International Journal of Computers for Mathematical Learning 9, 3, 309-326
situated abstractions of mathematical ideas that are being developed and expressed, and how
these abstractions are webbed by the available tools and shaped by the interactions with these
tools and with the community. Further, we need to consider what may be gained by
recognising explicitly the relationship between the diversity of individual and community
knowledge and the official mathematical knowledge of the curriculum.
Trouche would, we are sure, agree that thus far, the discussion of orchestration and what he
terms instrumented orchestration is still schematic. Perhaps the different unfolding histories of
our respective research efforts - including the choice of technologies to study - has resulted in
differences in emphasis between us rather than in core perspective, most notably regarding the
weight we accord to the role of interactions among learners, and the role of technology in
mediating these interactions. To assist further in sharpening the discussion, we suggest that it
might be essential to bring into any elaboration of the orchestration process an analysis of the
ways in which artefacts shape this process, and thus also effect the learning of individuals and
communities.
Relevant to this endeavour would be a consideration of the considerable amount of research
that has been undertaken on the role of group work in learning mathematics and the importance
of forming collaborative communities in classrooms that encourage sharing of different
perspectives, explaining and defending ideas and critiquing and debugging the developing
ideas of the group (see for example Healy, Pozzi & Hoyles, 1995). Computer-based
collaboration, in particular, has been found to encourage a shift in relationships between
teacher and student and - under suitably managed circumstances with appropriate tools -
enhanced task-based interactions between students and between students and teachers. More
generally, there has also been a substantial research effort that seeks to analyse the role of
connectivity in teaching and learning (see, for example, the literature on computer-supported
collaborative learning, CSCL, in Koschmann, Hall & Miyake, 2001). A major area of CSCL
research concerns knowledge building through shared construction and interaction - to avoid
the trap of the didactical paradox, in which students are assumed already to know what it is
they are supposed to learn.
A focus on collaborative learning and co-construction of knowledge leads us to envisage a
mathematical tool, such as CAS, as not only a cognitive tool but also a genuine mediator of
social interaction through which shared expressions can be constructed. Indeed, this goal is
now readily technically achievable given that handheld calculators and laptop or palmtop
computers can now effortlessly communicate with each other and with other digital devices. It