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During my self-study (and soon to continue at a university) of mathematics, one thing I have been interested in is how to to effectively learn the material.

An answer to a question provided by Orlandpm about reading a mathematics book has caught my interest.

In particular, for read $0$ he says:

"Read 0: Don't read the book, read the Wikipedia article or ask a friend what the subject is about. Learn about the big questions asked in the subject, and the basics of the theorems that answer them. Often the most important ideas are those that can be stated concisely, so you should be able to remember them once you are engaging the book."

I believe what would be help is more information on how to gather this information. For example, I am beginning a course in abstract algebra in September, but the wikipedia article for abstract algebra doesn't seem to quite get at what the Read 0 above is stating (of course, I recognize that abstract algebra is a broad field and this likely accounts for that).

My question is: What are effective ways to learn the big questions and basic theorems of a field in mathematics. Wikipedia is provided, but what other resources and methods exist?

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    $\begingroup$ You can get a lot from textbooks in the field. For example, most calculus books start with an introduction describing the velocity and area problems. Stein's Real Analysis begins in the introduction with several problems that require measure theory. Flipping through the table of contents of several abstract algebra books, you'll notice the importance of groups, Sylowe's theorems, and Galois Theory. That might take a little guess work, but it can get you started. $\endgroup$ Jul 9, 2013 at 20:22
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    $\begingroup$ Some concisely stated theorems you might want to look into: cycle decomposition theorem, hairy-ball theorem, Abel-Ruffini theorem, and of course the fundamental theorem of algebra. All of these are some questions that are, to some extent, answered through abstract algebra (there are many different proofs for that last one though). $\endgroup$ Jul 9, 2013 at 20:52
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    $\begingroup$ The best way I know: ask your professor. It might be at lecture or other classes, or his office hours, or even via mail. Ask for an overview, main results, big questions, famous open problems, connections to other areas, and also what he likes about the subject. $\endgroup$
    – dtldarek
    Jul 9, 2013 at 22:03

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I think The Princeton Companion to Mathematics would be a good resource for this sort of information. You can buy the book online (or, depending on your scruples, download it for free if you know the right websites).

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A very good resource for getting the big picture is videos and there are loads of lecture series available on youtube. Since you don't need to get all the details at this point, I think spending a couple of hours watching recorded lectures is very time efficient.

You brought up abstract algebra, which is a subject that can get very technical and focused on strange terminology, and there is a big risk that the big picture is lost in my opinion. Let this guy guide you through many of the concepts before digging into the details and I think you'll be well prepared for the course!

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