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?
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?
Me: Is zero a number?
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?
Me: Do you know what it means to have half of a cookie?
G: Yeah…sort of.
Me: Is one-half a number?
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.
Me: And not one half.
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.
Me: You’re playing with blocks there, right?
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.