Leone Burton

______________________________________________________________________________________

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.

Mathematicians' beliefs

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:

"Some of the time I am a Platonist. As mathematicians we want to believe that it is objective but I guess rationally I think it is socio-cultural and emotionally I want to believe it is objective."

"I think mathematics is the product of people. It does turn out to be extremely useful. But even utility is a cultural product."

"There is something too strong in it for mathematics to be considered as a cultural artefact."

A male expanded on this same theme:

"If somebody said to me is mathematics objective or subjective, I would say subjective. If I were asked if mathematics were Platonist or formalist, I would say formalist but there is a duality because the way we deal with mathematics is more Platonist than formalist. But the issue of fashion and fad and reputation by association, prestige - there is a game which is being played and you make it hard for yourself by not playing it. I do look at scientific facts as complete garbage but I do want it to be a 'truth' - I like the way it is. There is a desire to be a Platonist, to accept these ideas as 'truths' but there is a contradiction that I cannot really do that. On the one hand, I agree with the cultural dependency argument but, on the other hand, I do find appealing the idea that it might be possible to build something which would be meaningful to some other race of people or creatures."

And another presented a familiar argument:

"I do see mathematics as a coherent thing...I don't think mathematicians can just make up a set of axioms which will lead to something new. I see grand traditions, mathematical physics, number theory, geometry, things coming down to us and if you stay within that broad front you are more likely to do sensible things whereas if you go off on some specialised little track of your own, dreaming up some axioms and pursuing them, you probably will end up doing nothing worthy ...Being within the grand themes is a good test of worthwhileness."

Two pure mathematicians had surprising perspectives:

"The only thing mathematicians can do is tell a good story but those stories do uncover mathematical truths - mathematical truths are discovered. However, the mathematics that we know, write down, is made up by men and women. Statements which have, in today's society, been accepted as proofs, under the conditions of today's society, are what I am calling mathematical truths."

"The definition, theorem, proof style is sometimes necessary to the health of mathematics but it can be over-prescriptive. People think that is what maths is whereas I think it is about filling in gaps, making the map. Maths isn't what ends up on the page. Maths is what happens in your head. I don't think maths is about proving theorems. It is one constituent but maths is about mapping abstract ideas in your head and understanding how things relate."

A difference between pure and applied was captured by this applied mathematician:

"I see mathematics as a way of describing physical things using formulae that work, modelling the real world and having a set of rules, a language, which allows you to do that."

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:

Truth and knowledge, as fallible and tentative achievements, are manufactured by human beings who accomplish what they know and what they can know in common. (1999:4)

As Philip Davis and Reuben Hersh (1983) have so beautifully demonstrated, mathematicians are past experts at shifting their positions:

the typical working mathematician is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all (p.321).

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.

"Sometimes one comes across the missing link. A small idea closes up a circuit that has been around not closed up for a long time."

"Your work is not geared immediately to applications although it would be excellent if one could establish the bridgehead which allowed that."

"Some ideas that were understood by one group of people were exactly what another group of people needed to make something work.."

"You feel that what you have been doing is part of something bigger."

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.

Mathematicians' practices

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:

"I don't understand what people who have almost identical knowledge get from working together"

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.

"With a collaborator we make faster moves because my stupid ideas get sat on faster. We also generate more ideas between us."

"If you are collaborating ... you feel much less isolated...When you start working with someone else you discover that there are things that you can do, perhaps better than they can, and things that they can do that you can't."

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.

"There can be tremendously savage competition in our field and that is not something I feel particularly comfortable with." (Male Reader in applied mathematics)

"In the States, I worked in an Institute where the work was unsupervised but the atmosphere was highly competitive. I became disillusioned with that, and the sense of self-belief that Americans seem to have but also the fact that everything had to be instantly commercially viable." (Male applied mathematics Lecturer)

Once regarded as the norm, competition operates everywhere. For example, in seminars:

"There is a particular culture which we are not used to where a seminar is a defence of your result against the audience which interrupts all the time, questions you aggressively, it is highly competitive. I hate that. The seminar is based upon the following structure - here is somebody's results ... here is why it is wrong, aren't they idiots. Nasty. I can't cope with that." (Female Reader in pure mathematics)

or sometimes in the collaborative role:

"It was also too competitive and the wrong things were made too important. One of the collaborators was used to a very argumentative style and I hated it." (Female pure mathematics Lecturer)

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:

"I think that scholarly articles are often written to impress people how clever you are. I try not to do that, I don't like it. If you explain something well it can sound quite simple but you don't want people to think it was easy, because it wasn't. The same thing applies to seminars. So often they can put up hundreds of equations that you can't take in, you might have no idea what the person is talking about after the first five minutes. I think this is a waste of time."

Many of the participants emphasised how important it was to be clear and accessible when writing or speaking.

"Mathematical quality alone is not enough. I am very conscious of how the ideas are expressed. I like a story - a paper, a seminar are both stories. There is a beginning, a middle and an end. So, well-written, clearly explained in a manner which is easy to follow and logical."

But, it was also made quite clear just how normalised are editorial practices so that:

"...you learn that you certainly do have to write all the letters in exactly the form the editor wants or else you won't get to referee those papers and they won't referee yours."

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:

'Common knowledge' about the nature and process of mathematical writing (as represented, for example, in the MAA guidance mentioned above) suggests that there is only one standard way of presenting one's research. Yet our examination of this relatively limited sample of published papers has shown considerable variation and, in a few cases, wholesale flouting of the recognised conventions...Knowing the conventions and being able to use them may be one step towards establishing one's position; knowing how they work and how and when to break them in order to achieve a particular effect is, however, an important way to express and establish a more powerful position (forthcoming:).

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.

Conclusions

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:

"I have been getting very disillusioned with pure maths. and mathematicians;, the arrogance within the subject never mind about anyone who isn't a mathematician."

References

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.

Leone Burton

University of Birmingham, UK

L.Burton@bham.ac.uk

NOTE: This is a paper in progress. Comments are requested, but please do not quote.