General Concepts and Mathematics TeachingAuthor(s): David WellsSource: Mathematics in School, Vol. 17, No. 5 (Nov., 1988), pp. 27-29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214540 .

Accessed: 13/04/2014 04:10

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 82.164.94.28 on Sun, 13 Apr 2014 04:10:34 AMAll use subject to JSTOR Terms and Conditions

Fig. 1 A good bad helpful misleading model of a molecule.

General Concepts and mathematics teaching

by David Wells

First Perspective How can any ordinary reader appreciate an article in Scientific American? Certainly not because he or she has a technical familiarity with the subject of the articles. Even professional scientists do not have such a broad technical knowledge outside their own specialist field.

Yet neither can he or she, at another extreme, 'make sense' of the article without any prior knowledge at all. So what kind of knowledge is needed?

One kind of useful knowledge is provided by models of molecules in which the atoms are small spheres of different sizes or colours, joined by rods.

Such models are crude but effective, simultaneously true and false, helpful and misleading, brilliantly illuminating and grossly oversimplified.

How is this kind of knowledge relevant to mathematical education?

Second Perspective Discussions of content and process often seem to be about content versus process. Traditional mathematics teaching focussed on content exclusively. It must now be changed to focus on process.

This tendency naturally upsets those who argue, quite correctly, that pupils do need to know some content, and that indeed content is very important.

It is also undermined by the existence of many ideas which cannot be classified as either content or process.

For example, any mathematician knows that the sum of an infinite series of terms may be a different kind of object from the individual terms. Thus a series of fractions may sum to the irrational and transcendental number x.

1 1 1 1 + + + ~-+ .... 1 1.2 2.3 3.4 4.5

1 1.3 1.3.5.7 1.3.5.7.9.11 2Z --+ + + + -.... 2 2.4.6 2.4.6.8.10 2.4.6.8.10.12.14 2

1.2 1.2.3 1.2.3.4 1.2.3.4.5 n --+

+ - + +...

- 1.3 1.3.5 1.3.5.7 1.3.5.7.9 2

Fig. 2 How can fractions sum to x/2 of 2 2

Mathematics in School, November 1988 27

This content downloaded from 82.164.94.28 on Sun, 13 Apr 2014 04:10:34 AMAll use subject to JSTOR Terms and Conditions

This idea surprises many students, as it surprised some great mathematicians of the past, who naively assumed the opposite.

It is also a very important idea. Yet it is certainly not part of the content of mathematics as usually conceived: there is no mathematical theorem which states and proves this proposition. Neither is it an aspect of heuristics, of ideas of process, as usually conceived. It won't be found in Polya's How to Solve It, for example.

So what is the status of such ideas?

Third Perspective Teachers who monitor their own conversation with pupils or pupils' conversations with each other, will often hear remarks that are not easily classified as 'mathematical knowledge' in any usual sense, and yet which clearly are useful and effective.

Suppose pupils are finding the centre of gravity of triangles. Someone remarks acutely that any triangle has another centre, the centre of the circumcircle. Someone else, perhaps the teacher, asserts that 'A triangle has many centres.'

Fig. 3 What kind of statement is, 'A triangle has many centres'?

What kind of comment is this? It is at one and the same time subtle and profound, vague and imprecise and true without being a mathematical theorem.

(Pupils of course may not appreciate this difference. Why is the statement about centres not provable, while 'The lines of rotational symmetry of shape are equally spaced', is provable, though on the face of it also rather imprecise?)

Is the statement a bit of mathematical knowledge? Yes, but of what kind? Is it an example of an important kind of knowledge? Yes, but why?

28

Fourth Perspective Mathematics as everyone knows is beautiful, although some particular problems, methods, ideas and theorems have more aesthetic appeal to mathematicians, and some have less. Just to make the situation more complicated, math- ematicians by no means agree in their aesthetic judgements.

"It is true aesthetic feeling which all mathematicians re- cognise ... the useful combinations are precisely the most beautiful." (Henri Poincare)

"I think it is true to say that the mathematicians's criteria of selection, and also those of success, are mainly aesthet- ical." (John von Neumann)

If these aesthetic judgements only appeared after the work was finished, then they might be of relatively little importance, a mere decoration, a small reward for work concluded. But they do not. 'Mathematicians use their aesthetic judgement even while they are solving problems or developing new ideas, as Poincare and von Neumann imply.

The same is true to a lesser extent even of teachers and pupils who pay no explicit attention to aesthetic factors.

For example, trying to fit the numbers 1 to 9 into a 3 by 3 square to make it magic, a pupil might comment that the 5 goes in the central cell because it is the middle number of the given sequence.

9 1

7

4

5

6

2

8 3

Fig. 4 How useful is a feeling for elegance and beauty in solving this problem?

More precisely, the given numbers in sequence are symmetrical about 5, and the geometrical figure is sym- metrical about its centre cell, and the conditions to be satisfied have the same symmetry. So it's obvious, isn't it?

Yes, except that this is not a mathematical fact, only an expectation, a probability. Such vague arguments are not guaranteed to work, and often don't.

Yet they are very valuable, and indeed much of an active mathematician's thinking runs along such lines.

So what kind of mathematical knowledge is involved here?

Mathematics in School, November 1988

This content downloaded from 82.164.94.28 on Sun, 13 Apr 2014 04:10:34 AMAll use subject to JSTOR Terms and Conditions

Fifth Perspective Teachers often seem to assume that they are passing on to their pupils what they, the teacher, knows. Yet pupils always end up knowing far far less than the teacher. This is true even of those rare pupils who can reproduce what they have been taught perfectly. How is this? What is missing?

What is the difference between a good 5th year student's understanding of quadratic equations, and the teacher's understanding?

Example 1. Solve 3x+2y=5,7x-y= 1/3

Solution 3x+ 2y= 5 (1) 7x- y= 1/3 (2)

Multiply (2) by 2 14x- 2y= 2/3

Add (1) and (3) 17x=5 2/3 x=1/3

(Etc.)

Fig. 5 What knowledge and understanding is missing from this textbook explanation?

Resolution: General Concepts Traditionally almost every subject, including mathematics, has been taught as a collection of facts, or techniques.

But our knowledge of the world does not consist largely of facts. It is not certain and precise, but more or less vague, uncertain and probabilistic. We are guided by hunches, feelings, judgements, and guesses.

Some of these possibilities, promising courses of action, useful tips, are brought together under the title heuristics, revived by Polya in his famous How to Solve It, and they form one basis for arguments that mathematics should be taught as an activity, as a process.

However, such heuristics, exceedingly general in form and wide in scope, mostly applying to problem solving in general, and to other subjects apart from mathematics, are only one aspect of our non-factual knowledge. Concent- ration on such broad heuristics may produce as unbalanced a view of mathematics as the previous concentration on facts.

Most of our mathematical knowledge is neither as widely applicable as these heuristics, nor as narrow as the facts in the textbook.

Rather, it consists of what I call general concepts. These ideas do relate to content: the examples I have mentioned referred to infinite series, magic squares and triangles. Yet they are not mathematical facts in the way that theorems are. For example, they are not proved.

They do not, it must be admitted, all share the same features among themselves, but generally speaking they are vague, imprecise, require interpretation, are potentially useful as well as misleading (especially if taken too literally) and can often not be described as either true or false without qualification.

Nevertheless they correspond to an important aspect of the mathematician's or mathematics teacher's knowledge, which the teacher needs to pass on as far as possible to the students.

Mathematics in School, November 1988

Teaching which only gives pupils 'isolated facts' is rightly criticised. Yet a large part of the stuff and substance of understanding which relates facts together, embeds them in experience, and makes sense of them, is composed of general concepts, not factual in themselves. As such, gen- eral concepts occupy the broad continuum between the extremes of traditional facts, and recently emphasised con- cepts of process.

General concepts have other important characteristics, of which I will mention three.

In so far as they can be and are expressed briefly in language they can be thought of as proverbs expressing the accumulated wit and wisdom, the seasoned judgement, of the mathematician. Just as much traditional wisdom was preserved and transmitted through proverbs, saws and sayings, so can the wit and wisdom of mathematicians be passed on.

Small children give you headache, big children heartache

The more you squeeze a nettle, the less it stings Put a pattern in, get a pattern out

A rolling stone gathers no moss Strange things happen at infinity

There are more numbers than you think Good news travels far, bad news, farther

Fig. 6

Proverbs often employ figures of speech. So can general concepts, especially the figures of metaphor and analogy. (The use of the word 'centre' to describe several points associated with a triangle, is metaphorical in origin.) This appearance of metaphor and analogy is appropriate, given their importance in mathematics where they appear under such thin disguises as isomorphism and homomorphism.

General concepts are a wonderful stimulus to problem posing, just because of their combination of some portion of truth with vagueness which demands interpretation. (Many of the seminal developments in mathematics have been driven by attempts to make the imprecise, precise. For example, the ideas of limits and continuity.)

What does 'centre of a triangle' mean? Does a quadri- lateral have a centre? Several centres? Are there any re- lationships between the centres of a triangle? (An important general concept is that when a problem has several solu- tions, there is usually a relationship between them.) How can an equation which is verbal rather than visual, be symmetrical? How many whole numbers are there? Are there as many irrational numbers as rational?

Conclusion General concepts are a large part of the everyday currency of mathematical thought, talk and discussion, whether we are aware of them or not. An overemphasis on either facts or heuristics will distort and impoverish our and our pupils mathematical thinking. By being more aware of what we are doing and how we are doing it, we as teachers can teach, and help our pupils to learn, more effectively.

This article is based on Broad concepts and metaphor, in Three Essays on the teaching of mathematics (1982) and General concepts, and teaching, in Studies of Meaning, Language & Change No. 18 (March 1987), both published by Rain Publications, 6 Carmarthen Road, Westbury on Trym, Bristol BS9 4DU.

29

This content downloaded from 82.164.94.28 on Sun, 13 Apr 2014 04:10:34 AMAll use subject to JSTOR Terms and Conditions