*A propos* of nothing the other day, Tabitha asked a strange question.

**Tabitha** (six years old): Why are zero and half the same?

**Me:** They aren’t.

**T**: Like seven is one more than six, but zero and half are the same. They’re both nothing.

**Me:** One half? if you have half of something, that’s more than nothing.

**T**: But half, the **number**, that’s the same as the number zero.

Recall that last fall, she was not convinced that one-half was a number at all.

She now accepts that one-half is a number. But she hasn’t really dealt with the idea that there are numbers between other numbers. She is doing a bit of beautiful kindergarten logic here. Her premise is that there is only one number less than 1, namely 0. She has also accepted that one-half is a number less than 1. Therefore, one-half and zero are the same.

And—rightly—she is suspicious of this conclusion. The logic is sound, but it doesn’t make sense.

I go to work on that first premise.

**Me:** Oh. I see. Well, one-half—the number—is **between** zero and one.

I draw this picture, which I feel is certain to be totally unconvincing.

I was writing upside down. Forgive the crummy 2’s. Note the complex fraction. Take that, Common Core!

But then again, we hadn’t talked about one-half being a number since October. That last conversation seems to have been fermenting all this time, so maybe this one will do the same.

To be continued, I am sure.

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The great thing is that your daughter has really good wrong ideas and is able to articulate them. If we could get all kids to articulate their wrong ideas and pursue them with other kids and knowledgable adults, 90% of our national difficulties in mathematics education would vanish.

The number line approach seems a bit abstract for someone her age. How about drawing one whole pizza, one-half a pizza, and an empty circle with “no pizzas” to represent 1, 1/2, 0? Or make her a PB&J sandwich and cut it in half. Give her half: “You have 1/2 of a sandwich.” Take it away: “You have 0 sandwiches,” etc. 🙂

keyjames, you are absolutely right! The number line is way too abstract. Hence my certainty that it would be unconvincing.But we were working on

betweennessofnumbers. Tabitha had already acknowledged that half of something is more than none of it. She was talking about numbers in the abstract. (Which, can we agree, is amazing and delightful?)How else can we represent relationships among rational numbers in the abstract besides the number line?

Again, I am not in any way expecting that this conversation will answer the question for her. Instead, it addressed the root of her question–what is the relationship between zero and half when they are viewed as

numbersrather than asamounts?One very cool offshoot of my Talking Math with Your Kids posts is that we can trace the development of these ideas over time. Tabitha and I discussed zero a year ago. We talked about fractions last fall. She even insinuated that a fourth of a cookie is less than half of a cookie back in November.

So interesting how she views the number itself (as she emphatically said “number” in conversation) and its worth differently. I love Michael’s comment about Tabitha’s “really good wrong ideas.”

Wow. KeyJames stole my comment. I was just going to say, “I wonder if she’s recognize half of a food item as

a number between one and zero. I was even gonna use PB&J as my example.Man, I need to read your blog faster.

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Hi Christopher,

Maybe she thinks of numbers (numerals) as entities that exist separately from quantities, and hasn’t yet solidified the relationship between the two. You could try bridging the gap between the representation and the quantity by using a pack of gum alongside an empty number line. She can lay the gum pieces along the line, making a mark and a numeral indicating the end of each new piece (like a ruler), letting her create a concrete link between the two. Then, once she’s got them all laid out and marked one through (how many pieces of gum are in a pack, 5?) whatever, you can let her move them away. Then give her some to put back, determing the quantity in connection with the end of the last piece and the mark and numeral on the number line. Then give her a half piece. See what she does, and how she makes sense of the numerical representation. Then, maybe, ask her if zero is the same as 1/2. I miss kindergarten…

Of there are equal negative number ascendant there are positive numbered than zero is half

Excuse the spelling , the keyboard has a mind of its own.

There are as many negative and positive numbers than zero is

HALF Or half way.