International Journal of Computers for Mathematical Learning 9, 3, 309-326
engaging in —context-bound” mathematics and the community of teachers of —institutionalised”
mathematics. (See Newman, Griffin & Cole, 1989, for a somewhat similar perspective
developed within a Vygotskian, rather than an activity theory framework.)
In the CAS-equipped mathematics classroom, a boundary object might be the display of an
artefact (the CAS software) together with the —gestures” required to carry out certain
mathematical procedures (Trouche gives some examples in the case of graphics calculators
expressing equations to be solved, as discussed earlier). These, we suggest, can be thought of
as generative of situated abstractions: students construct and interpret procedures and teachers
(re)interpret their constructions. Situated abstraction thus becomes the concrete (visible and
audible) expression of the different communities‘ views of the boundary object.
Viewed in this light, orchestration becomes a mutual act, rather than something that one
community does to another. In other words, thinking in terms of boundary objects suggests a
kind of orchestration in which mutual negotiation and meaning-construction is the norm for
both sides of the boundary, rather than the preserve of one protagonist. In a pedagogic setting,
boundary objects are generally carefully designed, not simply objects about which
communities happen to coalesce. In the past, many of us who have researched in this area have
worked with learners interacting in computer-based microworlds that were designed to foster
mathematical meanings through mutual construction, interaction and feedback (see for
example Healy, Hoyles and Pozzi, 1995). The idea was that the students could web their own
thinking by communicating with and through the tools of the microworld and shaping them to
fit their own purposes, including the need to communicate with others. Thus it was through
careful design of tools and of the interactions planned to take place in activities around these
tools, that we as observers were able to trace and analyse the learners‘ thinking-in-change —
their construction of mathematical meanings.
We therefore would like to suggest that the notion of situated abstraction might complement
the instrumentation-orchestration theory, by focusing specifically on the abstraction process.
Orchestrating to foster situated abstractions as well as to negotiate common meanings is a key
objective of teaching. An important challenge therefore lies in finding the right kinds of
representational infrastructure (tools and linguistic frameworks) for computational
environments, which highlight (to the student) the key mathematical elements of a problem
situation and provide —natural” expressive power - the right things to talk about, and ways to
talk about them.
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