Engaging Minds: There’s a lot that inspires you in your everyday life that is connected to math. From where does this love for calculus, love for math, and love for teaching come?

James Stewart: Well, I don’t know. I think it started when my kid sister Sally had trouble with trig when she was in high school. And she came to me with problems. And I helped her with the problems. I didn’t do the problems for her, but I nudged her in the right direction. And that gave me a good feeling, you know, that I was able to help my little sister. Well you know Thanksgiving here in Canada was the other day and it was our family reunion there, I was

reminded of this and talking to Sally of how helping her gave me a good feeling. That feeling continued later on when I was teaching that if a student came

into my office during office hours with a problem, I refrained from doing the problem for the student. That’s not how you learn, just, you know,

watching somebody. Well, you can learn a little bit that way but [not everything]. So same thing, I kind of nudged the student by giving

them little hints and asking, “Well, had you thought about this maybe?” And then they’d say, “Oh, yeah, I see,” and when a

student says that, it gives me a good feeling right here. So that really inspired my love of teaching.

James Stewart There’s something to be said for hard work. In the last forty-plus years since

two of his students at McMaster University suggested he write his own calculus

book, world-renowned author and mathematician James Stewart has basically

never stopped writing. Having essentially devoted his life to mathematics,

Jim has published more than 70 textbooks, releasing a new edition almost every

year. But calculus isn’t Jim’s only passion; in addition to being a mathematician,

he is also a concert-level violinist and music philanthropist. These different

themes have played an integral role in his life as told in the upcoming feature

documentary, The Integral Man, created by Auratic Media.

There are many things that inspire Jim, who recently completed writing the latest

edition of his best-selling calculus book and his new biocalculus books: his students,

his love for mathematics, and his love for music. But when Stewart starts chatting

with Engaging Minds, we realize this: he is a teacher at heart.

An Interview with

EM: Is there a message in your books that you want students to grasp?

JS: At the beginning of the book, there’s “To the student.” At the very end of “To the student,” I want students to see that calculus is both practical and beautiful. Throughout the book, I try to incorporate that message with the practical aspect of mathematics, solving these problems or engineering problems or whatever, is very powerful. But that’s one aspect of calculus; I want them also to discover that calculus is beautiful. Now, that’s a little bit more difficult to convey, the beauty of calculus. You can’t explain [it and] you can’t realize that all at once. It’s cumulative. So that by the end of the course, my hope is that students see how calculus all fits together, and then they say, “Oh, that’s nice.”

I had one student who graduated a few years ago come back and talk to me, and she said that of all the courses she had taken—she was in the arts and science program, so a real variety of many, many different courses—but the most important course she said was the calculus course and the logic involved in how it all fit together. And in studying for the final exam, she saw how it all fit together, and so she thought that was the most important course she’d ever taken at the university. That’s the beauty of calculus; how it fits—the various parts fit together and form a complete whole.

EM: What topic do you think calculus students struggle with most?

JS: Oh, good question. Well, of course, the thing about math is this—this follows from this, which comes from this, which is a consequence of this. And so, you know, if you need to understand something, you have to go back to the beginning. That’s why instructors these days are always complaining about all the students don’t know the precalculus. They’re not ready for calculus because they don’t know the algebra. That’s a common complaint. But, you know, I think that’s overdone because once the students get back into the swing of things, they remember these things, you know, from

precalculus. So I take those complaints about students with a grain of salt that…in the process of studying calculus, their precalculus skills at the beginning may be pretty bad, but as time goes on, it will come back to them. They’ll say, “Oh, yeah, yeah. I remember that.” So…I don’t think the students are quite as bad as the common wisdom would have it now.

EM: Do you have any advice for students taking calculus?

JS: When I was living in San Francisco, I was invited to give a talk at Sacramento State University, and so I drove up from San Francisco to Sacramento. It was a nice sunny day, good weather. I arrived a bit early, so I decided to sit down on a bench outside the mathematics building, and along came a student who sat down on the bench beside me and he opened his book and started reading it. And I glanced over and saw that it was my calculus book. And so I introduced myself to him as the author of the book he was reading. Well, of course, that surprised him. It’s such a coincidence. But then I said to him, “I’m happy to see you reading the actual textbook because sometimes we [wonder].” I said, “Do you have a test coming up? That’s why you’re reading it?” He said, “No, but we’re covering this section this afternoon, the professor is, so I’m reading this section of the book before I go into class.” And I said, “What a great idea. Did it help you?” And he said, “Oh, yeah, it does.” So I said, “How can I get my own students to actually open the book and read it, you know, before a lecture?” And he said, “Tell your students that whenever I have time to read a section in the book before the professor covers it, I get so much more out of the lecture.” So every year, I tell that story to my own students and I don’t know if it’s had an effect or not. I hope it has. I hope it has.

EM: So what makes your books unique?

JS: Apart from anything else, you know, the essence of any calculus program are the problem sets; this is how students learn, and I think my problem sets are really good [brief laughter]. I have these wonderful problems

that I love that are challenging, the Problems Plus that are in the chapters. In the eighth edition of my calculus book, I have also selected a number of new ordinary exercises, and those are in the preface to the book. Well, if you take a look at those exercises that I have chosen there that I liked, I think that’s probably the best way to see the difference in my books. Because as I say, the essence of any book is the exercise sets. I mean there are other things, too, of course, that we’ve improved, but the exercise sets are, just as I said, the essence of the book.

EM: Why are those exercises your favorites?

JS: Oh, I don’t know, that’s hard to say why they’re my favorites. I don’t know, how they just appeal—they’re novel. They’re off the beaten track, some of them. I don’t think you’ll find them in any other book, and they’re just cute. If I can—if I may use that to describe a mathematics problem, it’s cute.

EM: How have your books changed the way instructors teach and students learn calculus?

JS: Oh, that’s a big responsibility. Well, I don’t know. I don’t know how to answer that.

EM: So how did you decide to become a writer?

JS: Way back in the—this was mid-70’s when I was first teaching calculus, I wasn’t really happy with the book we were using. It was research-oriented and it was extremely practical but there was an absence of pedagogy. So after one of my lectures, two of my female students came to the front of the auditorium and said, “Dr. Stewart, we have a suggestion. We suggest you write your own calculus book because the notes you give on the blackboard and your lectures, they’re better than the book we’re presently using.” That gave me the inspiration to write my book. That was the encouragement, and I wish I had stayed in touch with those girls. I would say that I started writing the book for my own students because the book we

were using had an absence of pedagogy. And how much pedagogy did I know I needed to put into my book? Well, when students came to my office for help on a particular topic, and other students came with kind of similar problems, I realized we need to explain that topic more. We need to explain that better. So at the beginning, when I was writing the first edition, all those years ago, I paid attention to my students. I paid attention to the questions they were asking, which ideas needed more explanation. And so I incorporated that into my book. And so, really, it was kind of student driven…what they found challenging, I tried to explain more and better.

EM: In your opinion, what makes a great textbook author?

JS: That’s a tough question. What makes a great textbook author? Well, you have to be patient. Like any good teacher, you have to be patient, and you have to be willing to invest a lot of time in writing it. You have to write at a level that the students can understand. And, well, for instance, the first edition of my calculus textbook…I estimated that it would take three years. It took seven years to write. And those were seven very intense years where I worked 13 hours a day, 364 and a half days a year. I took a half-day off for Christmas. Most authors who start writing a calculus textbook give up part way through. I have this on good authority. And I can understand that; there’s meticulous attention to detail. But as well, you want to be able to convey to the student the sweep of the subject. You need to be able to convince why is calculus regarded, rightly so, as the mathematics course. You know, you’ve got to convey why that is true.

In writing a calculus textbook, or really any mathematics textbook, you have to be aware of two people looking over your shoulders. One is the professor. And for the professor, you know, all the i’s need to be dotted and t’s crossed, and there’s going to be a lot of rigor there and to be absolutely correct mathematically. The other person looking over the other shoulder is the student. And the student has

a completely different point of view. What’s this all about? You know, you’ve got to convey to them the essence of a subject. It’s got to be simple enough to explain so the student can understand. And so, sometimes these two people looking over your shoulders have contradictory aims. So you’ve got to balance these two. And sometimes it’s a delicate balance. The only way to describe it is a delicate balance between the points of view of the student and the professor.

EM: Can you walk us through your typical day as an author?

JS: Oh, well, I generally start writing at nine in the morning. After breakfast. And I usually write for about four hours. And I’m one of these people who can just sit at a desk for four hours without having to get up. Although some people like to wander around. If I’m starting a new book or a new chapter or a new section, that’s a challenge, and I wonder how to get started…once I have that first sentence the rest will flow because it is mathematics. So sometimes I have to get up and like pace a little bit and think “How can I start this? How can I get that first sentence?” And, “Oh, yes, I’ve got it” and the rest flows, you know.

It’s mathematics. This flows from that, flows from that, flows from that.

EM: And how do you pick out the exercises or how do you go about creating them?

JS: They’ve got to be different types of exercises, of course. There’s got to be enough drill exercises to gain students’ fluency. But I like to start off an exercise set with some elementary conceptual questions. So before we get too carried away with the standard drill questions, let’s make sure that we understand the meanings of these symbols that we’re manipulating, you know. So everything has more meaning.

EM: I want to talk about accuracy.

JS: Oh, yeah. You know, in a mathematics textbook, accuracy is absolutely essential. I own calculus books, I won’t name them, that have failed because there were errors, other answers in the back of the book, or whatever. A mathematics textbook has to be accurate. All the answers in the back of the book have to be correct. Otherwise the students get demoralized when they work something out and then they look in the back of the book and it’s different from theirs. And maybe they’re right. You know, it’s possible. It’s rare, but it happens. So I claim that that doesn’t happen in my books because it’s such an important point. For that reason, students could become demoralized if it’s the other way around.

EM: Over the years you’ve drafted talent to help you write a number of your books. What role do your coauthors play going forward to carry on the legacy of these textbooks?

JS: Well, I have trained, if that’s the word, three young mathematicians who have helped me with various books and whom I trust implicitly. Saleem Watson and Lothar Redlin on the three precalculus books, who have done a marvelous, marvelous job, and they are really fantastic writers. They were former students of mine at McMaster University. So I asked them, I approached them to—they had taught these courses many times—to help me with those books. And the division of labor was equal on those books. And they’ve done an absolutely terrific job.

The other coauthor that I had been developing, if you want to call it that, is Dan Clegg, whom I’ve known for 18 years. When I was giving a talk in Southern California, after the talk, he approached me to chat with me and said how much he admired my calculus books. And I thought, hmmm, this guy’s enthusiastic. He said, “Do you have any materials that you give to your class that I might take a look at? Would you please provide me with those?” I thought this sounds pretty good, you know, so I engaged him to

do this, first of all…work on a solutions manual for the calculus book. And then we engaged in another task, like proofreading. Proofreading is a skill that not everybody has. And he is very good at it. And then we discovered that we could think the same way. And that was remarkable about Saleem and Lothar, too. We agreed on almost everything, which is kind of rare. And the same thing with Dan and myself. We agreed on just virtually everything. And so Dan and I coauthored an applied calculus book, but also in the big calculus book, he has played a role in the last several editions. He will sometimes suggest improvements and sometimes he will contribute material to it. Exercises for instance. So some of the exercises in the last few editions, sixth, seventh editions, have been suggestions from Dan. I trust Dan as well as Saleem and Lothar to carry on the legacy of my books. They are all just terrific writers.

EM: Tell us about your experiences working with Saleem and Lothar.

JS: When we were writing the first edition of Precalculus, Saleem was having trouble getting started so I took a sabbatical at Cal State Long Beach where he was teaching. Lothar and I had gone and worked with him. And so on those trig chapters we worked closely together and I monitored Saleem’s writing. You know, Lothar is a natural writer, Saleem was not originally a natural writer but he is wonderful now. And so he and I developed the trig chapters while I was on sabbatical there. Saleem has developed into an outstanding writer, which tells me that it is possible to teach people about how to write and he’s really an extraordinary writer.

EM: Tell us more about Dan. When you say you guys think alike, you have the same attitude. What comprises that? What makes it?

JS: Well, we look alike. [Laughter.] I mean I was going over my photos, old photos and when I was his age, my God. We looked the same, but of course it’s not skin deep. I mean we have the same kind of sensibility and I

don’t know how to express that. In a sensibility that there should be, maybe it’s this, that there should be sufficient explanation but not too much. You got to know when to shut up. And some authors don’t know when to shut up. Because, you know, if you go on and explain something again, maybe it’s different from the first time and the student gets confused, you know. So there needs to be a delicate balance—there should be enough explanation but not too much. And he and I are in sync with that.

EM: So what message do you have for them going forward? What advice and what message do you want to give them?

JS: Keep up the good work. The advice is, well, the message is, I trust you guys. I trust you guys to carry it on. And I can’t think of anybody better to do that than the three of you.

EM: You are an authority on calculus education and its application. Have you ever been asked to consult by friends or colleagues?

JS: One evening, I got a phone call from Richard Armstrong, who is an engineer, he works for [an] engineering consulting firm, and he was consulting on buildings like hotels and hospitals. Large buildings, in places like Beijing and New York City. In both of those cities, they have fire regulations that state that, well, in the first place, they have these emergency water containers on the roofs of the buildings and they’re cylinders. They’re in the shape of cylinders. And Richard said to me, “The local regulations say that I have to be able to guarantee a certain minimum water pressure for a certain amount of time for ten minutes after, you know, a fire starts, as an emergency water supply.” He said, “Now, I know it’s obvious that at the top, if it’s full, the water system is, if it’s full, then I can compute everything. But as the water level drops, the pressure is going to be less because the height of the water is less. And so I need to be able to tell my clients to build, you know, these cylindrical containers sufficient to guarantee those minimal requirements. So, could you solve that problem for me?” he said.

And I said, “Richard, well, I could, but I think it would be much more constructive if you did it yourself. I’m going to give you a copy of my calculus book and we have essentially that same problem here in this differential equations chapter, and I would suggest that you read this section and that will explain how to do the calculation.” Well, he did, and he was proud of himself, and I was proud of him, too.

EM: Let’s talk about technology. You’ve embraced technology and incorporated it into your books over the years. Why is that important to you?

JS: Well, it’s already revolutionized calculus education, starting a couple of decades ago, but the extent to which it has been embraced by different instructors is quite dramatic. I mean some instructors don’t want to have anything to do with it even now, such antediluvian attitudes, but there’s other instructors who embrace it with open arms—it really depends on the instructor. And I’m afraid to say [it] but…the majority of instructors neglect it, neglect the technology. Some courses are set up for that and that’s great, and that’s terrific, but others, they just can’t be bothered because those things weren’t around when they were in calculus, you know? Meanwhile others are absolutely terrific in their embracing technology because students can learn so much. Well, for example, in my books, what we call TEC, Tools for Enriching Calculus, a lot of instructors don’t even know it’s there. Dan Clegg and I co-authored those TEC modules and visuals and I think—I think they’re very good learning tools, I mean that’s one example of technology. We have to embrace technology in calculus education.

EM: Why do modules like your Tools for Enriching Calculus help students and instructors?

JS: Well, because the TEC brings the subject alive. You see, calculus can be regarded as the mathematics of motion, it shows things approaching other things and you know, it’s hard to convey that in a static object like a book, but on the screen, it comes alive.

You see this approaching this and that’s what calculus is all about; it’s the mathematics of motion so it’s just a natural combination—I mean, even talking about algebra or trig with technology, but its calculus where it really plays a role.

EM: What are your thoughts about the changes in calculus education; how students have changed, how instructors have changed the way they teach calculus, and how do you envision things changing with technology as a part of it?

JS: Well, it’s a cliché that many instructors bemoan the fact that students aren’t as good as they used to be. I’m not sure that’s true, it depends on the type of school you’re teaching at; in other words, open access or not, you know, it really depends on that…but quite more generally than not, many instructors will say, they think back to when they were a student at some elite institution and they say, “Wow, you know, kids today are just not as good,” I think they’ve forgotten how bad students were 10, 15, 20 years ago. You know, they’re reminded by the mistakes the students make now but, you know, they were making those mistakes back then, too, I think. So, I’m not sure, frankly, that students have changed a whole lot in the last, you know, 10, 15 years. I’m not convinced that they have changed dramatically.

EM: Now, they have shorter attention spans.

JS: Oh God, you mean with all these devices and stuff, oh boy, yeah, yeah, I know. That’s another issue entirely that I wasn’t even thinking of. Do the students have shorter attention spans? I don’t know, maybe they do and if so, I personally cannot solve that problem for the world.

EM: What advice do you have for instructors going forward?

JS: Wow, that’s a tall order. You’re so demanding. Do this and you’ll be—well, I would say, respect the students. We expect the students to respect us,

but we should respect the students, too; that when the students come to see you in your office hours or after class, they’re human beings and they make errors that may be like annoying or you think, “Oh, well, they don’t get that,” but no, you have to respect students and give them explanations that they can understand and be polite about it.

EM: What makes the content timeless?

JS: Well, that would be true in mathematics. It wouldn’t be true in any other subject. Because an equation in mathematics, it’s either true, or it’s false, you know. So that eiπ + 1 = 0 is what we discovered three centuries ago is equally true today. It is timeless. It’s timeless. So mathematics…it’s a different character from other subjects. Now, we add to that knowledge, but we don’t subtract from that knowledge. Oh, that’s really clever, if I do say so myself.

EM: So in the era of open source, what is the role and importance of mathematics authors?

JS: Well, my general opinion of open source is that people who create these open source materials are rather blasé. They don’t understand how much time and work goes into writing. So when Bill Gates had this marvelous idea of making things free and decided, “Oh, let’s have so-and-so write this. Let’s have so-and-so write that,” not just in math, but in other subjects, too, he said, “Oh, they may choose someone eminent, but someone eminent may not know how to write.” So, I mean, that is the problem with open source. There’s some marvelous materials there, but the authors or the writing team who are writing that material may not be as able, or may not be good writers.

EM: And why is that so important?

JS: Oh, well, is it not obvious that when you’re trying to explain something, and you write it down, that it be understandable?

EM: What are your thoughts around open courses?

JS: Well, we’re still finding our way around MOOC’s, and one MOOC is so different from another. Sometimes it’s a professor doing his own thing, his own eccentric thing, which I don’t think is very valuable or—but [on the] other hand, sometimes…it could be wonderful, but the thing I would emphasize is that you’re never going to replace one-on-one talking in the same room and looking into the eyes of the student to see that they understand, or that student looks into the eyes of the professor who is professing. It’s true that different students learn in different ways, but I think for the most part, students do appreciate a live human being talking to them and trying to explain something, and I, whatever the virtues of individual MOOC’s, I’m not sure one can ever replace one-on-one contact.

EM: What role do you think authors would play, textbook authors in particular, in a world, where there’s plenty of open source MOOC’s and so forth?

JS: Well, an experienced textbook author could author some of these MOOCs, sure. And some good new ones might emerge as well, but…I think open source shouldn’t be regarded as a magic thing. It’s fraught with difficulties, and hopefully it’s been written by somebody good.

EM: Do you have a message for instructors teaching calculus in high school and students regarding the college-level calculus?

JS: Well, in a sense, I envy high school teachers of calculus. Because they have more time to lavish on students learning a given topic. I mean if you compare the time allotted high school and in university, you know, it’s really double, I think in high school, and…you’ve got time to do little projects, you’ve got time to do the things and to lavish on the students. So I envy high school teachers of calculus. And I talked with some of them over the years. And they’re really dedicated to their kids. Which I think is wonderful.

EM: What inspired you to write your latest books, Biocalculus and the eighth edition of Calculus?

JS: Well, the Biocalculus book was particularly exciting for me because it went in a different direction and it rose from the fact that, you know, a lot of my students wanted to go into the life sciences and wanted to become doctors, or at least their parents did anyway, they wanted them to become doctors. So I organized a team of people to do this, including, I found a wonderful co-author, Troy Day. I’m very proud of that new book, Biocalculus. We spent a lot of time scouring the literature for applications to the life sciences and, it was quite exciting for me personally because I learned a lot about biology in the course of writing this book.

The eighth edition of my original calculus book, of course, it’s a little more traditional. And people wonder, “Why do you have to write a new edition?” Well, it’s because I keep thinking of good ideas. You know, I keep files of ideas for future editions. Some of them are good exercises, some are just interesting things that I think I can use in some way. And so, well, in particular, the eight edition has three new projects, which I think are really good. One deals with birds flapping their wings and then gliding, you know, should they be flapping more or gliding more? I thought, “How could mathematics come into that, how does mathematics enter into such a question?” But it does, and it’s marvelous to see. Another of the new projects deals with controlling red blood cell loss during surgery, which I think would kind of interest people—you don’t want to lose too much blood. You know if you lose a couple of liters of blood while you’re in surgery, that’s okay, but I wouldn’t go much further than that. There’s certain procedures, ANH procedure that we explained there and so, what happens if you remove some of the blood before the operation, and then you replace it after the operation, that is very helpful. And the third new project was the Speedo Racer. It cut down drag in the water and a lot of people broke swimming records, you know, time trials, and then they outlawed

it because it was…giving an unfair advantage. So there was a bit of an argument about that. Why does mathematics come into this at all? Well, we asked a student, using techniques from multivariable calculus, [if] you can see why. The students are asked to explain why a small decrease in drag could contribute to a record-breaking performance.

EM: I heard the exercises in your Biocalculus book were inspired by former students, how did that happen?

JS: Well, when I went into hospital last year I was in—I was not in good shape. And when the chief doctor came around who is in charge of my care, I recognized her as Lisa Hicks, who was my calculus student 20, well, at that time it would be 22 years ago, at McMaster University. And she recognized me, too. So I was delighted because I remember Lisa as being a brilliant student. I think it’s good to have someone smart in charge of your care. Anyway, I think it’s important to have good hard data in a book. And I figured as long as I’m in the hospital I may as well make use of my time here and so [to] my kidney doctor, Ron Ball, I asked him explicitly did he have any data for me. And he referred me to some websites, which are terrific perhaps of the kidneys. But also my former student, Louey Lou, now a professor at Gastroenterology at University of Toronto, he is likely to have absolutely great data of a blood alcohol concentration, and I mean a lot of exercises in both the Biocalculus and in the eighth edition of my calculus book [are] surrounding blood alcohol concentration because I think the students can relate to this. You know, drinking, the whole drinking and driving thing, you know, so I think the students can relate to that. And so he steered me on to these great data from this research paper so that was very fortunate. So there are two examples of my former students. But there have been several others as well.

EM: Your life has been influenced by math in several ways. Your love for architecture and your love for music. Do you have a few words to share how these different spheres of your life have impacted one another?

JS: Well, you know, I’ve gone back and forth through my life between mathematics and music. Because I love both of them and wouldn’t want to sacrifice one for the other. But they are both—and I have thought a great deal about it—it’s a long story. The connection between mathematics and music, most people think of mathematics as relating to rhythm in music. But, you know, if you divide music into four of its elements, rhythm, harmony, melody and form, then mathematics plays a role in all four of those aspects of music. And if you think about it, it kind of makes sense that—well, think of it this way. Mathematics is famous for being logical. This follows to this, which leads to that, which leads to that, which leads to that. Well, the same is true of music, isn’t it? That you start off with the melody and then you develop it. The point is that music evolves. Music takes place over a time period. Art is more static. So it stands to reason that mathematics, which is concerned with logic and the flow over a period of time, relates to music for which that is also true.