Zero=half revisited

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: 4\frac{1}{2} is a number.

Me: Oh? 4\frac{1}{2} is a number, but not one-half?

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 4\frac{1}{2}.

Me: Oh. So when we count count, we skip over 4\frac{1}{2}?

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 \frac{1}{2}; that it expresses a number between 0 and 1.

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4 responses to “Zero=half revisited

  1. 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?

  2. 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 \frac{1}{2} 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).

  3. 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!

  4. 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?

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