Commentary on Theories of Mathematics Education

610 N. Sinclair
in taking Walkerdine’s point of view of mathematics as a discourse—one that depends
on the evolving norms of the mathematics community—it becomes evident
that covert ways of knowing mathematics also depend strongly on that discourse.
Polanyi (1958) makes this clear in his description of the way in which the mathematical
values that underlie the public practice of mathematics—journal articles,
conference presentations, graduate student advising, and so on—are passed on to
new practitioners. Csiszar (2003) undertakes a close analysis of this process by looking
at the style of mathematical writing, and the way in which mathematical journal
articles endorse certain values that are not always explicitly articulated, including
the “discipline’s tendency to exclude all but the very few” (p. 243).
By focusing so much on the covert, I may have exacerbated the dichotomy between
it and the logical, cognitive, formal, rational, and, indeed, contributed to asserting
its “otherness” in mathematics education research. I have tried to avoid doing
so by resisting the usual glorification of these covert ways of knowing, and focusing,
when possible, on the dynamics between what we cannot tell and what we can.
The history of mathematics education research would suggest that studying what
we know by what we can tell is much easier than studying covert ways of knowing.
Yes, it is easier, but it markedly distorts the actual picture. The kinds of methods
that have proved useful in studying the latter may be quite different, the psychoanalytic
approach being a striking example. Others include the use of autobiographies,
which may offer singular ways of understanding the complex of passions that motivate
and circumscribe mathematical experience. Gesture analysis, which has grown
in sophistication in the mathematics education community, also offers a means for
describing what people are thinking. Close readings of student interactions further
provide a means of offering interpretations of knowing that can be compared and
evaluated. 10 Indeed, with the growing availability of video and audio records, the
possibility of devising different interpretations of student knowing becomes in some
ways easier. Some interpretations might be more ‘cognitive,’ and others more ‘psychoanalytic,’
but knowing that interpretations are, in an important sense, all we have,
and that different ones can be fruitful in different ways, might encourage research
that transcends such artificial dichotomies.
Lastly, much of my previous work has been focused on aspects of aesthetic experience
in relation to mathematical thinking and learning. In Sinclair (2009), I explore
the different meanings aesthetics has taken on for contemporary scholars in philosophy,
cognitive science, anthropology, and, not least, in mathematics education.
Some of them link aesthetics to embodiment (and thus gesture), others to feelings
and emotion (and thus passion and desire), and still others to axiological concerns
(and thus values and even ethics—see Lachterman 1989). After having written this
chapter, I now have a sense of a much wider arena in which aesthetic issues play
10 See Pimm (1994) and Tahta (1994) for examples of pursuing multiple interpretations of a single
classroom interaction, and the goals for doing so. This endeavour is not really like the ‘mixedmethodologies’
research that has grown in popularity in mathematics education research, where
the goal is, say, to use qualitative research to help explain findings made from quantitative data,
and to thus arrive at the meaning of the phenomenon under study.