On the Math Mindset I am a soon-to-be high school graduate with inclination towards mathematics. I enjoy doing math, and am relatively good at it, but I dislike the way I am being taught. I feel like I am being taught methods for solving a lot of problems, but I don't see how any of this fits together.
I was considering pursuing a career in mathematics, but I realized I don't really know much about it as a whole as opposed to how to solve specific problems that will appear on some test.
Are there any good books out there about the "math mindset", or whatever it is that it's called?
 A: I have spent a TON of time in the math library at Rutgers just thumbing through all the books, and I never really found a book about the "math mindset" that was half decent.  You could do a lot worse than to read stack exchange every day and become a regular on this forum; everyone here is really helpful and you shouldn't be afraid of posting a "silly" question out of fear of being embarrassed; the only thing that upsets people here is when someone doesn't put a sufficient amount of thought/effort into the problem before asking, or else when the question being asked is clearly off topic.
I think that talking with mathematicians about specific problems you are working on will give you a far deeper insight into how mathematicians think than reading some low level book for a popular audience will.
The most important thing in my opinion is not the book you finally choose, but rather the attitude that you should always try to solve EVERY SINGLE problem in whatever math book you read.  You should spend more time solving the problems than reading the sections.  
I recommend you pick up "A survey of modern algebra" by Birkhoff and Mclane.  The difficulty level is pretty low, and the problems are numerous and instructive.  One reason I so highly recommend this particular book for you is that "modern algebra" is not only fundamentally important for contemporary mathematics, it also is not taught in high schools generally, not to mention that many of the results are beautiful.  Another great thing about this specific book is that the author's are famous mathematicians, and you can figure out how they think about math just by seeing the organization of how the material is presented and how the results are derived.  The first chapter in this book deals with number theory, but actually contains a very nice introduction to writing logical and rigorous proofs.  
I think you should just read through books on specific subjects, and pick the subject you will study based on your current interests and level of knowledge.  Some subjects to start with are:
set theory, metric spaces, complex analysis, group theory, number theory
One thing you might want to try is just to go ahead and check out some books by the great masters.  For example, there is a great set theory book by Frankel (who has the axioms of set theory named after him), and a great book on number theory by Sierpinski; and I recommend both of these not only because they are really well written and basically at the high school level, but because this will give you an insight into how the great masters thought about math.  Finally, if you are feeling confident go ahead and check out Disquisitions Arithmetic by Gauss, just to try your hand at it.
Let me also mention a compilation of writings for a popular audience by Henri Poincare called "The Value of Science".  A large portion of this book deals with physics, but there are two particular passages you should pay attention to for understanding the "math mindset": the first is the discussion of induction, and the second is a fairly famous section where Poincare discusses the internal psychological process he experienced when solving a highly technical problem.
Best of Luck!
A: I think certain parts of The Princeton Companion to Mathematics might be just what you need.  While many of the topics will go over your head at the moment, there is a very lucid introduction to the history and basic points of view of modern mathematics in Parts I and II, as well as a lot of interesting historical overviews and essays in Parts VI and VIII.  And of course much of the book is filled with fascinating mathematics.
A: Here are some further classics that you may find inspiring and enlightening: Polya: How to solve it; Rademacher and Toeplitz: The enjoyment of mathematics; Kac and Ulam: Mathematics and Logic. 
A: Unfortunately, a very popular approach to teaching mathematics at the grade school level is to de-emphasize the concepts and focus more on the rote, procedural aspects of the subject.  I don't think many people really like this type of education but it's one of the simpler approaches to teaching. On the positive side, by focusing on the procedural you have a systematic way with which you can judge a students' progress.   In some sense this is a mode of teaching that is dictated largely by the desire for a convienient means of evaluation, rather than being dictated by holding the students' interest, or teaching them valuable things.  The hope is that by learning enough procedures by rote, the underlying concepts somehow sink-in, as if by magic.  I think this does work for many students but by-and-large it's unsatisfactory to almost all involved.  To teach mathematics in any other way with similar objective measures of success tends to be much more time and energy consuming for all involved, as far as I can tell.
To complement Sam Nead's very nice suggestions, I enjoyed reading Devaney's "Introduction to Chaotic Dynamical Systems".  I don't remember when exactly I picked it up, might have been grade 12 or 1st year of university.  Parts of it were over my head at the time, but I was able to digest much of the contents, slowly.  In that regard I think it's useful to take a non-linear approach to learning.  It's useful to have as a goal to understand some high-level topic, and to have a mental roadmap for how you're going to get there. You don't want to be too aggressive on this, though.  Around grade 12, Devaney's text was very useful for me.  It's a book that can be used both for graduate courses and for upper-level undergraduate courses.  So you'll probably see it as several steps ahead of what you're used to.  But it's well-written and well-motivated.  Digest what you can, slow down and think about aspects that appear hard, maybe come back to them a few months later.  Attempt some of the more fun-looking problems at the end of sections.  Do some computer experiments.  That kind of thing. 
