Michael Starbird is a University Distinguished Teaching Professor at the University of Texas at Austin and a member of UT's Academy of Distinguished Teachers. He received his B.A. degree from Pomona College and his Ph.D. in mathematics from the University of Wisconsin-Madison. He has been in the UT Department of Mathematics since 1974, except for leaves, including one to the Institute for Advanced Study in Princeton, N.J., and one to the Jet Propulsion Laboratory in Pasadena, Calif.

Starbird has received more than a dozen teaching awards, including the MAA's 2007 Haimo Award for Distinguished College or University Teaching of Mathematics and several that are presented to only one professor at UT annually. He is a popular lecturer, having presented more than a hundred invited lectures since 2000. Starbird's books include, with co-author Edward B. Burger, the award-winning mathematics textbook for liberal arts students *The Heart of Mathematics: An Invitation to Effective Thinking* and the trade book *Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas*. With David Marshall and Edward Odell, he co-authored the MAA textbook *Number Theory Through Inquiry*. His Teaching Company video courses include "Change and Motion: Calculus Made Clear," "Meaning from Data: Statistics Made Clear," "What are the Chances? Probability Made Clear," and "The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas." These courses reach tens of thousands of people in the general public annually.

**Ivars Peterson**: Where did you grow up? How did you get interested in mathematics?

**Michael Starbird: **I was born in Los Angeles, brought up in Southern California. My parents were both teachers. My mother taught remedial English in the barrio of East Los Angeles, and my father taught mathematics, physics, and astronomy at East Los Angeles Junior College.

Both of my parents had decided as a matter of child-rearing technique that at the dinner table they would not talk about mundane issues but instead only talk about other things. In particular, my father would often bring mathematical puzzles for me and my brother to work on. I was brought up thinking that mathematics was something that you talked about at the dinner table. Naturally, both my brother and I have Ph.D.s in mathematics.

One interesting thing that I sometimes worry about is that my own kids worked so hard in school, particularly in their math classes. It was so fast, such an intensive kind of education that I didn't feel any kind of enrichment at home would have been possible. They already had way too much of that kind of stuff. When I went to school, it was really quite easy. There was very little homework; it wasn't a hard, intensive kind of experience. For me, mathematics was always a joyful thing, partly because one of the great associations with it was doing these puzzles, thinking them through.

**IP**: Were you a math major in college?

**MS**: Yes, but I wasn't a very serious student in college. I had actually gone to Harvard for my first year, but I didn't like it there, so I went to Pomona. After Pomona, I went to graduate school in Wisconsin. That's where I really started to enjoy mathematics a lot. It began in a Moore method topology class. R.H. Bing was at Wisconsin, though he was not my teacher in any class. But one of his students, Russ McMillan, taught my introduction to topology class. He would pose questions, and we would go home and prove them on our own and then present our work to the class. To me, it was just like a treasure trove of puzzles—infinitely many, wonderful, delectable puzzles. I flourished in that setting.

For my thesis, Mary Ellen Rudin took me on as a student. She was interested in point-set topology, and she just gave me some questions, and I did them.

**IP**: What happened after graduate school?

**MS**: I've never applied for a job in my life. Bing was at Wisconsin when I arrived there, and then he was hired by Texas. When he left, he said, "Starbird, when you're finished, come to Texas." A year later, I went to Texas and worked with Bing. So I learned geometric topology.

My career has all been determined by people. The story of my life is people. Every step has been meeting somebody who I like, and then I would get involved in something that they encouraged me to do.

**IP**: A large part of your focus is on teaching mathematics. Was that interest there from the beginning?

**MS**: I always loved to teach. Both of my parents were teachers. My grandmother was a teacher. Teaching has always been and continues to be a real pleasure. I enjoy the students; I enjoy the idea of explaining ideas to people.

**IP**: How would you characterize your teaching style?

**MS**: I do different things in different classes. In upper division mathematics classes, I use a modification of the Moore method. I have the students do the bulk of the work proving the theorems on their own, presenting their work to the class. My number theory book is an example of that. It was a course designed when we were trying to get math majors to be better at proving theorems on their own, so that they would be more effective in upper division math classes. So this was an introduction to proof that we developed. But I had taught topology using that method from the beginning of my career at Texas.

**Listen to the rest of Michael Starbird's response.**

I also teach other classes in different ways. My class for liberal arts students, for example, has been very influential in my life. It started when a friend of mine, who was a professor of English and director of our honors liberal arts program, wanted to create a more interesting mathematics course for her students. We were offering the typical not-so-interesting mathematics courses that unfortunately are still a national problem, it seems to me. I don't understand how we can persist in teaching college algebra or calculus, for example, as terminal courses for students. There's so much fascinating mathematics to teach people.

I developed this course almost 20 years ago with the idea of teaching things that I find fascinating. I imagined the following scenario. Students, as they become educated, go to their literature class, and they read the greatest literature ever. When they go their music class, they listen to the best music. I wanted the same thing to happen in the mathematics class. Things that we mathematicians view as candidates for some of the greatest ideas, not just in mathematics but some of the great intellectual accomplishments of humanity, it seems to me that's what mathematicians have to offer.

After a rocky start and hard work, the class evolved into, for example, the textbook *The Heart of Mathematics*: *An Invitation to Effective Thinking*. We had the dual goal of presenting great ideas in mathematics and having students become better able to think clearly about issues well beyond mathematics.

**IP**: Do you have favorite topics?

**MS**: I could list many. One is the fourth dimension. I also love to talk about infinity, the concept of being able to reason about something that seemed historically beyond human comprehension. What an incredible story that is to tell.

There are ideas of topology that are so counterintuitive, the idea that you can physically take things in your hand and manipulate them in various unexpected ways. I love geometry, all sorts of geometry. And I learn new things. I did a Teaching Company course just last year on geometry. I had never taught the geometry course that we teach in college. I learned a lot of geometry that I had not known before, such as these amazing theorems about triangles.

**IP**: How did your collaboration with Ed Burger come about?

**MS**: Ed was a graduate student at the University of Texas. While he was there, he was in two of my graduate topology classes. He's a number theorist, though, so he wasn't my Ph.D. student. He and I have somewhat similar personalities, and we had common interests. Then he developed a mathematics class for liberal arts students at Williams independently, and I had been developing one at Texas. When he had a sabbatical in 1994, I saw him again, and we became reacquainted and discovered that we had been working on parallel lines on this kind of a course.

When a publisher came to my office and asked me about writing a book, I told him about this class. I started writing some stuff, then I called up Ed and asked, "Would you like to work on this together?" He agreed. That was the terrific beginning to a wonderful collaboration. We still collaborate.

We tried to make it fun and to lead students from one idea to the next. The concept was to expose them to methods of mathematical thinking. I think of it as implicit discovery. When I'm explaining an idea, I want people at every stage to think that's the natural way to do it. The only way to get to that stage is to build a foundation for it, either through examples or previous techniques that are similar, that lead in that direction. It's a natural progression.

The presentation of mathematics where you start with definitions, for example, is simply wrong. Definitions aren't the places where things start. Mathematics starts with ideas and general concepts, and then definitions are isolated from concepts. Definitions occur somewhere in the middle of a progression or the development of a mathematical concept. The same thing applies to theorems and other icons of mathematical progress. They occur in the middle of a progression of how we explore the unknown.

**IP**: What major projects do you have in mind for the future?

**MS**: I have many. For example, Katherine Socha and I are working on a project about fractions. This came up because I gave talks at three PMET workshops two years ago. At each of these workshops, someone else was talking about fractions. I asked myself, "What's this thing about fractions? Why are they talking about fractions?" Katherine and I are looking at how to present fractions. Why are they such an obstacle to so many people, and what's a good way to present them in an effective way?

The phrase that best captures our philosophy is the title: *Fractions Aren't Pizzas*. It seems to us that pizza is a particularly poor metaphor for the concept of fraction. For things such as adding or multiplying fractions, if you have in mind pizzas, the physical combination gets messy very fast. It's one of these cases where a well-meaning metaphor turns out to be an obstacle to the basic concept of fraction as a number, which is a good place to start in understanding fractions.

**IP**: Where would you like to see yourself, say, 10 years from now?

**MS**: My path through life has been following people—it's always been people who have gotten me involved in projects that have then blossomed in unexpected directions.

**Listen to the rest of Michael Starbird's response.**

I think that we mathematicians have a way of thinking that is interesting and has a great deal to contribute to society, besides and well beyond the mathematics itself. It's not the best way of thinking. I don't think mathematicians would be great leaders of society, in general. But together with other qualities, mathematical thinking can be a terrific addition to people of all walks of life. I would guess that 10 years from now my projects would be moving in the direction of seeing if we can bring some of those really interesting mathematical perspectives to broader questions.

**Read about Michael Starbird's Carriage House lecture.**