Professor W.W. Sawyer  




In 1957 I came to the United States on the invitation of some mathematicians who were launching the "Modern Math" campaign. They thought my book "Prelude to Mathematics" was a useful expression of the spirit of modern mathematics. Max Beberman was working on a series of textbooks expounding the approach of "Modern Math". These books were rather behind schedule and it was thought that as a writer I might help them to catch up by writing some books in the series. This was of course an illusion. I was appalled by their approach, which was the exact opposite of my idea of good mathematics teaching. (For details see The "Modern Math" Epoch in the United States of America. or Oscillations in Systems of Mathematical Education.)

It had been arranged that I should teach calculus to a class of first year students at Illinois University. There they accepted students who had graduated from high school. Sometimes this was not sufficient to ensure a successful college career and the first year to some extent served the purpose of an entrance examination.

The organization of Illinois University was interesting. Promotion depended on research achievement and was designed to give increasing scope for research work. A full professor gave three lectures a week. An associate professor gave six. Below these there were ranks in which one was required to give nine or twelve lectures a week.

The first year classes were taught by a group of people who did not expect academic promotion. I thought of them as extremely good school teachers. They of course could not gain tenure, but there was an understanding that they would be allowed to continue until they reached retirement age. My assignment was to cover the first ten chapters of a particular calculus textbook. I imagine the normal procedure was to go straight through the book, giving tests in which students would receive grades A,B or C for partial understanding.

I did not follow this procedure. The first chapter dealt with limits. No one sees any reason for thinking about limits before having some exposure to calculus, so I left chapter 1 for much later in the course.

It always seems to me a pity to spend time carefully explaining the ideas of calculus and then to have the first test pose such questions as
Differentiate x2 +2x+3" My first test dealt exclusively with calculus ideas. I cannot now remember exactly what the questions were, but they were something along the following lines;-

A car has travelled s miles after time t hours. Express in
terms of s, s', s" the following.

The car is travelling at 60 miles an hour. (s'=60).
The car is going forward. (s' positive.)
The car is going backwards. (s' negative.)
The car is going forward and accelerating (s' and s'' both positive. Accelerating in reverse is interesting but I do not think I used it.)

With y'=dy/dx and y"=d2y/dx2

what is the sign of y' for the graph / ?
for the graph \ ?
Sketch a curve with y' +ve and y" +ve. (etc) .

It may be thought that these concepts were too slight to be worth testing, but it was not so. I wish I had kept notes of the questions students raised in class. The discussion in class went on for weeks dealing with questions such as "If y' is positive, does this mean y must be positive?" This was answered by the question,"If you are going up, does this mean you must be above ground?" No, you might be coming up out of a coal mine.

Early on I said to them,"You don't get any credit for knowing 90 percent of the multiplication table. In the same way, it isn't much use being able to use calculus correctly most of the time. In all tests the pass mark will be 100 per cent, but you will be allowed to be tested as many times as you need to reach this."

Television advertisements suggested an idea. Surely no sane person sets out consciously to learn the jingles, but most people know then. They soak in by constant repetition independent of any desire or effort on the part of the viewer. So, at the beginning of every lesson, I put on the blackboard, as concisely as possible the results that had been covered.

From quite early on there was

y y'

x nxn-1

As time passed there appeared what I called (and nobody else did) the five architectural principles;-

I. (ky)' = ky'

II. (u+v)' = u'+v'

III. (uv)' = uv'+u'v

IV. (p/q)' = (p'q - q'p)/q2

V. dy/dx = dy/dz . dz/dx.

There was one youth in the class who was obviously much better than the others. I let him work ahead on his own. With the others I first went through chapters 2 to 10, with only the easiest exercises being considered. Then, when the class could see the course as a whole, we went through these chapters again in more detail. I do not know who set the examination at the end of the course. It was of the usual form, testing whether the students could carry out and use calculus operations correctly.

The majority of the students wrote essentially perfect answers. I cannot now remember if there were minor slips but there were certainly no mistakes in principle. I gave them grade A. There were three students who had had made a mistake such as integral of

(2x+1)(3x2 +1) is (x2 +x)(x3 +x)

- the old trap of the universal distributive law.

It is interesting that students seem most aware of the distributive principle in the situations to which it does not apply. I said to them,"I cannot give a grade A in calculus to a student who makes a fundamental error in principle such as this." They received B. They complained bitterly that I was being too severe.

I kept the students' answers to the exam for several years as I imagined the distribution of marks in most classes differed radically from that in mine, and that I would be accused of having much too low standards. However the question was never raised.

Copyright © W. W. Sawyer & Mark Alder 2000

Version: 22nd March 2001


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