Celebratio Mathematica

Morton Brown

Morton Brown

by David Roberts

Mor­ton Brown was born and edu­cated in the Bronx, New York. He re­ceived both un­der­gradu­ate and gradu­ate train­ing in math­em­at­ics at the Uni­versity of Wis­con­sin in Madis­on, com­plet­ing his Ph.D. in 1958 un­der the dir­ec­tion of R H Bing. After a year at Ohio State he took a po­s­i­tion in math­em­at­ics at the Uni­versity of Michigan in Ann Ar­bor, where he has been ever since. His early ca­reer was very much fo­cused on re­search in to­po­logy; he gained prom­in­ence in 1960 with his proof of the Gen­er­al­ized Schoen­flies The­or­em. (Pro­fess­or Brown tells an an­ec­dote about his in­ter­ac­tion with the noted Eng­lish to­po­lo­gist J. H. C. White­head on this top­ic.) Pro­fess­or Brown was not act­ive in edu­ca­tion dur­ing the New Math era, but has sub­sequently re­flec­ted on the les­sons of that time; among oth­er re­marks, he de­scribes his chan­ging as­sess­ment of the cri­ti­cisms Mor­ris Kline dir­ec­ted at the New Math and at the con­duct of math­em­at­ics edu­ca­tion gen­er­ally. In re­cent years, mo­tiv­ated es­pe­cially by his at­tempt to ac­com­mod­ate tech­no­logy in the un­der­gradu­ate math­em­at­ics classroom, Pro­fess­or Brown has been act­ive in the cal­cu­lus re­form move­ment. In this in­ter­view Pro­fess­or Brown also dis­cusses his ex­per­i­ence as a stu­dent in classes taught in the style of R. L. Moore (primar­ily by Bing, but also by F. B. Jones), and his sub­sequent modi­fic­a­tion of the meth­od in his own teach­ing; he com­ments on his in­debted­ness to Creighton Buck, a Wis­con­sin pro­fess­or who did not use the Moore Meth­od; and he de­scribes his in­ter­ac­tion with oth­er stu­dents of Moore, es­pe­cially E. E. Moise.

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.