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.

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.

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.

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.

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.

I would like to offer the following contribution to the discussion on the Choice of programming language for schools which will (I hope) ensue following F. R. Watson's and David Tali's article in the December 1987 Gazette. (lam happy to consider other points of view for publication: Ed.) As the authors observe, if the schools do not have a policy of their own, they may well have one they don't like foisted onto them.

A tree is a network of vertices connected by edges such that the removal of any edge disconnects the network. In the language of graph theory a tree is a connected simple graph in which every edge is a bridge. Here are all the trees with 4 vertices.

For a given positive integer n, the nth triangular number, Tn, is defined by

it is consistent to extend this definition by writing T0 = 0 and T-n-1 = Tn, so that the above formula holds for all integers n, whether positive, zero or negative.

Some mathematics problems, while easy to pose and visualise, can be deceptively difficult to solve. For example, suppose we want to find the area of a three foot wide footpath around the edge of an elliptical shaped swimming pool. The problem is trivial if the pool is circular, in which case we simply subtract the areas of two concentric circles. At first blush, the elliptical pool problem does not appear to be much more difficult, especially if we can recall that the formula for the area of an ellipse is A = πab where a and b are respectively the lengths of the semi-major and semi-minor axes. In any case, we might think that it would be a straightforward calculus problem involving the area between two curves.