International Journal of Computers for Mathematical Learning 9, 3, 309-326
The collective French research effort, exemplified by Trouche‘s paper, has provided a
theoretical way forward, and offers the notion of orchestration as central in the construction of
effective teaching and learning environments from the design point of view. As a next step, the
research community could usefully identify and present scenarios in use (as mentioned by
Trouche) with the inclusion of analyses of the trajectories of students‘ evolving situated
abstractions and how they are shaped and become taken-as-shared as a result of didactical
strategies followed by the teacher (see Healy, 2002, on —filling out” and —filling in” as a
systematic example of this approach). This might be one way to understand better the
instrumentation-orchestration process and to tease out the implications for classroom activity.
Our intention in this paper has, therefore, been to expand the notion of orchestration to include
a cultural dimension, a recognition that it is not merely the direct teacher-student interaction
that needs to be engineered, but also the —medium” in which this interaction takes place, and
above all the artefacts and objects that are shared between the communities of teachers and
learners.
We conclude with two important caveats. First, we recognise that pedagogy is closely tied to
cultural contexts and that the French and Anglo-Saxon mathematical cultures are distinctly
different. Perhaps there is a case for collective investigation through design experiments or
didactical engineering projects in several countries, which take our combined theories as a
starting point. We certainly do not presume to imagine that the challenges that face UK
mathematics educators map precisely onto those faced by our French colleagues, even though
we do believe that the four key problems outlined by Artigue at the outset of this paper are
indeed those that are generally faced by us all.
Second, we would like to caution against epistemological relativism, which our advocacy of
diversity may seem to suggest. Mathematical legitimacy is important to us. It is not denied by
recognising that there exists a diversity of knowledge webs in which people make connections
as a result of new computational tools: they see things they couldn‘t see; connect things that
couldn't be connected; represent things that were hitherto un-representable. Connection -
always a key mathematical objective - becomes a possibility. But connectivity is not a strict
sequence of steps that lead from disconnection to connection. Recognising the diversity of
webs, and their development over relatively long timescales, is key to bringing about
4 We borrow the French formulation here. Simply, didactic = pedagogy + knowledge. At a time when, in Anglo-
Saxon cultures at least, the epistemological dimension is so widely taken for granted, this formulation is helpful, even though
the connotations of the word “didactic” in English are potentially misleading.
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