I am teaching an intro to proof course for undergraduate math majors at a medium-sized american research university. I would like to provide my students with some incorrect proofs for the purpose of having them critique or correct the work of others. Can anyone recommend references or resources?
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1$\begingroup$ This is not an answer to your question, but to suggest something else for your course! I really suggest asking your students to read, understand, and summarize (in terms of the core and structure of the argument) some of the proofs given by Euler. As you can see here, Read Euler, Read Euler, I have started believing that proof learning should pass from reading Euler! $\endgroup$– Amir AsghariApr 8, 2016 at 10:49
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1$\begingroup$ Related: matheducators.stackexchange.com/questions/2206/… $\endgroup$– AerykApr 8, 2016 at 16:55
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1$\begingroup$ Recall $\sqrt{ab} = \sqrt{a}\sqrt{b}$, e.g., $6 = \sqrt{36} = \sqrt{4 \cdot 9} = \sqrt{4}\sqrt{9} = 2 \cdot 3$. Next, observe that: $1 = \sqrt{1} = \sqrt{-1}\sqrt{-1} = i^2 = -1$. Therefore, $1 = -1$. QED $\endgroup$– Benjamin DickmanMar 17, 2017 at 23:16
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3 Answers
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A Transition to Advanced Mathematics by Smith, Eggen, and Andre contains numerous (usually between five and fifteen) such exercises at the end of each exercise set. Since it is a regular feature of these exercise sets, topics in these proof exercises span properties of numbers, set theory, functions, groups, rings & fields, and even the completeness of the real numbers. The book contains hundreds.
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There are some "evaluation of proof" questions in Sundstrom's book, Mathematical Reasoning: Writing and Proof, typically at the end of each list of exercises (starting at 3.1, when proofs take centre stage).
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1$\begingroup$ Thanks for the link! Note: Version 2.0 of the same book is available. Or do you specifically recommend the first one and not the second? "There are no changes in content between Version 1.1 of this book and Version 2.0. The only change is that Appendix C, Answers and Hints for Selected Exercises, now contains solutions and hints for more exercises." $\endgroup$– M. VinayApr 13, 2016 at 17:33
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1$\begingroup$ @M.Vinay Thanks for reminding me to updated my bookmarks :-) $\endgroup$ Apr 14, 2016 at 0:40
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Late to the party to answer, but hopefully this will be helpful.
I would say that the best source of incorrect proofs can be your students themselves! My undergraduate real analysis professor basically built the course around this idea in a pseudo-Morse methodology. The breakdown would go like this:
- Assigned set of ~15 proofs from the text were due at the start of class every day.
- The first ~5 minutes of class were devoted to students writing one of their proofs on the boards around the classroom (worked well for a class of roughly 25 students with ample boardroom). Wasn't a particular grade for "thou shalt write X problems up each week," but the prof would encourage regular contributions from all the students.
- We would then go over each of the proposed proofs as a class and decide if they worked (~5-10 minutes). If not, we would decide what was necessary to fix them or if we needed to reject them outright.
A key point is that none of this was graded (FERPA!) and semi-anonymous. The focus was always on "is this a valid argument" and not "did Joe fail to prove something yet again."
As a student, the first week was certainly a touch awkward. Once we all settled into the routine and got over the stage fright, it ended up being one of the most effective courses I had. Just like learning to edit is a key component of becoming a competent writer, learning to evaluate (other people's) arguments is a key to becoming a competent mathematician.
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2$\begingroup$ In the United States, the "Family Educational Rights and Privacy Act" which forbids, among other things, disclosing a student's grades to anyone without their express permission (including parents of students over 18). ed.gov/policy/gen/guid/fpco/ferpa/index.html $\endgroup$– erfinkMar 16, 2017 at 7:04