(November 1, 2018) “That’s a pig!” said F, as we all walked over the picnic tables to start our Math Circle outside today.
“No, Penelope is not a pig. She is a pig puppet. There’s a big difference,” I replied as we sat down. This seemingly inane comment of mine captured everyone’s attention.
“You know how it is said that a year for humans is like 7 for dogs and 5 for cats?” I asked? Everyone nodded.
“There are also fox years,” added A.
“Yes, and there are pig-puppet years. But they work in the opposite direction as dog years and cat years. For every 10 years that a human ages, a pig puppet only ages 1 year. So even though I got Penelope almost 30 years ago, her age is really 3.”* We discussed this concept for a moment, and then I announced, “I’m having some trouble with Penelope that you can probably help me with. I’m trying to teach her about numbers, but listen to how she responds to my lessons:”
Me to Penelope the pig puppet: “If Grandma gives you five cookies and Grandpa gives you five cookies, how many cookies will you have?”
Penelope to me: “None, because I’ll eat them all!”**
DOES CONTEXT MATTER?
Penelope’s statement set off a huge mathematical conversation. Students had questions and comments about what numbers are, how to explain them, what counting really means, the difference between numbers and numerals, and how numbers first came into being. We talked about all of these things, and then returned to the original scenario. The students tried and tried to teach Penelope the math problem above by changing the context:
Student: “What do you get when you combine five blobs and five blobs?”
Penelope: “One, since blobs squish together when you combine them!”
Student: “What do you get when you combine five pieces of titanium and five pieces of titanium?”
Penelope: “One, since titanium is a metal and metals melt at high temperatures.”
Student: “What do you get when you combine five wooden blocks and five wooden blocks?”
Penelope: “Zero, because I like to play with matches!”
Student: “What do you get when you combine the numeral five and the numeral five?”
Penelope: “Fifty-five, since the fives are right next to each other now!”
Turns out that no matter what context the students came up with (probably 20 examples in all), Penelope had a way to make the problem not work. No matter what, five things plus five things didn’t equal ten.
A: “How can Penelope know so much about other things and not know anything about math?”
Me: “She’s a science prodigy.”
F: “But aren’t math and science related?”
Me: “Yes, but that’s another thing that’s special about pig puppets. We can take some creative liberties.”
By this time, the puppet Penelope had somehow moved from my hand to F’s hand, and the students had taken over both roles – coming up with new contexts and finding ways to contradict the hoped-for result. No one was able to come up with a context that Penelope (in most cases actually M) couldn’t knock down. I was just enjoying the show.
STRIPPING AWAY CONTEXT
“What’s the difference between the problem five plus five equals ten and the problem five things plus five things equals ten things? I asked. The students’ thinking even further intensified. They posited conjectures, debated them, rejected them until S*** said “My brain hurts!” The others agreed.
“What’s the difference between numbers and things?” I asked more directly.
“Well, numbers are something that we made up to talk about things,” answered A.***
“Do numbers exist as things in the natural world?” I asked.
“Yes,” said about half the students.
“No,” said the other half at the same time.
They all looked at each other. Then those that said Yes changed their answers to No.
“Are they ideas?” I asked.
“Yes!” everyone agreed. We talked about ideas versus things. How mathematicians use the word abstract to describe ideas that can then be applied to multiple scenarios.
“Like abstract art,” said S excitedly. Then she quickly reversed herself: “Actually, no, since abstract art is a thing.”
“My brain really hurts now,” said A.
“Do cookies behave logically?” I asked?
“No. People eat them!”
“So would mathematicians rather study things that behave logically or things that do not?” The students all agreed that “logical things” is the answer. I explained that mathematicians like to strip away the context to get at the underlying abstract structure of things. This can reveal similarities, I continued, like in that problem we did last week with the symmetries and arrangements.
But is it always mathematically sound to strip away all context? If a problem is totally abstract, will you arrive at a useful answer?
I presented the students a paraphrase of a problem from Eugenia Cheng’s book How to Bake π:****
You run a company that takes people on tours. You’re organizing a trip for 100 people. You’re renting minibuses and want to maximize your profit. Each minibus holds 15 people. How many do you have to rent?
The students started out by trying numbers: 10 busses – too many. 9 busses – still too many. Then S suggested dividing 100 by 15, yielding 6.6̅. After some discussion, they concluded that we need to rent 7 busses, so 100 ÷ 15 = 7.
“That’s it? That’s the problem?” said S, a bit disappointed. She was happy to hear that no, that’s just the first part. The problem continues:
Now you’re shipping some chocolates to a friend. You pre-paid for a stamp that covers the cost of mailing 100 ounces. Each chocolate weighs 15 ounces. How many pieces can you send to your friend without paying extra for shipping?
Immediately the students saw that it’s the same calculation but a different interpretation of 6. 6̅. They all were talking but not so much to each other or me. More like each was thinking aloud, simultaneously. S persevered the longest and gave a solid explanation of why in this case, 100 ÷ 15 = 6.
“So here’s the real problem,” I said to the students. “Why is the answer 6 when you’re talking about chocolates and stamps but the answer 7 when you’re talking about people and busses?”
“Context matters,” they all agreed. I quoted Cheng to them: “Be careful not to throw away too much… Category theory brings context to the forefront.”
“What would be the answer,” I asked, “to a person who looked at this problem purely abstractly, with all of the context stripped away?”
“6. 6̅” they all agreed. They definitely were grasping abstraction versus reality and some key points about context. But it was time for a break. People’s brains had started hurting 20 minutes ago.
FUNCTION MACHINES AND CAKE CUTTING
I gave everyone an apparent brain break by doing a function machine with them. (The students provide a number that goes “in,” I tell them what number comes “out,” and their job is to discern the rule.)
“Now try this one: If you slice a cake, what’s the function for the maximum number of pieces you can get with a certain number of cuts?” (I also worded it in the language of circles at F’s request: “What is the function for the maximum number of regions you can create with a certain number of chords in a circle?”)
I started sketching this on the board with the students’ verbal instructions (“2 pieces from 1 cut, 4 pieces from 2 cuts,” etc.). But almost immediately, the students were all at the board figuring it out for themselves. Once again, I sat back and enjoyed. Most of the students quickly got 7 pieces from 3 cuts.
“I got 10 pieces from 4 cuts,” announced M.
“Can you get more?” I asked. She tried, without success, and then went on to test 5 cuts and 6 cuts. By then, at least three students had diagrams with 10 pieces from 4 cuts. “You can get more,” I promised. “There’s something that all of you are doing that you could change to get more.” They kept working.
“Can you get more than 7 pieces from 3 cuts?” backtracked S.
“No one ever has,” I said.
“But just because no one ever has, does that mean it can’t be done?” she asked. “Has anyone demonstrated that it definitely can’t be done?”
“That is one of the key questions in mathematics,” I said, so excited by this question. A huge goal of our math circle is to teach kids to be doubters. “In math, it’s not enough that no one has ever done something. There has to be a proof that it can’t (or can) be done for us to believe anything. And yes, there is a proof that you cannot do more than 7.”
“I got 11!” announced A, who had been fervently trying to beat the class record of 10 from 4 cuts.
“Now you’ve reached the number that has been proven to be the maximum.”
My intended point of this activity had been to look for a pattern/function/rule to determine the number of slices. We had a nice sequence of numbers (2,4,7,11), but no one was interested in pattern-seeking. They just wanted to keep testing. So I played the big-number card: “How many pieces could you get from 500 cuts?”
“We would need bigger whiteboards and more markers,” said someone, defeating my attempt to redirect the approach.
I played the ridiculous-number card: “If you had to determine how many pieces you could get from 5,000 cuts, would you rather have a bigger whiteboard or know the rule?” Someone grudgingly said that the rule would be better in that case, but it didn’t detract from anyone’s enthusiasm for drawing.
We were out of time. Had we gotten to the point where students showed interest in determining a rule, I would have burst their bubble anyway with some talk about how patterns don’t mean rules without a proof. (Again, training doubters.) So we ended on a high note with me connecting this activity to the idea of abstraction.
A FEW OTHER THINGS
Early in the session, the students brought up the golden goose problem from last week. (Would you rather have golden eggs, a goose that makes golden eggs, a machine that makes those geese, or a machine that makes those machines?) Some students had talked about it at home and were reconsidering their answers from last time. We talked about the levels of knowledge you needed for each item in this hierarchy, and how that ties to mathematics. The students wanted to explore whether the answer to the question would be different if we removed the goose as a possibility. I explained that “What would happen if we changed the question a bit?” is exactly something mathematicians ask all the time. This came up later in the class, when S was doing the cake-cutting problem without realizing that the cuts had to be chords, not just random line segments. She made a quick shift from initial disappointment that she had misunderstood the problem to excitement to hear that this might a new way to do this problem that people hadn’t worked on before.
*I’m relaying this anecdote so that interested parents can have a jumping-off point to talk more about ratios and proportions at home.
** Eugenia Cheng, How to Bake π, p19. We also dramatized with Penelope the pig puppet Cheng’s examples of the difference between having memorized the sequence of counting numbers and actually understanding what they mean. (p20) Cheng discusses the cake-cutting problem on pages 33-34. You can find an algebraic explanation of the problem on Wolfram MathWorld and many other places. It’s a classic problem.
***We have two students whose names begin with S, and I’m using S for both. Ditto for A.