International Journal of Computers for Mathematical Learning 9, 3, 309-326
different conceptions of limit, and that - in our terms - they are, in general, likely to develop
different situated abstractions of the notion of mathematical limit that are intricately connected
to their use of the tools.
In moving from an individual to a more social perspective, Trouche distinguishes different
facets of the notion of a scheme and mentions its extension beyond the individual to what are
termed social schemes. He notes that the notion of social schemes may have certain closeness
to situated abstraction, although as far as we can tell, the idea of social schemes, like the
schemes of individual cognition, does not in itself take any explicit epistemic focus.
Clearly, it is vital to understand the connections between instrumental genesis (at the student
level) and orchestration (of a group of students learning in the classroom). The challenge,
therefore, is to theorise the mechanisms by which situated abstractions are developed in and
through a community, harnessing the discourse (shaped by the tools at hand) as a means for
student-student and student-teacher communication, as well as a means for generating
individual knowledge. Thus it is important to keep in mind that the word abstraction refers to
both an action (a collective action, in the classroom), and to the intellectual structure (within
individuals) that results from that action. More pragmatically, if we could understand more
precisely the roles that teachers are called upon to play in computationally-based environments
, then it would surely throw light on the nature of the mathematical knowledge involved, its
relationship with the priorities of the official mathematics curriculum, and help to clarify how
the process of integrating technologies productively into classroom practice might be
facilitated.
Orchestration and the role of artefacts
In Trouche's paper, the term orchestration refers to a process of —external steering of students’
instrumental genesis”, and through this to enhance their learning of mathematics. Hence
orchestration is an element of the broader didactic process of mathematics teaching.
Orchestration/instrumentation is a felicitous metaphorical pairing, which captures something of
the relationships involved2. One striking issue about the metaphor (although bearing in mind
that metaphors should not be interpreted too literally) is this: what is the analogue of music in
the orchestration of mathematical instruments? Presumably the answer is mathematics, but
2 Cf. in music: “Orchestration is the study and practice of arranging music for an orchestra or musical ensemble. In
practical terms it consists of deciding which instruments should play which notes in a piece of music.” [Source:
www.wikipedia.com]