Did this kid just get excited by a coincidence? Or is there something deeper going on here?
Squaring doesn’t make equivalent fractions.
Thanks again to Gregory Taylor for the submission.
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.
Oh man, this is going to be tough for kids. Good mistake.
What makes this so hard? Or am I over-estimating its difficulty?
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!)
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.