A side note on the psychology of independently studying math: for several years, I've been big on the idea that it's far better to praise people for working hard than for being smart. There's some independent evidence for this. People who have been praised for being smart eventually hit problems which are hard for them, and then they stop trying. People who have been praised for working hard view hard problems as an opportunity to demonstrate how hard they work.

Generally I feel this way about myself and math. My strength isn't that I'm smart (which is not to say I'm not). My strength is that I have an almost obsessive need to solve problems that are put in front of me. At least, that's what I tell myself most of the time.

However, I've been flipping through my real analysis text and noticing my emotional reactions to some topics that are coming up. There's basic cluelessness, such as topology, where I don't know the basic vocabulary. I look at the text and it's all jargon that I don't understand. There's a level on which I feel like I can't be interested in this because I don't even understand what it's about. On the other hand, I have confidence that I will figure it out when I get there.

Infinite series is a subject that's a little more complicated. I have a long standing feeling that I don't really understand them. However, I reviewed the chapter on infinite series in my calculus textbook recently, and I felt like I was on top of the material as far as it went. The presentation in my calculus textbook was more concerned with the question of whether a particular series would converge than with what particular value it would converge to. It presented a bunch of convergence tests, but I'm not even sure whether it demonstrated why the tests work. I can look at a series and use the tests, but I still feel unsatisfied. Maybe the issue is just that the calculus text didn't go far enough. In that case, real analysis may give me what I'm looking for.

And then there's vector calculus. I came out of multivariable calculus not understanding vector calculus at all. Last fall, I worked out enough understanding that I could correctly answer test problems, but I still feel like I have no idea what's going on. There's a section on vector calculus in my analysis textbook that fills me with dread. On a certain level, I think this is totally irrational. When I get there, I'll be well prepared, and I will take as long as I need to work out what's going on. In all likelihood, once I figure it out, I'll wonder what the big deal was.

I went through this process with trigonometric substitution. Completely failed to get it first time around in calculus, then tried it again recently and found it straightforward but tedious. At that point I sort of resented it, until I realized that it's actually useful for evaluating line integrals.

So I expect that similar patterns will play out in the future. Topics will pass from completely unknown to incomprehensible to doable but pointless or annoying to actually useful. They may not all go through every stage, or in that order, but as long as I keep working hard, the comprehensible and exciting stuff should keep growing.

This leads to what I feel is the big weakness of independent study. There's no feedback. I think I mostly sort of understand what's going on so far as I've been working on real analysis, but I'm not sure. When I try to work out a problem from the book, I can feel like I both overthinking the problem and missing the point. Am I assuming things which I haven't proved? Is my reasoning not rigorous enough? Or am I trying to unnecessarily reinvent the wheel? In a classroom, there's much more opportunity to see how others are approaching problems and to just ask the professor for clarification. I have the book and the notes, but mostly I have to figure it out on my own. I know I've missed it badly in some cases, but I hope most of time I'm doing well enough. I also hope that the farther I get, the easier it will be to determine what the important parts of the stuff I've already covered were.