## Textbook recommendations for undergraduate proof-writing class

I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows:

1. Logic, Sets, Equivalence Relations and Induction should be covered.
2. Price should be reasonable (say around $30 or less). 3. Distractions like "historical comments" and "mathematical perspectives" should be kept to a minimum. I plan to supplement such a book with "What is Mathematics" by Courant and Robbins. I would be pleased to hear some recommendations! - In this for math students? The fact that there is a proof-writing class makes me sad... That we have managed to dissociate so much proofs from learning calculus and algebra that a separate course is needed is quite a feat in absurdity! – Mariano Suárez-Alvarez Apr 22 2011 at 17:32 (There was a nice letter to the editor by Maclane on the AMS Notices. where he argued that the reason to teach calculus is to teach logic (I guess he included proof ẃriting in that...), something like "one shoul dteach enough calculus so that the logic comes across". I cannot locate it) – Mariano Suárez-Alvarez Apr 22 2011 at 17:51 @Mariano: unfortunately, we have to take the world as it is even while trying to make into what it should be. In other words, I don't think it's fair to suggest as you do that the need for a proofs course stems from a screw-up in curriculum design. I never had a "proofs" course, never taught one, and I don't relish the idea either. But for those of us who have calculus classes that mix math majors with people who intend from the beginning to stop at Calc 1 (pre-med anyone?), there is only so much we can do in those classes and the case for a proofs course more or less makes itself. – Thierry Zell Apr 22 2011 at 20:33 I have yet to understand why abstraction is commonly taught at the same time as rigorous proof-writing. They're the two most important skills for undergraduates to learn, and they're different skills. IMHO, combinatorics is an excellent subject for learning to write rigorous proofs, precisely because the definitions are easy to understand, and you don't have to spend a lot of time proving theorems which "look obvious". – Frank Thorne Apr 23 2011 at 6:20 @Frank: I find your comment intriguing, the more so because I can't decide whether I agree with it or not. ("Having an opinion" is not usually a problem for me!) If you're right, then we math pedagogues of the world are missing out on something rather important and fundamental. I encourage you to think and say more about this -- maybe via a MO question, maybe via email. – Pete L. Clark Apr 23 2011 at 19:06 show 8 more comments ## 16 Answers If you want a book which is priced under$30, write it yourself and put it on the internet: then it's free.

(This is not a quip or a dismissive comment: please do actually do this. I have done this sort of thing myself.)

Among books that the evil empires of publishing put out, I used one for such a course twice and -- apart from the price -- it was pretty good:

I'm not sure exactly why you are against historical comments (nor do I know exactly what "mathematical perspectives" means in this pejorative context), but so far as I recall this book is fairly businesslike. (Added: I just processed the part of your question where you mention supplementing the book with material from Courant and Robbins. That latter book is all about perspective, so I guess the idea is that you want to avoid duplication of content, which is very reasonable. Sorry if I sounded overly critical before.)

I was most pleased with the treatment of logic and sets in the first two chapters: just about the right amount, with just about the right level of formality and sophistication...to my taste, of course.

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I second that idea,Pete. – Andrew L Apr 22 2011 at 18:37
Our university has used the Pearson book now for a while, and it seems to garner general acceptance. I like it, although there were a few oddities here and there. Overall a nice book. – Pace Nielsen Apr 22 2011 at 19:15
I just might do this after I teach it a couple more times (and get tenure!) – Eric Rowell Apr 23 2011 at 16:44
An alternative to giving it away is to use print on demand (POD) publishers. I have been very happy with Createspace, who give a good royalty rate, as they are an amazon company, and are very efficient. Of course you have to do the publicity yourself. I discuss this on pages.bangor.ac.uk/~mas010/orderbook.html although some details are out of date, since Booksurge has become Createspace. – Ronnie Brown May 15 2012 at 20:53

This year, my colleague has been using the art of proof by Matthias Beck and Ross Geoghegan (Springer 2010). It's slightly below $40 I believe, which is still in the reasonable range, commendably short and I hear it's proved very satisfactory so far. I think it has the topics you're looking for. - This book is also one of the ones springer has already uploaded to their "SpringerLink" website, so some universities might even have a subscription making it freely available to university IPs. Here's a link: springerlink.com/content/978-1-4419-7022-0 – Rob Harron Apr 25 2011 at 15:31 I learned out of Dan Solow's How to Read and Do Proofs and it was great (this was about six years ago and the professor had used this book for many many years). It's also very cheap: http://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/0471406473/ref=ntt_at_ep_dpi_3 http://product.half.ebay.com/How-to-Read-and-Do-Proofs-by-Daniel-Solow-2001-Paperback/948996&tg=info -  I recommend anything but this book. When I taught out of it, prepared students liked it a lot, but borderline students (who need such a class more) struggled more than they might have without any book. It hurts weak students. It seems to feed an expectation in them that all proof-writing is done "step-by-step", with the precise sequence of steps dictated entirely by the formal structure of the statement to be proved, and the exercises do not carefully delineate between what math can be taken for granted and what cannot--- giving the impression that "proof writing" can be done in a vacuum. – anon Apr 26 2011 at 0:17 Hmm, I guess I can't comment about how the book is for "weak students" but I will say that when I took this course I had no experience with proof writing whatsoever. I was pretty hopeless when I started, but the instructor met with me many many times and this book was a great supplement for those meetings. In particular, I learned a ton from the "backwards-forwards method" of solving a problem...reducing your problem to simpler pieces and solving those. I can't remember well enough to comment on the exercises, but I do know the book covers all proof methods you see in early undergrad math – David White Apr 26 2011 at 14:27 In addition to those mentioned, here is a good book which is just under$30:

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 I like this book very much, but I wonder if it has too many "distractions" for the questioner. – Carl Mummert Apr 25 2011 at 17:15

http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?ie=UTF8&qid=1303491885&sr=8-1

This text was used in the "Math Structures" class at my undergraduate institution (basically an intro to proof writing) and I found it really useful for transitioning from calculus type problems to constructing proofs. I think it meets all your requirements (definitely the first two, and I don't recall there being a great deal of historical\philosophical digressions).

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 I second this: Velleman's book "How to prove it" is quite usable and not too expensive. I would say it focuses quite a lot on the mechanics of how to attempt a proof (how to prove a statement of the form if A or B holds then C holds). For strong students this is probably unnecessary, but for average students it's very useful. – Anthony Quas Apr 22 2011 at 20:35

I have used Velleman's How to Prove It with success.

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I am not so sure of the US system but one of the books we recommend at our university is Martin Liebeck's "A concise introduction to pure mathematics".

http://www.amazon.co.uk/Concise-Introduction-Pure-Mathematics/dp/1584881933

At least in the UK that book is pretty darn cheap.

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 I should add the "disclaimer" that in fact Martin Liebeck works at my university! I believe he wrote the book because he couldn't find "the book that he wanted", but this was before the days when one could self-publish so he couldn't follow Pete's advice... – Kevin Buzzard Apr 22 2011 at 20:32

I quote from a recent article by Brown and Porter in the De Morgan Journal, published online by the LMS. http://education.lms.ac.uk/wp-content/uploads/2012/02/Brown_and_Porter.pdf and commented on subsequently by David Wells. I feel the following idea needs advertising.

"A technique widely used by psychologists and trainers is error-less learning. This falls into two types. One is where large hints, props, and supports to a specific course of action are given, and the action is rewarded as a symbol of success. Then the various props are gradually withdrawn. The other type uses reverse chaining: the easiest way to see to this is to think of encouraging a child to put on a vest. You do not throw him or her a vest and say put it on; instead, you put it almost on, and then ask the child to do the final action. Subsequently, you gradually put the vest less and less fully on, till the whole action can be done.

One way of using the last technique in university mathematics is to write out a formal proof and then erase bits of it. The student has to fill in the bits, using clues from the rest of the proof. This has some analogies with the practice of a professional mathematician, who may have an idea and outline for a proof, but needs to work on details. The student also gets an idea of the structure of a proof. Such an exercise is also very easy to mark!

The general feeling about error-less learning is that it works like a dream!

In either method, the fact long verified by psychologists is used, that we learn from success. We can also learn to accommodate failure if that is gradually introduced, and strategies are available for dealing with failure."

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I've been really happy with Smith, Eggen and St. Andre:

Though that breaks your price requirement.

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Peter Eccle's "Introduction to Mathematical Reasoning: numbers, sets and functions" seems to fit the bill of what you are looking for. It is slightly higher than your preferred price of 30 dollars (it is 38). I would also check out the Google books preview.

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 I have taught out of this book and second the recommendation. (But yeah, the price.) – anon Apr 26 2011 at 0:01

A "book" that satisfies all of your criteria is a set of notes from the Journal of Inquiry Based Mathematics called "Introduction to Proof" by Ron Taylor. linky

The chapters are

1. Symbolic Logic
2. Proof Methods
3. Mathematical Induction
4. Set Theory
5. Functions and Relations

There are two appendices: one on mathematical writing and one on Style (By James Munkres).

It is a set of notes for an IBL class, so the assumption is that the students will be doing virtually all of the proofs themselves. I've never used this set of notes for teaching, but I've used others from the journal. I like them very much.

Their copyright notice allows free use and printing as long as attribution is given and no charge for the students other than printing costs. Similar sets of notes that I've used have cost the students about $6. Others from the journal's website about intro to proof/foundations are http://www.jiblm.org/downloads/dlitem.aspx?id=17&category=mathnerdscollection http://www.jiblm.org/downloads/dlitem.aspx?id=16&category=mathnerdscollection http://www.jiblm.org/downloads/dlitem.aspx?id=14&category=mathnerdscollection (These last three haven't been refereed by the journal, but they still gives links to them.) -  Thanks! I really like Taylor's notes. I will definitely include this as an extra resource if only for the advice in the appendices. – Eric Rowell Apr 26 2011 at 15:34 The Book of Proof by Richard Hammack is free online and available from Amazon for$12.95.

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Jimmy Arnold has a full book available online (An Introduction to Mathematical Proofs):

http://www.math.vt.edu/people/elder/Math3034/

Also, Michael Hutchings has a very nice 27 page manuscript on the subject (Introduction to Mathematical Arguments)

http://www.math.berkeley.edu/~hutching/teach/proofs.pdf

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http://www.amazon.com/Proofs-Fundamentals-Course-Abstract-Mathematics/dp/0817641114

This is written by my professor Ethan Bloch. It is slightly overpriced, though.

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 Still priced a lot more reasonably than the book by Douglas Smith et al. – Deane Yang Apr 22 2011 at 17:07 I am aware of that. But I think you can get a personal discount from him by 25$per book if you used it for your class, which is the cost we used to buy it. – Changwei Zhou Apr 22 2011 at 22:22 Not a book, but it's free. May I humbly suggest my DC Proof software. Using a very user-friendly proof-checker, students can work through a ten-part tutorial that introduces various methods of proof. For more information, free download, testimonials, etc. visit my website at http://www.dcproof.com - The book I used in my 'proofs' class was "Doing Mathematics: An Introduction to Proofs and Problem Solving" by Steven Galovich, here on Amazon. The class was called "Mathematical Structures", which is an apt name since the class wasn't solely about learning to prove things. It was learning to prove things in the context of learning about basic mathematical objects. It starts with basic logic, but after it introduces sets, relations, functions, equivlance relations and the like, it goes onto to develop the ideas of cardinality, including Cantor-Bernstein. It also has a couple other topics, like some basic combinatorics, the constructions of number systems, or looking at consequences of the field axioms. It was a great introduction to what math is "really about" coming after some mostly computational calculus and linear algebra courses. The price is about$50, so it is a little more than you were looking for. But it is absolutely a book worth having.

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