Logic for the Very Young
(February 26 and March 5, 2013)
“One afternoon, 2 children wander into a Kingdom unknown to them ….”
So began our two-week Math Circle for children aged 6-7. This story framed an exploration of logic games, questions, and strategies. I had lost my voice, so my assistant Rachel led the Circle while I sat quietly with the group.
In the first session, we played “Picking Fruit” and then “Wearing Hats,” two cooperative games in which children make deductions from process of elimination and other people’s statements. These games advanced the plot of the story of the Kingdom, as did the next puzzle – the tale of A Very Clever Prince.1 This puzzle requires logic to understand it, and incessant question-asking to shake out assumptions.
The puzzle stymied the kids. They made conjecture after conjecture, each of which was refuted by another student. Some people wanted to return to the story. One child said, “Maybe there is no answer.”
“Every question has an answer!” asserted B. At this point, I found enough of my voice to jump in and disagree. I showed the kids a handout describing open questions in mathematics that children can understand.2 S begged to try to solve one. The others joined in his enthusiasm. Unfortunately we were out of time, so I promised that we could look at one of these next week. Rachel and I asked the kids to think more about the Very Clever Prince during the week.
It was fortuitous that my laryngitis required a substitute leader this week. I’ve been inspired by the work of teen leader Daniela Ganelin of the Art of Inquiry, and had the pleasure of collaborating with her at the Julia Robinson Math Festival in Mayaquez, PR, this month. I’ve dreamed of a teen-led Math Circle here in Philadelphia. It finally happened today. Rachel has participated in nearly 50 Math Circle sessions, has read Teaching with your Mouth Shut, and edits my Math Circle reports. After I spent a few hours prepping her on today’s materials, and she was more than ready. The children were energized by the presence of teenager playfully engaging in mathematics.3
Rachel was not available the following week so I returned as the leader. “Leader,” however, is an inaccurate title. My role is more that of a secretary. I present a question, contribute no ideas, but help the kids keep track of their own thinking. We revisited the Very Clever Prince puzzle. No one had had any insight over the week, so we reviewed what we knew. This led to discussion, debate, and countless conjectures. The kids got so close to a solution, but were reluctant to follow their line of reasoning because it involved something “yucky.” C helped me to act it out in order to convince the group that this solution would work. Further convincing was needed to convince several of the members that something yucky was preferable to death, at least in this case, since it wasn’t really that yucky.
I then continued our tale of the Kingdom with a “whodunit” mystery: The police bring Alla, Belle, and Collard - three known criminals in the Kingdom – to the new king and queen for questioning. Someone has used a car to steal a bunch of loot from a store. The royals want to determine whether Alla was involved, as she has applied for a job in the palace kitchen and claims to be reformed. The police state the facts in the case:
1) No one other than Alla, Belle, and Collard was involved.
2) Collard never does a crime without his big sister Alla (and possibly others). He is too scared.
3) Belle cannot drive.
Was Alla involved?4
“That’s a hard one!” declared T. Everyone then simultaneously shouted out assorted wrong answers until I asked, “Why?” The kids paced for a moment, then came back together when one child produced a conjecture supported by the premises of the problem. That conjecture led to another child’s connected conjecture, and so on and so on and so on. It was a beautiful collaboration in which all 5 kids, T, E, C, B, and S, each contributed to the solution.
We then finished the session with an exploration of the “No Three in a Line” question, an open problem in mathematics. Of course we didn’t solve it, but we got to the point where the kids knew to look for a pattern. They devised a system for taking turns making conjectures and for testing conjectures. Everyone got to speak, everyone made and discussed conjectures, and I was no longer needed. On that note, we ended our session triumphantly.
Feel free to email me for more details on the problems that we explored.
1Thanks to the Berkeley Math Circle and Bay Area Elementary Math Circle for this puzzle.
2Thanks to Gordon Hamilton of Math Pickle for introducing these questions at the 2012 Math Circles on the Road Workshop in Washington, D.C.
3Rachel, too, was energized. As soon as the kids left, she commented on how hard it had been to Tell Kids Nothing. Later, she added, “Co-leading the math circle was really fun! In all of the math circles for younger kids that I sit in, I always want to talk! It's because even though the kids are younger, and the problems are at a lower level, the actual problems themselves are intriguing for any age. This time I got to talk, but it was really hard not to give the answers away! I had to only ask questions that might make the kids think about things so that they could come up with the answers themselves.”
4 Thanks for J for the names Alla, Belle, and Collard. Their story is a retelling of Raymond Smullyan’s puzzle 71 from What is the Name of this Book? (page 67, “From the Files of Inspector Craig”). After class, I referred those children who begged for more puzzles to this book. Email me directly for some suggestions of which of these puzzles might be solvable for young children.