Warwick Sawyer's articles

W W Sawyer published many articles at an elementary level. Below we list some of them, given in chronological order beginning with the earliest. Some of these articles, and various others by Sawyer, are available at https://www.wwsawyer.org/work-by-sawyer.html.

Let us say a little about the extracts from the articles we give. These follow no particular pattern; some give the introduction to the paper, others give an example or an ideas taken from the paper, while others are extracts from the conclusions. We hope this mixture gives a flavour of Sawyer's ideas about teaching mathematics.

W W Sawyer, Trigonometry Abstractly Treated, The American Mathematical Monthly 64 (10) (1957), 734-737.

Trigonometry when taught to thirteen year olds is (rightly) treated as a branch of physics. If a pupil asks, "What is an angle of 40º?" the teacher points to this angle on a protractor, that is, explains the idea by means of a physical object. Is it possible to make the ideas of angle, sine, cosine precise without any appeal to figures?

The natural way to explain angles would be by stepping off equal intervals around the arc of a circle. If $A, B, C, D, E$ ... are evenly spaced points on the circumference of a circle with centre O, we should recognise statements such as $AOB = BOC, AOC = 2 AOB$ . In marking out the points $A, B, C, D, E$ , we use compasses to make equal not the arcs $AB, BC, CD, DE$ but the direct distances $AB, BC, CD, DE$ . In drawing the circle $ABCDE$ with centre $O$ we also use distance. The starting point is therefore the idea of distance between points.
W W Sawyer, Using Your Mathematics Student Journal: Plans for the Journal, The Mathematics Teacher 51 (2) (1958), 138-140.

The situation of American mathematics is a great paradox. The general attitude of Americans to life is intelligent, enterprising, non-traditional. In mathematics teaching the opposite obtains; everything is traditional, based on rote learning, the acceptance of authority, the absence of understanding. It is only the exceptional efficiency of Americans in other spheres of life that has enabled America to survive with a mathematical syllabus in which the average child is two years behind his European counterpart and the able child at least four years. And the European system itself is far from being a model of how mathematics should be taught. Indeed in Europe, as here, the efficiency of mathematical teaching is not merely low, it is a negative quantity - it puts pupils against mathematics in a very large number of cases. Sooner or later, as mathematics became increasingly necessary for technology, a crisis was bound to come. Sputnik has only dramatised the situation already existing.
W W Sawyer, Things and un-things, The Mathematics Teacher 51 (1) (1958), 14-15.

Now, of course, since we cannot in this world actually have -3 apples, there must be some sense of unreality connected with negative numbers. I am about to describe a universe in which you could have -3 apples. This universe is a little fantastic, but, if anything, that should commend it to children. Once we have this universe imagined, there is no cheating. In it, children could learn to add and multiply negative numbers with the same apparatus and procedures that actual children use in Grade 1 for actual numbers.

Grade 1 arithmetic is really a series of experiments in physics. You put two apples and two apples together and you find you have four apples. Or, starting with four apples, you can break them up into two apples and two apples.

We now make our break with the accepted physics, and suppose the world contains not only things but also un-things. If an un-thing and a thing meet, they wipe each other out. An un-apple wipes out an apple, an un-dog wipes out a dog, and so on. It can work the other way round, too. If you have a box with nothing in it, a little later you may find it contains a tiger and an un-tiger, or a dollar and an un-dollar.
W W Sawyer, What About Science is Important to Teach?, The Science Teacher 25 (8) (1958), 430- 431; 433.

It is the author's opinion that the fundamental objective of any science course is to develop within the minds of students a broad understanding of how science functions. Reduced to simplest terms this is the scientific method. The cardinal principles discussed above are not thought of as being in any fixed order, nor do they prevent anyone from starting his scientific investigation at any level of this method. When based upon this objective it becomes of vital importance that the science offering be organised and centred around the curiosity of boys and girls. The "how" and "why" questions raised become the real problems upon which the members of the class apply the previously described concepts of scientific investigation. The learning situation is exciting and with most students a high degree of interest prevails. Factual information will also be acquired and in a manner which gives true understanding and long retention.
W W Sawyer, Why Is Arithmetic Not the End?, The Arithmetic Teacher 6 (2) (1959), 95-96; 99.

In primary school we learn to answer questions like, "What is 3 times 8?" and "If an object cost $8.23, what would 37 such objects cost?" After a bit, someone who was good in arithmetic might be able to add, subtract, multiply and divide any numbers that anybody mentioned. It seems there would be nothing more for such a student to know. But there is a whole subject that grows out of arithmetic, and yet is not arithmetic. Below we give some questions. Each question is a question in arithmetic. But if you answer these questions, you will notice something about the answers, something you did not expect, something that arithmetic does not explain. And you will be able to guess the answers to much more difficult questions, without doing any working out.
W W Sawyer, Proofs with a new format, The Mathematics Teacher 52 (6) (1959), 480-481.

If you want to make geometry an exercise strict logic, you must bring in axioms of betweenness and other refinements, and thus make geometry a topic for graduate school - incomprehensible to Grade 10. If you are not going to make it an exercise in strict logic, but rather in plausible reasoning, be honest about it. Geometry is particularly unsuitable for teaching logic. So many of its results are so clearly suggested by the figures that students do not see why a proof is needed; in any case, it requires profound philosophical skill to separate the logical argument from the evidence of our senses. But it is quite possible to produce reasonable arguments.
W W Sawyer, Pressures on American Mathematics Teachers, SIAM Review 1 (1) (1959), 32-37.

The research mathematician is particularly aware that no twentieth century mathematics is dreamed of by most teachers. The technologist is concerned with the industrial applications. The college professor of physics or engineering wants to teach mechanics to college freshmen and would rejoice if high schools gave even a seventeenth century intuitive understanding of calculus. The good teacher in school or college regrets the prevalence of rote learning and its destruction of initiative and curiosity. The school teacher may observe how bored the best students are by the long dragging out of elementary arithmetic, so that some of the ablest turn their backs on mathematics as a dull subject. Each and every one of these viewpoints reflects one aspect of the truth. The danger, in this age of specialisation and intellectual atomisation, is that each aspect tends to become a separate creed, instead of an element in our awareness of the whole situation. Any one of these partial views can defeat its own ends.
W W Sawyer, The concrete basis of the abstract, The Mathematics Teacher 52 (4) (1959), 272-277.

The most valuable thing any mathematics teacher can do today is to have the courage to face the situation as it actually is. There is an enormous amount of mathematics. It takes years to learn even a reasonable part of it. The only solution is to start extremely young. We have very few teachers of young students with the necessary background. Making the necessary changes will not be simple, but it must be done.

The following are aspects of our difficult situation:
1. The past failure to provide mathematical training for teachers on an adequate scale.

2. Partly as a result of this failure, the prevalence of rote learning, of lack of understanding, of meaningless memorisation of procedures (e.g., calculus) both in schools and in colleges.

3. The fact that "the high school syllabus corresponds almost exactly to the state of knowledge in the year 1640" (said in various ways by many mathematicians). This means:

a) High school work has little relation to modern research in pure mathematics.

b) It has little relation to the most recent demands of technology.
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How, then, shall axioms come into algebra? Different mathematical systems obey different sets of axioms. Axiomatics plays much the part in mathematics that comparative anatomy plays in biology. It is unlikely that comparative anatomy would ever have developed if only one kind of animal inhabited the earth. If, in accordance with our basic principle, we are not to tell students about axiomatics, but to lead them to develop the idea, it is essential that they should have met several mathematical systems so that they can compare them. But usually they have met only one system, arithmetic and the algebra growing out of it. Even if they know something about integers, rational numbers, and real numbers, these are still too much alike for comparison to be very exciting. It is possible to develop some miniature mathematical systems very quickly.
W W Sawyer, Algebra in Grade Five, The Arithmetic Teacher 7 (1) (1960), 25-27.

The teaching of algebra has not been part of the duties of a Grade Five teacher in the past. The difficulty in introducing algebra earlier is clearly that of finding teachers who can present it effectively rather than of finding students who can absorb it.
W W Sawyer, Algebra, Scientific American 211 (3) (1964), 70-79.

It would of course be beyond the scope of any one article to attempt to trace all or even most of the effects of these two new ideas on the course of modern algebra. For one thing, algebra itself has become so compartmentalised that each separate branch would have to be treated more or less in isolation. On the other hand, isolated snippets of information are unlikely to satisfy the general reader, who would remain ignorant of the overall framework that gives these fragments their significance. The only satisfactory solution seems to be to cut off a rather large chunk of one branch of modern algebra and explain it in detail. The reader will then be able to gain some idea of the general direction in which algebra is moving from a consideration of how this particular branch has developed in the years since 1800. I have chosen to devote most of this article to a detailed discussion of vector algebra and matrix algebra, two subjects that are just beginning to find their way into the high school curriculum. Both subjects have already made significant contributions both to pure mathematics and to science. Finally, I shall sketch more brie y several of the other departments of modern algebra in which interesting and significant work is in progress.
W W Sawyer, Notes on Matrices, The Mathematical Gazette 54 (387) (1970), 1-9.

It is very desirable when teaching mathematics not merely to state definitions and prove theorems, but also to show the train of thought that would lead us naturally to these definitions and theorems. Students are thus enabled to see mathematics as an alive, continuously developing subject. They are reminded of the interdependence of the various branches of mathematics; the memory of each branch is kept alive in the students' minds, because it is continually being revisited, and the student finds it easier to remember results when he understands the thinking that produced them. This approach also encourages a student to try to carry the train of thought further and to make his own discoveries.
W W Sawyer, Some thoughts on examinations, Mathematics Teaching 69 (December 1974).

The first question is whether there should be any examinations at all. When I worked in Britain I was aware of various unfortunate effects of examinations. In Ontario I have observed a system where students are admitted to university without examination and the consequences are even more unfortunate. It is no kindness to a student to admit him to a mathematics course for which he is not adequately prepared; he will endure a frustrating, harmful and purposeless experience. I would therefore like to see a variety of mathematics courses available to students and adequate testing to ensure that a student gets into a course for which he is suited.

One great defect of examinations in schools and universities throughout the world today is that they lend themselves to faking. A course is taught in such a way that students do not understand the basic ideas, but they learn to produce rigmaroles that look sufficiently like mathematics to justify a pass mark. After a few months the rigmaroles are forgotten; nothing of positive value remains in the mind.

Mathematics involves understanding, techniques amid ingenuity, but without the first of these time other two are impossible. Unfortunately it is the latter two that examinations tend to stress and it is only by accident that lack of understanding is isolated and exposed.
W W Sawyer, Does It Matter?: No, Not Really, The Mathematical Gazette 59 (410) (1975), 277-279.

A famous sneer describes university lecturing as answering questions that have not been asked. In no part of mathematics do we so easily lay ourselves open to this reproach as in analysis. We can justify analysis to a student only by showing him the errors into which he may fall, at his present mathematical stage, if he does not take correct precautions.

Clearly then, the analysis taught must depend on the maturity of the student, on the level at which he is working, and on the distance in mathematics he expects to go. If we have to choose between a presentation the student finds understandable and fruitful and one which would be preferred by a research worker at the frontiers of present knowledge, our choice must be for the former. A space suit is an essential piece of equipment for an astronaut; it is not very convenient for walking to the office.

It may seem paradoxical, but the fact is that only by observing the past can we cope with the future. If a student is shown how at each stage of mathematics it has been necessary to bring in new concepts and new safeguards, he can realise that our present equipment will have to be overhauled and refined in another 20 years.
W W Sawyer, The Case for Fractions, Mathematics in School 6 (1) (1977), 16-17.

It may be objected that only mathematical specialists and scientists need such a command of algebra. I am totally in disagreement with this view. For many years I have been convinced that the heart and core of mathematical education, not only for advanced students but also for technical workers and for the ordinary citizen who wants to understand something of the world around him, is the ability to use simple algebra with understanding and with reasonable facility. Neglect of this was the main mistake of the "new maths" even in the mild form (School Mathematics Project, SMP) that reached this country. SMP of course has many virtues, being a splendid example, of lively and vigorous teaching, far better than, for instance, anything produced in North America. But, as is now widely recognised, in its initial reaction against excessive drill, it made the mistake of essentially abandoning the teaching of algebraic operations, instead of seeking more effective ways to teach them.
W W Sawyer, Computer Preview, Mathematics in School 12 (3) (1983), 13-15.

The further you go in mathematics the more interesting it becomes. There are many simple and beautiful results in advanced mathematics which most learners never hear about because they do not have the necessary mathematical background. It is one of the virtues of the electronic computer that it helps us to show the nature of such results to pupils who are not ready (and perhaps never will be) to understand the underlying theory.
W W Sawyer, Algebra: A Vital Ingredient, Mathematics in School 17 (4) (1988), 45-47.

Now arithmetic abounds in mild surprises, which children enjoy spotting, and which it is the business of algebra to explain. These will come to light if we are careful not to set arithmetical exercises at random, but rather in a sequence chosen so that some pattern can be seen in the answers. For example, children may be asked to find the numbers $5 \times 5, 6 \times 4, 7 \times 3, 8 \times 2, 9 \times 1, 10 \times 0$ and examine the answers, which are 25, 24, 21, 16, 9, 0. Pupils may notice that the downward steps are 1, 3, 5, 7, 9, the odd numbers in order. We may go on to ask if there is anything interesting to observe in the differences between these numbers and 25. It is apparent that all the differences are perfect squares. Surely this cannot be an accident.
W W Sawyer, Vision in Elementary Mathematics, Mathematics in School 18 (2) (1989), 6-7.

In teaching mathematics there is always the danger of imagining that our pupils understand some idea, which in fact they do not. It is therefore always wise to begin a lesson by asking the class to illustrate the idea on which we propose to build. The present article gives an example of such an approach. It also shows how pupils can arrive, naturally and without anxiety, at an understanding of algebra. The central importance of algebra was stressed in the previous article.
W W Sawyer, Vision in Elementary Mathematics Part 3: Investigations, Mathematics in School 18 (4) (1989), 42-43.

In USA I once heard of a class where they spoke of "Mary Smith's Theorem" instead of "Pythagoras' Theorem". The idea was that each theorem was called after the pupil who first noticed it. This emphasised that mathematics was something that learners could arrive at, not just a series of rules that had to be memorised.

Arithmetic is even more suitable than geometry for such an approach. There are many patterns in numbers that can be observed, illustrated by drawings and perhaps eventually explained. This is more interesting than just making calculations and getting the right answer. In fact, in the course of an investigation many calculations will be made but they will hardly be noticed; attention is concentrated on finding and explaining the pattern.
W W Sawyer, Basic Tables and All That, Mathematics in School 18 (3) (1989), 43-46.

H G Wells' History of Mr Polly, published in 1910, gives a very fair picture of schooling at that time. Mr Polly had a lively interest in everything - except the subjects he had met in school. In mathematics "he was always doubtful whether it was 8 sevens or 9 eights that was 63 (he knew no method for settling the difficulty)."

The last eight words bring out the essential weakness of something taught purely by memory. If the memory becomes garbled, there is no way to recover the original message.

We are so used to the idea of most people dreading mathematics and not understanding it, that we do not realise what an extraordinary feat of mystification it was to turn something as simple as 7 eights into a mystery. If Mr Polly wanted to know the value of 7 eights, all he needed to do was to set out seven rows of eight objects and rearrange them as far as possible in groups of ten.
W W Sawyer, Vision in Elementary Mathematics: Part 4. Experiments with Graphs, Mathematics in School 19 (1) (1990), 21-23.

Drawing graphs to illustrate data is fairly simple and quite young children get this experience. It is a big step to graphs based on formulae. Sometimes these are introduced by choosing some simple equation and asking pupils to plot it. This can be tedious and may involve much correcting of mistakes. There is something to be said for doing things the other way round - starting with the graph and trying to guess the formula.
W W Sawyer, Vision in Elementary Mathematics Part 5: Mathematics and the Sense of Power, Mathematics in School 19 (3) (1990), 8-9.

The most important test of mathematics teaching is its emotional impact. Taught properly, mathematics gives learners devices which they understand and which enable them to cope with an ever growing number of real life situations. Badly taught, it appears mysterious and frightening. People taught this way, even if they may acquire sufficient rules and tricks to fool examiners, will neither expect nor be able to make practical use of mathematics.

A good attitude is encouraged if mathematics lessons are related to activities the learners consider worthwhile. It can also be encouraged by work within the subject itself, if learners have the experience of doing what a mathematician does - finding new results by using methods they already know.
W W Sawyer, The Importance of the Unbelievable, Mathematical Review (November 1991).

The enjoyment of a play or a film is said to depend upon the suspension of disbelief. In the development of mathematics also there are times when progress depends upon the acceptance of an idea that appears absurd or impossible.

Negative numbers were at one time such an idea. The common sense view reveals itself in sayings such as 'I couldn't care less'. The amount of something cannot be less than nothing. For many years mathematicians went along with this view. They did not accept a negative number as a value for a symbol or as the solution of an equation. If asked to solve $x + 1 = 0$ they would say that no solution existed.

It would be immensely inconvenient if this view had prevailed. In graphical work, for instance, we would need a different coordinate system for each quadrant of the plane.
W W Sawyer, Algebra, the Cement of Mathematics, Mathematical Review (November 1992).

Computers are rapidly making it possible for any purely mechanical mathematical operation to be carried out by a suitable program. This will raise an interesting problem for the learning and teaching of mathematics. In the past - say 70 years ago - young mathematicians became proficient in the mechanical processes of algebra by practising them (sometimes with boring and excessive drill) and in the course of this achieved some feeling for the strategy of algebra, how operations could be combined to give new and interesting results. Basically algebra enables us to derive from some collection of results, expressed by equations, other equations which may be far from obvious. How this feeling for algebraic possibilities is to be conveyed in the computer age is a question of increasing importance.

This article does not attempt to solve that problem. Its aim is simply to give an example of the way in which algebra can tie a new result to theorems we already know.
W W Sawyer, Vision in Elementary Mathematics. Part 6. Matrices in School and in Life, Mathematics in School 22 (3) (1993), 45-46.

I met a most unusual application of matrices when I was consulted on a problem in brain surgery. The surgeon wishes to arrive at a particular place in the brain. At one time the purpose was to destroy the pain centre, so that a terminally ill patient became incapable of feeling pain. To enable the surgeon to reach the correct spot a framework is placed around the patient's head. This contains three perpendicular bars, $OA, OB$ and $OC$ . A position in 3 dimensions can be specified by giving 3 numbers, $x, y$ and $z$ , the distances in these 3 directions. Data are available, specifying the position of various cells in the brain, also by 3 numbers. Naturally the axes used for the system inside the brain are not the same as those in the framework, and they may not even have the same directions. It is necessary to have a formula for changing from one system to the other. Matrices give a very simple way of doing this.
W W Sawyer, mathematics emotions and things, Mathematics teaching 142 (1993).

For many years I had believed that it should be possible to have a school in which pupils worked naturally and did not have to be continuously chivvied along by teachers. In the years 1937-1944 I saw such a school in action. It was in Prestolee, a little mill-town between Manchester and Bolton, a school for ages 5 to 14, under the control of the Lancashire County Council. It was organised as a library and a workshop. When you visited it, children were moving around, looking up information they needed or engaged in practical work, which had altered to whole appearance of the school building and the playground. This school gave the opportunity to all pupils to advance at whatever rate corresponded to their academic ability, and all were involved in a variety of practical pursuits. The school was ornamented with notices (made by the pupils of course) one of which read "We learn by doing". I visited this school frequently, and was struck by its extraordinarily peaceful atmosphere. These visits later influenced me very strongly in various attempts I made to help pupils who reacted rather to things than to words. This article describes some of these experiences.
W W Sawyer, Catering for the Extremes, Mathematics in School 24 (2) (1995), 28-30.

I am convinced that comprehensive schools could be quite as successful in producing first-rate mathematicians as the best public schools were in the past.

It may be objected that the resources available to schools are far below what they should be and that teachers already have too much to do. This is true. However, as will a later in this article, the point of the approach in question is that teacher involvement is minimal.

It is well known that Mozart showed remarkable musical ability at the age of four and in fact embarked on a career of performing in public at the age of six. It would clearly have been both cruel and stupid if some bureaucratic government had insisted that he should go to school at the age of five and do five-finger exercises along with the rest of the class. No syllabus can possibly meet the needs of such a pupil; what was needed - and what, fortunately, happened in fact - was to liberate him altogether from every kind of syllabus. Few people recognise that mathematicians can be equally precocious.
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I have been among those trying to lessen the gap in mathematical education. Since 1977 interested secondary school pupils have been coming to our home on Saturday mornings. They usually start at the age of 14 and finish when they leave the Sixth Form. The work has frequently been well up to university level, and some of the students have already begun distinguished careers.

I am however under no illusion that such individual efforts are in any way of adequate magnitude. This is apparent in two ways. On the one hand, the number one can help by such efforts is a minute fraction of the number who could benefit from it. On the other hand, with the morning sessions at intervals of a week, it is much harder for the students to recall the details of work done some time previously. What is required for mastery is that work should be done day in, day out, and learning reinforced by working many examples. Only in this way can the material become really ingrained in the mind. No one can offer this continuity of work except the schools.
W W Sawyer, What Use Are Abstract Spaces?, The Mathematical Gazette 80 (487) (1996), 167-175.

Fréchet took up this question. He examined the classical work on real and complex numbers, and observed that distance played a decisive role. In analysis we are much concerned with questions about limits. For both real and complex numbers, a sequence of numbers z_n, tends to a limit L if the distance of z_n from L tends to zero. He decided that, if it was possible to find a satisfactory definition of distance between two mathematical objects (of any kind), it would be possible to find theorems about these objects analogous to the theorems about real and complex numbers.

The first question then is: what is a satisfactory definition of distance? He looked at the traditional proofs and found the only properties of distance used were the following very simple ones:
1. Distance is measured by a real number, which is never negative.

2. A distance is zero if, and only if, it is the distance between a point and itself.

3. The distance from $A$ to $B$ is the same as the distance from $B$ to $A$ .

4. You cannot shorten your journey by breaking it. If you go from $A$ to $C$ , and then from $C$ to $B$ , the total distance cannot be less than the distance from $A$ to $B$ . (It may of course be equal, if $C$ lies on the direct route from $A$ to $B$ .)
The fourth property is known as the triangle axiom. It corresponds to Euclid's remark that the sum of the lengths of two sides of a triangle must exceed the length of the third side.

Fréchet's investigation was extraordinarily fruitful. It was found possible to give a satisfactory definition for the distance between two matrices, two transformations, two functions, or two operations that may involve differentiation and integration. At one blow, this opens the door to a whole series of results concerning the most varied situations.

Last Updated November 2020