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.