A-level courses and examinations are with us for some time to come. Even if Mevel is introduced, for most pupils who spend two years in the I sixth form A-levels will remain the basis of their studies. It seems to me that we need a major debate about A-level mathematics; we need to review it as it is and try to decide what developments are necessary. The discussion I paper ‘Mathematics at A-level’ has already raised a number of points, some of which are touched on here. This article is not a statement of official SMP policy; any opinions expressed in it are my own. The aim is to provoke discussion. I shall consider a number of issues which seem to me to be important, hoping that others will be stimulated to contribute comments and suggest other topics which need attention.

The rationale for the introduction of the system of three points for a win in the Football League in 1981-82 was, presumably, that more reward for winning games would encourage more positive attitudes from teams and that the consequent more attractive and attacking football would bring in bigger crowds. Certainly the philosophy of many away teams, or at least their managers, that they began a game at 0-0 with one point to hang on for had become commonplace in recent seasons.

A recent unfortunate incident in the medical world leads to some interesting mathematics [1]. The incident centred around the half-life of a drug used in treating arthritic people; elderly patients having longer drug half-lives than young people, e.g. due to impaired liver or kidney function. Doses were being repeated before the half-life period of the patient, with a dangerous build-up of the drug in the system, leading to death in some cases. A mathematical model can be developed to describe the normal drug concentration variation with time, and the cumulative effect of repeated doses.

One day, as I was struggling to construct examination questions on vectors in geometry, the following thought (1) occurred to me.

• (1) If the position vectors of points A, B, C are denoted by a, b, and c respectively, then the position vector of the centroid G of ΔABC is ⅓(a + b + c), but similar formulae for the circumcentre, orthocentre and incentre are not commonly known.

After the thought had drifted about in my mind, it prompted a second observation (2).

What is the most popular sport in Britain?—Fishing? Crown green bowling? Camel-racing? No. I conjecture that it is Bingo! No longer do Auntie Maud, Auntie Flo and Granny drag Mum off to the local dance-hall on a Saturday night—instead they troop eagerly down to the EMI Bingo-hall, with the likes of Vera Duckworth, Ivy Tilsley and Ida Clough.

In one dimension, a set of rods of equal length may be used to cover the line. In two dimensions, an area may be covered by joining up sets of identical symmetrical squares, equilateral triangles or regular hexagons. In three dimensions, there are at least four different symmetrical solids which can be packed together to fill space. Of these, the cube is most obvious, two others are moderately well known (see [1] for example), while I have been unable to find any record of the fourth solid.

The entertaining book, There’s one born every minute, by Harry Blackstone, presents many situations where most people would intuitively be prepared to wager even money on the outcome, whereas a proper study of the probabilities concerned shows that the bet is heavily biased. One particular problem is interesting because the author claims to have sought evaluation of the probability concerned, without success. It is as follows:

Most direct proofs of Morley’s theorem are very involved. Dick Hogg’s neat geometrical proof (Note 67.3) circumvents the problem with its slightly indirect approach, but is very pleasing, none the less. It may however not be generally known that there is a direct straightforward proof using complex numbers which is simple to those familiar with the technique involved; indeed the proof is almost inevitable, once the right start has been made.