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Effects of past experience on future mathematics learning: Consequences for teacher education

Eugene Kaminski
Faculty of Education
Australian Catholic University
Initial investigations into experiences in, views of and attitudes to mathematics and its learning were conducted with six second year student teachers through semi-structured, individual interviews. It was considered that a greater understanding of these non-cognitive dimensions of student teachers' mathematics learning would assist with later structuring of mathematics learning experiences and with contextualising student teachers' responses in tutorials. Planning for future mathematics learning experiences would need to incorporate opportunities for student teachers to engage in (a) reflective practices to examine their beliefs in, and attitudes to mathematics, (b) active mathematics knowledge construction and (c) explaining and communicating their mathematical thinking.


The need for understanding and successful use of mathematics has received constant recognition, attention and emphasis. This has been evident in varied reports, for example, 'Curriculum and Evaluation Standards for School Mathematics' (National Council of Teachers of Mathematics, 1989), 'National Statement on Mathematics for Australian Schools' (Australian Education Council, 1990) and 'Shaping the Future: Review of the Queensland School Curriculum' (Wiltshire, 1994). Indeed, issues of numeracy and student performance in mathematics have been consistently raised and debated with large scale studies providing data for interpreting such aspects nationally and internationally (NAEP, 1988; Third International Study in Mathematics and Science, 1996 ; US Department of Education and National Centre for Educational Statistics, 1996).

However, understanding of mathematics incorporates experiential and affective as well as cognitive dimensions in learning, an important interrelationship recognised in the previously mentioned surveys. Not only are attitudes to mathematics important in learning endeavours in this discipline (Grouws, 1992; Leder, 1987, 1993; McLeod, 1992; Schuck, 1996) but for student teachers the existence of a direct relationship between mathematics beliefs and mathematics teaching practice has been posited (Brown & Borko, 1992; Brown, Cooney & Jones, 1990; Thompson, 1992).

To interact productively with student teachers during their engagements with preservice mathematics education units and in their development as teachers of mathematics, it would be beneficial to gains insights into their past experiences in mathematics. Unfortunately, research has reported little student teacher engagement in making sense of their mathematical experiences (Cobb, 1988) and an over-reliance on rote learning approaches (Hiebert, 1988). The transmission model of teaching (Tobin & Fraser, 1988) appears to have been in prominence even though constructivist approaches to learning have had strong advocates (Cobb, Yackel & Wood, 1992; Steffe, 1988).

Discussions with student teachers provide opportunities for exploring their views of, attitudes to, and beliefs in, mathematics which are challenged during their teacher education courses. The importance of monitoring and shifting student teachers attitudes to mathematics from negative ones to more favourable ones has been reported by Relich, Way and Martin (1994) and Bobis and Cusworth (1997). Indeed, it has also been stated that beliefs play a critical role in how teachers teach (Pajares, 1992; Perry, Howard & Conroy, 1996).

But how do student teachers view or describe mathematics? What attitudes do they currently have to mathematics and what factors may have influenced these? What mathematical learning experiences have they had and what was interpreted and understood from these?

PURPOSE OF THE STUDY

This study explored the mathematical background experiences, views and attitudes of six student teachers prior to the commencement of a semester. This was done with a view that data thus gained may not only be useful in later structuring of mathematical teaching/learning experiences, but may also provide a relevant context for the researcher to appreciate student teachers' diverse responses and needs during the semester's work in a mathematics education unit.

METHOD

Sample, instrumentation and process

The research approach taken in this study was chosen to maximise the opportunity of interacting with each student and thus gaining a better insight into their experiences, responses and views. Each student completed a set of mathematical tasks prior to being interviewed. However, the analyses and findings of those tasks were to be reported at a later date and only findings from the interviews will be included in this report.

The set of selected interview questions was developed to explore student teachers' descriptions of mathematics, sense-making in mathematics, enjoyment of mathematics, use of mathematics, and the part attitudes and beliefs play in the learning of mathematics. The questions not only sought responses on specific issues but permitted the pursuit of interesting related aspects in student teacher responses, that is, semi-structured interviews were supplemented with probing questions which were not necessarily the same for each participant.

Six second year student teachers were approached and agreed to participate in semi-structured interviews - four female, (Penny, Jacqui, Joanne and Madie) and two male (Colin and Wendal). Madie and Wendal were mature age students. The six student teachers' mathematics results in their first mathematics education unit were categorised as 'excellent' (80% and greater), 'very good' (79% - 65%), and 'satisfactory' (64% - 50%). Thus, Penny results attained an 'excellent' rating, Joanne, Jacqui, and Colin 'very good,' and Wendal and Madie 'satisfactory'. It was anticipated that this selection, while not designed as representative, would provide a sufficiently broad student teacher perspective on issues raised and represent a diverse set of background experiences. These included recent Year 12 students, mature aged students, male/female students, and those in three broad levels of recent mathematical achievement. Prior to the commencement of Semester 1, the six student teachers were interviewed individually and at each session the author explored a set of eleven questions (see appendix). All interviews were audiotaped, transcribed and returned to student teachers for validation. Each interview lasted approximately forty minutes.

RESULTS

Examination of data from student teachers' responses suggested a presentation under three themes:

Student teachers' views of mathematics

From the interviews, it was evident that student teachers' descriptions of mathematics focused on functional and utilitarian perspectives, for example, 'using it in everyday life' and 'something you can use not something you can know and not use'. All student teachers indicated the need for mathematics to be relevant and oriented to real life situations. High school mathematics was generally seen as too abstract, not actively generated by learners as it was just straight out of a book, and with communication of mathematical thinking apparently occurring infrequently. No student teachers could recall occasions where they were involved with activities to create their own mathematics. While mathematics was seen as more than numbers and algebra, it was still perceived by most as generally consisting of discrete, compartmentalised areas (fractions, geometry, graphs). Such views reflect those reported in research (Herscovics & Bergeron, 1986; Lampert, 1986). In one divergent view however, Penny indicated the need to present mathematics in relation to other subjects:
Maths flows into everything and the other subjects flow equally, you just have to find ways in which they do and if you're teaching in separate areas I don't believe that children are going to learn as good as they could, whereas if you taught in a holistic way and things make sense, it's like a flowchart. If you look at things in basic, isolated things, they make sense by themselves but they won't make sense together.
Additional dimensions of participants' views on mathematics were also revealed. Three student teachers commented on the emphasis in mathematics on using rules and formulas to find the 'right' answer, while relying mostly on rote approaches, for example, 'rote learn, don't learn for understanding, just rote learn, short term learning' for passing tests and examinations. Joanne, one of the three students, recounted that:
All the times I've done mathematics I haven't really understood it so I've only ever relied on learning my rules and knowing my symbols to get through. You know what I mean? When I did advanced maths in grade 9 and 10 I never understood it and I ended up, I just got a VHA but it was only because I'd learnt my rules off by heart and I knew what my symbols meant but I didn't understand why the rules were there, I just knew I had to know that rule. You just learn them because you have to, to get the right answer.
The experiences of the three student teachers are consonant with research results in which it was reported that there had been little engagement by student teachers in making sense of mathematical experiences (Cobb, 1988; Schoenfeld, 1989; Steffe, 1988) in reflection on mathematical thinking (Gagatsis & Petronis, 1990) or in communicating mathematical thinking (Greeno, 1989). Past approaches focusing on completing text book mathematical exercises, frequent use of rote learning approaches and an emphasis on obtaining correct answers and passing examinations may have contributed to student teachers having few opportunities to construct mathematical knowledge and to engage in making sense of mathematical experiences. Such perspectives on symbol manipulation, rote learning and memorisation in mathematics reflect the findings of Hiebert (1988) and Van Lehen (1986).

What has emerged from exploration of student teachers' views of mathematics has been their commonly held perspective that finding exact, correct solutions in mathematics is of central importance and that this is mostly achieved by manipulation of symbols on paper according to memorised rules. Student teachers indicated, in their discussions, that none had been involved in organised or structured development of reflective practice or the use of reflective strategies in their past experiences in mathematics. The researcher observed an absence of connection between student teachers' mathematical understandings and use of rules. This observation, one consonant with the findings of Dossey, Mullis, Lindquist and Chambers (1988), and Hiebert (1988) occurred during exploration of mathematics exercises with the student teachers following their interviews but these results will be reported elsewhere. It appeared that practice in, and routinisation of, symbol manipulation rules persisted predominantly for the production of correctly written answers. Case (1986) noted that a disadvantage with such a practice was that students, particularly weaker ones, are more likely to permit symbols to become separated from their quantity and contextual referents. Unless addressed, students will continue to view mathematics, as Romberg and Carpenter (1986) reported, in terms of a static bound discipline, fragmented and divorced from reality and inquiry.

Thus, the six student teachers' views of mathematics generally focussed on the use and relevance of mathematics to everyday situations, and were characterised more by its parts, for example, algebra, arithmetic and geometry. An important aspect which appeared to contribute to the six student teachers' views of mathematics was that of their past experiences in mathematics.

These indicated:

Attitudes towards mathematics

Varying levels of, and reasons for, the enjoyment of mathematics were expressed by the student teachers. One view, 'if you have a teacher who is interesting, brings in a bit of humour, you know people will take interest and it does help', indicated enjoyment was related to the teacher's approach and ability to interest and motivate pupils. Humour was seen as a teacher characteristic, which provided greater levels of interest and enjoyment in students' learning. Indeed, the importance of the effects of affective factors on mathematics learning has been clearly acknowledged (McLeod, 1992; Leder, 1993).

Madie, however, indicated that enjoyment is not necessarily related to confidence in mathematics:

I enjoy it. It's just that I don't feel confident about it and I hate not feeling confident about something. You know that's just me. I hate that, not knowing it and I feel stupid or something but I'm not. I enjoy it ... I got a buzz from trying to get it right.
For Penny, enjoyment was directly related to success in mathematics and could be most rewarding when insights occurred unexpectedly:
Ah, I'll never get this, so I kept going and going and got it eventually. It just clicked, it was just - I thought, how could I have missed that and I was so glad that I got it, and I thought, yes, I've finally mastered this and when I can master something I can really enjoy it. I don't push myself but I have to understand everything but I always want to find the answer to something if I can and if I can get the answer to it I really enjoy it.
However, as McLeod (1992) highlighted, the existence of a link between attitude and achievement does not necessarily indicate improvements in the one will automatically produce improvements in the other. Rather, they interact with each other in complex ways.

For another student, though, negative feelings and fear of mathematics had resulted in her not attempting to take risks mathematically, by 'sticking to the most simple things I know', even when selecting mathematics options in Years 11 and 12.

Two student teachers had begun to consider the relationship of teaching approaches and non-cognitive dimensions, with one student teacher noting, 'the things we enjoy we do well in'. The 'mechanics' of teaching was still uppermost in student teachers' thinki ng when planning children's learning experiences, indicating that student teachers were still operating mostly at a technical rational level (Van Manen, 1977). If such teaching experiences were to be successful, then student teachers indicated attitudinal aspects needed to be addressed. The importance of teacher attitudes affecting instructional practices which affect student attitudes and achievements has been noted by Mayers (1994). Linda reflected on the possible relationship of enjoyment and understanding when she stated, 'This made me realise that maths, if approached in the right way, can be fun for children and through this enjoyment, children can learn quicker and with deeper understanding'.

It appeared though that no student teachers had reflected to any significant degree on their beliefs in mathematics. Four indicated that they were aware of some of their beliefs but there was no evidence of student teachers having previously been required to reflect on them and justify or defend them. All recognised, and many experienced, the detrimental impact of negative attitudes on learning, for example, 'if you've got a bad attitude towards maths, you are very wary of it and you don't take risks'.

With mature age students, negative feelings about mathematics appeared to be associated with lack of successful results. For Wendal, one of the mature aged students, negative feelings in mathematics were related to a lack of results, which appeared to be attributed to lack of effort or intensity rather than difficulties with mathematical understanding:

I used to have negative feelings about maths basically because I would never get good results but I guess I didn't try a lot.
In recognising the importance of positive attitudes to learning, interviewed students suggested the following as facilitating the development of student teachers' positive attitude towards mathematics: Interestingly, a number of the teacher characteristics outlined above closely mirror those noted by Jarvis (1987) in promoting more effective student learning. The benefits of attaining more positive attitudes to mathematics were seen by Joanne in greater levels of confidence and greater risk taking:
It's like with anything, if you are confident in dealing with something you'll take a risk, like if you are not confident you're sort of more likely to just stick with what you know and just do what you have to.
Whereas attitude was seen by one student teacher as related to self esteem, for example, 'you've got to have self esteem ... attitude, I definitely agree, plays a major role', it was also perceived by Madie that a teacher's own negative attitudes to mathematics can be detected by pupils and can impact negatively on their learning and enjoyment of mathematics:
I am beginning to realise the importance of covert messages from teachers influencing children's learning. If I hate maths then I'm going to pass that on to children so I am really trying to gain a better, more positive attitude to maths.
Not only may children detect teacher attitudes to mathematics but, as Ernest reported (1988), teachers' attitude to the subject may affect their attitudes to the teaching of the subject which impacts then on the culture of the classroom.

The interviewed student teachers' negative attitudes to mathematics appeared influential in the lack of development of themselves as risk takers in mathematics and in their past selections of mathematical options in secondary education. While some of the six student teachers had experienced enjoyment and interest in mathematics, this did not necessarily translate to confidence or success in mathematics. However:

Student teachers as a learners of mathematics

Responses to the ways students best learned and understood mathematics varied widely but for two students, an active, hands on approach using manipulative materials was felt to be effective. For one of those students, Joanne, an auditory approach was also valued.
I find it strange because I learn from physically doing things but I also sort of learn if I really listen hard and concentrate ... if I concentrate and listen I understand it well and I won't forget it, so I'm a listener and a doer.
For two other students, the traditional, direct teaching approach of 'teacher instruction first then having to apply heaps and heaps of questions' was preferred. Both mature aged student teachers were more in favour of being taught by an approach where the 'basics' and 'formulas' had to be known, where student teachers were exposed to a routine practice of 'step by step processes' with examples and where 'it was drummed in and I know it's rote learning but at least you knew'. Such a description can be associated with a transmission model of teaching (Tobin & Fraser, 1988) which has attracted criticism particularly by proponents of a constructivist approach (Cobb, 1988; Steffe, 1988; von Glasersfeld, 1987; Wang, Haertel & Walburg, 1993). Indeed all but one student teacher indicated a past emphasis in their schooling on routine practice of standard computational procedures.

The level of past results, one's own level of intellect, relevance, attitudinal factors, for example, confidence and presentation of lessons, were seen by student teachers as important influences on the level of difficulty they found in mathematics. However, Penny presented a perspective in which she felt that learners might experience difficulties in mathematics due the expectation of a correct answer when none may be forthcoming:

Probably because they [students] feel pressured to give an answer that they've got to get and that they know they can't get that answer, or they don't think they can get that answer - because in another subject like language - there is no right answer ... I don't know if it is because in maths and science there are rules and you have to follow rules.
In outlining approaches which may be helpful in overcoming difficulties experienced in mathematics, four student teachers favoured an initial approach to peers and listening to an alternative explanation. Such an approach reinforces the notion of communication being an essential element of mathematics activity (Cobb, Yakel & Wood, 1988) and embraces aspects of a cognitive social learning perspective (Vygotsky, 1978).

Only one student teacher noted the locus of responsibility for learning lay with the student. On the other hand, Wendal, a mature aged student, saw the teacher as the font of knowledge:

Teachers should be the conveyers of knowledge. In a lot of instances the teacher is the first thing that people have ever had as their introduction to maths. First impressions are pretty vital. The teacher has to be able to convey the knowledge and find the pupil who has a lack of understanding.
The role of the teacher as one 'knowing the correct answers', as arbiter on correctness of solutions was proffered by two other student teachers, suggesting some were operating from a basic dualism position as articulated by Perry (1981). The need for teachers to have a strong content base of knowledge was also sugg ested: 'would have to be one step ahead of you [the learner] all the time, heaps ahead of you.' There was no evidence to suggest student teachers believed that acquiring a strong content knowledge base would be useful or valuable in making sense of mathematical experiences. However, the importance of acquiring such knowledge and its influence in assisting student learning and shaping teacher practice have been outlined in research (Ball & McDiarmid, 1990).

Views on the preferred ways of being taught mathematics were consonant with those expressed by student teachers during the exploration of the ways they best learn and understand mathematics. One student teacher, who previously recognised the responsibility students had for their own learning, noted that understanding of processes was an important aspect of learning mathematics:

I do like the holistic process best of all and then the onus if put back onto us for our own learning to see how we'd go with it. If any queries or problems arise to be able to get them clarified by the lecturer.
Preference was expressed by four of the student teachers, for a relaxed working environment where small group work of two or three people was accommodated but not to the exclusion of quiet individual work, for example, `when you need concentration ... I don't want anyone bothering me at all'. Benefits of working in groups were recognised by five of the student teachers (for example, clarifying and consolidating understanding and actualisation of knowledge), which articulated with benefits outlined by Pontecorvo (1985). Student teacher development of reflection, that is reflection on mathematical thinking and solution strategies, had not received much emphasis during their past learning experiences in mathematics. Indeed, one student teacher indicated the opportunity for reflection had not been provided in past mathematics learning experiences, which seemed to suggest that student teacher reflection was the responsibility of the teacher, a rather curious perspective. This situation, however, may have been due to student teachers confusing reflection with revision. In Madie's case, reflection appeared to be the simple review and repetition of steps and procedures previously exercised:
I went back and refreshed everything so I thought to myself now if I do this straight after every single thing I'm going to do even better. So I've learnt that.
For some students, understanding of, and distinction between, reflection and revision may have lacked clarity, for Colin, a clearer notion had emerged:
No, no, it betters your learning if you sort of stand back and have a look at it. Once again it's trying to sort of understand the whole thing. I think it's good to sit back and have a look at it.
Exploration as to whether student teachers continue to use school taught methods and algorithms or personally modified, non-standard ones when operating with mathematics in everyday social situations provided a student response generally in favour of the former. It was noted that the mature aged student teachers and those whose previous mathematics education results were in the category of 'satisfactory' continued to use, possibly uncritically and unreflectively the standard school taught algorithms 'it's been built into me over the years'; 'when I learned it at school it was just given to me.' Research (Sowder & Markovits, 1989) indicated that students who are proficient in the use of school taught algorithms are resistant to changing to 'invented' ones. Only in Jacqi's case was there any consideration of and modification to school taught procedures: 'especially if you don't understand ... if you can just use your own way sometimes it helps'.

Approaches to mathematics learning and assessment experienced by student teachers do not appear to have fostered 'deep' learning (Biggs, 1988), possibly due to students' use of rote memorisation. Such past student teacher learning approaches may have mitigated against holistic understanding and development of a conceptual whole of mathematical topics being studied. Indeed an emphasis on application of rules and attainment of correct responses, without an appropriate conceptual framework can result in what Skemp (1989) described as instrumental understanding. However, three student teachers indicated a greater desire to develop more mathematical thinking and understanding of solution processes, for example, 'I want to get the answer so that's my aim but the process is also really important'. Students whose general or frequent exposure to teaching was that of the transmission model of teaching (Tobin & Fraser, 1988), or teaching by imposition (Bishop, 1985), have little opportunity to be active agents in developing contextually based shared meanings in mathematics.

Thus, the six student teachers' experiences as learners of mathematics indicated:

CONCLUSION

As the semi-structured interviews were conducted with only six student teachers, the researcher acknowledges the need for caution when interpreting the data and results. With this consideration in mind, however a number of important aspects have emerged from the analysis of data from the six student teacher interviews.

The results of the study provided the researcher with a deeper appreciation of the diversity of student teachers' views, prior mathematical experiences and learning contexts. This will have implications for the researcher's development of future mathematics activities and learning approaches for student teachers.

To provide a balance and contrast to the common past mathematics experiences of the student teachers, the researcher will focus upon and emphasise socio-cognitive and constructivist approaches to learning. Student teachers will be engaged in active knowledge construction, in making sense of mathematical experiences and in communicating their mathematical thinking.

Opportunities will be planned for exploring and developing non-standard mathematical procedures and for student teachers to engage in reflective practices to examine their mathematical learning, beliefs, assumptions and attitudes. Encouragement will be provided, directly and through supportive structured group activities, for student teachers to become 'risk takers' in mathematics and participate in shared meaning making, rather to see the lecturer as a dispenser of knowledge and arbiter on mathematical matters.

However, what appear as further important areas of investigation are student teachers' beliefs in mathematics and mathematics pedagogy, factors that influence such beliefs and how student teachers experience opportunities to examine, justify, defend and, where appropriate, alter their beliefs. The opportunity for changing beliefs is essential for teacher development (Lappan & Theule-Lubienski, 1994).

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Bobis, J. & Cusworth, R. (1997). Monitoring change in attitude of preservice teachers towards mathematics and technology: A longitudinal study. Proceedings of the Twentieth MERGA Annual Conference, Aotearoa.

Brown, C. & Borko, H. (1992). Becoming a mathematics teacher. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 292-302). New York: Macmillan.

Brown, S., Cooney, T. & Jones, D. (1990). Mathematics teacher education. In W. Houston, M. Haberman & J. Sikula (Eds), Handbook of research on teacher education (pp. 639-657). New York: Macmillan.

Case, R. (1986). The new stage theories in intellectual development: Why we need them; what they assert. In M. Perlmutter (Ed.), Perspectives on intellectual development (pp. 92-102). Hillsdale, New Jersey Erlbaum.

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Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2-33.

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Gagatsis, A. & Petronis, T. (1990). Using geometrical models in a process of reflective thinking in learning and teaching mathematics. Educational Studies in Mathematics, 21(1), 29-54.

Greeno, J. (1989). A perspective on thinking. American Psychologist, 44, 134-141.

Grouws, D. (Ed.) (1992). Handbook of research on mathematics teaching and learning. New York: Macmillan.

Herscovics, N. & Bergeron, J. (1984). A constructivist vs formalist approach in the teaching of mathematics. Proceedings of the Eighth International Conference on the Psychology of Mathematics Education. Sydney.

Hiebert, J. (1988). A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.

Jarvis, P. (1987). Adult learning in the social context. New York: Croom Helm.

Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3, 305-342.

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Perry, B., Howard, P. & Conroy, J. (1996). K-6 teacher beliefs about the teaching and learning of mathematics. In P. Clarkson (Ed.), Technology in mathematics education (pp. 453-460). Melbourne: Deakin University Press.

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Wiltshire, K. (Chair). (1994). Shaping the future: Review of the Queensland school curriculum. Brisbane: Government Printer.

APPENDIX

Student Teacher Interview Questions

  1. Maths is seen by some people as just use of rules and symbols. Do you agree with such a description ?

    Some students think maths is about finding the one right answer to situations or problems. What is your reaction to such thoughts?

  2. If someone was to ask you to describe maths, how would you describe it?

  3. In what ways has maths ever made sense to you?

  4. Do you feel maths situations and real life situations are in any way related?

  5. Do you feel maths is related to other subjects you do or have done ?

  6. Some people say they have found learning maths enjoyable. What has been your experience ?

  7. How did you feel about the subject of maths at school?

  8. Do you think maths games have any part to play in learning maths?

  9. Do you use the maths you learnt in school in a different way now e.g. when you go shopping or calculating costs?

  10. What part do you think one's attitudes and beliefs play in the learning of maths?

Author details: Dr Eugene Kaminski
School of Education (Queensland)
Australian Catholic University
53 Prospect Road
Mitchelton Qld 4053
Phone: 07 3855 7155 Fax: 07 3855 7247 Email: e.kaminski@mcauley.acu.edu.au

Please cite as: Kaminski, E. (1999). Effects of past experience on future mathematics learning: Consequences for teacher education. Queensland Journal of Educational Research, 15(2), 173-191. http://education.curtin.edu.au/iier/qjer/qjer15/kaminski.html


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