- A good deal of the strategy of teaching is rhetorical strategy, choosing words and images with great care in order to evoke the response: "I never thought of it that way before," or "Now that you put it that way, I can see it." What distinguishes, not simply the epigram, but profundity itself from platitude is very frequently rhetorical wit. In fact, it may be doubted whether we ever really call an idea profound unless we are pleased with the wit of its expression.
- --Northrop Frye
Strangely enough, despite the contributions I have made to my field, the original urge that drew me to mathematics was not to do research. The primal motive force for me has always been learning more mathematics, as well as figuring out how to explain it to other people. In my research, this paid off in that I was able to accomplish a lot of what I did simply because I knew things that other people working in my field did not. (And for a long time this led me to reject the judgements of mathematicians who called me creative. As far as I was concerned, what I was good at was stealing ideas from other people.)
In my teaching, this constant quest to learn more has led me to put an enormous amount of effort into teaching an incredible variety of different courses. In fact, there was a time when I considered it somewhat a dark secret how much of my time and energy went into my teaching rather than doing the mathematical research which is the normal focus of attention for a mathematician.
(Admittedly, another motivation for my teaching some of our more applications-oriented courses -- such as Operations Research, Numerical Analysis and Partial Differential Equations -- was simple economic fear. There was a time during the 1980's when all the signs seemed to indicate that it would soon be no longer economically feasible for me to continue in the academic world. It therefore seemed like a very good idea to improve my knowledge of those aspects of mathematics which might be considered useful outside academia. For a while, I was in fact considering taking the actuarial exams, as another tenured faculty member in the department did. And I spent my sabbatical in 1983-84 at Berkeley primarily sitting in on courses in computer science and applied mathematics.)
My compulsive urge to find ways of explaining things to people also led me for quite a while to have a special interest in Math 111 -- our course for elementary education majors. I was fascinated by the task of taking these future elementary school teachers, many of whom were terrified of mathematics and couldn't even add fractions, and finding ways of getting them to understand a little of the beauty of mathematics and realize that they were capable of learning it. (My stated objective was to teach them how to learn and how to be interested in mathematics, rather than covering any particular subject matter.) In teaching at other universities, I had developed a course of my own for elementary education majors. However here at UH the department was committed to another approach developed by our associate chair Ruth Wong which involved having students work in groups and go through a set of notes written by a former faculty member (Stan Kranzler). Although I was skeptical about this at first, I eventually found that Wong's approach worked better than the lecture-style course I had been used to giving before. I made some suggestions which Professor Wong used in thoroughly revamping the Kranzler notes, and gradually I also gradually began supplementing these with some materials of my own which were somewhat more ambitious. (In fact, some of the early versions were quite a bit too ambitious, especially when other faculty tried using them.)
Eventually, though, the incompatibility between the attitude towards mathematics of the College of Education and that of the Mathematics Department became irreconcilable. The educationists decided that Math 100 (the large lecture course that ought to be called General Math) would be adequate preparation for prospective elementary school teachers and Math 111 essentially died.
In all my classes, I stress to my students that what they are learning is more than a set of techniques for doing calculations. Equally important, they are learning a language and a conceptual framework which can be used to express relationships not easily describable without the use of this language. They are also learning certain patterns of thinking and approaches to problem solving which may be useful in contexts having nothing to do with such explicit mathematical paradigms as algebra or calculus. And finally, they are getting an introduction to a subject which is every bit as much a science in its own right as physics or chemistry or biology, a science where there is an existing body of knowledge and also many mysteries which are the subject of current active investigation.
Of course one of the facts of life about teaching mathematics, especially in lower division courses, is that a large part of one's audience is present under duress and considerably less than enthusiastic. In my younger days, when I was much concerned with such things as tenure and promotion, my attention was quite emphatically directed to the necessity of getting along well with those students whose objective was simply to get their three units credit and get out. I was made very aware of the fact that no one ever gets denied tenure for teaching a course that's too easy. I put a lot of energy into trying to force, cajole, and seduce students into actually learning some mathematics. And the outcome was often extremely depressing, since I was fighting against a resistance that seemed overwhelming.
Recently, I have realized that in order to preserve my own self-respect (and sanity) I need to face that fact that I as an instructor cannot create the desire for learning. The desire to learn has to come from the student, and in order to do a reasonable job of teaching I have to presuppose that desire. Now I have become quite upfront about telling students that my only purpose is to offer them the opportunity to learn. I find it more effective to simply tell students that a particular topic is especially important for them rather than trying to force them to learn it. If they prefer to be ignorant, that is their decision and is their problem and not mine.
And if I have a class where almost all the students just sit there sullenly, never ask any questions except ``Is this going to be on the test?'' then I simply accept the fact that I am not going to be able to be enthusiastic about teaching this class most of the time, will often be unprepared, and will screw up calculuations on the board even more often than I usually do. I just accept this fact about myself now and refuse to feel guilty. I don't get paid enough to be a saint.
The evaluations I have received from students have always run the gamut, with usually two or three students in a class being very enthusiastic, another two or three being quite negative, and the rest not really expressing any definite judgement. The overall tenor of evaluations from a particular class rarely seems to have much to do with how much energy I put into the class or how good a job I myself think I've done.
I think that the wide range in evaluations I get has to do in large part with the fact that my main interest is in teaching the underlying ideas of mathematics and giving students a good intuition for these ideas. I think that students who want to understand what mathematics is really about appreciate this and don't have much problem with those times when I write down 2+3=6. Students who have difficulty simply mastering the mechanics, on the other hand, find my approach less useful to them, and the careless blackboard errors that I consider unimportant actually get in the way of their being able to follow my presentation.