Thinking about Mathematical Thinking - heterogeneity and its social justice implications

Leone Burton

University of Birmingham, U.K.



School curricula persistently demonstrate an unresolved conflict between developing the mathematicians of tomorrow, equipping them with the knowledge and skills deemed necessary by university mathematicians, and providing an entry for everyone into mathematical culture adequate enough to meet the demands of society. Most societies fall between the two and, certainly in the UK, both industrialists and mathematicians have been articulate in their claims as to the 'failure' of school maths. From these it would appear that young people are neither knowledgeable and skilled enough to cope with the mathematical demands in the work place, nor ready to pursue further study at university. But the studies which were done for the Cockcroft Report, 1982, (particularly the Bath and the Nottingham studies) to some degree contradicted this in demonstrating that workplace-based mathematics was not the same as school-based mathematics and that employees developed an on-the-job facility with workplace-based mathematics (also borne out by the work of Mary Harris, see Harris, 1990). Nonetheless, many adults are quick to identify school mathematics as an area of failure for them. As far as degree level work is concerned, students have been voting against mathematics with their feet for some time and alongside this must be put a persistent critical theme emerging from studies done in universities listening to the voices of students (see, for example, Crawford et al., 1994) and experiences of attempting to interest mathematicians in innovative styles of teaching and learning (see, for example, Burton & Haines, 1997).

I believe that much of this confusion is exacerbated by a teaching obsession with content and, at the same time, an ignoring of the impact of epistemology and pedagogy on the mathematical experiences of learners. To take this further, I generated an epistemological model to describe the process of coming to know mathematics (Burton, 1995) and I have recently undertaken a study of research mathematicians to ascertain to what degree my model matches how they describe their own activities. The model understands coming to know, in mathematics, in terms of:

In 1997, I undertook a study of 35 women and 35 men in career positions as mathematicians in universities in England, Scotland, Northern Ireland and the Republic of Ireland. The female participants in the study were found by invitation and through snowballing (one person involving another) and each female was asked to find a male "partner" preferably in the same institution, to pair her for the purposes of the study. All 70 participants were individually interviewed, 64 face-to-face the remainder by telephone. The interviews ran from an hour to the longest which was two and a half hours. Most were between one and a quarter and one and a half hours.

My purpose in doing this study was to try and find out how mathematicians understand their researching practices in order to try to map the disjunction between mathematicians as learners, and mathematicians as teachers. I believed that the practices of mathematicians might be closer to the learning practices that many of us have been promoting in formal mathematics education for a very long time against a critical backdrop of some powerful university mathematicians.

I am convinced that the classroom experiences of mathematics learners are a result of a complex relationship between epistemology, pedagogy and the discipline of mathematics. I do not see any evidence that teachers have clarity of vision on any one of these three even though policy makers attempt to provide such clarity at least on the third. I believe that we damage both the learners, the discipline and ourselves as teachers when we fail to take this complexity into account by operating as if only one is important (usually the mathematics itself) and do not recognise that the mathematics itself is permeated by our epistemological and pedagogical perspectives. Hence the assertion of an epistemological model which attempts to include the who, and where with the what of the mathematics as well as invoking the senses with the cognitions. It was my belief that research mathematicians would use these categories in speaking about their working practices and that it might then be possible to relate them to mathematics teaching and learning experiences.



Homogeneity was not revealed by my participants when they spoke about mathematics, how they understand mathematics, how they think about mathematics, how they work in mathematics. Many public stereotypes were overthrown.

There is not ONE:

Only when it came to discussing ways of experiencing the world of mathematics, and the impact of sex, 'race' and class, was there any singularity of experience. A world of power was revealed and that power had all the overtones demonstrated in the literature (see, for example, Acker & Feuerverger, 1996, Aisenberg & Harrington, 1988, Morley & Walsh, 1996, Lie & O'Leary, 1990 and Seymour & Hewitt, 1997). It was acted out in their research and career patterns, in research supervision, and in the women's reporting of aspects of:


The inner battle that professional women fight [which] is particularly difficult because its terms are rarely clear. Unpredictably, women will encounter trouble that looks like a knot of circumstances that they seek to pull loose, not recognizing at its center - except possibly in retrospect - a profound conflict concerning their own identities.
(Aisenberg & Harrington, 1988: 8)
The role of Oxbridge in the training of future mathematicians was a particularly powerful factor in the allocation of access to influence.

Social Justice Implications in Classrooms

The celebration of heterogeneity is, I believe, something about which to be exceedingly joyous. For me, this study has provided evidence to support a mathematics, and styles of learning and teaching the subject, which emphasises humanity, vision, creativity. It has also provided evidence of where and why the teaching of mathematics has failed many generations of learners. To cling tightly to one 'true' mathematical path might provide a sense of security but it is also stifling and unreflective of the world we all experience. To open mathematics to multiple interpretations, multiple possibilities, provides opportunities for learners to experience what the mathematicians whom I interviewed described: "When I think I know, I feel quite euphoric. So I go out and enjoy the happiness. Without going back and thinking about whether it was right or not, but enjoy the happiness. When I discover something, I just enjoy the feeling."

"You can do all these interesting and exciting things without having to go out and do things with them. Whether what you are thinking about is new, research, known things or not, for you it is all new. When you understand a new proof, it becomes your own. Internally, it is as though you did it."

"It is just fun."

In the presentation, I will explore the implications of this approach for potential classroom confusion! In particular, I will look at:

Re-writing the mathematical experience

What are the powerful teacher questions to ask?


Acker, S. & Feuerverger, G. (1996) 'Doing Good and Feeling Bad: the work of women university teachers', Cambridge Journal of Education, 26(3) 401-422.

Aisenberg, N. & Harrington, M. (19880) Women of Academe: Outsiders in the Sacred Grove, Amherst: Uni. Of Massachusetts Press.

Burton, L. (1995) 'Moving Towards a Feminist Epistemology of Mathematics', Educational Studies in Mathematics, 28, 275-291.

Burton, L. & Haines, C. (1997) 'Innovation in Teaching and Assessing Mathematics at University Level', Teaching in Higher Education, 2(3) 273-293.

Cockcroft, W.H. (1982) Mathematics Counts, London: Her Majesty's Stationery Office.

Crawford, K., Gordon, S., Nicholas, J & Prosser, M. (1994) 'Conceptions of mathematics and how it is learned: the perspectives of students entering university', Learning and Instruction, 5, pps 331-345.

Harris, M. (1990) (Ed.) School, Mathematics and Work, London: Falmer Press.

Lie, S.S. & O'Leary, V.E. (1990) Storming the Tower: Women in the Academic World, London: Kogan Page.

Morley, L. & Walsh, V. (Eds.) (1996) Breaking Boundaries: Women in Higher Education, London: Taylor & Francis.

Seymour, E. & Hewitt, N.M. (1997) Talking about Leaving: Why Undergraduates Leave the Sciences. Oxford: Westview Press.