James Stewart - The Cengage Learning Blog

An Interview with
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.
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.
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.
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,
EM: Why do modules like your Tools for Enriching
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.
JS: Well, because the TEC brings the subject alive.
EM: What advice do you have for instructors
Calculus help students and instructors?
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.
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.