January 15, 2013
We lounged around the room, talking about math. M and I sat on the floor, slouched against the wall while E lay on the floor. J did the occasional unobtrusive headstand while D, V, and C sat in chairs. I never moved from my position, not even to get up and write something else on the board. The students never needed nor wanted a move to the table or a change in topic. Our scheduled 55-minute Math Circle continued beyond an hour, and could have gone much longer if parents hadn’t been waiting. It was magic. Or, maybe it was a good group size or dynamic. Most likely, though, it was the natural response to an accessible mystery.
One thing was written on the board as the students, ages 7-9, arrived for our first session:
All puddings are nice.
This dish is a pudding.
No nice things are wholesome.1
“What does that mean?” demanded M as soon as the group entered the room. The kids immediately started questioning terms in the words (pudding? dish? wholesome?). “The chalk board says no nice things are wholesome,” said M. “Whose words are those? Who wrote that?” J posited that I wrote them, but that they were the words of some old, dead mathematician from England, since so many mathematicians are old, dead, and English, “like Plato.” I asked how she could know for sure that I wrote them, and C stated he had seen me do it, so the group accepted that part of her conjecture as true. They begged to know who wrote it. I answered that, yes, it was someone old, dead, and English, and that I would ask them some questions to help them figure it out for themselves.
“Is it possible to lie without knowing it?”2 I asked. Voices filled the room, so I asked for 10 seconds of silence before answering. I repeated the question and counted to 10 with my fingers. This delay introduced doubts. Initially, about half of the kids thought yes, half thought no, and C wasn’t sure. After the 10, the distribution of votes was the same but some kids had changed their minds. C still wasn’t sure. Students debated this question in general terms, which introduced even more doubt and confusion. I gave a more specific hypothetical example: “Suppose that I asked V where his sister is, and he told me that she was at her after-school activity. But suppose he didn’t know she actually had felt sick during school and didn’t go to her activity. Instead she was home on the couch, watching TV. He wouldn’t have seen her since he came straight from school to Math Circle. Would V have been lying about where his sister was?” The commencing debate was immediately silenced with V’s answer:
“I know where she is. She’s at rowing.”
“What if she felt sick at lunch and went home and you didn’t know about it?”
“No, that wouldn’t have happened because she felt fine this morning.”
“What if she got food poisoning at lunch?”
“No,” replied V, enjoying the feeling of certainty, “my mom would have called my dad and he would have told me in the car.”
I’m clearly not adept at presenting hypothetical realities, so I said, “I’m having trouble coming up with an example of a situation where someone could be lying without knowing. Could you all help me think of one?”
“Well,” answered M, “there are white lies and black lies.” The group now wanted to discuss types of lies and reasons for lying, but I directed back to the matter at hand with the question “What is my name?”
“Rodi,” said D.
“How do you know?”
“You told me that when I first met you.”
“Did I say ‘My name is Rodi,’ or ‘I am Rodi’ and does that matter?” At this point, M revealed that she and J both knew that Rodi is what I am called but it is not my legal name. We talked about how, in math, it’s important to define the terms in a question, since many terms have more than one meaning. We also did a “lying” scenario about whether I was wearing glasses. E’s questions about this required me to refine the question to “Am I wearing glasses on my face?” When I asked for another show of hands for the lying-without-knowing question, we saw there was still no consensus. (At this point, 4 people out of 6 said that you must know it’s not true for it to be a lie.) I wrote down “unresolved” in my notebook, showed them the comment, then asked, “Do people ever lie about their names?” After we discussed this, I said that I had been online earlier today reading about the motivations behind the pseudonyms of Daniel Handler (a.k.a. Lemony Snickett) and Joanne Rowling (a.k.a. JK Rowling).
“How did you know that our conversation was going to lead to this topic?” asked E, shocked at my seeming prescience.
“She told us that she prepared the list of questions ahead of time,” answered V. He was right. At the start of our Circle, I told the group that I had a few questions somehow related to the Pudding question that I was going to ask them.
“When are you going to tell us who wrote it?” asked D.
“How did you know that our answers would lead to this point?” asked someone.
“Teachers never ask questions they don’t know the answer to,” replied V.
“I do sometimes,” I replied.
“Like what,” challenged the kids.
“Like our first question today – is it possible to lie without knowing it. I don’t know the answer to that.” After I gave that eye-opening response, I revealed that the writer of “All Puddings are Nice” was Charles Dodgson, a famous mathematician/logician who also wrote books.
“I love logic!” exclaimed D. “What did he write?” I told the group that they could ask me “yes/no” questions to try to figure out his most famous book. They got pretty close, but then I had to plant two questions to enable them to narrow it down to Alice in Wonderland. Then I asked what math skills they had used to answer this.
“Strategy!” announced E and D and J. I asked for them to be more specific.
“Each answer got us to know something that wouldn’t work,” said C. His comment led into a short conversation about narrowing possibilities with more information and process of elimination.
“But how will that help us figure out what the thing on the board means?” asked M. Good question. I suggested that we do an easier question first – one published by Raymond Smullyan, one of Dodgson’s biggest fans.
“Did he use a fake name?” someone asked. I explained why I thought he didn’t need to, then I read the question: “What happens if an irresistible cannon hits an immovable post?” 2
We had 5 minutes left. Several kids asked what “irresistible” and “immovable” mean, to which D swiftly replied “hits everything in its target” and “can’t get knocked down.” Then the debate started. Five minutes was clearly not enough time. We went beyond our time limit before I sent them home thinking excitedly about this question. Since some kids were absent, their friends in class promised to fill them in on what they missed. Parents of absent kids, you may want to read this report to your kids, or better yet, just ask the questions in bold, or even do nothing at all because it will be easy to catch up in the next session. Since we’ll have more kids at the next session, I’m very interested to see how this conversation (for that’s what a Math Circle is) goes.
Thanks to Lhianna Bodiford, Michael Paul Goldenberg, and Sandy Lemberg for pointing me in the direction of Smullyan and encouraging the exploration of logic with young students. Despite his fame within mathematics, I had never heard of Smullyan. I guess that’s why he didn’t need to use a fake name.
1 Charles Dodgson, The Game of Logic
2 Raymond Smullyan, What is the Name of This Book? p5, p8
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