A few years back, one of my elementary licensure students was trying to understand a conversation she been part of. In this conversation, an early elementary-aged child could not give a substantive answer to the question, *What does five mean?*

I suggested to my student that this was a pretty abstract question and I wondered what exactly would constitute a good answer to it.

And then I decided to show her what I meant by this. I made a video (embedded below) in which Griffin (who was five at the time) and I had the following conversation.

Me:Can you count out five blocks?

Griffin(five years old): [He does so flawlessly while mugging for the camera] What’s next?

Me:Is five a number?

G: Yes.

Me:What does five mean?

G: I don’t know. Do you?

There you have it. He knows what five *is*, but cannot articulate what five *means*.

There is more fun to be had here, though.

Me: Is one a number?

G: Yeah.

Me: Is zero a number?

G: Mmmmm…No!

Me: Why do you say that?

G: Because it’s…not necessarily a number. It’s not big…I don’t know. It’s just not a number.

Me: How about uh…

G: It’s not like any other number.

This is very common. Another math (and physics) teaching dad, Casey Rutherford, discovered this recently when discussing what *10 takeaway 10 *is with his five-year old daughter. She said *zero*. I asked him to pursue the question of whether zero is a number with her.

Having settled the status of zero, I ask Griffin about one half.

Me: Do you know what one-half means?

G: No.

Me: Do you know what it means to have half of a cookie?

G: Yeah…sort of.

Me: Is one-half a number?

G: No.

Me: Why do you say that?

G: No…because….No. It is not a number

Me: What is a number?

G: A number is like 2, 4, 6, 8, 9, 1…stuff like that.One hundred. A billion. Three hundred and ninety nine. Those are numbers.

Me: But not zero.

G: No.

Me: And not one half.

G: No.

Examples. Griffin tells me what a number is by offering me examples of numbers. Maybe this is because he has no other way of talking about the meaning of words? Maybe he *always* offers examples as the sole means of explaining what a category of objects is.

I investigate this hypothesis too.

Me: So I’m gonna ask you one more silly question.

G: OK.

Me: You’re playing with blocks there, right?

G: Yeah.

Me:What is a block?

G: A block is something that’s made of wood and it can be colorful or just plain. And you can build stuff with them. And it’s a toy that has…

There it is—the abstract nature of numbers.

*What is a number?* All he can give is examples.

*What is a block?* He is full of characteristics of blocks, uses for blocks, categories into which blocks fit, etc. He has a robust and explicit scheme for sorting blocks from non-blocks. He has no such thing for numbers.

The contrast with how he can talk about blocks in contrast to numbers is quite fascinating!!!

I’ll put on my linguist hat here.

Blocks are nouns, a thing, very easy to describe.

Numbers are quantifiers, and less specifically, determiners. Determiners are like adjectives, but instead of describing attributes, they describe things like whose and how many of an object.

So it’s not surprising that a person can describe a noun (since it’s a thing in this case) but it’s almost not fair to ask someone what the word that’s supposed to introduce a noun means. It’s basically impossible to talk about this word without the noun itself.

But if your main argument here is that numbers are abstract ideas, this is supported by the abstract nature of the types of words they are.

Zero is a very big issue in early childhood education. We talk about it here:

http://ccssimath.blogspot.com/2012/04/concept-of-none.html

Griffin is 5 years old, but the concept of fractions isn’t usually taught until Grade 3, so no worries.

http://ccssimath.blogspot.com/2013/02/fractions-are-numbers-too-part-1.html

It turns out numbers are pretty hard to describe for mathematicians as well. In order to describe numbers and their property of being “well ordered” for example, mathematicians have to use set theory and depending on which framework for describing numbers you use, the fact that whole numbers can be arranged in a specific order is either an axiom or something which requires a reasonably complex proof.

The idea of two-ness or three-ness is somewhat complex, I think, and so I’m not surprised that children find it difficult to explain. I wonder, if you asked math teachers to explain what the number 5 means, how many of them could?

Seems to me that Albert Cullum (of “A Touch of Greatness”) explored this with upper elementary students. He posed the very questions, “What is oneness, twoness, etc?” and they dove in. He helped them see that, yes, a number is a representative abstract concept. Then they struggled together to represent the abstractness outside the math realm, resulting in artwork capturing the -ness of each numeral.

It seems to me like he has defined “number” to be the “natural numbers”, that is integers greater than 0. Probably he has been taught numbers primarily through counting, which leads to numbers as the possible cardinalities of a finite set of things to count. The empty set is a special case, so zero is also.

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