A few weeks back, Tabitha asked Why are zero and half the same? I was curious to know whether that conversation had affected her thinking in any way. So I asked.
Me: Tabitha, do you still think zero and half are the same? Or have you not thought about that in a while?
Tabitha (six years old): I think…Half isn’t a number. I mean, it’s made of numbers put together, but it’s not a number.
Me: What is a number?
I love this question. How people answer it can be revealing. I asked a version of it of Griffin when he was in Kindergarten.
T:
is a number.
Me: Oh?
T: Yeah. But it doesn’t really get used.
Me: What do you mean by that?
T: Well, people say, 1, 2, 3, 4, 5, 6, but not
Me: Oh. So when we count count, we skip over
T: Yeah.
We are both silent for a few moments, thinking.
T: Zero, too. People don’t count starting at zero. They say 1, 2, 3…
Me: Yeah. Isn’t that funny?
T: It should go half, zero, 1, 2, 3…
It seems clear that has indeed been thinking about that conversation. She is struggling with the betweenness of 
Overthinking my teaching 
I wonder why it should go half, zero, 1, 2, 3… I wonder if Tabitha would consider that some people think it should go zero, half, 1, 2, 3… I also wonder if she’d rather have 0 tostadas or 1/2 a tostada. Or is there even such think as a half a tostada, to Tabitha?
Yes, Max, the relationship between half and zero is a surprisingly complicated one for her. In the earlier post, she accepted that half of something is different from (and more than) none of it. It is
as a number that is the issue here.
And I find it totally fascinating to watch this develop in her mind; the relationship between amounts (half of a tostada) and numbers (one half).
So the ordering 1/2, 0, 1, 2, 3… could almost be an ordering by kind of number, or could be random, but doesn’t have much to do with 1/2 and 0 as amounts of things. Cool!
When we count in reverse order, it generally feels right to include zero: 3, 2, 1, 0. Is this because we’re usually keeping track of time when we do this, as in zero seconds left? Any other reasons worth considering for this?