Disclaimer: I am not great at reading math textbooks. This post is not about how you, the reader, should read a math textbook. I present this merely as a catalog of my (still ongoing) personal journey of becoming efficient at self-studying, without compromising on my enjoyment of the learning process.
Having a good time while learning is hard
When I read a math textbook, I’m usually doing it because I think that I’m going to get enjoyment out of learning the material it contains. For me, it’s not about anything other than pure enjoyment; if it’s not fun, why bother?
Unfortunately, there are many hurdles to having fun while learning, and I think most of them are about pacing. The following two I think are the most straightforward:
- If I’m not being introduced to new ideas at a consistent pace, then I won’t feel like I’m learning. Therefore, I won’t feel like I’m having fun.
- If I’m not learning the concepts in enough detail to work out examples and solve exercises, then I won’t feel like I’m actually learning. Therefore, I won’t feel like I’m having fun.
These two things sort of quarrel: how am I supposed to go fast enough to see new ideas consistently, but also go slow enough to work out all the details?
The answer is to not work linearly. Reading a textbook line-by-line just doesn’t work, so instead, I like to work in a way which I like to compare to a chess algorithm. One of the most common algorithms used in chess computers is alpha-beta pruning, which in short tries to optimize the amount of “depth” vs. the amount of “breadth” considered when searching through a tree. By analogy, the “depth” is the level of detail I read at, and the “breadth” is the amount of content I read. The issue being, when reading a textbook on content which is unfamiliar to me, I probably don’t know at what level of detail I want to learn the content yet.
The solution I’ve found to this problem is to start as coarsely as possible, and from there, read in finer and finer detail. Working more coarsely at first gives me the freedom to peruse a larger chunk of content all at once, while also allowing me to make as little of a commitment to this unknown material as I can possibly manage.
So what’s the process?
Boiled down into steps, my approach to reading a textbook is approximately as follows, supposing I’m trying to read a single section at a time:
- On reading #1, I go through each section, only reading the statements of all definitions, lemmas, theorems, etc. and any bolded or otherwise emphasized text. Little attempt is made to actually understand anything; this is largely for the sake of knowing the names for things I’ll be reading about in advance. (~5 minutes)
- On reading #2, I go through each section and read all exposition, but I read none of the proofs. For each definition, I try and write down in my own words what that definition means. I try to spend more time on the things which I identify in reading #1 as more difficult or more central to the narrative of the text. Everything the author gives as information intended to clarify intuition, I read. However, I don’t worry too much about actually understanding every piece of exposition, but I at least make sure I read it so that it’s in the back of my mind on any further reading. (~30 minutes)
- On reading #3, I go through each section, reading in as much detail as I can muster. For every example and every proof, I do my best to work them out on my own, consulting the text only when I’m quite stuck. (unbounded amount of time, and dependent on the text, but usually between 30 minutes and 2 hours)
- Lastly, I do as many exercises as I practically can. If I find myself unable to do more than roughly 50% of the exercises, I do another pass through the content, similar to reading #2.
The third step is crucial for things which are particularly new to me; if I don’t do this with exceptionally new material, I’ll breeze over the kinds of basic arguments which are omitted because they’re standard. And if I do that, I’ll certainly be ineffective at doing exercises.