Bouncing between Rationalism and Empiricism
(February 4, 2014) I felt a smidge of trepidation coming to Math Circle today: Gina was not able to make it with the art/henna component, and I know that at least a few of the kids signed up more for the art than for the math. Would I be able to hold their interest in the math of sacred geometry for 90 minutes?
Fortunately, Gina provided some compelling mathematical art. We started out looking at the works of Heather Hansen (charcoal, using her body as the compass) and Andres Amador (sand on beaches, using, well, we didn’t know what). We spend some time examining both artists’ works, trying to figure out just how they created with such symmetry and precision. Finally, after many inventive conjectures, we had to resort to Amador’s FAQ page to discover that rope is his main tool.
It just so happened that I had some jump ropes and dog leashes with me. “If he can do that in sand, I bet we could do it in snow,” said I. Out we went with our tools. Kids worked cooperatively to draw circles – some isolated circles, some overlapping, and some concentric. But people got very cold very quickly, so we had to go in before any compass art emerged.
Once inside, we put empiricism (experimentation) to rest and brought out some rationalism (abstract reasoning). I drew some of our outdoor shapes on the board in a discussion of the parts of a circle. We conversed about the ethereal point, the newly-dubbed rope (formerly known as radius), and the motion-produced circumference. We discussed (1) whether all circles are the same shape (similar), and (2) whether factors other than the “rope” might affect the snow-circles size:
(1) The kids decided that there is only one way to draw both circles and squares, as they only require one measurement. Triangles and rectangles, however, are not all similar.
(2) The kids realized something I never thought of (I just LOVE when that happens!). Outdoors, the height of the people holding the rope and the speed of the person moving might affect the size of the circle. (For instance, a fast runner might stretch the rope.)
I continued to read from Schneider’s book, moving from the monad/point to the dyad/circle. With metal compasses/pencils, the students drew circles, symbolically creating a world. We discussed the illusion of area:1 how the circumference of a circle reminds us of the rotary motion and cycling of all matter. I demonstrated this with Doodle Tops, which are ink-containing tops. The students saw how the words on the top of the top blurred into an apparently solid ring. They knew something about chemistry and provided examples of the actual amount of space matter takes up when you don’t include the area between electrons.
At this point, the students were armed with paper, pencils, compasses, and doodle tops, and were creating their own mathematical art. After a bit, we dropped pebbles into a pan of water. Those of you who know me may think I was using this exercise as a contemplative practice to sharpen concentration, but actually the kids were already engaged in a contemplative activity with their tools. The purpose of the pebbles was to study the patterns naturally emerging in the ripples. Soon, the kids wanted to drop in two at a time, so we looked at the shape formed from the overlapping ripples. I asked them to construct this shape - called the vesica piscis (“bladder of the fish”) in the Western tradition and the mandorla (“almond”) in India - with their compasses. The students felt most comfortable calling it the almond, so thus it is dubbed. We then discussed the vast sacred and symbolic use of the almond.2
At some point, I asked “If you were a pirate and could take all of the square-shaped gold pieces that you want, as long as you could fit them on a plate, what shape of plate would you choose?” I gave the kids graph paper (representing the gold pieces) and yarn (representing the perimeter of the plate) to experiment with the relationship between area and perimeter.3 Most of the kids immediately got to work making shapes and counting the enclosed squares, but a few would not even try. It was unthinkable to them that the perimeter and area of a shape were not always directly proportional, no matter the shape. “It’s impossible to change the answer by changing the shape,” said one student. “It doesn’t make any sense,” said another. How fun - some people were attacking this question with reasoning, while others with experimentation. That classic struggle between empiricists and rationalists4 was playing out right here. The empiricists all came up with numerical evidence that circles maximize area. One rationalist accepted the evidence and changed her position, another experimented for herself before changing her position, and one held tight to her position, even when I informed the kids that mathematicians accept that circles do maximize area. All I can say is three mathematical cheers for her (the dogged rationalist) for demanding proof over pattern. I will try to find an age-appropriate proof of this.
At another point, I gave the kids a quick bathroom break, and no one took advantage of it, so I was relieved that things seemed to be going okay without Gina. After the break, kids worked semi-independently on their own things. G, M, M, and T showed each other how to create patterns on paper with the various tools, while X, J, and E explored whether the Doodle Tops work on human hands, in the hopes of using them with henna.
With a few minutes left, I showed the kids how to make a mőbius strip, which most had never done. I asked whether this strip would represent the Monad/circle/one, or the Dyad/line/two. Opinions varied. Schneider describes it as a “three-dimensional symbol of the Dyad, or one appearing as two.”5 I brought this up so that we could discuss whether you have to agree with conventional interpretations of symbols. Fortunately, everyone said “no.”
I noticed a definite shift in the room today, from the session being more teacher-led to more student-led. The kids are starting to ask more questions, put forth more conjectures, and initiate explorations on their own. Inquiry is happening. (Woo hoo!) Meanwhile, I am metaphorically sitting on my hands after duct-taping my mouth shut, in hopes of not revealing the definition of a circle. I have become very attached to the idea of them discovering its definition, so will just keep giving them the tools that might foster this. So that deep meaningful discovery can occur, I have to swallow the urge to go Socratic and ask leading questions. (In rare instances we just have to do some things Socratically in math circle, but the definition of a circle is just too delicious to diminish its discovery in any way.*) **
Anyway, click here to see photos of the activities from today, as well as some of the work people created.
1 Michael Schneider, A Beginner’s Guide to Constructing the Universe, p 13-16
2 ibid, p 31-35
3 I took Schneider’s idea of the graph paper and yarn (p17), but reworded it to include pirates since IMHO any math discussion is enhanced by the addition of pirates.
4 If you and your kids are looking to bring some healthy debate into your household, try googling “empiricists vs rationalists.”
5 ibid, p24
*I’d love for any of you math circle leaders who are reading this to throw some ideas my way. I really thought the dog-leash in the snow activity and subsequent discussion would be enough, but the kids are still thinking that the circle definition somehow lies only on the circumference.
** And as long as we’re talking about the challenge of facilitating inquiry-based learning, I’d like to mention that when I read aloud from Schneider, I am only reading the historical and cultural passages. I am rephrasing his math content as questions for the kids to contemplate.