## An Interview with James Stewart

**July 6, 2009**

James Stewart obtained his master’s degree from Stanford University and his Ph.D. from the University of Toronto. After two years as a postdoctoral fellow at the University of London, he became Professor of Mathematics at McMaster University in Hamilton, Ontario. His research has been in harmonic analysis and functional analysis. Stewart’s books include a series of high school textbooks as well as a best-selling series of calculus textbooks published by Brooks/Cole. He is also co-author, with Lothar Redlin and Saleem Watson, of a series of college algebra and pre-calculus textbooks.

Stewart was named a Fellow of the Fields Institute in 2002 and was awarded an honorary D.Sc. in 2003 by McMaster University. The James Stewart Centre for Mathematics was opened in October, 2003, at McMaster University. Stewart is now Emeritus Professor at McMaster and on the faculty of the University of Toronto. His newly built home in Toronto, Integral House, has received much attention for its innovative design.

**Ivars Peterson:** You grew up in Toronto. What got you interested in mathematics?

**James Stewart:** When I was in grade 11, my math teacher at Earl Haig Collegiate was Ross Honsberger. He was not your typical high school math teacher. He was always going off on digressions. In grade 11, he presented the proof on the board that the rational numbers are countable and that the real numbers aren’t. I don’t know what my fellow students thought, but I thought that was fascinating.

**IP:** What happened when you went to university?

**JS: **I was good at and interested in all subjects, including languages, history, and English, as well as music, because I played the violin, and I couldn’t decide what to go into. My [high school] guidance councilor said I should go into science. So, being an obedient sort, I went into the math, physics, and chemistry program at the University of Toronto.

I had studied the violin, ages 7 through 17, and I love music. At the end of second year, I came very, very close to switching from mathematics to music. In the end, I decided not to because I thought it would be better to be a mathematician whose hobby is music than a musician whose hobby is mathematics. That’s why I stayed in mathematics, though I have a real passion for music.

*Mathematician and textbook author James Stewart sits at a custom-made desk in his Integral House office*. Photos by I. Peterson.

**IP:** You went to Stanford for your master’s degree.

**JS:** I knew I wanted to specialize in analysis, and at that time virtually the entire math faculty at Stanford consisted of analysts. And there was the appeal of California weather. But I came back to the University of Toronto for my Ph.D.

I did my Ph.D. with Lionel Cooper. He was at the University of Toronto only two years—the two years that I did my Ph.D. I knew Cooper was leaving to take up the headship of Chelsea College at the University of London. So from the time I got the subject of my Ph.D. thesis to the time I defended it was one year. It was crazy, but I wanted to finish up before he left. I never worked so hard in my life.

Then I followed Cooper to London and did two years of postdoctoral work at the University of London. While I was there I took up the study of the violin again, studying at the Guildhall School of Music. This pull between mathematics and music came into play again. When I got my first job at McMaster University, in addition to playing chamber music, I was asked to become concertmaster of the McMaster Symphony. I also ended up playing for some years professionally in the Hamilton Philharmonic Orchestra.

**IP:** At that time, you were also doing research.

**JS:** I had a full research program in harmonic analysis and supervised some Ph.D. theses.

**IP:** What role did teaching play?

**JS:** I was passionate about teaching from the beginning. When I taught as a graduate student at Stanford, they gave the TAs leeway in the recitation sections. We actually taught some material. I knew I loved teaching from the moment I stepped into a classroom.

**IP:** How did your venture into textbook writing come about?

**JS:** The idea came from my students. After a calculus lecture at McMaster, two of my students came down to the front of the classroom and said, “Dr. Stewart, we have suggestion. We suggest that you write your own calculus book because the notes you give on the blackboard are better than the textbook we’re using.” I thought, “That’s an interesting idea. I never thought of it before.” It was their idea, and it changed my life.

Before I had a chance to do that, I was approached by two local Hamilton high school teachers, who had a contract with McGraw-Hill to write a series of high school textbooks. They asked me to collaborate with them. So I did. I found it to be a useful apprenticeship. Together we wrote grades 10, 11, and 12 textbooks that came to be used in a lot of high schools.

I discovered that I enjoyed this. So after we finished the grade 12 textbook, I thought that now is the right time to take up my students’ suggestion and write my calculus textbook, because now I know what students are supposed to know when they enter calculus. That’s when I started to write my calculus textbook, around 1979-80.

I thought I could write one in three years. Instead, it took me seven years—seven really, really intense years—while I continued with my teaching and research. With the writing, I spent 13 hours a day, 364 days a year at work during those seven years. Once I had started, I had to finish it.

*Stewart’s innovative Integral Houses perches on the edge of a ravine in Toronto*.

**IP:** Were you into another book project as soon as you finished the first one?

**JS:** I haven’t had a break since the first book. My editor at Brooks/Cole suggested that I write a pre-calculus book, and I said I would want some help with that. So I drafted two of my former students at McMaster, now teaching at American universities, to help me. That spawned a series of college algebra and pre-calculus textbooks.

There have also been variations of my original calculus textbook. The very first draft of the calculus book had transcendental functions in chapter 1. At that time, it was very rare to do that. I thought, “Why are we pretending that kids don’t know about exponential and log functions? Why are we waiting until second semester to define them as integrals when kids know about these functions? They study them in pre-calculus and high school.”

The first draft of my manuscript had transcendental functions in chapter 1. Of course, when it got reviewed, I was told that students weren’t ready for that yet. When we brought out the second edition, my editor said to me, “I’m hearing more and more about your idea of putting exponential and log functions in first semester.“ So in the second year of the second edition, we brought out the “early transcendental” version, thinking it would be a minority viewpoint. It caught on, and two-thirds of my sales are now of the “early transcendental” version.

In the 1990s, I had a great deal of sympathy for the principles of calculus reform. When the reform movement became prominent, my publisher asked me to write, more or less from scratch, a calculus book that went in that direction. It became my version of reform, which is moderate reform. Instead of being wildly experimental, my book was grounded in what I thought would work in the classroom. Now, the fourth edition of that book has come out.

Earlier, I had done a version of calculus with early vectors. It was an interesting experiment. I had been approached by Texas A&M. They integrate vectors into first-year calculus from the beginning of the course. That’s especially good for engineering schools. It made a great deal of sense to me, so I collaborated with them on the “calculus with vectors” book. A number of schools use it, but it’s not the huge movement that we thought it might become.

Later, my editor asked, “So many instructors are saying, ‘Why do calculus books have to be so big and so expensive?’ Do you think you could write a much shorter book—what people are asking for?” Of course, for years reviewers have been saying to me, “Calculus books are too big, but, by the way, could you please add this topic because it’s my favorite topic.”

My initial reaction was, “I don’t know how I would cut it down and still retain the essence of calculus.” A year later, I told my editor that I could see how to do it. I started writing the book, now called *Essential Calculus*, which is 800 pages instead of 1200 pages. It wasn’t just a matter of cutting this and that. I went back and rewrote the material. It may not have enough detail for the slower student. It doesn’t have quite as much motivation and not quite as many exercises, but I think it’s enough for many people.

**IP:** What are you working on now?

**JS:** At the moment, I am working simultaneously on three books. One is the seventh edition of my original calculus book. The other two are new books, both with co-authors. One is applied calculus for business and economics students. That has to be started from scratch. I’ve taught that course a few times, and it really has a different character.

The other new book I am writing is a very interesting one. It’s a reform college algebra book, which I think is more reform than anything else out there. This is my take on the reform algebra movement. It’s very much data driven.

I was actually approached by four instructors from Mercer County Community College in New Jersey, who requested such a book. A lot of schools have some math requirement, so you have these kids who are never going to take another math course, and yet they have to take a college algebra course. I’m not sure that’s a sensible policy, but given that it is a widespread policy, let’s make it more interesting for those kids to really try to draw them into it. That’s not an easy task. So I wrote a prolog, “Algebra and Alcohol,” for that book, which is now in production. No matter what your attitude is to alcohol, I thought this would get their attention.

In the prolog, I present data. If you take an alcoholic drink, your blood alcohol concentration goes up pretty rapidly, depending on how quickly you drink, and it declines at a rather slower rate. You can model the rates by various types of functions. Every chapter of the book has little blurbs and exercises that follow up on this.

**IP: **I have heard that you are interested in writing a book on mathematics and music.

**JS: **In my early years at McMaster, the students in the mathematics club approached me to give a talk on mathematics and music because they knew of my interest in both. That’s when I first started thinking about it. So I gave the talk and have since expanded on it. I’ve given my talk on mathematics and music dozens of times throughout the world. I explain some of the principles, concentrating on the analogy between form in music and structure in mathematics. I bring my violin and give demonstrations.

The theme that runs through the talk is: How do you explain the phenomenon that mathematicians tend to be musical? I observed this phenomenon when I played my violin in various university orchestras: Toronto, London, Stanford, McMaster. It always seemed to be the case that, among the student and faculty musicians, mathematics, science, and medicine were over-represented. There were not so many from the humanities.

This runs as a kind of theme through my talk: trying to explain why that might be so. My idea for a book is to expand on these lectures that I have been giving, with a little CD in the back, trying to explain this phenomenon. These correspondences are such a rich area that you could have totally different types of books exploring this. That’s a book I really want to write, but my publisher is keeping me busy with all my other projects.

*Integral House incorporates a performance space that can accommodate as many as 150 people. Angled wooden fins divide the curved glass walls into segments, giving viewers startlingly different perspectives on the wooded ravine outside as they move from place to place*.

**IP:** Are you still teaching?

**JS:** Although I am Professor Emeritus at McMaster, a year ago I was appointed professor of mathematics at the University of Toronto, and I have twice taught first-year calculus. Although I don’t teach fulltime anymore, I love teaching. Being an author is a pretty solitary, sedentary occupation, so I miss the social aspect—which is teaching. I do it partly to keep in touch with kids, because it brings out the best in me, and to give me new ideas for new editions of my books.

This fall I am introducing a new course at the University of Toronto on problem solving. I introduced such a course at McMaster quite some time ago.

When I was a graduate student at Stanford I fell under the spell of George Polya, who was retired but used to come in and give these problem-solving talks. He had all of us—teachers and students alike—literally sitting on the edges of our seats with mathematical excitement, presenting data, asking us to make conjectures.

The idea is: Suppose you’re faced with a problem that you have never seen before. How do you get started? The first few lectures introduce some basic principles of problem solving. The remaining lectures start with a “problem of the day.” How would you solve it? What strategy would you use? What about trying a special case or solving a simpler problem first? It’s my favorite course to teach.

I’m doing that this fall, working with some of the faculty at the University of Toronto so that they can carry on after me. It will be a kind of capstone course. You’re drawing on everything that you’ve learned up to that point, putting it together. There’s no new content whatsoever. But once you take a problem out of the context of a specific course, it becomes harder.

**IP:** When did you get a sense that your calculus books were a big success? Did it happen right away, or did it grow?

**JS:** I basically wrote the book to use in my own classes. I wanted a book that put the transcendental functions early. I had no idea that it would catch on. In the first edition, we sold fewer than 20,000 copies in the first year. In the second year, more; in the third year, more; in the fourth year of the first edition, more. That’s unusual; the trend is generally the opposite. By the second year of the second edition, in 1992, it had become the best-selling calculus book. I just shook my head: “How did this happen to me?” It took over my life. I was very surprised.

**IP:** Did you get a sense of why it happened?

**JS: **I’m not sure. I think it’s partly because I listen to my students. When I detected a trend—that maybe kids are having a problem with this or that—I would incorporate it into the book. I was also as strong in English as I was in mathematics, so I could combine some talents that I had. Whenever I teach and whenever I write, I cast my mind back to the first time when I was a student in calculus. I was able to recapture my own thoughts.

In retrospect, I think one reason for the success is accuracy. I’m a fanatic for accuracy. There can be no wrong answers. There are so many ways a wrong answer can creep into a book, but I’ve got systems in place that prevent any wrong answers. That’s part of it, but mostly it’s a mystery to me.

**IP:** I’ve heard that one reason for your success is your “middle-of-the-road” approach.

**JS: **That’s especially true of the [reform calculus] book. And I brought some of those ideas into the mainstream book, as well.

When I was toward the end of writing the reform book, I got a call from a [mathematics department] chair, and he asked me to come to his university. He explained that his instructors—the reform and those more traditional—were not speaking to each other. So he asked me to come as a peacemaker. He had me meet in separate rooms with the two warring factions. I went back and forth, and I asked the traditionalists, what it was they objected to, and so on. When I gave a talk to the entire department, the two warring factions sat on opposite sides of the room.

In the end, I was able to bring the two sides together. The chair asked me, even though the book wasn’t ready to be used, if they could use a preliminary version—in page proofs—in the course of unifying the calculus teaching, which they did. And they’ve been using the *Concepts and Contexts* book ever since. So I acquired a reputation as a peacemaker. I went on other peacekeeping missions. I think the rhetoric has cooled down now.

**IP:** How have mathematics textbooks changed over the years?

**JS:** Compared with the textbooks that I had as a student, textbooks are so much better now. I don’t know how kids learned from these old books. There was no motivation. It was very austere. You can go too far in the other direction, but the state of the exposition of mathematics is just so much better than it was three decades ago.

As an author of the high school textbooks in the 70s, I kept my eye on trends in education. The new math had been well ensconced by then. But what I observed and decried was the waves, the extremes, the pendulum going back and forth from the new math back to basics. You still see this, especially in the U.S., especially at the high school level, where it is much more virulent. At that time, I longed to get hold of that pendulum and stop it somewhere in a sensible middle. People get too dogmatic.

**IP:** Your books have been translated into many languages, including Chinese.

**JS: **When my publisher told me about the Chinese translations, I was surprised. The general wisdom is that the Chinese are pretty good at math. In China, I had a look at some of the Chinese books—I don’t read Chinese—but I could see what they were doing. They’re kind of like the textbooks of the 50s in this continent, very austere, and perfectly fine for the brilliant student, but not good for everyone else. The Chinese thought a translation of my book would help ordinary students who needed more help understanding the subject.

**IP: **Where do you see yourself 10 years from now?

**JS: **I will probably never stop writing books, but I hope I will have drafted enough talented help so that I can take things a little easier. Although I work incredibly hard, I love my work. It is very, very satisfying. Even now, when a new book comes out, a big smile comes over my face. The feedback that I get on my books and the fact that the majority of students use my book are sources of quiet satisfaction.

When I started writing my first book, I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost [$24 million] is double the original estimates.

**IP:** It isn’t very often that a private house is so striking, so path-breaking. It is a unique kind of structure.

**JS:** It is special. Aesthetics has always been important to me; my mother was an artist.

I’ve set out to do two major things in my life, but I didn’t think of them as major at the time. I just thought, “My two students suggested that I write a calculus book; I think I’ll write a calculus book.” Look what happened. And then I thought, “It would be nice to build a brand-new house.” I naively went about interviewing architects, and look what happened.”

“James Stewart and the House That Calculus Built,” *MAA FOCUS*, August-September, 2009.