William S. Mahavier


William S. Mahavier : A Remembrance
By G. Edgar Parker

My connections to Dr. Mahavier originally occurred through no fault of his. The exigencies of my personal pacifistic convictions juxtaposed with the Vietnam War led to my not going straight out of Guilford College in 1969 to graduate school at the University of North Carolina at Chapel Hill on the very attractive fellowship it had offered me. Despite the degree to which I enjoyed teaching and coaching high school students, when the war ended four years later, I decided I wanted to go to graduate school.

The support for a 26-year old was not forthcoming in the way that it had been for a 22-year old, but J. R. Boyd, my mentor at Guilford, was able to arrange for me the possibility of attending Emory University and Emory offered me a two-thirds assistantship. Mr. Boyd’s reason for referring me to Emory was that “Mahavier is there.”

A great irony is that Dr. Mahavier was absent from Emory during my first year there. Apparently, in the wake of some departmental issues, Trevor Evans had agreed to step down as chairman and Mahavier had agreed to do a year at the University of Houston. To this day, I have no idea what the realities were. But I had only a two-thirds assistantship anyway and that paid for only two courses, so I took analysis with David Ford and algebra with Dr. Evans. And this set the stage for my second year: topology with Dr. Mahavier and functional analysis and differential equations with John Neuberger. I was not aware at that time that they were something of a team.

Fall quarter was an interesting experience. I had done a fair amount of topology at Guilford; J. R. Boyd used the linear order axiom in his course for targeted freshmen and sophomore majors that would, in today’s jargon, be called “Introduction to Proof”, I had done a Moore space seminar and an analysis course built on metric space principles with Mr. Boyd as a junior, and Elwood had had David Cozart and myself use Greever’s Theory and Examples of Point-Set Topology as the launch point for our senior seminar in topology and analysis. Dr. Mahavier never let me go to the board the entire quarter. He never told me why, but I had heard stories of Moore from Mr. Boyd and knew the lore of his not letting people in his classes if they “knew too much”, so I assumed my undergraduate experience was the reason and at least he let me in the course. The only time I resented it was when I proved that the Heine-Borel Property followed from the right properties on a set in a conditionally compact Moore space. Nobody else got that problem that I know of and it didn’t show up on the exam (at least not mine), yet he made me show my argument to him in his office.

An incident in spring quarter personifies Dr. Mahavier as well as any description I can give. I thought I had proven a theorem on components of a set in a Moore space and went to the board to present my argument on a Monday. I was up again on Tuesday and finished (or so I thought) on Thursday, and left class that day having answered all the questions that had been raised. Dr. Mahavier had been his usual inscrutable self, watching the proceeding without facial or physical response, without approval or disapproval, and raising the occasional question when a detail needed clarifying. When we came to class on Friday, he began with “Mr. Parker, can you show me how your proof of Problem __ gets around this example?” Once I digested the example, I realized that my argument had been invalidated. I didn’t feel devastated, but apparently I appeared so (or else I mistook Dr. Mahavier’s handling of the situation as an act of humaneness rather than the equally plausible seizing of a teachable moment for the class). He said, “Mr. Parker, I don’t want you to think that you haven’t proven a theorem. You just haven’t proven this theorem. One day, you are going to be working on something and you’re going to say to yourself, ‘I’ve been here before’, and the work you’ve done here is going to have a nice payoff.” The next academic year, Becky May and I took the second year topology course with Dr. Mahavier. He started Becky off on degree theory and started me off on decomposition spaces. As I worked through the first few problems, I realized that I had proven three of them while “not proving” the problem from the year before. My proof was valid if “upper semi-continuous” had been added to the premise. Later in my professional career as a teacher, I often drew on the lesson I took from this: it is okay to start with your “own” problem sequence, but as the course progresses, tailor your problems to what you have observed about the way your students appear to be thinking. To this day, I do not know whether the flaw in my argument escaped him while I was presenting and he uncovered it after the fact, or whether he left it hanging on purpose in hopes that a student in the class would uncover it in going over the argument.

The aspect of Dr. Mahavier’s teaching that I found most awe-inspiring, and frightening, was his ability to get inside my head, that is, the ability to figure out what I would think next. First, understand that practically all of the time that I spent with Dr. Mahavier as a student was in class. Although he almost certainly cared about each of us as a human being, everything explicit about his behavior in our presence said “I care about you because you may become a mathematician and our personal interactions are for the purpose of making you a mathematician and mathematics is sufficient to adjudicate any discussions we may have.” That this may have been correct would have been underscored by a pronouncement I heard him make more than once (“I do not drink beer with students”), and the story he once related to Becky and myself about a graduate student whom he said he told, “Come back when you see me as the solution rather than the problem.” I accepted this separation; but, because of it, I did not visit his office unless I had a question that was both consuming me and which I thought I could articulate exactly. Thus, although I had nine quarters of work with him (Topology I, Topology II, and the topology seminar), I visited his office only six times as a student in three years. On five of these occasions, the visit ended with his posing a question different from the one I had asked about but which I was eventually able to answer, and this led immediately to my being able to answer the question I had originally asked him. As for the sixth occasion, I still have yet to answer both his question and the one I posed.

As you can see, I idolized Dr. Mahavier as a teacher and the personal distance between student and professor did nothing to diminish this adulation. On at least one occasion, he gave me advice as a mathematician. I had my thesis in hand and I was in Dr. Neuberger’s seminar, following up on a problem in dynamical systems that I had come across consequent to a talk I had heard at an AMS meeting. The direction I had taken had a strong topological flavor and Dr. Neuberger suggested I talk with Dr. Mahavier about it. I described the problem and the ideas I was investigating. Dr. Mahavier listened as he would in class. When I came up for air, he excused himself, walked over to a two-foot pile of what appeared to be off-prints, searched the stack and pulled a paper out from about one-third of the way down. He gazed at the paper for a few minutes, remarked to me, “This is why old mathematicians, even if they have lost some of their edge, know more than young mathematicians.”, and proceeded to make some suggestions about things that, if I knew them, might help. Mind you, he didn’t refer me to the paper or suggest what I read; rather, he directed me to ideas that were clearly pertinent to what I had articulated to him. With his actions, he had clearly told me “Reading is a good thing.”, but before I left, probably to make sure that I didn’t misunderstand the message, he offered me the following advice:

To my mind, the most difficult decision you have to make is when to go to the literature and find out what other people know. Once you clearly understand what the problem is, if you go to literature then, the only way you can improve on what has been done is to be smarter than everyone else who knows what has already been done, because your thought processes will be affected by the way they made their arguments. On the other hand, if you attack the problem the way you found it, you may think about it in a fresh way. But if the problem still eludes you and you care about solving it, the literature is your place to add another idea to get you going again. The tough call is to figure out when you need to be poked from the outside and how much to read. After three weeks? After three months? After three semesters? After three years? Read one paper? Read one book? Research an entire bibliography?

I do not know whether this was personal advice or a blanket attitude about research, but it certainly spoke to my condition. As a mathematician, practically every nice result I have gotten has been from a “non-standard” attack. In addition, when I have researched the areas after the fact to see if my results warranted an attempt at publication, I have to admit that if I had known how “it should be attacked”, I almost certainly wouldn’t have seen what I saw.

Once I completed my PhD, Dr. Mahavier treated me as a peer. I changed from Mr. Parker to Ed the instant I walked out of my thesis defense and over the course of my professional career, I have gone to him many times with inquiries about topological issues with which I was dealing in my research, and he never failed to respond. But in the area where I most admired him, I could never get him to respond. By the late 1980’s, my experiences teaching had given me enough anecdotal evidence to suggest that the gains from using Moore method at practically any level of the curriculum were substantial enough that any perceived drawbacks were relatively inconsequential. I perceived that it was time to try to re-infect the mathematics education literature with a dose of Moore method. So I contacted Dr. Mahavier about the possibility of his writing such a paper. He rejected the idea out of hand, telling me, “It is really quite simple; just listen to what the students say”, and grumbled some remarks about how obvious a choice it was to make. I didn’t let go, responding to his letter with a case for how different my experience with different Moore method teachers had been and why his brand should be the model for a new opening shot. But I was not able to persuade him, or the other persons to whom I floated the idea, and ended up doing it myself. I am grateful for all that Harry Lucas, Jr. has done to promote Moore method. But had his initiatives succeeded only in getting Dr. Mahavier to document some of his own experiences, Mr. Lucas’s work would have been a success. Thanks to Mr. Lucas and the Educational Advancement Foundation, we have Dr. Mahavier’s musings on his work and tape of him in one of his courses, as well as the personal experiences of those whom Dr. Mahavier mentored under the auspices of EAF in their initial implementations of Moore method.

Dr. Mahavier is one of many persons to whom I owe my professional life. But, as a teacher on whose practices one might reflect, his star shines brightest of them all.

G. Edgar Parker
Professor of Mathematics, James Madison University

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Latest revision: 26 April 2011