Pancakes and Ham Sandwiches
Browsing in our excellent, local second-hand bookshop a week or so ago I came across a book I should have known but didn’t: The Mathematical Experience (Study Edition) by Davis, Hersh and Marchisotto. Browsing in that book today I came across a theorem I should have known but didn’t: the “pancake theorem”, the two-dimensional version of the “ham sandwich theorem”.
The pancake theorem says that, given any two closed curves in the plane, there is a single straight line that bisects the area of both of them. The proof is straight-forward if you accept the concept of continuous functions and I see from the link above that it also appears in Courant and Robbins, an old favourite of mine.
However, the point of demonstrating the proof was that there is one step at which students tend to balk. They eat the lemma and the definition of two functions p(theta) and q(theta) and then the next step is “define r(theta) = p(theta) – q(theta)” and this is where they suddenly dig in their heels.
Of course, at heart you can’t argue with a definition. If I want to define r to be (p – q) then that’s what it is. What fascinates the author of the article is why the students block at that point. And, of course, it’s the arbitrariness of it in what he calls “proof by coercion” that causes the sticking point. As it says in the Mathematician’s Miscellany: “but please Sir, what if x is not the number of sheep in the field?”
The author then describes a different proof that avoids the arbitrary definition and reports on much greater success with students when using it.
I think that it’s a slightly different issue. Although I’d never seen the original proof (the one with r = (p – q)) before, I found that, when I was about 25% of the way through, I could see where it was going and only needed to skim-read the rest before I felt comfortable I could reproduce it: it fell into a well-known pattern and I didn’t need the details. For a student coming to this type of proof for the first time, the “let r = p – q” must feel like another item to be pushed onto an already over-loaded stack.
When a good pianist comes to sight-read a page of music she knows immediately what notes can be left out while retaining the musicality of the whole. I can’t remember ever having been taught how to read a proof—it was always assumed that one started at the top and worked through line by line until one reached the end. And arbitrary things like “let r = p – q” were meaningless and annoying: “WHY!?”. My basic mathematical education took place a long time ago and the book originally came out in 1981. Perhaps things are a lot better in secondary school mathematics these days.
Anyway, I think the term “study edition” in the title means that the book includes questions. And there’re a couple at the back that would reward some thought:
- “Suppose you were able to travel back in time to meet some like Polya, Descartes or Pythagoras. Suppose he was discouraged and was considering giving up the pursuit of mathematics. How would you encourage him?”
- “Discuss the relationship between belief and information.”