Plato, Codes, and Exploding Dots
“I must add how charming the science of arithmetic is and in how many ways it is a subtle and useful tool to achieve our purposes, if pursued in the spirit of a philosopher, and not of a shopkeeper!'”
This was the last line of a dialogue my assistant R and I read at the beginning of this week’s Math Circle. Despite some almost indecipherable language, the kids hung on every word, and then tried to make sense of it. J correctly guessed that we were quoting Plato. Last week, X had requested that we learn about him. A pointed out that the dialogue was about math, and asked, “Is he still alive?” I added that it’s also about philosophy, but no one knew what that is. After our reading and conversation, I told them a bit about Plato’s Republic and his pronouncements on just who should be allowed to vote and why. His opinions on this raised their ire, which I calmed by telling them that his ideas about Guardians didn’t stand the test of time as well as his other ideas did.
We then continued our signaling problem from last week. We were essentially creating a code for a code. Students tested last week’s creation to see if the code worked out arithmetically. It didn’t. V proposed a new system: each window could represent a different power of 10 (units for the parents’ rooms, tens for the hall, and hundreds for the children’s rooms). This sounded so “mathematical” that once everyone understood it, they were convinced that it must work. Sadly, it did not: some of the code numbers could not be expressed with only 5 candles. For instance, said S, “The 44 wouldn’t work, but 40 would, because you don’t have a candle as small as a 4.” At this point some students wanted to revisit “engineering” approaches to the problem (spies, distractions, seagulls, swimming pools, letters burnt into a sheet, and a burning banshee). I insisted that we not give up on using codes just because it’s challenging. I focused the group by starting lists of questions and conjectures on the board.
A new idea then emerged from V’s idea: let’s add in more numbers. D suggested that “one window could be worth 3, and then you could put a 6 in the one remaining window. Then you could do the 16 and the 13.” The group calculated that the numbers 6, 7, 10, 10, and 3 could be combined to make our signals 27, 16, 44, and 13, as long as we change 44 to 36. “Danger to the house: 36. All the windows lit up!” proclaimed X.
While some students checked these numbers, some others realized that each window could be a different signal without the need to calculate. This plan seemed to address C’s concern that “we only have 5 candles – do we need to show all these messages at once?” and S’s concern that the first plan would only work if the people in each room could communicate with each other.
It looked like we had 2 solutions. People wondered: do these both work? We discussed, deliberated, and calculated. No one saw a problem with the solutions. A asked multiple times, “Rodi, if we see no problems with it, it must be right, isn’t it?” I told them that it seems to be right, but that I wasn’t sure. Something didn’t sit right with me, I said, and I need time to think about it. She wanted to declare it solved right then: “We don’t have to be 100% sure.” I suggested we switch gears and come back to the problem so that we could be 100% sure.
“V’s idea of using units, tens, and hundreds reminds me of a game. Have you ever played Exploding Dots?” I moved the group to the other board to demonstrate what is basically a place-value function machine (but don’t tell the kids, please). I put the boxes on the board and asked them to make explosion noises when I indicated. I filled in dots, they made the noises, I moved the dots, and I asked for the rule. We did it a number of times until the kids understood what was going on. Then I asked them to describe what was happening mathematically. This task was tough, and we ended class trying to do it. We’ll continue with this next week. And speaking of next week, I may not email out a report until the following week. I’ll be away for Thanksgiving week with no computer access. I’ll tell you now what I “plan” (ha!) to cover, but know that since “math is freedom” (to quote Bob Kaplan), we might not follow the plan. The basic plan will be to continue with Exploding Dots and the signaling problem (a more advanced version), to talk about Navajo Code Talkers, and possibly discuss another Plato anecdote.
- For more about Plato, see http://www-history.mcs.st-and.ac.uk/Extras/Plato_on_mathematics.html and http://www.historyforkids.org/learn/greeks/philosophy/plato.htm
- For more about Exploding Dots, search online for “Exploding Dots Tanton.” You might enjoy both the middle school version and the advanced version. I plan to do more of this each week with our group, so please wait until the end of the course to have fun doing this with your kids at home. For now, ask them what it is, and how to play it. Thanks to Jim Tanton for inventing this intuitive approach.
With warm wishes to all of you for Thanksgiving,