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.

When I began teaching secondary mathematics after several years in industry, I hadn’t been trained as a teacher. Having been away from school mathematics for so long, I was sometimes nonplussed by pupils’ questions.

One day I was showing a mixed-ability first-year class how to obtain the decimal expansion of a vulgar fraction. After explaining the how and, I hoped, the why of the method I did several examples on the blackboard and concluded by telling them that the expansion would either terminate or recur (or words to that effect). At once a small boy demanded to be told how I knew this. This floored me: I hadn’t got a ready answer and I had to promise to think about it.

While all teachers of Mathematics should be interested in the paper read last September by Prof. Perry and in the debate which followed it, members of our own Association, founded as it was for the Improvement of Geometrical Teaching, should give it special attention and should take advantage of the upheaval at Glasgow to press some of their immediate aims. Although Prof. Perry presented his case with a certain amount of picturesque exaggeration, I feel sure that his contentions were in the main true, and that the mathematical training of the schools, however well fitted for those who are being prepared for a University course, turns out the ordinary boy with a modicum of Mathematics which he finds of little use for any professional purpose. I do not however think that we need look upon Prof. Perry as an opponent but rather as a valuable ally. I am convinced that much might be done, at any rate in the earlier stages of mathematical teaching, which besides going far to meet Prof. Perry’s views and to remove the reproach (in my opinion deserved) of the Engineer, would be a considerable gain from our own point of view. Our cause may be advanced considerably if we make a proper use of the present opportunity.

Given a mathematical operator, how should one reason about the outcome of its repeated invocation? This question is relevant in both mathematics and computer science, where iterative operator invocations are core, algorithmic elements.

An initial approach, which one may naturally follow, is to try the operator in diverse situations, observe the results, and suggest a general outcome. Such an approach embodies operational reasoning, where inference derives from ‘behaviours’ of invocation sequences. This may indeed reveal some behavioural characteristics, but is it sufficient for rigorous argumentation of the general outcome? Not quite.

Trigonometry , like most abstract sciences, has both an operational side and a quantitative side.

The operational side is dominated by the hypothesis that a function f(x) exists which has the property

It is from this that the usual rules for operating with the trigonometric functions are deduced.

When exploring some triangle geometry, we stumbled upon a set of Cevian axes: lines having certain properties in common with the Euler line. These axes yielded interesting results that we had not seen before. However, we have since learnt that many of them were discovered in the 19th Century. We were also unaware that John Rigby had covered much of this ground in unpublished work in the 1990s (private communication). We are therefore reluctant to claim any of our results as new, even though we hope that some may be.