relearning something in mathematics?

Question

I find that the following scenario happens to me quite a lot in my studies as an advanced undergrad: I take a relatively difficult course and do well in it. Eight months down the road a specific topic from the aforementioned course is brought up in my current course and I have completely forgotten said topic. I then find myself backtracking and taking close to an hour or two relearning the topic. Is this "normal" for those serious about mathematics? It happens to me more often than not and I am too afraid to ask my direct peers for fear of them finding out just how much I lack in memory management.

So does this happen to you all? And if so, how do you best make use of your "retraining" period?

Answer 1

As @Mick said Backtracking is quite normal. You cannot learn mathematics by just memorizing the instructor lecture and work problems. In order to learn mathematics, you must be actively involved in the learning process. I always follow three steps:

Do as much mathematics as you possibly can. This is not limited to textbook exercises but includes competition problems, finding alternate proofs of theorems, working out concrete examples of abstract theorems, etc.

Question everything. There are a few aspects to this. If something is unclear or unmotivated to you, ask yourself exactly where it becomes unclear or unmotivated. Find someone to explain it to you (for example, on math.SE!). Read a blog post about it. Write a blog post about it! Ask yourself how things generalize and how they connect to other things you know. (Again, math.SE is good for this.) The worst thing you can do is to accept what a textbook tells you as the Word of God.

Finally, teach as much mathematics as you possibly can. You would be surprised how much you can learn about something by teaching it.

Answer 2

In my experience, that is completely normal. No one can remember all the theorems and proofs there are and no one expects you to do so.

BUT it really shows how deeply you understood something. If you just reread all the theorems a while later and think "Oh yeah right, I now remember" then you understood it in the first place.

If you need to learn it again and go through everything your understanding was not very deep and you did not connect the proof strategies to your overall skills to solve future problems.

So if you notice this, try to challenge yourself if you ACTUALLY understand what you are reading right now and not just accept it as you read it.