My 5-year old daughter, Daphne, was looking out the car window when she laid this question on me:

“I’ve been wondering something. If there’s a whole of a tree, and half of a tree next to it, what do you call it? A *whole half*?”

At the next light, I turned on my phone and recorded our conversation. What follows is a word-for-word transcript. It’s one of my favorite math discussions ever, because I learned so much about her thinking.

“First of all, tell me about the half of a tree. How do you know it’s a half?”

“It’s a stump.”

“It’s a stump?”

“No. I mixed up the word. It’s a trunk.”

“It’s a trunk. OK. So tell me how you know something is half again. What does half mean?”

“Half means they are even pieces and you cut one of the even pieces up and you leave the other one, then they’re even, and that’s half. The one that’s down and the one that’s up without the other one is half.”

“OK. I have a whole lot of questions. Are you ready? This is the first one. Tell me what you mean by *even*.”

“Even? They’re both the same amount of tall.”

“They’re both the same amount of tall.”

“Yeah.”

“Whoa.” (Long pause.) “You’re talking about trees. What about wide? Do they have to be the same amount of wide also, or just the same amount of tall?”

“Just the same amount of tall.”

“Just the same amount of tall. OK. So here’s my next question about halves. How many pieces do you have?”

“Only two!”

“Only two. How come?”

“Because you cut one off and leave one on.”

“And that’s halves?”

“Yeah.”

“What if you cut it into three pieces that were the same amount of tall?”

“That would be quarters.”

“How come?”

“Because they just are!”

“What’s a quarter?”

“A little bit of something.”

“A little bit of something. What if I cut it into, like, 20 pieces of the same amount of tall?”

“That would be like…inches.”

“Inches! OK. So, it goes halves, the next smallest is quarters, and then the next smallest is inches?”

“Yeah.”

“What if I cut it into, like 1,000 pieces?”

“It would be just specks.”

“Specks…Hmm. And you said it doesn’t matter, like, I’m looking at that tree over there, and it’s fatter at the bottom–the trunk is fatter at the bottom and skinnier at the top–but the top has lots of leaves and the bottom doesn’t. Does any of that matter, or it’s just the same amount of tall?”

“Where?”

“That tree with the orange leaves.”

Long pause.

“That one has too many branches.”

“OK. So what kind of tree does it work for?”

“Trees…like that one!” (Pointing at a sapling.)

“OK, so what is it about that tree?”

“It’s smaller and it doesn’t have very many branches.”

“Hmm. What sorts of things can we cut in half?”

“Bread…flags…” She trailed off, and we were quiet for a while. I decided to revisit the thirds and quarters idea a little more.

“Can we think about cookies for a second? We made cookies yesterday. If you and Maya and I shared a cookie and we all had the same amount of cookie, how much cookie would we each have?”

“A quarter.”

“How come?”

“Because that’s a little bit of sugar and it’s all the same.”

“How many quarters does it take to make a whole cookie?”

“It matters how big the cookie is.”

“Tell me about that.”

“It would be a lot of pieces if it was a huge cookie, but just a little bit of pieces if it was small.”

“So, if I broke it into pieces so you got one piece, and Maya got one piece, and I got one piece of a *small* cookie, and then I took a big cookie, and I broke it so you got one piece, and Maya got one piece, and I got one piece of the *big* cookie, it would take different numbers of pieces, you’re saying?”

“Yes. Because they’re bigger.”

“So if it’s a bigger cookie that the three of us share evenly, we have more pieces?”

“Yeah.”

We kept going for a while longer, but you get the gist.

I love this conversation. I love it partly because it’s evidence that kids are mathematical thinkers who come to us full of ideas. Some of the ideas hold up to scrutiny and some don’t. Some are misconceptions we’ll need to address, and some are deep conceptual understandings and experiences we’ll want to build on.

What does Daphne know? We can divide wholes into parts. We can keep dividing into lots of pieces, and the different pieces have different names. We can use fractions to share. And, most importantly, we can look out the window and see mathematics, and wonder, and have conversations about our ideas.

What does Daphne not understand? Units, although I adore her sequence of halves, quarters, inches, and specks. She has a lot of fraction ideas intertwined. For example, even though she thinks we can have halves of big things, like trees, and halves of smaller things, like flags, she thinks a quarter is a size, and it’s small. There’s a lot of room to explore the distinction between the *number* of pieces and the *size* of the pieces. She has developing ideas about measurement and the idea of identifying a dimension: the “same amount of tall” goes in my math Hall of Fame. Volume didn’t occur to her here. That’s OK.

In case you’re wondering, I didn’t “address” any misconceptions, and I’m not worried about any of them. My goal here was not to have Daphne identify fractions with accuracy, or learn any definitions, or figure out how to measure something as complex as a tree, or answer my questions with an expected answer. My goal was to join her wondering, listen to her ideas, help her clarify her thinking here and there, and gain a deeper sense of how she’s seeing the world. I look forward to revisiting fractions and measurements with her many times, and to bearing witness as she makes sense of these ideas. It’s my privilege, as a parent and a teacher. See, math is fascinating. And Daphne is fascinating. Together? It’s the best.

I gave my sixth graders a fraction division problem from http://www.illustrativemathematics.org/illustrations/463 last week, the gist of which was, if you have 1 cup of rice and a serving is 2/3 cup, how many servings do you have? It’s accompanied by a picture of a box split into thirds. It worked really well to generate discussion, since students were split about evenly on whether this was 1 1/3 servings or 1 1/2 servings.

Anyway… students were working on this on individual whiteboards and I came around to Mia (not her real name). She had plunged in and had lots of work on her whiteboard, but it was puzzling, to say the least. I asked her what her thinking was so far, and she said proudly that she got an answer of 50. I asked her to explain her steps some more and it went something like this (although, truthfully, more garbled; I’m probably somewhat imposing my impression of where her ideas came from):

“I took the box and drew the three lines. So that made it into fourths. Each fourth is 25 so I wrote that in the boxes. And then the 2/3 serving says to color in 2 boxes, so I did, and that is 50!”

It was sort of surreal. It was kind of like she was imitating things she’d seen done, but without much (any?) real comprehension. One thing I asked her was, “Were you thinking of dollars when you wrote the 25?” but it was hard to tell if she even really knew where she got that. Maybe she was thinking of percents? Who knows?

This girl was already on my radar, but she’s missed a lot of school so it’s been hard to tell how much of her confusion was from that. This conversation definitely ramped up my concern level, and yet I was also extremely curious about where each idea was coming from. It was sort of Common Core by Picasso.

Julie, I love this example. She’s clearly trying to make sense, even though it doesn’t make sense. Lots to work with, though!

Part of this conversation reminds me of a pizza sharing question discussed over at Fawn Nguyen’s blog. The gist is that fractions work nicely for idealized math abstractions, pieces of a single whole and for things that come in standardized units. However, real physical pizza and trees may not be standard enough to sensibly talk about comparing fractional pieces from different starting wholes.

For your chat with your daughter, it was interesting how you engaged in this conversation. I would have found it difficult to avoid shifting into a teaching mode to try to get to some clarity on thirds and quarters.

Joshua,

Thanks for your comment. I have to confess, we did head into the thirds and quarters a little more than I blogged about. I never taught or brought clarity, but I kept going to understand her thinking more. I tweeted about it a bit here: https://twitter.com/Trianglemancsd/status/526749399754350593 and https://twitter.com/TracyZager/status/526781553217908736 Thirds are hard! She’s clearly living with some cognitive dissonance on it these days, and that’s OK.

What a fun conversation! As teachers, we are tempted to fix our students, but there should be a place in classrooms for this type of exploratory conversation. As we shift math instruction from “filling empty vessels” to “uncovering standards,” we must think about when and how learning occurs. We don’t have to address every issue right away. Over time, students will clarify these ideas, testing and retesting moving towards coherent understanding. Mathematics provides such a great opportunity to honor students thinking. Thank you for sharing!

Totally agree, Jennifer. We get way too focused on the short-term.