Table of Contents
- Teaching With Three-Act Tasks: Act One
- Teaching With Three-Act Tasks: Act Two
- Teaching With Three-Act Tasks: Act Three & Sequel
I taught using a three-act math task in Cambridge last winter. The good folks at NRich posted the video so I’m highlighting some of the pedagogy behind this kind of mathematical modeling. Ask questions and share suggestions.
- [07:36] “What information do you need from me? What information will be necessary here?”
- [08:36] “I want to go ahead and capitalize ‘stack’ here. Does everybody know what stack means? Tell me how stacks and layers are related.”
- [10:10] “Are all the stacks the same?”
- [10:30] “Did you use all the same coins?”
- [11:00] “What is your estimate of how many coins are in the stack?”
- [11:45] “I’m gonna add a question to the list here: ‘Why 13?’”
- [12:15] “How many on the base layer do you think?”
- [12:47] “So what’s on the next level up? 38 by 38? 39 by 39? What am I looking for if it’s 38 by 38?”
- [13:52] “That’s everything you said you needed. You asked for this info because you had some kind of fuzzy plan in your head. Might not have been a perfect plan. But you had some need for this information. So I want to see you put that information into play somehow.”
This is the guts of modeling right here. Try to find a framework for modeling in mathematics that doesn’t include a line like, students need to “identify variables that represent essential features.” If students aren’t grappling with the question, “What’s important here and how would I get it?” they may be doing lots of valuable mathematics, but they aren’t modeling.
We’re attending to precision. When students ask me for information, I press them on units or I press them to clarify what they’re after, exactly. We coin vocabulary terms like “stack” and “layer” and emphasize that we need those terms to communicate about the task.
Lots of different students get status in these tasks. We’ve done a great job convincing students that they’re good in math class if and only if they’re able to memorize operations and perform them quickly and accurately. That’s it. That’s the sum of mathematical proficiency as we’ve defined it in the US.
So I love moments when I get to compliment a student for coming up with a useful vocabulary word like “stack.” Or for asking an interesting question about the pyramid. And, for totally personal, subjective reasons, my favorite moment of the whole task comes at 10:10 when a student asks, “Are all the stacks the same?” (I explain why here.)
That is a kid who is totally unwelcome under traditional modeling curriculum. With traditional modeling curriculum all the information is given already. The problem is stretched tight. And then along comes this bored kid who amuses herself by poking at the problem, by asking about exceptions and corner cases. That kid has low status, generally. She irritates teachers.
But with actual mathematical modeling, when there isn’t any information given, we need that student’s input. Her questions about exceptions and corner cases are invaluable. And I get the chance to turn a classroom loser into a classroom hero, to compliment that student on her sharp eye, and to turn my reproachful stare on the other students and say, “Did the rest of you just assume all the stacks were the same size? You can’t just assume that stuff!”
Moments like that. What a job, teaching.
Look to the primary sources for answers and ask for guesses first. The students ask me “how many pennies are in each stack?” and “how many stacks are on the base of the bottom layer?” In both cases I could have just said the answer (“Forty stacks along the base. Thirteen pennies per stack.”) but instead I direct their attention back to the raw media, taking me out of their relationship to math and the world. I also ask for guesses on both questions. Because guesses are cheap and easy and motivating for a lot of students.
This is where I’d lecture. Because these are teachers and not students, I don’t have to do a lot of explanation. But I begin something of a lecture here, as the teachers get blocked up. They’ve done the creative work of conceptualizing the pyramid as a sum of forty squares. No one wants to crunch those numbers by hand, though.
In the last post, Yaacov asked when these kinds of problems are useful – before or after learning skills. I said they’re most valuable to me before learning skills, or rather as the motivation for learning skills. I don’t expect that students will just figure everything out on their own, though. Act one helps generate the need for the tools I can offer them here in act two.
What did you see in that clip that I didn’t talk about here? What was missing? What would you add? What would you have done differently? Go ahead and constrain your analysis to the second act of the task.