(June 6, 1996)
W.R.R. Transue wrote a Ph.D. thesis in topology under the direction of B.J. Ball, a student of R.L. Moore. He is Professor of Mathematics at Auburn University.
I first encountered this style of class at U. of North Carolina in 1958, as I began graduate school. The place where it made the biggest impression was in J.S. MacNerney's introduction to Complex Analysis. Though I never did come to like MacNerney, I feel that his course had more of an effect on me than any other. I had enjoyed a plane geometry course I had had in a French school in Italy but in two more years of high school and in four years of majoring in math at Harvard I did not really encounter anything that seemed to me of much note. I do recall a couple of experiences in college, which though perhaps not typical, point out some of the reasons I am now mostly in favor of teaching my courses "discovery" style.
In a graduate Algebra course, with about 60 students, Richard Brauer spent 3 days proving a certain theorem. The class dutifully made notes of what he said. Midway through the third day he realized he had misstated the theorem with an inequality backwards. He changed it. The class dutifully made the change, and the lecture proceeded. I don't believe any member of this rather large class at one of our supposedly distinguished universities had the faintest understanding of what the lecturer was talking about. In another class, this one a number theory class with 5 students in it, a young Englishman lectured, his back to the small audience, loud enough so that if we had been 300 instead of 5 we could easily have heard him. One of the students was a nun who was having trouble keeping up, and who felt that it was her obligation to God to do so. She despaired that it was not possible to get the teacher's attention in order to so much as ask a question. It seemed to me that in a class of 5 a teacher might have paid some attention to the students.
Through graduate school both at the University of North Carolina, and at the University of Georgia, many of my teachers were students or grandstudents of Moore and Wall, and although I always felt that, because I had learned some mathematics in college, I had become tainted in the eyes of these teachers, I nonetheless took their courses quite happily. In the non-discovery courses I took, I still felt that understanding of the material on the part of the students was a fortunate accident if it happened at all. There were, toward the end of my stay at Georgia, rumblings of a debate amongst the faculty concerning these pedagogical issues. An opinion of some of the graduate students was that it was important to get started in a "discovery" type course, and once started, to proceed in a more conventional way. I never got to the point where I could listen to a lecture very profitably, unless it concerned material with which I already had considerable familiarity.
As a faculty member at Auburn, I experienced a certain amount of frustration in that I didn't teach the graduate topology sequence until a couple years ago--after I had taught here for twenty five years. The courses I taught in the early time here tended to skirt this subject. Nonetheless I enjoyed teaching intermediate and advanced undergraduate classes in modern algebra and analysis, and graduate courses in algebraic topology and algebra (my graduate algebra course, populated by a motley group of which I recall only an electrical engineer, skittered off into a collection of considerably non-standard ideas in algebra--though I had had a fair amount of algebra in graduate school, I didn't really have enough of an idea of where things ought to be going to provide the sort of gentle leadership in the proper directions that I should have.) I am, in retrospect, gratified by the results of those earlier efforts. Our faculty experimented with an introduction to analysis course at the sophomore level, and although we abandoned the experiment, it developed much later that many of those students had gotten more out of the course than we realized. It has been a bit of a surprise to me to see that some of those students have made a contribution mathematically.
More recently I have been teaching a mix of roughly two parts calculus (I have never--well, hardly ever--tried to teach calculus discovery style) and one part advanced undergraduate courses. These latter I do discovery style, though this has recently been made somewhat more difficult by the fact of a crowd of quite mediocre math students. even so, I am told by my students that in many of their other courses, they and their fellows fail to comprehend anything, and that the standard way of getting through is to memorize a few proofs and not worry about the meaning of any of it.
I mentioned an exception to not teaching calculus discovery style. One summer, years ago in Maine, a young woman about to be headed off to college came to me--she felt her high school math was not adequate for her planned biology major and wanted to supplement it. I agreed to help and we started into calculus, discovery style. She discovered the product rule for derivatives and showed her preparation to be, finally, more than adequate. She went on to major in mathematics and pursue a Ph.D. in statistics. Though she didn't get the Ph.D., she is now working as a statistician and making a salary which led her brother-in-law, a lobster buyer, to wonder just what it was she does for the outfit which pays her.
In summary I would say that it is my belief that a great many people are capable of understanding some mathematics if it is presented to them appropriately. I believe that some people are clever enough to understand mathematics in spite of its being presented pretty much incomprehensibly. There are, I suspect, a fairly significant number, who could be pretty good who never get started. It is them that I think I help.