Kids were experiencing math collaboratively without my help.

In DC this past weekend, I gave a taste of our Math Circle to some of the 350 children who participated in the 2012 Circles on the Road Workshop. While our Circle informed that workshop, that workshop also informed our Circle, which I led 2 days later.

In DC, Blake Thornton of WU said something like this about a good Math Circle: if everyone is willing to do some mathematical work (computation, etc.), the group will make progress. The problems may be deep, continued Blake, and there should always be a “why.” In Philadelphia, we worked hard, made progress, explored a question deeply, and asked why.

“Is this a fractal?” I asked our group. They were looking at a circle inscribed in a square inscribed in a circle inscribed in a square etc, etc, etc.

As everyone was saying “yes,” K jumped up to the board and said, “Of course it is, since you can continue the pattern forever.” She drew more circles and squares inside the original, and sat down.

“You can also continue the pattern bigger,” added T. I asked whether expanding infinitely in both directions is necessary for something to be a fractal, and the consensus was yes.

Then I drew a square split into quarters and asked The Question again: Is this a fractal? People hesitated. A few said yes, a few said no, and a few said nothing. I drew a single square and asked The Question. All said no. “Why?” The students explained that it was too “vague” – you could repeat the square in various proportions, positions, and directions. “Could you do that with the quartered square?” I asked. Hesitation again. They thought that you could, but they didn’t like it.

“There should be a rule,” announced Z, “that a pattern must be repeated a certain number of times, like 5 times, before it can be definitely called a fractal.”

“But must you see this pattern 5 times to know that it’s a fractal?” They said no, but still didn’t like calling it a fractal without seeing the pattern repeat numerous times. So I quartered one of the quarters. “That’s better,” someone said.

“But wait a minute,” said Z, “That’s not really a fractal since it doesn’t actually go on into infinity the way you drew it.”

“But it could,” countered P.

“But it doesn’t, and fractals are supposed to be infinite.” (Z)

“So, what is it then, just a possibility of a fractal?” (P)

“Well,” R jumped in, “If something has the possibility of being a fractal, then it is a fractal.”

“If,” countered Z, “it’s written as an equation. Then the infinite nature of it is clear. For instance, have you seen the Mandelbrot set? There’s an equation that goes along with that.” She described what this looks like geometrically, and mentioned beautiful computer-generated images of it. At the mention of computers, more kids jumped into the fray. I asked whether computers could create actual fractals that graphically go on into infinity.

“Definitely,” said J, “computers can do anything.”

“But can they graphically display infinite images?” I asked. Silence.

“Yes,” said M after a moment.

“No,” retorted R. “In the time that people ‘discovered’ fractals, there were no computers that could do this. And so, for them to call what they found ‘fractals’, they couldn’t have been looking at infinite ones.” People pondered this.

T’s comment brought us back to pondering aloud: “Computers can do all math, so we don’t even need people for it.” This tangent generated even more dissent.

In DC, I had the pleasure of being on a panel discussion about Elementary Math Circles with Daniela Ganelin and Alan Du, high school students who run Math Circles for younger children. Daniela and Alan shared their experience that “Kids bring stuff in, go off on tangents, and sometimes learn more from the tangents.” That certainly is our experience in Philadelphia. None of this discussion was part of my alleged “plan,” I just wanted to make sure that all our kids think about fractals.

At this point (in Philadelphia), I put a new geometric pattern on the board: a circle inscribed in a square inscribed in two circles inscribed in square inscribed in three circles, etc. As I was drawing, with my back to the group, I asked The Question. At first, there was silence. I heard someone whisper loudly, “Don’t raise your hand!” This was refreshing to hear, as I had never given instructions on how participation should happen in our collaborative explorations. The kids had worked it out for themselves. P pointed out the pattern, and stated that this pattern would continue into infinity. Many voices agreed out that yes, it is a fractal.

“No, it’s not a fractal,” said someone, “It can’t continue infinitely on the inside. How many circles could there be going in that direction?” (Woo hoo! We were at the point where kids were asking more questions than they were answering.)

Several kids returned that it can continue infinitely in the middle with zero circles, fractional circles (quickly dismissed), or negative circles. “Can you draw negative one circles?” I challenged. They thought. Since all agreed that a pattern had to get both infinitely bigger and smaller to be a true fractal, this one was not one of those after all. The discussion moved back to an attempt to precisely define a fractal. I wrote some of their ideas on the board, but we did not end up with a conclusive defining statement. Following the example of Berkley Math Circle leader Laura Givental, I did not give them a definitive answer.

In DC, I worked on Laura’s team. This team led some math circles on logic games for very young children. Laura did not ask leading questions. I was awed when she dismissed our morning group without having reached The Solution to one of our problems. These were kids we would never see again. Even parents were begging her for The Solution. “Let her think about it; she’s on the right track,” said Laura to one parent at the end. “When she figures it out for herself, she’ll be the happiest child in the world.” Laura invited these families back to the afternoon session, when we would repeat the activity with another group. Some did come back and solved the problem with the second group. Another family came back after the second group had finished. Laura told those kids that she would whisper the solution into their mom’s ear, but not theirs. I tell kids in my Math Circles to email me later if some unanswered Math Circle question is driving them crazy. In the meantime, let them ponder.

In Philadelphia, this was our last middle school Math Circle until the fall. I wanted them to leave our session with not only an idea of what a fractal is, but also with the idea that a function can be represented both graphically and algebraically. So we did some Graphing Function Machines. Instead of calling out straight “in” and “out” numbers, I plotted points on a coordinate plane and asked them to name the function. The kids corrected my (and each other’s) axes labels and variable names. We started out with lollipops and prices as variables (for the function y=2x), then “in” and “out” (y=2x+2), and finally just x and y as variables – no words (y=x^2). The kids were confused by the order in which I presented them. G pointed out that the functions I present usually get harder, but this time the second rule was harder to discern than the third. “Good point,” I said. “You are right about how I use an order of difficulty with functions. So there must be a reason that I thought that the third could be trickier than the second.” The kids discussed this, produced conjectures, rejected conjectures, and finally realized that if you connect the points in the third function, you get a curve instead of a line. We then discussed why this happens.

“What’s that called?” asked M.

“One name for it is exponential growth.”

“That’s what I thought,” he said.

Now it was time to switch gears (again!) and watch a Vi Hart video. We switched gears a lot – today and in each of the 6 weeks of this particular Math Circle session. In general, I design the Talking Stick Math Circles to emphasize depth over breadth. The format of this course (weekly Vi Hart videos), however, has required a format emphasizing breadth over depth. This topic came up in DC at a panel on summer programs. The panelists were discussing summer math camps. Many, it seems, give kids a chance to explore interesting math questions very deeply. One panelist, however, made the point that there is value to be had from exposing kids to the rich variety of topics that math has to offer. That comment reassured me about our own Circle here. Steve Maurer was on this panel. He is a professor at Swarthmore College (in the Philadelphia area) and the Academic Director of the MathPath summer camp for 11-14 year olds. Please let me know if you are interested in this for your child; I have information and flyers. Steve had tracked me down via email over the winter. He had found our Circle on the mathcircles.org website, and was so excited that the Philadelphia area finally had a Math Circle. He told me that we were the only major metropolitan area in the US without one. In DC, Steve and I started to make plans for him to visit our Circle in the fall as a guest speaker.

Back in Philadelphia, we viewed Vi Hart’s video “Angle-a-trons.” As soon as it ended, K asked, “Can we see it again?” so we did. Then M declared, in response to something shown in the video, “You can’t fold a polyhedron out of a single piece of paper without cutting it. I know; I’ve tried it!” I challenged the kids to make the attempt, but no one took it up.

Z made another proposition to the group: “Anyone want to do the complementary thing?” No one took her up either. Everyone had chosen personal challenges from the video.

R said, “One thing I didn’t get is how she made the 180 degree angle-a-tron.” The other kids told her that you just use the edge, or fold a piece of paper in half. She tried, but couldn’t do it. It was hard to tell why this was confusing for her. I asked some deeper questions to elicit the nature of her confusion, and it turned out that she was attempting to make a device for drawing a 180 degree arc, not a 180 degree angle. With her help in explaining the difference, I put both on the board so that everyone could enjoy this delightful delineation. As we all know (but may be reluctant to accept), more real learning usually comes from mistakes, confusion, and frustration, than from systematic instruction. In DC, Sage Moore, a high-school math teacher from Oakland, CA, made this precise point: “Experience and struggle is what gets you to really get math, not talking kids into understanding.” Sage was on a panel called “Circles for Teachers.” These, I learned, are inquiry-based explorations of mathematics just for teachers, so that they can experience the way that mathematics is truly learned, and can therefore set up experiential situations for their students.

Back in Philadelphia, the kids continued their geometric challenges while discussing whether Legos are sexist (another conversational gambit fomenting dissent). Then P asked me, “How do you make the 45 degree one?” Before I could say anything, both K and M were rising from their seats to show her. I kept my mouth shut. Kids were experiencing math collaboratively without my help. I was curious what would happen if I left the room.

“I’ll be right back,” I said for the first time ever in a Math Circle. I left the room and closed the door behind me. When I returned, it seemed that no one had noticed my departure. Impressive, huh? Most kids were still conversing about math and working on their own challenges. But not G. She had looked deep in thought for the past 10 minutes. She didn’t know how to do anything shown in the video. She asked me if I could show her something. I said not now. She was interested in a particular doodle that appeared to be square Borromean Rings.

“Oh, I know how to do those,” offered M. He showed her on his paper first, using his angle-a-tron, and then at the board. While he was up there, I told him, and the group, that Vi Hart does have a video that demonstrates construction of Polygon-a-trons. Yes, it can be done without scissors. I suggested that they check it out at home, since we were out of time. I thanked everyone for coming, and look forward to continuing our middle-school explorations in the fall.

For any middle-schooler wanting more Math Circle experience before the fall, I am looking for an intern or two to help with our younger group that starts next week. I was so intrigued with the work that Daniela and Alan are doing, that I dream of that for us some day. They started out as students in the Art of Inquiry Math Circles, then helpers, then leaders. (I hope I have that right.) The intern responsibilities here would be photography, note taking, providing an extra set of hands, and eventually planning activities and presenting a problem. The job has 4 requirements: Math Circle experience as a student; affection for younger children; enjoyment of mathematics; and the restraint required to keep your conjectures to yourself. A benefit is exposure to math you might never see otherwise. I can provide some training. Email me if your child is interested.

Thanks to R, who has been doing some of these intern duties and also helping me with this report/blog all year.

Also, thanks to Dave Auckley of the MSRI for inviting me to the workshop, for financially supporting the Talking Stick Math Circle with an MSRI grant, and for encouraging me to include the workshop in my writings to you.

— Rodi