(April 22, 2014) Today was the first of a six-week math circle for ages-6-7. We began with a crime-solving logic puzzle involving 3 characters, Abigail, Bartram, and Carle. This puzzle was Smullyan’s puzzle #71 of his “From the Files of Inspector Craig.”1 The four kids in attendance solved it methodically, cooperatively, and with glee. We had bonus fun postulating on what “loot” was stolen from the store: Money? Antiques? Jewelry? Coins? Phillies shirts? Food? Books? Art? Mirrors? I used this puzzle as an icebreaker: not everyone knew each other, nor had all of the children done math circles before.
FEELING THE NEED FOR NUMBERS
Now that the ice was very broken, we moved into number-theory history. I gave the students choices of scenarios and (optional) puppets, and asked each to improvise a dialogue with me. The challenge: we were from pre-numeric history, so we were not allowed to use numbers.
- A’s chosen task was to invite me to a party. Come on over tomorrow, he said. How will I know when to arrive? The middle of the day. What if my middle is not your middle? The middle of the middle of the middle… Clarification was quite difficult, and fortunately fun for everyone.
- M cared more about which puppets she could use than which task she did, so she chose puppets for each of us and I chose her task. She was to hire me, a doctor, to come and tend to a sick someone at her home. She “wrote” her role with input from A and F. My sheep is sick. I need a veterinarian. How will you pay me? With wool. From your sick sheep? Yes. How much? A pound, said F. All of it, said N. The kids debated whether “a pound” was a number. F was convinced that “a pound” was allowed, but acquiesced to the rest of the group and agreed to N’s idea, since it’s much harder to argue against “all.” During M’s interaction, the students began to hesitate with the realization that the task of not using numbers is actually quite difficult. I acknowledged that I had given them near-impossible tasks, and they all breathed sighs of relief to hear that it was supposed to be hard. (Until I acknowledged this out loud, I imagine that every head was filled with the thought of “everyone knows the answer except for me.”)
- N chose to be an army general opposed to my army general. I won the war, my puppet told his, Look at all my living soldiers standing behind me! I won, replied his puppet. I won! No, I won! From prior math circles, I knew that N was well acquainted with the idea of proof, so I challenged him: Prove it! So he punched my puppet. That was unexpected. I said, Just because you can punch my puppet doesn’t prove that you won the war. Another student punched my puppet. (This was not a show of might makes right – it was done for humorous effect, and it was pretty funny.) Then the dialogue continued. We debated who had more living soldiers remaining. Someone suggested “a hundred,” which was quickly rejected as disallowed. (A number! Gasp!) N told me that I only had “a couple” soldiers left alive. The kids all looked satisfied – some because “couple” was a word and not a number, and others because they didn’t believe me that the word refers to one specific number. I brought up the concept of unitizing2 – that there are certain words for specific quantities, and that couple was one of them. Finally, N suggested using the phrase “a bunch” to make the point that his general had won the war.
- F was excited about her role as tax collector. I explained that the more land a person had, the more taxes one paid, and that people with no land paid no taxes. She suggested measuring my land in yards. No one in the class knew that a yard was a unit. I defined it. Everyone then agreed that “a yard” is the same as “one yard.” A pointed out that if all properties in the land were the same size then numbers or measuring might not be necessary. Then F made the argument that using the word “a” is allowed in this role-play because “a” is a word, not a number.” More specifically, “a” is a symbol which represents a number. (We did not yet discuss the idea of numbers themselves as symbols – coming soon.) The other kids thought that if 1 is a number, then “a” is a number too - an implied number. But then someone raised the possibility that 1 might not be a number. I told the kids that I read about an ancient society who wondered and debated the same thing.4
MOVEMENT AND THE TRIANGLE
I asked the kids to move to the taped shape on the floor. They immediately identified it as a triangle, and asked where to stand. We played Simon Says with commands such as
- Stand on a side of the triangle
- Jump in the triangle… out of it
- Squat on a vertex
- Tiptoe on the sides without touching a vertex
- And so on
We discussed the meaning of the word triangle by dissecting it into “tri” and “angle,” defining those terms too. Then I asked the kids to move to a table where I had blocks out.
MAKING TRIANGLES WITH BLOCKS
I had set up a triangle on the table for 6 blocks: a row of 3, a row of 2, and a row of 1. I asked students questions such as “Can you make a triangle with more than 6 blocks? With fewer? Are there numbers that are impossible to make triangles out of?” Within a minute, kids had taken more blocks from the box and were making their own of various numbers. They debated whether each other’s creations with four blocks were triangles. They argued that you cannot make a triangle with 1 or 2, but that with numbers greater than that, you can, “depending on how you count them.”
My original intention had been for the kids to discover triangular numbers and their sequence, and then to explore patterns. My hope here is to build an early connection between numbers and shapes. But the kids didn’t discover triangular numbers. And they were using a different definition of “triangle made from cubic blocks” than I was. So I didn’t even hint at my agenda, and instead left the kids satisfied with their own laudable conjectures based upon their own assumptions. Triangular numbers will wait for another day.
MOVING OUTSIDE FOR MORE LOGIC
We had been working hard for 50 minutes with class near an end. A few kids were getting antsy, running out the one door and back in the other. So I moved the group outside for one last puzzle. Moving outside didn’t help, by the way. I had essentially removed the boundaries that the walls were providing, and did not have a compelling enough question to keep people focused.
Actually, the question was compelling, but my lead-in to it was not. I had the idea to tie a logic puzzle about kings holding prisoners to the movie Frozen. My kids and their friends are almost obsessed with this movie, so I thought it would capture people’s interest here. It turns out, though, that the puzzle itself was compelling enough without the pop-culture tie-in, and that the kids had strong enough emotional reactions to the film that they each wanted to tell of their own reactions or interpretations of it. So we were out of time by the time I got to pose the question. The question was hard, no one could solve it, and I promised to begin with it next time.
Next time, I’m going to re-structure the physical space to help keep people focused, but still involved. If you’d like to read more about the necessity of physical movement to mathematical learning, I’d recommend Malke Rosenfeld’s blog Math in Your Feet5.
Thanks to J and L for helping out today.
1 Raymond Smullyan, What is the Name of this Book?, p67.
2 According to Maria Droujkova and Yelena McManaman, there are “four main ways toward building the basic concept of a number: subitizing…, counting…, unitizing (multiplication and division are based on equal groups or units), and exponentiating… Many US curricula do not have enough early unitizing and exponentiating.” (Moebius Noodles: Adventurous Math for the Playground Crowd, p4)
3 I don’t know whether there were tax–collectors in pre-numeric times. If you’re curious about this fascinating subject, though, google “number history.”
4Now that I’m writing this, I can’t remember any details about people uncertain about whether one is a number. Can someone re-enlighten me?