Inspector Craig Cases/Variables
(April 29, 2014) We began with Smullyan’s puzzle #72 (another Inspector Craig whodunit). Before class, I wondered whether to use his convention of calling the suspects in each case A, B, and C. The cases are distinct, and the variables are true variables – they represent different people in each equation… I mean in each case. I thought this would be confusing for the kids, but I want them to think. So I decided to give them different names each week. In our first session, I wrote on the board A = Abigail, B = Bartram, C = Carle. This week I wrote A = Alana, B = Blaze, C = Caddie.
The kids did think that the suspects in the case were the same people as last week. I had to explain several times that A, B, and C are different each time. Then they joyfully collaborated on solving the puzzle.
Role Playing Need for Numbers, part 2
We tried this again, as it was such as success last week, but today it backfired. I made the mistake of not giving the kids any time to play with the puppets in their own way before imposing rigid structure on their use. Next time, I’ll let kids just play with them for a few minutes first.
Can Crows Count?
We role-played the following real-life experiment:
A squire wanted to trap a crow that lived in his watchtower. Each time he came over with a trap, though, the crow flew into a nearby tall tree, and returned to the watchtower when the squire left. The squire decided to trick the crow by bringing along a friend, leaving that friend hidden inside the base of the watchtower, and then return to his manse. The crow, however, wasn’t tricked. The crow remained in the tree until the friend grew weary and left. The next day, the squire tried with two friends (one remaining in the tower). No fooling that crow. The next day, three friends. It wasn’t until the squire brought four friends that the crow was tricked, flying back to the tower while the person holding the trap was still hidden in the tower. Does that mean that crows can count?
I did not read the whole scenario to the kids ahead of time. G and N and the puppets acted it out event by event as I slowly revealed the plot, while A and M made predictions throughout. G asked in the middle of the drama, “Isn’t this a math circle?” I reassured him that it is. All of the kids made suggestions on changes to the story throughout, but I wouldn’t allow that since we were re-enacting something that really happened. The kids were enthralled, and enjoyed making predictions. My private prediction, by the way, was that the kids would think that the crow was able to count. I should have realized that these kids were more sophisticated than that; they are just beyond that age of anthropomorphizing and magical thinking. They all agreed that the crow was not counting.
I said, “So what was the crow doing? How could he tell, up to a point, that not every person had exited the watchtower?” Students were unsure. They posited then quickly rejected several conjectures. Then A announced, “He knew because it was less.” In other words, he used number sense. According to William Richardson and his students at Wichita State University*, “The number sense is not the ability to count, but the ability to recognize that something has changes in a small collection. Some animal species are capable of this.”
“Do you think that humans have a number sense that’s better than, worse than, or the same as a crow,” I asked the kids after we discussed number sense. Everyone agreed that humans would be much better at this. They were shocked to hear that ours is about the same (according to Richardson and other sources). Of course, we can count, and crows can’t.
We then did some experiments with blocks to test our own number sense and ability to subitize (recognize quantities without counting). We found that certain arrangements of the blocks facilitated the ability to instantly recognize larger quantities.
Another Attempt at Triangular Numbers
This week, again, I attempted to introduce triangular numbers. I drew dots on the board in a triangular pattern (first 1, then 3, then 6), and asked “What’s the next number in the sequence?” Someone called out 8, and the rest agreed. “Let’s draw it to check,” I suggested, and then everyone agreed that it was actually 10. “Do you notice anything interesting about these numbers?” I asked. They tried really, really hard because I think they want to please me, but didn’t have much to say and no one was greatly interested. One person did point out that you add one more dot than you did the last time to keep the pattern going, but even that didn’t seem as interesting to them as it does to me. So we moved on.
“This is a mathematical piece of music,” I said, as I played a recording of When Johnny Comes Marching Home.** “See if you can figure out why.” The kids listened attentively. A posited that the piece is mathematical because of the patterns of the stars and stripes of the waving flag image on the youtube video. “Yes, that’s something I didn’t even think of!” We listened more. Someone started whistling along.
“I’ve heard this before,” said someone. The others nodded in agreement. But no one remembered where they heard it. After more listening, G suddenly announced, “It’s a counting song!” Others agreed. No one knew what it was, so I told them about the original piece, then sang for them the newer version, The Ants Go Marching. The students listened and joined in. What a testament to the power of music. We were 50 minutes into the math circle at the end of a long day, and kids were antsy (pardon the pun) until the music started.
We spent our last few minutes positing conjectures on what it meant to go marching “one by one,” “two by two,” etc.
*I found the crow anecdote in their article The Number Sense, part of a website that states “Welcome to the Math 750J Project. This web page is the end result of the Math 750J Workshop in the History of Mathematics for Middle School Teachers. The goal of this workshop was to produce a web site which would be a source of information for teachers of mathematics.” Poke around on there – lots of the material can be adapted for kids younger than middle school. Thank you, Professor Richardson and students, for putting this together.
** I played this really nice arrangement of When Johnny Comes Marching Home (The Ants Go Marching) by the Canadian Brass. There are no vocals, the melody is obvious throughout, and in each verse, more instruments play.