Bramanti M., Travaglini G. Studying Mathematics: The Beauty, the Toil and the Method

Springer, 2018. — 402 p. — ISBN: 978-3-319-91354-4.This book is dedicated to preparing prospective college students for the study of mathematics.
It can be used at the end of high school or during the first year of college, for personal study or for introductory courses. It aims to set a meeting between two relatives who rarely speak to each other: the Mathematics of Beauty, which shows up in some popular books and films, and the Mathematics of Toil, which is widely known. Toil can be overcome through an appropriate method of work. Beauty will be found in the achievement of a way of thinking.
The first part concerns the mathematical language: the expressions “for all”, “there exists”, “implies”, “is false”...; what is a proof by contradiction; how to use indices, sums, induction.
The second part tackles specific difficulties: to study a definition, to understand an idea and apply it, to fix a slightly wrong argument, to discuss suggestions, to explain a proof.
The third part presents customary techniques and points of view in college mathematics.
The reader can choose one of three difficulty levels (A, B, C).The Language of Mathematics
An Introductory Test (Level A)
Quantifying (Level A)
Using the Sets (Level A)
Propositions and Properties (Level B)
Proofs, Implications, and Counterexamples
Negations and Indirect Proofs
Formulae and Indices (Level A)
Saturation of Indices and Syntactic Consistency of a Formula (Level B)
Induction and Natural Numbers (Level B)
Studying a Math Book
To Read a Definition
To Understand, i.e., to Know How to Apply
Learning to Correct
To Discuss the Ideas
To Understand, i.e., to Know How to Explain
Pages and Ideas
Majorizations
Uniqueness Proofs (Level B)
Functions and Set Theoretic Arguments
Tiles, Polyhedra, and Characterizations