More and more these days, when I look at student work I’m just using it as a jumping off point for anything that I find interesting. When we started this project last June, I was only looking to explain how the student ended up writing what she did, but these days that requirement seems sort of restrictive. Different pieces of student work are interesting for different reasons, and what interests us is going to vary anyway.

To me, this mistake raises the possibility that it was a reading error. Reading errors tend to get poo-pooed by teachers — along with procedural errors, “stupid” mistakes, and guesses — as the results of non-mathematical issues. Either the kid was rushing, or the kid wasn’t thinking, or the kid was sloppy, etc.

Maybe that’s right. But it also seems to me that as you get better at math you get better at noticing the structure of these sorts of questions. You know what details are crucial, you eyes start to dart in different ways, you chunk the expression differently.

In other words, you learn how to read mathematically. And while some people would prefer to distinguish between mathematical knowledge and mathematical conventions and language, such distinctions don’t really do much for me. Being able to parse mathematical language seems bound up with mathematical knowledge.

In summary: A lot of the things that we call “reading errors” or “sloppiness” are really issues in mathematical thinking.

In this case I’ll offer a testable hypothesis: People who don’t really get how negative numbers work don’t see a distinction between subtraction symbols and negative signs, and will tend to elide them in reading a problem. People who do get negative numbers immediately read the numbers, along with their sign, and then read the operation between them.

(Three cheers to Andrew for the submission!)

Thoughts?

I know the pic is a bit small, but can you see the mistake? It all has to do with what the exponent applies to. Somewhere on the internet one of you wrote about how you tell kids that “the exponent only sticks to one thing.” This mistake is about just that.

Thanks to Gregory for the submission.

Presented without comment, and with thanks to Andrew.

Offered to you without comment. Say something interesting in the comments.

(Thanks Timon!)

I just dug this up. It’s what I handed back students after a “pre-quiz” (i.e. a quiz at the end of the unit, but before their quiz). I had forgotten that during that first year I handed back these things with class performance percentages on them.

Anyway, the way those percentages break down is interesting to me. Is it surprising that kids had so much trouble with negative exponents in numerical context, but had such less trouble with variables?

Last week I posted a short video from a tutoring session I had with a kid. We were solving equations, and he had some interesting ideas, and it was nice to have those ideas and his mental workings become explicit.

Here’s another chunk of that video:

Comment on whatever you like, but here are some prompts:

1. Help me understand his thinking. How did he devise his test for whether his solution is correct?

Or jump in with whatever you like in the comments.

Comment on anything that you like, but here are some prompts:

• Do you like my questioning? (Because I didn’t. Notice that moment when I pause and try to unask my question?)
• Describe, as best you can, the way that this kid thinks about the Distributive Property.
• What would you say next, if you were me?
• Does this video have implications about the way that you’d teach this topic? (And, come on, don’t give me that “they need lots of practice stuff.” Of course they do. But what else?)
• Do you prefer having videos over images on this site? (Because I have about a half hour more footage from my work with this kid…)

Looking forward to a great bunch of comments here. Don’t let me down?

Update:

Here’s how I responded:

Why does this student answer $2a + 9$ for the first question, and $x + 49$ for the second questions?

Why does this student express more confidence on the first question than the second question?

This post is part of a series analyzing a bunch of survey results. For previous posts, go here, here and here.