Pre-Service Primary Teachers' Approaches to Mathematics Education Elizabeth Buckingham Deakin University - Burwood Campus This study was an attempt to identify pre-service primary teachers orientations on mathematics on completion of their First Year unit in Mathematics Education. The grounds for investigating epistemological orientations are based on the view that certain orientations are more likely to result in in depth learning than others. Wilkinson (1989) wrote of the relationship between the ways a student learns and his or her definition of knowledge. Royce and Mos (1980) regarded a student's epistemological orientation to be relatively fixed, depending on the learners "ability to symbolize conscious and unconscious experiences (Wilkinson, 1989)". The models preferred here are those which suggest that there is a progression in students' thinking about knowledge, elaborated in schemes by Perry (1968), and Belenky, Clinchy, Goldberger and Tarule, (1986). The Schemes of Epistemological Orientations explained as they might be applied to mathematics education Perry (1968) and others, as student counsellors, made a longitudinal study of undergraduates studying Humanities at Harvard. He classified students' learning behaviours and perceptions of knowledge, into a developmental scheme, which had three main categories of epistemological orientations, Dualist, Multiplist and Relativist. Those holding an early stage orientation were unlikely to be able to perceive knowledge with a more advanced orientation. The Dualist saw knowledge as compartmentalised, and in terms of "right" and "wrong"; he believed that there was a right answer to everything, determined by "Authority", whose role was to teach this to him. Diversity of opinion was seen as "unwarranted confusion". This view of the source and nature of "valid" knowledge would explain students' resistance to considering alternative views of knowledge; to allowing, for example, that there might be more than one valid theory to explain an historical event. This Dualist thinking is apparent in the mathematics classroom, where there is often resistance by learners to attempts to teach for relational rather than procedural understanding (Copes, 1982, White, 1991). In the pre-service course, the Dualist would be more likely to have a rule- driven approach to computation. Dualist students appear, at best, ambivalent about the need to understand the concepts underlying procedures, and, at worst, "confused" by explanations. In mathematics education, such a pre-service teacher might be capable of reproducing ready made curriculum activities, probably, holding the view that the "doing" of the activity , was adequate, but having little sense of the "construing" needed for children to construct their knowledge of mathematics. The Multiplist of Perry's scheme, accepts a multiplicity of solutions in certain circumstances (see Adam below) but still holds to Authority as the arbiter, who will determine the right way eventually. The Multiplist orientation is the prelude to one in which the studentr values his own view of knowledge, (Belenky, 1986) and makes a more sophisticated appraisal of relationships between aspects of knowledge. A student with this orientation now has the 'space' to make decisions about knowledge, and to learn with some purposes of his own. A student prepared to view some solutions as better than others, depending on the circumstances, and to decide between options on the basis of his/her own priorities, has a Relativist orientation. Authority no longer has the prerogative on "right" solutions. In mathematics education, being a Relativist could mean that a pre- service teacher has an appreciation of many factors contributing to the learning of mathematics, and of the possibilities of learning through mathematics (Copes, 1982). In planning for a mathematics class, a Relativist student could decide between a number of methods of teaching a topic, on the basis of the learners that he/she had in mind. He or she may view mathematics as a creative activity (Buerk, 1985), with an enhanced awareness of the possibilities of making connections between mathematical ideas. This orientation would necessarily involve a more interactive, higher quality of learning (see Saljo's Levels in Table 1 . ii below). Tables 1. i and ii, below, give descriptions, in the authors' terms, of the orientations and learning behaviours of Perry's (1968), Belenky et al.'s schemes and Saljo's concept of Quality of learning scale. I have simplified their schemes and aligned their categories in to three broad bands, Brackets, I, II, and III, that were appropriate for the differences found in this study., amongst students. Table 1 . i Paraphrase of categories from Schemes used and put into three Brackets for this study BRACKET IBRACKET II BRACKET III Perry's Scheme of Intellectual and Ethical Development (1968) Author's Positions: 1 and 2 3 and 4 5 (through 9)Dualist World viewed in polar terms, we-right -good vs other - wrong - bad. Right answers to everything ... Authority's role is to teach. Knowledge and goodness are viewed as quantitative accretions. Diversity in opinion seen as unwarrantedconfusion, by poorly qualified Authorities. Multiplist Aware of multiplicity of perspectives; everyone has a right to their own opinion.Relativist, . . . Perceives all knowledge as contextual and relativistic; Sees the need to orient himself in a relativistic world through some form of person Commitment. Belenky et al., working with women returning to tertiary education, regarded self-concept and the desire to communicate, as integral parts of a student's epistemological orientation, which I found particularly appropriate for the students doing the pre-service course. Their view of the growing assertiveness of women in relation to their own, subjective knowledge is useful in explaining the ways students approach mathematics. They used the concept of "Voice " to provide the imagery of language and the student's role in this. Bracket I Where a student views knowledge as a fixed body, external to his or her own thinking, waiting to be discovered and amassed, her part in making sense of this, is a passive one, associated with rote learning and replicating processes without questioning (see Alison below). The label for the early stage, "The Voices of Others", conveys the passivity in thinking about knowledge, in this stage. Bracket II Later, comes a recognition by the student, of her own Subjective Knowledge, and its value in making decisions, a sense of self-reliance, and a willingness to relate these to objectively known knowledge, and to communicate her ideas. A student's recognition that "Received Knowledge" is not the only "valid" knowledge, but rather, objectified knowledge, and a public aspect of knowledge, allows them to consider her own experiences of mathematics in relation to what is taught, and so make sense of it. Bracket III Here, Belenky et al. describe students "Knowledge viewed as contextual, ... valuing both subjective and objective strategies for knowing." It is here that a student might view mathematics as a language with which to think, to quantify and order data. This involves a more interactive relationship with knowledge, and the quality of a person's learning would be at the upper end of Saljo's Scale, "an interpretive process aimed at understanding reality". Table 1 . ii Paraphrase of categories from Schemes used and put into three Brackets for this study Belenky et al.'s Scheme of Personal and Intellectual Development (1986)BRACKET IBRACKET II BRACKET III Authors' Stages: 1 and 2 3, 4, and 5 6 and 7Silence Received Knowledge: Listening to the Voices of Others; women perceive themselves as receiving, even reproducing knowledge from all-knowing external authorities, but not being capable of creating knowledge of their own. Subjective and Procedural Knowledge: Knowledge perceived as personal, private .. . in which women have a growing sense of self reliance ... Invested in learning and applying objective procedures for obtaining and communicating knowledge. Procedural Knowledge Separate and Connected Knowing: Aware of the need for conscious, deliberate, systematic analysis, ... the most trustworthy knowledge comes from personal experience rather than the pronouncement of authorities; Knowledge viewed as contextual; subjective and objective strategies for knowing, both valued, ...Saljo's Scale of Quality of Learning (1984) Levels: 1 and 2 3 4 and 5Memoriser A quantitative increase in knowledge User of Procedures The acquisition of facts, methods etc. which can be retained and used when necessary. Generaliser Abstraction of meaning An interpretive process aimed at understanding reality The Context of the Problem The Mathematics Education component of the Bachelor of Teaching course is one of a range of challenges confronting students wishing to qualify as primary teachers at the Burwood Campus. Taking a broad view of the factors affecting students' learning at this level, there are the prevailing structures of the i university and community in controlling the qualification system; ii established curriculum, in determining what will be taught; iii employment prospects; iv cultural forces, such as social status including gender; and v individual's sense of all this and desire to control what they find. They each impinge on the student's lives, and the ways a student copes with them, are likely to be dependent upon his or her self-concept , and orientation on knowledge, its origins and its purposes. The First Year Unit, in Competence and Methodology in Mathematics at Burwood is designed to extend students understanding of the structure of our number system, concepts associated with the four operations and the development of algorithms, and to improve their procedural skills. For some, students the assessment is a major hurdle, and a small proportion of students, about 15%, fail this Unit, and return to repeat it , some a number of times. The reasons for failure are not obvious. Many students work hard to obtain procedural skills, and succeed in the Skills Test. Others are less successful, and often attribute their errors to "silly mistakes" and lack of hard work. They often mean by "hard work", drill and practice, rather than any reappraisal of procedures taught. They do not necessarily see their learning strategies as inadequate. The Questions Is this restricted view of mathematics learning more characteristic of students failing the unit C & M Mathematics 100, than of those passing the unit ? What indicators might be used to determine not just their feelings about the subject, but their concepts of what constitutes mathematical knowledge, where it originates, and how it might be used? Do those failing the unit hold different views about mathematics, which are in some way more constraining, and do those passing the unit appear to view mathematics in such a way that assists them to interact with it,? Method of investigating Students Orienation on Mathematics A combination of developmental schemes used by Perry and Belenky et al. are used here, as the developmental model seemed reflect informally observed intellectual growth in pre-service teachers over the three year training. A scale of quality of learning, as defined by Saljo (1984) was used in conjunction with the orientations to relate ways of doing mathematics to learning behaviours. Two Case Studies Two students failing the CM100 mid-year examination, worked through some division questions, involving decimals, in a pilot questionnaire, and were interviewed about their workings shortly afterwards. They expressed views about their learning, assessment in mathematics, and ways of doing mathematics. Interviewing was an effective way to gather the qualitative data needed to build a picture of the two students' orientations on knowledge and their approach to learning. Below are extracts from these interviews, which are reported on in more detail in Buckingham (1992). Bracket I : Alison's Orienation Alison's reponses to questions about the unit, and her learning of it, suggest orientations characteristic of Bracket I. She described how she had learnt her notes, "word for word" for the examination, although she said she had been advised to do otherwise. Her quality of learning seems largely restricted here to Bracket I behaviours. Alison: " . . . For exams and assignments I'll do very hard work whereas in the classroom I mightn't feel like learning anything, and I'll do it in my own time." Interviewer: "Does that mean that you don't have a lot of confidence about what's going on in the classroom?" Alison: " Oh, no! No I just _ some things I find irrelevant, like Base Five. I learnt to do it. But then I knew I wouldn't have to do it when I taught. It's notin the school curriculum." She had an instrumental approach to Base Five Arithmetic. She did not apparently have an orientation which would allow her to see through this topic, the complexity of the task that children are involved in, when learning algorithms. She talked of her own reading in Child studies, which indicated her interest in children's learning, and suggests that her "Received Knowledge" orientation is not restricted to mathematics: Alison: " I might do extra learning outside college on something that I find fascinating _ I might do extra reading ...Things like for Child Studies, for example, we did things like what does a child do at each stage. Piaget ... I read up on all stages in my own time, just to see what stage the child I had was at, ... so that I knew what to expect." Alison had difficulty with computation involving decimal division. She had her own ideas about solving the partition problem, where the divisor was a whole number: 0.36 / 4, (answer: 0.09) supplying, when prompted, a practical model: "A third of an apple, how can I divide it amongst four people? " and then using a halving strategy, clearly her own, she said: "0.36, 0.18, 0.09, 0.045" not realising that she had repeated the process once too often. The last number she gave indicates a level of competence with the decimal notation. (Some students would fail to give the correct order of the number, and write 0.45, intstead of 0.045). She could make no suggestions as to the next problem, involving quotition , (where the divisor is a decimal number) 3.6 / 0.09, (answer: 40) and had written on the questionnaire, "Not another decimal question. I didn't like it because of the 0.09. " This is an area of difficulty for many students. Just to explain the alternative quotition model of division is often ineffective for a learner concerned wholly with procedural skill. She was not in a connection-making mode, and did not mention her knowledge of 9 as a factor of 36, nor her experience of quotition division discussed in the unit, where she would have heard the language to describe such an operation, (eg. "How many lots of 0.09 are there in 0.36?"), and probably seen the number line model ( in Mathematics Qn a iii in questionnaire below). This suggests that she has made little association between different aspects of her learning in mathematics, that she has a compartmentalised view of processes, which have contributed to her rote learning strategies. She has a stressful but passive rather than interactive approach to learning. Bracket I and II: Adam Adam calculated efficiently, gaining 100% on the Skills Test, and enjoyed the variety of ways calculations could be done, particularly, the option to teach subtraction by either Decomposition or Equal Additions methods, because he felt it would give those who could not understand it one way, another way to manage it. He mentioned "other ways " of doing mathematics more than two dozen times in the interview. This is typical of Multiplist thinking. He was asked for an explanation of his working of the question, 3.6 / 4 ( answer 0.9) . He wrote down an estimation strategy: "3.6 is smaller than 4, however only just. Since 4 /4 = 1, the answer would be close to one." He alsowrote the relevant number fact: "36 / 4 = 9. " and "3.6, one number behind the decimal point, thus one number behind in the answer. 0.9" a rule-of-thumb, which he did not want to explain at interview: He managed to draw together several aspects of his learning. His focussed upon the use of procedures, rather than explanations and the merits of these. He had had difficulty with the nature of the questions on the examination, which he felt, were "curly": "I just like questions that are straightforward, direct and have an established answer." He had missed a number of classes, through employment, which would have affected the to the learning strategy he adopted, "I studied, I went through the book ..., ticked the pages, read the pages, read every little bit I could read ... all the methods. And then the way the questions were put on the exam, threw me.." He viewed mathematics as a series of procedural skills, which he could do well. He did not recognise the need to explain the rules that he used. He was not concerned with making decisions about better ways to do mathematics depending on circumstances, as the Perry Relativist might be. The next step involved developing a questionnaire to find out how the students who had passed the unit and those who had failed it and were returning to do it again, differed. Development of questionnaire The questionnaire was designed on the basis of i what two case study students had said about the ways they approached learning, and explanations they gave about their methods of working out mathematical problems; ii the findings of the developers of the two schemes, (the questions asked, were similar to those posed by Perry and Belenky et al.); iii the findings of Buerk (1984) and Copes (1982), who had used the schemes to explain different performance in mathematics by their students; iv that Saljo's concept of quality of learning was expected to be consistent with students' underlying orientations on mathematical knowledge. Saljo's Scale was a means of categorising the ways students appeared to have learnt their mathematics. There was the problem that much of mathematics is learnt procedurally and most students would reflect this, but it should be possible to distinguish differeing orientations, inspite of this, by asking a range of questions not immediately associated with the doing of mathematics. Establishing students' orientations on mathematical knowledge The students' ways of doing mathematics were used as pointers to their orientations. The mathematical questions set, ranged from work done in the unit to ones that had not been taught in the unit, but which students might be expected to cope with. Success in these indicated: i procedural knowledge, but not necessarily relational understanding, though explanations of workings might reveal this; ii an ability to make connections from one area of learning to another, particularly if there was evidence that students made connections between a) school taught and everyday mathematics, and b) the doing of mathematics and explaining mathematical operations. Error analysis of workings supplied information about nature of a student's understanding of procedures, and where this was based on misrecalled rules, it was assumed that the student probably lacked a holistic view of what he was doing, and that he probably had a compartmentalised view of mathematics, a Bracket I orientation. Adaption of skills, referred to as connection making, across areas, suggested that a student taken an interactive approach to knowledge, and the orientation was classed in Bracket III. Between these two levels of competence, are the students "invested in learning and applying objective procedures for obtaining and communicating knowledge" (Table 1 . ii Bracket II). Main Survey of 92 Second Year Students Of students, who returned to the Second Year of the course, about a quarter of those who had passed CM100, called here, the CM200's, and those who had failed the unit, called the Repeats, were given questionnaires. It was answered in class time, within a few days of beginning their Second Year. The CM200's comprised all 77, who were present, in each of three unstreamed classes. The Repeats (15) were part of a class of 18 repeating students, three of whom chose not to respond. At the end of their Third Year, one hundred of the same cohort of students answered the same questionnaire. A random sample of 20 of these, (every fifth returned questionnaire) is reported upon here, to note changes over the period. The Questionnaire The following describes the questionnaire and why each question was asked: A - C sought attitudinal information, D sought performance in other areas of the course, and E sought information about connection making in mathematical contexts. A Students associations of given items with mathematics Students were asked to mark the following items with an 'M' if they associated it with mathematics, and with an 'L' if they did not. . Group a Arithmetic, Measurement, Problem Solving: These were selected as likely to obtain similar responses between Repeats and CM200's, and to act as a check on the ways students answered the questions: Group b Ordering Ideas, Abstract Thinking, Speed, Boredom: These were selected as pointers to students' awareness of the nature and usefulness of mathematics. Group c Imagination, Creativity, Rote Learning, Difficulty, Fun, : Imagination and creativity were selected on the basis of studies made by Dorothy Buerk (1985) who found that the "Dualists" in her class, believed mathematics was not concerned with creativity and imagination, whereas those with "Relativist" orientations, did. It was expected that Repeats and CM200's would differ on all of these. B Preferred Types of Instruction Copes had found (1982) that Dualists preferred one clear demonstration of how to do a mathematical procedure (a), rather than an explanation. (b). Dualists would be unlikely to opt for types of instruction which required them to explore and choose between methods, (c). Questions When studying mathematics, do you like to know (answer yes or no to each question) a) how to do a problem but not why the method works b) one way to do it and why it works c) explore several ways a problem could be solved C Self-reports on Literacy Skills, Oracy in classes, and methods of coping when stuck doing mathematics These were asked to assess whether students felt able to perform a number of essential tasks in this academic environment. A negative result in any of these, would signal incompetence and/or lack of self-efficacy, a belief that the student has about his or her ability to manage a task, (Bandura, 1977, in Biggs and Telfer, 1987, pp.122-123). Questions i Are your literacy skills for the following, excellent, adequate or in need of improvement? a) to cope with academic reading b) to summarise a news article c) to find a reference in the college library ii When I am unsure of something, I find it easy to ask a question in a mathematics class. Agree Disagree Don't know iii How do you cope when you become stuck doing mathematics? D Formal Assessment in two other areas of the course. In the formal assessment system, Honours were distributed to about half of the students in each unit. If Repeats had significantly fewer than half of these, they might be seen as under achieving in more than just mathematics. Students were asked to self-report on their results in Organisation, Learning and Teaching, and Child Studies. Results A - D A Table 2 shows notable differences and similarities. The Repeats have clearly associated Difficulty with mathematics and dissociated Fun from it. CM200's have done the opposite, but the proportion associating Difficulty is still high, at 58%. A third or more of all groups associated mathematics with Boredom, which could be interpreted to mean that challenges in mathematics are not ones to which these students relate. Repeats do not stand out as finding Mathematics more boring than the CM200's and CM300's. Rote Learning, associated with Bracket I quality of learning, is marked "M" by a greater proportion of CM200's than Repeats, an unexpected result. The Repeats may have found rote learning as a strategy, not helpful. The CM200's might have been more successful as rote learners. No Repeats associated mathematics with creativity, which was intended to find those with a Bracket III orientation. 38% of CM200's marked Creativity with "M". Results for Imagination were similar. The three groups gave similarly responses for Arithmetic, Measurement and Problem Solving (about 98%). CM200's (46%) were more inclined to associate mathematics with Speed, than Repeats (40%) but this aspect seems to have reduced over the two year period, to 30%. Table 2 A) Association of Ideas with Mathematics (%)n = 15n = 77n = 40 TypeRepeats '90CM200's '90CM300's 91Boredom33.336.440Rote Learning4053.260Difficulty86.758.455Fun6.741.640Creativity037.730 B This question permitted people to tick more than one of the type of instruction. Many Repeats (40%) chose (a), 53% of Repeats like to explore several ways to solve a mathematics problem, much like the CM200's ( 57%). The shift to this category by the Third Years (80%) is marked, but 20% of learners still do not like this option. Table 3 B) Preferred Types of Instruction(%)n = 15n = 77n = 20 TypeRepeats '90CM200's '90cm300's 91a) Shown How Only401420b) Shown How & Why677755c) Explore Several Ways535780 C In the self-report on literacy skills, there were no marked differences between the Repeats and CM200's. About two thirds of both selected "adequate". Repeats might have been expected to have a lower self image here, but this was not the case. Two years later, the CM300's appear to be more conscious of shortcomings in this, which may be because they have used the Library more extensively as the course has proceeded, and become more aware of the difficulties. D In both subjects, Repeats had about half the number of Honours that CM200's had. Fail grades and withdrawals were twice as frequent in the Repeats (20%) as in the CM200's (10%). E Mathematics Questions Known Differences between Groups in Procedural Skills From observations of formal assessment, students were known to have difficulty with division involving decimals,. They were generally successful in whole number operations. The CM100 unit provided experience with concrete models (MAB) of decimals and operations on them. Many students appeared to have a greater command of the processes after this, but others did not. Division involving decimals: making connections between models The range of questions set was designed to assess whether there was evidence of knowledge gained in one context, informing another. If this was evident, orientations in Bracket II or even III were possible. Where students did not answer satisfactorily in all areas, connection-making or relational learning was assumed not to have occurred to any useful extent, and the orientation assumed to be below Bracket III. Where students had not grasped the procedural skills, Bracket I orientations were assumed. The models chosen were: a) i money and the purchase of stamps, as a practical model of whole numbers and decimals, and the operations of multiplication or division, and subtraction. Question How many 41c stamps you could buy for $3 and say what change you would get if any. (answer: 7 stamps, and 7 cents change) a) ii a similar abstract, symbolic problem, which suggests the use of school taught algorithms. Question Circle the answer to 2 / 0.4 (answer: 5 ) 2 0.2 0.16 1.6 8 5 0.5 a) iii a number line, which would draw on teaching of operations with such a model in the unit. Question Show how this division would look modelled on a number line. <-------------------------------------------------------------> a) iv the student was required to produce an everyday problem which could be represented by this symbolically represented code. 2 / 0.4 Question Write a real world problem that accurately reflects this mathematical sentence, 2 / 0.4 (a possible answer: How many lots of 40 cents on $2.00?) b) Difference, Ratio and Percentage in a context: This problem involved the use of some or all of the following: whole numbers and subtraction, proportion and scaling, fractions and percentage. The problem required three steps, finding the difference, the ratio and then the relevant percentage. Because of this, it was thought to be non- routine, and to require something more than compartmentalised thinking about mathematics. The third step, finding the percentage, might only have required the mechanical implementation of a procedure to achieve the percentage, similar to the keying in of values to a calculator. On the other hand, this step might have required the student to recognise the relationships between the numbers, and scale them down to eighths. This involves an adaptive process, more characteristic of orientations in Bracket II and III. Finally, when plotting the proportion of loaves left on a number line, the student needed to review the question, to check whether they had in fact labelled it with "loaves left" (12.5%) rather than "loaves sold"(87.5%). Lack of such reflection is considered characteristic of a Received Knowledge thinker. Question A bakery cooks 2000 loaves. Of these, 1750 were sold. What percentage of the loaves are left? Explain how you work this out. Results Question a) i 90% of all groups obtained the correct number of stamps. The few who did not were no more than one stamp out , in any case. Both groups were about equally capable of handling a practically based problem involving a relatively concrete model of decimals. a) ii In contrast to this rate of success, when the context changed to an abstract, symbolic problem, the two groups diverged. Half of the Repeats gave a wrong answer, with a decimal point in it, suggesting that they had not visualised either the quantities involved or the effect of division by a decimal. Lack of connection-making in those who failed to get both answers (a i and a ii ) right, was inferred in half of the Repeats and a quarter of the CM200's, Quality of learning: Bracket I a) iii In this question, the lack of connection - making was more obvious, particularly in the Repeats, none of whom managed it. 40% of CM200's were able to do so. a) iv 20% of Repeats and only 31% of CM200's were able to provide an everyday model of this problem. Nearly two years later, 45% of the CM300's could do this. Table 4 Question E. a Division involving Decimals(%)n = 15n = 77n = 20 Type of QuestionRepeats '90 CM200's '90CM300's 91i) Money Model: Buying Stamps$3.0 how many 41c?60 68 70 ii) Symbolic Representation2.0 / 0.4 =478375iii) Numberline Model0 38 55 iv) Real World Model203145Connection - making0 31 45 b) Loaves Difference found (2000 - 1750 =250) Most students managed this. Ratio made (250:2000 or some thing similar) Only 20% of Repeats managed this, and just over 70% of CM200's. By Third Year, with 85% did this successfully. Percentage found (Ratio X 100 or something similar) 13% of Repeats gave "1/8" as their answer, the correct proportion, but not the percentage asked for, "12.5%". 60% of Repeats answered, "25%", suggesting misrecalled procedural skills, a lack of experience in making ratios and in scaling down. 56% of CM200's achieved the correct percentage, and 13% , having used an algorithm, satisfactorily, to find a percentage, gave the one for "loaves sold" rather than "loaves left", suggesting an automatic mode of thinking, rather than a reflective one. 75% of Third Years managed this successfully. Table 5 Two-step Problem Solving involving Ratio (%)n = 15n = 77n = 20StepsRepeats '90CM200's '90CM300's 91Difference found878885Ratio made207185Fraction or % found1/8 or 12.5%1356751/4 or 25%601007/8 or 87.5%01310Connection making135675 Discussion The problem of deciding how to establish what a student's orientations on mathematical knowledge was, was more effectively done by interview, where interviewee's responses could be followed up. Use of a questionnaire to make comparisons between the mathematical orientations of the groups was partly successful. It has shown some differences between the groups, but across the groups, there appears to be a band of passing students both in Second and Third Year, who, like the Repeats, have Bracket I orientations. Inferences from attitudinal responses The attitudinal questions indicated that the Repeats were more pre-occupied with Difficulty, and that they did not associate mathematics with Creativity and Imagination. A substantial proportion of CM200's and CM300's responded similarly. The 40% or more in each group, associating Rote Learning with mathematics, may not have the same conception of this type of learning. Where these students were successful in doing mathematics, but rated Rote Learning highly, they may be making more useful connections between aspects of the mathematics and its uses, than the less successful students, while they are apparently rote learning, revising notes, practising skills and recalling algorithms for assessment. Without understanding more about the nature of this, it is not correct to use these results to say categorically that these students have Bracket I orientations. The emphasis upon Rote Learning, may say as much about what is expected in assessment as the students' orientations. The problem was expressed by a student from this cohort, evaluating her CM300 unit in a separate questionnaire, when she said that it was good that she had learnt procedural skills, but her difficulty was in knowing how to teach them effectively. She, and others like her, have recognised both the importance and the inadequacy of knowing the procedural skills, and are invested in developing contexts for teaching them. These students are asserting their views, and are aware that various aspects of their knowledge needs to be brought together. They have Bracket III orientations. The Repeats and CM200's were similar, in that about a third, found mathematics Boring. The conclusion from this is that even in Third Year, at least some of the time, for 40 % of students, mathematics is not challenging, engaging and Fun. The lack of interaction suggests at most, Bracket II orientations. The question on preferred types of instruction was one of the more useful questions, in providing indicators of orientation.. Half of the Repeats did not respond, as Bracket I thinkers, but wanted the opportunity to explore ways to solve a problem. This suggests that these students are more like Bracket II thinkers, and "invested in learning", probably other aspects of mathematics than procedural skilling. Creativity selected by 38% of CM200's is a useful cue to their more interactive relationship with mathematics, and the more advanced orientations of Bracket III. This seems to have decreased slightly, rather than increased by the Third Year. Interview data from students with Bracket III orientations needs to be gathered to explain this. The Repeats and CM200's did not differ noticeably, in their self-reporting on literacy skills, which suggests that self-efficacy with regard to this area of their work was similar. The Repeats' success rates in other areas of the course indicates that, as a group, they are not as weak in other areas, as they are in mathematics, but that they do not excel in them either. This suggests that their knowledge definitions and learning styles, do not differ much across other subject areas, as was the case with Alison and Adam. Inferences from mathematical responses As expected, those failing CM100, were grouped on overall results, in Bracket I, the majority of CM200's were in Bracket II, and a few showed Bracket III orientations. The Repeats' lack of success in procedural skills was known, but a surprisingly small proportion of the CM200's have made connections between different aspects of division of decimals, concrete and abstract. The connection making requiring setting a problem in a practical context, suggests that 20% Repeats have elements of Bracket II orientations. The lack of connection making in CM200's is noteworthy, with only 31% managing this. In Third Year, this has risen to 45%. The results indicate that the Repeats, and many of those passing the unit, have responded similarly to these questions, and that they probably have Bracket I orientations. About a third of students passing the unit appear to have the more interactive Bracket III orientations. Conclusion The recognition that Bracket I orientations might hinder relational understanding in mathematics, has provided me with grounds for reviewing how mathematics is contextualised in my classes. It is worth looking beyond the immediate target of competence, to a system where shifting students' orientations becomes a priority. The need for competence in procedural skills does not necessarily challenge students to relational thinking, and pages of abstract skills questions may signal to students that procedural skills is what mathematics educators value most in mathematics. The problem is that a student may regard mathematics as primarily a set of procedural skills, and consequently, think of "drill and practice" as the way to learn mathematics, virtually limiting his/her thinking about mathematics. Challenging students to shift their orientations, is something that should be addressed. One effective way to make this challenge, is to use students' expressed interest in learning and teaching, to raise debate on how children learn, so that students become aware of their own "Voice" and contribution to make to mathematical thinking. References Belenky, M. F., Clinchy, B. M., Goldberger, N. R. and Tarule, J. M. , 1986. Women's Ways of Knowing the Development of Self, Voice anfd Mind. Basic Books Inc., Publishers, New York. pp. 20 - 120. Biggs, J. B. and Telfer, R. , (1987).The Process of Learning, 2nd Edition. Prentice-Hall of Australia, Ltd. , Sydney. Biggs, J. B. and Collis, K. F. , (1982). 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