The Art of Proof: Basic Training For Deeper Mathematics is a book by Matthias Beck and Ross Geoghegan published by Springer in 2010.
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here.
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Among the core ideas in mathematics are: integer, induction,
algorithm, real number, rational
number, modular arithmetic, limit, and uncountable sets. And among the
core methods of
mathematics are: axiom, theorem, and proof.
The authors take the position that the student knows a significant amount: they call this knowledge "Sesame Street through Calculus." With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The book puts the instructor in control: some proofs are presented in detail, but the instructor decides which of the omitted proofs to present in class and which should challenge the student to discover a proof and write it down correctly.
"This book offers an approach well-balanced between rigor and clarifying simplification.
"This qualitative transition presents a most acute pedagogical challenge. [...] This book does feature definite mathematical content, contrasting with works that aim at decoupling purely logical apparatus from strictly mathematical concerns. [...] The authors write with the authority of research mathematicians and clearly mean to open that avenue to students. Summing Up: Recommended." (D. V. Feldman, |

"Logic moves in one direction, the direction of clarity, coherence and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. [...] It is the interaction between these different aspects that gives mathematics its power."

William Byers (*How Mathematicians Think*, Princeton University Press, 2007)