Speaking of books about calculus, I like Robert Strichartz's The Way of Analysis. Robert develops intuition and motivation of key concepts in an amazingly detailed way. He often starts with an intuitive but flawed idea, and then gradually revises the idea to a mature definition or proof. The whole book focuses on "why": why real numbers are defined that way, why limit uses the delta-epsilon notation, why we need to topology, etc and etc. Reading the book is more like an adventure than following a set path. The author explains his approach eloquently: "Mathematics is more than a collection of theorems, definitions, problems and techniques; it is a way of thought....My goal in writing this book is to communicate the mathematical ideas of the subject to the reader. I have tried to be generous with explanations....I believe that what you learn through a process of struggle is more likely to stick with you than what you learn without effort....Understanding mathematics is a complex process. It involves not only following the details of an argument and verifying its correctness, but seeing the overall strategy of the argument, the role played by every hypothesis, and understanding how different theorems and definitions fit together to create the whole. It is a long-term process; in a sense, you cannot appreciate the significance of the first theorem until you have learned the last theorem.”
"Why" is something that was always a noticable omission from my math classes in school. I got by simply by learning the mechanical motions of whatever topic we were learning but there was never any cohesion to the concepts. I've been working through this book (and necessarily brushing up on my algebra which has atrophied for the same reason) and I've been absolutely loving it.
I was a math tutor in college, and many of the tutees would come into the tutor lab completely downtrodden, dreading their homework. I found the most helpful thing I could do with them was to focus more on getting them to understand the motivation behind what they had gone over in class that day, and in particular saying things like "now why would we do this" as soon as I could see in their face that it was clicking. Getting them to understand the why (and getting them to feel like they understood the why) was incredibly effective for helping them feel less insecure when going to class.
When I was taking calculus classes, and struggling, a tutor helped me immensely in the exact same way; explaining both practical applications and often the historical context around the method he was teaching me.
