(November 11, 2014) For the past 2 weeks, I’ve been trying to figure out just how to facilitate the topic of infinity for 5-6 year olds in a way that kids could make some discoveries without me feeding them any facts. I had a bunch of ideas, so decided to bring them all to class. My plan was to throw them all up against the wall and hope that something would stick.
It turns out that the kids came in with their own ideas about infinity. (For some reason, I expected blank slates. What’s up with that? Expectations again…) Before class officially began, the students shared their knowledge about infinity. Their knowledge was diverse, a bit contradictory, and wonderfully infused with curiosity and excitement.
MATH RED LIGHT GREEN LIGHT
We went outside and played a quick game of traditional Red Light Green Light, led by my helper J. “Now we’re going to change things up. We’re going to play Math Red Light Green Light.”
“Does that mean you give us a math problem and we get sent back if we get it wrong?” asked V.
“No. In math, you often learn more from mistakes than from quickly getting a correct answer,” I replied. “Mistakes are to be treasured as long we learn from them.” (to paraphrase Julia Brodsky)
“So if it’s right, it’s wrong?” V asked, confused. Everyone looked confused by now. I see I had opened up a can of worms, and quickly explained how there’s value in both right and wrong answers. Back to explaining the game…
“This time, when you run, you may not go further than halfway to the end. So you could get sent back to start for 2 reasons: if J sees you running after she says ‘red light’, or if you go further than halfway.” They were excited to try it. Everyone lined up at the starting line to play. J was about to turn her back and call out “green light…”
“Uh, Rodi,” asked B tentatively, “how do we know where halfway is?”
Therein lies the rub.
Turns out all seven kids were wondering that. They discussed it. Several ran up the walkway to find the spot. Negotiation followed, then consensus. Everyone ran back to start again. All were eyeing the halfway point. J was about to call out “green light…”
“But how will we remember where the halfway spot is?” asked someone else.
Someone suggested drawing a line with sidewalk chalk. Someone else drew a line, and the game commenced. J called out “green light.” The kids ran. They stopped at, or close to, the line just as J called out “red light” and turned back to face them. Everyone was safe. J called out green light again. Some kids ran, but most stayed. Then she said red light and the questions began.
“Now that we’re at the halfway line,” questioned someone, “we can’t go any further, since you said we couldn’t go further than halfway.”
“Oops! I forgot to tell you the most important part of the game. Each time you run, you cannot go past the point that’s halfway between your new location and the finish line. We need to start over.” So we did. The kids first ran to or near the initial halfway line (midway between start and finish). Then they figured out their new halfway points, and marked them with chalk. Play resumed.
Several turns went by with full comprehension and a bit of pleasant debate among the kids about where halfway points should be marked. Point confused a group of 3 kids who arrived together at the initial halfway line on their first turn: they were no longer together after several turns. One of those kids was adamant that if the rules were being followed, the group should still be together. Another argued that they could end up separated if each runs at a different speed. I chose not to get involved in a discussion about rates at this point, as we were just learning (and inventing) the game. I just said, “It’s okay.” I didn’t want to lose the sky for the trees. Hopefully we can revisit this question at some point.
Now people were close to J at the finish line. “Anyone could reach out and touch her on the next turn,” said the kids. Would that end the game? We were at that point in a math problem where you realize you need to define your terms. How would we define finishing? J was standing next to a stump. Did she need to be on the stump, or could the finish line vary based upon where she was standing? Did reaching out and touching her with an arm qualify as finishing? Everyone agreed that reaching out with arms killed the spirit of the game, which to the kids at this point seems to be finding those halfway points. The kids proposed that finishing means your feet touch J’s feet. “But I’ll get knocked over,” protested J. So I drew a circle around her feet and said that the circle would be the finish line. Everyone liked this. It made things easier as the kids were still drawing lines indicating their goal spot on each turn. They demonstrated what finishing would look like, and resumed play.
But things got confusing with folks just inches from the end. When J said “green light,” a few kids jumped into the circle instead of advancing halfway to it. Other kids protested that. I tried to facilitate discussion about this, but we had been at it for 15-20 minutes, and some of the kids were ready for a change. I suggested we shelve the matter until next week, since this was just a sample game to figure things out. L wanted to spend time brainstorming a better name for the game than Math Red Light Green Light. I agreed with him that this working name for the game is pretty lame, but said that we should have this discussion once we all understood the game better.
Then we moved over to a picnic table.
To each person, I passed out a sheet of wax paper, a napkin, a plastic knife, and one noodle. “Now you have everything you need to feed the whole world with that noodle,” I announced. I asked them to watch and copy me. I cut my noodle in half, then took half of that and cut it in half again, then took half of that new piece and cut it in half again, etc etc etc. This probably sounds very simple and straightforward, but it wasn’t. All kinds of questions and problems came up. The most troubling were these:
- What to do with the other half after each noodle cut? (I suggested eating or discarding it, but some kids were cutting both halves in half, ending up in a different mathematical situation.)
- When the noodle pieces got very small, it became apparent to some kids that this was possibly not a linear problem. A linguine noodle is actually a very long skinny rectangular solid. We’re not just splitting line segments in half, but splitting solid shapes in half. Does this affect how you have to cut it to make halves?
We never resolved these issues. Some kids were ready to quit early on; others were captivated. Every time someone slowed or stopped, I replied with either, “I don’t think you have enough yet to feed the world,” or “Do you think you have enough to feed the world?” That always got people cutting again, more doggedly. Sadly, however, our instruments and noodles weren’t fine enough to make smaller cuts at a certain point. “Do you think we could get enough to feed the world if somehow our materials worked better?” I asked. Some said yes, some no. Some kids were getting pretty squirmy. No one was positing conjectures. The results (in terms of mathematical thinking) of the activity, therefore, were inconclusive. I hoped that this activity had at least planted some seeds of mathematical curiosity.
THE SONG THAT NEVER ENDS
I got out my Lambchop puppet and asked J to teach the kids “The Song That Never Ends.” The lyrics go like this:
This is the song that never ends
It just goes on and on my friend
Some people started singing it not knowing what it was,
And they will keep on singing it forever just because . .
(Don’t kill me, but I did tell the kids to search “LambChop Song That Never Ends” on youtube. I was surprised that none had heard this before. It was a big hit when I was teaching little kids 20 years ago. Times change!)
J sang it enough times that the kids were smiling and some singing along. Then I asked the question, “Does this song actually end?” Another surprise for me: some kids offered differing opinions, but no one seemed very interested. (Although S did comment that the song is like a chicken. How? You can’t tell which comes first, the chicken or the egg. He recognized the loop aspect of it.) That song has always interested me mathematically. Where does it start? Does it end? What’s the mathematical equivalent of the word because? Some musicians see infinity as circular – can it be? And many other questions. The kids, though, were still squirmy and, more importantly, not positing conjectures. Once again I hoped I had planted some mental seeds, and then we moved inside to a comfy couch for a story.
I LOVE YOU AS MUCH
J read them the book I Love You As Much, by Laura Krauss Melmed. After the reading, I asked all the kids for their favorite animals in the book. After each one, I asked the follow-up question “How much.” For instance, the text reads “Said the mother goat to her child, I love you as much as the mountain is steep.” So my question was “How steep is the mountain?” These questions really got the kids thinking mathematically. Of all the activities I threw at the metaphorical wall today, this is the one that really stuck.
We talked about the mother whale, who loves her child “as much as the ocean is deep.”
“How deep is the ocean?” I asked. Many answers. (Never ending. You can touch it. You can never touch it. Infinity.) After a lot of discussion, the kids seemed to reach a consensus that the ocean does have a bottom, so it’s not infinity feet deep, but there are parts too deep for humans to reach.
“What is infinity?” I asked. The responses were generally “never-ending” or “you can’t reach it.”
We talked about the bear, who loves her child “as much as the forest has trees.”
“How many trees does the forest have?” (100. 1,000. A trillion. L drew many overlapping circles with his finger on the tabletop. Infinity.) We discussed these possibilities. Each person’s idea stimulated another’s. The students came to consensus that infinity is the answer because there are so many trees that by the time you got near the end of counting them, new ones would have started growing. This discussion felt like collaborative proof-generating.
“If you can never reach the end of counting something, how do you know that they really go on forever?” I asked. The kids didn’t really understand the question. I clarified.
“Do you need to see something to believe in it?” Some said yes, some no.
“Can you make the choice to believe in something you can’t see?” Most said yes.
“Do you believe in infinity?” I asked. This question slowed the discussion as kids thought deeply. I explained that in math, some things you have to take on faith, or to make assumptions about. If you assume something, you’re deciding that it’s true, I said.1
I wasn’t sure whether the kids were following this, especially because of the vocabulary I was using. (It was hard to discuss this topic without using the words assume or assumption, two words I thought the kids wouldn’t really understand.)
“Do you believe in unicorns?” I asked the kids. They debated. Then R spoke up.
“Unicorns are real in Magicland.” This statement seemed to clarify for everyone what I was getting at. Now, since I’m writing this a week later, and didn’t jot down the follow-up discussion in my notes, I’m not sure exactly how M’s statement helped. I do, though, remember that it was a sudden moment of clarity for everyone.
Then we talked about the goose, who loves her child “as much as the endless blue sky.” V commented that his favorite animal in the book is a goose, because geese can both swim and fly. L also identified the goose as his favorite. So I said to him, “Is the sky really endless?”
“Yes,” he said. B to jump up out of his seat in a manner reminiscent of a TV lawyer objecting to some statement the prosecutor makes.
“Wait a minute,” he said to L. (And I loved that he said this to L and not to me. For most of the circle today, kids were talking to me and not so much to each other.) “Do you mean up or sideways?”
“What do you mean? replied L. B said that if you go straight up like a rocket, the sky is endless. But if you go sideways, like flying on an airplane, the sky is not endless. Several others agreed. Someone added that if you’re on an airplane, you can get off at any point, but on a rocket you can’t. B explained that an airplane could just go around and around the earth forever following the same path and that’s different from going into space forever. I was astounded. These kids were debating whether infinity is linear or circular (or both or neither). Exactly what I was thinking about earlier. Did the Song that Never Ends plant seeds that fostered this thinking? Would they have thought of this without that? No matter - deep mathematical thought and debate was happening, and that’s what matters. Too bad we were out of time. Parents were starting to come in to pick kids up as we discussed. To be continued.
1 James Tanton has a nice brief discussion about the need to take some things on faith in mathematics here: http://www.jamestanton.com/wp-content/uploads/2012/03/Curriculum-Newsletter_September-2012.pdf