Note 74.11 from Nick Lord in the March 1990 Gazette looks at the fascinating Simpson's paradox starting from a complex number situation; and in the same issue John Haigh, reviewing Cox and Snell's Analysis of Binary Data, gives a (fictitious) example; but the fascination of the paradox stems from the real occurrences.

The celebrated Steiner-Lehmus theorem states that if two internal bisectors of a triangle are equal then the triangle is isosceles. It beautifully illustrates the point that the converse of a theorem may be far more difficult to prove than the direct statement. The literature of the Steiner-Lehmus theorem is considerable with many contributions in the Mathematical Gazette.

An interesting variation on this theme may be found by examining the triangle formed by the second intersections of the medians with the circumcircle. Let the median point of a triangle ABC be G and let the medians meet the opposite sides at DEF and the circumcircle again at LMN.

The possible dangers and anomalies arising from the casual interpretation of “mean values” or “averages” are well known. The “velocity problem” is often cited as an example (see e.g. [1,2]). A typical version of this problem is as follows: A cyclist goes from A to B at 20 mph and returns from B back to A at 10 mph. What is the average speed? The casual answer is 15 mph, the arithmetic mean of the two speeds; this is wrong. The correct answer, obtained by dividing the total distance travelled by the total time taken, is obtained as 13.3 mph, and this latter figure is readily shown to be the harmonic mean of the two speeds. Langley goes on to say that “the harmonic mean is often appropriate when the data consists of ratios (miles per hour, . . .)”; this is true but not specifically informative. Reichmann comments that “the arithmetic mean cannot properly be used to average rates of speed” and “the harmonic mean is always the right one to use to average rates . .-.”; these generalisations are incorrect.

Robert Harbin, in Waddingtons Family Card Games, gives an item called Prediction. You ask a friend to name two values of cards and to shuffle a deck. Then you bet that his two values will come up together as the deck is turned over. Harbin says he has worked out the odds as 32 to 19 in favour of success— “but I could be wrong. See what you make it.”

Cyclic pairs

When you transpose the digits of a two digit number and obtain a multiple of the original, the original may be called a “cyclic pair”. I have before me, a computer printout obtained by Mr F. B. Satterthwaite of 439 East 51st Street, New York 10022, exhibiting all the cyclic pairs for which the multiple of the original obtained by transposing the digits is ⩽ 16. This computer printout exhibits patterns of many kinds. The most striking pattern that Satterthwaite identified was that cyclic pairs do not exist when the base is one less than a prime number, but do exist for every other base ⩾ 5. His claim may be substantiated as follows.

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.

Mathematical magic, properly used, can play an important role in the mathematics classroom. Students need to be motivated to study mathematics (or any subject), and unusual and unexpected results, though sometimes frivolous, often provide more effective motivation than the utilitarian ideas of real world applications. Such an idea was discussed by Anderson in his article on the twenty-one card trick.

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.

Ever since I was a schoolboy, my favourite polynomial has been

It is a wonderful polynomial, with lots of beautiful and useful properties which, alas, do not seem to be known to the present generation of mathematical students. To remedy this, we collect together some of the known properties of this polynomial and present some additional properties which appear to be new.

Perhaps the most extensive use of dimensional analysis in applied mathematics is in the application of the principle of similitude to obtain equivalence between different physical systems of unequal scales occurring in widely different branches of science and engineering. However, there is a widespread misconception that dimensional analysis can only yield the functional dependence of a physical quantity on various other parameters. In this article I shall therefore demonstrate some hidden strengths of dimensional analysis.

The array shown in the diagram, where each integer appearing shows the number of routes to that point, assuming there is a North-South wall that cannot be crossed, but otherwise constructed using the same one-step, no back-tracking rules as in the production of Pascal’s triangle, may be used as the starting point of an interesting investigation, provided one’s pupils know a bit about combinations, the binomial theorem and discrete probability distributions.

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.

In French law, the 12 members of a jury are not required to return a unanimous verdict. It was decreed in 1831 that a vote of 8 or more would suffice. Four years later, it was proposed to reduce the requirement to 7 or more. There was a heated discussion in the Chamber of Deputies of this proposal which was strongly challenged by Arago, a mathematical physicist. Despite his ardent use of probability theory, the law was in fact changed.

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.

The Russian Multiplication Algorithm is frequently discussed in history of mathematics courses and in math education courses for elementary and secondary school teachers. Students are intrigued by the algorithm as it allows one to multiply any two whole numbers knowing only the multiplication and division facts for the digit 2.

Unfortunately, most textbooks do not address the natural questions which students ask about the algorithm: “Why are the even numbers crossed out in the left-hand column?” and “Can this be done with numbers other than two?” In a survey of eight textbooks by eight different pubishers, only one textbook answered the first question and none of the textbooks addressed the second question.

A number of deep theorems and unsolved problems ask questions about the relationship of the additive and multiplicative structure of the natural numbers. The famous four squares theorem of Lagrange says that if we form all possible sums of four squares we get the entire sequence of nonnegative integers. Goldbach’s conjecture states that if all possible sums of two odd primes are formed we get every even integer greater than 4.

The depth and difficulty of such problems stand in surprising contrast to the similar question: “Which numbers are not generated when all possible sums of two or more consecutive natural numbers are formed?” The answer is all powers of 2 and the proof is accessible to anyone who can sum arithmetic sequences. This result is an excellent discovery exercise for students, particularly appropriate for teacher training courses.

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.