Commentary on Theories of Mathematics Education

614 D. Pimm
In this chapter, I will pick up on some of Sinclair’s many themes, focusing in
particular on gesture and the tactile on the one hand and the psychodynamic on the
other, before ending with a few thoughts about metaphor. But first, I found myself
thinking about her choice of terms.
What’s in a Word?
In deciding not to go with Polanyi’s term ‘tacit’, and offering instead ‘covert’, Sinclair
brings about a shift of focus. ‘Tacit’, from the Latin verb tacere “to be silent”,
refers to something being done without words, 3 but makes no commitment to how
or why it came to be silent or whether there is anything fundamental preventing
it from finding its voice. In particular, Polanyi makes no reference to the psychoanalytic,
a theme which Sinclair offers us with some force toward the end of her
chapter.
By proposing ‘covert’ (from the French verb couvrir “to cover”), Sinclair alludes
to the fact that, in some instances at least, there is a someone or a something
that is deliberately silencing (or attempting to keep silent) the knowing, rendering
it partially hidden or secret (‘furtive’ is André Weil’s gorgeously suggestive
term), or insisting it stay out of sight. ‘Tacit’ and ‘covert’ share the same advantage
as antonyms in not differentially marking the ‘other’ pole (explicit—“explained”,
overt—“open”) as being the preferred one, nor falling into the overworked ‘public/private’
distinction. In consequence, Sinclair alludes to a larger domain of possible
enquiry than Polanyi. But there is still the same interesting tension that is already
present in Polanyi’s work, namely attempting to be explicit about the tacit, trying to
be overt about the covert.
Sinclair’s paper is, in part, about the possibility of reclaiming that which is known
but untellable for a variety of reasons. Sigmund Freud, when talking about the
project of psychoanalysis, once described it as the cultural equivalent of ‘the draining
of the Zuyder Zee’. This enormous physical undertaking not only made visible
the formerly invisible (a task Paul Klee claimed as his characterisation of art), but
even more important, it made the result accessible for subsequent use (the reclaimed
land that had previously been submerged was immensely fertile). So one question
in response to Sinclair’s chapter is what is potentially made usable by work in this
area? If we were able to be more articulate about the knowledge we have, what
might we gain, as an individual, as a teacher in front of a class, as a culture?
Personally, although I continually find I have learnt far more than I intended, I
have forgotten much that I once knew. But just because it is ‘forgotten’ does not
mean either it is not still there or that it can never be reaccessed. Caleb Gattegno
repeatedly reminded us that our bodies once knew how to make ourselves. “I made
my brain”, I heard him assert on occcasion. Is the knowledge coded in my DNA
3Thus, the book Proofs without Words (Nelsen 1993) could equally accurately be entitled Tacit
Proofs.