The impact of secondary schooling and secondary mathematics on student mathematical behaviour

DAVID CLARKE

THE IMPACT OF SECONDARY SCHOOLING AND

SECONDARY MATHEMATICS ON STUDENT

MATHEMATICAL BEHAVIOUR

ABSTRACT. Findings are reported of an intensive study of ten students over three school years spanning the transition from primary to secondary school. The changes in mathematical behaviour associated with commencing secondary mathematics are discussed. It is argued that a comprehensive descriptive framework is required incor- porating consideration of the social context in which mathematics is taught and learned. Conclusions relate to the idiosyncratic nature of student response to secondary mathematics, and to the need for a broader conception of teaching goals. The findings are illustrated by excerpts from the case studies of two students: Cathy and Darren.

1. INTRODUCTION

The significance of the transition to secondary schooling in the mathematical education of British students was emphasized in the Cockcroft Report (1982, paragraph 429): “We believe that the greatest problems exist on transfer to secondary or upper school”. And the relationship is not uni-directional. In their study of transition in Queensland schools, Power and Cotterell (1981) found “major curriculum discontinuities in Mathematics” (page 18) and further,

In this study, the school curriculum turned out in the end to be one of the major factors in determining the shape of the transition problem . . . areas of overlap and mismatch creating particular difficulties at one or both levels seemed to exist in . . the Grades 7-8 Mathematics program.

(Power and Cotterell, 1981, page 36).

It is evident that if transition creates problems in a student’s mathematics education it is equally true that mathematics contributes significantly to the difficulties of the student in transition. A study of student mathematical behaviour during the period of transition from primary to secondary school seems overdue.

This study analysed detailed test, interview and observational data in an attempt to understand the impact of secondary school and secondary mathematics on the mathematical behaviour of ten students. Coming from four primary schools employing very different mathematics programs, the student responses to the same secondary mathematics class demonstrate the idiosyncratic nature of student reaction to that critical transition. In particular, the demands made on some students led to classroom dynamics which retarded the students’ learning, encouraged a dysfunctional conception of mathematics,

Educational Studies in Mathemutics 16 (1985) 231-257. 0013-1954/85.10 0 1985 by D. Reidel Publishing Company.

232 DAVID CLARKE

Mathematical Behaviour

Child-Adolescent Transition

a. abilities as

Understanding of Mathematics

a. fractions b. decimals and I----! place value c. proportion d. general

understanding

a. achievement

a. attributions b. expectations c. self-esteem

Mathematics

Practices of the Learning Environment L a. school b. peer grew c. home

‘rimary-Secondary Transition

PWSWld Environmental

Fig. 1. The elements of a model of mathematical behaviour.

and affected the students’ self-concept in a variety of important ways. The research reported here was carried out in Victorian schools, where the transition occurs from year 6 to year 7, and application of its findings to other education systems must be tempered by an awareness of any structural differences.

2. MATHEMATICAL BEHAVIOUR

Since it was Jhe effect of an external event, transition, on the student’s mathematical behaviour which was being studied, and the event is manifestly a social one, methods were devised for the collection of social data. In addition to the identification and description of the context in which this investigation was to occur, it was essential to specify the constituent elements of mathematical behaviour which would provide the descriptive framework for the study. Figure 1 displays the elements of mathematical behaviour as a two dimen- sional array. The model is entirely a product of the research reported here. Its possible application to other contexts is of less concern than the extent to which it facilitates the analysis of changing mathematical behaviour within the specific period of transition. Justification for the inclusion of each element is given below.

IMPACT OF SECONDARY SCHOOLING 233

Covington and Omelich (1979) suggested that students’ expectations are more dependent on the students’ initial self-concept of ability than on their attributions. This study assumed a dynamic interplay between self-concept and attributions, and took self-concept as the more inclusive term. This association parallels the argument of Meece et al, (1982) that “attributions, particularly attributions to ability, play a critical role in the formation of one’s self-concept of ability”, but also proposes that the nature of a student’s self- concept affects the attributions made. Another consideration relevant to the student’s response to criticism or encouragement is the importance accorded by the student to the person providing the criticism (for instance). It was necessary, therefore, to establish which members of the student’s learning environment constituted ‘significant others’.

Perceptions of the sex typing of mathematics should influence mathematical behaviour ‘to the extent that one’s sex-role identity is a critical and saZient component of one’s self-image’ (Meece et al., 1982). Ethnicity might similarly comprise a significant component of a student’s self-concept, but, while the possibility is acknowledged, it was not manifested in the mathematical behaviour of the students in this study.

In employing mathematical ability as an element of mathematical behaviour, a distinction was made between the demonstrable presence or absence of specific abilities (Krutetskii, 1976) as displayed on tests and in interviews, and the perceptions of a student’s ability professed by peers, teachers, parents and the student himself.

In as much as the student is a participant in the practices of the Zeaming environment, these structure and constrain the student’s behaviour. It is the practices of the learning environment that determine the ways in which the student may demonstrate competence, express dissatisfaction, or actively participate in the learning process, and these practices constrain the range and nature of the activities in which a student may participate.

The remaining elements of the model were devised to encompass those outcome variables which characterize mathematical behaviour and which an educational program might hope to influence.

The element conception of mathematics was taken to be a description of the associations mathematics has for the student. Assessment of a student’s conception of mathematics at any time during the study involved establishing the student’s response to certain key questions:

(a) What is mathematics? That is, what activities characterize mathematics? (b) Who is good at it, and why? That is, what specific skills or abilities

does the student see as essential, who posseses these, how are they acquired, and how is competence demonstrated?

234 DAVID CLARKE

(c) How is mathematics taught? That is, what is the role of the mathematics teacher? This question is different from the question, “How is mathematics learnt?” which is implicit in (b).

(d) How do you feel about it? That is, what is the student’s attitude to things mathematical?

The student’s mathematical perfokance in terms of achievement on a variety of mathematics tests and as seen in interviews, where different mathematical tasks provided information on student persistence and error patterns, was taken to be another element.

The understanding of mathematics which a student may have acquired is an additional distinct aspect of that student’s mathematical behaviour. Selecting three different areas of mathematics (Fractions, Decimal Place Value, and Proportion) on which to focus, distinctions were drawn between types and levels of understanding (for example, Biggs and Collis, 1982; Karplus et al., 1980; Skemp, 1976). The variations possible in the relationships between ability, performance, understanding and conception of mathematics are a continuing source of wonder and dismay to mathematics teachers. Their proposed existence as distinct (but not unrelated) elements of student mathematical behaviour should accord with the experience of practising teachers.

The remaining element, individual student classroom practices, is that aspect of student behaviour in which the interaction between the student’s personal attributes and the environment is most apparent. In addition to the influence of environment on student classroom behaviour, it is also suggested in this model that elements such as conception of mathematics and self- concept influence the classroom practices a particular student may employ.

The categories of the descriptive framework of mathematical behaviour are not mutually exclusive. Discussion of mathematical understanding and ability is necessarily derived from performance data. Descriptions of individual student classroom practices naturally comprise a subset of the practices possible given the constraints of the learning environment. There will be associations between the explanations a student provides for the competence of others under Conceptions of Mathematics and the attributions that student makes for success or failure. It remains to be shown that sufficiently clear relationships exist between elements for inferences to be made about likely changes to a student’s mathematical behaviour arising from initial change in a single element.

3. DATA COLLECTION

The students whose mathematical behaviour provides the focus for this discussion were two of ten students from four different primary schools whose

IMPACT OF SECONDARY SCHOOLING 235.

progress from grade 6 to year 8 was monitored through observations, questionnaires, interviews and tests. In secondary school all ten students were members of the same year 7 class and the experiences and perceptions of all ten provide a composite picture of the secondary mathematics classroom and illuminate through comparison and contrast the individual behaviour of any one student.

Data collection fell naturally into three phases corresponding to students’ participation in years 6, 7 and 8. Certain methods of data collection were common to all three phases and it was these which exemplified the orien- tation and character of the study.

Of these, the most important was the clinical interview (Easley, 1977; Erlwanger, 1975; Pines et aE., 1978). Over the two-year period each student was interviewed at least 15 times (in Cathy’s case 5, 5, 5 over the three phases; in Darren’s case 6, 6, 5), for a period of between 30 and 45 minutes, a typical interview format comprising an affective (attitude) component and a cognitive (task) component. The clinical interviews provided data for all seven elements of mathematical behaviour and varied in structure and strategy according to the specific focus.

Classroom observations were made during each phase and supplemented by interview accounts from teachers and students. Priority was given to obtaining the participants’ perceptions of the learning environment.

Three class tests were used in each phase: 1. The ‘Operations Test’ from the Mathematical Ability Profile Series

(MAPS), (ACER, 1978). 2. The Monash Assessment of Mathematical Performance (MAMP) Test,

Monash University, 1977. 3. The ‘Numbers, Shapes and Opinions’ (NSO) Test, which was a composite

test designed for this study employing items derived from Eke&am (1977) Stage et al. (1980), the MAPS ‘Space Test’ (ACER, 1978) Aiken (1979) and Fennema and Sherman (1976). Also included were multiple choice items whose alternatives included some erroneous responses given by students to similar items in clinical interviews. While the MAPS and MAMP tests provided data on Mathematical Performance, the NSO test also explored the extent to which particular attitudes and misconceptions were generally held by class members.

Questionnaires provided information on teacher, parent and student attitudes and beliefs.

Since the focus of the study was the impact of secondary schooling on student mathematical behaviour, the greatest detail was sought during the year 7 phase. Two strategies in particular provided this additional detail. First,

236 DAVID CLARKE

the year 7 mathematics teacher kept a diary in which she recorded details of lesson content, method of instruction, class and individual behaviour for every lesson taught throughout the year. Further, a semantic differential instrument was designed to assess differences and similarities in an individual student’s construction of 18 concepts which included “Mathematics”, “High School”, “My Grade Six Teacher”, “My Friends”, “The Person I Would Like To Be” and “Myself’. The specific bipolar scales employed in the instrument were obtained from the ten subjects in interviews employing a technique’ of ‘embedded emicism’ (Pelto, 1970).

The role of the year 7 mathematics teacher in this study was a crucial one, and the teacher involved was invited to participate because she was an experienced teacher who had previously taught year 7 mathematics. It was seen as important that in monitoring student mathematical behaviour during the transition to post-primary school any difficulties of adjustment should not be attributable to the teacher’s lack of experience or lack of familiarity with the content or the year level. Indeed, the teacher concerned was held in high regard by all members of the school community, as being both conscientious and highly competent.

Finally, the students themselves were selected by their grade 6 teachers under direction from the researcher. The only criteria for selection were that they should have indicated an intention to attend the selected secondary school and that they should be able to talk in a relaxed fashion with an adult. The students so selected exhibited no extreme behavioural problems or marked learning disabilities. Figure 2 shows their distribution according to performance on the MAPS and MAMP tests at the end of their grade 6 year. Empirical norms appropriate to their grade level are shown for each test.

Score 50 55 Q

vtv’vtvv

2 MAPS

Student yQ’ ‘“‘Q

Number , 2 3 4 5

I

6 7 9 10

* * 8

SCOW 20 MAMP

Student vav ”

Number 4 2 1 6 3 6 7 5 9

10 * cndicates appropriate test norm

Student Number 1 = Brian 6 = Al~ron 2 = Da”” 7 = Cathy

3 = Darren 8 = Andrea

4 = Bernie 9 = Cameron Note: Students w!th the same lnifcal letter

5 = Chris 10 = Annette attended the same primary rchool.

Fig. 2. Test performance of study students (exit grade 6).


4. RESULTS

In reporting case studies a major difficulty is to summarize the findings with- out sacrificing the detail which both characterizes and justifies the case study. The following findings draw on two case studies of children experiencing the transition to secondary mathematics: Cathy and Darren (Clarke, 1984a, b). The format employed facilitates comparison of the two cases for each of the seven elements of mathematical behaviour. The data presented are not complete, but simply indicative of the type of data collected and of each student’s responses.

4.1. Practices of the Learning Environment

1. Primary School

Cathy Cathy’s grade 6 class was a combined

class of 27 grade 6 students and 6 grade S students. During the period of data collection the total class size reduced to 30 students. Cathy believed she was among the ten best mathematics students in her class. Mathematics was also the subject she liked least.

Mathematics teaching was the responsibility of the classroom teacher. The class was divided into four ability groups: E, F, G and H, each group corresponding to a section of the mathematics curriculum guide published by the Victorian Education Department (1972). These ability groups formed the working units within the class and a typical lesson might have group G working with the teacher, correcting previous work and introducing and discussing new work, group H working from a text (Perrett and Whiting, 1978), group F working from a teacher-prepared worksheet, and group E working from the second of two teacher- prepared worksheets.

This organization was rotated throughout the week so that one week’s mathematics instruction for a particular group would comprise one lesson with the teacher followed by two worksheet lessons and one textbook lesson. While the two worksheets prepared for each group concentrated largely on number skills

Darren Darren was one of eleven grade 6

students in a combined (grades 5 and 6) class of 30 pupils, and felt himself to be the most able at mathematics. When asked to describe his grade 6 mathematics program Darren replied, “We work off trolleys and there’s books. And there’s equipment with the books, and then just to make it interesting they chuck in a game or two”.

Mathematics at Darren’s school was organized around an individualized program in use in many primary schools and some secondary schools throughout Victoria. The program set out a learning sequence of ‘concept’ books, worksheets, ‘applied’ books, games and puzzles. The use of concrete materials and equipment was incorporated in the program. Student progress from level to level within the program sequence was ‘self-paced’ and occurred upon satisfactory completion of the appropriate mastery test.

Darren’s teacher, Mr. Davidson felt that the strength of the program lay in its use of structural materials and in the capacity to cater for a range of ability. However the slow progress of the ‘weaker’ students, the demands on the teacher which led to some students being neg- lected, and a concern that the mastery tests were not detecting student mis- understandings prompted a decision to

238 DAVID CLARKE

appropriate to the group, the text used adopt a more teachercentred approach by all four groups contained only prob- the following year, retaining the mathe- lems with a high verbal content requiring matics program as a resource only. the application of number skills. Cathy was in group G.

2. The High School

Interviewer: Describe a typical High School Maths class.

Cathy She writes notes first if you’re starting on a new topic. And we copy them down, and then when she gets to the sums she’ll describe what we’re going to be doing. And then when she’s described it all and done a few examples, then we start doing the sums or whatever’s there.. . We’ll go on the sums for a fair while depending on how many there are. And then she’ll stop. and explain one that, say, somebody didn’t understand. And then keep going. Correct a few. Keep going. Correct a few. If she does give us some work from the book, which she sometimes does, after the black- board work, she just gives us the page number. And we do it. And if we can’t understand anything we ask her.

Darren Wait outside if she’s not there. Come in if she’s there. Sit down. And she tells us what we’re gonna do. And she’ll prob- ably write up a few examples and notes on the board. Then we’ll either get sheets handed out or she’ll write up questions on the board. Not very often. We mainly get a text book. We’ll get pages. She’ll write up what work to do, page number and exercise. And if you finish quick you may get an activity sheet. And that’s about what happens.

Casual observation of Cathy and Darren’s year 7 mathematics class produced an impression of a classroom efficiently organized by a conscientious, well-prepared teacher. Lessons observed included the use of discussion, practi- cal activities, and audio-cassettes, and the two descriptions above of a ‘typical’ mathematics lesson seemed unduly mundane.

The analysis, from teacher diary entries, of 110 year 7 mathematics lessons displayed as Figure 3 vindicated the students’ descriptions. Observations of 87 year 7 mathematics classes in 37 different schools produced strikingly similar findings (Clarke, 1984~) indicating that Cathy and Darren’s mathematics class was representative of current teaching practice at that level.

Other participants in the learning environment did not encourage a positive attitude to mathematics. The parents of both children had found mathematics difficult as students. Both sets of parents disliked the subject, however Darren’s parents (father: electrician - mother: school cleaner/home duties) felt that the subject was useful and communicated this belief to Darren. Cathy’s parents (father: secondary English/History teacher - mother: nursing sister) felt that the subject was of limited value.


Method of Instruction

Boardwork (B)*

teacher ( t) student (s)

Text

Worksheet (IV)

Manipulative activity (M)

Games ( G 1

Discussion (D)**

Percen tege

of Classtime

11

21 37 21

6 2

3

One Year’s Mathematics

136 timetabled lessons. 120 taken by the class

mathematics teacher.

Of these, 10 were Test lessons.

The table and diagram refer to the

remaining 110 mathematics lessons.

* The distinction here is between teacher-centred explanations using the chalkboard and students working

from problems or examples written on the board.

** Discussion of mathematics indepen- dent of any other method.

Fig. 3. Year 7 mathematics instruction for Cathy and Darren.

4.2. Mathematical Performance

In making comparisons between the mathematical performance of Cathy and Darren as detailed below, two facts serve as significant reminders of the dangers of drawing inferences from isolated test data. First, it must be remem- bered that only in year 7 were Cathy and Darren classmates receiving the same instruction in mathematics, and second, in an early administration of the MAPS test (September, grade 6) Cathy and Darren scored 50 and 52 respectively.

December grade 6 :

November year 7:

September year 8:

1. Test Performance

Cathy

MAPS MAMP 57 32 Score

eq. 6 4 Rank

54 34 Score

eq.11 eq. 3 Rank

55 33 Score eq. 9 eq. 1 Rank

Darren

MAPS MAMP 51 28

9 4

52 32 18 eq. 7

54 33 eq. 9 4

240 DAVID CLARKE

2. Error Analysis

Problem: Given 741 x 12 = 8892, find these products:

(a) 74.1-x 12 (b) 0.741 x 12 (c) 7.41 x 1.2

These three problems were answered correctly by Cathy in both years 7 and 8 using a standard rule for decimal multiplication.

Cathy (August, year 8) C: Well, when you do a times sum, you just do the times then you put in

the decimal point . . . . You move it how many times it’s been (pause) how many decimal places in the question (indicates on paper how to move the decimal point).

I: Is this for everything, addition, or just for multiplication? c: Just for multiplication. I: What about division? c: Division? No, you just put it straight down (shows on paper). I: So when did you do decimal points? Did you do them this year? c: We started in grade 6, just basic things, and then we just kept doing

them now.

Analysis of student errors employed an error hierarchy (Clements, 1980; Newman, 1977) which consisted of the categories: Reading, Comprehension, Transformation, Process Skills and Encoding. Darren’s reading and comprehension were generally very good. His difficulties arose in identifying the correct mathematical procedure to use (a Transformation error). For example, given that 741 x 12 = 8892, Darren answered the three questions as follows:

(a) 74.1 x 12 Answer: 88.92 (b) 0.741 x 12 Answer: 0.8892 (c) 7.41 x 1.2 Answer: 8.892

Darren explained his reasoning in this fashion (July, year 7) D: The first one was seventy-four point one times twelve, so I just put the

decimal point in between the eighty-eight and ninety-two of eight thousand eight hundred and ninety-two. Because [in] seventy-four point one there was two numbers before the decimal point so I made the two numbers before the decimal point in my answer.

I: What about (b)?

D:

I: D:

I: D:


That was nought point seven four one times twelve, so I just put a nought and a point in front of eight thousand eight hundred and ninety- two. ‘Cos you multiply something by nought it still comes to nought, and again there was a point was in front of the seven hundred and forty- one in the question and so I put the point in front of eight thousand eight hundred and ninety-two in the answer. What about (c)? What did you do there? There was a point in seven hundred and forty-one and a point in one point two. So that meant there was only one whole number in each part of the questions so (pause). What do you mean “one whole number”? The number that’s on the left hand side of the decimal point in the equation type thing. So I knew the first number in the answer was eight so I just put eight point eight nine two . . . . There was one point in both of the numbers but they were both in the same spot.

Despite the fundamental similarity of the questions, Darren described his reasoning very differently in each case. By August of year 8 Darren was able to correctly use the same multiplication algorithm described by Cathy.

A major class Number Skills Test for which Darren interpreted his score of 74% as failure (Clarke, 1984a) provided Cathy with confirmation of her competence since her score of 81% was sufficient to exempt her from further instruction in number skills.

The level of understanding Cathy developed at any time was consistent with the immediate test or performance requirements. Retention was limited where the skills or concepts were not in regular use. This ability to develop short-term operative competence distinguished Cathy’s behaviour from Darren’s. The pace of secondary mathematics lessons, which had proved such a handicap to Darren, was not a concern for Cathy. Cathy was quicker at “picking things up”, her school test performances were consequently better, she experienced more success, her confidence was heightened, her self-esteem enhanced. Yet consider Darren’s and Cathy‘s MAPS and MAMP test scores in September of year 8: Darren MAPS, 54; MAMP, 33 and Cathy MAPS, 55 and MAMP, 33. How can these almost identical test performances be reconciled. with the enormous differences evident in every facet of their mathematical behaviour?

4.3. Understanding of Mathematics

Improvement in Cathy’s performance following instruction in fractions in term 3, year 7, led to the successful completion of items like 314 + l/5, but

242 DAVID CLARKE

questions involving mixed numbers were answered incorrectly. The distinction between performance and understanding (or, in another framework, between “achievement” and “competence” (White, 1979)) was illustrated in Cathy’s poor performance on a similar item in February of year 8.

By July of year 8 Cathy’s performance on questions involving fractions had improved. In identifying diagrams which represented 3/8, Cathy included appropriate diagrams with more than 3 parts shaded: for example, 6/16. Cathy’s confidence in her ability to successfully manipulate fractions had certainly improved and she used fractions to demonstrate the validity of one of her answers to a decimal task.

The following interview excerpt is indicative of the development of Cathy’s understanding of fractions by mid year 8.

Cathy (July, year 8) c:

I: c:

I: c: I: c:

(reads) Which of these fractions is the smallest: 2/5 or 3/8? I think 3/8. How did you do that? This isn’t how I did it, but it would explain it. (Long pause while Cathy converts 215 and 318 to 16/40 and 15/40) But how did you get 318 at first? Well an eighth is smaller than a fifth and three is just next to two. The next question. (reads) Which of these is the largest: 4/12 or 6/14? Four-twelfths, I guess.

In this question both of Cathy’s methods were difficult to apply. Her intuitive approach of looking at the differences between corresponding numer- ators and denominators did not help, since the difference was two in each case. And the formal common denominator method had been made more difficult by the size of the two denominators.

I: You guess 4/ 12? C: Urn. I can’t think of a common denominator.

* * * * * *

Throughout the study Darren remained unable to identify


as representing 3/8. Yet by February, year 8, Darren was able to calculate l/2 + l/3 using an algorithm which employed equivalent fractions. It was evident from Darren’s discussion of the algorithm that it was a rote-learnt procedure, and one of limited applicability since his difficulties with equivalent fractions prevented its use with more complex fractions or mixed numbers.

The learning of this algorithm, lacking understanding of the mini-procedures it employs, taxed Darren’s mathematical ability and persistence during term 3 of year 7. Darren’s perceptions of the pace of mathematics classes and his difficulties and subsequent behaviour can be attributed in part to his weakness in this area and the apparent lack of remedial instruction.

4.4 Mathematical Ability

In grade 6 all ten students were introduced to the concept of directed numbers and the procedures related to the addition of integers. Directed numbers were not part of the mathematics syllabus of any of the students, and the one hour lesson was designed to provide data on student approaches to new concepts and procedures. One week later, a short test was administered. Cathy’s performance on this test was particularly good: successfully ordering positive and negative numbers, including fractions and decimals, and successfully adding three-digit directed numbers. This ability to quickly assimilate a new concept and related skill was indicative of one aspect of Cathy’s mathematical ability. Characteristically this new knowledge was not retained into year 7, as was evident from Cathy’s responses to MAPS items involving negative numbers. This ability to quickly grasp a new algorithm, successfully apply it in a test situation, then subsequently, over time, forget it, was demonstrated again late in year 7 with the addition of fractions algorithm. In this Cathy differed from Chris, Cameron and Annette, who both learnt and retained the new knowledge, and from Darren and Bernie, for instance, who never fully acquired it.

Cathy’s experience was the converse of Darren’s. Where Darren moved from an environment in which he was respected for his academic ability to one where he was not, Cathy joined a school community which consistently told her she was talented. In this study it is argued that these consensus opinions held by the community in which the child is embedded make a more significant contribution to the child’s behaviour than any supposed absolute attributes the child may possess: such as “mathematical ability”.

The interdependence of ability, self-concept, and response to others’ perceptions of his ability was apparent in this typically succinct response when Darren was asked whether he would like people to think he was “smart at maths”.

244

Darren (July, year 7):

DAVID CLARKE

Well, I’m not smart at maths. So it doesn’t matter. I know most of me stuff, but people don’t think I’m smart at it.

4.5 Conception of Mathematics

Within their individual conceptions of mathematics Cathy and Darren held interwoven beliefs relating the teaching and learning of mathematics and their associated feelings. Their evolving conceptions are evident in the following interview excerpts and in the questionnaire responses tabulated in Tables I and II.

Cathy (June, year 7) I: What is the job of the Maths teacher? C: Well (pause) they’ve gotta teach us

how to (pause) do the maths and (pause) so that, like when you go shopping and things like that, you’ve gotta know how to add up things (long pause). They’ve gotta be able to help you and be able to (pause) sort of explain, explain it to you. Like, some teachers don ‘t even explain it; they expect you to know it and just do it (emphatic) (pause) and they don’t explain through and tell you about it (pause) [or] even help you to do it a bit if you don’t know what’s going on.

Cathy (July, year 8) I: What sort of things do you need to

be good at to do Maths? C: To know basic things like your tables.

To be patient (laughs). I can’t think of anything else (pause). You have to be able to use your brains.

I: What sort of people are good at Maths? C: They’re usually quiet (long pause).

I can’t explain. I: Why are they good at Maths? C: The way they’re born (pause). Some

people are just capable, I suppose, it’s easy for them to understand. They concentrate and limit their time to assignments and maths homework.

I: Can anyone be good at Maths? C: I suppose (very hesitant).

Darren (February, year 8) I: You like Miss Hughes (year 8 Mathe-

matics teacher)? D: She’s better than Mrs. Hilhnan (year 7),

last year. She helps you more. Makes it more interesting, too. We have games more. Sort of maths games. We’re still learning stuff but it’s really good.

Dan-en (August, year 8) I: How do you (pause), do you feel

you’re doing better now than you were last year?

D: Yeah. I: Your dad does. He was saying he

feels you’re happier about your Maths now than you were last year.

D: Yeah. ‘Cos Mr. Hayes (term 2, year 8) is o.k. Oh, I used to hate Mrs. Hillman.

I: Yeah. Funny that. Why? D: Just didn’t know, didn’t like. I: What did she do that you didn’t like? D: I couldn’t understand her. And she

used to get in a big huff ‘cos I kept putting me hand up asking her how to do things and then after that, urn, (long pause).

I: Hang on, you couldn’t understand her because she went too fast or her explanations were (pause). You couldn’t understand her explanations.

D: Yeah. And she used to say I weren’t listening. And after that I just used to muck around all the time. ‘Cos I didn’t like it.

I: And that didn’t help, of course. Because when you mucked around.

D: I failed.

TABL

E I

Chan

ges

in re

spon

ses

to q

uesti

onna

ire

item

s re

lated

to

Conc

eptio

n of

Mat

hem

atics

: Du

rren

Item

No m

atte

r ho

w ha

rd I

stu

dy I

will

get l

ow m

arks

in M

aths

. I t

hink

I c

ould

hand

le m

ore

diffi

cult

Mat

hs.

I thi

nk

ever

yone

ca

n lea

rn M

aths

. M

aths

is a

lot

of r

ules

for

num

bers

. I’m

not

the

typ

e to

do

well

at M

aths

.

I ex

pect

to

have

littl

e us

e fo

r M

aths

wh

en I

get

out

of

scho

ol.

Othe

r su

bjec

ts a

re m

ore

impo

rtant

to

peo

ple t

han

Mat

hs.

Resp

onse

Nove

mbe

r, gr

ade

6

Stro

ngly

Disa

gree

Unde

cided

Stro

ngly

Agre

e Ag

ree

Stro

ngly

Disa

gree

Stro

ngly

Disa

gree

Stro

ngly

Disa

gree

Lear

ning

Mat

hem

atics

Usin

g M

athe

mat

ics

Febr

uary

, ye

ar 8

Unde

cided

Stro

ngly

Disa

gree

* Un

decid

ed*

Stro

ngly

Agre

e*

Unde

cided

Unde

cided

Unde

cided

*

Doing

Mat

hem

atics

-

Mat

hs is

enjo

yable

to

me.

Un

decid

ed

Disa

gree

* I’m

no

good

at M

aths

. St

rong

ly Di

sagr

ee

Unde

cided

* I d

o as

littl

e wo

rk

as po

ssibl

e in

Disa

gree

Un

decid

ed*

Mat

hs.

* Th

ese

item

s we

re a

lso a

dmini

stere

d in

Nove

mbe

r, ye

ar 8

. Res

pons

es w

ere

ident

ical

to t

hose

give

n in

Febr

uary

, ye

ar 8

.

TABL

E II

Cath

y’s r

espo

nses

to

Like

rt ite

ms

on c

once

ptio

n of

mat

hem

atics

Inte

rvie

w NS

O Te

st

Nove

mbe

r De

cem

ber

Grad

e 6

Grad

e 6

Inte

rvie

w Ju

ly Ye

ar 7

NSO

Test

No

vem

ber

Year

I

Inte

rvie

w M

arch

Ye

ar 8

NSO

Test

No

vem

ber

Year

8

I am

sur

e th

at I

can

lear

n M

aths

SA

SA

**

SA

A

A Ot

her

subj

ects

are

mor

e im

porta

nt

D SA

**

U

U SA

th

an M

aths

M

aths

is a

lot

of r

ules

for

num

bers

A

A Ye

s A

A A

Mos

t job

s re

quire

so

me

Mat

hs

A A

Yes

A A

A I w

ould

have

mor

e fa

ith i

n th

e an

swer

for

A

A **

D

U D

a M

aths

pro

blem

so

lved

by a

man

tha

n a w

oman

I’m

not

the

typ

e to

do

well

at M

aths

D

D **

D

D D

I do

as lit

tle

work

as p

ossib

le in

Mat

hs

D SD

No

SD

D

SD

,I th

ink

ever

yone

can

lear

n M

aths

-

SA

A **

A

A D

Mat

hs is

enjo

yable

to

me

A A

Yes

A A

U ‘I

thin

k I c

ould

hand

le m

ore

diffi

cult

A U

Yes

A A

U M

aths

I w

ill us

e M

aths

in m

any

ways

as a

n ad

ult

SA

A **

A

A U

SA =

St

rong

ly Ag

ree,

A =

Agr

ee,

U =

Unde

cided

, D

= Di

sagr

ee,

SD =

Stro

ngly

Disa

gree

.


The trend is quite clear. Darren had developed serious doubts about his ability to learn mathematics. His attitude to the subject had hardened, and he was less convinced of its general importance. Averaging Darren’s responses on sub-groups of 34 Likert items showed a drop in motivation and a marked drop in confidence (4.5 to 2.8 on a S-point scale).

4.6 Self Concept

The values which structured Cathy’s world view were evident in a variety of ways: from her responses on the semantic differential questionnaire in which she identified valued constructs with attributes like “useful”, “active”, “kind”, “successful” and “interesting”, and from her construct clusters which associated her parents, and Cathy herself, with ideals like “The Person I Would Like to Be”. In addition, Cathy’s descriptions of the participants in the learning environment, and her conception of the teacher’s role, revealed much about her values,

Mathematics remained distant from all that Cathy valued. In this Cathy was similar to Darren. Also both had resilient, stable, general self-concepts. The difference lay specifically in their relative views on their mathematics competence: Both considered Cathy to be the superior mathematics student (November, year 7). Both students were, of course, unaware of the similarity in their MAPS and MAMP scores.

Cathy was more inclined to persist both socially and mathematically. Social and mathematical rules existed to be used, and she had some confidence in her ability to learn how to use them. Unlike Darren, Cathy’s secondary experiences inclined her to optimism and a less defensive approach to schooling.

On examining Darren’s responses to success and failure, and to the encouragement and criticism he met, deterioration of his self-concept with regard to mathematics, increasingly negative attributions, and his lowered expectations (see Table I) present a picture of a student at the end of his first year of secondary schooling, whose response to a non-routine mathematical task was conformist, defensive and pessimistic. Darren’s persistently stated belief in the usefulness of mathematics was an indication of his respect for his father’s opinions rather than a personal conviction. In according a level of importance to the people in Darren’s learning environment, Darren’s father is the figure most consistently referred to in interviews. From the semantic differential a clear stratification emerged. Father, Mother and Friends were valued highly and Grade Six Teacher was closely associated with this group. But Primary Teachers, High School Teachers and the Mathematics Teacher were accorded importance in that order and grouped separately from more personal constructs.

248 DAVID CLARKE

The association of the Grade Six Teacher with valued personal and ideal constructs indicated that Darren did not entirely disregard teachers. The important question posed by Darren’s responses to his first year of secondary mathematics concerned the possibility of teacher action to correct the negative trends in his mathematical behaviour: to involve him as a participant in learning, to reduce the difficulties he was experiencing in class, and to increase the likelihood of his continued successful participation in the subject. The actions of Darren’s term one year 8 mathematics teacher appear to have been successful on two fronts: Darren’s understanding of basic concepts improved markedly and Darren’s enjoyment of mathematics classes increased. As a result Darren’s confidence recovered slightly and his performance improved. Darren responded to what he saw as a more caring approach to mathematics teaching and his recognition of the teacher’s qualities was evident in his close association of this teacher with other valued adults on the semantic differential.

Despite these gains Darren’s perception of the subject mathematics remained largely unchanged. Mathematics was still very difficult, very confusing and fairly boring. The positive experiences of term one were attributed to the teacher rather than to the subject. And even these positive experiences had not led to a greater commitment to mathematics or involvement in class. Darren’s term one mathematics report read. “[Darren] has shown on his tests that he understands this unit. However he wasted much time in class and did a minimum of work”.

4.7. Individual Student Classroom Practices

A picture emerged of Darren as a student who, while indicating a preference for male teachers, was not sensitive to the teacher’s approval and seldom con- sulted the teacher during a lesson. Of Darren’s year 7 teachers only the Science and Woodwork teachers were male. Darren’s behaviour in a subject was more likely to reflect his opinion of the subject than his opinion of the teacher. In relating these trends to Darren’s observed behaviour in mathematics classes two additional observations must be recorded. Despite Darren’s reported reluctance to consult the teacher he occupied a disproportionately large amount of the teacher’s time. In November, year 7, observation notes on one mathematics lesson concluded, “Darren required a lot of attention, assistance, encouragement and coercion”. Also Darren’s relationship with his male peers was positive and successful (as established August, year 7). Mathematics was not a significant element in Darren’s relationship with his peers (he was seen by his peers as neither an expert, a fool, nor a major disruptive influence) and Darren professed ignorance of his friends’ attitudes to mathematics.


Comparison of Table III with Table IV reveals a striking difference in teachers’ perceptions of the classroom practices of Cathy and Darren. The consistency of teachers’ descriptions of Cathy’s behaviour contrasts with the variation among the descriptions of Darren’s behaviour provided by the same teachers (“Work Style”, for instance).

In September of year 7 Cathy was given a deck of cards, each with the name of one of her year 7 classmates, and asked to rank the cards according to “how good they are at maths”. The resultant sequence was Cameron, Chris, Christine, Caroline, (Alison, Andrea, Amanda, Annette, Monica), Cathy, Mary, Murray, Deborah, Mark, Ben, Davy, Michael, Dan-en, Bob, Bernie, Dorothy, Damian, Brian, Margaret, Donald. Given the same task Darren produced the sequence: Cameron, Chris, Caroline, Murray, Amanda, Annette, Mark, Alison, Christine, Cathy, Mary, Anthea, (Bernie, Darren, Davy), Damian, Michael, Ben, Monica, Andrea, Dorothy, Brian, Bob, Margaret, Donald. The similarity is clear (p = 0.75) but the listings are certainly not identical, and Cathy and Darren perceived some students very differently (Andrea and Monica, in particular; but also Bernie, and Darren himself). The degree of consensus is sufficiently high to confirm that information concerning the relative competence of children is communicated very efficiently. Despite having known most class members for only 8 months, neither Cathy nor Darren saw the ranking task as difficult or inappropriate.

Cathy was then asked to group the cards “according to how they behave in class”. Figure 4 shows Cathy’s groupings, annotated with the characteristics she felt distinguished each group. These groupings, and Cathy’s perceptions

“They think they’re tough.

They’re cheeky, and they laugh at other people’s answers. Michael is a

big mouth. So is Darren.”

[Davy. 1 Murray, Mark, Ben

Dorothy, Andrea, Alison, Monica,

Amanda, Annette.

“Good. They sit at the back. Oh, they’re not

always good.”

Cathy, Mary, Deborah

“Normal. Sometimes

“Good.” “Cheeky.” good, sometimes bad.”

Chris, Cameron.

“Loud. About work. Always fighting about work . But

they hardly ever talk in English and Humanities.”

Christine, Caroline. IlNlargaretl

“Loud. Always “Seen, but

talking.” not heard.”

Fig. 4. Cathy’s perceptions of her year 7 mathematics classmates.

TABL

E III

Teac

her

respo

nses

to

a qu

estio

nnair

e on

Da

rren’s

cla

ssroo

m pra

ctice

s -

Septe

mber

, ye

ar 7.

Mathe

matic

s En

glish

Sc

ience

Hu

manit

ies

Lang

uage

Ph

ysica

l ed

ucati

on

Craft

Dr

ama/

b-2

photo

grap

hy

2

Appr

oach

to

class

work

Often

ha

sty

and

voca

l, inc

onsis

tent

easil

y dis

tracte

d

Poor

to

Fair

Hasty

co

-ope

rativ

e Ha

sty.

some

times

dis

rupti

ve

voca

l inc

onsis

tent

High

, bo

okwo

rk po

or

and

Indus

trious

, ha

sty

coop

erati

ve

Unint

eres

ted

disru

ptive

Ind

ustrio

us

enthu

siasti

c co

-ope

rativ

e

Enthu

siasti

c co

mpeti

tive

activ

e vo

cal

incon

sisten

t

Hasty

vo

cal

incon

sist-

ent

Abilit

y in

subje

ct Ab

le Hi

gh

Fair,

finds

it

hard

to

conc

entra

te an

d do

es

not

get

down

to

work.

Seldo

m

Able

Able

High

but

tests

good

.

Freq

uenc

y of

contr

ibutio

n in

class

, an

d na

ture

of the

co

ntribu

tion

Seldo

m.

Answ

ering

a

ques

tion

direc

ted

to hi

m

Reac

tion

to He

sitan

t, be

ing

aske

d em

bara

ssed

a

ques

tion.

indiffe

rent

Seldo

m.

Askin

g for

ins

tructi

on

to be

cla

rified

.

Indiffe

rent

Seldo

m.

Givin

g an

an

swer

.

Occa

siona

lly

Answ

ering

a

ques

tion.

Often

. Co

ntribu

tes

to al

l ac

tivitie

s

Seldo

m Se

ldom

Indiffe

rent

Hesit

ant

and

emba

rass

ed

Conf

ident

if

he k

nows

the

an

swer

, ind

iffere

nt if

he

does

n’t

Not

at al

l

Indiffe

rent

Very

confi

dent

Hesit

ant

Teac

her

cons

ultati

on

(per

lesso

n)

l-2

times

1-

2 tim

es

Not

at al

l 1-

2 tim

es,

very

occa

siona

lly

l-2

times

No

t at

all

l-2

times

Work

sty

le Re

quirin

g co

nstan

t su

pervi

sion

Requ

iring

Requ

iring

regu

lar

cons

tant

direc

tion

amx=

rvic

inn

Inde

pend

ent

Requ

iring

cons

tant

o....

n...i

oi^.

.

Inde

pend

ent

Requ

iring

cons

tant

Inde

pend

ent

TABL

E IV

Te

ache

r re

spon

ses

to a

que

stion

naire

ab

out

Cath

y’s c

lass

room

pra

ctice

s -

Sept

embe

r, ye

ar 7

Mat

hem

atics

En

glish

Sc

ience

Hu

man

ities

Lang

uage

Ph

ysica

l Ed

ucat

ion

Craf

t Dr

ama/

Ph

otog

raph

y

Appr

oach

to

cla

sswo

rk

Abilit

y in

subj

ect

Freq

uenc

y of

co

ntrib

ution

in

class

, an

d na

ture

of

the

co

ntrib

ution

Reac

tion

to

being

ask

ed

a qu

estio

n.

Teac

her

cons

ulta

tion

(per

les

son)

Wor

k st

yle

Indu

strio

us

enth

usia

stic

co-o

pera

tive

activ

e,

som

etim

es

voca

l

High

Indu

strio

us

co-o

pera

tive

High

Occa

siona

lly.

Ofte

n,

‘Ans

wers

, An

swer

ing

opini

ons,

ques

tions

, gr

oup

read

ing

disc

ussio

n.

aloud

.

Conf

iden

t, ke

en t

o di

spla

y kn

owle

dge.

l-2

times

Conf

iden

t.

1-2

times

Inde

pend

ent

Inde

pend

ent

Indu

strio

us

enth

usia

stic

co-o

pera

tive

com

petit

ive

Very

Hi

gh

Ofte

n.

Cont

ribut

es

in all

po

ssibl

e wa

ys.

Very

co

nfid

ent.

3-4

times

Inde

pend

ent

Indu

strio

us

enth

usia

stic

care

ful

co-o

pera

tive

Very

Hi

gh

Ofte

n.

Answ

ering

qu

estio

ns,

relat

ing

incid

ents

.

Conf

iden

t.

Not

at a

ll

Inde

pend

ent

Indu

strio

us

enth

usia

stic

care

ful

co-o

pera

tive

High

Ofte

n.

Answ

ering

qu

estio

ns.

Conf

iden

t, ke

en t

o di

spla

y kn

owle

dge.

3-4

times

Inde

pend

ent

Indu

strio

us

enth

usia

stic

co-o

pera

tive

activ

e

Very

High

Ofte

n.

Help

ful,

(ans

wers

qu

estio

ns,

follo

ws

inst

ruct

ions

,

Very

co

nfid

ent.

1-2

times

Inde

pend

enl

Indu

strio

us

enth

usia

stic

care

ful

co-o

pera

tive

activ

e vo

cal

High

Ofte

n.

Opini

ons,

answ

ers

Very

co

nfid

ent.

3-4

times

Inde

pend

ent

Indu

strio

us

co-o

pera

tive

High

Occa

siona

lly

Ques

tions

, di

scus

sion,

I a

nswe

rs

Quie

tly

hesit

ant,

care

ful.

3-4

times

Inde

pend

ent

but

requ

iring

som

e di

rect

ion.

252 DAVID CLARKE

of them had a fundamental validity in that they represented the classroom reality as Cathy saw it, and as such contributed to the form of Cathy’s participation in general classroom practices: alliance or identification with a particular- group possibly involving the adoption of a particular style of behaviour. Yet the accuracy of Cathy’s perceptions would be challenged by any of the teachers of 7H, particularly her description of Davy and Brian as “good”.

Whereas Cathy’s classroom behaviour in year 7 drew praise from all teachers, by term 2 of year 8 three-quarters of her teachers described her as “easily distracted”. The values advocated by Cathy in interviews conform throughout the study to conventional ‘success through effort’. However observations of Cathy in class made it clear that she was experimenting with a mode of classroom behaviour oriented towards social rather than academic success.

5. CATHY AND DARREN: A DISCUSSION

Cathy’s perception of the role of rules, both mathematical and social, was different from Darren’s, and this may be the result of her adroitness at picking up patterns and procedures. The rules need not necessarily make sense; their value is that “they always work”: they produce the desired outcome. The distinction between Darren’s position and Cathy’s is a fine one which, in a way, typifies other differences in their mathematical behaviour. Even after application of a rule Darren was not confident of the correctness of the answer and asked repeatedly, “Is it right?“. Cathy had greater belief in her ability to correctly apply a rule and this underlying optimism was expressed in other aspects of her mathematical behaviour: in her persistence, in her higher expectation of success, and in her more assertive approach to new work. By comparison, Darren was defensive and pessimistic.

How did this difference in approach arise? On what grounds did Darren and Cathy attribute greater ability and likelihood of success to Cathy? Certainly there was interview data and NSO test evidence that Cathy may have had a slightly better understanding of mathematics, but Darren and Cathy were not in a position to make that comparison, lacking access to that data. Their performances on the MAPS and MAMP tests were almost identical, and on the final class numeracy test in year 7 Darren scored 74% and Cathy 81%. But the mastery level was set at 80%. Other than proficiency in classroom participation, this was the most public demonstration of the difference in competence between the two children. It appears that both children accepted the verdict implied in this assessment and incorporated its implications into their evolving self-images.

Darren’s case study raised the issue of poor teacher-student communication,


and demonstrated the possible consequences. Cathy’s experience demonstrates above all the importance of social factors, and the implications of this pre- occupation for the classroom behaviour of some students.

Finally, what of the function of the transition experience? A detailed summation of the environmental changes associated with transition, and of Cathy’s and Darren’s responses to these changes follows.

(It is important to state that the points made in the summary which follows arise from a much greater body of data than the small illustrative sample provided in this paper. As a result some points may appear to lack empirical support. Given the volume of data collected and the confines of the medium it is difficult to see how this could be avoided.)

The most obvious characteristic of the phenomenon of transition is the change in learning environment. This change takes different forms and the experiences of Cathy and Darren suggested that some forms of environmental change were of particular significance.

Cathy (4 The need to make new friends;

to join or initiate new social net- works.

@I

(cl (4

(e)

(h)

(0

0)

The loss of status, responsibility and notoriety which primary school had afforded. The increased difficulty of the work. The change in teacher expectations and in the perceptions of the child’s ability. The use of instructional procedures which encouraged rule-based learning, which gave students little autonomy, and which expected very little of students with regard to co-operation, self-motivation and awareness of the rationality of the form of instruction. The change in parent satisfaction with the school environment. The change in institutional factors like size, organization and number of students. The change in relative competence of Cathy’s peers. The increased incidence of being “picked on” by peers. The increased experience of success.

Darren

(4 The change in competence of Darren’s peers relative to Darren’s own performance.

(b) (cl

(4

(e)

The change in the pace of instruction. The change in the teacher-student relationship, specifically in the variety of different adults with which Darren must interact and the small proportion of males. The change in teacher expectations, where these arose from subject- based instructional criteria rather than from knowledge of Darren’s abilities and experiences. The change in the form of instruction; teachercentred boardwork occu- pying a much greater proportion of instruction time.

(f 1

Cd

(h)

The increased stress on rules and terminology. The decreased responsibility and recognition accorded Darren by the school structure. Institutional changes including size, organization and the number of students.

0) The increased workload. ci) More frequent testing. (k) The lack of success.

254 DAVID CLARKE

It was the responses which Cathy and Darren made to these changes which provided the most significant results of this study.

Cathy (a) Enhanced social confidence. (b) The development of a sense of

independence. (c) A period of academic consolidation

followed by experimentation with less conformist classroom practices.

(d) The consolidation of a belief that the goal of mathematics learning is instrumental understanding.

(e) Regret over the lost security and closeness of the primary school environment.

(f) Enhanced self-esteem with regard to mathematics, leading to expectations of continued participation.

(g) The relatively low esteem accorded high school teachers.

(h) The lack of improvement in mathematical performance as evidenced by MAPS and MAMP scores.

(i) An increased tendency to attribute success to ability rather than effort. This change was less extreme than in Darren’s case, possibly because Cathy did not have the same experience of failure.

(i) An apparent reduction in student- initiated teacher contact associated with developing independence.

(k) The development of Cathy’s ability to achieve short-term operative competence in an algorithm sufficient to meet immediate test requirements, whilst providing no guarantee of retention.

(1) A lessening of care and precision. (m) The resilience of attitudes and values

relating to mathematics, teachers, teaching and acceptable behaviour apparently originating from the values of Cathy’s parents.

(n) The general lack of importance accorded to mathematics.

(0) me steady but unexceptional improvement in mathematical understanding. This improvement seemed to be less dependent on instruction than it was on Cathy’s personal development and maturation.

Darren (a) A loss of self-esteem in mathematics. (b) A more restricted conception of

mathematics. (c) An increase in disruptive behaviour

in class. (d) The lack of a sense of ‘belonging’

which had been present at primary school.

(e) The apparent stability (and resilience) of Darren’s general self-concept.

(f) The relatively low esteem accorded high school teachers, no teacher assuming the significance as a model attained by Darren’s grade 6 teacher.

(g) A withdrawal from active participation in learning.

(h) The lack of progress in mathematical understanding during year 7, minor improvements in understanding not being attributable to instruction.

(i) An increased tendency to make attributions to ability rather than effort.

(i) His tenacity in retaining a miscon- ceived rule despite instruction where this rule was based on a personally significant construct.

(k) The limited utility of some of his newly-learnt rules.

(1) The continued lack of reflection on the accuracy of his answers or the appropriateness of his methods.

(m) An increased reliance on known rules.

(n) An increased need for teacher assistance together with an exaggerated reluctance to ask for it directly.

(0) Lowered expectations with regard to his continued involvement in mathematics.

IMPACT OF SECONDARY SCHOOLING 25.5

In Cathy’s case the commencement of high school and secondary mathematics represented a series of social and academic challenges which she successfully overcame. The result was enhanced self-esteem and personal growth. For Darren, the transition demonstrated his inadequacy in the area of mathematics. Since his self-esteem was not predicated solely on academic success, and since mathematics was peripheral to the things Darren valued, the consequences for Darren’s self-concept were not critical. But within the particular area of mathematical behaviour it must be concluded that what was a positive and productive experience for Cathy was a destructive and personally-restric- ting one for Darren.

The preceding discussion demonstrates the essentially idiosyncratic nature of student experience of secondary mathematics, and the necessity for mathematics educators to frame goals which encompass much more than test performance. The virtual equivalence of Cathy’s and Darren’s year 8 test performances belied the dramatic differences in the impact of secondary mathematics on their mathematical behaviour. The elements of figure 1 provide a starting point for consideration of what these broader goals might be.

6. CONCLUSIONS AND A POSSIBLE COURSE OF ACTION

The two case studies reported suggest that greater recognition be given to the importance of the social context in which mathematics is learnt, and demonstrate the efficiency with which the school community reaches consensus on student attributes and abilities and communicates these consensus opinions to the student. It appears that the impact of secondary mathematics on a student’s mathematical behaviour may be determined during the first year of secondary school through the evolution of community opinions which the child comes to share.

The findings of this study confirm the need for a more comprehensive descriptive framework than the usual restricted consideration of attitudes and achievement if the process of student response to secondary schooling and secondary mathematics is to be usefully understood.

From these two case studies it is possible to argue that the variety in student background, the differences in personality, and the evident diversity in students’ responses to the same mathematics classroom, do not suggest that a single teaching style or administrative structure would be likely to meet the needs of children commencing secondary mathematics.

If secondary mathematics teachers are prepared to accept as an appropriate goal, “To increase the likelihood of a student’s continued successful participation in mathematics”, then the need is for a means whereby the teacher

256 DAVID CLARKE

can both monitor and respond to the changing needs of students. A research project has been initiated in 36 Victorian classrooms which seeks to establish such a mechanism (Clarke, 1984d). In this program a child commencing secondary mathematics is regularly given the opportunity to inform the teacher of difficulties experienced, success achieved, and sources of anxiety, and to reflect critically on the teaching and learning of secondary mathematics. The preliminary findings are encouraging.

REFERENCES

Australian Council for Educational Research (ACER): 1918,Mathematical Ability Profile Series, ‘Operations’ and ‘Space’ Tests, ACER, Hawthorn.

Aiken, L. R.: 1979, ‘Attitudes toward mathematics and science in Jranian middle schools’, School Science andMathematics 79,229-234.

Biggs, J. B. and Collis, K. F.: 1982, Evaluating the Quality of Learning: The SOLO Tax- onomy, Academic Press, New York.

Clarke, D. J.: 1984a, ‘The application of model of mathematical behaviour to one student’s experience of the primary-secondary transition. Case Study No. l’, Faculty of Education, Monash University.

Clarke, D. J.: 1984b, ‘Mathematical behaviour and the primary-secondary transition: Case Study No. 2’, Faculty of Education, Monash University.

Clarke, D. J.: 1984c, ‘Secondary mathematics teaching: towards a critical appraisal of current practice’, Vinculum 21(4), 16-21.

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Faculty of Education, Monash University, Clay ton, Victoria 3168, Australia

The impact of secondary schooling and secondary mathematics on student mathematical behaviour

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