## Thursday, April 25, 2019

### Guest Post: Ed Parker on Graduate-Level Math Teaching and IBL

SY :  I’ve had several people sympathetic to IBL methods suggest that as IBL methods become more widespread at the undergraduate level, they will be unnecessary in graduate mathematics programs.  Do you have any thoughts on this?
First, I would suggest that anyone interested in the issue watch the video of John Neuberger’s talk on graduate mathematics education, delivered at the 2007 Legacy Conference. Rather than focusing on technical math education issues, he began with the challenge of turning curiosity into passion for the hunt and then proceeded to relate some of his experiences relative to benefitting from, and implementing, IBL.  Even an impartial judge (which I am not!) would likely judge his career as a teacher a successful one.  In 1977, I became his 13th PhD student and he continued to produce productive PhD’s after his move to the University of North Texas the following year.  A notable aspect of his talk is his view that graduate teaching is a natural adjunct to a mathematician’s research.
At Emory, I took the algebra sequence and the analysis sequence my first year, taking only two courses since I had only a 2/3 assistantship.  David Ford’s introductory analysis course in Lebesgue measure and topological vector spaces was totally IBL.  Trevor Evans’s algebraic structures course was given from course notes.  He had us buy Herstein’s Topics in Algebra and the Schaum outline on group theory as resources.  Dr. Evans talked through the course notes three days a week, stating additional problems as he went, and the fourth day was student presentation day.  A typical presentation by me went like this: I presented.  Dr. Evans would stare at the board, stroke his chin, then put down his pipe and say,  “I suppose you are correct, Mr. Parker, but WHY DIDN’T YOU THINK OF THIS?”  Then he would take the chalk from me and show the class a “good” proof.
As a second-year student, I took the third first-year course, a topology course in Moore spaces given by William Mahavier through IBL.  I had come to Emory with the idea of studying foundations and Dick Sanerib was offering a course on Model Theory that year. However, Emory would not give me credit for the fall quarter due to a course in symbolic logic that I had done at Guilford, so I audited model theory which was done by straight lecture following Bell and Slomson’s text and took John Neuberger’s Functional Analysis and Differential Equations, which was given by IBL, for credit.  I solved three problems that quarter:
There is a single function, call it $f$, so that if $x$ is a number, then $f’(x) = f(x)$ and $f(0) =1$.
There is a single function, call it $f$, and a largest non-degenerate connected set containing $0$ that is the domain of $f$, so that if $x$ is a number in the domain of $f$, then $f'(x)=-f(x)^2$ and $f(0)=1$.
Suppose that $x$ is a number.  Then  $\Sigma_{n\in\mathbb{N}}\frac{1}{n!}*x^n=\Pi_{n \in \mathbb{N}}(1-\frac{x}{n})^{-n}$
Imagine, if you will, knowing that  expe was the answer to the first question but having no idea how to make it appear, or that you could solve the second differential equation by “separation of variables”, but realizing that assuming a solution existed begged the question.  Needless to say, I didn’t think I was doing very well.  Looking back on it, I’m kind of glad I didn’t think of producing a power series from thin air, then proving that it worked since the path I took led through the Fundamental Existence and Uniqueness Theorem.
Near the end of the fall quarter, Dr. Neuberger stated a list of eleven problems that I later found to be, if one took the collective hypotheses and conclusions, the Hille-Yosida Theorem.
At the end of the quarter, I had to decide whether to continue Model Theory or Functional Analysis and Differential Equations.  Neither professor recruited me and I still don’t know why I chose to continue FA&DE.  Did I mention that I didn’t think I was doing very well?
I finished Hille-Yosida in early March.  (It took my classmate only three weeks once he went to work on it!)  Within a calendar year of when I finished Hille-Yosida, I had the theorems that formed the core of my thesis although I had still not passed my algebra qualifying exam.
Heading into my third year, having passed my analysis and topology qualifiers and failing my algebra qualifier and having taken complex analysis in summer school, I was scheduled to take the second level topology course and Dr. Neuberger’s research seminar.   A note of comparison is in order here.  A student of Dr. Evans pursuing an algebra thesis was expected to spend his “year of preparation” reading the pertinent literature.  On the other hand, I didn’t even know I was beginning work on a thesis.  Dr. Neuberger gave me a paper of his on Lie Semigroups and a short paper of Tosio Kato’s that had distilled (brilliantly!) a very long paper of Miyadera’s which had originally proven the dense differentiability of non-expansive semigroups on Hilbert spaces to work through.  I was given no guidance of which I was aware about why or how.
Before continuing on this line, I should mention that I seriously considered dropping out after getting the news that I had failed the algebra preliminary exam.  The birth of our second child the day after news of having failed the algebra prelim rescued me psychologically, but it also added yet another level of family responsibility to my table.  I talked with Dr. Neuberger, who was teaching complex analysis and he said that it was fine for me to use the course time to write a master’s thesis and that Dr. Mahavier had described to him an example I had made in spring quarter of the first-year topology course that would likely provide the substance for the master’s thesis.  I talked to Dr. Mahavier and he agreed to supervise the thesis.  Ironically, Dr. Evans, with whom I had taken the first-year algebra sequence, whose second-level seminars I had attended, and who never seemed to like my proofs, suggested that I should continue.  That, together with my wife’s encouragement, won the day.  My assistantship was renewed and I embarked on my third year.
In the research seminar, I tried to work my way through the two papers.  I had never been good (as in quick) in following other persons’ arguments, but I dutifully slogged my way through, with a cognizance of the structures Dr. Neuberger had appropriated from Hille-Yosida.  The elegance of Kato’s argument made it easy (even for me) to follow, but I realized that I was just verifying details.  This caused me to set out on my own, mimicking Dr. Neuberger by thinking about Hille-Yosida structures in non-linear contexts.  The Cesaro mean (I later found out that was what it was called) was the vehicle to a theorem on non-linear semigroups that I formulated and proved.  In seminar, Dr. Neuberger listened to my argument without changing his expression.  When I finished, he gave me a copy of Glenn Webb’s landmark example of a non-expansive semigroup on a Banach space that contained an open set in its domain where it was nowhere differentiable and asked me to see if his semigroup satisfied the premise to my theorem.  That night my euphoria turned to despair; I could prove that the premise was satisfied, and once I understood Glenn’s example, the verification of the application was dirt simple.  Thus, because it was easy even for me, I was sure my theorem must be no good.  In seminar the next day, Dr. Neuberger asked if I had been able to do what he asked and I mumbled something like “It can’t be any good; it’s too easy,” and showed him my argument.  He became instantly animated and told me, “This is the sort of theorem that theses are built around.”
At this point, Dr. Neuberger gave me some entries into the literature through which I learned about the work of the Japanese school that Kato had consolidated and gained access to the (then) current work of Brezis, Pazy, Crandall, Martin, and Liggett, and Neuberger’s seminal paper which had ignited the Japanese school’s initial successes.  An ancillary aspect of learning to use the library was to make sure that my theorem and its application were original.
There was still the issue of passing my algebra qualifier.  At Emory, the rule was two strikes and you’re out.   I was auditing Mary Frances Neff’s first year algebra sequence which I continued for the year and passed the algebra qualifier on my second try.  The department was kind enough to expand the two 4-hour qualifiers from the year before to two one-day tests.  They kicked me out after 9 hours the first day and 8 hours the second.  Both days, there were still problems I thought I could do.
I finished the following year.
What inferences can I make if I add to the mix what I have learned from talking to colleagues about their graduate school experiences?