International Journal of Computers for Mathematical Learning 9, 3, 309-326
from a mathematical vantage point but without necessarily mapping it onto standard
mathematical discourse. The notion is particularly pertinent in computational environments,
since the process of instrumental genesis involving the new representational infrastructure
supported by the computer will tend to produce individual understandings and ways of working
that are divergent from standard mathematics.
In fact, both we and the French researchers are acutely sensitive to the discontinuities between
computationally mediated mathematical abstraction and the norms of traditional mathematical
curricula. Abstraction is a key concern within the field of mathematics education — a
characteristic it does not necessarily share with other disciplines — that has been discussed in a
variety of contexts by many researchers (such as Arcavi & Hadas, 2000; Dreyfus, 1993;
Nemirovsky et al, 1998). Our contention is that it is the process of abstraction of mathematical
properties and invariants that is key, and this is necessarily both situated and shaped by the
tools being used, the users‘ relationship to the tools — including whether the users judge them
to be expressive of their developing mathematical meanings — and ultimately whether these
meanings are valued and judged by the community (e.g. the classroom) to be mathematical.
One of the most remarkable characteristics of digital technologies, such as CAS, is that they
provide an unprecedented symbolic means of expression for mathematical abstraction as a
process — something which has been wholly embraced by professional users of mathematics
(e.g. in science and engineering, the use of multiple representations in the CAS to model and
design physical processes, or in mathematics using the visualisation power of a CAS to explore
abstract mathematical structures), but which, as we noted at the beginning, remains marginal to
the concerns of the mathematics classroom.
It might be helpful at this point to look again at several of the examples presented by Trouche.
He presents an example from Guin and Trouche (2002), which demonstrates how the tools
provided by a CAS/graphical calculator shape the ways in which a user might solve two
equations in two unknowns, and how one may interpret the meanings that the user constructs
concerning equations and their solutions. Later, in a discussion of students attempting to find
the limit of a function not directly computed by a CAS calculator, he describes how they build
specific schemes and notes the differentiation among students‘ behaviours (our emphasis).
These behaviours are then distinguished as different instrumentation processes, which are
closely connected with what Trouche calls —command process”. We infer from Trouche that
the students will develop different instruments. Given that the mathematical goal of the activity
is to learn about limits, we are sure that Trouche would agree that students will develop