What feedback would you write on this kid’s paper? Why?

(Thanks, KN!)

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!)

@mpershan Can’t tell you how many times I saw this mistake this quiz. Yikes! twitter.com/mr_stadel/stat…

— Andrew Stadel (@mr_stadel) March 17, 2013

What’s the fastest way to help this kid?

Incidentally:

@mpershan Yes, please post. Visually, if most of them graphed the line would identify +slope. Abstractly, they either rush or forget it’s +.

— Andrew Stadel (@mr_stadel) March 18, 2013

@mpershan I need to encourage them more to make a sketch of the 2 pts first in order to visualize and identify the slope first.

— Andrew Stadel (@mr_stadel) March 18, 2013

Where does this mistake come from? I mean, the kid knows that 1 – 1 = 0, right? So does the kid think that 1 – (-1) = 0 too, or does the kid misconstrue this as 1 – 1? What’s your theory?

Thanks to Chris Robinson for the work sample.

Say something smart in the comments, and then go hang out at Chris Robinson‘s place.

What’s something that interests you about this student’s work?

Another quality submission from John Weisenfeld.

Say something smart about this in the comments. When you’re done, go check out the blog from whence this mistake came.