Devlin's Angle

June 1999

The Greatest Math Teacher Ever, Part 2

In last month's column, I gave a thumbnail sketch of a person I claimed was arguably the greatest university mathematics teacher ever and asked you to name the individual. If you have not read that article, please do so now, before I give the answer.

Done that?

Ready to proceed?

My nomination for the greatest ever university mathematics teacher is Robert Lee Moore. Born in Dallas in 1882, R. L. Moore (as he was generally known) stamped his imprint on American mathematics to an extent that only now can be fully appreciated.

If you measure teaching quality in terms of the product -- the successful students -- Moore has little competition for the title of the greatest ever math teacher. (Incidentally, he would hate to be described as a "math teacher"; he always insisted on using the word "mathematics" rather than the nowadays more prevalent abbreviation "math".) During his long career as a professor of mathematics -- 64 years, the last 49 of them at the University of Texas -- Moore supervised fifty successful doctoral students. Of those fifty Ph.D.'s, three went on to become presidents of the AMS (R. L. Wilder, G. T. Whyburn, R H Bing) -- a position Moore also held -- and three others vice-presidents (E. E. Moise, R. D. Anderson, M. E. Rudin), and five became presidents of the MAA (R. D. Anderson, E. Moise, G. S. Young, R H Bing, R. L. Wilder). Many more pursued highly successful careers in mathematics, achieving influential positions in the AMS and the MAA, producing successful Ph.D. students of their own -- mathematical grandchildren of Moore -- and helping shape the development of American mathematics as it rose to its present-day dominant position. That's quite a record!

In 1931 Moore was a elected to membership of the National Academy of Science. Three of his students were also so honored: G. T. Whyburn in 1951, R. L. Wilder in 1963, and R H Bing in 1965.

In 1967, the American Mathematical Monthly published the results of a national survey giving the average number of publications of doctorates in mathematics who graduated between 1950 and 1959. The three highest figures were 6.3 publications per doctorate from Tulane University, 5.44 from Harvard, and 4.96 from the University of Chicago. During that same period, Moore's students averaged 7.1. What makes this figure the more remarkable is that Moore had reached the official retirement age in 1952, close to the start of the period in question.

In 1973, his school, the University of Texas at Austin, named its new, two-winged, seventeen-story mathematics, physics, and astronomy building after him: Robert Lee Moore Hall. This honor came just over a year before his death. The Center for American History at the University of Texas has established an entire collection devoted to the writings of Moore and his students.

Discovery learning

Moore was one of the founders of modern point set topology (or general topology), and his research work alone puts him among the very best of American mathematicians. But his real fame lies in his achievements as a teacher, particularly of graduate mathematics. He developed a method of teaching that became widely known as "the Moore method". Its present-day derivative is often referred to as "discovery learning".

One of the first things that would have struck you if you had walked into one of Moore's graduate classes was that there were no textbooks. On each student's desk you would see the student's own notebook and nothing else! To be accepted into Moore's class, you had to commit not only not to buy a textbook, but also not so much as glance at any book, article, or note that might be relevant to the course. The only material you could consult were the notes you yourself made, either in class or when working on your own. And Moore meant alone! The students in Moore's classes were forbidden from talking about anything in the course to one another -- or to anybody else -- outside of class. Moore's idea was that the students should discover most of the material in the course themselves. The teacher's job was to guide the student through the discovery process in a modern-day, mathematical version of the Socratic dialogue. (Of course, the students did learn from one another during the class sessions. But then Moore guided the entire process. Moreover, each student was expected to have attempted to prove all of the results that others might present.)

Moore's method uses the axiomatic method as an instructional device. Moore would give the students the axioms a few at a time and let them deduce consequences. A typical Moore class might begin like this. Moore would ask one student to step up to the board to prove a result stated in the previous class or to give a counterexample to some earlier conjecture -- and very occasionally to formulate a new axiom to meet a previously identified need. Moore would generally begin by asking the weakest student to make the first attempt -- or at least the student who had hitherto contributed least to the class. The other students would be charged with pointing out any errors in the first student's presentation.

Very often, the first student would be unable to provide a satisfactory answer -- or even any answer at all, and so Moore might ask for volunteers or else call upon the next weakest, then the next, and so on. Sometimes, no one would be able to provide a satisfactory answer. If that were the case, Moore might provide a hint or a suggestion, but nothing that would form a constitutive part of the eventual answer. Then again, he might simply dismiss the group and tell them to go away and think some more about the problem.

Moore's discovery method was not designed for -- and probably will not work in -- a mathematics course which should survey a broad area or cover a large body of facts. And it would obviously need modification in an area of mathematics where the student needs a substantial background knowledge in order to begin. But there are areas of mathematics where, in the hands of the right teacher -- and possibly the right students -- Moore's procedure can work just fine. Moore's own area of general topology is just such an area.

You can find elements of the Moore method being used in mathematics classes at many institutions today, particularly in graduate courses and in classes for upper-level undergraduate mathematics majors, but few instructors ever take the process to the lengths that Moore did, and when they try, they do not meet with the same degree of success.

Part of the secret to Moore's success with his method lay in the close attention he paid to his students. Former Moore student William Mahavier addresses this point:

" Moore treated different students differently and his classes varied depending on the caliber of his students. . . . Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. Most of us would not consider devoting the time that Moore did to his classes. This is probably why so many people claim to have tried the Moore method without success."

Another well known mathematician who advocates -- and has successfully used -- (a modern version of) the Moore method is Paul Halmos. He says:

"Can one learn mathematics by reading it? I am inclined to say no. Reading has an edge over listening because reading is more active -- but not much. Reading with pencil and paper on the side is very much better -- it is a big step in the right direction. The very best way to read a book, however, with, to be sure, pencil and paper on the side, is to keep the pencil busy on the paper and throw the book away."

Of course, as Halmos goes on to admit, such an extreme approach would be a recipe for disaster in today's over-populated lecture halls. The educational environment in which we find ourselves these days would not allow another R. L. Moore to operate, even if such a person were to exist. Besides the much greater student numbers, the tenure and promotion system adopted in many (most?) present-day colleges and universities encourages a gentle and entertaining presentation of mush, and often punishes harshly (by expulsion from the profession) the individual who seeks to challenge and provoke -- an observation that is made problematic by its potential for invocation as a defense for genuinely poor teaching. But as numerous mathematics instructors have demonstrated, when adapted to today's classroom, the Moore method -- discovery learning -- has a lot to offer.

If you want to learn more about R. L. Moore and his teaching method, check out the web site: www.discovery.utexas.edu.

But before you give the method a try, take heed of the advice from those who have used it: Plan well in advance and be prepared to really get to know your students. Halmos puts it this way: "If you are a teacher and a possible convert to the Moore method ... don't think that you'll do less work that way. It takes me a couple of months of hard work to prepare for a Moore course. ... I have to chop the material into bite-sized pieces, I have to arrange it so that it becomes accessible, and I must visualize the course as a whole -- what can I hope that they will have learned when it's over? As the course goes along, I must keep preparing for each meeting: to stay on top of what goes on in class. I myself must be able to prove everything. In class I must stay on my toes every second. ... I am convinced that the Moore method is the best way to teach there is -- but if you try it, don't be surprised if it takes a lot out of you."

References

Paul R. Halmos (1975), The Problem of Learning to Teach: The Teaching of Problem Solving, American Mathematical Monthly, Volume 82, pp.466-470.

Paul R. Halmos (1985), I Want to Be a Mathematician, Chapter 12 (How to Teach), New York: Springer-Verlag, pp.253-265.

William S. Mahavier (1998), What is the Moore Method?, The Legacy of R. L. Moore Project, Austin, Texas: The University of Texas archives.


Devlin's Angle is updated at the beginning of each month.
Keith Devlin ( devlin@stmarys-ca.edu) is Dean of Science at Saint Mary's College of California, in Moraga, California, and a Senior Researcher at Stanford University. His latest book The Language of Mathematics: Making the Invisible Visible, was published last fall by W. H. Freeman.