Commentary on Theories of Mathematics Education

Knowing More Than We Can Tell 609
(rarely doing mathematics); he gradually improves his mathematics schoolwork.
Weyl-Kailey argues that Gilles needed an outlet for his aggression, and that controlling
the sessions provided this. Remedial pedagogy would never have improved his
performance since, like so many students, Gilles saw school mathematics as imposing
constraints and strict rules: “That is the impression [of mathematics] that many
students have. It is the law of the Father, the law of Destiny, the LAW which is
forbidden to transgress and to discuss, and which can only restrict its influence by
ignoring it” (p. 24, my translation). Both mathematics and his anger were controlling
him.
Other analysts have use psychoanalysis to interpret the sometimes amazing mathematical
actions of patients, such as a refusal to write certain digits, or a block
against the idea of equations, or the use of two unknowns. Tahta (1994) points out
that these interpretations are necessarily unverifiable—was it a castration complex
or an oedipal conflict?—but some may be more fruitful than others. Indeed, Weyl-
Kailey’s work with children with learning difficulties attests to possible victories
that come from fruitful interpretations. A child who refuses to acknowledge 3 because
of an oedipal conflict might be rare, but knowing that it happens may help us
appreciate the role that unconscious processes such as condensation and displacement
(discursively seen as metaphor and metonymy) play in mathematical thinking
(see Tahta 1991, for more discussion of the processes of condensation and displacement).
The example of Gilles would seem to be much less rare, given the pervasiveness
of mathematical apathy and anxiety reported in the literature. Weyl-Kailey
treats the problem as stemming from Gilles’ need for control and, in this sense, doing
mathematics may well help Gilles find out about himself. However, Walkerdine
also invites us to question the extent to which the problem lies within the discursive
practices of mathematics itself.
Looking Back, Looking Forward
In this chapter, I have attempted to consider the wide range of constructs and phenomena
associated with what we can know without being able to tell, for which I
have proposed the word covert. My goal has been to suggest a way of creating distinctions
somewhat differently, more inclusively, and to try to understand some of
the similarities between distinct and highly specialised areas of research in mathematics
education such as gesture, intuition, anxiety, and aesthetics. I have done this
by starting within mathematics, rather than within psychology or sociology. The
creation of distinctions has resulted in three main focal areas, with one involving
sensory “subconscious” knowing, and another aesthetic “subconscious” knowing.
The third invokes both sensory and aesthetic “unconscious” knowing. In this taxonomy,
intuition and emotion are cross-cutting, whereas creativity and imagination
are seen as being products of rather than roots of knowing.
In examining the large terrain of covert ways of knowing, I have focused much
more on the individual knower than on more socio-cultural aspects. In particular,
the psychoanalytic perspective seems to orient one’s attention inwards. However,