Seventeen Pumpkins

There were seventeen pumpkins, all in a row.

I walked into the kindergarten room a few minutes into the math block, and I tried to get my bearings quickly. I learned that the students had each brought in their own small pumpkin. The classroom teacher, Ms. Lombardi, sat behind the pumpkins, calling students up one at a time to “make changes.”

“Are these all lined up? From the biggest to the smallest?” She asked.

One by one, students approached the line, and swapped positions for a few pumpkins: a round white one for a stout orange one, etc. It was like the Halloween Kindergarten edition of a “Price is Right” game, and the students were taking it very seriously, hedging their bets.

What does it mean to be the biggest?

“Does the stem count?”

Maisie moved a round, deep orange pumpkin towards the “big” end of the line.

Jonah protested: “wait, why did you put that one there?”

Maisie pointed to the stem. “It’s really big.”

Jonah protested, again: “it’s not that big. It’s just really tall.”

“Oh, Maisie, so you’re saying that the stem should count when we’re deciding how tall something is,” the teacher said. Maisie nodded.

“But I don’t think the stem counts!” Jonah stayed in his rug spot, but I could see that his fingers were slowly curling into fists. He was getting frustrated.

“What do you think, kindergartners? Does the stem count?” Ms. Lombardi asked.

Some students agreed, and some disagreed, and others looked perplexed. The task had seemed simple: order the pumpkins by size, and now they had to wrestle with some delicious ambiguities. They had to decide.

Aoi’s Cinderella Pumpkin: Size and Shape

Most of the pumpkins were squat. They had deep ridges that radiated out of the stem, carving lines of latitude down the gourd. Because many of the pumpkins had the same shape, they were easy to compare. A taller pumpkin also had a wider girth and larger sections. They were almost proportional. I started to lose myself in thoughts about our 7th grade work with scale copies and our 8th grade work with dilations, when I noticed that student after student came up to the line to move Aoi’s pumpkin.

Aoi had brought a pumpkin that looked like the “before” photo for Cinderella’s coach. It was beautifully round, and even, and smooth.

It was also taller than many of the squat pumpkins. Invariably, a student would come to the line and put it towards the taller side of the line, only for another student to place it towards the shorter side of the line at their next turn.

I paused the class. “Why do we keep moving Aoi’s pumpkin?”

Jonah was the first to respond. “Everyone think it’s big because it’s tall, but I don’t think that’s right. Aoi’s is tall but the other pumpkins are wide.”

“Okay, so for some of us big is tall. A pumpkin could be tall or short,” Ms. Lombardi said. “And then Jonah said that the pumpkins could be wide. So they could be wide,” she said, indicating the width of the squat pumpkin. “Or… narrow? Is that what you’d say, Ms. Laib?”

“Yes! We have tall and short, and wide and narrow. Opposites,” I affirmed. Both Ms. Lombardi and I gesticulated broadly, with the pumpkins as our props. “And then I heard Jonah say that Aoi’s might be tall, but the others are wide. So do we want big to mean tallest? Do we want big to mean widest?”

Farah, who was sitting near me, bit her lip.

“Okay, so point to the pumpkin that’s bigger if we think big means tall.” I prompted. The students pointed to Aoi’s. “And now point to the pumpkin that’s bigger if we think big means wide.”

We weren’t getting closer to consensus, and I could see Niran starting to shift anxiously in his rug spot.

“Okay, are there any other ways to think about bigness?” I asked. I lifted up the two pumpkins, Aoi’s Cinderella pumpkin and Anaya’s squat pumpkin, and confessed, “I can’t even tell from holding them.”

“I have a scale!” Ms. Lombardi chimed in.

“So: scales help us measure how heavy something is. That’s another way to think about how big it is!”

The Balance Scale

Ms. Lombardi retrieved a balance scale from the closet. I placed Aoi’s Cinderella pumpkin into one basket, and the scale dropped dramatically down.

“What do you predict the scale will look like if Anaya’s wide pumpkin weighs more?” Students positioned their arms tilted down, and up, and every which way. “What about if Anaya’s wide pumpkin weighs less?” The same wild predictions.

“What if they weigh the same amount?” Leah asked.

“Excellent question! So, using a balance scale, the heavy thing will be lower, just like how Aoi’s pumpkin dropped down to the ground. And the lighter thing will be higher, and, if they weigh the same, it’ll go straight across. Shall we check to see what happens if we add Amaya’s pumpkin?”

The kindergartners offered an enthusiastic “YESSS!!!” that surely the classes on either side of us could year.

I positioned Amaya’s squat pumpkin above the basket, and… plop.

“YES, Amaya’s weighs more! Let’s do it with another one!” Jonah cackled. Like a little bit like Pumpkin March Madness.

We compared Aoi’s Cinderella pumpkin to a few other squat pumpkins, adjusting its position down the line. We discovered that all but the two smallest pumpkins, belonging to David and Niran, weighed more than Aoi’s.

“What do you think is inside a pumpkin?” I asked.

“oh, oh, oh!” Several students eagerly cawed to get my attention. “I’ve opened up a pumpkin before! It’s full of seeds!”

“Yes! And I don’t think seeds weigh much. Is there anything else in there?”

“Smooshed up vines!”

“Anything else?”

The kids looked at me, and I finally offered up: “there’s also some air in there, which doesn’t weigh much.” It is true that when something appears to be nothing — like air — it’s hard to identify that it’s contributing something to the equation. Zero is a robust concept.

So what does it mean to be big?

Aoi’s pumpkin was now far down the line, sitting amongst the tiniest of our pumpkins.

“Does anyone want to change any other placements right now?” Ms. Lombardi asked, although the students generally seemed content.

“You came up with so many ideas about what it means to be big!” I gushed. “There’s whether something is tall, or whether it’s wide, or whether it weighs a lot. And if we use one way to measure something, like how tall it is, we might end up with a different answer than if we use something like how heavy it is. That’s beautiful.”

“And the stem doesn’t count,” Jonah insisted.

“And, if you wanted to talk about the order with someone else, you’d have to agree about the stems. You have so many big math ideas when it comes to the idea of big!“

Inviting, Celebrating, and Developing Student Thinking

I offer that the work of teaching, as I understand it, is simply to invite, celebrate, and develop student thinking. Give me another few years and I might be able to simplify that further. 

Dan Meyer (October 6, 2022)

This “simple” mission statement about math education from Dan Meyer — to invite, celebrate, and develop student thinking — has been percolating in my brain for these last few weeks. It’s elegant. It’s actionable. It’s brilliant.

The kindergartners would continue with their pumpkin measurement — using connecting links to measure how wide or how far around the pumpkin is, i.e. circumference — after I left, but I wanted to check in with the classroom teacher about the “big” classroom discussion first.

“I love how ambiguous the word big is!” I told her.

“You do?” She almost seemed surprised.

“Yes! I continued. That ambiguity promotes discourse. All of a sudden, in order to convince someone else, they need to be able to explain how they are defining and measuring bigness. It was amazing. It was a great way to invite the students to share their mathematical ideas. Then you celebrated the student ideas by allowing the students to physically move the pumpkins and also explain their thinking. Jonah had so much to say about his classmates’ decisions! And lastly, the class discussion developed student thinking about how we measure things, and how measurement can help us more precisely compare. We started with a pretty non-mathematical term — big — and ended up using a balance scale!”

I could talk for a long time about good things that I see in classrooms! The classroom teacher smiled through this, and then expressed an interest in continuing to work together on classroom discourse when I do my December residency in her room. While I didn’t name the invite, celebrate, and develop mission explicitly to her, I think she will connect with it. How can we create spaces that allow us to engage in those three actions?