I would like to spend a day with my students analyzing mathematical writing. One way I might accomplish this is to offer multiple proofs (some good, some poor) of the same simple statement and ask them to critique the writing. Is there a good reference for such proofs? It would be ideal if the proofs were a part of a textbook that itself contained some commentary, just in case we need a little help getting started.
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1$\begingroup$ Which theorems are you interested in? I think this is a great idea. How would you define a good proof? $\endgroup$– KarlCommented Jun 16, 2015 at 21:00
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$\begingroup$ The specific content doesn't matter to me, but it should be basic (numbers, sets, functions, etc.). The point is to dive right into the proof, not fuss over definitions. I think "good proof" is one of those things I can't define, but I know it when I see. Some (competing) qualities such proofs might have include correctness, rigor, clarity, and conciseness. $\endgroup$– Austin MohrCommented Jun 16, 2015 at 21:06
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1$\begingroup$ I remember giving this some thought a while ago, and wondering whether there was anything in the literature on writers developing "voice" that could be applied to the context of proof-writing in mathematics. I tracked down a book entitled Poetry and Mathematics (Buchanan) that was interesting but of no real help, and abandoned ship. I think your best bet is to solicit proofs from a few people in your department of a simple proposition, and then suggest categories (as you did: correctness, rigor, clarity, conciseness) to help get students started in examining them. $\endgroup$– Benjamin DickmanCommented Jun 16, 2015 at 21:15
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4$\begingroup$ Somewhat close work I know of is Amabile's Consensual Assessment Technique around having experts in a domain (e.g. mathematics) rate products (e.g. different proofs) for undefined attributes (e.g. creativity). One then checks for agreement (e.g. using the intraclass correlation method or Cronbach's alpha) and, in the case of consensus, operationalizes accordingly. For the case of creativity, validity has been pretty well established (see e.g. here). (This comment is mostly to note that there's probably a good Math Ed thesis hiding here.) $\endgroup$– Benjamin DickmanCommented Jun 16, 2015 at 21:22
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$\begingroup$ Related question: matheducators.stackexchange.com/questions/2206/… $\endgroup$– AerykCommented Jun 17, 2015 at 18:09
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3 Answers
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I found this an effective teaching technique.
I take a topic they know, and find a Wikipedia article discussing that topic. If you are specifically focused on proofs, as opposed to more generic descriptions, you can find many proofs in Wikipedia. E.g., of Sperner's Lemma, or Euclid's proof of $\infty$ # primes.
Then I project the text in class, and have the students interactively criticize a paragraph sentence-by-sentence and suggest improvements. And here is why they attend closely: I (or one of the students) actually edits the Wikipedia article directly. One has to be registered to edit Wikipedia, but this is easily accomplished. Changing Wikipedia makes it real; and it's satisfying to even $\epsilon$-improve the world.
answered Jun 17, 2015 at 22:05
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3$\begingroup$ So, to make it interesting, you really need someone else to go and sabotage certain wiki proofs before class ;) $\endgroup$ Commented Jun 19, 2015 at 2:09
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2$\begingroup$ @JamesS.Cook: No need for sabotage: There are plenty of opportunities for improvement. Or: They have been sabotaged from the get-go. :-) $\endgroup$ Commented Jun 19, 2015 at 21:43
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There are quite a few textbooks that have critiquing sample proofs as exercises. Here are three I know of:
A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre
The Foundations of Mathematics by Thomas Q Sibley
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow
This past year I used the Solow text in my "Intro to Proofs" course because it specifically approaches the logic and proof writing from a different angle than a standard math text. About 80% of my class time was spent critiquing proofs. Where did they all come from? The students of course. Since Solow focuses on writing and proof logic more than mathematical content, I pulled a bunch of standard theorems and propositions (and false statements) from other traditional texts and students signed up for problems individually. They then submitted their proof/counterexample to be critiqued anonymously in front of (and by) the class.
So my first piece of advice to you is that one day will not be enough. The first day was almost comical in how accepting the students were. They were so used to being spoon-fed perfect proofs that they just assumed everything was right. After about 8-10 classes, the bright students really caught on and became very good at critiquing (and, of course, at submitting better work themselves). It took the middle-level students about twice as long (and of course, some students never really got it). So it took about half of a 15-week semester doing this to get to the point where the students were appropriately skeptical of a proof's logic.
After that we tried to work on issues like voice, phrasing, exposition, etc. That was a lot harder (for me to teach and for them to get). At some point there's an amount of personal style that comes into the writing and, of course, not every student resonated with my style (which, I had to remind myself constantly, is okay).
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Go to Mathematics Stack Exchange or MathOverflow. There are many questions there looking for proofs and there are many different answers, some good, some bad (some are even wrong). Ask your students to criticize the proofs (Are the proofs they consider good the ones that are highly upvoted?). The commentary that you want is sometimes there (as comments).
Aside from looking at the quality of the answers, also look at the quality of the questions. Some posts go directly to the question without defining the terms, and you will see users commenting on how the question is unanswerable and voting to close these as unclear what you're asking. (In my opinion, good mathematical writing is clear and succinct, with all important terms defined and no unnecessary details included.)
(I'm assuming you have access to the internet in your class. If not, then just print out the posts of interest beforehand.)
