Student perspective on effective mathematics pedagogy: Stimulated recall approach study

STUDENT PERSPECTIVE ON EFFECTIVE MATHEMATICS PEDAGOGY: STIMULATED RECALL APPROACH STUDY

Berinderjeet Kaur and Low Hooi Kiam

Abstract The goals of this study were two fold. It was a special focus project of CRPP and complimented in some ways the research of Panels 3 and 4 of the core research programme (2004 – 2007). It was also a part of an international comparative study, the Learner‘s Perspective Study (LPS), led by David Clarke from the University of Melbourne. The study examined the practices of three secondary two (grade eight) competent mathematics teachers and their classrooms in an integrated and comprehensive manner. This qualitative study adopted the research methodology of the LPS. A significant feature of the methodology was the use of three cameras to capture the data during sequences of 10 lessons for each teacher. The data collected for the study is indeed very substantial. Analysis of the instructional approaches of the teachers, the role of textbook in their classrooms, nature and role of homework, the source and cognitive demands of mathematical tasks the teachers used, the type of teacher questions that were asked during the instruction, what teacher‘s attached importance to in their lessons, what students valued in their mathematics lessons and student‘s perspectives of good mathematics lessons have been carried out and the respective findings reported in several publications as well as this report.

Introduction

The goals of this study were two fold. On one hand it was a special focus project of CRPP‘s research program. It complimented in some ways the research of Panel 3: What Goes On in Classrooms part II? Coding Practice and Panel 4: What Goes On in Classrooms part III? which formed the core research programme of CRPP from 2004–2007. On the other hand it was Singapore‘s contribution to an international comparative study initiated by David Clarke at the University of Melbourne (Clarke, Keitel & Shimizu, 2006) namely, The Learner‘s Perspective Study (LPS).

Background of the LPS

A major premise of the LPS is that teaching and learning are not discrete activities present in a common context, specifically the classroom. This premise arises from a few considerations. The first of which is that learning is a social activity and that Vygotsky in his writings often considered the duality of the terms ―teaching‖ and ―learning‖ which is reflected in the myriad of translations of his seminal work. The following passage from Vygotsky‘s work:

―We propose that an essential feature of learning [teaching] is that it creates the zone of proximal development; that is, learning [teaching] awakens a variety of

FINAL RESEARCH REPORT Project No. CRP 3/04 BK

April 2009

FINAL RESEARCH REPORT

developmental processes that are able to interact only when the child is interacting with people in his environment and in collaboration with his peers‖ (Vygotsky, 1978, p. 90)

with the word ―learning‖ replaced by ―teaching‖ is key to the theoretical basis for the LPS. The second is that learning in classroom settings has centred on the negotiation of meaning (Clarke, 1996; Cobb & Bauersfeld, 1995). These meanings are not restricted to content-specific meanings but include the negotiation of the social meanings by which the practices of the classroom are constituted and enacted (Yackel & Cobb, 1993). Further, it has been argued that every aspect of the classroom experience is constructed by the participants, including both the situation or classroom context as well as any contexts invoked by the tasks used in instruction (Clarke & Helme, 1998). In the LPS, the documentation of such constructed or construed meanings is taken to be a major priority. Therefore, it is essential that research address the processes leading to learning in classroom settings. Without an understanding of these processes, attempts to improve teaching practices and learning outcomes in mathematics classrooms have little chance of success. The need to improve the quality of process as an essential precursor to the improvement of product is well understood in most other professional and industrial fields. The same principle needs to guide practices in education. The improvement of mathematics teaching must be founded upon an understanding of both teaching and learning and the relationship of both activities to student achievement. The LPS also draws on the weaknesses of the TIMSS Video Study (Stigler & Hiebert, 1997, 1999) and merits of fine-grained analyses of classroom video data and micro-analysis of classroom practices, such as Cobb and Bauersfeld (1995), to investigate the ―culture of secondary two mathematics classrooms‖. It is doubtless that the findings will provide important links between teaching, learning, understanding and achievement.

Objectives of the Present Study

The study examined the practices of secondary two (grade eight) mathematics classrooms in an integrated and comprehensive manner. Unlike, most studies on mathematics classrooms, a distinguishing feature of this study was the exploration of learner practices across a sequence of lessons.

The main objectives of the study were: 1. to document practices of competent mathematics teachers in secondary two

mathematics classrooms, 2. to study from the perspectives of students the roles of the textbook and homework

and what constitutes good mathematics lessons, and 3. to identify common classroom pedagogies from the perspectives of both teachers

and students that enhance the teaching and learning of mathematics.

Research Methodology

The study adopted the methodology of the LPS, described in detail in Clarke (2006). The methodology was largely qualitative in nature. A key characteristic of the methodology was the documentation of sequences of lessons, rather than just single lessons. Another important feature of the study was the documentation of the participants‘ perspectives (teacher and students) through post-lesson video-stimulated interviews.

The Method

Three secondary two (Grade 8) mathematics teachers participated in the study in 2005. One secondary two mathematics class taught by each teacher was videotaped for a


period of approximately three weeks which included three familiarisation lessons, followed by a sequence of ten consecutive lessons. Appendix A shows the process of data collection in the classroom. Three video-cameras: a ―Teacher Camera‖, a ―Student Camera‖ and a ―Whole Class Camera‖ were used to video-tape the lessons. The Teacher camera captured the teacher‘s actions and talk during the lesson. The Student camera focused on a group of students (3-5 students depending on the seat plan of the students in the class), known as the ―focus group‖ and captured their actions and talk during the lesson. Each group of students was only videotaped once. The Whole Class camera captured the whole class in action. After every lesson was video-taped two students from the focus group were interviewed separately after school. A split screen video record which was mixed on-site from the Teacher and Student cameras of the day‘s lesson was used as a stimulus for the student interview. Figure 1 shows the split screen image used for the student interviews. Appendix B shows the prompts used by the researchers for the student interviews.

Figure 1. Mixed on-site video image

A chronological record of all activities that took place during every lesson videotaped was recorded in the form of lesson tables by the researchers present in the classrooms. Appendix C shows an example of a lesson table. Student artefacts (e.g. worksheet, textbook pages and homework) from the focus group were photocopied after each lesson and stored as data. The written materials produced by the students during interview were also collected. The teacher was interviewed three times, once each week. The stimulus for the interview was a video record of a lesson the teacher taught during the week. Appendix D shows the prompts used by the researchers for the teacher interviews. Survey data was also collected from both the teachers and students. The teachers completed two substantial questionnaires before and after the video-taping period. They also completed a short questionnaire for every lesson video-taped. The three teacher questionnaires can be found in Appendix E. Students completed a questionnaire (see Appendix F) designed by the local research team. At the end of the study, students took the International Benchmarking Test (see Appendix G) as required by the LPS. Appendix H shows a summary of the data collected for every video-taped lesson of the study.

The Subjects

Teachers

Altogether three secondary two mathematics teachers from three schools participated in the study in 2005. All the three teachers were specialists in their field. These teachers are from a pool of teachers deemed as ―experienced and competent‖, where experience was a measure of the number of years they have taught mathematics in secondary schools and competency was a composite measure of their students‘ performance at examinations and their performance in class in the eyes of their students. All the three teachers met the requirements and volunteered to participate in the study. Written consent for participation was sought from all three of them (see Appendix I for consent form).

Student Image

Teacher Image


Teacher 1 (T1) from School 1 (SG1) was a female with 21 years of teaching experience across all levels from secondary one to five, of which seven years were spent in School 1. Teacher 2 (T2) from School 2 (SG2) was also a female with 27 years of teaching experience across all levels from secondary one to five, of which six years were spent in School 2. Teacher 3 (T3) from School 3 (SG3) was a male with 15 years of teaching experience at upper levels from secondary three to four.

Students

All three classes that participated in the study were Secondary Two Express classes and each class comprised boys and girls. Teacher 1 from school 1 taught a class of 37 students. The Primary School Leaving Examination (PSLE) aggregate score of students in school 1 were in the range of 245 – 267 with mean score = 250 and median score 249. These students were high achievers. Their mean International Benchmark Test (IBT) score was 42.8 (85.6%). The maximum score for the test is 50. Teacher 2 from school 2 taught a class of 40 students. The PSLE aggregate score of students in school 2 were in the range of 253 – 265 with mean score = 253 and median score 252. These students were high achievers. Their mean IBT score was 41.9 (83.8%). Teacher 3 from school 3 taught a class of 40 students. The PSLE aggregate score of students in school 3 were in the range of 188 – 253 with mean score = 207 and median score= 206. These students were of average ability. Their mean IBT score was 38.3 (76.6%). All the students volunteered to participate in the study and consent was sought from them (see Appendix J for consent form) and their parents/guardians (see Appendix K for consent form) to be videotaped during the data collection period.

The Research Questions

More specifically, the objectives of the study were explored through a number of research questions? The questions are: 1. What are the instructional approaches of competent secondary two mathematics

teachers? 2. How do learners view the roles of the textbook and homework? 3. What are the characteristics of a good lesson from the perspectives of students? 4. What are the common classroom pedagogies from the perspectives of both teachers

and students that facilitate the teaching and learning of mathematics?

The Data and Analysis

Lessons in SG1 and SG2 were about an hour each, while those in SG3 were a combination of 30 minutes and an hour each. The topics covered during the lesson sequences were Arithmetic Problems and Standard Form, Congruence and Similarity, Scales and Maps in SG1, Factorisation of Algebraic Expressions, Algebraic Manipulation and Formulae and Simultaneous Equations in SG2, and Quadratic graphs and their applications, Pythagoras theorem and applications and trigonometrical ratios (Sine, Cosine and Tangent of angles) and applications in SG3. The video data collected was digitized and all the classroom utterances and interviews were transcribed. Time-coding of the transcripts were carried out to synchronize the video records and utterances. Where necessary the video analysis software called Studiocode

TM was used. This software has been specifically developed for the LPS.

The data collected for this study is very substantial. Hence the analysis of the data collected would be an ongoing effort for a long time to come. However, at the time of writing this report, the analyses have focused mainly on classroom pedagogies. The following aspects of the data have been analysed. They are:


1. Instructional approaches of the teachers 2. The role of the textbook 3. Nature and role of homework 4. Source and cognitive demands of mathematical tasks 5. Nature and purpose of teacher questions 6. What teachers attach importance to in their mathematics lessons 7. What students value in their mathematics lessons 8. Students‘ perspectives of good mathematics lessons The analysis of each of the above is next presented together with the corresponding findings.

Instructional Approaches of the Teachers

Data

The video records of the ten-lesson sequences for each of the teachers were the main source of data analysed. The sequences of lessons enable us to analyse for the overarching features of the lesson structure within a sequence rather than for the lesson script (Stigler & Hiebert, 1997) of individual lessons, which may vary depending on at which stage the lesson was within a unit (Shimizu, 2002).

Analysis

The constant comparison method (also known as the grounded theory approach) (Glaser & Strauss, 1967) was used to identify for activity segments that characterised the three lesson sequences that were analysed. As an exploratory study, activity segments – ―the major division of the lessons‖, served as an appropriate unit of analysis for examining the structural patterns of lessons since it allows us ―to describe the classroom activity as a whole‖ (Stodolsky, 1988, p.11). According to Stodolsky:

In essence, an activity segment is a part of a lesson that has a focus or concern and starts and stops. A segment has a particular instructional format, participants, materials, and behavioural expectations and goals. It occupies a certain block of time in a lesson and occurs in a fixed physical setting. A segment‘s focus can be instructional or managerial (Stodolsky, 1988, p. 11).

For the purpose of this report, the activity segments were distinguished mainly by the instructional format that characterised them, although there exists other segment properties such as materials that differed among the various activity segments identified. Six categories of activity segments emerged through reiterative viewing of the video data. These mutually-exclusive segments were found to be able to account for most part of the 30 lessons from SG1, SG2 and SG3. Four categories were prominent in all three classrooms. They are: whole-class demonstration [D], characterised by whole-class mathematics instruction

that aimed to develop students‘ understanding of mathematical concepts and skills; seatwork [S] during which students were assigned questions to work on either

individually or in groups at their desks; whole class review of student work [R], during which the teacher‘s primary focus was

to review the work done by students or the task assigned to them; miscellaneous [M], a catch-all category during which the class was involved in mainly

managerial and administrative activities. The two other categories, one of which was exclusive to a single classroom and another to two classrooms, are:


group quiz [Q] found only in SG2, during which a representative from each group (usually four groups at a time) was asked to compete with other representatives to solve a problem on the board;

test [T] found only in SG1 and SG3. Such a coding scheme closely corresponded with another study on Singapore primary mathematics classrooms which had also identified similar activity segments (Ho & Hedberg, 2005). To code for the activity segments, Studiocode

TM, a video coding

software was used which allowed us to see the patterns in which the various segments were sequenced in the lessons. Exclusive to SG3 were interruptions denoted by ^. These interruptions were characterised by non-instructional segments which involved disciplinary and administrative activities such as announcements related to mathematics such as reminding students to sign up for ―Math Trails‖, etc. As suggested by previous studies on lesson structures, which have shown that the structure of lessons varied from lesson to lesson (Mesiti, Clarke & Lobato, 2003) and depended on the stage at which the lesson was located within a teaching unit made up of a lesson sequence of related content (Shimizu, 2002), the mathematical content is an important factor that accounts for the structural variations of lessons. As such, further analysis was carried out to identify the instructional objective that each segment was concerned with, based on the mathematical content that was covered. Table 1 below shows an abridged version of the analysis that combined both the lesson structure and the mathematical content.

Table 1. Analysis of lesson structure with mathematical content

Lesson no.

Activity segment

code Mathematical content

Instructional objective

Instructional cycle no.

1 [S] Practice task: 2x + 4y – 3(x + 2y)2

Factorisation by grouping

1

1 [R] Student wrote answers for practice task on board

1 [D] Worked examples: x

2 – 9, y

2 – 1/16, 9y

2 – 4z

2 Factorisation of expression in the form of difference of two squares

2

1 [S] Practice tasks: a

2x

2– 16y

2, 50x

2 – 2p

2

1 [R] Teacher and students worked out practice tasks on board

Findings

Pattern and structure of lessons. Coding of the video data revealed patterns of instructional cycles that comprised mainly of combinations of the three main categories of classroom activity, whole-class demonstration [D], seatwork [S] and whole class review of student work [R] for the sequences of 10 lessons each for T1, T2 and T3. Figure 2 shows the segment sequence for the 10 lessons each for T1, T2 and T3. Activity segments that served different instructional objectives were separated by a dotted vertical line. In an instructional cycle the mathematical tasks shared the same instructional objective. The instructional cycles of T1, T2 and T3 comprised of various combinations of the segments D, S and R as well as standalone segments of D and R. In addition, for T2, there were also combinations of D, S, R and Q. Table 2 shows the frequencies of the various instructional patterns for T1, T2 and T3.


Table 2. Frequencies of instructional patterns

Pattern Frequency

T1 T2 T3

D 2 1 3

R 4 1 2

DS 1 1 3

DR 1 - 2

DQ - 2 -

DSR 9 6 1

DSRS - 1

DSRDSR 1

DSRSR 4 1

DSRSRQ - 1

DSRSRSR 1

DSRSRSRD 1

DRSR - 1

DRSRSRS - - 1

SR - 2

SRS 1

SRSR 1

SDRSR - - 1

RSR - - 1

RSRS - - 1

QSR - 1

Total 26 18 15

For T1, the most typical pattern of an instructional cycle was composed of the three consecutive segments, whole class demonstration [D], independent student work [S] and whole class review of written work [R], i.e. [DSR]. At times it was followed by another round of [DSR], or by another one or two rounds of seatwork and review ([DSRSR] or [DSRSRSR] or [DSRSRSRD]). During these instructional cycles, T1 focussed on a specific instructional objective or part of it. The sequential cycles were incremental as mathematical knowledge was built up from one cycle to the next. The standalone segment, whole class review of student work [R] during which review of homework was carried out was also frequent in the ten lesson sequence of T1. As documented by Mesiti, Clarke and Lobato (2003) and Shimizu (2002), the structure of lessons varied depending on the stage it was within a teaching unit and hence the ten lesson sequence of T1, also showed other compositions of instructional cycles, such as [D], [DS], [DR], [SRS] and [SRSR]. Each of the cycle was guided by a specific instructional objective addressing the needs of the learners. For T2, the most typical pattern of an instructional cycle was also [DSR]. At times it was followed by [S], i.e. providing students with more tasks to complete in class under the watchful eye of the teacher, or another round of seatwork followed by whole class review, i.e. [SR], or another round of [SR] culminating with the group quiz [Q]. The ten lesson sequence of T2, also comprised of other compositions of instructional cycles, such as [D], [R], [DS], [DQ], [DRSR], [SR], and [QSR]. During each of these cycles, T2 focussed on a specific instructional objective or part of it. The sequential cycles were incremental as mathematical knowledge was built up from one cycle to the next. For T3, the instructional pattern of the sequence of 10 lessons had only one instructional cycle of the type [DSR]. The instructional cycles, whole class demonstration [D] and whole class demonstration followed by individual seatwork [DS] were relatively more frequent than whole class review of written work [R] and whole class demonstration followed by whole class review of written work [DR]. The other cycles unique to only the


instructional pattern of T3 were the cycles, namely [DRSRSRS], [SDRSR], [RSR] and [RSRS]. During each of the cycles, T3 focussed on a specific instructional objective or part of it. The sequential cycles were incremental as mathematical knowledge was built up from one cycle to the next. Compared with T1 and T2, T3 took a much longer time to build up the mathematical knowledge of his students as they were not only slower in grasping concepts but also had gaps in their past knowledge. The instructional cycles of all the three teachers shared several similar features. Firstly, it was not necessarily completed within the one lesson but could be resumed in the next lesson. In one instance for T1, six instances for T2 and five instances for T3 (shaded regions of Figure 2), the same instructional cycle continued into the following lesson. Another similar feature was that while the mathematical questions within the same instructional cycle were related to one another, they could also be related to the objective focused on during the demonstration segment in the preceding cycle. For all the three teachers, the lessons invariably began and ended with the miscellaneous segments [M], moments which were characterised by the obligatory greetings (‗Good morning/afternoon, class/[name of teacher]‘ and ‗Thank you, class/[name of teacher]‘) and instructions for homework collection and other administrative matters. The instructional models of all the three teachers were a composite of instructional cycles that were at times similar in format and repetitive. These sequential cycles were incremental as mathematical knowledge was built up from one cycle to the next. Each cycle focused on a single instructional objective, or part of an objective. Such an instructional model, characterised by its highly episodic structure with each self-contained instructional cycle making up an episode, tended to mask the developmental nature of the spiral curriculum (cf. Alexander, 2000). Besides the difference in the compositions of the instructional cycles between that of T1, T2 and T3, another difference was the frequency and duration of the activity segments. From Table 3, it is apparent that generally the duration of the activity segments and hence instructional cycles of T1 were shorter than that of T2 (except for group quiz (T2) and test (T1)). For T3, the durations of [D], [R] and [M] were the largest and the duration of [S] the smallest amongst all the three teachers. Also, the frequency of [D], [S] and [R] was the lowest for T3 amongst all the teachers, with the exception of [M]. This shows that generally, the instructional cycles of T3 were at times the longest in duration. Also, for T3 almost double the time as for T1 and T2 was taken up by the [M] segment.

Table 3. Duration of lesson segments for T1, T2 and T3

Activity Segments

T1 - SG1 T2 - SG2 T3 - SG3

Total Total no. of

Total Total no. of

Total Total no. of

duration instances duration instances duration instances

Demonstration [D]

111 min 22 129 min 14 142 min 13

Seatwork [S] 217 min 33 153 min 18 23 min 12

Review of student work [R]

135 min 33 143 min 19 155 min 16

Group Quiz [Q] - - 75 min 4 - -

Test [T] 38 min 1 - - 23 min 1

Miscellaneous [M]

32 min 22 34 min 22 68 min 24


Teacher 1 [T1] (School 1 [SG1])

L01 M D S R D S R D S R D S R D S R M

L02 M R D S R D S R S R D M D S R S R D S R M

L03 M R D S R S R S R S R *S M

L04 M R D S R D S M

L05 M D S R D S R S R *S M

L06 M T M D M

L07 M D R D S R D S R M

L08 M D S R S R S R *S M

L09 M R S M

L10 M R S D S R S R S R D M


L01 M D S R D S R S R Q D S M

L02 M R D S R D S R D S R D Q S R M

L03 M R D S M

L04 M R D Q S R M

L05 M D S R S M

L06 M R D R S R M *S M

L07 M D R M

L08 M R M Q S M

L09 M R D S M

L10 M D S R S M


L01 M ^R S R S D M

L02 M D M ^D S R D S M

L03 M T M

L04 M R M R D R M S D M

L05 M ^R S R M

L06 M R S R ^D M

L07 M ^D M

L08 M R ^D S D S D M

L09 M ^R M ^D M

L10 M ^R S R S R S D M

Legend

Represents the border between instructional cycles

* Time-filler

^ Segment with interruption

Shaded regions represent the same cycles across adjacent lessons.

Note: The lengths of segments do not reflect their duration within the lesson.

Figure 2. Structural patterns of SG1, SG2 and SG3 lesson sequences


Nature of classroom talk. To understand the instructional approaches further, we need to go beyond structural patterns of the lesson sequence. The following section describes the key features of the classroom talk through which the teachers realised their roles in not just the teaching of mathematics but also in engaging students to learn it. Given the exploratory nature of the study, description of the classrooms‘ discursive practices in this paper, specifically that between teacher and students, is confined to the main themes that emerged from the video analysis and complemented by focused transcription of relevant episodes and field notes recorded during classroom observations. As the content focus and interactional pattern of the teachers‘ talk were observed to be different across categories of activity segments, the following section describes the similar characteristics found in each category except those found in test and miscellaneous segments where the talk tended to be minimal or non-instructional. Whole-class demonstration Occupying more than one fifth of the total class time in all the three classrooms, these were the segments during which all three of the teachers played the most active role in expounding mathematical concepts and problem-solving skills mainly through the use of examples, both in the form of concepts and mathematical problems, as their teaching tool. The examples used were often carefully selected on the basis of systematic increase in complexity. The discourse pattern adopted by the teachers appeared to be very similar. The most common interaction pattern in the three classrooms was the initiation-response-feedback (IRF) discourse format (Sinclair and Coulthard, 1992). That is, the teacher asked a question, students responded and teacher provided feedback. Sometimes the teachers nominated specific students to answer their questions but usually students responded spontaneously to the teachers‘ frequent questioning. Student activities For the students of T1, the main student activities were individual or group seatwork while for the students of T2, there was the additional competitive activity group quiz during which representatives from the groups were asked to solve questions on the board. For the students of T3, the main student activity was individual seatwork. For T1 and T2 about 41% of the class time was spend on student activities alone, while for T3 only a mere 6% of class time was spend on student activity. During this activity segment, all three teachers were actively engaged in between-desk instruction whereby they moved from desk to desk either to monitor students‘ progress or to provide students with individual guidance. For both T1 and T2, longer periods of class time were devoted to student activities as their students were able to work through many of the tasks they were given with some guidance from peers and the teachers, while for T3 most of the time many of the students working individually were not able to complete the tasks without the teacher reviewing student work or demonstrating how to ―do it‖ on the board. Whole-class review of student work This segment followed independent student work (both homework and classwork), group student work or the quiz in the case of T2 and occupied about one-quarter of the class time of T1 and of T2, while nearly two-fifths of the class time of T3. Although it still appeared to be teacher-dominated in form, the main source of content for the discourse actually came from the students‘ work – mistakes, presentation of solutions and multiple solutions to a problem. During these segments, written input from the students was fore grounded in the discourse. One significant similarity of all the three teachers was the ways in which the discourse was built around student work, particularly the emphasis placed by all the teachers on learning from mistakes. The main thing that all the three teachers stressed was not what the final answers should be but the kind of mistakes students made while working through the steps needed to arrive at the final answer.


The Role of the Textbook

Data

From the corpus of data of schools SG1, SG2 and SG3, three sources of data, namely the lesson tables, interviews with teachers and interviews with students, were used to ascertain the role of the textbook in the classrooms. A lesson table is a chronological narrative account of activities that take place during the lesson. This table also details all the tasks (learning, practice and homework) that the teacher uses during the lesson, and their source. A learning task (Mok, 2004) is a task that the teacher uses to teach the students a new concept or skill. Practice tasks are tasks used during the lesson either to illuminate the concept or demonstrate the skill further, and tasks the teacher asks students to work through during the lesson either in groups or individually. Homework tasks are tasks assigned to be done at home or during out of class time.

Analysis and Findings

From the lesson tables of T1, T2 and T3 the number and sources of learning and practice tasks were traced. Similarly the sources of homework assignments were also traced. Table 4 shows the source of the learning and practice tasks while Table 5 shows the source of the homework assignments. It is evident from Tables 4 and 5 that the textbook is a significant source of the mathematical tasks that the teachers used in their sequences of lessons. It is also apparent from the tables that the textbook is not the only source that both the ‗competent‘ teachers rely on for their lessons. These findings concur with that of Zhu and Fan (2002) and Kaur, Seah and Low (2005). In Table 4, for SG1-L02, T1 used 10 learning tasks that were not from the textbook. These tasks were crafted by the teacher on the spot and they involved simple rearrangements of numerals and symbols such as 3^6, 2.1

5, 4.5 x 10

-6 and so on to

demonstrate the use of a calculator to compute. These 10 very simple tasks have resulted in making the percentage of learning tasks taken by T1 from the textbook and other sources over a period of ten lessons equal. As it was not so easy for both T2 and T3 to create learning tasks on the spot due to the nature of the topics they were teaching, this result may be considered as an anomaly. Unlike T1 and T2 who used learning tasks from other sources which were similar to those in the textbooks, T3 on two occasions used learning tasks categorised as ―other sources‖ which were created using technology (Geometer‘s sketchpad and powerpoint slides).


Table 4. Source of Learning and Practice Tasks

School -Lesson

Learning tasks Practice tasks Total Source Source

Textbook Others Textbook Others

SG1 – L01 5 3 11 - 19 SG1 - L02 2 10 3 - 15 SG1 – L03 1 - 11 1 13 SG1 – L04 1 1 24 - 26 SG1 – L05 4 1 14 - 19 SG1 – L06* - - - - - SG 1– L07 1 1 2 - 4 SG1 – L08 1 - 16 - 17 SG1 – L09** - - - - - SG1 – L10 2 1 5 - 8 Total 17 (50%) 17 (50%) 86 (99%) 1 (1%) 121

SG2 – L01 3 4 6 9 22 SG2 - L02 3 - 7 3 13 SG2 – L03 5 1 14 2 22 SG2 – L04 4 - 5 - 9 SG2 – L05 6 - 3 - 9 SG2 – L06 4 - 1 1 6 SG2 – L07** - - - - - SG2 – L08** - - 7 5 12 SG2 – L09 2 - 4 - 6 SG2 – L10 3 - 4 - 7 Total 30 (86%) 5 (14%) 51 (72%) 20 (28%) 106

SG3 – L01 2 - 8 - 10 SG3 – L02 1 - 8 - 9 SG3 – L03* - - - - - SG3 – L04** 1 - 7 - 8 SG3 – L05 2 - 4 - 6 SG3 – L06 2 1 - - 3 SG3 – L07 - 2 - 9 11 SG3 – L08 3 - 9 - 12 SG3 – L09 2 1 10 - 13 SG3 – L10 1 1 - - 2 Total 14 (74%) 5 (26%) 46 (84%) 9 (16%) 74

Textbook refers to the prescribed textbook the class and the teacher use Others refers to a source other than the prescribed textbook * class was having a test ** teacher went through the test corrections


From Table 5 it is evident that homework was either given from the textbook or other sources by both T1 and T3. T2 almost always gave homework from both sources.

Table 5. Source of Homework Assignments

Lesson

Homework Assignment

SG1 SG2 SG3

Source Source Source

Textbook Others Textbook Others Textbook Others

L01 no homework assigned √ - √ -

L02 √ - √ √ √ -

L03 - √ √ - no homework assigned

L04 √ - √ - √ -

L05 √ - √ √ √ -

L06 no homework assigned √ √ no homework assigned

L07 no homework assigned - √ √

L08 no homework assigned no homework assigned √ -

L09 - √ √ - √ -

L10 √ - √ √ no homework assigned

Textbook refers to the prescribed textbook the class and the teacher use Others refers to a source other than the prescribed textbook

The second source of data was the teacher interviews. It was during the fourth teacher interview for T1and T2 and first interview for T3 that the three teachers were asked specific questions related to their lessons, vis-à-vis selection of tasks for learning, practice and homework. The interview data was consistent with the lesson tables. Interview data has been transcribed and relevant episodes were examined. T1 stated that she used the textbook in planning her lessons and also as a source for learning, practice and homework tasks. T2 stated that her first source of learning and practice tasks was also the textbook. T2 stated that she takes about half her homework tasks from the textbook and the rest from other sources, such as other books and past year examination papers. T3 stated that the textbook for an adequate source and he used it as a resource for learning, practice and homework tasks. He felt that since the scheme of work was closely aligned to the textbook there was no need to source for tasks from other sources. The third source of data was the student interviews. The student interview data revealed that in addition to using the textbook to complete homework assignments, they also used it as a first aid to clarify their thinking and self assess their knowledge and application of concepts and skills the teacher has taught in class.

Nature and Role of Homework

The nature and role of homework assignments from the perspectives of both the teachers and students in the three classrooms are explored in this section.

Data

From the corpus of data of schools SG1, SG2 and SG3, three sources of data, namely the lesson tables, interviews with teachers and student questionnaires, were used to ascertain the nature and role of homework from the perspectives of the teachers and students in the three classrooms.


Analysis and Findings

From the lesson tables, the nature of homework was established. Table 6 shows the types of homework assigned to the students during the sequence of ten lessons each by the two teachers T1 and T2.

Table 6. Types of Homework Assignments

School - Lesson

Home Assignment

Type

I II III

SG1 – L01 No homework assigned SG1 – L02 √ - - SG1 – L03 √ - SG1 – L04 √ √ - SG1 – L05 √ - - SG1 – L06 no homework assigned SG 1– L07 no homework assigned SG1 – L08 no homework assigned SG1 – L09 - √ - SG1 – L10 √ - -

SG2 – L01 √ - - SG2 – L02 √ √ - SG2 – L03 √ √ - SG2 – L04 √ - - SG2 – L05 √ √ - SG2 – L06 √ - - SG2 – L07 √ √ - SG2 – L08 no homework assigned SG2 – L09 √ - - SG2 – L10 √ √ -

SG3 – L01 - √ - SG3 – L02 √ - SG3 – L03 no homework assigned SG3 – L04 √ - - SG3 – L05 √ - - SG3 – L06 no homework assigned SG3 – L07 √ - - SG3 – L08 √ - - SG3 – L09 √ - - SG3 – L10 no homework assigned

Type I - Same-day-content Type II - Amplify, elaborate & enrich previously learned information Type III - To prepare, in advance, material to be learned in subsequent lessons.

From Table 6, it is evident that the homework assigned by the three teachers T1, T2 and T3 was of Types I and II. None of them assigned Type III homework. On the two occasions when T1 gave her students homework assignments of Type II, they were intended in SG1-L04 to prepare the students for the upcoming test and in SG1-L09 to remediate their poor performance in the class test held during SG1-L06. Other than these two occasions T1 mainly gave her students homework of Type I – for practice of same-day-content. It appears that there are two main reasons for this. First, homework appears to be an extension of the lesson during which the students engage in seatwork, and second, T1 believes that ―reinforcement of memory‖ is important, hence the homework is meant to hone the concepts and skills taught during the lesson. Homework assignments given by T2 on many occasions comprised Types I and II. T2 gave Type I homework for very similar reasons as T1. She believed that ―practice makes perfect‖. However, she


assigned students Type II homework to enable them to consolidate their learning and engage in challenging tasks, taken mainly from non-textbook sources. She only graded assignments of Type II, to assess her students‘ progress and give feedback. T3 gave his students Type II homework on only one occasion. It was meant to prepare them for the test held during SG3 – L03. Like T1, T3 gave Type I homework on all other occasions. T3 too believed that homework helped hone the concepts and skills taught during the lesson. The second source of data was the teacher interviews. During the fourth teacher interview the teachers, T1 and T2, and during the first interview T3 were asked a specific question: ―Do you think homework is important to students? Why?‖ The interview data further established the role of homework from the perspective of the teachers that has been presented in the above paragraph. The third source of data was student responses to the question ―Do homework assignments given by [name of teacher] help you in the learning of mathematics?‖ which was a part of the student questionnaire. All 37 of the students from SG1, 38 of the 40 students from SG2 and all 40 students from SG3 completed the questionnaire. The qualitative responses to the question were analysed. Thirty-six of the students from SG1, all 38 students from SG2 and 39 40 students from SG3 indicated in their responses that homework assignments given by their teachers assisted them in their learning of mathematics. Shown below are three sample responses to the question. The phrases in italics were used to infer the functions of homework. SG1-S9 (Doreen)

The homework assignments given by [name of teacher] allows us to practice more questions, therefore having better understanding of maths, and is useful. However, I feel that homework should not only be given from the textbook, and the quantity of homework should be reasonable.

SG2-S26 (Raihanah)

It helps me in the learning of mathematics because I get to practice maths and learn new methods on how to do the questions.

SG3-S30 (Wong Hao)

Yes, like I said it is used as a practice and practice makes perfect. That‘s why it helps me.

A qualitative analysis of the all written responses was carried out. Table 7 shows the six functions of homework that were inferred from the students‘ responses. It appears that all of the six inferred functions of homework are direct consequences of the type of homework assigned by the teachers. The function ‗extension of mathematical knowledge‘ was solely inferred from the responses of students from SG2. Unlike students in SG1 and SG3, students in SG2 were exposed to ‗challenging‘ tasks taken from non-textbook sources. This may have provided them with opportunities to extend their mathematics knowledge.


Table 7. Functions of Homework

Function Descriptors

Improving/ Enhancing understanding of mathematics concepts

Help to improve understanding on the subject / Better understanding of maths / Understand topic taught / Understand well the subject / Help in understanding the concepts better / Improve maths / Better understanding / Understand more about the topic taught / Further understand formulas and concepts taught.

Revising/ practicing the topic taught

Help to revise daily / Practice topic taught / Recap the topic taught / Practice in areas that are unfamiliar/not good at / Practice makes perfect / Revise works / Revise and practice topic taught / Practice methods taught / Practice on the type of questions for that topic / As a revision / Give ample practice / Practice is important in mathematics / Refreshes memories so that can remember better / Remember the method of solving problems.

Improving problem-solving skills

Become more fluent in doing sums through practice / Able to solve problems / Help to master the skills of mathematics / Learn how to apply formulas in different questions / Able to do higher order questions and assignments / Reinforce the ways of solving the questions / Familiarize formulas taught / Help to understand the formulas and put in good use / Learn how to solve a problem using different approaches.

Preparation for test/exam

Practice for the tests / Know what kind of questions are coming out for exams / Will not panic if some challenging questions come out for exam.

Assessing own understanding/ knowing own mistakes

Able to learn from mistakes made in the homework / Assesses how much have learnt about the topic taught / Acts as a gauge to see whether can understand the concepts taught / Help to see whether understand the lessons / Assesses level of understanding through practice / Assesses level of understanding on how to apply certain formulas to some questions / It would determine whether we understand the topic or not. If not, we are ―forced‖ to learn it in order to do the questions / Challenges our mind so when confronted with easier questions, able to do with ease.

Extension of mathematics knowledge

Exposes to different types of questions / Overview of the setting of questions / Exposes to how different types of questions are being phrased / Broadens knowledge / Learn new methods on how to do the questions / Exposes to more challenging questions.

Source and cognitive demands of mathematical tasks

Tasks used by the teachers were classified as learning tasks, review tasks, practice tasks and assessment tasks according to the following functions. A learning task is an example the teacher uses to teach the students a new concept or skill. A review task is a task used by the teacher to review previously learnt concepts and/or skills so as to facilitate the learning of new concepts and skills. Practice tasks are tasks used during the lesson to either illuminate the concept or demonstrate the skill further and tasks the teacher asks students to work through during the lesson either in groups or individually or during out of class time. Assessment tasks are tasks used to assess the performance of the students. Based on these considerations, we have attempted to examine the tasks used by the teachers, in particular the source of the tasks and aspects of the demands the tasks make on the learners.

Data

For the purpose of this report, a topic of a particular textbook chapter that the teachers taught during the 10-lesson period was selected for study. The topic was defined by a sequence of lessons that captured the curricular coherence between lessons. For all the


three teachers, the topic selected was made up of learning tasks, review tasks and practice tasks. As assessment tasks were only administered during class tests we have included these ―test items‖ in our analysis to represent assessment tasks. Table 8 shows the contents of the lessons of which the mathematical tasks were studied.

Table 8. Content of lessons

SG1-T1 SG2-T2 SG3-T3

Topic: Power of Ten and Standard Form

Topic: Algebraic Manipulation and Formulae

Topic: Pythagoras‘ Theorem

Part I [L01] ordinary notation and power of ten Part II [L02] – standard form and the use of calculator Part III [L03] – Continuation of the use of calculator

Part I [L02] simplification of algebraic fractions Part II [L03] More methods of simplifying algebraic fractions Part III [L04] multiplication and division of algebraic fractions Part IV [L05] – Addition and subtraction of algebraic fractions Part V[L06] – Changing the subject of a formula

Part I [L02] introduction of Pythagoras‘ theorem Part II [L04] application of Pythagoras‘ theorem Part III [L05] continuation of application of Pythagoras‘ theorem

Class test [L06] – power of ten, standard form and problem solving strategies

Class test [L07] – power of ten, standard form and expansion and factorization of algebraic expressions

Class test [L03] – linear graphs and their applications

Analysis

Mathematical tasks can be examined from a variety of perspectives including the demands of the tasks and the presentation of the tasks. However it is not always possible to subject all the tasks to the same type of analysis. As learning tasks, taken from textbooks or other sources, are set up for specific goals of instruction during the instructional cycle, these tasks cannot be treated in the same vein as review, practice and assessment tasks because the corresponding classroom discourse [lesson event (Mok and Kaur, 2006)] has a lot to do with how the pupils engage with it. Drawing on the framework for the analysis of learning task lesson events proposed by Mok and Kaur (2006), three levels of the first aspect, differentiation of the learning process, namely

Level 1: introducing new concepts and skills Level 2: making connections between new and old concepts or skills Level 3: introducing knowledge or information beyond the scope of the curriculum requirement or textbook

were found relevant to the present study. Koh and Lee (2004) as part of a core project of the Centre for Research in Pedagogy and Practice (CRPP) at the National Institute of Education (NIE) in Singapore created and validated a set of standards for scoring teacher assignments

1 (practice tasks) and


assessment tasks in languages (English, Chinese, Malay and Tamil), mathematics, science, and social studies. Six standards were selected to analyse the practice and assessment tasks used by the teachers in the study. The first standard, depth of knowledge, is about the type of knowledge the task requires. The second standard, knowledge criticism, is about what students are required to do with the knowledge. The third standard, knowledge manipulation, is about the nature of thinking skills the task requires students to engage in. The next three standards: supportive task framing, clarity and organisation, and explicit performance standard or marking criteria are about the form of the tasks. Details of the standards are shown in Table 9.

Table 9. Relevant Standards and Dimensions (Koh & Lee, 2004)

Standard 1 - Depth of knowledge

Dimension 1 – factual knowledge Possible indicators are tasks that require students to recognise mathematical terms; state concepts, facts or principles; identify objects, patterns, or list properties; recall rules, formulae, algorithms, conventions of number, or symbolic representations; describe simple mathematical facts and computational procedures and perform routine arithmetic operations. Dimension 2 – procedural knowledge Possible indicators are tasks that require students to know how to carry out a set of steps; use a variety of computational procedures and tools; perform strategic or non-routine arithmetic operations and manipulate the written symbols of arithmetic. Dimension 3 – advanced knowledge Possible indicators are tasks that require students to expand definitions; relate facts and concepts; make connections to other mathematical concepts and procedures; explain one or more mathematical relations; understand how one major math topic relates to another and understand how a mathematical topic relates to other disciplines or real world situations

Standard 2 – Knowledge criticism

Dimension 1 – presentation of knowledge as truth or given Possible indicators are tasks that require students to accept or present ideas and information as truth or affixed body of truths; perform a well-developed algorithm; follow a set of preordained procedures and perform a clearly defined series of steps. Dimension 2 – comparing and contrasting information or

knowledge Possible indicators are tasks that require students to identify the similarities and differences in observations, data and theories; classify. Organise, and compare data, and develop heuristics to identify, organise, classify, compare and contrast data, observations or information. Dimension 3 – critiquing information or knowledge Possible indicators are tasks that require students to comment on different mathematical solutions, theories, and procedures; discuss and evaluate approaches to mathematics-related problems; make mathematical arguments, and pose and formulate mathematical problems.

Standard 3 – Knowledge manipulation

Dimension 1 – reproduction Possible indicators are tasks that require students to reproduce facts or procedures; recognise equivalents; recall familiar mathematical objects and properties; perform a a set of


preordained algorithms; manipulate expressions containing symbols and formulae in standard form; carry out computations; apply routine mathematical procedures and technical skills, and apply mathematical concepts and procedures to solve simple and routine problems. Dimension 2 – organisation, interpretation, analysis, synthesis

or evaluation Possible indicators are tasks that require students to interpret given mathematical models (equations, diagrams, etc); organize, analyse, interpret, present or generate data or information; interpret tables, graphs and charts; predict mathematical outcomes from the trends in the data; interpret the assumptions and relations involving mathematical concepts and consider alternative solutions or strategies. Dimension 3 – application or problem solving Possible indicators are tasks that require students to apply mathematical concepts and processes to solve non-routine problems; apply the signs, symbols and terms used to represent concepts and use problem-solving heuristics for non-routine problems. Dimension 4 – generation or construction of knowledge new to

students Possible indicators are tasks that require students to come up with new proofs or solutions to a mathematical problem; generalize mathematical procedures, strategies and solutions to new problem situations and apply modelling to new contexts.

Standard 4 – Supportive task framing The task provides students with appropriate framing or scaffolding (written or graphic guidance in view of the students‘ skill levels and prior knowledge) in order to support them to complete the task given Dimension 1 – content scaffolding Dimension 2 – procedural scaffolding Dimension 3 – strategy scaffolding

Standard 5 – Clarity and Organisation The task is framed logically and has instructions that are easy to understand. Standard 6 – Explicit performance standard or marking criteria The task is provided with the teacher‘s clear expectations for students‘ performance and the marking criteria is explicitly clear to the students.

Aspects of Stein and Smith‘s (1998) task analysis guide were also drawn on to establish the cognitive demands of the tasks used by the teachers in their classrooms. A brief outline of Stein and Smith‘s guide with adaptations made by the authors for the analysis of data presented in this report is shown in Table 10.


Table 10. Levels of Cognitive Demand

Levels of cognitive demand Characteristics of tasks

Level 0 – [Very Low] Memorisation tasks Level 1 - [Low] Procedural tasks without connections Level 3 [High] Procedural tasks with connections Level 4 – [Very High] Problem Solving / Doing Mathematics

- Reproduction of facts, rules, formulae - No explanations required - Algorithmic in nature - Focussed on producing correct answers - Typical textbook word - problems - No explanations required - Algorithmic in nature - Has a meaningful / ―real-world‘ context - Explanations required - Non-algorithmic in nature, requires understanding of mathematical concepts and application of - Has a ―real-world‖ context / a mathematical structure - Explanations required

Hence, appropriate aspects of the works of Stein and Smith (1998), Mok and Kaur (2006) and Koh and Lee (2004) contributed towards the analytical framework used for the analysis of learning tasks, practice tasks, and assessment tasks. However, the framework was not able to provide for the analysis of the review tasks which were a part of the data analysed and presented in this report. Both the authors of the report compiled all the tasks from the selected lessons and traced their sources, i.e. where they were taken from. Next they analysed the tasks using the appropriate frameworks elaborated in the first part of this section. The match of frameworks and task types was as follows: 1. framework proposed by Mok and Kaur (2006) to ascertain the role of the learning

tasks in the learning process differentiated by levels; 2. framework of Koh and Lee (2004) to examine against selected standards and their

corresponding dimensions the nature of practice and assessment tasks; 3. framework of Stein and Smith (1998) to establish the cognitive demands of the

practice and assessment tasks; 4. as none of the frameworks were suitable to characterise the review tasks an

exploration of these tasks was carried out to develop a possible framework. Using the appropriate frameworks, the authors analysed all the learning, practice and assessment tasks independently. The overall rate of agreement was 80%. Next, the authors jointly examined the ―disputed tasks‖ and after extensive discussion reached consensus on them. The review tasks were analysed differently. Both authors jointly examined the purpose of each review task by referring to the lesson during which it was enacted as well as the lesson prior to it.

Findings

It was found that:

The textbook appears to be a significant source of learning and practice tasks while past school and national examination papers appear to be a significant source of the assessment tasks. This alignment suggests that to a large extent the textbook tasks drive the implementation of the school curriculum, while the ―collection of past examination tasks‖ assists teachers in assessing the performance of their students benchmarked against ―examination standards‖.


All the learning tasks of T1 and T2, and four out of the six of T3 were used to introduce new concepts and skills as stipulated by the curriculum guides. 2 of the learning tasks used by T3 were for introduction of knowledge or information beyond the scope of the textbook requirement.

Based on the small sample of review tasks used by the teachers, it was found that they were used for the four following purposes: – To recall of prior knowledge – To provision of scaffolding for subsequent tasks – To connect newly acquired knowledge with past knowledge – To relate newly acquired knowledge to real life examples.

The practice tasks dealt predominately with procedural knowledge and required the lowest level of knowledge manipulation, i.e. reproduction. They were mainly of the memorisation type and required students to use algorithms to do typical textbook type of exercises which contextualise concepts and skills taught.

All the assessment tasks tested procedural knowledge. All except eight of the tasks were of memorisation type [level 0], i.e. required students to reproduce facts, rules and formulae without any explanations. The eight non-memorisation type of tasks were of the procedures without connections type [level 1]. These tasks were algorithmic in nature and required students to apply facts, rules and formulae to standard textbook type of problems.

Nature and purpose of teacher questions

Data

An analysis of the questions posed by the teacher during the lessons was carried out. The video-records of the teachers and the accompanying transcripts were the main source of data. For each teacher five lessons that were rich in classroom discourse were selected for the analysis.

Analysis

Using the selected categories: high-and-low-order questions and what, when, how, who and why questions from Cole and Chan (1994), a framework was constructed and used to analyse the teacher questions of T1, T2 and T3. The first level of analysis helped to categorise the verbal questions as ―Mathematical‖ and ―Non-mathematical‖. The Mathematical ‗M‘ questions were of the types shown in Figure 3, while the non-mathematical ‗NM‘ were one of the following types:

Type MR ‗Mathematics-Related‘ questions do not relate directly to the subject matter being taught, but were asked in the process of teaching. Example: ―Can you add in the question?‖ (SG2_L03).

Type NMR ‗Non-Mathematics Related‘ questions do not have no connection at all to the subject matter being taught or to the process of teaching. Example: ―Can you pick up the cover?‖ (SG2_L02).

The next level of analysis only concerned ―Mathematical‖ questions and they were categorised according to six levels ranging from low-order questions to high-order questions. These questions were also analysed for their intended audience, i.e. whole class or individual students.


QUESTION TYPE EXAMPLE

Mathematical Questions

Type 0 : Agreement Questions These questions often end with ‗isn‘t it?‘, ‗correct?‘, ‗alright?‘ or ‗right?‘

―You will get one, but this one is three over two, correct?‖ (SG2_L02)

Type 1: Factual (short) Questions These are basically lower-order questions which require knowledge of subject matter or the recall of facts and specifics. These questions usually begin with ‗what‘ or ‗which‘.

―What is the power of Y?‖ (SG2_L02)

Type 2: Factual (long) Questions These include procedural questions, that is questions that require the students to explain the workings or the steps leading to the answer (i.e. the processes or procedures). Also These questions often begin with ‗how‘.

―How do you simplify this one?‖ (SG2_L02)

Type 3: Explanation/ Justification Questions These questions require students to give reasons for given outcomes. They usually begin with ‗why‘.

―Why you expand it?‖ (SG2_L10)

Type 4: Opinion/ Evaluation/ Judgment Questions These questions seek students‘ own perceptions and views on concepts learnt and they are invited to voice their opinions through critical thinking.

―Is it advisable to expand it?‖ (SG2_L02)

Type 5: Conjecture Questions These are higher-order questions which require students to synthesise and think critically. The questions often involve the word ‗if‘ in it.

―What happens if it is zero?‖ (SG2_L02)

Figure 3. Types of Mathematical Questions

Findings

Table 11 shows the number of mathematical and non-mathematical questions asked by the three teachers for five of their lessons each. For each teacher more than half of the questions asked were Non-mathematical which shows that the classroom discourses centred on other concerns too, in addition to the mathematical content which was the focus of each lesson. Table 12 shows the distribution of the mathematical questions according to the six categories shown in Figure 3 for each lesson of the teachers. From Table 12, it is apparent that more than a quarter of the questions asked by T1 and T3 were of the agreement type, i.e of the lowest level. Such questions often mask the revelation of students‘ understanding and are often whole class responses. More than three quarters of the questions that each teacher asked were of ‗agreement‘ and ‗factual (short)‘ types. This shows that there was an emphasis on lower order thinking during the lessons. Table 13 shows the number of mathematical questions directed at the whole class and individual students. From Table 13, it is apparent that more than 65% of all the questions asked by each of the three teachers were directed at the whole class. Table 13 also shows that T1 and T3 asked significantly more individual students questions compared to T2.


Table 11. The Total Number of Mathematical and Non-mathematical Questions asked in the

three classrooms

Lesson (Duration) Mathematical (M) Non-mathematical

(NM) Total

SG1_L01 (55:41) 84 74 158 SG1_L02 (52:33) 79 108 187 SG1_L04 (1:00:20) 102 194 296 SG1_L05 (53:02) 88 130 218 SG1_L10 (54:26) 63 76 139 Total (4:43:33) 416 (41.7%) 582 (58.3%) 998

SG2_L01 (57:41) 52 47 99 SG2_L02 (57:11) 95 70 165 SG2_L03 (35:33) 43 49 92 SG2_L04 (56:36) 90 87 177 SG2_L10 (58:46) 84 123 207 Total (4:25:07) 364 (49.2%) 376 (50.8%) 740

SG3_L01 (33:01) 44 66 110 SG3_L04 (1:09:18) 75 140 215 SG3_L06 (31:38) 58 53 111 SG3_L08 (1:08:01) 74 194 268 SG3_L09 (40:38) 64 176 240 Total (4:02:36) 315 (33.3%) 629 (66.7%) 944

Table 12. The Frequency of Mathematical Questions by Teacher and Lesson according to the

Six Types

Lesson Question Types Total (M)

0 1 2 3 4 5

SG1_L01 24 23 7 3 23 4 84 SG1_L02 24 16 14 6 17 2 79 SG1_L04 33 48 3 11 6 1 102 SG1_L05 21 40 13 2 11 1 88 SG1_L10 19 17 9 2 12 4 63 Total 121

(29.1%) 144 (34.6%)

46 (11.1%)

24 (5.7%)

69 (16.6%)

12 (2.9%)

416

Total (M)


(11.3%) 177 (48.6%)

61 (16.8%)

21 (5.8%)

62 (17.0%)

2 (0.5%)

364

Total (M)


(26.4%) 160 (50.8%)

29 (9.2%)

22 (7.0%)

14 (4.4%)

7 (2.2%)

315

Legend Type 0: Agreement Type 2: Factual (Long) Type 4: Evaluation Type 1: Factual (Short) Type 3: Justification Type 5: Conjecture


Table 13. The Number of Mathematical Questions directed at the Whole Class (WC) and

Individual Students (IS)

Individual Students (IS) Whole Class (WC) Total (M)

SG1_L01 20 64 84 SG1_L02 10 69 79 SG1_L04 68 34 102 SG1_L05 35 53 88 SG1_L10 3 60 63 Total 136 (32.7%) 280 (67.3%) 416



What teachers attach importance to in their mathematics lessons

Data

Only three teacher interviews, one for each teacher, analysed so far were the source of data. These interviews were post-lesson video-stimulated interviews. The teachers were interviewed once very week, i.e., after every 3 or 4 lessons. The whole class video record was used as a stimulus for the teacher interview. Every week the teachers chose their video-records to comment on during the interview. They were asked to view the video-record and comment on sections they thought were important. The following are some of the prompts used by the interviewer for the teacher interviews:

Please tell me what were your goals in that lesson?

In relation to your content goal(s), why do you think this is important for students to learn?

What do you think your students might have said if I asked them this? {Here is the remote control for the video player. Do you understand how it works?}

I would like you to comment on the videotape for me. You do not need to comment on all of the lesson. Fast forward the videotape until you find sections of the lesson that you think were important. Play these sections at normal speed and describe for me what you were doing, thinking and feeling during each of these videotape sequences. You can comment while the videotape is playing, but pause the tape if there is something that you want to talk about in detail.

Would you describe that lesson as a good lesson for you? What has to happen for you to feel that a lesson is a ―good‖ lesson?

Analysis

An exploratory analysis of three teacher interviews was carried out. The teacher interviews analyzed were the first interview of T1, the second interview of T2 and the third interview of T3. Transcripts of the teacher interview data were scanned carefully and the lesson segments that the teachers chose to comment on were annotated. The interview videos were watched to ascertain what the teachers were commenting on and the respective transcripts were tabulated as shown in Table 14.


Table 14. Number and Aspects Importance was Attached to in Teacher Interviews

Teacher T1 T2 T3

No of segments

10 8 9

Segment Aspects of teacher / student practice importance was attached to

1 Ss: Setting their goals for mid-year examinations

T: Making correction to terminology (identities & formulae) she used in the past lessons

T: seeking feedback from students about their homework assignment

2 T: Giving feedback on graded assignments

T: Giving feedback for Q2 of test paper – students did poorly

T: Asking a student to draw the diagram on the whiteboard

3 T: Reviewing the past lesson

T: Highlighting that Q3b is a very good Q as it tested knowledge of operators and indices

T: drawing attention of class to the diagram and labeling of the vertices of a quadrilateral

4 T: Walking around the class - monitoring student work and providing between desk instruction

T: Giving feedback re: common mistake for Q 3a, c & d – multiply the base and exponent

T: Reinforcing the degree of accuracy of their calculations to arrive at answers correct to 3 significant figures

5 Ss: Presenting their work to the class

T: Giving feedback: Common mistake for Q4 – errors in expressing 64 as 4

3 and 125 as 5

3

without the use of a calculator

T: Prompting and guiding them in working through the solution process

6 T: Reviewing past knowledge

T: Giving feedback: Q5 is marked wrong if the answer is not written in a statement form

T: Checking if any student managed to find the solution

7 Ss: Working in groups S: Asking teacher for marks for Q5, as his numerical answer was correct?

T: Explaining how to find the solution once again

8 T: Using student work to highlight common mistakes

T: Giving feedback on common mistake for Q6 – 3

3 = 9 ?

T: Demonstrating solution on the whiteboard

9 T: Assigning homework T: Reviewing past knowledge

10 T: Demonstrating how the calculator works using the OHP.

Legend: T – teacher; S - student; Ss – students

Findings

It was found that what the teachers attached importance to was constrained by the nature and position (i.e. at the beginning, middle or towards the end of the study) of the


lesson they chose to view and comment on during their interviews. Teacher 1 chose the first lesson which started with some school-wide benchmarking practice, Teacher 2 chose a lesson she did around the middle of the study and was solely based on test corrections and feedback while Teacher 3 chose a lesson that was conducted towards the end of the project and was one in which he reviewed a particular problem he gave the students as homework and started on a new topic. The findings of the teacher interview data shows that collectively they attached importance to:

self assessment (setting of targets at the beginning of the school year for achievement in mathematics),

demonstrating procedures,

review of prior knowledge,

close monitoring of student progress in learning,

feedback on student work be it graded assignments, test papers or class work,

homework assignments.

A significant aspect of teacher / student practice that all the three teachers attached importance to was whole class review of student work.

What students value in their mathematics lessons

Data

In school 1 (SG1), school 2 (SG2) and school 3 (SG3) the number of students interviewed were 19, 20 and 20 respectively. To date, only 19 student interviews from SG1, 8 student interviews from SG2, 5 student interviews from SG3 have been analysed. The student interviews were post-lesson video-stimulated interviews. Two students from the focus group were interviewed separately after the lesson. The interviews were conducted immediately after school each day and the split screen video record, mixed on-site from the Teacher and Student camera images of the day‘s lesson, was used as a stimulus for the student interview. The students were asked to comment on sections of the lesson that they thought were important. The following are some of the prompts used by the interviewer for the student interviews:

Please tell me what you think that lesson was about?

How, do you think, you best learn something like that?

What were your personal goals for that lesson? {Here is the remote control for the video player. Do you understand how it works?}

I would like you to comment on the videotape. You do not need to comment on all of the lesson. Fast forward the videotape until you find sections of the lesson that you think were important. Play these sections at normal speed and describe for me what you were doing, thinking and feeling during each of these videotape sequences. You can comment while the videotape is playing, but pause the tape if there is something that you want to talk about in detail.

After watching the videotape, is there anything you would like to add to your description of what the lesson was about?

What did you learn during the lesson?

What are the important things you should learn in a mathematics lesson?

How would you generally assess your own achievement in mathematics? etc..


Analysis

Transcripts of the student interview data were scanned carefully and the lesson segments that the student chose to comment on were annotated. The interview videos were watched to ascertain what the students were commenting on and the respective transcripts were tabulated as shown in Table 15.

Table 15. Analysis of Lesson Segments that Student [SG1-1] Attached Importance to

Stop Transcript - Lesson segment Reason / Inference Remarks

1 Ah… this part. She teaches us the method

The teacher teaches us the method of doing the… standard form and the power of ten.

Teacher / Exposition/ Demonstration

2 Yeah, this part also She is teaching the method lah Teacher / Exposition/ Demonstration

3 Um this part where we do the question.

Uh, because like you get to do it lah … And then you can show it to the class but I didn‘t get to show it lah.

Student / Seatwork/ Groupwork

4 Ah, this one. //Ah, she is explaining clearly about the (points)

Because she teach us then I was thinking oh yeah hoh like that

Teacher / Exposition/ Explains clearly

5 Yeah this part. [friend is presenting soln on the board]

Ah we get to see the mistakes of other people so that we won‘t… do the same mistakes lah…. Because like um…the teacher will explain to you like more more detailed.

Teacher / Review of student work/ Student Presentation

6 Yeah, this paper. [paper refers to worksheet given by teacher]

Yeah. It gives us more practice and to identify the… um the ten to what power ah.

Seatwork/ Instructional Material

7 Ah…this one Because ah the the teacher explains clearly how to how ah about one problem that I‘m stuck on eh. Is the the ten.

Teacher / Exposition / Explains Clearly

8 Ah this one…. Standard form… [student making connections between what teacher is showing and what the student has seen in the textbook].

Um like you can write the the speed…./In a shorter way eh

Teacher / Exposition / New Knowledge

9 Ah, this part we are doing the practice again lah… Practice on the book.

Yeah, because if … like at home is like sometimes not enough time to practice eh.

Student / Seatwork/ Groupwork

10 Eh, this one [group member is presenting on the board].

We get to see our answers and check our answers.

Teacher / Review of student work/ Student Presentation

11 Ah, yeah, the the two ways Solving the answer and putting Teacher /


which is important lah…. The two ways of answering the question.

the answer whether in standard form or… not. Yeah, … then you can like pick the easier one eh.

Review of student work / Student Presentation

A fine-grained analysis of the aspects of the instructional practices students attached importance to, resulted in the following sub-categories. For exposition, the specific actions that students attached importance were: teacher explains / explains clearly (EC); teacher demonstrates a procedure, ―teaches the method‖ or shows using manipulative a concept / relationship (D); teacher introduces new knowledge (NK); teacher gives instructions (assigning homework / how work should be done / when work should be handed in for grading, etc.) (GI); and teacher uses real-life examples during instruction (RE). For seatwork, the activities or thing to which the students attached importance were students working individually on tasks assigned by teacher or making / copying notes (IW); students working in groups (GW) and material used as part of instruction (worksheet or any other print resource) (M). For review and feedback, the actions that students attached importance to were teacher reviews prior knowledge (PK); teacher uses student‘s presentation or work to give feedback for in class work or homework (SP); teacher giving feedback to individuals during lesson (IF) and teacher giving feedback to students through grading of their written assignments (GA). Table 16 shows the number and categorization of lesson segments for SG1, SG2 and SG3 student interviews.


Table 16. Number and Categorization of Lesson Segments for SG1 Student Interviews

Student ID

No of

segments

Instructional Practice

Exposition Seatwork Review & Feedback

EC / D / NK / GI / RE I W / GW / M PK / SP / IF / GA

SG1-1 11 2 / 2 / 1 / 0 / 0 0 / 2 / 1 0 / 3 / 0 / 0

SG1-2 8 1 / 1 / 3 / 0 / 0 0 / 1 / 0 0 / 2 / 0 / 0

SG1-3 5 0 / 1 / 2 / 0 / 0 0 / 0 / 0 0 / 2 / 0 / 0

SG1-4 8 1 / 2 / 0 / 0 / 0 0 / 1 / 0 2 / 2 / 0 / 0

SG1-5 0 0 / 0 / 0 / 0/ 0 0 / 0 / 0 0 / 0 / 0 / 0

SG1-6 1 0 / 0 / 0 / 0 / 0 0 / 1 / 0 0 / 0 / 0 / 0

SG1-7 5 2 / 0 / 0 / 1 / 0 2 / 0 / 0 0 / 0 / 0 / 0

SG1-8 5 1 / 1 / 0 / 0 / 0 0 / 1 / 1 0 / 1 / 0 / 0

SG1-9 2 1 / 0 / 0 / 0 / 0 0 / 0 / 0 0 / 1 / 0 / 0

SG1-10 3 0 / 0 / 0 / 0 / 1 1 / 0 / 0 0 / 1 / 0 / 0

SG1-11 3 1 / 1 / 0 / 0 / 0 0 / 0 / 0 0 / 0 / 0 / 1

SG1-12 3 1 / 0 / 1 / 0 / 0 0 / 0 / 0 0 / 1 / 0 / 0

SG1-13 3 1 / 1 / 0 / 0 / 1 0 / 0 / 0 0 / 0 / 0 / 0

SG1-14 1 0 / 0 / 0 / 0 / 0 0 / 0 / 0 0 / 0 / 1 / 0

SG1-15 3 0 / 1 / 1 / 0 / 0 0 / 1 / 1 0 / 1 / 0 / 0

SG1-16 2 0 / 0 / 0 / 0 / 0 0 / 0 / 0 0 / 1 / 0 / 1

SG1-17 3 1 / 0 / 0 / 0 / 0 1 / 0 / 0 0 / 1 / 0 / 0

SG1-18 7 2 / 1 / 2 / 0 / 0 0 / 0 / 0 0 / 2 / 0 / 0

SG1-19 1 1 / 0 / 0 / 0 / 0 0 / 0 / 0 0 / 0 / 0 / 0

SG2-1 7 0 / 1 / 2 / 0 / 0 0 / 1 / 0 1 / 2 / 0 / 0

SG2-2 8 1 / 1 / 0 / 0 / 0 1 / 2 / 0 0 / 3 / 0 / 0

SG2-3 4 0 / 0 / 1 / 0 / 0 0 / 1 / 0 0 / 2 / 0 / 0

SG2-5 3 0 / 1 / 0 / 1 / 0 0 / 1 / 0 0 / 0 / 0 / 0

SG2-6 6 1 / 3 / 0 / 0 / 0 0 / 1 / 0 1 / 0 / 0 / 0

SG2-7 7 2 / 0 / 1 / 0 / 0 1 / 0 / 0 0 / 2 / 0 / 1

SG2-8 7 1 / 1 / 0 / 2 / 0 1 / 1 / 0 0 / 1 / 0 / 0

SG2- 9 14 3 / 4 / 1 / 0 / 0 3 / 1 / 0 0 / 3 / 1 / 0

SG3-1 4 1 / 1 / 0 / 1 / 0 0 / 0 / 0 0 / 0 / 0 / 1

SG3-2 10 1 / 4 / 0 / 4 / 0 0 / 0 / 0 0 / 1 / 0 / 0

SG3-3 5 1 / 0 / 0 / 1 / 1 0 / 1 / 0 0 / 0 / 0 / 1

SG3-4 2 0 / 0 / 0 / 1 / 1 1 / 0 / 0 0 / 0 / 0 / 0

SG3-5 2 0 / 0 / 0 / 1 / 0 1 / 0 / 0 0 / 0 / 0 / 0

Total 153 26 / 27 / 15 / 12 / 4 12 / 16 / 3 4 / 32 / 2 / 5

Legend: EC – explains / explains clearly; D – demonstrates a procedure: ―teaches the method‖ or shows using manipulatives a concept / relationship NK – introduces new knowledge; GI – gives instructions (assigning homework / how work should be done / when work should be handed in for grading, etc.); RE – uses real-life examples during instruction; I W – students working individually; GW – students working in groups; M – material used as part of instruction (worksheet or any other print resource); PK – reviews prior knowledge; SP – uses student‘s presentation or work to give feedback for in class work or homework; IF – gives feedback to individuals during lesson; GA – gives feedback through grading of written assignments.


Findings

The findings of the student interview data show that collectively, they attached importance to several sub-aspects of the three main aspects of the instructional practice of their teachers. The three main aspects of the instructional practice were exposition, seatwork, and review and feedback. As part of exposition, students attached importance to:

their teacher‘s explanations which were simple and logical;

demonstration of mathematical procedures – showing them the ―method‖ or concrete representation of a concept with the use of a manipulative;

introduction of new knowledge – knowledge they were being exposed to for the first time;

instructions that guided them in their work, and

the use of real-life examples that helped them appreciate the use of math in life. As part of seatwork, students attached importance to:

individual work during class time that provided practice and an opportunity to check for own understanding;

group work during which they experienced teamwork spirit and peer support, and

the material (mainly in print form) given by the teacher to engage them in practice of concepts and skills they had learned.

As part of review and feedback they attached importance to:

review of prior knowledge which helped bridge past knowledge with the present and also construction of new concepts using past knowledge;

student presentations which resulted in the use of student work to highlight mistakes and demonstrate alternative approaches, and

feedback given to students individually during class time and also through grading of written assignments.

Students’ perspective of good mathematics lessons

Data

Altogether 59 students were interviewed as part of the study, 19 from SG1, 20 each from SG2 and SG3. The transcripts of their post-lesson video-stimulated interviews and in particular responses of 57 of them to the prompt: ―what has to happen for you to feel that a lesson was a good lesson?‖ is the main source of data for this section.

Analysis

The framework developed earlier for ―what students value in their mathematics lessons‖ was used to kick start the analysis of the qualitative data. The framework expanded in the process to include aspects of learner‘s engagement. Table 17 shows the analysis of sample student responses that collectively span the spectrum of all aspects of instructional practices of teachers and learner‘s engagement attributes that constitute a good lesson from students‘ perspectives.


Table 17. Analysis of student responses for aspects of instructional practices and learner’s engagement attributes

Student ID Student‘s responses Aspects

Instructional Practice

Learner‘s Engagement

SG3-16 Activities, interesting, not so boring like those graphs. Good to teach with their explanations very clear so the students will understand lor.

Activities (A) Explains clearly (EC)

Interesting topic (IT)

SG1-4 Mm there's explanation. There's practice. The teacher showed you the comparison between the wrong method and the correct method. The teacher correcting you…like when you're lost then maybe she's there to help you as a class or personally.

Explains clearly Seatwork Whole class feedback through review of work, Individual feedback

SG1-2 Ah I have to feel um… that I‘m…understand all the…lessons being taught lah.

Makes sense of what is taught

SG2-7 The teacher will recall back some of the some of the things she teach us on the at the previous lessons yeah and and show it to us and let us recall back. Then later she will teach us new methods.

Review of prior knowledge Introduction of new knowledge

SG1-5 The same thing lah learn more new things (and then)… revision work ah.

New knowledge gained Revision of past knowledge

SG3-3 Got group discussion. Test then… then maybe er…can see how much we learn so far.

Groupwork

Assessment of knowledge gained

SG3-7 Teacher to explain those important points which most student does not understand it. Then give more test so that student will remember the steps, most of the time. Tell us jokes ah give us some break or may be show some funny videos lor. To make us more alert instead of just talking then this will make us very tired.

Explain clearly difficult concepts Provide tests to hone skills

Interludes, such as jokes, video clips, to make lesson interesting

SG3-1 When I understand the whole thing ah. Like if I have ques any questions I can ask Mr Lee lah. Like if I try another question I can get the answer that means er… somehow get the grasp of it lah.

Makes complete sense of what is taught Approach teacher for help Self assessment

SG1-15 When we we never learn about this topic before, then the teacher um teach us about this topic then I find it‘s quite good ah essential for my life later on.

New knowledge Utilitarian value

SG2-12 Er…laughter, er…lesson in life and… knowing friends better ah. Knowing friends better. Er…sometimes er…after doing a question, I can find out that he is weak in that or he is strong in that next time I should go help him or next time I should go er ask him for the… how to do and he might actually teach me ah.

Through review of work in class by teacher gets to know his friends better

Knowledge of his friends ability in math so that he knows whom he can help or turn to for help


Findings

The fine-grained analysis of 57 responses, to the question

―What has to happen for you to feel that a lesson was a good lesson?‖ has resulted in the following findings. Collectively the students credited a lesson good if the teacher

explained clearly the concepts and steps of procedures,

made complex knowledge easily assimilated through demonstrations, use of manipulatives, real life examples

reviewed past knolwedge

introduced new knowledge

used student work/group presentations to give feedback to individuals or whole class

gave clear instructions, related to mathematical activities for in class and after class work

provided interesting activities for students to work on individually or in small groups

provided sufficient practice tasks for preparation towards examinations The students also credited a lesson to be good when they

found the topic interesting

enjoyed the lesson, environment was conducive

were able to make sense of what was being taught

found knowledge acquired relevant for future use

were able to do the tasks given by the teacher following the lesson

learnt from their mistakes or their friend‘s mistakes

regulated their learning through self assessment

felt empowered to help others with their work

felt they had friends who could help them with their difficulties

Significant Findings of the Study

The main objectives of the study were: 1. to document practices of competent mathematics teachers in secondary two

mathematics classrooms, 2. to study from the perspectives of students the roles of the textbook and homework

and what constitutes good mathematics lessons, and 3. to identify common classroom pedagogies from the perspectives of both teachers

and students that facilitate the teaching and learning of mathematics. More specifically, the above objectives were explored through the following research questions: 1. What are the instructional approaches of competent secondary two mathematics

teachers? 2. How do learners view the roles of the textbook and homework? 3. What are the characteristics of good lesson from the perspectives of students? 4. What are the common classroom pedagogies from the perspectives of both teachers

and students that facilitate the teaching and learning of mathematics? From the similarities of the instructional approaches of the three competent teachers it may be claimed that some of the characteristic features embedded in their instructional cycles were:


1. the very specific instructional objectives that guided each instructional cycle, with subsequent cycles building on the knowledge;

2. the carefully selected examples that systematically varied in complexity from low to high used during whole class demonstration;

3. the active monitoring of student‘s understanding during seatwork, as teachers moved from desk to desk guiding those with difficulties and selecting appropriate student work for subsequent whole class review and discussion; and

4. reinforcement of student understanding of knowledge expounded during whole class demonstration by detailed review of student in class work or homework.

From the perspectives of the students, the textbook was not only a source for their practice and homework tasks but also a first-aid for them to clarify their understanding and self-assess their knowledge and application of concepts and skills they were taught in class. The students who participated in the study were in grade eight and of above average ability. It is interesting to note that, during the interviews, a student reported ―reading the textbook‖. This student apparently had the mathematical language skills to make sense of the worked examples (tasks presented with detailed solutions) found in significant numbers in the textbook. ―Practice makes perfect‖ appears to be an underlying belief that drives the initiative of homework from both the perspectives of the teachers and students. The role of homework from the perspective of the students appear to derive to a large extent from the types of homework assigned by the teachers. Students perceive that the role of homework is to help improve/enhance their understanding of concepts taught, revise/practice what has been taught and prepare for tests/examinations, assess their own understanding, improve their problem-solving skills and extend their mathematical knowledge. It appears that the role of homework from the perspectives of the students match the intended outcomes of homework from the perspectives of the teachers. However, such an ideal student-teacher relationship may be a consequence of students having internalised what their teachers tell them about the usefulness of homework. From the perspective of students a good mathematics lesson was one where the teacher did one or more of the following: 1. explained clearly the concepts and steps of procedures; 2. made complex knowledge easily assimilated through demonstrations, use of

manipulatives, real life examples; 3. reviewed past knowledge; 4. introduced new knowledge; 5. used student work/group presentations to give feedback to individuals or the whole

class; 6. gave clear instructions, related to mathematical activities for in class and after class

work; 7. provided interesting activities for students to work on individually or in small groups;

and 8. provided sufficient practice tasks for preparation towards examinations. Finally by juxtaposing the findings of the teachers instructional approaches and interview data of the students it is hypothesised that good mathematics teaching in the three grade eight classrooms comprised of three main segments, namely whole-class demonstration (exposition), seatwork and review and feedback. Some of the actions that characterised good teaching in each of the segments were as follows:


Whole-class demonstration (exposition)

Teacher

explained clearly the concepts and steps of procedures,

made complex knowledge easily assimilated through demonstrations, use of manipulatives, real life examples

introduced new knowledge

Seatwork / Out of class assignments

Teacher

gave clear instructions, related to mathematical activities for in class and after class work

provided interesting activities for students to work on individually or in small groups

provided sufficient practice tasks for preparation towards examinations

Review and feedback

Teacher

reviewed past knowledge

used student work/group presentations to give feedback to individuals or the whole class

Acknowledgement

This paper makes use of data from the research project ―Student Perspective on Effective Mathematics: Stimulated Recall Approach‖ (CRP 3/04 BK), funded by the Centre for Research in Pedagogy and Practice, National Institute of Education, Singapore. The views expressed in this paper are the author‘s and do not necessarily represent the views of the Centre or the Institute.

Notes

1 Teacher assignments – this term is coined and means written assignments given by teachers to their students for

either in class or out of class follow-up work subsequent to a lesson.

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Publications arising from the study

Benedict, T. M., & Kaur, B. (2007, May). Using teacher questions to distinguish pedagogical goals: A case study of three Singapore teachers. Paper presented at Redesigning Pedagogy: Culture, Knowledge and Understanding, Centre for Research in Pedagogy and Practice, National Institute of Education, Singapore. Retrieved from http: //conference.nie.edu.sg.


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Kaur, B. (2007). Teaching and learning of mathematics—What really matters to teachers and to students. In C. S. Lim, S. Fatimah, G. Munirah, S. Hajar, M. Y. Hashimah, W. L. Gan, & T. Y. Hwa (Eds.), Proceedings of the 4th East Asia Regional Conference on Mathematics Education (pp. 10–16). Penang: Universiti Sains Malaysia. [Plenary Lecture at EARCOME 4]

Kaur, B. (2008, March). Developing procedural fluency in an algebra classroom - A case study of a mathematics classroom in Singapore. Paper presented at American Educational Research Association 2008 Annual Meeting, New York.

Kaur, B. (2008). Teaching and learning of mathematics—What really matters to teachers and students? ZDM—The International Journal on Mathematics Education, 40(6), 951–962.

Kaur, B. (2009). Characteristics of good mathematics teaching in Singapore grade eight classrooms—A juxtaposition of teachers‘ practice and students‘ perception. ZDM—The International Journal on Mathematic Education, 41(3).

Kaur, B. (2009, June). Pedagogical actions of mathematics teachers valued by Singapore students. Paper presented at Redesigning pedagogy: Designing New Contexts for a Globalising World, , Centre for Research in Pedagogy and Practice, National Institute of Education, Singapore. Retrieved from http: //conference.nie.edu.sg/

Kaur, B. (in press). A study of mathematical tasks from three classrooms in Singapore schools. [chapter of a book to be published by Sense Publishers]

Kaur, B. (in press). A study of two grade eight competent mathematics teachers: Instructional practice and student ability. [chapter of a book to be published by Sense Publishers]

Kaur, B. (in press). Developing procedural fluency in an algebra classroom - A case study of a mathematics classroom in Singapore. [chapter of a book to be published by Sense Publishers]

Kaur, B., Low, H. K., & Seah, L. H. (2006, May). What students value in their mathematics lessons? In Diversity for excellence: Engaged pedagogies. Paper presented at the annual meeting of the Educational Research Association of Singapore, Singapore.

Kaur, B. Low, H. K., & Seah, L. H. (2006). Mathematics teaching in two Singapore classrooms: The role of the textbook and homework. In D. Clarke, C. Keitel, & Y. Shimizu (Eds.), Mathematics classrooms in 12 countries: The insider’s perspective (pp. 99–116). Rotterdam, The Netherlands: Sense Publishers.

Kaur, B., Low, H. K., & Benedict, T. M. (2007, May). Some aspects of the pedagogical flow in three mathematics classrooms in Singapore. Paper presented at Redesigning Pedagogy: Culture, Knowledge and Understanding, Centre for Research in Pedagogy and Practice, National Institute of Education, Singapore. Retrieved from http: //conference.nie.edu.sg/

Kaur, B., Seah, L. H., & Low, H. K. (2005, May). A window to a mathematics classroom in Singapore—Some preliminary findings. Paper presented at the international conference on Redesigning pedagogy: Research, Policy, Practice, Centre for Research in Pedagogy and Practice, National Institute of Education, Singapore. Retrieved from http: //conference.nie.edu.sg/rprpp

Mok, I. A. C., & Kaur, B. (2006). ‗Learning task‘ lesson events. In D. Clarke, J. Emanuelsson, E. Jablonka, & I. A. C. Mok (Eds.), Making connections; comparing mathematics classrooms around the world (pp. 147–167). Rotterdam, The Netherlands: Sense Publishers.

Seah, L. H., Kaur, B., & Low, H. K. (2006). Case studies of Singapore secondary mathematics classrooms. In D. Clarke, C. Keitel & Y. Shimizu (Eds.), Mathematics classrooms in 12 countries: The insider’s perspective (pp. 151–166). Rotterdam, The Netherlands: Sense Publishers.


About the authors

Berinderjeet Kaur is an Associate Professor with the Mathematics and Mathematics Education Academic Group and Centre for Research in Pedagogy and Practice, National Institute of Education, Nanyang Technological University, Singapore.

Low Hooi Kiam is a research assistant with the Centre for Research in Pedagogy and

Practice, National Institute of Education, Nanyang Technological University, Singapore.

Contact us

For further information, please email: [email protected] You may also contact: A/P (Dr) Berinderjeet Kaur Office of Education Research National Institute of Education 1 Nanyang Walk Singapore 637616 Tel: +65 6790 3895

Centre for Research in Pedagogy and Practice

National Institute of Education 1 Nanyang Walk

Singapore 637616 http://www.crpp.nie.edu.sg

mailto:[email protected]


Camera 3 Teacher * DVR camera * tripod stand * radio mike

Camera 1 Whole Class * DVR camera * wide angle lens, * tripod

Camera 2 Focus Students (4) * DVR camera * wide angle lens * tripod stand * radio mike or flat mike

Camera 2 and Camera 3 would be connected to the Audio-Video Mixer and Audio Mixer to produce the picture in picture image. * Audio-Video Mixer * Audio Mixer * TV Monitor * Video Recorder (records picture-in-picture image)

Appendix A

Diagram of the use of equipment in class during data collection

Student Image

Teacher Image

Mixed Video Record

Video-Stimulated Interview with Focus Students * TV monitor * Video player * Audio-tape recorder

After every lesson video-recorded the following data set is available: * Tape from Whole Class Camera * Tape from Student Camera * Tape from Teacher Camera * Tape from Video Recorder [Composite image

from Teacher & Student cameras] * Audio-tape of interview with focus students


Appendix B

Protocol for student interview

Prompt

No. Question Remark

1 Thank you, [student‘s name], for attending this interview. Please tell me what you think that lesson was about? (lesson content/lesson purpose)

2 How, do you think, you best learn something like that?

3 What were your personal goals for that lesson? (What did you hope to achieve?) Do you have similar goals for every lesson?

Here is the remote control for the video player. Do you understand how it works?

(Allow time for a short familiarization with the control)

4 I would like you to comment on the videotape for me. You do not need to comment on all of the lesson. Fast forward the videotape until you find sections of the lesson that you think were important. Play these

sections at normal speed and describe for me what you were doing, thinking and feeling during each of these videotape sequences. You can comment while the videotape is playing, but pause the tape if there is something that you want to talk about in detail.

The student should not be led to comment on specific aspects of the lesson but rather he/she should be free to comment on any aspect of the lesson which he/she feels were important, nor should the interviewer decides for them what is meant by ‗important‘. The student may need to be reminded to elaborate on what he/she was doing, thinking and feeling whenever he/she chooses to play the sections at normal speed.

5 After watching the videotape, is there anything you would like to add to your description of what the lesson was about?

6 What did you learn during that lesson? (other questions to be prepared in relation to the topic taught to probe student understanding of this topic)

It is in the probing of student responses to this question that the advice of consultants will be of greatest benefits to the study. Once the topic(s) to be taught have been identified in consultation with the teacher, prior to the commencement of videotaping, consultant advice can be sought on prevalent student misconceptions with regard to this topic and a list of questions prepared to be used in interviews, as necessary, to probe student understanding of this topic. Whenever a claim is made to new mathematical knowledge, this should be probed. Suitable probing cues would be a request for examples of tasks or methods of solution that are now understood or the posing by the interviewer of these succinct probing questions related to common misconceptions in the content domain.


7

Would you describe that lesson as a good one for you? What has to happen for you to feel that a lesson was a ―good‖ lesson? [How would you describe the lesson in your own words?] Did you achieve your goals? What are the important things you should learn in a mathematics lesson?

―Good‖ may not be a sufficiently neutral prompt in some countries – the specific term used should be chosen to be as neutral as possible in order to obtain data on those outcomes of the lesson which the student values (it is possible that these valued outcomes may have little connection to ―knowing‖, ―learning‖ or ―understanding‖, and that students may have very localized or personal ways to describe lesson outcomes. These personalized and possibly culturally-specific conceptions of lesson outcomes constitute important data.

8 Was this lesson a typical [geometry, algebra, etc] lesson? What was not typical about it?

9 How would you generally assess your own achievement in mathematics? (or How would you describe/rate your own performance in maths?) [And how do you know that – can you give me an example of the sorts of things that let help you to tell how well you are achieving in maths?] [What methods/cues do you use to assess your own achievement in maths?]

10 Why do you think you are good [or not so good] at mathematics?

11 Do you enjoy mathematics and mathematics classes?

12 Do you do very much mathematical work at home? [Who do you approach to for help when you encounter difficulty with maths?] Have you ever had private tutoring in mathematics? Do you attended additional mathematics classes outside normal school hours?

Students may be asked to elaborate briefly if the answers to these questions are ―yes‖.

[13] [Do you think mathematics is important? Why is it important [or not important] to you?]

[14] [Please do this question for me. Can you explain how you solve this question?]

For lesson involving test

I. How do you find the test? Do you think you did well in the test?

II. How did you prepare for this test? Do you always prepared your tests this way?

III. How important is this test to you? Why is it [important/not so important] to you?

IV. Is there any question that you considered as important to you? Why is it important to you?

V. Is there any question that you found challenging?

VI. Do you think this test help you in your learning? How did it help in your learning? [or why do you think it did not help in your learning?]

VII. What factors do you think contribute to high performance in tests?


Appendix C

Structure and example of a lesson table

Format of a lesson table: School: Date:

Time Organisation of Interaction Description of Activity

Description of Content Framing Focus Group

Real time - may be changed to time on CD after digitizing

According to the teacher‘s intention

What the focus students are actually doing

Including remarks on materials used

Type of task in words – no shortcut

Typical example of the content/tasks in the lesson; source of task (book, teacher, student)

Note: A new row (and time) starts when the activity changes (see example) Sample of a lesson table: School: School 1 Date: 24/8/2000

Time Organisation of Interaction Description of Activity

Description of Content

Framing Focus Group <names of students>

03:07 Classwork Setting up Getting student materials ready to continue yesterday‘s activity Revision: Concepts

Measuring the circumference (C), diameter (D) and radius (R) of circles printed on textbook p174

07:29 Both responding to teacher question.

Sharing

12:48 Teacher Explaining investigative task

13:20 Discipline

14:13 Sharing: Others (non-maths)

1

st column: Description according to the teacher‘s intention (This does not mean that all

students act according to this intention)


2nd

column: Description of what the focus students actually are doing. If the organization of interaction changes, a new horizontal line appears Descriptors include:

I. Classwork II. Seatwork: Individual

III. Seatwork: Small groups IV. Working with partner V. Any other reasonable descriptors can be added; the aim is to use descriptions

that are mutually exclusive Description of activity:

- If the activity changes, a horizontal line appears - Time when new activity starts indicated in the first column

Activity codes (Based on an elaboration of codes used in TIMSS) include:

a) Teacher talk or demonstration to whole class (intention: students should listen) b) Teacher talk or demonstration and students participating (e.g., by writing in

books) intention: students should listen) c) Teacher talk to individual student d) Setting up of a task (mathematical ―instructions‖ given prior to the

commencement of work on a task) e) Setting up of organization (―instructions‖ not related to the content of a task) f) Students working on task g) Students working on homework h) Students working on a test i) Sharing (solution, solution methods or products of previous work are discussed,

compared or corrected – usually this is a shift in activity from ―working on‖) j) Any other reasonable codes can be added provided the meaning is clear; the aim

is again to use categories that are mutually exclusive. Description of content: The content can be a typical task (if available with reference to a worksheet or book used), a definition or a solution method.


Appendix D

Protocol for teacher interview

Prompt

No. Question Remark

1. Please tell me what were your goals in that lesson? (lesson content/lesson purpose)[elicit content and other goals]

2. In relation to your content goal(s), why do you think this is important for students to learn? What do you your students might have answered to this question? [What do you think the students might have said if I asked them this?]

Here is the remote control for the video player. Do you understand how it works?

Allow time for a short familiarization with the control

3. I would like you to comment on the videotape for me. You do not need to comment on all of the lesson. Fast forward the videotape until you find sections of the lesson that you think were important. Play these

sections at normal speed and describe for me what you were doing, thinking and feeling during each of these videotape sequences. You can comment while the videotape is playing, but pause the tape if there is something that you want to talk about in detail. In particular, I would like you to comment on:

a) Why you said or did a particular thing? (for example, conducting a particular activity, using a particular example, asking a question, or making a statement)

b) What you were thinking at key points during each video except? (for example, I was confused, I was wondering what to do next, I was trying to think of a good example)

c) How you were feeling? (for example, I was worried that we would not cover all the content)

d) Students‘ actions or statements that you consider to be significant and explain why you feel the action or statement was significant.

e) How typical that lesson was of the sort of lesson you would normally teach? What do you see as the features of that lesson that are most typical of the way you teach? Were there any aspects of your behaviour or the student behaviours that were unusual?

4. Would you describe that lesson as a good lesson for you? What has to happen for you to feel that a lesson is a ―good‖ lesson?

5. Do your students work a lot at home? Do they have private tutors?


Teacher interview questions (In addition to LPS protocol)

No. Question

1 How do you plan your lesson?

2 What are the factors that influence your planning?

3 As the lesson progresses, what are the things/factors that will cause you to change your original plan for the lesson?

4 How do you think students learn? [Note: this question is to find out which theoretical learning theory that the teacher most likely subscribe to]

5 How do your views of the way students learn affect how you plan and conduct the lesson?

6 How do you select the learning tasks for your lesson? What criteria, if any, do you use?

7 Where do you usually source for your questions for demonstration? For seatwork? For test?

8 When do you consider using individual work and when do you consider using group work and why?

9 What are the main considerations when you conduct a lesson? What are the few significant things you would do or get your students to do during most, if not all, lessons?

10 Do you think homework is important to students? Why? (or Do you see the need of assigning homework after each lesson?)


Appendix E

Samples of Teacher Questionnaires

Teacher Questionnaire 1 (Preliminary questionnaire – pre-videotaping) Teacher Questionnaire 2s [short version] (Daily post-lesson questionnaire) Teacher Questionnaire 2L [long version] (Daily post-lesson questionnaire) Teacher Questionnaire 3 (Post-videotaping questionnaire/interview) _______________________________________________________________________

Learner’s Perspective

The Learner’s Perspective

VIDEOTAPE CLASSROOM STUDY TEACHER QUESTIONNAIRE 1

Your Name: ____________________________________ Date: ______________ School‘s Name: _________________________________ Name of Class being videotaped: ________________________________

A. In this section we ask you to provide some details about yourself.

1. Age: ________________ 2. Gender: _________________ 3. Academic Study completed successfully (please show the awarding institution

for each qualification): Qualification Institution ____________________________ _______________________________

____________________________ _______________________________

____________________________ _______________________________

4. Years of Teaching Experience: ___________ 5. Grade levels taught during years of practice (giving no. of years): ________________( ) ________________( ) _______________( ) ________________( ) ________________( ) _______________( ) 6. Grade levels (and subjects) being taught at the time of the video (in addition to

Sec two mathematics): _______________________________________________________________

_______________________________________________________________ 7. Total hours of face-to-face teaching per week: ___________________ 8. Starting time of typical teaching day: ____________


9. Finishing time of typical teaching day: ____________ 10. Number of years at the present school: ____________ 11. Number of years teaching Sec two mathematics:

At all schools: _________ At this school: ___________ 12. Names of other schools at which you taught:

________________________________________________________________

________________________________________________________________

13. Professional experiences other than school teaching (please describe):

________________________________________________________________

________________________________________________________________

________________________________________________________________

B. In this section we will ask you a few questions about the lesson sequence we plan to videotape.

14a. How many lessons are in a typical topic sequence? _____________ 14b. How many such topic sequences would be taught in a single school year? __________ 14c. Please describe the subject matter content of the lessons to be videotaped. (Check as many as apply)

1. Whole numbers

2. Common and Decimal Fractions

3. Percentages

4. Number Sets and Concepts

5. Number Theory

6. Estimation and Number Sense

7. Measurement Units and Processes

8. Estimation and Error of Measurement

9. Perimeter, Area, and Volume

10. Basics of One and Two Dimensional Geometry

11. Geometric Congruence and Similarity

12. Geometric Transformations and Symmetry

13. Constructions and Three Dimensional Geometry

14. Ratio and Proportion

15. Proportionality: Slope, trigonometry and interpolation

16. Functions, Relations, and Patterns

17. Equations, Inequalities, and Formulas

18. Statistics and Data


19. Probability and Uncertainty

20. Sets and Logic

21. Problem Solving Strategies

22. Other Mathematics Content

15. What are the main things you want students to learn from the lesson

sequence? Why do you think it is important for students to learn these things?

16. For this class of students, will the content of the lesson sequence be

review, new, or somewhere in between? (Check one alternative only)

all review mostly review

half review/half new mostly new

all new

C. In this section we will ask you a few questions about the class and the way it is usually organized.

17. Was this class formed on the basis of students‘ mathematics ability? (Check one only)

Yes, this is a low ability class

Yes, this is an average ability class

Yes, this is a high ability class

No, this is a mixed ability class

Other (please explain)

_______________________________________________________

_______________________________________________________

18. Do the students in this Mathematics class work individually or in groups?

Mostly individually

Often in groups

19. How are any student workgroups usually organized?

Ability Groups

Mixed Ability Groups

Other (describe briefly):

_______________________________________________________

______________________________________________________

_______________________________________________________


20. Homework

20a. Do you regularly assign mathematics homework to this class?

no (skip to end) yes (go to 20b)

20b. Please describe the typical form of this homework (Check as many as apply – where more than one are checked indicate the approximate percentage occurrence of each in the homework you assign).

practicing new procedures taught in the lesson %

applying new procedures taught in the lesson %

open-ended investigation %

consolidating content from previous lessons %

completing work started in the lesson %

20c. How long would it have taken the typical student to complete this

homework? _______________ minutes.




VIDEOTAPE CLASSROOM STUDY

TEACHER QUESTIONNAIRE 2S

Your Name: ____________________________________ Date: _____________ School‘s Name: _________________________________ Class: ________ Topic taught: ________________________________

In this short questionnaire we will ask you a few questions about the lesson we just videotaped and the students in this classroom.

1. What was the main thing you wanted students to learn from today‘s lesson? Why do you think it is important for students to learn this? ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ 2. For this class of students, was the content of today‘s lesson review, new, or

somewhere in between?



all new

3a. Was there anything about today‘s lesson that did not go according to plan

or that you would have wanted to be different?

no yes (go to 3b)

3b. Please describe what did not go according to plan. ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________




VIDEOTAPE CLASSROOM STUDY TEACHER QUESTIONNAIRE 2L

Your Name: ____________________________________ Date: ______________ School‘s Name: _________________________________ Name of Course: ________________________________

A. In this section we will ask you a few questions about the lesson we just videotaped and the students in this classroom.

1. Please describe the subject matter content of today‘s lesson. (Check as

many as apply)

1. Whole numbers


3. Percentages


5. Number Theory















20. Sets and Logic




2. What was the main thing you wanted students to learn from today‘s lesson? Why do you think it is important for students to learn this? ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ 3. For this class of students, was the content of today‘s lesson review, new, or

somewhere in between?



all new

4a. Was today‘s lesson planned as part of a sequence of related lessons (e.g.,

a unit), or was it a stand-alone lesson?

stand-alone lesson (skip to 5) part of a sequence (go to 4b)

4b. What is the main thing you want students to learn from the whole sequence of lessons?

__________________________________________________________ __________________________________________________________

4c. How many lessons are in the entire sequence? _____________

4d. Where did today‘s lesson fall in the sequence (e.g., number 3 out of 5)?

______

5. How is the topic of this lesson related to other topics in the mathematics

curriculum?

____________________________________________________________

____________________________________________________________

____________________________________________________________ 6a. Did you previously assign mathematics homework that was due for today?

no (skip to 7) yes (go to 6b)

6b. Please describe the content of this homework. ____________________________________________________________ ____________________________________________________________


6c. How long would it have taken the typical student to complete this homework?

_______________ minutes.

7a. During today‘s lesson, were all students given the same work to do, or was

different work given to different students?

same work for all students (skip to 8a)

different work for different students (go to 7b)

7b. How was the work tailored for different students?

____________________________________________________________ ____________________________________________________________ ____________________________________________________________ 8a. Did you divide your class into groups for any part of today‘s lesson?


8b. Were the groups formed to be relatively homogeneous or heterogeneous with respect to ability and/or mathematical proficiency?

homogeneous heterogeneous

B. In this section we want to compare what happened in today’s lesson with what normally happens in your classroom.

9. The teaching methods I used for today‘s lesson were:

very similar to the way I always teach

similar to the way I always teach

somewhat different from the way I always teach

very different from the way I always teach

10. What, if anything, was different from how you normally teach? ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________


11. How would you describe your students‘ behavior during today‘s lesson?

very similar to their usual behavior

similar to their usual behavior

somewhat different from usual

very different from usual

12. What, if anything, was different about the nature and amount of your

students‘ participation during today‘s lesson? ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ 13. How would you describe the tools and materials (e.g., worksheets,

manipulatives, models, pictures, calculators) used in today‘s lesson compared to those you normally use?

very typical

mostly typical

not typical

completely atypical

14. What, if anything, was not typical about the tools and materials used in

today‘s lesson? ____________________________________________________________ ____________________________________________________________ 15. How would you describe today‘s lesson as a whole? Was it

typical/representative of the lessons you normally teach?

very typical

mostly typical

not typical

completely atypical

16. Do you think that having the camera present caused you to teach a lesson

that was better than usual, worse than usual, or about the same as usual?

better than usual

same as usual

worse than usual


17a. Was there anything about today‘s lesson that did not go according to plan

or that you would have wanted to be different?

no (skip to end) yes (go to 17b)

17b. Please describe what did not go according to plan.

____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________




VIDEOTAPE CLASSROOM STUDY TEACHER QUESTIONNAIRE 3

Your Name: ____________________________________ Date: ______________ School‘s Name: _________________________________ Name of Class: ________________________________

A. In this section we will ask you a few questions about the lesson sequence we videotaped and the students in this classroom.

1. Please describe the subject matter content of the lesson sequence just

videotaped. (Check as many as apply)

1. Whole numbers


3. Percentages


5. Number Theory















20. Sets and Logic




2. For this class of students, was the content of this lesson sequence review, new, or somewhere in between?

all review

mostly review

half review/half new

mostly new

all new

3. What was the main thing you wanted students to learn from the videotaped

lesson sequence? 4. Why do you think it is important for students to learn this? 5. The following questions are intended to determine how the videotaped

lesson sequence fits into the year‘s mathematics curriculum for this class

5a. Does the lesson sequence constitute:

(Check one only)

a single topic or unit?

part of a single topic or unit?

part of two topics or units (that is, the end of one and the beginning of

another)?

more than one entire topic or unit?

some other organization of content? (Please explain)

____________________________________________________________

____________________________________________________________

5b. How many lessons are in a typical topic sequence? _____________

5c. How many such topic sequences would be taught in a single school year?

__________

6a. During the videotaped lesson sequence, were all students given the same

work to do, or was different work given to different students?

same work for all students (skip to 7a)

different work for different students (go to 6b)


6b. How was the work tailored for different students?

7a. Did you divide your class into groups for any part of the videotaped lesson

sequence?


7b. Were the groups formed to be relatively homogeneous or heterogeneous with respect to ability and/or mathematical proficiency?

homogeneous heterogeneous

B. In this section we want to compare what happened in the videotaped lesson sequence with what normally happens in your classroom.

8. The teaching methods I used for in the videotaped lesson sequence were:

very similar to the way I always teach

similar to the way I always teach

somewhat different from the way I always teach

very different from the way I always teach

9. What, if anything, was different from how you normally teach? 10. How would you describe your students‘ behavior during the videotaped

lesson sequence?

very similar to their usual behavior

similar to their usual behavior

somewhat different from usual

very different from usual

11. What, if anything, was different about the nature and amount of your

students‘ participation during the videotaped lesson sequence?


12. How would you describe the tools and materials (e.g., worksheets,

manipulatives, models, pictures, calculators) used during the videotaped lesson sequence compared to those you normally use?

very typical

mostly typical

not typical

completely atypical

13. What, if anything, was not typical about the tools and materials used the

videotaped lesson sequence? 14. How would you describe the videotaped lesson sequence as a whole? Was

it typical/representative of the lessons you normally teach?

very typical

mostly typical

not typical

completely atypical

15. How nervous or tense did you feel about being videotaped?

very nervous

somewhat nervous

not very nervous

not at all nervous

16. Do you think that having the camera present caused you to teach in a way

that was better than usual, worse than usual, or about the same as usual?

better than usual

same as usual

worse than usual

17a. Was there anything about the videotaped lesson sequence that did not go

according to plan or that you would have wanted to be different?



17b. Please describe what did not go according to plan.

C. In this section we want to find out about the ideas that influence your teaching.

18. How aware do you feel you are of current ideas about the teaching and

learning of mathematics?

very aware

somewhat aware

not very aware

not at all aware

19. How do you usually hear about current ideas about the teaching and learning of mathematics?

20. What written materials (e.g., reform documents, publications, books, articles) are

you aware of that describe current ideas about the teaching and learning of mathematics? Please list up to three, and indicate whether or not you personally have read each one.

___________________________________ I have read: all of it

most of it

some of it

none of it

I have read: all of it

most of it

some of it

none of it

I have read: all of it

most of it

some of it

none of it


21a. To what extent do you feel that the videotaped lessons were in accord with current ideas about the teaching and learning of mathematics?

not at all (go to 22)

a little (go to 21b)

a fair amount (go to 21b)

a lot (go to 21b)

21b. Please describe one part of the videotaped lesson sequence that you feel exemplifies current ideas about the teaching and learning of mathematics and explain why you think it exemplifies these ideas.

22. To what extent did you attempt to draw students‘ attention to linkages

between lessons in the videotaped lesson sequence and


22a. Previous mathematics lessons in the same sequence

Several statements to the whole class

A few statements to the whole class

A few statements to individual students

Not at all

22b. Previous mathematics lessons other than those videotaped




Not at all

22c. Lessons in subjects other than mathematics




Not at all

22d. The students‘ experiences outside school




Not at all


Appendix F

Student Questionnaire designed by research team in Singapore

Name of School: _______________________ Name of student: _______________________ Class: _______________________________ Please answer the following questions as completely as possible. Q1: Describe the method/s your teacher, [Name of teacher], use in class to teach

mathematics that help you understand what she is teaching? Q2: What do you do when you do not understand the mathematics [Name of teacher]

is teaching? Q3: Does homework assignments given by [Name of teacher] help you in the learning

of mathematics? Thank you for your participation.


Appendix G

International Benchmark Test (IBT) Paper


Appendix H

Summary of data collected for each video-taped lesson

Data collected for each lesson

S/N Data Description of Data Qty

1 Digitized Videos Teacher video 1

Whole Class video 1

Mixed video 1

Student video 1

Student Interview video 2

2 Student Work Work done by all students in focus group

Varies

3 Lesson Materials Materials used by or referred to by Teacher

Varies

4 Interview Student Work Work done by students interviewees Varies

5 Interview Student Audiofiles Audiofiles of student interviewees 2

6 Student Interview Transcript Transcripts of student interviewees 2

7 Lesson Transcript Transcript for lesson 1

8 Lesson Table Records of times and activities of lesson

1

9 Teacher Questionnaire Two (TQ2)

Short questionnaire completed by teacher after the videotaped lesson

1

Additional data collected from the classroom

S/N Data Description of Data Qty

1 Teacher Interview Audiofiles Audiofiles of Teacher Interviews 4

2 Teacher Interview Videos Videofiles of Teacher Interviews 4

3 Teacher Interview Transcript Transcripts of Teacher Interviews 4

4 Teacher Questionnaires Questionnaires done by Teacher (Pre, Post and Lesson-specific)

5 Consent Forms Teacher consent form 1

Student Consent Forms 1

6 Fieldnotes Lesson fieldnotes 1

7 Test Scores Marks scored by students – IBT, Sec1, Sec 2(Test 1)

3

8 Class list Students‘ class list 1

9 Pseudonym list Pseudonym list of students 1

10 IBT Students‘ Answers Answers to IBT by individual students All students

11 Seating Plan Seating arrangement for different lessons

12 Timetables Class and teacher timetables 2

13 Syllabus Syllabus prescribed by Ministry of Education, Singapore

14 Textbook Textbook chapters used by class

15 Tests Test papers of all students


Appendix I

Teacher consent form

Title of project: Student Perspective on Effective Mathematics Pedagogy:

Stimulated Recall Approach

Principal Investigator: A/P Berinderjeet Kaur

Collaborators: Prof David Clarke (University of Melbourne)

Telephone: 67903895 Email: [email protected]

Address: National Institute of Education

1 Nanyang Walk, Singapore 637616

1. Purpose of the study:

A research team from the Centre for Research in Pedagogy and Practice (CRPP), National Institute of Education is conducting an MOE funded research project as part of an international study of secondary two mathematics classrooms (The Learner‘s Perspective Study).

A significant characteristic of this study is its documentation of the teaching of

sequences of lessons, rather than just single lessons. This project will look at secondary two mathematics classrooms in Singapore and compare these with the practices employed in other countries, by identifying similarities and differences in both teaching practice and in the associated student perceptions and behaviours.

2. Procedures to be followed:

If you consent to participate in this study,

Three video cameras from the project team will be positioned in your classroom for recording your mathematics class for at least 13 consecutive lessons.

The cameras are mounted on tripods and remain in the same position throughout the lesson. The intention is to be as minimally intrusive as possible.

One camera focuses on the teacher throughout the lesson. Teacher statements are recorded through a radio microphone throughout the lesson.

A second camera focuses on a group of four students sitting sufficiently near each other to be captured on screen. The public statements and conversation of this student group are recorded through a single ‗field microphone‘. The essential features of this approach are the on-site mixing of the images from two video cameras to provide a split-screen record of both teacher and student actions and the use of the technique of video-stimulated recall in interviews conducted immediately after the lesson to obtain participants' reconstructions of the lesson and the meanings which particular events held for them personally.

A third camera will record ―corporate‖ student practices – that is, the practices common to the whole class group.

Three to four researchers will usually be present in the classroom: one operating the teacher camera, one operating the student camera, and


one monitoring the mixed image and/or taking field notes to guide the subsequent student interviews.

Four 45 minute interviews are planned with you. One each week during videotaping and one after videotaping has concluded. During these interviews, video records of selected lessons will be used as stimulus for your reconstruction of classroom events. The interviews will be held at your convenience.

You will need to complete a questionnaire prior to videotaping and one after videotaping is finished.

An additional one-page questionnaire is completed after each lesson.

In addition, a focus ‗group‘ of four students is videotaped continuously for two consecutive lessons.

The word ‗group‘ is used only to indicate students sitting near each other. There

is no intention to require students to be actually undertaking ‗group work‘ as this is commonly interpreted. At least two students will be interviewed after each lesson, and all four ‗focus students‘ will be interviewed over the two lessons in which they are videotaped. In this study, students are interviewed after each lesson using the video record as stimulus for their reconstructions of classroom events. It has been the experience of the international teams over several years that both teachers and students benefit from the chance to view themselves on videotape and to reflect on their actions in class. Ideally, these interviews should occur immediately after the lesson, but this will inevitably vary according to local conditions affecting student availability. The interview will take no longer than 30 minutes and should this require the student to miss part of a class, this will only occur with the approval of the teacher of that class. Each student will only be interviewed once. Our aim in the interview is to try to see the classroom through the student‘s eyes in order to connect the activities occurring in secondary two classrooms with the meaning these activities have for students and the subsequent learning that results.

During the week following videotaping, the International Benchmark Test will be

administered to all students at a time convenient to you and your students. The purpose of the test is to locate the class in relation to the national secondary two student population. This test takes 60 minutes, including 10 minutes ‗reading time.‘ Data collected during this period will be analysed by members of the research team and shared with the educators communities from Singapore and around the world. 3. Confidentiality:

All personal details will remain confidential to the research team, subjected to legal requirements. No printed publications will identify the schools, teachers, or students in any way. However, it is anticipated that video clips drawn from the core data and the findings from this project will be utilised in a variety of professional development programs. Any use of the video material in conference presentations or for teacher professional development would be entirely dependent upon agreement from all videotaped teachers and students, obtained after being given an opportunity to view the video clips to be used.

4. Duration:

You are asked to help this study by allowing the project team to videotape one class for at least 4 weeks during July and August 2005.


5. Risks:

The only foreseeable risk to you participating in this study is temporary minor class distraction due to the presence of the cameras and observers.

6. Benefits:

The results of this study will provide knowledge of teacher and learner practices in mathematics classrooms and their relationship with the attainment of curricular goals. This study also aims to identify generic teacher and learner practices worthy of more widespread emulation. No direct personal benefit will accrue to you as a result of your participation in this study.

7. Participation is voluntary:

You are free not to participate and to withdraw your consent at any time without penalty.

8. More information: Please contact the Principal Investigator, Associate Professor Berinderjeet Kaur on 67903895 if you have any questions or concerns about this study. If you wish, you may also contact the Dean of the Centre for Research in Pedagogy and Practice, Professor Allan Luke, tel. 6790 3186, if you have any concerns about the project.

9. Consent:

This is to certify that I, ................................................................................................................. ,

a teacher at ................................................................................................................................ ,

hereby consent to participate in this project. I acknowledge that I have been informed of the purpose and contents of this research project.

Name of teacher: ........................................................................................................................

Signature of teacher .......................................................................... Date ..........................

I consent to the videotaping of my teaching during the lessons specified in the information provided to me. Teacher Initial: _________________ I consent to be interviewed on the occasions specified and for the interview to be audiotaped. Teacher Initial: _________________ I consent to the use of selected video clips for the purposes of conference presentations or teacher professional development, on the condition that I am provided with the opportunity to view and approve the video clips to be used.

Teacher Initial: _________________

This is your / CRPP's copy. Please sign and retain / return to CRPP.


Appendix J

Student consent form



Principal Investigator/s: A/P Berinderjeet Kaur

Collaborators: Prof David Clarke (University of Melbourne)




School: Class:

Description of project:

This project is about the teaching and learning of mathematics in secondary two classrooms. For a period of about 3-4 weeks, your mathematics lessons will be videotaped. One camera will focus solely on the teacher. Another camera will focus on a group of 4 students. Each group will be the focus for two consecutive lessons. After the lesson, 2 focus students will be interviewed. Each student will only be interviewed once. A third camera will focus on the whole class.

1. I have been informed about the nature of this project.

2. I am willing to participate in the project.

3. I am willing to be a focus student and be interviewed.

4. I consent to the use of selected video clips for the purpose of conference presentations and teacher professional development.

5. I understand I will not be individually identified.

6. I understand that participation is voluntary.

1 Name Signature : ________________ Date : ________



















41 Name Signature ________________ Date : ________

42 Name Signature ________________ Date : ________


Appendix K

Parent / Guardian consent form



Principal Investigator: A/P Berinderjeet Kaur




1. Purpose of the study:

A research team from the Centre for Research in Pedagogy and Practice (CRPP), National Institute of Education is conducting an MOE funded research project as part of an international study of well-taught mathematics classrooms (The Learner‘s Perspective Study). This international project looks at learning and teaching in secondary two mathematics classrooms in a more integrated and comprehensive fashion than has been attempted in previous international studies. The project will complement recent international comparative research into student achievement and teaching practices with an in-depth analysis of mathematics classrooms in Singapore and other countries such as Australia, Germany, Japan and the USA.

This project will look at well-taught mathematics classrooms in each country and compare these with the practices employed in other countries, by identifying similarities and differences in both teaching practice and in the associated student perceptions and behaviours. A significant characteristic of this study is its documentation of the teaching of sequences of lessons, rather than just single lessons. The basic method of data collection is videotape supplemented by post-lesson video-stimulated student interviews.

2. Procedures to be followed:

This study will involve observing the mathematics classroom in which your child may participate. Three video cameras from the project team will be positioned in his/her classroom for recording the mathematics class for at least 10 consecutive lessons. A ―familiarisation period‖ of up to one week is planned for each school to allow students and teacher to become accustomed to the presence of the researcher(s) (plus equipment) in the classroom, and to become familiar with the research routine. The video data collected in the familiarisation period is not intended to be analysed. The argument in favour of accessing a sequence of lessons (rather than just single lessons) is a strong one. The requirement of ―at least ten lessons‖ was chosen as the number of lessons likely to encompass all stages in the delivery of a mathematics topic.

The cameras are mounted on tripods and remain in the same position throughout the lesson. The intention is to be as minimally intrusive as possible. One camera focuses on the teacher throughout the lesson. A second camera focuses on a group of four students sitting sufficiently near each other to be captured on screen. The public statements and conversation of this student group are recorded through a single ‗field microphone‘. The essential features of this approach are the on-site mixing of the images from two video cameras to provide a split-screen record of both teacher and student actions and the use of the technique of video-stimulated recall in interviews conducted immediately after the lesson to obtain students' reconstructions of the lesson and the


meanings which particular events held for them personally. A third camera will record ―corporate‖ student practices – that is, the practices common to the whole class group.

Three to four researchers will usually be present in the classroom: one operating the teacher camera, one operating the student camera, and one monitoring the mixed image and/or taking field notes to guide the subsequent student interviews.

In each lesson, a particular ‗group‘ of four students is videotaped. The focus group will be videotaped continuously for two consecutive lessons. The word ‗group‘ is used only to indicate students sitting near each other. There is no intention to require students to be actually undertaking ‗group work‘ as this is commonly interpreted. At least two students will be interviewed after each lesson, and all four ‗focus students‘ will be interviewed over the two lessons in which they are videotaped. In this study, students are interviewed after each lesson using the video record as stimulus for their reconstructions of classroom events. Ideally, these interviews should occur immediately after the lesson, but this will inevitably vary according to local conditions affecting student availability. The interview will take no longer than 45 minutes and should this require the student to miss part of a class, this will only occur with the approval of the teacher of that class. Each student will only be interviewed once. Our aim in the interview is to try to see the classroom through the student‘s eyes in order to connect the activities occurring in well-taught classrooms with the meaning these activities have for students and the subsequent learning that results.

During the week following videotaping, the International Benchmark Test will be administered to all students at a time convenient to the students. The purpose of the test is to locate the class in relation to the national secondary two student population. This test takes 60 minutes, including 10 minutes ‗reading time.‘

With regard to the reporting of the results from the study, local research teams will prepare both conference papers and research papers for publication based on local and international data. In such papers, local issues and priorities will be addressed specifically. The combined international research team will also produce papers addressing issues that transcend local data and priorities, typically reporting analyses of the full international data set. It is also anticipated that video clips drawn from the core data and the findings from this project will be utilised in a variety of professional development programs. These papers will be made available to the classroom teacher and principal of the school. The smaller the sample size, the more difficult it is to ensure the anonymity of every school, class, student and teacher but the use of student and teacher pseudonyms is intended to protect the identity of individual students and teachers.

It has been the experience of the international team over several years of doing this sort of research that students derive significant benefit from the chance to view themselves on videotape and to reflect on their actions in class. The next page sets out the consent for your child to participate. Consent is required from both yourself and your child. Please discuss the contents of this letter with your child and encourage your child to ask about any points of interest or concern during our preliminary school visit (in which we will explain and discuss the research with the students).

3. Confidentiality:

All personal details will remain confidential to the research team, subjected to legal requirements. No individuals will be identified in any publications. However, it is anticipated that video clips drawn from the core data and the findings from this project will be utilised in a variety of conference presentations and professional development programs.


4. Duration:

You are asked to help this study by allowing the project team to videotape the mathematics class for a period of 3-4 weeks.

5. Risks:

The only foreseeable risk to your child participating in this study is temporary minor class distraction due to the presence of the cameras and observers.

6. Benefits:

The results of this study will provide knowledge of teacher and learner practices in mathematics classrooms and their relationship with the attainment of curricular goals. This study also aims to identify generic teacher and learner practices worthy of more widespread emulation. No direct personal benefit will accrue to you as a result of your participation in this study.

7. Participation is voluntary:

You are free not to participate and to withdraw your consent at any time without penalty.

8. More information:

Please contact the Principal Investigator if you have any questions or concerns about this study. If you wish, you may also contact the Dean of the Centre for Research in Pedagogy and Practice, Professor Allan Luke, tel. 6790 3186 if you have any concerns about the project.

9. Consent: This is to certify that I, (please print your name) .................................................................. ,

Parent/Guardian (please circle), consent for my child (please print child‘s name)

...............................................................................................................................................

to participate in this project. I acknowledge that I have been informed of the purpose and contents of this research project.

Signature of Parent/Guardian: ...................................................... Date .............................

I consent to the videotaping of my child during the research and familiarisation period and to the photocopying of their work samples consistent with the information provided to me. Parent or Guardian‘s Initial: ______________

I consent to my child being interviewed and for the interview to be audiotaped.

Parent or Guardian‘s Initial: ______________


I consent to my child sitting for the International Benchmark Test.


I consent to the use of selected video clips for the purposes of conference presentations or teacher professional development, on the condition that I and my child are provided with the opportunity to view and approve the video clips to be used.


This is your / CRPP's copy. Please sign and retain / return to CRPP.

Student perspective on effective mathematics pedagogy: Stimulated recall approach study

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