A puzzle which introduces simple ideas relating to deduction within a mathematical system in a concrete and problem-oriented way is the following:

The natural numbers are the elements of the set N = {1,2,3,…}, the integers are the elements of the set Z = {0,± 1,±2,…} and the rational numbers are the elements of the set Q = {p/q:pεZ, qε6 N}. A real number is called algebraic if it is the root of a polynomial equation

P(x) = anxn + an−1xn−1 +…+ a1x + a0 = 0

We are looking for all triangles with sides of integer lengths a, b, and c say, with an angle of 60° opposite the side of length a.

The Greeks were the first real mathematicians. Littlewood expressed this thought in the following terms; they were not clever schoolboys or scholarship candidates, but Fellows of another College. The sentiment is universally accepted by mathematicians, no matter how clever we consider the ancient Egyptians, nor how under-rated the achievements of the Babylonians (nor indeed how inscrutable the works of the ancient Chinese). But if the Greeks were the first in the field, who was the first of the Greeks?

If you were to ask a random sample of students from your school if they liked mathematics, what do you suppose their responses would be? Almost certainly most of them would respond negatively and some would do so with considerable vigour. But what of the others? Some students do like mathematics and we might well ask why this is so? Do you suppose it is because these students realise that knowledge of mathematics is a necessary survival skill in today’s increasingly technological society? Surely the answer is no. Do you suppose it is because they realise that mathematics has many useful real life applications or that lack of knowledge of mathematics will effectively prevent them for entering college in most disciplines? Again the answer is almost surely no. Certainly mathematics is relevant and we should teach our students that this is so and we should teach them useful and appropriate applications as well. At the same time, it is my experience that student attitudes towards mathematics are much like they are towards exercise—in general, they do not exercise because it contributes to good health, but because exercising is usually done in the form of a game or a competition that they find attractive in and of itself. Similarly, what turns students on to the study of mathematics and provides the needed motivation for learning are the remarkable patterns and surprising results with which our subject is so replete. Successful teachers realise this fact and utilise it to maintain a high level of interest and enthusiasm which enables them at the same time to teach ideas which, while useful and necessary, may be more prosaic.

I have just received my latest transfusion of fresh ideas. What teacher can be enthusiastic day after day, year after year, without pausing to renew his strength, revitalise and remotivate himself? We need occasionally to be fired by contact with better minds than our own, to observe excellent teaching and thinking to remind ourselves of what they are like. For such excellence is the ideal we all should seek. It is so easy to have one’s spark attenuated year by year by the dull monotony of ritual in a school’s monastic life—so that one hardly notices.

It has always seemed to me that one of the most fascinating parts of learning mathematics is the recognition of an inter-relation between what seemed before to be totally unconnected branches of mathematics. These relationships, which often help to cement ideas in both branches of the subject, are very often not discussed in school text books, perhaps because authors worry that they may be assuming too much background on the part of their readers.