(February 25, 2014) Kids arrived to see some yarn and 3 different-sized cans on the table: salt, coffee, and espresso. I asked the kids to point out the key parts of a circle on the cans – the circumference, the area, the (invisible) center, and the rope/radius,1 and the diameter. We hadn’t worked with diameter before, so we talked about how you can deduce its mathematical name from the roots in its name. Unfortunately, false etymology really kicked me in the butt, as I surmised that “di” came from the Greek prefix di- for two (two radii, get it?), but I found out later that it derives from the Latin and Greek dia- for across.
Then I asked, “How many diameters does it take to make a circumference?”
First we collected conjectures: 100? 2? 3? 6? 50?
Then we discussed whether we should try to answer this empirically (with experimental data) or rationally (to reason it out). The group was split about 50/50. Those who wanted to reason it out were able to eliminate 100 and 50 from the conjectures list, but didn’t have a clue how to proceed from there. I pointed out the yarn, which was pre-cut to match the diameters and circumferences of the cans.
“Listing conjectures is an inexact method,” declared the kids. “Figuring it out by measuring is exact.”
(Before I go on, an important aside here is the layout of our room. We have a very long narrow room with little space to move about. Gina tends to situate herself and her stuff at one end of the long table while I do so at the other. I mention this because it seems that more art gets done on Gina’s end of the table and more math gets done on my end, and we all sit in the same place each week. I always wonder whether things would be different with a bigger room and a round or square table. For now, though, I’m going to really dumb things down to make the writing easier and refer to an “art side” and a “math side” of the classroom.)
The kids on the math side of the room immediately picked up the yarn, began testing, and announcing results.2 The kids on the art side watched quietly while sketching patterns to be used with henna. Kids on both sides were adamant that we had to state assumptions about a properly-drawn circle.
The students’ declaration of measuring being an “exact” method went out the window. Estimates of the ratio of circumference to diameter ranged from 3 ¼ to 4 ½. We ran into big problems because the yarn stretched. Matching degrees of tautness to compare measurements was quite frustrating. Another challenge was estimating. Several students used the tape-diameters-along-the-can’s-circumference method to generate date. It looked like three diameters weren’t quite enough. So did we need more than three or fewer than three?
“We need a little less than three,” one student was convinced. Others thought more. We spent some time clarifying this. Finally everyone (reluctantly) agreed that is looked like we need a little more than three diameters to make a circumference.
“Let’s assume, then, that we do this same yarn test on the circle in the middle of a basketball court. Would we still need a little more than three diameters to make the circumference?”
“No,” said several students at once.
“Why not?” I asked.
“Because the people who made the court might not have used a compass. The circle might not be drawn right.” The kids wanted to explicitly state the assumption that the circle was, in fact, a perfect circle. (We still haven’t deduced the definition of a circle, so are not at the point where a circle is a circle period.) Once we made the circle assumption, though, everyone agreed that the number we’d be looking for is 3+.3
INTRODUCING π (PI)
“Have any of you ever heard of a number that’s always a little more than three (a constant), and means something in mathematics?”
No one had a clue.
“This number, a little more than 3, is pretty famous in mathematics,” I prompted.
Still people were quiet until a younger student who was sitting in said, “By the way, what is pi?”
I affirmed that yes, we are talking about pi, and how did you know that anyway?
“I was with you when Amanda told you how to do the yarn activity.”
The group, especially those at the math end of the table, got into a vigorous discussion about pi (vigorous to the point of the students taking over the white board at several points):
- the exact value of pi,
- the observation of “Pi Day,”
- possible methods for getting its value more exact than “a little more than 3,”
- how many digits people knew (and an anecdote from the Simpsons about this),
- how to memorize more digits, and, most importantly and disturbingly,
- the irrationality of pi. (Irrational numbers cannot be expressed as fractions of whole numbers.)
One student, E, immediately saw a troubling implication of this: “If the circumference is a whole number, you can never exactly calculate the diameter. And if the diameter if a whole number, you can never exactly calculate the circumference.” E was astounded by this realization, troubled even, and wasn’t sure whether to accept it.
I talked briefly about the Pythagoreans and their similar (supposed) reaction to the discovery of irrational numbers. I mentioned that some of the ancients (and possibly some people today) consider pi and other irrationals sacred for the reason that E expressed above. Those numbers represent the infinite, the unknowable.
E remained incredulous, as did (I suspect) the other people sitting near him. “If pi goes on forever, how can you get a definite measure for the circumference?!”
CIRCUMFERENCE = π x DIAMETER
The calculation would yield another irrational, students realized.
“Wait a minute,” exclaimed one of the students at the art end of the room, “That’s what pi really means?” Her eyes lit up and she rose up in her chair. Everyone became caught up in the emotional moment. Some were emotional about the discovery of what pi means (the ration of C to D), and others about its irrationality.
In this course, it often seems that the kids on the math side doing the math step-by-step work (calculating, testing conjectures, drawing figures, asking questions), but the kids on the art end often draw conclusions and announce the big-picture take-away messages. We can’t assume that people who are engaged in art (or, as some might say condescendingly, “doodling”) during mathematical discussions are not engaged in mathematical thinking. Just ask Vi Hart.
Discussion continued. “Is there any way that you could get a whole number for both the circumference and diameter?” I asked.
“I don’t know,” said R with wonder. We played around with numbers a bit (see photos for specific details), and the students gradually started to accept that the answer is probably no. E introduced a counterexample – the square. You can draw a “diameter” from the midpoint of one side to the midpoint of the opposite side to make a “radius.” You can assign a whole number to that, and calculate a whole number “circumference” (perimeter) from that. It kind of blows the mind that this seeming obvious approach doesn’t work for the circle. One thing I didn’t ask here (and probably should have) is whether a square really has a radius, and if so, that’s the only way to make a radius in a square.4
“How have people figured out pi to so many digits?” asked R. I promised to try to address that question next time.
MORE ON PI…
The pi conversation generated so much excitement that the kids – from both sides of the room – were coming up to the board and taking the marker out of my hand. When you see the boardwork in the photo gallery, keep in mind that the following (and more) were written by the students:
- the pie,
- the first bunch of digits in pi,
- the infinity symbols, and
- the expression of pi as a Greek letter.
Students also came up to the board (on their own) to point out things in the circle diagrams.
We briefly discussed Pi Day (3/14), which was new and intriguing to this group. FYI, Denise Gaskins’ blog Let’s Play Math has a quite thorough assortment of ideas on how you can observe Pi Day: Pi Day Roundup.
…AND SOME ART
During all of the above, students were using henna. Several students invited their parents into the session, and those students were applying it to their parents. Last week, Gina had mentioned that she sometimes gets commissioned as a sculptor to create belly casts of pregnant women. Today she brought in one of these, at the students’ request. The students were fascinated. We briefly discussed the topology of the hemisphere5, and then some of the students decorated the cast with henna. Gina offered to donate the cast to Talking Stick’s upcoming 5K Walk/Silent Auction fundraiser. The kids proposed using it as the Math Circle gift basket. We’ll defer to Christopher, the event’s organizer, on that.
A PEDAGOGICAL REFLECTION
Initially it was hard to know what to do with the students’ request for the belly cast. The name of this course is “The Nexus of Sacred Geometry and Henna” for a reason. Examining a belly cast didn’t obviously fit in with the topic.
But this is an inquiry-based course.6 And the students were curious. And there are some styles of inquiry that give students 100% freedom to follow their curiosity.
Wait a minute, though… By signing up for this course, students declared their curiosity in math and henna. Everyone knew the topic going in.
What to do?
Gina astutely observed that a big challenge in inquiry-based education is “staying grounded as a teacher.” Do we follow the kids’ interests, or stick with our agendas? I think it’s the facilitator’s responsibility to integrate both if/when possible. Therefore we decided on the above activity with the cast.
UPCOMING COURSE FOR YOUNG CHILDREN
I’ll take the time here to mention that we are offering a Math Circle for students who are about 5. This course begins 3/18 (postponed beginning due to snowstorms). Another course for kids 6-7ish will begin on 4/22. Visit the webpage for more details.
1 rope is a reference to our prior formation of circles in the snow with ropes.
2 thanks to Amanda Sereveny for the idea of the cans-and-yarn approach to pi
3 extension of this idea do with your kids (James Tanton’s idea): make the circle bigger and bigger – as big as, say, Texas. Will that ratio still be 3+? (Let me know if you want a hint.)
4 If I don’t ask this next week, in our final session, do ask your kids at home. One they work with the Pythagorean theorem, the answer to this question is another lead-in to irrationals. Moreover, the answer might negate the claim that squares behave differently from circles in terms of whole number measurements. (Email me if you want more details.)
5 In the original planning of this course, I had hoped to investigate topology more, as there is a rich opportunity to delve into it via henna-cone-folding. If you are exploring this on your own and have the time to integrate topology with henna, let me know how it goes!
6 The Talking Stick Math Circle most definitely falls under the umbrella of inquiry-based education, but not all math circles do. Math Circles exist to fulfill different goals. Check out the Wikipedia article on Math Circle to see how some other circles run. I think it’s a very well-written article except for one thing: I’d ever-so-slightly change the sentence “One feature all math circles have in common is that they are composed of students who enjoy learning mathematics” to also include students who want to enjoy learning mathematics.