Fundamentals

I'm working with a student this summer and wondering how to communicate "how to learn mathematics".  We don't have a class, a syllabus, a textbook, and I can tell that this student needs a bit of guidance on what to do. This is something professional mathematicians usually learn through osmosis in graduate school. Can I tease apart some good habits to suggest to this student now?

Many mathematicians have written about this: Terence Tao in particular has some great advice in small portions. On a personal level, his advice to write down what you've done is probably the first item I have on my how to learn new math list. I learn a lot through writing things down. I try to write for others I know (grad school friends, colleagues, students) and then put it aside for a month. When I come back to the notes I am often amazed by what needs to be rewritten to communicate well and reflect my greater understanding -- and the final product is much improved with these changes!

Fine, write down what you've done. But what do you do?

Read a lot -- find some good introductory work on the topic and go to town. You'll get a broad picture of the basics this way.

Write down your questions and subquestions and try to answer them. Write down your explanations.

Find some interesting articles, recent or older, and take them apart. What are the main points? What are the questions people are trying to answer? How did they develop from questions earlier in the area's history?

Why does anyone care about the results? Why do you care?

Prove things. Pay attention to details (this was hard for me to learn). Share with others so that they can ask dumb questions and sharp questions. Remember, it's not personal -- we all just want to understand.

Go back and repeat. It's a great spiral staircase. You go back to the basics of the new area and see them with new eyes because you've looked at the questions people are asking now.

Ok; off to take my own advice...