I would like students to be critical and not believe that every proof they see is correct. Lecturers make mistakes and students should not think: "That must be a valid argument/proof/syntax because it was given in the lectures/lecture notes/book." I hope that this kind of critical thinking would make them better at deciding whether the proofs they write themselves are correct.
To this end, I would like to present invalid proofs and results to the students. But how should I go about this? What kinds of problems have you had using false proofs? I have never really tried this or seen this used, so I would like to hear experiences and recommendations.
Context: University students that have received or are close to receiving a bachelor's degree. Possible courses might cover, for example, point set topology, measure theory or differential geometry. A typical course will consist of lectures and exercise sessions (for which the students have received the problems a week or so in advance).
Here are some scattered ideas I have:
- One option is to give a false theorem in (almost) every lecture and ask the students to find it. This may have some benefits, but the big drawback is that then there might be no lecture material that the students can completely rely on. It is often desirable that students be able to use the main theorems of the course even if the proofs are too complicated for them to grasp. I think this approach would cause too much confusion, especially when results are built on earlier results.
- I could give false theorems and "proofs" for them as exercise problems and ask them to figure out what went wrong. They could start by finding a counter-example to the claim and then proceed to finding a "corresponding" error in the proof. It might be too much of a burden to give the students five proofs or so and make the judge which are correct; this takes time away from other exercises.
- I want to point out to students that there are false proofs of correct statements. Such problems should be included.
- Maybe I could make the students read proofs from other students and evaluate them. This would require some rearranging of the work flow of the course.
There is already a question about examples of bad proofs. I'm not asking for examples but for good ways to make "critical proof reading assignments" for relatively advanced courses. Assume I already have a bunch of bad proofs; I want to focus this question away from finding good bad proofs.