American universities do a rotten job of teaching mathematics. That's the message of Morris Kline's book.
According to Kline, the reasons that we do so badly teaching mathematics have to do with what is taught, how it is taught, who teaches it, what books it is taught out of, and why it is taught.
Many of Kline's complaints will be fairly familiar to anyone who has had much to do with universities: professors neglect their teaching in favor of research, their interests are much too specialized and divorced from the real world, they know very little about how to teach, and they in large part entrust the teaching of elementary subjects to graduate assistants.
To a large extent, Kline's book expresses my own thoughts on the teaching of mathematics. (One thing which he somehow seems to have missed is the phenomenon of beginning classes taught by graduate assistants who can't even speak intelligible English.) However Kline is so hell bent on finding absolutely nothing good any any aspect of the situation that I think he makes himself an easy target for those who want to discredit him.
I'm not qualified to talk about American mathematical education at the university from an authoritative, objective perspective. (But then neither is Morris Kline, and he doesn't let that stop him.) What I can talk about is my own experiences -- certainly far from typical -- as a student, graduate student, and professor at several universities.
I started learning mathematics in high school, out of books. I was especially drawn to the editions put out in paperback by Dover Books, because they were cheap. It was many years later that I finally realized that the reason Dover could publish books so cheaply was that those books were dreadfully out of date. For me, though, they were very exciting books.
There was a book on vector analysis, which seemed like quite useful stuff and not too difficult, though I did have a hard time getting my mind around the idea of differentiating and integrating vectors.
There were Knopp's three volumes on The Theory of Functions. I knew what a function was, but I couldn't imagine how one could create a theory about them, and I was desperately curious to find out. (I didn't actually buy these until after my first year of college.)
Likewise I was extremely curious about a book by someone named Kamke, as I recall, on the Theory of Sets. I couldn't imagine how one could find much of anything interesting to say about sets. However, I think I never actually got that one at all. (I couldn't afford to buy very many, even at Dover prices.)
I didn't completely understand these books, but I worked very hard at them. My desire to understand as much as possible was passionate.
One book I devoted an immense amount of effort to was Hermann Weyl's Space, Time, and Matter. The title seemed to promise that if I could just master that one book, I'd understand the whole universe. Besides, it was largely about the theory of relativity, which I very much wanted to understand, because at that time it had the reputation of being the most difficult subject in the world. (Later on, as a graduate student and mathematician, I would make a valiant attempt, extending over several years, to understand algebraic geometry, for much the same reason.)
The difference between the mathematics I started learning at age 16 and 17 from these little Dover paperbacks, and the mathematics I would later learn in graduate school was as drastic as the difference between the art of the Impressionists and that of the Cubists and Abstract Expressionists. Or the difference between the literature of Dickens and Conrad that of Joyce and Eliot and Pound.
Between 1957, when I graduated from high school, and 1966, when I enrolled in graduate school, the mathematical world, along with the rest of the academic world, changed in other ways too. Before 1960, being an academic was a calling, something involving a major sacrifice, which one did because one cared passionately about one's subject. And, as far as I could tell as an undergraduate at the University of Arizona, the number one concern of academics was with their teaching. Grants were essentially unknown. Colloquia were so unusual at Arizona that they were announced in the undergraduate mathematics classes, and I and some of my friends went to some of them, assuming that there would be something of interest to us.
By the time I got to graduate school in 1966, being an academic had become simply a career choice. If one liked going to school, and didn't like the idea of a forty-hour per week job, then one went to graduate school. (Another major consideration was the Vietnam War. Having a wife and daughter had always kept me safe from the draft, and by the time I enrolled in graduate school I was already over the age limit, but many of my male friends were quite frank about the fact that they had no intention of getting their degree until they were 26 and no longer needed the student deferment. A few years later, the lottery would be introduced and the draft would become a little more sane.)
By 1966, professors' salaries had become quite good -- it was taken for granted that getting a Ph.D. would pay off in financial terms. (Those Ph.D.s who weren't good enough to get a job at a major university could always get one in industry, where the money was even better.)
As a freshman at Johns Hopkins, I was nominally an engineering major. I took analytic geometry and all three semesters of calculus my freshman year. Because of the way they were scheduled, this involved my taking Calculus 2 before Calculus 1, but that was not a problem, since I already knew differential calculus fairly well. Calculus at Hopkins was at that time taught by a professor named Morrill, who taught out of his own text and seemed absolutely passionate (if not fanatical) about his teaching.
When I went to the University of Arizona after my one year at Hopkins, I wanted to take only mathematics that was useful -- vector analysis, partial differential equations, matrix algebra. (Somehow I had missed out on ordinary differential equations, which hadn't been part of the calculus sequence at Hopkins, so I taught it to myself out of Schaum's Outline.)
The Math Department had recently added a requirement that all majors had to take the first semester of real analysis, and that was actually the first math course I took at Arizona. The class was mostly full of graduate students, who had an extremely hard time with all the epsilon-delta proofs, but for me it was mostly just a review of familiar material I'd learned out of Knopp. We were all, though, very puzzled by the question of just what all that stuff might be good for.
Matrix Algebra, which I took (along with two other Math courses) my second semester at Arizona, was taught by Evar Nering. He and Harvey Cohn had the reputation of being the top mathematical minds in the department, and the most demanding teachers. Nering anounced to us that he would not be using the text by Franz Hohn from the bookstore, but instead would be teaching us a subject called Linear Algebra from a dittoed set of notes he would hand out. I found Nering's talk about vector spaces and linear transformations to be quite fascinating, and I retained a fascination with linear algebra for years later. However I didn't actually learn the material very well, because for the first time in a mathematics course I was having a hard time forcing myself to actually read the book. Fortunately, Nering was so nervous about the fact that he was deviating from the prescribed curriculum and using this new abstract approach that he gave us essentially trivial tests, and I wound up with an A in the course. (The following summer, I did read through his notes and discovered the material to be for the most part fairly easy. I couldn't understand why I hadn't been more conscientious about reading the notes during the semester.)
By my senior year, I was aware that there was a sea change going on in the mathematics program at Arizona. Some new faculty had been hired who seemed to have a different, very modern attitude toward mathematics. The graduate program was being expanded and a number of new courses were added to the curriculum, including one which I was especially excited by: topology.
I had heard that topology could be considered to be the foundation of all mathematics, but I hadn't been able to learn very much about it. Some books described it as geometry done on rubber sheets, which seemed to me mildly interesting but hardly of fundamental importance. For the most part, though, I couldn't make heads nor tails out of the books on topology in the library.
I enrolled in topology the first semester of my senior year. Up till then, I had never taken notes in a mathematics course. I would just sit in class and follow what the professor was saying in the textbook, making occasional notes in the margin. But Louise Lim (a logician) came into class the first day, didn't tell us anything of what the subject was about but just started putting theorems and proofs on the board. I started leafing wildly through my book, trying to figure out where she was, but what she was doing didn't seem to be in the book at all, so I pulled out a notebook and frantically tried to catch up. She started talking about open sets and closed sets, which should have been familiar material from real analysis, but what she was doing didn't seem to relate to the concepts of openness and closedness I knew about at all. And what she was saying didn't seem to have anything to do with geometry, on rubber sheets or otherwise.
In this topology course,
I was encountering for the first time a very different attitude
towards mathematics.
Kline quotes the following words by Marshall Stone, approximately in 1960
(i.e. just a year or two before I took Louise Lim's topology course):
When I transferred to U.C. San Diego,
I found myself among a group of very bright students,
who were very enjoyable to hang out with,
and a group of professors,
many quite distinguished and also in large part fun to just talk to.
To my delight, I found that everybody at UCSD thought in terms
of categories and functors.
The graduate algebra course there was taught out of the new textbook
by Serge Lang and constantly used most of the ideas
which I had learned in the library at the University of Maryland
from Peter Freyd's little book on category theory.
When I left UCSD with a Masters degree to go teach at
Humboldt State University (in Arcata, California),
I found myself in a department comprised mostly of very young faculty
with degrees from excellent schools
(including several Berkeley Ph.D.s),
whose interests in mathematics were very much like my own.
We proceeded to revamp the mathematics curriculum.
We tried teaching calculus using an approach based on Dedekind cuts,
developed by one of the faculty there named Marshall Ruchte.
I taught the upper division course in set theory,
required for all mathematics majors,
using a book by Monk which was completely axiomatic and rigorous.
I taught multi-variable calculus out of a book by Lang
based on differential forms.
In the one-quarter course in Linear Algebra,
I covered more material (and more abstractly)
than Evar Nering would ever have dreamed of --
much more than I would subsequently ever manage to cover
in any semester course.
Kline writes:
Actually, the vaunted value of deductive reasoning
is grossly exaggerated.
In daily life, business, and most professions,
deductive reasoning is practically useless.
The liberal arts values of mathematics
are to be found primarily in what mathematics
contributes to other branches of our culture.
Mathematics is the key to our understanding
of the physical world; it have given man the conviction
that he can continue to fathom the secrets of nature;
and it has given him power over nature....
In fact, the entire intellectual atmosphere of our time,
the Zeitgeist, has been determined by mathematical achievements.
These are the liberal arts values of mathematics
and should constitute the essence of a liberal arts course.
I think that there will always be a conflict between those,
on the one hand, who see the value of mathematics primarily
as a set of powerful tools and a theoretical framework
for the physical sciences and engineering,
and on the other hand those who value mathematics
primarily for its own intrinsic beauty and depth.
In my own teaching, I have always been primarily interested
in trying to teach students mathematical reasoning
and showing them that there's more to mathematics than just solving
equations.
On the other hand, I have to remember that most of my students
are only taking mathematics courses in the first place
because they have been convinced that mathematics will give them
useful tools, and that for many of them the occurrence of the words
``mathematics'' and ``beauty'' in the same sentence is simply bizarre.
I do think that there's a lot of merit in what Kline says.
On the other hand, one can also see it as a manifestation
of an attitude that I've seen over and over again from mathematicians:
namely, the attitude that whatever field a particular mathematician works in
is the most important part of mathematics
and the one which ought to be emphasized in teaching.
An algebraist, such as myself, will see abstract algebra
as of vital importance to students.
But according to a logician, mathematical logic will be
what will benefit students most.
Kline is someone who has specialized in certain parts of applied
mathematics, and also in the history and philosophy of mathematics,
and so naturally these are the parts of mathematics
which he thinks should play the leading role in the curriculum.
His condemnation of abstract mathematics would be more convincing
if he had actually done some work in those areas himself
and had given it a fair chance to show him its value and beauty.
In the course for elementary education majors at Humboldt State,
instead of taking students laboriously and thoroughly through the basic
mechanics of arithmetic
I taught them about sets, one-to-one correspondences,
relations, functions.
I developed the rational numbers from the Peano Axioms
and then went on to the real and the complex number systems.
These elementary education majors,
many of whom were not very competent even in simple high school algebra,
were real troopers,
and for the most part they had complete faith that all this abstract
stuff was part of the ``New Math'' that they would need to know
to teach in elementary schools.
The axiomatic development of the real numbers from the Peano axioms
was something that had been highly stressed
in my graduate studies at the University of Maryland,
so it was natural that I would want to share it with my
elementary education majors at Humboldt State.
Klein sees this topic as highly artificial, contrived, and complicated,
and says that it destroys students understanding rather than enhancing it.
I'd have to say now that I mostly agree with Kline.
Developing the real numbers from the Peano axioms is a tour de force,
and is a good illustration of a particular kind of thinking.
However it is what might be called a set piece,
or it might be compared to the thoroughly rehearsed performance of a
musical work by a musical virtuoso,
or some very carefully choreographed number in a musical comedy.
It is not really that representative of the kind of thinking
one actually does in developing new mathematics.
As Kline says (overstating his case, as usual),
``The deductive presentation of mathematics is psychologically damaging
because it leads students to believe that mathematics
is created by geniuses who start with axioms
and reason directly and flawlessly to the theorems.''
As far as I can see,
being able to rigorously define the real numbers is not very useful
even to graduate students and mathematical researchers.
However at least one can defend the axiomatic development of the reals
on the grounds that it is pretty.
And at least it does show that one can prove an enormous amount
from an extremely minimal set of axioms.
I don't know of anything at all that can be said in defense
of another idea that was a universal fetish
in the teaching of mathematics
during the time I was a graduate student:
namely, the idea that a function is a set of ordered pairs.
Not only is this idea of no actual use, as far as I can see,
but in a way it's actually dishonest:
I have never known any mathematician who actually thought of functions
in terms of ordered pairs.
As far as calculus students are concerned,
I have long since resigned myself to the fact that no matter how long
a song and dance I give them,
they are going to think of a function as a formula.
And why not? It's a way of thinking that will get them through calculus
with no trouble, and the few that take more abstract courses later
can adjust their thinking at that time.
In a course like differential geometry or topology,
rather than making a big deal about the formal definition of a mapping,
I give them the example of
the correspondance between a portion of the surface of the earth
and a map of that portion in a book or hanging on a wall.
I certainly wish somebody had given me an explanation like that
before I studied complex analysis!
My teaching at Humboldt State was a very heady and rewarding time for me.
And mostly my students were willing to come along with me
as far as I asked them to go.
However occasionally, as for instance after I had managed to get
the math majors in one of my classes
to understand the difference between a countable and an uncountable set,
a student would stop me short by asking,
``Just what can this be used for?''
Mostly they'd be satisfied with my vague assurance
that this was really essential stuff for serious mathematics,
but it bothered me that I didn't really know the answer to that question
myself.
When I thought about it,
I realized that certainly countability is an essential concept
in measure theory and probability and topology,
but it occurred to me that maybe it was a bit bizarre to be
teaching this stuff to sophomores and juniors.
(And in fact, I seem to remember teaching my elementary education majors
about uncountable sets as well.)
The involvement of faculty in specialized research, Kline says,
tends to influence their teaching towards the specialized and modern,
instead of what students really need.
Furthermore, the interests of faculty are so distant from the material
appropriate for students,
that faculty have a difficult time understanding the needs of students
and are unable to communicate in language students can understand.
They have little insight into how to motivate students.
At Humboldt State I started to realize
that the biggest problem students had with my teaching
was not their lack of knowledge or manipulative skills,
but the enormous gulf between the language I used
and the kind of language that made sense to them.
Things that I took completely for granted,
like the use of subscripted variables,
were completely alien to some of my elementary education majors.
I once gave them a problem that started out with something like,
``Define a function f(x) by ...''
and had students say, ``We don't know how to do the first step.''
Since the first sentence was an imperative sentence,
they thought that they needed to actually write down an answer for it.
They didn't understand that it was just a way of saying,
``In this problem, f(x) will stand for the function which is defined as...''
Even in my classes for math majors,
if a problem began, ``Prove that for any prime number p,''
some students would interpret this as meaning
that they could choose any specific prime number they wanted for p.
At a point where most of my elementary education class
seemed to understand the ideas of identity elements
and inverses fairly thoroughly,
for a quiz I gave them the addition table for a small finite arithmetic
in which the elements were represented by capital letters
and asked them to find the additive identity
and also the additive inverses for several of the elements.
I expected this to be extremely easy,
but most of them complained bitterly
because they'd never had a problem like that before
and had no idea how to do it.
At the time, schooled in modern mathematics as I was,
it was bewildering to me that students could know the formal
definition of a concept and yet be unable to recognize
the occurence of that concept in a concrete example.
I was frustrated that many of my elementary education majors
couldn't manage to learn even the very simplest proof.
I would announce ahead of time a ten-minute quiz,
for which they would be required to write down a very specific
two or three line proof that had been given in class.
On the day of the quiz, more than half the students
would be unable to write down any remote semblance of the proof.
Some would turn in blank sheets of paper.
Later on, talking to some students in my office,
I realized that the extremely simply proofs I wrote on the board
made absolutely no sense to them.
Because of this, I started realizing that there is a big difference
between a proof and an explanation.
I started to realize that it's possible for explain general ideas
in terms of specific examples,
and that this made sense to a lot more of my students than proofs did.
For instance when I first started teaching, if I wanted to explain
how to find the sum of the first n integers,
I would say
Did I serve my students badly? Kline would certainly have it
that I did. It was if I were teaching a course in Art History,
and started the course with Van Gogh and then moved quickly
to Jackson Pollack, De Koonig, and Jasper Johns,
devoting the last half of the course to the 1970's and 1980's.
And yet I think that I was right in wanting to show my students
something of what contemporary mathematics is really like --
to show them that there's more to mathematics than just manipulating
equations.
I think that to some extent, students always benefit
by being taught something that the teacher is really
enthusiastic about.
And when I compare the modernistic and very overambitious courses
I taught at Humboldt State to the much more routine courses
I've taught here at Hawaii,
I can't help but think that the Humboldt State students
got the better deal.
When I think back on it now, I realize that what really interested me,
often to the point of obsession,
was learning mathematics.
Once I managed to really understand some existing piece of mathematics,
I would then become fascinated with reformulating it in various
ways and seeing its connections with other parts of mathematics
I knew and explaining all this to other people,
which is something I could do best in writing.
The various sets of notes on my mathematical home page
reflect this interest fairly well.
There's almost no new mathematics there,
but there's a constant attempt to take routine results
that were part of the classes I've taught over the years
and explain them in what seems to me like better ways.
Morris Kline complains,
When I look back on it now,
it seems that learning to teach at Humboldt State
was much more satisfying to me
than becoming a research mathematician would later be.
In that teaching, as in my web page,
I seem to be fulfilling my purpose in life:
to learn as much about the world as possible,
and then figure out how to explain it to others.
However that was not a path that was open to me.
Humboldt State was not willing to give me a regular position.
No matter what the Mathematics Department might have thought
about my teaching,
the concern of the administration was to be able to boast of
having the maximum possible number of Ph.D.s on the faculty.
Going back to school and getting a Ph.D. seemed to be the
only possible choice for me.
(I couldn't take leave from Humboldt State to do that,
because I didn't have a regular position.
In any case, by the time I graduated from New Mexico State,
the temptation to try a career as a research mathematician
would have become too strong.)
I think that if I could have been a mathematical scholar,
devoting my life to making known results in mathematics
more understandable and accessible to a wider audience,
I could have remained satisfied to be a mathematician.
But that opportunity never seemed to be there.
There is almost no journal where expository articles on mathematics
are welcome,
and such articles are considered of little value
by those who evaluate faculty for tenure and promotion.
Even the publication of books is often regarded as an activity
of secondary importance compared to the publication of research articles.
And, as Kline states, those books that do get written
are almost always directed at a super-specialized audience.
What's really strange is the contrast between Mathematics Departments
and English Departments in this respect.
In the English Department, one finds that the vast majority of the faculty
are scholars, and at most universities
the creative people in the English department
are second-class citizens.
(The University of Hawaii is a happy exception in this respect.)
And yet one would certainly think that there is a much greater need
for scholarship in mathematics than in literature.
I've sometimes found a great deal of value in literary criticism,
but still, the average person is pretty capable of reading
Mark Twain or Shakespeare and figuring out what's happening
for himself.
But even I, a mathematician, find it extremely difficult
to pick up a monograph on a mathematical topic outside my own field
and make any sense of it.
And for most physicists and engineers,
most mathematics done in the past fifty years
is an impenetrable mystery.
In fact, what has happened is that physicists and engineers
have redeveloped those parts of contemporary mathematics
that they need and written their own books
using their own terminology.
It seems to me that there's a real need for people
who will take the mathematics that has been done in the past thirty years,
some of which is actually quite worthwhile,
and explain it in a fashion that will be understandable
by those to whom it would have value.
For me, as a graduate student,
the combination of these words was quite terrifying.
Especially after my unsuccessful attempts at getting started
on a dissertation at UCSD,
the likelihood that I could actually discover something
that nobody else knew seemed to me quite slim.
What one does to write a dissertation is to work on a
``problem.''
And no, a problem is not an equation to solve.
And it's not like a homework problem in some course.
A problem is a question that nobody knows the answer to.
And the answer will take the form of a number of theorems,
plus preliminary theorems that will be required to prove these main
theorems,
and perhaps some new concepts needed in order to state one's results.
(I am really a lot more pleased with the concepts
I have introduced into my field than with the theorems
I have proved.)
For more than three centuries, ``Fermat's Last Theorem''
was an open problem.
``The Four Color Problem'' was another.
(``Can every possible map be colored using only four colors
in such a way that neighboring countries will never be the same color?''
It may sound like geography,
but to a mathematician it's geometry.)
Unfortunately, it turned out that the answer to this first problem
was already known.
In fact, I had somehow managed to overlook it in the
introduction to the paper my advisor had had me read.
Still more unfortunately, I didn't discover this until after
I'd taken up five or six hours of his time discussing the problem
with him.
Luckily, though, this didn't piss him off nearly as much
as I'd expected. (``These things happen.'')
The second problem I worked on, and did eventually solve
(shortly before Christmas, I remember)
takes longer to state.
``If A1, A2,... is a countable collection
of p-groups, each of which is a direct sum of countable groups,
is the torsion subgroup of their direct product
also a direct sum of countable groups?''
(Why would anyone care? you wonder.
Well, there was a reason why my advisor had once wanted to know that,
but the main thing was that it was a problem that my advisor
and some of the other faculty at NMSU had been unable to solve,
so they were impressed when I managed to solve it,
even if the result was of minor significance.)
I wrote my dissertation in the area of abelian group theory
not because the subject was especially fascinating to me,
but because that's what the algebraists at NMSU specialized in.
Whether the subject would fascinate me or not
was something I could only know after I'd done some research in it.
(It's basically the same principle as the fact
that you can't actually read the course description in the catalog
for a mathematics course unless you've had that course.)
As it turned out, certain aspects of the theory of abelian p-groups
without elements of infinite height were rather repulsive to me.
Furthermore, the subject was pretty well mined out.
To my enormous good luck, though, in the spring semester
a young assistant professor (David Arnold) offered
a course in another sub-specialty of abelian group theory:
finite-rank torsion free groups.
I found this much more to my taste
and most of my research for the rest of my mathematical life
was done in this area,
where I was able to make some major breakthroughs.
From my four hundred letters, I got three favorable responses
showing possible interest in giving me a regular position.
The least undesirable of these was from the University of Kansas.
I also got an offer of a temporary two-year lectureship
at the University of Illinois.
I was shockingly ill-informed as to what the stakes were
in making a decision between the Kansas offer and the Illinois one,
since I knew virtually nothing about either school.
In any case, even if I'd realized what a marvelous math department
the University of Illinois has,
I probably still would have chosen the permanent job
at Kansas,
because my primary concern was with being able to support
my wife and daughter (who was now in junior high school).
I'm sure the thought also crossed my mind
that in case I turned out not to be the superb mathematician
that the faculty at New Mexico State seemed to think I was,
it would probably be easier to get tenure at Kansas than at Illinois.
(Although not that much easier, as I later realized.
The math department at Kansas was actually quite snooty,
showing the clear symptoms of an inferiority complex.
They did like me, though.)
And in any case, the chairmen at Kansas and Illinois
worked out a deal whereby I would accept the regular position
at Kansas, but then after a year
take a leave of absence and spend a year at Illinois.
This arrangement, as it turned out,
made an enormous difference to the way people looked at me
when I got to the University of Illinois,
because during my first year at Kansas I proved some results
that the mathematicians in my field found enormously impressive.
The chairman at Kansas told me that unless something
quite exceptional happened,
I could feel pretty secure in my job for three years.
After three years, though, it would be important
that I had written at least a few reasonably good papers,
otherwise my contract would not be renewed.
I was also told quite bluntly
that I shouldn't put a whole lot of time and energy into teaching
my classes, since at this point in my career
what was really important was my research.
I wasn't at all sure that I was capable of doing mathematical research
without an advisor to help me.
Despite having written a dissertation and one additional paper
that people thought highly of,
I really didn't understand how one goes about finding an idea to work
on and how one gets started working on a problem.
(I never did learn this.
Starting on a new piece of research has always been a hopeless act
of desperation for me, except in those few cases
where I had a co-author.
A lot of the reason for my success in mathematics
has been perversely the fact that I've always worked on problems
that were unreasonable and that most mathematicians would consider
hopeless, because to my despair I've never been able to think
of the reasonable ones.)
In my hope of getting started on something
where I might find at least some sort of publishable results,
I started thinking about the simplest kind of torsion free groups
I knew about that wasn't completely understood --
almost completely decomposable groups.
And eventually I did manage to write
a paper on almost completely decomposable groups
that most people
thought was pretty good.
(Recently, a leading mathematician in my field
has referred to it as a ``historic'' paper,
which I consider a pretty good joke.
In any case, the word ``historic'' really has to do
with the fact that a lot of mathematicians
were later able to use my ideas to do things
that I'd never imagined myself.
At the time, the paper itself seemed interesting but of
somewhat minor significance.)
I was also at this time constantly mulling over in my mind
some ideas that David Arnold had started me thinking about
while I was a graduate student.
In retrospect, I realize this mulling process
has in fact always been my strength as a mathematician.
What I thought I was supposed to do, and always wanted to be able to know,
and never was able to learn how to do
was to think up a problem
and sit down and work on it systematically.
But what in fact I was good at was taking various ideas so vague
that the people I tried to explain them to would only shake their
heads,
and continue mulling over them, worrying them,
pushing and pulling at them,
wondering if there could possibly be anything useful in them,
until I finally realized something important.
In any case, I wound up solving, almost by accident,
the major unsolved problem for finite-rank torsion free groups
(``Can a finite-rank group have finitely many essentially different
direct sum decompositions?'')
and coming up with a new concept (``near isomorphism'')
which seemed to clearly have profound importance.
Later it would turn out that,
although some of my subsequent research was pretty good,
these three results I obtained
during my first year after getting my Ph.D.,
with the possible addition of another theorem
I proved the following year at Illinois,
were what people would always know me for.
It was these results which were responsible for my later getting
a job at the University of Hawaii,
and tenure, and promotion to full professor.
During my year at the University of Illinois
I sat in on two advanced graduate courses and two seminars
each semester
and taught one five-hour calculus course and one three-unit
baby upper division course each semester.
I devoted almost no time to my classes,
and yet these were some of the most successful teaching I've done
in my lifetime.
In the linear algebra course I taught there, for instance,
I started out having students write proofs for such
purely computational results as the fact
that the product of a matrix and its transpose is symmetric.
This was new and awkward for them,
but most of them seemed to get the hang of it after two
or three assignments,
and then later on when we moved to more verbal (``conceptual'')
proofs, I was astonished when they seemed to have
very little difficulty with them.
This can undoubtedly be attributed more to the quality of the students
than to my own effort.
Back at Kansas the next year,
it was suggested that I might want to teach
an advanced graduate course.
I choose to teach
a year's course in homological algebra.
I wanted to teach this mostly because I had never had a course in it
myself, but had been reading books on it
ever since the time
before I started graduate school when I was a computer programmer.
Also, I wanted to educate the other faculty at Kansas
on the subject,
because I was really annoyed by their attitude
that anything homological was much too esoteric
for them to have anything to do with.
I interpreted the topic of the course quite liberally,
and included a whole lot of things I'd learned the year before
from my courses and seminars at Illinois.
This included a lot of category theory
and parts of commutative ring theory
that had at least some homological aspect to them.
I devoted an enormous amount of time to this course.
I was taking material from half a dozen different books
and a number of journal articles.
I wrote out detailed notes for my lectures
on loose sheets of paper,
which came to about seven pages per hour lecture.
I actually worked on notes about three or four weeks
before I was to present them,
then went over them and rewrote them twice
before I actually presented them.
In class, I copied the notes onto the board about as fast
as I could write.
I was very shy that semester, and never looked at the class,
so that when Steve Butcher came up to me at the Catfish Tavern
several weeks later and introduced himself,
I didn't know who he was.
The truth was, I was really teaching the course
mostly for the faculty who were sitting in.
As far as I was concerned, the students were incidental.
I did assign some problems for them
(which, of course, the faculty did not condescend to work on)
and made some attempt to keep students from getting lost.
That spring, the graduate students gave me their award
for the best graduate course of the year.
This was especially noteworthy because up until then
that award had always been given to someone teaching
one of the beginning graduate courses.
Since I only had an enrollment of four,
my students must have had to lobby really hard
to get others to vote for me.
(Either that, or the graduate students that year
thought that the other professors really sucked!)
What bothered me was that I was spending so much time on this course,
obsessively, and not doing any research.
I successfully applied for a grant from the university research fund,
though, and did write a paper the following summer.
(I would have done the same research even without the grant.
The grant simply gave me a month's summer salary
so I wouldn't be tempted to teach summer school.)
For my undergraduate course those two semesters,
I chose the mathematics course for liberal art students,
which has always been more satisfying for me to teach
than something like calculus,
although most faculty avoid it like the plague.
The first semester I didn't put as much work in on it as I should,
and the shyness which had beset me for some reason that semester
prevented me from establishing a really good relationship
with my students.
The second semester,
I was, if I may say so myself, superb with my liberal arts students.
At the end of the semester,
some of them asked me if there were any more courses
like that they would qualify for,
and made comments such as,
``I never realized before that mathematics really is a liberal art.''
Unfortunately, the university decided not to do student
evaluations that semester.
That was one of several seemingly minor little accidents
that ultimately resulted in my being denied tenure at Kansas.
I would have liked to have followed up the homological algebra course
with a really good course in commutative algebra,
with a more modern orientation than the old-fashioned
Gilmer-style stuff the ring theorists at Kansas liked to work on.
And then I thought that maybe the following year
I could teach a course in algebraic geometry and algebraic number
theory.
At that point, we algebraists at Kansas
would have been au courrant with the mainstream
of contemporary commutative algebra
as I had seen it at Illinois.
It turned out, though, that my ambitions for the algebraists at Kansas
were much greater than their ambitions for themselves.
(This sort of thing has been very typical for me throughout my life,
both in my teaching and in realms.)
The Kansas ring theorists had found a nice little niche for themselves,
where they could manage to publish lots of papers,
and as far as they were concerned the length of their publication lists
was quite sufficient proof that they were leading mathematicians.
They saw no need to learn new things that were difficult and unfamiliar.
So the following year I taught the basic graduate algebra course.
At least I thought that I could give the students there
what I considered to be an honest algebra course,
such as had been taught at UC San Diego when I'd been a student there,
instead of the rather anemic course that had been previously customary
at Kansas (and undoubtedly was again after I left).
I only had an enrollment of seven.
One of these was an extremely bright undergraduate,
who was very appreciative of the material I was teaching
and went to Princeton to do his graduate work the following year.
The other six had an enormous amount of difficulty with the problems
I was assigning.
I spent an enormous amount of time with them in my office
teaching them to write the sort of routine proofs that,
in my opinion (then and still now) they should have learned
to do in their undergraduate work.
As I recall, three of them dropped out
and the other four all failed the comprehensive exam the following
year.
Since this comprehensive exam emphasized material that I had expecially
stressed in the course,
and problems that had been assigned as homework in the course,
I had a sense of futility about the whole effort.
Once again, as with the faculty,
my ambitions for my students had been much greater than their
ambitions for themselves.
In any case, through a combination of strange accidents,
some of them almost absurd,
the University of Kansas, over the vociferous objects of the
Mathematics Department, denied me tenure.
(Which is the polite academic way of saying that they fired me.)
The Mathematics Department had been concerned
that at the higher levels of the university,
my list of nine or ten publications might be considered a little short.
But instead I was informed that the university-wide committee
on tenure considered my research ``outstanding,''
but my teaching ``below average.''
Given the fact that I had been given a teaching award by the graduate students
for my homological algebra course,
my first reaction was that this was truly bizarre.
My second reaction (not expressed) was to say essentially,
``Fuck you!''
I wasn't anxious to spend the rest of my life in Kansas anyway.
I wanted to be at a better university,
and I wanted to be in a more acceptable place to live.
So I wrote letters to the well known mathematicians
who had given me glorious praise for my work
and told them that I was going to be in need of a job.
And essentially got back numerous copies of the same letter:
``We would absolutely love to have you in our department right now,
but there's a budget crunch going on and we're not hiring anyone.''
I thought, ``So this is the reward one gets
for doing outstanding research!
It looks like maybe this whole mathematics idea was a big mistake.''
I was taking some Russian courses at the time,
and the next year I had the opportunity to take a course on translating
from a professional translator (Galina Bismuth,
Armand Hammer's personal interpretor at the time, in fact),
and I started thinking that maybe I could improve my Russian
enough to work as a translator.
I thought it would be nice to actually earn a living
using one of my languages.
I also looked at a used furniture store and a tavern
that were for sale in Lawrence.
My daughter was now in high school,
and I until I got offered a job by Hawaii,
I was beginning to be fairly desperate about how I could
support her and my wife.
Financially, my life would have turned out to be enormously
better if I'd been able to stay at Kansas.
But when I look back at that time in Kansas,
I see it as an enormous black morass,
a pool of quicksand that was threatening to suck me down
and suffocate me.
I've always found it really odd
that the Mathematics Department at Kansas
is generally considered to be far superior to Hawaii.
I have plenty of bad things to say about the University of Hawaii,
but I found the mathematics department here
quite refreshing after Kansas.
The faculty are much more interesting as human beings
and much more interested in broadening their knowledge base
and trying things that are new.
The faculty at Kansas may have had much more impressive publication
lists (and a lot bigger egos), but in my experience they were
(with some exceptions, none of them algebraists) extremely limited.
In many cases, their success was simply do to their having found
a niche to work in where they could produce papers
on a fairly consistent basis.
But looked at in terms of abilities and attitudes
I think that for the most part the faculty here
are far better mathematicians than the ones I knew at Kansas.
Furthermore, I think there's a much better overall attitude
here at Hawaii that fosters mathematics,
whereas at Kansas there were many good individual researchers but,
whatever their strengths,
they did not contribute to a general community of mathematicians
in the department.
For instance here at Hawaii, the department puts a great deal
of effort into making sure that there is a colloquium
at least once a week.
Preferably this colloquium will be given by a visiting speaker,
but is none is available
one will be give "in house" --
by one of the regular faculty in the department.
Another very important strength in the department here in Hawaii
is the number of visitors -- both short term and long term ones.
Of course we are fortunate in that respect,
in that Hawaii is a natural stopping off place
for travelers on their way from the United States
to the Far East.
It is also seen as a very desirable place
for a mathematician to spend a semester or year,
quite unlike Kansas -- where almost no one ever visited
if they could possibly avoid it.
(Whether Hawaii is equally desirable as a more permanent
place to live is another question.
Certainly from an economic point of view,
it has many severe drawbacks.)
Nonetheless, the department here also deserves a lot of the credit
for the large number of visitors.
We put a lot of effort into facilitating that.
In my first years here at the University of Hawaii,
I really benefitted from some of the outstanding visitors
that came here for a semester or a year.
Even within one's own specialty, or sub-specialty,
it is really difficult to know what is going on,
and visiting scholars are one good way in which news is spread.
Reading journals is of limited help
because by the time an article appears in a journal
it is already a year or two out of date.
If one is lucky, one is on the mailing list for pre-prints
from some of the leading people in one's specialty.
(It helps in this regard if one already has a pretty solid reputation.
I used to get all sorts of pre-prints in the mail
that I had not the slightest interest in reading.)
Aside from visitors, what serves the purpose of scholarship
within a specialty is the conference.
The important thing at a conference is not so much
listening to all the papers that are presented,
most of which are of extremely limited interest,
or listening to the expository talks,
which are somewhat more valuable.
What's really important at conferences
is talking to the other people working in one's own field
at lunchtime or over beer in the evenings.
That's when one gets a sense of the general landscape of current research.
I didn't realize this until a few years
after I stopped getting grants and therefore no longer
had automatic travel money.
The University of Hawaii has the attitude
that there is no point in sending two people to the same conference,
since the one who goes can tell the one who stays home
about it.
Since there is another faculty member here who works in the same
specialty as myself,
and since he seemed to care about conferences
a lot more than me,
I stopped going.
It was only after a few years that I started realizing
that I no longer had any idea of what was happening in my field.
This just made it a lot easier from me to withdraw from
being an active mathematician,
which was pretty much the direction I was moving in
in any case.
(I do have to say, though, that on the whole KU
is a much better school than UH.
The University of Kansas is a real university,
with real intellectual life -- to which the mathematics department's
contribution while I was there was insignificant --
and at least some students who really care about their education.)
Here at the University of Hawaii,
over the course of about twelve years,
instead of putting my main effort into research,
I taught almost every undergraduate course we offer.
This took an enormous investment of time and energy
on my part,
but it seemed very worthwhile,
both because I was learning things I wanted to learn
(or thought that it would be prudent for me to learn),
and because it was a constant challenge.
Unfortunately, though, it was not at all clear
that I was giving my students something they found of value.
Over and over again the same pattern would repeat itself:
I would start out teaching a subject
which I usually knew a fair amount about
but had never taught before.
I would give a lot of thought to it,
going over the textbook and several other books on the same subject,
mulling it all over in my mind,
trying to figure out what the really important ideas
in this subject are,
and how these could be explained in a way
that would make the most sense to students.
Then for six or seven weeks in class I would explain
the subject as I understood it.
Students would seem reasonably attentive.
I'd assign homework,
which was non-routine and designed to make students really think about
the subject.
(In most upper division courses,
I consider the homework at least as important
as the lectures as a teaching vehicle.)
And then students would troop into my office,
saying, ``How are we expected to do these problems.
There aren't any examples in the book like these.''
And I would realize that the idea of actually
thinking about mathematics was quite alien for them.
Occasionally, when a student seemed somewhat intelligent,
I would refuse to give him the answer he was asking for
and instead just give him some hints and ask him some questions
that would set him on the right path.
But most of the time,
I could see that the idea of doing something without having
stop-by-step directions for it
was completely hopeless for the student,
so I'd just sigh and show him how to get the answer.
(The interesting thing is that I think that usually the first
type of students thought that they were being discriminated against.)
Aside from spending time with students individually in my office,
I'd usually also spend a lot of time
typing up answers to the homework problems to hand out in class.
Then I'd give a test and discover that students had no understanding
whatsoever of what I'd been talking about.
Despite all my efforts at showing them to do the homework problems,
they couldn't do the same problems
when when they showed up on a test.
At that point, I'd think: ``Damn! I let myself be suckered again!
I should know better by now.''
I realized that once again I'd taught a course
designed for students who were like me.
When I am a student,
a want to learn more than just how to be a seal
who can balance a ball on his nose.
I want to really understand the subject.
But my students were not me.
They had no objection to learning to be trained seals.
On student evaluations, I would get comments like,
``Why doesn't he just show us how to get the answers?''
and ``Why doesn't he just teach what's in the book?''
and ``He wasted lots of time on stuff that wasn't even
on the tests.''
What my students were looking for
was a course that was routine.
But routine is not something I've ever been very good at,
and it's not something that I find valuable.
Any more now, I try to explain this to students
on the first day of class,
so they can know what to expect
and change to another course or another section
if what I offer is not what they're looking for.
But it doesn't do any good.
Students stay in my course,
and then at the end of the semester they complain
about exactly the things I told them I was going to do
on the first day of the semester.
In some courses, such as Linear Algebra and Number Theory,
I have definitely accomplished something with many of my students,
in that they did eventually learn to write simple proofs.
Being able to do teach students to write proofs is
one of the things I'm most proud of in my life as a professor
-- more so than any of the research papers I've published.
I think that I'm fairly unique
in trying to teach students in a somewhat systematic way
the kind of thinking involved in
proving theorems.
I remember that when I was a student
my professors simply assumed that I knew how to write proofs,
and assigned proofs as homework.
Fortunate, I had learned the basic patterns by osmosis,
through having worked my way through so many books on mathematics.
Students who didn't have this skill
either somehow managed to learn it on their own,
or they were written off insufficiently intelligent for mathematics.
I'm pround of making the process of writing proofs
accessible to a large number of students.
As I tell them in courses like Number Theory or Linear Algebra,
this skill is a step forward in their education
almost as important as learning to read was
when they were in the first grade.
Being able to follow proofs and write at least basic proofs
gives students access to a whole world of mathematics
that they would have been shut out of otherwise.
But after a while,
I started being aware that writing proofs had never been something
that my students had wanted to learn.
It was something that I wanted them to learn.
And occasionally when I would see one of my better students
a semester or two later and ask,
``Has what you learned in my course about proving theorems
been helpful to you?''
the student would answer, ``Well, I have professor Jones for my course now,
and he doesn't make us write proofs.''
(I'd overhear other faculty saying,
``You can't assign proofs in our classes,
because these students are not capable of writing proofs.'')
It also occurred to me that what I was doing
might be a little like teaching a dog to play checkers.
In a way, it's a remarkable tour de force,
but after a while one can't help but notice
that none of the dogs plays checkers very well.
Maybe if a student has to be laboriously taught how to write
the simplest kind of proof,
that student just doesn't have the kind of mind
that will ever be able to prove anything non-routine.
It seems that the longer I teach, the more disenchanted I become
with the idea that a college education should consist
of forcing students, for their own good,
to learn things that they really don't want to know.
After I went through the NLP training in 1983-84,
I started trying to find a way to use NLP to make my teaching
of calculus and other lower division courses into something worthwhile.
One of the ideas in NLP is that anyone can learn any skill,
if only one can figure out what the basic cognitive strategies
involved in learning that skill are.
These basic strategies are taken for granted so much by the people
who have them that they are seldom explicitly taught to students.
So, taking this as a working hypothesis,
I thought that maybe I could
concentrate on teaching my students how to learn mathematics.
I would try to analyze whatever it is
that makes it easy for me to learn mathematics and then
teach that strategy to my students.
For about five or six years I attempted to do this,
but will very little success.
And eventually I realized that all my attempts in this direction
were mostly for my benefit.
This was my project, not my students'.
They had no particular desire to learn new strategies
for learning mathematics;
what they wanted was someone who would present mathematics
in a way that would make it easier for them to use their
existing strategies.
Besides, the more I thought about what it is that makes
me good at learning mathematics,
the more I realized that all my strategies involve
being really interested in mathematics and
really thinking about mathematics.
But the students I had who were really interested
and willing to think about mathematics
already had good learning strategies.
The rest of my students might in some cases
be willing to work fairly hard,
but they were adamantly opposed to being interested
and they had no desire whatsoever to think about mathematics.
What they wanted was someone who could show them how to solve problems
by sticking Tab A into Slot B.
I do have to say that, over the years,
I have gradually found quite a few ways
of reducing various mathematical techniques --
techniques of integration, or first order differential
equations, or even proving certain sorts of theorems --
to essentially a paint-by-numbers procedure.
This is the thing many students appreciate about my teaching.
But I have to say that, when I really think about
what I'm teaching -- enabling students to avoid thinking --
sometimes it's hard to avoid thinking of myself
as simply an academic whore.
I find myself with two conflicting values.
On the one hand, I think that if a university is to have self-respect,
then the courses it offers to its students have to have a quality
comparable to those taught at other universities.
And I believe that here at the University of Hawaii,
this is simply not the case.
(Although I have not sat in on any undergraduate math courses
at other universities since I came to Hawaii,
I have sat in on undergraduate computer science courses
at Berkeley, and I think that they're comparable to
what we teach in graduate courses here.)
But if I were to teach what I myself consider to be an honest course,
almost none of my students could cope with it.
And at that point, I have to ask myself:
if I'm teaching a course that none of my students can understand,
then for whose benefit is the course being taught?
There is something to be said for the point of view
that there is a value in helping students learn as much as they
are capable of, of helping them take their understanding and
abilities to a new level,
even if by objective standards it's not a very high level.
In the upper division courses that I think I teach reasonably well
(at least some of the time) -- Linear Algebra, Discrete Mathematics,
and Number Theory -- I've come to adopt pretty much the same
attitude that I had when teaching Elementary Education majors.
My main goal is to help my students increase their ability
to read and to write mathematics,
and give them the ability to do at least some rudimentary
sort of mathematical thinking.
I think that in this way a lot of my students
(maybe close to half) get some real benefit from the course.
On the other hand,
I can't help being bothered by the fact
that these students are going to be given a transcript
which will seem to indicate that they have been taught certain
subject matter,
and that I will in fact not have covered anywhere near
all this subject matter.
At Kansas, it was the department which applied to have a faculty member
tenured or given promotion.
The faculty member him/herself had ostensibly no standing in the matter.
Initially, I was uncomfortable with the fact that here,
at UH, it is I myself who was expected to present the justifications
for giving me tenure or promotion.
I was expected to blow my own horn.
However as I got into writing up an application for tenure
and promotion to full professor,
I started to realize that while it might be uncomfortable
about blowing my own horn,
on the other hand no on else could blow it quite so loudly.
It was really soon apparent, in fact,
that what the higher echelons in the university cared about
was not how good a teacher or researcher I really was,
but how impressive a description I could manage to put together
of myself.
On the application for tenure, there were sections where I was
to describe my Research, Teaching, and Service.
These all seemed pretty straightforward
(except that in the Service section
there was the challenge of making routine committee work
sound like something exceptional).
But then there was another section that drove me bats
for several weeks: I was supposed to describe my
Present and Future Value to the University.
I couldn't figure out what this was supposed to be about,
and nobody else seemed to be able to explain it to me.
Finally, I figure it out.
It was supposed to be about two pages.
Other than that,
I realized that it didn't make much difference
what it said.
Unfortunately, though, the one thing at the University of Hawaii
that was not under my control, nor under the Math Department's either,
whether by ``shibai'' or otherwise, was my salary.
I had known when I accepted the position here
that I was being offered a salary that was too low,
but what I had not realized was that, unlike Kansas,
faculty salaries at the University of Hawaii are controlled by a rigid
civil-service structure,
so that the salary one is hired at determines one's salary
for the rest of one's life.
But added to that,
the nation soon entered an extremely drastic phase of Reaganomics inflation,
and the State of Hawaii responded by giving only minimal salary raises
(0% one year and 2% the following),
and the entire faculty here at UH was becoming financially desperate.
One of the reasons I dropped out of active mathematical research
was a realization (even before Reagan came into office in 1981)
that it would soon no longer be financially feasible
for me to continue working here.
And as I stopped letting my life be taken over by doing research,
and after being promoted to full professor
had finally jumped through all the hoops that young academics
(although I was no longer exactly young)
are require to jump through,
more and more I was asking myself
exactly what it was that I want out of life,
and to what extent my career as a mathematician was providing it.
Maybe I would never have asked myself these questions
if it hadn't been for the financial situation,
but once I started asking them I realized that being a mathematician
was not giving me the life I wanted at all.
It wasn't the money, as such, that was the real issue,
because what I had wanted out of life had never been a lot of money,
and even the salary I had here would have been quite adequate
if I'd been living anywhere except in Hawaii.
(This was especially true after my daughter graduated from college
in 1982.)
What I wanted was to lead an interesting life
and to know interesting people.
If someone were to offer me a job, the number one thing they could say
to attract me is that I'd have really interesting co-workers. (I think
for one year in my life I had a job like that --- the year I spend as a
lecturer at the University of Illinois. The two years I taught at
Humboldt State University were also not overly bad in that respect.)
When I look back at my life now, one of the most depressing things I
see is an almost steady decline in the quality of the people who I'm
friendly with, starting from the time I was in high school.
Morris Kline writes:
In fact, one of the things that I realized
when I started assessing the life I had as a mathematician was that
the people I really got along well with and considered worth knowing
were not mathematicians, but wives of mathematicians.
In any case, I thought that by the end of 1985 I would be gone from
the academic world.
At that time, I had assumed that the best choice for me
would be to go back into the aerospace industry, which,
as I now realize, would have been completely disastrous.
Then during my sabbatical in 1983-84 I learned NLP (Neurolinguistic
Programming), realized that I was very good at it and really liked
doing it,
and thought that maybe I would do some graduate work in clinical
psychology and become a licensed psychotherapist.
(This was not a very realistic idea,
although not absolutely infeasible.
Alternatively, simply hanging out my shingle as an NLP practitioner
without any conventional academic credentials
would have been a very precarious way to earn a living,
but in retrospect maybe that's what I should have done.
At least I would have been using some of my talents
and doing something challenging.)
But the salary situation started rapidly improving at that point,
and I chickened out.
For more than ten years now, I've been a year away from quitting
this university, but have never quite done it.
At the moment, I'm just waiting for the real estate market
here in Hawaii to recover so I can sell my apartment
without losing an incredible amount of money.
But in the meantime,
I've shaped a more satisfying life for myself.
Being a college professor can actually be a wonderful job
and in some ways is ideally suited for for me.
It gives me lots of chance to spend time alone and read and learn
things and have lots of thoughts, which has always been one of the main
things I've wanted in life since I was six years old.
Once I stopped letting my research, and later my teaching,
completely take over my life,
my job gave me me a lot of personal freedom,
which is one of the things that most
consistently gives me pleasure. To a large extent, it allows me to
choose my own projects to work on, even when some of these projects
(like my web page, not to mention the English courses I take, or my
articles on usenet) seem to my colleagues to be virtually evidence of
insanity.
Having my summers free
enabled me to have a number of experiences
which I consider to have been the most noteworthy things in my life --
certainly more worthwhile than any of the theorems I got so much acclaim
for proving. For instance going to the
Clarion science fiction workshop
for six weeks in 1981, going through a lot of
NLP training,
doing
volunteer suicide-prevention work
(although that was not during
the summer), taking courses at the
Institute for Advanced Study of Human Sexuality, and taking the
training for
the San Francisco Sex Information hotline.
(Having my summers free also made my salary seem much more reasonable,
since now I was only working for nine months.)
As far as having interesting people in my life,
the answer to that has been to simply look outside my academic environment.
To some extent, I've done pretty well in that respect.
Hawaii in its way can be a very nice place,
once you adjust to the sort of place it is.
It's not a highly competitive environment that emphasizes high performance.
It's a laid back place where there are no rewards for performing well,
but there are also no penalties for performing badly.
To the extent that I teach my classes well
(which I certainly don't always do),
and create a lot of material for my students and my web page,
I do so because I personally get satisfaction from it.
It makes absolutely no difference to the way
the university treats me.
There's no reason for me to feel driven
except by my own inner desire to accomplish things that are worthwhile.
And when I want to take a writing workshop from the English Department,
or sit in on a course in literature or film, I do it.
When I look back over my life, I almost feel that in a way I
should be thankful to the University of Hawaii for having provoked a
crisis in my life by paying me such a pitiful salary for so many
years.
When I first became a mathematician,
the burning question for me was,
``Am I capable of doing this stuff or not?''
Once I proved that I was indeed capable of doing research,
the temptation would have been to just keep on doing it
as a matter of habit and conformity to institutional pressured.
Now, when I try to imagine what it would have been like to
have been a serious mathematician all those years, which for me would
mean being totally obsessed with my research, I find myself looking at
that alternative history almost with horror. I am very very glad that
I stepped off that path and am not that kind of person.
I've got to get out of this place.
Mathematics can equally well be treated as a game
which has to be played with meaningless pieces
according to purely formal and essentially meaningless rules,
but which becomes intrinsically interesting because
there is such a great fascination in discovering and exploiting
the complex patterns of play permitted by the rules....
I wish to emphasize especially that it has become necessary
to teach mathematics in a new spirit
consonant with the spirit which inspires and infuses the work
of the modern mathematician....
In fact, the construction of mathematical models
for various fragments of the real world,
which is the most essential business of the applied mathematician,
is nothing but an exercise in axiomatics.
When I enrolled as a graduate student at the University of Maryland
in January of 1966, the spirit Marshall Stone speaks of
had in fact completely taken over.
I was very fortunate that for the preceding three years
I had continued to read books on mathematics,
including Commutative Algebra by Zariski and Samuels,
and Algebraic Topology by Hilton and Wylie,
because the courses I had taken at Arizona
were very poor preparation for the graduate curriculum at Maryland
(which by then was fairly standard across the nation).
Louise Lim's topology course had been kid stuff
compared to the textbook by Kelly which was required reading
for the preliminary exam.
Likewise, the course in differential geometry I enrolled in
for my first semester
seemed to have no relationship at all
to the graduate-level differential geometry course I had taken at Arizona,
taught out of Struik.
In fact, this new differential geometry course
(taught out of the first chapter of Helgason's book on
symmetric spaces)
seemed to have almost no recognizable geometry in it,
and proved almost no theorems that seemed to have any real depth.
Certainly nothing remotely comparable to the Gauss-Bonnet Theorem,
which I'd seen in my undergraduate course.
But I had faith that if I could just take more courses in the subject,
eventually I'd get to something really profound.
Kline complains that mathematics has lost touch with its roots.
Mathematical research has become largely a matter of abstraction
for the sake of abstraction,
generalization for the sake of generalization.
For the most part, he says, those who study abstract systems
are not aware of the tangible problems which gave rise to them.
A liberal arts course that includes the most commonly chosen topics
[axiomatization, symbolic logic, abstract structures such as groups
and rings,
simple combinatorics, and graph theory]
must devote a great deal of time to convincing students
that they should learn what the entire mathematical world did not miss
for thousands of years,
and what very few mathematicians need even today.
The topics have about as much value as learning to dig for clams
has for people who live in a desert.
``For some n, consider 1 + 2 + ... + n.''
Later I learned to say,
``Let's suppose we want to figure out how to add up all the numbers
from one to, say, 13.
We could write that as 1 + 2 + ... + 13.''
This was a lesson which has been useful to me ever since,
even in teaching upper-division courses for math majors.
(However it's important that mathematics majors do learn to write proofs,
so in courses such as Linear Algebra or Number Theory
it's important to strike a balance between giving explanations in language
which is natural to students,
and being a good role model by giving the kind of formal proofs
that they need to learn to be able write themselves.)
The teaching I did at Humboldt State was very labor intensive.
It involved an enormous amount of time spent with students in my office
and a whole lot of time trying to think up material
that I hoped my students would be interested in,
figuring out how to present it in a way that would make sense to them,
and making up ditto masters for all the material I made up
to replace the assigned text.
(As far as I can recall, the department at that time
didn't even have a photocopy machine.
If they did, we certainly weren't allowed to use it
for making up classroom handouts.)
Universities do not encourage scholarship:
the writing of expository, historical,
and critical articles on mathematics, nor good texts at any level.
Books are considered meritorious
only when they are written for an extremely specialized audience
on very specialized subtopics.
Universities choose professors the way some men choose wives --
they want ones that others will admire.
I was able to finish my dissertation and get my Ph.D. at the
end of a year at New Mexico State.
I then sent out 400 letters to every possible sort of college
and university seeking a job.
The question as to whether I should take a position at a four-year
college like Humboldt State or at a research university
was resolved quite easily:
I didn't get any offers from four-year universities.
In any case, everyone except myself was taking it for granted
that I would want a job at a research university
if at all possible.
Morris Kline writes that
young faculty are pushed into doing research much too rapidly,
and consequently there is a pressure to do superficial research.
One thing I really liked about Hawaii was that
here I have the full range of courses (at least on the undergraduate
level) available to teach.
At Kansas, the algebra faculty controlled the undergraduate
and graduate algebra courses,
the analysis people controlled the analysis courses,
and the topology people could teach the few topology courses
that were offered.
If I had wanted to teach a course in analysis, for instance,
I would have had to find an analyst who wanted to teach an algebra course
and convince the algebraists to do some horse trading.
Furthermore, I couldn't teach the mathematics department's course
was prospective elementary teachers
(which I had so enjoyed teaching at Humboldt State),
because we had an Education Specialist
(treated as a second class citizen)
who took care of courses like that.
One thing I think is especially important at a university like UH
is to expand students' horizons,
try to let them know that there is more to the world,
especially intellectually, than what they know or see around them.
For this reason, I tell them about my own process of intellectual
development, the fact that my learning of mathematics
involved a lot of reading and not just taking courses,
and the fact that I was also constantly reading books
in history, philosophy, and literature.
I guess mostly what I want to do is just to let them know
that there are people in the world who really care
about intellectual matters.
For many of my students,
I'm sure that this is simply one more proof
that professors like myself are alien creatures
that might as well come from some other world.
But I hope that a few of them see in my accounts of my own
background and experiences and indication of something
desirable beyond the limitations they are accustomed to --
something they might want to emulate.
Lack of ability can be a very exciting challenge,
as in classes for elementary education majors,
which have been among the most satisfying courses I've taught.
But I find the lack of interest that students here show
completely demoralizing.
One respect in which the way the University of Hawaii
is organized is really different than the University of Kansas
is that Hawaii really places a blatant high priority on bullshit.
(The local word is ``shibai.'')
In general, research mathematics professors
are introspective and introverted;
they do not feel at ease with people;
they shy away from personal contacts.
They like to concentrate on their thoughts.
They choose mathematical research partly
because mathematics per se does not pose the complex problems
that are involved in dealing with human beings.
One can't help but feel saying this is Kline's attempt to settle
a few personal scores within his own circle of colleagues.
I've known many professors of mathematics, both here and elsewhere,
who are anything but introspective.
(One of the reasons I chose Fred Richman to work with
at New Mexico State rather than Elbert Walker
is that Walker seemed like such an alarming extrovert.)
And to the extent that what Kline says is true of some mathematicians,
the same could be said of many other academics, writers, performers,
and anyone whose craft requires an enormous
effort and discipline.
At the same time, though,
I have to realize that I'm simply wasting my life.
Yes, a lot of the things I've done during my summers and otherwise
have been personally very worthwhile for me.
But for the most part,
I'm not accomplishing anything.
Nobody here at the University of Hawaii
has any need of my talents.
6/3/97