The arbitrariness of variable names.
When cake is 20, pie is 5. When cake is 100, pie is 85. So pie = cake – 15, or p = c -15.
The students named these variables, figured out the rule, and created the equation after we had first done a traditional function machine and then discarded the conventional language (‘”in,out” and “x,y”). After all, the words and symbols are arbitrary. They debated and even voted on the names of the variables for the next function, but when “pancake” tied with “kombu” as candidates to correspond to “candy,” I gave G the power to name anything – anything at all. She declared it would be “chocolate.” The students struggled with discerning the rule for the candy/chocolate function. After a number of examples, only T knew the rule. So he said the rule and the rest worked together on writing the equation. Devising symbols was a bit tricker with two C words, and the resulting equation, Ch = 2Ca + 2, was deemed less than satisfactory because of its resemblance to a chemical equation involving calcium. I found it interesting that the students struggled to “get” 2x+2 after having no problems at all with our initial function, 3x-1. The struggle likely resulted from their choosing small input numbers to test their conjectures. With these functions, bigger numbers are more auspicious from a ballparking perspective – a concept worth visiting in the future.
The arbitrariness of variable names led us into a discussion of Brahmagupta, the Indian mathematican who is considered by some to be the creator of algebra. He used names of colors (among other nouns) to describe functions in mathematical ways. A fact even more interesting to this group was that it was he who first named and used the number zero. “Did he invent it or discover it?” I asked, and lively debate ensued. Z pointed out that the concept of “nothing” couldn’t have been new at that time. The group agreed that the concept of “nothing” would have been a discovery at some earlier point, but the actual mathematical measure of it was probably an invention.
“If I said to you all, ‘Could somebody get me a piece of chalk?’” would it matter who got it for me?
“No,” said R, “since ‘somebody’ is a variable.”
“What is a variable?” I asked the group.
“Something that changes,” replied L, as the rest nodded in agreement.
“What if somebody popped into the room from India and got the chalk? Would that still count as ‘somebody’?” asked N.
“It still wouldn’t matter,” answered A. “’Somebody’ doesn’t have to be a specified person.”
“What if I said, ‘It’s raining outside?’ What does that word ‘it’ refer to.”
“That’s another variable,” said R, since it could refer to anything outside.”
“No it couldn’t,” argued A. “It has to refer to the sky or the air or something like that.”
More debate followed. Z attempted to clarify by using grammatical terms. Other students who had been quiet until this point chimed in. The consensus seemed to be that “somebody” is a variable that could be anything, but that “it” is more limited, and might refer to only one thing.*
We then moved to the table for some drawing, as promised. As we sat down, P asked, “What is Pascal’s Triangle exactly?” (This had come up in a Vi Hart video 2 weeks ago.) L came to the board and demonstrated.
“What’s the point of doing it with numbers?” asked R.
Another answered, “What’s the point of doing it without numbers?” (Once again, we see evidence of how people use very different avenues into the enjoyment of mathematics.)
Several students began expanding Pascal’s Triangle in their notebooks while others worked on the fractals that we had discussed over the past few weeks. As they worked, I told the life story of Sierpinksi (the Polish mathematician who invented the famous triangle fractal). This biography did not seem to capture their fancy (no one asked any questions about him). At this point, a few students asked to see another Vi Hart video. After debate over which one to see, I showed “Squiggle Inception.” We had ten or fifteen minutes to draw at this point, and draw everyone did.
After a couple of minutes, I heard a quiet chorus of “squigga-squigga-squigga-squigga” from one part of the room. (Watch the video yourself for the significance of this chant.) Then from others came “snake-a-snake-a-snake-a-snake-a.” And yet another student quoted Hart repeatedly with “Don’t put too much space between squiggles or a monster will grow.” Most people talked as they worked – talked about Vi Hart, the nature of squiggles, the Hilbert Curve, gaskets, patterns, and triangles. When time was up, everyone showed their work to the group, and some took photos of their work. When time was up last week, no one left and I had to say, “I am officially kicking you out. Your parents are in the hall. Go to them.”
The kids beat me to it today when N asked, “Are you officially kicking us out?”
“Yes,” I answered, as I gently ushered them out as they continued to work.
*Thanks to Bob Kaplan for the pronoun/variable analogy.