How to determine the big questions in a field of mathematics?

Question

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?

Answer 1

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).

Answer 2

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!