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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?

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    $\begingroup$ Backtracking is quite normal. That is why we need re-search. $\endgroup$
    – Mick
    Mar 5, 2020 at 6:23
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    $\begingroup$ This happens to me quite often. That's life, no? Frankly, my analysis needs some dusting up. $\endgroup$
    – Rushabh Mehta
    Mar 5, 2020 at 6:40
  • $\begingroup$ "close to an hour or two" -- I have had to invest amounts of time into this that were bigger by several orders of magnitude. $\endgroup$
    – Torsten Schoeneberg
    Mar 5, 2020 at 16:04

2 Answers 2

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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.

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  • $\begingroup$ I've recently started questioning every theorem I see. It has really helped me in my statistics course so far. I have not tried using alternative proofs for the same theorem though. I've always thought that the one was enough. I may try my hand at it though. $\endgroup$
    – DataInTheStone
    Mar 5, 2020 at 8:00
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    $\begingroup$ @DataInTheStone I personally try to break event single line of the theorem and asking why it needs or what happens if I don't assume it. Soon I get the logic and that's helped me to bind this logic in my head. Which make sure I didn't forget the theorem for a while 😅. $\endgroup$
    – emonHR
    Mar 5, 2020 at 8:10
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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.

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  • $\begingroup$ I think I struggle very much with the idea of "understanding". If I got each exercise in the homework correct, then I thought that I understood. Yet here I am re-searching (@Mick TM) the very concepts whose exercises I went through. I think its time I sit down and formally write down for myself what "understanding" is as I certainly don't know how to grasp it. $\endgroup$
    – DataInTheStone
    Mar 5, 2020 at 7:56
  • $\begingroup$ You probably worked in a group while doing the homework, right? One thing is understanding the solution that someone else found. Another thing is coming up with the solution. The most important thing in these exercises is to understand the proof technique, not the proof itself! Generalize it, try it on something else, see its shortcomings and advantages. THEN you understand the proof :) $\endgroup$
    – Nurator
    Mar 5, 2020 at 9:31
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    $\begingroup$ @DataInTheStone There are different levels of understanding, such as: Do I know the meanings of the formulas and theorems and can I use them to solve routine problems? Do I know how to derive the formulas and prove the theorems from scratch? Do I know a thought process that would naturally lead someone to discover this formula or theorem or proof? Can I see intuitively why this theorem is true? Can I see why this result is secretly obvious? Am I able to convert my intuitive argument into a rigorous proof? Do I appreciate the mathematical beauty of what I'm learning and do I "hear the music"? $\endgroup$
    – littleO
    Mar 5, 2020 at 11:28

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