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Gardner’s Whys and Wherefores by Martin Gardner, Oxford University Press,
pp 261, Pounds sterling 5.95

The Puzzling World of Polyhedral Dissections by Stewart T. Coffin, Oxford
University Press, pp 191, Pounds sterling 17.50

How to Solve It by George Polya, Penguin, pp 253, Pounds sterling 5.99

QUIETLY, the teaching of mathematics in Britain is being transformed.
Teacher shortages notwithstanding, the message that Martin Gardner has been
preaching all his writing life is beginning to win through. In his view
‘the best way to motivate youngsters is to give them tasks they actually
enjoy. By first capturing a class’s interest with a good puzzle, paradox,
game, model, toy, joke or magic trick, the teacher can lead students into
significant ideas without the students even knowing they are learning.’

It is a rare teacher who can entrance a whole mixed ability class in
this way, but by providing individual children or groups with well-prepared,
correctly structured activities, the trick can indeed be turned. Over the
past few years publishers and teachers have produced much material of this
nature, and its use in the mainstream of mathematics lessons is growing
as a new generation of enlightened teachers enter the classroom. And pupils
entering for maths GCSE now have to submit miniprojects called ‘investigations’
that comprise the work they produce as a result of such activities.

This development owes much to Gardner, who in his Scientific American
column has month by month provided educators with many ideas for topics
to be included and methods of presentation. But his new compilation of essays
and reviews, Gardner’s Whys and Wherefores, yields little material of this
kind. It also lacks examples of his famous diatribes against the paranormal
‘scientists’ and other charlatans. The work presented in this volume indulges,
rather too heavily perhaps, Gardner’s own idiosyncrasies: a typical example
is the quite long discussion of the not particularly fascinating word puzzles
in Ulysses. Nevertheless, there are some gems, such as the tribute to Japanese
writer on mathematics, Mitsumasa Anno.

The book, in short, is for the Gardner fan; new readers should turn
to his earlier works such as Mathematical Puzzles and Diversions, Order
and Surprise or the Whys of a Philosophical Scrivener, depending on which
of his obsessions they find closest to their own hearts.

As a devoted follower of Gardner myself, I thought I knew something
about polyhedral dissections. That is the technical term for ways of cutting
up a flat or solid block of regular shape in order to reassemble it to form
a new pattern. The popular Tangram puzzle, for example, is a polyhedral
dissection of a square into simple geometrical shapes (five triangles, a
square and a parallelogram) that can be reconstructed into millions of different
pictures and symbols.

But Stewart Coffin’s book The Puzzling World of Polyhedral Dissections
soon showed me how much I did not know. It dismisses Tangrams in a few sentences
and goes on to discuss numerous variations, posing some challenges to the
reader along the way. But Coffin’s real speciality is three-dimensional
puzzles, of which he describes an enormous variety.

He suggests starting with a set of simply defined but often dissimilar
and asymmetric pieces, and then teasing out a way to construct a highly
regular whole. ‘An eye for symmetry,’ Coffin writes in one of his few brief
philosophical asides, is ‘one of the most basic and deeply rooted instincts
of mankind . . . a symmetrical solution represents a small but significant
human victory over the natural tendency in the world toward disorder and
randomness . . . it is the one point to which all paths lead upward and
from which one can go no higher.’

Most of us can identify with that. But be warned: Coffin is a true obsessive
and the bulk of the book is addressed to the specialist or would-be specialist
maker and puzzler. That said, those with only a casual interest might be
entertained – and even seduced – by a browse in library or bookshop. The
same is true of the other five titles in Oxford University Press’s ‘Recreations
in Mathematics’ series edited by David Singmaster of Rubik’s Cube fame.

The new trends in the teaching of mathematics make the reissue of George
Polya’s classic How to Solve It most welcome. As Ian Stewart points out
in his excellent new preface, Polya was ‘a first-class research mathematician,
a brilliant teacher and an able expositor’. His book aims to provide students
with food for thought about the basic strategies involved in tackling mathematical
problems. In doing so it is almost unique, and certainly uniquely successful.
This approach has its weaknesses. It’s rather like a ‘teach-yourself-to-drive’
book: helpful to start with, but ultimately useless without the practice
that, along with the instruction of a good teacher, makes perfect. And,
as Stewart points out, Polya rather loses touch with the question of how
to decide if the problem was worth solving in the first place.

It would be nice to be able to write that material based on Polya’s
ideas should be avail able to mathematically-inclined youngsters. A watered-down
form could be useful for GCSE investigations. But it is a waste of time
for A-level candidates, as the exams are based very much on bookwork and
offer little opportunity to develop the lateral thinking skills required
in problem-solving. There is still some way to go in making Gardner’s dream
come true.

Ros Herman is a teacher of mathematics and a science writer.