The Work of W. W. Sawyer
By David M. Clarkson
Warwick Sawyer, who retired in June 1976 as Professor jointly to the Department of Mathematics and the College of Education, University of Toronto, has made a consistent and impressive contribution to mathematics education in the world for the past three decades. From the publication of his most successful book (continuously in print since 1943, ten translations, and about half a million copies Mathematician's Delight to his latest An Engineering Approach to Linear Algebra, in his numerous journal articles and speeches, and perhaps most importantly in his personal encouragement of students and teachers. Sawyer has exercised a profound effect on the teaching of mathematics in and out of schools.
It is in some sense ironic that his consistency of view has sometimes placed him on the right wing and sometimes on the left of pedagogic controversy, for any rereading of his published work offers considerable evidence that his recommendations were based basic, perennial issues, carefully analyzed, and that his defense of them cannot properly be considered in the context of, say, the 'modern maths' wars of the 1960s. Indeed, the scope of Sawyer's contribution far exceeds curricular issues, and this is as evident in his first book as it is in his latest.
As a mathematician, Sawyer is concerned to identify those aspects of mathematics that are most useful, and to emphasize these in establishing curricula. He defines 'useful' as that mathematics which is actually most used these days as well as those topics which are most likely to be of use in future.
Sawyer found that by this criterion, classical mathematics (in particular, calculus), vectors and matrices, and functional analysis were of the most use, in that order. He proposed that some way be found to inform the teaching profession of the mathematics currently most useful, as the list and its order were bound to change. This whole approach is typical of Sawyer's practicality when dealing with sticky issues.
A second problem he addressed was the question of organization.
His basic reason for this view sounds very Piagetian today ". . . education consists in co-operating with what is already inside a child's mind". (p.27). The genetic approach attends to this need; the logical approach; especially the axiomatic, ignores it. His books abound with specific examples of this philosophy. For instance, The Search for Pattern he rejects theca common method of introducing negative and fractional exponents in algebra courses so as to preserve the laws which hold for positive integral exponents on the grounds that such arguments are unconvincing to students, and, noting that Napier's invention of logs preceded Wallis's introduction of fractional and negative exponents by nearly half a century, proposes the early discussion of logarithms in the simple seventeen century manner. It is then possible to use a logarithmic scale to explain fractional exponents as follows:
Sawyers inventiveness in finding physical illustrations of basic concepts is nicely illustrated in Chapter 6 (How to forget the multiplication table) of M.D. where he considers the logarithmic effect of winding a rope around a post. Suppose that the rope and post are of such a size that one full turn has the holding power of ten times the pull on the other end; two full turns will then have holding power of 10 X 10 or 102, having thus given concrete meaning to such abstractions 108 (eight full turns of the rope about the post), he defines "the number of turns required to get any number" corresponding to your holding power as "the logarithm of the number". Since 102is the magnifying effect of two turns, 101/2 is that of half a turn and "the logarithm of 2 will be that fraction of a turn which is necessary to magnify your pull 2 times". This is certainly a far cry from traditional methods of introducing logarithms!
The search for simple starting points, the essence of the genetic approach (since most mathematics starts from attempting to solve problems) was Sawyer's first and constant effort. Perhaps he was stimulated by his early teaching experiences. After a superior mathematics education at Highgate School (the Headmaster was a mathematician) he took his B.A. at St. John's College, Cambridge. But before assuming a lectureship at Dundee, Sawyer obtained permission from his old head to take some of the lowest classes at Highgate.
Sawyer was by no means alone in thinking of the usefulness of mathematics and a genetic approach in teaching it. During the height of the curricular controversy of the sixties the memorandum On the Mathematics Curriculum of the High School was signed by 55 leading North American mathematicians from a list of seventy-five invited to sign it, including such men as Garrett Birkhoff, Richard Courant, H. S. M. Coxeter, Marston Morse, George Polya and, of course, Sawyer, and asserted that "the introduction of new terms and concepts should be preceded by sufficient concrete preparation and followed by genuine, challenging applications . . .".
One of Sawyer's biggest contributions to the "concrete", though evident in examples from Mathematicians Delight, cane as a result of his work as Head of Mathematics at Leicester College of Technology from 1945 to 1947. The Departmental study of applications of mathematics to industry resulted in construction of numerous mechanical models for instruction.
Sawyer believes that much mathematical activity is a search for pattern, and that children often "fail to solve problems because they cannot understand what the problems are". (Vision, p.6) Hence much of Sawyer's writing and teaching is "based on the idea of the pupil discovering mathematics mathematics for himself". (Ibid.) In his writings for children (and their teachers) he makes extensive use of the rectangular array model for all kinds of multiplication, and introduces the basic algebraic concepts by a model, first used in M.D., of "pebbles in the bag". The influence of these approaches may be seen by the number of textbook writers who have incorporated them in their own work.
Sawyer subsequently published a pamphlet of these devices, Math Patterns in Science, and they appear in most of his books, especially in Chapter 1 (Mathematics through the hand) of for The Search for Pattern. Nor was his concern limited to secondary school students or college students. As early as Mathematician's Delight he is writing of concrete models for simple arithmetic, and an experimental approach for juniors is outlined in an early book with L. G. Srawley, Designing and Making. Later, in America, he spent the 1958-9 academic year regularly teaching a class of l l-year-olds in a public elementary school in Middletown, Connecticut. The results of this experiment were collaboration with Bob Wirtz and Morton Botel (joined later by Max Beberman) in a new kind of elementary program entitled Math Workshop for Children, An overview of the philosophy behind this method is to be found in Vision in Elementary Mathematics as well as in numerous journal articles.
On the secondary level, Sawyer emphasised elementary approaches to calculus, again based on work with children in the 12 to 15-year-old age range. First examples of this appear in Mathematician's Delight, especially Chapters 10-12, and later in What is Calculus About? He starts the discussion in M.D with - "The Basic Problem".
This definition of the "The Basic Problem" is repugnant to those mathematicians who want to start with what Sawyer called "the analysis or epsilon-delta stage" (What is Calculus About?, p108) but is consistent with Sawyer's genetic approach.
The results of the crisis created a calculus that was more rigorous but also more difficult to learn, and perhaps not everyone needs to begin "with the latest and most fashionable model" (p.94). So, Sawyer's first approach has always been intuitive. For example in M.D. he considers the problem of finding y' corresponding to the formula y = log x by use of the method of "rough ideas". By By filling out a table of values from x (holding power) = 1 to 10, with the corresponding columns for y= log v (parts of a full turn), Dy, and Dy/Dx, he leads the reader to form the conjecture, by pattern, that "y" corresponding to y=log x is something like 0.414/x. The inconvenience of 0.411- motivates discussion of natural logs, and leads to the result that if y=logex, then y=1/x. But he does not just stop with the intuitive approach. What is Calculus About? ends with a Guide to further study. Asserting that it is extremely important to read books about calculus in the correct order, he outlines a progression that starts with Durell and Robson's Elementary Calculus, through Klein to Courant's Differential and Integral Calculus, and a variety of texts for those with special interests.
His elementary university level work has been mainly in the field of linear algebra, starting with the general Prelude to Mathematics, and A Concrete Approach to Abstract Algebra which was an outgrowth of the course he taught for secondary school teachers at the University of Illinois in 1957. The article Algebra contained more ideas, and was followed by another general work, A Path to Modern Mathematics, and finally his latest, a textbook for first year university students, An Engineering Approach to Linear Algebra. This last book is a very carefully thought out course, developed with his classes at Toronto during 1970-72, and perhaps represents Sawyer's methods at their best. In the preface, Sawyer says that "the student learns to picture or imagine the things he is dealing with, and then, because his imagination is working, he is able to reason about these things". Opposed to the lecture method of instruction as technologically backward, Sawyer uses a text as something for students to read, and the classes as opportunities to argue about the meanings of what has been read.
For those who wish to assess Sawyer's written contribution to mathematics education, this book is probably the canonical form.
One hopes, of course, that in his retirement more books are forthcoming, for instance the missing Volume 2 of Introducing Mathematics, or a more complete library of his mechanical devices. But Sawyer's contributions are by no means exhausted by the printed word. As much as any major figure in mathematics education over the past three decades he has exerted a personal influence on students and teachers worldwide. After establishing a Mathematics Department at University College, (Gold Coast (now University of Ghana) in 1948-50, he went to Canterbury College in New Zealand for five years, 1951-56, the University of Illinois in 1957-8, Wesleyan University for seven, 1958-65, before settling down at Toronto in 1965. In each of these places he was confronted with hard new problems which he addressed with his usual practical methods, What kind of mathematics do bright young students of a developing country with no mathematics traditions need? (Ghana) How can the supply of good mathematics teachers be improved? (New Zealand) What curriculum for high schools? (Illinois) Problem solving in industry and government for bright university undergraduates (Wesleyan) and mobilising elementary school teachers in Canada to provide concrete experiences for their children in the late sixties; these were a few of Sawyer's crusades over the years.
In each case, the attempts at solutions were practical. Where do teachers form their attitudes towards mathematics? In schools. How do you improve teaching? Work on the teachers before they become teachers.
But, Sawyer noted, even at the university level "resources both of manpower and of finance" were extremely limited.
So, with backing from Canterbury College, Sawyer established a mathematics club for sixth form pupils in 1952. It ran for the six years he was there, and after, with a steady attendance of 100 to 150 enthusiastic members.
The club was clearly a successful effort as far as its members were concerned, and there is even some evidence that it may have had an effect on teacher recruitment.
In the States, as Editor of the Mathematics Student Journal (N.C.T.M.), he solicited articles and problems from high school students, and contrary to the general expectation of the time, whole issues were entirely written by students before his two years as editor were up (November 1958-May 1961). Teacher training? Go into the schools and work with teachers. As editor (with L.D Nelson) of MATHEX, Sawyer solicited a wide range of contributions from elementary school classroom teachers. At one point he proposed a novel method of teacher training.
Perhaps Sawyer's most important contribution to mathematics education as whole is his encouragement of individuals to try out new ideas and "buck the system if necessary". He did not just say "Once more into the breach" and let it go at that. On many occasions he followed through with attempts to enlist support for an innovative teacher from administrators. Sawyer has always been willing to offer his personal services, free, to such causes, and takes pains to endorse individuals by name in his articles and speeches. One can expect that he will not retire from these activities.
Recently, David Wheeler called on mathematics educators to "develop criteria . . . which will give everyone concerned with decisions, the tools for making better judgments" (MT No. 75). Sawyer devoted his professional career to asking, and attempting to form criteria for, such difficult questions as: What mathematics is important to teach? How can the curriculum be best organised? What methods work, and when, and with whom? There is no necessity to agree with his criteria, although the evidence in support of them is often impressive, but the continual asking of the questions is vital. Re-read Sawyer; he was never more relevant than today!
By David M. Clarkson
We are grateful for the permission of the Association to reproduce the above article.
Copyright © W. W. Sawyer & Mark Alder 2000
Version: 28th April 2001