The number of the Mathematical Association’s new headquarters in London Road, Leicester is 259.

This number has some quite interesting features. It is for instance the sum of the three squares

32 + 52 + 32 × 52,

and the sum of four cubes

23 + 23 + 33 + 23 × 33 and 13 + 23 + 53 + 53.

All teachers of mathematics are aware of variations both in notation and definitions between different books; in addition, the conventions used in science often differ from those used in mathematics. Readers may have seen the Royal Society booklet Quantities, units and symbols and the report SI units, signs, symbols and abbreviations issued by the Association for Science Education. A sub-committee of the Mathematical Association’s Teaching Committee has been studying this subject with the aid of examples and proposals submitted by members.

Over the past few years more and more schools have been able to gain access to computers. Many have their own terminals, and can work on-line, others send their material to a centre for batch processing. This article looks at some examples of the way such computing facilities can be used in the teaching of statistics. Clearly much more work can be done with interactive programs carried out on-line, but a great deal can be done with off-line facilities. Two difficulties with statistics are that experiments can take a long time and that the resulting computations can become tedious. In both cases the main statistical ideas can get lost in the sheer drudgery involved. One undeniable advantage of a computer is that it does things quickly. It is putting these two facts together that can make computers a useful tool in the teaching of statistics.

Suppose that Pn is some proposition about the integer n, which we want to prove for all n≥n0 (usually n0 = 0 or 1). The form of inductive argument most commonly taught in schools is the following:

A. Simple induction. If Pn0 is true, and Pn ⇒ Pn+i for each n > n0, then P" is true for all n > n0

When I started teaching real analysis to first-year college of education students, I thought that at least I ought to be consistent in what I said. When I told the students what a word meant, I thought they ought to hear the same story as they heard from my colleagues who were teaching other branches of mathematics. I even thought I ought to tell them something like what they would find in a consensus of the real analysis books they might look at. I knew that I ought not to use different meanings for the same word on different occasions. Alas . . .

Peter is a lively lad. All systems are at ‘go’. He will usually eat when he’s hungry, like any healthy boy.

There was no trouble about lunch last Saturday. It was roast chicken with bread sauce—one of his favourites. He tackled it with relish, and we were smoothly through to pudding at a brisk rate of knots.

But today things are different. He’s hungry all right, and it’s roast chicken again. But he won’t make a start. Why not? Now that he’s six he knows what he likes, and Mum has forgotten the bread sauce. Would you believe it? But it’s too late to make some now, so we shall all have to be satisfied without it.

In a game where there are two ways of adding to your score, but one way gives you more points than the other, some scores may be impossible to obtain. For example, you could never achieve 19 points in a game where each score was either 4 points or 9 points.

Some time ago, Geoffrey Matthews and others introduced the idea of ‘shopping’ matrices as a method of introducing matrix multiplication in schools, and it recently occurred to me that these shopping matrices might also provide a palatable realisation of something that undergraduates traditionally find pretty indigestible when they first meet it, namely the idea of dual space, and particularly the transpose or adjoint mapping associated with a given linear mapping between two vector spaces.

Some time ago I had the opportunity to pay a visit to Farawaystan to study how they treated different educational questions. In the following I am going to tell about my strange experiences. One of the first lessons I attended was in tool science. Here is a section from the textbook:

“3.1 definition. A screwdriver is a metal cylinder, of which one end is flattened and the other end is fixed in a cylinderlike, stiff material, whose surface is not smooth. If the material cylinder is rotated through an angle a, the metal cylinder will rotate through an angle b in the same direction and of the same size as a. The diameter of the metal cylinder is smaller than that of the material cylinder. The material cylinder is often called the handle.

In his Newtonian studies Alexandre Koyré remarked that, from the area law of planetary motion, Kepler erroneously deduced that the planets move in their orbits with speeds inversely proportional to the distances from the sun. It was this assertion, which I suspected to be false, that led me to make a detailed study of the discovery of the area law. Quite independently and unknown to me, Curtis Wilson had been making a similar study of the discovery of the elliptical orbit. Since the arguments leading to the two Keplerian laws are inter-dependent, our researches may be described as variations on the same theme. What I wish to do this evening is to explain in simple terms our reconstruction of Kepler’s path to the elliptical orbit and the area law.

The result obtained in this short paper was discovered by chance in the course of constructing some examples for the first-year students of numerical analysis. It may therefore be of particular interest to others who have similar duties to perform. The relevant example from those which I had set is:

Call the ends of a ruler A and Z, and (towards A) place on it a mark X. Draw a line BC the length of AX, and a square BCDE. Erect a perpendicular bisector of BC. With centre C and radius CE draw a circle. Now place the ruler (Fig. 1) so that AZ passes through B, with A on the perpendicular bisector of BC, and with X on the circle. Then ∠BAC = π/7.