TITLE: Is it not effective to learn math top-down?
QUESTION [65 upvotes]: By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving basic results.  Eventually you'll get there but is this a bad idea?  Is learning each required math area by textbook the better way?

REPLY [3 votes]: If you want only superficial understanding of the content in a paper then the top-down method is useful, but if you aim at contributing something, then the tortuous, self-studying and time consuming method is the only better way. Learning new maths through textbooks is a highly challenging for anyone. There are only 5% textbook authors in maths who have the gifted ability to present a thing in such a manner that any genuine reader can learn directly through reading their books upto appreciating-level.