This past quarter, I was a TA for Math 51 for the third time. This course, the most popular mathematics course at Stanford, covers linear algebra and differential multivariable calculus. It’s the only course I have TAed at Stanford.

Last year, after the second time I TAed this course, I wrote about my experiences. This year, things were a bit different.

The most striking and instantly observable difference between my classes this year and those last year was that of size. Last year, my class sizes were 16 and 9; this year, they were roughly 30 each. It’s not completely clear to me whether I should teach differently as a result of the change in class size, and if so, how. Of course, with more students, I tend to be asked more questions, so that slows the class down somewhat. Indeed, at the beginning of the quarter, my earlier class was asking so many questions that I struggled to get through the material I needed to cover. But, having said that, in the first few weeks of the class last year, I was behind and a bit panicked about it, although I eventually managed to get back on pace. It’s tougher to cater to the needs of specific students when there are a lot of them, with a wide range of interests and levels of understanding. I should think about how to do this better in the future; I welcome advice from others, as always.

One effect of the large classes that caused me much personal embarrassment this year was the difficulty I had in learning my students’ names. My method for learning their names was to have the students line up to collect their homework assignments and exams and tell me their names. (I always alphabetized their assignments by first name before handing them back so this activity would not be too time-consuming.) This method is horrendously inefficient: I only had a few seconds to try to make an association between a face and a name, and then it was time to move on to the next student. I managed to learn a few more names each week, until I had all the names of the regular attendees down, perhaps around week 7 (of 10). Even after that, I was worried about making errors and giving someone the wrong assignment. The previous year, with 25 students between my two classes, learning my their names was a breeze, and a task I mastered in only a few weeks.

Fortunately, my students didn’t expect me to learn their names and were surprised and pleased to discover that I was trying hard to do so. Several of them asked how I knew their names at all, with one going so far as to point out that their homework assignments didn’t have their pictures on them! Some of them were also surprised to find out that I generally remembered their exam scores, at least roughly.

So, at least when it comes to knowing about them, the students are far more forgiving of me than I am of myself.

Also troubling for me this year was that my students were, on average, much weaker than those last year. I’m fully aware that I had freakishly strong students last year, and that how well students do in the class is pretty much independent of the quality of teaching, but I still had hopes of repeating my success at getting students to master the material. This proved not to be possible to the extent I had wished.

Last year, after most of my sections, I felt that my explanations were about as good as they could possibly be, my examples were appropriate and helpful, and the students left class able to solve basic problems competently and confidently. At least one student from last year agrees strongly with this: he wrote me a fantastic evaluation and then an email echoing most of these points (which allowed me to identify him from among the anonymous evaluations) and told me again a few weeks ago (when I happened to run into him in the math building) how fabulous my teaching was, and that I’m his favorite TA. Naturally, I tried to emulate my teaching methods from last year much of the time, and I was disappointed to find that they were less successful. Students didn’t immediately understand everything I said, they had perhaps a bit more difficulty working out examples on their own, and they asked me to repeat things a lot more.

Now I suspect that the discrepancy wasn’t caused by inferior teaching on my part this year. Rather, any reasonable teaching would have made me look great last year: with a class of exceptional students eager to learn, how I could I possibly go wrong? And I might not have noticed last year when my explanations were subpar, if the students somehow understood them anyway.

I think the biggest difference between my teaching this year and my teaching last year is that I told more stories this time. I find increasingly that doing so is vitally important. Most of the students taking my classes are in their first quarter of college, and before arriving at Stanford, they might never have met anyone who likes math. And even if they have met people who like math, they very likely have never met anyone qualified to tell them what awaits them in their future study of mathematics. So, when more advanced topics enter my mind as I’m teaching, I need to talk about them. Topics such as Jordan form, representation theory, cohomology, complex analysis, partial differential equations, partitions of unity, differential geometry, and Bézout’s theorem all showed up naturally as part of our conversations. I also always act as though all my students are math majors, so that saying things like “in two years or so, you’ll be taking a class on abstract algebra, and you’ll learn about representations” is completely consistent with the general way I play the role. I think I stuck to the course material more faithfully last year.

They seemed to enjoy my stories, even though they take away a bit of class time from solving problems that might potentially help them to perform better on exams. I always have two goals for my students: one is that they learn the material, and the other is that they enjoy mathematics. But the second one is much more important to me, and I do nothing to hide that from them. In fact, I explicitly told that to one of my sections this year. If they all get A’s in the class, that’s very nice, and I’ll feel happy about that. But if they go home over the break, devour Tristan Needham’s *Visual Complex Analysis* and are too excited to sleep until they’ve learned more about the Cauchy integral formula, then show up to my office in the winter excited to discuss mathematics, that’s far better. (I can dream, right?) Surprisingly, one person asked for more stories in my evaluation!

I might try to delimit my stories a bit better from the main class material in the future. During the final exam review, one of my strongest students asked, with complete seriousness, whether I could work out a problem involving partitions of unity, apparently not realizing that they were not examinable. (I don’t actually know how to give a problem about partitions of unity. I know ones that involve them but also involve many other things they haven’t learned yet.)

In a most encouraging note, the reviews I got from my students improve every time I teach. Most of them found my sections to be very helpful, and many said that they were much more helpful than Rob’s lectures. It’s unfortunate that they so often think seem of me as the “good guy” who wants to help them by giving clean and easy explanations, while Rob is the “bad guy” who wants to confuse them with abstruse math-speak. I don’t think that Rob is out to confuse them or present them with complicated material that won’t be relevant to them; we both want them to understand as much mathematics as possible. How can we get the students to enjoy the main lectures more?

A difficult puzzle for me to solve is to determine how to reconcile the students who say something along the lines of “Simon covered much of the same material as Rob did in the lectures; this is a waste of our time” with those who say “It’s good that Simon talked about the same topics as Rob did because we didn’t really understand them the first time.” I wish I could side with the first group of students, but I think they’re being unrealistic. Perhaps a few of them do understand the material the first time around, but most of them probably don’t, and starting immediately on new material or problems about topics they aren’t comfortable with is likely to leave many of them in the dust. Fortunately, I don’t get too many comments of the first type.

So, I think things went pretty well. But, perhaps it’s time to try teaching a different class next year.

Hi Simon,

As far as learning names is concerned, I find the following technique useful: for the first two or three weeks (or, roughly, until it’s no longer necessary), every time a student makes a comment or asks a question, I ask for his or her name. When calling on a student who has already given me his or her name, I repeat it to help fix it in my mind. I also pause once or twice during class to repeat the names I’ve learned. I find that I can pick up about 7-10 names per 50 minute class this way, with some attrition over the course of the few days between classes. MIT also does us the favor of making student photographs available, so I can go home and look through the list of students to help reinforce (though in practice I rarely do this). Anyhow, thanks for sharing your experiences.