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?”

12 ÷ 5 = -3

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.”

12 ÷ 5 = 0

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

“But it’s TWELVE divided by 5!” Neil interrupted.

“Yes, it’s 12, not 15. So we take away the extra 3 from the answer. 3 – 3= 0.”

Leila raised her hand. “That’s not right,” she countered. “You can just take away the 3.”

“So what do you 12 divided by 5 is, Leila?” I asked.

“Zero,” she stated plaintively.

“Okay. So it sounds like you got the same answer as Luuk, but maybe in a different way? Why is it zero, Leila?”

“Because it doesn’t work. If it doesn’t work, it’s zero.”

I saw several other students nod, and heard some murmurs of agreement, but the -3 camp was still holding out strong. We did not have any consensus about 12 ÷ 5.

What IS 12 ÷ 5? 

We were at an impasse, so I asked the class to make connections: “how did we solve the cookie problems earlier this week?”

“Pictures!” “Cubes!” “Counters!” “Multiplication!” The responses were near simultaneous.

“So I heard pictures, cubes, counters, multiplication… let’s try one of those.” I wanted to use the green tiles we had been using the day before, but I didn’t think it would work given the students’ sightlines from the rug. The document camera was on the other side or the room.

“Let’s image that our problem, 12 ÷ 5, is about sharing cookies. How many cookies will there be? Think to yourself. Now whisper shout the answer. Ready, set, go.”

“12!”

“And how many people?” I paused for wait time. “Whisper shout. Ready, set, go.”

“5!”

“Okay, so that’s 12 cookies split among 5 people.” We had actually done a lot of problems where the phrasing was reversed: 5 people sharing 12 cookies. We want students to be able to interpret this information flexibly. 

I drew 5 circles to represent the 5 people, or the people’s plates.

Then we split up the cookies. The first cookie went to the first plate, the second to the second plate, and continuing, until everyone had 2 cookies. “They look like faces!” several students suggested. We moved on. There were 2 cookies leftover. 

“What do we do with those cookies?”

“They’re leftovers. You get to eat them,” one student suggested.

“Let’s cut them!” Yosef offered. 

“How should we cut them?”

“Into 5s. Then give out the 5s.”

Our third grade unit on fractions comes after division, so we hadn’t discussed the term “fifths.” Yosef is also not a native English speaker. Given those constraints, he expressed himself quite clearly. I drew fifths on the board, and modeled splitting them up. “Does this match your thinking, Yosef?”

“YES!

Several students looked like they were having moments of realization, so I moved us into writing the equation.

12 ÷ 5 = ….

“2 2/5!” Neil called out. 

Several students were nodding ever so slightly in agreement, while others looked bewildered. 

“You know, there’s another way to record this, without the fractions,” I started. “Cookies can be cut, but some things, like balloons, cannot be cut. So what if we had 12 balloons shared by 5 people? Everyone would get two… and then…?”

“There’s still two more.”

“Yes, and those are leftovers, or remainders.” I wrote the words “leftovers” and “remainders” on the board, along with the equation 12 ÷ 5 = 2 r 2

“Today we’re going to explore more of these problems that may have leftovers, or remainders. Let’s see what happens. I’ll put the tiles out at each table in case you want to use them to experiment.”

Some students were content with our equations, while others were still trying to figure out how it is we came to a point in math class in which letters and fractions started to enter equations. The classroom teacher and I knew that we would need to listen closely to students as they worked, to help them elaborate on their ideas and clarify their thinking.

We may have had some answers on the board for 12 ÷ 5, but there’s still a lot of learning to do.

The classroom teacher and I were keenly focused on the important mathematics in the lesson. We were also thinking about how students build these understandings — and how they build their own identities within the classroom. We modeled how to listen to ideas, and to consider them. We tried to show students that mathematics is about sensemaking.

Of course, Jasmine sat in the back of the class, twirling a pencil for most of the 12 ÷ 5 debate. Damien asked to go to the bathroom 3 times. There’s still a lot of learning to do about what it means to be engaged in mathematics — both for the students, and for the classroom teacher and I. There is a lot more listening we need to do.