A few years ago, an experiment was made at Marlborough (1) to begin the teaching of statistics with a discussion of rank correlation because “within a few periods and without any complicated arithmetic, it was possible to discuss the main items in a statistician’s philosophy”. What I want to do in this article is to show how the same can be done with one of the commonest situations in elementary statistics, when we wish to compare populations from which two samples are drawn and decide whether one is “better” than the other. In the classical approach, this is often done by Student’s t test which is highly sophisticated compared with the Mann-Whitney U test, to which I now turn.

In exploring the probabilities involved in a roll-a-penny game, some properties turned out to be a little more complicated than expected. The main aim here is to find the probability that the player will lose after a certain number of rolls.

Suppose we have a board upon which squares are drawn, and that the side of each square is four times the radius of a penny. (In practice, this means that the squares are of side 4 cm.) The first question is, “What is the probability that the penny lands entirely within a square, assuming that the dividing lines are of negligible thickness?”

The purpose of this note is to show how two first integrals of the equation of motion of a particle moving under the action of an inverse square central force can be obtained very simply using vector techniques. In fact it is shown that, amongst all central forces, only the physically important inverse square force can be treated using the technique given here, thus demonstrating one facet of the mathematical simplicity of that force.

The Report on “The Teaching of English in England” lays great stress on the important truth that the use of English concerns all teachers, but when it deals with the matter more closely it enumerates teachers of history, languages, geography and science, and says nothing about mathematics (e.g. § 110).

Geometry is gradually coming back into the school syllabus, but so far only 2-dimensional geometry. I would like to make a case for including some 3-dimensional geometry as well, because the latter is vital for describing the world throughout science, engineering and architecture. Higher-dimensional geometry also comprises a major part of modern research within mathematics itself. Also 3-dimensional geometry fosters both our intuitive understanding and our geometric imagination. It teaches us to see things in the round. It also trains us to see all sides of an argument simultaneously, as opposed to algebra and computing which emphasise thinking sequentially.

As a rule graphical methods are more tedious than an analytical treatment; but they are welcomed by the weaker student on account of the illumination which they shed upon relations that would otherwise remain somewhat obscure.

Two notes which have recently appeared in the Mathematical Gazette dealing with the graphical discrimination of the cubic tempt me to publish methods of discussing the roots of the cubic and of the quartic which have been in use here for some time and have stood the test of experience.