By Ivars Peterson
David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College. He served in the Peace Corps, teaching math and science at the Clare Hall School in Antigua, West Indies before studying with Emil Grosswald at Temple University and then teaching at Penn State for 17 years, eight of them as full professor. He has held visiting positions at the Institute for Advanced Study, the University of Wisconsin-Madison, the University of Minnesota, Université Louis Pasteur (Strasbourg, France), and the State College Area High School.
Ivars Peterson: Were you interested in mathematics at an early age, or did that develop later?
David Bressoud: It really started for me in seventh grade. I was extremely lucky to have a great seventh-grade teacher who realized I had an aptitude for mathematics. He encouraged me to come in after school, and he gave me good, challenging problems to work on.
Between my junior and senior years of high school I participated in Albert Wilanksy's NSF summer program at Lehigh University. I grew up in Bethlehem, Pa., so it was easy to get to Lehigh. I remember the program vividly. They gave us great problems to work on. You’d go in for three or four hours every day during the summer and spend the rest of the day just working on those math problems.
I was intrigued by mathematics, and I knew I was good at it. But when I got to college, I wasn’t certain I was going to major in mathematics. I was at Swarthmore, so I took lots of different courses in lots of areas. I went through a severe sophomore slump and decided that I wanted to get through college as quickly as I could, so I decided to finish up in three years. Having decided that, I didn’t have much choice other than a math major. That was the easiest way for me to get a degree.
Surprisingly, one of the most influential classes was studying probability theory out of Feller [William Feller, An Introduction to Probability Theory and Its Applications]. Dave Rosen taught that class, and he ran it as a seminar. Two of us were in charge each time. We had to read the chapter or section that was coming up, figure out what was going on, and present it to the class. Dave sat in the back of the room and listened to us do our presentation, prodding us at appropriate points. I learned that I could read mathematics and figure out what was going on for myself and then present it. Feller presents things so beautifully that it’s almost surprising I didn’t become a probabilist.
IP: Did you have support at home for your interest in mathematics?
DB: My mother was a musician, who got her degree at Eastman, and my father is an artist. He had studied commercial art at Syracuse University. They never quite figured out how they had turned out a mathematician. When I mentioned my background to my doctoral advisor, Emil Grosswald, he said something which I still treasure to this day. He said that he could see the fact that I was coming out of this artistic background in my mathematics. I’ve thought about that a lot over the years and tried to emphasize this aspect of mathematics. I think of myself in many respects as an artist who works in mathematics.
IP: After you graduated, you went into the Peace Corps.
DB: I wanted to get out of college. By the time I graduated the one thing I knew I would never do is go back to graduate school. I didn’t bother taking the GRE, and the Peace Corps was a great way of getting overseas and feeling like I was doing something useful. It also gave me two years to decide what I wanted to do. So I was in the West Indies on the island of Antigua teaching at the Clare Hall School. I was their first math and science teacher.
IP: At what grade level?
DB: Seventh and eighth grade, which in the British system was first and second form. It was a great experience. I got a new perspective on the United States and an understanding on how much we influence the rest of the world.
I found that I missed mathematics while I was there. I had stayed in touch with Dave Rosen. He kept sending me problems from The Monthly, and I’d taken a couple of my textbooks from Swarthmore and started working through them. Two of them in particular were very influential. I had taken a course in point-set topology using the Gemignani book. It's a little book but very nicely done. When I decided to do some mathematics on my own, the first thing that I did was to solve every single exercise in that book. That really helped me understand proofs.
I had never taken a course in complex analysis as an undergraduate, so I had picked up a copy of Nevanlinna and Paatero's book on complex analysis. I decided to work out every exercise, but it was so frustrating. I’d read a section and try to do a problem, go back and reread the section to try to figure out what was going on. I made virtually no progress. But when I did go to graduate school and took a course in complex analysis, suddenly all the pieces fell into place. I had done all that spadework in advance to try to figure out what’s going on. When I then got into a class it went beautifully for me, and I fell in love with complex analysis.
IP: Once you got to graduate school, your interests went more in the direction of number theory and combinatorics.
DB: They did. My undergraduate record was not that strong. Dave Rosen had suggested that I apply to Temple University and connect with Emil Grosswald. He and Jim Stasheff were the two outstanding mathematicians at Temple at that time. I took that complex analysis class in my first year with Grosswald, so it became natural to follow him. I knew nothing about number theory when I got to graduate school. I really went after the person, not the subject.
IP: What was it about number theory that captured your interest?
DB: I love the simplicity of the questions: how you can compose questions that are so easy to state and then draw on so many different areas of mathematics. That’s what I’ve loved my entire career working in number theory and other related areas of mathematics. You get to see how the different parts of mathematics tie together and how you can get an answer from an unexpected direction. I could tie my discovery of complex analysis to these questions of number theory and see how very concrete questions about the structure of the integers can be answered by doing sophisticated work in complex analysis.
Things like the prime number theorem excited me. So my initial intention was to work in analytic number theory because that’s the area that Grosswald worked in. But in my thesis I started moving toward enumerative combinatorial number theory, and that’s the direction I took off in.
IP: Was it in your thesis or in other early work that got you interested in proving some of Ramanujan’s results?
DB: That was my thesis. Bryan Birch had discovered a couple of loose sheets of paper in the math library at Oxford. The pages contained a list of 40 identities in Ramanujan’s handwriting, and nobody had any idea how they had wound up there. Bryan had found them around 1975 and published them. Grosswald saw the list and told me that proving these results would make a great problem for a doctoral thesis. Ramanujan had simply stated them without proof, and it looked like they could be proven fairly easily with the theory of modular forms.
I started to work on them. They turned out to be much more difficult to prove using modular forms than either Grosswald or I had expected. But I started looking at some similar identities that had been done by L.J. Rogers. He had taken a much more direct, manipulative approach, looking at them as generating functions. I started taking that approach and found that I could not only solve many of them but also generalize them and come up with large classes of identities of which these were special instances. That became my thesis, which launched me into thinking about generating functions and combinatorial number theory, and it got me interested in questions in partition theory, which has been the core of my research over the years.
IP: You ended up at Penn State.
DB: There’s an interesting story behind that. Work on my thesis went very quickly. I got the problem late in 1975, and by the summer of 1976 I had most of the work done. Grosswald suggested that I go to the [MAA] summer meeting, which was in Toronto that year, and do a 10-minute presentation on what I had found in my thesis. Very few people show up for these presentations, and as it happened my presentation got scheduled opposite the announcement by Appel and Haken of the proof of the four-color theorem. So who’s going to go to a 10-minute talk by a graduate student when there was the announcement by Appel and Haken? In the room, there was myself, the person who spoke before me, the person who spoke after me, and the moderator. That was about it.
But one other person showed up for my talk: Dick Askey. Dick had gone through the abstracts and seen what I was talking about and realized that my subject tied directly to basic hypergeometric series, which was the area that he was excited about at that time. He came and listened to me and spent the rest of the day with me telling me what I needed to read, what I needed to think about, how my work tied in to what other people were doing. He then went back and contacted George Andrews [at Penn State] and told him that there’s somebody at Temple that he needs to pay attention to.
George and Emil [Grosswald] were good friends because they had studied under the same doctoral advisor, Hans Rademacher. They arranged for George to come and give a talk at Temple, and I got a chance to meet George at that time. During his talk he threw out lots of open problems, fairly straightforward problems that he’d thought about briefly but had not spent enough time on to answer. At this point I was essentially done with my thesis, so I started working on those problems. In a few weeks I had solutions, which I sent to George. Then George started suggesting other problems—we were corresponding at this point—and he brought me to Penn State on a two-year visiting position, which then turned into a tenure-track position at Penn State.
A few years later I went to Wisconsin and did a two-year post-doc with Dick. I owe a tremendous debt to Dick Askey and also to George Andrews. We have often disagreed over the years about teaching mathematics, we have somewhat different points of view, but I have enormous respect for the two of them. I learned a lot from them—how to view mathematics, how to judge mathematics, how to appreciate mathematics.
IP: A lot of your interests now are in the teaching of mathematics. Was that interest there from the start?
DB: Teaching in the Peace Corps was something I loved doing. I realized that at heart I’m a teacher. I also realized during the Peace Corps that I missed higher mathematics, that I would not be happy teaching at the high school level for the rest of my career.
When I first went to graduate school, I didn't expect that I’d become a research mathematician. My plan was to get my Ph.D., and work with future teachers. Then I was bitten by the research bug. That was my focus at Penn State for many years. I was always a good teacher, and I put a lot of attention and effort into my classes, which were important to me. But the research was also there.
There came a point in the late 80’s when I realized that staying on the cutting edge of research was more of a drain than a reward for me. I wasn’t enjoying having to bring something new to every research conference, and I wasn’t enjoying putting together the proposals to NSF or NSA for funding. The search for grants was something that I worried about excessively.
I made a very conscious decision in the late 80’s that I would do what I really wanted to do. I was a full professor at this point. I was getting more interested in teaching; I was thinking about how to put together a number theory course that was based on factorization and primality testing. I wanted to write textbooks. I got involved with the Science, Technology, and Society program at Penn State. I started going to conferences that were looking at the calculus reform effort, and I found that I enjoyed that.
Penn State is one of the top research universities, and the department is big enough that they can tolerate people doing strange things like this. But I didn’t find the kind of support that I wanted. That became clear to me during the 1990-1991 academic year, when I did an exchange with a high school teacher at State College High School, Annalee Henderson. She taught a course at Penn State. They wanted her to show how graphing calculators could be used in college calculus. And I took over one of her classes, an AP calculus class at the high school.
Working with Annalee was phenomenal because she was so excited about teaching and had so many great ideas about how to get things across to students in the classroom and how to get the students engaged and make it an active undertaking. This made me realize that there was nobody at Penn State with whom I could have these kinds of conversations. So I began to look for more opportunities to talk about teaching and how to improve teaching.
In my last year at Penn State, 1993-1994, David Smith from Duke had a sabbatical there. That was a transformative experience for me. David was so enthusiastic about teaching and had such great ideas about how to do it effectively, to be able to spend an entire year with him was great. We regularly went to lunch together and talked about what was going on in our classes. I was trying out his Project Calc material, then talking over what was happening in the classroom directly with him and getting ideas from him. That’s what convinced me that Penn State was not where I needed to be.
I needed to be in a place that had a strong focus on teaching and a community of people for whom teaching was what they were most interested in. And that meant Macalester College. Macalester in many ways was a very natural fit because I knew Wayne Roberts; I had interacted with him through the calculus reform movement. Also, my wife is from St. Paul and her mother just happened to live a mile from the college.
It was the day after Labor Day when I opened up The Chronicle of Higher Education and saw that they were advertising for a senior mathematician at Macalester. I applied immediately. I later learned that mine was the very first application to come in for that position, and I got it.
IP: Have you continued your research, or at least aspects of it?
DB: I have. It’s nice being so close to the University of Minnesota. They’ve got a fantastic group in enumerative combinatorics. I’ve known Dennis Stanton since we both came out of graduate school in the same year, and I’ve done some joint work with him. Certainly the research has taken a back seat to my other interests. It’s now been a few years since I’ve published a research paper, but I’m still interested in it, I still keep abreast of what’s happening in partition theory and basic hypergeometric series, and of course all of the things that have come out of studying the alternating sign matrix conjecture.
IP: You have done quite a lot of writing, especially books.
DB: It started with my book on factorization and primality testing, which came out of teaching the course. For years I went to the West Coast Number Theory Conference. The people who were working on factorization and primality testing during the 80’s all went to that meeting. So I learned a lot about what was going on, and back in the early 80’s the methods were very simple. They were methods you could teach in an undergraduate course. So I got the idea of building an undergraduate number theory course around the problem of how you decide if a large integer is a prime or not, and if you know it’s not prime how you find its factorization. After teaching the course for a few years, I ended up turning my set of notes into a book.
I found that I enjoyed that, and almost immediately started working on Second Year Calculus, looking at vector calculus from a historical point of view. I had studied calculus using Apostol, volume two. That’s what was used for several-variable calculus, and it was totally opaque to me. I wasn’t able to appreciate it as a student. But going back to it later as a working mathematician, I could. It’s just so beautiful. So that was one of the books I used as a foundation for Second Year Calculus. Another was Edwards’ book on advanced calculus. I loved the way he dealt with differential forms. I wrote my Second Year Calculus because I wanted students to be able to use Edwards’ advanced calculus book, but I knew there were very few undergraduates who would be capable of handling that. So I wanted a book that was a little bit more accessible.
At the same time I was getting excited about the history of mathematics. I was very privileged to hear Chandrasekhar give a series of lectures on Newton’s Principia at Penn State. That got me excited about the Principia. I decided to tie that into the course and talk about vector calculus in terms of the progression from celestial mechanics through electricity and magnetism up to special relativity.
I enjoyed writing that book so much I couldn’t stop. I think the times when I am happiest are when I am working on a book, to be able to immerse myself in the research and the writing. I especially love the rewriting. I like to spend time on how the words sound as well as the structure.
I had taken real analysis from Royden's book as a graduate student. I learned it, but I didn’t understand any of it or where it was coming from. That led to my real analysis book. I especially like my book that’s just come out on Lebesgue integration, finally figuring out where measure theory is coming from, figuring out why people cared about it and the reasons for coming up with this particular set of definitions, and why it’s important for students to learn these definitions.
IP: You’ve done a set of videos for the Teaching Company on the history of mathematics.
DB: I’m not a historian of mathematics, and I don’t pretend to be a historian of mathematics, but I’m fascinated by it and I love to use it in my teaching to motivate what’s going on. I had been contacted by the Teaching Company to see if I would do a series of lectures for them. They bring you into the studio to see if you look reasonable in front of the camera, then they ask you to propose courses you might be interested in doing. So I proposed a course in discrete mathematics and courses in number theory, but they kept hinting that they would like to see a course on the history of mathematics. So I took the hint and proposed it. They ran with the idea.
I enjoyed doing the research and figuring out how to present the entire history of mathematics in 12 hours; I had 24 half-hour lectures. I started at the beginning with the Babylonians and the Egyptians, then ended up with 21st century mathematics, what is happening today and why everybody should be excited about mathematics. One of the things I found out for myself is how important the 17th century was. I see that as the focal point of the history of mathematics. All of the different threads come together in Western Europe in the 17th century, and that in turn lays the foundation for everything that happens after that point. So I spent a lot of time in these lectures on the 17th century.
IP: Another thread in your career is your involvement with the MAA and other professional groups.
DB: During the 90’s I got very heavily involved with the Advanced Placement program. I eventually was on the development committee, and in 2002 I became chair of the committee. Shortly after I went to Macalester, I was elected to the AMS governing board and served on their Committee on Education. My connections with the CUPM Curriculum Guide were a result of my working for the AMS because I was initially appointed as an AMS representative to the CUPM working group on the curriculum guide. But then very quickly I got involved on the MAA side and eventually became chair of CUPM.
IP: How do you see your role as President of the MAA?
DB: It’s a great opportunity to be able to talk about and have some influence on the issues that I care about, in particular undergraduate education. I love the fact that we’ve got this organization for which the emphasis is mathematics at the undergraduate level, communicating the excitement of this mathematics to a broad audience and thinking about how we can teach it more effectively.
We need to rethink what goes on in undergraduate mathematics. I think we’ve been tied exclusively to supporting the engineering colleges for too long. There are so many exciting opportunities and challenges out there for mathematics to broaden its involvement with the biological sciences and the social sciences. I want to see that pushed, and this is an opportunity for me to do that.
I’m also very concerned about the transition from high school to college. We have so many students now who are taking calculus in high school—about 16 percent of all high school graduates have taken calculus—and that’s changing the populations of students who are coming into colleges. The colleges need to pay attention to this fact. They need to rethink what they’re doing with these students. Heading the MAA gives me a chance to push for the MAA to help look at what can be done and to help promote the programs, the courses, the materials that can help individuals and colleges improve their teaching of mathematics.
IP: Are there any other interests at the MAA level that you’d like to pursue?
DB: The way people look for information is changing dramatically, and I applaud what the MAA is doing to make a lot more of its programs and materials available electronically. Things are changing in the ways people look for things, and look for support. I think of the model of Project NExT and the listserv that goes out to Project NExT fellows and how very, very helpful that is. We need to expand that kind of service so that everybody has access to broad expertise and to suggestions for how to do things locally.
IP: Where do you see yourself ten years from now?
DB: Ten years from now I hope I’m still writing books. That’s when I’m happiest. Writing—and continuing to travel and talk. I love talking about mathematics because it’s so exciting. I love talking about the story of the alternating sign matrix conjecture and the way mathematics really works and the wonderful surprises that come up in it. I hope I can continue to travel and talk and share the enthusiasm, but also to continue to explore the history of mathematics and write books that show people where the mathematics is coming from.