A simple lock contains a bolt and one or more levers which are lifted the correct amount by the appropriate key, thus allowing the bolt to be moved in and out. In addition there are often circular obstructions or “wards” to stop some keys being turned in the lock.

In 1906 the editor of this journal made a plea for more articles on “the purely pedagogic side of mathematics”. As he emphasised: “the object of the Gazette is to improve the teaching of our subject in our schools”. Since then such articles have appeared with varying frequency. What rarely appears is any article specifically devoted to research in mathematics education. The divorce between mathematicians and researchers in mathematics education, often made worse by the former’s distrust of the latter, would seem doubly unfortunate. In this article, based on part of a talk given in Madrid, I wish to look briefly at how research in mathematics education developed and at some of the difficulties associated with it.

The notion of Morse code is familiar to many school pupils through interests outside of mathematics and computer studies. How can this knowledge be related to algebraic coding theory? The difficulty is that mainstream algebraic coding theory deals with block codes—in which the codewords are of uniform length. Morse code has codewords of variable length in which short codewords are used to represent common characters and longer ones for representing characters which occur less frequently (like. for E as opposed to —..—for X).

For the 3 x 3 matrix A = [a ij], the trace t of A is the sum of its main-diagonal entries:

The characteristic equation of A is

a cubic equation in λ which can be written as

where Δ is the determinant of A and

A very appropriate use of a computer in a mathematics class is to provide investigation vehicles for formulating conjectures to be verified theoretically. In other words, first we notice something and then we try to prove it. In this paper we shall reverse the process, first arguing the plausibility of a very fascinating result and then obtaining confirmation by looking at some computer generated data. The argument is based on the work of the eighteenth century mathematical giant, Leonhard Euler (1707-1783).

The motivation for this article was the desire to make the cosine rule in some way geometrically “obvious”. What we have ended up with is a mixture of inductive and deductive approaches which we hope goes some way to illuminating the cosine rule, and to enhancing students' relational understanding of it. The germ of these ideas can be sown soon after students have met Pythagoras, with the question “O.K., suppose the triangle ABC is not rightangled: what can we say about a2, b2 and c2 now?”

In teaching a sixth form class recently it occurred to me that there must be a more gentle, perhaps more intuitive, approach to this result than is normally found in textbooks and that some Advanced level mathematics candidates really need a simpler approach. What follows is an account of one lesson. I make no claim to originality, for I find it hard to believe that others have not previously thought of such a simple introduction. However, other teachers did indicate that they had not seen this approach before.

A familiar result is that in any group of six people there are three who either all know each other or are all strangers to each other. This can be reformulated to say that, if six points are marked on a piece of paper and every pair of points is joined by a red or blue line, then there will be a monochromatic triangle, i.e. one which is all red or all blue. This result is related to a theorem which the Cambridge philosopher Frank Ramsey proved in 1930.

Two players A and B have a game of squash. Player A serves at the start of the game. They play a sequence of rallies. If the player serving the rally wins the rally, he scores a point, and serves the next rally. If he loses the rally, then the other player serves the next rally, with the score unaltered.

Generally the first player to reach nine points wins the game. However, if the score becomes eight-all then the first player to reach eight has a choice: he specifies whether the game is to be played to nine points or to ten ponts.

What is the correct choice? We propose to analyze the problem by use of an elementary probabilistic model.

The “real” in the title refers to real numbers rather than real life. I expect, and hope, that many articles on the future of A-level will expound the cognitive and mathematical necessity of making A-level more practical. While I applaud and try to practise this, classroom teaching and research have convinced me that the notion of the real number line has subtle conflicts that will not go away by simply not mentioning them; they need to be addressed. Moreover teacher-led explorations of the real number line are essential if we are to lead students to an understanding of higher mathematics.

As a mathematician involved in teaching students whose abilities range from barely numerate to MSc level, I am frequently concerned by the lack of basic understanding exhibited by pupils regarding the subject of integration. Invariably the most practical way of introducing the subject of integration is by thinking of it as “anti-differentiation”, so that the problem is to pick a function which “when differentiated will give what you first thought of”. This of course may be allied to the idea of the area under a curve to give the student a conceptual feel for the subject. Usually when simple rules for the integration of powers, trigonometric functions, exponentials, logarithms and so on have been mastered, the next step is to introduce the standard methods for proceeding.

It is my purpose to establish a set of inequalities between symmetric homogeneous polynomials in several real variables. That may sound a bit grand, but the simplest example of such an inequality is the familiar

A more complicated inequality of the same general type is the inequality

Again increasing the degree of complexity we have the pair of inequalities

I am looking forward with great enthusiasm to the arrival at advanced level of the GCSE student who has experienced a problem solving approach to mathematics through investigations. I hope examination boards are equally enthusiastic. Here for them is the opportunity to broaden their approach to syllabus content and methods of assessment. Can we hope that they are already working with this in mind? Advanced level mathematics courses, particularly in the options of applied mathematics, will need to nurture the good work of the GCSE courses and allow sixth-forms to produce a student capable of modelling real problems, in terms of sound and necessary mathematical techniques.

A ball is shot at an angle of 45 degrees from the corner of a rectangular billiard table of size m × n (where m and n are integers). It always bounces off from a side at 45 degrees until it lands in a corner pocket. Will it always land in a corner or will it keep bouncing for ever? And how many times will it bounce? And which corner pocket will it land in?

Try it for yourself. For example in the 7 × 4 table shown the ball will bounce 9 times and then fall into the top left-hand corner pocket.

This is a well-known problem but we now present a simpler proof than we've seen elsewhere of the fact that the ball always ends up in a corner pocket, and we derive a formula for the number of bounces and we determine the finishing pocket.

The set of whole numbers less than and coprime to n, with the operation of multiplication modulo n, forms an Abelian group (try to prove it for yourselves). We'll write Mn for this group, and call all such groups modulo groups or M-groups. M-groups are easy to compute. It is also particularly easy to calculate quotient groups of M-groups, making them a good source of examples and practice for the beginner. There are also some very hard, possibly unsolved, problems in classifying M-groups. In attacking such problems there is an interaction between number theoretical and group theoretical arguments which makes this a rich topic, for projects or personal pleasure.

The Open University has developed a model (Calculators in the primary school) which identifies two components in the mathematics curriculum. First there are lessons in which specific techniques or concepts are learned. Second, whether in mathematics lessons or in other subjects, opportunities arise for techiques to be used (often through some form of problem solving). The point is made that these two aspects are interdependent; the learning of techniques paves the way for future application, while the using of techniques provides a purpose for the learning of mathematical ideas.

Even the purest of pure mathematics can have a crucial influence on practical problems. In this article we show how a topic in pure mathematics (modular arithmetic) originally pursued for its own interest only, turns out to have unexpected application to an area of communication theory (cryptography). The fact that at the present time it is easy to construct large prime numbers but very difficult to factorise large composite numbers has made it possible to devise simple codes which are uncrackable by known methods.