Here’s a nice guide to what is a proof and how to write one:
How to write proofs: a quick guide by Eugenia Cheng.
This is brief, has a nice outline, and gives some good examples. I love the analogy made, that a good proof is like a good story: it has a beginning, middle and end. The point: a proof can be bad by having the parts in the wrong order.
A shortcoming of the article, in my opinion, is the explanation of an induction proof. It’s a pet-peeve of mine, but I dislike seeing “n = k” and “n = k+1” in the same paragraph. Problems with teaching and learning induction is a greater issue that is partially addressed in the May/June 2008 article Some Observations on Teaching Induction by Mary E. Flahive and John W. Lee.
Comments
2 responses to “Teaching Proofs”
Japheth, this is great! I already see parts I disagree with, and I think it will be fun to write a reply to her article at some point. Both her article, and the reply I am just beginning to imagine will be very helpful to my linear algebra students. Thanks!
>we’re given the beginning and the end, and somehow we have to fill in the middle.
The proof assignments my students are struggling most with begin with “prove or disprove”. They are not given the end, instead they are given two possibilities for the end, and must choose the one that works.
>One of the easiest mistakes to make in a proof is to write it down in the order you thought of it. This may contain all the right steps, but if they’re in the wrong order it’s no use.
She says writing a proof is like writing a story (or building a bridge or …). The proper story of the logic is the order she’s telling us to use. But logic can be foreign, and the human story that connects to the logical story does have a different order. I think one of the mistakes made by textbook authors is to give the final proofs, instead of thinking about how the idea is learned by humans.
I wonder if the article I wrote for the Journal of Humanistic Mathematics is relevant here. It tells too much of my story perhaps, but it also walks the reader through one person’s process for solving the problem described. For me, most proofs don’t tell enough of a story (unless they’re pictorial). A walk-through, that gets me thinking about the underlying ideas, so I can recreate the proof myself, is much better.
Sue, there’s a distinction that Eugenia Cheng makes that I feel is very important: What is a proof, vs. what is a good proof. Eugenia states that
In this sense, a proof is a “certificate of authenticity” of a mathematical statement that doesn’t depend on the audience. In practice, we (almost) never reveal the actual proof of a statement. Besides taking too much space, and being boring, the proof is likely not to share the real insights, and keep the audience engaged. A proof is a bad proof.
With my own students, I ask them to assume more, that is, to fold away those tiny things that we’ve already considered and focus on what is new. When they turn in written work, I want engaging writing. My guidelines include using paragraphs to reflect the logical structure of the argument, proper spelling and grammar, and not abbreviating full English words with mathematical jargon. I’m the audience, and this helps their work engage me as a reader. Of course, we throw out these guidelines when they present their proof to the class (a different audience), but I do ask them to motivate their proof. For example, telling the story of how they figured out the proof, or presenting a generic example (something I learned from working with Larry Zimmerman, but I just recently learned th term from Susanna Epp) that illustrates the full reasoning, but with less cognitive load.
I do appreciate your proof assignments to “prove or disprove”, and should probably use this device more often. It’s motivating not to know how the story will end! In working with Ben Blum-Smith last year, we had a lot of fun not even stating what might be true, but just throwing our students directly into the pool and giving them authority to decide what the story should be.