(May 28, 2013) Seventy-five minutes. No future sessions. Four kids. Five proofs. What to do?
This was the dilemma facing our Math Circle today. I’d been deliberating all week over how best to use our last session. So many proofs. So little time.
I gave the decision to the kids in attendance, R, P, N, and G. Our options:
- a little one-line algebraic ditty that would make obvious the invalidity of semantic shifts, therefore demonstrating the folly of the “Cat has Nine Tails” proof;
- a Pythagorean-theorem proof by a blind girl1 who utilized both visualization and algebra (I thought our group could follow this mathematically);
- an algebraic Pythagorean proof (I didn’t think we had enough time for this,2 since we’d have to explore the possibly unfamiliar concept of similar triangles);
- a geometry proof similar to last week’s vertical-angle-theorem proof (I told the kids that they’d have to prove this themselves); and
- another faulty proof for them to seek the flaw in.
We put #1 to bed handily since it was on the board already. Then the kids chose to do #4 and #5. They wanted to THINK, not to follow.
“x and y are interior angles in a triangle. z is the third angle’s supplement (see photos for diagram). Prove that x + y = z.”
The students dictated their first two steps to me. Next, R jumped up to the board to do her next few steps. Then everyone worked together to finish the proof algebraically. We thought we were done, but then P issued forth a zinger: “Would the proof still hold if z were a right angle?”
G immediately took the position that anything proven algebraically holds for all numerical values. P was unconvinced. N and R had no strong opinions. We decided to redo the proof with z = 90. It still worked. “But what if we drew it a different way, with y = 90?” asked R.
“What is algebra, anyway?” I asked. It turns out that even though we all knew how to wield a variable, we all did not know what variables did philosophically. Our conversation made it clear that algebra is a method for generalization. But still the kids wanted to see what would happen to our proof if y = 90. So we tried it.
The algebra was stickier this time. We reached a seeming impasse after a few lines. We applied the strategy of checking all of our steps. No mistakes. Then we applied the strategy of thinking through the implications of each step in relation to our diagram. This approach generated a realization on R’s part: we could use substitution a different way to end up with our desired result. Once again, Q.E.D., it was proven. Satisfaction. Time for Proof #5.
“I am going to prove to you that 1 = 2. There is a flaw in the proof. See if you can find it.”
After the students had some fun attacking the “givens” of the proof, I led them somewhat painfully through this proof. I say “painfully” for two reasons:
1) The algebra was challenging.
2) The steps felt unnatural, even pointless. The kids didn’t like how I was so clearly manipulating each move to mold the proof to my desired end.
But they stuck with it, and somehow, after great collaborative struggle, realized that the flaw was a line of algebra that really meant dividing by zero. “You can’t do that,” said N.
“Why not?” I asked.
“That’s what they taught us,” someone replied. I resisted the urge to inquire about the identity of “they,” and instead asked whether the kids wanted to know why.
“Yes!” they all called out in unison. So we had an edifying conversation about the meanings of division. We took a quick detour into factoring and the conventions for using 0 as a digit.
We then wrapped up with more conversation about proofs in general, and a big request for more.
Math Circle for students is taking a break for the summer, but this doesn’t mean that I am taking a break. Sadly, I will not be attending the Kaplan’s Summer Institute at Notre Dame again this year, but I am doing math at home. I am participating in the MOOC/citizen science project “Problem Solving for the Young, the Very Young, and the Young at Heart,” organized by James Tanton, Maria Droujkova, and Yelena McManaman. I am also working with Sue VanHattum to put the final touches on my chapter in her anthology Playing with Math. And of course, I’m brainstorming topics for the upcoming Talking Stick Math Circle year. Possible topics include Pascal, infinity, more proofs, counting, Cantor, more unanswered questions, set theory, and game theory.
Wishing you all a good summer. Hope to see you in the fall. Feel free to email me in the meantime if you want to talk about math.
1Emma Coolidge, whose biography we had discussed last week. Her proof and history is detailed in the Kaplan’s Hidden Harmonies, pp103-107.
2I offered proofs 2 and 3 and partial demonstrations because we did not have the time needed (weeks) for the students to come up with them using pure inquiry.