Book request: teaching proving and reasoning at an American university

Daniel Solow, How To Read and Do Proofs
Intended for abject beginners, unlike some of those other answers.

I know no better book for reasoning than

Thinking Mathematically, by John Mason, Leone Burton and Kaye Stacey.

It is superb in inspiring action and instilling methods of reasoning. Quoting from the introduction:

Experience in working with students of all ages has convinced us that mathematical thinking can be improved by

● tackling questions conscientiously;

● reflecting on this experience;

● linking feelings with action;

● studying the process of resolving problems; and

● noticing how what you learn fits in with your own experience.

The authors delineate the process of thinking mathematically, presenting a rubric for attacking problems. This rubric does not mean a recipe, but a set of words designed to get you thinking and prompt you toward action (in the spirit of Polya's questions). However, the book does not explicitly deal with problems to prove, and for that I recommend two other books:

How to Prove It, by Daniel Velleman,

and

How to Think like a Mathematician, by Kevin Houston.

These two books combined will help students discuss and learn:

• Logic (Sentential and Quantitative) - Velleman
• Proof strategies (negations, conditionals, quantifiers, existence and uniqueness, contradiction, induction, etc) - Velleman and Houston
• Relations - Velleman and Houston
• Functions - Velleman and Houston
• How to read definitions and theorems - Houston
• Divisors, the Euclidean algorithm and modular arithmetic - Houston

I believe the combination of these three more will result in an overhaul in your students' thinking habits. Their abilities of reasoning will vastly improve.

A book we often use in my department is Reading, Writing and Proving by Gorkin and Daepp. One feature of this book that distinguishes it from others mentioned in other answers is that it has chapters on $mathbb{R}$, its completeness, the convergence of secuences in $mathbb{R}$, and the Cantor-Schröder-Bernstein theorem. In other words, its focus is more on preparing students for a course in real analysis and less on discrete mathematics. Considering that the South American universities I have taught at often start with the completeness of the reals in first semester calculus, this might be the kind of book that will get students reading "non-American" textbooks quickly.

Another free option is Lehman, Leighton, and Meyer's Mathematics for Computer Science. It's written for an MIT introductory discrete math course that emphasizes training students in proof-writing.

Some things we're currently considering for a similar course at a large urban community college:

• Epp, Discrete Mathematics with Applications
• Artin, Algebra
• Gilbert, Elements of Modern Algebra
• Lay, Analysis with an Introduction to Proof
• Wade, An Introduction to Analysis

And also some OER (open educational resources) options:

• Sundstrom, Mathematical Reasoning: Writing and Proof (formerly a mass-market textbook) http://scholarworks.gvsu.edu/books/9/
• Hammack, Book of Proof http://www.people.vcu.edu/~rhammack/BookOfProof/
• Hutchings, Introduction to Mathematical Arguments (27 pages; I esp. like the "Logic in a nutshell" table on p. 9) http://math.berkeley.edu/~hutching/teach/proofs.pdf
• Turner, A Brief Introduction to Proofs (9 pages; meant as a primer to any course involving proofs) http://persweb.wabash.edu/facstaff/turnerw/Writing/proofs.pdf

Edit: Struck out Arten's Algebra from list above due to consensus in comments.