An Algebra Teacher I could Understand

Emmy award-winning journalist and bestselling author Cokie Roberts once said:

An Object of Pride

A mathematician's relationship with the general public most closely resembles bipolar disorder—at the same time they admire us and hate us. Almost everyone has had at least one bad experience with mathematics during some part of their education. Get into any taxi and tell the driver you are a mathematician and the response is predictable. First, there is silence while the driver relives his greatest nightmare—taking algebra. Next, you will hear the immortal words: “I was never any good at mathematics.” My response is: “I was never any good at being a taxi driver so I went into mathematics.” You can learn a lot from taxi drivers if you just don't tell them you are a mathematician. Why get started on the wrong foot?

The mathematician David Mumford put it this way:

A Balancing Act

The other most common response we get from the public is: “I can't even balance my checkbook.” This reflects the fact that the public thinks that mathematics is basically just adding numbers. They have no idea what we really do. Because of the textbooks they studied, they think that all needed mathematics has already been discovered. They think “research” in mathematics is library research. They have no idea that mathematicians can't balance their checkbooks either—although for reasons different from theirs.

Accounting to the Public

The public sees us as slightly mad geniuses since we take for granted things they cannot even imagine. They see us as aliens who are just visiting this planet long enough to make their lives miserable. They are not sure if they should be talking to us or running for the exit. They are pleasantly surprised if they discover we can hold a normal conversation with a mere mortal. We like appearing as “geniuses” to the public. If we must have a false mystique, this is probably the best we could ever hope for. The problem is that their definition of “genius” is quite different from ours.

Many of us who are mathematicians do not spend a lot of time fussing over why we became mathematicians. It was something that we evolved into, and it happened naturally.

But an invitation to write for this portion of our book gave several activemathematicians an opportunity to really dwell on this question. And the results are informative and inspiring.

Some people became mathematicians because they love to solve problems, others because they love to teach or communicate, others because they love to develop theories, still others because they love logic. In many, if not most, instances there was a key teacher or mentor who set an example and inspired the individual in question to pursue the mathematical life.

And this is an observation not to be downplayed. It would be virtually impossible for a person in complete isolation to develop into a mathematician. There must be some teaching or mentoring or instruction involved. Even the great Ramanujan noted a particular text that he found inspiring.

It is always instructive to pause and endeavor to determine how and why we became who we became. Such rumination helps us to develop a sense of ourselves, and also a sense of what mathematics is, what it means to us, and how it has played a role in our lives. It is unlikely that a carpenter would engage in similar cogitations. There is something special about mathematics that inspires us, and gives our lives meaning. One of the main purposes of this book is explicate this point, and to make it understandable to non-mathematicians.

Perhaps reading these words will attract some new people to consider taking up mathematics. That would be a happy outcome for all concerned.

What This Book is About

The mathematical life is a rewarding, satisfying, and fulfilling one. But it has its trials. As a clinical psychologist has observed, mathematicians and oboe players have a lot in common: they both do something very difficult that other people do not appreciate. We have all had our fill of the shopworn lament, frequently encountered in social situations, to the effect that “I was never any good at mathematics.” Or, perhaps worse, “I was very good at mathematics until we got to the stuff with the letters—like algebra.”

Fortunately, there is evidence that the tiresome observations in the last paragraph are not really indicative of the general public feeling about mathematics. Just as an instance, the popularity of the movie A Beautiful Mind and the television show Numb3rs is an indication of some public fascination with mathematics. The public responded strongly and enthusiastically to Andrew Wiles's solution of Fermat's last problem. There was genuine interest and curiosity about Grisha Perelman's proof of the Poincaré conjecture. Most anyone would like to know more about the shape of the universe. People respect mathematics; they are just intimidated by it.

At the MAA Summer MathFest in 2007, Peter Casazza, Steven Krantz, and Frank Morgan were invited to organize a special session entitled The Psychology of the Mathematician. The avowed purpose of that event was to discuss what mathematicians think of themselves and what others think of us. This was a well-attended session, with lively and heartfelt discussions. The speakers went into some detail, and were often quite emotional. As a consequence, MAA Editor Don Albers invited us to produce a volume inspired by our special session. Frank Morgan withdrew from that particular project, and Randi Diane Ruden joined it. So we now have a volume edited by three scholars with diverse interests.

We invited two types of articles for this volume. The plenary articles, generally by well-known mathematical figures, are putatively about the theme of the session: “What Do People Think About Mathematicians?” The secondary articles, rather more brief, are about “Why I Became a Mathematician.” Taken together, the two collections of articles paint a varied and multifaceted panorama of ways to think about our profession, our subject, and those who people it.

If one wants to become a mathematician, then one is faced with the prospect of a goodly number of years of schooling. Usually this culminates with a Masters Degree or a Ph.D. It is just a fact of life that most of that schooling concentrates on the learning of mathematics and, in the case of the Ph.D., on the discovery of new mathematics. Perhaps an insufficient amount of this time and effort is devoted to ideas about teaching.

It stands to reason, then, that a mathematician writing about “On Becoming a Mathematician” will rely on his/her own personal experience in graduate school, and as an Assistant Professor, to describe what this maturation process consists of. And a key feature of that genesis will be research. When we think about the path that we followed to become a mathematician, we think about papers that we struggled to write, journals that we worked with to get our work published, grants that we applied for, conference invitations that we hoped to receive, and promotions that we sought. Of course teaching was also a big part of our lives, and we worked hard to become excellent teachers. We would be disingenuous to suggest that teaching is not of utmost significance. And we are proud of our teaching. But, when someone asks me what is involved in becoming a mathematician, it is themathematics that looms large.

The essays in this part of the book reflect the philosophy described in the preceding two paragraphs. They certainly talk about teaching and communicating, and well they should. But the emphasis is on the subject matter itself. The reader should balance these pieces against those appearing in the “Who are Mathematicians?” part of the book in order to get a complete picture of what this work is about, and what message it is endeavoring to convey. Everyone who wrote for this volume loves mathematics, and wants to share that enthusiasm and passion.

Certainly one of the main focuses of our special session at the MAA Math Fest in 2007 was the question of who mathematicians are and how we think of ourselves. Also we talked about how others (non-mathematicians) think of us. This part of the book is squarely situated in that circle of ideas.

There is no question that many of us think of ourselves in terms of the mathematical work that we do—proving theorems, writing papers, writing monographs, authoring texts, giving talks. But we also think of ourselves as communicators and teachers. Certainly there is no meaning, and little value, to our mathematics if we do not effectively communicate it and teach it to others. And, thanks to Euclid of Alexandria, we have a well-developed paradigm for doing so. For two thousand years we have had logic and the axiomatic method and Occam's razor as the model for mathematical development. And it has served us well.

This mode of thought shapes not only the way that we write but also the way that we teach. For our job as mathematics teachers is to show our students how mathematical discourse works, and why it works. We want to teach them many particular ideas and techniques, but we also want to teach them the critical thinking skills that shape our lives.

The essays in this part of the book reflect the value system described in the preceding paragraphs. Several accomplished mathematicians, from many different fields and disciplines, describe their personal take on who mathematicians are and what they are trying to accomplish. Many of these essays are quite personal, and they all serve as role models for young people studying mathematics today—budding math teachers, math communicators, and math researchers.

This part of the book should be assessed and digested alongside the “On Becoming a Mathematician” part of the book; the two taken together form a whole that is greater than the sum of its parts.