By John W. Neuberger

Bill and I were together at five different universities:

University of Texas,

as students;

and

Illinois Institute of Technology,

University of Tennessee,

Emory University, and

University of North Texas,

as professors.

There were quite a number of graduate students who took work from both of us. For a number of these we would wonder in a casual way which of the two of us might end up being their supervising professor. There was never even a hint of competition between us.

When students showed up in one of my classes, undergraduate or graduate, I could quickly detect a trademarked attitude in them, if they were veterans of a class with Bill. They would speak carefully and clearly about mathematical matters. They expected hard work from my class. They expected to grow mathematically. For such students my job was mainly to keep good problems in front of them. It is widespread for teachers of mathematics to complain about the preparation of students they inherit from other teachers. There was nothing like that for me with students who had had a course from Bill. He also did not complain about the preparation of students he received from others. He accepted their state as a part of life and set about trying to make improvements.

When I no longer had students that he had worked over, my teaching substantially diminished.

One of Bill's outstanding abilities as a teacher was his clinical approach to bad attitudes in students. I doubt that he took such attitudes personally. They were, rather, a challenge. He would think about the state of individual students and look for ways to help them get rid them of unproductive attitudes. He always worked with good humor and substantial insight into their minds. He routinely would help bring about changes in indifferent students, leaving them with a new focus on a productive life. He claimed that R. L. Moore cured him of a bad attitude. He may well have been right.

Bill was a physics undergraduate major and had a life-long interest in technical and scientifc matters. During graduate school he worked in several jobs using mathematics, adding to his insights into what mathematics contributes below the resarch level. Later as a professor, he was excellent in dealing with departments which regarded mathematics as a service. When an engineering professor, for example, was givin g input about a prospective course such as advanced calculus, Bill asked `What problems do you want your incoming students to be able to work?' He would promise that his students would be able to work the kind of problems that the engineer laid out. A discussion of textbooks and chapters to `cover' were artfully avoided. Bill knew that these `problems' could be dealt with in a way that would still allow him to teach the kind of course he wanted - one that was a hunting ground for new mathematicians and also one that had a substantial intellectual impact upon students.

At one point Bill and some of his fellow mathematics graduate students were taking a graduate physics class in electromagnetic theory. They decided among themselves to ask for clear statements of mathematical matters in the course, almost unheard of in physics classes of the day. According to Bill this created some tension with the teacher, but the teacher came around to their point of view. This well-known physics professor became a supporter of the group of mathematics professors centered on R.L. Moore and H. S. Wall.

Bill and I spent four years together at the University of Tennessee. We were both doing, as usual, our Discovery Method/ Moore Method/ IBL teaching, but in those days this was not accepted by the mathematics department there. One day I had a session scheduled with the chair and indicated to Bill that I was considering resigning. He told me ‘If things get bad enough, you can resign for me.' Later I told Bill that at one point I might have said `That does it. Mahavier resigns.' Bill realized that what he had meant to say was that if I resigned, then I could resign for him too. It was a rare lapse for Bill to make a careless statement about something important.

At the University of Tennessee, Bill and I shared an offce. We were both teaching around twelve hours per week. It was a rare time when there wasn't at least one student in our office. Almost no silence ever in that office. It was a rare opportunity to observe, over four years, Bill's interaction with students. I couldn't help but hear what was going on and I learned a great deal from what I heard.

At one point there was a fellow who worked at Oak Ridge National Laboratory. The fellow was interested in getting a PhD degree and had run through most of the mathematics department faculty as prospective supervisors. It finally fell to Bill to work with him. The fellow had been considered a prodigy in mathematics, but it became clear that he knew about almost everything but could do almost nothing. Bill talked to him and finally found a part of mathematics that the fellow had not heard about - something far afield from Bill's usual interests. Bill devised a set of problems for the fellow to try to solve, but attempted solutions were more a stream of consciousness rather than anything coherent. Finally Bill set the a rule. If the student babbled even once, the session was over. This helped a little, but I am sad to report that this was `one who got away'. He was such a tough case!

Here is a story about Mr. J, who was a member of the first graduate course I taught. It was in the spring of 1958. It was a class in which most of the time was spent listening to students' presentations, my usual practice. However, I think the only thing that Mr. J. said the entire semester was `I am no mathematician.' He did write a nice term paper, which took the place of a final examination. The following summer, Bill was to teach the second half of a special advanced calculus class, special in that it had been designated as open only to those who had taken the first half of the course with Bill in the previous semester.

Mr. J. was a graduate engineering student and wanted an advanced calculus course to satisfy some requirement. Bill was working during registration and Mr. J. was quite persistent in wanting to register for Bill's class. Bill finally asked him more closely about courses he had taken. When Mr. J. mentioned my graduate class, Bill immediately said, `OK, you can take my course', leaving a satisfied but bewildered Mr. J.

Now Bill and I regularly discussed our classes and related stories about our students. A few weeks into his summer advanced calculus class he mentioned one of his students was doing a particularly fine job of proving and presenting theorems. He did not at first mention the student's name but it sounded like Mr. J. I wondered if Bill might take a Polaroid picture of the student, all the time thinking it couldn't be the same Mr. J. It did turn out to be Mr. J, who became both an eminent mathematician and an outstanding teacher. Mr. J. later related that in the first meeting of Bill's class, he thought `Oh no, not another one of these!'