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.
Act Three & Sequel
- [18:36] "This guy wants to make a pyramid out of a billion pennies. And I'm curious how big that would be. Help me with that if you're completely finished here. Or tackle some of the other questions we had up there earlier."
- [20:20] "Is that number in between your high and low from earlier? Does it fit in the range of possible numbers for you? If it didn't we should go back and ask ourselves 'do we trust the mathematics here?'"
- [20:45] "I'm going to show you the answer here."
- [21:00] "Who guessed closest to that? Margaret or Eddie. Let's all give one clap to Eddie."
- [21:16] "Who got the closest guess overall? Who is closer? 250,000 or 300,000? One clap for these two."
- [21:50] "Let's look at other questions we had back here."
- [23:00] "How could we figure out how long it would take?"
Show the answer. There's the bombastic, visual element, the part that results in students cheering the answer to their math problem. It's hard for me to overvalue that reaction.
But there's another reason why students ought to see the answer to modeling tasks. (I'm not picky about answers to other tasks.) The Common Core's modeling framework asks students to "validate the conclusions" of their models. Showing the answer acknowledges the messiness inherent to mathematical modeling and allows students to discuss possible sources of error and then account for them with newer, better models.
Make good on the promises from act one. Earlier I asked students for numbers they knew were too high and too low so I asked them here to check their answer against those numbers. I said I was curious who had the closest guess so I had to find out who did and show them some appreciation. I said I hoped we would get to everybody's questions by the end of the day so I returned to those questions. If I fail to make good on any of those promises, I know they'll seem awfully insincere the next time I try to make them.
Good sequels are hard to come by. The goals of the sequel task are to a) challenge students who finished quickly so b) I can help students who need my help. It can't feel like punishment for good work. It can't seem like drudgery. It has to entice and activate the imagination.
I have one strategy I'll try on instinct: I flip the known and the unknown of the problem and see if the resulting question is at all interesting. In this case, I originally gave students the dimensions of the pyramid and asked for the number of pennies. So now I'll give them the number of pennies (one billion) and ask for the dimensions. Then I try to activate their imagination around the sequel, asking "Would you be able to build it in this room? Would it punch through the ceiling?" Etc.
In some cases, the initial task just serves to set an imaginative hook for the sequel, which is much more demanding and interesting. Once students have a strong mental image of the pyramid of pennies, I can ask them to manipulate it in some flexible and interesting ways. (Nathan Kraft has written about this recently.)
Formalize the math. Because I'm working with adults, I gave the math a brief treatment here. In general, act three is where the math is formalized and consolidated. Conflicting ideas are brought together and reconciled. Formal mathematical vocabulary is introduced.
Title the lesson. Lately, taking inspiration from this Japanese classroom, I ask students to provide a title that will summarize the entire lesson. Then I offer my own.
All of this happens at the end of the lesson, not the start. I'm not defining vocabulary at the start of the lesson and I'm not greeting students at the start of class with an objective on the board. Those moves make it harder for students to access the lesson, lofting interesting mathematics high up on the ladder of abstraction.
Here's my best guess how this kind of task would look in a print-based textbook. How does it differ from the task I did in Cambridge? Try to resist easy qualifiers like, "It's more boring," etc. How is it more boring? How is the math different? What are the downsides? What are the upsides? (I can think of at least one.)
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?
As soon as I know I have all the data, the exploring side of my brain just checks out. I go straight to my brains list of formulas and start looking for ones that will fit together to solve the problem. When I don’t have the numbers yet, I can almost feel synapses firing all over my brain.
That first sentence is sure a doozy. “A pyramid is made out of layers of stacks of pennies.” If you have a picture of what that means, then sure, it makes sense, but if you don’t, it doesn’t exactly give you a lot of clarification about what it means.
2013 May 25. James Key has created a nice visual proof of the formula for the sum of squares.