I took the kids camping this past weekend. Fall along the Mississippi River, mid-70-degree days and 50-degree nights. Pretty much perfect. Having read information about our state park together earlier in the day—including the park’s acreage—Tabitha posed a question.

Tabitha (five and a half):How many acres or miles is our campsite?

Me: It’s only a small fraction of a square mile, but it’s about of an acre.

Tabitha:What’s a fraction?

Me: It’s like when you cut something up. It’s a number bigger than zero, but less than one.

Tabitha:Huh? That doesn’t make any sense!

Me: Well, let’s say you, Griffy and I had three s’mores, and we wanted to share them equally. We would each get one, right?

Tabitha:Yeah.

Me: But what if we only had 2?

Tabitha:Well, then you’d have to cut them in half.

Me: Right. So is a fraction

Tabitha:Oh.[later]

Tabitha:But what’s the number? You said a fraction was a number bigger than zero, but less than 1.

Me: One-half is more than zero, but less than one.

Tabitha:Half of what?

Me: Half of anything is more than nothing, but less than the whole thing.

Tabitha:But what’s the number? A half isn’t a number!

**
**I have been thinking about the moment when there is a choice to talk math with my kids. I have been trying to understand what I need to know in order to

*recognize*that a choice exists and in order to

*pursue*a mathematical conversation.

Fractions are tough because there really is a lot of specialized knowledge about how people learn them. I have been reading and teaching from the book *Extending Children’s Mathematics *over the last year or so. The authors make the argument that *fair sharing* is the best entry point for children’s sense-making about fractions. Not part-whole. Not number line. Fair sharing.

Notice how this plays out in my conversation with Tabitha. I start with part-whole, move to (arguably) number line and she protests that these ideas make no sense.

But as soon as I go with fair sharing, she’s on it. She gets that things sometimes need to be cut up in order to be shared equally.

She also understands—**and this is crucial**—that halves are meaningless without a referent whole. “Half of what?” is a brilliant and essential question.

So what did I need to know in order to pursue this conversation? I needed to know that there are multiple ways of thinking about fractions, and that *fair sharing* is going to be helpful for a young child to think about. And that *part-whole *and *number line* are going to be dead ends.

Tabitha learns from the conversation that fractions have to do with fair sharing. She doesn’t understand one-eighth—the fraction that initiated the conversation. She doesn’t understand anything more about the size of our campsite, nor about acres, miles or even square miles.

She learns that fractions have to do with sharing. That’s a pretty good Kindergarten-level idea, right there.

She’s a bit young for the number line as an entry point (no matter what Professor Wu has to say on the matter). But after she asks that last question, I wonder where you can go? Somehow, she needs to make the connection that counting numbers aren’t the full story of what counts (no pun intended) as a number. Talking about the number line definitely did nothing for her, but you weren’t actually looking at a number line, were you?

Obviously, I haven’t the first clue how I thought about such matters at her age. But kids DO know about fair sharing because they experience it. However, you need to be very careful with “half.” Kids often confuse it with “two pieces of the same thing, not necessarily equal to each other.” Hence, the expression, I’ll take the bigger half.”

This just in from

Andrew Stadel(via email):Great post Chris. I can relate. You could even extend the fair share down to my two-year old. He understands the concept of sharing as in Dad will share some of his noodles with me at dinner. He loves noodles and knows I can break my dinner into smaller parts for him to receive. Even better is breakfast, and I’ll make us eggs. If the pieces of scrambled eggs are too big on his plate, he’ll take his tiny fork and say, “I’m going to cut this in half.” Whether or not he cuts it in half is a different story. Ha! However, he’ll extend it to say, “I’m cutting it down the middle, Dad.” He loves to give us the play by play. I love hearing it. Yes, he’s mimicking his mom and dad, but he understands fractions (a part between 0 and 2) in such a basic way and through fair share. He’s also building his vocabulary. The best part after cutting his food in half, “I have two, Dad!” He’s excited to have created a larger counting quantity, but doesn’t understand it’s still the original amount.

I wonder if Tabitha honed in on: “It’s A NUMBER bigger than zero, but less than one.”

I wonder what would have been different if you had said “fractions are numbers that are bigger than zero and less than one”

I wonder if there’s a way to relate fair shares to the number line, if you had a lot of s’mores and a sharp knife. For example, cut a s’more in half, and another in thirds, and another in fourths.

Line ’em up, 2 s’mores, 1 s’more, 1/2 a s’more, 1/3 of a s’more, 1/4 of a s’more, etc.

Ask “how many s’mores is this?” pointing to the 2 s’mores. “2”

Then point to the 1 s’more “1”

Then to the 1/2 s’more “1/2”

She wouldn’t have names for the other ones I guess, but she could see that they are smaller and smaller amounts of s’mores. Less than a whole s’more but more that 0 s’mores. Based on her previous conversations, I bet she would confidently say “zero!” when you asked her how many s’mores there were when pointing to a pile of no s’mores.

Somewhere an idea would be building that “Fractions measure an amount of something that’s there but there’s not a whole one of it there.”

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