How to teach

Disclaimer: All the statements here are my own opinion. It does not reflect any organizations.

Also, this post will be continuously updated (maybe…)

Although I’m just a graduate student, I have years of experience teaching mathematics. I taught my colleagues and students in university (both when I was an undergrad and grad) and did some private tutoring. From these experiences, I have my own teaching tips, which I will describe here.

TL;DR: you need to answer the following question:

What is the main goal of the course?

What ARE you teaching?

This is the most basic but important question to answer when teaching some courses (and I also found that some people don’t know what they are teaching). For example, assume you are preparing a calculus course for first-year students. Obviously, you need to know about calculus (differentiations, integrals, etc.) to teach them; otherwise, you’ll say random words in your class and waste their time. The ideal teacher should be able to answer any questions they ask (related to the topic and at an appropriate level), and even if you don’t know the answer, you should be able to answer it later.

Not all calculus courses are the same

On Twitter X, I saw some controversy about changing the curriculum of calculus courses. The answer highly depends on the classes. The calculus course for math and other majors would be different because their goals are different.

For example, the calculus courses for math majors (especially the honor’s calculus) are designed to let students get used to rigorous mathematical proofs. Proving a limit using $\varepsilon$-$\delta$ definition could be one example of the first rigorous proofs they face as newbie mathematicians. Even when you are computing integrals, justifying each step (e.g., what kind of theorems are you using) is a training for being a mathematician.

However, if you teach a calculus course for biology majors, the ultimate goal could be different (and often, they use different textbooks). Usually, the goal is to let students get used to computations that appear in their future studies (and this principle applies to other courses and majors, too). For instance, differential equations show up a lot in biology when modeling a biological process. At least you need to know about differential equations and how we can solve some simple ones analytically. Of course, some of them cannot be solved analytically, and you may need to find solutions by computers numerically. But it still helps when you know how the differential equations are designed and how the computer solves them numerically (at least, there are some old techniques like the Euler method to approximate solutions of differential equations).

If you are doing private tutoring, then the student’s main goal would be a good exam score. Then you can teach some problem-solving skills to boost up the scores (like the substitution $t = \tan(x/2)$ for computing integrals of rational functions of $\sin$⁡ and $\cos$⁡), rather than explaining the history of the calculus.

Not all students in the classroom are the same

All the students have different backgrounds and knowledge. Some may already know all the course contents (but are taking it for graduation requirements or just for fun), and some may need help understanding the course’s prerequisites. Hence, giving a lecture that fits all the students is impossible, but at least you can try it. For example, you can propose challenging questions to the students who understand the materials well and need something else. But it is essential to know that some of your students are following your lecture and some are not so you can help the second group later (e.g., during office hours).

Follow the references

Someone may think that the notations you use are unimportant if you teach the correct concepts. But, I found that this is wrong, based on my personal experiences. For example, you can denote vectors as

\[v, \quad \vec{v}, \quad \mathbf{v}, \quad \vec{\mathbf{v}}, \quad \mathbf{x}\]

and norms of them as

\[|\mathbf{v} |, \quad \|\mathbf{v} \|.\]

These may not be confusing at all. But for the students who learn vectors for the first time, they get confused when you use different notations for vectors. Also, the textbook may use dot product, but you may use inner product, which can confuse them, too. Hence, the best way is to follow the textbook or professor’s notations to eliminate possible confusion.

Motivate students

This is the hardest part of teaching. Even if you are a great teacher and know everything about what you teach, some of your students may need more motivation because math is boring. You should not force them to concentrate on your lectures, but at least you can try to motivate them to study the subjects. For example, I usually give a brief overview of the whole course for the first lecture to let them know what is essential and the goal of the class. If they don’t know why they are learning eigenvalues and eigenvectors, they won’t concentrate on your class, even if you are a master of eigenvalues and eigenvectors. (In fact, this principle applies when you are giving seminars or writing papers - without motivation, people won’t do anything.)

Tags: math