Commentary on Theories of Mathematics Education

616 D. Pimm
fact that the ancient Egyptian hieroglyphs for addition and subtraction are stylized
pairs of legs, moving in the opposite direction to one another.) So we have agency,
motion and cause all wrapped up together into a single drawing/drawn mathematical
hand.
There is much going on at the same time when someone is doing mathematics.
As has been well documented (for one account of this, see Pimm and Sinclair 2009),
only a minute fraction of this is conventionally recorded as being the mathematical,
making itself available to be communicative. The active implication of the hand in
the mathematics can be found in the following (non-exhaustive) list: the writing
(or Rotman’s scribbling) hand, the calculating hand, the drawing hand, the haptic
or pointing hand on the mouse, the gesturing hand. But it is not just a gesturing
means placed into the diagram, nor even a pointer; there is also the desire (and I
use this word advisedly) to touch the mathematics directly, a desire which is forever
to be placated, subverted, frustrated, denied. There is frequently a compulsion with
children to touch, to grasp, to hold: the hands too are an organ of sight and insight,
of learning and knowing.
The ‘silencing’, the prohibition, the ‘no’ I mentioned earlier, need not only be
with the verbal. A tactile mathematical example perhaps relevant to Sinclair’s comments
on gesture comes from many children in certain countries and cultures being
explicitly told not to count on their fingers after a certain age (though many Japanese
children, educated in an intense abacus tradition, frequently use just such gestures
to support considerable feats of mental arithmetic—see Hoare 1990). Do not touch!
reads the sign in the museum of mathematics.
Psychoanalytic Tremors
The opening quotation from mathematician David Hilbert speaks of the ‘rage’ to
know in mathematics. Where does such a compulsion come from? Gestures become
frozen in diagrams and in so doing lose contact with the body. The temporality in
which we all live must be denied in mathematics and then repressed. Some of the
challenge around the calculus is that it is shot through with temporal metaphors, in
the language and even in the notation (what is that arrow doing in denoting the limit
as n ‘tends to infinity’, for instance?). What are the costs of repressing time, for
mathematicians, for teachers, for students?
In Nimier’s exchanges, I have no doubt because of the subtle listening and particular
attending he brought to bear during the conversations, he succeeded in bringing
to the surface feelings and broader awarenesses. One can but marvel at the articulateness
and psychological acuity of these seventeen-year-olds. But they also had
the good fortune to grow up in a country in which the psychodynamic is a core and
explicit cultural reality. The anglophone desire (of the mathematician, of the mathematics
educator?) to dwell only in the full glare of the ‘cognitive’ begins to look
slightly suspect in this light.
Nimier’s remark that Sinclair quotes about the student having apprehended
‘mathematics-as-object’ for herself and her own use evoked for me both the Zuyder