The early composers of change ringing music for English church bells were not mathematicians, yet they developed intricate algebraic ideas more than a century before mathematicians independently discovered them. We introduce the mathematical concepts of permutation group and symmetry group by means of elementary change ringing compositions of the 17th and 18th centuries.

The subject of polygon loops was introduced by Gerry Price, a County General Inspector in Hampshire for Mathematics, and Roger May, Teacher Adviser for Mathematics, Southampton, during an evening meeting for teachers of Mathematics in 1991. The question posed was, “Can you always make a loop with a set of congruent regular polygons?” Since then various articles have appeared in which polygon loops of various kinds have been discussed. Ann MacNamara and Tom Roper [4] described how Year 9 students worked with octagon loops in a trial Standard Assessment Test, and Jackie Davies [2] used a particular family of square loops with primary children. Grunbaum and Shephard [3, pp. 104-106] show plane tilings in which one can spot polygon loops of equilateral triangles, squares, and regular hexagons, octagons and dodecagons.

Most of us who teach “complex numbers” (especially to classes where the emphasis is primarily mathematical) take the opportunity at some stage to place the theory firmly on the concept of ordered pair (of real numbers), assuring the students that they have long been familiar with this notion (e.g. plane cartesian coordinates) and its manipulation (e.g. rational numbers as ordered pairs of integers and their arithmetic) without explicitly recognising it. Later we might ask (or even be asked!): How about ordered pairs of complex numbers? This article, consisting of chips from several quite old blocks, offers some forwardlooking ideas for investigation and development. An excellent general reference is [0].

A standard application of calculus is to model some quantity which changes and to find the circumstances under which it takes some optimal value. There are a number of stock examples which are often used to show this in the teaching of calculus. This article considers how the extension of one of these can lead into a rich area of exploration.

I have always been intrigued by the Pythagorean triple (20, 21, 29). It interests me for two reasons. Firstly it looks really improbable and I am fascinated by the things that remain counter-intuitive in mathematics despite my repeated verification of them. Secondly apart from the basic (3, 4 ,5) triangle, it is the only small triple in which the first two numbers are consecutive. For ease of reference I shall call these special triples. I had never even been sure how many more of these there were, although I had tracked some down by computer at one point. In the middle of preparing some follow-up work for a lecture, I decided to look at these again to see whether there was anything suitable for the students to do. I uncovered a connection that certainly I had never known before.

Many sports such as rugby, American football, soccer, lacrosse, ice and field hockey have as their objective the placing of an object (a ball or puck) through some designated portion of a vertical plane. In rugby and American football, for example, the conversion or place kick is a very important aspect of the game and frequently is the difference between winning and losing.

The relative merits of GCSE and O-level were extensively debated at the time of the introduction of GCSE in the late 1980s. Various aspects have been under discussion since, especially in changes in grade distributions in various subjects. The ‘knowledge gap’ was identified as a particular problem in so called ‘linear’ subjects, in particular mathematics and the physical sciences. The gap referred to was the one created by the policy of the government in reforming the curriculum up to the age of 16 without making matching changes post-16, for A-level in particular. Indeed, a report (see [1]) from the School Examination & Assessment Council to Kenneth Baker as Secretary of State in My 1989 referred to this gap. That report was compiled by SEAC in the Spring of 1989 on A/AS levels. It referred to “ … the claim made by many respondents that the gap between GCSE and A-level was too great for them to provide appropriately for a sizeable proportion of 16 year olds who might otherwise continue full time education beyond the compulsory period.” Sixth form mathematics teachers have been well aware of this problem and have taken appropriate measures. Some of these have consisted of ‘remedial algebra’ or ‘crash courses’ in algebra at the beginning of the A-level course. Some of the problems are discussed in detail in [2] and [3].

An interesting application of mathematics is that pertaining to the idea of preferential voting. By this we mean that if you don't get your first choices elected then you should still have some input into those who candidates who are. For example, suppose there are 20 candidates running for 5 positions. In this case it is assumed that you will rank all 20 candidates from 1 through 20 so that your 6th and later preferences get some weighting as well as the first five.

The ideas are illustrated by considering the elementary case in which three of six candidates are to be elected. In each example it assumed that the voters must rank the candidates in order of preference, with 1 being the first choice and 6 the last choice.

Over the last twenty years or so there have been numerous trials and curriculum development exercises in Mathematics involving the use of calculators of various levels of sophistication; more recently, work has been carried out on the use of computer algebra systems (CASs), which perform much of the routine algebra and calculus currently examined at A-level. At the same time, there has been a move in certain examinations to allow candidates access to reference materials (such as set texts) or other forms of support. This paper provides an overview of these developments together with their consequences and considers the possible effects on Mathematics examinations of affordable hand-held devices which run CASs. It has been written largely with regard to A-level, although much of it is also relevant to other examinations.