There are certain topics connected with the teaching of mathematics which seem to present themselves periodically for review. The aims or objectives of mathematical teaching is one of them ; the foundations of school mathematics is another. I have selected for review to-day one of the fundamental concepts of mathematics, that of functionality.

May I begin my discussion with a brief historical survey ? The idea that the function concept should have a prominent, even a dominant, place in the mathematical programme of the schools may be said to have originated with Klein. Others before him, in France, England and America, as well as in Germany, had advocated the inclusion of “ variables ” and “ functions ” in the school programme, but Klein was the first to press the view that “ functional thinking ” should be made the binding or unifying principle of school mathematics. Klein was equally distinguished as a mathematician and as a leader in the pedagogy of mathematics in Germany, but some years elapsed before his views became generally accepted, even in his own country.

Coincidences are a source of amusement and occasional unease in our lives. They often appear to defy the laws of probability – even to demand a supernatural “explanation”. Probability theory reveals the root cause of this disparity between expectation and reality. As I shall show, it stems from the fact that our intuitive estimate of the probability of a coincidence occuring becomes increasingly inaccurate the more “outlandish” the coincidence seems to be.

The roots of a polynomial can be represented as points in the complex plane. The time value of money (TVM) equation that is commonly used in finance is a polynomial equation. (See the appendix for a short description of the TVM equation and an example of its use in finance.) In [1] and [2] it is shown that concepts from financial mathematics can be obtained from the pattern of the roots of the TVM equation. The concepts are given in terms of distances between the roots and other salient points in the plane. This note shows that this particular polynomial, and the technique, can be applied more generally. When a series of data is fed into the coefficients of the polynomial, the mean and standard deviation of the data are seen in the complex plane as combinations of distances between the roots and other salient points. The results are aesthetically pleasing as well as mathematically interesting.

The theorems that follow are mostly well known; the object of this note is to show that they all follow simply from a single argument, whereas they are usually reached from different starting-points.

The function of the mathematical laboratory can best be understood in the light of the experience of many mathematical teachers. Anyone who has devoted even a comparatively short time to the mathematical training of workers in any branch of applied science must have had the experience that many past students return at frequent intervals for advice and consultation. The difficulties with which they are confronted appear invariably to be associated with the application in more or less concrete form of the mathematical principles and methods which they have learned. I find, as a matter of actual experience, that the complaint, when it is voiced, concerns itself not so much with the width of the mathematical field covered during training as the inability to know when to apply this or that mathematical method, how to attain the so-called solution, and how, when the mathematical solution has been obtained, to derive from it the desired information with reasonable expenditure of labour. It is partly this gap that the mathematical laboratory seeks to fill. An engineer confronted with a problem in heat conduction will not be satisfied should you provide him with an expression for the distribution in temperature in terms of a series of Bessel functions, unless he has at hand the means of immediately calculating to so many decimal places the values of these functions, either from a set of tables or by some direct mechanical process which he can apply. The form of solution provided may be to him more difficult to interpret than the original problem. It is partly because in the past mathematical teachers have been content to regard a problem as solved when, somehow or other, an analytical solution has been obtained, or to regard a problem as insoluble when such a solution cannot be obtained, that the failure exists on the part of the students of that training to utilise the mathematical weapons at their disposal, to modify them according to circumstances, and to interpret their results in actual cases.

The relationship in this article was discovered by Alan Tomkins and the proof supplied by David Pitt. The original inspiration was the statistical study of gambling systems—one method of attempting to win being to increase the amount staked each time you lose. The idea of this is that when you eventually win the amount won is sufficient to more than offset the losses on the preceding string of losers and put you back into the black. The snag is that this string of losers can leave you with insufficient funds to keep increasing the stake as necessary. The question which comes to mind is: in a given number of races, on how many occasions are we to expect a run of losers of a certain length?