The concept of the ordinary binomial coefficient

can be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers , where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define corresponding to and For example,

As an introduction to elementary programming for digital computers, it has been found advantageous to make a start with a desk machine. A very simple code was devised for this purpose. Interested sixth-formers tackled problems prominent in their ordinary mathematical work by this method and were soon able to make programmes and get results themselves. They were later able to visit a digital computer installation and to try their hands at using an established “autocode” (Sirius). They found that their previous experience with the desk machine was most valuable in initiating, very quickly, the kind of thinking required for the more sophisticated procedures. The total experience, although very brief, appears to have been well worth while, as a means of showing them something of the potentialities of modern computers in relation to problems within their own competence.

The result obtained in this short paper was discovered by chance in the course of constructing some examples for the first-year students of numerical analysis. It may therefore be of particular interest to others who have similar duties to perform. The relevant example from those which I had set is:

I am continually surprised by the mathematical structure which surfaces in unexpected situations, and even more so when arranging a tenth birthday party for my elder son. The invasion of the home by some eight rather energetic boys on a cold February afternoon was viewed with some trepidation. The instinct of self preservation was uppermost in the search for activities which would allow the afternoon to pass with the minimum of discomfort and the maximum enjoyment. A family friend is very good at organising team games at parties, but with a space problem (our house does not have a large living room as so many modern houses have these days) it was important to split the children up for their activities, but yet devise a programme which was coherent.

A teacher can make a great deal out of some of the questions which pupils and friends put to him from time to time. In an article in Mathematics in School a few years ago [1] I listed a number of such questions and discussed one, which involved a geographical application of matrices, in detail. The present article continues this theme, but its main aim is to stimulate discussion of the place of “mathematical modelling” in the curriculum. For it is felt by some that, in spite of the opportunities offered by (say) linear programming, mechanics and probability, insufficient emphasis is placed on modelling in the secondary school. It has been said too that we have not been notably successful in finding good applications of topics new to our syllabuses and incorporating them into our teaching [2].

The scope of the additive prime number theory is evident from its name. In this part of the theory of numbers we are concerned with the representation of integers as sums of primes; and here the central problem consists in the proof (or possibly refutation) of a celebrated conjecture made by Goldbach in 1742 to the effect that every even integer ≥4 can be represented as the sum of 2 primes. The truth of this conjecture is still unsettled; but though progress in this field has been slow, attempts to deal with the problem have led to a whole series of striking results.

Imagine a horizontal rectangular tray with vertical sides, and a disc lying flat in the tray and rolling around the inside of the tray in such a way that its circumference is always in contact with the edge of the tray. We shall suppose that the disc has been rolling for ever in the past, and that it will continue to roll for ever in the future. What can we say about this physical system? Our aim is to illustrate the complexity and the discontinuous nature of the system under arbitrarily small changes in the size of the tray (a feature that is typical of many dynamical systems).

In spite of the great increase in the last fifteen years in the use of the Calculus in the higher forms of schools, and of its introduction into the Intermediate Examinations of many universities, it has not yet been used as much as it could be in the teaching of elementary Applied Mathematics.

It has, of course, long been realised that in the kinematics of the motion of a particle in a straight line we have some of the best examples of the ideas that are the basis of the Calculus, and the notation and machinery of the Calculus is now extensively used in dealing with this part of the subject.

(1) The institution of Local Education Authorities in 1902. These Authorities have been able to survey over a wide range of Secondary Schools the working of the examination system and to estimate to some extent its effects on Secondary Education as a whole. Consequent on this, but not wholly the result of it, came some general reforms, e.g. the formation of the Secondary Schools Examination Council. But L.E.A.’s are also involved in the examination system as examiners, since they test candidates for scholarships and free places in Secondary Schools. Their examination view-point is therefore a wide one, embracing that of both teacher and examiner. The full effect of this dual point of view has not yet matured, but some innovations or improvements in junior scholarship examinations have already resulted from it; for instance, the development of the oral examination and the use of the teacher’s assessment of a pupil. In some areas, too, teachers participate in marking scripts. Examinations of School Certificate standard are so far little affected, but it would be in accord with the trend of events if the wide powers of University Examining Authorities were further modified and the school record of the candidate made to play a more important part in the examination than at present.