Table of Contents

1. Teaching With Three-Act Tasks: Act One
2. Teaching With Three-Act Tasks: Act Two
3. Teaching With Three-Act Tasks: Act Three & Sequel

I get nervous when I see long-time blog readers in my workshops on mathematical modeling with three-act tasks. I tend to assume they’ll be bored. I assume that the pedagogy around these tasks has been self-evident or overly blogged-about these last few years. I should know better. It’s one thing to read about these kinds of tasks. It’s another to do one as a student. After a Saskatoon session last week, for instance, Nat Banting said that the process seemed tighter, and more engineered than he assumed from reading about it.

@dcox21 Very precise implementation. I think teachers view it as "wing it" an open ended. Very calculated.

— Nat Banting (@NatBanting) May 3, 2013

More than a few people have approached me with the impression that you simply show a photo or a video and then pursue student questions in any direction they take you. Sean Geraghty just asked me to script one of these tasks out with every question I’d ask. I’ll seize that opportunity to post some video of a session I facilitated with teachers this winter around Penny Pyramid in Cambridge and clarify what I think are the important teacher moves in a three-act math task, starting today with act one.

Act One

• [00:43] “Here it is. First, I just want you to watch this very brief video.”
• [01:27] “Would you go ahead and write down the first question that comes to your mind, if any? No question? That’s perfectly fine.”
• [01:45] “Would you introduce yourself to your neighbor and share your question? See if it’s the same question, or a different question.”
• [02:28] “I’m really curious what questions are out there. Just toss one out. Who else finds that question interesting?”
• [03:04] “I like that you coined a vocabulary term there for us. ‘Layers.'”
• [04:24] “I would love to get to all these questions but given limited time we’ll start with these ones up here.”
• [04:43] “I want you to write down on a piece of paper your best, gut-level guess for how many coins there are. I’m curious who can guess the closest.”
• [05:32] “Would you also write down a number you know is too high — there couldn’t possibly be that many pennies — and a number you know is too low — there couldn’t possibly be that few pennies. Share them with your neighbor.”
• [06:09] “I’m very curious in here who has our highest guess. “
• [06:53] “What’s our lowest guess in here?”

Post-Game Analysis

Act one attempts to lower barriers to entry. It’s visual. It requires very little literacy from the student. (Notice that I’m using very little formal mathematical vocabulary.) It’s perplexing.

Now look at the student tasks. Students are asked to to watch a video. Students are asked to pose a question. (But if you don’t have one, that’s okay!) Students are asked to decide if they find someone else’s question interesting. Students are asked to guess at a correct answer. Students are asked to decide what an incorrect answer would look like. No one is throwing a hand up saying, “I don’t know where to start.” I don’t know how to make it easier to start a modeling task than this.

I make three promises during act one.

1. I tell students I’m very curious who guessed closest to the answer.
2. I tell students I hope we’ll get around to answering all the questions on their list.
3. I ask students to set an error check on their answer.

I’ll need to make good on each of those promises by the end of act three.

I ask for student questions, but that doesn’t mean you have to. (You don’t have to do any of this of course.)

I have two competing goals in my head in act one. One, I want students to answer the question, “How many pennies are there?” Two, I want to know what questions students have when they see that stupid-huge pile of pennies.

I want to know their questions because students are interesting creatures and, while they spend a lot of time answering questions, they don’t get a lot of opportunities to pose their own. Asking for student questions orients our community around curiosity as a shared value.

But those goals are in conflict. How do you ask students for their questions while knowing, in the back of your head, the question you’re going to pursue. I know some teachers will ask for student questions and then “wait for” or “nudge students towards” the question they want to ask. I suspect this drives students crazy. It drives me crazy, this sense that there’s some question the teacher wants me to ask even while she’s insincerely asking me for my questions.

The quick way around this is to say, “Great. Love these questions. I hope we get to all of them. Here’s one I’ll need your help with first.”

Your Analysis

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 yourself to the first act of the task. We’ll pick up tomorrow where I say, “What information do you need here?”

2013 May 9. As usual, a pile of great follow-ups in the comments. Kate Nowak points out a few details that I missed in my discussion. James Cleveland suggests asking for a high and low range before the more precise guess. Great call! Lots of commenters struggle to balance asking for student questions with their curriculum objectives and I respond. So does Math Forum Max. Elaine Watson maps this task to the Standards of Mathematical Practice.

2013 Jul 15. Kevin H:

One thing I do when I ask students to guess some of the given information (like the fact that each stack is 13 pennies) is to have each student write their guess on the whiteboard and then have everyone simultaneously show one student, “Bryan.” Then Bryan is tries to ball-park an average of the numbers everyone showed him. It takes about 45 s., but they seem to enjoy the process.

Nathan Garnett, via e-mail:

I showed students a few pictures of the Luxor Hotel in Las Vegas, and then I asked them how long they it would take to wash all of those windows. We did lots of math. I had a student clean a 2ft by 5ft section of the board (with windex sound effects) so we could get a cleaning time per 10 square feet. It was a blast.

Great work. We can do something similar with Pyramid of Pennies. Students are often curious how long it took to make the pyramid. So we give students enough pennies to make the top two layers. We time them. Then they use proportions to answer how long it would take them to make the entire pyramid.

And then we list all the reasons that answer is wrong.

It doesn’t account for bathroom breaks. For sleeping. For eating. For the fact that the top two layers are small and easy but the bottom ones require scaffolding and much, much more care.

Those exceptions aren’t reasons to not ask the question. Those exceptions make the question more messy, more meaningful, more like actual modeling, and less like textbook modeling where air resistance is neglected, the rates are constant, the men are strong, and the women lithesome.

We need more messy modeling tasks like Nathan Garnett’s.