(March 9, 2017)

“The math circle teacher is here!” announced S to the other students who were playing outside, as I walked onto the Talking Stick grounds. I was early, but before I even had my bag open and the whiteboards set up in the classroom, everyone was sitting on the floor waiting to start. So we started. Immediately everyone wanted to do Function Machines

### FUNCTION MACHINES

We got to work with a student-designed machine that looked like a “round robot” with 2 heads, 20 hands, and 3 wheels. The rules that the students had to figure out were much trickier than last week’s and took a lot of time. (See photos to see if you can figure them out.)

Students were begging for a chance to lead their own machines, so we scheduled different children on different weeks. Fair scheduling was tough since so many wanted to be first. I used a “guess a number” strategy at first, but that was taking too long, so the kids suggested we switch to Eeny-Meeny-Miney-Moe. That was much more efficient.

The results: next time (class 3), C and female M get a turn; for class 4 it’s A and S; for class 5 it’s L and M. Not everyone definitely wants to do this, so I told everyone it’s totally optional – we’re just making sure everyone has a chance if desired. I instructed everyone to make sure they knew their rules beforehand and to get my help when they lead if needed. Parents, I suggest they try them out with you at home first.

### THE OGRE PROBLEM, CONTINUED

Most were excited to get back to the Ogre Problem from last week. I told the students that I forgot the question, so they had to recap it for me. If you strip away the narrative, the problem boils down to “Can 10 people lie in a circle so that in exactly 5 places, two heads meet?”

“So how can we go about solving this?” I asked. Silence. They all looked to me for instruction. I wasn’t about to tell them, of course. I reminded them that the title of the course is “Problem Solving,” and that we were using as our guide the techniques in the book Avoid Hard Work. “Do you remember any of the problem-solving strategies we discussed last week?” They didn’t.

“One technique is to say how the problem makes you feel.” Their faces brightened, and they recalled and acknowledged their feelings.

“Another technique is to do something about it.” We did that in one way by making the problem into a story.

“Does having the story about the Ogre make it any easier?” The kids thought not, but felt that having the story made them more motivated to try.

I also reminded them that they had suggested acting it out last week with their own bodies. They remembered this, but didn’t seem excited to try it. M then remembered that we had little paper pins left over from last week when we were discussing the “Penny” problem. One of the students pointed out that the Penny problem was similar to the Ogre problem. (The students still don’t seem to realize that the problems are mathematically identical.) She suggested that we dramatize the Ogre problem with the “pins.” There was not, however, much enthusiasm about this either. More silence.

### A CONJECTURE

“I have something that might be an idea,” said L. “If there are 10 people, and we need 5 places where heads are together, then there needs to be half as many places where heads touch as there are people. I think that means we need half of the people to be touching someone else’s head.” I wrote this on the board and labeled it conjecture. Maybe there was hope. But still no one could think of a strategy for how to proceed, even armed with a conjecture ready for testing.

### TECHNIQUE: START SMALL

I finally gave them some new information in the form of another problem-solving technique: start small. “If we don’t want to get 10 people on the ground to see if we can do it, one approach mathematicians use is to test the same conjecture with a smaller number. Do you think we could try that?” The students were now enthusiastic. They had a starting point. Someone suggested starting with 7.

“But we don’t have 7 students,” objected someone else.

“We can’t even get half of seven. We can’t chop people in half!” said someone else. Hmmm… What to do?

“Let’s start with 8,” suggested another.

Some of the kids started lying down to test this. But not everyone wanted to lie down, and still, we only had 6 students (and 1 sibling – no one suggested recruited parents as they had suggested last time). The students felt they were at another impasse.

### REFINING THE TESTING METHOD

“Is 8 the only number smaller than 10 that we can test?” I asked.

After some discussion, the students decided to start with 2. Two students lie down, head-to-head. So there were two students, and half that number of places where heads touch. The students were excited – it seemed that if this was possible for one example, it would be more likely to be possible for other examples too. They then eliminated 3, tested 4, eliminated 5, then ran out of participants. (One student had the job of “people counter” so was not on the ground.) Another roadblock. This one was easily overcome, however, when the students suggest that I participate and that we use the little paper pins as people. They worked to show that an arrangement with half as many head-meeting places as people was possible with both 6 and 8 people. This was hard work, and energy was flagging. Several people were distracted. I suggested that we take a break and do a different problem, then return to this one next week. “Mathematicians sometimes need to take a break and see things fresh another day,” I explained. The students agreed, provided that I promise to resume this problem later or next week.

### GRAPH THEORY PROBLEM

“I have another problem that I could give to you with a story, or just as straight math. Which do you want?” The students were equally split, so we decided to try it without a story first, and then layer on the story if we needed it to increase understanding or generate enthusiasm. I set up 2 copies of letters A, B, and C taped to a whiteboard in a particular way and gave the problem:* “Can you connect A to A, B to B, and C to C, without crossing any lines or going off the board?”

The students studied it for a bit. L grabbed a marker and started drawing. It didn’t look promising. “It’s impossible,” declared C.“Impossible?” I asked.

“Impossible?” I asked.

“Yes. It can’t get done.”

I told the students that mathematicians would not look at a problem for a total of 2 minutes and declare it impossible. “They would work for hours, days, weeks, months, or years to be sure.” Other people took up markers too. Each person’s contribution triggered another person’s idea. After about 10 minutes, they had solved it! True collaboration had produced a solution. What a great way to end the session!

**— Rodi**

**Avoid Hard Work, p37 (I have a photo of the students’ solution, but don’t want to spoil anything for you by posting it here. Ask your children what the problem was and see if you can figure it out. Email me directly if you want to see their solution.)*

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