Henry Pollak Interview, Part 2 of 4

  • Pollak: I was disappointed in physics because the true understanding of the problem and the appreciation of the approximations that were made in going fromthe real world to mathematics never happened. I think probably that's what began my feeling that a major purpose of teaching mathematics is understanding and that in doing applications of mathematics youalso ought to be doing understanding. You should be understanding something outside of mathematics. That has influenced me in various ways which will come up I think later in the conversation ifwe get that far. Roberts: In Mathematical People you mention the effect of Ed Begle's teaching approach, what one might call the positive effects of a certain lack ofpreparedness. Now, surely there's a fine line here. I've encountered mathematics teachers whose lack of preparedness became more than a little exasperating. Do you feel it's possible to codify this situation,to train a teacher to attain just the right amount of spontaneity? Pollak: The answer to that really is to think about what the purpose of the spontaneity is and what is lacking in a typical polished presentation thatdoesn't have any. What's lacking is any understanding of why you're doing what you're doing. When Ed [Begle] or a number of other people that I've seen got stuck, you could see the basic principle, the understanding, the idea behind it that they wereapplying, and if, as is so typical in analysis, you start out a proof by saying well consider the following function, [laughter] you don't get any of that. So, yes, you can, of course, expect maybe of your graduate students that some of them are going to look at the proof when it's finished and say tothemselves, now why did that work? What is it that's really going on here? That's good for becoming a researcher in mathematics, but I think that it would be good for everybody to do more of that in the course of education than weever think of doing. That's what this takes the place of. When a person gets stuck you're seeing how they actually work, how their mind operates in this situation, what the principle is that they were thinkingabout, what they had in mind, and that is the thing that I got. I like that so, yes, I think you could substitute by explaining at various times now look, here's what we're going to do and here's the difficulty, and hereis why we're going to do what we're going to do to try to get around this difficulty. All of this makes perfectly good sense, and that could take the place of it, but if you never do that, then I'd rather have you get stuck. I try to do thatwith my students at this point by saying, all right, we want to prove the following. Now, what in the world can we do to involve what we think the answer is into the problem? How do we ever get that phenomenon that we think is going onmirrored in an analytical way? How can you make it come up? You talk about that, and yes, I think that takes the place of getting stuck. Roberts: Is any of this applicable at lower levels of instruction? Is it appropriate for a high school teacher to teach like Ed Begle or an elementary schoolteacher? Pollak: I think all teaching involves a certain amount of acting. You have to pretend that you're seeing this stuff for the first time, and it is only rarelythat that is actually the case I hope. So, yes, I think it makes sense to get kids to think about what are the alternatives here? What's the difficulty here? What are you going to have to tryto get around? What are alternate ways of perhaps getting at the subject? Which is what this is trying to do. I have certainly seen professors who deliberately put an error into something they were doing, in the hope, not always realized, that one of the studentswould catch them because that was part of the education that they were trying to get across. So, I think that there is a place for this, but it's a place for the larger aspect of it which is an emphasis on understanding, not just an emphasis onmanipulation and routine, and so forth. That's what you're trying to get, and one of the ways of getting understanding is when somebody gets stuck. Roberts: I'd like to pursue this just a bit further. At one extreme we have the perfectly polished mathematical lecture, then we come to the Begle case wherethe student learns by watching the teacher struggle Pollak: And this didn't happen all the time, it happened occasionally. Roberts: And finally we come to what some call discovery learning, often associated with R. L. Moore of Texas wherein the teacher doesn't lecture at all, butinstead induces the student to do the struggling. Did you have any course in your mathematical education which could be described as using a discovery learning approach? Pollak: Probably not. I thought about that, and the first time that I can remember somebody actually deliberately doing something like that was a course inso-called currents, distributions on manifolds, taught by a visiting professor Georges DeRham at Harvard at around 1949 or so. On the first day I remember he defined continuity of these distributions. He derived a bunch of properties, and then he came up with a counter example,something that was undesirable, and then he said, now how did we get into this mess? His English wasn't like that, but that was what he said, and we decided that hey, the definition of continuity was wrong, that it was not the one that oneshould use, and so the definition was changed and then a different definition was used for the rest of the semester. I remember that happening and it obviously made an impression on me, so the first time I can remember it explicitly happening was in graduate school andwasn't particularly associated with R. L. Moore or any of his offspring. I think also during graduate school Ray Redheffer was an instructor at Harvard, a post-doctoral instructor, now a professor at UCLA and probably retired atthis point too. He did more of this open-ended kind of instruction. I was only auditing the course and I don't remember the details but I had that impression at the time, so no it probably was done many times and I justwasn't aware of it. Roberts: Do you have any general positive or negative opinion of discovery methods? Pollak: I think that there's an awful lot of room for that. People have caricatured the notion of memorizing and regurgitating and nobody defends that as theonly thing to do. Other people have gone to the other extreme and said never do any of this memorization, and that doesn't make any sense either. I think that getting kids tothink in this way is really an important thing to do. Roberts: Just a couple more questions about - do you have any positive or negative feelings about R. L. Moore and his school of mathematics? Pollak: Certainly I know many of his former Ph. D's. They have been extremely active in both the MAA and the AMS; they are wonderful people, they're welleducated, but that's all I can say. The number of presidents of MAA among his students is enormous. [Roberts concurs] Roberts: Were there any other mathematics teachers at Yale besides Begle who stand out in your memory? Pollak: There are different people who stand out in different ways. After my first semester they had me earn my board, I guess, by grading mathematicspapers. Everybody had to have a so-called bursary job if he was a scholarship student, and that's what I did, so I got to know more people than just in the coursesthat I was taking. There were some very interesting things that happened. I remember one semester grading papers for a professor by the name of Egbert J. Miles I think it was, M-I-L-E-S, and he was teaching calculus. I found out that the students had a habit of getting together for the hour before class and talking over the homework and what was going on, and so I thinkalmost all of the students happened to have that hour free, and I took to meeting with them and just regularly met three times a week before their calculus course class and talked abouteverything and so forth. The main effect of this was that, of the twenty-five students in class, which was about right, eighteen of them earned A's and Professor Miles turned ineighteen A's and the administration at Yale said you can't do that; we grade on the curve and you cannot give eighteen A's. Because of the time on task it was completely clear to me that eighteen students had earned A's in that class, and they really knew their stuff, but theyspent twice as much time on it as anybody else. It just happened, and I spent the time with them. I think he was a full professor and he could make it stick. But it was a very great lesson to me, not onlyabout the misuse of the normal curve, but also about how important time on task really is in the learning of mathematics. So, that was one of the interesting things that happened. I've already mentioned Hille. I took complex variables from him and I remember how tremendouslyshocked he was when I gave a wrong answer once in class. He asked what kind of singularity is something and I said it was essential, and he said, My God, this is only a first order pole and he was very disappointedthat I had done that. Another sort of amusing incident was [William R.] Longley, old man Longley who at that time had taught the differential equations course for many decades andhe had a grader, and the math department said, look you don't want to spend your time taking that course, you can learn that stuff by grading it. So, that's exactly what happened. Longley had a notebook which had all the answers to every problem in the book; it was his book of course that was beingused. I caused enormous consternation one day by finding that one of the answers was wrong in his notebook, and I can remember that as sort of an incident while Iwas there. The others are just anecdotes; I don't think that they are important for your purpose here. But, those are some of the people that I remember. I remember W. A. Wilson who was not a very pleasant person. He took it as his job in teaching second yearcalculus to catch students that were unprepared and really show 'em up. He did it to me once; I had something else and hadn't really prepared for that class, and boy, he let me have it for the whole hour and that never happenedagain. But, those are some of the stray things that I remember. Roberts: What were your dates of study at Harvard? Pollak: I started in the fall of '47 and got my Ph. D. in the spring of '51. Roberts: Who was your doctoral advisor? Pollak: Lars Ahlfors. Roberts: How would you describe the interaction between the two of you? Pollak: There was not a heck of a lot. Ahlfors was a tremendous mathematician and wonderful fellow, but really instinctively very shy. I was pretty shy too.I think maybe once during the whole time I may have visited his house, but he was a very good man to work with. I remember he gave me one of his papers to read, and it was a very important paper on extremal length and contained an absolutely critical lower bound and I,being sort of, at least in part at heart, an analytical manipulator, started playing with an upper bound and got one which in may cases was pretty darn close, and so you know I worked on thatfor several months and went to see him and showed him the latest thing. I said, How are things coming? to him and he said, Well, you've got enough. Start writing it up. [both chuckle] That's how it went. I remember that I had two different upper bounds and I couldn't decide which was better and I was sitting in the father's waiting room at Boston Lying-InHospital waiting for my oldest daughter to be born, tremendous tension, and during that time I figured out that the two upper bounds were independent, that one was sometimes better and theother was sometimes better, and that was sort of a key point that I hadn't settled in my dissertation. I don't remember anyone else that really stands out. I had liked topology very much which is what I had learned from Ed Begle as an undergraduate and hadreally sort of had the idea that I would study topology when I came to Harvard and found out that there was no one there teaching the subject. Hassler Whitney was in his last year I think before moving to the Institute for Advanced Study, and the only thing he was teaching that year was the modernalgebra course out of Birkhoff and Mac Lane, [G. Birkhoff and S. Mac Lane, A Survery of Modern Algebra (New York: Macmillan, 1941)] and so I had to look around for something else and thecomplex variables appealed to me very much and that's what I ended up working on. Roberts: Did you take anything besides mathematics at Harvard? Pollak: I don't think so. Roberts: Are there any other facets of your Harvard experience which were especially valuable to your later career? Pollak: In my last year at Harvard I was a teaching assistant and had the experience of teaching for the first time. One of the things that they did atHarvard which was a very good idea is that they gave you a grade on your teaching. On some unannounced day a senior professor would arrive, sit down in your class, and at the end would determine a grade to be put into your permanent recordas to how you did in your teaching. It was the one and only time that I saw the senior Van Vleck who arrived in my class one day and, at the end, all I remember him saying is Very good, Pollak,except for one thing - you've got to learn the names of your students. And, I have still not learned how to learn the names of my students as people at Teacher's College can attest. So that difficulty I never solved, but thishad a tremendous influence on me. I was very conscious of it at the time. What I discovered, or at least that's what I thought I discovered, so whether it was true or not doesn't matter because I believed it, is that I was notsupposed to take the teaching seriously even though I was being paid to do that. My job was to work on my dissertation and I found that in a given class, say twenty-five students, there would be twenty-five different, absurd hiddenreasons why they weren't understanding what I was saying at the moment. In an average of maybe four hours per student, I could dig back with them individually, find out where the problem was, fix it, and send them on theirway. The system had no interest in giving me four hours per student to try to take care of this kind of a problem, so I remember quite deliberately saying tomyself, I don't like this. If what you're going to do is to pay me actually to do research, even though you're pretending to pay me for doing teaching, then I'm going to go some placewhere what they pay me for and what they pretend to pay me for are the same thing, so I decided to go to Bell Labs. Then Bell Labs came around to interview. Deming Lewis was the Harvard math recruiter and I went and talked to him, and it looked like that was a betterbet. Of course, this observation is undoubtedly not as simple and extreme as I've made it, but there's truth to it. So, I spent thirty-five years in industry, andthen after I retired have now gone into teaching, in fact, have gone into teaching teachers. [chuckles] But, that's why I never seriously considered going into teaching, because it seemed to me that it was a little hypocritical as to just what I was supposed todo. Now, undoubtedly there would have been many places where this wouldn't have arisen as a difficulty, but I just didn't do it. Roberts: Were you at all involved with pre-college math education before SMSG? Pollak: Nope. I got involved with the SMSG, one, because I'd been a student of Ed Begle, two, because I had occasionally gone down to Al Tucker'sCombinatorics Seminar at Princeton, and so Al Tucker who was probably chairman of the committee that helped to organize SMSG also knew me. Begle and Tucker were good friends. So, I think they decided between them, what the heck let's invite this guy and see what happens. No, I asked questionsabout, not so much teaching, as exposition in Tucker's Combinatorics Seminar. I remember some things coming up in there and I didn't go very often, but that's probably how it happened. Roberts: When you became involved with SMSG what was your level of concern about the state of mathematics education? Pollak: I don't think my concern was as much for the state of mathematics education in the country as for time to give something back to education. Educationhad been very good to me, and at Bell Labs I was doing all sorts of interesting things and contributing in many ways, but not contributing anything back to education. I had a little conscience about that and, in fact during the years that came after that, I found that many of my colleagues had the same kinds of feelings.Just an anecdote that has nothing to do with this - there was a project that you may or may not have heard of called Project Seed. There have been many, many projects called that because the initials lend themselves very conveniently to all sorts of titles. This meant Special ElementaryEducation for the Disadvantaged. It was a project by Bill Johnson, professor of mathematics education, I think at Berkeley. His idea was to have professional mathematicians go intoelementary classrooms and teach algebra, abstract high-powered algebra because of its prestige value. The reason he wanted mathematicians was that one of his rules was that you must never say a kid is wrong. He wanted his people to know enough mathematics sothat they could figure out what they were thinking about when they gave that answer. So, at some point I invited Johnson to do a demonstration in the auditorium at Bell Labs. We bussed up some kids from Plainfield, gave them a good lunch, andhe demonstrated his methods. A number of my people decided, hey I'd like to do this a few times a week before I come in to work in the morning, working with the kids. Again, it filled aneed. I think myself that industry, as any other form of employment, should fulfill that kind of a need. I'm sorry. I went off on a tangent to what you were asking. I felt that I ought to contribute. Yes, I think everybody was worried about Russian technologicaladvances at the time, and SMSG was aimed at some upper fraction of students. I think probably the original courses were aimed at something like upper thirty percent of all kids. It broadened out to something more like fifty or more asit went along, but that was part of the technological competition that we were talking about. Roberts: Did you have at this time any personal experiences with pre-college math through your children coming back and being unhappy with their education orany such thing? Pollak: Nothing serious. I want to be honest about the times. I don't think the children were old enough so that I would have had this experience. Icertainly noticed things. My daughter had the dubious pleasure of having to use the textbook that my name appeared in along with others at various places during her high schoolcareer, but I don't think there was anything else that comes to mind. Roberts: Could you describe your role in SMSG? Pollak: Well, when we started out in 1958 I roomed with Henry Swain who was head of the mathematics department at New Trier High School outside of Chicago,Winnetka. And I was on the ninth grade algebra team. We started out by really trying to understand what ninth grade algebra is all about and it was jointly professional research mathematicians or universitymathematicians and mathematics educators. Also on the team was Martha Hildebrandt who was a past president of NCTM from Proviso High School in Maywood, Illinois. We just had to question what is allthis about? It was an amazing discovery for me and for many of the mathematicians involved that the intellectual content of high school mathematics is very real, veryserious, very difficult to organize and to understand. What are you really doing when you solve an equation? What really is a variable? All sorts of questions like that. What really are negative numbers? What really is an absolute value? There are both mathematical and pedagogic issues thatcome up. Okay, we could easily name six different ways that you could define an absolute value. Now how do you pick among them? Well, one thing that goes into pickingamong them is pedagogy. Which one are the kids likely to get right? Well, wait a minute. What do kids usually get wrong in absolute value? Well, they get the idea that "in order to find the absolute value of a number you drop the sign", unquote.The sign is an essential part of the name of a number and you don't drop it, so it's not a good habit to get into. But, also as soon as you ask about absolute value of x you don't have a sign to drop; you don't know, so it's wrong. Some of the definitions are much morelikely to lead to this problem than others. So, that's an example out of hundreds of the kinds of questions that you have to ask. What should you do first in an algebra course? Well, when you really think about it, it's quite clear that there are three or four different things, each ofwhich you must do before you do the other. Now, that's impossible, and the question is what are you going to do about it? That is what's algebra about? Algebra is about variables. So, the first thing you should do is introduce them. It's about the laws, what we recognize as thefield axioms that underlie arithmetic and about those in a more structural formal kind of way. So, you want to introduce those. It's about negative numbers because at that time, for many kids, that was the first introduction to negative numbers. It's about word problems. That's whatit's famous for.