Commentary on Theories of Mathematics Education

600 N. Sinclair
gesture. While it is in a sense the body that knows, the communicative act (to ourselves
and to others) is that of metaphor and gesture, so that these become, in effect,
the tacit ways of knowing that give rise to feelings of intuition, insight, creativity,
or ‘aha’. 2 I can hardly use the word ‘tacit’ without calling upon the work of
Michael Polanyi in his book Personal Knowledge (1958), in which he undertakes
a ground-breaking ‘post-critical’ examination of the non-explicit dimension of scientific
knowledge, and its relation to formal, public knowledge. To Polanyi, tacit
knowing involves subsidiary awareness, in contrast with the focal awareness of explicit
knowledge, which, as he shows in his studies of scientific and mathematical
discoveries, is governed in large part by both personal and communal values and
commitments. In the next section, I explore the more psychological dimension of
covert ways of knowing.
On Passions and Pleasures
In the well-known “Shea number” episode described initially by Deborah Ball
(1993) and studied in numerous subsequent papers, a group of grade 3 students
are working with the concept of odd and even numbers, and arguing about the status
of different numbers. Shea proposes that the number 6 “would be an odd and an
even number because it was made of three twos,” but is challenged by several other
students, who invoke their definitions of evenness to try to show that 6 is in fact
only an even number, despite the fact that it contains an odd factor. In the face of his
apparent stubbornness, Mei asks Shea whether the number 10 is also both odd and
even, hoping to make evident his erroneous ways. But Shea thanks Mei for “bringing
it up” and claims that 10 “can be odd or even.” Mei then says to Shea, “What
about other numbers? Like, if you keep on going on like that and you say that other
numbers are odd and even, maybe we’ll end up with all numbers are odd and even?
Then it won’t make sense that all numbers should be odd and even, because if all
numbers were odd and even, we wouldn’t be even having this discussion!”
Although the episode may be read in terms of Shea’s “misunderstanding” of the
concept of even numbers, and Mei’s sophisticated reasoning about them, it can also
be read in much more psychological terms. I propose one possible reading here,
based on Polanyi’s epistemology. Polanyi takes the existence of tacit knowing for
granted, tracing its roots in the urge, shared by all animals, to “achieve intellectual
control over the situations confronting [us]” (p. 132). His main project, however,
is to use examples of scientific and mathematical discoveries to examine the relation
between the tacit and the explicit in terms of how the tacit functions, how that
which is ineffable works and what it accomplishes in scientific inquiry. According
to Polanyi, tacit knowing has selective and heuristic functions that are based on aesthetic
response and goal-driven striving. While many commentators, especially in
2 See Liljedahl (2008) for an extensive investigation of the role of “aha” moments in the work of
both mathematicians and mathematics learners.