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I learn math in my free time. I do it mainly as a hobby, but it also helps me as a software engineer. My main resources are MathTutor site and Stroud books. The problem I have is I can't work on math everyday, and I always feel I forget what I learnt in the past - from simple things like how "factor by grouping" works or trig identities to a bit more complicated things like what integration technique should be used for a given problem. So, I go over algebra, pre-calc and basic calculus over and over again. I mostly watch videos, read and do the exercises "in my head".

Do you have any tips that could allow me to remember what I learn and continue to more advanced math? For example, is it important to do the exercises on paper or keep you workbooks so you can come back to it later? Do you use concept maps flash cards and things like that?

Any tip would be greatly appreciated,

Thx

Pierre Tammet
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    Writing down things is key -- you won't really get practice with thinking mathematically unless you actually write down proofs of your own. – hmakholm left over Monica Nov 08 '13 at 20:51
  • Could you give some examples of things you have difficulty remembering? – Jack M Nov 08 '13 at 21:37
  • @Jack M - I edited my question. Thanks. – Pierre Tammet Nov 08 '13 at 23:50
  • @Henning Makholm - Writing on what purpose? Come back later to what you wrote, or make your thinking more "concrete"? – Pierre Tammet Nov 08 '13 at 23:52
  • @PierreTammet I think most of us recall things easier later on when we have written them down initially. I rarely physically flip through my notes for something I forgot because the process of deriving that thing is somewhere in mind, so when I need it I tend to be able to rederive it in the same way without necessarily recalling specifics. Possibly one reason why paper and pen works so well is that when taking notes you force yourself to look away from the source and think about it--for this reason I believe direct copying should be avoided in lieu of using your own words. – Karl Kroningfeld Nov 09 '13 at 01:03
  • @KarlKronenfeld, my thoughts exactly. It is much easier to recall when you've written it down. That's what lecture forces you to do. With videos, you don't feel that same need. – Christopher K Nov 09 '13 at 05:53
  • Does this answer your question? How to deal with the temporary nature of my knowledge? – user1147844 Oct 07 '23 at 09:40

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Personally, I've never had the patience to watch videos (I'll either fast forward through them or not watch them and opt to read the book instead) and I don't think they are all that effective. Lectures, on the other hand, are in real time and interactive, which is better. So I recommend auditing or enroling in a course at university. Outside of lectures, reading key texts is important. By reading, you've really got to "own the material" (this means definitions, theorems and proofs). Now, when you said you did exercises "in your head", I got a little bit concerned. If you can do the problem "in your head", then the ones you chose were too easy (probably chug and plug) and not worthwhile. To really learn material well, and thus remember it, you have to experience the frustration of not being able to solve a problem, but then still have the tenacity to unravel and ultimately solve it. The key to mastery is doing exercises of increasing difficulty (up to and including harder problems) on a given topic. You should keep record in a notebook. As for "flash cards", no self-respecting student of mathematics uses them; they only work for 'memorizing' but not 'understanding'. I conclude by saying that learning math is a 'marathon' and not a 'sprint'. It is much more about racking your brain about harder problems, then it is about doing the computation for trivial ones as fast as possible.

DISCLAIMER: These views are larged based on personal experience, but there is no one size fits all method for studying. So please hold my opinions at face value.

Christopher K
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  • Flash cards are great. If you find yourself "memorizing but not understanding", you need to do more problems. But memorizing definitions and theorems is not going to hurt your ability to solve problems. If anything, it makes it easier, since you don't have to do slow things like looking for definitions and theorems in the book. – nomen Nov 08 '13 at 23:05
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    @nomen, I would never fail to emphasize the importance of knowing your definitions and theorems. Your point is valid. However, flash cards are, at least personally, a waste of time. If I've read and understood the proofs, I then move on to exercises. If I get stuck on a question, then I review and (re-read if necessary) the proof. You may think you understand, but the exercises will invariably emphasize certain nuances. However, copying or regurgitating has NEVER helped me. If I apply something enough times, then and only then do I seem to remember it. – Christopher K Nov 08 '13 at 23:31
  • @Chris K. Thanks for your answer. The thing is I don't think I'm an efficient reader, so I favor videos. Maybe I should keep reading to feel the "frustration" you describe. I must admit I don't experience it doing math, which could mean something as you suggest... P.S. what do you mean by "key texts"? (English is not my mother tongue) – Pierre Tammet Nov 08 '13 at 23:54
  • If you stick with videos then stop them every few minutes or after every major idea and write down what's going on in your own thinking. If a problem is presented or a question is asked then stop it and try to work it out on your own before the solution is presented. – R R Nov 09 '13 at 00:07
  • @PierreTammet The efficiency of reading (new) math is not the usual "pages/time" because math concepts are by nature difficult to understand, and based on your own knowledge and experiences some ideas will be easier to grasp than others. This means some pages might take a few minutes to read and others might take hours. (No, I am not exaggerating, and I am referring to pages of prose and not problem sets which vary based on the tastes of the author.) – Karl Kroningfeld Nov 09 '13 at 00:53
  • @PierreTammet What I mean by key texts is something like Spivak's Calculus for an introduction to single-variable analysis (if you haven't seen $\delta-\epsilon$ proofs before, this is a good place to start); or Munkres/Spivak for Analysis/Calculus on Manifolds; Dummit and Foote's Abstract Algebra... the list goes on. As for videos, the funny thing is I usually fast forward through them, instead of taking my time like when reading. But each person has their own flavour of studying, right? – Christopher K Nov 09 '13 at 05:46
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My approach is typically to find books that are challenging enough, and have lots of exercises. Then I get the hang of the material through two approaches (besides careful reading):

First, do any exercise you can understand. Some category theory books, as an example, often have exercises drawn from areas I'm totally unfamiliar with, so I don't aim to do all of them; but I try to get as close as reasonable.

Second, if you're reading a proof in the text, and you have a hunch that you know how it goes, put it down and work out the proof yourself. I've also done this reading papers aimed at a better educated audience; if the details are sketchy, I try to flesh them out on paper.

I absolutely recommend having a notebook (virtual or otherwise) to work things out on. In one's head, it's easy to skim over details based on what seems sensible. Only when you actually write them out will you discover (to your enrichment) that you don't have quite the grasp you thought you did on some point. Better to get those cleared up sooner than later.

I also recommend keeping around material that's well over your head, but sounds interesting. They can be a great resource for suggesting study programs (by working backwards to figure out what's required to understand it), and as benchmarks to see what you understand now that you didn't before.

Malice Vidrine
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