What Is Calculus About?

It all began one winter's night. Forty years ago and one month on the February winter night of the lunar eclipse in North America in 1968 was the day I began to read W.W. Sawyer's "What is Calculus About?" And my life was forever changed. I had never read a style of writing up to that point on any topic that exuded as much confidence, as much philosophy of learning and conveyed as much mystery, the mystery of how to handle infinity as that deceptively small book. I had known about the concept of infinity but up to that point did not know that you could calculate with it. That is take limits. Take derivatives. Then work backwards. But when Sawyer explained it it was like watching a professional magician. There was it could be said a "fresh rebelliousness" in his attitude toward Mathematics understanding and its teaching. It was as if reading his words got you directly in contact with the minds of Newton and Leibnitz and other mathematicians of that "happy-go-lucky" era that Sawyer talks about in his book. Everyone had warned me at the time that I wouldn't be able to learn Calculus on my own. That it was this mathematics that was so special that it was for many impossible to learn especially if you studied it by yourself. That is was somehow an unattainable goal. That I would somehow "ruin" my learning. It was amazing the degree of prejudice that at that time prevailed. But they were all wrong. I ended up mastering it. And then the "fur really began to fly". I wasn't supposed to have understood it. I was as one science ( later to be my physics teacher) teacher said: "in way over my head...and if I didn't watch it I'd wind up ruining my career...life..you name it." It was the sixties, the 1960's and my parents generation were total reactionaries. My mother got all hysterical and dragged some of my mathematics books ( unbeknownst to me ) down to the school to the principal and was jumping up and down insisting that I might need psychiatric help. The mathematics teacher of the high school went absolutely berserk with much the same tirade as the physics teacher had. It was to say the least embarassing. All over something they reaqlly didn't understand. And that is exactly what humanity at it's very best is like. If they don't understand it, "watch out". Because it's like Einstein said:" always the brilliant minds are persecuted by those minds who are of a simple, pedestrian nature." ( that's not a perfect quote but it will do) But through it all there was W.W. Sawyer. You might say he helped me "get through" the sixties. ( I had other heroes too ) In a strange way you could compare him to Dylan at that time ( my other big hero). (And it could be rightfully argued that Sawyer was in fact part of that 1960's "beat generation" of scholars who were trying to tell society and the Education Establishment that maybe as far as learning they the accepted teaching establishment had got it wrong.) Then there was his bibliography and references at the b

The book was required for a master's in middle grade mathematics. It is a good introductory book into calculus.

Whenever I teach calculus, my emphasis is always on the fact that the basic ideas of calculus can be understood by almost anyone. While I am not always successful in proving this to the students, Sawyer certainly would be. His explanations of the basics of a derivative are the clearest, most understandable that I have ever seen. There are many diagrams, and each one has a specific purpose and they are well integrated into the textual explanations. This is not a book that you could use to teach a college calculus course. The mandatory epsilons and deltas that form the backbone of basic calculus are mentioned only as an incidental. Sawyer sets out to explain the foundation ideas of calculus in terms of everyday occurrences and for that reason it is better suited to someone who is curious about calculus. However, it could be used as a supplemental text in the foundations of science, as calculus is used in all areas of change, which describes almost all of nature. While the notation of mathematics is concise, abstract and often appears esoteric, many of the ideas expressed in that notation are quite easy to follow. In this book, Sawyer explains what calculus is all about in terms that anyone who understands motion can follow. There needs to be more people like him writing books like this.

When I was reading this book all along, I felt comfortable. I heard it as if my grandfather told an interesting story. When the story was over, I became a real university student. I had learned a vacant caculus, but I have known that the calculus 'is' our life.