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By now the advice I give to students in math courses, whether they are math majors or not, is the following:

a) The goal is to learn how to do mathematics, not to "know" it.

b) Nobody ever learned much about doing something from either lectures or textbooks. The standard examples I always give are basketball and carpentry. Why is mathematics any different?

c) Lectures and textbooks serve an extremely important purpose: They show you what you need to learn. From them you learn what you need to learn.

d) Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle". You have to do the work yourself, but you need someone else there to either help you over obstacles you can't get around despite a lot of effort or provide you with some critical knowledge (usually the right perspective but sometimes a clever trick) you are missing. Without the prior effort by the student, the knowledge supplied by a teacher has much less impact.

A substitute for a teacher like that is a working group of students who are all struggling through the same material. When I was a graduate student, we had a wonderful working seminar on Sunday mornings with bagels and cream cheese, where I learned a lot about differential geometry and Lie groups with my classmates.

ADDED: So how do you learn from a book? I can't speak for others, but I have never been able to read a math book forwards. I always read backwards. I always try to find a conclusion (a cool definition or theorem) that I really want to understand. Then I start working backwards and try to read the minimum possible to understand the desired conclusion. Also, I guess I have attention deficit disorder, because I rarely read straight through an entire proof or definition. I try to read the minimum possible that's enough to give me the idea of what's going on and then I try to fill the details myself. I'd rather spend my time writing my own definition or proof and doing my own calculations than reading what someone else wrote. The honest and embarrassing truth is that I fall asleep when I read math papers and books. What often happens is that as I'm trying to read someone else's proof I ask myself, "Why are they doing this in such a complicated way? Why couldn't you just....?" I then stop reading and try to do it the easier way. Occasionally, I actually succeed. More often, I develop a greater appreciation for the obstacles and become better motivated to read more.

WHAT'S THE POINT OF ALL THIS? I don't think the solution is changing how math books are written. I actually prefer them to be terse and to the point. I fully agree that students should know more about the background and motivation of what they are learning. It annoys me that math students learn about calculus without understanding its real purpose in life or that math graduate students learn symplectic geometry without knowing anything about Hamiltonian mechanics. But it's not clear to me that it is the job of a single textbook to provide all this context for a given subject. I do think that your average math book tries to cover too many different things. I think each math book should be relatively short and focus on one narrowly and clearly defined story. I believe if you do that, it will be easier to students to read more different math books.