Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on Knowing More Than We Can Tell 615
tacit, covert? Could I reaccess it if I needed to? Polanyi’s examples are often about
physiological and perceptual knowledge, and in his 1962 Terry Lectures he came
out with this very clear assertion about embodied knowing: “by elucidating the way
our bodily processes participate in our perceptions we will throw light on the bodily
roots of all thought, including man’s highest creative powers” (1966, p. 15).
A number of Sinclair’s examples point to the potential significance of gesture
for mathematics and for teachers of mathematics to attend to, both in themselves
and in their students. What does the body know about the object being drawn, for
instance? What does the hand know (or what is it supposed to learn) from the use
of so-called ‘manipulatives’ (with the Latin word manus for ‘hand’ lurking right
inside this word—see Pimm 1995)? We have two words for ways of using the ear
(listening and hearing), two for the eye (looking and seeing), and two for the hand
(feeling and touching). Listening, looking and feeling all refer to going out to the
world, searching. But ‘feeling’ has a far broader metaphoric connotation than either
listening or looking.
TheTactoftheTactile
The connections Sinclair makes between gesture and diagram (which, up to twenty
years ago or so, also entailed a distinction between the dynamic and the static) drawing
on the work of Gilles de Châtelet and Brian Rotman are intriguing to say the
least. It brought to mind a not-much-referenced part of a chapter in the early work
of Martin Hughes (1986) in his book Children and Number. This work involved
children aged between three and seven engaging individually in a number of tasks,
initially concerned with their making of marks on paper attached to identical tins to
help the child know which tin had which number of cubes in it. In Chap. 5, Children’s
Invention of Written Arithmetic, he classifies young children’s written number
symbolism (all very distanced, with no human representation in the main) into four
categories. But towards the end, in a section entitled ‘Children’s representation of
addition and subtraction’ (pp. 72–75), the pages suddenly explode with drawn active
hands, reaching for or pushing, holding or touching the objects (the drawn ‘cubes’
that are the objects of the arithmetic).
Just like the Italian Futurists, endeavouring to evoke speed and motion in their
paintings as characteristic of the new age in the early twentieth century, so these
children imbue their work on paper with the dynamic of human hands (sometimes
disembodied, sometimes attached to arms; sometimes from the left, sometimes the
right and even a couple operating from the top of the page). In one instance, the
undifferentiated arm extends for an impossible length right to the edge of the paper,
the drawer apparently unwilling to chop off the hand to exist by itself.
Human mathematical agency abounds in these drawings/diagrams: there is little
doubt in my mind these are the drawer’s own hands we are being shown. The very
last image in the section is one where the arithmetic ‘objects’ are human soldiers,
marching from the left (with bearskin hats) for addition and from the right (Légionnaires!)
for subtraction. These people are presumably self-motivating, no puppeteer’s
hands are present. (Their signifying directionality also brought to mind the