#### by David Roberts

**Roberts:***
This is David Roberts speaking with Professor Morton Brown on
January 4, 2000. I’m speaking from my home in Laurel, Maryland.
He is in his office at the University of Michigan in Ann Arbor.
Professor Brown, when and where were you born?
*

**Professor Morton Brown:**
I was born in 1931 in the Bronx, New York.

**Roberts:***
Okay, first I’d like to ask you about the educational background
of your parents.
*

**Brown:**
My father had a sixth-grade education, and then went to work, and
my mother graduated from high school.

**Roberts:***
Did you receive from them or from any other family members any
specific encouragement to go into academics?
*

**Brown:**
Certainly from my mother. My mother was very encouraging for me to
excel in academic activities and certainly to go to college. My father
supported that, but it was my mother who was more proactive in that
direction.

**Roberts:***
Any specific encouragement towards mathematics?
*

**Brown:**
Only because that’s what I wanted to do.

**Roberts:***
Okay, could you describe your pre-collegiate schooling? Where was this,
and …*

**Brown:**
I went to a public school in the Bronx, New York, and then to what
we called junior high school then — it would correspond to middle
school — and then to a high school also in the Bronx. All of these
were pretty standard type of schools, no specialty schools.

**Roberts:***
Do you recall any particular experience with mathematics in any of
these schools, positive or negative?
*

**Morton:**
I remember cheating in sixth grade on an addition examination,
one of those times we had to add columns of numbers as rapidly as
possible, and I wasn’t very good at that. But, then I do remember
in about seventh or eighth grade, I’m not sure which it was, when
algebra was introduced, the subject just completely turned me
on and I found it absolutely beautiful. From then on, I wanted to
be a mathematician. I got some encouragement from one of my junior
high school teachers, and then the next important event in terms of
mathematical education and teachers would have been in high school
when I took an honors course in — in those days calculus was not
taught in the high school, so this was called a college algebra — but
it was much more advanced than modern college algebra is now, and
covered a number of topics that were challenging and very interesting,
and the class was run in a highly competitive way. I’m not sure if
you’re familiar with the New York high school system fifty years ago,
sixty years ago, but it was highly competitive. People got grades
like 91 and 93 and 94 and so on, and it was not A, B, C, D. I do
remember quite distinctly that there was another student in the class
who was the son of John Mott-Smith.
Or maybe his name was John, the
famous Mott-Smith may have been
Geoffrey1
but he was the son of Mott-Smith, and the teacher was a friend of
Mott-Smith, and he was my main competition. He was very well trained
and very bright, and we worked in different styles. I worked harder.
He knew more. [Roberts chuckles] I still remember with some choler
that I got a 96 in that course and he got a 97. [both laugh]

**Roberts:***
During any of this time, did you read any books outside of class
designed to popularize mathematics?
*

**Brown:**
No, I was never very interested in popularizations of math at that time.
I became interested in that later, largely for interesting other people.

**Roberts:***
Okay.
*

**Brown:**
I certainly never read
[E. T.] Bell
or
[Lancelot] Hogben,
and of course,
Martin Gardner.
I’m not sure he was even around…

**Roberts:***
No, he wouldn’t have been around then. Are you much interested now in
recreational mathematics?
*

**Brown:**
Not overly, other than as a teaching tool.

**Roberts:***
So you do see some role for it for interesting people in mathematics?
*

**Brown:**
Oh, yeah, absolutely. I mean, people love it, and a lot of people
loved Martin Gardner’s material. I always resented a little bit the
fact that in the magazine [*Scientific American*] there would be
all sorts of articles about science, but the mathematics section was
always called recreations. Now, that was good in the sense that maybe
it attracted people to the recreation part, but it was bad because it
demeaned the subject. But, I’ve met loads of people who find that area
particularly interesting.

**Roberts:***
Have you done much reading in the history of mathematics?
*

**Brown:**
I would say from your perspective, no. I actually did not get very
interested in the history of mathematics until relatively recently, and
in fact, I reread in the sense that I read it twice in the last couple
of years,
Morris Newman’s
book on the history of western mathematics.
As a matter if fact, at the moment I’m trying to get a used copy of that
somewhere. I may be able to get one from Powell’s Book Store.

**Roberts:***
Are you thinking of Morris Kline here?
*

**Brown:**
I’m sorry — did I say Morris Newman?

**Roberts:***
Yes.
*

**Brown:**
Morris Kline.

**Roberts:***
Okay.
*

**Brown:**
I find his perspective on the history of mathematics good. He’s
an excellent writer. I had first, of course, come across him as a
critic of the New Math.

**Roberts:***
Yes, I’ll want to get into that a little later, yeah.
*

**Brown:**
We’ll get into that later, because I more recently reread some of
his books in that area.

**Roberts:***
Do you see a role in math education for the history of mathematics?
*

**Brown:**
Yeah, that’s why I read the books. I’ve taught some courses where
there are future K‑12 teachers, and that led me into trying to give
some historical background for some of the more abstract things that
we have to talk about in class, and I found reading about the history
was very informative both for me and helpful for them.

**Roberts:***
What do you think of the notion that mathematics education ought to
roughly recapitulate historical order of development of concepts?
Does that appeal to you at all or is that…*

**Brown:**
Do you mean in terms of children being educated?

**Roberts:***
Yeah, or at any level really.
*

**Brown:**
I don’t think there’s a formula that’s going to work K-12, and so
recapitulation is one of a number of things that might be useful,
good things to do. I’m grappling with the question of how important
it is to recapitulate Euclidean geometry and two-column proofs, and
so on. So, I have no answer.

**Roberts:***
Okay. Where did you get your undergraduate education?
*

**Brown:**
I went to Wisconsin, the University of Wisconsin, for undergraduate
and graduate.

**Roberts:***
Okay.
*

**Brown:**
I went there for two reasons. One is it had a reputation for a good
mathematics department, at least from the high school counselors I
talked to, and that turned out to be correct. The other is I wanted
to get out of New York. I should say there was a third one. The third
one was it was almost impossible to get into a college in New York
State because of anti-Semitism at that time.

**Roberts:***
Okay. What undergraduate degree did you get, and what
*

**Brown:**
It was a bachelor of science at Wisconsin.

**Roberts:***
What year would that have been?
*

**Brown:**
Let’s see. I was class of ’52. I think I actually got my degree in ’53.
And then just continued on as a graduate student and got my Ph.D.,
as you know, in ’58, although I left in ’57 to work on — you know,
I took a job working on my dissertation.

**Roberts:***
Okay, were you much interested in other subjects besides mathematics?
*

**Brown:**
Not as subjects. I mean, I had a number of interests. Music was a
particular one, but not as anything to study.

**Roberts:***
Did you take much science?
*

**Morton:**
No, not much. I didn’t like much science. I took a course in physics;
I took a course in chemistry. I didn’t like either of them very much.
Physics I found very difficult to understand. Now, that I look back at
the teaching of it, I see that maybe a little bit better, why that
happened.

**Roberts:***
Now, you got your Ph.D. in ’58 with Bing as your advisor. Is that
correct?
*

**Brown:**
Correct.

**Roberts:***
How did you happen to choose Bing as your advisor?
*

**Brown:**
I’m not sure it was that way around. I originally had already decided
I was going to major in math, and I had had a different advisor, but
then, as a sophomore, I took a calculus course, second semester calculus
from Bing, and after that course, he invited me to take a course that he
gave, a very famous course at Wisconsin, in undergraduate topology.
That was it.

**Roberts:***
Okay, so you had Bing both as an undergraduate and as a graduate
instructor?
*

**Brown:**
That is correct. Bing and I arrived at the University of Wisconsin at
approximately the same time, of course in different positions.

**Roberts:***
Okay. Did Bing ever speak much about his own advisor, R. L. Moore?
*

**Brown:**
He did when I took graduate courses from him. He would speak about Moore
from time to time. He certainly was copying the Moore Method; he
certainly had modified the Moore Method. So it was less dog eat dog than
Moore’s method was, although if people were not doing the job, if they
were not working sufficiently hard or diligently, he was quite tough on
them, so in that sense, he was like Moore. But, I’ve seen the Moore tape,
and he was not anywhere near as harsh. I think of somewhat a different
clientele and also a different personality. I never did meet Moore.

**Roberts:***
Okay. What was Bing’s policy, for instance, on reading? What he very
strict on that?
*

**Brown:**
Bing pretty much followed Moore’s teachings in that direction. He did not
encourage people to read in the subject when he was teaching until,
you know, later graduate years. Then we would read papers, but in the
first two or three or four courses that I took from him those were
largely out of Moore’s book, and we were not encouraged to try to get
a copy of that book. In fact, we were just given theorems, and we were
told to work on them, and that’s what I did. And that was my style of
learning, which I adapted to other courses, sometimes less successfully.

**Roberts:***
So, you tried to do this sort of on your own in other courses?
*

**Brown:**
Yeah, that was my understanding of how to learn. If I couldn’t learn
it by myself, if I couldn’t prove the theorem by myself, then I hadn’t
proved the theorem.

**Roberts:***
I see.
*

**Brown:**
I would refuse to recognize knowing the theorem if I couldn’t prove it.

**Roberts:***
Okay.
*

**Brown:**
Which led to some difficulties when I took an honors test as a graduating
senior, because I was asked about some [phone disconnects]

**Roberts:***
Let’s get going again. You were talking about a problem that you
encountered on a
*

**Brown:**
Yeah, I don’t think that it was particularly important.

**Roberts:***
Okay.
*

**Brown:**
I tend to get too much interested in my own history rather than anything
that’s relevant to anything that you would want to do. At any rate,
I had totally bought into that as a method of learning, and it’s always
been my method of learning. I think that it’s not that I chose that as
the best way of learning. I think it was the best way of learning for me.
Over the years, as I’ve had my own students, I’ve found that, and I’m
sure it will shock you, that different people learn in different ways,
[both laugh] and that some people learn extremely well by just absorbing
information from other people, building what I think of as a thin layer
of ice on this large lake of knowledge, and skating on it. Some fall
through. Bing thought they all fell through. [Roberts chuckles] But some
just manage to thicken the ice more and more and have a very solid
foundation. Of course, there’s always been lots of objection to the Moore
Method on the ground that the people who use that method don’t, quote:
cover enough material, and that’s been a standard criticism, and there’s
some truth to it. Bing’s view, of course, as Moore’s view, was that the
deeper learning was the more important one. You could always learn these
other things for yourself as time went by, but after I got my degree
I did realize, when I went out into the larger world of mathematics,
that my background was limited, and when I came to Michigan I found that
I was starting to teach courses whose material I really had never seen
before as a graduate student. So, that was a challenge, and I was pretty
well able to meet it. I should mention that late in his life,
[Edwin E.] Moise
came to a different view of Moore, and Moore’s not
allowing people to read books. It was a much darker view. Moise’s view,
in those later years, and this is probably about fifteen years ago, was
that Moore wanted his students to live in any level, any type of
ignorance, that he shared. He didn’t want the students to know things
that he didn’t know, and that he didn’t know much, that he had dug
himself this deep little hole, consisting of his book and the things
around it, and that was all he knew, and he was not familiar with any
of the modern developments in mathematics, and feared that the students,
if they learned some of those things, might veer away from the faith
so to speak. That was Moise’s view. I could not speak to whether that
was true or not. It was definitely not true of Bing, and it was
definitely not true of
Wilder
or of Moise themselves, or
Anderson.
However, when I first came to Michigan, Moise was one of the people
I came for, and we had a number of talks about Bing and about Moore.
Moise pointed out to me, at that time, that all of Moore’s best students,
with one exception, immediately left the area that Moore specialized
in as soon as they got their degrees, and the place they made their
reputation was in those newer areas: Wilder going into algebraic topology,
Moise going into three-dimensional manifolds, Bing fairly rapidly following
Moise. I don’t know if you’re familiar with those careers, but in fact,
Bing followed Moise for maybe five to ten years, followed Moise’s path,
reproving theorems that Moise proved, looking for mistakes that Moise may
have made; they were very competitive. It was not a great friendship.
But, at any rate, Bing’s important contributions — he proved some
well-known results, and had made a small reputation doing what we
now call continuum theory, which is the remainder of the Moore-type
topology, but his important work was in three-dimensional topology,
and all of that was following Moise. It was Moise that broke the ground
there.

**Roberts:***
Who was the one exception of Moore’s students?
*

**Brown:**
[Gordon T.] Whyburn.

**Roberts:***
Okay. Now, when Bing would teach one of these courses, would he explain
to the students why he was teaching the way he was teaching?
*

**Brown:**
Oh, yeah. When you teach that way you’ve got lots of spare time in the
class, and so he would explain why he was doing it and what he was doing
and what he expected people to do. There was lots of that. We had no
doubts about what his goals were and what it was that we were supposed
to do.

**Roberts:***
What were the reactions of the students? Can you remember some general
reactions?
*

**Brown:**
Well, the students that remained [both chuckle] — we joined right in.
I don’t remember any students not being happy with this, but of course,
I really was not very conscious of the students that were not part of the
action. There was not a huge amount of interaction between the students
anyway, other than competitive, so it’s hard to say. The only person
I can remember distinctively, who kind of dropped out of it, was someone
who dropped out maybe more for reasons that were social than for
mathematical reasons, although I’m not sure. It was a person who just
didn’t make a good fit with Bing’s style of teaching or manner of speech.

**Roberts:***
Now, how exclusively did Bing use the Moore Method?
*

**Brown:**
I would say totally exclusively in graduate. In undergraduate, I only
took that one — well, the course he invited me into was taught in the
Moore style, so that one, of course, was completely Moore style. The
calculus course was, I remember him explaining things about “absalum”
and delta. [laughs] So, he certainly did explain in that course. He was
an excellent lecturer. He didn’t do it much, he did it very slowly, very
carefully, without mistake, and you could see him; he would stop and
think and consider, not only the proof, but how to present the proof.
So, he knew how to do that.

**Roberts:***
What’s your overall assessment of the effectiveness of Bing’s teaching?
Has it changed at all over time?
*

**Brown:**
No, my assessment is the same, although I have modified his methods in
my own teaching, and when I first started teaching graduate courses
I used the Moore Method, and it was quite popular. There were some
criticisms from within the department that the material — I was not
covering enough, the usual thing, the students might not be prepared for
their examinations. But that turned out not to be the case. So, the one
thing that I did drop in more recent years, completely drop, was the
competitiveness, and now I use a modified Moore Method. I have not used
it in graduate courses simply because I have not taught those graduate
courses in the last couple of years, but I use a modified Moore Method
for advanced calculus and advanced undergraduate courses where the
students prove theorems, but they work in teams, homework teams. And
this has turned out to be very, very fruitful because one of the worst
problems of the Moore Method would be a student who gets up in front
of the class and wastes forty minutes with a proof that is just not
going to work, and everybody sits there and you have this unresolved
situation at the end of the hour where people walk out and it was a
totally wasted time. At the very best, the students or the teacher are
just cutting the poor victim to pieces, but very frequently it would
just be rambling on the part of the person at the blackboard. By having
them prepare proofs for which the whole team is responsible, and they
show the proofs to the other people on the team, there’s a much greater
level of responsibility and a much greater view and examination of the
proof by other students before it goes to the blackboard. So it’s
actually quite rare that you wander into twenty minutes wasted and
suddenly a horrible mistake. That just doesn’t happen too frequently,
and that’s really eliminated what I always thought was the most serious
drawback of the Moore Method, and one of the things that leads to so
little material being covered. So, I’ve found that very effective, and
I’m going to be giving a talk of some sort at the AMS meeting. I’m
not sure whether — I think it’s at a NEXT meeting, so that’s not
being advertised in the journals. Do you know NEXT?

**Roberts:***
I’ve heard of it.
*

**Brown:**
They have some meetings that are just for the NEXT people, and I’ve been
asked to talk about this method for teaching advanced calculus, and I’ve
used it now about three or four times for advanced calculus, for abstract
algebra a couple of times, and it’s worked quite well.

**Roberts:***
Do you see any — this leads me to a comment that is occasionally
made, that the Moore Method is only workable in certain areas of
mathematics. Do you just disagree with that?
*

**Brown:**
I think it would be a mistake for people to be using the Moore Method
in all areas of mathematics, for a student to be only seeing the Moore
Method, and for example, there was a big episode at Auburn University
where some faculty were teaching all courses that way, and there were
enough of them doing it so that some students weren’t seeing any other
kinds of mathematics taught in any other way, and so there were complaints
that their background was becoming extremely limited. I’m not sure that
the Moore Method works for everything, but I haven’t found the place where
I know for sure it doesn’t work. I’m teaching linear algebra, I’m teaching
some versions of linear algebra, and I’ve never found it a particularly
attractive method in linear algebra. But on the other hand, for abstract
algebra, advanced calculus, courses that require real understanding of
deeper concepts and very complicated statements, it’s really quite well
designed for those.

**Roberts:***
When I spoke to*
*
Gail Young*
*a month or so before he died, he thought that it didn’t work very well in
complex analysis.
*

**Brown:**
I think that might very well be the case. Complex analysis — I
learned it by — I took a course in complex analysis, but I didn’t
learn any complex analysis — I never really mastered the subject
as a graduate student in those courses. I think I eventually found a
textbook, and I forget the name of the author, but it’s quite well-known,
whose emphasis was on the geometry of complex variable and I found
that much more amenable, but it was not a particularly good course to
try to learn the theorems by myself because the proofs are elegant and
surprising almost at the very beginning. I would agree with that view.
There are subjects where it’s probably not going to work.

**Roberts:***
Were there other professors at Wisconsin who especially influenced you
other than Bing?
*

**Brown:**
I guess the other professor that influenced me was
Creighton Buck,
who was almost the exact opposite of Bing. He was a marvelous lecturer,
extremely fast, very subtle, deep, committed, and he gave me a really good
idea of how elegant presentations can be, and it was a good contrast with
Bing.

**Roberts:***
Did you ever take a Moore Method class from anyone other than Bing?
*

**Brown:**
Yes, one summer I took a Moore Method class from
F. B. Jones,
a visitor,
and it was in basically two-dimensional topology, even more Moore Method
than Bing’s courses, and I enjoyed that course very much. When I look back
on it, it was a fairly weak course in terms of mathematics, but it was a
very strengthening course in terms of working on theorems and recognizing
how close theorems can be between being true and being false. Bing used to
emphasize that a lot. His mantra was that a really good theorem was one
that’s almost false.

**Roberts:***
Were there any fellow students at Wisconsin who particularly influenced
you?
*

**Brown:**
Yes, this was
Lee Rubel,
who died about four years ago. He was a couple
of years older than I was. We’d actually met once in high school on
competing chess teams, and then we remet when I was taking this
undergraduate course in topology. For some reason he was in that course
even though he was a graduate student. We got to be friendly, and I
learned a lot of mathematics from him. As I say, he was a couple of years
ahead of me, and we became very good friends, and I can still remember
particular things that I learned from him that were very important for me.
The whole attitude — he was the person who taught me to think about
mathematics without a pencil in my hand.

**Roberts:***
Did you do any teaching at Wisconsin?
*

**Brown:**
I was a T.A., yes. I taught — in those days you could not teach
calculus until you had passed prelims, if you want to think back to the
old times, so that the people who did teach calculus at least knew a
little bit more mathematics than the students we throw into the classroom
now.

**Roberts:***
Yeah.
*

**Brown:**
So, most of what I recall was teaching precalculus.

**Roberts:***
Any special recollections of that, any experiences?
*

**Brown:**
No, other than that I was interested in teaching, but I don’t remember too
much about it.

**Roberts:***
Okay, you said that you hadn’t ever met R. L. Moore himself. Would you
say that you had much interaction over the years with his descendants?
*

**Brown:**
Oh, yeah.
Dick Anderson
is a friend of mine.
B. J. Ball
I knew,
certainly not as well. Anderson I knew a lot. Moise and I became very
friendly when we came to Michigan because we had this common interest,
among other things our points of view about mathematics were similar and
we had this common interest of finding out more about Bing. [both laugh]
Wilder
was here, so I got to know Wilder better, but not well because
at that time he was already getting close to retirement.
Mary Ellen [Rudin],
who when I was in my last year as a graduate student,
Walter [Rudin] and Mary Ellen came to Wisconsin. Walter came as a faculty
member, Mary Ellen of course, they had at that time nepotism
rules — she was [tape ends]

**Roberts:***
Okay, we’re rolling again.
*

**Brown:**
Gail Young, I knew also, was at Michigan when I came.

**Roberts:***
Ah, yes.
*

**Brown:**
But, I knew him. The person I was closest to at Michigan was Moise, and
it was during that time Moise was working on the Poincaré conjecture,
which, of course, he did not solve, and then made his transition into
teacher education, and then left and went to Harvard in their teacher
education program.

**Roberts:***
Maybe I should ask about this now. What is your understanding of Moise’s
motivation for going into teacher education?
*

**Brown:**
I’m not sure. I think he was interested in teacher education, and we had
a number of discussions about it. He had done a study or an examination
of textbooks and believed that there was a cycle of textbooks, that every
hundred years some great mathematician would examine all the old proofs
which had degenerated into nonproofs and then rewrite a textbook, and
that would last for a while, and then every __\( n \)__ years, that would have
to be redone. I guess he must have been feeling that it needed to be
redone. At this time we’re talking probably 1961 or 1960, something like
that.

**Roberts:***
Was he frustrated by his research at that point?
*

**Brown:**
I think so. I think he had done this marvelous work on the triangulation
of three-dimensional manifolds, which was really quite impressive work,
and then the next thing was Dehn’s Lemma which was solved by
[C. D.] Papakyriakopoulos
and
[Tatsuo] Homma,
and so the remaining big theorem
was the Poincaré conjecture, which he worked very hard on, and I think
eventually started realizing that he was coming across just a theorem he
could prove, which was not the strength of the Poincaré conjecture. It
was kind of like that line between truth and falsity, and he couldn’t
cross that line. But, I think he did bang his head against the wall once
too frequently and decided to give it up, which I think he did.

**Roberts:***
Now, going back to my line of questioning here, do you believe it’s
meaningful to speak of an R. L. Moore legacy or teaching tradition?
*

**Brown:**
Absolutely, absolutely, yeah. I meet people from time to time who are
students of Bing. Bing must have had sixty students. Many of them
continue some sort of legacy of that type, and if they don’t continue it,
they still remember it, and often it’s people who only took a course from
Bing. As to who else was continuing that legacy, Whyburn, I think, was
much less successful, possibly because the mathematics he was doing did
not grow. It was just a continuation of R. L. Moore methods, and
R. L. Moore topology, and it was an area that eventually became more
and more isolated from the rest of mathematics, and that just got worse
and worse,
and for a while, that type of mathematics was only found south of the
Mason–Dixon line, distributed among a number of lesser schools and was
becoming rather ingrown. I remember a discussion with Mary Ellen, maybe
you should get this information from her, about her views as what she
would do if she were made chairman of Texas after Moore retired. If you
ever interview her, if you haven’t already, you might ask her what her
views were. [chuckles]

**Roberts:***
Okay, I haven’t talked to her.
*

**Brown:**
But there was a perception on the part of the more successful Moore School
people, and those are the ones who went into other areas of mathematics or
topology, that the area had become over refined and was stagnant. I was
interested to discover that when I got interested in dynamical systems,
which was about fifteen to twenty years ago, that at one point, dynamical
systems, differential equations, and topology were joined in more or less
the same area, and that somehow maybe between just after World War I or
around that time, that join seemed to have been cut, and the people in
one area seemed to be in ignorance of what the people in the other areas
were doing, so that the early *Fundamenta Mathematica* journals
seemed to make no reference to the fact that many of the things they
discussed actually were done by German mathematicians and Dutch
mathematicians fifteen years before, or had any relevance to what they
were doing. That was an absolutely brilliant and marvelously inventive and
fertile kind of mathematics being done in those first three, four, or five
copies of *Fundamenta*, but there seems to be no reference to any of
this earlier work from a different school. That was rather peculiar, and
that was part of the beginning of the isolation of the Moore School type
topology. There’s another branch to the Moore School type topology and
that’s what we now call general topology. We always were involved not only
with proving theorems but in proving theorems in non-metric spaces and in
non-normal spaces, and I’m not sure if you’re familiar with all of these
axioms, but the axiomatics…

**Roberts:***
Yes.
*

**Brown:**
In 1955, there was a conference at the University of Wisconsin. Are you
familiar with that, a Summer Set Theoretic Topology Conference?

**Roberts:***
No.
*

**Brown:** You should look it up because, at that conference, people
like Bing, Anderson, Moise, all gave talks, and they were doing their
major work, and Bing was just beginning to get into three-dimensional
topology, but the emphasis was on point-set topology at that time. And
a gentleman, a mathematician named
Ed Hewitt,
who
was from this school and left it entirely, I believe, gave a lecture
in which he criticized what he called beta T-spot axiom analysis,
which he thought were arid exercises in axiomatics.2
He was correct, I believe. Nevertheless, Bing gave a
response speech,3
and I think you might enjoy reading those.

**Roberts:***
Yeah, that sounds interesting.
*

**Brown:**
I may even have a copy here somewhere. I’m not sure I still do, but it
was the University of Wisconsin; it was maybe the second NSF conference
in Madison

**Roberts:***
Yeah
*

**Brown:**
and I believe it was 1955.

**Roberts:***
Okay. Here’s a question I skipped over. What was your dissertation topic?
*

**Brown:**
A continuous decomposition of Euclidean
__\( n \)__-space minus the origin into
__\( n-1 \)__-dimensional hereditarily indecomposable continua

**Roberts:***
Okay, and how did you choose that topic?
*

**Brown:**
Bing assigned that.

**Roberts:***
Bing assigned it, okay.
*

**Brown:**
It was the last paper I wrote in that subject.

**Roberts:***
Okay.
*

**Brown:**
Yeah, so I followed the kind of path that Bing followed, namely getting
out of that field as soon as he got his degree. He actually stayed in it
a little bit longer. I do remember that Bing and I did not have many
discussions about mathematics after I got my Ph.D., and I remember hearing
from somebody that Moore did exactly the same thing with his students,
that he never discussed mathematics with them after they got their
degrees. In the case of Bing, he certainly was happy to discuss
mathematics, but he did not pursue mathematical interests with his
students in the sense of suggesting problems and corresponding and so on.
It was pretty much *you go on your own*. He was probably busy with
his current students. Anytime you talked to him at a conference or
something like that, he’d always be full of questions; that was not the
problem.

**Roberts:***
You mentioned Wilder. How well did you get to know him?
*

**Brown:**
Not well at all. For one, he had become seduced by algebraic topology,
and what he had basically done in his book was take all of the Moore
ideas and put them in a setting of algebraic topology. One can read
through his book and see that he is reinterpreting Moore’s notions so as
to be able to talk about higher dimensional things and using this version
of algebraic topology that was current at the time, a very cumbersome one,
and so it was a difficult subject. The students of Wilder had a geometric
lineage; I think particularly of
Frank Raymond,
who was a student of Wilder’s and who also came on the faculty
at Michigan. The approach was, although the background was algebraic
topology, the approach was still very geometric. But Wilder was not
teaching Moore Method.

**Roberts:***
Okay, I’m interested in that.
*

**Brown:**
You might check with someone like Frank Raymond if you want to get some
information about Wilder.

**Roberts:***
Okay.
*

**Brown:**
He’s probably the best person to talk to. He’s retired and in Ann Arbor.

**Roberts:***
Okay. I did ask Gail Young a little bit about Wilder, and he made the
statement that Wilder hardly ever used the Moore Method. I’m interested
in that. Let’s see. When did you leave Madison?
*

**Brown:**
1957.

**Roberts:***
and you went
*

**Brown:**
I put in a year in Columbus, Ohio, and then went to Ann Arbor the
following year.

**Roberts:***
Where were you at the time Sputnik was launched?
*

**Brown:**
When was that, ’58?

**Roberts:***
That was in October of ’57.
*

**Brown:**
In October I had probably just arrived in Ann Arbor.

**Roberts:***
Did that make any particular impression on you at the time?
*

**Brown:**
In terms of mathematics?

**Roberts:***
Yeah, and I just wondered if you were much aware of the sudden surge of
interest in math and science education at that point.
*

**Brown:**
Oh, yeah, I was aware of the fact that Bing’s salary tripled in a
two-year period, [Roberts chuckles] and that my starting salary was
almost what Bing’s was as a full professor two or three years before
that. This was a time of tremendous shortage of mathematicians and
growth in the colleges. Salaries were shooting up, teaching loads were
dropping. When I first came to Michigan, someone told me — I can’t
swear that this was the case, but someone said that your teaching load
would be 16 hours minus your rank. That is, if you were a professor,
you taught 13 hours, or 12 hours. If you were an instructor, you
taught 16. A four-course teaching load was standard. The year before at
Ohio State the teaching load was lower. It was three five-hour courses,
so it was a 15-hour teaching load that was the standard. In the next
couple of years, it was quite rapidly dropped because people would
simply say, we’re not going to come unless we only have to teach three
courses, and then eventually only have to teach two courses, so we were
all quite aware of that. I still remember a colleague of mine,
Nicky Kazarinoff
who was at Michigan and had been an undergraduate, I think, at
Michigan, or maybe a graduate student at Michigan, saying that. He was
one of the very first mathematicians invited to visit the Soviet Union.
He spoke Russian. And he talked to some scientists there who were living
rather nice lives, and he said that the mathematicians told him that the
best thing that could happen to Russian mathematics would be if the
Americans would put up something better than Sputnik.

**Roberts:***
Did it seem to you or did you feel at the time that Sputnik demonstrated
some weakness in math and science education?
*

**Brown:**
No, I was totally unaware and uninterested in those things. I was just
working on my research.

**Roberts:***
Okay. Were you aware of the activities of the School Mathematics
Study Group?
*

**Brown:**
Only very little. I knew of one or two people that were involved with it.
I knew that Moise was certainly involved with it. What the nature of that
involvement was, I did not know. I was not particularly interested in
that area. I didn’t know too much about the New Math, although it looked
like the right thing to do, and that is an indication of how little I knew
about the whole subject. But, it was dominated by people who really did
not know very much about teaching, interestingly enough. I shouldn’t say
dominated; some of the people I knew involved, like for example, Moise.
Moise did not know much about teaching. He was not a good teacher in the
sense of classroom teacher, and I think he did not have an extremely good
feel for how people learned. I think they were interested in syllabus,
[Roberts concurs] and less aware of how people learn, rather than
probably what we might now call a constructivist view, that you learn by
constructing your own mathematics, constructing your own learning, which
is probably the real legacy of R. L. Moore. So, it’s a kind of
generalized constructivism without any of the details and without any of
the frills.

**Roberts:***
Did it seem to you, at the time, or does it seem to you, in retrospect,
that there was a large participation of Moore students in the New Math?
*

**Brown:**
Yeah, and I think there in general, there is this time around too. I’ve
heard people make comments about how many of the folks who have gotten
involved in the reform of the last ten years actually come from the
Moore School. Now, that may be partially this view about constructivism,
but it’s also an interest in education. I wrote a paper, which was
actually a talk I gave at a conference in San Marcos [Southwest Texas
State University] about ten years, fifteen years ago, basically a
conference in Bing’s honor after he died, and one of the comments I made
at the end of that paper was that one of the things that distinguished
Bing from many other mathematicians was that he never separated teaching
from research. He would start talking about one, and end up talking about
the other. To him it was one thing. So, he, I think, had a better
perception about teaching than most other people did, than most other
people from the Moore School. I think there is a legacy, and I’m
repeating myself. The legacy is probably this — if you learn it by
yourself, you’ll learn it much more deeply.

**Roberts:***
Now, Bing, himself, was involved in SMSG to some extent.
*

**Brown:**
Yeah, I didn’t know much about that, other than that he gave lots of
high school lectures, public lectures, and things of that sort. He was
a marvelous expositor and gave good dramatic talks. He was very good,
so if he wanted to lecture, he could. [laughs]

**Roberts:***
I take it you did not participate in SMSG yourself in any way.
*

**Brown:**
That is correct.

**Roberts:***
and you didn’t participate in any other program that would have been
labeled a New Math program?
*

**Brown:**
Not at that time, no. I was certainly not interested in that.

**Roberts:***
Did you have any personal interaction with any New Math critics?
I guess now is the time to ask about Morris Kline.
*

**Brown:**
I remember reading Morris Kline’s book and not being — what was
it — [Roberts and Brown together] *Why Johnny Can’t Add*
[(New York: St. Martin’s Press, 1974)] and not being overly impressed
by that. There was something about it. I think I was prejudiced against
Morris Kline, and I’m not sure why. My guess is that I had just a general
feeling of support for the New Math, although I could see it was working
very bad locally in the high schools. I think when I read Kline’s book,
at that time, I thought that this was just a blowhard. I’ve recently
reread it, and his criticisms were right on, and his book
*Why the Professor Can’t Teach*, [(New York: St. Martin’s Press,
1979)] you could publish that right now. It would maybe be even more
to the point than it was at the time he wrote it. So, I very strongly
subscribe to his criticisms of both the New Math and the structure of
mathematics in college departments, their tilt toward research and its
effect on teaching. So, I’m a big follower of Morris Kline now.

**Roberts:***
I see. Now, the New Math is often identified with several notions: its
emphasis on set concepts, emphasis on the function concept, emphasis on
the distinction between numbers and numerals, doing arithmetic in bases
other than ten, algebraic structures. And critics, of course, including
Kline, complained that these were very much overdone. Do you agree with
that?
*

**Brown:**
Absolutely. I even remember, if for no other reason, the fact that the
teachers couldn’t understand the material to teach it. At that time I had
friends who had kids in school who were being subjected to some of the
horrors. I mean, if you can imagine the mathematician who goes to one of
these mini-classes and hears a teacher who doesn’t understand the
difference between the associative law and the inverse law and so on, has
them confused explaining these concepts to the parents. [Roberts chuckles]
That was fairly common, and more of these fine technical differences
between cardinal and ordinal numbers for eight year olds. There were a
lot of very weird things. I think there’s a nice summary by — gee, I
would have go look up the book. I think it’s Mathematics That Works, or
something like that,4
one of the AMS publications of the last year or two on MAA Notes
summarizing that the syllabus for the New Math was very well oriented
towards having a seamless link from elementary school to graduate school,
and that it was very good there. But also, pointing out that a number of
things that were introduced at that time have actually remained quite
successfully in the syllabus like Venn diagrams and elementary set theory.
The doing arithmetic bases other than ten, you can still see that stuff
reading the NCTM Journal [sic], people having speed tests — how fast can
you add in base seven and various things like that. The reasons for it
have been lost. I think there were a number of these attempts to get the
students to understand much deeper algebraic structures, were just
wrong-headed for that age group or for the teachers. I think Kline, who
always believed, or seemed to believe, that the fundamental approach of
mathematics should be applied mathematics would have been critical of
taking things that were really substantial and had substance and
replacing them with abstractions that were much harder to understand and
had no clear reason.

**Roberts:***
It’s been occasionally proposed that the New Math is a result of the
influence of*
*
Bourbaki* *
on mathematics. Do you subscribe to that at all?
*

**Brown:**
No, I don’t think Bourbaki has had that much influence on American
mathematics. I can see some similarities, and I can see someone guessing
that there might be a relationship. I think it was more the influence of
professional mathematicians going back not to Bourbaki, but to an earlier
time of trying to have mathematics not have contradictions and not have
circularity. I think, at that time, there was a belief that one could
develop a program for K-12 which would work that way, and I don’t think
anybody’s ever developed a particularly good one for Euclidean geometry
that works that way. [laughs] I think that was doomed to failure, and I
think it was the — you know, someone pointed out rather recently that
one of the reactions of the NCTM, one of the reasons — okay, I’m
recalling this a little bit better. You know there’s been a fuss recently
with letters to Secretary of Education Riley?
[David Klein
et al.,
“An Open Letter to United States Secretary of Education
Richard Riley,”
*Washington Post* (Nov. 18, 1999)]

**Roberts:***
Oh, yes.
*

**Brown:**
The main complaint was that there were no, quote, mathematicians, in
this group. [Roberts concurs] Someone pointed out to me, and I don’t
know the answer to this, but it’s an interesting observation, that this
very likely was a reaction of the K-12 community, the K-12 leadership to
professional mathematicians who had their day, who screwed up, and they
were just going to leave them out of it. Now, I think that’s not entirely
true, but there may be an undercurrent of truth there, and a justifiable
one. [both laugh] So, I was actually one of the signatories of another
letter that went to Riley of a much smaller number of people and not
publicized, trying to point out that there were quite a number of people
in the mathematics community who had been taking an interest in education,
supported it, who were not going to support some particular ones of the
seven or eight math programs, but felt that to just discard them
automatically was a very poor and uninformed idea. But, in terms of math
reform, I’ve been very much involved in the new reforms. I’m not sure if
you were aware of that.

**Roberts:***
No, I did want to ask you more about that. Let’s see if we can cover a
little bit more here on the New Math. Were you much aware of the work of
**
George Polya*?

**Brown:**
I remember reading Polya’s book [**which one?**] [sic] twenty years ago,
twenty-five years ago, and thinking, *okay, well this is okay, but
it’s not the Moore Method*. That it was much less deep in its approach
and much less general, and much more oriented to solving a certain kind
of problem, and what he basically did was collect various kinds of
problems and suggest solving them this way or that. I think there’s a
lot to Polya’s materials. It’s one of very few books if students want to
know *how do I go about learning how to solve problems*. There
aren’t many places you can send them. Unfortunately, Polya’s staff is a
little bit too advanced for most of them. So, there’s not a lot in that
area. So, I liked Polya’s work, but I thought it was a somewhat different
direction from Moore’s.

**Roberts:***
Okay, and were you at all aware of Polya’s — he was another critic
of the New Math.
*

**Brown:**
No, I was not aware of that. My understanding of the New Math is coming
from doing my own historical research, and if I had about five minutes
I could look up some of these references that you might be interested in.
There’s a very nice attempt to justify the place of the New Math and the
so-called New Reform math and the different roles that they play and
their historical relationship. But, I’d have to go over to my bookcase.
However, I would like to suggest — about how much more time do
we have?

**Roberts:***
Well, whatever time you have. I’m completely flexible.
*

**Brown:**
If you’d like to do it for about another ten or fifteen minutes, I need
to take a break. So, do you want to call me back in about five minutes?

**Roberts:***
Sure. That would be fine, okay.
*

**Roberts:***
Okay.
*

**Brown:**
I did have time to look up the article I wanted to refer to. It’s a
report called *Assessing Calculus Reform Efforts* by the MAA [1994].
It’s called *A Report*, by [Alan C.] Tucker and
[James R. C.] Leitzel,
and there’s what I think is a — you, as a better informed
historian of this area might disagree, but I think there’s a very nice
description of the period from the New Math to the Reform Math written by,
I think, Al Tucker called “The Modernization of Collegiate Mathematics
Beginning in the Fifties.”

**Roberts:***
I’ll have to look at that. Good, okay. Now, another figure who was
prominent during the New Math Era was
**
Max Beberman* *
of the University of Illinois.
*

**Brown:**
Yeah, I never heard of Max Beberman. This is the first I’ve heard of him.
I was just rereading your questions five minutes before you called so as
to be definitely well-prepared, and no, I don’t know anything about that,
so I will have to look that up.

**Roberts:***
Yeah, he was not a Ph.D. mathematician. He got a doctorate at Teacher’s
College, but was at the University of Illinois. They had a program there
beginning in the early 1950’s, well before Sputnik, which had a number of
the features that people think of in association with the New Math, and
in particular, he was noted for his emphasis on discovery as a teaching
tool. But, if you don’t know anything about him, you can’t answer a
question whether there was any relationship between his discovery methods
and Moore’s discovery methods. But, I can ask you, how do you feel, in
general, about the use of discovery-type methods at the pre-college level?
*

**Brown:**
As soon as you used the word discovery, I had to start thinking about
what my views on discovery are. I have not had a chance to really clarify
my views on discovery, but I have some thoughts, which run something like
this. First, there’s a question of what does one mean by discovery?

**Roberts:***
Yes.
*

**Brown:**
and how structured is it? When you were talking about Beberman
vs. Moore and Moore vs. Polya, those are really the issues that come up.
Polya is much more structured than the Moore, but the Moore Method, as
Bing did it, and I’m pretty sure Moore did it in a very similar way, you
have laid out for you a collection of definitions and a collection of
theorems. That’s a lot of structure, and well, Euclid
does the same thing.
[chuckles] He gives you five axioms, of which only three are really any
use, and then a collection of theorems. So the criticisms of discovery,
the people who like to criticize discovery, usually don’t present it that
way. They present it as *okay, class, we’re going to invent algebra.
Everyone sit around and do something with a pencil and talk to each
other, and after fifteen minutes of discussion, we’ll decide what
algebra is*, and of course, that’s the extreme that nobody subscribes
to, so there is a question of how much structure is there. And I think in
every discovery course that one would develop the issue of how much
structure should be used and how much is most effective, when should
you not have structure, and so on, is crucial in how that course would
go, and how that learning would go. I don’t think anybody has a solution
to that problem. I’m not sure there is a single solution.

**Roberts:***
Just in terms of, say the undergraduate and graduate curriculum, should
there be sort of a transition where you have more Moore Method as you go
along in your undergraduate and graduate training?
*

**Brown:**
No, I haven’t felt particularly that way. I think that for people who are
going to be involved in understanding concepts, developing intellectual
concepts, to have been exposed to a Moore type method, at least once, is
very useful and important. That it be used for everything is undoubtedly
a mistake, and how much it should be used is really a function of the
individuals that are quote, teaching, and the individuals who are quote,
learning.

**Roberts:***
Okay.
*

**Brown:**
When I think back — you know you asked me about who else influenced
me — I think, I’ve often thought about the fact that Bing and Buck
were very different and they were both very important for me. And actually
one of the best courses I can remember was [phone connection lost]

**Roberts:***
You were saying there was one particular course that [connection lost
again]
*

**Brown:**
Hello.

**Roberts:***
Yes, I’m here.
*

**Brown:**
There was another professor at Wisconsin named
[Richard] Bruck,
straight
lectures from beginning of the hour to the end of the hour, absolutely
beautiful, completely worked out, very carefully worked out so as to be
as understandable as possible. It was just gorgeous. As a matter of fact,
I took a course from him as an undergraduate in what was then called
theory of equations because I wanted to be a number theorist originally.
It was after I met Bing that I changed. But, I still remember those
lectures. I did not understand the subject as well as when I took a
course from Bing, but I still know that material because of that, which
is not quite the same as understanding the subject, so I think that a mix
is what everyone should have. I think it would be a terrible mistake, as
some institutions have, to have all of one or none of one.

**Roberts:***
Okay, you’ve been at Michigan then since
*

**Brown:**
’58.

**Roberts:***
since ’58. How would you describe the relative importance in your career
of research, graduate instruction, undergraduate instruction, and
administration?
*

**Brown:**
I always was interested in teaching, but, of course, during the earlier
parts of my life, research was 99% and teaching was only 5 or 10%
since we all have 36 hour days. [Roberts laughs] At this stage of my life,
I’m sixty-eight now, I’m much less interested in research in mathematics,
in pure mathematics, and I’ve gotten much more interested in the almost
new career that I’ve constructed beginning around eight years ago with
the math reform group, which has led me into more careful analysis of what
teaching is all about. I have a kind of pilgrim’s progress from the
original goal of changing the syllabus — I’m sorry, the original goal
was introducing the new technology. I had an epiphany with the TI-81
calculator, and I thought I suddenly saw the way that we could make
calculus teachable, and it was really an epiphany. Then I developed a
course using the TI-81. Is that right, the 80, was that the first one?
Then I discovered that you could not use that TI-80 with a standard
calculus textbook, that the calculus textbook was doing everything it
could to defeat you, so I realized that the technology issue then led to
an issue of syllabus, and then discovered that the issues that started
arising were not just syllabus, but teaching, and eventually I got in on
the secret that the real issue was learning.

So, it was kind of a pilgrim’s progress that took about four or five
years and, at any rate, as a result of this feeling that I could do
something with the TI-80 to completely transform the teaching of calculus,
I got a lot of support from the chair and wrote an NSF grant, and we have
totally revamped the calculus around 1992 at Michigan. You can read about
that in an article I wrote in something called
*Calculus, the Dynamics of Change*, it’s another MAA Note
[published 1996] by probably
[A.] Wayne Roberts,
I think was the editor.

**Roberts:***
Okay.
*

**Brown:**
I have about a ten-page report on the history of that, how it worked, and
what the aspects of the program were. But, what we have is — we
adopted the Harvard book at that time, which had just been published in a
pre-publication form, and team homework and cooperative learning in the
classroom, and we started getting more involved with having a more
extensive training program for the new instructors. Things like
mid-semester feedback where someone comes into the class and does
a — are you familiar with that?

**Roberts:***
No.
*

**Brown:**
SGIDS is the technical name. Somebody, not even necessarily connected
with mathematics, comes and visits the class for say, half the hour,
and then the instructor leaves, and then that person asks three questions
with the students in moderate-sized groups, maybe 4–6, and tries to get
a consensus on a) what’s going well with the course, b) what would you
suggest for improvements — maybe those are the two things, and then
collects those and then has a discussion with the instructor. But while
observing they would be counting how many people are in the discussion,
how many people are actually participating, and various things like that.
So, these would be people who have had some training in observing
what goes on in a classroom. Then that feedback would go, in a completely
secret fashion, back to the instructor. It’s called mid-semester because
there’s time then for the instructor to change gears and adjust to the
changes. We found that very effective. We found that that helped, not only
that, but we found that people, with almost no exception, people found it
a very helpful and useful thing to do and did not at all feel like their
sovereignty was being trod upon. That was one of the features. And some
of the features here then were syllabus, some were we were using the
graphing calculator throughout, and then there were the educational
ideas of cooperative learning, which we introduced before we knew
anything about what it was.

**Roberts:***
Okay. Now, have you been at all involved in pre-college math education?
*

**Brown:**
Not very much. I’m starting to get interested because of the — I
have to learn more about it because I’m now the director of our
elementary program, which includes the first two years, and the nexus
between high school math in the last year or two and college math in the
first year or two is an issue that keeps coming up. We are expected to,
quote, tell the high schools, unquote, what they should be doing about
this or that. With these new NSF-sponsored math curricula in the high
schools, this is causing lots of problems for those students and the
colleges that are accepting them, especially when one of them comes from
a reformed high school curriculum into a non-reformed college curriculum.
Well, there are four possibilities; you can see all the difficulties;
only two of the possibilities are good. So, I’ve had to get more involved
in that. I’ve also, while teaching some of these middle level math
courses, started realizing that the qualifications of the teachers in
high school — really, we need to do something about that. The
question is what. That involves re-examining our undergraduate program,
so these issues all come together.

**Roberts:***
Do you have any particular comments you’d care to make about the
*
NCTM Standards?

**Brown:**
No. [chuckles]

**Roberts:***
Okay. What’s your general assessment of the relationship between
professional mathematicians and other math educators, teacher educators,
school teachers?
*

**Brown:**
Well, I think it needs a lot of improvement. I remember giving the first
talk — You seemed to have noticed in your note that I was a member of
AMATYC [American Mathematical Association of Two-Year Colleges]. I think
I’ve let my membership slip there, but I joined AMATYC because I felt
that was a place where the teachers were dealing with calculus on a
large level, whereas the NCTM, of course, has a different orientation,
and the teachers would be more experienced than our teachers were.
Because our calculus teachers are frequently very inexperienced. They’re
beginning graduate students or beginning Ph.D.’s who have gone to very
prestigious places where they didn’t do that much teaching. So that
joining the AMATYC was a way for me to find out what some of the issues
were in the teaching of calculus. I remember giving a talk to an AMATYC
group, holding up my card, saying *I’m a card-carrying member of
AMATYC* and getting this huge applause. [both laugh] I know I was the
only member of the faculty in math [at the University of Michigan] who
had ever been a member of AMATYC. Now I’m a member of NCTM, but I think
my AMATYC membership has slipped, and I think it’s an important
connection that has to be made. Right now, at Michigan we’re trying to
put together some relationship between our math ed, which is the school
of ed, and the math. I think, at Michigan, as at probably many other
places, during the post New Math period, and maybe in the New Math period,
the school of ed math people basically got out of college math education
and went into, I guess, administration, so that left a big gap. There was
almost no relationship between the mathematics department and the math
ed people. We’ve had one or two or three people in math who had a
connection with them, but then those people were marginalized within the
math department. They’re now all retired, but Michigan has hired some
stronger people in the school of ed who are doing what is considered to
be quite respectable research even in the math department in terms of
what they’re doing, and have strong research credentials. So, there’s
been an attempt to move together and have a joint seminar. We do have a
couple of joint seminars now. There’s a school of ed seminar, but we also
have an educational [phone disconnects]

**Roberts:***
Hello.
*

**Brown:**
At any rate, the short version is that there is much more of a getting
together of people in the school of ed math and a subgroup of people in
the math department who are interested in math education issues. So,
we’ve got both formal and informal arrangements, and we’re getting some
support from places like deans and so on, so that’s moving. The math
department and school of ed just made a joint hiring of a major figure in
the subject in math,
Hy[man] Bass,
who really cements this, because he comes with tremendous mathematical
prestige in the math department, and he’s interested in education. So,
that part is changing here, quite a bit. What the ultimate effect will
be is not that clear.

**Roberts:***
Now you’re also an AMS and MAA member. Any comments on the effectiveness
and appropriateness of the roles taken by these organizations in
promoting math education?
*

**Brown:**
Well, the AMS has changed a lot. Fifteen years ago the AMS’s view was
that none of these are issues for the AMS. There was a committee called
The Committee on (I was chair of it and presided over its demise — I
mean, I helped kill it.) I forget what it was called. It was called
something like the Committee On Education And Hiring or something like
that. It combined both the issues of education and the problem of the
hiring of mathematicians — two issues that were going to become
rather important — and by the fact that they were relegated to a
committee was
kind of proof that no one was interested in it. Shortly afterward, the
AMS set up its own committee on education with a bunch of higher brass
in it. There was talk about the AMS and the MAA joining together, because
their roles were not so clearly different. I think that’s not going to
happen, and I think the AMS will drift back towards a stronger emphasis
on research, but I don’t think it will ever be one hundred percent
research again. The AMS has changed. I think it’s going to change back a
bit, and I think there still is a large majority of so-called research
mathematicians, which defines itself as mathematicians, that will
distance the AMS from educational issues. Two years ago, the
*Notices* of the AMS was just replete with letters and articles
about educational issues. In the last couple of months I don’t think I’ve
seen any. So, I think there is a retreat going on there, and I think that
will continue. That may be partially the fact that the MAA will just, you
know, that those things will be seen as more properly with the MAA, but
maybe with official support by the AMS. A lot more offices of the AMS are
connected with educational issues than in the past.

**Roberts:***
There are several issues that have produced controversy in pre-college
education in recent years. I wondered if you would care to comment at all.
The role of calculators and computers in the classroom, for instance.
*

**Brown:**
Yeah. I don’t know the answer. We don’t even know what the role of
calculators and computers in the undergraduate classroom should be. We
are struggling with the problem — when we started the new calculus
course the position we took was whatever technology can do, you can
have it all the time in class, outside. If you want to roll in a huge
computer, fine. We were going to worry less about the disadvantage or
advantage that certain students would have over others, against the
alternative of simply trying to ban these things. And we believe that
you can always develop courses where — that would not defeat the use
of these machines, but where the use of these machines would smoothly
interact with the questions, or where you could design questions that
were so conceptual in nature that the machines were not that relevant.
So, we didn’t see that as a bad thing. The idea was to incorporate
technology, recognize that it was going to be changing, and that it
would become more and more powerful, and gradually adjust to that. The
thing that has happened is that we have discovered that various basic
kinds of things that we currently believe everyone should know like
~~\textit~~{what is the derivative of the sine and the cosine or what does
the graph of __\( e \)__ to the __\( x \)__ look like} that there is more and more
of the so-called calculator dependency. Students very frequently will not
know what
__\( e \)__ to the __\( x \)__ looks like without hitting a button, and this is
disturbing to a lot of us. We haven’t worked out the answers to that, and
it’s the same kind of problem that teachers are having in K-12. So how do
you use the calculator to help understanding and how does it defeat
understanding, and does it defeat understanding and what kind of
understanding are we looking for? The technology change has been so rapid
that we haven’t had time to figure those things out, if we ever will.
So, to say keep calculators out of the classroom is crazy, to say use
calculators to add __\( 2+3 \)__, right now most of us would think is crazy;
I don’t know what people will think in ten years. Long division — you have that in a question here. I think there is something that one can
learn by writing out the algorithm for long division, but it’s not a
hell of a lot. And it’s nice, for example, to prove that every fraction
has a repeating decimal expansion by appealing to long division. That’s
a rather thin prize for the pain that long division has led to, [laughs]
and you know, I can remember as a student, learning about mantissas and
things like that, how to calculate logarithms, and then someone told us,
hey, you know there’s this thing called a slide rule. [both laugh]
So, I think one’s view changes when you have a whole generation that is
going to add __\( 7+5 \)__ on a calculator, and those people start teaching the
courses, their attitudes may be very different, and they certainly won’t
be emphasizing long division, as we no longer emphasize extraction of
square and cube roots, although there is something to be learned from
that too. So, I think we just have to work on those things, and we’re
victims right now of rapid technological change.

**Roberts:***
What about the notion of segregating math students according to ability
or career goals in K-12?
*

**Brown:**
We do it in college, and I’m unhappy with it. I don’t like it, but
I don’t know an alternative that is both educationally and politically
acceptable. So, for example, we have calculus reform in our standard
calculus course, but we don’t have very much reform in our honors courses,
and I believe that’s got less to do with deep educational thought about
honors students than it does with the attitudes of the people that teach
the courses. They don’t know what the new stuff is, they don’t want to
know what the new stuff is, and they want to just keep teaching what
they’ve always been teaching. I think there is a lot of that. Some of
the people do have philosophical backing, but it’s frequently not clearly
thought out. It all hides under words like *rigor*. So, in terms
of the segregation in K-12, I think the same kinds of issues come up.
Maybe less that the teachers are intellectually lazy in terms of their
learning how to teach, but the issues of segregation of students
according to ability and career goals, I think, is often more of a
political issue within the schools than it is an intellectual issue.

**Roberts:***
Is there a core of mathematical knowledge that you feel that everybody
ought to know, all students even if they’re not going into a mathematical
field?
*

**Brown:**
I don’t know the answer to that. I don’t know the answer to that, and
I know that the amount of mathematics that’s available is so broad now,
if you take a look at math majors in many, many schools, but particularly
large universities, it is so spread out that there are fewer and fewer
things that one can call core courses, and those which are core are
starting to give way at their core. Students who go through a core area
will just find that a great amount of mathematics that they could have
learned or could have used effectively was just not made available to
them. So, this is another area that I just don’t know enough about, and
my perspective is largely from within the University of Michigan, and
it’s something I’d like to learn about, and we are getting together with
other people and trying to work out whether there should be a core. As
you know, the MAA has a committee that’s trying to develop a new core
program. The last time they did a core program in the 1960’s it was
perfectly geared to graduate school. What they’re going to suggest this
time is less clear, and all of that really, I think, exposed itself with
the calculus reform issue. I think that was just the first place that
that kind of issue opened, and now we see it’s much more general.

**Roberts:***
Now, I take it that not all mathematicians would agree with your views
on education.
*

**Brown:**
Oh, really? [Roberts laughs]

**Roberts:***
What I’m particularly interested in is are you able to detect any
pattern in the background or training of those you find congenial
compared with those with whom you might disagree?
*

**Brown:**
I would say that the people who are most sympathetic to calculus reform
in our department, for example, and in other departments, have been
more connected with applied mathematics. There’s almost a natural
welcoming of those ideas in applied mathematics, possibly those people
interact more with other people who use mathematics as a tool rather
than set it up as an icon. Also, people who were connected with, in
their own education, where there was perhaps a more serious interest in
education. For example, people coming from English, British education
seem to be more amenable to looking at some of these reform issues. When,
in mathematics, I think the most critical have often been people whose
mathematics is very pure, and most of the objections to the various kinds
of reforms in the calculus have been not from the engineers, not from
the physicists, but from mathematicians, who are afraid that the
engineers and physicists are going to get shortchanged. So, that’s the
only kind of distinctions I’ve been able to see.

**Roberts:***
Now, you’ve — *

[Interruption as tape is changed.]

**Roberts:***
So are you somewhat of an outlier here, being a pure mathematician
interested in these reform ideas.
*

**Brown:**
Yeah. To me it’s a question of lots of new learning and trying to,
if not change my attitude, at least recognize other attitudes. Let me
give you an example. Morris Kline felt that mathematics should be, at
its base, applied. I remember reading
Courant
saying the same thing and thinking…
*Oh, my God, if that had been — I would never have gone to
graduate school if that were mathematics*, and that’s probably true.
I think that may reflect the attitude of a lot of people, including math
majors I’ve advised and talked to. There are those who just love the
purity part of the mathematics and who are totally uninterested in the
applied part, and there are those who like and need the applied part,
and they’re not particularly bowled over by the pure part. I think this
is one of the real problems we have in developing something like a math
major that has any coherence. From my own part, I was referring to Morris
Kline and thinking of his comment in attacking the New Math. This is
probably — I don’t know whether this was in
*Why Johnny Can’t Add* or
*Why the Professor Can’t Teach*. He said, *One of the most
important issues in science — is it disease, is it war, is it
famine? — According to mathematicians it’s the Königsberg
Bridge problem*.
[both laugh] That’s Morris Kline for you. So, he was quite against
mathematics that was not based upon some real applied mathematics.
I can’t buy that. I can understand it, and I see it as a very valid view,
but it’s not necessarily the reason that loads of people love
mathematics, and it’s not the reason that people read
Martin Gardner.

**Roberts:***
Okay. We’re coming down to the end here. You’ve already mentioned a
couple of articles that you’ve written about math education.
*

**Brown:**
Well, I’ve only mentioned one because I’ve only written one.

**Roberts:***
Oh, okay.
*

**Brown:**
about it, and that’s

**Roberts:***
the one in the volume edited by
**
Wayne Roberts*?

**Brown:**
Yeah. That’s basically a description of the program, a history of the
program. I’m actually planning to start doing some writing on issues,
but there’s nothing right now.

**Roberts:***
Okay. You also mentioned a talk that you’d given at this conference in
Bing’s honor after he died.
*

**Brown:**
Oh, yeah.

**Roberts:***
Is that published?
*

**Brown:**
Yeah. That is published. It’s in a journal — what the heck is the
name of the journal? It has the word topology in it probably and it was
published at Auburn University, and the editor was
Donna Bennett.
If you contacted her, she could — she lives in Auburn and I think
she may have retired, but she was the editor for many years.

**Roberts:***
Okay.
*

**Brown:**
It was a publication of the talk that I gave. If you can wait a second
I’ll see if I have a copy of it.

**Roberts:***
Sure. [pause]
*

**Brown:**
Nope, I gave it away, so it is in that journal, and I’m sure she could
locate it for you, or if you need to get that reference, anybody at the
University of Auburn in their topology department — they have two
departments, an algebra or something like that, and a topology
department — can steer you to the journal. [Morton Brown,
“The Mathematical Work of R.H. Bing,”
*Topology Proceedings* 12 (1987): 3–25.]

**Roberts:***
All right. Then, my last question then which is whether you possess
unpublished material which might in the future be profitably studied by
historians of mathematics or math education. Do you keep your
correspondence, for instance?
*

**Brown:**
I do, but I don’t think any of it is worth saving. I don’t have that
view. I have not started collecting all of my correspondences and so on,
but I think there would be very little that would be of use or interest
to you.

**Roberts:***
I’m not speaking necessarily of me, but sometime in the future somebody
else might be interested in your specific areas of research, for
instance.
*

**Brown:**
Oh. Well, in terms of research, I have some unpublished materials,
yeah, and they’ll probably remain that way for awhile.

**Roberts:***
Oh, well, just curious to know whether — do mathematicians at
Michigan, for instance, donate their papers to the university when they
retire?
*

**Brown:**
Some do. Very frequently they just leave their stuff out in the hall.
[Roberts laughs] I think that Germanic attitude of feeling of the
importance of one’s papers I think — I think if
Riemann
were living next
door, we would probably say, *okay, well let’s collect his stuff*
[both laugh], but I think most of us don’t have such an exalted opinion
of our unpublished work. The one thing I do have is a letter, which I
picked up out in a hall, but I’m not sure if this is of any interest to
you because it’s got nothing to do with education, is a letter that
Henry Whitehead — is that a familiar name to you?

**Roberts:***
Sure, sure.
*

**Brown:**
That Henry Whitehead wrote to, now who was it at Princeton, at any rate,
he was explaining to this person what the flaw was in his false proof of
the Poincaré conjecture with a picture, and I’ve always felt that that
was a kind of valuable thing, and I have it stashed away somewhere.
[Roberts agrees on value] But, just as an indication of the attitude
towards history, it was just lying out in the hall with some of
Whitehead’s papers. I was a big fan of Henry Whitehead, so when I
saw that I grabbed a couple of the papers, and there was this letter
stuck inside one of the preprints.

**Roberts:***
That’s interesting. No, I interviewed
**
Peter Hilton* *
earlier, and he, of course, was a student of Whitehead.
*

**Brown:**
Yes, yes. Yeah, I know Peter. Whitehead was also a somewhat important
figure for me, although I knew him for a very short time. When I proved
the Schoenflies theorem, which was the theorem that made my name when
I was a young person — actually I went, I had proved the theorem
here at Michigan, and then showed it to Moise and
Samelson,
and neither of
them believed it. [Roberts laughs]. It was very simple and very small,
and as Moise later described it, it looked like it was all done with
mirrors and smoke [Roberts laughs], and eventually they agreed that it
was correct, and fortunately Moise was editor of the *Notices*,
or at least wherever it was that bullet announcements were made, and
convinced me to get this out right away, and I did get it out right
away, and that was lucky because it was not too much longer after that
that
Marston Morse
came up with a paper that had resolved the part of
[Barry] Mazur’s
results that would have also given a complete conclusion.
There is a historical thing there because
Frank Raymond
claims to have a
letter from Morse giving me precedence, but I’ve never seen the letter.
At any rate — I think I just lost my train of thought.

**Roberts:***
You started out talking about your admiration for Whitehead.
*

**Brown:**
Right. So, there was a November meeting, I think I got the result in
October, and there was a November meeting of the American Math Society
somewhere around Cornell or something like that, and I went to that
meeting. And I was not scheduled to present a paper; I didn’t have it
in in time. But Whitehead was at that meeting, and someone introduced
me to Whitehead, and said,
*this young gentleman has proved the Schoenflies theorem*, and
Whitehead, said, *oh, really?* and he just dropped these three
other distinguished mathematicians he was talking to and came over and
asked me what the proof was. And I outlined the proof with four pictures,
and he looked at it, and he said, *that’s terrific; that’s fabulous!*
and I’m thinking to myself, *this guy is faking*, and then he went
over and completely explained the proof to these other three men.
[both laugh] So, I’ve never forgotten that.

**Roberts:***
That’s a wonderful story. [laughs] That’s great.
*

**Brown:**
And then he came and visited Michigan for a couple for weeks and gave
some lectures and then went back to Princeton and died only a few
months later. He was a person that burned the candle at all three ends.
I’m not sure if you heard about Whitehead, but he was a fine
mathematician, and I had read several of his papers, very, very deeply,
and one very long paper with a student, so that was an influence.

**Roberts:***
Oh, good, okay. Well, that really does bring me to the end of my
questions unless you have anything else you want to add.
*

**Brown:**
No. [laughs]

**Roberts:***
I do appreciate this, and did you get that release form that I mailed?
*

**Brown:**
Yeah, I’ve got the release form, and I’ll send that out to you. By the
way, do you know that at San Marcos they have this little hagiographic
area for Bing?

**Roberts:***
No.
*

**Brown:**
There are two great graduates from San Marcos, one is Bing and the other is
Lyndon Johnson,
and they have done some collecting of Bing memorabilia there too.

**Roberts:***
I see.
*

**Brown:**
And one of the things that happened at this conference we had in Bing’s
honor — I think at this time Bing was still alive for this one, this
was a conference in 1990, they did some videotapes of various people
talking about Bing and topology, that rotated about Bing and so on, if
you’re interested in that. You could check on that. But, there is so
much R. L. Moore School stuff that I’m sure you’ve got plenty.

**Roberts:***
Yeah. Okay, well, I hope maybe I’ll run into you in Washington.
*

**Brown:**
Okay, great. Introduce yourself. I will. Would you go to the history
of math meetings?

**Roberts:***
Yeah, I’ll be definitely at the history of math sections. I’ll be
talking at one of them.
*

**Brown:**
Okay, well, I’ll see if I can look you up if I’m free for that meeting.

**Roberts:***
Okay.
*

**Brown:**
It was nice talking at you.

**Roberts:***
Okay, very good. Thank you very much.
*

**Brown:**
Good-bye.