For four consecutive years I taught a fourth-semester course called “Introduction to Analysis,” in which we looked at differential calculus for a second time, stressing the foundations, the logical structure, and proving all the key theorems. We used Stephen Abbott’s excellent book, *Understanding Analysis*.

The course was intended primarily for math majors, although we had some interested students from other programs. Becuase we had 8 – 12 students each time I taught the course, I thought it would be a pity to lecture. So I ran the course as a seminar. I would lecture briefly to begin and end each chapter, and then I assigned problems from the textbook, for which the students had to present solutions in class. I would sit at the back of the class, and observe the discussions that took place after the presentations. Typically, if an error were made in a proof, the students wouldn’t necessarily notice right away. Instead, someone would ask, “Could you explain again how you got from line 3 to line 4?” or some such question. The presenter would typically struggle to explain the point, and within a few minutes everyone could see that there was something wrong. If the group could patch up the proof on the spot, great. If not, I would send them away, sometimes with a hint, with the task of fixing it up and presenting it again next time.

As long as all the points that I wanted to be discussed were actually discussed, I would stay silent. Of course, if not all the relevant points were brought up, I would ask questions to move the class in the direction I wanted them to go.

The feedback I got from students was interesting. They told me it was much more work than a regular class, but they learned a lot more than in a regular class, too. I think this has implications for all of education … it’s a key reason why lecturing to 500 students is largely ineffective, no matter how brilliant the lecturer. I took this feedback as evidence that this method of teaching proofs can be effective.

Secondly, a comment about calculus/analysis textbooks: the vast majority of them provide no training in the kind of thinking needed for creating proofs. (This is the reason that commenters refer to specialist books (Polya is great, as is *Proofs and Refutations* by Imre Lakatos), not standard calculus texts.) They simply provide finished products, often very tersely, without any sense for the thinking that goes into shaping a proof. Abbott’s book is a lovely counterexample. At a higher level (for analysis, anyway), T.W. Korner’s A Companion to Analysis is beautiful. The first few pages of this book provide tremendous motivation for the need to prove seemingly obvious theorems.

My point here is that writers of standard textbooks can learn a lot about how to make sections on proof much more effective. Part of the problem is that publishers want to please everyone, to maximize profits, and so they tightly restrict page count while trying to cram in as much content as possible. As in classrooms, cramming in as much content as possible is counterproductive to good teaching and learning.

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