Here are my recommendations for self‑studying pure math, based on 20+ years of watching students (and myself) struggle and improve.
1. Have fun (on purpose)
Set things up so you actually enjoy studying. We learn mathematics to use it in future research. If some concept is tied to stress, your brain will happily forget it. You’re much more likely to use ideas that came with curiosity and “oh, that’s cool” moments. Slow down enough to feel cozy and confident with each new concept instead of speedrunning the book.
2. Serendipitous pondering > grinding
A lot of the best work happens when you’re walking, daydreaming, or spacing out on the bus, asking yourself little questions about a construction and trying to answer them. Discussing ideas with friends is also extremely useful. Passive reading and memorizing proofs is not only ineffective, it can be harmful to your understanding.
3. Always have a “background” question
Try to keep at least one open question in your head that you want to think about when you get a quiet moment. Letting a problem simmer in the back of your mind often leads to the kind of understanding that forced concentration can’t reach.
4. Try to solve everything yourself
When you read a textbook, pause after each definition and come up with your own examples, non‑examples, and little statements you’re curious about. After each theorem, stop and try to prove it yourself before you look. Yes, this can take days. Yes, your proof might be wrong. The learning mostly happens in the attempt.
5. Train proof‑writing on purpose
You only get good at proofs by writing a lot of them and getting feedback. That can easily take a year or more of steady practice. This is one of the hardest parts of self‑study, because finding competent feedback isn’t trivial.
6. Celebrate small daily progress
It often feels like you’re going nowhere. Make a point of noticing even tiny wins: understanding a tricky definition, spotting your own mistake, or finishing one clean paragraph of a proof. That’s what real progress usually looks like.
7. No gaps in the foundation of your math tower
Before diving into pure math, be reasonably comfortable with some basic objects: a bit of elementary combinatorics, elementary number theory, calculus, and matrix algebra. Then learn propositional logic and quantifiers, sets, functions, and basic operations on them. After that, the three foundational subjects are waiting for you: Real Analysis, Linear Algebra, and Abstract Algebra. Reading more than one book on the same topic almost always helps. (What comes after that, and which textbooks to use, probably deserve their own posts.)
8. Find a study buddy if possible
Having at least one person to discuss things with is huge. Explaining ideas to each other and struggling together cements things in a way solo work usually doesn’t.
Please share your thoughts, questions, and critique.
What’s one thing you wish you’d known when you started learning math on your own?