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-

A more important point concerns the idea of expressing aspirations and ideas.
We are accustomed to thinking of computers as precise, detached, accurate. We are
less used to the idea of computers screens to express ideas, especially half-formed
ones. In fact, with the advent of web 2.0, social networking, YouTube and so on, the
conception of computers in popular culture is changing, and becoming more akin to
the infrastructural role that, say, paper and pencil have historically played as a
medium that is capable of supporting multiple modes of expression. But in education,
and mathematical education particularly, this transformation has yet to become
commonplace, and computers in formal educational settings are still largely
associated with activities some way removed from sketching half-formed thoughts, or
fostering creativity or inspiration.

By way of illustrating the point, we will give an example of how, using digital
technologies, students can produce an accurate sketch of the solution to a problem.
Here we use ‘sketch‘ in a technical sense: it is accurate in that it meets the
requirements of the problem situation but it is a sketch in that the necessary invariants
of the mathematical structure of the problem are not formalised, see also Noss &
Hoyles (1996). However the accuracy of the sketch means that by reflection on and
manipulation of the sketch, the students can more easily come to notice what varies
and what does not, and thus are more likely to become aware of what to focus on
(Mason, 1996).

An example of this phenomenon is taken from Healy and Hoyles (2001).
Here, two students are using a dynamic geometry system - Cabri Geometry in this
case - to work on a task to construct a quadrilateral with the property that the angle
bisectors of two adjacent angles cross at right angles. The students were asked that
when they were convinced that they had constructed a quadrilateral that satisfied
these initial conditions, they should seek to identify other properties of the
quadrilateral that had of necessity to be satisfied.

Below, we reproduce part of a description of the pair successfully using the
software to solve the challenge (for more detail see Healy & Hoyles, 2001). They
exploited a mixture of creation and construction tools (this distinction is expressed by
menu choices) to produce an accurate sketch of the quadrilateral required, explored it
and through this exploration conjectured about the necessary geometrical
relationships involved.

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