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So far, I have taken the calculus sequence, introductory differential equations, and some proof-based courses. I'll be moving into the introductory analysis courses (what my institution calls "advanced calculus," which is just before real analysis at my school) in the fall, but before that I would like to return to the very foundation of calculus and differential equations: Algebra. Most of us know that algebraic manipulations in integral calculus and differential equations are (usually) more difficult that the integral and differential "operations," simply because many of these manipulations require stokes of genius, mathematical maturity (and perhaps some of theses insightful moments are based on mood and good timing). Just look at how to solve the integral for sec(x) and you know what I mean. I'm certain that I would have never come up with such a manipulation. However I'm also certain that these people also have a superior understanding of algebra. Not just in a calculation sense, but almost an intuitive sense of just how numbers interact with each other.

Alongside my school's required intro analysis textbook (Rosenlicht's Introduction to Analysis), I will be reading Spivak's calculus. So what I'm looking to do before the fall is to get an book whose primary goal is to teach strategies and general problem-solving principles, perhaps not just limited to algebra or even math at all. For example, I am reading Gelfand's "Algebra" text. Those Russians make mathematics so crystal clear. I also read Gelfand's Trig book, and I instantly saw great improvements in my integral calculus skills. Gelfand does not focus on rote computation like most of the textbooks in the general American math curricula , but rather on properties functions and problem solving skills. Perhaps I should also think about getting a book in basic number theory.

I hope you more experienced mathematicians can help me out here when you were hungry-for-understanding math major. Thanks!

-John