Teaching, Writing, and Mathematics
By Bobby Neal Winters
I have been thinking recently about the writing of mathematics. This is rendered to being a highly academic exercise as I have not produced any original mathematics to write about for quite some time. I don’t bring very much credibility with me to this endeavor because during the brief interval I was producing original mathematics I cannot say that I was a very good expositor of it.
What has happened in to bring my thoughts to the writing of mathematics? One thing is that I have begun writing myself. I began writing a weekly column for the local news paper about 10 years ago. Figure 52 columns a year--I never miss a week--and seven hundred words a column--I usually overrun that barrier--and this comes to about 364,000 words. I write things besides my column so I will say that I’ve produced about half a million words in the last ten years. If repetition is the mother of learning--and I believe it is--then I may have learned something about how to write.
In addition to this, it has been my pleasure to teach out of a pair of very well-written books: A Radical Approach to Real Analysis and A Radical Approach to Lebesgue’s Theory of Integration, both written by David Bressoud. These books are, as I said, well-written, but I think more important to their effect on me is they are written as historical approaches to their particular subjects.
The historical approach has been very enlightening to me. Mathematics is a land populated by optimizers and those seeking efficiency through brevity. A result is discovered and proven with great difficulty. Then time is spent in organization working out theory wherein the proof of the result seems not only easy but inevitable. The historical approach allowed me to appreciate that what has been rendered as clear as glass to students of the subject today was once a riddle in a mirror.
As a teacher, it reawakened the excitement of the subject in me.
But in presenting this material to my classes, I’ve reproduced the older proofs of various results that have been included and in doing so I’ve noted a difference in style with much of modern mathematics. There is a tendency in modern mathematics to drift toward the abstract. This is a reasonable tendency as the abstract proofs tend to be cleaner and tend to give broader results, i.e. if I prove something about metric spaces, then I prove it about the real numbers, the complex plane, and spaces of functions at the same time.
That is something of an illusion, however. The abstract structures are out-growths of specific examples. The examples were originally things that were interesting in themselves and drove the construction of theory to discuss them of abstract structures to more easily prove results about them.
It is somewhat ironic that my experience as a mathematician helped me to be a better writer of things besides mathematics. As a topologist who studied 3-manifolds, I took trips in my head to places no one had been before and then attempted to explain them to those who’d remained at home. I learned the art of description as I described these strange places. In the construction of proofs order is important. In good writing, some things must be explained before others can be understood.
As a writer, it seems to me that in well-written mathematics, well-constructed examples serve the same purpose that metaphors do in good writing. Much writing seeks to create in you a picture that I have seen in me. This is true whether I am writing about house cats, building a computer, or mathematics. If I am to be successful, I must reach you by beginning with a thing you already understand and build from it. Our shared experiences make communication possible. Mathematically, the creation of a well-chosen example gives writer and reader a common, shared experience from which the writer may then deviate in order to build.
This technique is especially evident in the proofs in analysis that begin with the proof of a simple, very special case and proceed through stages of increased generality until the most general case is proved. At its best, one can see the whole of the problem is in that very special case and the succeeding generalizations are simply minor modifications of the original proof.
There have been times when I’ve complained that too much of the time we who teach mathematics treat it as a murder mystery. We withhold or downplay certain very important details. We keep things to ourselves as our own little secrets.
I think I know why we do this. We are trying to teach our students how to think for themselves. There is a reluctance to “lay out everything plain” because it will deny the students the joy of the gestalt that we ourselves experienced, the joy that led us to become mathematicians.
I must say that there is much virtue to this point of view. The problem is that it is so easy to do very, very badly. There is also a tendency--among some in the profession--to use this as a technique to build up their own egos at the expense of their students’. This is far from universal, however. Much more of a problem is judging difficulty. What is easy to a professional might be insurmountable to a student. The teacher’s job is to lay out bread crumbs to tempt the student to the point of gestalt but not so many as to rob the student of the joy of that gestalt. The fat cat will catch no rats. Not that a cat eats bread crumbs, but sometimes you have to mix your metaphors.
In our teaching, in our writing, we must free ourselves to ape literature. We we must foreshadow the great mathematical truths to come with smaller, more easily digestible truths. When the student finally comes to the climax, he must be so prepared that the final step to him is nature, the final joy is true, but he should then be able to reexamine his steps and realize his arrival at this particular point was no accident.
I mentioned at the beginning of this article that when I was producing original mathematics I wasn’t a particularly good expositor of it. There are those who will read that sentence and recognize me as a master of understatement. While a careful study of my articles would undoubtedly produce additions to this list, I believe I had two major problems: impatience and an over-reliance on notation.
I believe we can all agree that taking one’s time to do a good job is a virtue. A lack of patience can cause a lack of proper care.
Creating notation is a way around the difficulties we sometimes encounter in natural language. Creating a well-chosen metaphor, a well-created example is another. The example has the advantage that someone might name it after you one of these days.
If I were suddenly given my mathematical life to live over, I hope that I would choose to grasp onto mathematical exposition as the true art that it can be. To first live the mathematical adventure and then tell the tale, and to understand the telling of the tale is as at least as important as the adventure because its purpose is to convince others to have adventures themselves.
(Bobby Winters is Assistant Dean of the College of Arts and Sciences and Professor of Mathematics at Pittsburg State University. He now holds the title of University Professor.)