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?
Ed Parker:  Since I’ve never taught in a department that offered the PhD in mathematics, I’m probably not the right person to be asking.  But I’m certainly willing to respond.  You just need to understand that any expertise I may have is either historical or based on my limited teaching experience with master’s level students or my experiences as a graduate student.  It is somewhat ironic that the issue has arisen.  When I wrote Getting More from Moore back in 1988, IBL was, at least grudgingly, accepted as a viable option for graduate mathematics education while undergraduate IBL was pretty rare.  Several programs such as Auburn and North Texas had committed graduate programs and others such as Emory and the University of Texas had a visible presence.  When discussing IBL’s possibilities for undergraduates, I remember well being told by multiple persons from multiple places that IBL might work with well-prepared upper level majors, but that the students needed to be “ready for rigor” before it could have a chance to work.  And demonstrating and having the students reproduce what they had seen, then apply the theory to examples was apparently the way to make the students “ready”.  Readiness seems to be a given at the graduate level due to admission requirements.
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.
I entered Emory University’s graduate program in 1973 after a four-year hiatus following my graduation from Guilford College, during which time I had taught secondary mathematics at Bayside High School in Virginia Beach, Virginia.  My draft board had decided public school mathematics teaching was in the public interest and granted me an occupational deferment in lieu of processing my application for conscientious objector status.  (That draft board had not dealt with a CO hearing since my twin uncles during World War II.)  My undergraduate education at Guilford was decidedly IBL.  In the core courses, we mostly read stuff, talked in class about what we read, and then wrote about what we had read and talked about.  With the exception of the calculus sequence and differential equations, none of my mathematics courses had a textbook. Although Mr. Boyd had us buy Heider and Simpson’s Theoretical Analysis for analysis and Greever’s Theory and Examples of Point-Set Topology for topology and Mr. Walker had us buy Birkhoff and MacLane’s A Survey of Modern Algebra for algebra; all were used as reference points and problem sources.  We did not reproduce textbook proofs in any of these courses. 
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?
Broadly, IBL at the graduate level, if the goal of an advanced degree is to certify readiness for original problem solving and ability to pass the mathematical canon of one generation to the next, is a super-charged version of what happens at the undergraduate level.  In contrast to the undergraduate entry, at the graduate level a four-year baccalaureate mathematics degree is in place as well as an entry test result that gives comparisons with other such students and may show some level of breadth and recall of some curriculum.  The main question probably should be, “How can this base best be nurtured?”.  The traditional response has been to give a “graduate level” broadening and strengthening by “mastering” carefully selected texts and/or the arguments of the professors’ lectures to create a base, then to certify the students’ readiness with a battery of barrier examinations.  Those deemed worthy are then given a second dose of the program in more concentrated contexts, usually the advisors’ research specialties, and embark on their own research missions, often as colleagues of their advisors.   In IBL, the students recreate the canon by solving the problems fundamental to it as seen by their professors and realize some breadth as they are held accountable for the work of their peers.  The battery of barrier exams appears as an institutional commitment, but, according to what the students have demonstrated in their individual trips through the canons, the turn toward research is not much more than a continuation of what they were already doing, the major difference being that the questions are chosen closer to the frontiers of the subjects of the courses and the classes are smaller. [WARNING: The above characterization is the author’s and may not represent any consensus opinion!]   As results are achieved, the students are directed to the literature with the goal of broadening their knowledge of others’ efforts and increasing the effectiveness of the students’ abilities to find their own problems.
My prejudices in favor of the IBL model are likely too strongly held to give the traditional model a fair hearing.  However, a late ‘80’s/early 90’s tome out of our professional societies exhorted us to make mathematics education “a pump, not a filter”.  The traditional model, in its insistence on early graduate education being a preparation for its barrier examinations certainly looks like a filter to me.  A colleague (whom I greatly respect) who spent the bulk of his career at an urban state university in the same city as an elite private university once told me, “If we could just get the students that Elite U blows away, we would have a better graduate program than they do.”  Perhaps this was an idle boast; perhaps it was not.  The pump effect stands out in Lee May’s recounting of the “sheep and goats” parable in his book on IBL methods.  Lecture and test does not provide for the opportunity William Mahavier seized to split his topology class and recombine it two quarters later as a class of peers.  At a Legacy Conference a year or two after Robert Kauffman of University of Alabama-Birmingham had died, a former colleague spoke to the gathering.  Robert had fought what IBL practitioners might call “the good fight” for many years, standing on the principle of academic freedom to teach in the way he considered most effective.  The colleague, who admitted he was reluctant to become an ally of Robert at first, recounted how he and many of his peers, often after decrying the lack of preparation of their graduate students, would remark how lucky Robert was to get so many good students in his classes.
Stan has written thoughtfully and insightfully on the coverage issue, which is often used to justify criticism of IBL instruction at the graduate level.  Udayan Darji, at a Legacy Conference in the early 2000’s, used his time at the podium to remind the audience that, if there were gaps in what they “should” know, part of their research time should be spent in filling them.  If one looks at my experience in Neuberger’s Functional Analysis and Differential Equations course, it should be clear that anyone capable of doing graduate mathematics could “master” proofs of the three theorems I proved by reading them with the investment of less than a week’s work time rather than the two months it took me.   Similarly, one could likely slog through the Hille-Yosida treatment of the Hille-Yosida Theorem in less than a month rather than the three-plus months it took me.  But would the appropriation of other people’s ideas, at the expense of nurturing your own, get you to a thesis the following year?
Where then, might an IBL student get her/his breadth?  I would first point out that the library will always be there.  But budding mathematicians need not master its entirety before beginning to think on their own.  Considerable breadth is achieved in being handed the responsibility of verifying the veracity of classmates’ presentations.  I still remember Tom Pate’s proof of a theorem in Fourier Analysis, for which I had a “brute force” argument, using soft analysis.  I have not viewed linear algebra the same since that day.  I owe similar debts to Margaret Francel, Everette Mobley, and Terry McCabe, to name just three.  Lessons I learned from them gave me alternative outlooks when I would work through textbook proofs as I put together my own courses.   And, as one teaches with IBL, the students will direct you to “natural” lines of reasoning.  Accumulated experience as well as preparatory learning can also build a mathematician’s repertoire.
In conclusion, I return to Neuberger’s talk:  Your continued commitment to research will fire your teaching and your teaching will abet your research.  So let your students in on the hunt from the get-go.
No tome of mine is complete without a baseball analogy.  Cy Slapnicka became a legendary scout for discovering Bob Feller.  My question is, “Who could have seen Feller throw and not realize that he would become a star?”  In the modern game, the same could be said for Bryce Harper.  But they are the baseball equivalents of the students that Harvard, Duke, or Chicago recruits for its Putnam team and it is doubtful that any form of instruction in graduate school will keep them from succeeding.  There is another group of students that clearly has big-league possibilities and the minor league experience is expected to build into a body of players producing major league level play.  Certain organizations are known for “growing” these players while others let the cauldron of competition weed out the “weak”.  I would suggest that there is a strong analogy here with schools that admit only the testibly top students and then still blow many of them away.  But, in baseball, these two categories of players are not enough to fill all of the rosters.  Finding latent talent and nurturing it is responsible for developing the rest of the big-leaguers.  There are lots of mathematics majors out there with highly developable tools.  I suggest that IBL  does not inhibit the development of super-stars and is likely superior in the development of a far larger number of students.