Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on Knowing More Than We Can Tell 617
Zee and Weyl-Kailey’s elegant and skillful decisions as to whether to offer some
mathematics in order to work on a broader issue (say experience of equality) or
whether to work somehow on a bigger psychological issue in order to resolve a
mathematical difficulty that, so to speak, sorts itself out. Interestingly, she does not
talk about such tacit knowledge in her own book.
Although Polanyi does not explicitly mention psychodynamic issues, he makes
many resonant remarks in his discussion of the structure of tacit knowing. One such
is, “All meaning tends to be displaced away from ourselves” (p. 13). This directionality
is part of how he brings depth to his account of how tacit knowing comes about,
an account redolent of Gattegno’s notion of subordination. But in this psychoanalytic
spot, I am reminded of Bruno Bettelheim’s magnificent and tragic little book
Freud and Man’s Soul (1984) about the misdirection of working from the English
Standard Edition’s of Freud’s work. Just one instance here concerns the decision to
translate the very direct and familiar trio of German words ‘ich, über-ich, es’ (literally,
‘I, above-I, it’) by the distanced and ‘objectified’, Latinate and unfamiliar,
‘ego, superego, id’). In Poincaré’s French you see him referring to le moi (‘the I’ or
‘the me’).
Mathematics, though, is the great displacer of meaning, away from the body,
away from ourselves. In his work on mathematical proof, Nicolas Balacheff (1988)
writes about (for him) necessary discards from language on the way to formal mathematical
proof: decontextualisation, detemporalisation and depersonalisation. I call
them the three ‘de-s’. Sinclair in her discussion of Valerie Walkerdine’s book evoked
two of them: no people, no time. In my closing section, I take a brief look at each.
Frontiers and Boundaries: Mathematical Intimations
of Mortality
Sinclair’s image of circle inversion is a powerful one, not least as it projects what
was previously inside outside, and vice versa. The circle then acts as both a double
frontier and a double boundary. And we know from hydrodynamics that turbulence
occurs at boundaries, which is one reason why boundary examples are often interesting
places to look. What are some of the boundaries of mathematics and where
does the boundary between mathematics and non-mathematics lie? (This can be
evoked, for instance, in the Borwein and Bailey (2004) book title Mathematics by
Experiment, orintheBulletin of the AMS discussion around the paper by Jaffe and
Quinn 1993.)
One interesting boundary comprises a vapour barrier between mathematics and
the rest of the world, trying to keep its messiness, hybridity and change at bay. It
tries to maintain itself by its denial of time and the temporal, by its refusal to accept
mortality. Poet-philosopher Jan Zwicky (2006) writes of the ‘acknowledgements of
necessity’ that can come from poetry as well as mathematics, and she argues there
are core metaphoric insights inside both. However, she goes on:
Their differences stem from the differences in the necessities compassed in the two domains:
mathematics, I believe, shows us necessary truths unconstrained by time’s gravity;
poetry, on the other hand, articulates the necessary truths of mortality. (p. 5)