Home > Math, Teaching > Preparation vs. Winging It

Preparation vs. Winging It

October 12, 2008 Leave a comment Go to comments

In an earlier post, I described my worst recurring dream: I am standing in front of a classroom of students without having prepared for class. I know in general terms what topic I’m supposed to cover, but haven’t thought about it and have no idea what to say. What brought this nightmare to my consciousness was the experience of watching Katie Couric interview Sarah Palin.

One thing I learned as a student was that it wasn’t always a bad thing to have an instructor who was not fully prepared. How the instructor deals with the situation can be enlightening. The student gets to see how an expert on the subject works through the issues spontaneously, thereby (on occasion anyway) learning a lot more than might be learned from a perfect exposition with all the real thinking hidden from view. And the student gets to see how an older adult handles real-life stress.

I had an eye-opening experience along these lines in December 1969, the first semester of my freshman year at Harvard. I will describe it below, but first let me give some background on the nutty course I was taking, Math 55. To provide the proper context, I need to say a few words about calculus.

Most universities offer roughly a three-year sequence in calculus, or mathematical analysis. In the first year, a student might study what’s called single-variable calculus, learning about differentiation and integration of functions of one variable and some standard applications. An example of a “function of one variable” is the measurement of temperature in a particular location at any given time. As time varies, the temperature varies. We say temperature is a function of time. And the tools of calculus allow us to study this variation.

If we wish to study how temperature varies not just with respect to time but also with respect to location, then we must study temperature as a function of more than one variable — the variables of time and position. We might be interested in position along a line, on a flat surface, or in space. Thus position might itself be given by one or several variables, and temperature becomes a function of these position variables plus the time variable. It’s easy to think of many examples in which we wish to study one property of a physical system in terms of several others. This leads to multivariable calculus, which requires tools of another subject called linear algebra. (Linear algebra is fundamental to single variable calculus too, but it is one-dimensional linear algebra, which is essentially trivial and therefore hidden from explicit view.) Thus, a second-year curriculum typically consists of some mix of multivariable calculus and linear algebra.

I’m simplifying a bit. One may well see multivariable calculus in the first-year course. One will also study differential equations in these first two years. But in any case, there’s a standard body of material that students planning programs in mathematics, science, or engineering typically study.

A much narrower set of students goes on to the third year of the calculus or analysis curriculum, whose goal is the foundations of calculus. Why are the results that one uses in the first two years true? How does one prove that the techniques work? This turns out to be an elementary enterprise, in the sense that one starts from scratch with no prior knowledge of anything. But it’s a very sophisticated enterprise as well, requiring a new kind of thinking and an appreciation for the need to justify even the “obvious.”

In graduate school, one continues the study of analysis, often in a still more general framework. Rather than studying subsets of the plane or space, one might study analysis in the context of a geometric generalization called a manifold, which “locally” looks like the plane or three-space or perhaps higher-dimensional space. One might also generalize linear algebra to study infinite-dimensional spaces. These generalizations are not fanciful. They are the right settings to develop many of the most important results in mathematics and its applications to science.

Forty years ago, Harvard offered a three-year sequence more or less as I described above. It began with a first-year course on one-variable calculus, followed by a second-year course that integrated multivariable calculus with linear algebra and a third-year course on the foundations of calculus. An alternative was the two-year honors sequence. One could imagine designing such a sequence for the more highly motivated (and perhaps more talented) students that covers much the same ground as the three-year course, but manages to do it in two years by integrating reasoning and proofs from the beginning. The first year of the two-year honors sequence followed this approach. But the second year honors sequence, Math 55, was way more ambitious. As designed by Lynn Loomis and Shlomo Sternberg, and preserved in their textbook with the deceptively non-threatening title Advanced Calculus, Math 55 covered all of multivariable calculus and linear algebra with the maximum level of generalization. Why not cover infinite-dimensional spaces right away? Let’s study differentiation on Banach spaces. Let’s study linear algebra on Hilbert spaces. Let’s study manifolds which locally look like not just finite-dimensional spaces but also perhaps infinite-dimensional Banach spaces. As some readers will appreciate, this is borderline nuts. There might be a few astonishingly well prepared students who are ready for this, but very few. Certainly not me.

I arrived at Harvard in September 1969, at the age of 17 1/2, eager to study mathematics at a high level. In my senior year of high school, I had taken year two of a two-year calculus sequence at a college not far from our home — C.W. Post College. The students there were not the strongest, so the Math department compensated by taking a standard three-semester curriculum and spreading it over four semesters. The course was not demanding. I got 100 on every test, and on the last class day of each semester, the professor told me not to bother showing up for the final. I wasn’t challenged. At Harvard, I wanted to take the most demanding course I was prepared for.

During orientation, I showed up at 2 Divinity Avenue, the home of East Asian Studies and of the Math department, to speak to the professor who would be teaching Math 55, in order to find out if I should be placed into it or in something else. The natural alternatives were the usual second-year course in multivariable calculus and linear algebra (which would cover some material I already knew) or the first-year honors course (which would would cover lots of material I already knew, but at a high level of sophistication). The instructor was a young guy. I expected to have to take some sort of placement exam, but he asked me a single question on the spot, which he wanted me to answer orally. I realized that if I could answer it, I would be eligible to take the course, and if I didn’t, I would have to choose between the other two. Well, I kind of got lucky, if that’s the word. I could answer it. I can imagine many questions of comparable difficulty that I wouldn’t have been able to handle, but this one I knew. (For those who know about such things, I had to state whether the series 1 + 1/2 + 1/3 + 1/4 + … converges or diverges, and show why.) I was accepted into Math 55 on the basis of some fairly flimsy evidence.

I was pretty good at abstractions. I managed to do reasonably well in the course. But I never for a moment had any idea what we were doing in a more fundamental sense. I lacked the study of the traditional examples — the classical cases motivating the infinite-dimensional generalizations we were looking at — that would have allowed me to appreciate the point of it all. Like many advanced mathematics courses, this one started from scratch, proving everything as needed. I could see how each result followed from the ones before. I could see the logic of the statements. In this sense, I could understand what we were doing, step by step. But it was a foolish exercise. I should have taken a more concrete course. And no doubt so should many of my classmates.

Let me say a few more words about the instructor. He was very young. It was his first year at Harvard and despite his youth, he had been appointed at the level of associate professor. I knew only tidbits about him, but somehow I knew that he had received his Bachelor’s degree at Harvard a few years before (as befitted someone with his surname and implied ancestry), had gone to another prestigious school for his Ph.D., had already proved some famous result, had spent two postdoctoral years abroad, and had returned to Harvard that fall, just two years after his Ph.D., to teach for the first time.

Class was held in Sever Hall, the great brick H. H. Richardson building from about 1880 where just about all math classes were taught. I took several classes in the room in the northeast corner of the second floor, on the right in the photo below, looking out toward the Fogg Art Museum on the other side of Quincy Street.

Sever Hall, Harvard University

Sever Hall, Harvard University

The instructor would enter a coatroom that was behind the classroom, hang up his coat, hat, gloves, umbrella, or whatever, and enter the classroom through a door on the same wall as the blackboard. It was all rather charming in retrospect, though I was more intimidated than charmed at the time.

One of the standard results of an undergraduate linear algebra course is the spectral theorem for a self-adjoint linear operator. I won’t try to explain it, except to say that it tells you that certain transformations of a finite-dimensional space can be described as simple processes characterized by stretching the space along each axis by a certain factor. Had I taken a traditional course that year, with this traditional result, I would have had no trouble appreciating its importance and understanding its proof. But that’s not the way of Math 55. In Math 55, anything worth doing is worth doing in maximal generality. So instead of this result, we went straight to an infinite-dimensional generalization, the spectral theorem for a compact, self-adjoint operator on a Hilbert space. There’s a logic to doing this. The ideas are largely the same. But pedagogically, jumping to this for students who never studied the finite-dimensional case is absurd. Not that I had trouble understanding it in the abstract. I knew what the words meant. I just didn’t see the motivational antecedents.

Now I come to the story that this has all been preparation for. It was December. Winter break was approaching. The semester would continue in January, with first-semester finals late in the month, but with the long break over the holidays, it was desirable to reach a good endpoint in December. We were closing in on the spectral theorem for compact, self-adjoint operators on a Hilbert space. Our brilliant neophyte instructor had stated it at the end of one classroom session. Our story begins the following Monday, the start of the final week of classes before the break. He came in that magical door in the wall from the coatroom. Winter weather had set in. It was bleak outside. The leaves were gone. He re-stated the spectral theorem to begin Monday’s class and began the proof. He wrote maybe a few lines, then moved over to the window. (You can see it in the photo above, third from the right, second floor.) He looked out in silence, deep in thought. We sat there. He thought. We sat some more. A minute passed. Two minutes passed. Silence. Another minute passed. A fourth. I’m sure a fifth too. Then he turned to us and said class was dismissed.

This is when it dawned on me that through the whole semester, he hadn’t been preparing for class. He would simply look in the book to see what the next results were, then walk over to Sever Hall and improvise. I was stunned. I was in awe. I was puzzled too. He didn’t write notes, a sketch of the proofs, an indication of the key details. He just looked it over, or so I imagined, and then did his best to perform, to wing it. Wow!

Well, okay, one lost class. The next class day came. Wednesday. He apologized for not being able to prove the theorem. He re-stated it. He started the proof again. He got to the point where he had been stuck on Monday and explained that what you need to do here was, well, whatever it was. He continued. He stopped a little farther along, stared at the board, thought. He started again, stopped again. He walked to the window, stared out. We sat. He stared. He apologized once again, said he’d get the proof right on Friday. Class was done.

I really didn’t know what to think at that point, aside from the fact that the proof must be pretty darn subtle. Nothing had defeated him until now. It was kind of cool to see one of the most brilliant young mathematicians in the country be completely defeated. A lesson we were all learning is the one I stated in my earlier post: “One thing about mathematics: you can’t fake it. The subject is unforgiving. If you have a statement you wish to prove, either you have a proof or you don’t. And if you don’t, saying everything you can think of that is germane is no substitute for a proof. Sometimes silence really is golden.” This was a striking way to learn this lesson. I respected his silence. I couldn’t help wondering why, after his first defeat, he hadn’t bothered the second time around to bring some notes, or even a fully-written proof. But on another level, we were learning a lot.

Friday he proved the theorem. Maybe he even brought notes as a safeguard. I have a vague memory that he did, but I’m not sure. That was that. We were off to winter break.

1. He left Harvard a few years later for a position at the university where he got his Ph.D. He’s still there.

2. The spring semester was a disaster, but that’s a story for another time. Nixon. Cambodia. Kent State. Classes ending prematurely. Final exam deferred to the following December. And those crazy manifolds modeled on Banach spaces.

Categories: Math, Teaching
  1. dcollingwood
    October 13, 2008 at 8:42 AM

    what a great story; what a great lesson. enjoyed reading it.

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