When kids are learning to give fractions meaning, I think they often struggle to figure out how the numerator and denominator are coordinated. Here we see a middle step in understanding, maybe: it’s not that the numerator and denominator are totally disconnected. They’re just coordinated in a way that doesn’t really correspond to how they actually work together (i.e. denominator tells you the “unit” and the numerator tells you the “quantity.”)

Maybe the progression of learning looks like this:

- 2/3 means “2 and 3,” nothing to do with each other. Totally baffling notation.
- 2/3 means “2 by 3” or “2 times 3,” some more familiar situation where two numbers can be coordinated in a relation.
- 2/3 means “2 thirds,” which is a productive way to coordinate the numerator and denominator.

Thoughts? Am I overinterpreting this as a middle step in a progression, when it’s actually just a totally uncoordinated interpretation of the fraction?