Currently finishing my junior year of undergrad in math. I'd give different advice to a casual enthusiast than I would a full-time math student, but here are so

Currently finishing my junior year of undergrad in math. I'd give different advice to a casual enthusiast than I would a full-time math student, but here are some tips that have helped me: - Reread everything. The first time, only read definitions and theorems; proceed steadily and focus on high-level ideas. The second time, try to recall definitions and focus on details of theorems. The third time, try to recall definitions and recall proofs of key theorems; work on memorization and fluency. The fourth time, try to modify the conditions of theorems and see what changes. If you are struggling with reading, get as far as you can and then go around again - it gets easier every time. Take as much time as you can afford before reading the same section again to let the material sink in. - Do lots of problems - start with trivial examples and slowly scale up the difficulty. Lock down the basics before moving on to harder problems. Ask for recommendations for books that have particularly good problems. - Practice proving theorems and claims. If you have a question while reading, make a conjecture and try to prove it formally. It took me too long to realize that understanding abstract math is not the same as being able to do proofs and formalize intuition. Following along while reading doesn't mean you can quickly or easily prove facts about the material, and the proof is what counts in the end. Knowing why something is true is not helpful if you cannot formalize it. - Hone intuition. Proving things is just performing a (directed) graph search on a set of facts with a well-defined start and end goal. Having good internal heuristics will help the search proceed quickly, in the right direction, while avoiding dead-ends; the only way to develop this sense is by practice.