Published in Nunes,T (ed) Special Issue, ‘Giving Meaning to Mathematical Signs: Psychological,
Pedagogical and Cultural Processes‘ Human Development, Vol 52, No 2, April, pp. 129-
tool requires design and its integration into mathematical expression is worth close
consideration (see, for example, Ruthven‘s analysis of the role of squared paper in
mathematical pedagogy, Ruthven, 2009, in press), digital tools - by virtue of their
infinite malleability - have encouraged researchers to consider not only how best to
adapt tools to the learning of mathematics, but how to adapt the mathematics-to-be-
learned in the light of new tool-rich possibilities. Thus design moves even more to
centre-stage. Of course, this perspective leads to difficult questions of cultural
legitimacy and what, in other contexts, one might call ‘transfer‘.
This last point needs a little elaboration. If our focus is on understanding how
mathematical cognition evolves in tandem with the fluent use of digital tools
embedded in learning situations, we will need at some point to ask whether and how
such cognition generalises beyond the context in which it was developed. In this
respect, we will borrow from Papert‘s analysis of how one can foster the development
of a —Mathematical Way of Thinking” that goes beyond the teaching of specific
content of mathematical topics. He asks:
—Psychologists sometimes react by saying, ‘Oh you mean the transfer problem”. But I
do not mean anything analogous to experiments on whether students who were taught
algebra last year automatically learn geometry more easily than students who spent last
year doing gymnastics. I am asking whether one can identify and teach (or foster the
growth of) something other than algebra or geometry, which, once learned, will make it
easy to learn algebra and geometry. No doubt, this other thing (let‘s call it the MWOT)
can only be taught by using particular topics as vehicles. But the —transfer” experiment
is profoundly changed if the question is whether one can use algebra as a vehicle for
deliberately teaching transferable general concepts and skills. [...] Yes, one can use
algebra as a vehicle for initiating students into the mathematical way of thinking. But, to
do so effectively one should first identify as far as possible components of the general
intellectual skills one is trying to teach, and when this is done it will appear that algebra
(in any traditional sense) is not a particularly good vehicle.” (Papert, 1972, pp. 251).
Papert‘s focus on algebra, though pertinent given its hegemonic role as a
modern medium of mathematical expression, should be seen as an instance of a more
general insight that could equally reference geometry, number, statistics and calculus.
Papert‘s position has not lost any of its force in the intervening three-and-a-half
decades. It raises two major issues, each of which we will touch on below. First, and
most obviously, it challenges us to conceptualise not only the design of pedagogic
approaches and tools, but to understand more clearly what kinds of knowledge may be