“Is zero a number?”: Interviews with a Whole Class of Kindergartners

When I first became a math specialist in my district, I was assigned to work with upper elementary, middle school, and… kindergarten. I had never worked with kindergartners in an academic setting before, and, cute as their little faces are, they terrified me. Sometimes you don’t know what you don’t know — and sometimes you are fully aware of what you don’t know.

Thank goodness for kindergarten teacher Tanya Paris. Tanya is best described as “magical.” Her curiosity and sense of wonder are infectious. She is playful, and also deeply knowledgable about kindergarten content. I scheduled myself to be in Tanya’s room twice weekly. We would meet together to plan a lesson, alternate who would lead, and debrief afterwards. Even though she had not been formally assigned to mentor me, I called her my coach. She’s the reason I feel comfortable walking into room after room of wiggly five year olds to teach math.

My school is moving into a new building in a few weeks, and so I’m slowwwwwly packing up all of my professional belongings. Among my files, I found some of the work I did with Tanya: notes and notes about student thinking. I found one page titled “is ZERO a number?” The date at the top is October 2010. This was less than two months into my first year as a math specialist/coach. I was 26 years old.

I remember this lesson. One of the kids — maybe Arjun? — had stated confidently in class that “zero isn’t a number. It’s nothing!” In our Tuesday debrief, Tanya had asked me, “do you think other kids have this misconception?”

“Well, there’s only one way to figure that out,” I stated, and set to work on creating a template for us to record student thinking.

Interviewing Students

On Tuesday, Tanya and I spent time gathering student information. During a time in the schedule that she called “Outdoor Explore,” we tag teamed kindergartners, asking them is zero a number? We recorded their thinking.

	10/19 (Before Lesson)
Aadi	“No. If you have zero, there would be nothing!”
Ahmad	“When you have a 1, you put a zero in the back. First you say 0 and then 1.”
Akane	“I don’t know. I know that zero is nothing.”
Amelie	“I don’t know. Well, I know that 0 and 1 are 1, and 0 and 4 are 4.”
Camila	“No, because it’s nothing.”
Colby	“No, because it is lower than any number there could ever be. It means nothing.”
Ethan	“Yes, because my dad told me. He said some babies are zero years old.”
Hwan	“Yes, because you can add it to other things.”
Jacob	“No, because it does not have any numbers. It’s not big enough.”
James	“No, zero is nothing! It comes before 1.”
João	“No, because it means ‘nothing.’ It’s just an oval.”
Sakura	“No. I know 0 + 2 is 2. It doesn’t change.”
Kyleigh	“No, because zero is a number before the first number.”
Lina	“Yes, because they say it is. It’s kind of like a circle, but not really because a circle is just a shape. Wait! I know that it’s part of the number 10. So zero is a number.”
Lucas	“Yes, because you can start at 0 and go to 1.”
Lyra	“Zero isn’t a number. It means nothing. None at all.”
Olivia	“I don’t even know.”
Priya	“No, because it means nothing.”
Savannah	“No, but 2 is a number! 3 is a number, too.”

This meant that:

• 11 students said no, zero is not a number.
• 4 students said yes, zero is a number.
• 4 students were unsure.

More importantly, we were fascinated with how they wrestled with the idea. They said no, because it’s nothing. They said no, because it’s smaller than any number. They used positional words, like before.

They also had diverse ways of explaining why zero is a number. They offered an understanding of positionality (“because you can start at zero and go to one”), and also operationally (“you can add it to other things.”) Meanwhile, another student used similar logic to explain why zero isn’t a number. (“0 + 2 is 2. It doesn’t change.”)

The Lesson

Well, it’s been 13 years, so I can’t say that I remember the lesson perfectly. I do remember starting with all of the kids gathered on the rug, talking about how numbers tell us how many. Something like:

“When we ask ‘how many,’ the answer is a number. How many? Four. How many? A hundred. How many? zero. It wouldn’t make sense to say, How many? Apple! How many? Tarantula.”

This is, of course, a limited interpretation of number. It’s not often that the answer to “how many?” is something -1, but negative numbers are still very much numbers. Also this limits conceptions of numbers to quantity, and not measurement or labels, but that did feel like too many ideas for one day in kindergarten.

I think we played a counting game where the answer could be zero. I’m pretty certain that we didn’t read the book Zero, by Kathryn Otoshi, which had been released only a month before, but we did read that later.

The Second Interview

In older grades, I use written cool downs or exit tickets after some class discussions. Written work is a lot harder in, say, October. of Kindergarten. Instead, Tanya and I re-interviewed the students in her class to see if their thinking had changed as a result of their learning experiences. Here are the results:

	10/19 (Before Lesson)	After
Aadi	“No. If you have zero, there would be nothing!”	“Yes, we learned that in class. You can have nothing, but it’s still the number you have.”
Ahmad	“When you have a 1, you put a zero in the back. First you say 0 and then 1.”	“Yes. It is a number.”
Akane	“I don’t know. I know that zero is nothing.”	“Yes, when you ask me how much, I can say zero.”
Amelie	“I don’t know. Well, I know that 0 and 1 are 1, and 0 and 4 are 4.”	“Zero is made of circles and it is the first number. It comes before one.”
Camila	“No, because it’s nothing.”	“Yes, because you can write it.”
Colby	“No, because it is lower than any number there could ever be. It means nothing.”	“Yes, you can count with it, so it must be a number! We learned that in class.”
Ethan	“Yes, because my dad told me. He said some babies are zero years old.”	“Yes, you asked this, and I said my dad told me and he said some babies are zero.”
Hwan	“Yes, because you can add it to other things.”	“Yes, because it’s a number.”
Jacob	“No, because it does not have any numbers. It’s not big enough.”	“Yes, you already asked me! It’s a number for nothing. You can use it to make 100.”
James	“No, zero is nothing! It comes before 1.”	“Yes, it’s a small number.”
João	“No, because it means ‘nothing.’ It’s just an oval.”	“No, zero means nothing.”
Sakura	“No. I know 0 + 2 is 2. It doesn’t change.”	“No, because 0 + 1 is 1. It doesn’t change. 8 + 0 is 8.”
Kyleigh	“No, because zero is a number before the first number.”	“Zero is the number for nothing at all. I remember the rhyme.”
Lina	“Yes, because they say it is. It’s kind of like a circle, but not really because a circle is just a shape. Wait! I know that it’s part of the number 10. So zero is a number.”	“Yes, you already asked this! It’s in the number 10, so it must be a number. We don’t mix shapes like this.”
Lucas	“Yes, because you can start at 0 and go to 1.”	“Yes, you can count from 0 to 10 like 0, 1, 2, 3… you can also start at 10. 10, 9, 8,…”
Lyra	“Zero isn’t a number. It means nothing. None at all.”	“Yes, it’s a number that means we have nothing.”
Olivia	“I don’t even know.”	“Yes, you can go on 0, 1, 2, 3, 4.”
Priya	“No, because it means nothing.”	“Yes, we can count 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…”
Savannah	“No, but 2 is a number! 3 is a number, too.”	“Yes, and 10 is, too. Also zero is part of 8.”

This meant that:

• 2 students still said that no, zero is not a number.
• 17 students said that yes, zero is a number.
• 0 students were unsure whether zero is a number after the lesson.

Some of them gave very similar explanations to their earlier interview. Four students pointed out that I had already asked that, or they learned about this in class. There are more students mentioning counting sequences than in the “before” the lesson interview.

I remember meeting with Tanya to discuss what students in the “after” interview. We were fascinated by how many students maintained similar reasoning, like Sakura’s examination of the identity property of addition. () We must have taught the students how to write the number zero explicitly, because more students were mentioning the shape (including Savannah’s “zero is part of 8.” I think she means 8 looks like it’s written with two towering 0s.”

Tanya and I used this before/after interview format several times that year. It pushed our thinking about student mathematical conceptions, and helped us reflect on what success looks like for a lesson. What did students take away?

In Reflection

This kindergartners were actually the high school graduating class of 2023. I imagine many of them are now settling into their freshman year of college, only thinking about zero in the contexts that matter to them (how close their bank account is to zero, zero calorie sodas, how many zeroes are in the total dollar amount of their student loans, etc.)

So why am I still thinking about this?

This entire sequence, of interviews and learning, came out. of listening to a student comment during math class. We then listened to other students, following the threads of their thinking over. time. Finding these notes from Tanya’s class reminds me of how powerful this is, and how it can shape rigorous mathematical learning for our youngest learners. Look at all of the profound things these 5 an 6. year olds had to say!

This also reminds me of a research article I recently came across. In it, Kelly McGinn and Julie Booth explore precision and mathematics communication, ultimately finding that students “benefit the most when they attempt to describe the targeted mathematical concepts, regardless of the type of language used.” Through this interview-lesson-interview, or listen-learn-listen, structure, we were giving students space to wrestle with ideas, often using informal language, so that we could then build a collective understanding of zero as a number. Even if students ultimately changed what they were thinking after hearing what we had to say about numbers, they were mapping this new understanding onto their schema that we had previously activated. What is a number? That’s a big idea for kindergarten. It’s a big idea for me. How do we continue to build our thinking?

McGinn, Kelly & Booth, Julie. (2018). Precise mathematics communication: The use of formal and informal language. Bordón. Revista de Pedagogía. 70. 165. 10.13042/Bordon.2018.62138.

For more about kindergartners and the number zero, check out Five Little Pumpkins: Kindergarten Conjectures.