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I originally asked this over at MathOverflow, but it was suggested I might also find some good answers over here.

My university department is in the very early stages of developing a "bridge course" or "introduction to proofs" course, motivated by our lower-level courses not currently doing a good job of preparing our majors to go on to more advanced topics, which in this case means primarily abstract algebra and real analysis. The various service needs of those lower courses precludes making them more mathematically rigorous, so we feel the need to add something else. I know that other universities have instituted such courses, but it's not so easy to find information about them through brute force web searches. Therefore, I'm hoping to gather some information more directly. Ideally, I'd love to get some links to course syllabi and some suggestions for textbook choices, but I'd also very much welcome any other general resources concerning such courses, including any input concerning what's worked well (or what hasn't) at other universities.

I think that right now the faculty preference for such a course would be for it to focus more on getting the students practice with actual advanced material than on simply being a litany of propositional logic and proof techniques, but, again, any information about what's worked elsewhere would be a big help to us.

Thanks!

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In other areas (computing) I've seen lots of "learn <tools>" courses. They were discontinued before too long: during the course, the students don't "get" it; when the uses come around they'd all but forgotten the matter. Integrate this with real uses, e.g. split some of the first courses into two, emphasizing proof strategies, becoming familiar with manipulating sums/products (a favorite stumbling block) into <something> I, leave much of the meat for <something> II. –  vonbrand May 13 at 10:52
    
See also mathoverflow.net/questions/51891/… on MathOverflow. –  J W May 13 at 14:06

5 Answers 5

I taught such a course about 5 years ago. In principle you are ready to attempt such a course after calculus I, however, for reasons of crowd-control etc.. we make calculus II the prerequisite. Honestly, this course could follow college algebra for the right student and that would be ideal in a more perfect world. You can see my meager course notes at:

and my course webpage

I'm certain there are better resources out there. This much I would like to see stressed in such a course:

  1. methods of proof. How to write math both formally and informally. (with and without the shorthands $\forall, \in, ...$
  2. properties of functions
  3. definition of equivalence relation, partitions of sets
  4. math in the modern age, what's the big picture? math as an active, creative enterprise

For our particular program, I think it is also a good place to work out the particulars of modular arithmetic, but, in another institution perhaps the $\epsilon,\delta$ proofs would be a good fit here. In any event, enjoy your task, it's a great problem to solve!

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Going for the subject more than the question itself, I like Hammack's "Book of Proof" for it's organization and explanation of how to prove stuff. Knuth, Larrabee and Roberts' "Mathematical Writing" (MAA 1989) (a partial version is here) is a must. Perhaps Aigner and Ziegler's "Proofs from THE BOOK" (Springer, third edition 2003) would be a good complement.

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3  
Another popular option: Dan Velleman's How to Prove It: A Structured Approach. Now 2ed. –  Joseph O'Rourke May 13 at 12:42
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I like the suggestion of integrating this class with specific content better than the suggestion of Proofs From the Book. It seems misleading to show standards of elegance when the point of the class is standards of rigor. But if you've taught this class making good use of Proofs From The Book, I'd be interested to hear about it. –  Matt F. May 13 at 14:21
    
@MatF, I'm saying it could be a nice complement (i.e., "here you can see outstanding proofs, if you are really interested", not as a text/guide). –  vonbrand May 13 at 16:50

Highly recommended: Robert Talbert's blog, chronicling his adventures in an inverted intro-to-proof course:

blog

videos

Extremely well-written, free textbook

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Hey, thanks! :) Also, I put all my materials from the flipped proofs course up on Github: github.com/RobertTalbert/proof They're aligned with Ted Sundstrom's book that is at the link above. –  Robert Talbert May 15 at 1:12
    
@RobertTalbert Thank you! My intro to proof classes this term are a big hit, and your material has been an absolutely invaluable resource –  David Steinberg May 15 at 17:04

When I was a starting graduate student (note: in computer science), I found the following text book very helpful in filling in the gaps in my mathematical knowledge that you describe (well, the first part you describe: abstract algebra and proofs -- not real analysis).

Bloch, Ethan. Proofs and Fundamentals: A First Course in Abstract Mathematics, Springer 2011.

It is organized into 8 chapters (the last two of which are "extras"). The first six are:

  1. Informal Logic
  2. Strategies for Proofs
  3. Sets
  4. Functions
  5. Relations
  6. Finite Sets and Infinite Sets

The preface section entitled "To The Student" contains a great summary of the author's goal:

This book is designed to bridge the large conceptual gap between computational courses such as calculus, usually taken by first- and second-year college students, and more theoretical courses such as linear algebra, abstract algebra and real analysis, which feature rigorous definitions and proofs of a type not usually found in calculus and lower-level course.

He goes on for another page or so, essentially describing a student with my background. The preface also includes a similar section entitled "To The Instructor":

There is an opposing set of pedagogical imperatives when teach a transition course of the kind for which this text is designed: On the one hand, students often need assistance making the transition from computational mathematics to abstract mathematics, and as such it is important not to jump straight into water that is too deep. On the other hand, the only way to learn to write rigorous proofs is to write rigorous proofs; shielding students from rigor of the type mathematicians use will only ensure that they will not learn how to do mathematics properly.

The entire preface (4 or 5 pages) contains a good description of the design and goals of the text.

Oh, and the short set of notes compiled from Knuth's seminar on Mathematical Writing mentioned in vonbrand's answer was very helpful too (but it is about writing mathematics, not necessarily about mathematics itself).

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(I mentioned this in a reply to another comment, but it might be hard to see.)

Two years ago I redesigned our proofs course using a flipped classroom format, and all the materials for that are free and available online:

David Steinberg also linked to one of my blog posts about the design and rationale of that class.

I think there's a lot of room for improvement in the course, but I was pretty pleased with the way it worked out.

One thing to remember about proofs courses is that they are going to be hard to pull off no matter what you do. Students hit a wall in these courses because they force them to leave behind some immature notions about what mathematics is -- it's no longer about getting the right answer to simple computations in the shortest amount of time. There's a cultural and professional barrier that students have to get over, and many don't make it. The research literature is all over the place regarding the proper handling of these courses. It's a major course design and teaching challenge, but the successes IMO make it well worth the effort.

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