Fractions Negative Numbers The Number System

Negative numbers and fractions

photo (2) (800x600)


Would this student also say that 1/2 is equal to 2?

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!)

The Number System

2 and 1/5 squared makes…



This comes via Andrew. Thoughts?

Fractions The Number System

1 divided by 2/3 is…

1 divided by two thirds

I’m not sure that I agree with the teacher’s diagnosis that this sort of mistake is procedural. Presumably, she means that the mistake was the result of remembering some sort of algorithm, a wrong algorithm. But what sort of division algorithm would lead you to multiply 2 and 3? in this situation?

On twitter, someone suggested (was it you?) that the issue here was that 1 divided by 1/3 is 3. And then twice that would be 6.  This would then be an instance of a more common pattern of error, the “it’s always linear” error. (Was it Dave who said this? Maybe this was Dave.)

This actually fits pretty well with the student’s explanation. It’s not a bad take.

Do you agree? The first three times that I wrote this post I said something like “The ‘linearity’ hypothesis is a pretty good one, but it doesn’t quite fit with what the student said. I’d suggest that the student had an association between the numbers 1, 2, 3 and 6, and it’s that instant judgement that her explanation is aiming to justify.” But now I’m not sure if I can really find any fault with the linearity hypothesis.

People who have more experience with fractions than I have: is the linearity explanation one that resonates with your experience of kids learning to divide fractions?

(P.S. I’ve got 4th and a 5th grade assignments, along with my high school classes this year. I’m excited to bulk up my understanding of little person stuff. Any of your submissions would help me test the theory that analyzing student work is a solid way to help build pedagogical content knowledge. SUBMIT!)

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.


Fractions The Number System

Chocolate Bars and Bad Friends



What’s the fastest way to help this 9th grade student?

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.

Arithmetic The Number System

Sad faces and Negative Numbers

There’s some context that’s necessary for this post. So go read Nathan Kraft’s post about arithmetic with negative numbers.

What’s the mistake? Thoughts?

This piece of student work is kindly submitted by 3-act maven Chris Robinson.