Reflections on my First Course
I still haven’t taught a university course yet, and won’t for several years, but I’ve taken opportunities to teach where I’ve found them. I just finished teaching a week-long seminar for Chinese high-school students, entitled The Art of Mathematical Proof but mostly consisting of an introduction to group theory, at the Harvard Summit for Young Leaders in China (HSYLC). This post is my attempt to write down what I’ve learned, on the grounds that I would have had a much easier time if someone had pointed me to a post like it beforehand.
One lecture, one concept
If you look at the outline of a really well-done math course, like my first course in topology, you’ll see that all of the lectures can be summed up in one line and many of these lines contain only one noun phrase. Two is usually the maximum. This is not because it is impossible to introduce many, many new concepts in one lecture, but because, without taking time to write up examples, do small proofs to establish basic properties, and take questions, you will lose your students very quickly. I had planned to spend a day proving Lagrange’s Theorem and then using that to prove Fermat’s Little Theorem and the infinitude of primes. In fact, we exactly got to Lagrange’s Theorem by the end of the day, and I had to take time out of the next lecture to look at the consquences.
The exact definition of “one concept” is highly subjective, but I am now a big believer in making sure that the core content of every lecture can be summed up in one sentence with no subordinate clauses. If nothing else, this makes it easier for students to look back over their notes when they find out that they don’t understand.
Everything takes longer than you think
Explaining permutations to a friend who knows a lot of other math, and who understands how to think mathematically on the level that you do, might take about five minutes. Explaining permutations to a class of high-school students who had never seen them before took about twenty. It’s entirely natural to underestimate how long it will take to teach something, since by the time you’re teaching something you probably already know a lot about it. Learning new concepts takes time.
Sometimes students leave work to the last minute
If you say homework is due by 9pm on Thursday, you will be missing some homeworks if you go to collect them at 5pm on Thursday. Don’t be that professor who assumes that your course is your students’ top priority.
You are not your students’ only source of information
A number of students came up to me and asked if I could point them to some resources, such as books or articles, that explained more about the material. I was initially upset about this, since it implied that my own explanations were insufficient, but I complied, and the results turned out to be extremely encouraging. The students who asked for more material went on to participate in class, ask questions, and get very good grades. Asking for alternative viewpoints on a subject is not the hallmark of a student who is hopelessly lost, but rather of one who understands that a teacher is just one of many resources with which to carry on their education.
This has implications for me both as a student and as a teacher. As a student, I plan to rely on my professors slightly less in future and instead combine going to lecture with reading textbooks, speaking to other students, and generally trying to use all available sources of information at once. As a teacher, I plan to offer up other resources to my students proactively, at the very least by putting a “Further reading” section on my syllabi.
You can’t reach everybody
One of my colleagues, who was teaching a different seminar, sent a message to the Slack on the final day saying that he had set a multiple-choice quiz as his final exam and that one student, whom he selected at random to grade first, had got only two qusetions correct out of ten. One of the administrators responded that this sort of thing happens every year. I bring up this exchange because it made me feel much better about my class’s results, which ran the gamut from very high to very low indeed.
If none of your students pass your class, then you have failed as a teacher. This much is clear. However, if you are teaching a class that stretches your students, then at least some of them will end up being stretched too far. In a perfect world, I would have as much time with each student as I needed to ensure that they were up to speed, but this is not always possible. I taught two classes at HSYLC, one half the size of the other. The material was identical, but, on average, the smaller class did better. I attribute this partially to the luck of the draw but also partially to being able to think more about each individual student. In truly large lectures, this becomes impossible, which perhaps contributes some to the number of would-be mathematicians who get lost in intro calculus.