My instincts told me that this group was not going to get The Solution to this problem in the time allotted. Is that okay?
Falling off your chair, Pokemon figures, a shoe, 1 strand of hair, reading, anger, candy, God: this was the list of ideas and things that kids brought into Math Circle to challenge my conjecture that everything is somehow related to math. Before presenting their items, I told them that I’d need help defending my conjecture. So the kids (not me) argued for or against each item. I listed them on the board. E suggested a notational system to show the group’s opinion for each item: the letter M with a box around it. This symbol would have a check next to it for items definitely connected to math, a question mark next to it for items without consensus, and an X over it for items unanimously agreed unrelated. The verdict:
- Falling off your chair, the shoe, the strand of hair, and candy are definitely related to math.
- Anger, reading, and Pokemon might be related. When we voted, none of these were unanimous, although the majority of kids said yes. The objection was that you might have to somehow artificially connect them to math (you pay for them, or can somehow measure them). Because I want the kids to keep thinking about this, I didn’t even suggest production processes or other possible connections.
- “Look,” said H, when our list was complete, “none of them got an X.” Nothing was definitely unconnected.
But our list was not complete; I limited the list to one response per person in the interest of time. People kept calling out more and more ideas, triggered by the ideas already proposed. Ask your kids for more ideas, and ask for their justifications for the above list, to continue the discussion over the summer. Since today was our last Math Circle of this school year, I wanted to ensure time to discuss our path-arranging problem.
Before Math Circle began today, the early kids noticed that the back of my shirt looked like “The Answer” to that problem. V, J, P, M, and L grabbed paper, instructed me to stand still, and copied the shirt while talking out loud about what it might mean. V and L both advised me that if I want people to solve problems on their own, I shouldn’t wear shirts with the solution on the back.
When everyone had arrived, we engaged in an attention-focusing activity, then returned to the question.
“So,” I asked, “what is the answer to the problem?”
“I don’t really understand this.”
“It says that there are 5 possibilities with 4 bricks.”
“No, it says that there are 4 possibilities with 5 bricks.”
People were confused. We returned to our board work from last week, which clearly showed, from hard work the kids had done, that there are 5 arrangements with 4 bricks. “Does it make sense to have only 4 arrangements for 5 bricks then?” The kids agreed that their conjectures about my shirt didn’t make sense. E wanted to return to figuring out the number of arrangements with more and more bricks. No one else did. Most were weary of this problem. We talked about whether it’s okay to take a break from a difficult problem for a while, and work on something else. Everyone said yes. “For how long? A week? A month?”
“At least,” said M. “Some mathematicians have taken breaks for a year or more. Sometimes you have to to,” she explained. Both she and P remembered talking about this in the fall, and someone added that breaks can be important to preserve your sanity. The group agreed to take a break from this problem for a little while, and work on it at home over the summer if desired. E was fine with this so long as I email her a photo of our work so far, and a written description of the problem. Then we joyfully (and quickly) finished the story of the sorcerer challenge.
For the past few weeks my instincts told me that this group was not going to get The Solution to this problem in the time allotted. Is that okay? Should I coach them, ask leading questions, give little hints? The temptation to do so was tugging at me. I resisted, and asked the kids the question I was asking myself (“Is it okay to not solve a problem when you know there is a solution?”). I’m glad I resisted, and will be so curious to hear from you about whether your kids talk about this problem over the summer, and what they say about not solving it conclusively. I think that mathematics and mindfulness educator Richard Brady expresses well what we are trying to accomplish in Math Circles:
“When my students encounter obstacles, their first impulse is usually one of two extremes: they try to overcome them or give up. The approach of welcoming obstacles, sitting with them, and seeing what gifts of understanding they have to offer is foreign to my students. Yet it is one that could serve them well in life. I ask myself how I can do a better job of modeling this way of relating to difficulties in the classroom. I realize I could begin by curbing my impulses to diagnose and suggest remedies for students’ problems and learn how to just be with the students and their problems.”*
And speaking of Richard, it was from both him and our student A that I got the idea of using the game Jenga to focus attention in our group. I saw Richard’s post online recommending this game for an auspicious start to math instruction. But I don’t have the game. So I brought in the game pick-up sticks instead. When we played it, A pointed out that it works just like Jenga. She offered to bring it in. She brought it today, so we used it as our focusing activity. We discussed different ways to set it up, and, as usual, did not play competitively.
I am not using competitive games in our Math Circle. The goal here is what Bob and Ellen Kaplan call a “minds on, hands off” experience. We used the Jenga to cultivate attention, but not to compete or even play. According to Bob, games may give kids the impression that they are not doing something important or revelatory. We want kids to leave with a sense of “wow, I learned something new today.” According to Ellen, the “minds on, hands off” approach also leads to more conversation, one of the ultimate goals of Math Circles. Games and manipulatives have a place in mathematics education, and are occasionally needed as we explore interesting questions in Math Circles, but they are not the point, so we minimize their use. Remember, we’re here to discuss and think deeply about interesting questions. We are not teaching mathematics (to paraphrase Bob), but how to think mathematically. We may not see the results soon. Kids won’t always have new knowledge after 55 minutes. What’s important is how kids do 2 or 3 years later, after they have learned to figure things out for themselves.
I’m wishing you a summer full of interesting mathematical questions to ponder. I hope to see you and your kids in the fall. We will continue to offer Tuesday afternoon Math Circles at Talking Stick. Expect to hear from me (via my email distribution list or on the blog) a couple of times this summer. I have a few ideas brewing that I want to share.
PS Many thanks to V and her mom from bringing in the delectable “Math-Kabobs” (as D named them, since they were arranged in the Fibonacci sequence).