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 agree. Thank you for articulating how people with good mathematical thinking also read math well. And your negative/minus hypothesis is an excellent case in point.

• kmorrow-leong

Maybe. Just maybe, the notation for subtracting negatives has poor ergonomic design. Perhaps people who have learned to read math well are similar to those who have learned to dash off Haikus or read in IPA at the drop of a hat.

• Aran

This seems to be more of a place value issue than a negative number issue. It is possible the student was correctly thinking (-9) -(-4.8) = -(9-4.8) although perhaps more intuitively than symbolically. The issue then is it not negatives but in understanding place value and/or the standard algorithm for subtracting decimals.

• Eric

I think the student was absolutely thinking correctly on this one. . . .obviously. I don’t see this as a mistake as I would be extremely joyful to see this on a math test with my ‘low’ 7th graders. They just read that the problem was -9 + 4.8 in their minds, and then did what they were probably instructed: subtract their absolute valus when their signs are different and keeping the sign the same as the number with the highest absolute value.

Whew!

• Except that -9 + 4.8 = -4.2, not -4.1. Hard to tell where the error is without more data about the student…