Why I left math

Reading how shocked Doron Zeilberger is at the state of modern mathematics reminded me of why I left the subject.

Math departments regularly have visiting mathematicians come and give talks. Or at least the one I was at did. For the visiting professors these talks were a confirmation of success, all of these people came to hear about their research. So they would talk about their research and get quite excited about what they were describing.

As a grad student I attended. I quickly noticed that most of the professors in the math department went out of politeness. However they knew they wouldn't understand the talk, so they brought other things to do. If I looked around about 15 minutes into the talk, I'd see people reading books, grading homework, and otherwise not paying attention. At the end of the talk the speaker would ask whether there were questions. Inevitably the mathematician who invited the speaker would have some. Occasionally a second mathematician would have some. But the rest of the room wouldn't.

This was supposed to be the high point of the life of a mathematician? That's when I decided that, no matter how much I loved mathematics, I wanted a different career. Unfortunately my wife was in grad school as well, and we were in such a small town that I didn't have any immediate employment options. Therefore I remained a rather unmotivated grad student. In the end my wife switched to medical school just before I would have finished the PhD. I'm mildly disappointed that I didn't finish, but it really has been no loss.

Why do mathematicians put up with this? I'll need to describe a mathematical culture a little first. These days mathematicians are divided into little cliques of perhaps a dozen people who work on the same stuff. All of the papers you write get peer reviewed by your clique. You then make a point of reading what your clique produces and writing papers that cite theirs. Nobody outside the clique is likely to pay much attention to, or be able to easily understand, work done within the clique. Over time people do move between cliques, but this social structure is ubiquitous. Anyone who can't accept it doesn't remain in mathematics.

It is important for budding academics to understand this and get into a good clique. This is because your future career and possible tenure is based on your research. But the mathematicians making those decisions are unable to read your papers to judge your work. Therefore they base their decisions on the quality of journals you get your papers into, and the quality of people you get writing recommendations for your work. But both of those come down to getting into a group that includes some influential mathematicians who can get your papers accepted in good journals, and that can write strong letters of recommendation.

In fact if, like me, you are someone who likes to dabble in lots of things, you will be warned (as I was by multiple professors) about the dangers of not focusing on one small group. You will be told plenty of cautionary tales of mathematicians who published a number of good papers, but who didn't publish enough in any specific area to get good mathematicians to stand behind them. And therefore the unlucky generalist was unable to get tenure despite doing good work.

For a complete contrast, look at the situation in biology. A motivated advanced biology undergrad is both capable of, and expected to read current research papers. When biologists go to a talk they both expect to understand the talk. And biologists have no trouble making tenure decisions about colleagues based on reading their papers.

I subscribe to the belief that the difference is mainly cultural. Biology is fully as diverse and complex as mathematics. Furthermore what I have read about the history of mathematics suggests that the structure of the mathematical community was substantially different before WW II. For example David Hilbert was known for stopping speakers and forcing them to define anything he found unclear. (Amusingly he once had to ask Von Neumann what a "Hilbert Space" was.) But after WW II an explosion of university math departments and a focus on solving concrete problems lead to a fragmentation of mathematics. And once mathematicians came to accept that they couldn't be expected to understand each other, there was nothing to prevent mathematics from splintering into fairly small cliques. Which has happened, and this is unlikely to ever be reversed.

PS I'm amused at the fact that a number of comments at Y-combinator thought that the situation with programming was worse than mathematics. Yes, there are divisions within programming. But they are nothing compared to the fragmentation in mathematics. I've done both and there is simply no comparison.

Teaching linear algebra

In a recent Hacker News post I made reference to an interesting teaching experience I had in the mid-90s. This is a longer explanation of the same.

I was a graduate student in math at Dartmouth College. I wound up teaching an introduction to linear algebra course that was also the first course where students were asked to do proofs. The class was somewhere in the range of 15-20 students. If I remember correctly, this was in the fall of 1996.

In preparation for the class I set myself goals around how well the students would learn the material taught. After some thought I settled on four ideas that I would use:

1. Homework not present at the start of class would not be accepted. However students were only graded on the best 20 out of 27 possible homework sets.
   
2. All homework sets were cumulative. Generally 1/3 was the current day's material, 1/3 from the last week, and 1/3 from anywhere in the course. Those thirds were in increasing order of difficulty.
   
3. Every class would start with a question and answer session to last no less than 10 minutes.
   
4. Every student could expect to be asked at least one question every other class.

These ideas may seem odd, but there was a method to my madness. Here is each idea explained.

1. Homework not present at the start of class would not be accepted. However students were only graded on the best 20 out of 27 possible homework sets.
   
   The point was to make sure that class started on time, with everyone ready to pay attention for question and answer time. I also didn't want to deal with people doing homework during lecture, evaluating sick excuses, etc. The leniency of not having to turn in 7 homework sets compensated for the rigidness of the policy. And cumulative homework sets meant that I didn't have to worry about students not practicing any given day's material.
   
   This worked even better than I hoped. The downside was that I had an argument on the second day when someone came in 2 minutes late and was not allowed to turn in his homework. But the first complaint was the last, and the students liked the freedom to decide when something else took precedence over doing homework.
   
   
2. All homework sets were cumulative. Generally 1/3 was the current day's material, 1/3 from the last week, and 1/3 from anywhere in the course. Those thirds were in increasing order of difficulty.
   
   This was the most important idea I wanted to try. I had long been aware that research on memory had demonstrated that when you're reminded of something as you're forgetting it, it goes into much longer term memory. As a result periodic review at lengthening intervals is very effective in increasing long term recall. A typical effective study schedule being to review after half an hour, the next day, the next week, then the next month.
   
   Now of course you can tell students this until you're blue in the face - but they won't do it. However when the study schedule is disguised as homework, they don't have a choice.
   
   This really seemed to work. What I noticed on tests is that students were noticeably shaky on material they had learned in the previous week, occasionally didn't remember stuff for a half-month before that, but absolutely nailed every concept that they'd first learned at least 3 weeks earlier. I credit the forced review schedule from cumulative homework sets for much of that.
   
   
3. Every class would start with a question and answer session to last no less than 10 minutes.
   
   For me this was the most important part of the class. The questions that came up in this session were my opportunity to refresh people on what they were forgetting, and were how I kept track of what topics should come in for more review on future homework sessions. Given my knowledge of how critical review is to learning, I honestly felt that time spent answering questions was more valuable than lecture. As long as there were questions, there was no maximum on how much time I was willing to spend on this.
   
   Of course the challenge is getting students to ask questions. My strategy was simple: I told them that someone will ask questions and someone will answer them, but they don't want me to be the one asking questions. On the second day nobody asked me any questions and I had to demonstrate. I picked a random person and asked her to explain a key point from the first day's lecture. She couldn't. I asked another student the same question. Again difficulty. I asked if everyone was sure that they had no questions. Someone asked me the question that I had been asking everyone else. I answered the question, answered the follow-up, and the point was made. I never again had to ask a question during question and answer period. :-)
   
   
4. Every student could expect to be asked at least one question every other class.
   
   My goal here was to be sure that every student was awake and following the lecture. It was never my goal to embarrass anyone or put them on the spot. To that end I developed a rhythm. Every few minutes I'd stop, say, "Let's make that a question," ask the question, pause so everyone could think through the answer, then ask a random person the question. I made sure to rotate people around so that everyone got their turn fairly.
   
   The questions I'd ask were always straightforward. They were things like, "What is the result of this calculation?" Or, "Why is this step OK?"
   
   I treated failure to get the answer as my failures, not theirs. If they couldn't get the answers then they weren't following the lecture, and I needed to slow it down, figure out the rough spots, etc. It might seem that the constant interruptions were slow. But I found that having everyone pay attention more than made up for it. The class as a whole moved as fast as any other class - but with far greater comprehension. And the interactivity made the class become very open about asking questions.
   
   As a bonus I managed to convince the entire class that taking notes was not worthwhile. I learned this lesson about math in first year undergrad. What you do is read ahead in the textbook. If you really want a set of notes, you can make them from the textbook before class. Then show up at class having read the day's material and ready to pay attention. Then if anything that the professor says doesn't make sense to you when you're paying attention and have already read the day's lesson, then ask the question then and there. If you don't understand it, then probably nobody else does either. Add to that periodic reviews, and you'll have a huge edge in any math courses.
   
   Nobody ever believes that that works. But this class had no choice because there is simply no way to take notes and pay attention at the same time. Which meant that the note takers couldn't answer questions. But within a few days they learned to not take notes, and I believe did much better for it.

So how well did this package work? As far as my goals were concerned, much better than I had dreamed possible. What really brought this home was the final exam. Based on class performance I drew up a test that I though was a fair test of what I thought they understood. I showed it to some fellow graduate students. They thought I was crazy. They thought the class would bomb, and were willing to bet me on whether anyone would get the bonus question.

The class aced the test. That bonus question? 70% of the class got it. I don't remember what the bonus question was, but I do remember another one that I thought was cute. It went like this. Let V be the vector space of all polynomials of degree at most 2. a) Prove that d/dx is a linear operator on V. b) You can put a coordinate system on V by mapping p(x) to (p(0), p(1), p(2)). (Please imagine that flipped 90 degrees so it is a column.) Find the matrix that represents d/dx in this coordinate system. My fellow grad students got me worried that this might be too advanced for an introductory linear algebra courses. But I needn't have worried - the only significant errors were minor arithmetic mistakes in the calculation. And I think I dinged someone for not having enough detail in the proof.

Furthermore I was lucky enough to talk to some of my students about the experience a few months later. The general consensus was that the material really stuck. Furthermore nobody studied for the final. No joke. As one girl said, "I tried studying because I thought I should, but I gave up after a half-hour because I already knew it all." That is how I think it should be - if you study properly through the course, then you won't need to study for the final. Because you've already learned it. And you'll have a leg up on the next course because you still remember the material that everyone else has forgotten.

So were there any downsides? Unfortunately there were some big ones. I had set goals around learning. I failed to set any around happiness. Having to pay attention during class was hard on the class. Also it motivated them to work hard. Since everyone worked hard and they thought that I was going to grade them on a curve, there was a lot frustration that they wouldn't properly be recognized for their work. (In fact I gave half of them A's in the end.) This frustration showed up the teacher evaluations at the end of the course. :-(

Therefore if I had to do it over I'd ask somewhat fewer questions, hand out a lot more compliments, make it clear that I would not grade on a curve, and if they performed anything like that first class, I'd be even more liberal with good grades. Of course the point is moot since I've found myself profitably displaced from math to software development. But if anyone decides to replicate my experience, I'd recommend paying more attention than I did to those issues.