Exponential Functions Interpreting Functions Linear, Quadratic, and Exponential Models* Rational Expressions

Seeing Exponentials Where They Aint

Photo (4)


Did this kid just get excited by a coincidence? Or is there something deeper going on here?

(Thanks Tina!)

Expressions Rational Expressions Seeing Structure in Expressions

Squaring doesn’t make equivalent fractions



Squaring doesn’t make equivalent fractions.

Thanks again to Gregory Taylor for the submission.


Expressions and Equations Scientific Notation

Does Video Help Diagnose Mistakes On the Internet?

Can you see how to help this kid from this picture?


Does this video help?

What are the advantages and disadvantages of pictures and video, as far as presenting student work online is concerned?

Thanks again to Jonathan for the submission.

Equations of Parallel and Perpendicular Lines Linear, Quadratic, and Exponential Models* slope

Slope of two parallel lines

Tina says: “Two students have done this so far. Not a mistake, but still curious what these kids are thinking:”

2013-10-06 18.53.11 (1024x768)

She’s talking about the 4/6 thingy. Any ideas, people?

exponents Expressions and Equations

A tough exponents question



Oh man, this is going to be tough for kids. Good mistake.

What makes this so hard? Or am I over-estimating its difficulty?

Thanks Matt!


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!)

exponents Exponents Expressions and Equations

Another chapter in this site’s continuing chronicle of the ways exponents are hard for kids




exponents Expressions and Equations

What does the exponent stick to?



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.

Expressions and Equations Feedback Ratios and Proportions Uncategorized

Cross addition isn’t a thing

stadel proportion


Presented without comment, and with thanks to Andrew.

Expressions and Equations Fractions Solving Linear Equations

Fractions and Solving Equations



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

(Thanks Timon!)