Commentary on Theories of Mathematics Education

Knowing More Than We Can Tell 597
mathematics and mathematics education), I will take as starting points mathematicsspecific
phenomena, and endeavour to reach into covert ways of knowing that may
lie beneath.
On Furtive Caresses
In his 1999 paper, Efraim Fischbein complains that psychologists have paid far too
little attention to the phenomenon of intuition: “The surprising fact is that in the
usual textbooks of cognitive psychology, intuitive cognition is not even mentioned
as a main component of our cognitive activity” (p. 12). He acknowledges the fact
that the very concept of intuition often eludes definition, but suggests that it is characterized
by self-evidence and contrasts with the logical-analytical. While recognizing
its importance in mathematical thinking, Fischbein also warns of the potential
harm of incorrect intuitions for learners.
In her commentary on a recent special issue devoted to the role of gesture in
mathematics learning, Anna Sfard (2009) also tries to douse the enthusiasm communicated
by authors about the importance of gesture in mathematical thinking, by
reminding readers that the use of gestures may also lead to negative effects and undesirable
outcomes. Of course, gut feelings may also be wrong, as may metaphors
and analogies. Indeed, these devices are productive only because they are uncertain,
tentative and sometimes even murky. But, as many have argued, they are necessary
to mathematical thinking. Not only necessary for discovery, but, as André Weil
writes, for motivation: “nothing gives more pleasure to the researcher [than] these
obscure analogies, these murky reflections of one theory in another, these furtive
caresses, these inexplicable tiffs” (1992, p. 52). He clearly covets the covert.
The challenge lies in figuring out how such uncertainties eventually lead to the
greater certainties of formal, finished mathematical theorems. In fact, in her commentary,
Sfard also point out a lacuna in the set of articles, namely, how we might
encourage the movement from gestures—what might be described by knowings
of the body—to mathematical signifiers. By starting in the domain of mathematics
itself—rather than in philosophical and psychological domains that theorise
embodied cognition and gesture studies—Gilles Châtelet (1993) provides a complementary
counterpoint to mathematics education research on both gesture and
metaphor—and, more generally, on the consequences of embodied cognition. He
argues that mathematics offers two modes of converting the “disciplined mobility of
the body” into signs: metaphor and diagram. As we shall see, these means of conversion
draw together a number of often-disconnected areas of research in mathematics
education, including gesture, kinaesthetics, visualization, metaphor, and intuition.
Further, metaphors and diagrams constitute principal components of mathematics
discourse. I examine below first diagrams and then metaphors.
Diagrams, for Châtelet, “capture gestures mid-flight” (p. 10). The gestures made
by the boy working on the conservation task are frozen in diagrams such as shown in
Fig. 1. Diagrams transduce the mobility of the body; they are “concerned with experience
and reveal themselves capable of appropriating and conveying ‘all this taking