International Journal of Computers for Mathematical Learning 9, 3, 309-326
or which straightforwardly 'affords' this or that entailment). All too often, students are left to
their own devices, exploring mathematical tasks for themselves via their computer interactions
as teachers struggle to interpret what they do in mathematical terms. Furthermore, the
transformations of mathematical knowledge in the presence of technology may or may not be
judged as desirable from an educational point of view: it depends on the value of the
transformed knowledge as perceived, perhaps differently, by teachers and students. Thus, even
though students may be engaged and feel that the activity is legitimate from their point of view,
it may still pose a problem of legitimacy for the teacher, since what is expressed by students —
and how — has to be recognised as mathematical within the discourse of the institutional
learning system for mathematics.
Instrumental genesis and situated abstraction
The analytical framework of the French research is based on the notions of artefact and
scheme, where the latter is the psychological component of an instrument, a psychological
construct that a person operationalises in activity with an artefact in order to carry out some
task. Instrumented activity - i.e. activity that employs and is shaped by the use of instruments -
has a twofold outcome (Artigue, 2002). There is a process of instrumentalisation, in which (in
our terms) the subject shapes the artefact for specific uses, and simultaneously a process of
instrumentation, in which the subject is shaped by actions with the artefact. This dialectic by
which learner and artefact are mutually constituted in action is referred to as instrumental
genesis (see VёrШon, 2000, for a summary of these ideas; also Vёrillon & Rabardel, 1995).
A key challenge, then, for the integration of technology into classrooms and curricula is to
understand and to devise ways to foster the process of instrumental genesis. On a theoretical
level, Trouche makes good use of the notion of scheme for this task, recognising the
possibilities of reconciling the individual and social perspectives most obviously linked to the
work of Piaget and Vygotsky, and building on the work of Rabardel and Samurgay (2001). He
draws our attention to a still relatively undeveloped area, in which he (and we) are concerned
to map the important specificities of mathematical knowledge that are not necessarily part of
the general psychological constructs which support —knowledge generation” by individuals.
Although schemes of instrumented action recognise the crucial shaping of the learner by
interaction with tools, their very generality makes it all the more important to take account of
the specific way mathematical knowledge might be developed. This is what the notion of
situated abstraction seeks to address, by providing a means to describe and validate an activity