The technological mediation of mathematics and its learning

Published in Nunes,T (ed) Special Issue, ‘Giving Meaning to Mathematical Signs: Psychological,
Pedagogical and Cultural Processes‘ Human Development, Vol 52, No 2, April, pp. 129-

allowed different grain-sizes of interaction with the key mathematical ideas. We
conjecture that much the same could be true of the classroom: in order to benefit from
the pedagogic gain of outsourcing calculation to, say, the calculator, some attention
must be given to providing glimpses of the process in the interests of learning and
debugging. Opening access to some layers of the system while achieving an optimal
grain size is a matter of careful and expert iterative design.

4. New semiotic tools and representational
infrastructures

We begin with an example drawn from Seymour Papert (2006). He invites us to join
him in a thought experiment at an undefined time when the Roman numeral system
was in use. We are to imagine that the restricted number of experts versed in doing
multiplication suddenly became insufficient for the needs of their society, and that
mathematics educators were asked to remedy the situation. Naturally, they adopted a
range of carefully designed teaching experiments and their efforts were rewarded:
more people than before were able to multiply. But ‘something else did this far more
effectively: the invention of Arabic arithmetic turned an esoteric skill into one of —the
basics”.’ (ibid., p. 582).

It was Kaput who coined the term ‘representational infrastructure‘ to refer to
the kind of cultural tool epitomized by the Arabic numeral system (his work in this
regard and its implications for mathematics learning is summarised in Hoyles & Noss,
2008). One characteristic of such a representational system is that it is taken-for-
granted: this ubiquity and invisibility are critical facets of tool systems that become
infrastructural. A key point here is that students of mathematics learning need to be
aware not only of how mathematics is learned but also what is learned and the
language in which this is expressed. Multiplication, like Newton‘s laws, or
elementary calculus, is learnable, precisely because we have Arabic numerals, the
machinery of simple equations and Leibniz‘s calculus notation respectively. What is
to be learned depends on the representational forms with which it is expressed,
shaping and sometimes defining what can be considered as learnable. Thus we
would argue that those who study mathematical cognition ignore semiotic mediation
at their peril!

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