In a recent study, I interviewed seventy research mathematicians about their epistemologies. (For more details see, for example, Burton, forthcoming.) I was interested in the process of their coming to know mathematics, which meant that part of our conversation engaged with their thoughts on mathematics itself. I was trying to find how good a match to the practices of mathematicians and their understanding of those practices was an epistemological model that I had developed theoretically (see Burton, 1995). The model described knowing in mathematics in terms of five categories, its person- and cultural/social-relatedness, the aesthetics it invokes, its nurturing of intuition and insight, its recognition and celebration of different approaches and its connectivities. In other words, I was conjecturing that mathematical knowing is a product of people and societies, that it is hetero- not homogeneous, that it is inter-dependent with feelings especially those attached to aesthetics, that it is intuitive and that it inter-connects in networks. This differentiates my focus from a knowledge-based enquiry.
The model proved to be remarkably robust when matched with how the seventy mathematicians talked about mathematics. But in analysing the very large data base, it became clear to me that I had to distinguish between the mathematicians' beliefs, that is what they think mathematics is, their practices, that is how they approach their research, and the seductive nature of their social positioning into what everybody knows. Finally, although I had not set out to discuss teaching and learning with them, inevitably there were aspects of our conversations which provided data on how they understand their roles as teachers and as learners and the data contain a lot of material which can be interpreted (with all the caveats necessary to that) to throw light on the links between epistemology and pedagogy.
I began the study with a conjecture that the socio-cultural system producing mathematicians is far stronger than differences in sex. I expected to find a high degree of convergence in the beliefs of the mathematicians I interviewed about the nature of the discipline. In fact, what I found was both. Great heterogeneity was disclosed in the many different beliefs about coming to know mathematics, including some, a minority of about 10%, who did not adopt an objectivist, positivist stance. Sex did not seem to feature as relevant to these differences. At the same time, whether espousing Platonism or formalism, there was a uniform philosophical commitment to an absolutist view of mathematical knowledge as fixed, certain and a-personal, a view of mathematics without the 's'. So, within this philosophical homogeneity, the mathematicians to whom I spoke adopted a breadth and diversity of epistemological positions. Some female participants said:
The existence of the kind of diversity represented in the above quotes leads me to expect that a range of different beliefs exists across the discipline as a whole. As pointed out above, these beliefs are held within a philosophical position which Sal Restivo has called "transcendental" although, at the same time, he points out that even the transcendental is a human construction:
As Philip Davis and Reuben Hersh (1983) have so beautifully demonstrated, mathematicians are past experts at shifting their positions:
However, this mental flexibility also enables a tight hold to be kept on the absolute nature of "truth" and the power and legitimacy of the objects of mathematics, whether "discovered" Platonistically, or derived through game playing of a formalist kind.
Diversity was also exemplified on the other dimensions of the model so whether or not a mathematician held absolutist views about the discipline, their positionings on the role of aesthetics and intuition were likely to be equally divergently spaced along a continuum from unimportant to crucial. Only when it came to speaking of connectivities within and without mathematics was there almost universal agreement as to their importance, whether or not their current work could be so connected.
However, diversity does not appear to affect the wider, societal view on mathematics, most especially the view that influences how mathematics is understood in schools. Indeed, the epistemological diversity to which I am drawing attention co-exists with a philosophical uniformity, a strong commitment to The Big Picture in mathematics together with an image of an individual's contribution of one more piece to the mathematical jigsaw puzzle. (Newell & Swan, 1999 point out the connection between the jigsaw metaphor and a "traditional" approach which they describe in positivist terms. They call for innovation through knowledge sharing and dispersal. (See below). Nonetheless, it is very encouraging that the uniformity which so often features in the public pronouncements of mathematicians, or of non-mathematicians, about mathematics, is not matched by the voices of my participants as they describe their experiences. Furthermore, the spread of positions raises, for me, the importance of making dialogue about the discipline a necessary feature of its learning. I will return to this below.
Yet another contradiction appeared when the mathematicians started to talk about how they engage in research. What one might call the Andrew Wiles' effect, the mathematician locked away in an attic room working alone over a long time period, was not the practice the majority described. Of the seventy mathematicians, four, three males and one female, claimed only to do individual work. I am not speaking here of the cultural practice of giving seminars on ongoing work but of the organisational form these mathematicians described as dictating how they went about their research. They used the word "collaboration" to describe this although I would distinguish between the collaborators, those whose work was so inter-dependent that individuals could not identify who had contributed what to the final paper, the product of their study, and the co-operators, those whose contribution to a study was disciplinarily discrete from the other members of the team. Co-operation was frequent, for example, between statisticians and sometimes pure mathematicians and those in non-mathematical disciplines but even in these circumstances they sometimes described the processes using very collaborative language. Despite one mathematician saying:
those who did describe their work as collaborative listed thirteen different reasons for why such work was beneficial (see Burton, 1999). For mathematics educators, it was reinforcing to find the mathematicians identifying the same reasons to support collaboration as can be found in the educational literature.
So the epistemological heterogeneity was supported by a very interactive, and collaborative working style even where the mathematicians themselves referred to an overall competitive spirit within the discipline itself. Knowledge sharing was spoken of in very positive terms. Knowledge dispersal, in mathematics, invokes the traditional journal publishing routes which themselves carry particular community constraints discussed below.
Mathematicians' social positioning
I wish to describe the characteristics which define, bound and direct the practices of mathematicians as the disciplinary culture. Within that culture, their appear to be recognisable social practices which are well known, sometimes mocked (see Davis & Hersh, op. cit. for example) and certainly not always applauded. Of these, some of the mathematicians spoke very negatively about the effects of competition on work practices.
Once regarded as the norm, competition operates everywhere. For example, in seminars:
Although resisted by a number of my participants, especially the females, the competitive spirit appears not only to be pervasive, but to define and distort the experiences people have, again especially women. One female pure mathematics lecturer, in speaking about writing practices said:
Many of the participants emphasised how important it was to be clear and accessible when writing or speaking.
But, it was also made quite clear just how normalised are editorial practices so that:
In a paper written with Candia Morgan (forthcoming), we presented the result of a discursive analysis of 53 papers obtained from the participants in my study. We found close links between their epistemological positions and features of their writing style, and connections between philosophy and presentational style which are also noticeable in texts. We said:
I am suggesting that in these subtle ways, power and consequently social positioning is operated by means of the conventions of the discipline which are established and maintained through the social practices, for example with respect to publishing, but also in the rules by which the game of mathematics is played, especially interpersonally.
If an epistemology is a theory about knowing, and the model which I developed robustly describes knowing, in the experiences of these seventy mathematicians, as socio-culturally based, as being aesthetic and intuitive, heterogeneous and holistically inter-connected , the gap between this view of mathematical knowing and that encountered by learners is monstrous. It could be said to be a strong indicator as to why so much teaching of mathematics fails in that it comes from philosophical and epistemological perspectives that are disconnected from the enquiry experiences of research mathematicians even though such learning experiences are not different from those experienced by more naive learners when they want to know, i.e. understand, mathematics as opposed to pass a test or gain a necessary certificate. But it can also be seen that the unacceptable social practices within the discipline have been translated into similar practices in classrooms with an emphasis on individualism and competition. However, this inaccurately represents the research practices of the greater number of mathematicians and could therefore legitimately, in my view, be said to be a distortion of the conditions under which effective learning communities are created and maintained. That such a distortion supports, even possibly enhances, unequal access to the discipline seems to me to be self-evident. One of my participants remarked:
Burton, L. (forthcoming) The Practices of Mathematicians: what do they tell us about Coming to Know Mathematics? In Educational Studies in Mathematics.
Burton, L. (1995) Moving towards a feminist epistemology of mathematics. In Educational Studies in Mathematics, 28: 275-291.
Burton, L. & Morgan, C. (forthcoming) Mathematicians Writing.
Davis, Philip J. & Hersh, Reuben (1983) The Mathematical Experience, Harmondsworth: Penguin Books
Newell, Sue & Swan, Jacky (1999) "Knowledge Articulation and Utilisation: Networks and the Creation of Expertise" paper given at the User Workshop, Knowledge Management and Innovation, Royal Academy of Engineering, London, April 23.
Restivo, Sal (1999) "What does mathematics represent? A sociological perspective". Paper given to the fourth seminar on the Production of a Public Understanding of Mathematics, Birmingham, UK and available at http://www.ioe.ac.uk/esrcmaths.
University of Birmingham, UK
NOTE: This is a paper in progress. Comments are requested, but please do not quote.