Two Avenues to Advance in Mathematical Education
W. W. SAWYER, FIMA
In an earlier article  I quoted a very outspoken report of the Scottish Education Department and some comments of Wilf Flemming.  This report spoke of "the dullness and futility of much school teaching of the subject" and contrasted it with the progress made by pupils in the mathematical work of the Air Training Corps. (The report appeared in 1947 shortly after the end of the war.) Flemming described how teachers were amazed at the progress their former pupils made when they needed mathematics for their practical work in the armed forces.
The Scottish report said that its remarks applied to "the great central mass of boys and girls, ranging from the Cs well up into the B group". The report, therefore, is not saying that accepted teaching methods are ineffective with some exceptionally difficult or non-academic group of pupils; the criticism relates to those whose needs determine, or should determine, a large part of a school's organisation.
Towards the end of the stone age, hand axes of great beauty and efficiency were being produced. Understandably it never occurred to their makers that better axes could be made through the application of fire to certain rather drab-looking metallic ores.
The Scottish report suggests that our accepted chalk-and-talk method of teaching may have some analogy with the stone axe, and that some entirely different approach, with mathematics embedded in some purposeful, real situation, involving making and doing, might lead to an altogether higher level of achievement for the average learner. I stress "average"; chalk-and-talk is entirely appropriate for the most theoretically minded students. They readily imagine the situations involved, and may be impatient or clumsy in practical work. With them I use, and shall continue to use, chalk-and-talk. But for the bulk of the students, from the good and average students, mentioned by the Scots, right through to those needing remedial or special attention, it does appear that there should be careful experiment and a search for an alternative approach.
The wartime situation embodied two great advantages for the teaching of mathematics. The matheniatical symbols were related to actual objects, so the students could see what it was about; the students were in a real situation, where they wanted to succeed, so their wills were concentrated on achieving success. Both these factors are essential. If the student cannot see what the mathematics neans, he or she must fail even though the desire to learn is there. If the desire is not there, in a subject as active as mathernatics, failure is inevitable.
If we are able to find a peacetime way of realising these two conditions, a remarkable rise in rnathematical standards will result, but this wiIl not be the greatest benefìt. That wiil lie in the redirection of the energies of the young. Youth has abundant energy and expects life to be not merely tolerable but exciting. Any civiiisation that wishes to continue must find some constructive, or at least harmless, outlet for that energy. Any classroom in which an apathetic or restive class feels that its life is being wasted represents an abdication of adult responsibility. It leaves to chance the direction youthful energy will take, and if the result is delinquency, vandalism, violence or crime, there is no occasion for surprise.
Of course, it is one thing to be convinced that an effective new approach to mathematical education exists and quite another thing to realise it within a given situation. The most painful stage in such an endeavour is when you know, in general terms, what ought to happen but have no idea how to bring it about.
The aim is to create a situation in which pupils work at mathematics in the same way that enthusiasts work at their hobbies. The aim has been achieved when the teacher can go quietly out of the room and no one notices this until there is some need for advice or help.
Some notes now follow on experiences gained in attempts to achieve this ideal.
The Leicester experience, 1945 - 47
About the same time as the Scottish report but quite independently, a development in the same direction was happening in Leicester. W. H. Joint, head of the mathematics department in Leicester College of Technoiogy, had come to the conclusion that we were trying to teach mathematics to engineering apprentices in a foreign language. Mathematics appeared in words and symbols, but the apprentices thought, so to speak, with their hands, by appreciating the sizes and shapes of actual objects. He began introducing physical objects into mathematics lessons. For instance, the apprentices had to know that the area of a circle was proportional to the square of its diameter, not simply to the diameter. He had circular discs made from a sheet of metal. The disc with l-inch diameter would be weighed, and the class asked to guess the weight for a diameter of 2 inches. They guessd "twice as much". When the disc was weighed and found to weigh four times as rnuch, this produced an impression that months of sermons would not have done. Shortly afterwards, I took over the department and together with some excellent colleagues carried the development to the stage where we could present almost every topic of the ONC course by relating it to actual objects. [3-5]
Our guiding principles were:
We were, of course, dealing with an easier situation than that in many schools. The class consisted entirely of boys with the same occupation and more or less similar interests. The problem of motive was not too acute; many of them genuinely wanted to get the certificate, though of course there were some who felt that a day away from the factory should simply mean a marvellous holiday.
My general impression is that, by the time we had fully worked out our programrne, the approach was successful and led to a very harmonious relation between the class and the teacher.
The context of crisis
This may seem a very unsuitable time for proposing innovations in education. Barring a fundamental change in the economic systern, we can anticipate inadequate finance of education for the foreseeable future. This means large classes, with shortages of books and materials. It may be thought that we have our work cut out simply to cope at all with this situation. But the argument can also work in the opposite direction. We may get by with ineffective methods in normal times; in times of difficulty, it can be disastrous if we do not find the most efficient means.
One of my own ventures arose from a small local crisis. When I was at Leicester, a decision was taken by an industry that employed extremely unskilled workers to send their youngest workers to the College for one day each week. We believed that most of the boys and girls concerned had been very glad to escape ofrom school and would not at all welcome a return to classrooms. On the first day of the new scheme, two of my colleagues were ill with influenza, so I had to take three classes of boys in a large room for the entire afternoon. In these circumstances I saw no prospect of teaching much arithmetic. I felt I would have done my duty to the College if I made sure they did not burn the place down, so I opted unashamedly for an activity period. 1 collected all the things I could find that would keep them busy and interested. Their reaction to this was so striking that when things returned to normal and I had only my own class to deal with, I had not the heart to return to chalk-and-talk. The name of the course was changed from Arithmetic to "Science and Calculations". I must admit that for some time it was science with little or no calculation. Then, spontaneously and quite unexpectedly, the class raised the question of how you can find the height of a building, and we were into surveying and the elements of trigonometry. A delay of this kind is usually to be expected. A disorder that has been built up over the ycars is not cured by any treatment overnight.
I also taught girls from the same industry, and for a long time was puzzled to find a way of bringing them to life. Thc turning-point was entirely non-mathematical. One day 1 brought my typewriter into thc room. They became very excited and in turn sat down and typed their own visiting cards. The value of this was that it establishcd communication: they saw that I was looking for something that would interest them. Previously I had heen trying one thing after another in the hope of raising some spark. They could not see any pattern in it and probably thought 1 was mad. Incidentally, they could not have told me that a typewriter was the stimulus required. As one girl said to me a little later, "You know, Mr. Sawyer, when we first came here you asked us what we were interested in. There was no point in doing that. We had never been allowed to do what we wanted, so we didn't know what we were interested in."
At one stage I tried to find mathematical activities related to traditional feminine interests. After a while, the girls said to me, "Forget we are girls. We want to do what the boys are doing", so they also ended up messing around with home-made electrical gadgets.
The situation in my classes had an interesting psychological side-effect. The teacher who taught these students in the hour following mine said they came like blotting paper ready to absorb anything that was put before them. Thus, even at the stage where they were not learning mathematics in my class, a positive educational service was being rendered. This effect has been observed in other settings. Shortly after the war, HMIs told me of a physical education teacher who had raised the level of performance in every class in his school. He had obtained equipment previously used for training commandos. This appealed very much to the boys and girls who used it, with the result that the school acquired an aura of glamour. The resulting sense of well-being expressed itself, quite unconsciously, in better work in all lessons. An occasional phenomenon of this kind has been noted in books on industrial management. The potential of this factor is rarely recognised. We become satisfactory when we are deeply satisfied.
The search for an approach that will release hidden springs of energy in pupils is not an easy one. Fortunately, it is not necessary to make the transition to a new system in one jump. If a teacher can periodically find an activity that arouses enthusiasm, this creates an atmosphere in which the class works willingly through a few rather conventional lessons. The question arises - what should you do if you have only one idea that seems really effective? The answer is - play your ace at the first opportunity. The first lesson with a new teacher (like the first day in a new school) often fixes the emotional attitude to everything that follows.
There is no blueprint for success; we have to adapt to local and even individual situations. However it may be worthwhile to mention two devices of some interest.
The first of these is due to Richard Carothers, a student who took part in Canadian service overseas. He was in an African village and wanted to show them that mathematics could help an everyday task like cooking. He designed a parabolic bowl with supporting ribs, all made frorn cardboard, and pasted aluminium foil to the bowl. The diameter of the opening of the bowl was about a metre, and it made an effective solar oven. Later, on teaching practice in Northern Ontario (which is well on the way to the North Pole) he had a class make one of these solar ovens. There was snow on the ground. A mug was filled with snow and placed at the focus of the paraboloid. In 5 minutes, they had boiling water and made coffee with it. 
The second device appears in the books as "an electronic metronome". It contains a couple of transistors, a capacitor and a loudspeaker and works from a small battery. It has two terminals, between which a resistor can be connected and then emits clicks. The number of clicks per minute is given by a simple law, perhaps 6 000 000 divided by the resistance in ohms. This law can be discovered or a graph drawn. The resistance of the human body lies in the appropriate range, so pupils can find their own resistance by grasping the terminals and counting the clicks. A Toronto teacher, after a successful hour with a class of youths, ascribed the appeal of this device to two things;
The electronic metronome is a multivibrator and can be adapted to produce rather harsh musical notes. An exploration of its musical (and unmusical) potentialities might provide a theme for investigatîon.
Electronic bits and pieces are quite cheap today. In general, obtaining the necessary material for an activity is clearly a problem. One way of coping with this is to form a link with a department that already uses material, such as wood or metal-work. It represents a wasted opportunity if pupils are given a blueprint and told what to make. It would be far more instructive if they made their own designs and drawings in the mathematics lesson. It may be difficult to arrange this; a woodwork course may require the pupil to make fifty dovetails (or some other set of standard items) and leave no time for individual designing. However in some schools co-operation between mathematics, science and workshop departments is possible and can be most valuable.
Another possihility is to make use of wasted material, which our civilisation produces in great profusion. Thirty-odd years ago I talked with men, training to be teachers, who had been in a prisoner-of-war camp in Germany. The only fuel they had then had was newspaper. From old tin cans they made a blower that enabled them to boil a kettle in a minute. They also wanted some matting to play cricket. They broke down the string from parcels into fibres, spun these into thread and wove the thread into a mat. Without going to such extremes of improvisational virtuosity, schools should be able to find material that is wasted and could be used constructively. A survey of what is thrown out from homes, shops and factories in one's locality could prove instructive.
Training in innovation
In all of this we are venturing into unexplored territory, which calls for determination and caution - determination to find a way and caution to avoid a spectacular disaster that would produce a violent reaction. It is particularly important that teachers in training get help in developing new ideas and propagating them tactfully. When I was training teachers in Toronto, I used to suggest to my students that they should identify some weakness in the educational cystem and undertake a small experiment to see if they could improve matters. Once, after some discussion of the child's right to movement and the plight of physically active children who were required to sit still, two or three students decided to run a club for restless boys. The principal of a school, with which we had established good relations, carefully chose six restless boys for us. The idea was to have a sports programme, running, jumping, throwing balls at targets and the like, keep elaborate records with graphs and averages, and see how much mathematics could be brought in naturally. However, it did not work out quite like that. The students soon discovered that several of the boys were restless, not because of unused physical energy, but because they were intellectuaily capable of work 3 or 4 years ahead of the official syllabus. They were suffering the kind of boredom and frustration that an average 10 year-old would experience if confined to work designed for 6-year.olds.
This brings us to a controversial issue. There is a widespread feeling today against making special provision for the academically abler students. In part this arises from a perfectly justifiable reaction against past practices, in which attention was concentrated on the ablest with neglect of the others. But in part the present attitude involves ideas which are incorrect and harmful. It is necessary to disentangle these.
Flexibility or elitism?
By flexibility I understand trying to ensure the healthy development of all pupils, from the quickest to the slowest; by elitism I understand sacrificing the mass of the pupils for the sake of the ablest.
Any excessively competitive system, whether in study or sport, leads naturally to elitism. If a schools interest was concentrated on scholarship winners to the exclusion of other considerations, or on the success of the school teams without concern for the weIl-being of all pupils, a very unsatisfactory situation resulted. I would unhesitatingly condemn both attitudes. There is an understandable fear among teachers that special attention to gifted pupils may gradually let us slide back to such former practices. There is also some kind of identification of abler pupils with those who secure an unfair share of the worlds goods and end up as millionaires. This identification is sheer fiction, as we can see from the history of countless inventors who died in poverty, or by comparing the £4000 in Einstein's will with the sum left by any newspaper owner.
There is indeed a great misunderstanding of what it is that most giftcd pupils want, not wealth but the opportunity for self-development and self-expression. A country that wants to attract the best scientists should strive to provide excellent laboratories and libraries. A scientist wants the chance to work freely in these, not to own them. Some American companies have found that they attract good mathematicians by allowing them to spend part of their working time on problems chosen by themselves, rather than on the firm's business. This shows psychological rea1ism.
If those who regard thernselves as forward-looking go out of their way to frustrate the abler pupils, the effect can only be to make a present of the most intelligent section of the population to the cause of reaction.
There is a fear that attention paid to abler pupils means large allocations of teachers' time. The reverse is true. Slow students require a lot of teaching time. The greatest service teachers can do for the gifted is simply to get out of the way, not to tie these students down to a routine syllabus. Let bright students take an occasional test or examination. to provide evidence that the regular syllabus has been understood. Otherwise leave them free to read ahead on their own, with occasional consultation, which may perhaps be with someone outside the school - pupils in primary school perhaps meeting a secondary school teacher. and those in secondary school a university or professional mathematician.
The stratcgy of change
As has been inentioned already, great care is necessary in initiating any departure frorn established
practice, so that the least harm is done if the experiment does not come up to expectation. Very little risk is
involved in allowing the most gifted and most committed pupils to work ahead on their own
Tactically it is wise if, as in this article, special attention to the gifted is proposed at the same time and by the same people as the call for a more imaginative approach to the less academic pupils. Indeed, quite apart frorn tactical expediency, there is great value in having both developments taking place at once, since there should be interaction between them. The theoretically minded pupils can help if a knotty mathematical problem arises (as well it may) in the design of some practical project. On the other hand, one practical activity that should be included in any programme is the making of visual aids and other apparatus for academic courses. Theorists and craftsmen complement each other. A healthy society will accord them equal status and encourage communication between them.
1. Sawyer. W. W., Bull. I.M.A, 1978.
This article first appeared in The July 1980 Bulletin of the Institute of Mathematics and its Applications published by The Institute of Mathematics and its Applications Telephone 01702 354020
We are grateful for the permission of the Institute for permission to reproduce the above article.
Copyright © W. W. Sawyer & Mark Alder 2000
Version: 22nd March 2001