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-
The important point is this: the key (correct) conjecture was triggered by
reflection on an accurate sketch. During the process of dragging the sketch so that it
corresponded by eye to the constraints of the problem, the students became aware that
they should be keeping an eye open for possible relationships between the other
elements. Without the dynamic aspect expressed through dragging, this would have
been extremely difficult, as accuracy, as well as interactivity (through hand/eye
coordination) is essential to the process of noticing such relationships. Notice too that
this property of being dynamic is quite different from the sense of dynamic that
characterises, say, animated diagrams. The key factor is the interplay between
dynamic (while dragging) and static (stop when some relationship seems evident) and
that this is crucially in the control of learners - so they can pause, reflect, go back and
test in the light of feedback from the graphical image.
We conclude this section by noting another major function of the use of
accurate sketches such as these in learning mathematics, which is to produce the
motivation to hypothesise a theorem to account for the figures on the screen, prior to a
conjecture and also subsequent to it3. This is a constant challenge in mathematics
education: to motivate students to ‘keep an eye on the general‘ in all that they do.
3. Outsourcing processing power
We would propose, alongside Jim Kaput in his seminal paper of 1992, that a
fundamental property of digital technologies - one that distinguishes it from all other
technologies - is its affordance in ‘outsourcing’ processing power from being the sole
preserve of the human mind, to being capable of being undertaken by a machine.
Kaput‘s basic argument is that human history is entering a new phase, a virtual
culture based on the externalisation of symbolic processing4. We will not elaborate
the argument here (see Kaput, Hoyles & Noss, 2002). Instead, we will ask what kinds
3 It is worth noting here that only did the tools afford mathematical learning but also a teacher
is granted a way of appreciating the geometrical intuitions that the students had - and can model them
again by positioning parts of the construction by eye with a group or students. This is another example
of the crucial role of the computer as a window on mathematical meaning.
4 Obvious exemplars of external processing are computer algebra systems.