(November 16, 2022) For about 20 minutes we had been working on the famous puzzle from the book Gödel, Escher, Bach when Z asked, “What’s the point of this?” (besides the obvious problem-solving). I explained that I wanted to talk about axioms. “What are axioms?” the students asked, and I gave a very superficial explanation. I mentioned formal systems, showed the book, and said that axioms are “because I said so” answers to the question “why.” I’m very fortunate to be able to leverage the assets of the students: Z and D are very interested in mathematical philosophy and pedagogy so I didn’t even have to introduce the idea of axioms in any kind of pedantic way!
U: My brain hurts!
Me: Your brain is growing!
Students: Did you bring the ropes again today?
MORE VENN DIAGRAMS
I heard one of the students mention fried chicken.
Me: Stand in the red circle if you like fried chicken and the blue if you like fried, uh, ah…. I’m trying to think of something fried that not all of you would have tried before.
D: Fried pickles!
Me: Yes! Stand in the blue if you like fried pickles.
This opened up a whole wonderful can of worms.
Students: I don’t know/I don’t like either/I know my answer about one food but not the other.
Fortunately, they noticed that there was now a green rope in the bag. They debated extensively on whether to make an “I don’t know” circle with the green. If so, where to put it? Outside the red and the blue? Overlapping one or the other or both?
U: My brain hurts!
Me: That means it’s growing!
Z: Can I do some?
Once again, the students’ curiosity and excitement spared me from having to give direct instruction or information. I had arrived today with a plan to let the students take turns being the “Caller.” Z took over as Caller. Gave various prompts regarding holidays. “If you celebrate this holiday go in red and that holiday go in blue.” At first it was straightforward – red, blue, intersection, or universal set without being in red/blue. Other students took turns as Caller too. I asked for another turn.
Me: This is a yes/no. Stand in blue if you celebrate Bastille Day and red if you don’t celebrate Bastille Day. No one had heard of Bastille Day. What is it? (I answered.) Does it count if your parent is French and probably observes it somehow?
Students: What do you mean by “celebrate?”
Spoken like true mathematicians! They discussed what it means to “celebrate” and came up with a more specific definition. They debated how the ropes should be rearranged and moved them. Everyone took turns as Caller, trying to think of unfamiliar holidays.
U: I’d like to give a riddle. Can I do that now?
Me: I think that’s fine as long as you can turn it into a Venn Diagram activity.
Somehow this was possible. Go into the red if you know the answer and the blue if you don’t. Then someone standing in the red gave the answer to the riddle. Everyone was satisfied. (And I must admit that I was self-satisfied that I had started using the term Venn Diagram without ever having told the students what it was.)
V showed the group the solution to the Make 37 Puzzle, which students had worked on last week. V had figured it out at home with Cuisinaire Rods. V also gave an explanation that proved the answer. Most (or all – ?) of the others understood and appreciated this.