A Breakthrough, and Inventing our own Math (Eye of Horus 4)
(October 8, 2013)
As students arrived, they saw three cubes of different sizes positioned, left to right, on the table with a “do-not-touch-my-number” sign. By the time everyone was there, S and V had concluded that “There are 153 blocks there! The number is 153!” The rest of the students were mystified. I asked for an explanation of S’s and V’s counting method. V explained to everyone how he had counted the largest cube “by fives,” since every row had 5 cubes in it.
“How do you know that every row contains 5?” I asked. “You can’t see the center of the cube.”
“Well, they’re not glued together,” said someone. “They would fall otherwise.” (I love it when students are exasperated by my cluelessness – how empowering for them.)
Since everyone had been respecting my “do-not-touch-my-number” signs*, I was able to ask, “How do you know there’s no glue?”
“We’re assuming that there’s no glue,” said M. Ah, a nice opportunity to discuss assumptions. During that discussion, O accidentally bumped the pile of cubes, verifying the truth behind the assumption. Vindication for the kids!
Once everyone agreed that the table held 153 blocks, I granted that “Yes, when we’re standing on this side of the table, the number is one hundred fifty three.” Then I asked the kids to move to the other side. “Now the number is not 153,” I announced. After a small initial protest, curiosity took over. How could these piles of blocks represent a different number when nothing changed except the viewer’s position?
Students produced numerous conjectures. Could this seeming paradox involve
- color? (It looks like a Rubic’s Cube, said Z)
- stacking method?
- angle of the piles relative to the edge of the table?
- and many more
I moved the group around the table several times, which became tedious to a few kids. J became frustrated and exclaimed several times, “I have no idea what is going on here!” She even walked away from the group. However, several minutes later, after listening to the others’ conjectures, posited, “Might it have something to do with the position of the digits?” Actually, it did. (This is a great example of how acknowledging your emotional reaction to a math problem can open up possibilities.)
This breakthrough energized everyone. Their perseverance had been rewarded and ideas started flowing. Students suspected that the digit position was interacting with the number of digits, the number of dimensions, and the size of the cubes. We differentiated between sides and edges, between squares and cubes, and between 2-D and 3-D. I drew diagrams on the board, using chalk arrows to imbue the diagrams with movement to dramatize this interaction. Many students were getting great ideas and then immediately losing them. The solution seemed to be right on the edge of the group’s collective awareness. At this point, students had been exploring the question for at least thirty minutes.**
“This is a Narcissistic Number,” announced E suddenly, seeming to surprise himself. Everyone looked at him, expecting an explanation. He couldn’t easily explain it, so some of the other kids tried to assist. I swung my arms melodramatically, pointing to the diagram, the numbers on the board, and the actual blocks, à la Vanna White, during the attempted explanations. No one really understood fully what a Narcissistic Number is. R asked for a mathematical definition. I put several variations on the board, but these just confused the matter. What worked better was an idea proposed by one of the students: counter-examples. If you can’t easily explain something, it helps to apply the converse of “I know it when I see it.” Students were able to explain why 351 is not Narcissistic, even though 153 is. Of course, once they explained it, they forgot. At this age, the grasping of multifaceted concepts is ephemeral. You might want to play around with these numbers at home. For the kids who want to explore the question further, I suggested trying to discover whether any 2-digit Narcissistic Numbers exist.
The discussion wrapped itself up with the students accepting that they might not be able to fully fathom Narcissistic Numbers, but they are on their way. They were excited to hear about the discipline of Recreational Mathematics, and to know that they could just make this stuff up themselves. In fact, K had come to class today with a new kind of number of his own creation: Bodyguard Numbers, which have identical first and final digits protecting the inner digits. We had some fun proposing numbers to him for approval. R too had come in today with an interesting number: googol. She had written out the whole number for us to see. And after class, M pointed out a rock formation outside that could be a number.
WHAT ELSE WE DID TODAY
In keeping with our Riordan/mythology theme, I told the story of the Egyptian god Horus. This story led to the kids’ conjecture that the Egyptian gods were more genteel than those of Greece. (These gods had a boat race to settle a dispute, wrote a letter seeking advice from the head gods, and refused to let a murderer have the highest position.) Of course, there was plenty of bloodshed. I know little about mythology, so I’d appreciate if some more knowledgeable folks piped in here about the legitimacy of this conjecture.
In anticipation of moving into the math of the Horus story, I presented a function machine that produced an out number that was one over the in number. We debated how to describe this function in words. Descriptions varied widely***, all were accurate, and each built upon the prior – great teamwork!
*I earned this respect by introducing some drama and levity. When kids put their hands close, as if to touch the blocks, I intoned “I am the Keeper of the Blocks! Do not touch!” I lamented that my title didn’t sound that impressive or mathematical. L suggested an improved title: “Keeper of the Squares.” For the rest of the session the kids called me “Madam Keeper,” and we began a list of Math Circle names for everyone.
**It may seem surprising that a bunch of nine-year-olds might wrestle with a single math question for a long time at the end of a long day. Many people think that math has to be dressed up with games to interest kids, but a good question is inherently interesting. When a student finds math boring, I suspect that the problem is that we as teachers are not asking the right questions. We all make this mistake at least sometimes - how can we always know which questions will be inherently interesting to kids? I try to choose questions that are interesting to me, and then try to be flexible and reframe or even ditch the question when student reaction instructs me to.
***(1) “You’re writing a one over the number.”
(2) “You’re making a fraction of it.”
(3) “You’re splitting it up into portions.”