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.
Ah, and speaking of history, one other detail about PCK y’all might like to know:
I introduced the idea to CFAR back in the summer of 2012 during an internal colloquium talk series. It helped define a lot of how we thought about unit creation thereafter. We talked explicitly about PCK for various units, and for running the workshop as a whole, for the rest of the time I was there.
The “Seeking PCK” unit didn’t exist until quite a bit later. I don’t remember when we introduced it honestly. Maybe 2016?
But the basic idea is part of CFAR’s memetic DNA at this point. We even talked explicitly about how to transfer PCK between people when we were handing a unit off to a new teacher.
Seeking PCK was a full (hour or longer) class at every mainline workshop since October 2016 (sometimes called “Seeking Sensibility” or “Seeking Sense”). After you left it was always a full hour+ class, almost always taught by Luke, and often on opening night.
The concept of PCK became part of the workshop content in April 2014 as a flash class (as a lead-in to the tutoring wheel, which was also introduced at that workshop). In October 2016 we added the full class, and then a couple workshops later we removed the flash class from the workshop. Something very close to this chapter made it into the first draft of the CFAR handbook in May 2016, when PCK was still just a flash class, and I guess the chapter didn’t ever get expanded or moved.
After the class was transferred from Val to Luke, Luke was involved in teaching it until the last pre-covid workshop in January 2020. I’m pretty sure he kept the subtraction exercise (that exercise wasn’t removed from the handbook, it just never made it in). A couple other people also taught the class at some point (including Duncan in April 2017), I’d guess at workshops where Val or Luke was absent.
At the January 2020 workshop a new instructor was learning the class & taught some of it along with Luke. I suspect that’s why it was part of the day 1 rotation that workshop rather than being opening night (since it’s helpful for a new instructor to have repetition & smaller groups, and a new instructor’s version of a class hasn’t necessarily cohered enough to be ready to set the tone for the workshop on opening night).
(This history mostly based on records I looked up, supplemented by memory.)
I honestly don’t know how to make the subtraction example work in a handbook format. It really does best as something interactive. A lot of the punch of it evaporates if folk don’t get a chance to encounter their confusion after getting their own prompts answered.
Both I and Luke Raskopf tried our hand at teaching Seeking PCK as a full class, and (in my opinion) did a decent job—perhaps 85% as effective as what you were doing.
After that, though, it began to shrink. EDIT: See Unnamed’s comments above.
If you’re interested in fleshing out the writeup à la the other full-class entries, I would happily include it as the full class that it indeed was. I just discovered that Turbocharging was also skipped over in similar fashion because of having been moved to a “retired” section of the version of the handbook I’ve been working from.
The point was usually to illustrate a variety of different mental motions that a young mathematician might be making, OTHER than the one intended by the algorithm. So all sorts of examples are possible, and there were a number of things found in clinical interviews. Not just instances of one single pattern.
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.
Ah, and speaking of history, one other detail about PCK y’all might like to know:
I introduced the idea to CFAR back in the summer of 2012 during an internal colloquium talk series. It helped define a lot of how we thought about unit creation thereafter. We talked explicitly about PCK for various units, and for running the workshop as a whole, for the rest of the time I was there.
The “Seeking PCK” unit didn’t exist until quite a bit later. I don’t remember when we introduced it honestly. Maybe 2016?
But the basic idea is part of CFAR’s memetic DNA at this point. We even talked explicitly about how to transfer PCK between people when we were handing a unit off to a new teacher.
Seeking PCK was a full (hour or longer) class at every mainline workshop since October 2016 (sometimes called “Seeking Sensibility” or “Seeking Sense”). After you left it was always a full hour+ class, almost always taught by Luke, and often on opening night.
The concept of PCK became part of the workshop content in April 2014 as a flash class (as a lead-in to the tutoring wheel, which was also introduced at that workshop). In October 2016 we added the full class, and then a couple workshops later we removed the flash class from the workshop. Something very close to this chapter made it into the first draft of the CFAR handbook in May 2016, when PCK was still just a flash class, and I guess the chapter didn’t ever get expanded or moved.
After the class was transferred from Val to Luke, Luke was involved in teaching it until the last pre-covid workshop in January 2020. I’m pretty sure he kept the subtraction exercise (that exercise wasn’t removed from the handbook, it just never made it in). A couple other people also taught the class at some point (including Duncan in April 2017), I’d guess at workshops where Val or Luke was absent.
At the January 2020 workshop a new instructor was learning the class & taught some of it along with Luke. I suspect that’s why it was part of the day 1 rotation that workshop rather than being opening night (since it’s helpful for a new instructor to have repetition & smaller groups, and a new instructor’s version of a class hasn’t necessarily cohered enough to be ready to set the tone for the workshop on opening night).
(This history mostly based on records I looked up, supplemented by memory.)
Yep, this all sounds right to me.
I honestly don’t know how to make the subtraction example work in a handbook format. It really does best as something interactive. A lot of the punch of it evaporates if folk don’t get a chance to encounter their confusion after getting their own prompts answered.
Both I and Luke Raskopf tried our hand at teaching Seeking PCK as a full class, and (in my opinion) did a decent job—perhaps 85% as effective as what you were doing.
After that, though, it began to shrink.EDIT: See Unnamed’s comments above.If you’re interested in fleshing out the writeup à la the other full-class entries, I would happily include it as the full class that it indeed was. I just discovered that Turbocharging was also skipped over in similar fashion because of having been moved to a “retired” section of the version of the handbook I’ve been working from.
FYI I think even the current form of PCK described here feels large enough to be it’s own post.
Don’t leave us hanging… Could you please provide one or two more examples?
34-16=12?
63-17=11?
The point was usually to illustrate a variety of different mental motions that a young mathematician might be making, OTHER than the one intended by the algorithm. So all sorts of examples are possible, and there were a number of things found in clinical interviews. Not just instances of one single pattern.
34–16 yields 13
63–17 yields 16
For the four examples of
24-16=12, 53-25=25, 34-16=13, 63-17=16
is this the pattern?
ab-cd=ca