Summer’s over. Let’s get back to work here.

A sprinkling of thoughts:

- Nicora, on a recent post, came out against using “x out of y” to introduce fractions. She writes: “One of the most common ways to introduce fractions to young students is to talk about ¾ as
And it’s part of the reason why students have so much trouble with fractions later on.” I wonder whether this is the sort of mistake that you’d see more of if you use this sort of language in class.*“3 out of 4.* - I bet that this student would have trouble if asked to place these fractions on the number line.
- I don’t really get the task. Are these supposed to add up to 1? What’s the unit here?
- I don’t think that the student believes that 3/5 is more than 2/3. I’d bet that this student just isn’t paying care to the relative sizes of boxes. This aspect of the representation just doesn’t strike her as significant.
- I also notice, though, that within each activity there’s an attempt to make the boxes the same size.

I think that this last observation might be a way into a line of questioning that could help. I’d point to a shaded in box (maybe the kickball one) and ask, “What does this mean?” And then I’d point to another box and ask the same question. This would force us to bring out the unit, and the comparisons between the shaded boxes would force us to have a conversation about the relative amount of time spent at each activity. This would naturally bring us into ranking, which I think would be a good follow-up activity.

There are a bunch of interesting things here — please comment on them — but one moral I’ll take out of this is that learning math often involves becoming sensitive to nuances that would otherwise seem irrelevant.

## 2 replies on “Summer’s over, and we’re back with some fractions”

I, too, was wondering whether these were supposed to add to 1. If so, I’d then wonder whether Winston ever ate, slept, talked/sang, etc.

The 3/5 vs 2/3 distinction is WAY too close to distinguish in this kind of space. I think I’d definitely add a number line (with noticeable space between 0 and 1 but also space before 0 and after 1)

The 3 out of 4 type of language is one of the really troubling things about teaching younger kids. I know why we want to simplify these ideas when people first encounter them, but we don’t want to prejudice their view of these ideas. For someone who firmly ‘gets’ the fraction language of 4 out of 7 bananas or 3 out of 5 apples, how do we help them transition to a fraction such as 111/25 ?

Borrowing Dan Meyer’s language this is a classical pseudocontext type of problem. Just get to the point and ask which of these fractions is greatest, or more helpfully, just ask the student to order these fractions from least to greatest.

Distinguishing between 3/5 and 2/3 is too precise of a task to be done by hand, on what amounts to a single numberline. And this particular suggested representation does not encourage a folding/partitioning strategy because the unit bar can not easily be partitioned when they are in adjacent strips. Starting with the flaw in the test item is a good place to begin. Give kids a chance to use the manipulative or strategy given to them or don’t include it, implying they should use it.

I hesitate to throw out the baby with the bathwater by not ever using “out of” language with fractions. 3 out of 4 is a highly appropriate mental model, but it is not complete. If this is complemented with an intentional investigation of what, for example, 111/25 might mean, it is appropriate. It also needs to be connected to the point on the number line as well. The key here is balance. The problem with the “out of” mental model is its exclusivity of other mental models.

Similarly with the problem context. Challenge the students to respond to the context of the problem as well. What is the unit? Hours of summer leisure time? Sure. Why not? Hours in the summer overall? So what is missing? Summer camp activity days? Absolutely. Sure it is pseudocontext, but if the problem can be made meaningful and is worthy of the time, we can make it happen. If it’s not worthy, well, it’s simply not worthy and we move on.