Commentary on Theories of Mathematics Education

Knowing More Than We Can Tell 607
a distance from his loving mother, being allowed to explore without fear, and being
free to use the objects around him in symbolic ways:
The background of rigour on which the researcher’s scientific works depends plays a fundamental
role that conditions every movement and every initiative. It is analogous to maternal
love for the child: it is simultaneously a reference, a presence and a security. (pp. 163–164,
my translation)
The other explanation offered by Maher also draws on Winnicott, and the notion of
the mirroring role of the mother, to explain why mathematicians seem to be characteristically
visual. He proposes that “doing mathematics involves the gaze of the
mind on transitional objects (here, mathematical objects) in potential space (here,
mathematical reality)” (p. 138). He acknowledges the potential skepticism of his
readers regarding the epistemological problems of psychoanalytic methods, but remains
firm in believing that such methods “offer the most realistic insights into the
working of the human mind and hence into the experience, and activity, of mathematics”
(p. 139). Others have gone even further in suggesting some “alignment”
between mathematics and psychoanalysis insofar as they both help us find out about
ourselves through a “mixture of contemplation, symbolic representation, and communion”
(Spencer Brown 1977,p.xix).
If the desires, needs and fears of mathematicians, as expressed by in the above
discussion, seem far removed from the concerns of mathematics education, then
the mathematician René Thom (1973) provides a glimpse into the continuity that
may exist between mathematics and school mathematics, and between the psychodynamic
experiences of professional mathematicians and learners. In his plenary
lecture, delivered at the second ICME conference, Thom criticises Piaget (and his
followers) for placing “excessive trust in the virtues of mathematical formalism”
(p. 200). He goes on to characterize Piaget’s “reflective abstraction” as the process
of extracting conscious structures from the “mother-structure” of unconscious activity,
where the teacher’s task is to “bring the foetus to maturity and, when the
moment comes, to free it from the unconscious mother-structure which engenders
it, a maieutic role, a midwive’s role” 8 (p. 201). For Thom, abstract, logical notions
are the surgeon’s “brutal” and “feelingless” tools that extract the foetus to early, too
quickly, and are responsible for “losing the infant” and “killing the mother.” 9 The resulting
separation, loss, and even death, could not be further away from the pleasure
8 Poincaré’s (1908) description of mathematical invention is replete with similar imagery around
birth, and also conception. He refers to the combinations of ideas formed during conscious work
as being “entirely sterile.” Despite being sterile, the “preliminary period of conscious work which
always precedes all fruitful unconscious labor.” The unconscious mind is thus fertile, capable of
giving birth. Drawing explicitly on a scientific analogy, he describes the elements of combinations
during conscious work as being like “hooked atoms” that are “motionless.” In contrast, during a
period of rest, they “detach from the walls “ and “freely continue their dance.” Then, “their mutual
impacts may produce new combinations,” within a “disorder born of change.” The explicit
metaphor may be atoms, but the themes and images of fertility and conception seems quite pronounced.
9 See Sinclair (2008b) for a slightly different exploration of childbirth and midwives as a way of
comparing the discipline of mathematics education research with that of obstetrics.