On the Integration of Digital Technologies into Mathematics Classrooms

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

Conclusions

Research in mathematics education evolves through the design of classroom interventions,
feeding back into theoretical analysis. The notion of engineering as described in French
didactics is powerful: it suggests that the artefact (in this case, the didactical⁴ artefact) exists in
terms of a specified outcome. This has proved helpful since a general goal of mathematics
education research is to be able to specify the engineering of situations which, taken together
on a reasonable time scale, will enhance mathematical learning. A similar notion to didactical
engineering that we find appealing is that of —design experiments” — see for example see
Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003. However, on the shorter temporal scale
that typically governs change within the learning domain of an individual classroom teacher,
we suggest that Trouche‘s perspective might benefit from a somewhat more explicit
acknowledgement of the diversity of student (and teacher) response, and by implication, the
need to orchestrate at much higher levels of the system. .

Our two perspectives are united in recognising that if orchestration aims merely to bring about
—convergence” of mathematical expression with official mathematical discourse, the potential
of the technology will almost certainly be missed. Yet convergence is important, so two
qualifications of the previous comment are in order. First, convergence for the students may
take time - significant time over years rather than days or months. Second, the process of
orchestration must take place at different levels, separated in time: the first level of
orchestration being to foster the growth of situated abstractions (or, instrumented social and
individual mathematical schemes) which establish a —cognitive scaffolding” for a second level
of orchestration to bring about convergence or at least alignment through discussion of
boundary objects. This second phase might be expected to take place over an extended period,
and through a combination of collective activity in the classroom and individual work by
students. We suggest that a fundamental conjecture of Seymour Papert might usefully be
revisited with the insights provided by the theoretical apparatus we have tried to elaborate:
namely, that computationally-based activity should be seen, in the first stage, as a preparation
for this kind of convergence rather than a means of effecting it: indeed, Papert argued
convincingly that a key outcome of such activity is that it should make it —easier” at the later
point where formal teaching takes place (Papert, 1972).

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