International Journal of Computers for Mathematical Learning 9, 3, 309-326
The ways in which mathematical knowledge is transformed by the computer‘s presence have
been a preoccupation of ours for many years, beginning with projects with Logo in the 1980s,
and we have increasingly come to focus on the epistemological transformations of students‘
knowledge in interactions with computers in mathematics classrooms (see also Hoyles and
Noss, 2003, for a discussion of the types of software that have the potential to afford such
transformation). For example, learners‘ experiences with mathematical variables can be
transformed by first encountering them as —inputs” to Logo programs whose values can be
easily changed with immediate observable effects; intensive quantities such as rates —
notoriously difficult because of their abstractness — can be made —objects” to be manipulated
in many computer environments; and conjectures can be tested by making constructions in
dynamic geometry systems, thus transforming early encounters with proof from procedural
exercises in validation to exploratory exercises in explanation. These transformations of
mathematical meanings generated in —contexts of integration”, necessitate a conception of
mathematical understanding and of mathematical knowledge, which properly accounts for the
specificity of situations and the contingencies of mathematical expression on tools and
technologies and on the communities in which they are used.
This developing view of mathematical knowledge has helped us make sense of activities we
have researched in a range of quite different situations: activities with adults in workplaces, as
well as with students and digital technologies (Noss and Hoyles, 1992, Hoyles, Noss & Pozzi,
2001; Noss, Hoyles & Pozzi, 2002; Kent & Noss, 2002). We have sought, therefore, to
elaborate a unified theoretical account of how conceptions of mathematics might be situated —
in terms of language and connectivity and with the context of their genesis, means of
expression and use (that is, with the artefacts, goals and discourse that form part of the activity)
— and yet are abstract in that they extend beyond immediate concerns to more general
conceptions of knowledge, that is, they can be —mapped onto” parts of formal mathematics. It
is from this basis that we developed the twin theoretical notions of situated abstraction and
webbing (see Noss & Hoyles, 1996), ideas to which we will return below.
The marginalisation of technology by educational institutions has also turned our attention to
the need for a more precise analysis of the role of the teacher in these new and changing
didactical contexts. As Artigue points out, the computer‘s legitimacy has often been justified in
terms of its ease of use, the relative simplicity of its adoption, and the overstatement of its
potential (as if, incidentally, —it” - that is, the computer - is a thing which can be considered
unproblematically as an —element” in an otherwise static and unresponsive educational context,