In percolation today, we needed to prove that, if you have an infinite connected graph (where every vertex has finite degree) and you take a subgraph of it in which you keep every edge with probability p, then the probability \theta_v(p) that a vertex v is in an infinite component is either zero for every vertex or positive for every vertex. The following exchange ensued, as nearly as I can remember it:

Béla Bollobás (paraphrasing this line) “If x and y are two sites, take a path of n bonds between them. Then every bond is open with probability p, so the whole path is open with probability p^n.”

BB: “So p^n \theta_x(p) <= \theta_y(p).”

BB: “Is that obvious? Raise your hand if you think that’s obvious.”

(I hesitated and raised my hand. I was sitting in front, so I didn’t see how many people did.)

BB: “No, it’s not obvious. Why isn’t it obvious?”

(Confused mutterings from the class. I didn’t hear anyone propose anything.)

BB: “You need Harris’s lemma, that any two upsets are positively correlated.” [An “upset” is the amusing terminology for a set of sets closed under supersets.]

(Lots of students muttering in protest. BB stopped for a moment.)

BB: “Baby form of Harris’s lemma.” (And moved on.)

There are so many things wrong with this episode that it’s hard to list them all.

-“Sites” and “bonds” mean precisely “vertices” and “edges.” Apparently the founders of percolation theory fit a stereotype of certain mathematicians: graph theory isn’t real math, and they’re real mathematicians who don’t do graph theory. Accordingly, when they needed to define things that are exactly graphs in their work (already an indication of how silly they were to think they’d avoid graph theory), they made their own notation for it, and a few decades later, it’s still standard percolation terminology. Not as bad as http://fliptomato.wordpress.com/2007/03/19/medical-researcher-discovers-integration-gets-75-citations/, but we mathematicians have higher standards.

-If, as a teacher, you make a point of that something your students think is obvious isn’t, then make sure that something actually isn’t obvious. (I guess this is just a specific instance of the general principle that you should make sure what you’re teaching is correct.) Ideally you should be able to prove it’s not obvious, perhaps because the claim is actually false as stated or false without using some conditions that the claim assumes but the proof didn’t use.

-Even if Harris’s lemma had been required, it’d’ve made a bad example—although it’s less obvious than the above, its proof is still just a matter of unpacking definitions and doing simple algebra, not anything worth talking about. (I know because I hadn’t heard of Harris’s lemma before today, but understood and had proved it in a few minutes after BB stated it.)

-If, as a teacher, you want to test your audience’s understanding of something, make sure the answer isn’t obvious by lecture theory alone. The intended answer to “Is that obvious? Raise your hand if you think that’s obvious” is no, hands down. (Pun intended.)

-Don’t shame the audience.

-If, as a teacher, you realize you’ve made a mistake, admit it. BB must’ve realized something was wrong, but calling it a “baby form of Harris’s lemma” probably didn’t satisfy the people who knew Harris’s lemma was unnecessary or enlighten the rest. (This is particularly the case if, like Béla Bollobás, no one would doubt your reputation as a mathematician if you did.)

I’m not going to Percolation next week; I have plenty to do in the other classes.