I find this fascinating. This student clearly knows how that multiplying the base and the height of a rectangle gives you its area. She even knows how to multiply fraction. But when it comes to part (d), she adds the numbers instead of multiplying them.

In earlier writing I hypothesized that, when put in unfamiliar situations, students often default to an “easier” operation. This idea now seems problematic to me. What, after all, is an “easier” operation any way? And what exactly would trigger this default to some other operation? And how do we explain why competent adults — like me — make similar mistakes on my own work?

It now seems more likely to me that we associate certain pairs of numbers with certain operations. Think about the numbers 100 and 1/2. I’d suggest that most people have an association of “50” with 100 and 1/2. After all, how often have you been asked to add 100 and 1/2 together? How often have you been asked to subtract 1/2 from 100? In contrast, how often have you been asked to find 1/2 of 100?

How often have you been asked to multiply 5 1/2 and 2 1/4 together? My guess is that you — and the student above — have been asked to add these sorts of mixed numbers more often than multiply them.

The idea here is that the pairs of numbers themselves come with associations.

There’s a hard version of this claim that I don’t mean to make. I don’t mean to say that, no matter the context, you’d expect a student to add 5 1/2 and 2 1/4 together. I think a division problem with mixed numbers is unlikely to trigger associations with addition. Maybe I’m moving towards a two-part model? The sorts of mistakes we make with numbers depends both on the associations with the operation and also associations with the numbers? And things get really bad when these two associations point in the same direction?

This theory feels very testable, but at the moment I’m having a hard time articulating a possible test of it. But we should be able to mess with people’s associations with numbers and see if that changes the sorts of mistakes that they make. Ideas?

In a previous post, lots of commenters said that they didn’t feel that you could give helpful, written feedback because there wasn’t enough evidence of student thinking on the quiz. Given that complaint, it might be interested to see how those same teachers would give written feedback on a quiz that gives significantly more evidence of how a student is thinking.

Here’s another quiz: what sort of written feedback would you give? (The checkmarks are from the student, who was provided with an answer key and checked her own work, ala this.)

As before, imagine that you don’t have to write a grade on this paper. Some things I’m wondering about:

• Would you give comments on every solution, or only some of them?
• Will you ask kids to “explain why you said _______”? When is it helpful to ask for an explanation? When isn’t it?
• Will you give your kids specific next steps, or will you mostly point out the good and the bad of their work?
• Will you throw up your hands and say “You really need to have a conversation with the kid!” for this sort of quiz also?

Do you see the mistake? How would you help this student?

I keep on seeing this in my Geometry classes this year. Tasked with finding the area of a right triangle, kids move toward the hypotenuse even if two of the other sides are given. Then they end up stuck looking for a height that they can’t find.

I’m pretty convinced — based on talking to kids and looking at their work — that this is all about how they see right triangles. These kids must be seeing hypotenuses as bases, and it must feel weird for them to treat the legs as bases. Or maybe instead it’s about the height? Maybe it feels strange to them to use a leg as a height?

Lots to notice here, including the formula that the student is using for the area of a triangle.