To design a sixth form mathematics course suitable for grammar schools one must keep in mind the nature of secondary school education. In earlier days, grammar schools gave a specialized academic training to a small number of boys, whereas nowadays, with a much larger population, such schools are expected to train boys as useful citizens in a more complex social and economic world, giving them a breadth of knowledge and a depth of understanding.

In technical colleges one is often called upon to introduce advanced mathematical techniques to students whose background is not very extensive. Such a method is the use of the Laplace Transform for the solution of linear differential equations with constant coefficients. Textbooks which deal with this topic, even those specifically written for engineers, derive the transform from the Fourier Integral, or from Heaviside’s Operational Calculus, or just brusquely define the process. Then certain standard forms are evaluated, the rules for application to differential equations are laid down and a selection of worked examples follows. But such a treatment can mystify a student who wants to know why, in following these rules, he should, in effect, firstly, multiply each term of the equation by e-px, secondly, integrate the resultant products from 0 to ∞, and, thirdly, obtain in this way the complete solution.

A short course on the simple mathematics of satellites and space travel was tried as an experiment with boys who had just completed their “A” level examinations. They were “B” stream boys who had taken the Cambridge G.C.E. in Mathematics as one subject. The contents of the course may be of interest to others with time to occupy after examinations, or as useful illustrations of normal work in both pure and applied mathematics.

Schools differ very widely, even if we limit ourselves to that section of them dealing with pupils from 12 to 16 years of age which we commonly call “main” The pupils in the schools differ even more, and it is difficult to make generalizations about them, but nevertheless some guiding principles may be laid down in considering the kind of mathematics to which they are suited.

Suppose there are n towns every pair of which are connected by a single one-way road (roads meet only at towns). Is it possible to choose the direction of the traffic on all the roads so that if any two towns are named there is always a third from which the two named can be reached directly by road?

Several lines of thought led up to the present experiment in cooperation which has just started in Abingdon. I would like to start by trying to summarize them.

We are apt, in the grammar schools, to criticise the universities. We may say that their influence on our syllabus is bad. We may think that the teaching they give to our prize pupils is poor, the supervision unsuitable, the courses wrongly devised. These things we say and think and I am not concerned here with the rights and wrongs; but do we realize that these are precisely the things that primary teachers are saying and thinking about us?

In the concluding chapter of “On Teaching Mathematics” (1), we said (a) Mathematics is a subject which must be taught well at an early stage of a child’s schooling, and (b) the youngest children should not be taught by those insensitive to the subject. How much do we in the grammar schools know of the primary teacher and of what is being taught today in the primary schools?

The method of writing , … as recurring decimal fractions by cyclic permutation of the six figures 142857 is well known, but it is perhaps not so well known that this is but a tiny piece of a general pattern of behaviour of all fractions with denominators which are prime to the radix.

When a point source of light is placed low over one side of a gramophone record and the eye is diametrically opposite it at about the same height, then the bright reflections on the record take the form of a rectangular cross, symmetrically placed. The direct image of the source is coincident with the centre of the record.

Recent changes in the content of some Secondary School courses in mathematics have been brought about chiefly to bridge a discontinuity which is often found between the mathematics presented to undergraduates and the examination requirements for sixth forms. It is mathematicians who have been the leading advocates of change at this stage. There has been no such pressure from Secondary Schools to Primary, though the discussions on new Secondary programmes have roused a considerable interest among Primary teachers and have already begun to influence the changes which are taking place in so many Primary Schools and at such great speed.

Figure 1 shows a vertical cross-section of a bell floating freely in water in a circular vessel, both the gas pressure inside the bell and the atmospheric pressure above being p. The problem, which appeared in the Mechanical Sciences Tripos, Part I, 1957, is to find an expression for the distance δy the bell rises when the pressure in the bell is increased δp by the admission of gas through a small pipe.

Our major Colleges of Technology, and in particular, the Colleges of Advanced Technology, have had long and successful experience in preparing students for the General Honours and Special Honours degrees of London University. In some Colleges these are still the principal Mathematics courses which they offer, but in others significant developments have occurred. To put these in their proper setting, it is necessary to review briefly some of the events which have occurred during the past 20 years.

Suppose E is a vector space over the field of complex numbers with a complex valued scalar product ( , ), with the properties and (x, x) ≠ 0 when x ≠ 0, defined over it.

If a, b, and c are the sides of a triangle then

We shall prove an inequality in the reverse direction.

We shall also prove that

The object of this article is to draw attention to a point in connection with the above topic which, as far as the memories of the books I read and of those which I have more recently read vicariously or had summarised to me, seems to have been entirely overlooked. This is the existence of what I term the “index interval”, associated with, and clearly discernible in, the differential equation itself, the recognition and use of which both simplifies the presentation and removes the obscurity surrounding points in connection with the intrusion of logarithmic terms into the solution of some equations of this type.

If a1, …, an and b1, …, bn are two sets of real numbers, then (a1b1 + … + anbb)2 ≤ (a12 + … + an2)(b12 + … + bn2), the equality holding if and only if either 3 a real number λ such that ar — λbr (r = 1, …, n) with some br ≠ 0 or b1 = … = bn = 0. Write a = (a1, …, an) and b = (b1, …, bn). Suppose a ≠ 0 and b ≠ 0, for otherwise the inequality is trivial. Write с = a — λb.

Choose λ so as to make b, c=0, i.e take

the equality holding if and only if e=0, i.e if and only if a=λb,

2. Suppose x1...,xn and y1 are two sets of linearly independent vectors in the space E defined on the previous note.

The first session of this Institute was in the summer of 1950. At the 1949 Vancouver meeting of the Canadian Mathematical Congress a Committee on Research was set up with R. L. Jeffery, who was then Head of the Mathematics Department of Queen’s University, as Chairman. He raised with the National Research Council the question of a central location at which Mathematicians could carry on their programmes of research during the summer. Several reasons for such an arrangement were presented.

The role of Training Colleges and University Departments of Education in the training of mathematics teachers makes their attitude towards modern mathematics of more than parochial interest. The working conditions of these two types of institution, however, are so very different that it is impossible to discuss them as a single unit. This article has therefore been prepared by collaboration, with two main sections. In spite of these differences of structure, however, the general attitudes of mathematics staff in the two types of institution have so much in common that it has been possible to write a concluding section expressing these views. It may be that this final section will be the most characteristic part of this contribution to the symposium.