The reports of the London University on the General School Examinations throw an interesting light on the knowledge of candidates of the subject of approximations. The 1928-30 reports make it quite clear that the subject is badly taught and that glaring errors occur very frequently The 1929 report states that this fault is general and is not confined to particular schools. When it is considered that the University examines annually about 25,000 pupils from schools all over the country there is justification for the suggestion in the title of this paper.

The title of my address is “Mathematics for the Study of Frequency Statistics.”

By frequency statistics I mean statistics which give the result of counting. A statement as to the number of teachers receiving annual salaries in excess of a certain amount would come under the head of frequency statistics : a statement as to the total amount paid annually to these teachers would not. The sort of thing that I have mainly in mind is a table giving the numbers of boys of specified heights, the heights being taken to the nearest inch, or the numbers of girls who, having worked a paper with full marks 100, get under 5 marks, 5 and under 10, and so on. I may mention that the word “frequency” is usually applied to the actual number in a specified category; the proportion of this number to the total is called the “relative frequency.”

To the mathematician, at least in the field of elementary mathematics, it is immediately apparent what is true and what is false, and for the teacher of mathematics it is the truth, the correct answer, the right reasoning, that are all-important.

In this article, I propose to examine the other side of the picture, to consider the mistakes made by our pupils and to see how they can be turned to advantage in our work as teachers.

This article starts with a problem in probability and leads on to some natural connections with Fibonacci numbers. Details of some of the proofs are left for interested readers (or their students!).

In 1952 I was invited, with teachers of mathematics from many other schools, to a reception hosted by the Cambridge University Press. The guest of honour was A. W. Siddons, and the purpose was to celebrate the fiftieth anniversary of the initiation of Godfrey and Siddons’ first book: Elementary Geometry. There can be few authors who survive to enjoy such an occasion; fewer still whose book remains in print and continues to sell well after fifty years. But this was a book which marked a turning-point in the teaching of mathematics, and set a standard for mathematical education which remains relevant as we approach the end of the century.

Euclidean geometry and fractions not only make unlikely bedfellows, but would seem to be a sure recipe for boredom. However, in this paper, with the help of a little history, we hope to prove the opposite.

Given a (rectangular) lattice, then a lattice polygon is a polygon whose vertices are lattice points. Pick's theorem [1] - the area of a simple lattice polygon is given by ½b + i – 1 where b is the number of lattice points on the boundary of the polygon, and i is the number in its interior – is well known and has been proved many times and in many ways.