As the person who introduced CFAR to PCK and who created the "Seeking PCK" class, I want to add two comments.

First, Duncan, this is a dynamite write-up. I like how you even dug up the detail of where the terms "partitive" and "quotitive" came from. I actually didn't know that! It's kind of obvious in retrospect.

Second, when I was at CFAR this class was a full 1+ hour class, usually on the first night. I pulled from my own experiences with clinical interviews to create an exercise that seemed to stick with quite a few participants. I'd write 24–16=12 (in vertical form, like from an American elementary subtraction algorithm) on the board, pointing out that "the student made a mistake" didn't actually result in understanding what the student did. Rather, it described what the student didn't do.

Then I'd inform them (truthfully) that I was actually running an algorithm I'd seen in my clinical interview days. (My Ph.D. is in math education.) I pointed out that if they actually understood what the student was thinking, they should be able to predict how they would perform on other similar problems.

So from that point I'd invite the whole group to toss out example problems they'd want to give this student in order to probe their understanding. We'd do them in two batches: I'd ask for problems, give everyone a chance to make predictions about how the student would answer each one, and then I'd go through and answer them. Then we'd do it one more time.

Lots of people assume the student was just doing columnwise subtraction of the larger number from the smaller number, ignoring order. So usually it kind of blows everyone's mind when they see something like 53–25=25.

Often people will object that the student "isn't being consistent" — which, again, describes what the student isn't doing rather than what they are doing.

It's sort of forehead-slapping when I reveal the algorithm. Suddenly most of the data makes tremendous sense. (I say "most" because sometimes the kid had to be creative when their method didn't apply directly to the example.) I thought it was a great demo in understanding others' minds, asking relevant questions, and epistemic humility. If I remember right, it was an example that stuck with a lot of participants too.

I guess after I left CFAR, the crew stopped using this exercise. I'm honestly not sure why. Based on Duncan's intro here, I'm guessing they just felt it wasn't as central a point for participants to experience as other things, and they figured the 80/20 here was just pointing out the existence of PCK.

(The connection, in my mind, is that the kind of precise curiosity needed to navigate the puzzle is exactly of the type needed to gather PCK.)

But really, I don't know.

I just figure folk here would like to know a bit of the history of that tidbit.

Quadrivium

Quadrivium

Quadrivium

Quadrivium