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I am a European postdoc who recently teaching at a large public university in the United States. I will have to teach a course for undergraduate students that introduces them to proving and reasoning in mathematics.

The students have possibly no exposure to mathematical reasoning in general. At the end of the course, they should be able to read and write proofs, and use the standard logical/set-theoretical notation.

What books or lecture notes can you recommend (or dis-recommend) for such a course? I am particularly interested in material that gets them as close as possible to being able to read non-American textbooks.

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  • $\begingroup$ Has this course been taught before at your university? If so, what text has been used? What prerequisites does the course have? What students typically take it (only math majors? everyone?) $\endgroup$ – Michael Lugo Oct 23 '17 at 18:19
  • $\begingroup$ 1) How selective is your university? There is a big difference even between students at UC Berkeley and UC Riverside, never mind Arizona State or Northwest Southeast Central State U. $\endgroup$ – Alexander Woo Oct 23 '17 at 18:21
  • $\begingroup$ 2) Roughly speaking, do you want to optimize how well your best students do, or how well your average student does, or how well your worst students (only counting ones putting in reasonable effort) do? $\endgroup$ – Alexander Woo Oct 23 '17 at 18:23
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    $\begingroup$ 3) There are dozens of textbooks for this, and this is really a shopping question. $\endgroup$ – Alexander Woo Oct 23 '17 at 18:23
  • $\begingroup$ I strongly dis-recommend a book called "Chapter Zero: Fundamental Notions of Abstract Mathematics" which I had to buy once for an undergraduate course. It is badly over-written and uses weird notation that makes it quite hard to understand. $\endgroup$ – Flounderer Oct 23 '17 at 20:41
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Another free option is Lehman, Leighton, and Meyer's Mathematics for Computer Science: https://courses.csail.mit.edu/6.042/fall17/mcs.pdf . It's written for an MIT introductory discrete math course that emphasizes training students in proof-writing.

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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.

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

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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:

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

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    $\begingroup$ Artin is hardly an introduction to proofs and mathematical notation... $\endgroup$ – darij grinberg Oct 23 '17 at 6:21
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    $\begingroup$ @TommiBrander "Open Educational Resources" $\endgroup$ – Steven Gubkin Oct 23 '17 at 12:07
  • $\begingroup$ Wait, Mike Artin's algebra book? Am I missing something here? It's awesome but ... $\endgroup$ – kcrisman Oct 24 '17 at 14:08
  • $\begingroup$ @StevenGubkin Thanks. I suggested an edit. $\endgroup$ – Tommi Oct 24 '17 at 18:17
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    $\begingroup$ I actually learned to write proofs in a class using Artin's Algebra (because I enrolled without much background). If I hadn't had some very helpful TAs (who are now very good mathematicians in their own rights) I would never have survived. This is not an appropriate book for the purpose of learning basic mathematical argumentation. While I like it a lot, it suppose a certain sophistication of the reader. $\endgroup$ – Dan Fox Oct 25 '17 at 6:37
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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.

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