the group?

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Amy: Yeah right now ‘cos they’re out of the group. And ‘3’ means 3 and these three

[sticks] means 3, and ‘7’ means 7 and this seven [sticks] means 7.

Interviewer: Right. But what about these here? [Points to remaining 27 sticks]

Amy: They’d mean how many there are now. And these [remaining sticks] are still

in a group because they haven’t left the group.

Interviewer: Right. They haven’t left the group.

Amy: Yeah like … you tooken some away …

Interviewer: Right.

Amy: … I suppose. (I2, Qu. 7b)

It appears that Category II responses indicate an intermediate stage of place-

value understanding possessed by some participants, between believing that each

digit represents only its face value (Category I), and understanding that a tens digit

represents a number of collections of 10 units (Category III or IV). Evidence for this

idea comes from the fact that several of the participants who gave a face-value

interpretation for the digits in the first interview changed their responses to Category

II responses at the second interview.

4.5.3 Category III: Correct Total Represented by Each Digit, but Tens not Explained

In a Category III response the participant knew that the tens digit represented

the remaining objects, once the referents for the ones digit were removed, but could

not explain why that digit represented a number of objects greater than its face value.

In the following excerpt, Yvonne (h/c) indicated that the ‘2’ in ‘24’ represented all

the sticks apart from the four represented by the ‘4,’ and knew that there were 20 of

them, but could not explain the connection between the digit ‘2’ and 20:

Interviewer: Can you explain that for me, ‘cos that’s just a ‘2’ isn’t it? — Does this ‘2’ here

stand for all of those, or just some of them?

Yvonne: All of them.

Interviewer: How does ‘2’ stand for so many? Can you explain that?

Yvonne: [Shakes head]

Interviewer: But you’re sure it does stand for that many? Do you know how many there are

here?

Yvonne: 20. (I1, Qu. 7c)

141

Daniel (h/c) also had difficulty explaining the relationship between the tens

digit and the number of objects to which it referred. He proposed an interesting

explanation based on the efficiency of writing just a ‘2’ instead of the digits ‘20’

before the ones digit, but like Yvonne did not connect the ‘2’ with 2 tens, despite a

series of questions from the researcher, some of which are shown in this excerpt:

Interviewer: Can you tell me why that is a ‘2’ and that’s standing for all those?

Daniel: — They have to uh be ‘2’ instead of like being ‘20’ then a ‘4’ or otherwise it

would be two hundred and four.

Interviewer: — Uh-huh, but why do you write ‘2’ if it’s 20? — Can you explain it?

Daniel: Uh, because there’s … I forget … there’s a ‘2’ and there’s a ‘0’ at the end so

they just wanted it, just put it as a um ‘2’ to make it quicker? (I1, Qu. 7b)

Responses such as those from Yvonne and Daniel indicate knowledge of

numbers that is more advanced than a face-value construct, but still do not meet the

criteria for a conventional understanding of multidigit numbers, Category IV,

described next.

4.5.4 Category IV: Correct Number of Referents, Tens Place Mentioned

Category IV includes responses stating a correct number of objects for each

digit, explaining that the tens digit represents the number of groups of ten. The

following transcript excerpt shows that Rory (h/c) knew what each digit in ‘13’

represented, even in the face of misleading cues of three cups and one remaining

bead (see Figure 3.10):

Interviewer: Does this part [‘3’] of your ‘13’ have anything to do with how many beads

you have? Can you show me?

Rory: Yes. [Takes three out of one cup]

Interviewer: All right, that’s a good answer. Let’s put them back in there again. Now this

part [‘1’] of your ‘13,’ does that have anything to do with how many you have

here? Can you show me?

Rory: [Takes out 10 beads]

Interviewer: — OK. Can you explain to me how that [‘1’] stands for these [10 beads] here?

Rory: ‘Cos it’s 1 ten. (I1, Qu. 8b)

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4.5.5 Summary of Responses to Digit Correspondence Tasks

A summary of the categories of response demonstrated by participants at both

interviews is provided in Table 4.13. Again, there is clear evidence of the generally

superior place-value understanding of the high-achievement-level participants.

TABLE 4.13. Response Categories for Interview Digit Correspondence Questions

Question 7 Question 8

Participant Group Interview 1 Interview 2 Interview 1 Interview 2

Amanda High/Blocks III III II III

Craig III IV I IV

John IV IV IV III

Simone II II I I

Belinda High/Computer IV IV IV III

Daniel III IV IV III

Rory IV IV IV IV

Yvonne III III I II

Clive Low/Blocks I I I I

Jeremy I I I I

Michelle I I II II

Nerida I III I III

Amy Low/Computer I II II II

Hayden I II II II

Kelly II I II II

Terry I I I III Note. Categories: I – face-value interpretation of digits; II – no referents for digits; III – correct total for each digit; IV – referents for tens digit correctly explained.

Table 4.13 shows that several participants improved in the accuracy of their

response to Questions 7 and 8 from the first to the second interview, though others

achieved scored less in the second interview. It is also interesting to note that for

some participants their responses to Question 7 were quite different from their

responses to Question 8. Response categories of all participants as a group are

summarised in Table 4.14.

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TABLE 4.14. Summary of Digit Correspondence Response Categories

Category Interview 1 Interview 2

I 44 25

II 22 25

III 13 28

IV 22 22

Table 4.14 shows that, overall, participants in the study improved in

responses to digit correspondence questions between the two interviews; fewer

participants gave Category I responses and more gave Category IV responses at the

second interview, compared to the first. These data are compared in Table 5.1 with

figures for performance on similar tasks quoted by S. H. Ross (1989).

4.6 Errors, Misconceptions, and Limited Conceptions One clear pattern in the data from both interviews and teaching sessions was

the large number of errors, misconceptions, and limited conceptions evident in

participants’ responses. In this section these errors are categorised and described

separately: Counting Errors (section 4.6.1), Blocks Handling Errors (section 4.6.2),

Errors in Naming and Writing Symbols for Numbers (section 4.6.3), and Errors in

Applying Values to Blocks (section 4.6.4).

4.6.1 Counting Errors

Counting sequence errors.

The use by participants of counting approaches in responding to interview

questions is described in section 4.4.2. The use of counting approaches to work out

answers to questions involving multidigit numbers requires accurate use of counting

sequences for success. Difficulties in this area for some participants led to problems

in answering interview questions. One common problem was in naming the next

decade in a counting sequence. For example, Kelly (l/c) used the following sequence

when counting one-blocks: “… 40, 51, 52, 53, 54, 55, 56, 57, 58, 59, 30, 31, 32” (I1,

Qu 1b). Another common mistake of this sort is illustrated in this sequence used by

Terry (l/c) when counting tens: “10, 20, 30, 40, 50, 60, 70, 80, 90, 20” (I1, Qu 1c).

Terry evidently knew that this was not correct and restarted the count, only to repeat

the same error. It is likely that children sometimes make this error because of the

144

similarity of the two number name sequences “seventeen, eighteen, nineteen” and

“seventy, eighty, ninety.”

Counting errors were revealed in responses to Question 4 in both interviews,

which required participants to skip count by 1, 10, or by 100 with two-digit or three-

digit numbers. These tasks proved to be among the most difficult for the participants

and resulted in a low level of success (see Table 4.2). Four common errors made by

participants answering Question 4 are illustrated in the following transcript excerpts:

(a) Mistakes in the new number at a change of decade or change in the

number of hundreds:

Kelly (l/c): 73, 72, 71, 60, 69, 68, 67, 66, 65, 64, 63, 62, 61, 50, 59, 58. (I1, Qu. 4a)

Yvonne (h/c): 273, 283, 293, 203, 223, 233, 243, 253, … (I2, Qu. 4c)

Terry (l/c): 681 … 671, 661, 651, 641, 631, 621, 611, 501, … 591, 581, 571, 561, 551,

541, 531, 521, 511 … 491, 481, 471, 461 … (I2, Qu. 4d)

(b) Omitting numbers, especially numbers with a “teen” component, or 1 ten:

Yvonne (h/c): 52, 62, 72, 82, 92, 102, 122, 132, 142, 152 … (I2, Qu. 4b)

Daniel (h/c): 65, 75, 85, 95, 105, uh 125, 135, 145. (I1, Qu. 4b)

(c) Using an incorrect increment or decrement when asked to count on or

back by 10:

Hayden (l/c): 75, 80, 85, 90, 95, 100, 105, 110 … (I1, Qu. 4b)

Amanda (h/b): 452, 562, 672, 892, … I don’t know the one after that. (I1, Qu. 4c)

Michelle (l/b): 204, 205, 206, 207, 208, 209, 210. (I2, Qu. 4c)

(d) Omitting the ones part of each number:

Simone (h/b): [Asked to count by 10 from 463] 270, 270, 280, 290, … (I1, Qu. 4c)

Nerida (l/b): [Asked to count by 10 from 681] 670, 660, 650, 640, 630 … (I2, Qu. 4d)

Lack of knowledge of larger numbers.

The difficulties that some participants had with counting sequences were

compounded by a lack of knowledge about larger numbers, and a lack of familiarity

with hundred-blocks, or both. For example, Terry (l/c) evidently knew the name of

the hundred-block, but did not know how to read 2 hundred-blocks, counting them as

“100, 1000.” Amy (l/c) made similar errors when trying to count 16 tens, clearly

being unsure of how to count beyond 100. She rapidly ran out of number names for

places as she tried to apply a new place name to each new block:

145

Amy: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 … hmmm … 100, 1000, about 2000.

Um, infinity. [Laughs] Just gets up to infinity and then it gets harder.

(I1, Qu. 1c)

The following transcript excerpt shows an attempt to count a block

arrangement that was hindered by lack of knowledge of larger numbers. Kelly (l/c)

was attempting the task of reading 5 hundreds, 13 tens, and 2 ones, but was unable to

complete the task successfully because of difficulties with both the values

represented by the different blocks and the sequence of three-digit cardinal numbers:

Kelly: [Counts hundred-blocks] 100, 200, 300, 400, 500 … I’ve worked out a easy

way. There’s a hundred there [counts out 10 tens and puts them together]

there’s a hundred there … so it’s 100, 200, 300, 400, 500, [counts group of 10

tens] 600. [Counts individual “ones” on the next ten-block but miscounts, then

moves it next to the 10 tens] 207, [puts the next ten-block across as she counts

each individual “one” on it] 208, 209, 300, 301, 302, 304 …

The researcher stopped her and asked her to restart at 600:

Kelly: Oh, go on from 600 … [counts each individual “one-block” on the ten-block]

601, 602, 603, 604, 605, 606, 607, 608, 609, 700. [Gets the next ten-block and

again counts individual “ones”] 701, 702, 703, 704, 705, 706, 707, 708, 709,

800.

Interviewer: Can you count aloud? That’s 800 now is it?

Kelly: Yes. [Gets the next ten] 900, um 800, 900, 901, 902, 903, 904, 905, 906, 907,

908, 109 … 1000 … [puts 2 ones next to the other blocks] 1002. (I2, Qu. 1c)

Knowledge of the sequence of cardinal numbers is fundamental to

development of understanding of the base-ten numeration system. The difficulties

illustrated here would clearly cause further difficulties in learning about the base-ten

numeration system unless they were remediated.

4.6.2 Blocks Handling Errors

General handling errors.

Mistakes made when handling blocks were very frequent during interviews

and teaching sessions. Errors reported in this section are closely related to the

counting errors described in the previous section, and to mistakes made in assigning

values to blocks (section 4.6.4). However, the errors in this section are apparently

due to mistakes made in handling the blocks, rather than to either an inability to

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count or ignorance about the number of tens and ones in a number. Handling errors

made while using blocks to represent numbers included the following:

1. Simone put out 5 tens and 8 ones for 48. (h/b S3, T 8)

2. Clive counted out 6 tens when showing 70. (l/b S5, T 9)

3. Michelle counted blocks to show 75, but included 8 tens. (l/b S6, T 12)

4. When showing 75 with blocks, Nerida miscounted the first 3 tens in her

hand as “20,” finishing with 8 tens and 5 ones. (l/b S6, T 14)

5. When Craig and Simone showed 627 and regrouped a hundred into

tens, Craig put the 10 tens on top of the other blocks. Some blocks fell

off, unnoticed by the two children, resulting in the representation being

short by 1 ten. (h/b S9, T 32b)

Note that the above list includes only handling errors that went unnoticed by

participants for a lengthy period. During the counting of blocks many other handling

errors were temporary, as they were checked and corrected quickly by the participant

concerned.

Trading errors.

The process of trading blocks is an important one for students using base-ten

blocks to model the subtraction and addition algorithms. The participants in the study

had learned about trading with blocks previously. This was confirmed by Amanda

who said “We do it all the time - ‘Swap the Bank’,” to which Craig responded, “I

thought that it was called ‘trade’” (h/b S5, T 14). As Amanda and Craig were from

different classes, this indicates that both teachers of participants in the study had

taught previously about trading with blocks. However, errors made by several

participants indicated that their learning of this process was far from complete.

Participants’ trading errors are described briefly in the following paragraphs, grouped

into three categories: trades to 10, trades of 10 for 1, and trades of 10 for other

numbers.

Trades to 10. One faulty idea relating to block trading that appeared in the

teaching sessions several times was that trading was done up to 10, rather than

trading 1 larger block for 10 smaller blocks. On several occasions participants were

observed to remove a ten and replace it with sufficient ones so that there were 10

ones in all. For example, if there were 5 tens and 8 ones, this error would be revealed

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by the action of removing a ten and adding just 2 ones, to make 4 tens and 10 ones.

This was the process used in the following examples:

1. Clive traded a ten in 255 for some ones, then counted the ones,

removing extras so that there were only 10 ones. (l/b S10, T 31a)

2. Amanda traded a ten in 255 for 5 ones. Later she wrote that the new

arrangement represented 260 [sic], and did not equal 255.(h/b S8, T 31a)

3. John traded a hundred in 340 for 6 tens and wrote that there were 2

hundreds, 10 tens, and 0 ones. (h/b S9, T 32a)

4. Clive traded a hundred in 340 for 6 tens, resulting in 2 hundreds and 10

tens. (l/b S10, T 32a)

5. Daniel, when asked to trade a ten in 77, asked twice if he should add

just 3 ones. (h/c S1, T 4a)

The last example shows a participant using the computer demonstrating the

idea that trading is done up to 10. This example shows that the trade-to-10 idea was

independent of the representational format provided to participants, at least at first.

Daniel asked about making the ones up to 10 before he had used the saw tool in

completing a task, and while he and his fellow group members were considering how

to effect the trade. However, the researcher reminded the participants that they could

use the saw tool incorporated in the software—which they had used in their initial

training session—to carry out the trade correctly. After this task, the trade-up-to-10

idea did not recur in this group. One purported advantage that the software has over

conventional base-ten blocks is that users can use electronic decomposition and

regrouping tools to produce automatic trades that are always carried out correctly; it

may be in using the electronic tools, computer participants were able to recognise the

fallacies in errors such as trade-to-ten.

Trades of 10 for 1. A number of times participants traded a ten or a hundred

for a single one or ten. For example, in Session 1, carrying out the first trading task

of trading a ten in 77, every participant in the low/blocks group attempted to trade a

ten-block for a single one-block (l/b S1, T 4a). After some discussion, the four

participants agreed that the number represented by the blocks after trading was 68.

The researcher then corrected the participants and showed them that the trade must

always be done so that the blocks swapped were equal to the original blocks. Despite

this, at the low/blocks group’s second session Jeremy again started to trade 10 from

23 for 1, until Clive corrected him:

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Jeremy: [Moves a ten away]

Clive: [To Jeremy] Swap one of the tens for a one. [To teacher] A one?

Jeremy: Just get a one.

Teacher: No, it doesn’t say “a one.” It says “for ones.”

Jeremy: Just get a one. You get a one. Just get a one.

Clive: [Ignoring Jeremy, counts ones into his hand. Then he checks how many he

has:] 2, 4, 6, 8, 10. There. [Adds ones to the other blocks.] (l/b S2, T 4b)

Clearly even after having the correct trading procedure explained in the

previous session, Jeremy still believed that a ten could be traded fairly for a one. This

belief recurred among members of the low/blocks group later when trading of a

hundred-block for tens was introduced, when Jeremy and Michelle both stated that a

hundred-block should be traded for a single ten-block (l/b S10, T 32a). The actions of

the participants are consistent with a view that blocks were merely counters, and that

no matter what their size, any block was equivalent to any other. This idea is

discussed further in section 5.3.

Trades of 10 for other numbers. On at least two occasions participants traded

a ten for a number other than 10 ones, and did not trade up to 10. In the first incident,

Simone (h/b) traded a ten in 77 for 7 ones. The other participants in her group all said

that trades must be done for 10 ones:

Craig: [Quietly] 10. 10 for 10. 10, 10, 10. You swap it for 10.

Amanda: You have to swap it for 10, ‘cos otherwise it’s not the same.

John: Well, then [if a ten was traded for 7 ones] it’d just be 17 … no, then it’d just

be 70 [sic]. You need 77. (h/b S1, T 4a)

When the researcher asked the group if a ten could be traded for numbers

other than 10, there was some uncertainty to start with, with John and Craig saying

that they were not sure, and Simone asserting that “We can swap it for other numbers

too. — Like um, you can swap it for 7s, and 9, and 10, and the other numbers.” In the

ensuing discussion the children all eventually agreed that a ten must always be traded

for 10 ones, “or it wouldn’t be the same.” This question did not recur with this group,

though there was a later incident when trading for a hundred in which John traded up

to 10 tens (see previous discussion).

The second example of a participant trading a ten for other than 10 ones

involved Clive (l/b), who, in attempting to use blocks to calculate 83 - 48, traded a

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ten for 8 ones. This appears to have been related to the subtraction operation and how

it is modelled using base-ten blocks. Clive started with 8 tens and 3 ones, separated 4

tens, then removed a ten and traded it for 8 ones. He then put the 8 ones with the

removed 4 tens, making a representation for 48, leaving 3 tens and 3 ones which he

believed show the answer to be 33 (l/b, S8, T 21). When the researcher talked Clive

and Jeremy through the block transactions again, Clive said that the ten should be

traded for 10 ones, indicating that he had previously been told that, but had decided

otherwise when attempting to calculate the answer to the question.

4.6.3 Errors in Naming and Writing Symbols for Numbers

Participants made many errors in either naming or writing the symbol for a

number represented by collections of blocks. Some of these errors were due to a lack

of knowledge of names of larger numbers, such as when Clive, attempting to read the

symbol ‘932,’ said “Ninety-th … 9 … 109 … no… Can’t read hundreds; can only

read ones and tens” (l/b S10, T 31b). In other instances participants attempted to name

a number or write a numerical symbol, but applied the knowledge they had of

smaller numbers in incorrect ways. Such errors are described in the subsections

following.

Naming incorrectly concatenated number symbols.

There were two instances in teaching sessions in which participants read two-

digit non-canonical block representations as three-digit numbers. In each case the

participant evidently concatenated the symbols for the number of each size of block

and then named the resulting number:

1. Nerida said that 6 tens and 17 ones showed “six hundred and

seventeen.” (l/b S1, T 4a)

2. Daniel said that 1 ten and 13 ones showed “one hundred and thirteen.”

(h/c S2, T 4b)

Note that in example 1 Nerida was looking at the physical blocks, with no

written symbols available. In example 2 there were column labels available, which

may have helped Daniel to visualise the written symbol “113.”

Concatenating tens and ones names.

Similar naming errors were made by several participants who named non-

canonical block arrangements using the name of the number represented by the tens,

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followed by the name for the number of ones. As discussed further in section 5.3,

participants naming block arrangements this way were applying a method that will

work for canonical arrangements of blocks, but which gives non-standard number

names for non-canonical arrangements. This method treats each place as independent

of the other, and is evidence of the “Independent-place construct,” described in

section 5.3. The following examples of this type of error were noted:

1. Jeremy counted 8 tens and 11 ones, and read them as “eighty-eleven.”

(l/b S2, T 4c)

2. Yvonne looked at 5 tens and 10 ones, and read them as “fifty-ten.”

(h/c S4, T 14)

3. Clive said that 2 hundreds, 4 tens, and 10 ones represented “two

hundred and forty-ten.” (l/b S10, T 31a)

4. Clive and Michelle both said that 9 hundreds, 2 tens and 12 ones

showed “nine hundred and thirty-twelve.” (l/b S10, T 31b)

5. The researcher asked Daniel what the next number would be after 492

in a sequence adding tens, and he answered “four hundred and ten-

two.” (h/c S10, T 41)

Errors in writing three-digit numerical symbols.

Several times participants made mistakes when writing symbols for three-

digit numbers. The participants had not been taught about numbers beyond 99 in

their regular mathematics classes, and so it is not surprising that they exhibited

difficulties writing and reading them.

The errors in writing three-digit numbers usually resulted from concatenation

of values for the three individual digits; in other words, participants wrote the

symbols representing the value in each place one after the other. For example,

Jeremy, attempting to write the sequence of numbers counting in tens from 100,

wrote ‘10010, 10020, 10030, 10040’ (l/b, S8, T 24). A similar method was used by

Michelle, who wrote the number 538 as ‘500.30.8’ (l/b, S10, T 29a). It is interesting

to note that Michelle inserted full stops between the symbols for adjacent places; it

appears she believed that there should be something to distinguish each place from

the next.

An incident involving Amanda (h/b) is interesting because it shows that she

was able to tell that her first attempt at writing ‘204’ was incorrect, though she

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needed assistance to finally write the correct symbol. In the following excerpt,

Amanda had just added 170 and 34 using blocks, and wanted to record her answer:

Amanda: Two hundred and four. [She writes in her book ‘24,’ stops.] Whoopsies. Two

hundred and four - That’s twenty-four. How do you write that? [She looks at

the teacher, but he does not respond.] Oh, yeah. [She changes what she has

written to ‘240.’]

Teacher: You’ve written ‘240.’

Amanda: Oh, yeah, “zero four.” [She corrects her answer to ‘204.’] (h/b S10, T 36)

In the examples of errors made in writing symbols described here, the

participants appeared to consider each place of the number whose symbol they were

writing separately, rather than combining the places to form a composite number

from the separate places.

Perseveration errors.

The psychological term “perseveration” refers to a response to a stimulus that

continues after the stimulus is removed. Fuson and Smith (1995) used the term to

refer to a particular type of error made by children in which they continue to use a

certain place name or value after a change of place or block value. Examples of this

error included the following:

1. Michelle counted 3 hundreds, 6 tens, and 9 ones. She continued

counting in hundreds after 300 while counting the tens: “300, 400, 500,

600, 700, 800, 900 …” (l/b S9, T 28a)

2. When adding tens together, Kelly stated that the number 10 more than

100 was 200. (l/c S8, T 24)

3. Nerida counted 5 tens and 4 ones: “10, 20, 30, 40, 50, 60, 70, 80, 90.”

(l/b S6, T 12)

4. John counted 5 tens and 1 one: “1, 2, 3, 4, 5, 6.” (h/b S4, T 10)

5. When attempting to count 5 hundred-blocks, 13 tens, and 2 ones, Amy

(l/c) counted every block as a hundred: “100, 200, 300, 400, 500,

[continues counting tens] 600, 700, 800, 900, 10 hundred, 11 hundred,

12 hundred, 13 hundred, 14 hundred, … 15 hundred, 16 hundred, 18

hundred [sic], 19 hundred, [continues counting ones] 20 hundred, 21

hundred.” (I2, Qu. 1c)

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Though at first glance, it appears that these participants did not know the

correct values represented by blocks of different sizes, this is unlikely to be the case.

These same participants were able to count blocks correctly at other times, and

clearly did know the name assigned to each block. It seems that what happened in

examples like those here is that the participant continued (persevered) with an

auditory counting pattern, without changing the place of the verbal number names

when counting blocks of another size.

4.6.4 Errors in Applying Values to Blocks

The largest and most diverse category of errors made by participants in the

teaching sessions was that of errors made in referring to values represented by the

blocks. These errors included referring to blocks using incorrect values, using blocks

of the wrong size to represent a certain digit, referring to the value of the tens as the

number of tens in a block arrangement, attempting to combine numbers of different

places, and perseveration errors. These types of place error are described in the

following subsections.

Size and position misunderstandings.

Some participants were confused about two aspects of blocks used to

represent different places in a multidigit number: their size and their position. The

task in Interview Question 3 (c) (see Appendix I & Appendix J) was deliberately

framed to target the incorrect idea that the value assigned to each block is determined

by its position relative to other blocks, rather than by its size. Participants were asked

to say whether a given arrangement of blocks matched a printed three-digit

numerical symbol. The blocks were arranged so that the number of blocks of each

size matched the digits of the printed symbol, in spatial order. However, the sizes of

the blocks did not correspond to the values of the places. The following transcript

shows that Jeremy (l/b) had some uncertainty in his mind about the two aspects of

the blocks, size and position:

Interviewer: Do these blocks [1 ten, 7 hundreds, 2 ones] show that number [172]?

Jeremy: … Yes it does.

Interviewer: … So that ‘1’ is for the one there and ‘7’ is for seven and the ‘2’ for two?

Jeremy: [Nods]

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Interviewer: OK, what if I turn it round like that? [Arranges blocks in order of 7 hundreds,

1 ten, 2 ones.] What number is shown there? Is that still the same as that?

Jeremy: Yeah, because it’s changed around.

Interviewer: Right, but it’s still the same number?

Jeremy: [Nods]

Interviewer: So these blocks here show 172? Is that right?

Jeremy: [Puts hand on top of the hundreds pile] But that’s not a ten. It’s a hundred.

Interviewer: Right. So does it show the same as that or not?

Jeremy: No.

Interviewer: It doesn’t. All right, do you know what number this is here with the blocks?

Jeremy: 71 … seventy-hundred … 1, 2. [The correct answer was 712.]

Interviewer: Uh-huh. What if I turn it round that way [rearranges blocks as 1 ten, 7

hundreds, 2 ones from left to right].

Jeremy: 172.

Interviewer: OK, so when it’s like that it’s the same as that, but if I turn it round it’s

different?

Jeremy: Mmmm. [Nods] (I2, Qu. 3c)

Initially Jeremy agreed that the “face-value” representation was correct. The

researcher swapped the hundreds and tens blocks, to which Jeremy said that the

number represented was still the same. However, when the researcher mentioned the

name of the number in question he changed his mind, arguing that the hundred-

blocks were not tens. When the blocks were returned to their first position Jeremy

agreed that the number represented had changed back, though it seems that he was

not entirely convinced, as he merely nodded in response to the researcher’s question,

rather than verbalising an affirmative response.

The following excerpt shows a similar confusion between block sizes and

their relative positions when deciding the value they represented. Michelle (l/b)

accepted a face-value interpretation of blocks almost straight away. However, when

explaining her answer, she used block names and values that did not agree: She

referred to the 7 hundred-blocks in the middle position as hundreds, but counted

them as tens when attempting to confirm the value they represented:

Interviewer: Do these blocks [1 ten, 7 hundreds, 2 ones] show that number [172]?

Michelle: — [Counts the pile of tens] Yes.

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Interviewer: OK. How can you tell?

Michelle: Because there’s one block, 1 ten [puts hand on the 1 ten]. There’s supposed to

be … 7 hundreds, [counts the 7 hundreds one by one] 10, 20, 30, 40, 50, 60,

70, [touches the 2 ones] 2.

Interviewer: OK, so this number shown here [blocks] is 172?

Michelle: Yeah.

At this point in the interview the researcher explored Michelle’s

understanding of the block values, starting with a hundred-block:

Interviewer: What’s that block [holds up one of the hundreds blocks]?

Michelle: Hundred.

Interviewer: All right, what’s that one [continues to hold up hundreds one at a time]?

Michelle: 200, 300, 400, 500, 600, 700.

Interviewer: Keep going.

Michelle: 700, [touches the 1 ten, now in middle position] eighty-hue … 800, [touches

the ones 1 by 1] 802, 803.

Interviewer: Right, OK, that’s fine. Thank you.

Michelle: You changed it around.

Michelle’s comment about the order of placement of the blocks, which had by

this time been altered, prompted the researcher to probe her beliefs further about

values assigned to blocks:

Interviewer: I did … well actually, I’ll ask you about that. ‘Cos I did turn them around

[arranges blocks as they were initially]. You said before that that [1 ten, 7

hundreds, 2 ones] is the same as that [‘172’]?

Michelle: Yeah.

Interviewer: So that’s 172. But if I change them around [7 hundreds, 1 ten, 2 ones], it’s

different?

Michelle: Yeah.

Interviewer: OK. So it matters which way round you put the blocks.

Michelle: Yeah. (I2, Qu. 3c)

As with the instances of confusion about the values assigned to blocks,

confusion about block size and position also appears to have links with face-value

interpretations of digits. However, this cannot be the complete explanation, since

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confusions described in this section are more to do with values represented by

blocks, than with values represented by digits.

Counting ten-blocks as fives or twos.

On a few occasions, participants assigned incorrect values to base-ten blocks

in answering interview questions. For example, both Kelly (l/c) and Hayden (l/c)

counted ten-blocks in 5s, implying a value of 5 for each block. Kelly counted 4 tens

and 12 ones in this way, reaching the answer 32 (4 x 5 + 12). Later in a teaching

session Kelly counted 9 tens as 45 (l/c S4, T 7c). In a similar incident, at the second

interview Hayden counted 6 tens and 7 ones and arrived at the answer 37 (I2, Qu. 1a).

When the researcher queried this answer, he changed his answer to 67, admitting that

he had been counting the ten-blocks “in fives.” Jeremy made a similar error,

apparently because of a misunderstanding about how Clive used counting by twos to

speed up the counting of blocks. Jeremy, hearing Clive count two blocks at a time,

counted 9 tens as if each represented a value of 2: “2, 4, 6, 8, 10, 12, 14, 16, 18” (l/b

S2, T 4c).

Reversing values of tens and ones.

A different sort of error in assigning values to blocks was made by Kelly (l/c)

when she was asked to show 28, and then 134, with the blocks:

Interviewer: Can you show me 28 with the blocks, please?

Kelly: [Puts out 2 ones, then 8 tens]

Interviewer: Okay do you know another way of showing 28? Can you show me a different

way?

Kelly: [Puts out another 2 ones & 8 tens, arranged differently] There. (I1, Qu. 2a)

Interviewer: OK, now can you show me 134 using the blocks?

Kelly: [Puts out 1 hundred, 3 ones & 4 tens]

Interviewer: Uh-huh. Is there another way of showing 134, do you think, can you show

me?

Kelly: [Puts out another hundred, on right, then 4 tens on left, & 3 ones in the

middle] (I1, Qu. 2b)

Apparently, for some reason, Kelly had some confusion in her mind at the

time of the first interview regarding the values represented by each size of block, and

she decided to use one-blocks to represent tens, and ten-blocks to represent ones.

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Nevertheless, whatever the difficulty she had with tens and ones it did not seem to

affect her use of the hundred-blocks, as she used the correct block to show the

hundreds digit. This may be due to the fact that the hundreds place was new to her

and also she had never used those blocks before in mathematics lessons, and so she

guessed correctly that the new block represented the new place. At the second

interview Kelly used the blocks to show the numbers in Question 2 using a correct,

canonical, representation in each case, implying that whatever misunderstandings she

had in Interview 1 were corrected during the teaching phase. This error made by

Kelly is another apparent example of a participant considering places independently,

without regard for values of places relative to each other. Further discussion of the

independent-place construct is in section 5.3.

Applying incorrect values to blocks.

Participants were observed on many occasions to refer to blocks verbally or

in writing using incorrect values. For example:

1. Michelle, Daniel, Terry, Amy, and Kelly all read 4 tens as “4.”

(l/b, h/c, & l/c S1, T 1d)

2. Simone stated that 8 ones followed by 2 tens showed “82.” (h/b S1, T 2a)

3. Michelle was asked to write the blocks she had before and after trading

both tens in 21 for ones, and wrote “2 ones, 1.” (l/b S3, T 5a)

4. Simone wrote “36 = 3 ones 6 ones.” (h/b S2, T 5b)

5. Clive attempted to convince Jeremy that 51 is greater than 39. In his

explanation, he asked Jeremy to compare the digits ‘5’ and ‘3.’ Several

times Clive referred to the ‘3’ as “3 ones.” (l/b S5, T 10)

6. John read 9 tens and 6 ones as “960.” (h/b S6, T 16)

7. When attempting the task asking participants to add 31 and 28, John

held some ten-blocks in his hand, and twice referred to them as

“hundreds.” (h/b S6, T 18).

8. Amy read 12 tens as “112.” (l/c S8, T 24)

9. Jeremy counted 1 hundred and 1 ten as “101,” then 1 hundred and 2

tens as 102. Clive counted 1 more ten than 110 as “111.” (l/b S8, T 24)

10. Michelle stated that 8 hundreds and 2 tens showed “eight hundred and

two.” (l/b S10, T 29d)

11. Terry stated that 14 tens showed “104.” (l/c S10, T 32a)

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12. Kelly stated that 16 tens and 7 ones showed “607.” (l/c S10, T 33)

Similar errors were evident when participants chose incorrect blocks to

represent a particular digit. For example:

1. Belinda put out 4 tens to represent the number 4. (h/c S4, T 13)

2. Amy started to put out only tens when trying to represent the number

99. She stopped when Kelly pointed out that she had 39 tens.(l/c, S8, T 25)

3. Jeremy chose 2 tens to represent 200. (l/b S9, T 27)

4. Clive added 8 tens to 2 hundreds and 4 tens when attempting to

represent 248. (l/b S9, T 27)

As in the case of the previous type of error, this error indicates that the

participant had assigned an incorrect value to certain blocks. Note that in examples

(a) and (b), Belinda (h/c) and Amy (l/c) demonstrated this error, although they were

using the software, which would have indicated in the column labels and the number

window, if visible, that incorrect blocks were being used. Since this error was

observed even when contrary evidence was available to the participants, it appears

that the error is quite a common one among children of this age.

A number of times participants referred to the value represented by the

blocks when asked the number of blocks in a block arrangement. For example:

1. Asked how many tens there would be if a ten in 62 was traded for ones,

Belinda responded several times “fifty tens.”(h/c S2, Supplementary task)

2. Comparing representations for 73 and 29, Kelly said that there were “70

tens” and “20 tens.” (l/c S4, T 9)

3. Nerida said that 6 tens was “60 tens.” (l/b S9, T 28a)

4. Asked how many tens would be needed if a hundred-block in 340 was

traded for tens, Jeremy said “a hundred of the tens.” (l/b, S10, T 32a)

The phrase used by Jeremy in the last example is interesting, and seems to

indicate the confusion he was feeling between the number of blocks and the value

that that number of blocks represented. By saying “a hundred of the tens,” Jeremy

may have been indicating that he meant “a hundred-worth of the tens,” rather than

“100 tens.”

In another example of applying incorrect values to blocks, several participants

were observed to combine values from different places without converting or trading

them. By combining quantities represented by blocks or digits in different places, the

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participants were combining numbers from different places as if they were alike.

Examples included the following:

1. Amy attempted to work out the number represented by 16 tens and 7

ones, and said it was “23 hundred.” (l/c S10, T 33)

2. Terry said that 15 tens and 17 ones represented 32. (l/c S10, T 33)

3. Daniel wrote that 14 tens and 11 ones represented 25. (h/c, S9, T 33)

4. Daniel and Rory both said that 41 tens and 9 ones represented 50.

(h/c, S10, T 34b)

4.7 Use of Materials to Represent Numbers Materials used to represent numbers may be used in a variety of ways, some

of which were not intended by either their developers or teachers using them with

their students. One sub-question of this study, given in section 1.3, is “Which of

these conceptual structures for multidigit numbers can be identified as relating to

differences in instruction with physical and electronic base-ten blocks?” This section

contains descriptions of two aspects of the participants’ use of physical or electronic

materials: counting and use of trial-and-error methods.

4.7.1 Counting of Representational Materials

In order to use materials, whether physical or electronic, to represent

numerical quantities, a count must be made of the number of the various materials

present. In the case of the software used in the study, counting by participants was

not necessary, as the software keeps a continuous check of the number of blocks in

each place displayed in a text box at the head of the column; the column counters are

always visible to users. On the other hand, users of physical base-ten blocks must

count the blocks themselves when using them to represent a number. This subsection

includes descriptions of participants counting both electronic and physical blocks,

and discussions of links between participants’ favoured approach to numeration

questions and their counting of materials.

Counting justifications for answers.

As in the individual interviews, several participants referred to counting when

justifying answers to questions during teaching sessions. For example, in the

following transcript the researcher asked the high/blocks group how they could be

sure that 62 is greater than 48:

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Teacher: Mmm-mm. How do you know that he has more? That’s my question. I mean,

we know 62 is more than 48.

John: Yeah.

Amanda: You count.

Teacher: Why is it more than 48? — Counting is OK, but supposing you didn’t have the

blocks, and we were just talking, and someone said “Well I’ve got 62, and

you’ve got 48,” How do you know 62 is bigger? How can you prove that it’s

bigger?

All: [Many children talking together]

Amanda: Because when you were … we knew how to count, and we’ve got bigger.

John: Because there’s only this many, and they think it’s bigger because this,

because the ones are more [in 48], but the tens are less, and he reckons his is

bigger because the tens are more [in 62]. (h/b S3, T 8)

It is interesting to notice that John recognised for himself that the idea could

be given that the digit ‘8’ indicates that 48 is larger, but that the tens digit is of

greater value. On the other hand, Amanda referred twice to a counting approach to

decide which number is larger.

In the previous example, participants used their knowledge of the counting

number sequence to justify an answer. Counting was also used to justify answers on

occasions when participants counted physical or electronic blocks. For example, the

researcher asked members of the low/blocks group to justify their belief that 2 tens

and 8 ones could be placed in spatially different arrangements and still represent 28.

Clive responded by counting the blocks:

Clive: 10, 20, 21, 22, 23, 24, 25, 26, 27, 28. (l/b S1, T 2a)

One participant’s preference for counting.

One particular participant, Hayden (l/c), showed a clear preference for a

counting approach at both interviews, and during the group sessions. This case is of

interest because it demonstrates how a student’s preference for a counting approach

can apparently influence that student’s use of representational materials. Hayden’s

preference for counting materials came to the author’s notice because Hayden was in

a computer group, and yet he frequently counted on-screen blocks by pointing to or

touching the computer screen.

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Table 4.8 shows that Hayden used a counting approach more often than any

other participant did, apart from Kelly (l/c). This subsection includes a brief

discussion of Hayden’s use of a counting approach in the interviews, and evidence

that his preference for this approach had a bearing on incidents in the teaching phase

of the study. First, Hayden’s responses to a question at the second interview show his

reliance on counting to answer questions about numbers. The question required the

participant to compare 38 and 61; excerpts from the relevant transcript follow (see

Appendix O for the full transcript). To start with Hayden used a counting approach to

explain why he believed that 61 was larger:

Hayden: Because it’s six … 38 takes shorter and 61 takes longer.

Interviewer: If you’re counting, you mean?

Hayden: Yeah.

The researcher continued to ask Hayden about the numbers 38 and 61, asking

him specifically about the digits in the two numbers. It is interesting to note that

Hayden was unable to use the information contained in the respective digits of the

two numbers to understand which is larger:

Interviewer: OK, what about the numbers that are in [points to ‘38’ & ‘61’]? Does that tell

you anything?

Hayden: No, it doesn’t … like it still doesn’t mean that it’s got an ‘8’ on the end and

it’s got a ‘1’ on the end [points to respective digits] … because that’s um …

like that … like if you get 1, 2, 3, 4 … like 10, 20, 30, 40 … no 10, 20, 30 and

you just count to 8 … in the 30s, like it’s only the 38.

Interviewer: Uh-huh.

Hayden: And if you count the 61 it’s a “60 one.” (I2, Qu. 6a)

It is interesting that when the researcher asked further questions about the

digits in the two numbers Hayden rejected the suggestion that the digits indicate

anything on their own about the size of a number, particularly because the ‘8’ would

give an incorrect result if a face-value interpretation was used. There is a clear sense

that Hayden’s concept of numbers was based around the sequence of cardinal

numbers, based on his frequent use of counting-based justifications for his answers.

From the previous transcript excerpt it is evident that he was able to say quickly

which number was larger just by looking at the digits, implying some knowledge of

the place-value system. However, when the reasons behind his answer were probed

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further he responded in terms of where the two numbers were in the sequence of

cardinal numbers.

Excerpts from the teaching phase of the study show that Hayden’s ideas about

numbers influenced how he approached questions when using the computer.

Although the software displays counters that indicate the number of blocks in each

column, Hayden still chose to count blocks himself. For example:

1. He counted 4 tens put out to represent 40 to check that the

representation was correct. (l/c S1, T 1d)

2. He started to count the blocks after trading a ten from 77, before Terry

told him the computer could do the counting. (l/c S2, T 4a)

3. He started to count 6 tens and 11 ones to check that they represented 71.

(l/c S6, T 15)

The preference that several participants had for either a counting approach or

a grouping approach is one of the clearest findings of this study, discussed in section

5.2.1. Because a grouping approach is more efficient and more useful with larger

numbers than a counting approach, a relevant question is whether or not use of either

representational format might help a participant with a preference for counting to

switch to a grouping approach. In Hayden’s case, his use of counting was quite

successful, and it may have been some time before he considered changing his habit

of counting to answer numerical questions. The last example above supports this

idea. The researcher had asked Hayden’s (low/computer) group how many tens

would be present after a hundred in 340 was traded for tens. No other participant

could work out the correct answer: Amy and Terry said the answers “3” and “4,” and

Kelly did not reply. Hayden counted the blocks on screen, using the marks on the

picture of a hundred-block to count the tens that would result from a trading

procedure, then counted on the 4 tens in the tens column. After counting Hayden was

sure that the answer was 14 tens, and tried to convince the others in the group that he

was correct. The fact that Hayden’s answer in this instance was correct, and that

other participants were unable to work out the answer, would presumably have acted

as a reward to Hayden for using the counting approach, making it more likely that he

would do so again when the opportunity arose.

Apart from Hayden, very few computer participants actually counted the on-

screen blocks for themselves. It appears that Hayden’s strong preference for using a

counting approach led him to count on-screen blocks, although the software could

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have done it for him. Details of participants’ use of counting to gain feedback about

their answers are found in Table 4.16; it shows that participants in blocks groups

received much more feedback about their answers from counting blocks than did

participants in the computer groups. Feedback received by participants is discussed

in section 4.7.7.

4.7.2 Use of Trial-and-Error Methods

One aspect of use of materials to represent numbers that emerged in the data

was the incidence of trial-and-error methods by some participants when putting out

blocks to represent a number. Though this appeared a few times in blocks groups

(see transcript excerpt later in this section), it was most noticeable in transcripts of

computer group sessions. Computer participants were frequently observed to use

backtracking when clicking the relevant buttons to place blocks in the three places.

Analysis of the transcripts did not initially reveal this behaviour, as the videotapes

merely showed students using the computer to place blocks on-screen, without

indicating clearly how many blocks of each size were put out, and in which order.

However, the software audit trails revealed that on several occasions participants

overshot the number of blocks needed in a place and backtracked (see Appendix E

for a sample audit trail, showing the trail generated by Hayden and Kelly’s computer

at session 9). For example:

1. When representing 538, Terry put out 5 hundreds and 5 tens [550], took

away 3 tens [520], added 2 tens again [540], removed a ten [530], and

added 8 ones [538]. (l/c S9, T 29a)

2. When representing 712, Kelly put out 7 hundreds and 2 tens [720],

removed the 2 tens [700] and added 12 ones [712]. (l/c S9, T 29c)

3. When putting out blocks to show 147, Amy put out 1 hundred [100],

removed it [0], put out 1 ten [10] and 7 ones [17]. She then removed the

ten [7], added 1 hundred again [107], and added 4 tens [147].

(l/c S10, T 30a)

4. When representing 516, Yvonne put out 5 hundreds and 6 tens [560],

removed 6 tens [500], added a ten [510], and 6 ones [516].

(h/c S8, T 30c)

As mentioned earlier, this category of response was noticed first in software

audit trails, which logged participants’ mouse clicks when using the software. When

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consideration was given to whether participants using blocks exhibited the same

behaviour, it became obvious that blocks participants frequently used checks of the

blocks they put out, by checking the number required or by recounting the blocks.

The example below shows Jeremy (l/b) using both these checking methods as he

used blocks to show 95:

Jeremy: When representing 95 with physical base-ten blocks, uses the blocks left-over

from the previous question [94], removes the ones [90], adds a ten [100],

recounts the tens and removed the ten again [90], rechecks the card, counts out

5 ones and adds them to the representation [95]. (l/b S7, T 20)

This example is typical of participants using blocks, particularly with more

difficult numbers including three-digit numbers. It seems clear from transcripts that

participants had some difficulty remembering three-digit numbers after reading the

task instructions, and there were frequent examples of participants checking the

number to be displayed after starting to put out blocks, especially after completing a

place or two. This observation is not very surprising; clearly three-digit numbers are

cognitively more demanding on students than two-digit numbers, and students may

require more support when asked to carry out procedures using these larger numbers.

This point is discussed later in section 4.7.3.

Despite the apparent similarity between examples of trial-and-error behaviour

by participants using physical or electronic base-ten blocks, it appears likely that the

two representational formats had different effects in this regard. In the case of

participants in blocks groups making intermediate checks of the numbers of blocks

put out, it is likely that the participants were having to refresh their memory of the

number required, and to check that the blocks put out matched the number(s) in the

instructions. One does not get a sense that these participants were actually trying out

arrangements to see if they fit the requirements of the task at hand; rather, they were

using a “feedback loop” to check their progress as they chose blocks for each place.

However, in the case of computer participants’ frequent backtracking when showing

some numbers, it appears that the method they adopted was genuinely one of trial-

and-error: They appeared to be testing their ideas by putting out blocks and looking

at the available on-screen numerical symbols to see if they were correct. Because the

electronic blocks may be put out very quickly by clicking with the mouse, it does not

take a user long to put out some blocks to see if they are correct, looking at the

column counters and number window that are updated continuously as each new

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block is put out. Then it is a simple matter to remove blocks quickly by clicking on

various buttons. As the feedback from the software about the number of blocks put

out is virtually immediate, it enables, or affords (Salomon, 1998) such trial-and-error

methods, whereas to do this with physical blocks would be much more time-

consuming and cumbersome.

It may be noted in the examples given in this subsection that trial-and-error

methods were used most often by the low/computer group; examples were found of

every participant in that group using trial-and-error methods at some time. However,

in the high/computer group few examples were evident, and all by Yvonne, the

participant with the lowest interview scores in that group (see Table 4.3).

Furthermore, the examples of this behaviour all involved three-digit numbers: No

examples were found of participants using trial-and-error methods when showing

two-digit numbers. It appears that this behaviour was related to uncertainty in the

minds of participants who had weaker understanding of the base-ten numeration

system about which blocks they needed for the more complex numbers, and that the

participants used the screen blocks, counters, or both, to revise their decisions as

each block was put out. It is assumed that high-achievement-level participants using

the computer did not need to use trial-and-error methods because they had a

knowledge of base-ten numbers that was sufficiently sound that they did not to have

to make several attempts to show the numbers using on-screen blocks.

4.7.3 Handling Larger Numbers

The transcript examples cited in the previous subsection, showing participants

using trial-and-error methods to handle larger numbers, were all from the last few

teaching sessions, involving three-digit numbers. As mentioned earlier, larger

numbers involve more complex mental mapping between symbols, number names

and representational materials (section 2.4.3; see also Boulton-Lewis & Halford,

1992). Thus, it is to be expected that participants would have more difficulty with

these numbers than the cognitively simpler and more familiar two-digit numbers.

One aspect of participants’ attempts at tasks involving three-digit numbers that is

quite noticeable in the transcript data is the apparent difficulty some participants had

with holding the necessary information in their minds all at once. This was observed

in both blocks and computer groups.

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Difficulties with three-digit naming tasks.

One apparent example of the cognitive demands imposed by a task being too

great for participants to manage at once is found in the transcript of the low/blocks

group attempting to complete Task 28 (a), which required them to say the number

represented by 3 hundreds, 6 tens and 9 ones put out by the researcher (see Appendix

P for the full transcript). Every participant in the group appeared to have

considerable difficulty managing the task. The participants’ behaviour was consistent

with the conjecture that they were attempting to reduce the amount of information

they had to remember all at once. The following behaviour was observed:

1. saying part of the number name aloud,

2. counting blocks aloud,

3. touching the blocks,

4. separating blocks of different sizes from each other,

5. writing an answer down immediately after stating it,

6. looking at the answer of another participant, and

7. using the digit in each place and the place name to state the number

represented (i.e., “3 hundreds, 6 tens, 9 ones”), rather than saying the

complete number name (“three hundred and sixty-nine”).

(l/b, S9, T 28a)

The behaviour described here was quite common among participants in the

blocks groups when the numbers involved were larger. Whereas they could manage

to count blocks and write the number they represented in one step when the blocks

were in a canonical arrangement for a two-digit number, when three-digit numbers or

non-canonical arrangements were involved, they generally broke the task of checking

the number and writing its symbol into several steps. Typically, this involved

counting the blocks of one place, recording the appropriate digit, then moving to the

next place, and so on. Participants in the computer groups exhibited similar

behaviour, except that they did not need to count the blocks. On occasions they were

observed instead to check the place counters one by one, writing each digit before

checking the next counter. It is suggested that participants used these strategies

because otherwise they were unable to manage the cognitive demands imposed by

the tasks.

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Handling a non-canonical arrangement task.

One task in particular caused participants more difficulty than any other,

apparently due to the cognitive demands it placed on participants. Task 33 required

participants to put out more than 9 tens and more than 9 ones, then say what number

the blocks represented. The researcher asked participants to first work out the

number represented without regrouping, then to regroup and check their answer. This

task caused all 3 groups that attempted it some difficulty, in some cases to a

considerable amount:

1. John and Amanda had difficulty counting the number of tens and

reaching a consensus about the number. They eventually agreed that

there were 24 tens and 15 ones [255]. John said that the blocks

represented 219, which Amanda agreed with until the blocks were

regrouped. They counted the blocks, and John wrote 245, but Amanda

wrote 255. (h/b, S9, T 33)

2. Craig and Simone put out 19 tens and 52 ones [242]. Craig calculated

mentally that the number represented by the blocks was 242. When the

pair regrouped and counted the blocks, they reached an answer of 232,

which both participants accepted. (h/b, S9, T 33)

3. Daniel and Rory put out 14 tens and 11 ones [151]. Rory wrote that this

showed 151, but Daniel wrote that it represented 25. (h/c, S9, T 33)

4. Hayden and Terry put out 14 tens and 22 ones [162]. Terry said that this

showed 32, then revised it to 162. (l/c, S10, T 33)

Responses shown here of participants using electronic blocks were quite

similar to responses of participants using physical blocks, as they apparently

attempted to manage what to them was clearly a difficult thinking task. However,

this similarity in responses of participants using both representational formats is not

generally seen in the transcripts; it appears that the features of this task somehow

made the experiences of participants using either material very similar. Because of

the numbers of blocks involved, participants using either physical or electronic

blocks had to manage two trades at once to determine the number represented by the

blocks. The researcher did not permit the participants using the software to regroup

blocks or to use the number window, which would have told them the number

represented by the blocks. Thus, they were forced instead to use other methods that

they evidently found difficult. It appears that the level of support provided by the

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software under these conditions did not give participants in the computer groups

much of an advantage over participants using the physical blocks. In comparison,

when completing other tasks involving only a single trade participants in the

computer groups were often successful in working out the number involved from the

block pictures and the column counters.

Handling skip counting tasks.

Table 4.2 reveals that the tasks that caused participants the most difficulties in

both interviews were the skip counting tasks in Question 4. The four subtasks

required participants to count (a) back by 1 from a two-digit number, (b) on by 10

from a two-digit number, (c) on by 10 from a three-digit number, and (d) back by 10

from a three-digit number (see Appendix I & Appendix J). In each question, the

participant was asked to continue the sequence past the first necessary regrouping.

For example, in Question 4 (d) at Interview 1 the participants were asked to count

back from 496 by 10, and encouraged to continue until they were past the number

400. Results in Table 4.2 indicate that of all the interview tasks, performance on the

skip counting tasks showed the greatest difference between participants who used

physical blocks and participants who used electronic blocks. Based on the criteria

adopted for assessing skip counting performance (see Appendix M), participants of

the computer groups improved their combined score on skip counting skills by eight

criteria at the second interview, whereas the combined score of participants of the

blocks groups was unchanged. Reasons for these differences may be related to the

way that numerical symbols can be shown by the software when it is used to carry

out skip counting; this idea is discussed in section 5.5.5.

4.7.4 Interpreting Non-Canonical Block Arrangements

When blocks are used to represent “trading” actions, in which a block in one

place is exchanged for 10 of the next place to the right, the result is a non-canonical

arrangement of blocks that cannot be mapped onto the corresponding numerical

symbol by merely counting the number of blocks in each place. At the interviews the

participants were asked to interpret several non-canonical block arrangements and to

answer a question involving collections of 10 and more than nine single objects (see

Questions 1 (b), 1 (c), 3 (a), 3 (b), & 9 (b) in Appendix I & Appendix J). Though

most participants were able to work out the numbers represented by non-canonical

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arrangements of blocks, either by arranging blocks into groups of ten or by counting

all the blocks, several participants were unable to interpret correctly non-canonical

arrangements. Difficulties with non-canonical block arrangements were also

observed during teaching sessions. The following paragraphs describe the actions of

two participants which indicated that they found non-canonical block arrangements

difficult to understand.

The high/blocks group worked on Task 5 (a), representing 21 with the blocks

and then trading both tens for ones. Simone did the trade, then counted the ones

before recording her answer, which she wrote as ‘24.’ In the subsequent discussion

between the researcher and the participants, Simone found that her answer was

incorrect. This may have prompted her to change the way she handled such

questions, as described in the following paragraph.

On several subsequent occasions Simone (h/b) correctly traded blocks, but

kept the blocks from the original arrangement separate from the 10 “new” traded

blocks. When she had recorded her answer, which was generally correct, she

immediately swapped the blocks back to the original canonical arrangement. An

answer Simone gave to a question from the researcher supports the idea that she had

difficulty understanding non-canonical block arrangements: When the researcher

questioned her group about why, after a ten-block in the representation for 255 was

traded, the new arrangement still showed 255, Simone responded “You don’t count

the ones.” When the researcher queried her about this statement, she responded “You

swap 10 [ones] for a ten” (h/b S8, T 31a). It appears that Simone was saying that the

new arrangement of 2 hundreds, 4 tens, and 15 ones represented 255 only because it

could be swapped back for the original canonical arrangement of blocks, and not

because the non-canonical arrangement also represented 255.

Another participant, Michelle (l/b), was also observed to keep traded blocks

separate from other blocks of the same place. When answering Task 5 (b) Michelle

traded all 3 tens in 36 for ones, but kept the 30 traded one-blocks separate from the

original 6 ones. She then counted only the 30 traded ones and recorded her answer as

‘30 3 tens 0 ones.’ By this answer Michelle may have meant “30 = 3 tens 0 ones.”

Thus she did not record the number of ones after the trade, but gave a standard,

canonical, answer based on the values of the digits in ‘30’ rather than on the number

of blocks before her. During another task soon after this Michelle made specific

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mention of her method of separating traded blocks from the blocks originally present

when trading all the tens in 64 for ones.

4.7.5 Face-value Interpretations of Symbols

One conception of numbers that has been widely reported in the literature is

the “face-value construct” (see section 2.4.2). Students who hold this conception

regard each digit in a multidigit number as representing only its face value, and not

that number of tens, hundreds, thousands, and so on. There is evidence in the

teaching session transcripts that participants using both representational formats

demonstrated face-value interpretations of digits at various times, shown in the

following paragraphs.

Face-value interpretations among users of physical blocks.

There is evidence that when calculating answers to questions some

participants used blocks as counters, and in so doing gave each digit a face-value

interpretation. For example, see the following excerpt, in which Jeremy (l/b)

suggested that 39 was greater than 51 because its block representation comprised

more blocks:

Clive: [Puts out blocks to show 39 and 51.] That’s … 39, and that’s 51.

Michelle: Which one is most?

Clive: This one [51], because it’s 51 … 5 tens is most, and 3 ones [sic] is less.

Jeremy: But what about this one [39]? [Implied: it has a greater number of blocks.]

Clive: I talked about that one.

Jeremy: [Moves the blocks in 39 a little, so that all blocks are visible. Perhaps he

thought that Clive was mistaken because he couldn’t see all the blocks]

Teacher: What do you think, Jeremy? It has got a lot of ones there, hasn’t it?

Jeremy: [Nods.]

Clive: [Re-counts ones. Perhaps he thought the teacher believed that there were too

many ones]

Teacher: So, do you think this one [39] might be bigger than that one [51]?

Jeremy: That one [39] would be bigger, because it’s got heaps of ones. (l/b, S5, T 10)

Table 4.10 shows that Jeremy consistently used a face-value interpretation of

digits in both interviews: It was clearly a persistent idea that he had about numbers.

This transcript is interesting in that to Jeremy, his face-value interpretation of the

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digits in 39 and 51 appeared to be supported by the blocks themselves. In the

transcript, Jeremy referred to the blocks when arguing for a face-value interpretation

of the two symbols. It is clear that Jeremy was not looking at the size of the blocks at

all, but merely at the number of blocks. This implies that he did not see, or that he

ignored, the markings on ten-blocks indicating the shape and size of ten ones joined

together, and instead used each block as a counter. By counting the number of

blocks, no matter their size, he apparently believed he could make judgements about

the size of the number they represented.

Face-value interpretations among users of electronic blocks.

There is evidence from a teaching session with the low/computer group that

the software may have had the effect of supporting face-value ideas. This is revealed

by written responses to Task 27 (b), which required participants to state the meaning

of each digit in the number 248 after representing it with blocks. Transcriptions of

the participants’ written answers are shown in Table 4.15. Three of the four

low/computer participants gave a face-value interpretation of the digits in ‘248,’

which no other participant did. Two participants in the low/blocks group gave

partially incorrect responses—Nerida and Michelle wrote ‘200 hundreds’ or ‘2

tens’—but even so they still interpreted the ‘4’ and the ‘8’ correctly.

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TABLE 4.15. Participants’ Written Responses to Task 27 (b)

Group Participant Workbook Responsea

High/computer Belinda [no written response]

Daniel it mens are 2 Hundreds and 4 Tens and 8 ones

Rory every number is even / 2 Hundreds 4 tens 8 ones

Yvonne The 2 is hundred 4 is tens 8 is ones.

Low/computer Amy 2 means 2 4 means 4 8 means 8

Hayden 2 means 2 4 means 4 8 means 8

Kelly 2 = 2 H 4 = 4 Tens 8 = 8 ones

Terry 2 mns 2 cow / 4 m 4 bes / 8 m 8 horse [2 means 2 cows, 4 means 4 bees, 8 means 8 horses]

High/blocks Amanda two hundred 4 tens 8 ones

Craig The 2 stands for 200 and the 4 stands for 40.

The 8 stands for 8

John H t o

2 4 8

Simone 2 hander and 4 tens and 8 ones

Low/blocks Clive 2 h 4 tens 8 one

Jeremy 2 h 4 Ten 8 one

Nerida 200 tens 4 tens / 200 Hundreds and 4 tens 8 ones

Michelle 2 tens 4 tens 8 ones / 200 Hundreds Note. Task 27 (b): Explain the Meaning of the ‘2,’ ‘4,’ and the ‘8’ in ‘248.’ a “/” indicates new line started by participant.

It is possible that participants using the electronic blocks were influenced by

the presence of single-digit counters in the software, resulting in different responses

to Task 27 (b). It may be that participants in the high/computer group were able to

interpret the digits correctly without referring to the counters, but that the participants

in the low/computer group used the counters to guide their responses to the task.

Further evidence of face-value interpretations of symbols being held by

participants in the low/computer group is found in the following descriptions:

1. When answering Task 29 (c), low/computer participants were asked by

the researcher about the three digits in ‘712,’ after both pairs had put

out 7 hundreds, 0 tens and 12 ones. The researcher referred to the ‘7’

representing 7 hundreds, then pointed out that there was no ten-block.

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Terry responded “Oh no, 1 one! Means 1 one, and ‘2’ means 2 two, and

‘7’ means 7 seven.” (l/c S9, T 29c)

2. Hayden wrote about 80, ‘9 [sic] means 9 and 0 means 0.’ (l/c S8, T 26)

3. Amy wrote about 80: ‘8 means 8 and 0 means 0.’ (l/c S8, T 26)

4. Terry wrote about 126: ‘1 means 1 cow, 2 means 2 beds, 6 means 6

horses.’ (l/c S8, T 26)

5. Hayden wrote about 126: ‘1 means 1 and 26 means 26.’ (l/c S8, T 26)

There is insufficient evidence in the data from this study to decide with any

confidence the extent to which the software may have influenced participants to hold

a face-value interpretation of digits in written symbols. Results in Table 4.10 show

that low-achievement-level participants generally used face-value interpretations of

written symbols more often than did high-achievement-level participants, but it is not

possible to point to a definite effect by either representational material on

participants’ face-value interpretations. Whereas at the second interview there was a

reduction in the incidence of face-value interpretations given by Nerida (l/b), and

Terry and Hayden (l/c), other participants in the low/blocks group showed no

improvement, and Amy and Kelly (l/c) showed some deterioration on the relevant

questions at Interview 2. Any statements about the influence of each representational

format on participants’ face-value ideas must therefore be tentative.

4.7.6 Predictions About Trading

When blocks are used to trade a block for 10 blocks of the next smaller size,

an important concept for students to grasp is that the quantity represented does not

change. This concept is particularly important for learning how to handle each of the

four operations with multidigit numbers, each of which can include the need to

regroup from one place to an adjacent place.

Data from the first task involving trading.

Transcript data from all 4 groups showed that in every case at least one

participant in each group noticed when completing the first task involving trading,

Task 4 (a), that the number represented by blocks after regrouping a ten in 77 was the

same as the number represented before the trade. Transcripts show that there was a

clear difference in how participants using the two representational formats responded

to this discovery. Specifically, participants using electronic blocks made frequent

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mention in later sessions of the equivalence of block arrangements after trading, but

participants using physical blocks did not do so. Excerpts from these transcripts are

shown next (see Appendix Q for full transcripts of Task 4a for each group):

High/computer:

Belinda: [Looking at screen] 77! There's still 77! Cool.

Yvonne: No, there’s sixty …

Rory: It’s 77.

Yvonne: Oh, yeah it is [laughs].

Computer: 77. (h/c S1, T 4a)

Low/computer:

Hayden: [Uses mouse to have computer read representation.]

Computer: 77.

Hayden: [To Terry, with surprised expression] 77!

Terry: Oh! We’ve still got … Oh, cool, that’s easy! [Writes in workbook] Seventy …

77! [To teacher] How does it do that? It’s still got 77. [Teacher looks at him,

but does not respond] Oh yeah! [Look of recognition. Bangs himself on his

head with his hand]

Hayden: [Points to screen] It’s still … You cut it up, and it’s still 77! [Looks at Terry]

Terry: Mmm. [Pencil in mouth, apparently thinking.] (l/c S2, T 4a)

High/blocks:

Teacher: The last question I have to ask you is: Are the two amounts [indicates the two

block representations for 77: 7 tens 7 ones, 6 tens 17 ones] the same?

Simone: No.

Amanda: No. Y … [Stops, seems unsure]

Craig: No.

Simone: No.

Teacher: And I want you to discuss that with each other. I mean, you know what

number that is [7 tens and 7 ones]. Is that [6 tens and 17 ones] the same

number?

Amanda: — Yeah, ‘cos those ones, just for ten. Still the same. Make ‘em for ten.

Craig: — [Counts ones under breath] … 10. 1, 2, 3, 4, 5, 6, 7. Oh yeah, they’re the

same.

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Teacher: [Separates 7 ones from the other 10 ones and the 6 tens] So what number is

shown by these blocks [6 tens and 17 ones]?

Amanda: 77.

Craig: Er … 77. (h/b S1, T 4a)

Low/blocks:

Michelle: [Counts just the regrouped ones] 17.

Clive: Wrong!

Teacher: Well, you better count them, Clive. Michelle’s saying it’s 17.

Clive: 2, 4, 6, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, … What

was that one? 76, 77! Huh. — So we got double 77s. Mmm? That was tricky.

(l/b S1, T 4a)

The transcripts for this task show that not all participants made mention of the

equivalence of the two representations for 77, and that even after spending extended

periods of time looking at the block representations some participants still did not see

that the numbers represented by each were the same. For example, Simone (h/b) was

convinced that the traded blocks did not show the same number as the untraded

blocks had, and did not change her mind until the researcher demonstrated the

equivalence of the two arrangements for her.

Place-value software and understanding of trading.

The responses of several participants to the discovery that blocks still

represented the same number after a trading process indicated that they were

surprised that this should be the case. This was particularly the case for Belinda in

the high/computer group, and Terry and Hayden in the low/computer group. Both

Belinda and Terry used the colloquialism “cool” at the information provided by the

software that after trading a ten the blocks represented the same number. The voices

and faces of participants in both computer groups indicated a degree of both surprise

and pleasure, apparently showing that (a) they did not already expect blocks to

represent the same quantity after trading, and (b) they enjoyed having the computer

reveal this discovery to them. These reactions are in contrast to reactions of

participants in the two blocks groups that indicated that they were less convinced

about the equivalence of values.

The difference in the reaction of participants using physical blocks or

electronic blocks to the discovery that traded blocks represent the same quantity

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apparently extended beyond the incidents quoted earlier. Participants in both

computer groups mentioned the equivalence of traded blocks repeatedly during later

teaching sessions, without prompting by the researcher. On the other hand, the same

was not true of blocks participants: Not one participant in a blocks group mentioned

the equivalence of traded blocks after the initial discussion recorded in the excerpts

cited in this subsection. Appendix R contains many statements, made by at least 6 of

the 8 participants in the two computer groups, regarding the equivalence of traded

blocks. The transcripts show that participants using the electronic blocks commented

often about the fact that trading blocks produces representations that show the same

value. It is of particular interest that participants using the software were also able to

predict accurately the numbers of blocks that would be in each place after the trade,

and when questioned about this frequently mentioned that the new blocks would

represent the same number. The transcripts indicate that the participants had started

to develop considerable confidence in the fact that the equivalence of traded blocks

was always true, no matter what the number being represented.

Physical base-ten blocks and understanding of trading.

Whereas participants using electronic blocks demonstrated confidence in

traded blocks representing the same amount, such confidence was not evident among

participants using physical blocks. It appears that these latter participants were still

developing their understanding of trading, and that whereas participants using

electronic blocks were able to develop a generalisation that traded blocks are always

equivalent in represented value, physical blocks provided much less support for this

construction, and so did not help participants using them to develop the same level of

understanding. Members of the high/blocks group at times showed an awareness that

traded blocks represented the same quantity, but at other times made mistakes when

trading that indicated that their knowledge of trading was still not completely secure

on this point. For example, when trading a ten in 255, Amanda (h/b) initially traded a

ten for 5 ones, making the number of ones up to 10, and wrote in her workbook that

the new representation did not show the same number.

In the case of participants in the low/blocks group, they were clearly quite

confused in many instances about what to expect when manipulating blocks or

quantities. One characteristic of their use of the physical blocks that became quite

evident was that they had few expectations about the results of numerical processes.

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The frequency of errors in counting and handling blocks made by low/blocks

participants was so high, and their knowledge of the base-ten numeration system was

so weak, that their manipulations of blocks frequently produced incorrect answers.

At the same time, these participants often expressed confidence in the answers they

derived from manipulating the blocks, a confidence that was often misplaced; this

reaction to numbers revealed by one’s own counting is discussed further in the

following section. In this situation, it is not very surprising that they did not appear to

develop the understanding that blocks represented the same value before and after a

trade. There were many instances in which participants in the low/blocks group

argued about an answer, due to one or more errors made by different participants,

that ultimately had to be resolved by the researcher because the participants were

unaware of the errors made in the course of working out their answers.

4.7.7 Feedback

As mentioned earlier, an analysis was made of the data from teaching

sessions looking for essential differences between the responses of participants using

physical blocks and of those using electronic blocks. One super-category that

emerged was that of feedback: the receipt of information about an answer, indicating

whether or not it was correct. It became clear from the transcripts of the teaching

sessions that many of the interactions among participants, the researcher, and the

materials could be interpreted as feedback provided to participants regarding their

answers. Appendix L contains a description of the method used to identify and

analyse incidents of feedback in teaching sessions.

It should be noted that, as defined here, feedback includes information

derived from blocks, in the sense that by counting or otherwise manipulating blocks

a participant could have an answer confirmed, disconfirmed, or provided by the

blocks. Thus certain incidents of feedback were the result of participants’ actions that

led to their receipt of information about how to proceed. For the discussion in this

section, in such incidents the source of the feedback is considered to be the materials,

as the result of actions by the participant.

Sources of feedback.

Five sources of feedback became evident in the videotape data as the analysis

was conducted: the researcher, other participant(s), self-checking, counting of

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physical or electronic blocks, or electronic feedback from the software. These

sources of feedback are referred to in this section as Teacher, Peer, Self, Materials,

and Electronic, respectively. A list of descriptors used in analysis of feedback is

provided in Table L.1 in Appendix L. A summary of feedback data, indicating the

number of feedback incidents from each source for each group is presented in Table

4.16.

TABLE 4.16. Incidents of Feedback of Each Source per Group

Group

Feedback Source High/ blocks

Low/ blocks

High/ computer

Low/ computer

Teacher 96 130 70 103 Peer 89 115 63 59 Self 9 1 5 8 Materials 40 73 17 33 Electronic 119 104 Total incidents 234 319 274 307

The data in Table 4.16 are presented in Figure 4.4 in the form of a stacked

column graph. This graph makes it clear that, although the total numbers of incidents

of feedback received by the 4 groups are different, the proportions of feedback

received by each pair of groups using the same material were very similar.

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Figure 4.4. Proportions of feedback from each source for each group.

Possibly the most striking aspects of the data in Figure 4.4 are (a) the

similarity of each pair of adjacent columns representing groups using the same

representational format, and (b) the difference between the two pairs of columns. It is

clear that though there were differences in the feedback received by high-

achievement-level and low-achievement-level participants using the same

representational materials, there were many similarities between participants using

the same materials. On the other hand, comparing participants using physical blocks

and participants using electronic blocks, there are quite marked differences in the

patterns of feedback received. Participants using the software received proportionally

less feedback from the teacher, from each other, and from counting electronic blocks

themselves; however, they received a large number of instances of electronic

feedback from the computers.

One obvious question about the data in Figure 4.4 is: Did participants using

the software receive electronic feedback in addition to feedback received from other

sources, or instead of that feedback? In other words, were the actual numbers of

incidents of feedback from non-electronic sources similar to those of participants

using physical blocks? This question is answered by data presented in Table 4.16,

which contains numbers of incidents of feedback rather than percentages of the

totals. These figures show that the actual incidence of feedback from each non-

0%

20%

40%

60%

80%

100%

High/Blocks Low/Blocks High/Computer Low/Computer

Group

Electronic

Self

Materials

Peer

Teacher

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electronic source was lower for computer groups, compared with the equivalent

blocks groups. If electronic feedback is ignored, the high/computer group received

79 fewer incidents of feedback than the high/blocks group. Similarly, the

low/computer group received 106 fewer incidents of non-electronic feedback than

the low/blocks group. It is interesting to note that this difference is made up almost

exactly by the number of incidents of electronic feedback received by participants in

these groups. As electronic feedback very nearly makes up the “shortfall” of

feedback incidents among computer participants, it appears that electronic feedback

provided by the software was not simply added to the interactions that would

otherwise have existed among participants and the teacher. Rather, the availability of

electronic feedback appears to have reduced the incidence of feedback from both

humans and materials.

Effects of feedback.

Feedback provided by physical base-ten blocks is limited, as mentioned

previously. Participants may count blocks for themselves, but other than that, there is

no direct feedback possible from the blocks. This point is borne out by the data in

Table 4.16. Participants were able to count or recount blocks, but other sources of

feedback had to be human: another child, the teacher, or themselves. On the other

hand, participants using the software were able to access electronic feedback that

gave information about the number of blocks in each place, a numeral expander

representation of the written symbol, and the written symbol and number name for

the number represented by the entire block arrangement.

Feedback and answer status.

Incidents of feedback were coded to show the status of the answer of the

participant receiving the feedback and the effect(s) that the feedback had for the

participant receiving it. On some occasions, participants received feedback that was

in response to their correct or incorrect answer. On other occasions, the participant

had either no answer or an incomplete answer, and feedback was accessed to provide

an answer. In a few incidents, no judgement was possible about a participant’s

answer before receiving feedback; in these cases the answer status was coded as

“unknown.” Table 4.17 summarises feedback received by participants in each group,

summarised according to the status of the answer held by each participant before

receiving the feedback.

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TABLE 4.17. Percentage of Feedback Compared With Answer Status

Answer Status High/blocks Low/blocks High/computer Low/computer Correct 33 22 54 48 Incorrect 44 51 27 35 Incomplete 7 6 7 2 Nil 15 21 11 13 Unknown 2 0 0 2 Note. Values represent percentages of feedback incidents for each group.

The greatest differences between groups using the two representational

formats regarding status of answers prior to feedback being received are evident in

data for correct and incorrect answers. Overall, low-achievement-level participants

received proportionally more feedback for incorrect answers, and less feedback for

correct answers, than high-achievement-level participants did, whether using

physical or electronic blocks. When results for computer participants are compared

with blocks participants, an interesting pattern emerges. Whereas users of physical

blocks received on average more feedback for incorrect answers, participants using

electronic blocks received more feedback when their answers were correct.

Another aspect of feedback that is important when considering the assistance

that it provides to students is its quality: This is defined as the likely effect that

feedback would have on the participant receiving it, with regards to the recipient’s

confidence in the answer. In other words, if the feedback is likely to have encouraged

a participant to retain the answer, whether correct or not, then the feedback is said to

be positive. If, on the other hand, the feedback is considered likely to have

encouraged its recipient to reject the answer, then it is said to be negative. Table L.2

provides a list of feedback effects and the associated quality descriptions. The quality

of feedback is compared to the answer status of its recipients in Table 4.18, for

blocks and computer groups.

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TABLE 4.18. Quality of Feedback Provided for Correct or Incorrect Answers

Feedback Qualitya Groups Answer Statusb Positive Negative Neutral Total Blocks Correct 43 36 22 146 Incorrect 2 80 19 263 Computer Correct 87 6 7 294 Incorrect 4 77 19 181 Note. aValues in each row represent percentages of the total in the right-most column. Lists of feedback categories coded for each quality category are described in Appendix L. bFeedback coded with the following categories of answer status are not included, as feedback quality is not considered relevant in these cases: Incomplete (63 incidents), Nil (174), Unknown (13).

Table 4.18 reveals several interesting aspects to the feedback experienced by

participants using physical or electronic blocks. Participants using physical blocks

received many more incidents of feedback for incorrect answers than for correct

answers. On the other hand, computer users received more incidents of feedback for

correct answers than for incorrect answers. Cells of the table that reveal the most

dramatic differences between blocks and computer participants are those recording

feedback for correct answers. It appears that computer participants received a greater

proportion of positive instances of feedback for correct answers than did blocks

participants: Almost 90% of feedback for correct answers received by computer

participants was positive. For blocks participants, less than half of their feedback for

correct answers was positive, with 36% of feedback negative, and 22% neutral. In

other words, participants using physical blocks were less likely to receive

confirmation for correct answers than were participants using electronic blocks.

Interestingly, feedback for incorrect answers by both blocks and computer

participants had a very similar profile, with about 80% of instances of feedback for

incorrect answers being negative. To summarise these figures: Participants using

electronic blocks seem to have received much more feedback when they had a

correct answer than participants using physical blocks, and proportionally much

more feedback for correct answers given to computer participants was positive than

was the case for blocks participants.

These figures do not reflect the number of mistakes made by participants

using either type of materials. Mistakes were not specifically counted in analysis of

the data, but interview data lead to the conclusion that computer users were no better,

overall, than blocks groups at understanding numbers (see Table 4.3). Thus it is

assumed that when answering questions during teaching sessions the computer users

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had just as much difficulty understanding the concepts as their peers using physical

blocks. However, the feedback received by each set of participants is substantially

different. Blocks groups received more feedback for their mistakes than for their

successes, and more negative feedback overall. Moreover, when their answers were

correct they received almost as much negative feedback as positive. On the other

hand, computer groups received feedback that was mostly positive, and very few

instances of negative feedback given for correct answers.

Given these differences in feedback provided to participants using physical or

electronic blocks for correct or incorrect answers, it is relevant to inquire of the

source of the feedback in each case. The sources of feedback for correct answers and

incorrect answers are shown in Table 4.19 and Table 4.20, respectively.

TABLE 4.19. Percent of Feedback for Correct Answers from Each Source

Quality Groups Source Positive Negative Neutral Totals Blocks Teacher 20 1 16 37 Peer 10 32 2 44 Self 0 0 2 2 Materials 12 3 2 17 Totals 42 36 22 100 Computer Teacher 17 0 4 21 Peer 5 6 1 12 Self 1 0 1 2 Materials 3 0 0 3 Electronic 61 0 1 62 Totals 87 6 7 100 Note. Values represent percentages of the total feedback for either blocks or computer groups, rounded to the nearest percent.

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TABLE 4.20. Percent of Feedback for Incorrect Answers from Each Source

Quality Groups Source Positive Negative Neutral Totals Blocks Teacher 0 43 15 58 Peer 0 32 2 34 Self 0 0 0 0 Materials 1 5 2 8 Totals 2 79 19 100 Computer Teacher 0 43 12 54 Peer 2 24 6 31 Self 0 0 0 0 Materials 0 0 0 0 Electronic 3 10 2 15 Totals 4 77 19 100 Note. Values represent percentages of the total feedback for either blocks or computer groups, rounded to the nearest percent.

Data in Table 4.19 and Table 4.20 show that incorrect answers received

similar patterns of positive and negative feedback for both computer and blocks

groups, from the teacher, from peers and from materials. Furthermore, most feedback

for incorrect answers received by participants using either representational material

was accurate; very little positive or neutral feedback was provided for incorrect

answers. The exception to this is feedback from the teacher, which was often neutral

for incorrect answers. The reason for this is pedagogical: The researcher deliberately

gave neutral responses to incorrect answers on occasions to encourage participants to

reconsider the question before the researcher provided the correct answer. The

feedback category that makes up the bulk of the difference between the two pairs of

groups is electronic feedback for correct answers: Over 55% of all feedback given to

computer participants for correct answers came from the software. Feedback from

the teacher and from materials was in similar proportions for the groups using

physical or electronic blocks, but feedback from peers was less common for users of

the software. However, the experience of users of electronic blocks appears to have

been dominated by positive feedback from the software for correct answers.

Responses to feedback.

One further aspect of feedback of interest in the data analysis was the

category of response labelled “expressing satisfaction.” This category was introduced

to categorise a large number of instances of feedback in which the participant

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responded by expressing by either body language or verbal utterances that the

participant was pleased with the feedback. In many instances, this satisfaction was

expressed by the participant saying “Yes!” sometimes accompanied by a gesture with

the arms reinforcing the impression of pleasure and satisfaction. Overall, 95

instances of participants expressing satisfaction in response to feedback were

recorded. Of these 95 incidents, 10 were in response to feedback from the teacher, 4

to feedback from a child, 14 to feedback from materials, and 67 to feedback from the

computer. It was the frequency and character of participants’ responses to electronic

feedback that led to the creation of this category in the data analysis, and it is

considered interesting enough to describe in more detail at this point.

Selected examples of the expression of satisfaction made by computer

participants are given following:

1. After having computer trade a ten in 58 then read the name of the

resulting representation, Belinda said “Yep, that’s true.” (h/c S2, T 4d)

2. Belinda predicted that after trading a ten in 62 there would be 12 ones.

When the computer showed this was true, Belinda said “12! Yep, I was

right.” (h/c S2, T 4d)

3. Hayden checked the number represented by the computer blocks with

the verbal number name, smiled, and said “Yeah! We got it.”

(l/c S1, T 2d)

4. Kelly checked the symbol for the number 90, and said “Yep. Yeah, it's

right. That's how you write it.” (l/c S1, T 2d)

5. Amy put out blocks to show 15, had the computer read the number

name, and said “Yes,” and made a gesture of “success” with two fists

and bent arms. (l/c S1, T 3c)

6. Hayden put out blocks for 77, used the computer to check the verbal

name. Terry commented, “Yep. I believe it’s 77.” (l/c S2, T 4a)

7. After Amy put out blocks for 23, Kelly commented “Yep. We got it”

when the computer confirmed the verbal name. (l/c S2, T 4b)

8. Terry checked the block representation after a hundred had been traded

from 340. As the computer read the number name, he held his hand

behind his ear, and commented “Good! Just to make sure!”

(l/c S10, T 32a)

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Participants expressed satisfaction at feedback they received from the

software on many occasions. Often the participants were pleased merely to have the

computer “read” the block representation. At other times they were pleased to see

their answers confirmed by the number of blocks after a trading procedure, the

symbol for a number, or a number of ones equivalent to a multidigit number. Early in

the teaching phase the researcher encouraged participants to use the facilities of the

software to check the block representations they formed. The researcher made

frequent reference to the fact that the computer had the capacity to confirm the

number represented by the blocks on screen. The participants were quick to accept

this idea, and after a while used these facilities without any prompting by the

researcher. It was assumed by the researcher that once the participants had the idea

that the blocks represented numbers as they expected, they would stop using the

verbal name and number symbol features except when beginning a new type of task,

or a task with larger numbers. However, it was evident that the participants enjoyed

hearing and seeing the computer confirm their block representations repeatedly, even

when to an adult the confirmation was no longer needed. Participants frequently

accompanied their response to the computer feedback with comments like “Yep,

that’s right,” “Yep, we’ve got that number,” “I believe that,” or “Yep, that’s true.”

It is relevant to ask if the same category of expressing satisfaction to feedback

was observed among users of the physical blocks. There were just a few instances:

1. The researcher told John that his answer regarding the blocks left after

trading of a ten in 77 was “a good way of doing it.” John was clearly

pleased, and showed the other participants his book. (h/b S1, T 4a)

2. Amanda confirmed that her answer was the same as Craig’s, saying

“Yep, that’s what I got.” (h/b S2, T 4b)

3. The researcher told John that his answer was correct. John told the

others that he was correct. (h/b S5, T 14)

4. The researcher said that Amanda, Craig, and Simone were correct in

saying that 75 + 19 = 94. They expressed satisfaction at the researcher’s

comments. (h/b S6, T 19)

5. Simone recounted the blocks that she and Craig put out to show 394,

and said “Yep.” (h/b S8, T 30b)

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6. The researcher told John that his answer of 12 tens resulting from

trading a hundred in 627 was correct; John expressed his satisfaction at

being correct. (h/b S9, T 32b)

7. Craig used a ten-block to check the height of a stack of 10 hundreds,

and was clearly pleased to find that he was correct. (h/b S10, T 45)

The examples given here show similar reactions of participants using physical

blocks to positive feedback for correct answers. Interestingly, this feedback was

usually from the researcher, who told participants that their answers were correct. In

the case of computer groups, when participants expressed satisfaction in response to

feedback, the feedback always came from the software. The author considers it likely

that the role of the researcher in the examples given in the previous list was in some

way the same as that of the software when it confirmed participants’ answers.

4.7.8 Using Blocks To Discover Number Relationships

For someone who possesses enough familiarity with numbers, blocks may be

used to confirm or illustrate a numerical relationship. However, if this familiarity is

missing, and in the absence of other sources of information, use of physical blocks is

likely to prompt the user to count the blocks in order to discover the result of a

numerical process. One use of base-ten blocks reported in the literature (e.g., Fuson,

1992), which in this context may be extended to the use of suitable software, is to use

them merely as calculating devices for finding the answers to computational

questions. Rather than using their knowledge of number facts and the base-ten

numeration system to work out what an answer should be, some children use

materials in an attempt to discover the answer. Note, in relation to the approaches

identified in analysis of the interview data, this behaviour often involved a counting

approach (section 4.4.2): Blocks were manipulated to replicate the numbers and

processes involved in the question, then counted to discover the answer.

The expectation by participants that the blocks would reveal numerical

relationships was evident in several transcripts. In each of the following examples, it

would have been possible for a student with sufficient number fact or place-value

knowledge to answer the question without using blocks at all. In the examples here,

however, participants using physical blocks gave no indication of knowing what the

answer was until they had manipulated and then counted the blocks:

187

1. Amanda and Simone each separately used blocks to calculate 31 + 28.

(h/b S6, T 18)

2. After trading a hundred from 627 for tens Simone counted the ten-

blocks before writing her answer. (h/b S9, T 32b)

3. After trading a ten from 23 for ones Clive counted the one-blocks,

finding that the number represented was still 23. (l/b S2, T 4b)

4. Michelle used blocks to calculate 95 - 23. (l/b S7, T 20)

Counting blocks to discover answers is an example of feedback received by

participants during teaching sessions. During teaching sessions there were five

different sources of feedback available to participants, one of which was the physical

or electronic blocks. The relative frequency of counting blocks to discover answers

by participants in each group is shown in Table 4.21. It can be seen that in both

blocks groups participants’ favoured source of answers was the blocks themselves

(“Materials”), whereas in both computer groups the favourite source of answers was

“Electronic,” via on-screen number counters.

TABLE 4.21. Feedback Providing Answers from Each Source for Each Group

Group Source High/Blocks High/Computer Low/Blocks Low/Computer Teacher 9 0 1 3 Peer 27 32 39 18 Self 18 11 1 18 Materialsa 46 14 59 18 Electronic 43 44 Note. Values in each column represent percentages of feedback incidents used to discover answers for each group. aIncidents were included if representational materials were counted to discover an answer; incidents in which materials were counted only to make a block arrangement have been excluded.

Table 4.21 shows selected data from analysis of feedback incidents observed

on the videotapes, showing only incidents in which participants accessed a source of

information to find an answer that they did not already have; this represents

approximately 18% of all feedback incidents recorded. Clearly, the participants using

physical blocks counted the available representational materials to discover answers

much more often than did the participants using electronic blocks. However, the

proportions of feedback of this category are much closer if electronic feedback is

included in the figures for the computer groups. This seems to indicate that there was

a similar reliance on the available materials to provide answers when the participants

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could not work out answers themselves, by participants using both representational

formats, though the actual uses of the materials were very different. As mentioned

previously, without other inherent mechanisms for feedback in the blocks,

participants using physical blocks sometimes counted the blocks to reveal answers.

On the other hand, computer participants occasionally counted the on-screen blocks,

but more often used other forms of electronic feedback to discover answers. Note

that when the purpose of the question was to find out the number represented by an

entire arrangement of blocks, the researcher did not permit computer participants to

have the number window, displaying the number represented by the entire block

collection, visible.

Confidence expressed in the results of counting.

One feature of data from teaching sessions involving blocks groups was the

confidence expressed by several participants in the accuracy of physical block

representations. On several occasions participants were faced with two conflicting

answers to a question, one resulting from their count of the blocks and one resulting

from another source such as another participant’s count, or their own mental

processing of the numbers involved. Often a participant resolved the conflict by

expressing trust in his or her own counting. In light of the frequency of counting

errors made by participants, discussed in section 4.6.1, it would appear that such

confidence in their own counting accuracy was unwise. Indeed, on several occasions

participants were forced to retract their answers when they found that their count was

mistaken. For example, in the following excerpt Clive (l/b) miscounted the tens and

ones after trading a ten in 58. He decided that there were 17 ones, insisting that he

was correct until the researcher and other participants convinced him that the ones

were made up of the original 8 plus the extra traded 10, and therefore the correct

answer was 18. Following is a shortened excerpt; see Appendix S for the full

transcript.

Clive: [Writes in book] 58 equals 4 tens and … [counts ones] 3, 4, 5, 6, 7, 8, 9, 10,

11, 12, 13, 14, 15, 16, 17! 17 ones.

Teacher: The boys and girls have two different answers again. Clive? You have

different answers again.

Clive: — Youse are wrong.

Michelle: — We traded it for …

189

Nerida: … for ten ones and we kept our 8 ones already there.

Teacher: And would that make 18, or would that make 17?

Nerida: 18.

Clive: [With arms folded; in the previous dialogue of the girls, he has not been

showing agreement with what they said, or any apparent willingness to listen]

17.

Teacher: — If you had 8 to start with, and then you swapped and had another ten, what

number would that make, without counting?

Clive: 17.

Teacher: Ten and 8?

Nerida: [Shakes head] 18.

Clive: 18, I think. Think. (l/b S2, T 4d)

Jeremy (l/b) demonstrated an extreme example of confidence in the results of

his own counting, shown in the following example. The task was to regroup all the

tens in 21 and record the resulting number of ones. Jeremy was clearly unsure about

what to do, and watched the other two participants to see what they did (Clive was

absent that day). Jeremy took away 2 tens from his representation for 21, then put out

several ones. He watched the other participants carefully for a time, apparently trying

to produce the same arrangement as them, but without counting the blocks. He failed

to remove 1 of the ones, so did not actually carry out the trade. He counted the ones

he had put out, and wrote ‘21 = 17 one’ (l/b S3, T 5a). In trying to copy the other

participants, Jeremy evidently did not know how many ones they had, so he

estimated the number. When he counted his one-blocks he found that he had 17, and

so he wrote that for his answer.

In a similar incident, Michelle and Jeremy (l/b) miscounted the blocks in a

task asking them to add 10 to 26. The children chose to add 10 ones rather than a

single ten. In the process, one of the traded ones became mixed up with the initial 6,

and was removed by Jeremy. When Michelle counted, she reached the answer 35,

which she accepted. In response to a statement by Clive that the answer was 36, she

commented “36? It can’t be 36” (l/b S7, T 16).

Participants used physical blocks to find answers on many occasions in

addition to those incidents already mentioned. Though mistakes in general were

more common among low-achievement-level participants than high-achievement-

level participants, two incidents in which participants in the high/blocks group made

190

errors are particularly revealing, in that they demonstrate that the more able

participants also placed considerable trust in the results of manipulating blocks. On

each of these two occasions a participant in the high/blocks group correctly worked

out an answer to a question using mental computation, but then counted the blocks

and found the two answers to differ. In both cases, the participant rejected the earlier

mental answer in favour of the incorrect counted answer. The following example

shows an incident in which John counted the ten-blocks after trading a hundred in

627, after Amanda accidentally removed a ten-block:

Amanda: [Puts out 6 hundred-blocks and 2 tens, and starts to remove a hundred.]

John: [Counts 7 one-blocks into his hand, puts them on the table.]

Amanda: [Removes a hundred-block, and adds 10 ten-blocks to the representation. She

starts to count the ten-blocks. She starts to write in her book, absent-mindedly

picking up a ten-block and putting it on her book. Then she pauses to count

the ten-blocks. She finds that there are 11 ten-blocks, which she writes into

her answer.]

John: [Picks up the ten-blocks to count them] I don’t think there are 12. I mustn’t

have counted them properly. [He briefly looks at the floor as he replaces the

tens with the rest of the representation on the table. He writes his answer as 11

tens.]

Teacher: [To John & Amanda] Do you both say the answer is 11 tens? [They both nod.]

Well, I’m sorry, but you are both wrong.

John: [Immediately] I had ‘12,’ but I wrote ‘11.’

Teacher: Did you expect the answer to be ‘12’?

John: Yeah, but there were only 11 blocks.

Teacher: Well, you should expect 12, because that is the correct answer. [John is clearly

pleased that he was correct.] (h/b S9, T 32b)

It is evident that John was not completely happy with the answer found by

counting the blocks, but he still accepted it in preference to the answer he had

calculated mentally. A similar incident occurred when Craig and Simone (h/b) were

working out the number represented by a handful each of tens and ones blocks. The

children put out 19 tens and 52 ones, which Craig correctly calculated to total 242.

The researcher asked the pair to justify their answer, at which they proceeded to trade

groups of 10 ones or tens until they had a canonical arrangement. However, in the

191

process of trading a handling error was made, resulting in the answer shown by the

blocks being 232:

Craig: [With Simone checks the total representation, now that all the trades have

been done; they find it is 232. Craig is obviously surprised that this is the

answer.]

Teacher: What do you think?

Craig: I added an extra ten onto it. [He starts to rub out his previous answer of ‘242.’]

The answer is 232. [He writes ‘232.’]

Teacher: Where was the mistake made?

Craig: I put in an extra ten. I thought there was 142, because, 50, no, um, 190 plus

50, I um, it was um 40, 2 hundred and um 42, but instead, um, I forgot I have

to count an extra 10 ones.

Teacher: OK, so the right answer is 232? How do you know that’s right, and not 242?

Craig: [Looks very unsure.]

Simone: Because we did it with the blocks.

Craig: ‘Cos we traded …

Teacher: And that proves that it’s right? Could you have made a mistake, do you think?

Simone: [Shakes her head slightly.]

Teacher: Could you have got it right in your head, and wrong with the blocks?.

Simone: [Shakes her head.] No.

Teacher: Which do you trust — your heads or the blocks?

Simone: [Points] Trust the blocks.

Craig: Blocks.

Teacher: Well, with the numbers you started with, the correct answer was 242. You

may have made a mistake with the blocks, and missed a ten. [Craig & Simone

look quite surprised.] (h/b S9, T 33)

It seems clear that participants’ levels of confidence in the results of mental

computation were related to their computation abilities. In the previous example,

Simone was evidently unable to calculate an answer mentally, whereas Craig did so

correctly. It is not surprising, therefore, that Simone had greater confidence than

Craig in the result reached by counting the blocks.

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4.8 Chapter Summary This chapter includes description of a wide range of topics relevant to this

study, regarding the use of the two representational formats in the group teaching

sessions. The size of the study and the type of data collected constrain the

conclusions that may be drawn from the data. Without a strict experimental design,

and with a small number of participants, conclusions from the data have to be made

tentatively. Nevertheless, there are a number of trends in the data collected in the

study that are worthy of serious consideration in discussions of how different

representational materials are used by children learning about the base-ten

numeration system. The following chapter contains discussions of the findings

reported in chapter 4.

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Chapter 5: Discussion

5.1 Chapter Overview This chapter comprises discussion of results from the interviews and from the

teaching sessions, divided into four major sections, corresponding with four major

findings of the study:

1. Many participants demonstrated a preference for either grouping or

counting approaches to place-value questions (section 5.2);

2. A new category of conceptual structure for multidigit numbers, the

independent-place construct, is needed to explain evidence of

participants’ ideas about numbers (section 5.3); and

3. Several participants were evidently constructing their ideas about

numbers in light of new information and their prior knowledge about

numbers (section 5.4).

4. The different features of physical and electronic base-ten blocks

apparently caused the two types of material to have differing effects on

participants’ actions and conceptual structures for numbers (section

5.5);

5.2 Participants’ Ideas About Multidigit Numbers The literature search conducted prior to this study indicated several

conceptual structures for multidigit numbers identified by other researchers (section

2.4.2). These conceptual structures were used in initial analysis of data in this study.

However, the conceptual structures described by other authors were found to be

rather unhelpful in considering the responses of participants in this study. The

interview data in this study, rather than revealing neat, clear-cut categories of number

conceptions held by participants, show somewhat untidy patterns in participants’

ideas. Although some participants clearly had well-developed conceptions of two-

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digit and three-digit numbers, the responses of many other participants indicated

mixtures of correct ideas, incorrect ideas, and incompletely formed opinions about

numbers. In this section the conceptual structures described in chapter 2 are

compared with the analysis of this study’s data. Sections 5.2.1 and 5.2.2 contain

discussion of participants’ preferences for grouping or counting approaches in light

of their responses at the interviews and in the teaching sessions. Section 5.2.3

contains a critique of previously published schemes for categorising children’s place-

value understanding in light of this study’s data. Lastly, the “face-value construct”

described in the research literature is compared with this study’s findings in section

5.2.4.

A note about conceptual structures.

At this point, it is appropriate to explain the use in this thesis of the term

approaches rather than the more common “conceptual structures” or “concepts”

when describing children’s thinking about numbers. The term “approaches” has been

adopted here in light of the data collected in interviews and teaching sessions. As

explained later, the data collected in this study does not support the idea that the

participants had stable, coherent ideas about numbers, as is implied by the term

“conceptual structures.” On the contrary, the overwhelming impression given by the

data is that many participants adopted one of two clearly-distinguishable approaches

to number questions, counting or grouping, which individual participants used with

varying levels of consistency when answering various questions. Furthermore, the

approaches used by many participants appeared to be guided by often creative and

flexible use of a variety of numerical knowledge possessed by the participants. This

knowledge was often misapplied or misunderstood, but the fact that participants

attempted to apply several different pieces of information to a numerical question

indicated that their ideas about numbers were not fixed, and so could not be

described simply as belonging to a particular category.

Analysis of previously-described conceptual structures.

The conceptual structures described in section 2.4.2 can be compared with the

findings in this study. To reiterate, there were two sets of conceptual structures found

in the literature. Firstly, four conceptual structures were identified as being necessary

for the development of place-value understanding: (a) the unitary construct, (b) the

tens and ones construct, (c) the ten as a unit construct, and (d) the flexible

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representations construct. Secondly, three conceptual structures that constituted

limited understanding of base-ten numbers were listed: (a) a unitary concept of

multidigit numbers, (b) a face value construct, and (c) a counting sequence concept.

Descriptions in section 4.4 of participants’ approaches to interview questions agree

with descriptions of conceptual structures summarised in section 2.4.2 from the

literature. Grouping approaches (section 4.4.1) clearly show evidence of both the

“ten as a unit construct” and the “flexible representations construct”; counting

approaches (section 4.4.2) include both the “unitary concept of multidigit numbers”

and the “counting sequence concept”; and face-value interpretations of symbols

observed in this study (section 4.4.3) agree with descriptions in the literature.

However, as already discussed in this section, the idea that any participant

exhibited one of these conceptual structures as his or her principal model for

multidigit numbers was not demonstrated. For example, there was no participant who

was found to use a “unitary construct” generally when answering interview

questions. There were examples of participants using single one-blocks to represent

numbers (e.g., see section 4.4.2 for a description of Daniel’s [h/c] and Amy’s [l/c]

use of multiple one-blocks), but these participants did not show a unitary construct

model for multidigit numbers for other questions, or even as their preferred model of

numbers. Similarly, some more able participants used the “flexible representations”

construct, as shown by many examples of the grouping approach. However, that

construct could not be applied to the thinking of any particular participant, as that

participant’s primary conceptual structure for numbers. The following section

describes the preferences exhibited by participants for grouping or counting

approaches.

5.2.1 Participants’ Preferences for Grouping or Counting Approaches

This study has revealed a preference held by some participants for one

approach or another to answering number questions; the consistency with which

these approaches were adopted varied among the participants. The incidence of the

three main categories of responses to interview questions—grouping approaches,

counting approaches, and face-value interpretations—is summarised in Table 4.12.

This table shows that though a few participants were observed to adopt just one type

of response to the interview questions, the majority of participants used two or three

of the types of response during the course of a single interview. A comparison is

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made later in this section of the effects of grouping or counting approaches among

the participants. Face-value interpretations are discussed separately from the

grouping and counting approaches (section 5.2.4), because its use did not fit the idea

of a preferred approach in the way that counting and grouping approaches did.

Preference for grouping approaches.

The type of approach used most consistently by participants was grouping

approaches. Six of the eight high-achievement-level participants—Amanda, Craig,

John, Belinda, Daniel, and Rory—each used a grouping approach far more often than

either counting or a face-value interpretation of symbols. Based on observations

reported in Table 4.12, these 6 participants between them used grouping approaches

100 times at the two interviews, and used either counting approaches or face-value

interpretations of symbols only 8 times. The place-value criteria scores of these 6

participants ranged from 15 to 19 at Interview 1, and 17 to 20 at Interview 2; Figure

4.1 shows a clear tendency of participants with better place-value understanding to

use grouping approaches.

The prevalence of use of grouping approaches by high-achievement-level

participants (Table 4.7), and the apparent correlation between the use of grouping

approaches and interview scores (Figure 4.1), are quite striking. It seems likely that

there was a relationship between participants’ place-value understanding and their

use of grouping approaches. In order for a student to use a grouping approach in a

meaningful way, the student must already possess a certain understanding of the

base-ten numeration system that takes into account the groups of 10 at its foundation.

It appears that the more able students had previously developed meaningful, accurate

conceptual structures for multidigit numbers, which enabled them to develop and use

efficient methods when answering mathematical questions. A corollary of this

conjecture is that less able participants did not possess the knowledge of the base-ten

numeration system to enable them to use a grouping approach.

Preference for counting approaches.

The data for use of the other majority approach, the counting approach, are

far less clear-cut. Counting approaches were used most by 4 low-achievement-level

participants; Clive (l/b), Amy (l/c), Hayden (l/c), and Kelly (l/c); each of these

participants used a counting approach at least five times during at least one interview

(Table 4.8). However, these 4 participants together used counting on a total of only

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41 occasions, and with the exception of Kelly each of these participants also used

both grouping approaches and face-value interpretations of symbols, on a total of 17

occasions. Clearly, participants who favoured a counting approach did not use it to

the exclusion of other approaches. Furthermore, Figure 4.2 shows that counting

approaches were used both by participants with relatively poor place-value

understanding and participants with moderate or good place-value understanding,

indicating at best only a weak correlation between the use of counting approaches

and performance on the interview tasks. The discussion in section 5.2.2 includes

comments about the use of counting by children when first learning about single-

digit numbers, and its implications for learning place-value ideas. In light of that

discussion, it is quite possible that participants using counting approaches had not

changed the approach they learned when using single-digit numbers. Some of these

participants used counting approaches quite successfully on interview tasks, but

others clearly had many confusions about place value. It appears that the use of

counting approaches by participants was affected by several factors, and that simple

conclusions about levels of place-value understanding and the use of counting are not

justified.

Inconsistency of preference for approaches to problems.

Apart from the 6 participants listed previously who favoured grouping

approaches, few other participants showed much consistency in their approach to

answering interview questions, as shown in Table 4.12. Of the remaining 10

participants, 2 used only counting approaches or face-value interpretations, 1 used

only grouping or counting approaches, and the remaining 7 used all three response

types—grouping approaches, counting approaches, and face-value interpretations of

symbols—at least once during the two interviews. These figures show that

categorising the place-value understanding of these participants is not a simple task.

Each participant’s approach to answering each question could be described, and the

apparent lacks in knowledge of the base-ten numeration system noted; chapter 4

contains many such descriptions. However, participants’ use of multiple approaches

in answering different questions implies that they did not have a single idea of

numbers, that could be labelled for example as a “unitary construct” or a “face-value

construct,” that they used consistently in their thinking about numbers.

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Conclusions about preference.

Conclusions that may be drawn from this discussion of participants’

preferences for approaches to place-value questions include the following:

1. The most successful participants had developed an understanding of

multidigit numbers in terms of groups of ten, and used this

understanding to assist them in answering a range of mathematical

questions. The understanding of numbers held by these participants was

characterised by the ability to use the grouped nature of the base-ten

numeration system to answer a variety of questions involving

representational materials, written symbols and oral questions about

numbers. These participants rarely used counting approaches to answer

questions, and were not often convinced by incorrect counter-

suggestions, including face-value interpretations, offered during

interviews.

2. A few participants had developed a clear habit of using counting

approaches. These participants were often correct in their answers,

though their favoured approach is less efficient and more difficult to use

with larger numbers.

3. The least successful participants often held a variety of ideas about

numbers, including the incorrect face-value interpretation of symbols,

and used a variety of approaches to answering questions.

4. Few participants, except for the most mathematically able, could be

classified as having a single, particular, concept of numbers. Most

participants answered several questions incorrectly, and drew on a

variety of information they had learned about numbers in attempting to

answer mathematical questions. Even the most able students were

observed at times to use inefficient or incorrect approaches.

5.2.2 Comparison of Grouping and Counting Approaches

Whereas participants who used a grouping approach were usually successful

in answering interview questions, those using a counting approach often had

difficulties. Reasons for this appear to be related to two main factors: the generally

better knowledge of the base-ten numeration system of participants using grouping

approaches, and the fact that grouping approaches are easier to use successfully. This

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subsection contains a comparison of these common approaches of participants, and a

discussion of features of each approach that are relevant for teachers.

Counting is the first approach used when learning about numbers.

It has been pointed out by several authors (e.g., Booker, Briggs, Davey &

Nisbet 1997; Fuson et al., 1997; Resnick, 1983) that counting approaches are the first

methods used by children when learning to link single-digit numbers, number names,

and their referents. Thus, it is not very surprising to find that many children of the

age of the participants in this study use counting for managing two-digit numbers.

The principal way to discover the number of objects in a small collection is to count

them, and young children learn to associate symbols, number names and collections

of objects using counting-based methods. If a child persists with counting to make

sense of larger two-digit numbers, then that child is using a “unitary concept” of

numbers that merely continues the earlier approach. There are variations of this

approach, such as a “decade and ones” conceptual structure (Fuson et al., 1997), but

for the present discussion they may be considered together as counting approaches

that regard numbers as collections of single items that are apprehended by counting

them one by one.

Cognitive demands of counting and grouping.

To understand single-digit quantities, children need merely to associate each

of nine individual symbols with one of nine small words, and learn to apply correct

counting procedures to groups of less than 10 objects to determine the cardinality of

the group. Each process involves only a single mapping, between the objects and the

symbol representing them, the objects and the number name, or the symbol and the

number name (Boulton-Lewis & Halford, 1992).

If children extend their use of the unitary concept to numbers greater than 10

then the cognitive demands imposed by counting become greater. Two-digit symbols

and their associated number names, except for multiples of 10, all involve two parts,

representing the tens and ones parts of the multidigit whole. This is true even in the

case of numbers 13 to 19, in which the tens and ones parts of the number names are

obscured, and in 11 and 12 in which they are missing entirely. By retaining a unitary

concept for multidigit numbers, in which the number name and symbol are

considered to apply to an undifferentiated collection of single items, a child is

effectively attempting to retain a single mapping between the collection and its name

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or the collection and the related symbol. However, as explained by Boulton-Lewis

and Halford (1992), “place value because it rests on a binary operation, is at the

system-mapping level” (p. 5; see also section 2.4.3). Thus, even if a collection of

more than 9 items is regarded as a single group, the symbol and the number name

applying to the collection necessarily each involve at least two parts. Added to this

difficulty is the extra cognitive demands imposed on those using counting

approaches by the rules of the base-ten numeration system and the English language.

As described in section 4.6.1, counting across changes of decade, changes in the

number of hundreds, or from tens to hundreds involves simultaneously keeping track

of the numbers in several places. A grouping approach is often simpler to execute

because it allows a student to (a) count blocks in each place separately before

combining them, and (b) count only a single-digit number of blocks per place. In the

case of non-canonical representations, the extra blocks can first be regrouped,

leaving only single-digit counting to be done. This is precisely the approach adopted

in the interviews by several mathematically-able participants. Claims made here of

higher cognitive demand placed on those who use counting approaches are supported

by transcript excerpts showing participants’ counting errors, including counting

sequence errors (section 4.6.1) and perseveration errors (section 4.6.3).

Use of counting by children with poor knowledge of the base-ten numeration system.

Many participants who had difficulties with skip counting (section 4.6.1)

were the same participants who generally chose counting approaches when

answering interview questions. Students with difficulties remembering the counting

sequence correctly who choose a counting approach rather than a grouping approach

are then in a double bind: Firstly, their approach is less efficient and more likely to

lead to errors, and secondly, limitations in their knowledge of counting sequences

mean that they are also more likely to make counting errors before they reach the

answer.

Counting of larger numbers is inefficient.

A counting approach is clearly less efficient than a grouping approach, taking

the student more time and introducing the potential for more errors. The relative

lengths of previous transcript excerpts from interviews with Kelly (l/c) and Belinda

(h/c) answering the same question (3 tens + 17 ones), repeated here, reflect the

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difference in time that it takes a student to use either counting or grouping

approaches:

Kelly: [Touches each packet of gum] 10, 20, 30. [Counts on fingers by touching them

one by one on table] 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43.

43 pieces of gum. (I1, Qu. 9b)

Belinda: 47. There’s um, three of them and then there’s a one, which would make a 40,

and then you put a ‘7’ on the end and it equals 47. (I1, Qu. 9b)

As already indicated, there are at least two difficulties facing students who

regularly use counting strategies. Such strategies are vulnerable both to counting

errors and to higher cognitive demands. Students may either lose count and arrive at

the wrong answer, or they may not be able to hold all the parts of a question in their

heads, and so be unable to complete the task. Such problems will surely become

more pressing as students progress in school and the numbers involved become

larger. It is highly unlikely that children could continue using such strategies, once

the questions facing them involved three-digit numbers.

Limitations of using counting to understand numbers.

Though there are other reasons for using grouping approaches rather than

counting approaches, the most serious limitation of counting approaches to number

questions from a teaching perspective is that counting is much less helpful to

students to help them see the wider picture of the base-ten numeration system. The

system of base-ten numbers is made up of a number of repeating patterns, the most

fundamental being the repeated pattern of groups of 10 from place to place. The

sequence of counting numbers also contains patterns, but unlike the totally consistent

patterns of the symbol system, counting patterns include a number of inconsistencies

making them harder to follow and remember. In particular, the “grouped tens” aspect

of two-digit numbers is obscured “because the spoken numerals lack reference to the

tens and ones that are contained in them (e.g. eleven, twelve, thirteen, etc. and

twenty, thirty … one hundred)” (Boulton-Lewis & Halford, 1992, p. 9).

Every primary-age student is expected to learn the sequence of counting

number words in their spoken language, no matter how complex or difficult that may

be. Several writers have pointed out that, because of the inherent inconsistencies in

all European languages, learning to count and to use multidigit numbers to solve

mathematical problems is much more difficult for European-language-speaking

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students than for their Asian-language-speaking counterparts. Thus, the route to

understanding the base-ten numeration system for speakers of European languages

who habitually use a counting approach is likely to be circuitous and difficult.

Changing a child’s preferred approach from counting to grouping.

The reasons underlying a child’s use of one particular concept of numbers to

answer a variety of mathematical questions are not directly discernible from the data

in this study. However, it seems likely that the use of a certain approach to thinking

about numbers is the result of habit, of having successfully used that approach in the

past to answer mathematical questions. This certainly seems to have been the case

for certain participants who preferred the counting approach. As mentioned

previously, counting is often the first method used by young children when dealing

with single-digit numbers. Unless a child is assisted to see other, more efficient,

approaches, if that child experiences success with counting approaches, it will not be

surprising if the child continues to use counting when asked a range of mathematical

questions.

There is evidence from transcripts of interviews with one participant, Hayden

(l/c), that habitual use of a counting approach may actually have prevented him from

using other approaches (section 4.7.1). Hayden was one of the more successful

participants at the interviews, and often he was the only one in his group to have a

correct solution to a question in the teaching sessions. This is notable as he was also

one of the participants to use counting approaches the most at the interviews. It is

relevant to consider how easy it would be for a student like Hayden to develop an

understanding of the grouped aspect of the base-ten numeration system, and to alter

his favoured approach from counting to grouping. Considering the two types of

approach, there is little common ground between them, implying that there may need

to be a fundamental shift in thinking about multidigit numbers before a child could

change from counting to grouping. There is evidence in this study and from Fuson et

al. (1997) that students sometimes use more than one concept of numbers at different

times. However, it appears that participants using counting will need support from

teachers to understand grouping concepts and to adopt grouping approaches to

number questions.

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Summary of comparison between counting and grouping.

There are various interrelated difficulties likely to be faced by students who

favour a counting approach:

1. By focussing on a unitary conceptual structure for multidigit numbers,

either a single continuous number line or a sequence of cardinal

numbers containing all numbers in order, participants using counting

approaches are very unlikely to see the repeated grouped-by-10 pattern

inherent in the base-ten numeration system.

2. The seemingly random rules of the sequence of counting numbers

obscure the regularities in the sequence of numerical symbols. For

example, thirty, forty, and fifty do not clearly relate to three tens, four

tens, and five tens; furthermore, thirteen, fourteen, and fifteen are

similar in sound, but very different in meaning.

3. Sequences of number names become much more complex and more

difficult to manage mentally as the numbers involved become larger.

4. Counting of blocks representing multidigit numbers involves skip

counting with changes of increment at each new place. For example,

counting 5 hundreds, 8 tens and 2 ones: 100, 200, 300, 400, 500,

[switch to adding tens] 510, 520, 530, 540, 550, 560, 570, 580, [switch

to adding ones] 581, 582. By contrast, a grouping approach to the same

task involves counting three separate sequences of single-digit numbers

before combining them. For example: 1, 2, 3, 4, 5 hundreds - 500; 1, 2,

3, 4, 5, 6, 7, 8 tens - 80; 1, 2 ones – 2; 582.

5. Errors made when counting can prevent a student from reaching a

correct result. Frustration and an inability to continue with a task are the

likely immediate results; in the long term, learning of place-value

concepts is likely to be slower because of a lower incidence of success

on number tasks.

6. Finally, the habit of using counting approaches may blind a student to

the possibility of using other, more efficient methods. Switching to a

grouping approach is going to be beneficial in the long term, but it

seems likely that a student who has used counting approaches for a long

time, and who has not recognised the grouped aspect of multidigit

numbers, might find the change quite difficult to manage.

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5.2.3 Difficulties With Existing Conceptual Structure Schemes

It appears from a perusal of the literature on research into children’s

understanding of place-value that children’s thinking can be categorised according to

their thinking about numbers (e.g., Cobb & Wheatley, 1988; Miura et al., 1993; S. H.

Ross, 1990). The implication of the various proposed schemes is that individual

children possess stable ideas about numbers that influence their performance on

number-related tasks. Thus, several authors have devised models that comprise

stages or levels of understanding according to which an individual child’s

understanding of number concepts may be categorised. There is a good deal of

overlap among these schemes, with several conceptual structures being identified in

more than one study (see section 2.4.2). For these reasons, the data collected in this

study were expected to replicate many of the earlier findings, which could then be

analysed in relation to the two representational formats used in the study.

The intention to use the list of conceptual structures synthesised from the

literature search was found difficult to carry out, however. Results from this study

disagreed with those cited by other authors, on at least three grounds: (a) the great

variety in the responses of individual participants to different interview questions,

(b) a lack of confirmation in this study of both the frequency and the character of

certain conceptual structures previously reported, and (c) the generally limiting effect

that placing a student into a certain category was felt to impose on a researcher’s

understanding of students’ place-value concepts. The principal research on place-

value reported in the literature that was reviewed for this thesis was conducted by

Ross (S. H. Ross, 1989, 1990; S. H. Ross & Sunflower, 1996), Miura and colleagues

(Miura & Okamoto, 1989; Miura et al., 1993), Cobb and Wheatley (1988), Resnick

(Resnick, 1983, Resnick & Omanson, 1987), and Fuson and colleagues (Fuson &

Briars, 1990; Fuson et al., 1992; Fuson, et al., 1997). The research reported by each

of these groups is discussed in the following paragraphs and compared with the

findings of this study.

Ross’s five-stage model of digit correspondence performance.

Ross’s research on place-value understanding (S. H. Ross, 1989, 1990) has

received much publicity; in particular, her digit correspondence tasks have been

replicated in several other studies (e.g., Carpenter et al., 1997; Fuson & Briars, 1990;

Miura et al., 1993) or described by other authors (e.g., G. A. Jones & Thornton,

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1993a; C. Thompson, 1990). As described in section 5.2.4, there is broad agreement

between Ross’s five-stage model and this study’s proposed four-category model in

several descriptions of participant responses to digit correspondence tasks. Where

results of this study differ from Ross’s, however, is in the categorisation of students

according to their purported stage of development of place-value understanding.

Ross categorised the 60 students participating in her study according to the stage in

her five-stage model to which their thinking apparently belonged, stating that

each of the sixty students in the reported study was assigned to one of the five stages according to performances on six digit-correspondence tasks and a positional-knowledge task in which students were asked to identify, in a two-digit numeral, which digit was in the “tens place” and which was in the “ones place.” (pp. 49-50)

Thereafter in the paper, Ross referred to students as being “at” a particular

stage, or as “using a stage-n understanding.” Results of this study, on the other hand,

revealed students whose responses varied as they attempted different tasks. The

students themselves were not “at” a certain stage, in the narrow sense described by

Ross; the researcher was unable to neatly categorise their thinking based on

responses on one particular type of task. This applies particularly to what Ross

labelled the “face value” stage. As discussed in section 5.2.4, face-value

interpretations of symbols were demonstrated in this study, but were not

demonstrated on all relevant tasks by even one participant. Based on results of this

study it is suggested that face-value interpretation of symbols is but one symptom of

a general confusion about numbers possessed by many students of this age, rather

than an identifying characteristic of their mathematical thinking generally. The

results of asking children a variety of place-value questions in this study were messy,

often inconclusive, and not easily summarised. The contribution Ross has made to

the place-value literature is significant, and her descriptions of interpretations for

digits in multidigit numbers given by students are very valuable. However, results

from this study indicate that describing students’ mathematical thinking may be more

untidy and difficult than Ross suggested.

Miura’s three categories of place-value conceptions.

Miura and her colleagues (Miura & Okamoto, 1989; Miura et al., 1993), who

used some of Ross’s ideas, developed another scheme by which students’

understanding of place-value could be categorised. In both studies Miura et al.

investigated children’s perceptions of base-ten numbers via tasks that focussed on

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how the children represented multidigit numbers using base-ten materials. Results in

the two studies were used to categorise participants’ “cognitive representation of

number” as belonging to one of three categories. Like Ross’s research, Miura et al.’s

research has been widely reported by other authors, and there is apparently wide

support for her ideas about the effects of number names in European and Asian

languages on children’s conceptions of numbers (though see Saxton & Towse, 1998,

for a critique of Miura et al.’s findings). However, Miura et al.’s findings are open to

some of the same criticisms directed at Ross’s findings in the previous paragraph,

and again results in this study did not reproduce Miura et al.’s discrete categories of

place-value conceptions. Via the use of a narrow range of tasks and a limited set of

categories with which to describe student understanding of base-ten numbers, Miura

et al. claimed to have identified significant differences that existed between the

thinking of U.S. and other (principally Japanese) children with regards to numbers.

There is no doubt that differences in the thinking of children of different nationalities

and backgrounds do exist, and questions of effects of culture and language on

children’s learning of mathematics are worthy of investigation. However, it is

entirely possible, and based on this study seems quite likely, that the thinking

investigated in such studies is far more complex and less tidy than Miura et al.’s

three categories of student thinking would suggest.

Cobb & Wheatley’s three levels of children’s ideas about ten.

Cobb and Wheatley wrote an influential paper (1988) describing in some

detail the conceptions of ten held by young children. Their research is particularly

useful in pointing out the difference between children thinking of ten as a collection

of 10 single items, ten as a single unit, and ten as a collection of 10 that can be

counted as an item. Similarly to the research by Ross and Miura reviewed above,

Cobb and Wheatley also claimed to have categorised participants in their study

according to their performance on certain number tasks: “On the basis of their

performance on the counting-by ones, thinking strategy, and subtraction tasks, the

fourteen children were placed at three levels with respect to their addition and

subtraction concepts” (p. 10). Space does not permit a lengthy discussion of Cobb

and Wheatley’s research. In brief, however, it appears that the small sample size in

their study gives cause for questioning the descriptions of identifying characteristics

of categories, which seem unnecessarily rigid and even somewhat arbitrary. In fact,

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in several places the authors “hedged” over descriptions of a participant’s responses,

claiming that variations in participant responses did not invalidate the authors’

classification of the participant’s place-value understanding. Cobb and Wheatley’s

research, like that of S. H. Ross (1990) and Miura (Miura & Okamoto, 1989; Miura

et al., 1993) discussed earlier, has made a valuable contribution to the place-value

literature. However, like Ross and Miura et al., Cobb and Wheatley claimed to have

developed a scheme by which the place-value thinking by children in general can be

categorised. Results of this present study do not support such claims, implying

instead that children’s place-value thinking often defies researchers’ efforts to place

it in a stage- or level-based scheme.

Fuson et al.’s six conceptual structures.

The researcher whose work on place-value understanding most closely agrees

with the findings of this study is Fuson. Her work includes detailed analysis of the

base-ten numeration system and the necessary skills needed for children to learn to

use it proficiently (Fuson, 1990a, 1990b, 1992), as well as research into children’s

thinking in a variety of number tasks (Fuson & Briars, 1990, Fuson et al., 1992).

Fuson and her colleagues (Fuson et al., 1997) proposed a model of six conceptual

structures used by children, including five more or less accurate conceptions and the

incorrect “concatenated single-digit conception,” or face-value construct. However,

unlike other authors Fuson et al. (1997) did not try to fit the six conceptual structures

into a scheme by which a child may be categorised, or a stage model purporting to

show how each child’s thinking develops over time. Instead, the authors noted that

children may hold more than one conception at one time and that the conceptions are

used in ways that depend on the child’s background and the particular situation in

which they are accessed:

Children who have more than one multidigit conception may use different conceptions in different situations. . . . Furthermore, not all children construct all conceptions; these constructions depend on the conceptual supports experienced by individual children in their classroom and outside of school. Therefore, children’s multiunit conceptions definitely do not conform to a stage model [italics added]. (p. 143)

The findings of this study agree with the above statement by Fuson et al., in

(a) finding that individual students use a variety of number ideas rather than one

main idea, and (b) rejecting the idea that students move through various clearly-

defined stages or levels.

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5.2.4 Face-value Interpretations of Symbols

As discussed in the previous section, results in this study give only qualified

support for many of the conceptual structures identified in the literature search

carried out before this study, described in section 2.4.2. One conceptual structure that

is especially common in the literature is the “face-value construct” (e.g., S. H. Ross,

1989), or “concatenated single digits” conceptual structure (Fuson & Briars, 1990;

Fuson et al., 1997). There is evidence in data collected in this study for this

conceptual structure; however, this author believes that there are pertinent aspects of

this construct that have not been identified in previous literature. Firstly, descriptions

in the research literature of apparent evidence for the face-value construct do not

agree entirely with findings in this study, especially in light of participants’ responses

to the digit correspondence tasks in the interviews. S. H. Ross (1989, 1990) described

a five-stage model of children’s interpretations of two-digit numerals. In this thesis, a

four-category model is proposed to describe participants’ understanding of similar

two-digit symbols (section 4.5). Data from the two studies are compared in the

following subsection.

Comparison with Ross’s digit correspondence test data.

S. H. Ross (1989) reported the data from one particular task given to Grade 3

participants, upon which this researcher based Question 7 in each interview. In each

study, the researcher asked participants to count some sticks (25 in the case of Ross’s

study; in the present study, 24 in Interview 1 and 37 in Interview 2), and then to write

the symbol for the number. The researcher then asked the participants to say which

sticks corresponded with each of the two digits. Since participants in the two studies

were of similar age and school experience, it is valid to compare the reported results

of the two studies (Table 5.1). Such a comparison leads to three conclusions. First, it

appears from descriptions given by each researcher that the behaviours observed

were broadly similar; second, each researcher identified a category of behaviour not

mentioned by the other; and third, researchers applied different interpretations to the

results from the two studies. These conclusions are discussed in the following

paragraphs.

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TABLE 5.1. Comparison of Results of Digit Correspondence Tasks Between This Study and Ross (1989)

This study (Question 7)a Ross’s (1989) studyb

Cate-gory

Interpretation of digits in “24”

% (N = 16)

Interpretation of digits in “25”

% (N = 60)

I “2” meant two sticks, “4” meant four sticks. 38 “2” meant two sticks, “5”

meant five sticks. 13

Invented numerical meanings: e.g., that 5 meant half of ten.

23

II Individual digits had nothing to do with how many sticks were in the collection.

16 Individual digits had nothing to do with how many sticks were in the collection.

20

III “4” represented four sticks, “2” represented 20 sticks. 22 “5” represented five sticks,

“2” represented 20 sticks. 43

IV “4” represented four sticks, “2” represented 20 sticks, “2” meant “2 tens.”

25

Note. Results on the same row represent similar descriptions of students’ interpretations of digits from the two studies. Blank cells indicate that no equivalent category was described matching the category opposite in the other study. aResults from interview Question 7 are quoted, as this task matches the one used by Ross. See Table 4.13 for a summary of results from digit correspondence tasks for each participant. bFrom S. H. Ross, 1989, Parts, wholes and place value: A developmental view. Arithmetic Teacher, 36, p. 48.

S. H. Ross (1989) identified a number of behaviours that broadly match

observations made in this study, as shown in Table 5.1. At the lowest levels of

performance, both Ross and this author identified participants who gave face-value

interpretations of digits. At higher levels of performance, both researchers observed

participants explaining the values represented by each digit in terms of tens and ones

language. In between the two extremes, there were participants who did not accept

face-value interpretations of symbols, but who also did not explain digit

correspondence in terms of groups of ten.

S. H. Ross (1989) and this author each included a category of response not

identified by the other. Firstly, Ross distinguished between participants who gave a

straight-forward face-value interpretation of the digits and participants who

responded with “invented numerical meanings, such as that the 5 meant ‘half of ten,’

that the 5 meant that groups contained 5 sticks, or that the 2 meant ‘count by twos’”

(p. 48). It is apparent that the “invented numerical meanings” category does not

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exclude a face-value interpretation of the digits. If Ross’s lowest two categories

(face-value and invented meanings) are added, their combined incidence (36%) is

close to each incidence of face-value interpretations in this study (44% and 31% at

Interviews 1 and 2, respectively). Secondly, this author distinguished between

responses explaining the value of the tens digit in terms of the name of the multiple

of ten (20 or 30), and responses referring explicitly to the number of tens (two tens or

three tens). It appears that Ross’s highest category of participant response might

include both of this author’s two highest categories, if Ross did not make the

distinction described in the previous sentence. If the assumptions described in this

paragraph are accepted, this then adds support to the claim that the two studies have

identified similar patterns of response. However, Ross’s and this author’s

interpretations of these responses differ markedly, as explained in the following

subsection.

Differing interpretations of face-value responses.

Interpretations of children’s face-value behaviour made by this author differ

from S. H. Ross’s (1989) interpretations of similar behaviour (see Appendix T), for

two reasons. First, there is evidence of an inconsistency in Ross’s interpretations of

children’s responses indicating that a two-digit symbol represents the entire referent

set, without referents for each digit. Second, this study does not support that idea that

individual children possess stable mental models for numbers that can be used to

describe their number understanding generally, as explained in section 5.2.3.

S. H. Ross (1989) and this author give different interpretations for the face-

value responses made by participants in their respective studies. The first column of

the table in Appendix T includes descriptions of the four categories of digit

correspondence task response from section 4.5. Adjacent to most descriptions is an

excerpt from Ross’s paper that apparently describes similar behaviour. When the two

columns are compared, a striking difference emerges between the categories defined

by the two researchers. In particular, Ross and this author disagree regarding

participants’ responses indicating a belief that the entire collection of objects was

represented by the entire symbol, but that each digit did not have its own referents.

Ross defined this type of response as being at the lowest level of understanding of

digits, and in particular, below the face value stage. In analysis of participants’

responses in this thesis, this type of response was categorised as Category II, above

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Category I, face value. The author considers a Category II response to demonstrate

superior understanding of the digits to a straight-forward face-value interpretation of

digits, for three reasons: (a) it rejects the incorrect face-value interpretation, (b) it

explains correctly that the entire symbol represents the entire collection of objects,

leaving none out, and (c) participants exhibiting this category of response generally

demonstrated superior performance on other interview tasks than participants who

answered with face-value interpretations (see Table 4.13). If indeed the two

categories (Stage 1 / Category II) essentially describe the same behaviour, then it

appears that this study has revealed an aspect to children’s thinking about digit

correspondence that has not been widely reported before. This aspect is that, as

described in section 4.5, some participants were not comfortable with a face-value

interpretation of the digits, and appeared to operate at a higher level of thinking about

digit correspondence in rejecting face-value interpretations. This was despite the fact

that they did not fully understand the grouped aspect of base-ten numbers and were

not able to explain numbers in terms of place value.

Despite the similarities in data collected in S. H. Ross’s (1989) study and the

present study, one particular aspect of the data, already alluded to, points to a

difference in interpretation of children’s place-value understanding. Whereas Ross

categorised the children themselves, in this study it is the children’s responses that

are classed as belonging to a particular category. Furthermore, there is compelling

evidence in this study that such categories were not fixed, but altered with the

particular context in which the response arose. In short, results of this thesis did not

demonstrate even one participant who held a consistent belief that each digit

represents only its face value. The participant who was the most likely candidate for

possessing a face-value construct for multidigit numbers is Jeremy (l/b). He had one

of the lowest scores at both interviews, and he was observed to use a face-value

interpretation of symbols at least eight times during the two interviews, in every

instance unprompted by the researcher. However, despite this pattern of responses,

he clearly rejected a face-value interpretation of symbols on several occasions. For

example, the following excerpt shows Jeremy’s response to the question asking

which is bigger, 183 or 138:

Jeremy: That one [‘183’].

Interviewer: Okay, what is that number?

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Jeremy: one eighty ... 183.

Interviewer: Uh-huh, and why is that one bigger?

Jeremy: Because it’s got a ‘1’ and it’s a ‘8.’

Interviewer: What about the other number?

Jeremy: It’s got a ‘1’ and a ‘3.’

Interviewer: Okay why does this one ... see this has got an ‘8’ as well as that one and it’s

got a ‘3’ like that one. Why is that one bigger than that one if it has the same

numbers in it? — Is there a chance these two are the same, do you think?

Because they’ve got the same numbers … or is this [‘183’] going to be

bigger?

Jeremy: That one [‘183’] will be bigger.

Interviewer: — If you were counting would … do you know which one of these numbers

you’d come to first?

Jeremy: That one [‘138’].

Interviewer: Uh-huh. Do you know why you’d come to that one first?

Jeremy: Because it’s down lower. (I1, Qu. 6b)

Considering Jeremy’s comments about the face values of single digits (e.g.,

“It’s got a ‘1’ and a ‘3’”), rather than the values represented by the digits, it might be

inferred that he was using a face-value interpretation of the two symbols. However, if

Jeremy believed that digits only represented their face value, he would not have

rejected the researcher’s counter-suggestion that 183 and 138 are equal because they

have the same digits. Even though Jeremy did not know what the two numbers were,

and could not read them, he still believed that the values they represented were

different, and that the order of the digits indicated which one was bigger. Clearly

Jeremy’s thinking about these two symbols cannot be summed up with the label

“face-value construct,” even though at other times he clearly demonstrated a face-

value interpretation of digits; this observation is repeated many times in the interview

transcripts.

This author contends that a new category is needed to describe children’s

numerical thinking that may help to interpret response patterns such as those

described in this section. The following section contains a description of such a

category, the independent-place construct.

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5.3 Independent-Place Construct Results of this study indicate the presence of a previously unreported

conceptual structure for numbers in the minds of some participants, here named the

independent-place construct. Discussion of the independent-place construct in this

section is arranged in the following subsections: a description and definition (5.3.1),

comparison between the independent-place construct and the face-value construct

(5.3.2), evidence for the independent-place construct in this study (5.3.3) and in the

research literature (5.3.4), and the effects of the independent-place construct on

written computation (5.3.5) and on place-value tasks (5.3.6). Implications of the

independent-place construct for teaching are discussed later in the final chapter, in

section 6.3.3.

5.3.1 Description & Definition of the Independent-Place Construct

As explained later in this section, the independent-place construct includes

aspects of face-value interpretations of symbols and the use of materials as “column

counters.” Use of this construct is indicated by participants’ actions indicating that

they regarded individual places in multidigit numbers, block representations, or both,

to be separate and unrelated. In other words, they did not see any link between

“hundreds,” “tens,” and “ones” places, but regarded them as independent categories

of quantity with separate names, separate digits, and separate block representations.

In doing so, though participants were able to complete certain simple tasks, it appears

that they were not able to appreciate the value represented by an entire multidigit

symbolic or block representation in any meaningful way. For the reasons discussed

in following sections, the author contends that the independent-place construct is

essentially different to face-value interpretation of symbols, and provides a better

explanation of patterns of computation behaviour previously labelled by other

authors as examples of the face-value construct.

The following definition for the independent-place construct is used for the

subsequent discussion:

The independent-place construct occurs when a student treats symbols or

concrete materials representing values in one place in the base-ten numeration

system as separate from other places, and does not attempt to relate one place to

another.

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5.3.2 Comparison of the Independent-Place Construct and the Face-Value Construct

The independent-place construct is proposed here as a means of explaining

apparent anomalies in this study’s data when considered in the light of previously

published research in the field. As discussed earlier, although certain responses by

participants indicated that they believed that individual digits in multidigit numbers

represented only their face values, data in this study do not support the idea that

participants held these beliefs consistently as they answered place-value questions.

The author asserts that much behaviour previously identified as revealing a face-

value construct is better understood as demonstrating a perception that digits are

independent of each other.

The independent-place construct proposed here and the face-value construct

widely reported in the literature are similar and yet distinct from each other.

Although both constructs have the effect of leading a student to ignore the values

represented by individual symbols, the essential natures of the two constructs are

quite different. Whereas a person possessing a face-value construct denies that each

digit in a multidigit number represents anything other than its face value, a person

with the independent-place construct considers each place separately from other

places, while taking no account of what each digit actually represents. Furthermore,

whereas the presence of a face-value construct indicates a serious misunderstanding

of the base-ten numeration system, and rightly attracts attention from teachers and

researchers, the independent-place construct is consistent with computation practices

that ignore actual values represented by digits for the sake of efficiency.

5.3.3 Evidence for the Independent-Place Construct in This Study

Evidence of an independent-place construct is found in several patterns of

participants’ responses reported in chapter 4, including (a) trading 1-for-1, (b)

choosing incorrect blocks, (c) number naming errors, (d) use of place names merely

as labels, (e) errors made in writing numerical symbols, and (f) a reluctance by some

participants to consider non-canonical arrangements of blocks.

(a) Trading blocks one-for-one.

Firstly, as described in section 4.6.2, several participants proposed to trade a

block of one size for one block of another; that is, to trade a ten-block for a single

one or a hundred-block for a single ten. This idea may be an example of believing

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that blocks are merely counters, and that each has a “value” of one, no matter what

size it is; it is difficult to understand how children could think that a ten and a one

could be swapped unless they perceived each to be merely a single block.

(b) Choosing incorrect blocks to represent numbers.

The error of choosing the “wrong” blocks to represent places, was shown by

Kelly (l/c), as described in section 4.6.4. Kelly used 2 ones blocks and 8 tens to show

28, then used 1 hundred, 3 ones and 4 tens to show 134. It could be argued that

Kelly’s response to this question indicated a face-value interpretation of digits, since

she did not realise that the quantity represented by the tens digit was in groups of 10

ones, and was happy to use blocks that were ten times the size of her “tens blocks” to

represent ones. However, two aspects of this incident make it appear that this was not

the case: (a) When asked to show each number in another way she retained the same

block-value assignments, merely changing their relative positions; and (b) the

numbers were given to her verbally, so there were no written symbols for her to

interpret.

In light of the current discussion, Kelly’s response can be interpreted as

demonstrating an independent-place construct: In Kelly’s thinking there was

apparently no relationship between the ones and tens places, with regards to the size

of the blocks representing digits in each place. Furthermore, Kelly’s consistency in

using small cubes (“ones”) to represent tens digits and long blocks (“tens”) to

represent ones seems to indicate that she did not believe that the blocks could be

applied arbitrarily to any place, which presumably would have been the case with a

true face-value construct. On the contrary, when asked by the interviewer if there

was any other way to represent each number, Kelly consistently used the same

blocks to represent digits in both the ones and tens places four times over the two

questions, changing only the spatial arrangement of blocks of exactly the same sizes.

(c) Errors naming non-canonical block arrangements.

The third evidence for the independent-place construct is provided by certain

examples of participants mis-naming numbers represented by non-canonical block

arrangements. For example, some participants incorrectly stated the number

represented by a non-canonical collection of different blocks by applying a name to

each size of blocks in turn. In such incidents participants “read” the block

arrangement using an “x-ty y” form, where x is the number of tens blocks, and y the

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number of ones. For example, as reported in section 4.7.1, Jeremy (l/b) read 8 tens

and 11 ones as “eighty-eleven,” and Yvonne (h/c) read 5 tens and 10 ones as “fifty-

ten.” Fuson et al. (1992) commented on such non-standard number names that “one

can easily say more than nine of a given multiunit and such constructions have a

quantitative meaning even though they are not in standard form” (p. 42). Despite this

point, since such constructions are not standard English number names, it seems

likely that, if asked, the children themselves would regard these number names as

incorrect. If this is so, then it appears that the participants were merely applying a

linguistic procedure that is successful with canonical block arrangements, of naming

the tens and then the ones, without taking account of the meaning of the resulting

number name.

(d) Use of place names merely as labels.

The fourth type of evidence for the independent-place construct in the study

is participants’ use of place names “hundred,” “ten,” and “one” with no apparent

notice paid to the numerical basis for the names. For example, section 4.6.4 includes

mention of an incident in which Clive attempted to explain to Jeremy (both l/b) why

51 was greater than 39. It is interesting that Clive, who clearly knew that 51 was

greater than 39 because of their respective tens digits, could not explain why the tens

should be regarded as having greater value than the ones. When asked by the

researcher he replied that “the tens are first on the tens mat, ten sheet, so . . .” but he

was unable to say why the digits’ positions on a place-value chart made a difference

to the values represented. Clive was correct in noting the relative positions of tens

and ones, both on place-value charts and in written symbols. However, he apparently

did not make a connection between the name “ten” and the idea of 10 ones (see

Fuson et al., 1992; NCTM, 2000).

(e) Errors in writing numerical symbols.

The fifth type of evidence in the study for the independent-place construct

relates to certain attempts by participants to write numerical symbols for multidigit

numbers, either represented by blocks or spoken verbally as a number name. For

example, section 4.6.3 includes a description of Amanda (h/b) having some difficulty

in writing the symbol for 204; writing ‘24,’ then ‘240’ before writing the correct

symbol. It is possible that Amanda had become used to writing a single digit for each

part of a two-digit number’s name, and tried to use the same process with three-digit

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numbers. Such a method would work with all two-digit numbers except those ending

with zero. Thus, fifty-six can be written by recording a digit for the fifty [5] and

another for the six [6]. It would also work with many three-digit numbers, such as

two [2] hundred and ninety [9]-eight [8]. However, the method fails if there is a zero

in the tens or ones place: two [2] hundred and four [4] has only two place number

words, resulting in just two digits if the “each-number-word-is-a-digit” method is

used. Nerida (l/b) used a variation of this method, writing ‘617’ when attempting to

write the symbol for the number represented by 6 tens and 17 ones. She then read the

symbol and stated that the blocks represented “six hundred and seventeen.” This

incorporates the idea of independent places, since writing ‘617’ for 6 tens and 17

ones involves concatenating the symbols for the two subsets of like-sized blocks

without regard for their respective values.

(f) Reluctance to consider non-canonical block arrangements.

As already mentioned in this section, the independent-place construct appears

to be linked with various behaviours associated with non-canonical block

arrangements. Further support for this is provided by observations of participants

who were apparently reluctant to consider non-canonical arrangements of blocks

(section 4.7.4). When asked to trade a block for 10 of the next place, at least two

participants, Simone (h/b) and Michelle (l/b), attempted to keep traded blocks

separate from non-traded blocks so that a canonical arrangement could be made as

soon as an answer was recorded. When considered alongside other difficulties

participants had with non-canonical arrangements, the desire of these participants to

revert to canonical block arrangements is consistent with an understanding of

multidigit numbers that relies on counting blocks and naming and recording numbers

in each column separately.

5.3.4 Evidence of the Independent-Place Construct in the Literature

Observations made by several other researchers lend support to the proposed

independent-place construct. Evidence is given in this section of reports of students

constructing tens as abstract singletons, calculating answers column by column,

choosing incorrect blocks to represent places, and choosing misleading independent-

place materials to represent two-digit numbers.

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(a) Tens as “abstract singletons”

Cobb and Wheatley (1988) made an important discovery in their study of

children’s abilities to manage a variety of tasks involving tens and ones. The authors

found that several children in their study perceived of tens and ones as “abstract

singletons” and “abstract units,” respectively. These children were evidently unable

to perceive of a ten as comprising a collection of ten ones, but instead saw it only as

an abstract, indivisible unit that could be counted separately from ones units. This

finding gives clear support to the idea that some children operate on numbers using

an independent-place construct. The independent-place construct encompasses this

and other behaviours, as described in this section, and thus is considered to include

the abstract singleton and abstract unit constructs described by Cobb and Wheatley.

(b) Column-by-column computation.

A number of authors (Cobb & Wheatley, 1988; Fuson & Briars, 1990; Fuson

et al., 1997; Nagel & Swingen, 1998) have noted students adding or subtracting

numbers by considering numbers in each column separately. Cobb and Wheatley

(1988) asked second-grade children to add pairs of numbers such as 16 and 9,

presented either horizontally or vertically. The authors commented that “the children

seemed to operate in two separate contexts: (a) pragmatic, relational problem solving

and (b) academic, codified school arithmetic” (p. 1). When researchers presented

numbers horizontally, the children typically used a counting-on procedure that

necessarily incorporated some notion of the sizes of the two numbers. When the

same numbers were presented in a conventional vertical algorithm format, several of

the same children made concatenation errors of the type described in an earlier

paragraph, with several students writing that 16 + 9 equalled 115. Fuson and Briars

(1990) also noted arithmetic performance of this type, and referred to it as addition

“column by column: . . . The sum of each column was written below that column

even when the sum was a two-digit number (e.g., 28 + 36 = 514)” (p. 189). Fuson et

al. (1997) commented on this phenomenon:

The vertical presentation elicits an orientation of vertical slots on the multidigit numbers that partitions these numbers into single digits. The physical appearance of the written multidigit marks as single digits and the nonintuitive use of relative left-right position as a signifier may combine to seduce children to use a concatenated single-digit conceptual structure even if they have a more meaningful conception available. (p. 142)

219

Further evidence of independent-place thinking is provided by Fuson et al.

(1992), who investigated the effects of using base-ten blocks with groups of second-

grade children to investigate written symbols, number names, and base-ten blocks.

Researchers presented groups of participants with four-digit addition problems, and

asked them to solve the problems using base-ten blocks. The authors noted that

every group immediately added the like multiunit blocks. After making each addend with blocks, they . . . pushed the addend blocks of each kind together and counted all the blocks of a given kind. . . . Evidently the visually salient collectible multiunits in the blocks supported the correct definition of multiunit addition as adding like multiunits. . . . All groups also added two four-digit written marks addends by adding together the marks written in the same relative positions. (p. 76)

Fuson et al. (1992) commented that it was difficult to tell if children linked

the idea of adding like digits with the place names, or if their actions were only

“based on a procedural rule and did not imply understanding of adding like

multiunits” (p. 76). This author suggests that at least some of the children may have

had an independent-place construct that enabled them to correctly add column

amounts separately even though they had not been formally taught procedures for

adding four-digit numbers prior to the study. Support for this suggestion is found in

Fuson et al.’s (1992) comments that some children found difficulty when attempting

to add numbers when trading was needed.

(c) Students choosing the “wrong” blocks.

Another example of apparent independent-place thinking in Fuson et al.’s

(1992) paper is that when asked to add pairs of three-digit numbers, many

participants chose to use incorrect blocks, starting with the largest available block,

the thousands block, to represent the first digit, the hundreds. By starting from the

left-most digit and the largest block size, the children were able to make a

representation for numbers that was mathematically sound, providing that the “ten-

block” was given a value of 1. This also is consistent with independent-place

thinking, as each place is mapped to a block size, without regard for the “absolute,”

“correct” value represented by each block.

(d) Students choosing independent-place materials.

An unintended illustration of the effects of certain representational materials

on students’ actions is provided by a recent report of place-value research. Saxton

and Towse (1998) designed their study to test the central claim by Miura et al. (1993)

220

that a child’s spoken language affects the way the child represents numbers using

base-ten materials. Saxton and Towse introduced an important change to the method

used in the earlier research by Miura et al., in making a critical alteration to the

representational materials provided to participants. Whereas Miura et al. provided

participants with standard base-ten material to represent numbers, Saxton and Towse

asked 6- and 7-year-old children to represent two-digit numbers using orange and

green cubes, arbitrarily assigned to represent tens and ones digits. Saxton and Towse

justified this change to the test procedure used by Miura et al. by arguing that base-

ten blocks “concretised” the abstract relationship between tens and ones material:

In principle, a child could represent a multi-digit number with blocks of ten units without any clear understanding of place value, simply by counting the component units in each block. The use of single cubes to represent tens avoids this possibility, and moreover, ensures that block counting is not prompted by the increased perceptual salience of large blocks over single cubes. (p. 69)

In view of the large volume of literature on children’s faulty face-value

conceptions of number, and the present discussion of the independent-place

construct, Saxton and Towse’s (1998) argument for using cubes of the same size but

different colours to represent tens and ones seems particularly problematic. Rather

than forcing their participants to focus on the abstract relationship between tens and

ones, as they intended, the researchers may instead have prompted the participants to

use face-value or independent-place interpretations of the digits to represent the

numbers asked of them. Instead of “avoiding [the] possibility” of children counting

units in ten-blocks, this method is likely to promote face-value or independent-place

ideas about what each digit represents. Tellingly, when the researchers did not model

the use of “tens” and “ones” cubes, most participants (over 90% of some cohorts)

used only ones cubes to represent two-digit numbers. However, when the researchers

demonstrated how cubes of two colours could be used to represent numbers, the use

of both “tens” and “ones” increased dramatically. It seems quite possible that

children to whom the researcher modelled the use of two arbitrary colours to

represent tens and ones were thereby encouraged to use an entirely false and

misleading face-value construct or independent-place construct. Furthermore, using

such materials, the responses of children who possessed good understanding of the

base-ten numeration system would be completely indistinguishable from responses

of children who thought either that each digit represented its face value, or that the

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tens and ones places were independent and could be represented separately by

“unitary” material.

5.3.5 Written Computation and the Independent-Place Construct

There is an apparent contradiction between the teaching of place-value

concepts and the practice of competent users of written or mental algorithms. On the

one hand, students are taught to recognise the different values that digits assume

according to where they are found in a number; on the other hand, efficient use of

computational algorithms requires the user to ignore actual values represented by

individual digits and to focus instead on their face values. These differing

conceptualisations make recognising the conceptual structure possessed by a student

who is carrying out written computation very difficult, especially if the computation

involves no regrouping. For example, a student adding 47 and 31 may arrive at the

correct answer merely by adding pairs of digits in each place, without any regard for

the values represented by the tens digits. A student doing so may have a good

understanding of place-value concepts, or may be operating from an independent-

place construct; an observer would be unable to distinguish one from the other

without further questioning. On the other hand, evidence of faulty or limited

conceptions of number may emerge in examples requiring regrouping, as illustrated

by earlier-mentioned examples reported by Cobb and Wheatley (1988), Fuson and

Briars (1990), and Nagel and Swingen (1998).

It is relevant to point out that regarding symbols in written algorithms as only

composed of single digits, and adopting a face-value interpretation of them when

carrying out the computation, is not inherently incorrect, and may have merit when

compared with counting approaches. Clearly it is more accurate to use counting on

to arrive at an answer which, even with minor counting errors, is close to the correct

sum, than to add columns separately and arrive at an answer that could be several

times too large (such as “16 + 9 = 115”). However, there are other factors to consider

before judging an independent-place method too harshly. Counting approaches may

give results that are more or less accurate, but they are prone to errors, inefficient,

and cumbersome for large numbers. Independent-place methods, on the other hand,

take advantage of the power of the base-ten numeration system to represent

quantities with a small number of written digits by considering each place in turn. A

child who says that the sum of 16 and 9 is 115 needs help to see why that is not a

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reasonable answer, and to interpret, add, and record the partial sums 1 and 15

correctly. However, the method of separating places is quite sound, and is the basis

of conventional computation procedures, providing that the separated partial sums

are correctly interpreted.

5.3.6 Place-Value Tasks and the Independent-Place Construct

Like the face-value construct, the independent-place construct can be difficult

to recognise in responses to many place-value tasks. Firstly, the independent-place

construct does not preclude the use of terms such as “tens” and “ones,” if they are

used only to name places and not to refer to values represented by objects or symbols

in places. As mentioned by other authors (e.g., S. H. Ross, 1990), such terms can be

perceived merely as labels, with no particular meaning with respect to value. As C.

Thompson (1990) pointed out,

having students mechanically put numerals in columns is of no value if the complex and difficult grouping concepts have not already been constructed by the students. There is little doubt that young children can count the number of sets of ten sticks and write that number in a box labeled TENS and similarly count single sticks and write that number in a box labeled ONES. But such activity does not help students construct the relationships between tens and ones or the concept of representing larger quantities by using groups of ten and singles. (p. 90)

Secondly, the presence of the independent-place construct does not

necessarily cause students to arrive at incorrect answers, depending on the nature of

the questions asked (see also Reys & Yang, 1998). If tasks given to students rely

only on the student being able to link each place with a block size or with a set of

number names, or both, then students can consider places to be independent of each

other with no detrimental effect on task performance. A student may (a) consider

separately each digit in a symbol, (b) consider separately each block size in a block

arrangement, (c) use the intuitive mapping that exists between digits and subsets of

like-sized blocks, or (d) state a number name by considering each place separately,

and will often receive correct answers as a result. The following comment by S. H.

Ross (1989) regarding the face-value construct applies equally to the independent-

place construct:

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Students who use a … face-value interpretation of digits succeed on a … [wide] variety of tasks, including many that use manipulative materials. In many instructional tasks students are asked to make correspondences between digits and materials. If a collection is already grouped into a standard place-value partitioning of tens and ones, a student who is asked to make correspondences for the digits in 52, for example, need only look for “five of something and two of something else.” (p. 50)

To this observation, we may add that if asked to represent a number using

base-ten blocks, a student need only choose the “right” block size to represent each

digit to be considered correct. If there are only two sizes of blocks to choose from,

and two places to represent, the only remaining problem is to know which block

represents which place. Simple training to associate two block sizes with labels “ten”

and “one,” and teaching of the number names for multiples of ten, would be

sufficient to ensure that many students could correctly show blocks to represent a

number while having no real idea of the values represented by the digits or the

blocks. An example from section 4.6.4 illustrates the importance of the point that

correct task behaviour does not necessarily indicate correct numerical understanding.

In her first interview Kelly (l/c) was asked to show a two-digit and a three-digit

number with blocks. In response to both questions she consistently used one-blocks

to represent the tens, and ten-blocks to represent the ones. By making the mistakes

that she did Kelly drew attention to her faulty ideas. However, the fact that other

participants generally chose conventional block sizes for each digit to represent

multidigit numbers does not rule out the possibility that they may have had similar

misconceptions about digit referents to those apparently demonstrated by Kelly. The

task given by Saxton and Towse (1998), described earlier, illustrates this point:

Students were asked to represent two-digit numbers using green and orange cubes to

stand for tens and ones digits, according to the researchers’ arbitrary assignment of

each colour to represent a place.

5.4 Participants’ Construction of Meaning One prominent feature of the data in this study, mentioned several times

previously, is the noticeable changeability of participants’ responses to questions,

both in interviews and in teaching sessions. This general observation has led the

author to the view that, in the majority of cases, the participants’ number conceptions

were not characterised by fixed conceptual structures. On the contrary, the

participants appeared often to be weighing up the evidence before them and making

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the best sense of it they could, altering their answers as and when inconsistencies

appeared between their responses and other information. This sense-making

character of participants’ responses is believed to show that for many participants

their conceptions of numbers were still in a “construction zone,” subject to influence

by outside information such as visual cues provided by representational materials or

the interviewer’s questions.

5.4.1 ‘Organic’ Understanding

Results of this study show a picture of children whose ideas about numbers,

symbols, and representational materials fluctuated with the introduction of further

data to challenge those ideas. The understanding of many participants could perhaps

best be described as organic, rather than as belonging to a particular fixed category:

Participants’ understanding appeared often to be in a developing state, subject to

various influences in the surrounding “environment.”

One particular type of question that elicited frequent changes of opinion was

the digit correspondence questions. As demonstrated in section 4.5.2 and elsewhere,

several participants who believed that each digit represents only its face value

nevertheless understood that there was a contradiction between their explanation of

digit referents and the evidence of the objects before them, and apparently tried to

resolve the contradiction by generating other explanations. This phenomenon is

captured in response Category II in digit correspondence tasks; participants giving

these responses rejected face-value interpretations of symbols in favour of a different

explanation, that individual digits had no meaning in the multidigit symbol, but

together represented the whole collection of objects. S. H. Ross (1989) believed that

this type of response represented the lowest level of understanding of digits, showing

that the child had little idea at all of what the symbol meant. However, as explained

earlier in this chapter, it is believed by this author that, to the contrary, such

responses show a willingness on the part of the child to forgo the immediate

suggestion that each digit represents just what it would represent if it were on its

own, and instead to find some other explanation for two symbols each of small value

representing a comparatively large collection of objects.

Even in the case of participants holding apparent face-value interpretations of

the symbols, there was evidence of construction of meaning about the symbols.

Section 4.4.3 includes a series of statements made in the interviews by 9 participants

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who gave a face-value interpretation of digits, explaining the reason why the

remaining sticks “left out” of their face-value interpretation of digits apparently had

no written representation. It is clear immediately on watching videotapes or reading

transcripts of these responses that the participants did not appear to be troubled by

the question. In most cases participants stated the answers without hesitation,

apparently indicating that they had already decided on an interpretation of the digits

before being asked by the researcher. In fact, there was not one participant holding a

face-value interpretation of the digits in Question 7 who did not offer an explanation

for the remaining sticks. The second thing that is surprising to an adult observer is

that the illogicality of their view either did not occur to the children, or at least did

not trouble them. They accepted a situation in which the symbol ‘24’ represented 24

sticks, and simultaneously the ‘2’ represented two sticks and the ‘4’ represented four

sticks, with 18 sticks not represented by any symbol at all.

The acceptance by participants of two mutually exclusive propositions is a

characteristic of several responses made by participants that, again, supports the idea

that the participants were actively trying to make sense of a situation about which

they did not have fully-formed opinions. Section 4.4.5 includes a transcript excerpt

showing Jeremy (l/b) attempting to explain which is larger, 27 or 42. This account

typifies several transcripts showing participants weighing up various pieces of

information in justifying their responses. Jeremy did not merely accept every

suggestion made or implied by the researcher, but considered each one in turn. When

the researcher finally convinced him that 42 was later in the counting sequence,

Jeremy mentioned again the sizes of the digits and supported his answer by referring

to the relative order of the digits in the two symbols. He was also able to defend his

revised belief about the two numbers against another incorrect face-value suggestion

from the researcher about the larger ones digit in ‘27.’ This pattern of responses is

indicative of on-going development of number conceptions, not simply of a fixed,

incorrect face-value construct.

5.4.2 Participants’ “Invented” Answers

One feature of the construction of meaning evident in transcripts is that

participants often referred to ideas that they had evidently invented in order to answer

questions about numbers. This is shown in the discussion of Terry’s (l/c) explanation

about 27 and 42 in section 4.4.5 (the full transcript of which is in Appendix N).

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Terry’s idea that even numbers are larger than all odd numbers is a good example of

an invented response. No teacher would teach this idea, but it is conceivable that if a

teacher presented diagrams showing odd and even numbers in a certain way, a child

might arrive at Terry’s rule in the absence of further information to challenge it. It

appears that some of the participants had been taught about even and odd numbers

shortly before the study, as several participants referred to odd and even numbers in

the interviews, though there was no mention of these numbers in any question.

Some participants found they could not interpret three-digit numerical

symbols using their existing knowledge of two-digit symbols, leading to some

interesting ideas. Clive (l/b) and Daniel (h/c) both suggested that ‘138’ had 1 ten and

38 ones, and ‘183’ had 1 ten and 83 ones. Daniel clearly attempted to interpret these

symbols using his knowledge of two-digit symbols. He said that the ones column

was “round here somewhere,” indicating rather vaguely the tens and ones digits, and

seemed amused to find that it had “two numbers in it,” the ‘3’ and the ‘8’

(I1, Qu. 6b).

It is evident that the changing of opinions when responding to interview

questions was exhibited most often by low-achievement-level participants. It is to be

expected that low-achievement-level participants would have ideas about numbers

that are less fully developed than high-achievement-level participants. Thus, it may

be deduced that low-achievement-level changed their ideas about numbers more

often than high-achievement-level participants did. However, such an observation

may be misleading. As discussed earlier, apparently correct responses to

mathematics questions can obscure faulty understandings if the questioner does not

probe the reasoning behind responses. In cases in which participants responded

quickly with correct answers, the researcher often did not probe their thinking to any

great extent, assuming that their correct answer represented a sound understanding of

the topic. However, such quick, correct responses may hide very similar processes of

testing tentative theories that are taking place mentally, and therefore out of sight.

Thus, it is quite possible that high-achievement-level participants may also have been

engaged in the construction of understanding of numbers, and considering multiple

interpretations before giving their answers.

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5.4.3 Teaching, Learning, and Constructing Meaning

The idea that participants in this study were engaged in meaning construction

closely matches the literature on constructivism. Constructivist ideas focus on the

individual nature of understanding, and on the notion that each student constructs an

understanding of each concept that is unique to that student. There is no place in a

constructivist pedagogy for a teacher to try to give information to a student, because

it is not possible to transmit ideas directly from one person to another, or from

another source of information to a person. Instead, teachers are exhorted to

encourage each student to develop ideas personally, to allow students space to

develop unique understandings of each topic in the curriculum.

Evidence in this study’s data of participants apparently thinking actively

about numbers and what they mean, implies that ideas that teachers present to

students may not be received as the teachers intend. In fact, depending on how an

idea fits with a student’s already-existing concepts about numbers, the student may

interpret it in ways that the teacher could hardly imagine. This thesis includes many

examples of such unusual ideas held by children that an adult is unlikely to have

predicted. In some cases, these involved the simultaneous acceptance by participants

of contradictory or inconsistent beliefs as participants attempted to make sense of the

information available to them. There are clear implications in these data for how

teachers present mathematical information to students and how teachers ascertain

their students’ understandings of numbers. These points are mentioned again in

section 6.3.4.

5.5 Effects of Physical or Electronic Base-Ten Blocks on Place-Value Understanding

This study has explored a wide range of issues relating to children’s

understanding of numbers when using materials. In this section the effects of

physical or electronic base-ten blocks on how children represent numbers are

analysed in light of the study’s findings. Four aspects of the relevant results are

discussed in this section: (a) minor differences that were evident between the

learning that occurred among participants who used physical blocks and learning

among those who used electronic blocks; (b) the sensory impact of both types of

material; (c) the facility that each material offers to aid the development of links

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among blocks, symbols, and numbers; and (d) the support that each material provides

for the development of number concepts.

5.5.1 Differences in Learning of Participants Using Physical or Electronic Blocks

As reported in section 4.3, though there were clear differences between the

learning of high-achievement-level and low-achievement-level participants during

the course of the study, differences in learning that occurred among participants

using blocks and learning that occurred among participants using software were

minor. Based on performance on place-value tasks at the interviews before and after

the teaching sessions (Table 4.3), individual participants such as Simone and Nerida

did show improvement in their understanding of place-value concepts. However,

aggregate scores of the 4 groups show no differences in learning about place-value

that could be attributed to use of one material or the other. It must be pointed out that

the small scale of this study does not support strong claims of such differences,

which on the basis of the data here appear to be rather subtle. Similar studies

conducted with larger numbers of participants and over longer periods of time would

be needed to confirm initial ideas of differences between the use of physical and

electronic number models mentioned here.

5.5.2 Sensory Impact of Physical or Electronic Blocks

One apparently common view of teachers when it is suggested that software

could be used to take the place of base-ten blocks to teach place-value concepts is

that children using software would be somehow missing out because of a lack of

tactile contact with the medium. As noted by Clements and McMillen (1996)

“manipulatives are supposed to be good for students because they are concrete”

(p. 270). However, as pointed out by other authors (Hunting & Lamon, 1995; Perry

& Howard, 1994; P. W. Thompson, 1994) the mathematics is not contained in the

material, and so benefits from conventional blocks’ physical attributes may be more

imaginary than real.

Though students have no direct physical contact with computer-generated

blocks, they do have access to other sensory input that differs from that offered by

conventional blocks. First, there is the auditory input of the audio recordings of

number names used to read the numbers represented by blocks, the number

represented by the blocks of one size, and the numeral expander. As shown in section

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4.7.6, participants who used the software enjoyed using the audio capabilities of the

software to gain confirmation of their block representations. Furthermore, section

4.7.1 demonstrates that the audio recordings in the software acted as one source of

feedback available to users of the software. No such features are available to users of

blocks; other sources of auditory feedback such as a teacher, if available, must be

accessed instead.

The second source of different sensory input provided by the software is the

visual arrangements of blocks, coupled with counters, labels, number window, and so

on. Though the on-screen blocks appear to be quite similar to their physical

counterparts, there are several differences that the study data showed to be important.

One difference is the juxtaposition of several representations of a number

simultaneously. It has been mentioned several times that the blocks and the on-screen

numbers changed at virtually the same time, to provide a continually updated set of

parallel representations for numbers, in close proximity to each other. Users of the

software were able to take in visually the various representations for numbers with

little effort, and watch changes occur in all representations at the same time. The

other major visual difference between blocks and software was mentioned by

Clements and McMillen (1996), writing about computer manipulatives in general:

“[computer] representations may also be more manageable, ‘clean,’ flexible, and

extensible” (p. 272). There is no question that the computer representations of

numbers were much neater than representations made with physical blocks. As

mentioned in several places in the thesis, counting and handling errors with the

blocks were quite common. These errors mostly resulted from difficulties with

managing the material so that numbers and processes could be correctly represented.

Some participants were generally careful when handling blocks to check counted

arrangements to ensure that the correct quantities were put out. However, other

participants were less careful and made frequent handling errors that led to incorrect

answers. Errors made with computer blocks were less frequent, apparently because

of the feedback provided by column counters and the number window.

Related to the sensory impact of the computer blocks is the ease with which

they may be used to demonstrate numbers and numerical processes. Not only does

the software provide on-going feedback about the number of blocks displayed, it also

enables very rapid placement of blocks via clicks with the computer mouse. Each

click of the mouse on the appropriate button results in the placement of a block in a

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particular place. As the participants using the software became familiar with it, they

became adept at placing blocks very quickly. On occasions participants using the

software overshot the number they required, but they were able to assess this and

correct it without much delay. The ease of use of the software is particularly

noticeable when it comes to trading processes. The regrouping (10 for 1) and

decomposing (1 for 10) tools ensure that each trading action is done accurately and

quickly. In the case of trades performed with conventional blocks, mistakes were

quite frequent, particularly with low-achievement-level participants, and the

researcher had to step in to correct errors before they caused the participants further

difficulties.

5.5.3 How Numbers Are Represented by Physical or Electronic Blocks

As described in section 2.3, the base-ten numeration system has a number of

features with which primary school students need to become proficient. These

features include the place-value system underlying the written symbols, the system

of naming numbers, and the trading processes necessary for multidigit computation.

Students learning place-value concepts in their second or third year of schooling face

several difficulties, for a number of reasons: (a) the “collected multiunit” idea

(Fuson, 1992) is far more complex than single digit representation of numbers up to

9, (b) the system of English number-naming words contains many inconsistencies,

and (c) trading processes produce non-canonical arrangements of tens and ones that

temporarily break the normal rules of the base-ten numeration system.

Base-ten blocks and place-value software incorporate features that may

support or hinder students as they face these obstacles to understanding base-ten

numbers. For example, it is important to consider how each representational format

helps students to (a) represent numerical quantities, (b) name quantities and written

symbols, (c) carry out trades, and (d) recognise different numerical representations,

such as non-canonical arrangements of blocks. These considerations are addressed in

the following subsections.

Physical base-ten blocks and the independent-place construct.

As explained in section 2.4.3, base-ten blocks are a type of analogue of

numbers. The relative sizes of the blocks map directly onto the relative values

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represented by the first four places of the base-ten numeration system (English &

Halford, 1995).

However, despite the apparent transparency of the mapping between blocks

and places, data in the study show quite convincingly that some children do not

regard the sizes of base-ten blocks when they use them, but merely use them as

“place counters.” As described in more detail in section 5.3, children possessing the

independent-place construct match block sizes, written digits, and number names

without regard for the values involved. Such children are able to use blocks quite

successfully on routine tasks; however, it is probable that they could be just as

successful using materials that did not act as a proportional analogue for the base-ten

numeration system, such as coloured chips or other materials arbitrarily assigned to

represent each place. On more difficult tasks involving trading the independent-place

construct leads to errors, such as writing concatenated place symbols, like “215” for

the sum of 17 and 18. However, if early work with two-digit numbers does not

involve trading or non-canonical representations, children with an independent-place

construct can use materials such as physical base-ten blocks without revealing any

errors in their thinking.

Electronic base-ten blocks and the independent-place construct.

Place-value software such as Hi-Flyer Maths can be used in similar ways to

physical base-ten blocks, and may also fail to challenge students who possess the

independent-place construct. Though the various counters incorporated in the

software were designed to assist students to make connections among numbers,

written symbols and block representations, there is some evidence that, like physical

base-ten blocks in the previous discussion, they may have helped support face-value

interpretations of symbols among low-achievement-level participants. The software

used in the study incorporates a counter at the top of each column of hundreds, tens,

and ones blocks that displays a continuous tally of the number of blocks in that

column. Because these are only counters of the number of electronic blocks, in

themselves they do not indicate anything of the represented values. A label is

included below each counter to indicate the place name “hundreds,” “tens,” or

“ones”; but a student could possibly see these as words only, rather than as numerical

values. Thus, for example, if there are 2 hundreds, 4 tens and 8 ones on the screen, it

is possible that a student may notice only the digits ‘2,’ ‘4,’ and ‘8’ (see Figure 5.1),

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and interpret the symbol ‘248’ as being composed merely of a concatenation of these

face values.

Figure 5.1. Column counters in software representation of 248.

It is clear that students who possess an independent-place construct could use

the software to represent numbers, noting the column counters and the number

window, and listening to the number name read to them by the software without their

independent-place ideas being challenged. On the other hand, certain features of the

software are likely to cause some conflict with an independent-place construct. The

on-screen numeral expander will show different ways of grouping digits to form

symbol-based non-canonical representations for numbers, having the effect of

transferring a digit into an adjacent place. For example, the number 267 can be

shown on the expander as “26 hundreds 7 ones,” “2 hundreds 67 ones,” or “267

ones.” Each of these representations for 267 breaks the central idea behind the

independent-place construct by showing different ways of interpreting the values of

the digits. Similarly, the software will quickly and accurately demonstrate either non-

canonical arrangements or trading processes that could be used to challenge an

independent-place construct held by a student. Though physical blocks and numeral

expanders could be used with the same effect, the speed and accuracy of the software

provides extra convenience. Using electronic blocks, a student could witness many

more examples of number representations to challenge the independent-place

construct than could be shown by physical base-ten blocks in the same time.

5.5.4 Development of Links Among Blocks, Symbols, and Numbers

Section 2.5.2 includes a discussion of difficulties students have in using

materials to model numbers that have been identified by several researchers, and

focuses particularly on the idea that there is a conceptual gap in children’s thinking

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between symbols and number material such as base-ten blocks. Data gathered in this

study provide information that adds to the available knowledge of how children use

representational materials, and may help inform discussions of why some material is

not always successful in teaching students about the base-ten numeration system.

This study also provides the opportunity to compare physical base-ten blocks with

electronic blocks, to see if the different features of the two representational formats

make a difference in helping children make connections between symbols, numbers,

and the material.

An important issue that has a bearing on how well children develop

conceptual links among numbers, symbols, and blocks when using physical base-ten

blocks is the accuracy of the block representations formed by the children. As

mentioned previously, one major difference between physical and electronic blocks

is the facility of the software for providing accurate counters for the number of

blocks in each column and an accurate symbol for the entire number represented by

the blocks on screen. To generate equivalent symbols when using physical blocks it

is necessary to count the blocks; if sufficient care is not taken with counting, errors

can be introduced that require remediation before correct ideas can be gained from

the blocks. In light of the large number and variety of errors made by participants,

described in section 4.6, clearly users of physical blocks need to take great care when

counting blocks to ensure that counting or handling errors do not give an incorrect

impression. Later in this section the tendency of some participants to trust their own

count of the blocks, even in the face of other contradictory information, is discussed.

It is clear that one solution to difficulties that students have in using physical blocks

to understand numbers is for the teacher to stress the importance of care in handling

the blocks, and the frequent use of checking procedures to attempt to trap errors.

However, such procedures will only assist students when the errors made have been

counting or handling errors, and if the students already have enough understanding of

numbers not to introduce incorrect ideas, such as trade-up-to-10 (section 4.6.2).

Conceptual errors, as opposed to counting or handling errors, cannot easily be

checked by the person possessing them; they require another person or agent to point

them out before their effects can be countered.

Thus, support provided by a representational format for the development of

accurate conceptions of numbers depends to a large degree on the accuracy and

correctness of the manipulations of the material carried out by the user. No material

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is going to demonstrate correct ideas about numbers if the user makes fundamental

errors in using the material that remain uncorrected. In the teaching phase of this

study the researcher was always present to “pick up the pieces” if participants made

errors they were unable to correct themselves. The researcher also made sure that any

errors were corrected and faulty ideas challenged before participants started new

tasks. In a busy classroom with 30 or so students, a teacher does not often have time

for this sort of management of the learning environment, and so errors and

misunderstandings can easily go unchallenged.

5.5.5 Support for the Development of Number Concepts

The one component of the two interviews in which there was a notable

difference between the performance of participants who used physical blocks and

those who used electronic blocks was skip counting. It is noted in section 4.3.2 that

interview results appear to indicate a higher performance on skip counting tasks by

participants who used electronic blocks than by those who used physical blocks (see

Table 4.2). The following transcript excerpt illustrates some skills required to skip

count successfully:

Daniel: 681, 671, 661, … 661, 651, 641, 631, 621, 6 hundred and … uh … 11, 601, 6

hundred … no, so that must be … 5 hundred and … 91, 581, 571, 561 …

(I2, Qu. 4d)

In order to complete this task correctly, Daniel (h/c) had to keep track of (a)

the number of hundreds; (b) the number of tens; (c) the names of each decade,

including the “teen” number 11; and (d) the rules of the base-ten numeration system

that define how to count 10 less than 601. In completing this task successfully,

Daniel was able to use a regular “six hundred and n-ty-one” pattern in naming the

numbers to 621. However, this pattern is not used for the number 611 or the numbers

less than 600, causing Daniel to pause in his counting while he thought about those

numbers.

The features of the software available to its users may help explain why

participants from the two computer groups were better able to skip count after the

teaching phase than were participants who had used physical blocks. During the

teaching phase the researcher encouraged participants to use the number window

when doing tasks that involved skip counting (such as Tasks 13-17 and 40-43; see

Appendix H). The effect of the number window during these tasks was to show a

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counter that changed instantaneously when a block was added or subtracted, showing

clearly the changing digit and how the numbers changed at the change of a decade or

a hundred. For example, using the number window while completing Task 41 would

show the symbols 462, 472, 482, 492, 502, 512, 522, and so on. Experiences with

this “odometer effect” may have helped participants from the computer groups to

improve their skip counting abilities.

5.6 Place-Value Understanding Demonstrated by High- and Low-Achievement-Level Participants

5.6.1 Similarities in Place-Value Understanding of High- and Low-Achievement-Level Participants

Many of the observations made in chapter 4 and earlier in this chapter apply

to both high-achievement-level and low-achievement-level participants. First, though

high-achievement-level participants in general performed much better on the place-

value tasks they were set, at times they also demonstrated similar misconceptions and

errors to the low-achievement-level participants. Specifically, at various times a

small number of high-achievement-level participants used inefficient counting

approaches (Table 4.8) or face-value interpretations of digits (Table 4.10), and gave

the lowest categories of response (Category I or II) to digit-correspondence tasks

(Table 4.13). High-achievement-level participants also made similar errors to low-

achievement-level participants, including each of the types of counting, block-

handling and naming errors described in section 4.6.

Second, a few low-achievement-level participants at various times showed

similar abilities to high-achievement-level participants. For example, Table 4.2 and

Table 4.3 show that certain low-achievement-level participants demonstrated similar

numbers of numeration skills as certain high-achievement-level participants. Also, as

shown in Table 4.6, Table 4.8, and Table 4.10, there was some overlap regarding the

frequency of use of counting and grouping approaches and face-value interpretations

of digits by low-achievement-level and high-achievement-level participants.

Similarities in responses of low-achievement-level and high-achievement-

level participants may partly be due to the nature of the Year 2 Net (Queensland

Department of Education, 1996) and how students’ scores are determined, as

intimated in section 4.3.2. Another factor is the small number of available children at

the school. It can be seen in Appendix F that differences in mathematical

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achievement between low-achievement-level participants and high-achievement-

level participants were not very great; in a school with a larger pool of Year 3

students from which to select participants, it may have been possible to have

participants who demonstrated more widely separated mathematical achievement

levels.

More importantly, similarities in response patterns of high- and low-

achievement-level participants may point to important factors regarding the learning

of place-value concepts by Year 3 students generally. As discussed in section 5.4,

one notable feature of the data in this study has been the changeability of

participants’ ideas about numbers. As already discussed, though changeability of

ideas was more commonly exhibited by low-achievement-level participants, it is

quite possible that high-achievement-level participants also used their existing

knowledge of numbers to test hypotheses regarding questions put to them, before

responding to the researcher’s questions. This idea is supported by incidents in which

high-achievement-level participants changed their answers or accepted incorrect

counter-suggestions offered by the researcher. These observations are important

because they show that even the high-achievement-level participants at times

demonstrated ideas about numbers that were being developed and subject to change,

rather than being fixed and immutable.

5.6.2 Differences in Place-Value Understanding of High- and Low-Achievement-Level Participants

In spite of the similarities in responses of high-achievement-level and low-

achievement-level participants reported in the previous section, a number of clear

distinctions were observed between them. Several tables in chapter 4, including

Table 4.13, show dramatic differences between responses of participants in high-

achievement-level and low-achievement-level groups. Table 4.2 and Table 4.4 show

that in the interviews high-achievement-level participants outperformed low-

achievement-level participants by an average of about 8 or 9 place-value criteria.

Table 4.6, Table 4.8, Table 4.10, and Table 4.12 show clear differences between

high-achievement-level and low-achievement-level participants’ approaches to

place-value questions, with high-achievement-level participants adopting grouping

approaches much more often, and using counting approaches or face-value

interpretations of digits much less often. Table 4.13 shows that high-achievement-

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level participants generally gave more advanced interpretations of digits in digit

correspondence tasks. These tables together appear to illustrate important distinctions

between the two groups of children. High-achievement-level participants

demonstrated more effective, accurate approaches to place-value questions, and

demonstrated better grasp of place-value concepts, than low-achievement-level

participants. It is not difficult to believe that these two observations are related. The

participants who exhibited best knowledge of the base-ten numeration system, and

the best understanding of place-value concepts, also demonstrated more efficient and

accurate strategies for answering place-value questions. As mentioned earlier in the

discussion of the use of counting, it appears likely that use of more accurate and

efficient strategies enable students to perform better on place-value tasks, and

understand place-value concept better, than their peers who use less accurate or

inefficient strategies, or both. In effect, by having better knowledge of numbers and

the base-ten numeration system, more able students have access to better strategies,

that in turn give quicker, more accurate results, leading to further improved

knowledge and skills. This apparent “Matthew effect” (Burstall, 1978), leading to

improved understanding and performance by those students who start in front, has

clear implications for teachers who attempt to provide equal opportunities for

successful learning to all their students; this point is taken up later in section 6.2.4.

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Chapter 6: Conclusions

6.1 Chapter Overview This chapter is divided into three major sections. Section 6.2 includes

discussion of emerging answers to the study’s research questions, section 0 addresses

implications of the study findings for the teaching of place-value concepts, and

section 6.4 outlines recommendations for future research into place-value

understanding.

6.2 Conclusions About Answers to Research Questions This study was conducted to address the following research question: How do

base-ten blocks, both physical and electronic, influence Year 3 students’

conceptual structures for multidigit numbers? To answer this question, four sub-

questions have been addressed within the context of Year 3 students’ use of physical

or electronic base-ten blocks. Each of the four following subsections addresses one

of these four questions.

6.2.1 Conceptual Structures for Multidigit Numbers Evident in Participants’ Responses

1. What conceptual structures for multidigit numbers do Year 3 students

display in response to place-value questions after instruction with base-

ten blocks?

As a result of the literature search conducted early in this study, certain

conceptual structures were identified as part of a sequence of essential conceptual

structures adopted by children as they developed their place-value understanding. It

was hoped to be able to analyse the interactions observed in the interviews and

teaching sessions and compare participants’ conceptual structures for the two

representational formats, blocks and software.

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However, as discussed in section 5.2.1, what emerged from this study’s data

was a pattern of preferences held by participants as they answered a range of

questions, rather than stable conceptual structures. In some cases, participants’

preferences for a particular approach were quite well defined, especially for those

participants who favoured grouping approaches; there is good evidence that these

participants had well-developed conceptual structures for multidigit numbers that

included the grouped, multiplicative aspect of the base-ten numeration system.

However, many participants did not use a single approach to the questions, and

appeared not to have developed stable ideas about multidigit numbers. Their

responses were characterised by the adoption of a variety of approaches and a

marked changeability of opinion about the questions asked.

To summarise this section, conclusions about Year 3 students’ conceptual

structures drawn from the data in the study are as follows:

1. Conceptual structures described by other authors (e.g., Miura &

Okamoto, 1989; S. H. Ross, 1990) were evident in participants’

responses.

2. However, in many cases conceptual structures were not held firmly, but

were altered in response to further information or further questioning.

3. Thus, in light of the changeability of students’ opinions, it is more

accurate to categorise a student’s response, than to categorise the

student per se.

6.2.2 Misconceptions, Errors, or Limited Conceptions Evident In Participants’ Responses

2. What misconceptions, errors, or limited conceptions of numbers do

Year 3 students display in response to place-value questions after

instruction with base-ten blocks?

As mentioned in chapter 4, the large number and variety of misconceptions,

errors, and limited conceptions of numbers evident in the study data have important

implications for how place-value topics are taught in primary school. These

implications include, but are not limited to, a lower likelihood of success that errors

impose on those making them. Other implications are the greater difficulty added to

the learning of topics and the greater cognitive load certain errors cause. Such errors

are related to broader topics in this study, including the preference of some

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participants for counting and the independent-place construct. In fact, many errors

observed in the study would be less serious if one could be sure that the participants

making them were doing so accidentally. However, in many cases participants

making errors appeared to have such deep-seated confusions about numbers that they

were unable to carry out successfully all but the most basic of place-value tasks.

Such errors evident in this study as trading a ten-block for a one-block, or counting a

collection of ten- and one-blocks as if each represented only one, lead the author to

the conviction that children making them had little real understanding of the base-ten

numeration system. If a teacher ignores such mistakes in the belief that children

demonstrating such errors are merely being careless, the children will be denied help

they need to develop accurate understanding of base-ten numbers.

The diverse errors made by participants in the study are summarised in

chapter 4 as being errors of counting, handling errors, errors in trading, errors in

naming and writing symbols for numbers, and errors in assigning values to blocks.

The root causes of such errors can be summarised as being of one of three

fundamental problems: (a) lack of knowledge of base-ten number naming

conventions (such as the pattern “x hundred and y-ty z,” etc.); (b) lack of familiarity

with base-ten blocks; or (c) or the independent-place construct, characterised by a

lack of understanding of the relationship between each place and the places either

side of it. This latter misconception is particularly difficult for the teacher to

recognise, as it is often disguised by the tasks typically given in textbooks and some

classrooms, that ask children merely to state which digits, number names, places, or

base-ten blocks to associate with each other. If a question about hundreds, tens, and

ones can be answered by making single-dimensional associations between a digit, a

place-specific number name, and base-ten blocks of a particular size, then the

response of a child possessing an independent-place construct will be entirely

indistinguishable from the response of a child who understands the groups of 10

behind multidigit base-ten numbers.

One other aspect of the data in this study has relevance for helping children

overcome erroneous ideas they have about base-ten numbers. The evidence of many

incidents of participants inventing or mis-applying ideas to explain features of base-

ten numbers leads to a certain confidence that children will be able to understand the

base-ten numeration system for themselves, provided that their teachers give them a

logical basis for understanding the relations that exist between numbers and their

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symbolic and concrete referents. Since participants were evidently comfortable in

applying knowledge about numbers to novel questions, this gives good reason to

believe that with accurate information in a form that is accessible to them, children

will be able to develop correct, coherent, sensible conceptual structures for base-ten

numbers.

Evidence in the study for children’s construction of knowledge of numbers

has relevance for the view taken of children’s errors in understanding numbers. Since

it is evident that children are prepared to use a wide variety of information to help

them make sense of numbers, consistent with current advice that children should be

encouraged to develop their own understandings of the world, it is likely that in the

process errors will eventuate. Furthermore, it is clear that if children are taught

merely to follow procedures with blocks or written symbols, their attempts to make

sense of numbers are likely to be frustrated, and may result in the development of

faulty ideas.

Teachers must recognise the great leaps in conceptualisation that have to take

place at various points in the teaching of mathematics topics. In particular, the step

from recognising one-digit symbols as standing for collections of so many single

items, to seeing that two-digit symbols stand for collections of 10 items and left-over

single items, can be accurately labelled a “conceptual leap” (labelled by Baturo,

1998, a “cognitive leap”), rather than a mere progression based on previous ideas.

This study set out to investigate the teaching of the hundreds place; what has

emerged is a clear problem in the learning of the tens place for many children of this

age group. The majority of the low-achievement-level participants had such a limited

understanding of two-digit numbers that questions involving hundreds were really

beyond their abilities. Even many of the high-achievement-level participants had

limited understanding of the base-ten numeration system, and though they could

often answer a question successfully, their understanding of what is represented by

tens digits was based often on independent-place ideas.

Based on the context in which limited and faulty ideas about numbers

emerged in this study, it is clear that the mathematics tasks presented to students

have a strong bearing on the mental models for numbers that they will accommodate.

As mentioned several times, tasks based on matching numerical symbols, number

names and base-ten blocks can often be answered without addressing relationships

between places, and with very limited understanding of the base-ten numeration

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system. Thus the type of task given to students has an important bearing on the

place-value understanding that is revealed by student responses. Teachers and

curriculum writers need to be aware of these points, and to limit tasks that rely only

on knowledge of place names, block names and number names. Rather than

questions such as “Show me the tens part of this number,” questions which challenge

children, such as “What is another way to use base-ten blocks to represent this

number?” will help to distinguish between children who understand the grouped-ten

aspect of base-ten numbers and those who do not.

6.2.3 Effects of the Two Materials on Students’ Learning of Place-Value Concepts

3. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in instruction with physical and

electronic base-ten blocks?

The central question of this study is how each of the representational formats,

physical or electronic blocks, affect Year 3 students’ learning of place-value

concepts. This question is addressed principally via the data from the two interviews,

discussed in chapter 4. Results from the two interviews show that many participants

did improve their understanding of the base-ten numeration system over the course

of the study. However, there was no marked trend that could be identified to compare

differential effects of the two representational materials on student learning. Gains of

conceptual understanding of the base-ten numeration system were quite conservative,

and neither cohort using physical or electronic blocks appears to have done

significantly better than the other. Certain individual participants showed pleasing

improvement on interview questions over the course of the study, but others showed

no improvement or even deterioration in place-value understanding.

Use of physical base-ten blocks to learn place-value concepts.

The clearest finding about the use of blocks in the data collected in this study

is that students using physical blocks need support to represent numbers and

numerical processes. In this study, this fact has come into sharp focus, as

comparisons with the use of electronic blocks to represent numbers show that

physical blocks lack certain features that appear to have made electronic blocks

easier for participants to use. Specifically, physical blocks lack any sort of counting

device to inform the user of the number of blocks present, or the number represented

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by a blocks arrangement. Furthermore, unlike electronic blocks, physical blocks do

not have any mechanism for carrying out trading actions that will ensure that such

actions are done correctly. These facts, coupled with the high incidence of errors in

counting and handling blocks, meant that there was the potential for the participants

using physical blocks to face many difficulties in learning about the base-ten

numeration system. In this study’s teaching sessions the researcher was able to give

physical blocks users feedback about their use of the blocks, and thereby to correct

mistakes and misconceptions before they could become entrenched in the

participants’ thinking. This might not be the case in a typical classroom, as it is

unlikely that a teacher with an entire class to supervise would be able to monitor the

use by individual students of base-ten blocks very closely.

Transcripts of blocks groups indicate cause for some concern about how

useful physical base-ten blocks are for teaching number concepts. The approaches

taken by both high-achievement-level and low-achievement-level participants using

physical blocks often were not conducive to the generation of understanding of base-

ten numbers. Firstly, high-achievement-level participants showed reluctance to use

physical blocks to illustrate numerical processes that they evidently understood; it

appeared that at times these participants regarded the block representations as

redundant because they felt they already understood the concepts illustrated by the

blocks. When the researcher required high-achievement-level participants to use the

blocks, however, on a number of occasions participants expressed greater confidence

in the blocks than in their own thinking, and accepted incorrect answers produced in

mishandling the blocks in preference to correct answers they had worked out

mentally. Secondly, low-achievement-level participants typically used the base-ten

blocks as calculating devices, and evidently had no idea of the answers to many

questions until they counted the blocks. The large number of errors made, both in

counting and handling blocks and in thinking about numbers, meant that in many

instances low-achievement-level participants received misleading information from

the blocks, and the researcher needed to correct their mistakes.

The difficulties described in the previous two paragraphs resulted in many

instances of feedback being provided to participants using physical blocks by the

teacher and by their peers. Rather than thinking about the numbers involved and

attempting to work out answers to questions mentally, on many occasions

participants in the blocks groups used other sources of information to tell them

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answers. Often the source of information was the blocks themselves, counted by a

participant. On other occasions participants received help from their peers, or relied

on the researcher to tell them if they were correct or not. The researcher attempted to

reduce the amount of feedback he gave to blocks participants, to encourage them to

use other resources, including their own thinking, to come up with answers.

However, in many instances, this was not successful, and the only source of accurate

information available to the participants was the researcher.

Use of electronic base-ten blocks to learn place-value concepts.

As noted in the previous section, the software incorporates features to provide

users with feedback about the blocks on the screen, and the numbers they represent.

Descriptions in chapter 5 of interactions among the researcher, participants, and the

software indicate that these feedback-providing features influenced the ways

participants used the materials, and the frequency with which they accessed feedback

from non-electronic sources. Specifically, these features include an uncluttered view

of blocks, electronic counters of three types that keep a track of the numbers of

electronic blocks present, audio number name recordings, and accurate trading

transactions. The software was designed to incorporate these features in the hope that

they would assist students in learning about the base-ten numeration system; though

results are far from conclusive, there are positive indications of the effects of the

software on student thinking about numbers. Specifically, compared to users of

physical blocks, participants using electronic blocks received considerably less

feedback either from the researcher or from each other, instead using the software to

inform them about the numbers they were representing. In the process they received

far more positive, more accurate feedback overall, which implies a likely positive

effect on students who use similar electronic blocks for learning about numbers.

On one aspect of number processes in particular, trading operations,

participants who used electronic blocks demonstrated great confidence in the

equivalence of traded blocks, after observing accurate trades many times, supported

by electronic symbols (section 4.7.6). Though the same information was available to

participants using physical blocks, and though the researcher pointed out the

equivalence of traded blocks, blocks participants did not exhibit any marked

awareness that traded blocks always represent the same quantity. This positive

learning effect evidently resulting from the use of electronic blocks implies that such

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software could be very useful for teaching such concepts to students, provided the

software was designed to incorporate carefully-planned support for the concepts.

Comments about learning effects observed in this study.

The comments in the previous paragraphs seem at odds with other aspects of

the results reported in this chapter. In particular, it seems reasonable to expect that

participants using electronic blocks would improve their understanding of certain

aspects of numbers more easily than would participants using physical blocks, based

on descriptions of the apparent effects of using the software.

A number of comments may put this in perspective. First, at both interviews

the students were provided only with physical base-ten blocks with which to answer

questions regarding use of blocks to represent numbers. It is possible that participants

who had used the software were at some disadvantage at the second interview,

having just spent 2 weeks using only electronic blocks to represent numbers; on the

other hand, participants from blocks groups had just had practice in using physical

blocks for the same 2 week period. Some evidence for this is found in an interesting

excerpt from Hayden’s (l/c) second interview, mentioned in section 4.7.1. In the very

first question of Interview 2, Hayden counted 6 tens and 7 ones, counting the ten-

blocks as five each, reaching the answer 37. He quickly corrected himself when the

interviewer asked him if he was sure, but it is possible that Hayden had momentarily

forgotten how to use physical blocks, after having used only electronic blocks for a

fortnight.

Secondly, the time for this study was quite short. If lasting effects were to be

produced by the use of either representational format, it is likely that it would require

a longer period for these effects to become evident. Since the participants had all

used physical base-ten blocks in class for a considerable time prior to the study, it

may be less likely that electronic blocks would produce a marked effect without a

longer period of exposure, given that the electronic blocks constituted a novel

representational format to these students.

Thirdly, the teaching phase comprised group sessions in which there was one

teacher for four students. This favourable teacher:student ratio provided participants

with accurate, timely feedback from a teacher regarding their deliberations about

numbers that is unlikely to be available in a normal classroom. This would have

particularly helpful to participants who used the physical blocks, in light of the

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frequency of feedback they received from the researcher (section 4.7.7), apparently

because of a lack of other sources of accurate information. Users of the electronic

blocks, however, could receive similarly accurate feedback from the software in

place of that from the researcher, and so were able to attempt the tasks set for them

with less need for adult intervention. Thus in a classroom with group activities being

conducted, electronic blocks may prove to be more useful than physical blocks for

helping students understand numbers, because of the software’s capacity to provide

feedback without the need for constant adult supervision. Further classroom-based

research would be needed to test this idea.

Effects of feedback provided by physical or electronic blocks.

The differences in the effects of physical or electronic blocks appear to be

rather subtle, except with regard to the provision of feedback. As discussed in

chapter 5, participants who used physical blocks received their most accurate

feedback from the researcher, with less accurate feedback coming from their peers or

from the blocks. On the other hand, participants who used electronic blocks received

less feedback from the researcher or their peers, compared to feedback from the

representational materials available to them. In the sense that the software presented

information to the children, it provided users with feedback for their ideas that was

much more accurate than similar information available to users of physical blocks.

The computational facility of the computer running the software provided nearly

instantaneous feedback about the number of blocks on screen, the number

represented by the blocks, the symbol for the number, and the verbal name of the

number. Clearly, physical blocks offer none of these facilities, meaning that such

information must come from some other source.

It could be argued that much of the feedback provided by software could

easily be provided to users of conventional blocks by well-designed worksheets or by

a careful teacher or adult helper. However, the reactions of participants to feedback

that they received indicates that there were important differences in the participants’

confidence in the feedback, with important implications for use of materials to

represent numbers.

To summarise this section, conclusions about the learning effects produced by

physical or electronic base-ten blocks are as follows:

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1. With a teacher available to provide assistance and correction, Year 3

children are able to learn place-value concepts using either physical or

electronic base-ten blocks.

2. Children using electronic blocks are able to rely on the material for

accurate feedback regarding their ideas about numbers, whereas users

of physical blocks need other sources of accurate information.

3. During the short time of this study little difference was evident in the

place-value learning by participants using either material. Over a longer

period of time, and with more limited teacher assistance in a regular

classroom, students using electronic blocks may have an advantage in

learning place-value concepts over students using physical blocks.

6.2.4 Differences Between Place-Value Understanding of High- and Low-Achievement-Level Participants

4. Which of these conceptual structures for multidigit numbers can be

identified as relating to differences in students’ achievement in

numeration?

As discussed in section 5.6, there were both similarities and differences

between the performance of high-achievement-level and low-achievement-level

participants. Similarities in responses related especially to the idea of knowledge

about numbers being constructed by participants; high-achievement-level

participants were observed to change their responses in light of further information

or challenges to their initial answer, much as low-achievement-level participants did.

This supports one of the main contentions of this author in this chapter, that the

participants’ knowledge of the base-ten numeration system did not fit into any neat

set of categories, but was marked by flexible, changeable ideas that the children

altered in light of further information.

Differences between responses of high-achievement-level participants and

those of low-achievement-level participants were quite pronounced, as demonstrated

by several tables in chapter 4 (see section 5.6 for discussion). These tables reveal

clear and significant distinctions between both the conceptual structures and the

place-value task performance of the two groups of participants. Not only did high-

achievement-level participants demonstrate clearly better understanding of place-

value concepts, as a group, than their low-achievement-level counterparts; they also

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adopted more accurate, efficient and useful strategies for answering place-value

questions.

In summary, these points are noted in comparing the overall performance of

high-achievement-level and low-achievement-level participants:

1. There was some overlap of performance levels achieved by high-

achievement-level and low-achievement-level participants, so that some

low-achievement-level participants achieved higher results than some

high-achievement-level participants did.

2. In general, high-achievement-level participants achieved more place-

value understanding criteria in interviews, on average achieving more

than 8 more criteria on each interview.

3. High-achievement-level participants demonstrated the use of more

efficient and more accurate approaches to place-value questions, and

adopted the incorrect face-value construct for multidigit numbers much

less often, than low-achievement-level participants did.

4. High-achievement-level participants on average demonstrated much

better performance on digit correspondence tasks than did low-

achievement-level participants.

5. Despite their better performance on place-value questions generally,

high-achievement-level participants still exhibited similar changeability

of answers and ideas about numbers.

6. It appears that a “Matthew effect” (Burstall, 1978) existed, by which

those participants who had better understanding of the base-ten

numeration system used more accurate and efficient strategies when

answering place-value questions, leading to further improvements over

participants with more limited place-value understanding to start with.

6.3 Implications for Teaching

6.3.1 Implications of Using Physical Base-Ten Blocks to Teach Place-Value Concepts

One of the biggest hurdles to overcome in teaching with physical base-ten

blocks may be the knowledge that teachers themselves have about the base-ten

numeration system, and the apparently transparent way in which blocks represent

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that system. Cobb and Wheatley (1988) and Clements and McMillen (1996) have

pointed out that teachers should not assume that children see numbers and block

representations of numbers the way that adults do; evidence of a number of unusual

ideas held by participants in this study has supported these statements. A related

point is that children’s ideas about blocks and about numbers are often not made

visible by typical classroom mathematics tasks. Both the face-value construct,

previously identified and extensively discussed in the literature, and the independent-

place construct, proposed in this study, are ideas apparently held by children that are

not revealed if mathematics questions are kept simple. Routine questions such as

“show me the number in the tens place” can be answered with very limited

knowledge of the way symbols represent numbers, and can be answered quite

successfully while holding any of a number of faulty or limited conceptions for

numbers. Thus, it is important for teachers to attempt to find as much as possible

about how children perceive the “mathematical objects”—including written

numerals, base-ten blocks, and electronic blocks—used in the classroom. One way to

foster this is to assign tasks that are likely to reveal incorrect thinking, including digit

correspondence tasks, trading tasks, and tasks requiring the production and

interpretation of non-canonical block representations. The other aspect of this

recommendation is for teachers to monitor children’s use of the materials quite

closely. This study revealed a large number of errors which, except for the presence

of the researcher, would most likely have gone unnoticed by the participants. Left

alone, children are going to make errors in manipulating materials and answering

mathematical questions. There needs to be some procedure in place in a classroom to

identify and remediate these errors in a timely way.

Provision of tidy, structured working spaces.

Another clear aspect of block use revealed in this study is the “messy” nature

of physical block representations (Clements & McMillen, 1996). When using blocks

to represent two-digit numbers this problem is not likely to be very serious, but with

three-digit numbers and beyond the sheer number of blocks can cause significant

difficulties for children if they do not adopt orderly practices. There were many

instances in this study in which participants in blocks groups made counting or

handling errors that were likely to have been at least partly due to this problem. One

particular error noted at times during the teaching sessions was a difficulty

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participants had with keeping the current block representation under consideration

separate from the rest of the available blocks. With a large collection of blocks on a

desk, participants in the study sometimes found it difficult to remember whether a

certain block, “found” near others that were being counted, was part of that counted

set, or if it was an “extra” from the other uncounted blocks. Such situations appeared

to cause several errors by participants.

One idea to assist children in keeping track of blocks is to provide containers

for extra blocks, and to use some sort of structured “mat” on which to place block

representations. This idea appeared to have been used by the classroom teacher of

some of the participants, as a “tens mat” was mentioned by participants during the

teaching session. This mat could be as simple as a place-value chart on a piece of

paper marked “Tens” and “Ones,” with a vertical dividing line between the places.

More complicated structured material on which to place base-ten blocks could be

devised that assists students in counting the blocks. A similar idea is commercially

available for use with Unifix™ cubes in the form of a shallow plastic tray that is the

right size to contain a certain number of cubes. If such a device was available for use

with base-ten blocks, having counters to judge how many blocks of each place were

present, it could overcome a major disadvantage for users of blocks over users of the

software, the fact that blocks have to be counted frequently to determine the number

present. This idea may be useful, but is likely to add to the cost of the material, and

may introduce other unforeseen difficulties of interpretation. Whatever method a

teacher adopts for use of base-ten blocks, it is recommended that students be

encouraged to work neatly, to count blocks carefully, and to recheck answers if they

seem unusual. Had participants in the blocks groups routinely used such an approach

they may have made considerably fewer errors.

Base-ten blocks are no substitute for number sense.

The use of base-ten blocks by participants in this study revealed a number of

difficulties if teachers believe that the base-ten blocks show children accurate models

of numbers and associated processes. Participants often did not appear to regard a

block representation holistically, but place-by-place. Rather than engaging with two-

digit or three-digit numbers as complete entities, participants often seemed to use an

independent-place construct that enabled them to manage task demands with less

cognitive effort necessary. As revealed in the transcripts, participants often seemed to

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use base-ten blocks as “place counters,” mapping each number of like-sized blocks

onto a place digit or onto a number name. Kamii et al. (1993) did not use base-ten

blocks or any other representational material in their study, arguing that base-ten

blocks promote the idea that mathematical knowledge is somehow contained in the

blocks, rather than in a person’s “mental action” (p. 201). This author agrees with the

basic thrust of this argument, but would urge better use of base-ten material rather

than a complete abandonment of it. Nevertheless, Kamii’s argument is supported to

some extent by the results of this study, in that participants on many occasions

seemed to be “missing the point” that the blocks were supposed to illustrate,

manipulating blocks in procedural, unthinking ways that did not appear to assist

participants in developing better concepts about numbers. The solution to this

problem might be either to follow Kamii et al.’s advice and stop using the blocks, or

to interrupt children’s manipulations to ask pertinent questions about the quantities

they are modelling. The point made several times in this thesis and elsewhere is that

blocks themselves are only a means to understanding numbers, not the end purpose

for their use. They are no substitute for having an internal understanding of numbers

that includes knowledge of number facts, computation skills, and number sense.

One implication of the independent-place construct and its potential to render

invisible many errors in interpreting values represented by base-ten blocks is that

students’ use of base-ten blocks must be closely monitored. For the sake of

children’s development of number ideas, teachers cannot afford to allow students to

use materials such as base-ten blocks without checking the children’s interpretations

of the representations produced. This may involve greater use of questioning of

students to probe what they believe the blocks demonstrate about numbers, and the

earlier introduction of questions involving trading and other non-canonical

representations of numbers (see Fuson, 1990b, for similar recommendations).

Summary of teaching recommendations for use of base-ten blocks.

The following recommendations to teachers are made for the use of base-ten

blocks, and particularly physical blocks, in primary classrooms:

1. Challenge students’ ideas about numbers by asking them a variety of

non-routine place-value questions that include non-canonical

representations of numbers and trading.

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2. Closely monitor students’ use of base-ten materials to identify various

counting, handling, or conceptual errors that can be made.

3. Provide help for students to keep their block arrangements neat and

orderly. Use place-value charts or other materials to help add perceptual

structure to the block arrangements.

4. Be prepared to interrupt children’s use of base-ten blocks to challenge

possible faulty concepts about numbers. If necessary, stop children

using blocks for a time and challenge them to think about numbers in

different ways.

5. Do not use base-ten blocks to teach numeration concepts to young

children. Use material that includes grouped single material instead,

such as bundling sticks, at least until children understand the grouped

aspect of the base-ten numeration system.

6. Be aware of and alert for signs of common misconceptions held by

children about multidigit numbers, in particular the face-value construct

and the independent-place construct.

6.3.2 Implications of Using Electronic Base-Ten Blocks to Teach Place-Value Concepts

Though many software titles to teach mathematics are currently available, it

is not clear how many of them are designed specifically to teach place-value

concepts, nor how many include representations for numbers similar to the software

used in this study. Furthermore, there are no data available to the author of the

proportion of primary teachers who use such software in their teaching of

mathematics. There is clear anecdotal evidence, however, that the use of computers

generally in Queensland primary schools has increased rapidly in recent years, and it

seems likely that the trend is similar in other school regions. The recommendations

in this section are directed towards designers of mathematics software for teaching

place-value concepts, and towards teachers in the position of choosing software for

use with their class. As the designer of software (Price & Price, 1998) that is used in

primary schools in Australia, the author is aware of the wide range of skills needed

by software designers and programmers, and the need for up-to-date information

about children’s mathematics learning; it is hoped that results of this study will lead

to further software for use in primary schools.

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Generally, the results of participants’ use of the software in this study are

encouraging. Results of the two interviews (Table 4.3) indicate that learning occurred

both for participants who used physical base-ten blocks and for those who used

electronic blocks. It appeared that participants using the software were content to

regard the pictures of blocks on screen as actual entities, and to manipulate them

using on-screen tools to represent numbers and number processes. In particular, the

representation of trading processes seems to have been very successful; participants

were very confident in the idea that traded blocks are always equivalent in value to

the blocks before the trade. There were few aspects of the software that appeared to

introduce misconceptions in participants’ thinking, except perhaps for column

counters. It is believed that counters above the three columns on screen may have

promoted or supported either face-value constructs or independent-place constructs.

On the positive side, feedback mechanisms incorporated in the software were used

often by participants to confirm their ideas. It appears that participants enjoyed

having their answers confirmed by the various electronic means of feedback, and that

the feedback received was more accurate and more encouraging than the feedback

received by users of physical blocks.

Certain tasks were more difficult to manage with electronic blocks than others

were, prompting recommendations for further features to be incorporated in place-

value software. In particular, the software does not easily represent two quantities

simultaneously, as there is just one set of column counters and one number window.

Tasks involving the comparison of two numbers were handled in the study by having

participants at each computer represent one of the numbers, which would not be

possible if there was only one computer available. Similar difficulties were evident in

the representation of arithmetic operations. Participants were able to carry out

addition by adding extra blocks to a representation, and subtraction by taking blocks

away from a representation. However, the software has no facility for keeping a

record of the blocks added or subtracted, making the modelling of these operations

difficult for children to visualise. It would be useful to have a feature that enables

two quantities to be represented at the same time, and to keep a record of

manipulations made in the course of carrying out an operation.

If classroom teachers are in the position of choosing software to use with their

students to teach place-value concepts, it is recommended that teachers choose

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software that includes features that appear to have been successful with the software

used in this study. Among these features are

1. linked block and number symbol representations of numbers, so that

both blocks and symbols always show the same number;

2. dynamic representation of trading processes that demonstrate the

decomposition of a ten into 10 ones and the recomposition of 10 ones

into a ten; and

3. audio recordings of number names that can be accessed to compare

with block and symbol representations.

One other feature if incorporated in the software could enhance the software’s

versatility and enable its use by children unattended; if children’s tasks were

presented by the software itself on the screen, children could use the software with

less adult supervision.

6.3.3 Implications of the Independent-Place Construct for Teaching Mathematics

As discussed previously, students’ responses to place-value questions may

not reveal faulty conceptual structures, including the independent-place construct, if

students can answer by considering only one place at a time. If, however, students

have to deal with relationships between places, the independent-place construct will

not help students get correct answers. Questions that do foster thinking about places

in relation to each other include questions involving the interpretation of non-

canonical arrangements of blocks.

Another type of question that may help students to overcome an independent-

place construct is computation questions involving regrouping. Each of the four

arithmetic operations involves thinking about adjacent places whenever regrouping is

carried out. For example, when answering “43 – 28,” one method is to regroup one

of the 4 tens into ones, making 3 tens and 13 ones, and then subtracting each place in

turn. Similar examples could be given for addition, multiplication, and division. If

students are faced with problems that require regrouping when they first learn about

an operation, they will find that independent-place thinking will not allow them to

find a solution. Thus, it is recommended that teachers present to their students

operations that require regrouping from the very first examples, to avoid reinforcing

any independent-place thinking that the students may have. An associated

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recommendation for the teaching of operations is to assist students to visualise the

quantities being used, through the use of blocks or place-value software.

The alternative, of teaching students to follow a rote written procedure, does

not equip students to handle further examples of that operation in other, perhaps

more efficient, ways. For example, in the case of the addition operation 49 + 35, one

efficient method would be to adjust the addends to 50 and 34, making the task a

simple mental arithmetic question. A child with an independent-place construct,

however, may only be able to answer the question 49 + 35 strictly as written, by

calculating 9 + 5 ones, and 1+ 4 + 3 tens.

In summary, recommendations to teachers that may reveal and remediate

examples of the independent-place construct among their students are to:

1. Give students place-value questions that involve non-canonical

arrangements of materials.

2. Challenge students to think of a digit in terms of the adjacent places, to

regroup quantities in different ways.

3. Give computation examples that require regrouping from the start.

4. Encourage students to develop creative methods of answering place-

value and computation questions, based on flexible regrouping of

numbers.

6.3.4 Implications of Construction of Meaning for Teaching Mathematics

Results of this study show that participants’ thinking was sometimes difficult

to interpret from an adult perspective. The results also show that one possible reason

for the difficulty in interpreting the actions or statements of the participants was that

they were attempting to make sense of questions posed to them using a wide range of

knowledge that they had about numbers. During that process, participants were

observed to make statements that appear to be illogical or absurd. One temptation for

teachers hearing such statements might be just to tell the student what the answer

should be, or to move on to new material. However, results in this study agree with

statements appearing in the writing of other authors, to the effect that many

apparently nonsensical statements made by children are actually the product of

rational thinking processes, using whatever knowledge the children possess at the

time. In this regard, Cobb and Wheatley’s (1988) advice to researchers is equally

relevant for teachers when responding to children’s sometimes unusual ideas:

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A fundamental assumption of conceptual analyses is that children’s actions are always rational given their understandings. We have all seen children who, from our adult perspective, do some strange things as they attempt to solve mathematical tasks. One reaction is to wonder how the children could be so stupid or to ask what is wrong with them. . . . An alternative approach is to readily admit the inadequacy of adult mathematics for understanding children and for planning instruction. From this perspective, children’s apparently strange actions are viewed as problems for the observer to solve. The trick is to develop an understanding of children’s mathematics so that their actions can be seen as rational and sensible. (p. 2)

It is recommended that teachers take this attitude in attempting to make sense

of what children are thinking. If they are successful, teachers will take on the role of

a researcher, making sense of children’s thinking in order to tailor instruction to help

them understand the realm of mathematics. In conclusion, the following

recommendations are offered for teachers to manage the demands of teaching

students who are making sense of what they experience in the classroom:

1. Teachers must recognise the changeability of students’ ideas, and the

fact that children will think quite rationally about number concepts

based on their perceptions of the realm of numbers.

2. A teacher may have a useful role to play in introducing new information

that conflicts with a student’s incorrect stated belief. Without a person

with expert knowledge pointing out the inconsistencies in a student’s

conception of numbers, that conception may remain unchallenged for

some time; if a faulty belief is accepted for a long time it is likely to be

more difficult to correct than if its inconsistencies were pointed out

earlier.

3. Without asking the right sort of probing questions a teacher is unlikely

to discover what students actually believe about the base-ten

numeration system and about how numbers are represented by it. As

already mentioned, some types of school mathematics questions are

easily answered by a student who has incorrect conceptions about

numbers, such as face-value constructs or independent-place constructs

(section 5.3.6; see also S. H. Ross, 1989).

4. A teacher is unlikely to have a meaningful impact on a student’s

number conceptions by merely repeating the procedure to use in

answering a certain type of number question. Some students definitely

appear to accept procedures that they do not understand, and of which

they cannot make sense. Such unthinking acceptance of taught

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procedures does nothing to help a student tackle novel problems, and

will allow the student to answer only questions of the type to which the

learned procedures applies.

5. In view of the “head start” that more able students seem to have in the

area of understanding place-value concepts and answering place-value

questions, it is important to give extra support to less able students to

enable them to understand place-value foundations and to adopt

efficient strategies that utilise the grouped aspect of the base-ten

numeration system. Without this extra support it appears that those

behind in understanding place-value concepts will fall further and

further behind as they continue to use inefficient, labour-intensive

methods of dealing with multidigit numbers.

6.4 Recommendations for Further Research This is an exploratory study, designed to explore a wide range of factors in

two versions of a particular learning setting. The results of this study have led to

several proposals for explaining what appeared to be happening in the situations

investigated. Each of these proposals is a possible topic for further research to

increase knowledge of children’s learning of place-value concepts.

As described in chapter 2 of the thesis, the children’s number conceptions

have been heavily researched in the past 20 years or more. Nevertheless, results in

this study suggest that some schemes for classifying children’s number concepts may

have other interpretations that need investigating. In particular, the trend towards

classifying children’s number concepts based on certain limited number tasks seems

particularly problematic. It is suggested that further research should be directed

towards finding out more about children’s knowledge of numbers. In particular, in

light of this study it would be appropriate to test the range and character of number

conceptions held simultaneously by individual children.

Associated with research into children’s number conceptions, research into

children’s understanding of base-ten blocks and other representational materials can

be pursued further. There appears to be a range of opinions about the use of base-ten

blocks to teach about the base-ten numeration system. Some writers do not advocate

their use, perhaps because of the observed mishandling of blocks by children. Other

writers caution teachers about allowing children to believe that mathematical

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knowledge is contained within the material. There appears still to be considerable

faith in the mathematics education community in the use of representational

materials, and clearly many teachers use base-ten blocks. However, it is also clear

that the use of base-ten blocks by many children is error-prone and based on faulty

ideas about the base-ten numeration system. These varying opinions about

representational material point to a dilemma regarding advice to give to teachers,

which in the context of published research papers is contradictory. There is

promising work being done by researchers, including Fuson and her colleagues

(Fuson et al., 1997), who are involved in various research projects investigating this

important topic and collaborating in reporting the results. This present study points to

a need for such research to continue.

This thesis contains a proposal for the existence of a concept apparently held

by some children, named the independent-place construct. Difficulties applying the

face-value construct to certain responses in the data led to the proposal of the

independent-place construct; the author felt that the differences between the two

concepts could not be overlooked, and so the new label was proposed. The

independent-place construct may be misleadingly similar to the face-value construct,

which has received considerable attention in the literature in recent years. However,

evidence presented in section 5.3.2 suggests that the independent-place construct is

different to the face-value construct, and equally difficult to identify in responses to

certain routine number tasks. In the final analysis, it may be found that the

independent-place construct is so similar to the face-value construct that it can be

considered as a variant of it. However, the status of the independent-place construct

cannot easily be judged without further research.

Finally, this study points to a theme in the data that appears to have great

relevance for the use of educational software, the place of feedback. Results of this

study show that participants using the software were able to access information about

represented numbers from more sources than participants using the physical blocks

could. In particular, electronic forms of feedback were used by the participants to

assess whether their block representations were accurate, and apparently also to

check their ideas about the numbers. Electronic feedback appears to have taken the

place of feedback from human sources, in particular the researcher, that were

accessed more frequently by participants using physical blocks. Implications are that

software that incorporates feedback mechanisms can give students valuable

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information about their use of electronic blocks that helps students adjust their ideas

and their manipulation of software artefacts being used to present information or

answers. There is a place for continued research to test the relative effectiveness of

various forms of electronic feedback; in light of these results research should be done

to assess the usefulness of feedback specifically for the purpose of representing

mathematical knowledge.

Summary of research recommendations.

The following recommendations are made for research topics that may

continue addressing certain issues discussed in this thesis:

1. Research into children’s learning about the base-ten numeration system,

and in particular the use of multiple conceptions by individual children.

2. Research into the effective use of base-ten representational materials,

including physical and electronic base-ten blocks.

3. Research into the independent-place construct, to investigate if it is a

separate category of children’s responses to number tasks, and to find

out how it is influenced by the use of various representational materials

and teaching practices.

4. Research into the effects of feedback mechanisms contained in

representational software for teaching mathematics, and how positive

effects from the feedback could be maximised.

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Appendix A – Design of Software used in the Study This appendix comprises a description of the software designed for use in this

study and a comparison between it and other software applications that share similar

design features.

Computer Software Incorporating Base-Ten Blocks Several researchers have described, or developed themselves, software that

generated pictorial versions of base-ten blocks for students to manipulate; three such

programs are discussed here (Champagne & Rogalska-Saz, 1984; Clements &

McMillen, 1996; P. W. Thompson, 1992). The various computer programs have at

least four features in common: Each one (a) represented numbers primarily as

pictures of base-ten blocks; (b) could present written symbols for numbers; (c)

allowed manipulation of the blocks, especially to regroup blocks in 10-for-1 trades,

and (d) modelled basic actions taken with physical base-ten blocks. In this section

comparisons are made between the software used in the study, called Hi-Flyer

Maths, and three similar computer programs: untitled software described by

Champagne and Rogalska-Saz (1984), Rutgers Math Construction Tools (1992), and

Blocks Microworld (P. W. Thompson, 1992).

Champagne and Rogalska-Saz (1984) described software that consisted of 15

lessons presented on-screen, each comprising an instructional and a practice

component. The computer presented users with questions, such as “How many cubes

are there?” (Figure A.1). Pictures of blocks could be “regrouped” to show

equivalence of different groupings, to answer the questions. The authors referred to

three representations used by the software: pictorial, verbal, and numerical. It needs

to be noted that the “verbal” representation was number name displayed on screen as

text; this is generally considered by mathematicians to be another form of numeral.

The software used in this study uses audio facilities of computers that have become

readily available only since the time when Champagne and Rogalska-Saz conducted

their study.

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Figure A.1. Screen view of on-screen tutorial question with block representations.

Note. From A. B. Champagne and J. Rogalska-Saz, 1984, Computer-based numeration instruction. In V. P. Hansen & M. J. Zweng (Eds.), Computers in mathematics education: 1984 yearbook, p. 48. Reston, VA: NCTM.

Rutgers Math Construction Tools (1992) presented pictures of base-ten

blocks on-screen that could be dragged to any position in the main working area,

“broken” into 10 of the next smaller block, or “glued” together to form a next-larger

size block. Symbolic representations available were the standard numerical symbol

(e.g., 3428), an expanded numerical symbol (e.g., 3000 + 400 + 20 + 8), or a number

and place name symbol (e.g., 3 thousands + 4 hundreds + 2 tens + 8 ones; see Figure

A.2). The screen could be divided into two sections, each with a symbolic

representation, for addition of two numbers or partition of one representation into

two subsets.

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Figure A.2. Partial screen image from Rutgers Math Construction Tools, showing block and symbol representations of a number.

Note. From Rutgers Math Construction Tools [computer software], 1992. NJ: Rutgers University.

P. W. Thompson (1992) used a computer program called Blocks Microworld

to investigate “students’ construction of meaning for decimal numeration and their

construction of notational methods for determining the results of operations

involving decimal numbers” (p. 125). Thompson’s software included pictures of

base-ten blocks and a symbolic representation in the form “1 cube 1 flat 11 longs 1

single = 1211” (Figure A.3). By using a “unit menu,” a user could nominate any one

of the four block sizes as the unit, so making possible the representation of decimal

fractions. The previous example then became “1 cube 1 flat 11 longs 1 single =

0.1211” (p. 129).

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Figure A.3. Screen view of Blocks Microworld showing block representation of a number, nominating a cube as one.

Note. From P. W. Thompson, 1992, Notations, conventions, and constraints: Contributions to effective uses of concrete materials in elementary mathematics. Journal for Research in Mathematics Education, 23, p. 127.

One particular feature of Blocks Microworld sets it apart from the other

software reviewed here. Other programs allowed users to manipulate the blocks on

the screen, by using the computer mouse; changes were then reflected in the

numerical symbol, where available. In contrast, P. W. Thompson’s (1992) software

allowed users to manipulate only the numerical symbol displayed on the screen,

which was then mirrored in the blocks; blocks themselves could not be directly

manipulated. Thompson justified this as an example of “constraints on students’

concrete actions in places that are likely to draw their attention to relationships

among meaning, notation, and expression” (p. 127). In light of many findings that

students are prone to manipulate symbols without reference to the numbers they

represent (e.g., Hart, 1989), Thompson’s idea may be brought into question.

However, Thompson’s results with students using the software were generally

encouraging, in that they seemed to make better sense of decimal fractions than the

group using only physical blocks.

Rationale Behind Hi-Flyer Maths As explained in chapter 2, there is widespread support for the idea that the

key to the development of higher level conceptual structures for numbers is making

connections between numbers and their referents (Fuson & Briars, 1990; Hiebert &

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Carpenter, 1992). Despite the popularity of base-ten blocks among teachers for 40

years, research shows that learning effects from their use are equivocal (Hunting &

Lamon, 1995; P. W. Thompson, 1992); this may be due to difficulties that students

have in making links between symbols and the blocks.

Clements and McMillen (1996) listed several advantages that computer

manipulatives can have over their physical counterparts. These included avoidance of

distractions, flexibility, dynamic linking of representations, encouraging problem

solving, and facilitating explanations. Clements and McMillen advised teachers to

“choose meaningful representations then guide students to make connections

between these representations” (p. 278). They recommended a broader view of

manipulatives than just physical materials and stressed the potential advantages that

computer software could offer to counter some of the problems of conventional

materials.

It is not surprising, given the dates of the papers reported in this section, that

the assumptions underlying the design of Hi-Flyer Maths and the methods of this

study are closer to those of Hunting and Lamon (1995) and Clements and McMillen

(1996) than those of Champagne and Rogalska-Saz (1984). This study is based on

the belief that students have to actively engage in learning activities in order to

benefit from them (Baroody, 1989). Students need to construct their own conceptions

of numbers through interacting with learning resources. As several writers have

pointed out (Clements & McMillen, 1996; Hunting & Lamon, 1995; P. W.

Thompson, 1994), mathematical meaning is not inherent in materials themselves.

Rather, the source of meaning is located “in students’ purposeful, socially and

culturally situated mathematical activity” (Cobb et al., 1992, p. 6).

The other major assumption underlying this software’s design is that in order

to build up accurate conceptual structures of numbers students need to make

meaningful connections between numbers and symbols, and between symbols and

referents. This topic has been adequately covered in chapter 2. This assumption is

operationalised in the inclusion of at least five different representations of a number

possible with the software, described in detail later in this appendix.

Features Though Hi-Flyer Maths shares a number of similarities with the other titles

described in the previous section, it incorporates several innovations that were not

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seen in any other software evaluated for use in the study. As these features are an

integral part of the rationale for the central investigation of this study, they are

described in some detail in the following section.

General Description The main screen (Figure A.4) presents a workspace with “source blocks”

from where blocks may be taken to form representations of numbers. Next to the

source blocks are buttons that enable access to various features. Most of the screen is

presented as a “place-value chart” with three columns, labelled “hundreds,” “tens,”

and “ones.”

Figure A.4. Main screen of Hi-Flyer Maths.

As blocks are placed on the place-value chart, the labels above the three

columns simultaneously show the number currently in each column. Similarly, if the

number symbol box or numeral expander are visible, changes to blocks are reflected

immediately by changes in the relevant text boxes.

Numerical Representations Any number from 1 to 1599 can be presented by the software in five different

ways: (a) as a canonical arrangement of base-ten blocks, (b) as a variety of non-

canonical arrangements of base-ten blocks, (c) as a written symbol, (d) as a numeral

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expander, or (e) as an audio recording of its verbal name. An important characteristic

of the different representations is that changes in the block representation are

mirrored by changes in the written symbol representations (if shown) virtually

simultaneously. In other words, as an extra ten-block is added to a representation, for

example, the written symbol(s) for the represented number change to reflect the

change in the blocks. The rationale behind designing the software in this way is that

in order to enable students to make necessary links between numbers and their

various representations, it is desirable to present changes (such as trading a ten for 10

ones) in all representations at the same time. This idea embodies the approach using

base-ten blocks recommended by Fuson (1992), that every change in written symbols

be mirrored in the concrete materials as close in time to the change as possible.

Block arrangements.

Base-ten blocks can be added to the display either by dragging a copy from

one of the source blocks, or by clicking on the relevant “plus” button next to the

source blocks. If a plus button is clicked, a new copy of the associated block is added

in the correct column, in an ordered arrangement: ones and tens are placed in rows of

10 blocks and hundreds are placed 3 across. In that way, every block placed is visible

and none are overlapped. If blocks are dragged, then the user can overlap them.

However, the block arrangement is “cleaned up” if blocks are regrouped.

Non-canonical representations are achieved by adding blocks to a column

until there are more than nine in a place. A temporary non-canonical arrangement is

also achieved by clicking on either the “show as tens” or the “show as ones” button.

When the show as tens button is clicked, any hundred-blocks present on the place-

value chart are changed into a representation of 10 ten-blocks (Figure A.5). When the

show as ones button is pressed, hundreds and tens blocks are changed into

representations of 100 or 10 ones, respectively. Simultaneously, the column labels

are altered to indicate the new represented number of blocks. When the mouse is

clicked again, the pictures of blocks and column labels are reverted to their previous

state.

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Figure A.5. “Show as tens” feature activated.

Written symbols.

The “show number” button may be clicked to reveal a “number name

window” showing the written symbol for the number represented by the blocks. The

symbol always refers to the total number shown by the blocks, whether they are

arranged canonically or non-canonically. For instance, if 3 hundreds, 14 tens, and 17

ones were placed on the place-value chart, the number name window would show

“457.” This would not change if the blocks were regrouped, or if the show as ones or

tens buttons were pressed.

A variation of the numerical symbol that may be displayed is the numeral

expander (Figure A.6). This is an on-screen version of a device made from light card

used in many primary schools to show equivalence of various representations of a

number. In the non-electronic version a number is written in blank spaces on the

card, with the names of the places hidden by folding a section of the card behind the

number spaces. Then the expander may be pulled open to reveal one or more of the

place names. For example, the number 518 could be shown as “5 hundreds 1 ten 8

ones,” or “51 tens 8 ones,” or as “518 ones.” The software reproduces this with a

picture of an expander that may have the place names hidden or revealed one place at

a time, by clicking on a place with the cursor. As with the number name window, the

numbers on the numeral expander do not change if the blocks are arranged non-

canonically; the numbers shown represent the value of the entire display of blocks.

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Figure A.6. Number name window and numeral expander displayed.

Verbal number names.

The software incorporates 36 audio files that are accessed by the computer to

enable any number from 0 to 1599 to be “read” aloud. For example, to read the

number name of 324, the computer plays the audio files for the words “three,”

“hundred,” “and,” “twenty,” and “four” in succession. Though the speed at which

each successive file can be accessed depends on the computer used, the gap between

words has been found acceptably short on most machines.

Verbal names can also be accessed for the numeral expander, so that the

numbers and place names shown are “read” as they are shown. Similarly the column

labels can be read individually if desired, as “four hundreds,” for example. Lastly,

clicking the “speech bubble” cursor onto one of the blocks can access the number

represented by blocks in each column alone. For example, if there were 5 ten-blocks

and 7 one-blocks, the computer would read the tens blocks alone as “fifty.”

Regrouping Blocks A primary use of base-ten blocks by teachers is to model each step in

computational algorithms, especially those for addition and subtraction. The

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processes involved as numerical quantities are altered in the steps of a computation

algorithm can be modelled using base-ten blocks. In common with other place-value

software, Hi-Flyer Maths will also demonstrate combining and separating of

quantities with the on-screen blocks, though the column labels will show only the

total number of blocks in each column, rather than two separate quantities.

When using physical base-ten blocks, regrouping in both directions is carried

out as a “trade” or a “swap.” In other words, to change 10 of a block for one of the

next larger block, or to change a block for 10 of the next smaller block, the blocks

must be traded or swapped for other blocks from a supply container. It is appropriate

to note at this point that some teachers and authors (e.g., Resnick & Omanson, 1987;

P. W. Thompson, 1992) refer to this process as “borrowing”; however, this is not an

accurate description of what is represented. As there is no “paying back” (as there is

when the equal addition algorithm is used), the use of the terms “trading” or

“swapping,” with their connotations of an equal transaction seem much more

appropriate. Consequently in the teaching phase of this study these terms were used

when referring to regrouping.

This process of 10-for-1 trading of blocks has been pointed out as precisely

the point where students may misunderstand what is happening in the numerical

realm. In the world of numbers 10 of one place is equivalent to one of the next larger

place: Thus one may imagine, for example, 38 being regrouped into 2 tens and 18

ones without pause. However, when using physical blocks this same transaction

would require the physical act of removing a ten-block to another place and replacing

it with 10 one-blocks. This process does not accurately mirror what happens with

numbers, and in children’s minds confusion may exist about what the trading means

in the numerical realm. The same problem does not exist with some materials where

ten or hundred material are not pregrouped (Baroody, 1990), such as Unifix™ cubes

or sticks, that may be combined or separated by the child, without having to do any

trading.

Hi-Flyer Maths, in common with each of the other computer programs

reviewed at the start of this chapter, allows combining and separating of blocks to be

dynamically displayed on the screen. Each program handles this process differently.

The software described by Champagne and Rogalska-Saz (1984) required users to

type instructions to initiate regrouping actions on screen. For example, typing “TH”

caused the display to regroup 10 ten-blocks to form a hundred-block. Rutgers Math

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Construction Tools (1992) includes a “hammer” tool that can be clicked on a block

to cause it to change into 10 of the next smaller piece and a “glue” tool that has to be

clicked onto 10 blocks of the same size to cause them to join together. P. W.

Thompson’s (1992) software required users to click on a digit (for example, the 6 in

“2 hundreds 6 tens 4 ones”) and then click on a “borrow” button, causing the display

to “explode” the relevant block into 10 blocks of the next smaller size.

The software in this study was designed to show dynamically on screen the

processes that are understood mathematically when quantities are regrouped. To

regroup a hundred-block or ten-block into 10 blocks of the next smaller place, a

“saw” tool is used that causes the computer to display the block being progressively

sawn into 10 pieces. The 10 new blocks are then moved into the correct place and the

column labels are changed to show the new number in each place. For example, in

Figure A.7, after the ten-block is sawn up, the 10 ones move to the ones place and

the labels change to show “5 hundreds, 2 tens, 14 ones.”

Figure A.7. A block is “sawn” into 10 pieces.

It is hoped that by showing an on-screen block being progressively sawn, the

software will provide a useful analogue for what happens with abstract numbers. It is

crucial that children understand that a ten is both a separate entity and a composite of

10 ones. This point is not necessarily clear to children when using base-ten blocks, as

they have to be traded. By showing blocks broken up and recombined, the software

shows an analogue of the regrouping process on numbers.

Regrouping in the other direction, from 10 smaller blocks to one larger block,

is achieved with a “net” tool that is placed over either the ones or the tens place and

clicked. If there are at least 10 in that place, the first 10 blocks are highlighted with a

272

surrounding red line, and then progressively moved together to form a new block that

is then moved to the next place to the left. If there are fewer than 10 in the place, a

message is given that there are insufficient blocks to regroup.

Addition and Subtraction In order to model addition and subtraction, the software allows up to nine

blocks to be added or subtracted consecutively in any column. The user presses

either the “add blocks” or the “subtract blocks” button, and the software responds by

displaying a box asking how many to add or subtract, and which place (Figure A.8).

If, when subtracting blocks, there are insufficient blocks to remove the chosen

number in that place, a message is displayed to that effect. Otherwise, the software

adds or subtracts the requested number of blocks. No further action is taken by the

software, so if a non-canonical arrangement results, then the user has the option of

regrouping blocks if desired.

Figure A.8. “Add blocks” requester.

The addition and subtraction features are designed to enable accurate

modelling of the written algorithms for these operations. Rather than simply adding

or subtracting the entire amount of the second addend or the subtrahend, the software

allows for working on one column only, as is done using the written algorithm. For

273

example, to model the algorithm for 72 - 34 using the software, the student has to

first regroup a ten-block for 10 one-blocks, before using the “subtract blocks” button

twice to subtract separately the 4 ones and 3 tens.

The previous paragraph raises the question of whether or not students should

invent their own algorithms, as recommended by several authors (e.g., Kamii et al.,

1993). The process described above mirrors the standard written algorithm, in which

the right-most place is subtracted first. However, research has shown that students

inventing their own addition or subtraction methods invariably choose to start on the

left (Kamii et al.). The software will support either method (and others also), as with

the blocks representation there is no need to record intermediate calculation steps

(the “carry marks”) of the written algorithms.

The question of how to teach students computation is beyond the scope of this

thesis. However, it is important to point out that a question is raised in presenting

students with a (physical or pictorial) block representation of the amounts to be

added or subtracted of why students should always start on the right, when the blocks

indicate that starting on the left would work as well.

Other Features

Requesting a number representation.

The usual method for putting out blocks to represent a number with the

software is by dragging a source block, or by clicking on the “add” buttons once for

each block. Once a user is familiar with this procedure, another quicker method is

available. There is a menu item named “Choose number” that brings up a text box

requesting a number up to 999. If the user types a number and clicks “OK,” the

software will put out a canonical display of blocks representing the typed number,

block by block. This feature is convenient for representing a number quickly for

further investigation.

A similar feature accessed via the menu bar is a random number requestor.

The user can choose a range from which the software will choose a random number

and then display a block representation of that number. The ranges available are 1-

19, 11-99, 101-499, 101-999, and tens from 10-990. This feature was not used in the

study.

If the number name window, the numeral expander, or both are visible when

either of the number-requesting methods are used, the symbols displayed change as

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each block is added. For example, if the number to be displayed is 126, the number

name window will display the numbers 100, 110, 120, 121, 122, 123, 124, 125, and

126 as each block is placed on the screen, starting with the left-most column.

Sounds.

Sounds are used in the software in three different ways: (a) as motivational

devices, (b) as reinforcement of metaphors, and (c) as information sources.

A few sounds are provided to add to the appeal of the software for children.

The opening screen shows a balloon picture, which when clicked causes an aeroplane

to move across the screen accompanied by the sound of a plane. There is a “bomb”

tool on the main screen, which when clicked on a block causes it to return to the

source area as a whistle sound is played. To remove all blocks from the screen at

once, a button is clicked; a short “reveille” is played as the software removes the

blocks. These sounds are assumed to add to the novelty effect of the software, but do

not have any other educational purpose.

The second category of sounds has a much more important role, in

reinforcing metaphors shown pictorially. The regrouping features of the software are

designed as an essential part of the modelling process to indicate block

transformations. As explained above, the software demonstrates block regrouping in

a dynamic way not possible with physical base-ten blocks. The idea of sawing a

(wooden) block is reinforced by the button icon (see Figure A.4), by the animation

shown as the block is changed (Figure A.7), and by the sound effect played. As each

smaller block is “sawn off” the larger block, the computer plays a short sawing

sound. Thus as a block is regrouped into 10 smaller blocks, the sound is played nine

times as the transformation takes place. In a similar way, as 10 blocks are placed next

to each other to form one larger blocks, a “gluing” or “zipping” sound is played as

each one is placed. This use of sounds is in accord with advice by Hereford and

Winn (1994), that sounds may be used to “refer metaphorically to qualities of

objects” (p. 217). One aspect of this study is to investigate whether the use of these

sounds assists students to develop accurate understandings of the numerical

processes and relationships.

The third group of sounds used in Hi-Flyer Maths is audio recordings of

numbers’ verbal names. It is common practice for teachers to ask students to link

verbal names, written symbols, and concrete materials representations as place-value

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concepts are taught (Fuson, 1992). Students need to be able to move among these

three forms of representation to develop accurate understandings of the numbers. The

software will play audio recordings of the number names in a variety of forms, as

described above. Again, it was hypothesised that this feature adds richness to the

information presented to students to assist them in constructing understandings of

multidigit numbers.

The other software reviewed in this chapter did not include this feature;

Champagne and Rogalska-Saz (1984) mentioned “verbal representations,” but

referred to text displayed on screen only. As mentioned above, the lack of this

feature in other programs may be due to technical restrictions. The current program

requires a computer sound card for the sounds to work, which was not widely

available until comparatively recently.

Summary Several computer programs have been developed that display pictorial

representations of numbers in the form of pictures of base-ten blocks. Each of the

programs includes written symbol representations and permits manipulation of on-

screen blocks to model regrouping of numbers. The programs differ in style of

presentation and the means by which users manipulate the blocks and symbols.

An original computer program has been written specifically for this study,

incorporating the same basic features of the other programs reviewed, as well as

several features not previously seen in such software. These innovations include

presentation of several novel representations of numbers and the use of animation

and sound to reinforce analogues of number processes. It is hypothesised that the

incorporation of these features is beneficial to students in developing accurate

conceptual structures for multidigit numbers. Specifically, the inclusion of features

not available in physical base-ten blocks is expected to produce different and better

conceptions of numbers and associated processes.

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Appendix B - Overview of Teaching Session Content for Interviews and Teaching Phase of Pilot Study

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Appendix C – Summary of Pilot Study Teaching Program Session Activities Session 1 [ ] Register students’ details on screen.

[ ] Record students’ details on paper. [ ] Introduce basic software features to students, allow students to experiment with them. Compare software with base-ten blocks. [ ] Show base-ten blocks to students. Revise the use of base-ten blocks to represent numbers. [ ] Ask students to show two-digit numbers with base-ten blocks, and the same numbers with the software. [ ] Ask students to show two-digit numbers with base-ten blocks. Show students numbers represented in one of three forms (Verbal name–Concrete representation–Written symbol), ask them to give the other two equivalent representations.

Session 2 Revise questions about two-digit numbers using the blocksa, asking students to make translations among three representations (Verbal name–Concrete representation–Written symbol). Add sequence of ones to a two-digit number. Regroup 10 ones for 1 ten. Discuss with students the idea of regrouping of two-digit numbers, in both directions. subtract sequence of ones from a two-digit number. Trade a ten for 10 ones.

Session 3 Revise questions about two-digit numbers using the blocks, asking students to make translations among three representations (Verbal name–Concrete representation–Written symbol). Regroup two-digit numbers in various ways. Introduction of numeral expander (on paper and on screen). Show two-digit numbers as tens and ones, or just as ones. Discuss with students pairs of numbers and how they are represented in symbolic form or with blocks. Compare and discuss which is larger, and why. Ask students to order three or more two-digit numbers in each representational form.

Session 4 Revise use of numeral expander. Compare pairs of two-digit number sense, discuss which is larger, and why. Order three or more two-digit numbers. Show students representations of two-digit numbers, ask students to count on or back by ones. Ask students to add pairs of two-digit numbers with and without regrouping, using blocks and written symbols.

Session 5 Revise regrouping of two-digit numbers. Revise use of numeral expander. Ask students to count on and back from chosen two-digit numbers, by ones and tens. Add two or more two-digit numbers. Subtract two-digit numbers with and without regrouping using blocks and written symbols.

Session 6 Add two or more two-digit numbers. Subtract two-digit numbers. Have students make blocks up to 100, by starting from a number of tens and adding tens one by one to reach 100. Prompt the students to see that

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Session Activities regrouping of 10 tens is needed when 100 is reached. Discuss with students the size of 100 and situations where it is used. Introduce written notation for hundreds place. Ask students to show three-digit numbers between 100 and 200 using the blocks. Ask how each is represented using both blocks and written symbols.

Session 7 Revise addition and subtraction of two-digit numbers with and without regrouping, using the blocks. Revise notation of three-digit numbers to include hundreds beyond 200. Ask students to make translations among Verbal name–Concrete representation–Written symbol with three-digit numbers.

Session 8 Revise translations among the three representations of three-digit numbers. Ask students to regroup three-digit numbers, regrouping tens or hundreds, in various ways.

Session 9 Revise translations among the three representations of three-digit numbers. Ask students to regroup three-digit numbers, regrouping tens or hundreds, in various ways. Re-introduce numeral expander for three-digit numbers. Compare pairs of three-digit numbers, using blocks and written symbols.

Session 10 Ask students to regroup three-digit numbers, regrouping tens or hundreds, in various ways. Compare pairs of three-digit numbers, using blocks and written symbols. Order three or more three-digit numbers. Count on and back from three-digit numbers, in ones, tens and hundreds. Ask students to add pairs of three-digit numbers, using written symbols and blocks.

Note. [ ] – Computer Group. [ ] – Blocks Group. a“Blocks” refers to on-screen blocks, or base-ten blocks, for Computer and blocks groups, respectively.

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Appendix D - Excerpt of Teaching Script of Pilot Study: Session 1

1-1. Hello, N__ and N__. You and I are going to spend some time together in the next few weeks, doing some interesting activities with MAB blocks. I will ask you some questions, and I would like you to answer them as well as you can. If you don’t understand anything, please ask me to explain it again. I think you will enjoy the activities I have for you. I will be asking you to reach answers together as a team. I am interested in how you reach your answers, and what you understand and don’t understand, so please ask me questions if you have any. Do you have any questions before we start?

1-2. Introduce students to software on computer. Help students to log on to software, entering name and date of birth.

Ask students for name and date of birth, write on record sheet.

1-3. Show base-ten blocks to students. Ask them what they know of them, and how they are used. If necessary, revise the value of each size block, and how to use them to represent a number.

1-4. Can you tell me what these blocks are called? Do you use them in class? What do you use them for? Can you show me how to use MABs to show the number 25? (Correct if necessary.)

1-5. Introduce the students to the basic features of the software: How to drag a block with the mouse, how to clear all blocks from the desktop.

1-6. This computer program shows pictures of MABs, and will help you to learn about numbers. You will find that the computer can do different things from the base-ten blocks, as we go through the lessons. Do you have any questions about the program?

1-7. (Both blocks and computer groups) Ask students to show a series of two-digit numbers with base-ten blocks: 16, 38, 60, 82.

1-8. e.g., Can you show me the number 16 using the base-ten blocks? . . . Explain what you have shown.

1-9. Ask students to show the same numbers with the software. e.g., Can you show me the same number using the computer? Help students if necessary, by showing how to drag blocks from the “source blocks” on the left of the screen.

1-10. Ask students to write symbols for various two-digit numbers: 73, 91, 45, 27, 13. Have students compare each other’s answers. Correct if necessary. e.g., Write the number 73 for me.

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1-11. Show students numbers written on cards, ask them to represent them using blocks: 57, 39, 84, 22, 17. e.g., Look at this number written here. I want you to show this number using the blocks. Are you sure that you are correct? Explain it to me.

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Appendix E – Audit Trail Example Note that each line in the audit trail records the time, the mouse action, the number represented by the software and the number of blocks in each column. Date Today: 17 June 1997 9:36:37 AM Session 9 Heron Group - Kelly & Hayden Session Start. 9:40:40 AM Click: 1 on Sounds Number: 0 Blocks: 0 0 0 9:41:14 AM Click: 1 on PickHun Number: 100 Blocks: 1 0 0 9:41:15 AM Click: 1 on PickHun Number: 200 Blocks: 2 0 0 9:41:21 AM Click: 1 on PickTen Number: 210 Blocks: 2 1 0 9:41:21 AM Click: 1 on PickTen Number: 220 Blocks: 2 2 0 9:41:22 AM Click: 1 on PickTen Number: 230 Blocks: 2 3 0 9:41:22 AM Click: 1 on PickTen Number: 240 Blocks: 2 4 0 9:41:25 AM Click: 1 on PickOne Number: 241 Blocks: 2 4 1 9:41:26 AM Click: 1 on PickOne Number: 242 Blocks: 2 4 2 9:41:26 AM Click: 1 on PickOne Number: 243 Blocks: 2 4 3 9:41:27 AM Click: 1 on PickOne Number: 244 Blocks: 2 4 4 9:41:27 AM Click: 1 on PickOne Number: 245 Blocks: 2 4 5 9:41:27 AM Click: 1 on PickOne Number: 246 Blocks: 2 4 6 9:41:28 AM Click: 1 on PickOne Number: 247 Blocks: 2 4 7 9:41:29 AM Click: 1 on PickOne Number: 248 Blocks: 2 4 8 9:41:46 AM Click: 1 on Show Number Number: 248 Blocks: 2 4 8 9:46:11 AM Click: 1 on Subtract Number: 248 Blocks: 2 4 8 9:46:20 AM Click: 1 on Expander Number: 248 Blocks: 2 4 8 9:46:29 AM Click: 1 on hunEx Number: 248 Blocks: 2 4 8 9:46:37 AM Click: 1 on tenEx Number: 248 Blocks: 2 4 8 9:46:41 AM Click: 1 on Number: 248 Blocks: 2 4 8 9:46:43 AM Click: 1 on tenEx Number: 248 Blocks: 2 4 8 9:46:48 AM Click: 1 on oneEx Number: 248 Blocks: 2 4 8 9:46:52 AM Click: cursor "speak" on Speech Number: 248 Blocks: 2 4 8 9:46:53 AM Click: cursor "speak" on hunDrop Number: 248 Blocks: 2 4 8 9:46:58 AM Click: cursor "speak" on Speech Number: 248 Blocks: 2 4 8 9:47:05 AM Click: 1 on hunEx Number: 248 Blocks: 2 4 8 9:48:14 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 9:48:20 AM Click: 1 on hunEx Number: 0 Blocks: 0 0 0 9:48:21 AM Click: 1 on tenEx Number: 0 Blocks: 0 0 0 9:48:24 AM Click: 1 on oneEx Number: 0 Blocks: 0 0 0 9:48:30 AM Click: 1 on Expander Number: 0 Blocks: 0 0 0 9:48:33 AM Click: 1 on Show Number Number: 0 Blocks: 0 0 0 9:49:52 AM Menu Item Selected: ChooseNumber, alias MakeNumMAB Number: 0 Blocks: 0 0 0 Number requested: 369 9:50:21 AM Click: 1 on Show Number Number: 369 Blocks: 3 6 9 9:50:26 AM Click: cursor "speak" on Speech Number: 369 Blocks: 3 6 9 9:50:31 AM Click: cursor "speak" on hunDrop Number: 369 Blocks: 3 6 9 9:50:50 AM Click: 1 on Expander Number: 369 Blocks: 3 6 9 9:50:51 AM Click: 1 on Expander Number: 369 Blocks: 3 6 9 9:51:15 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 9:51:18 AM Click: 1 on Show Number Number: 0 Blocks: 0 0 0 9:51:21 AM Menu Item Selected: ChooseNumber, alias MakeNumMAB Number: 0 Blocks: 0 0 0 Number requested: 541 9:52:12 AM Click: cursor "speak" on Speech Number: 541 Blocks: 5 4 1 9:52:13 AM Click: cursor "speak" on hunDrop Number: 541 Blocks: 5 4 1 9:52:25 AM Click: 1 on Show Number Number: 541 Blocks: 5 4 1 9:52:40 AM Click: 1 on Show Number Number: 541 Blocks: 5 4 1 9:52:41 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 9:52:53 AM Menu Item Selected: ChooseNumber, alias MakeNumMAB Number: 0 Blocks: 0 0 0 Number requested: 215

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9:53:23 AM Click: cursor "speak" on Speech Number: 215 Blocks: 2 1 5 9:53:25 AM Click: cursor "speak" on hunDrop Number: 215 Blocks: 2 1 5 9:54:12 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 9:54:15 AM Menu Item Selected: ChooseNumber, alias MakeNumMAB Number: 0 Blocks: 0 0 0 Number requested: 670 9:54:53 AM Click: cursor "speak" on Speech Number: 670 Blocks: 6 7 0 9:54:54 AM Click: cursor "speak" on hunDrop Number: 670 Blocks: 6 7 0 9:55:02 AM Click: 1 on Show Number Number: 670 Blocks: 6 7 0 9:55:08 AM Click: 1 on Show Number Number: 670 Blocks: 6 7 0 9:55:33 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 9:56:55 AM Click: 1 on PickHun Number: 100 Blocks: 1 0 0 9:56:56 AM Click: 1 on PickHun Number: 200 Blocks: 2 0 0 9:56:56 AM Click: 1 on PickHun Number: 300 Blocks: 3 0 0 9:56:56 AM Click: 1 on PickHun Number: 400 Blocks: 4 0 0 9:56:57 AM Click: 1 on PickHun Number: 500 Blocks: 5 0 0 9:57:01 AM Click: 1 on PickTen Number: 510 Blocks: 5 1 0 9:57:01 AM Click: 1 on PickTen Number: 520 Blocks: 5 2 0 9:57:02 AM Click: 1 on PickTen Number: 530 Blocks: 5 3 0 9:57:05 AM Click: 1 on PickOne Number: 531 Blocks: 5 3 1 9:57:05 AM Click: 1 on PickOne Number: 532 Blocks: 5 3 2 9:57:06 AM Click: 1 on PickOne Number: 533 Blocks: 5 3 3 9:57:06 AM Click: 1 on PickOne Number: 534 Blocks: 5 3 4 9:57:07 AM Click: 1 on PickOne Number: 535 Blocks: 5 3 5 9:57:07 AM Click: 1 on PickOne Number: 536 Blocks: 5 3 6 9:57:07 AM Click: 1 on PickOne Number: 537 Blocks: 5 3 7 9:57:08 AM Click: 1 on PickOne Number: 538 Blocks: 5 3 8 9:57:18 AM Click: cursor "speak" on Speech Number: 538 Blocks: 5 3 8 9:57:19 AM Click: cursor "speak" on Number: 538 Blocks: 5 3 8 9:58:29 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 9:58:40 AM Click: 1 on PickHun Number: 100 Blocks: 1 0 0 9:58:42 AM Click: 1 on PickTen Number: 110 Blocks: 1 1 0 9:58:42 AM Click: 1 on PickTen Number: 120 Blocks: 1 2 0 9:58:43 AM Click: 1 on PickTen Number: 130 Blocks: 1 3 0 9:58:43 AM Click: 1 on PickTen Number: 140 Blocks: 1 4 0 9:58:43 AM Click: 1 on PickTen Number: 150 Blocks: 1 5 0 9:58:43 AM Click: 1 on PickTen Number: 160 Blocks: 1 6 0 9:58:46 AM Click: 1 on TakeTen Number: 150 Blocks: 1 5 0 9:58:47 AM Click: 1 on TakeTen Number: 140 Blocks: 1 4 0 9:58:49 AM Click: 1 on PickTen Number: 150 Blocks: 1 5 0 9:58:51 AM Click: 1 on PickOne Number: 151 Blocks: 1 5 1 9:58:52 AM Click: 1 on PickOne Number: 152 Blocks: 1 5 2 9:59:39 AM Click: 1 on Toolbox Number: 152 Blocks: 1 5 2 9:59:41 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 9:59:52 AM Click: 1 on PickHun Number: 100 Blocks: 1 0 0 9:59:52 AM Click: 1 on PickHun Number: 200 Blocks: 2 0 0 9:59:53 AM Click: 1 on PickHun Number: 300 Blocks: 3 0 0 9:59:53 AM Click: 1 on PickHun Number: 400 Blocks: 4 0 0 9:59:53 AM Click: 1 on PickHun Number: 500 Blocks: 5 0 0 9:59:54 AM Click: 1 on PickHun Number: 600 Blocks: 6 0 0 9:59:55 AM Click: 1 on PickHun Number: 700 Blocks: 7 0 0 9:59:58 AM Click: 1 on PickTen Number: 710 Blocks: 7 1 0 9:59:58 AM Click: 1 on PickTen Number: 720 Blocks: 7 2 0 10:00:12 AM Click: 1 on supply Number: 720 Blocks: 7 2 0 10:00:13 AM Click: 1 on supply Number: 720 Blocks: 7 2 0 10:00:14 AM Click: 1 on TakeTen Number: 710 Blocks: 7 1 0 10:00:14 AM Click: 1 on TakeTen Number: 700 Blocks: 7 0 0 10:00:32 AM Click: 1 on supply Number: 700 Blocks: 7 0 0 10:00:32 AM Click: 1 on supply Number: 700 Blocks: 7 0 0 10:00:33 AM Click: 1 on PickOne Number: 701 Blocks: 7 0 1 10:00:34 AM Click: 1 on PickOne Number: 702 Blocks: 7 0 2 10:00:35 AM Click: 1 on PickOne Number: 703 Blocks: 7 0 3 10:00:35 AM Click: 1 on PickOne Number: 704 Blocks: 7 0 4 10:00:36 AM Click: 1 on PickOne Number: 705 Blocks: 7 0 5 10:00:37 AM Click: 1 on PickOne Number: 706 Blocks: 7 0 6 10:00:37 AM Click: 1 on PickOne Number: 707 Blocks: 7 0 7 10:00:37 AM Click: 1 on PickOne Number: 708 Blocks: 7 0 8 10:00:38 AM Click: 1 on PickOne Number: 709 Blocks: 7 0 9

285

10:00:38 AM Click: 1 on PickOne Number: 710 Blocks: 7 0 10 10:00:38 AM Click: 1 on PickOne Number: 711 Blocks: 7 0 11 10:00:38 AM Click: 1 on PickOne Number: 712 Blocks: 7 0 12 10:01:03 AM Click: 1 on Show Number Number: 712 Blocks: 7 0 12 10:01:11 AM Click: 1 on Show Number Number: 712 Blocks: 7 0 12 10:01:16 AM Click: 1 on Show Number Number: 712 Blocks: 7 0 12 10:02:35 AM Click: cursor "net" on Regroup Number: 712 Blocks: 7 0 12 10:02:39 AM Click: cursor "net" on Number: 712 Blocks: 7 0 12 10:02:55 AM Click: 1 on Toolbox Number: 712 Blocks: 7 1 2 10:02:58 AM Click: 1 on Restart Number: 0 Blocks: 0 0 0 10:03:06 AM Click: 1 on Show Number Number: 0 Blocks: 0 0 0 10:03:26 AM Click: 1 on PickHun Number: 100 Blocks: 1 0 0 10:03:26 AM Click: 1 on PickHun Number: 200 Blocks: 2 0 0 10:03:27 AM Click: 1 on PickHun Number: 300 Blocks: 3 0 0 10:03:27 AM Click: 1 on PickHun Number: 400 Blocks: 4 0 0 10:03:27 AM Click: 1 on PickHun Number: 500 Blocks: 5 0 0 10:03:28 AM Click: 1 on PickHun Number: 600 Blocks: 6 0 0 10:03:28 AM Click: 1 on PickHun Number: 700 Blocks: 7 0 0 10:03:28 AM Click: 1 on PickHun Number: 800 Blocks: 8 0 0 10:03:32 AM Click: 1 on PickTen Number: 810 Blocks: 8 1 0 10:03:32 AM Click: 1 on PickTen Number: 820 Blocks: 8 2 0 10:03:32 AM Click: 1 on PickTen Number: 830 Blocks: 8 3 0 10:03:34 AM Click: 1 on TakeTen Number: 820 Blocks: 8 2 0 10:03:46 AM Click: 1 on Show Number Number: 820 Blocks: 8 2 0 10:03:57 AM Click: cursor "speak" on Speech Number: 820 Blocks: 8 2 0 10:03:59 AM Click: cursor "speak" on hunDrop Number: 820 Blocks: 8 2 0 10:04:55 AM Click: cursor "glue" on Show Ones Number: 820 Blocks: 8 2 0 10:05:04 AM Click: cursor "glue" on Show Ones Number: 820 Blocks: 8 2 0 10:05:05 AM Click: cursor "glue" on hunones Number: 820 Blocks: 8 2 0 10:05:10 AM Menu Item Selected: , alias exit Number: 820 Blocks: 8 2 0 10:05:10 AM Session Ended.

287

Appendix F – Results of The Year Two Diagnostic Net, Used to Select Participants for the Main Study

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Note. Students are ranked roughly in order of criteria achieved on the previous year’s Year 2 Net (Queensland Department of Education, 1996). Numbered columns include criteria from Phases C & D of the test. All students in this population achieved all criteria in Phases A & B. Cells with solid shading indicate criterion was fully achieved. Cells with line shading indicate criterion was partially achieved. All judgements about student achievement were made by the Year 2 class teachers at the time. aIndicates unnamed student from the available population of Year 3 students, not selected for inclusion in the study. bThe results for Hayden were unavailable. Hayden’s Year 3 teacher assessed his mathematical achievement, relative to other students, to be at the level indicated on this table.

289

Appendix G – List of Participants

Pseudonym Gender

Mathematics Achievement

Level Number

Representation Age at start of study (yy:mm)

Amanda F High Blocks 08:08 Simone F High Blocks 08:03 Craig M High Blocks 08:03 John M High Blocks 08:02 Belinda F High Computer 08:01 Yvonne F High Computer 08:04 Daniel M High Computer 08:02 Rory M High Computer 08:00 Michelle F Low Blocks 08:02 Nerida F Low Blocks 07:07 Clive M Low Blocks 08:06 Jeremy M Low Blocks 08:06 Amy F Low Computer 07:09 Kelly F Low Computer 08:01 Hayden M Low Computer 08:02 Terry M Low Computer 07:08

291

Appendix H - Main Study Teaching Program

TABLE H.1. Overview of Teaching Program Tasks

Two-digit numbers Three-digit numbers

Introduce notation 24, 25, 26, 27 Representations of numbers: Concrete to Verbal

1

28

Concrete to Symbolic 1 28 Verbal to Concrete 2 29 Verbal to Symbolic 2 29 Symbolic to Verbal 3 30 Symbolic to Concrete 3 30 Regrouping 4, 5, 6 31, 32, 33 Use of numeral expander 7 34 Comparing 2 numbers 8, 9, 10 35, 36, 37 Ordering 3 or more numbers 11, 12 38, 39 Counting on and back by 1, 2 13, 14, 15 40 Counting on and back by 10 16, 17 41, 42 Counting on and back by 100 43 Addition 18, 19 44, 45 Subtraction 20, 21 Grouping ones 22, 23

Introduction [Instructions to participants:] For each task, show the numbers with the blocks. Discuss the question and the answer with the others in your group. Make sure everyone in the group agrees with each answer. Ask Mr Price if you need any help.

Task 1 - Representing numbers Look at the blocks put out by Mr Price. Say what number the blocks show. Write the number in your workbook. [Numbers were not printed on task cards provided to participants.]

a) 25 b) 61 c) 13 d) 40

Task 2 - Representing numbers Listen to the number given by Mr Price. Show the number with the blocks. Write the number in your workbook. [Numbers were not printed on task cards provided to participants.]

a) 28 b) 31 c) 19 d) 90

292

Task 3 - Representing numbers Show each number with the blocks. Say the name of each number:

a) 38 b) 72 c) 15 d) 80

Task 4 - Regrouping Show the number with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

a) 77 b) 23 c) 91 d) 58

Task 5 - Regrouping Show the number with the blocks. Now swap all of the tens for ones. How many ones do you need? Record what you have done in your workbook.

a) 21 b) 36

Task 6 - Regrouping Show the number with the blocks. If you were to swap all the tens for ones, how many ones would there be? Write your answer in your workbook.

a) 64 b) 89

Task 7 - Use of numeral expander Write the number on the numeral expander. Show the number with the blocks. Use the

expander to show the number in different ways. Write the number in two ways in your workbook.

a) 34 b) 96 c) 52

Task 7 - Use of numeral expander Show the number with the blocks. Turn on the numeral expander. Use the expander to

show the number in different ways. Write the number in two ways in your workbook. a) 34 b) 96 c) 52

Task 8 - Comparing 2 numbers Tommy and Billy were arguing about who had more marbles. Tommy had 48 marbles, and Billy had 62 marbles.

a) Who had more marbles? Show the numbers with the blocks. Explain your answer in your workbook.

293

Task 9 - Comparing 2 numbers Suzie and Margie were collecting stickers. Suzie had 20 in one book and 9 in a packet. Margie had 70 in one book and 3 in her pocket.

a) Who had more stickers? Show the numbers with the blocks. Explain your answer in your workbook.

Task 10 - Comparing 2 numbers GameBoys cost 51 dollars and basketballs cost 39 dollars.

a) Which is more expensive? Show the numbers with the blocks. Explain your answer in your workbook.

Task 11 - Ordering 3 numbers Kellie wanted to put her books in order of size on her bookshelf, so the book with the most pages was first, then the middle one, and then the one with the least pages. One book had 82 pages, one had 37 pages, and one had 88 pages.

a) Show the numbers with the blocks. Show in your workbook how Kellie should put the books on her bookshelf.

Task 12 - Ordering 3 numbers Simon has 75 toy soldiers, 57 toy cowboys, and 54 toy animals.

a) Show the numbers with the blocks. If Simon puts the group of toys with the most on the top shelf, the next group on the middle shelf, and the smallest group on the bottom shelf, where should he put them?

b) Show your answer in your workbook.

Task 13 - Counting on by 1s Penny is writing down the dates until her birthday on the 28th of the month. Today is the 4th.

a) What are the dates before her birthday? b) Show the numbers with the blocks. c) Say them, then write them in your workbook.

Task 14 - Counting back by 1s The Sunny Surfboard Company has 75 boogie boards left.

a) If one is sold, how many are left? b) Then how many if another is sold? c) Say all the numbers in order from 75 back to 60. Show the numbers with the

blocks. Write them in your workbook.

Task 15 - Odd and even numbers Fern Street has the even-numbered houses on one side, and the odd-numbered houses on the other side.

a) The Smith family lives at number 30 Fern Street. What are the house numbers on either side of their house?

b) The Jones family lives at number 71 Fern Street. What are the house numbers on either side of their house?

c) Show the numbers with the blocks. Write your answers in your workbook.

294

Task 16 - Counting on by 10s Michelle has a collection of 26 football cards.

a) If she buys another packet of 10 cards, how many will she have? b) How many with another 10? c) Show the numbers with the blocks. Keep adding tens until you reach ninety-six.

Write the numbers in your workbook.

Task 17 - Counting back by 10s Mr Walker has made 82 bread rolls to sell in his shop.

a) He sells a packet of 10 rolls. How many rolls are there now? Show the numbers with the blocks.

b) Then he sells another packet of 10 - how many are there now? c) Keep taking away tens. Write the numbers in your workbook.

Task 18 - Addition Classes 3L and 3M went in a bus to the zoo. There are 28 children in class 3L and 31 in class 3M.

a) How many children went on the bus? Show the numbers with the blocks. b) Discuss how to work it out with your group. Show how you work it out in your

workbook.

Task 19 - Addition A Space Race video game costs 75 dollars, and a set of batteries costs 19 dollars.

a) How much will the game and the batteries cost? Show the numbers with the blocks.

b) Discuss how to work it out with your group. Show how you work it out in your workbook.

Task 20 - Subtraction There are 95 soldiers on parade. The sun is hot, and 23 soldiers faint.

a) How many soldiers are still standing? Show the numbers with the blocks. b) Discuss how to work it out with your group. Show how you work it out in your

workbook.

Task 21 - Subtraction Mrs Perry has 83 dollars in her purse. She buys a coat costing 48 dollars.

a) How much money is now in Mrs Perry’s purse? Show the numbers with the blocks. b) Discuss how to work it out with your group. Show how you work it out in your

workbook.

Task 22 - Grouping ones Pat wants to buy some mints. Mints are sold in packets of 10, or one by one.

a) Will Pat have more if she buys six packets, or 45 single mints? b) Discuss how to work it out with your group. Show the numbers with the blocks.

Show how you work it out in your workbook.

295

Task 23 - Grouping ones Trent’s mum is making lolly bags for Trent’s birthday party. She has 2 rolls of 10 toffees, and 16 single toffees.

a) Show the number of toffees with the blocks. How many toffees are there? b) Does Trent’s mum have enough toffees to give 4 guests 10 toffees each? c) Discuss how to work it out with your group. Show how you work it out in your

workbook.

Task 24 - Hundreds place Show the number 40 with the blocks.

a) Put out another ten, and say the number’s name. b) Keep adding tens. Stop when you have 10 tens. c) What is this number called? Can you trade 10 tens? d) Keep adding tens. Stop when you reach two hundred. e) Write the numbers you made in your workbook.

Task 25 - Hundreds place Donna and James have picked 90 apples.

a) Show that number with the blocks. b) Donna and James pick more apples, one by one. Show each number with the

blocks. Stop at 99. c) Write the numbers 90 to 99 in your workbook. d) What is the next number? Write it in your workbook. e) Show the number with the blocks. Can you trade any blocks?

Task 26 - Notation Write four numbers between 10 and 99 in your workbook.

a) Choose one number. Explain what each digit means. b) Now write the number that Mr Price tells you. Explain what each digit means. c) Show what the number means with the blocks.

Task 27 - Notation Show the number 248 with blocks.

a) Write the number in your workbook. b) Explain the meaning of the 2, the 4 and the 8.

Task 28 - Representing numbers Look at the blocks put out by Mr Price. Say what number the blocks show. Write the number in your workbook.

a) 369 b) 541 c) 215 d) 670

296

Task 29 - Representing numbers Listen to the number given by Mr Price. Show the number with the blocks. Write the number in your workbook.

a) 538 b) 152 c) 712 d) 820

Task 30 - Representing numbers Show each number with the blocks. Say the name of each number:

a) 147 b) 394 c) 516 d) 470

Task 31 - Regrouping tens Show the number with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

a) 255 b) 932 c) 314

Task 32 - Regrouping hundreds Show the number with the blocks. Now swap one of the hundreds for tens. How many tens do you need? Record what you have done in your workbook.

a) 340 b) 627

Task 33 - Regrouping tens and ones Put out a handful of tens and ones blocks.

a) Write the number of tens and ones you have in your workbook. b) What number is shown by the blocks? Do you need to regroup any blocks? c) Write the number shown by the blocks in your workbook.

Task 33 - Regrouping tens and ones Put out more than 9 tens and more than 9 ones.

a) Write the number of tens and ones you have in your workbook. b) What number is shown by the blocks? Do you need to regroup any blocks? c) Write the number shown by the blocks in your workbook.

Task 34 - Numeral expander Write the number on the numeral expander. Show the number with the blocks. Use the

expander to show the number in different ways. Write the number in two ways in your workbook.

a) 381 b) 419 c) 158

297

Task 34 - Numeral expander Show the number with the blocks. Turn on the numeral expander. Use the expander to

show the number in different ways. Write the number in two ways in your workbook. a) 381 b) 419 c) 158

Task 35 - Comparing 2 numbers Mary and Harriet were arguing about who had more insects. Mary had 341 insects, and Harriet had 289 insects.

a) Who had more insects? Show the numbers with the blocks. Explain your answer in your workbook.

Task 36 - Comparing 2 numbers Frank and Lenny were collecting stamps. Frank had 250 in one book and 7 in his pocket. Lenny had 170 in one book and 34 in a packet.

a) Who had more stamps? Show the numbers with the blocks. Explain your answer in your workbook.

Task 37 - Comparing 2 numbers Super Fast computer games cost 432 dollars and Ultra mountain bikes cost 419 dollars.

a) Which is more expensive? Show the numbers with the blocks. Explain your answer in your workbook.

Task 38 - Ordering 3 numbers A zoo keeper wants to put three baby animals in three pens: a large pen, a middle sized pen, and a small pen. The animals’ masses are shown in the table below:

Panda 419 kilograms Elephant 823 kilograms Giraffe 485 kilograms

a) Show the numbers with the blocks. Discuss with your group how the animals should be penned. b) Show in your workbook how the keeper should put the animals in the pens.

Task 39 - Ordering 4 numbers At a sports meeting, Fiji has 79 competitors, New Zealand has 607 competitors, Indonesia has 398 competitors, and Australia has 624 competitors.

a) If the four teams march in order with the largest team first, and the smallest team last, which order should they be in?

b) Show the numbers with the blocks. Show your answer in your workbook.

Task 40 - Counting on by 1s As people arrive at the fun park, they are given a number that counts how many people have arrived that day. Jenny is given the number 283.

a) Show the number with the blocks. What will the next number be? Which number is next after that?

b) Keep adding numbers. Stop at three hundred and twenty-three. c) Say the numbers, then write them in your workbook.

298

Task 41 - Counting on by 10s At the cinema each person pays 10 dollars to see the film. There is 462 dollars in the cash drawer.

a) Show the number with the blocks. How much money will there be in the drawer after the next person pays their 10 dollars? Then how much after the next person?

b) Keep adding tens. Stop at 600. c) Write the numbers in your workbook.

Task 42 - Counting back by 10s The supermarket is selling Easter eggs in packets of 10. There are 380 eggs left, and one more packet is sold.

a) Show the numbers with the blocks. How many eggs are there now? How many when another packet is sold?

b) Keep taking away packets. Stop at 200 eggs. c) Say the numbers, and write them in your workbook.

Task 43 - Counting on by 100s Helen has 264 stamps in her collection. Her mother gives her a packet of 100 stamps.

a) How many stamps does she have now? Show the numbers with the blocks. b) Keep adding hundreds. Stop at 900. c) Say the numbers, and write them in your workbook.

Task 44 - Addition Children from two schools compete in a sports day competition. One school has 274 students, and the other has 315 students.

a) How many students are at the sports day? Show the numbers with the blocks. Show how you work out your answer in your workbook.

Task 45 - Addition A cattle farmer has 624 cattle in one paddock, and 193 in another. How many cattle are in the two paddocks? Show the numbers with the blocks. Show how you work out your answer in your workbook.

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Appendix I - Main Study Interview 1 Instrument

Year 3 Place Value Interview 1

Name:....... . .....................................................Class: ..............................

Date:......... / ........ / ........

Number Representation

Qu. 1 Place base-ten blocks on the desk in front of student, randomly arranged. Ask for the number that is shown by the blocks.

a) 3 t & 8 ones. b) 4 t & 12 ones. c) 2 h, 16 t, & 1 one.

Qu. 2 Place base-ten blocks on the desk in front of the student, randomly arranged. Ask the student to show the number with blocks. Then ask the student to show the number another way.

a) 5 t & 30 ones: show 28 b) 3 h, 16 t, & 60 ones: show 134.

Qu. 3 Show the student the written symbol for 136. Ask student to look at the following block representations in turn, and to say whether it equals 136:

a) 1 hundred, 2 tens, & 16 ones. b) 13 tens & 6 ones. c) 1 ten, 3 hundreds, & 6 ones.

Counting

Qu. 4 Ask the student to count on or back in ones or tens. In each case continue beyond the next necessary regrouping.

a) 74 - 1 etc b) 65 + 10 etc. c) 342 + 10 etc. d) 496 - 10 etc.

Number Relationships

Qu. 5 Ask student for numbers, and for explanations: a) a little bigger than 56. b) much bigger than 56. c) a little smaller than 56. d) much smaller than 56.

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Qu. 6 Show the student the numbers written on paper. Ask student to point to the larger number, and explain.

a) 27; 42 b) 174; 147

Digit Correspondence

Qu. 7 Show the student 24 bundling sticks. a) Ask the student to count them (correct if necessary), and to write the symbol for the

number. b) Circle first the “4,” and then the “2,” and ask “Does this part of your 24 have

anything to do with how many sticks you have? Please show me. How do you know?”

Qu. 8 Show the student 13 lollies. a) Ask the student to count them (correct if necessary), and to write the symbol for the

number. Ask the student to share the lollies evenly among four cups. b) Circle first the “3,” and then the “1,” and ask “Does this part of your 13 have

anything to do with how many lollies you have? Please show me. How do you know?”

Novel Tens Grouping

Qu. 9 Show the student some packets of chewing gum. Tell the student that each packet contains 10 sticks of gum. Ask the following questions:

a) If Carla has 6 packets and 4 other pieces of gum, how much chewing gum does she have altogether?

b) If Bruce has 3 packets and 17 other pieces of gum, how much chewing gum does he have altogether?

c) If Sam buys 5 packets and eats 8 pieces of gum, how many pieces does he have left?

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Appendix J - Main Study Interview 2 Instrument

Year 3 Place Value Interview 2

Name:....... . .....................................................Class: ..............................

Date:......... / ........ / ........

Number Representation

Qu. 1 Place base-ten blocks on the desk in front of student, randomly arranged. Ask for the number that is shown by the blocks.

a) 6 t & 7 ones. b) 3 t & 16 ones. c) 5 h, 13 t, & 2 ones.

Qu. 2 Place base-ten blocks on the desk in front of the student, randomly arranged. Ask the student to show the number with blocks.

a) 5 t & 30 ones: show 38. b) 3 h, 16 t, & 60 ones: show 261.

Qu. 3 Show the student the written symbol for 172. Ask student to look at the following block representations in turn, and to say whether it equals 172:

a) 1 hundred, 6 tens, & 12 ones b) 17 tens & 2 ones c) 1 ten, 7 hundreds, & 2 ones

Counting

Qu. 4 Ask the student to count on or back in ones or tens. In each case continue beyond the next necessary regrouping.

a) 96 - 1 etc. b) 42 + 10 etc. c) 263 + 10 etc. d) 681 - 10 etc.

Number Relationships

Qu. 5 Ask student for numbers, and for explanations: a) a little bigger than 73 b) much bigger than 73 a) a little smaller than 73 b) much smaller than 73

Qu. 6 Show the student the numbers written on paper. Ask student to point to the larger number, and explain.

a) 61; 38 b) 259; 295

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Digit Correspondence

Qu. 7 Show the student 37 bundling sticks. a) Ask the student to count them (correct if necessary), and to write the symbol

for the number. b) Circle first the “7,” and then the “3,” and ask “Does this part of your 37

have anything to do with how many sticks you have? Please show me. How do you know?”

Qu. 8 Show the student 26 counters. a) Ask the student to count them (correct if necessary), and to write the symbol

for the number. Ask the student to stack the counters evenly on six circles on a card.

b) Circle first the “6,” and then the “2,” and ask “Does this part of your 26 have anything to do with how many counters you have? Please show me. How do you know?”

Novel Tens Grouping

Qu. 9 Show the student some packets of clothes pegs. Tell the student that each packet contains 10 clothes pegs. Ask the following questions:

a) If Julie has 4 packets and 8 other clothes pegs, how many pegs does she have altogether?

b) If Frank has 5 packets and 13 other clothes pegs, how many pegs does he have altogether?

c) If Sarah buys 7 packets and loses 6 clothes pegs, how many pegs does she have left?

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Appendix K – Letter Requesting Consent by Parents or Guardians of Prospective Participants

13 May, 1997 Dear Mr/Mrs _______

re: MATHEMATICS RESEARCH STUDY

I am a PhD student and part-time lecturer in mathematics education at QUT.

Your son/daughter, _______, has been selected from the year 3 students at ___________ State School to take part in a research study being conducted by myself, from 2nd to 20th June 1997. Each child will be needed for 12 teaching sessions of approximately 20 minutes each, during school time, in a room separate from the classroom. The children will be learning about 2- and 3-digit numbers.

The school principal, Mr ________, has given his approval for the study to be conducted in the school. Anonymity of the students will be maintained in all reports of the study, except in reporting results of the study to the class teachers.

Please indicate your agreement for your child to take part in the study, or otherwise, on the pro forma below, and return it to the school as soon as convenient.

Yours faithfully

Peter Price PhD Student/Lecturer QUT -------------------------------------------------------------------------------------------------------------- Mathematics Research Study __________ State School, Semester 1, 1997 I hereby {give / do not give}* my permission for my son/daughter ________ to be involved in the above study. I understand that his/her anonymity will be protected, and that he/she may leave the study at any time. Signed: .........................................................

Parent/Guardian * Cross out whichever does not apply

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Appendix L – Coding Teaching Session Transcripts for Feedback

Coding of transcripts for feedback.

It is necessary at this point to describe how categories of feedback were

developed and how decisions were made about how to categorise each potential

incident involving feedback. When the videotapes from each session were

transcribed, initial readings of the transcripts revealed a large number of categories of

participant responses. It gradually emerged that one defining difference between the

two representational formats was the ways in which participants were able to find out

whether or not their answers were correct using each type of representational

material. In light of this finding, the transcripts were re-analysed in more detail,

looking specifically for incidents of feedback. A computer-based database file

designed by the author was used to record and categorise each incident indicating the

occurrence of feedback in the 40 teaching sessions (see Figure L.1). Feedback was

defined for the purposes of this thesis in this way:

Feedback is considered to have taken place if a participant received

information from another source indicating whether the participant’s thinking

about numbers was correct.

It is important to note that feedback is considered to have occurred whether or

not the recipient of the feedback requested it: The primary consideration is to note

incidents in which participants received information about their answers. It is

presumed that feedback helped participants to decide whether their ideas were

correct, and whether or not they should change their answers.

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Figure L.1. Data entry screen for feedback database.

As the analysis of feedback was conducted, a number of aspects of each

incident were coded for later analysis, as illustrated in Figure L.1. Firstly, details of

the participant, session number, and task were noted. The feedback itself was

categorised according to its source, the effect of the feedback for the participant

receiving it, and the response of the feedback recipient. Added to this was an

assessment of the status of the answer for which the participant was receiving

feedback: in other words, whether or not the recipient of the feedback already had an

answer, and whether or not the answer was correct, at the time the feedback was

provided. Finally a note was made of the transcript reference that referred to the

same incident, and a text comment was added briefly describing the incident.

Aspects of feedback that emerged from the data analysis—sources of feedback,

effects of feedback, and responses to feedback—are listed in Tables L.1, L.2, and

L.3.

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TABLE L.1. Source of Feedback

Feedback Description Source Teacher feedback Teacher Peer feedback Peer Check peer writing Peer Count on fingers Self Mental computation Self Check own writing Self Count/recount blocks Materials Count computer blocks Materials Check computer counter Electronic Check computer symbol Electronic Check computer verbal number name Electronic Check peer computer Electronic

Table L.1 reveals the range of feedback types available to participants. At

least 12 types of feedback were observed in the data, including 4 types of electronic

feedback provided by the software. It is interesting that the “count on fingers”

category was used by participants in computer groups only, though only twice per

group. Note that feedback accessed by counting computer blocks is considered to

have “Materials” as its source, because it involves merely looking at pictorial

representations of blocks, much as physical blocks may be counted. Other electronic

feedback requires the computer’s computational facility to provide feedback

information.

TABLE L.2. Effects of Feedback

Effect of feedback Quality Confirm answer Positive Contradict answer Negative Explain a wrong answer Negative Ask a question in return Neutral Provide an answer Neutral Give directions Neutral Counter-suggestion Neutral No effect Neutral

Table L.2 lists the evident effects of feedback observed during this analysis,

and determinations made of the quality of each effect: This is defined as the likely

effect that feedback would have on the participant receiving it, with regards to the

recipient’s confidence in the answer. In other words, if the feedback is likely to have

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encouraged a participant to retain the answer, whether correct or not, then the

feedback is said to be positive. If, on the other hand, the feedback is considered

likely to have encouraged its recipient to reject the answer, then it is said to be

negative in quality.

TABLE L.3. Responses to Feedback

Responses to feedback No visible response Change answer Reject feedback Seek further feedback Express satisfaction Reconsider question Laugh or smile Repeat answer Write or represent answer State answer Explain answer Follow directions

Table L.3 lists the various responses to feedback observed in this phase of

analysis. The purpose of descriptions of responses to feedback is to consider the

effect of each incident of feedback on the recipient’s actions. It is important to

consider how likely children are to accept feedback provided by either blocks or

software, based on their actions after receiving such feedback.

The following steps describe the method used to analyse videotapes for

incidents of feedback:

1. An incident in which a participant received information regarding an

answer was noted. This included occasions in which a participant

received an answer from another source, such as another participant or

from the computer column counters.

2. The status of the participant’s answer prior to the feedback was

determined: was the answer correct, incorrect, incomplete, or non-

existent? In some cases, the status had to be coded as “unknown,” as it

could not be determined from the data available.

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3. The source of the feedback was determined. In most cases, this was

obvious once the identification of a feedback incident had been made,

as the source of the feedback is a necessary part of the incident itself.

4. The effect of the feedback on the participant was determined. Again,

this was quite simple once an incident had been identified, as the effect

that the feedback had on the participant was generally clear from the

feedback itself. For example: Did the feedback confirm what the

participant had already stated or otherwise demonstrated as the answer?

Did it contradict the previous answer?

5. The main response the participant made to the feedback, if any, was

noted. If a participant responded in two different ways, the incident was

coded as a single incident, and the principal response of the recipient

noted. The category “no visible response” had to be used in incidents

where the response of the recipient of the feedback could not be

determined.

6. Note that if two participants each contradicted the other, then two

incidents of feedback were coded: one for each participant receiving the

feedback. Similarly, if the researcher gave feedback to an entire group

or to more than one participant, then that feedback was coded separately

for each recipient of the feedback.

As the analysis of feedback was conducted, it was discovered that participants

received feedback many times, and often with great frequency. Over the 40 teaching

sessions, lasting a total of approximately 1000 minutes, 1134 incidents of feedback

were identified. Whereas this indicates an average of little more than one incident per

minute, there were periods of time in which little feedback was evident, and others in

which feedback occurred rapidly over a short period. This is demonstrated by the

transcript excerpt in Appendix U, showing a short period of time in which several

incidents of feedback occurred, as the low/blocks group attempted to work out

75 + 19.

Clearly coding for feedback is not an exact process, as it requires subjective

judgements to be made about what feedback a participant received, what effect the

feedback had on the participant, and what the participant’s response was to the

feedback. Nevertheless, it is clear that incidents of feedback did occur during the

teaching sessions, and the author contends that its occurrence can be described

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reasonably accurately using the methods detailed here. It is important to point out

that some incidents of feedback must necessarily be missed, no matter what method

is used to identify them. At times during teaching sessions participants were almost

certainly thinking about the information presented to them without any outward sign

of the character of that thinking. The incidents of feedback described here can

include only incidents in which participants made some outward sign of receiving

some feedback from their environment. During periods when participants were not

obviously receiving feedback or responding to their environment, it was difficult to

decide if feedback was occurring. For example, participants using the computer spent

much of the sessions looking at the computer screens; similarly, participants using

blocks spent much time manipulating and looking at the blocks. Though the learning

environments for all participants clearly provided almost continuous feedback of

various types, it is only possible to identify aspects of incidents that are visible on the

videotapes. However, by careful and consistent use of observation techniques it is

asserted that valid and reliable descriptions of numbers and types of feedback have

been made that can be compared among participants and among groups.

An attempt was made to check the reliability of the identification and coding

of feedback incidents. A second experienced teacher was asked to view a selection of

four videotapes of teaching sessions, one of each group, and compare events

observed on the videotapes with the coding already carried out by the researcher. The

second observer confirmed every single incident coded by the researcher, but also

felt that other incidents were evident on the tapes that had not been coded. As the

reliability of the feedback identification and coding is not high when considered by a

second observer, it must be considered as just an indication of feedback activity that

took place rather than an objective measure of it. As in much qualitative research,

judgement about feedback incidents is necessarily subjective, relying on

interpretations of many subtle interactions that took place among participants, the

researcher and the materials. The identification of feedback incidents reported here is

the interpretation of the researcher that would be certain to differ from interpretations

made by other observers of the same data.

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Appendix M – Descriptions of Numeration Skills Targeted by Interview Questions and Criteria for Their Assessment

No. Skill Sub-Skill Description

Inter-view Ques-tion(s) Assessment Criteria

1a State number represented by blocks

Two-digit canonical block representation

(Q 1a) Correct number name stated; single miscount allowed.

1b Two-digit non-canonical block representation

(Q 1b) Correct number name stated; single miscount allowed.

1c Three-digit non-canonical block representation

(Q 1c) Correct number name stated; single miscount allowed.

2a Show blocks to represent given number

Two-digit number (Q 2a) Correct number of blocks shown, non-canonical representation allowed.

2b Three-digit number (Q 2b) Correct number of blocks shown, non-canonical representation allowed.

3a State whether block representation matches numerical symbol

Three-digit non-canonical block representation with >9 ones

(Q 3a) Correct number name stated; single miscount allowed.

3b Three-digit non-canonical block representation with 0 hundreds, >9 tens

(Q 3b) Correct number name stated; single miscount allowed.

3c Reject incorrect three-digit face-value block representation

(Q 3c) Blocks counted correctly; statement that blocks represented the stated number rejected.

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4a Recite verbal

number sequence

Count back in ones through two-digit numbers

(Q 4a) Number sequence correctly counting past change of decade; no omissions or insertions.

4b Count on in tens from two-digit number, past 100

(Q 4b) Number sequence correctly counting past 120; no omissions or insertions.

4c Count on in tens through three-digit numbers

(Q 4c) Number sequence correctly counting past change of hundred; no omissions or insertions.

4d Count back in tens through three-digit numbers

(Q 4d) Number sequence correctly counting past change of hundred; no omissions or insertions.

5a Nominate numbers relative to a single given number

Nominate numbers close to a given two-digit number

(Q 5a, c) Numbers stated that are each within 10 of the given number, or such that difference is no more than 25% of difference in related “far” example.

5b Nominate numbers far from a given two-digit number

(Q 5b, d) Numbers stated that are each at least 30 away from the given number, or such that difference is more than 4 times the difference in related “near” example.

6a State which of two written symbols represents the greater number

Two-digit numbers, ones digit of smaller number > either digit of larger number

(Q 6a) Correct number stated as greater, incorrect counter-suggestion(s), if any, not accepted.

6b Three-digit numbers, respective tens and ones digits reversed

(Q 6b) Correct number stated as greater, incorrect counter-suggestion(s), if any, not accepted.

7a Count collection of 10-40 objects, show referent for each digit

Two-digit number (Q 7) Correct number of objects shown as referent for each digit, incorrect counter-suggestion(s) rejected.

7b Two-digit number with misleading visual cues

(Q 8) Correct number of objects shown as referent for each digit, incorrect counter-suggestion(s) rejected.

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8a Mental computation in a novel tens grouping situation

Add a number of tens and a number of ones

(Q 9a) Correct answer stated; un-prompted self-correction allowed.

8b Add a number of tens and 11-19 ones

(Q 9b) Correct answer stated; un-prompted self-correction allowed.

8c Subtract a number of ones from a number of tens

(Q 9c) Correct answer stated; un-prompted self-correction allowed.

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Appendix N – Transcript of Interview 1 Question 6 (a) with Terry

Qu. 6 (a) Show the student the numbers ‘27’ and ‘42’ written on paper. Ask student to point to the larger number, and explain.

Terry: I’m still … Oh! … I can tell by the even numbers. This one [‘42’] is bigger

because it’s even and this one [‘27’] is smaller because it’s odd.

Interviewer: OK what is this number [‘42’] here?

Terry: 42.

Interviewer: And this one?

Terry: 27.

Interviewer: And 42 is bigger because it’s even?

Terry: Yup.

Interviewer: And that’s [‘27’] smaller because it’s odd?

Terry: Yup.

Interviewer: Supposing this one here was … 57.

Terry: Yeah.

Interviewer: That’s still odd.

Terry: Yeah.

Interviewer: Would it still be smaller than that?

Terry: Yeah.

Interviewer: OK. So all even numbers are bigger than all odd numbers?

Terry: Yup.

Interviewer: How do you know that’s even?

Terry: Because it’s got a ‘2’ at the end.

Interviewer: Uh-huh. And how do you know this one’s odd?

Terry: Because you got a ‘7’ at the end.

Interviewer: All right, what if they were both even? Supposing this one [‘27’] was 26,

which would be bigger then?

Terry: This one. [‘26’] ‘Cos it would be 26.

Interviewer: 26 would be bigger? Why is that?

Terry: ‘Cos they are both even and this one is only 42 and the other one’s 26.

316

Interviewer: OK, and why would 26 be bigger?

Terry: ‘Cos it’s got a ‘6’ at the end.

Interviewer: All right, and the ‘6’ is bigger than …?

Terry: Oops! It’s this one [‘42’] because this one is away from 20.

Interviewer: I think we’d better write these down.

Terry: Ah.

Interviewer: Because we just want to be sure we both know what we are talking about.

[writes ‘26’ and ‘42’] 26 and 42. Which one is bigger?

Terry: 42.

Interviewer: Uh-huh. Why?

Terry: ‘Cos … ‘cos it’s one way [pause] it’s … like 46 is in the 20s …

Interviewer: 26.

Terry: 26 [correcting himself] is in the 20s, and 42 is in the 40s, and it’s all missing

the 30s so … so 42 is bigger?

Interviewer: How do you know the 40s are bigger than the 20s?

Terry: ‘Cos if you count in 10, 20, 30, 40, it’s a long way away from 20.

Interviewer: Comes after it you mean?

Terry: Comes after it.

Interviewer: OK, that’s a good answer, but with this one [‘27’] because that’s odd …

Terry: Yeah.

Interviewer: That’s smaller because it’s odd?

Terry: Yup.

Interviewer: All right. [pause] Right, right, right, right. What about … all right let me ask

you this one then. [writes ‘57’ and ‘42’] Look at those two numbers. What’s

this one here now? [‘57’]

Terry: 57.

Interviewer: And? [‘42’]

Terry: 42.

Interviewer: Which one is bigger now?

Terry: Oh, you just gave me an odd number. This one [‘42’] because it’s even.

Interviewer: 42 is bigger because it’s even?

Terry: Yep.

317

Interviewer: Uh-huh. But this [‘57’] is in the 50s isn’t it?

Terry: Yeah.

Interviewer: Don’t the 50s come after the 40s?

Terry: Yeah … oh yeah?! So this one is actually even … um … more ‘cos it’s after?

Interviewer: Right.

Terry: So are we actually talking in after and not before?

Interviewer: [Laughs] We’re talking about which one’s bigger. Which one means the

bigger amount.

Terry: Oh.

Interviewer: 57 or 42.

Terry: 57.

Interviewer: Because …

Terry: Because it’s got … it’s one way away from … it’s right after 40.

319

Appendix O – Transcript of Interview 2 Question 6 (a) with Hayden

Qu. 6 (a) Show the student the numbers ‘61’ and ‘38’ written on a card. Ask the student to point to the larger number, and to explain.

Interviewer: Can you tell me which of these numbers is larger?

Hayden: [Points to ‘61’]

Interviewer: What number is that?

Hayden: 61.

Interviewer: Uh-huh, and how do you know that’s bigger?

Hayden: Because it’s 6 … 38 takes shorter and 61 takes longer.

Interviewer: If you’re counting you mean?

Hayden: Yeah.

Interviewer: OK, what about the numbers that are in it? Does that tell you anything?

Hayden: No, it doesn’t … like it still doesn’t mean that it’s got an ‘8’ on the end and

it’s got a ‘1’ on the end [points to respective digits] …

Interviewer: Uh-huh.

Hayden: … because that’s um … like that … like if you get 1, 2, 3, 4 … like 10, 20, 30,

40… no 10, 20, 30 and you just count to 8 …

Interviewer: Uh-huh.

Hayden: …in the 30s, like it’s only the 38.

Interviewer: Uh-huh.

Hayden: And if you count the 61 it’s a 60 one.

Interviewer: All right, so if you are counting to 60 it would …

Hayden: … like take longer.

Interviewer: Uh-huh, that’s a good answer.

321

Appendix P – Transcript of Low/Blocks Group Answering Task 28 (a)

Task 28 (a) Look at the blocks put out by Mr Price [369]. Say what number the blocks show. Write the number in your workbook.

Michelle: [Puts one hand on top of the hundreds blocks, says immediately] 300.

Teacher Please don’t say them aloud, Michelle.

Clive: [Looks dejected again, is not moving to touch or count the blocks.]

Teacher: Girls, please keep your hands off blocks, so that boys can see them.

Michelle: [Puts her hand on the hundreds again.] 300 …

Teacher: Work out the whole number, Michelle.

Clive: [Moves hundreds so he can see the tens and ones. Frowns, apparently counting

the blocks.]

Teacher: Girls, please take your hands off the blocks. (It appears that the girls were

finding this too difficult, that they needed to put their hands on the blocks in

order to keep track of their count.) [Puts out another copy of the block

representation for the boys.]

Nerida: [Whispering, as she counts the ones] … 362, 363, 366, 367, 368 …

Michelle: [Does her own counting.] 3 hundreds… [She puts her hand to her forehead,

and apparently finding the next step difficult.] 300, 400, 500, 600, 700, 800,

900, 101, 102, 103, 104, 105, 106, 107, 108, 109. 109. 109. [To teacher,

quietly] We got 109.

Clive: [Counts hundreds, then tens. He starts to count ones, stops and frowns, moves

hundreds] That’s 300 … [He counts the tens quietly aloud by tens, then counts

on by ones to 69.] 369 [Quietly; writes in his workbook. He writes ‘3,’ counts

tens again, writes ‘69.’]

Teacher: Jeremy, what is the number?

Nerida: [Finishes counting the blocks under her breath, reaching ‘359.’ Writes her

answers straight away in her workbook, apparently before she forgets what the

number is.]

Michelle: [Looks surreptitiously at Nerida’s book. She starts to re-count the blocks,

apparently confused about what she should count after 300.] 300 [Very

quietly; looks away. She starts counting the ten-blocks, adding them to the

322

hundreds, counting by ones. It is very difficult to hear the numbers that she

stands, but she includes the numbers “7, 8, 9, 22, 23, 24.” She stops.] Oh, no!

Jeremy: [Looks at blocks for a while.] 300 … and … [counts, while nodding his head,

without touching blocks. Stops for a long while.] 30 … thirty one hundred …

thirty one hundred … thirty one hundred and …

Michelle: [Distracted by someone entering the room, stops counting.]

Teacher: Children, what are your answers?

Michelle: 30 …

Nerida: 359.

Clive: [Confidently] It means 3 hundreds, and 6 tens and 9 ones.

Girls: [Say they have a different number.]

Teacher: [Asks children to confirm that there are 3 hundred, 6 tens and 9 ones. The

boys agree, but Nerida shakes her head, and Michelle moves her head slightly,

in a nondescript sort of way, neither shaking it nor nodding. The teacher asks

what they disagree with.]

Nerida: We have a different number of tens. [The girls recount their ten-blocks by

ten.] There are 60 tens.

Michelle: Now, do you …

Michelle: 60.

Nerida: 60.

Teacher: Do you mean 60, or 6?

Michelle: 6 tens.

Nerida: There are 6 tens.

Teacher: What is the number altogether?

Clive: [Loudly] 369.

Nerida: 369.

Michelle: [Shakes her head.] I don’t reckon.

Teacher: Michelle, what do you think the number is?

Michelle: 369.

Teacher: Girls, don’t just change your minds just to agree with Clive.

Michelle: [Is having trouble reading the blocks. She starts to count on from 300:] 300,

40 …

Teacher: [Stops her.]

323

Michelle: No, 140, 150, …

Teacher: [Stops her again] It is not 30, but 300.

Michelle: 300, 4 … [She counts the tens alone, then counts on ones, to 69.]

Nerida: [To Michelle] It is 369.

Michelle: [Writes in her workbook. She repeats the first place of the number several

times as she starts. It appears that Michelle could not keep the three numbers

in her mind at once. Clive also appeared to have this same difficulty.]

Teacher: Now, write the number 369, rather than just the values in each place.

325

Appendix Q – Transcripts of Task 4 (a) from 4 groups

Task 4 (a) Show the number 77 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

High/computer: Yvonne: [Reads card]

Belinda: I don’t understand that. [Smiles]

Teacher: Do it one step at a time.

Belinda: [Reading card again, holding mouse] “Show the number with the blocks.”

Yvonne: Can I just use your pencil for a minute? [Borrows Belinda’s pencil to draw a

line in her workbook.]

Yvonne:? 77

Daniel: 77

Belinda: [Adding ten-blocks] 1, 2, 3, 4, 5, 6, 7

Yvonne: The first one … [indistinct] [Watches both computers, shifting gaze from one

to the other as Rory uses mouse]

Belinda: [Reading card again] Now swap one. Um, 70.

Yvonne: Task …

Belinda: [Adding one-blocks] 1, 2, 3, 4, 5, 6, 7

Computer1: 77

Daniel: [Pointing to screen] OK, put … [indistinct]

Belinda: [Reading card again] “Now swap one of the tens for ones.”

Teacher: Don’t start again, Rory, you haven’t finished.

Belinda: “How many ones do you need? Record what you …” I don’t get that. Now, go

to “Take away” [Looks back at teacher] mmm …

Yvonne: Take away one from the ones. … Don’cha? [Looks back at teacher]

Belinda: Oh, no.

Rory: [Clicks on “Start again,” then starts to add ten-blocks again]

Teacher: No, no. Don’t take the number away. Put the 77 back.

Rory: [Puts blocks back]

Yvonne: Swap one of the tens for ones.

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Teacher: OK, do you know how to do that?

Belinda: Nope

Yvonne: Nope… We done that before, didn’t we?

Daniel: Can we put only 3 on?

Yvonne: [Points at screen] Look!

Belinda: Oh! [understands] “remove block.”

Yvonne: No, you can’t. It’s only got 7.

Belinda: Oh.

All: [Unsure of how to continue. All look at teacher]

Teacher: You know what “swap” means, don’t you?

All: Yeah.

Teacher: OK, well you have to swap.

Daniel: It’s like trading.

Teacher: Yes. Swap one of the tens for ones.

Yvonne: Oh!

Belinda: I don’t get that.

Teacher: Make a swap. Swap a ten for …

Yvonne: [Points and taps pencil on screen on buttons, then on blocks]

Belinda: Huh?

Yvonne: [Points at screen in two places again] For ones [indistinct]

Belinda: Yeah. [Still looks puzzled]

Daniel: Do you add 3 onto the ones?

Teacher: No, we don’t want to make it into ten ones, we’re going to take one of the tens

and swap it. [Computer says there are too many blocks on screen, which will

be cleared] Oh, no. Too many blocks on the page. Click “OK.” I think we’ll

start yours again. [restarts program]

Daniel: [Points at screen] OK, use the saw. Saw one of them big ones.

Belinda: We need to put a bigger number on.

Yvonne: I know.

Rory: No.

Daniel: You have to. [Nods]

Rory: [Clicks saw on ten-block]

327

Yvonne: See if you can … [Makes buzzing noise with mouth]

Daniel: We’ve done ours. [Looks at teacher]

Computer2: [Saw sound]

Belinda: Done it.

Teacher: OK, can we have the sound turned off if you don’t mind? Just not to disturb

the other class. Is that what the instructions were?

Belinda: Yep, swap one.

Teacher: Mmm? Read your instructions here.

Yvonne: [Looks at teacher] Nope, nope [shakes head] nope.

Daniel: Swap one, yep.

Belinda: [Looking at screen] 77! There’s still 77! Cool.

Yvonne: No, there’s sixty …

Rory: It’s 77.

Yvonne: Oh, yeah it is [laughs].

Computer: 77.

Belinda: It sounds like Mr Price!

Yvonne: So do we have to write 77 in here?

Teacher: Well, you’ve got to answer this question here, now. Have you done what it

says “now swap one of the tens for ones”?

Yvonne: Yep.

Belinda: Yep.

Teacher: “How many ones do you need?”

Belinda: I don’t get that.

Teacher: [Under breath] Neither do I. Who wrote that? [laughs]

Belinda: Who writ that?

Teacher: When you swap a ten for ones, how many tens do you swap it for?

Daniel: 10.

Belinda: 10.

Teacher: Yeah. The computer does it for you.

Belinda: Cool.

Teacher: The other group only use the MABs and they have to work it out for

themselves.

328

Belinda: So it’s going to be 10.

Teacher: — OK. Record what you have done in your workbook.

Belinda: So, you’ve got to record the 10.

Daniel: So I’ve done …[indistinct]

Teacher: What did you have to start with?

Belinda: 77.

Teacher: Right, and what have you got now?

Yvonne: [To Daniel, looking at his book] Why did you write 77?

Belinda: 77!

Teacher: So what’s the difference?

Belinda: Nothing.

Rory: Nothing.

Teacher: Well, there’s a difference in the blocks, isn’t there?

Belinda: [Starts to write in workbook] Oh, 6.

Teacher: I want you to write it down somehow, to show how it’s different now from

what it was before.

Daniel: [Looking at teacher] Write ‘77’?

Belinda: Write 60 … 6 …

Teacher: Well, what’s the difference on the screen?

Belinda: 6 blocks …

Daniel: There’s 6 tens and 17 ones.

Yvonne: Yeah.

Teacher: And what did you have before?

Yvonne: 77 one …

All: 7 tens and 7 ones.

Teacher: OK. Can you write that down so it makes sense, that you had 7 tens and 7

ones, and now you’ve got 6 tens and 17 ones?

Belinda: How would you write that? You could write “six plus seventeen.”

Teacher: It’s not just 6, is it? It’s 6 tens plus 17. You could do that. Write down what

you had before, though, first.

Belinda: Oh. [Rubs out previous writing]

Yvonne:? Aw, now it won’t … [indistinct]

329

Belinda: Or 6 plus ten.

Yvonne: [To Belinda] Can I swap pencils? Mine’s not sharp.

Belinda: No.

Yvonne: Oh. [Keeps writing]

Belinda: [Writing] 77, full stop.

Daniel: [To Yvonne] Mine’s the same with yours. Mine’s the same.

Belinda: [Writing] Seven tens [indistinct] equals 77.

Yvonne: [Looking at Belinda’s workbook] 77, zero six nine [indistinct].

Belinda: That’s a full stop.

Yvonne: Oh, is it?

Belinda: 6 plus 17 equals 7, 77!

[Lots of indistinct speech from several children]

Teacher: [to Belinda, pointing to her workbook] Now, is that right?

Belinda: [Shrugs shoulders] I don’t know.

Teacher: Six plus 17 - is that 77?

Belinda: Uh-huh [confirming correct] ‘cos I checked it.

Yvonne: [Looks at Belinda]

Belinda: [Pause for 6 seconds] [Not so sure] I think. [Clicks on mouse to read number.]

Teacher: OK. Now write down what you did.

Computer: 77

Belinda: Yep. It’s right.

Daniel:? We … oh.

Teacher: You don’t have to write it all in a sentence, but write down what blocks you

had, and what you have now.

Yvonne: [sighs]

Belinda: I shouldn’t write these things. I reckon it looks too silly. [Uses rubber]

Yvonne: [Watching Belinda] Here, I need one.

Teacher: OK. Now what have you written? I want you to show each other and see that

you all agree with what you’ve written, because this is a group question. Well,

they’re all group questions.

Yvonne: Well, shouldn’t we all do it the same?

Belinda: [Looking at Yvonne’s workbook] We should be all … nuh.

330

Teacher: Well, you all have to think for yourselves. And you have to check to see

whether you all agree.

Belinda: [Swapping two workbooks] You check my answers, and I check your

answers.

Teacher: No, no, no, I don’t mean that. I mean just show it to her. You’re not going to

mark it. I’ll mark it when I go home.

Yvonne: [Laughs]

Belinda: Aw! [disappointed]

Belinda: [Comparing two girls’ workbooks] Yeah, that’s good. OK, that’s good.

Daniel: I agreed with yours, Rory.

Teacher: That’s very good.

Daniel: He’s got that. [Pointing at Rory’s book]

Teacher: He’s writing it in a sentence - that’s a good way of doing it.

Yvonne: [Looks bored, flops in chair] Do you have to start again?

Teacher: OK, well I’m going to have to challenge you children, because I don’t think

that 6 plus 17 is 77.

Belinda:? I do.

Teacher: I think 6 plus 17 is 23.

Yvonne: Hey?

Teacher: [Counting on fingers] 17, 18, 19, 20, 21, 22, 23. It’s not 77.

Yvonne: Oh. [seeing problem with answer]

Belinda: Aw. [disappointed] Oh, yeah but it says 6 plus 17.

Teacher: It doesn’t say 6 plus 17.

Daniel: Oh no, I’ll do it the same way as Rory now. [laughs] ‘Cos that’ll be the same

as Rory.

Teacher: Well, the idea isn’t just to be the same as Rory, it’s to make sense of what

we’re doing. You see, the whole point of this is, Do you understand the

numbers? Do you understand what the blocks are showing?

Yvonne: I need your rubber.

Teacher: [Laughs]

Belinda: It’s 14.

Teacher: I’m sorry; what’s 14?

Belinda: [Points at screen] That.

331

Daniel: Can I borrow the rubber after you?

Rory:? I’m going to write 36.

Yvonne: I’m going to write 23.

Teacher: I think you’d better talk about it between yourselves. I don’t want to tell you

the answer, unless I have to. I want you children to work it out yourselves.

Belinda: I want my rubber. [Reaches across to Daniel]

Yvonne:? I just changed mine to 23.

Teacher: Children, I need you to discuss the question, and work out what you’re going

to do.

Teacher: You need to talk about it with the others. Don’t just write down something on

your own. I want you to make sense of this, otherwise there’s no point going

on to hundreds, if we’re having trouble with the tens and ones.

Yvonne: What is it? [Reads card] “Show the number with the blocks.”

Belinda: It does not equal … something. It does not equal something …

Teacher: Well, first of all are the blocks showing 77?

Yvonne: Yes.

Daniel: Yes, they still are.

Teacher: They still are.

?? mmm [confirming]

Teacher: So, when we did that swap, it’s still 77?

Daniel: Yeah.

Belinda: Yeah! [sounding surprised that it should be questioned]

Teacher: You agree with the computer, that it’s still 77?

Daniel: Yes.

Teacher: You can see the 77 still there?

Daniel: Yeah.

Belinda: Hang on, hang on. Six…ty

Teacher: You explain to me how that’s 77, ‘cos I don’t see 70, and I don’t see 7.

Belinda: I do. ‘Cos just that’s one… There’s a ten there, and there’s another 7, and

that’s 60, 7, 7! So it’s right.

Teacher: OK. Can you write that down so it makes sense? Look, I tell you what. Let’s

go backwards. We’ll get these back to 7 tens and 7 ones again, by regrouping

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with the net, OK? [Uses “net” tool to regroup ten ones on each computer]

Right, that’s where we started.

Belinda: Yeah.

Teacher: OK. Now do the swap again, the trade, with the saw.

Daniel: [Both boys go to use mouse; Daniel continues] Can I do it, this one?

Belinda: Depose [sic].

Teacher: Now watch, ‘cos I want you to see what happens.

Belinda: Look, it’s right. It’s right.

Yvonne: 77 again?

Belinda: No, six… Yeah, it’s right. It is. [She appears to feel intuitively that it is still

77, but is unable to explain it to Yvonne.]

Yvonne: [Pointing at computer] Yeah, but it hasn’t got the zero! [Looks at other

computer also.]

Belinda: Yeah!

Teacher: Why doesn’t it have the zero?

Belinda: It’s, it’s… I, I think it’s … Hang on, 17, 18, 19, 20. 21, 22, 23. 23!

Yvonne: [Laughs]

Belinda: Mmmm. No, but that’s 60. [Looks at teacher]

Yvonne: It’s not - it hasn’t got the zero. It’s supposed to have the zero.

Belinda: [Pointing at ten-blocks] No - 10, 20, 30, 40, 50, 60.

Yvonne: [Looks at teacher] Yeah!

Daniel: Oh, yeah! [understands; looks at teacher]

Teacher: Don’t keep looking at me. Is that right? Does it make sense?

Daniel: Yeah.

Belinda: Yes. [Definite]

Yvonne: Mmmm. [Confirming agreement]

Daniel: But what does it all mean, though? [Raises hands palm up.]

Belinda: See.

Teacher: Well, what does the 17 mean?

Belinda: 17. No, that’s 60, 17. [Pointing at screen.] Group one, put it there, and it’s 77!

Yvonne: Yeah. [Understands; smiles]

Teacher: You look a bit confused, Rory. Is that alright?

333

Rory: [No verbal response.]

Teacher: You are confused?

Belinda: I aren’t.

Teacher: Girls, can you explain it to Rory?

Belinda: [Stands up] Well, …

Yvonne: Here, use it …

Daniel: I’m confused as well. I don’t understand.

Teacher: Well, explain it to Daniel as well. No, don’t get up. Stay there.

Belinda: There’s 60 there, because there’s 6 tens there. That equals 60. Then there’s a

ten over here, which, that you put that back there, which makes 70, and then

there’s 7 there. [laughs]

Yvonne: I don’t think he understands.

Daniel: Oh, what does 6 plus 6 [indistinct]

Teacher: [Uses mouse for computer 1] Here. Here, here, here. This label at the top says

‘6,’ doesn’t it?

Belinda: Yeah.

Teacher: But it’s 6 tens. What are these 6 tens showing?

Rory: 60.

Daniel: 60.

Teacher: It’s showing 60, isn’t it?

Belinda: And there’s another ten there. [Points at own screen]

Teacher: Tell you what, if you click on the “Show as ones” button, it will show you all

the ones. [Clicks on “Show as ones”] OK, can you see the 77 ones?

?? Yep.

Daniel: Oh yeah. [understands]

Teacher: All these here, and all those there make 77. That’s 60, and these here are 17.

Belinda: [Looking at her workbook] I’m right. Excellent!

Teacher: You need to write that down somehow in your book.

Belinda: Well, if I have … [indistinct]

Teacher: OK, you’re on the right track. We started with 7 tens and 7 ones, didn’t we?

Belinda: Oh! [understands] 60, plus 17, is 77!

Teacher: Now it makes sense, doesn’t it?

334

Yvonne:? Mmm. [confirming]

Teacher: But what we had before was different, wasn’t it?

Yvonne: Yep.

Daniel: Yeah.

Belinda: But then I [indistinct]

Teacher: OK. Write down what we’ve got now.

Daniel: Oh, yeah, 6 [indistinct]

Belinda: I’m so clever. Can we put, click the net and [motions with two hands] put it

back to 77?

Teacher: If you wish.

Belinda: Wh-who [pleased]

Teacher: You’re just about to start again with this one, so you can put the net and make

it go back to 7 tens and 7 ones.

Belinda: Whooo

Teacher: No, don’t drag it, just click it, Rory. Do it again, click on the net, and don’t

drag it. That’s it. Oh, it didn’t work. Sorry - do that again. [Uses mouse]

Sometimes the net doesn’t show up. There.

Daniel: There you go. (h/c S1, T 4a)

Low/computer:

Amy: [Reads card]

Hayden: [Helps her to read unknown words]

Teacher: First thing: “Show the number with the blocks.”

Hayden: [Starts to use mouse]

Amy: [Starts to use mouse] 77.

Kelly: No we did, we did … oh.

Hayden: [Using mouse, looking at screen] Do we press that?

Teacher: No, that’s the hundreds. You wanted ten, Amy. Make the number 77. 77.

Terry: I know how to do that.

Hayden: [Checks card, returns to using mouse]

Terry: [Pointing to screen with pencil] Keep on doing it until it’s 7. No, put, do this

to help ya’.

Hayden: What?

335

Terry: You can do that and it’ll tell you what number you’re on to. [Display number

window, presumably] 5, 6, 7. Now you need 7 ones.

Hayden: [Puts out too many one-blocks] Whoops! [Laughs]

Terry: Seven! You went up to 8. Get the bomb out. [Hears Kelly say “Take one

away”] Oh, take one away! I forgot that!

Hayden: [Laughs]

Terry: [Laughs] Oh, man!

Hayden: [To teacher] I made 77. Oh, I’ll have to check it [Starts to use mouse].

Terry: I made … 78.

Computer: 77.

Terry: Yep. I believe it’s 77.

Kelly: [Watching screen as Amy works, giving vocal encouragement. Moves her

head as Amy does something on screen] It’s not working. 2 … I’ll count them.

2, 3, [shakes head] um 2, 3, 4, 5, 6, 7.

Amy: [Intake of breath; apparently made a mistake]

Kelly: No, just take one away. Er-er. No, not one of them. Now you’ve got six.

Amy: No, er …

Kelly: Now, put one of them [points at screen] back on.

Computer: 77.

Kelly: [Moves hand toward mouse]

Amy: Now, I’m pressing it.

Teacher: Kelly, you can do the next one.

Terry: [Looking at girls’ computer] Wow! They’ve got 78!

Hayden: 77.

Terry: Hah. I thought it was 78.

Computer2: 77.

Teacher: Now what’s the next thing you have to do?

Kelly: — OK. [Starts to use mouse]

Teacher: You remember how you can do that? Just a minute, Kelly. Stop. You can get

the computer to do it for you in one go.

Terry: [Intake of breath] Oh yeah, [points to screen] the saw! The saw!

Computer: [Saw sound]

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Hayden: [Looks at Terry and smiles]

Terry: [Laughs] That’s easy! We’ve done it already! Now put 6 …

Amy: [Pointing to screen] There, there, there.

Teacher: I want you to swap one of the tens.

Kelly: Oh, one.

Teacher: OK, now you have to write down what you’ve done in your book.

Terry: OK. [He and Hayden start to write] 6, 17.

Amy: [Starts to use mouse straight away] I know how to …

Teacher: [Starts to use mouse, with hand on top of Amy’s hand] You’re clicking in the

wrong places. Right. Try again? Now, how are you going to swap a ten for ten

ones?

Amy: [Looking at screen] There.

Teacher: No, that doesn’t help.

Terry: [Watching girls’ computer] Nuh, it’s a saw. I bet it is. We already done it.

Hayden: [Watches girls’ computer]

Amy: Take …

Teacher: No, no. Amy, click on the saw.

Amy: [She does so]

Kelly: [Nodding] Now, press it.

Computer: [Sawing sound]

Amy: Now press “OK.”

Teacher: No, now you have to write something down in your book, to show what

you’ve done.

All: [Write in books]

Kelly: Mmmm, do you cut ten up? [Looks at teacher]

Terry: [Writing in book, stops to talk to Hayden] I pressed the saw! [Laughs]

Hayden: I pressed it! [Reads from workbook] We swapped the ten for a one.

Teacher: [To Hayden] OK, now can you write down the numbers of blocks you’ve put

out?

Terry: I already have: 6, 17.

Teacher: What do you mean, “6, 17”?

Terry: Uh, ooh, forgot to add it up!

337

Hayden: [Laughs]

Terry: Can we add it up?

Teacher: Yes.

Hayden: [Pointing to screen as he counts] 10, 20, 30, 40, 50, 60, … [Puts up fingers on

his left hand as he continues] 61, 62, 63, [Goes back to pointing to screen] 64,

65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77.

Terry: [As Hayden gets to 70] Hey, no! Why didn’t you ask … [To teacher] There’s

an easier way to do it.

Hayden: [Laughs] Oh, yeah. [Apparently realising that the computer will read the

number]

Terry: No, [points to screen] there’s an easier way to do it.

Hayden: [Starts to use mouse]

Computer1: 77.

Hayden: [To Terry, with surprised look] 77!

Terry: Oh! We’ve still got … Oh, cool, that’s easy! [Writes in workbook] Seventy …

77! [To teacher] How does it do that? It’s still got 77. [Teacher looks at him,

but does not respond] Oh yeah! [Understands; bangs himself on his head with

his hand]

Hayden: [points to screen] It’s still … You cut it up, and it’s still 77! [Looks at Terry]

Terry: Mmm. [Pencil in mouth, apparently thinking]

Amy: Cut … cut …

Kelly: We cut ten up [Indistinct]

Amy: Blocks … with …

Kelly: [Shows workbook to teacher]

Teacher: OK, now I want you to write down how many blocks there are.

Kelly: [Looking at screen] Um …

Amy: [Quietly] Six …

Kelly: I know! [Starts to use mouse]

Amy: 176.

Kelly: Which one?

Amy: It’s … [points to screen]

Kelly: Oh … [Turns to teacher] Do you press that, and ask?

Computer: 77.

338

Kelly: 77. Do you write how much you have?

Amy: [Puts book down] Now this is my go.

Teacher: No, just a minute. We haven’t finished.

Amy: I know that!

Terry: Start again.

Hayden: [Uses mouse to restart]

Computer1: [Reveille]

Teacher: No, I don’t want you to start again. I want to talk about this one a bit longer.

[Uses mouse to reset representation]

Hayden: Whoops! [puts hand to mouth]

Teacher: Alright. Girls, look up here at this one please. This is how it started off,

alright, with 7 tens and 7 ones. And you know that’s 77, don’t you?

Amy: Yeah.

Kelly: Yes.

Teacher: When you cut one up …

Computer1: [Sawing sound]

Hayden: … it’s still 77 [turns to look at teacher].

Teacher: It’s still 77.

Computer1: 77.

Teacher: If you show the number here, it still says ‘77.’ Now, how can that still be 77?

Because we’ve only …

Amy: ‘Cos we’ve just cut the same ones up again. Just put the …

Teacher: Mmm-mmm. So if we have 6 tens and 17 ones, can you see that that’s still 77?

Amy: Yeah.

Hayden: [Nods]

Kelly: Yeah, bec…

Teacher: I want you to write that down in your book, that you’ve got 6 tens and 17

ones.

Kelly: I did.

Amy: Sixty …

Kelly: Write beside that? [Indistinct]

Teacher: Write it underneath. Oh, it doesn’t matter. You can write it there.

339

Amy: 17 ones. [Puts down book] Finished.

Terry: [Writing in workbook] 6 tens and 17 ones.

Teacher: And what does that equal? Don’t keep putting your book down. You haven’t

finished, Amy. That equals how much?

Terry: 77.

Amy: [Looking at what she has written] One hundred and … Oh no, 77!

Teacher: Well write it down: “equals 77.”

Amy: “e,” “q,” “q.”

Teacher: Just an “equals” sign: two lines.

Amy: [Writing] equals … equals 77.

Terry: Whoops. Equals?

Teacher: Equals. Now let’s see what you’ve got

Terry: [Shows book to teacher] 6 tens, 17 ones.

Teacher: Yeah, you’ll have to write “tens,” though, as a word. It looks funny if you just

write “6 t 17 ones.”

Kelly: [Shows book to Amy] [Indistinct]

Amy: Put them … [Indistinct] Oh, I took that as well.

Teacher: OK, I asked you “Did it make sense?,” someone said “No,” before. [Boys are

looking as Terry writes in his book, and not really listening]

Amy: It did.

Kelly: Well, what do you mean by that?

Teacher: Does that make sense to you, that that’s 77?

Amy: Yes.

Kelly: Well, yes, because it um, if you add them up together, it makes 77. ‘Cos 17,

16, … (l/c S2, T 4a)

340

High/blocks:

John: [Reads Task 4 card]. Huh? Don’t make sense.

Teacher: OK. Do it one step at a time. “Show the number with the blocks.”

Simone: [Starts to count ten-blocks. Moves a ten-block at same time as Amanda moves

7 blocks]

Amanda: [Counts ten-blocks without counting them. Pushes Simone’s ten-block away,

counts out 7 one-blocks in a group, preventing Simone from contributing any.]

Simone: [Puts hand on top of one-blocks, moves them from side to side]

Amanda: 77. [Stops and watches Craig.]

Simone: [Watches Craig]

Craig: [Starts to put out ten-blocks] 70. 1, 2, 3, 4, 5, 6.

John: [Adds a ten-block to those already out, then another.]

Craig: [indistinct] [Removes John’s second ten-block, starts adding one-blocks] 61,

62, 63, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, …

Amanda: What are you doing, Craig?

Craig: 5, 6, 7, 8, 9, 10. [He had put out 6 tens and 20 ones altogether.]

John: Mmm? That’s 80.

Craig: No, 60 … 10, 20, 30, 40, 50, 60. 77.

Teacher: You’re trying to do too much at once, Craig. Do it one step at a time. Show 77

with the blocks first.

Craig: OK.

John: [Starts to count one-blocks from table into hand] Oh [understands]. [Picks up

a ten-block, pushes some one-blocks away] Move all these.

Craig: OK.

Teacher: Girls, you can move onto the next part, if you’re ready. [see later part of

transcript]

John: Put that there. [Counts ten-blocks, throws one away. Counts ten-blocks again -

there are 6. Starts to count one-blocks, loses count. Recounts, makes sure there

are 7 ones] 77.

Craig: [Starts to remove a ten]

Teacher: [Stops him, turns ten-blocks around so they are in vertical orientation] No, just

put those down. That’s 77.

John: Yes.

341

Teacher: Now do the second part - “Now swap one of the tens for ones.”

Craig: [Makes silly noise with mouth, moves ten-block away, adds the one-blocks

that were previously removed - making 80 altogether again] OK.

John: No. [Counts one-blocks carefully, until he has 17 in his hand, leaving 3 on the

table.] There. There’s 17 there [pointing to ones] 60 there [pointing to tens]

Teacher: What about these 3 here? They’re extras? [Removes 3 ones]

John: There’s 60 there [pointing to tens], and 17 there [pointing to ones].

Craig: [Using silly voice, nodding] Yes. Yes.

John: [Copying Craig] Yes.

Amanda: [Moves some ten and one-blocks to leave 2 tens and 3 ones] 23 [sounds bored,

has head on hands]

Teacher: No, no, no, we’re still on 77. Do this part - “Now swap one of the tens for

ones.”

Amanda: Oh.

Simone: [Starts to add 4 tens]

Amanda: [Reaches across to move ones] No, put the 7s behind.

Simone: [Starts to move a ten]

Amanda: [Counts ten-blocks, takes ten out of Simone’s hand and puts it back. Removes

3 ones, counts out new 7 one-blocks under her breath. Adds them to 7 tens]

Simone: [Picks up a ten, takes it away] [indistinct] … for ones. For 7 ones. [Adds 7

ones]

Teacher: [Taps on card to remind girls to go on to next step]

Amanda: [Counts tens and one-blocks] D’we have to put these [one-blocks] for ten?

Teacher: No, no.

John: Oh, well then we got it wrong.

Teacher: Read what it says there: “How many ones do you need?” How many ones did

you need when you did the swap?

John: 17.

Craig: 17.

Amanda: 7.

Teacher: How many ones did you swap the ten for?

Craig: 10.

Amanda: 7. She [Simone] swapped ‘em for 7.

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Teacher: Oh. [To Simone] Is that right?

Simone: [Nods]

Amanda: [Shakes head]

Craig: [Quietly] 10. 10 for 10. 10, 10, 10. You swap it for 10.

Teacher: You don’t think so? Why not?

Amanda: You have to swap it for 10, ‘cos otherwise it’s not the same.

Teacher: Is that right, Simone?

John: Well, then it’d just be 17 … no, then it’d just be 70. You need 77. Is it?

[Looks at card] Yep.

Craig: [Silly voice] 77.

Teacher: Do you understand, Simone?

Simone: [Nods]

Teacher: When you swap a ten, you’ve got to swap it for 10. Do you do that all the

time, or can you swap it for other numbers?

Amanda: No, we have to do it all the time.

Teacher: What do you think, boys?

John: Mmmm, I don’t know. [Shrugs]

Craig: I don’t know neither. [Shrugs]

Amanda: [Sure] Do it all the time.

Teacher: You always swap it for ten?

Amanda: [Nods]

Simone: [Shakes head]

John: I don’t.

Teacher: What do you think, Simone?

Simone: We can swap it for other numbers too.

Amanda: [Shakes head]

Teacher: Like what?

Simone: Like um, you can swap it for 7s, and 9, and 10, and the other numbers.

Teacher: Do you boys think that’s right?

Amanda: [To Simone] No we don’t.

Craig: [Looks unsure]

John: [Looks unsure] Well, …

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Teacher: Do you agree with it? ‘Cos Amanda’s saying you have to swap it for 10,

Simone’s saying you can swap it for 7 or 8 or 9 or other numbers; what do you

think?

Amanda: Uh-uh [Disagreeing].

Craig: I agree with Amanda. [Looks at John]

John: I agree with Amanda.

Teacher: Why?

Amanda: Otherwise it’s not the same.

Teacher: The same as what?

Amanda: As 10.

John: Because if you swap it for 7, there’s 10 [picks up ten-block and places it down

on its own].

Teacher: Why does it have to be the same?

Amanda: Otherwise it won’t be the same number.

Teacher: This is a good point, but why does it have to be the same? [Pause 3 seconds]

John: No, I don’t get it. [Sits back in seat]

Amanda: It has to be the same.

Teacher: Let me just show you. If we have a 10 [puts down a ten-block], or we have 10

ones [puts a line of 10 ones, parallel with ten-block], we all agree that you can

swap that for that [moves hand over ten and ones in turn], don’t we? You can

swap it? When we’re doing some sort of maths, sometimes the teacher will

say “Righto, swap that for that,” OK. Craig, do you understand or not? You’re

looking a bit …

Craig: Mmm, yeah, I understand a little bit.

Teacher: You do? Alright. We know we can swap that for that [indicates ten and ten

ones with hands]. Now if I take, say, three of those away [removes 3 ones],

and just leave 7, can I swap that for those, 7?

All: [Shake heads]

Amanda: [Definite] No.

Craig: No.

John: No.

Teacher: Why not? Simone?

Simone: Um, because that 10’s more than those.

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Teacher: It is, isn’t it? It wouldn’t, you couldn’t make that a fair swap. I mean, if this

was money, and someone said I’ll give you a $10 bill [picks up ten-block] and

you can give me 7 $1 coins

Craig: [Under breath, smiling] Cool.

Teacher: You’d be silly to do it, wouldn’t you? Because you wouldn’t have as much.

You need, you must have the 10 ones, and we can put next to each other, and

then that’d be the same. So, are you happy with that now, Simone?

Simone: [Nods]

Teacher: So, it’s always a swap of ten for ten. Now you [to boys] did that just now,

didn’t you?

Craig: [Nods] Yes.

Teacher: So how many ones do you have now?

Amanda: [Counts girls’ ones]

John: [Straight away] 17.

Craig: [Silly voice] 17. No, it’s only 6 … 17. [Looks at John]

Teacher: Now stop being silly, Craig, and work out the answer, please.

Amanda: 17.

Craig: 17.

Teacher: Are you just guessing, or … are you just copying John, or what?

John: No, ‘cos we counted … [indistinct].

Teacher: Is it really 17?

Craig: [Counts blocks, looks at John] It’s 17.

Teacher: OK, the last thing you have to do was stop after this one. This is what you

have to do: It says “Record what you have done in your workbook.” I want

you to write down what you had before we did the swap.

John: Huh?

Amanda: [Quietly] 77.

Craig: Oh yeah, 77.

John: 77 [writes in workbook]

Teacher: And then …

Amanda: Do the next one.

Teacher: … somehow I want you to come up with a way of doing it. I’m not going to

tell you how to do it. I want you to write down what we did to change it.

345

Craig: [Looks at Amanda’s workbook a couple of times, changes what he has

written, looks at John]

John: [Watches what Craig is writing]

Craig: Mmmm … [puts hand on forehead] [indistinct] … change that? [writing in

workbook] … that … we changed it.

Teacher: Well, how did you change it, Craig? It’s not enough just to say “We changed

it.” We want to know the numbers that you changed. Look up here. Girls, you

can look here too [indicates block representation for 77]. This is how we

started, we had 7 tens and 7 ones. And then you traded one of these for ten

ones and it turns out like that [indicates block representation showing 6 tens

and 17 ones below the first representation]: with only 6 tens, and 17 ones.

How can you write down what that change is?

Craig: We changed the … no, no … [indistinct] … got it.

Teacher: But what did you change it to, Craig?

Craig: Mmm, I changed one ten … to …

John: There [shows book to teacher] You supposed to do it like that?

Teacher: That’s a good way of doing it.

John: [Holds workbook up and shows it to others]

Teacher: Tell the girls, ‘cos they won’t be able to read it.

John: “6 tens and 17 ones.”

Teacher: OK, can you write down 7 tens and 7 ones? ‘Cos that’s what you had the first

time. That’s right, isn’t it?

John: Under here?

Teacher: You can write it underneath, or you can write it over the top. The last question

I have to ask you before we must go back to your class, is: Are the two

amounts [indicates the two block representations for 77] the same?

Simone: No.

Amanda: No. Y … [Stops, seems unsure]

Craig: No.

Simone: No.

Teacher: And I want you to discuss that with each other. I mean, you know what

number that is [7 tens and 7 ones]. Is that [6 tens and 17 ones] the same

number?

Simone: No.

346

Amanda: Yes.

John: Yes.

Craig: [Counts, nodding head]

Teacher: … and how can you be sure? I want you four to talk about it.

Amanda: [To others, definite] It’s the same.

Simone: [Shakes head]

Craig: [Still counting blocks]

Amanda: … ‘cept for one.

Craig: [Shakes head strongly] No.

Amanda: Yeah, ‘cos those ones, just for ten. Still the same. Make ‘em for ten.

John: [Looks carefully at both representations, points to both with pencil]

[indistinct] … they’re both the same.

Simone: [Shakes head again]

Teacher: Don’t talk to me, talk to each other. ‘Cos people are disagreeing.

Amanda: [To Simone] They’re both the same.

Simone: [Nods]

Teacher: How can you prove they’re both the same?

John: [Counts two sets of tens, touching with pencil. Looks puzzled]

Amanda: Swap … ‘cos there’s ten, and you swap them for ten it’s still the same.

Craig: [Counts ones under breath] … 10. 1, 2, 3, 4, 5, 6, 7. Oh yeah, they’re the

same.

Teacher: [Separates 7 ones from others and tens] So what number is shown by these

blocks [6 tens and 17 ones]?

Craig: Er …

Amanda: 77.

Craig: … 77.

John: 70.

Amanda: [To John] 77!

Craig: [Whispers to John] 77.

John: He said by these blocks [6 tens & 10 ones].

Teacher: We all agree this [7 tens & 7 ones] is 77 … Oh, no, no. I mean those [7 ones]

as well. I mean those as well, John. Sorry. All of those.

347

John: Oh. 77.

Teacher: So we haven’t changed the number, have we?

Amanda: No.

Teacher: We still have 77. So 7 tens and 7 ones is really the same as 6 tens and 17 ones.

John: [Nods] Yep. (h/b S1, T 4a)

Low/blocks:

Michelle: [Reads first part of task from card]

Teacher: Let’s just do that part. Show the number with the blocks. The first one.

Clive: [Pushes tens away]

Michelle: [Whispering] 77. [Pushes some tens back to Clive] You get the tens, … [Starts

counting out ones]

Clive: 2, 4, um, I lost count [laughs]. 5, 6, 7, and 7 ones.

Michelle: [Hiding ones in hand] I don’t have the ones!

Clive: What’s in there?

Michelle: [Laughs, adds ones, counts them] 1, 2, 3, 4, 5, 6, 8.

Clive: 8 [laughs].

Michelle: [Laughs] We got 8!

Clive: No, we got 7.

Michelle: [laughs]

Nerida: [Adds a ten, then adds ones] 2, 4, 6, … hang on. [Counts ones again] 1, 2, 3, 4,

5, 6, 7. [indistinct] … 77.

Teacher: [To Nerida and Jeremy] You look like you’ve got too many over there.

Jeremy: [Counts tens]

Nerida: [Counts tens from opposite side from Jeremy, removes two of them]

Teacher: OK.

Michelle: Look what we done. We done 1, 2, 3, 4, …

Teacher: OK. Next part. Alright, concentrate, ‘cos this is getting harder now. Are you

listening?

Clive: [Pushes blocks away]

Teacher: [To Clive] No, no, no, no, don’t put them away. We’re going to use those for

the next part.

Clive: [Puts hands over face]

348

Teacher: That part was easy. Now.

Nerida: Do you write it down?

Teacher: No, not yet. “Now swap one of the tens for ones.”

Michelle: [indistinct] … one of these …?

Nerida: [Picks up one of the tens] So you have to …

Jeremy: I’ll get it. [Picks up a one, adds it to blocks] … for one. So that’s 6 … 66.

Nerida: [Checks count, nods]

Jeremy: 66.

Michelle: [Swaps a ten for a one]

Clive: [making verbal noises]

Michelle: [indistinct] ‘cept … [?]

Teacher: [To Michelle] You had better concentrate, OK. You’re starting to be a bit

silly.

Clive: I know [how to carry out instructions].

Michelle: How could it be 66?

Teacher: How could it be 66?

Clive: [Counting tens] 2, 4, 6, …

Michelle: Shh! [Stops Clive from counting]

Jeremy: 68.

Teacher: 68. [To Michelle and Clive] Have you got 68?

Michelle: Yep.

Clive: [Counting ones] 2, 4, 6, 8.

Teacher: OK, well I’m going to have to ask you something. You’ve both done the same

thing, but you’re both wrong. [Removes a one, and puts back ten] Now,

there’s our 7 tens and 7 ones that we started with, and the instructions say

“Swap one of the tens for ones.”

Clive: [Starts to push away blocks]

Teacher: Now, let me do it, let me do it. We take one of the tens, and we’re going to do

a swap, for ones. Not for 1 one. [Holds up a one and a ten] Is that a fair swap?

Nerida: [Shakes head]

Jeremy: [Shakes head]

Michelle: No. Oh, yeah. [Understands] [Puts a ten back with their blocks]

349

Nerida: [Quietly picks up the one that was added earlier, starts counting ones to add to

it]

Teacher: Oh. No, no, no. Don’t put that back. We’re gonna swap that one for ones. But

we want it to be a fair swap.

Clive: Oh, ten ones.

Jeremy: [To Nerida] Ten ones.

Nerida: [Finishes counting, adds to representation]

Michelle: [Counts ones, adds to the one previously used in the trade, then adds to

representation]

Clive: [Pushes some ones toward Michelle’s collection, but she ignores them. When

she has got ten of her own, she pushes Clive’s blocks away again.]

Nerida: [Pushes ten ones with others, counts them all. Then she counts the tens]

Jeremy: My brother went to the sports.

Teacher: Did he? Not now, we’re concentrating. [To Michelle and Clive] OK, don’t get

mixed up. Make sure you’re doing the right thing. Why did you say ten ones,

Clive?

Clive: Um, because there’s one ten what all of them are glued, and there has to be …

Michelle: 20 there. [Puts hand on top of all ones in representation]

Clive: … and another te … them ones for another ten.

Nerida: [Finishes counting tens and ones] [Whispering, to Jeremy] 60, 17. [Looks at

Jeremy’s workbook] [Quietly, to teacher] 117.

Teacher: OK. [To Nerida and Jeremy] Do you agree with Clive that you have to have

ten of these [pointing to ones] to make one of those [picks up ten]?

Nerida: [Nods]

Teacher: Jeremy?

Jeremy: [Nods]

Clive: Swap one of them.

Teacher: When we do a swap, could you swap it for a different number? Could you

swap it for 8, or 7, or 9, or something?

Clive: No.

Michelle: No, you gotta swap ten.

Nerida: It’s gotta be ten swap [indistinct].

350

Teacher: … or 11, or 12? It’s got to be 10, hasn’t it? Always got to be 10. Next part.

Now this is where it gets tricky, ‘cos I want you to work out a way of doing it.

It says “Record what you have done in your workbook.”

Clive: Record?

Teacher: Write it down. Now I want you … Now let me put this out, because it’ll be

easier if I show you this. You started with that [puts out 7 tens and 7 ones

above Michelle and Clive’s representation], didn’t you?

Clive: 25.

Teacher: You started … No, it wasn’t 25. 7 tens and 7 ones. Jeremy, pay attention. It’s

nearly time to go, I know. Just this last one. We started with that [puts hands

on top representation], and then you did a swap with one of the tens and we

finished with that [puts hands on second representation]. Now I want you to

write what you’ve done there, in your book. Somehow.

Clive: Oh … Just …

Teacher: You do it how you think, so it makes sense, to explain what you did.

Michelle: So you gotta write it in words?

Teacher: Not necessarily words. You could do numbers, ‘cos you’ve got 7, and there

are other numbers that you can write as numbers.

Michelle: Oh, yeah.

Nerida: Can I just … Can you write the answer?

Teacher: Well, it’s not just the answer that I’m interested in. I want to know what

you’ve done, and what you’ve got in front of you.

Clive: [Counting] … 60, [Counts tens, pointing with pencil] 1, 2, 4, 6, [Counts ones]

61, 62, 63, 64, 65, 66, 67, 68. [laughs] 68!

Michelle: [Counts tens] 1, 2, 3, 4, 5, 6,…

Teacher: Hang on, hang on. Write it over here.

Michelle: Uh oh! Don’t worry, I …

Clive: Now I lost me counting.

Michelle: [Laughs] I just crossed that out.

Nerida: [indistinct; about how to write answer in workbook]

Teacher: OK, you can do that. That’s a good way of doing it.

Nerida: [Writes in workbook]

Jeremy: [Watches Nerida write]

351

Clive: [Touching tens with pencil] 2 … twen … 2, 4, 6

Michelle: 6. [Starts to count ones. Moves Clive’s hand away, puts both hands on ones]

Clive: [Waves pencil at Michelle in frustration]

Michelle: [Laughing] Clive!

Teacher: No, Clive wants to count them, and you keep moving them across.

Michelle: [Counting under breath]

Clive: [Watches Michelle for a bit, stops and puts hand under chin] I’ll just let the

girls do the work! [Laughs] [When Michelle finishes] What is it?

Michelle: 17.

Clive: Wrong!

Nerida: [indistinct] … to get 17. [Closes workbook] I done it.

Teacher: Well, you better count them, Clive. Michelle’s saying it’s 17.

Clive: 2, 4, 6, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, … What

was that one?

Teacher: 76.

Clive: 76, 77! Huh.

Michelle: Uh-uh. You’re supposed to count them [holds tens] in ones, and them [puts

hand over ones] in … ones.

Clive: So we got double 77s. Mmm? That was tricky.

Teacher: [To Nerida and Jeremy] What do you think? Have you got 77? ‘Cos Clive’s

saying that that [second representation] is 77, and that [first representation] is

77. Is that right?

Clive: Mmm-mm [confirming]

Michelle: [Writing in workbook, whispering] Oh! [Speaking aloud] Always get that

mixed up. Keep doing it. There - did it.

Teacher: Does that make sense?

Nerida: The top one’s 77 …

Teacher: Right. And the Bottom one?

Nerida: And the Bottom one … is …

Clive: [Starts to recount blocks] 2, 4, 6, 61, 62. That’s 62 …

Teacher: You just did that. We’ll have to cut it short, Clive, but you did count them all

before and you got 77, didn’t you?

Clive: Mmm-mm.

352

Teacher: Does that make sense? To have 77 again?

Nerida: [Shakes head]

Clive: Mmm-mm.

Michelle: Mmm … Yeah.

Teacher: After doing that swap? Look over here, and let me show you, ‘cos it is nearly

time to go, but I want you to see this before we finish. OK. If I put ten of the

ones together, like that [puts ten ones of second representation together in a

line], that looks like a ten again, doesn’t it?

Clive: Mmmm

Teacher: So that would look like our 7 tens and 7 ones. And we did a trade - one of

these for ten of those, and [jumbles up 17 ones] just push them all together.

Clive: They broke.

Teacher: Yeah, it’s like it got broken up. Now we’ve got 6 tens. Do you know how

many ones there are there? Without counting them?

Nerida: 17.

Michelle: 17 …

Teacher: There are 17 - you counted them before, didn’t you?

Michelle: Yep.

Nerida: I never.

Teacher: Well, is 6 tens and 17 ones the same as 7 tens and 7 ones?

Michelle: No.

Nerida: six hundred and 17

Teacher: Does it make 77?

Michelle: Yeah [not very confidently]

Nerida: [Shaking head] No.

Michelle: No!

Clive: Yes.

Michelle: No.

Clive: Yes.

Michelle: No!

Clive: Mmm-mm.

Teacher: Well, 10, 20, 30, 40, 50, 60. Six tens are 60, aren’t they? 61, 62, 63, 64, 65,

66, 67, 68, 69, 70. There’s our other ten that we used to have [pushes ten ones

353

into a line like a ten-block], but they’re now ten ones. 71, 72, 73, 74, 75, 76,

77.

Michelle: Oh yeah, that’s right. [Looks at Nerida and Jeremy]

Clive: See. Told ya. I’m a genius. [laughs] I came out of a lamp. (l/b S1, T 4a)

355

Appendix R – Transcript Excerpts Showing Participants Predicting Equivalence of Traded Blocks.

Task 31 (a) Show the number 255 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Teacher: What will the number [255] be after one of the tens is regrouped?

Hayden: 555 still.

Terry: 255.

Teacher: You mean 255?

Hayden: Yeah, 255.

Teacher: How do you know it will be the same number?

Hayden: Because, if you cut up a ten, it’ll, um … yeah, cut up a ten, and you swap it for

a one, it’ll still be 255.

Teacher: — Are you sure that the blocks still show 255?

Terry: [Confidently] Yeah, because that’s [points to the ones] still a 10, and that’s

15, 5. (l/c S10, T 31a)

Task 5 (a) Show the number 21 with the blocks. Now swap all of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Teacher: You’ve shown the number with the blocks, now do the next part. Amy, Amy,

look at it: “Now swap all of the tens for ones.” You haven’t done that.

Terry: Oooh!

Amy: One tens …

Kelly: [Starts to use mouse]

Terry: I know how to do that.

Amy: Start again.

Terry: So we’ve got … we’ve still got something, I know.

Amy: 21 …

Teacher: Terry, read your instructions. You haven’t done it yet. “Now swap all of the

tens for ones.”

Terry: Oh, all of the tens.

Kelly: And we still have 21. So do we write that down?

356

Teacher: Write down what you’ve done in your book.

Terry: Now we’re starting to get into the heart of the stuff that I liked. … I still know

what it is, ‘cos you just told us. Now let’s check, that it’s right.

Amy: 21 … 1, 2.

Computer: 21.

Terry: It disappeared!

Teacher: Now you have to write in your book what you’ve done.

Terry: Right.

Amy: OK. [Starts to use mouse] Start again?

Terry: I already have. Oh, no.

Amy: Oh, I haven’t. It’s …

Kelly: 5 … task …

Terry: chopped …

Kelly: I wrote what I did.

Teacher: Have you written down what you’ve got there on the screen now, that equals

21?

Terry: … ten …

Kelly: Um, I’ve got …

Terry: I did something very easy. “I chopped the tens up.” Is that all you really have

to do?

Kelly: “I got 21 ones. I cut 2 tens up, and I still got 21.”

Teacher: Did you write down what you’ve got on the screen now?

Amy: [To Kelly] Hey, [Indistinct] …like that, Kelly.

Terry: Yep. [Reads from book] 2 tens, 1 one. 21.

Teacher: [Points to screen] That’s not 2 tens and 1 one, though.

Terry: What is it?

Teacher: There aren’t any tens at all now.

Terry: Oh yeah! [understands] I get you now.

Amy: I cut … cut …

Teacher: Write it on the next line.

Kelly: Like this, Mr Price? [Shows her book]

Amy: Cut … cut …

357

Teacher: Mm-mmm. I’d like you to write down “21 equals,” and then write down how

many you’ve got there [points at screen].

Kelly: Could I do it on the next one, ‘cos I haven’t any …

Teacher: Of course. No, put “equals,” Terry. “Equals.” Now what have you got [points

at screen]? How many tens and how many ones?

Terry: 21!

Teacher: 21 what?

Terry: 21 ones.

Teacher: That’s what you write down.

Kelly: 21 equals …

Amy: I still have … have … [She writes in her book I kute the bloes I still have 21.

(I cut the blocks. I still have 21.)] (l/c S3, T 5a)

Task 5 (b) Show the number 36 with the blocks. Now swap all of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Kelly: [Writes briefly in book. Starts to use mouse] 36 …

Terry: Do we chop the other one up too?

Teacher: It says “all of the tens,” doesn’t it?

Computer: 36.

Kelly: Now I gotta chop up … Now I gotta chop ‘em up.

Amy: Um, 36.

Kelly: Tut.

Teacher: You have to get the “saw” again, Kelly.

Kelly: OK.

Amy: [Whispering] 36 … equals … I like the saw the best. Oh, the bomb is the best!

[Watching Kelly] Do it. Oh no, the bomb takes away.

Kelly: Whoa! Now that …

Amy: It’s still 36, see. [points at screen]

Kelly: Watch out! Must’ve pressed something wrong.

Computer: 36.

Kelly: Yep.

Amy: There we can use your rubber.

Kelly: 36. 36. 36 is …

358

Teacher: Mmm, I see what you were thinking of the first time.

Amy: [Writes in her workbook] We … split … blocks … up … We ended … ended

… ended with … thirty, 36 … again. [Writes: we cat the bloes pu we end with

36 one a gen.]

Terry: split … the blocks … I split the blocks up … [Writes: 36 = 36 one. I splet the

blos up an I hed 36 one]

Kelly: … cut the … blocks up … 36. [Writes: i cut 3 tens up and i till got 36] I fitted

all mine on.

Teacher: How are you going? All finished? 36 what, Amy?

Amy: 36 ones. Ones.

Teacher: Mm-mmm. So 36 can be 3 tens and 6 ones, or it could be 36 ones, couldn’t it?

Amy: Yep. (l/c S3, T 5b)

Task 6 (a) Show the number 64 with the blocks. If you were to swap all the tens for ones, how many ones would there be? Write your answer in your workbook.

Teacher: Now, it says “If you were to swap all the tens for ones, how many ones would

there be?” Now I don’t want you to swap them, but if we did, how many ones

would there be? All of the tens.

Amy: Still the same. Um, it’d be 64 again! — Because, if you have the sa… ‘cos it’s

6 tens and 4 ones, and if you chopped all them up with the saw, it’d still be the

same, but they’d all be ones! (l/c S3, T 6a)

Task 6 (b) Show the number 89 with the blocks. If you were to swap all the tens for ones, how many ones would there be? Write your answer in your workbook.

Amy: [Writing] 89 … [looks at screen] … 89 …

Kelly: Equals … Yep, we’ve still got 89.

Amy: … ones.

Kelly: Yeah.

Amy: [writing] 89 ones equals … [Writes: 89 = 89 ones] (l/c S3, T 6b)

359

Task 24 Show the number 40 with the blocks. Put out another ten, and say the number’s name. Keep adding tens. Stop when you have 10 tens. What is this number called? Can you trade 10 tens? Keep adding tens. Stop when you reach two hundred. Write the numbers you made in your workbook.

Terry: [To teacher] It’s a ten, so we can take it away and put a ten, hundred! [Terry

seemed to realise straight away that 10 tens could be treated the same as 10

ones, and regrouped for a larger block.]

Teacher: [Reads next part of task:] “What is the number called [all participants say it is

100]; Can you trade 10 tens?”

Terry: Yes!

Kelly: Yes, but you’ll end up with none left.

Terry: [To Kelly] Yes, you’ll end up with a hundred still! (l/c S8, T 24)

Task 31 (a) Show the number 255 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Teacher: [Talks the girls through the regrouping process.] How many tens and ones will

there be after the trade? Amy, how do you know it will be the same?

Amy: You’ve only cut up one ten.

Kelly: And it’s still the same. (l/c S10, T 31a)

Task 31 (b) Show the number 932 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Kelly: [Puts out the blocks to represent 932.]

Amy: Now cut one up. Cut that one there.

Kelly: It will be 12 there, wouldn’t it?

Amy: [Points to the screen] It’ll still be the same number, though.

Kelly: Yeah, it’ll be 12. Ten plus 2 is 12. [Uses the mouse to cut up a ten-block.]

(l/c S10, T 31b)

Task 31 (c) Show the number 314 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Teacher: How many blocks will be in each column after the trade?

Amy: There will be, 3 hundred, and 14 …

Kelly: There’d be 14 ones, and zero tens, and that will equal 314, still. (l/c S10, T 31c)

360

Task 32 (a) Show the number 340 with the blocks. Now swap one of the hundreds for tens. How many tens do you need? Record what you have done in your workbook.

Teacher: Stops Hayden, asks the boys to predict what the number of hundreds, tens and

ones will be after the trade.

Terry: 314.

Teacher: How many blocks will there be in each column after the trade?

Hayden: 3 hundreds, and 1 ten, and 4 ones.

Terry: 14, 14… [indistinct] …

Hayden: 3 hundreds, no tens,… and, um …

Terry: There still will be tens!

Teacher: Are you sure?

Hayden: There will be 14.

Teacher: You both agree that there will be 14 ones, but how many tens will there be?

Hayden: None.

Terry: w… [he was apparently going to say “1”] none … no, 1, 1..

Teacher: Will there? How many tens there will be a after the trade; 1, or 0?

Terry: [Pointing to the screen] There still will be ten, ‘cos, but it will only be cut up

into here [ones column].

Teacher: But how many tens blocks will there be?

Terry: None.

Teacher: [Asks the girls the same question, about how many blocks will be in each

column after the trade.]

Amy: There will be, 3 hundred, and 14….

Kelly: There’d be 14 ones, and zero tens, and that will equal 314, still. (l/c S10, T 31c)

Task 32 (a) Show the number 340 with the blocks. Now swap one of the hundreds for tens. How many tens do you need? Record what you have done in your workbook.

Terry: [Uses the mouse to show the number 340 with the blocks. He has the

computer read the number.]

Computer: 340.

Terry: 340. Oh, I got a odd number. I did an odd number. [After a while …] 340, if

we cut one up there’ll still be 340!

361

Hayden: Yeah!

Kelly: Mr Price, d’you want me to cut up one of the hundreds?

Teacher: Not yet.

Hayden: [Uses the mouse to swap one of the tens for ten ones.]

Teacher: Terry, stop. The card says to swap a hundred for tens.

Terry: Oh! I didn’t do it.

Teacher: How many tens will be swapped for 1 hundred?

Kelly: 100…

Amy: Still the same number. You’ll get 3 hundred and for … 340. (l/c S10, T 32a)

Task 32 (b) Show the number 627 with the blocks. Now swap one of the hundreds for tens. How many tens do you need? Record what you have done in your workbook.

Hayden: [Looking at the screen] So, there will be … 12 tens?.

Teacher: Terry, do you agree with Hayden?

Terry: [Looks at the screen, apparently thinking.] 12 tens, if you cut one up … Yep! I

agree!

Teacher: And how many hundreds will there be?

Hayden: 5.

Teacher: And how many ones will there be?

Hayden: 7.

Terry: 7.

Hayden: No, there’ll still be 6, um, hundreds, because there will be a hundred in the

tens [points at screen].

Teacher: OK, but in the actual hundreds column …

Hayden: Yeah, there will be 5. (l/c S10, T 32b)

Task 14 The Sunny Surfboard Company has 75 boogie boards left. If one is sold, how many are left? Then how many if another is sold? Say all the numbers in order from 75 back to 60. Show the numbers with the blocks. Write them in your workbook.

Teacher: Well, you’ve gone back to 70, now take away another one.

Belinda: You can’t.

Teacher: If you think about it, there’s a way to do it.

Daniel: There is a way. Just can’t work it out. 70, …

362

Belinda: Put another one in! Can’t! Regroup! No, can’t regroup. Depose! Decompose.

‘Cos then, you’d have one, and you can take away. [Nods to Rory] Do it!

Rory: But it’ll be, it’ll still be 7.

Belinda: No it won’t. [Puts hand on top of Rory’s to use mouse, but he keeps using it.]

Rory: Yes, it will. [Nods]

Belinda: Here, I’ll show ya’. [Puts hand to left of mouse as if to take over, but again

Rory keeps control of it.]

Rory: Look, it’ll still be 7. [Regroups a ten]

Belinda: Now saw one. Now you can take away some, more. [points at screen]

Rory: What do you mean?

Belinda: [Puts hand by screen to stop others from seeing what they are doing.] Here, let

me do it. [Removes Rory’s hand, starts to use mouse] Now, you can take

away. [She takes away ones until 60 is left] (h/c S4, T 14)

Task 24 Show the number 40 with the blocks. Put out another ten, and say the number’s name. Keep adding tens. Stop when you have 10 tens. What is this number called? Can you trade 10 tens? Keep adding tens. Stop when you reach two hundred. Write the numbers you made in your workbook.

Teacher: [Reads question] Can you trade 10 tens? [Belinda and Rory nod, say that they

can.]

Belinda: You could decompose one … that’s trading.

Teacher: Mmm … Does that help?

Belinda: Yeah. [Uses mouse to cut a ten into 10 ones.] It’s still a hundred. (h/c S7, T 24)

Task 31 (a) Show the number 255 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Belinda: [Shows the number 255 without hesitation. Uses saw to cut up a 10, shows

number window] It’s still 2 … yep, I thought so. (h/c S8, T 31a)

Task 31 (b) Show the number 932 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

All participants: [Show the number with the blocks.]

Teacher: Can you say what the number of blocks will be after regrouping? Write in

your books.

363

Belinda: Easy - You’ll still have 9 hundreds, but you’ll have 2 … tens, and … whoa! …

[Finishes writing, puts book down] I bet I’m right. I know everything like that

… I don’t know - it’s easy. (h/c S8, T 31b)

365

Appendix S – Transcript of Task 4 (d) from Low/Blocks Group

Task 4 (d) Show the number 58 with the blocks. Now swap one of the tens for ones. How many ones do you need? Record what you have done in your workbook.

Clive: [Counts ten-blocks] 2, 4, 5. [Puts tens down] 8! 58, 58. [Counts out ones] 2, 4,

6, 8. [Does a little “victory” gesture with arms. Writes in workbook] 58 equals

5 tens and 8 ones. I am a genie [genius]!

Teacher: [Laughs] OK, do this. This [indicating blocks Clive has put out] is 58 now,

Jeremy. And Clive’s just doing the swap.

Clive: [Swaps ten for ones, counts them in his hand] 2, 4, 6, whoops, 8, 10. [Puts

them on table] Now that means … [Counts ones] 40, 41, 42, 43, 44, 45, 46,

47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58. 58, again. [Pauses, blows air with

finger in mouth, looks briefly toward Nerida’s book, taps pencil, smiles,

pauses.] I need some help.

Teacher: You’ve done 5 tens and 8 ones, which you’ve got to write down, Jeremy. How

many tens and ones do you have now, Clive?

Clive: Ah, ooh. That’s what I missed. [Starts to count one-blocks]

Teacher: Write the tens down first. You know how many tens there are.

Clive: [Writes in book] 58 equals 4 tens and … [counts ones] 3, 4, 5, 6, 7, 8, 9, 10,

11, 12, 13, 14, 15, 16, 17! 17 ones.

Nerida & Michelle Nerida: [Looks at card] 58. [Removes some tens, counts out ones to show 5 tens and 8

ones]

Michelle: Finished!

Teacher: Do the swap now, please, Michelle.

Michelle: [Looking at book] We’ll have 4.

Nerida: [Picks up several blocks, puts down ones and picks up a ten] One ten …

Michelle: [Picks up some ones]

Nerida: [Puts out two hands side by side] Put them in my hand.

Michelle: Hang on! [Takes ones away and counts them] 5. There’s 5. [She starts to add

them to other blocks, Nerida puts her hand under them to take them in her

hand. Michelle starts to count ones on table

366

Nerida: [Stops her by putting her hand on top of them] We’ll lose count!

Michelle: [Counts on from 8 under Nerida’s hand] 9, 10, 11, 12, 13.

Nerida: [Looks dissatisfied with this, keeps 8 ones separate and counts 5 added ones.

Then she continues to count as she adds more ones to make up to ten]

Michelle: [Starts to count with Nerida, then sits back and folds her arms] Finished!

Nerida: [Carefully re-counts added ten-blocks, then starts with original 8 ones, and

counts all ones to reach 18. She writes in her book]

Michelle: [Watches what Nerida writes, then writes in her own book, then looks at

Nerida’s again. Nerida looks at her]

Nerida: [Quietly] 4 tens and 18 ones.

Michelle: [Writes in her book]

Teacher: The boys and girls have two different answers again. Clive? You have

different answers again.

Nerida: [Smiles at boys]

Michelle: We have 18. [Laughs]

Clive: Youse are wrong.

Michelle: No, we’re right!

Teacher: Well, explain it.

Nerida: We put out …

Michelle: [Touching ten-blocks] Five …

Nerida: We had 5 …

Michelle: 5 tens.

Nerida: … tens and 8 ones, and then …

Michelle: We traded it for …

Nerida: … for ten ones and we kept our 8 ones already there.

Teacher: And would that make 18, or would that make 17?

Nerida: 18.

Clive: [With arms folded; in the previous dialogue of the girls, he has not been

showing agreement with what they said, or any apparent willingness to listen]

17.

Michelle: 18. Look [starts to count blocks, starting with tens] 5, [continues with one-

blocks] 6, …

Nerida: [Stops her; touching tens] 4 …

367

Teacher: You had 8 to start with, Clive. Hang on, girls. Can we do it without counting?

Can you work it out, and say what’s sensible? If you had 8 to start with, and

then you swapped and had another ten, what number would that make, without

counting?

Clive: 17.

Teacher: Ten and 8?

Nerida: [Shakes head] 18.

Clive: 18, I think. Think.

Michelle: 18.

Teacher: What’s ten and another 8?

Michelle: [Counts sub-vocally; smiles] 18!

Clive: … 18.

Teacher: It is 18, isn’t it?

Nerida: Clive, Clive!

Clive: Doh!

All: [Laugh]

Teacher: It’s easy to miscount one, it’s very easy.

Clive: [Changes answer in workbook] No, it isn’t, it’s hard!

Nerida: That’s why I do my counting twice.

(l/b S7, T 19)

369

Appendix T – Comparison Between Ross’s (1989) Model and a Proposed Model for Categories of Responses to Digit

Correspondence Tasks Four-Category Model of Responses to Digit Correspondence Tasks

Five-Stage Model of Children’s Interpretations of Two-Digit Numeralsa

Category I: Face-value interpretation of digits. Category I thinking was evidenced by a participant’s statements that each digit represented only its face value, and that remaining objects in the set represented by the two-digit symbol as a whole were not represented by either digit.

“Stage 3: face value Students interpret each digit as representing the number indicated by its face value. The set of objects represented by the tens digit, however, may be different from the objects represented by the ones digit. They may verbally label as “tens” the objects that correspond to the tens digit, but these objects do not truly represent groups of ten units to students in stage 3: students do not recognize that the number represented by the tens digit is a multiple of ten.”

“Stage 2: positional property Pupils know that in a two-digit numeral the digit on the right is in the ‘ones place’ and the digit on the left is in the ‘tens place.’ Their knowledge of the individual digits is limited, however, to the position of the digits and does not encompass the quantities to which each corresponds.”

Category II: No referents for individual digits. Category II responses indicated that a participant accepted the two-digit symbol as representing the entire set of objects, but rejected the idea that each digit has separate referents, on the basis that some objects would be left out.

“Stage 1: whole numeral As pupils in our culture construct their knowledge about quantities up to ninety-nine and their symbolic representation as two-digit numerals, their cognitive construction of the whole comes first—the numeral 52 represents the whole amount. They assign no meaning to the individual digits.”

“Stage 4: construction zone Students know that the left digit in a two-digit numeral represents sets of ten objects and that the right digit represents the remaining single objects, but this knowledge is tentative and is characterized by unreliable task performances.”

370

Four-Category Model of Responses to Digit Correspondence Tasks

Five-Stage Model of Children’s Interpretations of Two-Digit Numeralsa

Category III: Correct referents for digits, tens not explained. A Category III response is one in which the participant knew that the tens digit represented the remaining objects, once the referents for the ones digit were removed, but could not explain why that digit represented a number of objects larger than its face value.

“Stage 5: understanding Students know that the individual digits in a two-digit numeral represent a partitioning of the whole quantity into a tens part and a ones part. The quantity of objects corresponding to each digit can be determined even for collections that have been partitioned in nonstandard ways.”

Category IV: Correct referents for digits, tens place explicitly mentioned. Category IV includes responses stating a correct number of objects for each digit, explaining that the tens digit represents the number of groups of ten.

Note. aRoss’s stage descriptions that are judged to be equivalent are placed adjacent to this author’s category descriptions (section 4.5). From S. H. Ross, 1989, Parts, wholes and place value: A developmental view. Arithmetic Teacher, 36, p. 49.

371

Appendix U – Sample Coding of Transcript for Feedback Note that incidents of feedback, their presumed effects, and the responses of recipients are noted in bold type inside square brackets. Clive: [Puts out 7 tens & 5 ones, adds a ten, then counts on from 85.] 85, 86, 87, 88,

89, … [He gets stuck at 89, apparently not knowing the next number.] … 100,

101, 102, 103, 104. [Count blocks/Provide answer]

Nerida: You’re wrong. [Peer feedback/contradict answer] [She counts blocks again,

getting 94.] [Count blocks/Provide answer]

Clive: [Does not listen to or watch Nerida as she counts.] [Reject feedback]

Jeremy: [While Michelle reads, reaches over her arm to show 75.]

Michelle: 75, and 10 more, makes … [Picks up tens, counts in tens to 100. She puts a ten

back straight away.] No, don’t need 100.

Teacher: Look at the card again. [Teacher feedback/Ask a question]

Michelle: Oh, I’m wrong. [Puts tens back] [Change answer] [She keeps a ten, counts on

9 ones. She re-counts the ones by two, removes one or two ones. She puts the

tens and ones together, counts tens, and starts to count ones from “81.”]

[Count blocks/Provide answer]

Jeremy: [Counts blocks from 91.] [Count blocks/Provide answer]

Michelle: Hang on, Jeremy. [She re-counts tens, continues with ones to 94.]

Jeremy: [Not satisfied] You missed some blocks. [Peer feedback/Contradict answer]

Michelle: Clive, it’s 94. [Peer feedback/Contradict answer]

Teacher: Have you all got the same answer? [Teacher feedback/Ask question]

Clive: [Re-counts blocks, changing to “100” after “89,” getting “103.”] [Count

blocks/Provide answer]

Michelle: [Laughs.]

Clive: I can count straight, but you can’t! [Peer feedback/Contradict answer]

Nerida: [Begins re-counting the blocks. Again Clive does not watch her.]

Teacher: Clive and Nerida, count them together. [Teacher feedback/Give directions]

Clive & Nerida: [They do so, arguing after “89” about whether it is “90” or “100.”] [Peer

feedback/Contradict answer]

Teacher: Is it 90, or 100? [Teacher feedback/Ask question]

Michelle: 90. [Peer feedback/Contradict answer]

372

Clive: 100. [Peer feedback/Contradict answer]

Teacher: What number comes after 89? [Teacher feedback/Ask question] [Clive and

Michelle repeat their respective answers.] Do you have 9 tens, or 100, which

is 10 tens?

Clive: 8.

Teacher: Make sure you include the ones. Is there an easier way to show the number,

since there are so many ones? [Teacher feedback/Ask question]

Clive: You can swap for ten.

Nerida & Michelle: [Do so.]

Teacher: Nerida, don’t swap all the ones, just 10 of them. [Teacher feedback/Give

Directions] OK, re-check how many there are. [They do so, and Clive agrees

that it is 94.] [Change answer]

373

References Akpinar, Y., & Hartley, J. R. (1996). Designing interactive learning environments. Journal

of Computer Assisted Learning, 12, 33-46. Alexander, P. A., White, C. S., Haensly, P. A., & Crimmins-Jeanes, M. (1987). Training in

analogical reasoning. American Educational Research Journal, 24, 387-404. Alexander, P. A., Willson, V. L., White, C. S., Fuqua, J. D., Clark, G. D., Wilson, A. F., &

Kulikowich, J. M. (1989). Development of analogical reasoning in 4- and 5-year-old children. Cognitive Development, 4, 65-88.

Alexander, P. A., Wilson, A. F., White, C. S., Willson, V. L., Tallent, M. K., & Shutes, R. E. (1987). Effects of teacher training on children’s analogical reasoning performance. Teaching and Teacher Education, 3, 275-285.

Australian Association of Mathematics Teachers. (1996). Statement on the use of calculators and computers for mathematics in Australian schools. Adelaide, South Australia: Author.

Australian Education Council. (1990). A national statement on mathematics for Australian schools. Carlton, Victoria: Curriculum Corporation.

Babbitt, B. C., & Usnick, V. (1993). Hypermedia: A vehicle for connections. Arithmetic Teacher, 40, 430-432.

Baggett, P., & Ehrenfeucht, A. (1992). What should be the role of calculators and computers in mathematics education? Journal of Mathematical Behaviour, 11, 61-72.

Ball, S. (1988). Computers, concrete materials and teaching fractions. School Science and Mathematics, 88, 470-475.

Baroody, A. J. (1989). Manipulatives don’t come with guarantees. Arithmetic Teacher, 37(2), 4-5.

Baroody, A. J. (1990). How and when should place-value concepts and skills be taught? Journal for Research in Mathematics Education, 21, 281-286.

Baturo, A. R. (1998). Year 6 students' cognitive structures and mechanisms for processing tenths and hundredths. Unpublished doctoral thesis, Queensland University of Technology, Brisbane, Australia.

Becker, D. A., & Dwyer, M. M. (1994). Using hypermedia to provide learner control. Journal of Educational Multimedia and Hypermedia, 3, 155-172.

Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school. Educational Studies in Mathematics, 13, 33-57.

Bednarz, N., & Janvier, B. (1988). A constructivist approach to numeration in primary school: Results of a three year intervention with the same group of children. Educational Studies in Mathematics, 19, 299-331.

Beilin, H. (1984). Cognitive theory and mathematical cognition: Geometry and space. In B. Gholson & T. L. Rosenthal (Eds.), Applications of cognitive-developmental theory (pp. 49-93). Orlando, FL: Academic Press.

Bell, G. (1990). Language and counting: Some recent results. Mathematics Education Research Journal, 2(1), 1-14.

Best, J. W., & Kahn, J. V. (1993). Research in education (7th ed.). Needham Heights, MA: Allyn & Bacon.

Bisanz, J., Bisanz, G. L., & LeFevre, J. (1984). Interpretation of instructions: A source of individual differences in analogical reasoning. Intelligence, 8, 161-177.

Bobis, J. F. (1991). Using a calculator to develop number sense. Arithmetic Teacher, 38(5), 42-45.

374

Booker, G., Briggs, J., Davey, G., & Nisbet, S. (1997). Teaching primary mathematics (2nd ed.). Melbourne, Victoria: Longman.

Bottino, R. M., Chiappini, G., & Ferrari, P. L. (1994). A hypermedia system for interactive problem solving in arithmetic. Journal of Educational Multimedia and Hypermedia, 3, 307-326.

Boulton-Lewis, G. M. (1993). An analysis of the relation between sequence counting and knowledge of place value in the early years of school. Mathematics Education Research Journal, 5(2), 94-106.

Boulton-Lewis, G., & Halford, G. (1992). The processing loads of young children’s and teachers’ representations of place value and implications for teaching. Mathematics Education Research Journal, 4(1), 1-23.

Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32-42.

Bruner, J. S. (1966). Toward a theory of instruction. New York: Norton. Burns, R. B. (1990). Introduction to research methods in education. Melbourne: Longman

Cheshire. Burstall, C. (1978). The Matthew effect in the classroom. Educational Research, 21(1), 19-

25. Carpenter, T. P., Fennema, E., & Franke, M. L. (1993). Cognitively guided instruction:

Base-ten number concepts and multidigit operations [Draft paper]. Unpublished manuscript.

Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1997). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3-20.

Champagne, A. B. & Rogalska-Saz, J. (1984). Computer-based numeration instruction. In V. P. Hansen & M. J. Zweng (Eds.), Computers in mathematics education: 1984 yearbook (pp. 43-53). Reston, VA: NCTM.

Chinnappan, M., & English, L. (1995). Students’ mental models and schema activation during geometric problem solving. In B. Atweh & S. Flavel (Eds.), MERGA 18: Galtha. Proceedings of the 18th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 156-162). Mathematics Education Research Group of Australasia.

Clark, R. E. (1994). Media and method. Educational Technology Research and Development, 42(3), 7-10.

Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2, 270-279.

Cobb, P. (1995). Cultural tools and mathematical learning: A case study. Journal for Research in Mathematics Education, 26, 362-385.

Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83-94.

Cobb, P., & Wheatley, G. (1988). Children’s initial understandings of ten. Focus on Learning Problems in Mathematics, 10(3), 1-28.

Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175-190.

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33.

Cockcroft, W. H. (Chairman) (1982). Mathematics counts: Report of the committee of inquiry into the teaching of mathematics in schools. London: HMSO.

375

Cohen, S., Chechile, R., Smith, G., Tsai, F., & Burns, G. (1994). A method for evaluating the effectiveness of educational software. Behavior Research Methods, Instruments, and Computers, 26, 236-241.

Confrey, J. & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education. Mahwah, NJ: Erlbaum.

Creswell, J. W. (1994). Research design: Qualitative and quantitative approaches. Thousand Oaks, CA: Sage.

Cruikshank, D. E., & Sheffield, L. J. (1992). Teaching and learning elementary and middle school mathematics (2nd ed.). New York: Macmillan.

Davis, R. B. (1992). Understanding “understanding”. Journal of Mathematical Behaviour, 11, 225-241.

De Laurentiis, E. C. (1993). How to recognize excellent educational software. New York: Lifelong Software, Inc. (ERIC Document Reproduction Service No. ED 355 932)

Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson Education. Duffin, J. (1991). Mathematics for the nineties: A calculator-aware number curriculum.

Mathematics Teaching, 134, 56-62. Edwards, L. D. (1995). The design and analysis of a mathematical microworld. Journal for

Educational Computing Research, 12, 77-94. English, L. D. (1993). Children’s strategies in solving two- and three-dimensional

combinatorial problems. Journal for Research in Mathematics Education, 24(3), 255-273.

English, L. D. (1997). Children’s reasoning processes in classifying and solving computational word problems. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 191-220). Hillsdale, NJ: Erlbaum.

English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Erlbaum.

Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3-17.

Fletcher-Flinn, C. M., & Gravatt, B. (1995). The efficacy of computer assisted instruction (CAI): A meta-analysis. Journal for Educational Computing Research, 12, 219-242.

Fox, M. E. (1988). A report on studies of motivation teaching and small group interaction with special reference to computers and to the teaching and learning of arithmetic (CITE Report No. 31). Bletchley, Bucks, England: Open University. (ERIC Document Reproduction Service No. ED 327 385)

Fraenkel, J. R., & Wallen, N. E. (1993). How to design and evaluate research in education (2nd ed.). New York: McGraw-Hill.

Fuson, K. C. (1990a). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7, 343-403.

Fuson, K. C. (1990b). Issues in place-value and multidigit addition and subtraction learning and teaching. Journal for Research in Mathematics Education, 21, 273-280.

Fuson, K. C. (1992). Research on learning and teaching addition and subtraction of whole numbers. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 53-187). Hillsdale, NJ: Erlbaum.

Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place value and multidigit addition and subtraction. Journal for Research in Mathematics Education, 21, 180-206.

376

Fuson, K. C., Fraivillig, J. L., & Burghardt, B. H. (1992). Relationships children construct among English number words, multiunit base-ten blocks, and written multidigit addition. In J. I. D. Campbell (ed.), The nature and origins of mathematical skills (pp. 39-112). Amsterdam, The Netherlands: Elsevier Science.

Fuson, K. C., & Smith, S. T. (1995). Complexities in learning two-digit subtraction: A case study of tutored learning. Mathematical Cognition, 1, 165-213.

Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., Carpenter, T. P., & Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28, 130-162.

Gentner, D. (1983). Structure-mapping: A theoretical framework for analogy. Cognitive Science, 7, 155-170.

Gentner, D. (1988). Metaphor as structure mapping: The relational shift. Child Development, 59, 47-59.

Gentner, D. (1989). The mechanisms of analogical learning. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 199-241). Cambridge University Press.

Gentner, D., & Toupin, C. (1986) Systematicity and surface similarity in the development of analogy. Cognitive Science, 10, 277-300.

Gilbert, R. K., & Bush, W. S. (1988). Familiarity, availability, and use of manipulative devices in mathematics at the primary level. School Science and Mathematics, 88, 459-469.

Glaser, R. (1982). Instructional psychology: Past, present, and future. American Psychologist, 37, 292-305.

Gluck, D. H. (1991). Helping students understand place value. Arithmetic Teacher, 38(7), 10-13.

Gold, R. L. (1969). Roles in sociological field observations. In G. J. McCall & J. L. Simmons (Eds.), Issues in participant observation: A text and reader (pp. 30-38). Reading, MA: Addison-Wesley.

Goswami, U. (1992). Analogical reasoning in children. Hove, UK: Erlbaum. Grandgenett, N. (1991). Using LOGO programming to teach analogical reasoning in science

and mathematics education. Journal of Computers in Mathematics and Science Teaching, 10(3), 29-36.

Greene, J. C., Caracelli, V. J., & Graham, W. F. (1989). Toward a conceptual framework for mixed-method evaluation designs. Educational Evaluation and Policy Analysis, 11, 255-274.

Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22, 170-218.

Guba, E. G., & Lincoln, Y. S. (1989). Fourth generation evaluation. Newbury Park, CA: Sage.

Halford, G. S. (1993a). Children’s understanding: The development of mental models. Hillsdale, NJ: Erlbaum.

Halford, G. S. (1993b, March). Experience and processing capacity in cognitive development: A PDP approach. Paper presented at the Annual Conference of the Society for Research in Child Development, New Orleans, LA. (ERIC Document Reproduction Service No. ED 355 028)

Harley, S. (1993). Situated learning and classroom instruction. Educational Technology, 33(3), 46-51.

Hart, K. (1989). There is little connection. In P. Ernest (Ed.), Mathematics teaching: The state of the art (pp. 138-142). Lewes, UK: Falmer Press.

377

Helms, C. W., & Helms, D. R. (1992, June). Multimedia in education. Association of Small Computer Users in Education (ASCUE) Conference (pp. 34-37). Greencastle, IN: ASCUE. (ERIC Document Reproduction Service No. ED 357 732)

Hereford, J., & Winn, W. (1994). Non-speech sound in human-computer interaction: A review and design guidelines. Journal for Educational Computing Research, 11, 211-233.

Hiebert, J. (1984). Children’s mathematics learning: The struggle to link form and understanding. The Elementary School Journal, 84, 496-513.

Hiebert, J. (1988). A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.

Hiebert, J. (1989). The struggle to link symbols with understandings: An update. Arithmetic Teacher, 36, 38-44.

Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23, 98-122.

Holt, J. (1964). How children fail. Hammondsworth, Middlesex, UK: Penguin. Hope, J. A. (1987). A case study of a highly skilled mental calculator. Journal for Research

in Mathematics Education, 18, 331-342. Howard, P., Perry, B., & Conroy, J. (1995, November). Manipulatives in K-6 mathematics

learning and teaching. In Directions: Yesterday, Today, Tomorrow: Annual conference of the Australian Association for Research in Education, Hobart, Tasmania.

Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36, 6-11. Hughes, P. (1995, August). Difficulties with understanding place-value. In NZAMT4: The

many faces of mathematics education. Biennial conference of the New Zealand Association of Mathematics Teachers, Auckland, New Zealand.

Hunting, R. P. (1983). Emerging methodologies for understanding internal processes governing children’s mathematical behaviour. The Australian Journal of Education, 27, 45-61.

Hunting, R. P., & Lamon, S. (1995). A re-examination of the rôle of instructional materials in mathematics education. Nordisk Matematik Didaktik [Nordic Studies in Mathematics Education], 3(3), 45-64.

Hunting, R. P., Davis, G., & Pearn, C. A. (1996). Engaging whole-number knowledge for rational-number learning using a computer-based tool. Journal for Research in Mathematics Education, 27, 354-379.

Hunting, R. P., & Doig, B. A. (1992). The development of a clinical tool for initial assessment of a student's mathematics learning. In M. Stephens & J. Izard (Eds.), Reshaping assessment practices: Assessment in the mathematical sciences under challenge (pp. 201-217). Hawthorn: ACER.

Irons, C. J., & Burnett, J. (1994). Mathematics from many cultures (Teachers’ notes). Hawthorn, Victoria:: Mimosa.

Janvier, C. (1987). Representation and understanding: The notion of function as an example. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 67-71). Hillsdale, NJ: Erlbaum.

378

Jonassen, D. H. (1995). Supporting communities of learners with technology: A vision for integrating technology with learning in schools. Educational Technology, 35(4), 60-63.

Jonassen, D. H., Campbell, J. P., & Davidson, M. E. (1994). Learning with media: Restructuring the debate. Educational Technology Research and Development, 42(2), 31-39.

Jones, G. A., & Thornton, C. A. (1993a). Children’s understanding of place value: A framework for curriculum development and assessment. Young children, 48(5), 12-18.

Jones, G. A., & Thornton, C. A. (1993b). Vygotsky revisited: Nurturing young children’s understanding of number. Focus on Learning Problems in Mathematics, 15(2 & 3), 18-28.

Jones, G. A., Thornton, C. A., & Putt, I. J. (1994). A model for nurturing and assessing multidigit number sense among first grade children. Educational Studies in Mathematics, 27, 117-143.

Jones, G. A., Thornton, C. A., & Van Zoest, L. R. (1992, April). First grade children’s understanding of multi-digit numbers. Paper presented at the meeting of the American Educational Research Association, San Francisco, CA. (ERIC Document Reproduction Service No. ED 353 141)

Jones, K., Kershaw, L., & Sparrow, L. (1994). Number sense and computation in the classroom. Perth, Western Australia: Mathematics, Science and Technology Education Centre, Edith Cowan University.

Kamii, C., & Lewis, B. A. (1991). Achievement tests in primary mathematics: Perpetuating lower-order thinking. Arithmetic Teacher, 38, 4-9.

Kamii, C., Lewis, B. A., & Livingston, S. J. (1993). Primary arithmetic: Children inventing their own procedures. Arithmetic Teacher, 41, 200-203.

Kozma, R. B. (1991). Learning with media. Review of Educational Research, 61, 179-211. Kozma, R. B. (1994a). A reply: Media and methods. Educational Technology Research and

Development, 42(3), 11-14. Kozma, R. B. (1994b). Will media influence learning? Reframing the debate. Educational

Technology Research and Development, 42(2), 7-19. Lobato, J. E. (1993). Making connections with estimation. Arithmetic Teacher, 40, 347-351. Marchionini, G. (1988). Hypermedia and learning: Freedom and chaos. Educational

Technology, 28(11), 8-11. Mason, J. H. (1987). What do symbols represent? In C. Janvier (Ed.), Problems of

representation in the teaching and learning of mathematics (pp. 73-81). Hillsdale, NJ: Erlbaum.

Mason, S. A. (1993). Employing quantitative and qualitative methods in one study. British Journal of Nursing, 2, 869-872.

Mayer, R. E. (1996). Learners as information processors: Legacies and limitations of educational psychology’s second metaphor. Educational Psychologist, 31, 151-161.

McArthur, D. (1987). Developing computer tools to support performing and learning complex cognitive skills. In D. E. Berger, K. Pezdek, & W. P. Banks (Eds.), Applications of cognitive psychology: Problem solving, education, and computing (pp. 183-200). Hillsdale, NJ: Erlbaum.

McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2-8.

379

McNeal, B. (1995, October). Fact families as socially constructed knowledge. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH. (ERIC Document Reproduction Service No. ED 389 563)

Merseth, K. K. (1978). Using materials and activities in teaching addition and subtraction algorithms. In M. N. Suydam & R. E. Reys (Eds.), Developing computational skills: 1978 Yearbook (pp. 61-77). Reston, VA: NCTM.

Miles, M. B., & Huberman, A. M. (1984). Drawing valid meaning from qualitative data: Toward a shared craft. Educational Researcher, 13(15), 20-30.

Misanchuk, E. R., & Schwier, R. A. (1992). Representing interactive multimedia and hypermedia audit trails. Journal of Educational Multimedia and Hypermedia, 1, 355-372.

Miura, I. T., & Okamoto, Y. (1989). Comparison of U.S. and Japanese first graders’ cognitive representation of number and understanding of place value. Journal of Educational Psychology, 81, 109-113.

Miura, I. T., Okamoto, Y., Kim, C. C., Steere, M., & Fayol, M. (1993). First graders’ cognitive representation of number and understanding of place value: Cross-national comparisons—France, Japan, Korea, Sweden, and the United States. Journal of Educational Psychology, 85, 24-30.

Mullis, I. V. S., Martin, M. O., Beaton, A. E., Gonzales, E. J., Kelly, D. L., & Smith, T. A. (1997). Mathematics achievement in the primary school years: IEA’s Third International Mathematics and Science Study. Chestnut Hill, MA: TIMSS International Study Centre.

Nagel, N. G., & Swingen, C. C. (1998). Students’ explanations of place value in addition and subtraction. Teaching Children Mathematics, 5, 164-170.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1998). Principles and standards for school mathematics: Discussion draft. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press.

Nevin, M. L. (1992). A language arts approach to mathematics. Arithmetic Teacher, 40, 142-146.

Newby, T. J., Ertmer, P. A., & Stepich, D. A. (1995). Instructional analogies and the learning of concepts. Educational Technology Research and Development, 43(1), 5-18.

Nitko, A. J., & Lane, S. (1990). Solving problems is not enough: Assessing and diagnosing the ways in which students organize statistical concepts. (ERIC Document Reproduction Service No. ED 329 597)

Ohlsson, S., Ernst, A. M., & Rees, E. (1992). The cognitive complexity of learning and doing arithmetic. Journal for Research in Mathematics Education, 23, 441-467.

Parkes, A. P. (1994). A study of problem solving activities in a hypermedia representation. Journal of Educational Multimedia and Hypermedia, 3, 197-223.

Patton, M. Q. (1990). Qualitative evaluation and research methods (2nd ed.). Newbury Park, CA: Sage.

Payne, J., & Rathmell, E. (1975). Number and numeration. In J. N. Payne (Ed.), Mathematics learning in early childhood (pp. 125-160). Reston, VA: NCTM.

380

Pea, R. D. (1985). Beyond amplification: Using the computer to reorganize mental functioning. Educational Psychologist, 20, 167-182.

Perry, B., & Howard, P. (1994). Manipulatives - constraints on construction? In G. Bell, R. Wright, N. Leeson, & J. Geake (Eds.), Challenges in mathematics education: Constraints on construction. Proceedings of the 17th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 2, pp. 487-495). Mathematics Education Research Group of Australasia.

Peterson, S. K., Mercer, C. D., McLeod, P. D., & Hudson, P. J. (1989). Validating the concrete to abstract instructional sequence for teaching place value to learning disabled students (Monograph #20). Gainesville, FL: Florida University, Department of Special Education. (ERIC Document Reproduction Service No. ED 355 728)

Peterson, S. K., Mercer, C. D., Tragash, J., & O’Shea, L. (1987). Comparing the concrete to abstract teaching sequence to abstract instruction for initial place value skills (Monograph #19). Gainesville, FL: Florida University, Department of Special Education. (ERIC Document Reproduction Service No. ED 353 744)

Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595-610.

Price, P. S. (1996, July). The computer as a generator of analogues. Paper presented at the 19th annual conference of the Mathematics Education Research Group of Australasia, Melbourne.

Price, P. S. (1997). Teacher presence as a variable in research into students’ mathematical decision-making. In F. Biddulph & K. Carr (Eds.), People in mathematics education (Proceedings of the 20th annual conference of the Mathematics Education Research Group of Australasia, Rotorua, NZ, pp. 422-428).

Price, P. S., & Price, P. M. (1998). Hi-Flyer Maths - Decimals (Version 1.04) [Computer software]. Brisbane, Queensland: Hi-Flyer Software.

Putnam, R. T., Lampert, M., & Peterson, P. L. (1990). Alternative perspectives on knowing mathematics in elementary schools. Review of Research in Education, 16, 57-150.

Q.S.R. NUD*IST (Version 3.0.4) [computer software]. (1994). Melbourne, Victoria: Qualitative Solutions & Research.

Queensland Department of Education. (1996). The Year 2 Diagnostic Net. Brisbane, Queensland: Author.

Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 109-151). New York: Academic Press.

Resnick, L. B. (1987). Education and learning to think. Washington, DC: National Academy.

Resnick, L. B. (1988). Treating mathematics as an ill-structured discipline. I n R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (Vol. 3) (pp. 32-60). Reston, VA: NCTM.

Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3) (pp. 41-95). Hillsdale, NJ: Erlbaum.

Reys, R. E., & Yang, D. C. (1998). Relationships between computational performance and number sense among sixth- and eighth-grade students in Taiwan. Journal for Research in Mathematics Education, 29, 225-237.

Ross, R., & Kurtz, R. (1993). Making manipulatives work: A strategy for success. Arithmetic Teacher, 40, 254-257.

Ross, S. H. (1989). Parts, wholes and place value: A developmental view. Arithmetic Teacher, 36, 47-51.

381

Ross, S. H. (1990). Children’s acquisition of place-value numeration concepts: The roles of cognitive development and instruction. Focus on Learning Problems in Mathematics, 12(1), 1-17.

Ross, S. H., & Sunflower, E. (1996). Place-value: Problem-solving and written assessment using digit-correspondence tasks. The Math Forum - Place-Value: Introduction [On-line]. Available: http://forum.swarthmore.edu/mathed/nctm96/construct/ross/

Rossman, G. B., & Wilson, B. L. (1985). Numbers and words: Combining quantitative and qualitative methods in a single large-scale evaluation study. Evaluation Review, 9, 627-643.

Rossman, G. B., & Wilson, B. L. (1991, March). Numbers and words revisited: Being “shamelessly eclectic”. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL. (ERIC Document Reproduction Service No. ED 377 235)

Rutgers Math Construction Tools [computer software]. (1992). NJ: Rutgers University. Salomon, G. (1988). AI in reverse: Computer tools that turn cognitive. Journal of

Educational Computing Research, 4(2), 123-134. Salomon, G. (1998). Technology’s promises and dangers in a psychological and educational

context. Theory Into Practice, 37, 4-10. Saxton, M., & Towse, J. N. (1998). Linguistic relativity: The case of place value in multi-

digit numbers. Journal of Experimental Child Psychology, 69, 66-79. Schielack, J. F. (1991). Reaching young pupils with technology. Arithmetic Teacher, 38(6),

51-55. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,

metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). New York: Macmillan.

Schwier, R. A., & Misanchuk, E. R. (1990, June). Analytical tools and research issues for interactive media. Paper presented at the Annual Meeting of the Association for Media and Technology in Education in Canada, St John’s, Newfoundland, Canada. (ERIC Document Reproduction Service No. ED 321 751)

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.

Sfard, A., & Thompson, P. W. (1994). Problems of reification: Representations and mathematical objects. In D. Kirshner (Ed.), Proceedings of the 16th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 3-34). (ERIC Document Reproduction Service No. ED 383 533)

Shepard, R. N. (1978). The mental image. American Psychologist, 33, 125-137. Silver, E. A. (1994). Treating estimation and mental computation as situated mathematical

processes. In R. E. Reys & N. Nohda (Eds.), Computational alternatives for the twenty-first century: Cross-cultural perspectives from Japan and the United States (pp. 147-160). Reston, VA: NCTM.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.

Simons, P. R. J. (1984). Instructing with analogies. Journal of Educational Psychology, 76, 513-527.

Sinclair, A., & Scheuer, N. (1993). Understanding the written number system: 6 year-olds in Argentina and Switzerland. Educational Studies in Mathematics, 24, 199-221.

Sinclair, K. E. (1993). The versatile computer: Both tutor and cognitive tool. Australian Educational Computing, 8(2), 21-24.

382

Skemp, R. R. (1982). Communicating mathematics: Surface structures and deep structures. Visible Language, 16, 281-288.

Sowder, J. T. (1988). Mental computation and number comparison: Their roles in the development of number sense and computational estimation. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 182-197). Reston, VA: NCTM.

Sowder, J. (1992). Estimation and number sense. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 371-389). New York: Macmillan.

Sowder, J. T. (1994). Cognitive and metacognitive processes in mental computation and computational estimation. In R. E. Reys & N. Nohda (Eds.), Computational alternatives for the twenty-first century: Cross-cultural perspectives from Japan and the United States (pp. 139-146). Reston, VA: NCTM.

Sowder, J., & Schappelle, B. (1994). Number sense-making. Arithmetic Teacher, 41, 342-345.

Sowell, E. J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20, 498-505.

Sparrow, L., Kershaw, L., & Jones, K. (1994). Calculators: Research and curriculum implications. Perth, Western Australia: Mathematics, Science and Technology Education Centre, Edith Cowan University.

Stedman, L. (1995). Technology in teaching. Unpublished manuscript, Queensland University of Technology at Brisbane, Australia.

Steffe, L. P., Cobb, P., & von Glasersfeld, E. (1988). Construction of arithmetical meanings and strategies. New York :Springer-Verlag.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage.

Sweller, J. (1999). Instructional design in technical areas. Camberwell, Victoria: ACER. Thompson, C. (1990). Place value and larger numbers. In J. N. Payne (Ed.), Mathematics for

the young child (pp. 89-108). Reston, VA: NCTM. Thompson, P. W. (1992). Notations, conventions, and constraints: Contributions to effective

uses of concrete materials in elementary mathematics. Journal for Research in Mathematics Education, 23, 123-147.

Thompson, P. W. (1994). Concrete materials and teaching for mathematical understanding. Arithmetic Teacher, 41, 556-558.

Touger, H. E. (1986). Models: Help or hindrance? Arithmetic Teacher, 33(7), 36-37. Trafton, P. R. (1992). Using number sense to develop mental computation and computational

estimation. In C. J. Irons (Ed.), Challenging Children To Think When They Compute (pp. 78-92). Proceedings of Conference, Queensland University of Technology, Brisbane.

Ullmer, E. J. (1994). Media and learning: Are there two kinds of truth? Educational Technology Research and Development, 42(1), 21-32.

Van Engen, H. (1949). An analysis of meaning in arithmetic. The Elementary School Journal, 49, 321-329, 395-400.

Vosniadou, S., & Ortony, A. (1989). Similarity and analogical reasoning: A synthesis. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 1-17). Cambridge University Press.

Wiersma, W. (1995). Research methods in education: An introduction (6th ed.). Needham Heights, MA: Allyn & Bacon.

383

Williams, M. D., & Dodge, B. J. (1993, January). Tracking and analysing learner-computer interaction. In Proceedings of Selected Research and Development Presentations at the Convention of the Association for Educational Communications and Technology, New Orleans, LA. (ERIC Document Reproduction Service No. ED 362 212)

Wilson, F., & Shield, M. (1993). Solving the problem of problem solving. The Australian Mathematics Teacher, 49(3), 10-12.

Winn, W. (1993). Instructional design and situated learning: Paradox or partnership? Educational Technology, 33(3), 16-21.

Yin, R. K. (1994). Case study research design and methods (2nd ed.). Thousand Oaks, CA: Sage.