Still forget even if theorem-proof "self-discovered"; Importance of intuition/proficiency of concepts in research work...

Question

It is widely said if we go through concepts/theorems/proof on our own by actively doing instead of passively reading, the idea will be ingrained in mind. I agree with that, it really often helps. Particularly if we re-construct the knolwedge in mind, see the big picture of what the goal it is and also dive into details for how to do it.

However, I found a phenomenon in my own experience, such that sometimes, even if I do derivations/proofs on my own and writing it down on papers, I still forget.

e.x. 1.

During the course in discrete mathematics, prior to learning it, I was playing sketches on paper for fun and "self-discover" the formula for the number of Hamiltonian path for a complete undirected graph.

e.x. 2.

During other courses, it occasionally happens such that something was self-revealed before learning it for the first time.

Thus, the concepts at least for me is "genuine", it should probably very much ingrained in my brain.

BUT, very strangely, I often found that I'll still forget them at all, after several months (even shorter) since the end of the course. In later time, when the concepts in front of me again, not only do I not remember what the formula in exact details, but also sometimes even forget what it is really doing or the idea how to derive the formula.(Even for the parts which were self-discovered)

Question 1: Does it happen to a working mathematician for the parts without "daily-use" ? Even one did works/results on one's own, but still forget with time ?

Question 2: How should it be well-resolved ? Should I use flashcard frequently after learning a new definition/theorem/technique by writing very brief cue on it, and repeat doing all details when it comes to leisure fragmentary time (walking to school/home, transportation, lecture breaks, lunch etc...) ?

Question 3: If a working mathematician also forget definitions/theorems/proofs, but they do generate good new works, so that what important thing could we conclude from it ? Precisely, should we focus more on what the concept is really doing (the goal/motivation), and how it is connected to other concepts ?

Question 4: Ex. from Terence Tao who have good works in many branches, in one of the interviews, he said "I always have many things on my plate". He even have works in signal processing (compressed sensing), cryptography... He might not be really expert/highly-proficient in signal processing, but still do great works, at least enter the field very quickly. What potentially good habit we could learn from this ? Can we say that it might better to be question-oriented learning/holding big pictures for what the things really doing rather than over-immersed in technical details ?

Question 5: Will it be recommended to re-think, summarize and organize the knowledge we've learned after each chapter on the textbook ? Since in daily reading, we might more focus on the technical details, why this step, how to do this calculation etc... But after each chapter, even with exercises, I might be able to solve types/sets of problems, prove some certain theorems, describe some definitions when it is in front of me, but not always have a clear "full map" after finishing the chapters. Maybe it is useful to "filter" the chapter after reading/solving problems, to hold the idea of important things, and "transparentize" the less-important ones ? Will a research mathematician necessarily/helpfully have this "structured full map" in mind, or just simply problem-in-solution-out is more common ?

Answer 1

Question 1: Everyone forgets things they saw long ago as they age. This is normal. Even if you are less likely to forget something you did yourself (which I don't think is necessarily true), you will still forget it at some point if you never look at it again.

Question 2: Stay involved in math. Reread your textbooks and your notes, read papers, read Wikipedia, and participate in MSE.

Answer 2

Question 1: Everyone forgets, there's nothing you can do to stop from forgetting something. There are things you can do, brain games, and excercises to help, but after time we will all forget things. Even my professors who know more about math then I hope to know in my whole lifetime come across problems that they know they should know what to do, but they can't think of the exact formulas or theorems that they need to use. I have even emailed people from mathematical journals about their article and have had some email me back and tell me that they would have to look back over that particular article as he couldn't remember the answer to the question I asked off the top of his head and would need to refresh themself. So even the best of us forget. On a side note, I've heard that you have to do something anywhere from 7 times to like 56 before you truly have it memorized. Which leads into the next question.

Question 2: The only way I can see to reduce the stuff you forget is by constant repetition. So I'd say that flash cards are a great thing to do. I use them all the time. For instance I try to learn $10$ German words at once one week. Then I'd learn another $10$ the next week. Then after that I'd go back to the ones from the first week and add $5$ more from the second week and constantly mix it up different ways, but constantly adding more words to the list. This is really hard to do with physical cards. I was wearing them out way to fast and they got time consuming to make. So I went looking for computer applications. I came across an application called Anki, which does everything for me, but it does it all electronically and it also lets me choose whether; I knew the word really fast, It took me a minute to remember the word, or that I didn't know it at all. These different buttons affect how often the card is shown. I also drive $2$ hours one way to school so I like listening to things in my car as I drive. MSE has also been a great help for me. Even the ones I get wrong I always learn something from it.