I’ve been thinking lately about what it means to be a number, for kids. My guess is that, for kids, something is a number if you can perform arithmetic operations with it. So, for example, when fractions are first introduced they aren’t seen as numbers, since you can’t count by them, add, subtract, multiply or divide them. Being a number involves living the life of a number, so to speak, and that life involves counting and arithmetic, mostly.

(Related: maybe measuring should be in that mix.)

If this idea is true, it would go counter to the conventional math education wisdom. When math education speakers or thinkers want to help kids realize that fractions are numbers, they usually recommend **number line activities**. The thinking seems to be that if we can show that fractions mingle with whole numbers, this will lend them legitimacy.

I think that kids find this unconvincing, though. I think being a number has more to do with whether the *actions *you can perform with numbers can also be performed with this new thing.

Or maybe it’s a 50/50 proposition? And both are important? Anyway, I wanted to look into this.

I asked my 3rd Graders, who have been studying fractions, whether they thought fractions were numbers. I asked them to write a few sentences explaining their thought. (I had to work to get those sentences. A bunch of them were very unhappy about writing.)

So, are fractions numbers?

Here are some of my favorite responses:

- Yes, because to make a fraction you need numbers.
- Fractions are a certain category of numbers because without numbers fractions would just be lines.
- I do not think this is a number (1/2) when the 2 and the 1 are numbers, but not together, for 1 + 2 = 3 number, but this is not a number it is a half.
- Yes a fraction is a number because a fraction is a number of pieces of something.
- Fractions are numbers because I don’t know to explain it.
- Fractions are numbers because you could add up to a number like 1/2 + 1/2 = 1 and a whole is a different way of saying 1.
- NO a fraction is not a number a fraction is only part of a number.
- Yes, because a fraction is a part of a whole number.
- I think fractions are numbers because all math is numbers.
- NO I think fractions are half of numbers. 1/2 is a fraction. 1 is a number.
- A fraction: a problem of numbers in groups. Therefore not really a kind of number, but closely related.
- Are fractions numbers?! Yes because 1/4 = 6 ’cause fractions are numbers but the bigger the number the less it is.

I have no idea what that last one means either.

So, how did my hypothesis hold up?

One kid used the fact that you can add fractions to make a whole number to decide that fractions are numbers. Though it wasn’t the mere fact of addition that helped this kid — it was the way that fractions could relate to whole numbers.

Reading these responses leads me to revise my hypothesis. Kids decide that fractions are numbers when fractions/whole numbers co-mingle in equations and problems.

The number line still, I think, is not necessarily relevant unless you *count along the number line*. Meaning, it’s not about the number line, it’s about the number line as a representation of counting that involves both whole numbers and fractions.

After asking my students to write their sentences, I led the class in some counting that I marked up on the number line: 0, 1/4, 2/4, 3/4, 1, 5/4, 6/4, 7/4, 2…

This, along with other arithmetic operations that involve *both *fractions and whole numbers, will help my kids increasingly see fractions as numbers.

**UPDATE: **I talked to my 3rd Graders about this again today. I just reminded them of the question, and then the room sort of exploded with arguments. Here are the arguments I heard.

*When two parents have a kid that kid is still a person. So that’s like when you take a 1 and cut it up into thirds it’s still a number.*

*You can make numbers out of fractions like 1/2 + 1/2 so those have to be numbers too, because otherwise how could they make a number?*

I like that these two arguments are sort of the flip sides of each other. I told my class that to me one of these sounded like “fractions are parents of numbers” and the other “fractions are children of numbers,” which a couple kids found hilarious.

One girl said that it really depended on the fraction. People talk about having half of something all the time, so half is a fraction. Other things — like 7/13 — that wouldn’t be a fraction because nobody talks about those. Later, I tried to put pressure on this position: what if I made a recipe that called for 7/13 of something? She said it really depended on what everybody did, not just me.

Her take reminds me of debates between prescriptivists and descriptivists about language.

Hi Michael,

In terms of the ‘counting along the number line’, try Choral Counting. I have a lesson here that uses it. It went fabulously from my perspective (it was helpful that this was not the first time we’d done Choral Counting together, it was more like the 5th).

David

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Yeah! I watched a video about choral counting on tedd.org and it’s been a part of my 3rd and 4th Grade teaching ever since. The “counting along the number line” I was trying to describe was using my half-assed version of choral counting with the class. (By “half-assed” I just mean that I don’t know if I was actually following the routine with any fidelity, I didn’t ask for a reflection after the count, etc.)

But yeah, I totally agree. It makes multiplication with fractions by a whole number much more available much earlier than what I used to do, I’ve noticed.

I notice you used just one number line. I used a series of number lines, one on top of the other. This made equality of different fractions along the count hard to notice, though, so I’m going to try your one-number-line version next with my 3rd Graders.

I wonder if that will make a difference?

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I think that the jump from “whole numbers as symbols” to “fractions as symbols” is missing out on the literal meaning of fractions, where the “thing” is one half, or two thirds. The “of” is often ignored altogether.

The time to move to fractions as numeric constructions must be much later, when the ideas are solid. Then the question — are fractions numbers? — is “yes, of course.”

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Unless we’re using them to describe amounts, I think it’s hard to get the idea of fractions. It looks like two numbers instead of a number of units.

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I love this deep discussion about what a number is! I am teaching a foundational math class for students that are freshmen not yet ready for algebra, this would be a great type of discussion to have with them! We are also going to use “clothesline” number lines this year. I believe there are resources on Estimation180 if you are interested!

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