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Math Circle Blog: Popcorn, Nose Rings, and Parameters

This Week Our Group Invents Parameters.

SEPTEMBER 25, 2012: Here’s what I expected to happen in Math Circle today: The students would walk in, discuss A’s discovery of an important calculation error last week, and then continue the Popcorn Problem. I had placed unpopped popcorn kernels on the table before I told them the life story of Enrico Fermi. Hopefully, they would realize from their manipulations that you can count how many kernels per linear inch, and then calculate per square inch, and then per cubic inch. Then it’s simply a matter of multiplying this number by the volume of the room, and, voila, there’s the number of kernels you need! Then I would ask Question Two of the Popcorn Problem: “What would it take to fill the room next door with popcorn?” The idea here is that kids will realize that they don’t have to start from scratch again, and will develop a formula, thereby creating a need for variables. It’s a nice way to realize a desire for and understanding of variables.

Fortunately, something much better happened.

Unfortunately, A and S were absent. So we never did mention the calculation issue. But fortunately, no one did measure the length of kernel strings as I spoke of Fermi. Instead, the discussion immediately turned to an idea that P had first posited last week: we could use the weight of the popcorn to answer the question. Not everyone agreed that weight was relevant; D and R wanted to solve the problem with volume only. J, a younger child on the other side of the room, had been listening in. She announced that Talking Stick does indeed own a scale. This announcement moved the group’s collective intellectual momentum toward weighing the popcorn.

Fortunately, the scale was not sensitive enough to register the weight of a single kernel. The conversation yielded a strategy to weigh “a handful” of kernels in a box with easily-measurable dimensions. Once that was done, I asked, “So, should we do it? Raise your hand if you’d like to use all these numbers and calculate exactly how many pounds of corn we need.” None one moved. No one said anything. No surprise; last week’s calculations were tedious and awkward.

“Could you do it if you had to?” The group said yes.

“So then let’s move on to part two. What would it take to fill the room next door with popcorn?” A lively discussion erupted. Would our assumptions still hold, kids wondered. What would change? What would stay the same?

A generalized procedure emerged. Step one was to take the volume of the other room and divide it by the volume of the box of kernels to get the number of boxes needed. Step two was to multiply the number of the boxes by the weight of a single box. I wrote the procedure on the board. Then I asked, “What would it take to fill the bathroom with popcorn?” … “What about the hall?”

At this point, P was whispering the word variables. The group decided that variables would be a good approach. “What symbols should we use for these things?” A few people called out some letters (x, y, v…) but then our fly on the wall J yelled out, “Smiley faces!” The group liked this idea, and R refined it with the suggestion that we use smileys for the things that don’t change, and frownies for the things that do change.

“Are they really variables if they don’t change?” I asked.

“No,” replied everyone.

“So what are they?” Several group members called them “controls.” Most everyone concurred. I threw into the mix the term “constant” as a possibility. A few others agreed with that. On the board I labeled the smileys “control/constant” and the frownies “variable,” and the group dictated equations for the two steps. Then I frowned.

“I’m so confused. I don’t understand these equations at all.”

Several people pointed out the problem: it’s ambiguous to only use 2 symbols when there are more than 2 variables/constants/controls. “So give one of them fangs!” called out J. Some folks groaned, but with C’s encouragement, the group got into the spirit, assigning distinct visual features to clear up the ambiguity. I read their equations back to them:

“Frownie with a nose ring divided by smiley with fangs equals frownie with a mohawk. Frownie with a mohawk times smiley with glasses gives you the weight of popcorn needed to fill any room.”

The kids concluded that while it is not easy to use, this set of equations sure was fun to create. Next week we’ll see how it would look with letters as variables.

Then we returned to a question from last week: what is data? Students mentioned various definitions and applications of both data and statistics. They debated whether data must be numeric. I asked, “What’s the difference between data and statistics?”

R blurted out a quick example: “A statistic is something like 20% of people in Philadelphia have nose rings.” After some lively discussion (Is that true? Could you find out for sure? If so, how?), I asked another question:

“Does that statistic, if true, mean the same thing as when I say 20% of people at that table there are wearing black hoods?” I pointed to the table where 5 students, including N in his black hoodie, sat. Most students, after some thought, said yes. I questioned them about this just enough to introduce a bit of uncertainty. Then it was time to go, so I gave them 3 optional questions to think about for next time:

  1. Do you, as a group, want to collect some data?
  2. If so, what data interests you? (Some found the nose ring question “gross.”)
  3. How would you figure out the exact percentage of people in Philadelphia who have nose rings?

My editor (and student) R does not like it when I wax pedagogical in these reports, but two things have come to mind as I reflect on this week’s session. First, reflect on your teaching. During the 2+ decades I’ve been an educator, I’ve been encouraged to reflect, but have not done it enough. Every time I do reflect, something very important comes bubbling up. This week, I realized after class that our group had invented parameters, those algebra symbols that are a hybrid of variables and constants. (A “parameter” or “smiley” or “variable that doesn’t change” stays constant within a given function but can change when the function changes. A true constant never changes. The students had it right when they called it a controlled variable.) Since I had my eye on the clock as we moved quickly through the material toward the end of class, and since I had preconceived notions on how this session would go, I didn’t even notice the significance of this creation until I reflected upon it later. Days later, I am still excited that the students came up with this concept, and look forward to discussing it further.

Freedom also comes to mind. Our group’s creation of parameters was possible because the students were given the freedom to explore a question in their own collaborative way. Because they debated, made mistakes, questioned assumptions, took their time, and considered alternative methods, they delved deeply into the rich structure of the problem. Had they been in a competitive or accelerated math environment, this probably would not have happened. Those environments, according to the Kaplans, “pervert the savoring of beautiful insights into opportunities to put one another down.” Math is an art. “Works of art are to be savored rather than dashed past. Better an hour in front of a Monet than five museums in a day of Paris. The beauty, the meaning, the structure of a piece of mathematics unfolds only with reflection… We’re not intensifying a standard mathematical diet by adding mental vitamins to it… we’re engaged in the wholly different enterprise of developing the architectural instinct.” *

Lastly, I’d like to mention the aspects of Fermi’s story that particularly interested the group:

  • his personality,
  • his defection,
  • his philosophy of scientific research,
  • the ethical decisions he had to make,
  • his connection to the recent discovery of the Higgs boson,
  • and the origins of “Fermi problems” in mathematics, such as our problem.

Students might still be thinking about this question: If you were Fermi, would you have worked on the creation of nuclear weapons?

Until next week,

-- Rodi

*Out of the Labrynth, Bob and Ellen Kaplan, pages 105 and 161