When I began teaching secondary mathematics after several years in industry, I hadn’t been trained as a teacher. Having been away from school mathematics for so long, I was sometimes nonplussed by pupils’ questions.

One day I was showing a mixed-ability first-year class how to obtain the decimal expansion of a vulgar fraction. After explaining the how and, I hoped, the why of the method I did several examples on the blackboard and concluded by telling them that the expansion would either terminate or recur (or words to that effect). At once a small boy demanded to be told how I knew this. This floored me: I hadn’t got a ready answer and I had to promise to think about it.

A very appropriate use of a computer in a mathematics class is to provide investigation vehicles for formulating conjectures to be verified theoretically. In other words, first we notice something and then we try to prove it. In this paper we shall reverse the process, first arguing the plausibility of a very fascinating result and then obtaining confirmation by looking at some computer generated data. The argument is based on the work of the eighteenth century mathematical giant, Leonhard Euler (1707-1783).

“A derivative is a linear transformation which approximates a function near to a point.” I heard this definition for the first time in 1964, in a lecture at Oxford on the calculus of several variables (the lecturer was D.A. Edwards). The words came as a revelation, and from then on I had a new conception of differential calculus. At school, I thought that I knew about calculus because I could write out proofs of differentiation ‘from first principles’. I enjoyed going through the hoops and proving that δy/δ tends to a limit, but somehow the idea of differentiation never became ‘part of me’.

It is well known that a curve has only one equation in Cartesian co-ordinates When referred to a given pair of lines as axes, but it does not appear to be so well known that a curve can have many equations in polar co-ordinates, all referred to the same pole and initial line. This fact, although of considerable importance in the study of curves by means of their polar equations, appears to be consistently ignored by most text-books.

The purpose of this article is threefold. First to suggest other approaches to the ‘Glass rod problem’ (Gazette note 65.6, March 1981). Second, to show the unexpected connection between that and the article ‘Triangle of triangles’ by H. B. Griffiths on page 10 of the same issue and with the article on ‘Random triangles’ by T. Easingwood in the December 1981 edition. And third, to explore further problems which arise from a consideration of the above.

Since the introduction in 1978 of the idea of a “common-core” for A-level mathematics, there was been a continuing debate over the structure and content of the subject at this level. Much of this debate remains pertinent in 1989: see, for example, [10], [15] and [28]. However, the setting-up of the Higginson Committee to review A-levels in March 1987 had the effect of putting much of the discussion specific to mathematics on the back-boiler. It is well known that the committee’s report of May 1988 recommended a wider subject-spread, roughly speaking on the model of the International Baccalaureate, and that its immediate rejection by the government created considerable furore in educational circles.

Integer Geometry is the geometry of points that are pairwise integer distances apart. An integer polygon is a convex set of n points in the plane such that no three points are collinear and the distance between any two points is an integer. Similarly an integer polyhedron is a convex set of n points in space such that no three are collinear, each face is an integer polygon, no two faces are coplaner, and the points are all integer distances apart. The measure used to order such polygons and polyhedra is perimeter-plus which is the sum of all of the edges added to the sum of all of the diagonals [1]. The purpose of this paper is to provide pictures of some fundamental examples as well as a brief explanation of some of the constructions. We begin by looking at polygons and then use those to build polyhedra and conclude by examining two closely related examples of sixfaced polyhedra.

There are hints of unique factorisation in Greek arithmetic. Indeed, some commentators have seen the Fundamental Theorem of Arithmetic (FTA), that the natural numbers can be expressed as products of primes in a unique way, lurking in Euclid’s Elements (c. 300BC). What can be said with certainty is that the history of the FTA is strangely obscure. It is not too much of an exaggeration to say that the result passed from being unknown to being obvious without a proof passing through the head of any mathematician.

Trigonometry , like most abstract sciences, has both an operational side and a quantitative side.

The operational side is dominated by the hypothesis that a function f(x) exists which has the property

It is from this that the usual rules for operating with the trigonometric functions are deduced.