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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.

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    $\begingroup$ If being critical is a goal, try giving problems that say "Prove or disprove..." and then a random mix of true and false statements. Often students are sloppy in their proofs because they know the book would never ask them to prove something that isn't true. $\endgroup$ – Aeryk Nov 21 '15 at 19:05
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    $\begingroup$ @Aeryk, that is true. Such "prove or disprove" problems are valuable and serve a purpose similar to what my question is trying to achieve. My focus here was on critical reading of whole proofs, though. $\endgroup$ – Joonas Ilmavirta Nov 21 '15 at 19:31
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    $\begingroup$ We discussed some bad proofs in my intro proofs class in college -- like the proof that every horse is the same color using a poor induction step. Are you referring to incorporating proofs like that one into more courses? $\endgroup$ – Opal E Nov 21 '15 at 22:32
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    $\begingroup$ @OpalE, in a sense yes. But the false theorems should be purely mathematical (no horses) and it should not be obvious at a glance whether or not they are true. (Example of a false theorem: A connected complete Riemannian manifold is uniquely determined by the pullback of the metric to a tangent space via the exponential map.) The problem is to find good ways to incorporate such false theorems (and true theorems with false proofs) into advanced courses. And these good ways should be backed by experience. $\endgroup$ – Joonas Ilmavirta Nov 21 '15 at 22:43
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    $\begingroup$ One possibility would be to present the students with two proofs of contradictory results, and have them decide which proof is correct and which is incorrect with justification. Edit: but because I am not a university professor or TA I can't speak with experience; I teach high school. $\endgroup$ – Opal E Nov 21 '15 at 22:48
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I would not recommend putting false proofs onto the board unless you immediately (within the same class period) point out their falsity, and make an assignment to find it.

For smaller mistakes that I make unintentionally, I give the students 'donut points'. 30 donut points (over the semester) and I bring in donuts for the class. I tell them that my main purpose is to get them to question what I say.

For a proof-oriented class, like linear algebra, I like giving them lots of practice with "prove or disprove". At each test, I give them two smaller tests, one on the content, and one that is a 'prove or disprove'. The three proof tests count as one test. Before each test we do what is often called 'speed dating' among math educators. The end of each chapter (in Lay) has a whole bunch of true false questions, which make great 'prove or disprove' problems.

I have the students put the desks in a U-shaped configuration, so the outside students can stay put and the inside students rotate. (Our room is tiny, and I was delighted to find a way to make this work.) Everyone discusses the first problem with a partner for two minutes, then moves. With their second partner they try to write a proof, again in two minutes. Then we discuss as a class. On to their next partner for the next problem.

The students work much harder and more actively than they would if they stayed in their own seats. When I did this last semester, there were students who thought it was juvenile and they kind of wrecked the atmosphere of it. This semester they love it, and are getting a lot out of it.

I have made a handout of about 6 student proofs after a test, and asked them to evaluate. They didn't do very well, but I think it was helpful. I should probably do it again. If you can get them to seriously evaluate each others' proofs, that could be a powerful learning experience.

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    $\begingroup$ Speed dating is an interesting idea, and it doesn't need to be restricted to "prove or disprove" problems. I want to try and implement that at some point... I fear that many students here would find it childish, so I have to be careful. $\endgroup$ – Joonas Ilmavirta Nov 22 '15 at 11:04
  • $\begingroup$ Coming up with another name for it might help. There's no real reason to think of it as childish. The research shows that when students actively participate they learn more, so it's clear this is a valuable technique (if students accept it and dive in). $\endgroup$ – Sue VanHattum Nov 22 '15 at 18:45
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Wondering Why are induction proofs so challenging for students?, I thought of trying this as a possible assignment after introducing induction:


Find the flaw in this induction proof.

  • Claim: $3n = 0$ for all $n \ge 0$.
  • Base: for $n=0$, $3n = 3(0) = 0$.
  • Assume Induction Hypothesis (IH): $3k = 0$ for all $0 \le k \le n$.
  • Write $n+1 = a + b$ where $a > 0$ and $b > 0$ are natural numbers each less than $n+1$.
  • Then $3(n+1) = 3(a + b) = 3a + 3b$.
  • Apply IH to $3a$ and $3b$, showing that $3a=0$ and $3b=0$. Therefore, $3(n+1)=0$. ∎


The advantage here is that the Claim is obviously false, but the flaw in the proof is not (I think) obvious.

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    $\begingroup$ That's a clever one! The proof fails in an unusual way, and that's what makes an illuminating example. $\endgroup$ – Joonas Ilmavirta Nov 22 '15 at 10:50
  • $\begingroup$ I should say this "proof" is not original with me, but I do not recall where I heard it. $\endgroup$ – Joseph O'Rourke Nov 22 '15 at 12:58
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    $\begingroup$ This is essentially the same flaw as in the "all horses are the same color" proof. It's definitely a good exercise in whatever variant you offer it, because it forces to really think critically about the structure of an argument and how one subtle flaw can lead to an absurdly false claim. $\endgroup$ – Michael Joyce Nov 22 '15 at 13:53
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There is a common type of exercise in French's mathematical secondary educators curriculum: give the student a (true or alleged) student's answer to a typical problem in your class, and ask them to analyze it. Is the answer correct? Near correct? What is good or wrong about it?

The nice thing is that they have to think critically even more than they would when asked to debug a faulty proof: here the proposed answer can a priori be perfect, or ok with a harmless error, or completely wrong even if the result is right, or the result might be wrong. Make sure all these possibilities turn out, and even the smallest set of basic exercises turns into a very complete training (just write down many genuine student's answers for later use if you don't want to forge them yourself).

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    $\begingroup$ This is an excellent idea! I could use student answers from previous years (assuming I can get such data for a course that I haven't taught before) or from previous weeks. In the second option the topics would drag behind the course schedule, but maybe a combination of repetition and analyzing peers' proofs wouldn't be that bad. $\endgroup$ – Joonas Ilmavirta Nov 24 '15 at 14:40

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