Building Functions Negative Numbers Sequences and Series Series The Real Number System

Arithmetic Series and Subtracting Signed Numbers


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

(Thanks, KN!)

Negative Numbers The Number System

(-9) – (-4.8) = …



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!)

Expressions and Equations Feedback linear functions Negative Numbers The Number System

Slope and Division of Negative Numbers


What’s the fastest way to help this kid?



Negative Numbers The Number System

1 – (-1) = …


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.


Negative Numbers The Real Number System

1 – (-1)



Thanks again to Chris Robinson.

Negative Numbers The Number System

Adding and Subtracting Negative Numbers

Wha? Explain in the comments.

And thanks again to Chris Robinson for the submission.


Algebra 1 Negative Numbers The Number System

Positive and Negative Integers

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

Feedback Negative Numbers The Number System

Negative Numbers

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

Another quality submission from John Weisenfeld.

Arithmetic Negative Numbers The Real Number System

Order of Operations, Negative Numbers

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

Negative Numbers The Number System

Arithmetic with negative numbers

Why is this mistake so attractive for students to make? If you saw this mistake in your classroom, what would your next step be?