Learning new mathematics 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?
 A: I read blog posts about it and then I blog about it.  The main thing this does is supplement reading a textbook, especially one that doesn't provide motivation or connection with other branches of mathematics.  
A: 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.
A: Writing a paper with someone who does understand the area can be fun; I've learned a lot that way.  
A: My procedure is: First looking around for motivations, e.g. how the theory I may learn is connected with other, already interesting, themes and what the motivations behind the new issue may be; then I look for survey articles and short expositions in other articles, finally I look for books on the new theme and basic articles of the work done on it. The sometimes big problem is step 1, because motivations/background intuitions are often not written up or taken as well known among the specialists on the theme, sometimes the context is unclear, e.g. I wonder if reading Sullivan's MIT lectures were interesting for me, if connections with algebraic geometry exist, or if one should better read less old expositions.
A: Find an interesting problem in an area that I do know about that needs techniques from an area that I don't know about.
A: Hang out in the lunch room and wait for my brain to get violated one of the professors.
A: First try to understand why the experts of the topic are interested in it (Blogs, papers, books, personal contact, whatever). Look for books on the subject, test-read each one, settle for at least two that are written on a level that seems accessible to you (if there are any, i.e. if the topic is not too hot).
Write down every formula, mark every sentence that seems important. (Note that different kinds of peaople need different kinds of techniques, some need graphics, some need to speak out loud, I need to write. Works for me in everyday life as well, as soon I have writen something down I have memorized it, well, at least for a while).
Try to explain everything you read to yourself. Repeat this until you are sure you could do a talk without any notes, anticipating all sensible questions.
A: Many have remarked that they first understood a subject only when they first taught it.
A: I try and find a simple example to play with that until I am comfortable. Then I pretend that all examples are like the toy. Soon I realize that's not true, but by then it's easier to see where the new behavior comes from. Then it's time to repeat the process.
