This come via Lois Burke on twitter, and immediately Max shows up with a possible explanation.

Dave has a different idea. Maybe the student was thinking in words — “5 and minus 1” — and this turns into its homonym “5-1.”

Personally, what I have the easiest time imagining is that the student just had “combine 5 and -1” on their mental ledger. When it came time to address that ledger, there was so much other stuff they were paying attention to that they slipped into the most natural sort of way to combine numbers they had, which is adding. (I like the metaphor of slipping. You’d very rarely see a kid slip in the other direction — from 5 + (-1) to 5 x (-1) — I think. There is a direction to this error.)

Here are the activities we came up with to help develop this sort of thinking in class. Ideas for improvement? More ideas? Other explanations of the student’s thinking?

UPDATE:

Pam Harris has an idea:

Love it. Here’s a digital version.

John Golden point out that there might be issues with the Which One Doesn’t Belong puzzle, so I offer this as an alternative.

John also offers a different problem string: “I’d be curious to see 5+i, 5+i^2, 5+i^3, 5+i^4, 5i, 5i^2, 5i^3, 5i^4.”

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

(Thanks, KN!)

One of my little obsessions is teaching complex numbers, but it’s really hard to find genuine instances of complex number mistakes. You start looking around at the most pernicious complex number mistakes, and they’re a lot like this one here: essentially algebraic. These mistakes, to my mind, are indistinguishable from the sorts of mistakes you’d expect from $(4-6x)^2.$

That observation is probably helpful in itself, though. The mistakes that we see from kids working with complex numbers are essentially algebraic mistakes. That means that kids aren’t really seeing much of a difference between the algebra that they’re usually asked to do and their work with complex numbers. Complex number arithmetic is just algebra with a twist.

These mistakes are mine. I was tasked with checking the associative property with matrix multiplication, i.e. checking that (AB)C = A(BC).

This was my first attempt. See the problem?

By my estimation, this is a fairly straightforward mistake. I was explicitly trying to remember a routine, and I remembered the wrong one. I don’t have any particularly deep appreciation for how matrix multiplication works, so there was nothing except memory of the routine for me to draw on.

(I have vague recollections of the concept that matrices represent functions from various real spaces, but nothing close to deep enough to ground this algorithm.)

I looked up a few examples, and got myself back on track. Still, I’m not particularly adept at matrix multiplication at the moment, and I found it cognitively taxing. (That’s a feeling that I can distinctively recognize. I feel it most often when I’m doing calculations or attempting to recall lots of numbers. It’s a feeling of overflowing.)

I made a mistake that I’d characterize as a working-memory issue. Do you see it below? I even circled it.

So far we have two sorts of mistakes in this problem: a mistaken routine mistake and a overtaxed working memory mistake. (Those names aren’t great. But you know what I mean.)

What else is there to discuss?

Well, what about the particular working memory mistake that I made? In my calculations I put a “10” where an “8” ought to be. In my mind, I was attempting 1*2 + 2*3. Why did I come up with 10 instead of 8? Was it mere chance? Could I just as easily have come up with 7, or 23?

I have a pet theory, one that I have no clue how to prove. I believe that there are certain numbers that we remember as being especially connected. To our minds, I’d suggest, the numbers 100 and 1/2 are especially connected with 50 — and not with 99.5, 200 or 100.5. Certain numbers are clustered with others, with varying degrees of strength, and in various moments our responses can reveal these deep connections between the numbers themselves.

Of course, I have no evidence for this pet theory. I’m not sure how to test it, though I’m wondering if some variant of what I tried with my friend last week might be workable.

Thanks to Tina, we’ve got this great example of a tiny little error that crops up during complex numbers. Here’s my take on it: there’s no way that this kid would make this mistake if their problem was just “Simplify the square root of negative 4.”  When the skill is laid forth is such a direct way, it’s very clear what the student is supposed to do. But when the skill is embedded in a much more complex problem, the student “handled” the negative root by realizing that this was a context that deserved a complex number. Happy and satisfied that they noticed and “handled” every aspect of the problem, the student moved on.

I like calling these sorts of mistakes “local maxima” mistakes, and I think they’re fairly common. To me, the importance of these sorts of mistakes is that they reveal the problem with testing any skill in isolation of others. I’m <i>absolutely sure</i> that this student could simplify the square root of negative four if plainly asked to. But  that didn’t mean that this student was able to use that skill in this context, when there are many more things to juggle.

To me, this means that you can’t really assess any individual skill in that sort of isolation. Instead, I’d prefer an assessment system that gives students a bunch of chances to use a skill — unprimed — in the context of a fairly difficult problem. If the student can simplify negative radicals in 3-4 more involved problems, then I’m pretty confident that this kid has that skill down.

What make you of this?

Thanks again, Tina!