They wept like anything to see
Such quantities of sand.
“If this were only cleared away,”
They said, “it would be grand.”

It would be, and if those young airmen of whom we heard in the February Gazette had not spent so much lost endeavour in weaving ropes of this sand, perhaps they would not now be in such need of the help which we are so eager to give them. For although the seven-times-seven mops of our teaching committees have been sweeping for far more than half a year, there is no doubt that school Arithmetic is still cumbered with a lot of stuff that is of no use outside the class-room. In spite of the fact that teachers mostly teach so well, and pupils learn so well, we must confess that children going out into the world often cannot do the calculations that they need, that Training College lecturers find that many students have to learn Arithmetic all over again (though we camouflage it as “Method”), and that intelligent well-educated folk seem to delight in telling anyone whom they suspect of mathematical leanings that “they never could do Arithmetic”. Surely it is the Arithmetic that is wrong, not the teachers or taught.

One of the fundamental difficulties in teaching the calculus is that we are obliged to introduce the concepts of the Theory of Functions and yet make no use of the real number theory on which the Theory of Functions is based. We must talk of limits, continuous functions, derivatives, definite integrals and of existence theorems like Rolle’s and Taylor’s, and must endeavour to hide the deficiencies of our reasoning beneath the kindly cloak of geometrical intuition. The Calculus course is usually a compromise, completely rigorous proofs of the basic theorems being deemed too difficult for the type of student for whom the course is planned, but I believe a far more rigorous account than the customary one is possible without greatly increasing the strain on the students’ powers of understanding, this can be achieved, however, only by a new approach to the subject and it is with the first steps in such an approach that we are here concerned.

It was quite clear that the company were much more interested in what the Baron had to say about the marriage customs of the people of Octonaria than in his remarks about their system of numbering. So I bided my time until he left the hospitable parlour of The Dolphin and, attaching myself to him, with little difficulty persuaded him to accompany me to my lodgings. Here, comfortably ensconced in my own armchair, he proceeded to fill the great octagonal pipe which had excited such curiosity at The Dolphin, while I set about collecting the ingredients for a bowl of punch.

The question of the length of masters’ holidays in a reception area is a somewhat thorny one, the essential factor being that the boys must be looked after the whole year round. In order to fix our ideas it is convenient to consider two extreme cases. First, let us suppose it is decided that the masters are to have the same length of holiday period as the boys. If a programme of work can be arranged with two-thirds of the staff on duty, and if the boys can be properly supervised when on holiday by one-third of the staff, then diagram I shows a possible solution for a holiday of four weeks.