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[Kernel Sohcahtoa]

I've just finished reading Daniel Solow's How to Read and Do Proofs, 6th edition.  Overall, this book really helped me get underneath mathematical ideas and propositions: after reading this, I gained the confidence to get my hands dirty and try additional proofs in some of my other books, which previously had me perplexed.  More importantly, Solow's book  gave me greater insight into the mathematical thought processes that are involved in writing and reading proofs (IMO, completing proofs is analogous to solving a puzzle and reading proofs reminds me of following origami diagrams, as multiple steps are often omitted between sentences).  For example, the condensed proofs that are often seen in textbooks are often the result of a detailed analysis of the proof by the author, where he or she asks a series of key questions and utilizes definitions, previous knowledge, facts, etc., in order to develop a logical series of statements that eventually produce the conclusion.  Oftentimes, the author works forward from facts given in the hypothesis of the proposition and backward from the conclusion, so that the forward process and backward process meet somewhere in the middle (Solow calls this the "forward backward method"), which then allows the author to write the complete proof from beginning to end .

With that said, if you are interested in gaining a better understanding of why math works or are just interested in proofs or problem solving, then I highly recommend this book, as there is an online solutions manual along with online lectures, which often elaborate on areas that may have been covered too vaguely in the text. 

Live long and prosper.