Echo Edie
Number Talks restating — repeating the most recent kid's idea before the group moves on
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The four kids at Sam's circle were arguing.
They were arguing about a problem involving four people and a boat. The boat could only hold two people at a time. Some of the people walked slowly and some walked fast and the question was how to get all four across in the shortest total time. The kids had been working on it for almost twenty minutes and they had several ideas going at once, and the ideas were starting to step on each other.
Owen had just said: "I think the slowest person has to go first."
But before he could explain why, Bex had jumped in: "No, the slowest person has to go LAST, so they only cross once."
And before Bex could finish, Joon had said: "Actually wait — what if two slow people went together?"
Owen, who had been about to explain his reasoning for why the slowest person had to go first, looked deflated. The thing he had been about to say was leaving his head. He was forgetting it. He could feel it slipping.
"I had — I had a reason — " he said.
That was when Echo Edie appeared.
Echo Edie was a small woman with a kind face and an unhurried way of leaning forward when she spoke. She had been quietly present on the screen, but until now she had said almost nothing. She turned to Owen.
"You said the slowest person has to go first," she said. "Tell us why. Before we move on to anyone else's idea, let's hear yours all the way through."
Owen blinked.
"Oh," he said. "Um. Because the slowest person is going to be the bottleneck. So if you send them with someone fast, the fast person can come back with the flashlight. So the slow person crosses once with the fast person, and then the fast person handles the rest of the trips."
Echo Edie nodded.
"So your idea is: send the slowest person first, paired with a fast person, so the fast person can shuttle the rest."
"Yes."
"Now," Echo Edie said, turning to Bex. "What were you saying?"
Bex said, "I was saying the slowest person should go LAST, so they only cross once."
"And your reason?"
"Because every time they cross, the whole crossing takes as long as the slowest person. So we want to minimize how many times they're walking."
"So your idea is: keep the slowest person out of the boat as long as possible, because they slow everything down each trip."
"Yes."
Echo Edie nodded again.
"And Joon," she said. "What were you saying?"
"I said: what if two slow people went together?"
"Why?"
"Because if both slow people cross together, that's still just as fast as one slow person crossing, since they walk at the slow speed anyway. So you might be saving a trip."
"So your idea is: pair the slow people together to use the bottleneck once for two slow people."
"Yes."
Echo Edie looked at all four kids.
"You have three ideas," she said. "Owen's: send the slow with a fast for shuttling. Bex's: keep the slow out as long as possible. Joon's: pair the slow people. They might all be right. They might combine in a way none of you have noticed yet. Do you want to draw the problem out and try all three?"
The four kids stared at her.
"You... you remembered all of that," Sam said.
"Yes."
"Even Owen's? He said it first and we were about to move past it."
"Especially Owen's. The first idea is the one most likely to get lost. I tried to catch it before it disappeared."
She faded.
The four kids looked at each other.
"Okay," Bex said. "Let's try all three."
The kids worked for another half hour.
In the end, they figured out something none of them had thought of alone, but each had contributed a piece to. The answer used Owen's shuttling idea, but it modified the shuttling to handle Bex's keep-the-slowest-out rule, and it added Joon's pair-the-slow-people trick in the middle. The answer was seventeen minutes.
It was the smallest answer any of them had been able to find.
When Echo Edie reappeared, she said: "Tell me what each of you contributed."
The four of them looked at each other.
Joon said, "I had the pairing idea."
Owen said, "I had the shuttling idea."
Bex said, "I had the keep-the-slowest-out idea."
Sam said, "I had... I had the idea of drawing the time on a number line so we could see when each trip started and ended."
"Yes," Echo Edie said. "And which idea was the one that made the answer work?"
The four of them thought for a moment.
"All of them?" Joon said.
"All of them," Echo Edie said. "None of you would have gotten there alone. The answer is not Owen's, or Bex's, or Joon's, or Sam's. The answer is the circle's. Your circle. Together."
She paused.
"This is why I do what I do," she said. "Ideas can get lost in a fast conversation. Especially first ideas. Especially first ideas from kids who don't say them loudly. My job is to catch the ones that are about to be lost and hold them up so the rest of you can decide whether they were useful. Most of the time they were."
She faded.
The four kids stayed at the table for a while.
Owen, who had spoken quietly, said: "Thank you, Edie."
Even though Edie was gone, he said it out loud.
The MathCircle ensemble
Echo Edie is part of MathCircle's distributed-narrative cast. Each character embodies a different curricular primitive; together they teach the full subject.
