July 1990. The Pop Maths Roadshow comes to town. “Chaos” Rules OK.

28 July. Receive invitation to post-Pop Maths party in September. One of the conditions of eligibility to attend is to tackle four problems listed on the invitation. Surely only mathematicians do this sort of thing? I glance at the problems. Two of them I know how to tackle, and a third I haven’t a clue how to begin; but it is the first problem that intrigues me most.

Among the ways in which a teacher can kindle a pupil's interest in mathematics is by invoking spirits from the subjects past: spirits of men full of imagination and determination; enterprising individuals brimming over with new ideas; pioneers who not only changed their particular discipline, but who in some cases even altered our perception of the world. The Scots have a heritage of which to be proud in science and mathematics. This article discusses four pioneers in statistics.

Before the last General Election politicians were arguing about the effects of a minimum wage policy. One was suggesting that we should have a minimum wage that was a certain percentage of the mean wage, while another claimed that such a policy would be inflationary, with wages spiralling up like a dog chasing its tail. What struck me was if you give some people an increase in salary to a fraction of the mean salary then you’ve increased the mean, so you’ll have to do it again. Might this process not lead to wages spiralling up to infinity?

The following activity is one of a series developed for junior high school students (age 12-15). Each of the activities involves a historical discussion of a subject related to a certain curriculum topic, and takes the form of a teacher directed presentation/discussion with accompanying transparencies and worksheet.

The choice and development of the activities was guided by certain general principles. The connection between the regular curriculum and the selected subject is meant to give relevance to the history and also, to motivate and deepen student understanding of subject matter. The structure of the activity enables the teacher to change, omit, or to add if he wishes. Points for discussion and questions to be worked are deliberately inserted at intervals to activate students, because in previous trial versions which were closer to the lecture format we found that students lost the thread and hence, also interest. Reactions to the present format were much more encouraging.

It has been my practice for many years to introduce the concept of a vector space to students by looking at families of magic squares and triangles. Consider, for example, all possible magic triangles like that shown in figure 1(a), where each circle is to be filled by a real number so that the totals along each edge of die triangle are constant. Then scalar multiples of such triangles, together with sums of them, will also be magic. Once four numbers are put into the circles, such as a, b, c, and d in figure 1(b) then the other two entries are uniquely determined. Because the other four numbers can always be freely chosen, there are four degrees of freedom.

Everyone knows that x−x = 0, but how many people realise the power of reversing this equality and writing 0 = x−x? What I mean is that we can always add and subtract the same quantity to and from any algebraic expression without altering its value. In general, we may choose infinitely many values of x, and the skill lies in choosing the most suitable x so that progress can be made in a specific problem. Beginners in mathematics are often mystified when a particular x is chosen and almost invariably respond with something like “I agree that what you are doing is correct, but why on earth did you pick that particular x?” There is no answer to this except perhaps “because it works!”.

Some three weeks into the academic year I presented, off the cuff, the following morsel to engage some lower-sixth formers who had polished off some routine derivatives all too quickly:

Sketch the curve whose equation is given by x3 + y3 + 3xy – 1 = 0 , and find the gradient at the point (–1 , –1).

Since this was near the end of a double period I thought no more of it. At the end of the next double period one student remarked that he had been having difficulty in sketching it. “It seems to be a straight line”, he said. I rashly suggested that he could well be mistaken, but promised to look at it. If my faith in the student had been greater I could have saved myself some labour searching for simple solutions of the cubic. I had noticed the solution (½ , ½) by the way. I began to have delusions of grandeur, imagining myself to be the natural successor of Diophantus, as I found the solutions (5,–4) and (–4,5).

The candidates have sat the examination. The examiners have marked the scripts. Everyone has gone on holiday … except the chief examiners, for whom one more task remains. Before the Board’s computer can start to churn out the results slips, they must decide the formula for converting the marks into grades.

When there is just a single paper, and no course work, as with many AS examinations, this is straightforward enough. The chief examiner, making a judgement of the quality of the scripts against grade descriptions, backed up by past experience, will propose a set of grade boundaries – that is, the lowest marks which should achieve each of the available grades – and the conversion is then automatic. For example, if the grade boundaries are A 76, B 68, C 59, D 50, E 41 and N 32, then candidates with marks of 61, 99 and 30 will be graded C, A and U respectively.

During the first few weeks of the A Level course (SMP Mathematics) in September my pupils at Prudhoe County High School (a 13-18 mixed Comprehensive in Northumberland) are faced with the problems of finding a minimum value without having met any calculus. This stems from the first chapter of Revised Advanced Mathematics Book 1 (New Edition) [1], which provides an excellent opportunity for some practical mathematics with a historical approach.

Pupils from the age of 12 to 16 can occupy themselves for hours with their skate-board, walkman or the video-recorder, while their concentration during a lesson of mathematics vanishes often within a few minutes. What could be the cause? Isn't it possible to keep their attention during the lessons? Of course, I'm not the only one posing this question. Many solutions have been proposed in the past for this concentration and motivation problem. In this article I just want to add another one. I do not have the illusion that I am presenting the ultimate solution, simply because it does not exist. For a good teacher having one solution only is not enough. He or she has to combine and diversify the various ideas continuously. The keyword for the lessons of a good teacher is variation.

While musing on the graceful sylvan scene
Depicted on a classic Grecian urn,
John Keats declared that “truth” and “beauty” mean
The same, and here this statement’s worth we learn.

“Thou still unravish’d bride of quietness!
Thou foster-child of Silence and slow Time”,
Wrote Keats [1], the um’s calm beauty to express,
While he, by permutations of the rhyme,
Did generate its group of symmetries,
Which is dihedral and of order six.
The um and its inversion therefore match
And all its patterns replicate in threes:
So triplet dancers demonstrate their tricks
While silent pipers play a triple catch.

Twelve golf balls appear identical, but eleven weigh exactly the same while one ball is either lighter or heavier than the others. Determine the “odd ball” and whether it is lighter or heavier than the others in just three weighings of the balls on a balance scale.

Puzzle problems like this abound and experience has shown that they are quite attractive to students. But this is particularly true in the present case since a solution exists that appears little short of magic. This problem provides excellent motivation for developing reasoning and communication skills. It also provides strong motivation for learning arithmetic in a base other than base 10 and for developing skill at doing mental arithmetic quickly, easily, and accurately.

Augustus De Morgan (1806-1871) was a prolific writer, a notable mathematician and logician, and an excellent teacher. His work in logic and his invention of the logic of relations are well-known. However he also thought and wrote extensively about mathematics teaching pedagogy, and his approach is still relevant to mathematics teachers today. In this short note I offer selected quotations from De Morgan's writings on mathematics education, preceded by a short biography taken from.

While looking at old copies of the journal Nature I chanced upon correspondence regarding the origin of the word radian. Textbooks today simply state it is an angular measure and have no interest in, nor space for, the history of the word. The Oxford English Dictionary states that the word first appears in print in 1879 in the second edition of the Treatise on natural philosophy by William Thomson (later Lord Kelvin) and Peter Guthrie Tait. In their discussion of angular velocity they wrote: “the usual unit angle is … that which subtends at the centre of a circle an arc whose length is equal to the radius. For brevity we shall call this angle a radian.”

An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York

In researching the history of recreational mathematics, one finds that the Propositiones ad acuendos juvenes of Alcuin of York is frequently cited as the earliest problem collection in Latin. The text contains 56 problems, including 7 major types of problem which appear for the first time, and 2 major types which appear for me first time in the West. By any standard such a collection is of major historical interest, so it is surprising that a critical edition was not prepared until Folkerts' edition of 1978, and yet more surprising that no English translation has ever been produced.