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.

The Royal Statistical Society and the Institute and Faculty of Actuaries have published a report, written by Roger Porkess [1], which draws urgent attention to the issue of statistical education: is it fit for purpose? Statistics is an ever-present part of real life, and there can be no doubt that for most pupils it is statistics that will be the most directly used part of their mathematical education.

The report follows an investigation into the perceived statistical needs of various subject ‘communities’, and is a follow-up to The future of statistics in our schools and colleges [2] by the same author and publishers. It is aimed at a wide readership, from policy-makers through curriculum providers and teachers (fairly low in the pecking order!) to authors and other providers of resources. There is no doubt that all such interested parties should be considering the issues covered by this study, and it should be read widely. The title of the report implies suggestions for cross-curricular work, but it is more essentially a plea to rethink the role of statistical education – a plea articulated more fully in [2]. It is the purpose of this article to further the debate.

“Find the modulus and the argument of each root of the equation:

(University of London A level, part question.)

We were investigating iterative methods of solving the equation, using a micro-package (in BASIC) which performs complex arithmetic and displays the results in an Argand diagram. The facility for “doodling” which this provides led to some interesting conjectures and results, of which this note is an account.

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.

How many ways are there of making up £1 using coins? Omitting the use of the £1 coin, we are asking for the number of ways of ‘partitioning’ 100 using parts of 1, 2, 5, 10, 20, 50. For n ≥ 0, let f (n) denote the number of solutions to the equation

in non-negative integers xi, so that our required number is f (100). It is not difficult to see that f(n) is the coefficient of tn in the product Π(1 + tk + t2k +… ) over the parts k = 1, 2, 5, 10, 20, 50.