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,
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.