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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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    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.2011-07-22
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    When a teacher asks you to show your work, they want to make sure you're doing the problems correctly. If you do everything in your head, then you might make a mistake and your teacher wouldn't know where to help you.2011-07-22
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    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.2011-07-22
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    I second and echo @Matt's comment: it is stunningly easy to deceive oneself, and I think this is just human nature. While optimism is necessary, one must run a separate, highly-critical thread also. It's hard to do both, I think, although with much practice it becomes more feasible.2011-07-22
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    @Arturo: What do you mean by "very good?" Doing problems in your head is a skill that can be trained over time right? I guess writing out a solution first and then doing it in your head to compare.....would be good training?2011-07-22
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    @Alex J: Very good means "very good at doing mathematical problems, not making mistakes, keeping track of everything, not making unwarranted assumptions", etc.2011-07-22
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    @Arturo: It just seems like a lot of work. I see people's blogs and they seem to just copy stuff from textbooks. So it seems more an exercise on pretty LaTeX and formatting then the actual math.2011-07-22
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    @Alex J: I'm sorry; are you asking what to do to publish a blog, or are you asking how to learn something? I can only say that if you don't do a lot of work, chances are you won't learn much.2011-07-22
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    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).2011-07-22
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    I don't have enough +1's for @Theo's comment.2011-07-22
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    @Arturo: Is there a difference between typing it up instead of writing it up (problems)?2011-07-22
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    I think it is possible to do exercises without writing their solutions down but even if you are good enough not to make errors in this process, consider the following: (1) You will likely forget the solutions to the exercises if you do not write them down. I know that this has happened to me; I am certain that I have solved an exercise correctly but yet I might not remember the solution because I did not write it down. In fact, this is the same for many things one does in mathematics. (2) You do not need to type up your solutions by LaTeX; writing it down by hand should be sufficient ...2011-07-23
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    ... Also, consider buying a smart pen if you have the money; it is very useful and allows you to convert handwritten notes and solutions into a PDF file. (3) If you write down all of your exercises, you will have your own "solution manual" so to speak. I love the idea of solving many exercises from mathematics books and keeping a notebook of my solutions. I do not believe in maintaining solutions online for obvious reasons (e.g., if the textbook is used in an university course) but it is still nice to have your personal solution manual!2011-07-23
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    I agree with @Theo's comment to some extent: I know that I have tried (unsuccessfully) to prove a deep result in finite group theory and I recall thinking that I had done it at least five times. Each time, I wrote down my "proof" and discovered a very small flaw even though the proof seemed convincing when it was sitting in my head. Of course, with "easy" textbook exercises, this might not be a problem but in general it is better to err on the side of caution. Of course, an alternative to writing down your solutions is to discuss them with another mathematician ...2011-07-23
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    ... I have done this in the past and it is usually an excellent learning experience; even if you have solved all the exercises on your own, another mathematician might be able to point out simpler solutions and insights that you might not have thought about. Finally, you might even be able to find neat solutions yourself and impress other mathematicians!2011-07-23
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    @Alex J: Of course there is a difference between "typing it up" and writing it up. Typewriters (and by extension, computers) were not designed to write out mathematical notation; our hands, on the other hand, are very good instruments. It's certainly a lot easier to *write out* a matrix than to type it, no matter how good you are with $\LaTeX$. Unless you are planning to publish, or have severe problems reading your own handwriting, why bother typing?2011-07-23
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    @Alex J: I also have to say that the only time I don't write things out is when I have done it in the past and I'm just trying to remember the basics, e.g., when going over material that I studied before.2011-07-23

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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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    I do not get this. So Bourgain proves theorems without making computations?2011-07-22
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    @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.2011-07-23