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Looking for math books that teach both the how and why of the topic at hand
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I question some of the more specific claims here.  Analysis isn't the "why" of calculus; you could even argue that it's a more precise form of how to do calculus,  along with a bunch of other things...  Saying that pure math books rarely contain the "hows" is a strange slant on what a "how" is.  You could even turn things around and say that "why"s are reasons we do something, i.e. the applications of pure math concepts, but that's just semantics at that point.  

Without knowing your motivation, I find that it's common for students to feel they have  FOMO ("fear of missing out")  on some aspect of math that their education seemingly failed to provide.  I have seen precalculus students who want to see proofs for everything, when it's pretty unlikely that that would enrich the precalculus experience for most students.

I will borrow an example of what rigor does in general.  It doesn't make the standard cases more meaningful. It does clarify the exact boundaries of what works and what doesn't work.  For example, if we are dealing with infinite conditionally convergent series, we can get into some trouble if we don't understand the problems with trying to sum that series.  But in most walks of life, you can go weeks if not years without ever having to deal with that eventuality.  (Or, you know, encountering a function that is continuous everywhere but differentiable nowhere.)  Most of rigorous math is like that.  If you care about it, it matters, but if you don't, then it probably doesn't.  (It's more like chess strategy or literary analysis,  as opposed to traffic laws or personal hygiene, which are things you can't safely ignore in most cases.)
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I also take issue with the ways some things are stated, but I think it's true that the "whys" are very important. It's not so much a matter of every statement needing to be proved, but more of providing justification to an extent that is reasonable taking into account the level of development of the students. I think the U.S. tends to fall short even on this measure.

What's not often appreciated in this discussion (at any rate in the way the question is discussed on internet forums) is that there is a very close connection between an appreciation for proofs in mathematics and work on problems where the only approach that will work is one based on careful reasoning similar to that found in proofs of theorems. I think most mathematicians probably enjoyed little number and logic puzzles as children (cryptarithms, magic squares, etc.), and proofs in mathematics are a natural outgrowth of the intellectual attitude you need to have to solve those kinds of problems.

What books to recommend sort of depends on what your current background in math is and how you see your level of ability. I think a good starting point for someone with a good high school background is elementary combinatorics, as in "Mathematics of Choice" by Niven. For someone with an interest in CS, a good high school background and high ability, "Mathematical Thinking: Problem-Solving and Proofs" by D'Angelo and West can be a good place to start.

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