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:

• its person- and cultural/social-relatedness i.e. it locates knowing rather than regarding it as 'objective' and free of influence from the individual or their society;
• the aesthetics of mathematical thinking it invokes i.e. how coming to know and knowing is described in terms of feelings;
• its nurturing of intuition and insight i.e. how the pathway to knowing is understood;
• its recognition and celebration of different approaches particularly in styles of thinking i.e. how the knowing is achieved;
• the globality of its applications i.e. not only applicable maths but what I have come to call the connectivities both within and across mathematics.

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.

Heterogeneity

There is not ONE:

• mathematics - depending upon the research area, it was differently understood as

• a 'rigorous' proof process;
• empirical;
• uncertain;

• way of understanding mathematics

• as well as the invention/discovery split, I found socio-culturalists and those who understand mathematics as a language;
• role of intuition/insight - there were those who denied and those who affirmed the importance of intuition and those who wanted to talk about the meaning of the different terms;

• way of thinking about mathematics

• three, not the conventional two, different thinking styles were described;

• way of working in mathematics

• individual/co-operative/collaborative were all described with an emphasis on the latter two;

Social Justice Implications in Classrooms

Challenging teachers' assumptions - doing it "my" way

- respecting the many different routes

- giving time to making mathematical maps

The celebration of differences - pedagogy

- meaning making

- links between informality and formality

Re-writing the mathematical experience

What about learning time? What should be acceptable and how long is given?

In the light of curriculum constraints, what is possible? What is feasible?

Can we re-write the mathematical experience? What is your emphasis?

Bibliography