Reflections on teaching


I just taught my last class of linear algebra and multivariable calculus yesterday, so it seems like a good time to summarize my thoughts about teaching. I taught the same class last year, but I think I did so quite differently.

One of the first things I asked myself about teaching was whether it is acceptable to teach while barefoot. I didn’t do so last year, but not because of any objection to doing so; rather, last year I was teaching in the winter, and I rarely wanted to be barefoot anyway. But in the fall, things are different. Until the summer, I didn’t see any problem with it at all. Then, I read an article that suggests that sometimes students get distracted by peculiarities in the teacher’s behavior, and that may make learning a bit more difficult. I wasn’t sure about that, so I decided that it would be safer to wear shoes to classes, and I did so.

On the other hand, it’s a good idea for the students to believe that the teacher is approachable. One of the best ways to seem unapproachable is to display a lack of personality. Going to class barefoot would be a reasonable way of displaying personality, but that one was out, so I had to come up with others.

I think that appearing interesting was one thing I did poorly last year. My classes were usually very serious, almost business-like. That was unfortunate. This year, I felt that my classes were more of a conversation between the students and myself. I think they asked a lot more questions this year than last year, although it’s hard to remember that well what happened last year. Therefore, the class was more geared toward things that students thought would be helpful, rather than what I would teach by default. Given that the class is supposed to benefit them primarily, and not me, that’s a good thing.

After a while, I think the conversational approach that my classes took on helped me to let my personality come through. It is important in all classes for the teacher to demonstrate that the material is interesting, and that the students should care about it. But this is especially important in the first math class that students take in college. Most of them probably have never had a math teacher who actually likes math before, so I felt responsible for convincing them that interesting people can do math and really enjoy it. Naturally, I have to convince them that I am an interesting person if that’s going to work.

I’m not entirely sure how I did that, but I think I did. Perhaps it helped that I occasionally ranted about the unfortunate situation of having to do linear algebra over a non-algebraically closed field. (Okay, so maybe it wasn’t so occasional. But I had to stop myself mid-sentence when trying to say that the determinant is the product of the eigenvalues way more times than I want to think about!) But however it happened, I think they believe that I’m an interesting person even though many of them don’t know that I’m vegan, militantly atheist, opposed to monogamy, don’t like to wear shoes, don’t use telephones, don’t drink, don’t drive, wake up at 5:30 every day, and so forth. But I guess to most people, being passionate about mathematics is enough to be far away from the vast majority of people they know.

It was also interesting to me to see how a single student can radically alter the dynamic of a class. I had intended for my two sections to run rather similarly, but this was very far from what actually happened. The presence of one very outgoing student in my afternoon section made me much more inclined to joke around and be informal in that section as compared to my morning section.

So, how does that affect class performance? I’m not sure how much the quality of my teaching influences students’ scores. I tend to believe that students will study a certain amount anyway, regardless of how well the teacher teaches. At the high and low ends, this is almost certainly true, but perhaps if I teach well I can help the students in the middle do better than they would have otherwise. My morning section was extremely strong; on the first exam they had an average 9 or 10 points above the entire Math 51 average. That was very encouraging, but it’s a small enough sample size that I’m not sure if I can give myself credit for that. (Both my sections did better than the class average on both exams though (and in three out of four cases, far above the class average), and if that continues on the final, maybe that’s enough to be statistically significant.)

My focus was quite different from last year. Last year, I taught the section something like the way I would have wanted to see it had I been a student, with most things proven. But proving things takes up a lot of time, so that cut out time from examples, which seems to be what the students would rather see in sections. So, this year, I gritted my teeth and presented more magic formulae than I would have liked and explained them only if anyone asked. Mathematicians tend to get a bit brainwashed by the presentation in graduate classes and books, so we think that everyone wants to know how to prove things, but this is false. So, I got to do more examples. The way I usually handled examples was to do a problem on some topic myself, and then give another problem for the students to work on, and then have someone present to the class when it appeared that most people were done. This is good for them because explaining how to do a problem is helpful for clarifying their thoughts, and of course “mathematics is not a spectator sport.” But I didn’t want to put anyone on the spot, so I always asked for a volunteer. Some people volunteered a lot, and some people never did, so I never got to see whether the shy ones knew what they were doing.

Another thing I did that surprised some of the students was to tell them (only occasionally!) that they should refuse to do certain homework problems. I always prefaced such remarks by explaining that I was acting in my role as a civilized human being rather than as their teacher in such cases. But once they reach a stage in which they have to do a lot of hideous symbol manipulation and no more actual math, they should stop working on problems. If they’re learning math for the “real world” (to which I assured them from time to time mathematics has no applications), they can find a CAS to finish off the problem. If they’re doing it just because they are interested, then there’s no more interest left in symbol manipulation. So the only reason I can think of for why they’d finish such a problem is because someone else told them to do so. But one should never do anything just because one is told to do so!

Also, I really like teaching! I don’t have that much confidence in my ability to solve hard problems (like my thesis problem) or to understand the literature, but I know that I understand basic linear algebra very well. And, since I get really excited by learning certain things, I like to pass on that excitement to others as well. It’s always nice to give back something to mathematics and the mathematics community, which have provided me with so much enjoyment.

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About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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6 Responses to Reflections on teaching

  1. zebrallama says:

    I agree with you that personality makes you more approachable. To use an example you’ll feel at home with, when I was an undergraduate we had occasional lectures from John Conway. He pretty much always had bare feet, even in the snow, and his lectures were much more chatty in style than other people’s. Also I remember him throwing a gown onto the chalky floor to use as a prop. Despite being utterly famous (which we knew even as undergrads), he seemed more approachable then the more run-of-the-mill professors.
    The least approachable professor was [name on application], who taught all his classes facing the blackboard instead of the students.

    Most of them probably have never had a math teacher who actually likes math before
    When I started (university) teaching, my brother had already been (high school) teaching for more than ten years, so I asked him for advice. He wouldn’t give me any except this:
    Be enthusiastic.
    So I followed that, and even though I did lots of things wrong I immediately had good student feedback, both formally and in obviously inspired students. So I think it was good advice! I’ve been teaching for 10 years now, and this advice is still the main thing I think about before a class.
    Another piece of advice I’ve been given is to always keep your office door open, and your desk facing the corridor so that your face is visible to people as they go by your office. I haven’t kept this one religiously because sometimes I want privacy more than I want to be a good teacher! But I have always kept my desk facing the door, so that at least when I don’t mind being interrupted it’s just a matter of opening the door. This only works if you’ve got an office on campus, of course.
    the class was more geared toward things that students thought would be helpful, rather than what I would teach by default
    In tutorials, I mostly refuse to tell the students what I think we should discuss, at least at first. I try to get them to tell me what they want to discuss, instead. I believe they find this very annoying! Almost all of them would rather be as passive as possible. But I think it’s worth it. (I’m just talking about tutorials here. I assume that either I or someone else has given them a lecture which has stuck to some sort of syllabus.)
    maybe that’s enough to be statistically significant
    Off topic here, but statistical significance is meaningless (in several senses). A good party trick is to ask people what it means: you’ll find that even professional statisticians won’t be able to say anything sensible about it. I guarantee they won’t even get the definition right. That’s assuming you mix with applied statisticians. Theoretical statisticians (a much rarer breed) will at least get the definition right, and often they’ll agree with me that it’s meaningless.
    I’ve written almost a whole book on this which I’ll point you to if you’re interested.
    one should never do anything just because one is told to do so!
    And that’s a good lesson in itself, isn’t it?

  2. amacfie says:

    I think flexibility and willingness to spend time to save others time are most important in a teacher. Different students learn differently. (Ahem – some do like proofs!) Students (or at least, I) don’t always care whether the teacher has a lot of enthusiasm. It was freshman material! I didn’t expect a lot from the prof. I think the rate at which the prof went was slow enough that it was impossible to expect him to be excited. Anyway it sounds like you are very competent and don’t need my advice.
    how do we compare?
    vegan – no
    militantly atheist – yes
    opposed to monogamy – yes
    don’t like to wear shoes – or socks (saves laundry)
    don’t use telephones – yes
    don’t drink – yes
    don’t drive – i do, but don’t like it
    wake up at 5:30 every day – no
    I think the monogamy issue is most interesting. My ideas on the matter were mainly stimulated by Andre Gide, Bertrand Russel and David Friedman. Do you consider it immoral? If so, what aspects of it? When did you become atheist?

    • Simon says:

      I don’t consider monogamy to be immoral, but I do consider it to be very unnatural and unsuccessful. If monogamy is supposed to be a universal part of human nature, it has an awfully bad track record, with a divorce rate exceeding 50%. So, it would appear that monogamy isn’t some structure that succeeds in making people happy. For some people, it works really well, and I congratulate them on making it work. But for society to insist (for all practical purposes) that everyone be monogamous is unreasonable. In fact, the number of divorces even understates the number of unhappy marriages; some people just live with unhappy marriages due to various social stigmas against divorce which are present in certain circles.
      So, that’s part of it. Another part is that monogamy is the most blatant example of materialism I can imagine. We usually think of certain things as being interchangeable with other things, but we don’t think of people as being interchangeable with other people. So if you like my computer, your reaction probably won’t be to steal my computer; rather, you can go and buy one just like it. But if you want my (hypothetical) significant other, you may be inclined to steal em. Thus, monogamy leads to the objectification of people; in some sense, people own their significant others in a monogamous society. I can’t tolerate the thought of another person being my property, and neither can I tolerate the thought of someone else owning me.
      On a practical level, I am not sure if I’d be interested in having relationships with several people at once (or even with anyone at all). But if I did, I would have to set up ground rules which state that we do not own each other. It might not mean anything in practical terms, but at least setting up such rules and language would allow my actions to be consistent with my beliefs.
      I don’t know when I became atheist. I started referring to myself as an atheist toward the end of my undergraduate years (probably between 2005 and 2007). But my religious background was a bit weird. I went to a Jewish elementary school, so we had Judaica classes in which we’d study various portions of the Tanach. I never had the impression that we were supposed to believe it; rather, it seemed that our discussions of these books in the Judaica class were completely parallel with our discussions of the novels we read in English class: in English class, we discussed some work of fiction, and in Judaica class, we discussed another work of fiction. I didn’t phrase it in that language back then, but that’s how it seems to me now.
      It wasn’t until I got to college that I realized that some people actually believed in god. Before that, I didn’t think of people who go to a church as being different from people who go to a synagogue or people who go to a mosque. I guess they just felt like different places to go on weekends and holidays. But in college, there were sometimes Christian extremists holding up signs about how everyone was going to hell, and then it finally occurred to me that people actually believed this stuff. I didn’t, so I started calling myself an atheist. After doing more reading on the subject, I realized how disastrous religion has been to our society, and so I became more militant about my atheism.

  3. amacfie says:

    Here’s something I always wondered: If shown a face of a random student from a class you taught this year, how much could you say about him/her? Name? Attendance record? Grades? Kinds of questions he/she asks? What if you were given a random name instead of a face?
    I suppose I would need to know the size of your classes for the answer to mean much.
    When you say you like teaching, do you mean a) you’re surprised how enjoyable it is but would still rather just do math or b) you’re happy to do some teaching even if you could be doing math instead?
    (There’s lots to be said about modern monogamy but I’ll save it for some other time.)

    • Simon says:

      I could definitely identify all of those given a face. With a name, it would be even easier. For many of my students, I could tell you their exam scores exactly as well. My classes are small enough for this to be easy (16 and 9), but I think I could do it if I were to have significantly more as well.
      I mean (b).

  4. Pingback: Math 51, take 3 | Quasi-Coherent

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