It’s Halloween! This morning, the kindergartners were reciting and acting out “Five Little Pumpkins.”

The kids stood up, five at a time, and assumed their assigned roles. The first student pointed at an invisible watch to show that it was getting late. The second one traced an open hand across an imaginary sky full of witches. And so on.

There is a lot of mathematical possibility for kindergartners in this task.

Students were linking **ordinal** names (e.g. first, second, third) with numbers. They each held up a pumpkin to indicate their placement, and thus connected the written numerals.

And then there was the possibility of mathematical modeling.

## How can we use numbers to show what happened?

**“We can use numbers to tell stories!” **I told the kindergarteners, who were surprisingly attentive given that we’d just watched a parade of costumed upper elementary students. “There was a beginning, some action, and then the end,” I continued.

With a dramatic flourish, I picked up a dry erase marker. “How many pumpkins were there at the beginning of the poem? Let’s whisper shout on 3, 2…”

I hadn’t even made it through the countdown1 before the students let loose a rippling chorus of “five!” However, one student — Yuan — called out “one!”

“Okay, so I heard people say five, and also one. Who wants to share why they think it’s five or one?”

“One is a yellow zone answer,” Farah said. *Yellow zone* refers to the “zones of regulation.” Green is peaceful, calm, and happy. Yellow is approaching dysregulation, perhaps silly or anxious or frustrated.

“Maybe,” I considered. “But also I think there’s a good reason someone might say one. Yuan, do you want to share your thinking?”

He smiled — betraying a hint of mischief — and then said, “at the beginning, one pumpkin.”

“Yes! The first pumpkin is the one who says ‘oh, my, it’s getting late! And how many pumpkins were there in all at the beginning of the story?”

“FIVE!” Like thunder.

“And then the pumpkins all said something. And then the five little pumpkins rolled away. When we take things away, we might write *minus*. That’s called subtraction.” (Was I losing them?) “So we had five pumpkins, and then five pumpkins went away…”

Towards the far part of the rug, Jonah and Noah had turned to one another, and were using their fingers like miniature swords. This was solid yellow zone territory, inching towards red zone dysregulation. I called everyone back to attention by tapping my shoulders. “Match me!” I crossed my arms so that my left arm tapped my right shoulder. “Match me!”

We could continue. “Okay, this time, we’re going to show the answer with your fingers instead of shouting out. Get ready with your fingers,” I cued. *“How many pumpkins were left at the end?”*

Simone curled her fingers into a round O. Jonah balled his fist. Others offered various permutations of these two actions. (And one or two called out. We aren’t perfect.)

“I notice that Simone is doing this, and Jonah is doing this,” I said, miming the two responses. “What number is that?”

“ZERO!!!!”

“We can write that like 5 minus 5 equals 0. That equals means ‘is the same as,'” I started. I paused, as I considered whether or not to tell students about the real power of the equals sign. It doesn’t have to signify *the end of the story*. Equality is a beautiful and nuanced concept.

But it was 9:15am on Halloween. We could address this later.

## Generating More Cases

“Hmm,” I paused dramatically. “What would happen if we had seven little pumpkins. And they all said something cute and hilarious, and then they all rolled away. Now there are…”

“ZERO!!!”

“How can we show that with a number sentence? Let’s see… 7… minus 7…”

“Equals zero,” Simone said.

“What if there were ten pumpkins, and the ten little pumpkins rolled away…”

“ZERO!!!!”

“Hmm. What if there were seventeen pumpkins, and the seventeen pumpkins rolled away…”

“Did you know there are seventeen kids in our class?” Farah asked. “That’s one pumpkin for every kid!”

“Precisely! Okay, are you ready for a wild number? Really big?”

Jeremiah nodded impishly, his hands clasped together.

“One hundred! What if there were one hundred pumpkins, and the one hundred pumpkins rolled away.”

A few peals of laughter echoed across the rug, and Niran raised his hand.

“100 minus 100 equals zero!” He trumpeted.

## “But I want a hard one.”

Jonah raised his hand, and then started talking immediately, without waiting to be called on. “I want to do a smaller number. Something harder.”

“Like three and a half?” I grinned.

“What?”

“Three and a half pumpkins, and then three and a half pumpkins rolled away. How many are left?”

“Zero!”

I recorded it on the board.

“Actually, I meant a smaller number to take away. Like keep the 100 and then the other number is smaller.”

“Oh, like 100 – 52?” I wrote this on the board. “You’re right, that does feel trickier. Meanwhile, you were able to solve a problem that had halves in it! We don’t usually subtraction with fractions until you’re into third or fourth grade.”

“But I want a hard one,” Jonah pressed.

“Okay, okay, I get it. That does sound fun! Maybe in a few minutes! But also we’ve come across a really, really important math idea here. All of a sudden you can do 100 minus 100, because you noticed a pattern.”

The kindergartners started talking, overlapping one another.

“Because the answer is zero.”

“It’s always zero.”

“We keep getting zero.”

“Because it’s all gone,” Farah said. I chose to amplify her voice: “say that again, Farah.”

“Because it’s all gone! You start with some pumpkins and they all roll away. They’re all gone.”

“So all of these zeroes…” I drew arrows from where I’d written “all gone” to the zeroes. “They show that all of the pumpkins are gone.”

“Yes!”

## Approaching a Generalization

“What other number sentences will make zero?” I said. I wasn’t certain how to get that idea out, but there’s no time to analyze my phrasing when facing seventeen kindergartners with Halloween energy.

“Six minus six,” Niran offered.

“A million minus a million,” said Noah.

“Yes!” I clapped my hands together. “So! What can we say about all of these problems? When will a subtraction problem equal zero?”

Crickets. The students looked at me.

“Twenty minus twenty equals zero,” Simone said.

Okay, yes. That’s true.

And also it was time for me to go upstairs to meet with a fourth grade teacher, and more and more students were entering that slightly dysregulated yellow zone. Halloween!

## Making Generalizations in Math Class

There is a lesson sequence about making generalizations in the book *But Why Does It Work?: Mathematical Argument in the Elementary Classroom* (Russell, Schifter, Bastable, Higgins, Kasman: Heinemann, 2017). I confess that I haven’t read the whole book, but, after seeing Susan Jo Russell and Deborah Schifter speak at NCTM Annual in LA last month, I’m motivated.

The Teaching Model the authors propose is as follows:

- Phase I: Noticing Regularity
- Phase II: Articulating a Claim
- Phase III: Investigating Through Representations
- Phase IV: Constructing Arguments
- Phase V: Comparing Operations

In this spontaneous moment with the kindergartners, we had moved through Phase I (Noticing Regularity) and into Phase II (Articulating a Claim). I can easily envision how we might continue the work — using cubes or other physical objects to investigate, and constructing more durable arguments — but this felt like a great start. We’re planting the seeds of algebraic thinking. Maybe next time we’ll build in time to get through more phases.

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