*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

**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.**

__might__
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 13

^{th}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
exp

_{e }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.