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-
We give two examples. The first is derived from the WebLabs project8, in which
we employed ToonTalk as a programming ‘language‘ for children to build models of
mathematical phenomena9. Our aim was to design tasks that would, we thought, be
relatively unrealistic for 13/14 year-old students with only conventional
infrastructures for expression. Or, to put it another way, to see if we could design new
representations that would make relatively unlearnable mathematics more learnable
for these students. For example, we designed tasks on infinite sequences and series,
and engaged students with the sum of sequences like 1, ½, ¼, 1/8, ... and 1, ½, 1/3, ¼,
....
In such a scenario, there are several difficulties with the conventional
representation. The first is evident with the use of ellipsis to denote "and so on". Not
all students see that, for example, 0.1428571... is an infinite decimal, preferring
instead to see the 1 on the right as the "last" digit. Indeed, the fact that it takes an
infinite number of digits to represent a tangible entity like 1/7 is a paradoxical situation
for many students - the difference between a number and its (various) representations
is far from obvious. So a second difficulty - more serious than the first - is that it is,
in conventional representations, impossible to write down an equation like 1/7 =
0.1428571 without some convention peculiar to the representational infrastructure
(such as judicious placing of dots either at the end, or above some of the digits).
To design our new representation we had, therefore, to eliminate rounding
errors. We achieved this by the implementation of exact rational arithmetic in
ToonTalk. In ToonTalk, it really is the case that there is an exact decimal expansion
of a rational number, and moreover, that this is recognised by the system (1/7 =
0.1428571... is "true").
But how to represent the "..." to the right of the decimal expansion? Clearly
this is a serious design challenge: no truncation should return 'true', yet there is a
decimal expansion of 1/7 that is exactly equal to it. We remark in passing that we met
8 Grant IST 2001-3220 of the Information Society Technologies Programme of the European
Commission.
9 We have put quotation marks around the word ‘language‘ to underline that ToonTalk is far
from a standard representational infrastructure for programming. Instead of the standard lines of code,
Toontalk is a programming system in which programs are instantiated as ‘robots‘, trained what to do
by - literally - being shown by the user‘s avatar, present in the form of its (your) hand.
14