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PROBLEM SOLVING #1: A New Group of Students

March 2, 2017

As I prepare for today’s math circle, I realize that I am in a privileged position of utmost responsibility: only one person in this group of six has ever been to a math circle before. For years, the majority of students in our circles have participated for years. I hope I am up to the task.

FUNCTION MACHINES

“I know something we can do while we’re waiting for everyone to arrive,” I said. “Do you know what a machine does?” I asked. The students discussed the question, then helped me draw a new machine on the board. I drew an enclosed region with curvature in its border (a.k.a. a blob) and asks the kids to suggest machine parts to add. After three rounds of play, it ended up named the “Bobby 3,000” and it had hands, a head, a propeller, lights, 3D glasses, a pail of poison, and more. In round 1, when you put in 1 out came 2, when you put in 100 out came 101, and so on. The students’ job was to figure out the rule that was being applied to these numbers, and they did so with glee.

WHAT IS A PROBLEM?

Soon everyone had arrived. I asked the students some questions: How were words invented? When? Why? After that discussion, I asked “How and why do you think the invention of the word problem come about?” The students agreed that the word is to describe difficult situations, “like when you get a thorn in your thumb,” added S.“This class is called Problem Solving. Why do we use the word problem in math?” The students said that in math, you can get into troubling situations that are unpleasant. “But the Function Machines we just did were problems, right? You had to figure out the rule. That was a problem. But it didn’t seem like having a thorn in your thumb. Doing function machines was kinda fun.” Eyes opened wider. Ah!

“This class is called Problem Solving. Why do we use the word problem in math?” The students said that in math, you can get into troubling situations that are unpleasant. “But the Function Machines we just did were problems, right? You had to figure out the rule. That was a problem. But it didn’t seem like having a thorn in your thumb. Doing function machines was kinda fun.” Eyes opened wider. Ah!“There’s another kind of problem,” proposed C. “When it’s fun to try to figure out the solution to something.” Everyone agreed. I showed them the book we are going to use in this course: Avoid Hard Work. I read to them the definition of

“There’s another kind of problem,” proposed C. “When it’s fun to try to figure out the solution to something.” Everyone agreed. I showed them the book we are going to use in this course: Avoid Hard Work. I read to them the definition of a problem from the book: “The word problem comes from the word probe, meaning inquiry.” We talked about that.“Why would you want to avoid hard work in math?” L laughed out loud. It was obvious to the kids.

“Why would you want to avoid hard work in math?” L laughed out loud. It was obvious to the kids.
I read a few playful mathematical inquiries from the book, then told them that I was going to give them two different problems and that their job would be to choose which one they would rather work on.

Technique: HAVE AN EMOTIONAL REACTION

Every problem in the book lists as the first step in the solution “React emotionally to the problem.”* The first problem was about pins. After we discussed what pins are, I read“A pin has two ends: one called a head and one called the point. Penny likes to arrange pins in loops For example, here is a picture of six pins in a loop. This picture has two places where a head and a head meet, two places where a point and a point meet, and two places where a point and a head meet. One time Penny arranged ten pins in a loop. She told me that in her picture there were FIVE places here a head and a point met. Is she remembering correctly?” (p29-30)

“A pin has two ends: one called a head and one called the point. Penny likes to arrange pins in loops For example, here is a picture of six pins in a loop. This picture has two places where a head and a head meet, two places where a point and a point meet, and two places where a point and a head meet. One time Penny arranged ten pins in a loop. She told me that in her picture there were FIVE places here a head and a point met. Is she remembering correctly?” (p29-30)

I immediately asked the students how they felt, and wrote in on the board:
• Good
• Bad
• Crazy

Technique: DO SOMETHING

Then I read the other problem. I didn’t tell the students that both problems are identical mathematically. Avoid Hard Work lists the second step to problem-solving as “do something about it.” Specifically here, “Make the problem into a story. Children need a very strong reason for why something needs to be done (as do we all.) It also has to be personal to them: that is, it must touch upon their interests, hobbies, or favorite objects.” (p30) The challenge here, of course, is that I don’t know these kids at all. I have no idea what’s important to them. The older my own children get, the harder it is for me to know what’s important to kids of a certain age.**

An ogre lives in an enchanted wood. The ogre is not friendly because he has no friends. He wants to feel special and important, but no one treats him as though he is special or important. (What should his name be?) He devises a plan to get some friends. He steals something very special from some people in the town and locks what he steals into a vault. (What would he steal from you?) He creates two portals with challenges to get the people to come to him to retrieve their belongings. He plans to have a party for them at the end.

The challenges involve passing through portals. At the first portal, to get through, 6 people have to lie on the ground in a circle. At 2 of the places where the people touch, a head and a head meet. At 2 of the places where the people touch, feet and feet meet. At 2 of the places where people touch, a head meets feet. There is a sign posted showing how it can be done.

The second portal says this: to pass through, you must arrange 10 people in a circle. At 5 places, a head meets feet. How can this be done?***

As soon as I said the question, M started counting the people in the room. We had 10, including adults and siblings. “We’re ready to act this out!” said L.

“Before we do anything,” I said, “let’s talk about how it made you felt when you read it.”
• Terrible
• Good
• Terrified
• Creeped out
• Scared
• Happy
• It made me want to run into a wall and die

These varying reactions made it hard to tell whether the storytelling enhanced the problem or created more barriers to entering the mathematics of it. Did I go too far with the ogre? “If you had to choose one problem to solve, which would it be, the Penny Problem or the Ogre Problem? The Ogre problem definitely got more votes, but the Penny problem was attractive too. It already had a bit of a story embedded in it, so it wasn’t just pure mathematics.

“Which one are we going to do,” asked the students excitedly, as we ran out of time for today. We’ll start with the Ogre problem next week.

REALLY OWNING THIS PROBLEM

The students seemed to be having a great time as I read the Ogre problem, especially the part about what the ogre would have taken from you. What do you value most? What is so important that you would go after the ogre in order to get it back? After suggesting some things that got crossed off the list quickly (like pizza), the students’ list looked like this:
• Gold
• Cake
• Detergent
• $1,000
• Golden glass
• T-rex tooth
• Computer
• Nerf gun

Looking forward to continuing problem solving next week!

-- Rodi

* One of my goals for the six weeks of this math circle is for students to make a habit of first acknowledging their own feelings about math problems. That really can unleash a flow of problem-solving energy. I work with a lot of high school students who have never been taught this. I think many of them could have had more successful and pleasant math experiences over the years had they known that it’s normal to have an emotional reaction to math.

**Many math circle leaders over the years have asked me how to know what’s important to kids of a certain age. It used to be easier for me. Now I would recommend going to a book store to peruse the children’s section as a good first step.

***You may notice that I changed the problem from can it be done to how can it be done? I did this because it’s our first session and I didn’t want too much frustration at first.

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