I think/hope this is okay for MO.

I often find that textbooks provide very little in the way of motivation or context. As a simple example, consider group theory. Every textbook I have seen that talks about groups (including some very basic undergrad level books) presents them as abstract algebraic structures (while providing some examples, of course), then spends a few dozen pages proving theorems, and then maybe in some other section of the book covers some Galois Theory. This really irks me. Personally I find it very difficult to learn a topic with no motivation, partly just because it bores me to death. And of course it is historically backwards; groups arose as people tried to solve problems they were independently interested in. They didn't sit down and prove a pile of theorems about groups and then realize that groups had applications. It's also frustrating because I have to be completely passive; if I don't know what groups are for or why anyone cares about them, all I can do is sit and read as the book throws theorems at me.

This is true not just with sweeping big picture issues, but with smaller things too. I remember really struggling to figure out why it was supposed to matter so much which subgroups were closed under conjugation before finally realizing that the real issue was which subgroups can be kernels of homomorphisms, and the other thing is just a handy way to characterize them. So why not define normal subgroups that way, or at least throw in a sentence explaining that that's what we're really after? But no one does. 

I've heard everyone from freshmen to Fields Medal recipients complain about this, so I know I'm not alone. And yet these kinds of textbooks seem to be the norm. 

So what I want to know is:


  Why do authors write books like this?


And:


  How do others handle this situation?


Do you just struggle through? Get a different book? Talk to people? (Talking to people isn't really an option for me until Fall...) Some people seem legitimately to be able to absorb mathematics quite well with no context at all. How? 

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.
