The technological mediation of mathematics and its learning

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-

this paper is to focus on a few tools that exemplify particular tool-use, and to describe
with some illustrative examples, how mathematical meanings - both pedagogical and
epistemological - are shaped by their use.

The discourse of mathematics is inevitably expressed within a set of semiotic
tools, so it is reasonable to conjecture that mathematical cognition evolves alongside
the representational systems afforded by these tools (for related work on the shaping
of representations, see Nunes, Schliemann & Carraher, 1993). The tools are cultural
in the sense that they have evolved historically in response to the demands of
mathematics itself, and, of course, the historical demands of the societies that gave
rise to new mathematics. Modern mathematics in particular is intimately tied to
algebraic expression - but this was not always so: consider, for example, the
geometrical (and to us now, baroque) way that Newton expressed his laws of motion
in Principia (diSessa, 2000).

Our interest in this paper will be on virtual tools and the computer will play a
central role in what follows. There are two reasons for this. First, the relative novelty
of digital technologies has offered us a chance to rethink the ways in which
representations shape learning. In particular, it has fostered a sharper focus on trying
to understand the role of tools more generally and how students‘ conceptions of
mathematics are shaped, not only by the actions and attitudes of the teacher, but also
by how far the students master what the French school of researchers term ‘the
process of instrumentation‘: the extent to which the learner is aware of the system,
and is able to look through it as well as look at it (Artigue 2002).

This strand of work entails a more sensitive realisation that a fine balance is
needed between the ‘pragmatic‘ and ‘theoretical‘ (or ‘epistemic‘) roles of calculation,
a point closely related to the dual nature of mathematics as both tool and object
(Douady, 1991). The simple, and initially at least, widely-held assumption that
technology could relieve the student of the need to calculate (in the broadest sense)
and allow a sharper focus on structure and relationships, has given way to a more
nuanced understanding that calculation and structure are intimately connected, and
that an acute awareness of their relationship should guide the design of the
technological artefacts intended for mathematical learning.

The second rationale for a focus on digital tools is their increasing ubiquity in
mathematics classrooms together with their multi-faceted functionality. While any

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