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I'm freshman, bachelor of Math major. When I read and learn the textbook, there're lots of formulas, laws, definitions and so forth. But what is the better way to handle it ?

I mean, is it necessary to remember all of them ? Because, I've tried to undertand it, and try to prove it by hand by myself, but it can't work all the time, sometimes, even if I prove it by hand, and somehow 'undertand' it, but still to forget it several days later.

I'm not very sure how to balance between , putting all the focuses just on think about why it works & grabbing the main idea & thinking method, and putting some energy on memorization. Sometimes, I am thinking about how those famous Math Masters in the world to learn the math most efficiently, are they just concentrating on thinking only, or proving it by hand for lots of times to remember it pretty naturally ? (I love math very much, but I think maybe I need to have a more efficient way to do that.)

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I am no maths major but every learning field reduces to some basic values. In engineering we are taught dont answer first, find the question first. It sounds philosophical but it isn't. I would like to share what i do myself i first of all try to find the reason i am learning anything/subject.Then i try to make clear that what is my motivation for that thing.

After writing all this i am starting to get dizzy but my point is that whenever i try to do anything i try to make it clear why i am doing anything like

  • why should i learn this? (because you love maths)
  • why should i practice the exercises?
  • what do i want from this thing?
  • What i am going to do with this in future?

All these questions are not clear at once or maybe not at all. but At last what is important is that you keep loving maths and it would compel you to learn more and consequently practice more (hope i do that myself too)

I am no maths but i love it in the form where things are superficial

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You can read every book ever written on chess, but if you never play you will still be, at best, a middling player. Even if you memorize every rule in every book on chess you still won't become a particularly good player. You must play!

The same is true of math. You must solve problems!

I was always a natural with math, and I almost always grasped concepts the first time my professors covered them. For a long time I believed that this was enough. But as the topics became more and more abstract and more and more complex, I started to fall behind. It didn't feel like I was so good any more. I had developed the belief that practice was somehow beneath me. But practice is exactly how you get good at math.

Understanding is enough at a basic level. You can hold everything in mind and, if you understand it, you'll see the solution. But math continuously builds on itself. As layer upon layer of complexity is added, nobody in the world can hold all the pieces in their mind at once. Nobody! If you practice enough though, you no longer need to think about it. If you've practiced every layer below the one you're working on to the point that it's pure instinct, you don't have to hold any of the lower level stuff in mind anymore. You can focus all your attention on the high level content.

That's exactly like chess. You may understand basic tactics. You may have memorized all of them. But if you still need to look for forks and pins you have a long way to go. Put the books down and go play a few hundred games. Eventually, seeing the basic tactical elements will become as natural as breathing. Now you're ready to begin seeing the deeper elements of the game.

You asked whether you should focus on memorization or understanding. I'm saying neither. You didn't memorize your native language. You don't need to memorize math. Immerse yourself in it. The remembering will happen automatically. In terms of understanding, unless you completely master each step by practicing it until it's instinct (you really just need to get 90% of the way there; as you reuse the concepts down the road you'll go the last 10%), you'll never see the deeper elements of the game. The understanding you get from reading the text is shallow. The understanding you get from practice is deep. It's fluency.

Do not be seduced by the apparent superiority of problems over exercises (if you know what tools you're going to use from the outset, it's an exercise; if you have no idea where to start and need to puzzle it out, it's a problem). Problems are great, and, ultimately, you should definitely test your knowledge on them. But exercises should be the bread and butter of your training. Sure you know how to do them. They almost seem demeaning. But if you regularly work your muscles on these seemingly trivial tasks (like jogging or weight lifting), the challenging, novel, exciting tasks (like climbing a mountain) will get easier and easier.

Do not read the chapter a second time until you've attempted most of the problems at the end. If you solve several of them you'll find that the chapter makes way more sense the second time around. Go back and solve the rest of the problems and if you read it a third time it'll seem painfully obvious. If the book has all problems (at the freshman level all of your books are probably chock full of exercises, but in a year or two you'll start seeing books like this) and no exercises you must find as many problems with solutions as you can from other sources. Many high level texts have a handful of challenging high level problems at the end of chapters (frequently without solutions). Each problem will be unique and be solved in a different way. The lack of repetition means that it's very difficult to attain the "instinct" level. Find more problems elsewhere and solve them. As much as possible, only work on problems you have a solution to (the feedback is essential).

If you can't get additional problems. Just solve the ones you've got over and over (this works good for proofs, just make a list of proofs you want to know and work through it once every night or two with a blank stack of paper). In fact, if you couldn't solve a problem the first time, always re-solve it after you've seen the solution. Keep coming back to it until you can solve it without even a peak at the chapter or the solution. If you get stuck on a new problem for an hour or two, go back and solve similar easier problems for a little while and come back to it.

This may sound like rote memorization, and, beyond that, like a heck of a lot of work. It's not about memorization. Just try it for a while and I guarantee you'll find that your understanding goes through the roof (even if you think it's pretty darn good to begin with). And, well, yeah, it is a lot of work. But maybe less than you think. Doing one-hundred problems is not ten times as much work as doing ten. Problems eleven to thirty probably take about as much time and effort as the first ten. So, probably, do the last fifty. At the beginning new tasks are often unpleasant and frustrating, but with practice they become, if not fun, at least satisfying. Just like jogging. Most people stop practicing just when the learning curve is getting steep (that's the good part, even though it sounds like the bad part).

That's probably more than you expected. I've made my way through a lot of math though, and this is what I've learned. Meta-learned would be more accurate I suppose. If somebody had explained this to me clearly when I was a freshman, I'd probably have gotten quite a bit more out of my education.

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I find this holds true with anything I have ever tried to master. –  juxstapose Apr 20 '11 at 17:42
    
I couldn't agree more with the last sentence. –  Harshdeep Dec 19 '13 at 23:20
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As you progress, the need to memorize things will disappear. Formulas which you use often will just find their place in your memory - that's how I "memorized" Stirling's formula.

Most of the stuff that really needs "memorization" - or rather internalization - is definitions and their elementary consequences, and basic theorems. The best way to tackle these is to solve exercises. You don't necessarily need to solve exercises "by the pound", though it's a good idea when you're just starting a new subject; as you progress, it's more important to solve challenging exercises, which really put what you've learned into perspective.

There are higher skills you need to tend to, such as problem solving and concept formation. For the former, difficult exercises will work. For the latter, you should reflect on what you've learnt, try to think of interesting questions, and then try to solve them. When you've reached this level, you're really doing mathematics.

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Could I say it's not such necessary to learn new subject very systematically? Focusing more on essential concept and understanding,not memorizing, but re-read and re-think it for many times, it could be in the memory pretty naturally? Because in my recent experience, I always want to master one prerequisite very well before moving to next one, (eg. master single-calculus very well before moving to multi-calculus), but I realize that it never be possible to master one thing 'perfect', so result's staying one place for too long time.Because I always forget details more or less. –  Xingdong Apr 20 '11 at 0:40
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Try to remember and focus on concepts and ideas. For example, a group is some set with some operation satisfying some axioms. Instead of memorizing those axioms, think about what concept the axioms try to capture. Instead of memorizing the definition of an $R$-module, just realize that it's an $R$-vector space (which is nonsense, but there is an obvious way to make sense out of it), or that it's just a ring acting on an abelian group $R\to \text{End}(M)$ (if that wasn't already your definition). Ideally, the concepts and ideas will become so natural to you, that the definitions and details are 'obvious', or 'inevitable'.

Also, I think it's good to have a stock of examples in your mind for each definition, especially when you're dealing with multiple definitions that have only subtle differences. Then sometimes you can patch these together. A louzy and contrived example might be `I remember that $\mathbb{Q}$ is a field but $\mathbb{Z}$ is not. Then probably you need to be able to divide (have inverses) in a field.'

Of course you can't avoid blindly memorizing some things. In fact (in my experience) most of the time such an understanding only comes later, after struggling, forgetting, and rereading. As has already been mentioned, doing a lot of exercises and making up your own exercises can be quite helpful for that.

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Could I say it's not such necessary to learn new subject very systematically? Focusing more on essential concept and understanding,not memorizing, but re-read and re-think it for many times, it could be in the memory pretty naturally? Because in my recent experience, I always want to master one prerequisite very well before moving to next one, (eg. master single-calculus very well before moving to multi-calculus), but I realize that it never be possible to master one thing 'perfect', so result's staying one place for too long time.Because I always forget details more or less. –  Xingdong Apr 19 '11 at 21:23
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No better way than doing lots of problems and exercises. After a while, most of it hopefully will become second-nature. Then go back to the text and try to prove things yourself, without looking at the proof in the book. Better yet, get a different text and prove its theorems.

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