Pushin’ Too Hard? (Problem Solving 4 and 5)

(March 30 and April 6, 2017)

I’ve had the song “Pushin’ Too Hard” in my head for the past two weeks. Math Circle is to blame. (Or to credit – I like that song!)

The students seem to want to really challenge themselves. Especially since they’ve been acknowledging their feelings about problems and said (several weeks ago) that the Balance Scale Problem makes them feel “smart.” (In my experience, this is typical of students this age.) So I’ve been trying to give the people what they want without burning them out. The narrative from the past two weeks is really an exploration of how to balance pushing appropriately versus pushing too hard.

SHEEP

We revisited the “Sheep” problem in session 4 – I showed it to kids who were absent and asked everyone for an alternative solution. While 5 students were arguing about it in the foreground, M solved it in the background without the other kids noticing until I pointed out his work. Then the students solved the same problem with 4 sheep instead of 3. Lots of “solutions” turned out not to actually work when a second mathematician took a look at it (just like real world mathematics – so we talked about collaboration).

FUNCTION MACHINES

C led a function machine.

Then M’s took a loooooooong time because it turned out to be a special type of function in which the number of the term was involved in the rule. She had to give a lot of hints. (See photo.)

OGRE

The kids recruited siblings and parents from outside and finally solved the Ogre Problem. “So we also solved the Penny problem too,” said someone, and they all agreed that the two problems were the same thing. (They didn’t realize this until today.)

TECHNIQUE: Make It Small

I presented a new problem from Avoid Hard Work: “Arrange four pencils on the table so that each pencil touches each of the other three.” (p50)

The book uses this problem to illustrate the technique of starting with a smaller example to build up to a solution for a larger number. This worked well for our group, except that they couldn’t remember solutions they were coming up with. We didn’t talk about this, but here’s where recordkeeping becomes important. I did tell them that a mathematician would solve it for 1, then 2, then 3, then 4 (as the kids did), but then ask “Is there a maximum number of pencils, and why?” After working on the Pencil Problem for a while, and having seen multiple demonstrations from the kids on how it can be done with 3 and 4 pencils, not everyone agreed that it could be done with 3 or 4. (Even though they had just seen it!) So I told the kids that if they want to do something at home, to work on this problem.

By the way, I am taking huge liberties in these reports when I refer to the problems by name. The authors of Avoid Hard Work do not give the problems names. Calling them “The Pencil Problem,” “The Penny Problem,” etc is my doing. Once problems start spreading, they tend to get universal names by default. (Some of my favorite named problems from math folklore are The Monty Hall Problem and The Missionaries and Cannibals Problem –

TIRED BRAINS

With three minutes left in session 4 I said, “Let’s do another quick problem.”

“Not another problem! My brain is tired,” said C, a normally huge math enthusiast. Everyone agreed. This 5-8-year-olds had been doing math for an hour straight at the end of the day on a Tuesday. So I filled the time with some little mathematical factoid or anecdote. Hopefully, brains will get enough rest by next week to do more problems.

THE HARDEST FUNCTION MACHINE EVER

We started session 5 with a function machine that I led. No matter what number they put in, out came “1.” This confused everyone. They conjectured that it was the type M had produced last week, where the order mattered. It was not. Finally, they all agreed that it involved subtracting something. “What do you think you get when you put in zero?” I asked.
“One?” they posited incredulously. Yep.“So sometimes you add and sometimes you subtract?” they asked. Yep.

“So sometimes you add and sometimes you subtract?” they asked. Yep.“But what’s the rule? Can you name one thing that’s being done all the time?” No one could. “What if you put in one?” One comes out.

“But what’s the rule? Can you name one thing that’s being done all the time?” No one could. “What if you put in one?” One comes out. Oh my. I pushed and pushed them to come up with a generalized rule. Finally, M said, “It destroys what you put in.” I told her no, but that she was closer than anyone. “How can I be close when I didn’t even mean that? I was just saying it out of frustration,” she said.“You are close because you just named one single thing that happens to everything you put in,” I explained to her. Then I asked everyone, “What happens if you put in a sock?” (One comes out, they agreed.) I asked about putting in some other non-numeric objects. They all agreed that one comes out. “What if you put in 9 houses?”

“You are close because you just named one single thing that happens to everything you put in,” I explained to her. Then I asked everyone, “What happens if you put in a sock?” (One comes out, they agreed.) I asked about putting in some other non-numeric objects. They all agreed that one comes out. “What if you put in 9 houses?”“Then one house comes out,” replied S, with everyone’s agreement.

“Then one house comes out,” replied S, with everyone’s agreement.“No,” I said, “just one comes out. The number one, not one house. So what do you think the machine does?”

“No,” I said, “just one comes out. The number one, not one house. So what do you think the machine does?”“It turns everything into one!” replied several kids, excitedly. They were happy to have finally described a single rule.

“It turns everything into one!” replied several kids, excitedly. They were happy to have finally described a single rule. Now, of course, they were technically correct when they said sometimes you add, sometimes you subtract, and sometimes you keep the thing the same. But what I was going for here was a generalization. This is something nearly impossible for kids younger than eight. I’m certainly not a neuroscientist, but I suspect that if you provide opportunities for a brain to think this way (in other words, push kids out of their comfort zone), you might be able to facilitate the kind of thinking that leads to the ability to generalize. And algebraic thinking is all about generalizing.

WHY SHOULD WE STRUGGLE?

I then gave another problem from Avoid Hard Work: “A teacher draws horses on a sheet of paper and holds the sheet up to the class. She asks Lashana, ‘How many horses do you see altogether?’ Lashana replies, ‘THREE.’ The teacher says, ‘Correct!’ She then asks Juan how many horses he sees. He replies, ‘FOUR’ and the teacher responds, ‘Correct!’ How many horses did the teacher draw on the paper, given that both Lashana and Juan were indeed correct with their answers?”This problem has multiple choice answers, and the problem-solving technique illustrated is to “eliminate incorrect choices.” The students just agonized over this problem. One student (and maybe more) got worried that there was something wrong with their brains because they couldn’t quickly solve it. I explained (as I often do) that the really good math problems take a long time to solve – hours/days/weeks/months/years. S said, “Hey, wait a minute. How is it fair that we have to sit here and struggle with this

This problem has multiple choice answers, and the problem-solving technique illustrated is to “eliminate incorrect choices.” The students just agonized over this problem. One student (and maybe more) got worried that there was something wrong with their brains because they couldn’t quickly solve it. I explained (as I often do) that the really good math problems take a long time to solve – hours/days/weeks/months/years. S said, “Hey, wait a minute. How is it fair that we have to sit here and struggle with this problem when you get to just sit there with the book and the answer?”“Excellent question,” I replied. I then told them about how struggling can make their brains grow. They were skeptical.

“Excellent question,” I replied. I then told them about how struggling can make their brains grow. They were skeptical.“Here’s another question,” said M to the other students, “why is she making us wiggle our arms like this?” (M was holding her arms out to the side and shaking/wiggle them.)

“Here’s another question,” said M to the other students, “why is she making us wiggle our arms like this?” (M was holding her arms out to the side and shaking/wiggle them.)“Rodi’s not making us do that with our arms,” another student told her.

“Rodi’s not making us do that with our arms,” another student told her.

“I’m not making you do that, but that is a really good idea. Let’s all do what M is doing with her arms.” So we all starting shaking our arms. “Are your arms tired from the struggle to keep them wiggling?” (Yes!) “I assure you that working your arm muscles like that makes your arm muscles grow. Do you feel like they’ve gotten stronger? Your brain works like a muscle – work it and it grows. So let’s get back to the problem.” The arm-shaking was somewhat cathartic and the students were eventually able to solve the problem. I don’t know whether my explanation of arm-shaking being analogous to brain-working did anything for the kids since, at this age, they are barely beginning to be able to transfer concepts from one context to another, but since this type of transfer is important in mathematics, I pushed them to do it.

At this point, it was raining very hard outside and the sky was dark and rumbly. All during class the kids were checking out the door and windows. They wanted to go outside in it. Finally, I said we could go outside and do more math on the covered porch so that they could feel the air and see the rain.

MY MISTAKE

We put on shoes and raincoats and went outside with a box of wooden cubes. I read them a problem from Avoid Hard Work that involved arranging blocks. “Can we just play with them,” asked the students. We didn’t have much time, so I (foolishly) said no. I realize that of course if I had let them have time to explore the blocks on their own first that they would have been happy to explore the question I was posing, even if that had to wait until next week. I did ask what’s special about these blocks and one child answered. Another child tried to do the problem by setting up 3×3, 4×4, and 5×5 arrays. But no one was even able to understand the problem because of the rain, the excitement about the blocks, and the tired brains.

SCREAMING CONTEST

“Let’s have a screaming contest!” said someone suddenly. All of the participants started screaming together at the top of their lungs. I just let them, even though the parents and siblings nearby were holding their ears. I thought the screaming would release some of their pent-up energy from the rainy day. I waited it out and then started whispering. They immediately paid attention.I very quietly recited for them another problem from Avoid Hard Work (a problem that I like to call “Three Statements”) but the students couldn’t follow it. “Never mind,” I said. “I changed my mind. Instead of three statements, I’m going to whisper one statement to you, and you tell me whether I’m telling the truth or lying. Here’s the statement: I am lying.” Finally, something the students could wrap their brains around. They discussed it and concluded that the statement is true and false at the same time. But by then everyone was cold. We had about 10 minutes left. “Let’s go in,” I said, and we did.

I very quietly recited for them another problem from Avoid Hard Work (a problem that I like to call “Three Statements”) but the students couldn’t follow it. “Never mind,” I said. “I changed my mind. Instead of three statements, I’m going to whisper one statement to you, and you tell me whether I’m telling the truth or lying. Here’s the statement: I am lying.” Finally, something the students could wrap their brains around. They discussed it and concluded that the statement is true and false at the same time. But by then everyone was cold. We had about 10 minutes left. “Let’s go in,” I said, and we did.

THE HIP-POCKET CONCESSION

Many years ago, I worked in the business world. In my training, I learned about something called the “hip-pocket concession” – something that you don’t reveal, that you’re willing to contribute, and you save it as a last resort in a negotiation. This concept is something that I have transferred to my math education life. I always like to go into a math circle or class with a mathematical hip-pocket concession, just in case nothing else catches students’ fancies. Usually, in a math circle, it is function machines. But I had already fried their brains with one of these today. So I resorted to the thing I keep the deepest in my pocket (since it can only be used once ever with any group): The Very Clever Prince.

This is a logic problem that probably every student in the Talking Stick Math Circle (since 2011) has worked on. I presented the problem, discussed it a bit, then the kids went home. Ask them to share the problem with you! We’ll continue that, and some other unfinished business, next week.

NATIONAL MATH FESTIVAL

The National Math Festival is coming up in DC in a few weeks. I’ve been to it before, and it’s a lot of fun.
Rodi

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