Mathematics as Patterning.
"A mathematician, like a painter or a poet" wrote G.H. Hardy, "is a maker of patterns"(1). Pondering from patterns in shapes to patterns in symbols gave rise to a series of reflections; following these, I decided to undertake the somewhat unusual and. eccentric task of proposing a sort of philosophical programme for understanding what Hardy once referred to as "mathematical reality"; in particular to provide a programme for a mathematical education.
I call my story "philosophical" to underline that it is tentative, speculative and incomplete. If something like it has been said before, I apologise for breaking an "intellectual copyright", but I plead, ignorance of other attempts like this. Perhaps my critics could put the record straight?
Mathematics has its roots in morphology - the study of shape. This begins with "imaginings" or "mind-pictures" (to use the phrase of Mary Boole(2)). Here we are asked to imagine shapes and- the motion of shapes - transformations. This gives us a sort of "mental film" upon which to work.
Further to this we may focus upon the patterns that are "traced" by the motion; for instance, a moving line of finite length, rotating about one end, will trace out a circle. But an alternative to this attention upon the trace is to focus upon two or more frames in the sequence of the film; for instance, the moving line at 12 o'clock and the moving line at 3, 6 and 9 o'clock.
In the former we are emphasising the process of transformation; in the latter we are emphasising the product of transformation, the event of change. Both are legitimate and necessary for mathematics as morphology, both have correlatives in quantitative mathematics.
Rene Thom, creator of Catastrophe Theory "believes that the provision of some kind of picture, at least to the mind's eye, is of primary importance"(3). Certainly it would help mathematics education, for Margaret Donaldson in "Children's Minds" has called attention to a domination of the visual in "both children and adults (4); this would also provide what Imre Lakatos refers to as the "quasi-empirical" in mathematics (5). There would appear to be pretty good grounds for grounding mathematics in a study of shapes - in morphology.
Now morphology is merely Qualitative, i.e., it consists of patterns and shapes - of pictorial representation. But mathematics is more than this; the next step is to measure (quantity) those representations.
This initial step is morphology, where the concern is to create measures for our forms and transforms; later there can "be a shift of focus from the process of measurement to the measures themselves.
By this further focus of attention, the background morphology "fades out" until we are unaware of its presence although it still functions indirectly in guiding our manipulation of symbols. This manipulation of measures without direct reference to forms and transforms is algebra.
If we lose sight of the underlying morphology and morphometry (which together may be called "geometry"), then there may be a breakdown in communication because there exists no shared background geometry between teacher and pupils.
It is important to remember this, because the uncritical assumption of background geometry will often result in empty "arm waving".
I should like, at this point, to propose a strong similarity between symbols in mathematics and metaphors in English; for information on the latter, I shall be referring to C.S. Lewis's "Rehabilitations", with particular reference to an essay therein entitled "Bluspels and Flalanspheres" (6).
When we are use a metaphor, "we are often acutely aware of the discrepancy between our meaning and our image"(7) similarly, there is a "gap" between the use of a symbol and the geometric background from which it is derived.
When we are given a metaphor, "we are entirely at the mercy of the metaphor. If our instructor has chosen it badly, we shall be thinking nonsense. If we have not
got the imagery clear before us, we shall be thinking nonsense. If we have it before us without knowing that it is metaphor.. then again we shall be thinking nonsense"'(8)
When we are given a symbol, or symbolic group (e.g. for the area of a circle), we are at its mercy. If it is given just as an algebraic formula, then we are likely to misunderstand it. If we have no clear conception of the geometry (e.g. enlargements), we shall be confused. And if we forget that the symbols do refer to a geometric background, we shall be as much at sea as Hillbert.
However, the metaphor can later lose its original meaning; this is "a temporary dress assumed by my thought for a special purpose, and ready to be laid aside at my pleasure; it did not penetrate the thinking itself, and its subsequent history is irrelevant." Likewise, we need not continually be conscious of an underlying geometry; once this has given us adequate control of the symbols, it can be relegated to the recesses of memory.
What is important about this process is that the metaphor can be utilised irrespective of original imagery only if I continue using it in the context (subject) in which it first arose. In other words, I can dispense with the underlying imagery if I have gone on to acquire such new knowledge that enables me to dispense with it. However, if this is not the case, and. if I lose touch with the original imagery, then the metaphor becomes "only a noise" (10).
I think this is the same with symbols. If we explore the geometry further, we can dispense with geometry that underlies our starting symbols; but if we don't explore the geometry further, then if we lose touch with the initial geometry, our symbols become only alphabet soup.
Returning now to the main discussion, I would, like to consider the shifting of attention (which I refer to as focusing). This involves a stressing of some things and ignoring of others. But I am at a loss whether this should be done directly (by teacher) or indirectly (by pupil). Obviously, the teacher cannot depend, upon a necessary focusing by the pupil, although this may happen; on the other hand, it would seem that too swift a change of focus by the teacher might result in a loss of comprehension. The problem is certainly a complex one and has no easy answers.
On a practical level, morphology may be initially difficult, but it should not be neglected for this reason! For it is needed as a background (which can gradually fadeout later). It can be aided by visual helps such as models and films.
The morphometry should, "fit" the morphology, i.e., be appropriate and not indulge in ad hoc jumps (e.g. area of circle as enlargement or plausibility). Where possible, it might be aided, by free-ranging formation of equivalence classes (11), followed, by a focus directed within the equivalence class (as the human decision of the teacher).
The decision of the teacher about where to focus attention is understood as a valid, form of academic autonomy; other interests can be pursued in the pupils' spare time, although if they pursue other lines and these seem interesting for the whole class to look at, then. the teacher may allow their ideas to receive public recognition.
The principle of academic autonomy that I am following holds that the pupils have a right to criticise the teaching of mathematics and to have their criticisms heard, but do not have the right to control the teaching of mathematics; obviously this will be more valuable for sixth forms than first forms (12)
This concludes a preliminary survey of mathematics as patterning.
1: "A Mathematician's Apology" (1976) p. 84
2: "A Boolean Anthology"(19 ) ed. D. Tahta p.
3: "Catastrophe Theory"(19?8) A. Woodcock p. 13
4: Op. Cit. p,65 (1979)
5. "Mathematics, science & epistemology", Lakatos (1978) pp. 30-35
6: Op Cit. Lewis tends to confuse metaphor as referring to a word and to an imagery; I always use it in the former sense and have accordingly adapted, his points.
7: Op. Cit. p.138 (1939)
8: Op. Cit. p. 140
9: Op. Cit. p. 143
10: Op. Cit. p. 146
11: That equivalence classes are fundamental to comprehension, see "The Development of Mathematical Activity in Children: The Place of the Problem in this 'Development" (1966;ATM) where D.G. Tahta &W.M. Brookes place it at the "genesis of mathematical activity" pp. 3-8
12: "Mathematics, science & epistemology"(Lakatos) pp. 247-253 for a discussion of issues involved.
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