This student — let’s call her Alice — is in 4th Grade. She did some work with fractions in 3rd Grade, but clearly isn’t comfortable with them.

I went over to Alice and noticed that she wrote “0.5” for point A. I asked her to read that number, and she said “a half.” Then I drew a half-filled circle and I asked Alice to tell me what fraction of the circle was filled in. She said “a half.”

Me: Can you write “a half” as a fraction?

Alice: Why do you have to? This way is so much easier.

[I show her how I write a half.]

Alice: Oh, a one and a two.

[I draw two more circles, one with a quarter filled in, the other with three quarters filled in.]

Me: What part of the circle is filled in in these two circles?

Alice: A quarter. Three quarters.

Me: How would you write those numbers down.

Alice: Umm…so this would be one-four?

Me: Yes, though I’d read this as one-fourth.

Alice: And this would be one-three.

This is interesting in all sorts of ways. First, because you can really see in Alice’s work the difference between written and spoken language. Alice can tell you what a half is. She can even tell you how much is shaded in on the other circles, but she can’t write it. Attention needs to be given to both verbal and written language, and we teachers tend to focus on our students written work.

Also, “one-four” and “one-three”? That’s so interesting. Alice sees “three” as the most important part of “three quarters,” and tentatively thinks that fractions are just always “one-something.” That’s a pretty strong tell.

The other remarkable thing is how strongly Alice prefers decimal representations to fractions. Alice showed this preference consistently in her problem solving.

The kindly Professor Danielson argues that, in a curriculum, fractions ought to precede decimals. But it’s also true that decimals are addictive. In my high school classes, kids use their calculators to transform fractions to decimals as a defensive measure. You know the easiest way to help (most) kids solve equations with fractions? Point out that they can convert those fractions to decimals.

Decimals are absolutely enticing to people, even to this kid who is just getting started in this whole mess.