“…Well, the 14 here is the diameter. Remember, diameter? … Diameter is all the way across the middle of the circle. In our formula, there’s an r. What does r stand for? … What’s an r-word we’ve been talking about with circles? … Okay, well, R stands for radius. The diameter goes all the way across the middle of the circle, but the radius only goes to the center.”
We’ve been finding the volume and surface area of cylinders, spheres, and cones for more than a week now. The rest of my class is completely off task (mostly because the work their teacher gave them is BORING). My lowest student, an easily offended JROTC cadet with huge hoop earrings and lots of lip gloss, looks at me with raised eyebrows and waits for me to hurriedly tell her the answer and rush off to put out fires.
… but some days I just want to teach someone something. So I shut out the craziness going on behind me and dive in.
Even now, even in December, it’s a deeper dive than I’m expecting.
Me: “So if we have the diameter, how do we get the radius?”
Cadet: “… subtract something?”
Me: If the diameter goes all the way across the circle, and the radius only goes halfway, how could we find what the radius is?”
Me: “How do I find half of something?”
Cadet: “… add? Multiply?”
Me: “We want to split 14 into two equal pieces.”
Me: “Okay. If I had 2 cookies, and I wanted to give you half of my cookies, how many would I give you?”
Cadet: “Um, a half.”
Me: “I would give you half a cookie? Hmm, I don’t know about that. … Let’s say I had 4 cookies. I want to split them up evenly with you. We’re dividing them between 2 people. How many cookies would I give you?”
Me: “Really? If I gave you three cookies, how many would I have left?”
Cadet: (checks on her fingers) “… one.” (yesss!)
Me: Yes! I would only have one left. Would that be fair? Three and one?”
Cadet: “No. Two and two!”
Me: “Exactly! We have to divide them into 2 equal parts. Perfect. Now, what if I had … 10 cookies? How many would I give you then?”
Cadet: “um… seven.”
Me: “You sure?”
Cadet: “No, five and five!”
Me: “Yes! I’d have five and you’d have five. We split it in half: half of 10 is 5. Now. Back to our problem. What if I had 14 cookies? How many would I give you?”
Me: “Thirteen?!? You get thirteen and I get one?”
Cadet: “No! No. Um… hold on.”
She digs in her backpack and produces a notebook. She flips past various cartoon figure doodles and what looks like a warm-up from Spanish class, finds a fresh page, and proceeds to draw 14 circles. I’m impressed.
First, she tries to count the 14 circles by pointing with alternating fingers of her two hands. But she gets to five and five and gets stuck.
Next, she draws a two-column table with ‘W’ (that’s me) and ‘A’ (that’s her). She tallies on each side (“one, one, two, two…”) but doesn’t stop at 7 and 7… she goes until she reaches 10 and 10, then says, “There. 10 and 10 is 14!”
Me: “10 and 10 is 14? Are you sure?”
Me: “How about you count them and see. Are there 14 marks there?”
She counts the 20 tallies (one by one, not by fives), but does so hurriedly and comes up with 15. “I have too many.”
Me: “Yep, you do. But I see more than 15.”
She counts again, correctly this time. I remind her that she needs 14 tallies, not 20—she erases one set of five, then counts again. “Ohhhh…” She says, and erases one more tally mark. Now there are 10 on the ‘W’ side and 4 on the ‘A’ side. “10 and 4 is 14!”
Me: “Yes! 10 and 4 is 14. But is that two equal parts? Would that be fair?”
I watch as she thinks for a second, then returns to her 14 circles. Now, she marks one with a ‘W’, the next with an ‘A’, the next with a ‘W’ again, and so on. ‘Yess,’ I think. ‘This is it.’
Then she counts them up—but she counts all of them. “14.”
Me: “… 14 what?”
Cadet: “Oh. Hold on.” She counts just the ‘W’s. “Seven.” And then, “ … Oh yeah, because 7 and 7 is 14!”
Me: “YES! Yes!! 7 and 7 is 14. So half of 14 is…?”
Me: “So if our diameter is 14, the radius is… ?”
Cadet: “Seven! See? If you talk about food, I get it!”
There are so many things about this exchange that I want to pick apart and understand. I’ve worked with her a lot before, leading her to figure out the answer, but I’ve never before let her pick her own method like this. What made her keep coming up with new things to try, especially when she’s used to being left behind and not understanding? Why is division so much harder than multiplication? Why did she say every operation but division, even when I emphasized that we were dividing cookies between 2 people? Why was half of 10 so easy and half of 14 so much harder, if she knew 7 + 7 = 14? What made her draw out 14 circles—did she just know I was about to tell her to do so? Why didn’t she know to count the tallies by fives? When she wanted to get from 20 to 14, why did she erase 5 and then count again, instead of counting backwards from 20 or something? And why did she count up all 14 circles, after she obviously knew she was splitting them into a ‘W’ group and an ‘A’ group?
… and why did she pass ninth grade algebra?