People enjoy math from many different pathways.

“I worked on my Flower of Life at home,” said Z as the students entered the room. As she showed me her work, the others began to draw compass designs long before our circle officially began. I unwrapped a tantalizingly mysterious object, and asked for surmises on its identity. When G guessed a bell, I told them it was a type of bell called a Tibetan Singing Bowl. I demonstrated it, and we brainstormed a list of how it and music in general are connected to math – to circles in particular. J suggested wavelength; other thoughts included drum circles, the circle of fifths, and circle dances.

The children were excited to discuss the definition of a circle that Z had brought in. The definition went on the board, and then underwent intense scrutiny for circular reasoning. None was found, but the terms in the definition needed clarification. Terms that came readily to the group included continuous, defined, diameter, and circumference, but radius was tough. I pushed hard for a definition that did not use the words diameter or line. R finally came through with “line segment” and someone else suggested center, which moved us into a debate about whether the term “point” could be defined. Numerous attempts were made, as I read them Bertrand Russell’s commentary on its evolving definition. More terms were bandied about (dot, sphere, disk, location) as I told more about Euclid and his attempts to define geometric terms. When I read them his definition of a line, A exclaimed, “What?!” and we all concurred. We concluded with a consensus that while a circle is definable, a line and a point are not.

They discovered how to use a compass to construct triangles, and to make 6-pointed stars within the circle. We inscribed stars within stars until they got too small or frustrating. At this point, the students diverged in their approaches. N, who hadn’t even been in class last week, immediately produced a design of numerously inscribed stars with no help. (Once again demonstrating how people enjoy math from many different pathways – numerically, geometrically, logically, etc.) Most of the kids, however, wanted help in finding the points to connect to make the design. I gave hints about using perpendicular and parallel lines within the design to guide them.

Once they had made a few stars within stars, some kept going until the thickness of their pencils prevented further progress. We observed how infinity can go in more than one direction. A experimented with concentric circles. J began coloring his creation, exploring symmetry with color. G had seen in my notes a colored compass design that she liked. I told her that it was a Euclidian construction of a square (compass and straightedge only). She wanted to attempt this on her own with no hints. I went against my better judgment and gave her a hint anyway (“You might need to draw some arcs to make it, and here is what an arc is”). She cheerfully called my bluff by constructing an apparent square without the need for any arcs. Once again I was reminded of the human architectural instinct; we inherently want to discover the structure of things. And when we’re given the freedom to explore, we do. (We then talked about how she could check for sure to see whether it’s definitely a square. I wonder if she will.)

For the most part, I kept my “teachering” instinct in check. I let go of my attachment to any particular agenda by telling math history stories as the kids worked on their constructions. I told of Euclid’s famous reported quote to Ptolemy I (“There is no Royal Road to geometry.”), of the ancient library of Alexandria, and of Galileo. I had planned to talk about the beginning of his persecution, but the students knew something about the end of it and therefore discussed his trial and death. We wondered why it took so long (until 1992!) for the Church to officially pardon him. “Did he have no family trying to clear his name over the centuries?” “Did they forget?” “And by the way, could I try that singing bowl?”

We had been having so much fun that we realized that our time together was officially over 5 minutes ago. We wound down with singing bowl attempts, a discussion of the paradox if its name (and the difference between a paradox and an oxymoron) and an update of our math vocabulary list. We’re all looking forward to next week, when we will attempt to define the term “angle.”