Geometry is not just the basis of language, but is a language in itself.

“Oh no, now she’s going to ask us the definition of the word flat,” said Z, after the whole group had agreed that a plane can be defined as a “never-ending flat surface without thickness.” I asked, “But aren’t you asking yourselves what flat means?” and the kids had to agree that yes, they were. Questioning and doubting is becoming automatic – wonderful!

In order to define “flat,” we revisited the Bear Problem from last week. While I try not to use a leading question in Math Circle, sometimes it’s necessary. Today it was. We talked about how a triangle has 180 degrees, but the triangle walked in the Bear Problem had 270 degrees. “Can you use that fact to define flat?” I asked, and they did. “But the earth doesn’t really have enough curvature for a 2-mile walk to put you back where you started. I tried it when we walked our dog,” objected M. This comment gave rise to a discussion of the arbitrariness of longitude, and a revised version of the original problem.

We then discussed Euclidian (ruler and straightedge) constructions. I passed out a list of figures that can be constructed with these tools, and asked the kids to try some. The challenge of copying a line segment was met with quick success by all, but then hands slowed and faces frowned at the rest. I gave them time to make some attempts while I recounted some anecdotes involving Euclid, Euclid’s Elements, and Abraham Lincoln. The kids had fun coming up with expressions of confusion in response to some Euclidian definitions. (“Come again?” offered J.)

I hinted that some of the constructions can be created with the Flower of Life. This hint was enough to get most of the kids going, but a few stared at their papers and then worked on their own compass designs. M was determined to use his tools to find the center of a circle since “they have a tool at Home Depot that can do that.” G helped me counter discouragement with the reminder that she had constructed a square two weeks ago. Some kids asked me to check whether their various attempts worked, while others did not want to see or hear what their colleagues were doing. To balance the pleasure of the individual challenge with our goal of collegiality, we constructed one figure collaboratively.

So far, no one had been able to create a perpendicular bisector of a line segment, and some demanded another hint. I asked all who wanted a hint to come to the chalkboard on the other side of the room to consult privately. Five followed me, while two stayed in their seats. I showed them the first step. “She accidentally showed us how to do it!” R called unhappily to the two across the room. It turns out that I had shown them more than the first step. That teachering instinct had come up involuntarily again (oops!). To return to them the joy of discovering for themselves that I had inadvertently stolen, I gave them the new challenge of completing the same task in fewer steps.

As work continued, I told of the view that geometry is not just the basis of language, but is a language in itself. I read aloud some descriptions of geometric symbolism in the field of Sacred Geometry. Then we talked about extra-terrestrials. How could we send a message into space if they don’t know anything about us? I showed them a modest version of the message sent into space on Pioneer 10. The kids were able to figure out the meaning of the planetary order, but needed explanations for the rest of the message. They debated the existence of extra-terrestrials for a moment before returning to their constructions.

As Math Circle ended, some kids were a bit frustrated that they hadn’t successfully constructed what Euclid had. I showed those children a web address at the bottom of their handouts and advised them to look at that site to get help with one or two only, and then to attempt some independently. (This is not required in any way, but those who are curious will want to try.) Anyway, just in making these attempts they had familiarized themselves with a whole passel of new geometric concepts: parallel, tangent, etc. To loosely quote Math Circle founders Bob and Ellen Kaplan, “If you want them to learn A, ask them to do B, which requires an understanding of A in order to complete the task.”

As people left the room, A and G ran up to Z, who had successfully constructed a perpendicular bisector in a very concise manner. A, who had been very quiet during the circle, demanded in an admiring tone, “Show me how to do it!”

“Don’t show me,” ordered R. Z led A and G into the corner to show them privately.

“I can hear you!” protested R, who exited, notebook in hand, still determined to figure this out for herself.

– Rodi