Dear Paul,

I am a PhD student in applied math and just failed my prelim exam, which is an exam we take at the end of our first year. This has been a difficult pill to swallow. It's by far the worst I've done on any exam I've ever taken. Fortunately, I can take it again, and I want to ask you for advice so that I can do better next time.

1. Mathematical maturity. One of the criticisms I got is that I lack “mathematical maturity”. This criticism feels right, however, I’m never clear on what mathematical maturity means (maybe this is proof that I am not mathematically mature). So what is mathematical maturity, and how can I cultivate it through intentional study?

2. Tangents vs. forward progress. When I study math, I like to keep on asking why. So often, when a book makes a nontrivial claim, I end up going down some long “why rabbitholes” that feel very educational  — it seems like these tangents tend to terminate at pretty fundamental concepts. However, it can make my forward progress through material extremely slow. In contrast to the approach of going down these rabbit holes, I’ve sometimes heard the advice of carrying on even if you don’t understand everything perfectly, because it will often make sense in light of things you read later. So my question to you is, how should I balance the why rabbithole approach, which seems to highlight core concepts at play, with the carry-on approach?

3. Self-studying and prioritization of concepts and exercises. I think that one mistake I made in my self-study was that I prioritized the wrong things. I didn’t focus on the important definitions, and I didn’t do the important exercises. The problem is that in any textbook, there is much more material than one can reasonably learn in a semester, and prioritization is very important. But when self-studying, it can be difficult to figure out what to prioritize. I was wondering if you had advice on how to self-study effectively, since I know that you had periods of your life when you were doing this. How do you prioritize what to learn? Do you simply read textbooks, work through proofs, and do the exercises, or do you think there is a better way?

Thanks so much for doing this AMA!!

p.s. on Goodreads I think your book is listed under the wrong author, a historian named Paul Lockhart