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When I go through textbooks should I write out solutions to the exercises? Or is it fine if I just do it in my head? I mean either way you are still doing the problems right?

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    $\begingroup$ This is entirely too subjective; assuming you aren't required to turn them in, some people get more benefit out of writing things out than just thinking them through, while others don't. However, unless you are very good (and you may want to have third party verification of this), you are far more likely to make lots of mistakes (or invoke unstated and invalid assumptions) by doing them in your head than by writing them out carefully. $\endgroup$ Jul 22, 2011 at 20:36
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    $\begingroup$ Write it out! You will quickly find that an unbelievably large number of times you think you have a full solution in your head when in fact you have steps that take a lot more effort to justify than you thought or are just plain incorrect. This isn't an attack on your abilities, it is a general statement about the fallibility of the human brain. $\endgroup$
    – Matt
    Jul 22, 2011 at 20:38
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    $\begingroup$ @Alex J: Very good means "very good at doing mathematical problems, not making mistakes, keeping track of everything, not making unwarranted assumptions", etc. $\endgroup$ Jul 22, 2011 at 20:45
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    $\begingroup$ I'm in a less charitable mood than Arturo and insist: YES! There's no way around it. Otherwise you'll just lull yourself into believing you actually understood something (humans excel in this respect in general). $\endgroup$
    – t.b.
    Jul 22, 2011 at 20:53
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    $\begingroup$ I don't have enough +1's for @Theo's comment. $\endgroup$ Jul 22, 2011 at 21:15

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I can't tell you what to do, but for me, I cannot overstate the benefits of writing things out! The things I learn seem to stick much better when I write them out on paper. I take notes all the time when I'm reading as well. Even if I don't save what I've written, it helps.

Maybe this is just me... I don't know, but as I said, for me this really works. Since I started doing this I remember things a lot better, especially definitions and conceptual stuff.

Try it!

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I'd recommend going through it in your head for the mental exercise. Then, write it out so you can prove to yourself you did it right. Also, if you write it out, it'll be easier to recognize patterns you can use to solve future problems.

Finally, you'll need to get used to showing your work because when you get to the real world, people are going to count on you to show them how you arrived at your conclusions. Writing out your work now is good practice.

Another note, you may want to be able to show your work to prove you didn't make the error that crashed the Mars orbiter. It's pretty hard to prove your math was right when you can't show someone your work.

Oh, a final final note, if you are good at writing out your work, you're more likely to get partial credit on tests. If a teacher sees that you understand the process but you accidentally transposed a few numbers or something, you'll probably get a handful of partial credit instead of missing the entire question.

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The following is from Bill Johnson's answer to the Famous Mathematical Quotes question at Mathoverflow:

Jean Bourgain, in response to the question, "Have you ever proved a theorem that you did not know was true until you made a computation?" Answer: "No, but nevertheless it is important to do the computation because sometimes you find out that more is there than you realized."

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    $\begingroup$ I do not get this. So Bourgain proves theorems without making computations? $\endgroup$
    – Alex J
    Jul 22, 2011 at 22:05
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    $\begingroup$ @Alex J: I interpret the quote as follows: For each theorem that Bourgain proved, he did not need to do a computation to convince himself that it was true. Often, there are structural methods outside computation that you can use to make guesses. $\endgroup$ Jul 23, 2011 at 6:43

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