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  Road Signs
    Mathematics and teaching, beauty and uselessness
By John Little

John LittleBeing selected as this year’s recipient of the Holy Cross Distinguished Teaching Award is a great honor. Thank you all for this wonderful recognition. As some of you know, both my parents were teachers in the Pennsylvania state college system and that had a huge influence on me. From high school on I knew I wanted to follow in the same profession. I feel extremely fortunate to have “grown up” as a teacher here at Holy Cross in an exceptional faculty of able teachers and scholars, mentors unusually devoted to the intellectual and personal development of their students, and uncommonly thoughtful, conscientious, and supportive colleagues. I believe this award truly celebrates all of our achievements, and I am proud to accept it in that spirit.

Being singled out in this way has been a great surprise, too (and for some of the same reasons I mentioned before). Frankly, it has been extremely difficult for me to reconcile your expression of esteem with my own perceptions of where I am in understanding effective teaching, and especially of the many things that I have not mastered. Rightly or wrongly, I feel that this award is taken as an indication that the recipient has to some extent found an “answer.” Most of the time, though, I have many more questions than answers. In fact, much of my effort in the past 15 years or so has been devoted to unlearning the style and attitudes I learned from many of those who taught me, and that I relied on at the start of my career.

While research mathematics has made great strides in the past 25 years, most mathematicians would agree, if they were being honest, that our educational efforts have not been successful to anything like the same degree over this time (if indeed they ever have been). Nationwide, there is concern about an ongoing decline in the number of students pursuing undergraduate majors in mathematics and a growing realization that different approaches to teaching may be necessary to capture and retain the interest of today’s students. Moreover, our understanding of how to harness the available computational power for educational purposes is, to be charitable, still in its infancy.

However, trying to change college and university mathematics teaching is a bit like trying to steer the Titanic. Mathematics teaching has a tremendous amount of inertia and many entrenched attitudes and competing pressures actively resist change. At colleges like Holy Cross, of course, teaching is a much higher priority than it is at many universities. But in order to get tenure, college mathematics teachers must also be (at least to some extent) research mathematicians, and the ways we are formed as professionals have often been unhelpful in reconciling the sometimes conflicting demands of teaching and research.

"TRYING to change college
and university mathematics
teaching is a bit like trying to steer the TITANIC."

One unfortunately common attitude among university mathematicians is exemplified by the following quote from the Cambridge don, Godfrey Harold Hardy, one of Great Britain’s leading mathematicians in the first half of the 20th century: “I hate ‘teaching,’ and have had to do very little, such teaching as I have done having been almost entirely supervision of research; I love ‘lecturing’ and have lectured a great deal to extremely able classes; and I have always had plenty of leisure for the researches which have been the one great permanent happiness of my life.’”

This candid and most unapologetic sentiment comes from Hardy’s famous little book titled A Mathematician’s Apology, in which he claimed that one justification for his life’s work in mathematics was its “uselessness.” He meant by this that even though he thought his mathematics was not likely to be of practical utility, it was at least “clean and innocent”—not likely to be of harm to any one. He would probably be horrified to learn that today the useless number theory he especially loved is the basis for encryption systems that make it possible for you to communicate securely over the Internet with an online merchant, or for terrorists to do the same thing with one another. The biggest single employer of number theorists in the United States (and of mathematics Ph.D.’s more generally) is the National Security Agency.

Hardy’s views here are extreme, of course. But for many of the mathematicians I know, myself included, what drew us to the subject was, if not its “uselessness,” then the almost otherworldly, crystalline beauty of its logical structure and the excitement of solving challenging problems within that structure. To some, mathematics can even come to seem like a refuge from the “messiness” and illogic of interacting with other people, and the intractability of many problems that matter in the real world.

It should come as no surprise that these attitudes carry over into traditional approaches to mathematics teaching, starting right from the single most influential mathematics book ever written, the Elements of Euclid. Much of our present conceptions of what mathematics is and what it is about (especially the idea of axiomatic, deductive presentations of mathematical results) can be traced directly back to Euclid. Indeed, some historians have called the Elements the second most influential book in Western civilization. And up until about 15 years ago, if you had walked into almost any high school geometry class, you would have found the textbook, the selection of topics, the style of presentation, and so forth to be recognizably derived from Euclid. John Little

As a ninth grader in Phil Brennan’s geometry class at Mansfield High in Pennsylvania, I absolutely fell in love with that kind of mathematics, and that experience (plus the five or six other courses I took from him) is certainly the main reason I am a mathematician. Mr. Brennan was one of those teachers who leave indelible impressions (not always positive ones) on generations of students. On a superficial level, he might have been the model for the photo next to the dictionary entry for “nerd.” On a deeper level, he really knew the structure of geometry proofs and could get enthusiastic (in his own understated way) about a clever construction or an unexpected way to solve a problem. He awoke a passion for mathematics in some of his students, but lost or alienated many of the others. I never knew him well enough on a personal level to know whether that discouraged him or whether he just accepted it as a part of being a mathematics teacher.

Almost all of the courses I took in college and as a graduate student were also taught in a deductive Euclidean style: the inexorable succession of “definition— theorem—proof’” that might be familiar. One of the teachers who influenced me the most here (and, I have to say it, in some of the least productive ways) was the longtime chair of the Haverford College mathematics department, Dale Husemoller. Dale always looked as if he had just been roused from a sound sleep, but he could give a beautiful hourlong lecture on any topic from measure theory to cohomology operations at the drop of a hat. All he asked you to do as a student was to learn the proofs of the theorems he carefully wrote out on the blackboard and repeat them on the exams. I was very good at it, and thought that was what mathematics teaching was supposed to be for a very long time. Dale’s approach to teaching was also an extreme case. You could say that it was extremely successful in its way. Haverford alumni are over-represented among Ph.D. mathematicians (as indeed are Holy Cross alumni).

What is wrong with Euclidean-style mathematics? I would argue that the principal lesson that those who have thought seriously about these issues have learned is that it is a mistake to rely too much on the final product of a mathematical theory (the Euclid-style deductive presentation) as a model for how that body of mathematics should be taught and learned. When they only see the final product, students have a difficult time appreciating the process by which human beings actually do mathematics—the guesses, the false starts, the backtracking, the exhilaration of finally getting a difficult argument to work, and finally the sometimes even more difficult job of explaining your ideas in a comprehensible way to another person. They don’t understand that they can participate in that process themselves.

When I began teaching in graduate school, and in my early years at Holy Cross, I was mainly concerned with giving clean, “elegant” lectures that would efficiently impart a high-level understanding of the subject to my students. More recently, stimulated by my colleagues Dave Damiano and Margaret Freije, I have gained a fair amount of experience in two complementary approaches to structuring classes—computer laboratories and small-group collaborative discussion exercises—that I now use together with lectures. Both laboratories and discussions have given the students more responsibility for what happens during the class meetings. I think they have also helped prepare students better for the ways they will need to use mathematics and perhaps even develop new mathematics themselves in their future careers. To put it a different way, I believe that my approach to teaching has definitely moved in the direction of making some of the assignments in almost every course more like really doing mathematics. But make no mistake, constructing appropriate discussion questions and truly educational lab activities is hard work and I cannot claim that I have been uniformly successful there.

To summarize, probably the most important things I have learned about teaching are first that students learn best when they are as actively and personally involved as possible, and second that learning happens most readily in a general atmosphere of encouragement of students as individual human beings. To be honest, though, it is in this area that I think I have the most work left to do. There have been times when I feel I have been too focused on the ideas I find interesting in mathematics.

"LEARNING happens
most readily in a general
atmosphere of encouragement of
students as INDIVIDUAL
human beings."

Before closing, I want to tell one more story related to teaching that means a lot to me and to correct an oversight. Toward the end of the spring semester of my fourth year at Haverford, I was asked to meet with a panel of faculty members who were charged with awarding College honors. One of the members of the panel was Brad Cook of the French department, who had taught five different semester-long French literature courses I had taken.

When it was his turn to ask questions in the interview, knowing that I wanted to be a college teacher, Brad gave me a scenario—I was a teacher of an average French class assigned to discuss Molière’s Le Malade Imaginaire. How would I structure the class discussion? What would I do to pique their interest?

I gamely proposed: “First, I would have the class see a production, point out the physical comedy and farcical situations to show how funny so much of it is (or can be).”

He responded, “Well, what if they don’t ‘get’ that aspect of it, what if they still don’t see why they should be bothered?

I pressed on, suggesting a number of other ideas. Each time, Brad responded with the same question, “What if that doesn’t work?” Eventually, we came around to the idea that it might very well be the case that nothing would work; successful teaching depends both on the skill and dedication of the teacher and on the receptivity of the student.

I am not sure that I appreciate the full wisdom of what he was trying to tell me even now. I do know that I have only very rarely encountered that sort of stalemate situation here. Almost all of the students I have taught at Holy Cross have responded very well to what I have tried to do in my classes, and I know I wouldn't be here giving this talk without them. I would like to take this opportunity to thank them also.

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