It is a great pleasure to welcome the Mathematical Association to Cambridge, which since Newton has always been one of the most important homes of mathematics in this country. Cambridge is a traditional home of advanced and research mathematics, and it therefore gives me particular pleasure to have the honour of giving my Presidential Address to the Association in Cambridge on the topic of primary mathematics the foundation and growing point of the ‘mathematics for all’ on which our increasingly scientific and technological culture makes greater and greater demands.

Scout‘s pace, according to my old boy scout manual, consisted of alternately walking and running 22 yards. I believe it is also used by military scouts and messengers who wish to travel at a moderate speed without too much effort. As I have also discovered, it is useful for elderly, overweight athletes with a low pain tolerance.

Now that transformations are no longer new in geometry teaching, students of mathematical education not brought up on the new orthodoxy may soon become extinct. However, I do have the good fortune to teach some members of this rare breed —mature B.Ed, students specialising in Craft, Design and Technology, and very refreshing it often is. Part of my job is to teach them about approaches to mathematics which are now common in schools, so that they know what sort of background it is reasonable to expect of their pupils.

Currently in Northern Ireland we have two GCE Syllabuses and examinations at Advanced Level in the mathematical domain—Mathematics and Further Mathematics. In 1982 a Further Mathematics Project began work on an alternative syllabus and examination (Mode 2) for Further Mathematics. A firm proposal has now been submitted to the Northern Ireland Schools Examinations Council and has been approved by the Secondary Examinations Council.

In the traditional approach to the teaching of vectors, the scalar product of two vectors u, v is usually defined to be

|u| . |v| . cos θ

where θ is defined to be the angle between the two vectors. In a similar way, the vector product of two vectors is defined to be a vector that makes a right handed triple with u and v, and whose length is given by

|u| . |v| . sin θ

From these definitions the familiar co-ordinate expressions for dot and cross products are derived.

A very simple way of generating a sequence is to take two starting values, a1 and a2, and construct a3 as their arithmetic mean, a4 as the mean of a2 and a3 and so on. This sequence always converges to the limit and seems to be fairly widely known as a topic for investigation in schools and colleges. It can immediately be generalised to the case of k starting values, and this forms a good subject for investigation using a micro-computer. It seems likely that such a simple set-up has arisen independently many times. One emergence occurred about ten years ago at a conference of the Association of Teachers of Mathematics, where the originator introduced it under the fictitious name ‘Littov sequences’. It has recently resurfaced at a Department of Education and Science conference.

Sometimes nǃ is very divisible by n. For example, 12ǃ is divisible by 125 while 720ǃ is divisible by 720178. On the other hand, if p is prime then pǃ is divisible by p but not by p2.

In general it is quite difficult to calculate the maximum exponent m(n) such that nm(n) divides nǃ

In this short article I will locate some values of n, for which nǃ is very divisible by n, find a tight upper bound for m(n), and point the reader in the direction of related problems.

It is an interesting exercise for a lower sixth form to investigate how muchquantitative information about the solar system can be obtained using crude apparatus, which means objects such as sticks, stones, string, coins. For the sake of saving time the concept is extended to include a ruler (which is only a graduated stick, the unit being arbitrary) and a watch (which could be replaced by keeping up continued observation of a simple pendulum).

Consider any point P inside ΔABC. The trilinear coordinates of this point are p1, p2 and p3, which are the perpendicular distances from P to the sides BC, CA and AB respectively.