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?
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: 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.