Commentary on Theories of Mathematics Education

Knowing More Than We Can Tell 599
in particular, the case in which the real values of an infinite sequence do not get
closer and closer to a single real value as n increases, but “oscillate” between two
fixed values. When he utters the word “oscillate,” his right hand, with the palm towards
the left, and with his index and thumb touching, moves his high right arm horizontally
back and forth. Núñez argues that the gesture and linguistic expression are
telling different stories, with the gesture referring to fundamentally dynamic aspects
of the mathematical idea. Harris might argue that the word “oscillate” is already
metaphorical and already points back in time to a spatio-temporal conception—and
that therefore, the linguistic and gestural expressions are actually quite similar. In
fact, Harris suggests the need for further gesture research involving mathematical
ideas and concepts that are not already as temporal as limits and continuity. 1
Brian Rotman (2008) sums up Châtelet’s position as follows: “If one accepts
the embodied—metaphorical and gestural—origins of mathematical thought, then
mathematical intuitions becomes explicable in principle as the unarticulated apprehension
of precisely their embodiment” (p. 38). For Rotman, intuition thus becomes
a “felt connection to [the] body” (p. 38). And maybe Weil’s pleasure is one of reconnecting
to the body. Certainly, in ruing the inevitable take-over, Weil suggests that
there is as much pleasure in this as there is in solving a problem or finding a proof:
“A day comes when the illusion vanishes: presentiment turns into certainty [...]
Luckily, for researchers, as the fogs clear at one point, they form again at another”
(p. 52).
Drawing on interviews with over seventy mathematicians, Leone Burton (2004)
identifies “intuition and insight” as one of the five principal ways in which mathematicians
“come to know.” In a previous piece, which also drew on these interviews
(1999), she examines why intuition and insight have not been nurtured in mathematics
education. She suggests that one way to nurture intuitions in problem-solving
contexts is to ask students to engage in “ponderings, what ifs, it seems to me that’s,
it feels as thoughs” (p. 30). The metaphoric emphasis of Châtelet would demand
that we extend this list to include “it’s like, it’s similar to, it reminds me of.” This
kind of activity may help bring to the fore the verbal, imagistic, kinetic associations
that Pimm (1994) argues are so important to meaning: “I believe meaning is partly
about unaware associations, about subterranean roots that are no longer visible even
for oneself, but are nonetheless active and functioning” (p. 112).
If intuitions are precisely the unarticulated consequences of embodiment, then it
becomes even more interesting to explore the ways in which our bodies experience
mathematics and, in particular, the extent to which technologies teach our bodies
how to think and feel. To focus on just one particular example: how does the link
between gesture and diagram change in dynamic geometry environments in which
the gesture no longer needs freezing and, more, in which the human gesture follows
from the machine one?
I have rearranged and compressed my list of covert ways of knowing by phasing
out “intuition” as a primary category and subsuming it instead to the specifically
mathematical endeavour of passing from body to sign, through metaphor and
1 Sinclair and Gol Tabaghi (2009) report on interviews with mathematicians that involve looking
for evidence of fictive motion in non-temporal settings, such as eigenvectors and multiplication.