Listening to Early Understandings of Division: “12 ÷ 5 is either -3 or 0”

This blogpost is a part of the (2019) HUMANIZING MATHEMATICS CONVENTION: Virtual Conference on Humanizing Mathematics hosted by Sam Shah and Hema Khodai
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I love, love listening to early understandings of an idea: students worm their way through a concept, pushing forward and circling back, and looking up with furrowed eyebrows. It’s a time of electric vulnerability. Allowing students space to mess around with the mathematical ideas is what helps build agency and identity.  It’s what ‘humanizes’ the mathematics classroom. 

I wrote this post while engaged in a residency in a third grade classroom. We had recently launched a unit on division

Because everyone arrives in the formal classroom with these various experiences, it’s especially important for the teacher to listen to students, and to allow students time to formulate and express their ideas. These introductory lessons are also the ones that I feel most anxious about. They set up the trajectory for future learning. The explicit teaching that we do in these early lessons — clarifying terms, building concepts, and connecting ideas — is so important. Learning is fragile.

In this third grade division unit, the classroom teacher and I wanted to give everyone the chance to develop some common understanding, so we launched with a context: sharing cookies, a lá The Doorbell Rang, by Pat Hutchins. (Lesson ideas courtesy of the inimitable Marilyn Burns/@mburnsmath) In the story, two kids are hoping to share a batch of 12 cookies (12 cookies shared by 2 people yields 6 each) but are repeatedly interrupted by more friends and neighbors at the door. (12 cookies shared among 4 people, then 6, then…)

(above: a photo of a student retelling the mathematics of the story on paper)

The next day, we recounted the story in a problem string. We focused on understanding division as sharing. 12 ÷ 2 is like 12 cookies shared among two people. I drew two circles (plates), and then distributed the cookies out

All of the situations in the Doorbell Rang involve whole numbers — whole people and whole cookies. Are there times when we can’t share evenly? What happens then?

Enter: 12 ÷ 5.

I presented this problem to the students in 3F. There was some uncharacteristic silence in this typically chatty classroom. Then, Neil said:

“Oh. I get it. It’s impossible. You can’t do that.”

Other students murmured in agreement. No one seemed annoyed that I had given them an impossible problem; perhaps they were too proud that they had figured it out. I was no match for their wits. They became distracted.

Towards the back of the rug, behind Neil, I saw Yosef squinting at the board. “12 ÷ 5… 12 ÷ 5… it’s… it’s…” Yosef raised his hand to speak.

“I think it’s -3,” Yosef stated coolly.

“Do a ‘me too’ [signal] if you agree that it’s -3,” I directed. A few students stared at me with their lips upturned in confusion. “Raise your hand if you have a different answer.”

Perhaps emboldened by Yosef, Luuk decided to share, too. “I think it’s 0,” he shrugged, as if apologizing for taking up space with his body or ideas. Thankfully, other students heard him, and started to use American Sign Language for “same.” “Me, too!”

I recorded 0 on the board below the -3.

“Does anyone who chose either -3 or 0 want to explain how they determined their answer?”

Yosef spoke first, to justify his answer of -3.

“Well, I took away 5 cookies to give one to each person. Then another 5. Then another 5.
12 – 5 is 7.
7 – 5 is 2.
Then 2 minus the last 5 is -3. So it’s -3.”

Luuk then explained his answer of zero. “15 ÷ 5 = 3. That problem is easy for me because it works.”

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