What Is Calculus About?
by W. W. Sawyer
New York, Random House, 1961
(Distributed by the Mathematical Association of America)

Comments by Lee Lady

This book was originally published as part of the SMSG (School Mathematics Study Group) project during the Sixties, which was the secondary education version of the New Math.

In its 118 pages, this book goes through material which in a typical calculus course (such as the one here at UH) is covered (if you can call it that) in four or five days.

This is a book about the ideas of calculus, not the techniques of calculus. You will not find the product rule, or quotient rule, or chain rule here. But you will find a rather detailed discussion of velocity, acceleration, and the slope (and direction of curvature) of graphs. The fundamental theorem of calculus is explained very clearly, but never named as such.

This book does what, in my opinion, most current courses in calculus fail to do, viz. to show that calculus is something interesting and worthwhile.

Here are a couple of quotes from the introduction which say things that I think almost all modern writers of books on mathematics need to listen to.

How should calculus be taught then? Should we bother the beginner with warnings that only become important in more advanced work? If we do so, then the beginner will be confused because he will not see any need for these warnings. If we do not, we shall be denounced by mathematicians for deceiving the young.

I believe the correct approach is to do one thing at a time. When you take a student into a quiet road to drive a car for the first time, he has plenty to do in learning which is the brake and which the accelerator, how to steer and how to park. You do not discuss with him how to deal with heavy traffic which is not there, nor what he would do if it were winter and the road were covered with ice. But you might very well warn him that such conditions exist, so that he does not overestimate what he knows.

In this book I begin with the simple ideas of calculus, with country driving. I do not look for awkward exceptions. In the main, I look at things as mathematicians did in the 17th century when calculus was being developed.

The theorem-proof-theorem-proof type of book does, in a certain limited sense, explain mathematics to the student. Theorem 1 is at least followed by a proof of Theorem 1, which may throw some light on why Theorem 1 is true. But very much is still left hidden. How did the writer decide that Theorem 1 should come first? How did he decide which theorems to include and which to omit? What is the book trying to do? What is the line of thought that lies behind it? How did all these theorems come to be discovered? What should the student do if he wishes to discover further theorems for himself? This last question is perhaps the most important of all. It is a very strange thing that many eminent mathematicians, who think the only thing really worth doing in life is to discover new theorems, often write books which give no hint at all of how a student should try to make his own discoveries.

There are at least four stages in mastering a mathematical result.

(1) You must see clearly and understand what the result states.

(2) You should collect evidence which shows that it is reasonable that this result should be so; you should feel that this result agrees with your evidence of mathematics.

(3) You should know what you can do with the result. It may have applications in science, or it may simply lead to other interesting theorems in pure mathematics. You ought to know what these are.

(4) You should know and understand the formal proof of the result.

I want to make it quite clear that this book does not attempt to provide formal proof of any result whatever. I have not attempted to deal with stage (4) at all. My concern is entirely with stages (1), (2), and (3). I want you to see that the ideas of calculus arise quite naturally, and indeed I want you to discover them for yourself. If we were in a room together, I would confine myself to asking you questions, and you would find that you arrived at calculus by clarifying ideas that you already have in a vague and shadowy form. Between the covers of a book I cannot follow that procedure. But I keep to it as nearly as I can. I am not trying to tell you any particular result. I am trying to call your attention to particular things that you can experiment with for yourself. The evidence that you collect will suggest certain conclusions to you. More than that, I do not claim. I do believe, however, that this experience will make it easier for you when you begin to learn calculus in real earnest. You will have some idea of the direction in which you are traveling.