TITLE: Learning new mathematics
QUESTION [20 upvotes]: While many of us have had the experience of learning mathematics informally by osmosis or more formally in classes, there are times when we have to sit down and systematically learn, without the benefit of a class, large amounts of mathematics.  For instance, there might be a technique that we need from a field we are not familiar with.
When you find yourself in such a situation, what are your best tricks for teaching yourself new mathematics?

REPLY [9 votes]: There are many good answers here already, but I'll just add an observation on my own technique once I have decided on a book or paper to read. I like to read fast at first to get an idea of the lay of the land, so to speak, and keep on reading until I am lost due to not having absorbed previously introduced concepts. Then I backtrack and try to tease out the essential definitions before proceeding. Also, rather than reading proofs line by line I take a quick look and then try to fill in the details myself. That's time consuming, though, so it is reserved for proofs I really want to learn. This method is probably not suited for everybody, though. Everyone has to find their own method for absorbing new material.