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-
accessed through such tools. Second, it maps a research agenda to try to understand
how mathematics can be expressed - and by implication, how mathematical
knowledge such as MWOT can be developed.
If the central challenge of mathematical learning is to express mathematical
abstraction, then we need to move beyond abstractions expressed only in traditional
algebra. We have used the idea of situated abstraction as an orienting framework to
describe and explore how interaction with semiotic tools shapes the development of
mathematical meanings and in turn is shaped by the conceptions and social context of
the students (see, for example, Hoyles & Noss, 1992; Noss & Hoyles, 1996). The
distinction between conceiving abstractions as situated and the traditional view of
abstraction that sidesteps the framing of representation tools, is both powerful and
problematic. It is powerful because it seeks to legitimise forms of mathematical
expression that are novel, and which may access precisely the alternatives to algebra
that Papert sought. But it is problematic as it is easily misunderstood as a kind of
pseudo-mathematics, falling short of traditional pedagogic practice, and too easily
erecting a barrier rather than a doorway between situated and traditional abstraction.
A theoretical corpus of work relevant here is the analysis of ‘instrumental
genesis‘ that seeks to elaborate the mutual transformation of learner and artefact in the
course of constructing knowledge with technological tools (Artigue, 2002; Trouche,
2005). Yet, as we have argued elsewhere (Hoyles, Noss and Kent, 2004), this
instrumental genetic analysis leaves relatively unexplored the texture of the meanings
evolved - the situated abstractions of mathematical ideas that are being developed and
expressed, and how these abstractions are knitted together or ‘webbed‘ (Noss &
Hoyles, 1996) by the available tools and shaped by the interactions with these tools
and with the social context.
This point is important because, although schemes of instrumented action
provide an effective means for conceptualising tool-learner interaction, there remains
a need to elaborate the kinds of mathematical knowledge that develop in such
interactions. This knowledge, or at least its visible traces, may not look or sound like
standard mathematical discourse. It is no coincidence that the idea of situated
abstraction was born in the context of studying students‘ mathematical expression
with computers, for example, by recording children expressing relationships, variants
and invariants through a Logo program or a spreadsheet. It is in the nature of