The kind of real mathematical thinking that a math circle encourages.
Photos of a Julia Set crop circle and a Cissbury Ring crop circle whetted our appetites for circles this week. Everyone had many questions about how the crop circles got there, particularly how they got to be “so perfect.” Then the kids wanted to continue the process of defining terms that we started a few weeks ago. Each week, the process becomes even more collaborative. With each attempted definition of “angle,” the kids automatically questioned the definition of terms within the definition, and refined their statements. R added in some math vocabulary. They ended up somewhat satisfied with the definition “2 line segments coming out from a common vertex.” Since this definition said nothing about measurements of angle, and the students intuitively felt that the definition must somehow talk about “distance,” they were not totally satisfied with it. (This is exactly the kind of real mathematical thinking that a math circle encourages. I let them know that they were thinking like mathematicians with all their questions and doubts.)
I asked them to consider defining an angle, in a less formal way, as a measurement of turning. I used 2 pencils to show a full circle of movement and asked “how many degrees in a full turn?” “360” said several voices. “How about half a turn?” “180 degrees.” “A quarter of a turn?” “90 degrees.” “And what is a degree and why do we use 360?” Now the voices were silent.
I started to tell them about how this number originated with the Babylonians. “I love that name Babylonians,” said M, and we said it a few times. He knew that we use 360 because of something about the earth’s rotations. I told them that it takes the earth approximately 365 days to travel around the sun. “Approximately?” asked Z. So then we talked about how leap years work before getting back to the ease of using 360 instead of 365.25 to measure a circle. I mentioned how the Babylonians looked at math from a practical point of view, as opposed to the later Greeks, who applied the rigors of reasoning to it. I asked if they saw the practical problem of using 360 for a circle if we were to travel to some other planet. R explained how another planet would have a different number of days for its orbit. I told them how mathematicians have tried to avoid that earth-centric approach by measuring angles in a unit other than the degree, such as the radian (based on a circle of radius 1), or the gradian (a measurement where ¼ turn is called 100 gradians). A seemed to really appreciate the idea that a number of things in math have been created, not discovered, by people. I asked if they would like to design their own unit of measure for angles. Most of the kids, however, seemed to really like the gradian – except for M, who would like ¼ turn to be called 1,000 whatevers so that even more accurate measurement would be possible.
“How many degrees are in a triangle?” “360,” said at least 3 assured voices, while the others remained silent. Then I asked each child to draw a large triangle on a piece of paper, and follow its angles with a pencil. Each noticed that when the pencil returned to its starting point, it had undergone half a turn. “So how many degrees are in a triangle?” “360! Er, uh, 180!” They thought about what they had just done, and then all agreed that triangles actually have 180 degrees. Z asked why would we go to the trouble of proving this when we could simply apply a memorized algebraic formula to much more quickly figure out angle measures. The group then had a very brief conversation about the value of knowing why something works in addition to how to do it.
I then gave them the classic Bear Problem.
A woman goes for a walk. She walks 3 miles south, then 3 miles west, then 3 miles north, and ends up where she starts. At this point she sees a bear. What color is the bear?
“Huh?” I assured them that there is a mathematical concept behind this, even though it sounds like simply a riddle or joke. There were several insightful and creative attempts at scientific answers involving the angle of the sun and black shadows, and the 3 directions representing 3 colors being mixed. But alas, those were not the answer. I told them that I would give them a solution strategy by showing them James Tanton’s Mathematician’s Salute.
I don’t want to tell you much about the Mathematician’s Salute because some kids are still mastering it and all will probably want to teach it to you, but I will say that it demonstrates the “working backwards” strategy in a math problem: start at the end of the problem and you may discover another pathway to the answer.
This was enough of a hint for G to realize how the end of the Bear Problem could be used to get to the answer. With help from her math circle colleagues, they did figure out together the solution.
Without giving away the solution to the Bear Problem, I’d like to say that this naturally led us back to our definitions. The Bear Problem made apparent a flaw in an attempted definition from last week, and I congratulated the kids for once again thinking like mathematicians. At one point in our mathematical conversations, I held up a flat clock as an approximation of a sphere. Several students started looking for a more spherical object to use, but I stopped them, explaining that in our goal to think mathematically, we’d like to move more toward abstraction.
At this point we were out of time. As we cleaned up, J showed me the numerous compass designs he had been creating at home all week to use as targets. “I really want a BB gun, and if I get one, I’d shoot it.” So, while we strive toward the abstract in mathematics, what fun it can be to put it towards a practical application. Not only can we apply the abstract to the practical, we can study the practical to draw out the abstract. Ever since N visited our math circle last week, his mom reported that he has been seeing circles, diameters, and radii in everything. He told her, “My mind will never be the same again.”
(NOTE: Credit is due to mathematician and teacher James Tanton, from whose works I’ve drawn on immensely. I used his approach to almost every concept we discussed this week.)