Thomas Kuhn made his reputation by pointing out that science is not just a steady series of rational developments, but is subject to paradigm shifts – revolutions in which the entire point of view changes. A less flattering word for the process is ‘fashion’. But what is fashion other than a paradigm shift in a rather trivial area of human activity? It is often thought that mathematics is immune to paradigm shifts. After all, mathematics consists of perfect truths, proven by impeccable logic. Doesn't it? Well, no, not really – but even if it did, that wouldn't prevent paradigm shifts. Paradigm shifts are not about the internal logic; they are about the external viewpoint. They are about seeing the same facts in a new light, incorporating them into a new framework.

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.

For a long time prime numbers have attracted the attention of mathematicians, especially those primes that possess some sort of a symmetry. The mysterious repunits An = 111… 1(n ), whose decimal representations contain only units, form an important class of them. For a repunit to be prime the number n of its digits must be also prime. But this condition is far from being sufficient: for instance, A 3 = 111 = 3.37 and A 5 = 11111 = 41.271. Some of the repunits are nonetheless prime: A 2, A 19, A 23, A 317 and A 1031, are the only known examples. The question of primeness of the repunits was discussed by M. Gardner and later in [2–4]. It is not clear whether the number of prime repunits is finite or infinite.

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].

We all have experience of waiting in a queue. We have waited in the dentist's surgery for treatment, we have queued at the supermarket check-out with our trolley or stood in line at the Post Office to buy stamps. No doubt we have had frustrating experiences in such situations, but before we rush to criticise the discipline of a queueing system we should think about the mad stampede for the next bus that can occur in some parts of the world.

With so much changing almost daily in schools and education, it has been difficult to decide on a title for this address and even more difficult to finalize a text. Some of the titles I’ve considered are shown here.

The reader should first try the following exercise, which motivates the later work.

Exercise: Prove that the set of real matrices

forms a group under the usual matrix multiplication.

This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?

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.

A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.