How I Learn
How do You?

How do you learn best? That's important because the purpose of my teaching is your learning. If you have a good sense of how you learn, you should tell me. If enough students tell me similar things, I will modify how I teach your class accordingly.

Below I present how I learn. Reading this may help you think about how you learn; it may cause you to realize certain things or lead you to observe yourself in ways you hadn't, from which you may later realize new things. I certainly realized things about myself better as I thought about this essay.

As a student, I now realize I found lectures of marginal value!

Instead, what I remember as being really useful was reading the text carefully (trying to prove each theorem before reading the proof given) and doing problems. Problems which I couldn't get at first are the ones I learned the most from. Sometimes they were simple problems but I thought the answer was wrong; on these occasions I misunderstood the concept, and so the problem straightened me out. On harder ones, there was some method I didn't see, and so when I finally figured it out, or was shown the answer, I thought hard about what was the key idea or step I hadn't seen.

An important source of problems was tests. I especially learned from problems I got wrong on tests, because I always went over them to make sure I never made that mistake again. I realize not everyone will have such a positive attitude towards getting things wrong on tests as I had, but please consider it.

The next most valuable things were writing and speaking. I didn't have to write much math as an undergraduate (we didn't hand in much homework in advanced courses and there were few papers, except the senior thesis I did), but since then I've done a lot. Often, while writing up a proof, only then did I discover that it wasn't complete, or was wrong. Similarly, talking out a solution in an honors seminar often led to corrections, and learning.

I also learn things by experimenting with computer programs. But usually I do this to learn how commands in a language work, rather than to better understand mathematical ideas.

I did not learn much by talking stuff out with other students (except in seminar) because I didn't do it much, but I am aware this method works well with many people, and is much more encouraged now than it used to be.

Back to lectures. There were two reasons I didn't get much from lectures at Swarthmore. First, I usually understood the material already, or could absorb it faster than the lecture. (This wasn't always true later; in graduate school and as a mathematician attending colloquia or conferences, I have sometimes attended lectures where I understood almost nothing. However, I have also attended some gems of talks. Anyway, as an undergraduate, I sometimes didn't get much out of a lecture because I was preoccupied with something personal and wasn't paying attention!)

Second, I don't think my professors always made the best use of lectures. Detailed presentation of theorems and proofs, or even of many examples, is not the best use because the pace won't be right for most people. However, lectures can also provide an overview of the material that is very worthwhile. A lecture can't cover as much material as the corresponding sections of the book, so the professor has to distill the material. I have learned lot over the years from lectures in which the speaker emphasized what s/he thought was the essence of the material, and then provided an open window on how s/he thought about it.

Lectures are important for another reason. They represent a commitment to set aside regular times during the week to fully concentrate on the subject matter.

In light of the above, here is how I run classes absent contrary information from you about how you learn best.

I also have expectations of you outside of class. Namely, Note taking. In a class that does have considerable lecture, you need to think about the material afterwards. This requires some notes to refer to, but interestingly, not a whole lot. There is a skill to math notetaking that most students haven't acquired. Please consult my handout on notetaking. It is adapted from a section of my manuscript for undergraduates, a Short Guide to Writing Mathematics.
Last updated: 8/17/01
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