(December 2, 2014) You may recall that last week everyone realized that “snails” were too big to use as pawns in this game – the piece could be on the finish line and not on the finish line at the same time – a contradiction. So this week I brought in “worms.” (The kids claimed they were colored grains of rice.) We played as usual – on each turn the worm could advance no more than halfway. A few interesting developments:
>>>L set new parameters for his worm: “Mine is going to move ¾ of the way on each turn.” Having different kids using different parameters made the game richer. OTOH, not everyone knew what ¾ means. And even with a smaller pawn, L ran into the problem of being finished and not finished simultaneously.
>>>Snails, almost by definition, are slow. Using them enabled the mathematical ideas to unfold for everyone almost simultaneously. I forgot to specify that these worms, like snails, are slow. So we ran into the problem of some kids racing to the finish line, while others remained behind, not able to see the action near the end.
>>>A serendipitous problem was that the whiteboard kept getting erased inadvertently by people’s sleeves as they moved their worms. The kids were still using dry-erase markers to pre-mark their “halfway points” for each turn. These marks kept getting erased. I told everyone to use their “mind’s eye” to make the marks and see the marks. Two of the oldest kids had already abandoned markers for their minds’ eyes. Most, but not everyone, understood what I meant, but the youngest kids protested this challenge – it was just too hard (or maybe not fun).
>>>We’ve been playing different iterations of this game each week. While the students are getting closer and closer to a huge insight about infinity, one or two are losing interest. We might not do it next week. (I’ll write in the report, though, how to further explore it at home for those who want to delve deeper.)
Once again, my assistant J helped to lead this activity, and L took some great photos – thank you! Our first function, adding one, was accessible to everyone. It got challenging, of course, when the input was infinity. What is ∞ + 1? There was some debate, but most agreed that it is still, somehow, infinity. The challenge really ramped up, though, with the next function: doubling the input. No one could understand how with each new input, a different number was added. Surely there must be just one single number that’s being added every time! We tried and tried so many numbers going in – big ones, small ones, crazy ones, etc. – and people just got more confused. Finally, N’s face lit up and body straightened. “You copy the number you’re putting in, and then add it to the number.” We did more examples applying his logic with success. It seemed that everyone at least sorta understood.
“But what’s the number you’re adding each time?” asked another of the older kids. Someone posited that maybe it was 4. We tried adding 4 to various numbers that we had already done, and got different results. Couldn’t be 4. So what was it? N tried to explain his reasoning using different words. J, my assistant, made an attempt. Then I tried, but to no avail. People were antsy, confused, and frustrated. We decided to do what most wise mathematicians do when they become antsy, confused, and frustrated (and mathematicians do feel this way, quite often!): come back to it. I wrote “Come Back To It” on the board, in the hopes of revisiting this problem another day.
We handled this disconcertedness with one of my favorite mindfulness activites – the Hoberman Sphere. “I can make this sphere expand without you actually seeing me moving my body. You’ll see it get bigger, but not see me move.” Then, of course, they concentrated really hard on my hands and claimed to see me move.* Everyone wanted to try this, so on the way out of class at the end, J lined everyone up to do it as they exited.
CAT IN NUMBERLAND
We read another joyful chapter of this book. I hope we finish it next week, our final session – still 2 chapters to go!
* BTW I did this activity last year with another group in a slightly different way – click here and read the 6th paragraph if interested.