## 1. Write solutons of problems down carefully

When you’re doing a problem or filling in a proof, you should write the solution down very carefully. For example, with a proof, it would be better to slowly write down each step in complete sentences rather than scrawling a bunch of shorthand and symbols. By doing this, you automatically check every step mentally, because the extra time to put down a statement or reasoning step in full sentences will force you to do so.

## 2. Read the foundations

Sometimes, it’s possible to get through advanced material, especially papers, without knowing the basic foundations. And eventually, you will have to take some mathematics by faith because it’s simply impossible to know every detail.

However, by spending as much time as possible on foundational material, you will have a deeper understanding of more advanced material when you read it. Things that appear out of nowhere will suddenly make sense. A classic sign that you don’t know enough foundations is always looking up the same facts again and again while trying to understand something.

For example, if your research area is commutative algebra, it would be a really good idea to go over at least one book on commutative algebra line by line, working out every proof and exercise. That type of foundational reading will pay off continuously every time you need to do research.

…read the rest of this post!