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Minimums, Maximums and Rates

When math questions don’t always have a single answer...

On the rug, the children sat around a green tape square which enclosed some toy cars, office supplies, a plastic dog, a book, and other sundry items. I explained that in our continuing zoo story, the owner had four children - Ginny, Ron, Fred, and Percy - who needed to be brave to feed some of the animals. These four played games to enhance their bravery, and this set-up on the rug was one of them. I asked the kids to look at the assortment for 10 seconds and to then close their eyes while I removed something; then they opened their eyes and tried to discern what was missing. The game went quickly until there was a bit of debate over whether the book was “green with 3 men on it” or “green with 4 women on it.” The book was Women in Mathematics by Lynn M. Osen. I called on the children who were not calling out answers and found that as usual, just because certain people were not volunteering answers didn’t mean that they weren’t successfully engaged. We played “The Take-Away Game” a few more times before I asked them what this might have to do with math. We recalled the feelings of frustration encountered last week when we attempted to determine a chromatic number. Our group realized quickly that this game helps with noticing, which helps increase attention, which helps increase frustration tolerance, which hopefully increases bravery (and gets that tree pixie and shark and mermaid fed). We talked about how math requires the bravery of Neville Longbottom of Harry Potter fame: the courage to make attempts without certainty of success.

And speaking of feeding these animals, this zoo had some pretty tall ones. The carnivorous vacuum, the unicorn, and the dragon each had a feeding slot that required a height of 7 feet . I reported that the four children were 3, 4, 5, and 6 feet tall respectively, and that they had stepstools of heights 1, 2, and 3 feet. From a lively discussion about which person could use which stool emerged the math concepts of maximum and minimum. When asked which stool 6-foot-tall Percy could use, the participants said “any of them.” “So is there only one answer to that question?” “No!”

I next asked whether math questions always have just one answer, and discovered that not everyone in the group realized exactly what “math” is. The group decided that this stepstool question is identifiable as math because it deals with numbers. We will surely expand this discussion in the future. This would have been a nice place to discuss what numbers are, but alas, we only have 55 minutes each week. The idea that math questions don’t always have a single answer felt like a turning point in our circle, which has been characterized by frequent musings about what “the answer” could be.

All the talk about feeding the animals got pretty exciting, so we refocused our attention with an interlude of bobble-head doll on the floor, then continued the zoo story. I reported that the 4 children also liked to play with toy cars. Their favorite games involved driving the cars at different but constant rates. “Rate” was a new word for everyone in the room, so we talked about it first, and then got out the cars. We took turns driving the cars at different rates. While 2 people drove, the rest of counted (in various ways). We practiced covering our set distance in 5 seconds and in 10 seconds. We talked briefly about how we were measuring rate in way similar to, but not exactly the same as, “the signs on the highway.” Our final exercise was to imagine that the blue car (“the fast one” that could drive in 5 seconds) was glued to the gold car (“the slow one” that could drive the same distance in 10 seconds). I asked what was the minimum amount of time they could cover the distance together. Some said 5, some 10, and some 15. Then we acted it out: 2 children held the cars together while the rest counted. The drivers drove at the maximum speed they could, considering the circumstances. When the cars crossed the finish line at the count of 10, the kids proclaimed, “They did it in 10! Now let’s do it with people!” We ended our circle with a promise that next week, we will.

I’d like to thank Maria Droujkova and Dmitry Sagalovskiy for direction in planning this session.