The invention of writing, and then of printing, has had a great impact on mathematics. More precisely, it has had an impact on what we are prepared to accept as valid mathematics - on what we choose to include in the mathematics curriculum. Because it is easier to write down,
2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 . . .
is felt to be a "correct" representation of the two times table, whereas [Graphic not available] is "only a picture". Again, the "correct" formula for the sum of the first n counting numbers,
1 + 2 + 3 + ... + n, is 12 n (n + 1)
whereas [Graphic not available]
is "only a picture", though it actually tells us far more than the formula about the structure of the mathematical idea being expressed. The sum 1 + 2 + 3 + ... + n may be thought of as an over-growing staircase: [Graphic not available] Two such staircases may be put together to make a rectangle, which will always be n by (n + 1): [Graphic not available] So one staircase, or half the rectangle, is half of n times (n + 1) - or 12 n (n + 1).
All this is very tedious and longwinded to write down. The formula grows directly out of the model, and it is far quicker, and more meaningful, to demonstrate it with Multilink.
Other ideas are harder still to put into print: the sum of the first n triangle numbers, for example, or the properties of angles in a circle, really do require dynamic, manipulable models.
But the pictures and models are not given the same status as the written representations, even though they convey a much more powerful, generalisable message, which is akin to proof. This decision to value the printable symbolisation, and to devalue the spatial representation, of mathematical ideas is a value judgement.
In a society which had not developed two-dimensional writing and printing, but had developed a way to record and communicate about three-dimensional and dynamic images - a society which had got moving holographic images, but not books or paper, say - the view of what is and what is not valid mathematics would be quite different. What we choose to say about mathematics is controlled tightly by the way in which we are able to say it.
This value judgement affects everyone, not just adult mathematicians. Pupils who cannot access printed symbols ("slow readers" and the like) are in any case disadvantaged in our schools by the predominance of a book-based approach to every area of the curriculum. A startling number of very able spatial thinkers end up in special schools for emotionally and behaviourally disturbed pupils: if they are forced to communicate through printable symbols which they cannot grasp, then such pupils may respond by ripping up the books and throwing the pencils at the teacher. They cannot say anything about mathematics, because they cannot write it down.
But computers are going to offer us an alternative way to say things in the classroom. Already the nature of the programs we use is having an impact on the sorts of questions we ask and the sorts of ideas we allow ourselves to think about and to accept as valid mathematics.
A pupil who has always used a graphical calculator may think of "sine x" as "how far up you go on your way round the circle", rather than as a ratio between two lengths on a static right angled triangle. Again, it is not totally meaningless to talk about a "four and a half sided polygon" in the context of a Logo procedure for drawing n-sided polygons.
But this sort of example only scratches the surface: soon the computers - the tools we use to think and communicate about mathematics - will have a far greater impact upon that thinking and that mathematics.
The alternative way to think and communicate which computers offer may be one reason for the rather surprising manner in which they seem to trigger changes in the ethos of the classroom, and motivate otherwise unco-operative and challenging pupils. Another factor involves the relationship of the pupil to the computer.
Pupils are commonly put into sets for mathematics in secondary schools. The rationalisation is that they lie along a continuum of ability: [Graphic not available] Pupils in a high set are more likely to take it on trust that they need to learn about, say, Pythagoras' Theorem, and to practise it.
Pupils in a low set are more likely to present challenging behaviour: they won't do their maths just in order to please the teacher and get good marks.
They do not see "pleasing the teacher" or "getting good marks" as worthwhile objectives. So there is another correlation: [Graphic not available] This implies a third correlation: [Graphic not available] But this correlation may be false. If so, then it could give some further insight into the reasons behind the surprising impact of computers on the ethos of many classrooms: computers are not upset by challenging behaviour.
Rather, with well-written software: * Computers give you instant feedback * Computers will let you try things out a different way * Computers never put you down if you get it wrong * Computers will always let you have another go * Computers do not mind people swearing.
Pupils who have been handicapped in the conventional classroom by their refusal to co-operate with the teacher are much more likely to respond positively to work with a computer. This may apply in any classroom, but it is particularly relevant in the case of many statemented emotionally and behaviourally disturbed pupils. Such pupils often have great difficulty learning to trust and relate to any adult - and particularly teachers, who confront their challenging behaviour and may seek to control it. The computer helps to put the pupil in control of their learning, and does not get involved in behavioural issues.
Tandy Clausen is a member of the executive council of the Association of Teachers of Mathematics. She has taught in a number of mainstream and special schools, and now works at the National Foundation of Educational Research