On the Integration of Digital Technologies into Mathematics Classrooms

International Journal of Computers for Mathematical Learning 9, 3, 309-326

behind the other with little if any connectivity, and contrast this with one where students have
wireless-connected laptops and can enjoy a hugely greater freedom to collaborate and share
their ideas, both face-to-face and virtually. This process of breathing life into technology
necessarily differs from one individual to another, if only because of differences between the
kinds of things a student takes for granted, already knows, or is trying to understand. In so far
as the artefact’s properties can be thought of as —orchestrating” the actions and expressions of
an individual or a group of learners, this orchestration is not invariant across different
individuals. Viewed in this light, individual difference is clearly not something to be minimised
or avoided, it is an inevitable part of orchestration itself.

Situated abstraction revisited

Recognising the diverse ways in which individuals use and communicate with technologies to
express mathematical ideas is not without difficulty. In what follows, we will suggest further
that the notion of situated abstraction as a category of mathematical expression may adequately
complement the idea of instrumental genesis, while effectively describing the kinds of
mathematical knowledge that arise in collective as well as individual instrumented activity.

One helpful starting point is the notion of boundary object (Star, 1989; Star & Griesemer,
1989). A boundary object names an important class of knowledge artefact, an object which is
shared between different communities of practice, which can be used differently by the
communities, but can also provide a means to think and talk about an idea without the
necessity of any one community adopting the perspective of the other, or even requiring one
community to understand the detail of what is already understood by another. This is the key
point for meaning-creation: a boundary object provides a generalised mechanism for meanings
to be shared and constructed between communities — for example, the community of teachers
and the community of students.

From a specifically mathematical point of view, boundary objects can provide the means by
which a common discourse involving situated abstractions might be fostered within a
community of mathematical learners, as well as a means by which different communities might
come to an agreement (often implicitly) that they are talking about —the same” mathematical
abstraction or set of abstractions. It does not matter whether or not the abstractions really are
the same (indeed, it is a moot question as to what —the same” might mean in this sense). What
does matter is that different situated abstractions can be brought into association and alignment
through negotiation around a boundary object: between, for example, a community of learners

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