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.

Nothing can convince us more that “things are not what they seem” than a comparison of the descriptions by two men of what they remember of Cambridge in the seventies ! I should of course recognise my friend Professor Forsyths picture of the Cambridge of that time and am grateful for his calling to my mind so many memories of the past. But for me he has painted them in far too drab a colour, so that one wonders whether his undergraduate days could have been like mine some of the happiest in his life. The reading of his paper enforces upon me the truth of the principle that environment and personalities are not things in themselves, but are our own “constructs”—each of us giving our own “reality” to such evasive phenomena.

The tide of technological advancement nearly swept away this article. I have had to rewrite this first section in the light of new developments, which I suspected might beat me to publication. For several years I have wondered why ‘pocket’ calculators, however well specified in other respects, have not had the pre-programmed ability to give values of the cumulative distribution function of a standard normal variable, commonly written as Φ (x), and its inverse. The A level mathematics student must be a prime target in the marketing of calculators and there can’t be many who don’t have to thumb through statistical tables a few hundred times in the course of their studies. The very latest generation of machines do include these functions, but it will, I suspect, be quite a few years before everyone is so well equipped.

Doublet tasks consist of changing one word to another by altering single letters at each step to make a different word. The two words at the beginning and end of such a chain must, of course, be the same length, and they should be related to each other in some obvious way. They must not have identical letters in the same positions. All words in the chain should be common English words, proper names excluded. A perfect solution has a number of steps equal to the number of places the given words differ. For example: COLD, CURD, CARD, WARD, WARM. If a perfect change is not possible, the best solution is the shortest chain. For playing doublets as a game with two or more players, Lewis Carroll invented a set of scoring rules to determine who wins.

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).

Some of my distinguished predecessors have ably dealt in their Presidential addresses with questions that are most closely related to the aims of our Association, namely, the teaching of Mathematics in Schools. Others have sought to speak on subjects in which they were actively interested. I have followed this precept and accordingly I propose to address you upon a subject closely related to my own scientific interests, namely, the exploration of our universe. Naturally, I cannot claim to speak with authority over its whole range.

A number of authors have considered the following (and related) problems:

In the March 1981 edition of Mathematics Teacher Lee Markowitz gives the following full set of solutions (though not the considerable calculations involved):

(5, 12, 13) (6, 8, 10) (6, 25, 29) (7, 15, 20) (9, 10, 17).

However, in many formulae involving the triangle, it is the semi-perimeter rather than the perimeter that arises naturally.