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I'll begin graduate school next year and I am very impatient to learn new things such as theories, ways of thinking and so on (I enjoyed reading about category theory on my own and I find Galois theory very interesting for example).

However, during my undergraduate - and I am worried that it would do that again in graduate school - I was often stuck because I had to remember old results and/or had to figure out basic things - that I didn't study - (and exemple of this "basic things" can be found in my previous post where I was stuck with a simple set theory equality involving complements) and I find this very hard to manage (in terms of energy and time consumption). When I finally understand what I missed, I am discouraged (it breaks my motivation and I am very tired). Who knows how many of these things I forgot ? What if I have to use one of these tricks during an exam / while learning new material ? How can I better learn to avoid this situation in the future ?

I would like to know how do you react while dealing with similar problem (those who have good - or fresh - memory are not in this position). Do you have a way to quickly find the good reference (except SE) ? Does that break your motivation like me ? If yes, what do you do to "get back on the rails"?

Thanks for your answers.

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    $\begingroup$ If you can, get a university job tutoring/leading a recitation section. It forces you to rethink and clarify material. $\endgroup$
    – Simon S
    Mar 14 '15 at 16:09
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    $\begingroup$ It is natural to forget things, don't let your motivation drop because of that! You need constantly repeat things in order to keep them fresh in memory. It also helps to make as much connections between different fields as possible and to have multiple ways to view at one thing, in order to keep remembering the facts. If you can give some real meaning to things, other than just having remembered it (e.g. a physical meaning), you tend to forget them last fast. Also, repeat, repeat and repeat. $\endgroup$
    – Pedro
    Mar 14 '15 at 16:11
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    $\begingroup$ Yes I understand. And of course you studied well enough, otherwise you would not be able to get where you are now. And it is true that you can prove things without saying why. :) And you do not always need to know the big picture to answer question, although it helps, and a lot of professors even don't expect you to know the big picture (and they don't tell this picture also very often), although you tend to forget things then. If you are interested in the big picture and you have time, lectures of MIT opencourseware help a lot. $\endgroup$
    – Pedro
    Mar 14 '15 at 16:21
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    $\begingroup$ I for example reviewed everything from linear algebra with the MIT opencourseware lectures of prof. Strang see here, because the way I was thought this material was very abstract and didn't give much intuition. If you learn the material from MIT professors you will never forget what you learned. You can do this if you have time of course. ;) PS: if your professors didn't gave you much big picture, they probably didn't capture the picture either. So you don't need it, but is helps to memorize. $\endgroup$
    – Pedro
    Mar 14 '15 at 16:24
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    $\begingroup$ @Pedro Yes I loved these lectures, as well as "Abstract Algebra" from Harvard Extension School (I love the way the world is sharing high educational knowledge and experience). And one more time, thanks for your answer and encouragements! $\endgroup$ Mar 14 '15 at 16:27
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In my experience, any material that you do not regularly brush up on, you forget. Sometimes partially, sometimes completely. You don't learn things once, then know them forever. You learn things once, then you remember the stuff that you need again and again. So don't get discouraged when you have to brush up on old material in order to understand new material, think of it as a chance to brush up on material in order to keep that material within your grasp. At some point, when you have had to brush up on certain material enough times, it will stick. This is part of a natural process where you forget material that you don't need, whereas the stuff you need is set in stone.

Always remember that the stuff you need is probably different from the stuff some other mathematician needs. So the process is unique to you, and in the long run defines the type of mathematician you become.

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  • $\begingroup$ You are totally right about the learning process, maybe I am too tough on myself... it is not normal though that it breaks my motivation and make me feel tired $\endgroup$ Mar 14 '15 at 16:30
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    $\begingroup$ My parents always told me that you need to repeat things your whole live, or you forget them. So I always tried to, from time to time, repeat old things. It is just the way it is, all people need to do it, or they forget things. I am quite sure that the other students in your group will also have forgotten the things they learned, if they wouldn't refreshed it. To tell you a funny fact, I am a computer scientist. And I once someone at my university who studied math something about differential equations. She just didn't know anything about it anymore. $\endgroup$
    – Pedro
    Mar 14 '15 at 16:48
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    $\begingroup$ In fact, all mathematicians at my university seem to forget their knowledge, so I guess most professors didn't gave much of a big picture during their lectures and also that most people didn't repeat that much. :p So you are not alone, you are just as every one else. ;) $\endgroup$
    – Pedro
    Mar 14 '15 at 16:48
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    $\begingroup$ @AlanSimonin To be honest, I think it is quite normal to feel exhausted after spending time recalling material that was probably hard to learn the first time around. Perhaps you should simply get into the habit of allowing yourself some time for this type of preliminary work, so you don't feel that you wasted your time and didn't get anything done. Be aware of the fact that you did not waste your time, you merely had to spend your time on other things than what you had hoped for, and this may have been for the best. Your feeling of discouragement has to do with failed planning only. $\endgroup$ Mar 14 '15 at 21:25
  • $\begingroup$ @user161825 Thank you for your comment. I am curently preparing myself for graduate school therefore I have plenty of time. When I am stuck because of things I have to remember it's ok. But when I have to know something that was not cover during my undergraduate school, I find that a little bit discouraging. I know that I'll have to do this research work later - especially if I attempt a phd - but I feel a litte bit irritated that the teacher assume we should know that. I should maybe take that more like a research than a lecture... $\endgroup$ Mar 14 '15 at 22:44
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This also has to do with how you used to study. I always forced myself to be able to reproduce everything from scratch. Deriving everything from first principles with all books closed on a blank piece of paper is a bit more time consuming, but this yields superior results.

To this date I can reproduce most of what I learned decades ago without needing to look up anything. The only reason I do look up things is because that saves me time. When I have to teach and I want to refresh my skills, I'll first reproduce the entire theory on a black piece of paper just like I did when I was a student. Only then will I look at the books. I'll then know what I should pay attention to.

What helped me here was the fact that I started studying from advanced math and physics topics from university books when I was still in high school. I therefore had plenty of time on my hands, not under any pressure to submit homework of to be prepared for exams. And then when I went to unversity I was so far ahead that I could just go on with that routine.

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  • $\begingroup$ Thank you for your comment. I like the idea to be able to build everything from scratch, and I would be happy if I could do the same. However, it needs very good bases that I don't (yet) possess. Maybe one day. $\endgroup$ Mar 14 '15 at 22:47
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Yes, been there a hundred times myself, the "Oh God, do I need to go through all that stuff AGAIN!" feeling. What helps me is to start a different section of a course, with new material to provide motivation, but to choose a section that depends for its understanding on the old material you have already covered.

So you are forced to go back a bit, then you dig out your old notes and find that after an hour or two, it's a lot easier to remember because, I think, its always still somewhere in your unconscious and I find myself saying, hey yeah, this is easier that I anticipated. I have the difficult parts already explained to myself in the notes.

I also try to keep an good index of my notes, it helps because part of the lack of motivation is the feeling you don't know what exactly you covered and you usually assume the worst. Also depends on your moood that day, I don't over think it or force it , I just let the motivation return which it does naturally when you see some new stuff or stuff you know that you already understand.

Like a golfer tired on practicing his swing a million time, gets fed up then the next day , sun is out, and he is invited to a new course with his friends. Will his motivation return, I reckon it would.

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  • $\begingroup$ Thanks for your comment. Sometimes my motivation takes a few weeks before it comes back, that is all the problem... $\endgroup$ Mar 14 '15 at 22:46
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    $\begingroup$ @alansimonin You asked an excellent question that I think would resonate with an awful lot of people who really want to learn. I hope you don't mind me saying so, it took, well courage is too strong a word, honesty is probably a better one, to post it. I learned a lot from reading the answers. My area is physics and i think that's a lot easier to recover after time than pure maths. Best of luck with your study in the future. Thanks and regards $\endgroup$ Mar 15 '15 at 2:32
  • $\begingroup$ You are right! The sensation was there for a long time, but being able to put it in a question was another challenge. My learning also includes reading the answers. You can't be stuck forever in a small exercise because of a litte misunderstanding. However, I made the mistake to look at ALL the answers instead of trying to solve the most by myself. Thank you again for your answer and your comment and good luck as well! Regards $\endgroup$ Mar 15 '15 at 10:41
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Some very good points are made in previous answers. I would just add that it's also good when you learn new definitions and results to dissect and interpret them to the maximum degree possible. It helps to remember theorems if you have a strong intuition for why they are true, what basic/familiar results they generalize (e.g. many things in functional analysis are generalizations from familiar results in linear algebra), what sorts of pathological counterexamples a definition is meant to exclude, etc. It's always good to try to see what would happen if certain hypotheses were dropped from a theorem - why would it fail?

Personally, I find that this kind of reasoning both sticks better in my mind and is more fun if I talk it out with a colleague for whom the idea is also interesting. A colleague can challenge your understanding by asking questions you wouldn't have thought to ask, and somehow the act of verbalizing always forces me to make my ideas more precise than if I just ran through them in my head. Sometimes thinking by myself, I have an over-inflated impression of how concrete my understanding is. Partly this is because I'm always impatient to learn things fast so may try to rush forward, and partly because I have a bad habit of agreeing with myself. This means that a good colleague to talk with is one whose intrinsic nature combats these tendencies. These might or might not be similar to your personal patterns, but maybe the principle generalizes even if not the particulars!

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    $\begingroup$ Thanks for your answer. I find that this is a very good way of learning too. That is the one I am trying to use this year (and being able to read another point of view from another book is very helpful too). For the second part of your answer, I don't have colleagues to talk with because it is an online formation and there is not a lot of people responding on the forum of the University. It forces me to find answers by myself though, which is a good practice. $\endgroup$ Mar 15 '15 at 10:48

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