By John Little
Being selected as this year’s recipient of the
Holy Cross Distinguished Teaching Award is a great honor.
Thank you all for this wonderful recognition. As some
of you know, both my parents were teachers in the Pennsylvania
state college system and that had a huge influence on
me. From high school on I knew I wanted to follow in
the same profession. I feel extremely fortunate to have “grown
up” as a teacher here at Holy Cross in an exceptional
faculty of able teachers and scholars, mentors unusually
devoted to the intellectual and personal development
of their students, and uncommonly thoughtful, conscientious,
and supportive colleagues. I believe this award truly
celebrates all of our achievements, and I am proud to
accept it in that spirit.
Being singled out in this way has been
a great surprise, too (and for some of the same reasons
I mentioned before).
Frankly, it has been extremely difficult for me to reconcile
your expression of esteem with my own perceptions of
where I am in understanding effective teaching, and especially
of the many things that I have not mastered. Rightly
or wrongly, I feel that this award is taken as an indication
that the recipient has to some extent found an “answer.” Most
of the time, though, I have many more questions than
answers. In fact, much of my effort in the past 15 years
or so has been devoted to unlearning the style and attitudes
I learned from many of those who taught me, and that
I relied on at the start of my career.
While research mathematics has made great
strides in the past 25 years, most mathematicians would
agree, if
they were being honest, that our educational efforts
have not been successful to anything like the same degree
over this time (if indeed they ever have been). Nationwide,
there is concern about an ongoing decline in the number
of students pursuing undergraduate majors in mathematics
and a growing realization that different approaches to
teaching may be necessary to capture and retain the interest
of today’s students. Moreover, our understanding
of how to harness the available computational power for
educational purposes is, to be charitable, still in its
infancy.
However, trying to change college and
university mathematics teaching is a bit like trying
to steer the Titanic. Mathematics
teaching has a tremendous amount of inertia and many
entrenched attitudes and competing pressures actively
resist change. At colleges like Holy Cross, of course,
teaching is a much higher priority than it is at many
universities. But in order to get tenure, college mathematics
teachers must also be (at least to some extent) research
mathematicians, and the ways we are formed as professionals
have often been unhelpful in reconciling the sometimes
conflicting demands of teaching and research.
"TRYING to change
college
and university
mathematics
teaching is a bit like trying to
steer the
TITANIC."
One unfortunately common attitude among
university mathematicians is exemplified by the following
quote from the Cambridge
don, Godfrey Harold Hardy, one of Great Britain’s
leading mathematicians in the first half of the 20th
century: “I hate ‘teaching,’ and have
had to do very little, such teaching as I have done having
been almost entirely supervision of research; I love ‘lecturing’ and
have lectured a great deal to extremely able classes;
and I have always had plenty of leisure for the researches
which have been the one great permanent happiness of
my life.’”
This candid and most unapologetic sentiment
comes from Hardy’s famous little book titled A Mathematician’s
Apology, in which he claimed that one justification for
his life’s work in mathematics was its “uselessness.” He
meant by this that even though he thought his mathematics
was not likely to be of practical utility, it was at
least “clean and innocent”—not likely
to be of harm to any one. He would probably be horrified
to learn that today the useless number theory he especially
loved is the basis for encryption systems that make it
possible for you to communicate securely over the Internet
with an online merchant, or for terrorists to do the
same thing with one another. The biggest single employer
of number theorists in the United States (and of mathematics
Ph.D.’s more generally) is the National Security
Agency.
Hardy’s views here are extreme, of course. But
for many of the mathematicians I know, myself included,
what drew us to the subject was, if not its “uselessness,” then
the almost otherworldly, crystalline beauty of its logical
structure and the excitement of solving challenging problems
within that structure. To some, mathematics can even
come to seem like a refuge from the “messiness” and
illogic of interacting with other people, and the intractability
of many problems that matter in the real world.
It should come as no surprise that these
attitudes carry over into traditional approaches to mathematics
teaching,
starting right from the single most influential mathematics
book ever written, the Elements of Euclid. Much of our
present conceptions of what mathematics is and what it
is about (especially the idea of axiomatic, deductive
presentations of mathematical results) can be traced
directly back to Euclid. Indeed, some historians have
called the Elements the second most influential book
in Western civilization. And up until about 15 years
ago, if you had walked into almost any high school geometry
class, you would have found the textbook, the selection
of topics, the style of presentation, and so forth to
be recognizably derived from Euclid.
As a ninth grader in Phil Brennan’s geometry class
at Mansfield High in Pennsylvania, I absolutely fell
in love with that kind of mathematics, and that experience
(plus the five or six other courses I took from him)
is certainly the main reason I am a mathematician. Mr.
Brennan was one of those teachers who leave indelible
impressions (not always positive ones) on generations
of students. On a superficial level, he might have been
the model for the photo next to the dictionary entry
for “nerd.” On a deeper level, he really
knew the structure of geometry proofs and could get enthusiastic
(in his own understated way) about a clever construction
or an unexpected way to solve a problem. He awoke a passion
for mathematics in some of his students, but lost or
alienated many of the others. I never knew him well enough
on a personal level to know whether that discouraged
him or whether he just accepted it as a part of being
a mathematics teacher.
Almost all of the courses I took in college
and as a graduate student were also taught in a deductive
Euclidean
style: the inexorable succession of “definition— theorem—proof’” that might be familiar.
One of the teachers who influenced me the most here (and,
I have to say it, in some of the least productive ways)
was the longtime chair of the Haverford College mathematics
department, Dale Husemoller. Dale always looked as if
he had just been roused from a sound sleep, but he could
give a beautiful hourlong lecture on any topic from measure
theory to cohomology operations at the drop of a hat.
All he asked you to do as a student was to learn the
proofs of the theorems he carefully wrote out on the
blackboard and repeat them on the exams. I was very good
at it, and thought that was what mathematics teaching
was supposed to be for a very long time. Dale’s
approach to teaching was also an extreme case. You could
say that it was extremely successful in its way. Haverford
alumni are overrepresented among Ph.D. mathematicians
(as indeed are Holy Cross alumni).
What is wrong with Euclideanstyle mathematics?
I would argue that the principal lesson that those who
have thought
seriously about these issues have learned is that it
is a mistake to rely too much on the final product of
a mathematical theory (the Euclidstyle deductive presentation)
as a model for how that body of mathematics should be
taught and learned. When they only see the final product,
students have a difficult time appreciating the process
by which human beings actually do mathematics—the
guesses, the false starts, the backtracking, the exhilaration
of finally getting a difficult argument to work, and
finally the sometimes even more difficult job of explaining
your ideas in a comprehensible way to another person.
They don’t understand that they can participate
in that process themselves.
When I began teaching in graduate school,
and in my early years at Holy Cross, I was mainly concerned
with giving
clean, “elegant” lectures that would efficiently
impart a highlevel understanding of the subject to my
students. More recently, stimulated by my colleagues
Dave Damiano and Margaret Freije, I have gained a fair
amount of experience in two complementary approaches
to structuring classes—computer laboratories and
smallgroup collaborative discussion exercises—that
I now use together with lectures. Both laboratories and
discussions have given the students more responsibility
for what happens during the class meetings. I think they
have also helped prepare students better for the ways
they will need to use mathematics and perhaps even develop
new mathematics themselves in their future careers. To
put it a different way, I believe that my approach to
teaching has definitely moved in the direction of making
some of the assignments in almost every course more like
really doing mathematics. But make no mistake, constructing
appropriate discussion questions and truly educational
lab activities is hard work and I cannot claim that I
have been uniformly successful there.
To summarize, probably the most important
things I have learned about teaching are first that students
learn
best when they are as actively and personally involved
as possible, and second that learning happens most readily
in a general atmosphere of encouragement of students
as individual human beings. To be honest, though, it
is in this area that I think I have the most work left
to do. There have been times when I feel I have been
too focused on the ideas I find interesting in mathematics.
"LEARNING happens
most readily in a general
atmosphere of encouragement
of
students as INDIVIDUAL
human beings."
Before closing, I want to tell one more
story related to teaching that means a lot to me and
to correct an
oversight. Toward the end of the spring semester of my
fourth year at Haverford, I was asked to meet with a
panel of faculty members who were charged with awarding
College honors. One of the members of the panel was Brad
Cook of the French department, who had taught five different
semesterlong French literature courses I had taken.
When it was his turn to ask questions
in the interview, knowing that I wanted to be a college
teacher, Brad gave
me a scenario—I was a teacher of an average French
class assigned to discuss Molière’s Le Malade
Imaginaire. How would I structure the class discussion?
What would I do to pique their interest?
I gamely proposed: “First, I would have the class
see a production, point out the physical comedy and farcical
situations to show how funny so much of it is (or can
be).”
He responded, “Well, what if they don’t ‘get’ that
aspect of it, what if they still don’t see why
they should be bothered?
I pressed on, suggesting a number of other
ideas. Each time, Brad responded with the same question, “What
if that doesn’t work?” Eventually, we came
around to the idea that it might very well be the case
that nothing would work; successful teaching depends
both on the skill and dedication of the teacher and on
the receptivity of the student.
I am not sure that I appreciate the full
wisdom of what he was trying to tell me even now. I do
know that I have
only very rarely encountered that sort of stalemate
situation here. Almost all of the students I have taught
at Holy
Cross have responded very well to what I have tried
to do in my classes, and I know I wouldn't be here giving
this talk without them. I would like to take this opportunity
to thank them also.
