Transferring and generalizing math concepts are important skills.
“Something is in the air today,” said Talking Stick co-director Angie. The kids came in brimming with energy, and most came early. As we waited for the last child to arrive (still early), four of the kids were at the table writing newspaper articles. Soon I asked them to put their papers on the windowsill. They complied a bit reluctantly, and I pulled out a small musical instrument in the shape of a triangle. I asked, “Who knows the name of this instrument?” “A wind chime!” guessed J. No one knew for sure so I gave the hint that its name is a shape name. “Triangle” called the group in unison. I instructed, “When I strike this and you think the sound has ended, it will not have and you’ll be wrong. Listen harder. Then put your head down when it’s really done.” I struck it, heads went partially down, back up, and then down again. M asked whether eyes should be open or closed, and I said “whatever you think – you could even try both ways.” O said “You mean like this?” and closed one eye. N said “I can’t do that,” so I suggested covering an eye with a hand like a pirate’s eye patch. We focused our attention with three triangle chimes before I asked them to recall what was happening in our zoo story last week.
"They were feeding the animals,” replied V. What else? “We were racing cars,” said someone, to which M added, “and we’re going to do it with people today!” I had the kids line up behind the green tape line and repeat several of last week’s exercises. “We’re being spectators,” said P and O as they returned to the table, newspapers in hand. J sat on the windowsill as all kids tried to predict what speed cars glued together could do. Most had forgotten the conclusion from last week and simply added the rates, but once we did dramatized it with cars the kids understood that the slower car determines the rate when the cars are glued together. Then we talked about attached people. A number of the kids did not transfer this concept of “attached” rates from cars to people and once again added the rates. So M stood in between P and V and walked at her maximum but slow rate, while P and V held hands with her and walked at their faster maximum rate. M asked why she had to be the slow one and I said “because you are strong.” The need for strength became apparent when P and V pulled forward and M pulled back and they all finished at the slower rate. Then the whole group was able to generalize the concept to “things can do less than their maximums but not more than their maximums” and “slow things hold back fast things.” Everyone agreed. Transferring and generalizing math concepts are important skills, so I was curious whether they’d be able to now apply the same concepts to our zoo family.
"Something bad happened at the zoo,” I told the kids. “Someone freed the unicorn, not knowing that its magic required 4 people holding hands around it to be able to eat. Now it was on the other side of the river, starving, with only 17 minutes left to live.” Immediately eyes lit up as those who attended our demonstration class remembered this story: “There is a unicorn dying at the end of a bridge. He has 17 minutes left to live, unless 4 people can join hands around him and recite a magical spell. There are 4 people on the other side of the bridge, but it is very dark, they have one flashlight, and the bridge can only hold 2 people at a time. Ginny can cross in 1 minute, Ron in 2, Fred in 5, and Percy in 10. Can they save the unicorn?”
"Those numbers add up to 18, so it can’t be done,” announced V. “Maybe it can,” someone suggested, since the people can cross two at a time. I pressed the kids on how we could know for sure, and they came up with a plan to act it out as we did the rates problem. It took a few minutes to come to something close to agreement on who would play each part. A few kids wanted to return to their newspaper writing at this point, but became spectators instead. The rest of the kids suggested that the two slowest walkers cross together first. The kids were able to accurately predict and demonstrate the rate of crossing (math success!) but were sorely disappointed that the unicorn would die with this method. The spectators started rolling around on the floor now, and J announced, “And the Nimbus 2000* whisked in and carried the unicorn to safety!”
With a few kids rolling around on the floor, another flying on an imaginary broomstick, and the rest still wanting to further test possible solutions to the unicorn story, I quickly drew a broom on the board and said “Actually, the Nimbus 2000 is also a function machine.” I told them how it worked and what sound it made and immediately everyone was sitting around the table, totally engaged.
The 60 seconds of disorder in out classroom reveals something about how people enjoy math: different people enjoy it in different ways. We all have what the Kaplan’s refer to as “the architectural instinct” – a natural desire to pursue revelations about structure. But some of us seek structure numerically (i.e. the stepstool problem of last week), others geometrically (i.e. map coloring), others logically (i.e. finding loopholes in a scenario to allow more possibilities), and so on. In our group, different styles are emerging, and it will be fun to see if and how they change over time.
And speaking of loopholes, let’s get back to our Nimbus 2000 function machine. At first, the machine did a function that the kids said did “minus one.” The kids suggested adding feet to the machine and increasing the sound, and the result was a machine that the kids said “figured out what half of the number is and then minuses that from the number.” (In my mind I was dividing by 2, but there are multiple approaches to every math problem, which is one reason that math is an art.) A few of the kids were calling out the functions before I was able to get multiple inputs and outputs on the board, so I decided to try to really stump some of them when they asked me to add ears to the machine. In went 10, out came 6. After a conjecture about “minus 4,” in went 100, out came 51. After another conjecture was disproved, in went 4 and out came 3, then in went 20 and out came 11. P posited that the rule was changing for each input, but I assured them that a function always does the same thing no matter what. The room was quiet; almost everyone had a smile yet a furrowed brow. I gave the hint that the machine with ears did the same thing as the machine with feet but with one extra step, and quickly several students called out the function. One said, “I don’t get it,” so I demonstrated and encouraged that excellent strategy of counting on fingers (and toes if needed). Once it appeared that everyone understood the function, M said, “But you changed the rule. You can’t change the rule.” I explained that when a technological advance comes along (such as feet or ears on a function machine or a camera on a cell phone) a machine can perform a new task, or even a task involving multiple steps.
At that point our time was up so I passed out snacks and told them they were done. No one left. They drew, wrote, ate, and continued to discuss function machines. O talked about a “function box” he has seen at school, and we figured out how that differed from the function machine. One parent asked what progress we had made on the unicorn problem, and I told him (so that the kids could hear this recap) that we now are convinced that a slower person’s rate determines a faster person’s rate when walking together, and we know that the unicorn dies when the 2 slowest people go first. One child said that next time we should try the fastest people first. J suggested doing it with “imaginary people.” I asked the group if this problem would be easier to solve in their heads instead of by acting out. The divergent replies showed that once again, there is more than one way to enjoy math. Finally, I ended by collecting supplies and promising to return to feeding the animals at the zoo next week.
*Harry Potter’s broom
(Photography and editing credits this week go to Rachel Steinig)