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.
3. Don’t study all at once
Once I had an exam in ring theory and I didn’t study until the night before. I think we’ve all been there. Actually, the exam went really well because I have a good memory. But then two days later I forgot most of what I studied.
By breaking up study over longer periods of time, you continually reinforce your knowledge and repeat it by using it. Previous theorems are used in new ones, and by using results in different ways over time you will remember the material much better.
The disadvantage of studying all at once aside from forgetting is that you will have to learn it more than once. And I always find trying to learn the same material a second time a little more difficult than the first time.
4. Explain the material to someone else
Find a friend in class or a colleague and explain the material you are learning. There are a few reasons to do this. First, you are really forced to go over the material, even things you might gloss over because they are not the most interesting. They will ask you questions also and therefore you really need to know the material to explain the answers properly.
Secondly, social learning is quite motivating. If someone else seems interested in anything, it will automatically make you more motivated to learn, especially if you are already piqued in the first place. So many times I have been asked questions when I was teaching tutorial sessions, and I felt afterwards much more interested in that specific line of reasoning simply because I saw someone else interested.
5. Do a sufficient number of exercises
If you are learning from a book, it probably has exercises. Do as many as you can, and select the questions that look a little tricky. There many be a few problems at the very start that are quite easy, so you need to skip those and do the harder ones.
For doing exercises, I would recommend spending a fair amount of time just thinking about it if it’s hard. Don’t look up a hint right away. If it’s a fairly tricky problem, at least give yourself half an hour to come up with a starting point. Even if you don’t, you might want to sleep on it before looking up a hint or solution.
6. Pay attention to the small details you don’t understand
Take some time to identify the tiny things you don’t understand or take for granted without really knowing why they are true. These gaps in knowledge are often where forgetting what you know will start to seep through. Filling them in is often also much more productive than learning random new material.
For example, when I was an undergraduate, for the longest time I just took for granted that finite fields have powers of a prime number of elements. I basically knew why that was but I never really sat down and wrote down a proof. Then one day someone asked me why that was true, and I had to go through it in very fine detail to explain it. After that it really stuck in my mind and formed an anchor for understanding finite fields.
In fact, I still have some gaps in my knowledge about how a spectral sequence works, so I better go fill in those details…
7. Do computations
If you’re learning about matrices or matrix multiplication, I would say start out by just multiplying dozens of matrices together. This is slightly different than doing exercises because here I am emphasizing just doing basically the same computation over and over again with different numbers. A professor of mine used to say magic starts to happen when you do computations, and he’s right. The mind is superb at patterns and if you do computations, you will start to notice things that will become anchors in your mind to absorb the theoretical material.
If you are studying something numerical, then write simulations in Python or Matlab. The same applies to groups or rings for which there are computational algebra packages like Sage or Mathematica. By doing computations, you will learn a lot of cool stuff and have a bunch of examples to illustrate every theorem.
8. Choose the right book or source
I’ve mentioned this before, but I really think it’s important to choose the right source to read from. All your friends might be reading one book, but your style might match another book much better. Personally, I can’t stand fraktur letters so I avoid the books that use too many of them.
Don’t be afraid to just stop reading a book or take a different one out of the library, even if the book you are reading is the textbook for the class.
Papers are a little harder to avoid when they’re bad because often a paper is the only source with the specific results in it. However, some older papers have their results explained in books as well, which is helpful.
9. Allow yourself not to understand everything
When math gets a little harder, a lot of things you are reading will be incomprehensible at the start. So, it’s important to be able to let some of it be that way until you have time to learn it. Yet at the same time, it is important to learn enough to be able to work with something.
In reading really complex papers, I find it helpful to simplify the paper at first by introducing strong assumptions. For example, I might assume all the rings are Noetherian or that the field is algebraically closed and see if I can simplify some of the proofs.
10. Take breaks
It is so important to give your mind a break when you become mentally exhausted. Don’t hold yourself to a rigid schedule and force yourself to keep working even when you are tired. I typically find that after 2-3 weeks of intense mathematical work I just need to take 3-4 days of not doing any math at all.
Not resting takes its toll. Eventually, you could become totally frustrated with math and switch into biology. To prevent that from happening, take breaks!