It would not be good to start, for example, with the question
How do mathematicians prove theorems?...
The question is not even
How do mathematicians make progress in mathematics?
Rather, as a more explicit (and leading) form of the question, I prefer
How do mathematicians advance human understanding of mathematics?What an amazing perspective!
Research and teaching We've all heard about some purported level of disrespect researchers have toward mere teachers and expositors. Some make it explicit; G.H. Hardy said "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done." This attitude is not useful for the advance of human understanding of mathematics.
I'm learning about myself: I want to spend my time creating mathematics, if I'm honest, and then communicating it. But the work of appreciating and communicating mathematics is crucial to our collective mathematical future. We need fans and enthusiasm and the idea that we don't is just the self-justification of a nerd whom no one likes reassuring himself that it's ok that no one like him.
Teaching and teaching All is not well, though, in the rosy land of lovey-dovey liberal arts teachers and expositors. With the glut of PhDs on the market, schools can demand ever more "research" from their excellent teachers without changing service demands. Why not? Now even the schools that respect teaching the most are putting research on the agenda, whether it be with undergrads or other professional mathematicians. In addition, the job market is ridiculous. Ridiculous! A SLAC can say they're looking for a representation theorist with statistics teaching experience and they can find exactly that -- the geometric representation theorist with statistics teaching experience will be out of luck -- and why not? It is, after all, every school's responsibility to choose the best person for their department. This is moral and right. What is not right is telling the geometric representation theorist they are not qualified to teach representation theory, or telling the person with phenomenal teaching evaluations and three years post-graduate teaching experience at two schools that they just don't have enough experience, or any number of similar things I've seen. Be honest. It's luck and it's a buyer's market. As it is, we on the market feel incredibly disrespected. Fine, it may be a blessing in disguise; the friend who was told he wasn't qualified to teach freshman stats got hired to do statistics for a large insurance company for three times the salary and the friend who was told she wasn't qualified to teach financial math got hired by a finance company to do financial math for twice the salary. They've got very nice financial packages and far more geographic freedom than academia affords. Maybe I too will leave and do the things I'm not qualified to do for much more money. But it leaves a bad taste in the mouth to have one's strengths and accomplishments belittled as they sometimes are in the job search. Once again, it's the faculty member on the hiring committee coming up for some justification for why things are as they are to make themselves feel better. It's simply not true.
Economic rewards Our current academic system is, like all systems, primarily interested in maintaining the status quo. Times are changing, though, and the status quo involves economic incentives that will no longer serve us well. Don't defend it: think about. We still reward incredible specialization and high paper counts. We still reward a certain slipshod approach to teaching at some R1 universities. We still reward behaviors that lead to burnout at some SLACs, and we're changing incentives there to reward even more unsustainable patterns. We reward a system that uses people up and throws them out rather than cultivating talent or helping people prepare for jobs in industry and government from the beginning if that's what they want. The disrespect involved in all these relationships and transactions does not advance human understanding of mathematics.