# Comparison of texbook for "how to write proofs"

I posted this question in the math stackexchange https://math.stackexchange.com/questions/4681694/comparison-of-textbooks-on-how-to-write-proofs and one person suggested that I cross-post it here. I'd be most apprecicative for your inputs. Here is the original post, slightly edited.

Next fall I will teach a class on "How to write proofs". Prerequisite is first-year calculus. The three textbooks I am considering are

• Hammack, Book of Proof (3rd ed)

• Sundstrom, Mathematical Reasoning: Writing and Proof (3rd ed)

• Cummings, Proofs: A long-form mathematics textbook.

This will be my first time teaching such a course, and I'd be most appreciative for feedbacks and comments from those who have taught such a class using these books -- why do you pick one over the others? feedback from students? advice on how to teach such a class/use these books? To give you an idea the issues I'm considering (incorrectly, perhaps!), here are my first impressions of the three books, based on a quick reading:

• Sundstrom and Hammack provide free downloads and Cummings is inexpensive, so cost is not an issue.

• Sundstrom is the only the one that provides instructor solution manual. I can obviously solve all the problems, but having solutions available does have me in picking problems for homework (e.g. length of solution, complications/tricky spots etc) and lecture preparation (e.g. be sure to go over certain examples before I assign a particular problem).

• Sundstrom is (relatively speaking) the most 'old-fashion' of the three. There is not a lot of motivation of the materials, and the "beginning activities" at the start of each section, while helpful for lecture planning, in a way "dilute" the materials, and I worry that it might discourage students from reading the text. I also wish that (very basic) set theory were presented in chap 1, and I find it a bit strange that congruence is covered before equivalence relations (please remember that these are my first impressions).

• Cummings is the opposite of Sundstrom: Very fresh and lively, and the presentation/examples are interesting. At the same time the exposition seems too chatty, and the jokes etc get tired after a short while. The problems also seem a bit more challenging that Sundstrom.

• Hammack is in a way half way between the two. It does not get to proofs until page 113, which seems a bit late --- speaking as someone who has not taught a "proof class" before. Again congruence is introduced before equivalence relations. I really wish there is a solution manual.

Again, for those who have used these books: Why did you pick one over the others? Feedback from students? Advice on how to teach such a use/use the book? Order of topics? (e.g. congruences before/after equivalence relations)

Thank you!

Note: I am aware of prior posts such as https://math.stackexchange.com/questions/3316114/book-recommendation-for-proof and Book request: teaching proving and reasoning at an American university The posts there are either lists of books and/or recommendation for a specific book. What I am looking for are comments and comparison about the three specific ones listed above. THANKS!

• I don't know the books, so can't comment more fully. Since the first two are free, you might think of a 4th option of using both. Apr 19 at 20:45
• I would strongly advise soliciting the opinions of your colleagues rather than those of strangers on the Internet, because the answer will depend on your students. I don't know any of these books, but I would be shocked if the best answer was the same at University of Michigan, Grand Valley State University, Kalamazoo College, and Adrian College (to pick a range of places in Michigan) Apr 20 at 1:24
• To get things started, you might consider a freeware proof-checker to learn the most basic methods of proof, e.g. introducing and discharging a premise, proof by contradiction, manipulating quantifiers, etc. Apr 20 at 17:56
• I assume you are aware that Sundstrom's textbook has screencasts? I have used them several times in conjunction with the textbook. Worked for me :-) Apr 20 at 20:33
• "congruence is introduced before equivalence relations." That is normal. We also introduce integers before general Euclidean rings. The idea is to give a non-trivial example of an equivalence relation one can work with by checking everything by hand first and to introduce the abstract scheme that works straight from the axioms by pure deductive reasoning later. Given that the students haven't been exposed to the pure deduction much before, it makes sense, though, of course, it is not the only possible order of exposition. Apr 21 at 13:27