(September 24, 2015) I decided to do something today that I rarely do:   to demonstrate to the students how to do something.  Last week the kids learned a whole bunch of geometry, but the group did not feel cohesive.  Everyone was pursuing different leads and goals, with simultaneous conversations about different topics.  I wanted to move the group more toward collaborative inquiry, but everyone’s geometric interests were varied.  So I decided to demonstrate how to construct the Flower of Life (FOL).

I started it with chalk on the board.

FIRST BIG QUESTION OF THE DAY

The first step in the construction of the FOL is to draw a circle.  Then you put the point of your compass on the circumference of that circle and draw another circle.  You end up with 2 overlapping circles.  If you connect the centers of both, you get a single line segment that is a radius of both circles.

“Why do they match up?” J wondered. I didn’t understand what she meant, so she spent some time clarifying the question, guided by questions and attempts at paraphrasing by me and other students.

“Never mind, it’s not important,” said J quietly.

“What did you say?” I asked her. I realized that we were at an important crossroads, and wanted her to repeat this loudly.  She was surprised, but complied.  It seems to me that “never mind it’s not important” often really means “I’m having some thoughts that are so deep and complex that I don’t even know how to verbalize them” – in other words, “this is really important.”  I didn’t say that exactly to the class (just wish I had), but encouraged her to proceed with getting her question out because it was probably important.

“Why do they line up with the middle?” she rephrased.  I still didn’t quite understand, and explained that mathematical things are often easier to talk about if they have names.  I asked the students to name each circle.  They were christened Bob and Henry.  I then suggested that we name Bob’s center “B” and Henry’s center “H.”  Then it became much easier to phrase the question in an understandable way:  “Why is B both the center of Bob and on the circumference on Henry, at the same time that H is both the center of Henry and on the circumference of Bob?”  That seemed to be quite a coincidence!  The question finally became “Why is the circumference of Henry on the center of Bob?” and then “How are Bob and Henry related?” 

Now everyone understood the question, even me.  It was important.  Various students offered conjectures about the answer to this.  The prevailing opinion was that the circles are the “same size.”

“How do you measure the size of a circle?” I asked.  More conjectures, finding consensus with “the distance from the center to the edge,” aka the radius.  We then worked together to measure the radii of each circle and conclude that they do in fact match up.

SECOND BIG QUESTION OF THE DAY

The above discussion morphed into a debate about the definition of a point.  Does a point have size?  A shape?  Is a dot the same thing as a point? What happens if you zoom in and in and in to examine tiny points?   I took a poll to see if we had any consensus about the answers to any of these questions, and there was none.  About half the kids adamantly held a position, and others felt very uncertain.  Students also discussed the possible definition of a circle.

In the mean time, I showed the next step in the FOL:  construction of a circle with same radius with center at an intersection point of the other two circles.  (See photos of boardwork.)  From there, I let the students figure out how to complete it on their own.

We ended class with me promising to bring in different definitions of “circle” and “point” for next week.

OTHER STUDENT QUESTIONS DISCUSSED:

• Why did we name the circles Bob and Henry? That doesn’t seem very mathy.  (We discussed the conventions of naming circles after their center points, and ended up renaming these circles, not just their center points, B and H, much to the chagrin of a few students.)
• Why is the lead in pencils called lead if it’s really graphite?
• What do you do if you run out of room on your paper after only drawing 3 circles in the attempt of the FOL?
• How can you expand the FOL and retain overall roundness?
• Is this the only math circle that exists? (This question led to a brief discussion about Eastern European math education.)
• How is what we are doing actually math?
• What is math?
• Does math necessarily involve using numbers? (We discussed how the whole Bob/Henry thinking process didn’t have to use numbers.)

MATH VOCABULARY BANDIED ABOUT

• Radius/radii
• Constant
• Circumference
• Point
• Circle

MATHEMATICAL ART PRODUCED

Please take a look at the photo gallery of student art.  Angie came in and took a lot of photos, of the kids at work and of their work.

Rodi

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