Commentary on Theories of Mathematics Education

602 N. Sinclair
that the values of the mathematical community should be make much more explicit
in the mathematics classroom, offers one possible response. However, Polanyi’s use
of the word passion, instead of just value, draws attention to the emotional and psychological
side of tacit knowing, and, in particular, the pleasures of knowing, controlling,
and finding certainly, relevance, and interest. One can make values explicit
relatively easily, but it is much harder to evoke passion and pleasure (see Sinclair
2008a, for some further discussion of this).
Recent work in the study of gifted and creative scientists bears some similarities
with Polanyi’s notion of intellectual passions, and my general theme of covert ways
of knowing. For example, in a book subtitled Extracognitive aspects of developing
high ability (2004), psychologists Larisa Shavinina and Michel Ferrari define extracognitive
phenomena as “specific feelings, preferences, beliefs” that include “specific
intellectual feelings (e.g. feelings of direction, harmony, beauty, and style),”
as well as “specific intellectual beliefs (e.g. belief in elevated standards of performance)”
and “specific preferences and intellectual values” (p. 74). The authors use
biographical, autobiographical and case-study methods to illustrate the importance
of the extracognitive in gifted and creative thinking. Their focus on the elite may
give the impression that it is the presence of these extracognitive phenomena that
gives rise to gifted and creative thinkers. Such an assumption has been common
in mathematics (and mathematics education) with writers such as G. H. Hardy and
Jacques Hadamard, for example, talking about the special aesthetic sensibility that
only great mathematicians have. Seymour Papert (1978) criticises this assumption,
and argues that non-mathematicians show productive aesthetic sensibilities as well
(see also Sinclair 2001, 2006 and Sinclair and Pimm 2009 for a more protracted
critique). The interpretation offered of the Shea and Mei interchange suggests that
even young students act on their specific values and commitments in the mathematics
classroom.
I would like to return to Polanyi’s notion of the heuristic function of tacit knowing
by considering an example that might more clearly illustrate it than my interpretation
of Shea and Mei. Despite their focus on elite mathematical thinking, Silver
and Metzger’s (1989) study of problem solving helps connect aesthetic values to
the idea of “specific feelings” and what one does with them. A mathematician is
asked to prove that there are no prime numbers in the infinite sequence of integers
10001, 100010001, 1000100010001,.... In working through the problem, the
mathematician hits upon a certain prime factorisation, namely 137 × 73, and decides
to investigate it further as a possible pathway to the solution. When asked
why he fixates on this factorization, the mathematician describes it as being “wonderful
with those patterns” (p. 67), apparently referring to the symmetry of 37 and
73. The lens of tacit knowing draws attention to the source of the mathematician’s
hunch (which turns out to be wrong 4 ): he expects and anticipates that symmetry
is productive, perhaps because the existence of a pattern must entail an underlying
4 See Inglis et al. (2007) for more examples of hunches that turn out to be wrong in the mathematical
work of graduate students, and that are expressed tentatively through linguistic hedges.