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?

## 8 replies on “2/3 as a “2 by 3””

2/3 started life as the ratio 2 : 3, read two is to three and equal ratios became equal fractions. Eating two of three pies has nothing to do with numbers less than one, yet you have eaten 2/3 of the pies.

2/3 means take 2 of the three equal quantities, which may or may not be parts.

The Japanese curriculum makes this clearer than the West.

Ratios are also part to part… So 2 to 3 could mean 2/5… Which causes no end of confusion for students who don’t have a firm foundation.

The initial approach to the teaching of fractions is a mess. Jonathan Crabtree is right, but I would go further, and further back. At the heart of the problem is the desperation to use “fraction notation”. If the questions were written in words, “…She gives three-quarters of them to Ethan..” then no misunderstanding of the notation is possible because we haven’t used it. And if they can’t get this right they could try “half of them”.

The notation should be avoided until the kids have got a clue about the meaning of the words.

“Which is more, half a pizza or three-quarters of a pizza?” makes sense. “Which fraction is the larger of 3/4 and 1/2 ?” means nothing without a firm grasp of “reality”.

to me this is even more encouragement to start with fractions as a way to record portions of a group in a discrete context. Introducing the number as a way to record an idea is powerful, especially when the notation for the number is such a mess.

I wonder what the students would have said 1/2 is, since most students have a better intuition for that. And then if they could find a connection for 1&2 in what they said was a half.

I appreciate the idea of a progression. I have never seen this use of the numerator and denominator. Thank you for sharing. I’ve been introducing fractions with words instead of numbers and symbols for a few years now. The students write two thirds in order to understand that 2/3 is one number or one quantity.

It appears the error was to see 2/3 as 2 threes and 3/4 as 3 fours. not sure how to fix it.

I agree that when I first started learning about fractions I had no idea how the numerator and the denominator connected because I was always taught that it is a division problem. I see a fraction as a division problem that will turn into a decimal. I understood that it isn’t a whole number yet. The children who is doing problem number 2 interpreted the picture as a 3 by 4 because the answer 3 fourths aren’t 12. It doesn’t make sense that there are 24 toys but half of the toys Ethan took it. 3 fourths are more than ½ and Ethan should’ve taken more than 12 toys. The child didn’t seem to understand the meaning of what the fraction is asking but they interpreted it as a 3 by 4. The children don’t understand the meaning of what a fraction is yet but solved it as 4 rows or car toys by 4 columns and doesn’t understand that ¾ is more than half.

I think that the understanding of what a fraction means that the child displays here is just a sidestep along the progression of learning fractions. I was always taught that 2/3 means “2 thirds”, but I was not always sure what that means in relation to other numbers. I don’t think I made the same mistake as this growing up, but I had only an instrumental understanding of how to do fractions for a long time. I find it really interesting that the child interpreted the fractions like 3/4 as 3 by 4 on an array of the whole number they are trying to take a fraction of, but I do think it is an uncoordinated interpretation of the fraction at least compared to how my understanding of fractions progressed.