International Journal of Computers for Mathematical Learning 9, 3, 309-326
what kind of mathematics? The production of —official” mathematics which the teacher (i.e.
conductor) wishes —to hear” in order to judge the students’ performance, or the more personal
and community-based norms tied to students‘ own developing conceptions and personal
inclinations? This question is not intended to raise general issues of learning style or
—epistemological preference” (cf. essay on epistemological pluralism, by Turkle and Papert,
1990), but rather, the complex set of issues which emerge wherever technology has a presence
in a mathematics classroom, and the dialogue that must take place between standard
(—official”) mathematical knowledge, knowledge about the tool, and the —computational
transposition” of the mathematical knowledge involved (cf. Balacheff, 1993). For example,
Defouad‘s (2000) research, described in Artigue (2002), illustrates the difficulty with which
(CAS-based) instrumented techniques gained mathematical status within a classroom and the
manner in which this tended to reduce the value of these techniques. Even when fully
legitimated, the techniques retained a sort of intermediate status in the classroom culture:
Defouad introduced the notion of —locally official techniques” in order to give an account of
this phenomenon. There is no simple way to solve the problem of integrating technology:
situated abstractions by their nature are diverse and interlinked with the tools in use, so how
can meanings be shared in the classroom, interconnect with each other and with standard
mathematical discourse?
The roots of the notion of instrumentation are cognitive, and if instrumented activity is
(strictly) judged from this perspective, then the process of instrumentation is one of cognitive
—overhead”, that is, a process of achieving a mastery over tools that is required in order for a
learner to begin to perform the —useful” task of learning mathematics. However, from a socio-
cultural point of view, instrumentation could be regarded as part of the process of developing
participation within a community of practice, a process by which individual understanding and
behaviour develops from and contributes to the collective activity.
Trouche describes an example of orchestration in which the architecture and organisation of
the classroom is configured to help —connect up” students, with attention to where key students
should sit, and which technologies should be switched on or off in order that individual
instrumented actions can become the object of collective as well as individual reflection and
discussion. This pedagogical strategy has considerable potential and is certainly facilitated by
appropriate technical infrastructure3. Left relatively unexplored, however, is analysis of the
trajectories of students‘ evolving (instrumented) mathematical knowledge, in terms of the
3 In England, this sort of scenario is nowadays most likely to be around an interactive whiteboard.
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