In this article I shall take a problem, the quadrature of the cissoid, and compare a hard (but brilliant!) pre-calculus solution of 1658 with the easy (and routine) calculus solution of only 13 years later. Afterwards we shall perhaps see the calculus through different eyes and maybe even teach it differently. Calculus is mechanical, but that it is mechanical is a marvel.

If the above phrase appeared in an A-level paper the Board responsible would receive many protests: “a contradiction in terms”, “impossible”. But is it? Our attention was drawn to this via this A-level problem from the J.M.B.

(Further Maths, Paper II, 1988)

People seem to regard the draw for the quarter-finals of what appears to have been the 1988 Benson & Hedges Cup in cricket as having been bungled: a quote from the relevant authority is that it was “an embarrassing cock-up”. Well, I'm not so sure! To start with, the draw as made—unlike some others!— was perfectly fair. The draw had to be restarted twice (this is what concerned observers) and so there is a question of efficiency of the chosen method of drawing. In fact, in seeking alternatives, there arise interesting questions of balancing efficiency, “excitement” and the (generally innumerate) public's perception of fairness. Let me explain!

On the surface the UK SMC does not look much like a “celebration”. It consists of a one hour multiple choice paper aimed at 30-40% of 13 and 14 year olds, and is taken each year on the first Thursday in February.

Well, what would one be willing to accept as a celebration? The London Marathon has certainly captured the public imagination. Few of us can deny the emotional appeal of watching, or joining with, other human beings who are pushing themselves far beyond what is normal in everyday life. Mathematics usually evokes an equally strong emotional reaction, except that most people respond negatively. And among those few who respond positively, even fewer (in the UK) have traditionally relished the harsh glare of mathematical competition. How is it then that the UK SMC, now in only its third year, attracts a far larger entry than the London Marathon and has plans to expand to five times its present size? (The 1990 entry was roughly 42 000 from 770 schools.) Quite apart from any mathematical advantages, the scale of such an event offers youngsters a unique opportunity to participate in something truly national—a rare instance of school mathematics overcoming its instinctive introversion by becoming conscious of a challenge shared with others.

GCSE mathematics and Sixth Form Mathematics are now out of step. Sixth Form Mathematics and University Mathematics may also soon be out of step. These are two far reaching statements—but what has this got to do with rugby balls and a minibus?

By now many of us currently or recently involved in GCSE mathematics, whether as students or teachers, have become used to mathematical investigations, applied problems and project write ups all lasting about two weeks. How times have changed. It is now quite common for 15 and 16 year olds in their GCSE mathematics lesson to work in groups on open ended problems. Gone are the days (familiar to some of us!) of solving endless lists of quadratic equations and stopping without hesitation the first time b2 is less than 4ac. Generations of pupils will now, if the National Curriculum does not completely move “the goal posts”, be growing up on a new range of discoveries made possible by the introduction of fresh ideas and new starting points. “222 and all that” and “Watch out” for example are two standard pieces of coursework encountered by many pupils in their GCSE training.

The development in recent years of software systems (“computer algebra” systems, or “symbolic manipulators”) which, in effect, automate large areas of mathematics presents an immediate and huge challenge to mathematics education, especially at A-level. For many students current practice in A-level mathematics seems largely to consist of the assimilation, rehearsal and implementation in stereotyped contexts of a more-or-less well-defined set of standard algorithms in short, “plug-and-chug” mathematics, as Philip Davis has described it. With the aid of computer algebra systems demonstrations of “A-level papers in ten minutes” have recently been possible, and this clearly illustrates the essentially “plug-and-chug” nature of the assessment tasks.

In July 1988, six small groups of teachers, from Bury, Essex, East Sussex, Leicestershire, Northamptonshire and Suffolk, met together to launch the Graphic Calculators in Mathematics project, sponsored by the National Council for Educational Technology. Each group was locally coordinated by an adviser or advisory teacher, while I was on 0.2 secondment to act as national coordinator of the project. Over the following two years, the project teachers worked with classes of students who had permanent access to graphic calculators throughout their advanced-level mathematics courses. Rather than following any prescribed programme of calculator activities, the teachers were free to plan the work of their classes within the normal syllabus constraints, meeting together from time to time to exchange ideas and review progress. In this article, I want to draw on experience from this project to discuss two major issues that new technology raises for advanced-level mathematics.

Writing in the Mathematical Gazette, MacKinnon [3], pinpoints several factors which are bringing pressure to bear on the development of the sixth form mathematics curriculum for the 1990s and beyond. Some of these factors are not new and their development has been chronicled by Bibby [1] and Wilson [9]. However these recurrent issues are still important and still unresolved. The changes in education pre-sixteen are prompting a reassessment of sixth form mathematics and this provides an ideal opportunity to develop a mathematics curriculum which addresses and hopefully resolves these issues. To achieve this goal I propose a modular approach to sixth form mathematics.

Much debate about A-level mathematics centres on organisational, assessment and content issues, whilst another major focus of interest concerns the role of the microcomputer in the sixth form classroom. Many of these issues have been discussed at length in a recent article in the Gazette by Bibby [1], but he does also seek to draw attention to the rather limited success there has been in directing attention to the importance of “process” as well as “content” in A-level mathematics.

It is my contention that in the teaching of calculus at A level we have, all too often, been producing an understanding of calculus—and, implicitly, a view of mathematics—that is distorted and unrealistic. I am glad of the chance to air these criticisms in this contribution to the debate about the future of sixth form mathematics. (In this article, “sixth form” is synonymous with A level, whether studied at school or college.) Nevertheless, as I maintain later, the changes now being wrought in sixth form mathematics generally may well prove beneficial.

The undergraduate mathematics course at Cambridge is called the Mathematical Tripos, and it has a long and distinguished history. The word Tripos refers today to a Cambridge degree course in any academic subject, which is examined by written papers. In fact, mathematics was the first Cambridge Tripos. It started in the 18th century, although its origin dates from earlier times when oral examinations were held (conducted by an examiner known as Mr Tripos, allegedly because of the three-legged stool on which he sat!). Periodically, radical changes have been made in the Mathematical Tripos, when perceived to be desirable in the light of either external events or changing emphases within the subject. A famous revolution took place in 1910, one of whose consequences was the abolition of the order of merit in which the Wranglers (candidates awarded first class honours) were placed in numerical order. The last major overhaul occurred in 1964, when the present structure was created. A similar such wholesale review has been taking place here over the last few years, with the result that a completely new course will begin in October 1991. This short article aims to describe some of the thinking behind these proposals.

This article outlines possible changes to the examinations in Mathematics offered by the Scottish Examination Board as part of the Certificate of Sixth Year Studies. It must be emphasised that, at the time of writing, these changes are still only proposals. Furthermore, any views expressed in the article are the personal views of the author.

The existing CSYS courses were designed as the third stage of a progression in which pupils sat examinations at Ordinary Grade (‘O’ Grade) in S4, at Higher Grade in S5 and at CSYS Level in S6. However, the old order changeth. The Ordinary Grade is being replaced by Standard Grade and the transition will soon be complete. For articulation with Standard Grade courses, Higher Grade is being replaced by Revised Higher Grade. Finally, CSYS courses are being revised to articulate with Revised Higher. Some of the proposed revisions concern content, a good example being the situation regarding matrices. Presently pupils meet matrices in Higher but there is no mention of matrices in the formal contents of Revised Higher, so that CSYS must be adjusted accordingly. Other proposed revisions arise as a result of changes in the teaching approach. At Standard Grade and Revised Higher Grade, one of the aims is to develop pupils' skills in problem-solving. Furthermore, investigations form part of the assessment in both cases. It seems sensible that CSYS courses should continue this philosophy.

Roger Heath-Brown (37) is a Reader in Pure Mathematics at Oxford University, and a Tutorial Fellow at Magdalen College. He works in analytic number theory, with interests covering the distribution of prime numbers, Waring's problem and Fermat's Last Theorem

This term I'm giving a first year lecture course—Groups, Rings and Fields. I'm not at my best at 9 am and despite my efforts the undergraduates find me rather dull. The weaker ones have to have everything spelled out in full, with the result that the interesting things are too sparsely scattered for the stronger students. I'm back in college by 10.30 am having collected a pile of mail. There's a request from the Mathematical Gazette for a “day in the life of” style article, and I toy with the idea of describing an occasion on which a journal asks me to describe a typical day's work …. I also receive a referee's report on a paper submitted to me as an editorial advisor to the London Mathematical Society. The report is not encouraging and I reject the paper—a frequent enough duty, but not one I relish. The associated paperwork fills the next three-quarters of an hour. This particular manuscript is rather too similar to something published in 1963. While double-checking this in “Reviews in Number Theory” I notice an abstract of a quite unrelated paper, by a Finnish mathematician.