Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's main source of mathematical interaction. Even a slow course, or a course which covers material which one already knows to some level, can be highly stimulating. However, there are usually a few months in the year when mathematics slows down socially; in the summer, one might not be taking any courses, for example. In this case, one might find themselves reduced to learning alone, with books.
It is generally acknowledged that learning from people is much easier than learning from books. It has been said that Grothendieck never really read a math book, and that instead he just soaked it up from others (though this is certainly an exaggeration). But when the opportunity does not arise to do/learn math with/from others, what can be done to maximize one's efficiency? Which process of learning does social interaction facilitate?
Please share your personal self-teaching techniques!
Best Answer
I think the crucial distinction here is not between books and people (after all, books are written by people!) or between physical contact and electronic contact, but between non-interaction and interaction. In my view, the crucial benefit that interaction provides is the ability to have your questions answered and your ideas critiqued. MathOverflow, of course, demonstrates the value of being able to ask a focused question and have an expert reply to it. In a non-interactive setting, you are limited to whatever answers have already been written down somewhere (plus your own ingenuity in answering your own questions).
Having your own ideas critiqued is also important. You may have a good idea (or a bad idea!) but not recognize it as such. An expert can often give you a quick assessment of your idea; you cannot get this without interaction. (Of course, expert assessment is a double-edged sword because sometimes the expert is wrong and you might have been better off without the incorrect feedback!) Even if you have a good idea that you recognize as good, you may have difficulty articulating it properly until you are forced to communicate it to someone else and get them to understand it. Some people are naturally gifted at expressing their ideas clearly and logically, but most people need to be taught communication skills interactively. And even the best mathematicians benefit from the exercise of teaching others what they know (or think they know!). The process of explaining something to someone else is of great value in clarifying your own understanding.
Finally, as for improving your efficiency if you have limited access to interaction, I would try to write down, as succinctly as possible, the questions and ideas that you want to get feedback on. Then you can efficiently send off a batch of questions and ideas and get feedback in a batch.