Motivating Mathematics

The following sections are included:

• Also by David Wells
• Preface: Themes and Sources
• Acknowledgements
• Contents
• Preamble: Young on The Teaching of Mathematics (1907)

Mathematics is a notoriously hard subject that many adults are happy and unembarrassed to admit they flunked at school and which many current students avoid in making their examination choices, preferring ‘easier’ subjects. So it is obvious, is it not, that motivation must be a key theme in any discussion of school maths? Yes, and no. Yes, it is obvious but no it is not a key theme, in fact it gets very little attention at all.

The following sections are included:

• What's the point?
• A nasty experience
• Science in mathematics textbooks: science as a motivation
• Absence of proof for most pupils
• Anxiety
• Syllabuses crammed for the sake of stronger pupils
• Time pressure and lack of time
• Vicious circles of decreasing expectation (and motivation)
• ‘Maths is easy’
• Sets that are too simple
• ‘Maths is fun’
• Steep curves of difficulty
• The steep ability curve in mathematics
• Matching curves of difficulty and ability
• The end results of mathematics teaching
• Bill Thurston

The following sections are included:

• Philosophy
• Extrinsic and intrinsic motivation
• Motivating pupils through the pupil's own activity
• Maximum motivation
• When is a problem solved?
• Puzzles, problems and exercises
• Problems, puzzles and motivation
• Changing expectations
• Choosing your own level of difficulty
• Taking things slowly: turning a steep cliff into a gentle slope
• Dynamic versus static
• Surreptitious learning in rich environments
• ‘Slow learning’
• Meaningful learning, understanding, and intuition
• Understanding vis à vis practice
• Levels of understanding and applications
• Intuition

Normal school teaching nods at conceptual structures but ends up by teaching facts and skills (poorly), largely ignores ‘general strategies’ and totally ignores appreciation. A grave mistake.

A basic principle of all teaching is to put the subject in context: this does not mean making it so directly relevant to the pupils' lives that they are prevented from understanding anything ‘beyond their ken’, as some right-wing critics of education would claim. It does mean making lots of meaningful connections…

Mathematicians have frequently lauded the beauty of mathematics and their best known quotes have become clichéd, so here are just two of the most famous: I think it is correct to say that (the mathematician's) criteria of selection, and also those of success, are mainly aesthetical. (John von Neumann 1947) It is true aesthetic feeling which all mathematicians recognise. The useful combinations are precisely the most beautiful. (Poincaré 1930/2003:59) Poincaré also gave a warning which will be recognised by anyone who has enjoyed problem solving: When a sudden illumination invades the mathematician's mind … it sometimes happens … that it will not stand the test of verification … it is to be observed that almost always this false idea, if it had been correct, would have flattered our natural instincts for mathematical elegance. (Ibid.) Pure mathematicians especially have enthused over the extraordinary beauty of mathematics — and sometimes been led astray as Poincaré warned — but the use of mathematics in the physical sciences has also astonished and delighted, the more so as the sciences generally have become more and more mathematical…

Which of these claims is correct?

Proof is a central concept in mathematics: indeed, it is the feature that distinguishes mathematics (and abstract games) from science, which proves nothing, though scientists often claim, for example, that they have proved that the background radiation they have detected really is from the Big Bang. No, scientists can never escape from the possibility that tomorrow evidence will be discovered to undermine their pet theory, even if it does not completely demolish it…

Three very general concepts or similes — or metaphors — illuminate the nature of mathematics: it is like an abstract game, a science, and is a question of seeing.

As an abstract game or collection of games, mathematics requires the player to see and observe, to make moves, to plan ahead, to develop an understanding of tactics and strategy and appreciate the power and elegance of mathematical arguments much as a good game of chess can be appreciated. Unlike the chess player, however, mathematicians repeatedly change the rules of the game, add new rules, new pieces and so on…

For nearly 2,000 years, Euclid and his Elements of geometry were seen as the perfection of mathematics and mathematical reasoning. The deep method by which he presented his conclusions, starting from a few simple and apparently obvious (with one exception) assumptions, seemed to be supremely logical. The wealth and variety of his conclusions from the very simplest to the extremely complex — and the far greater number discovered by his successors — was astonishing, as was their beauty and elegance…

Mathematicians have always experimented, observed, and then drawn conclusions. Philip Davis asks, How were the theorems of triangle geometry discovered? I can only conjecture that as with much of mathematics, it emerged from long hours of ‘playing around’. (Davis 1995:206) The counting numbers also invite ‘playing around’ and conjecture, so it is no surprise that according to G.H. Hardy, The theory of numbers, more than any other branch of pure mathematics, has begin by being an empirical science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they have been proved; and they have been suggested by the evidence of a mass of computation. (Hardy 1920:651) Needless to say, ‘playing around’ is a very enjoyable and motivating activity, and spotting patterns in a mass of data is especially rewarding: it certainly provides the ‘kick’ that Hardy used to explain the attractions of puzzles…

Very complex processes of perception lie behind game playing and science and indeed all our activities. The game player is said to have more-or-less quick ‘sight of the board’ as well as seeing — or failing to see — structural connections between the features of a position which guide him in selecting his moves…

It might seem obvious that the hard sciences which most pupils study in parallel with mathematics should be a major source of motivation for maths students. Amazingly, this is not the case. As we have seen, maths textbooks and syllabuses have little to say about applications in general and almost nothing about science in particular. Teachers and educators in these subjects seldom talk to each other and professional subject conferences are notable for the lack of cross-fertilisation…

The following sections are included:

• Extrinsic and intrinsic motivation in other countries
• Keeping pupils together
• Accepting differences in ability
• Differences in expectations: Continental and Oriental classrooms
• Ability versus effort
• France
• Hungarian classrooms
• Japanese classrooms
• The USA
• Problem solving in the USA
• English mathematics education
• Russian lessons
• Paul Andrews on TIMSS

Rich complexity — provided it can be grasped — is much more attractive than a false simplicity which is claimed to be ‘easy’ (and maybe ‘basic’ too) but which can only be appreciated as such by a sophisticated taste. Mathematics that is too simple or falsely simple especially invites the response, ‘What's the point, miss’…

The following sections are included:

• Appendix: Jennifer Kano's Letter
• Author's Bibliography
• Bibliography
• Index