I often discuss mathematics with my six-year-old son. As a teacher, my goal is to try to pinpoint where it is that student understandings go astray. As a parent, my goal is to teach my son some mathematics.

We’ve discussed division before, and I was inspired to explore it again because of some multiplication he’d done. When I asked him what three four times is, he said it was twelve. He explained it’s because that’s what the Magic Grow capsule packaging said. I asked if the package said that three times four equals twelve, and he said, no, it only said twelve, but there were three rows of four colors, so that means that three four times is twelve.

I decided to reverse this around. “What is twelve divided by three?” He didn’t understand the question. He works well with math concepts supposedly several years above his age level, but he struggles with the language. I clarified: “Let’s say you have twelve cookies, and you and I are going to share them. How many cookies will we each get?”

“Six,” he said. He understands sharing between two people very well. I repeated the exercise for three and four people, and he gave appropriate answers.

Finally, knowing where this would take me but not sure how he’d respond, I asked: “And what if your other Grandma was here, too? What about then?”

He thought a little longer, and said, “Well, everyone would get two cookies then.”

“And how many would be left over?”

“Two.” He said.

“How could we split those up?”

He thought about it, then said, “Well, we could break each cookie into five pieces, and then give each person two pieces.”

“Hm. So how many cookies would each person get in total?”

“Two whole cookies and two fifths of a cookie.”

A little bit later in the conversation, I decided to try another division problem. “What if you had four cookies and you were going to split them between you, me, and Mommy? What would you do?”

He thought about it. Apparently his earlier strategy was found lacking, because this time he said, “I could break each cookie into three pieces, and give everyone one piece from each cookie.”

In the first case, he’d built an argument for a mixed number: He’d given each person two whole cookies and two pieces of cookies broken into fifths, that is, \(2\frac{2}{5}\). In the second case, he’d built an argument for an improper fraction: \(\frac{4}{3}\). I’m not entirely sure why he chose to break all the cookies in the second scenario but only as-needed in the first case. Also, he didn’t seem to notice or care that it’s very difficult to break a cookie into five equal pieces, but that’s a separate issue.

Hearing him talk through each case, I reflected on how conceptually different improper fractions and mixed numbers are. They refer to the same quantities, certainly, but an improper fraction involves a quantity of pieces of a single unit, while a mixed number involves quantities of two different units. \(\frac{4}{3}\) is easier to say than \(2\frac{2}{5}\), but the concept referred to by the second expression is more effective.

I also encouraged him to align the two forms of expression by asking him what would happen if he took the pieces he’d broken up and made full cookies out of them. He struggled with this, but did eventually realize that he could take three third-pieces and make a whole cookie out of them. This was a mental exercise; I imagine it would have been easier for him had we actually been using manipulatives.

This told me that, first off, children his age can certainly understand the rudiments of fractions. What was interesting to me, though, was the difficulty he had moving between the two forms of fractions greater than one. Once he had decided to either break all the cookies up equally (improper fraction) or only break up the cookies as needed (mixed number), he struggled with realizing that people were getting the same amount of cookie either way.

His initial challenges also reinforced that teaching fractions in terms of abstract units might contribute to student misunderstanding. My son didn’t have trouble thinking in terms of complex units, such as fifths of a cookie, but he did have trouble thinking in terms of dividing abstract unit counts where there’s a remainder.