dy/dan

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 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 Two

• [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."

Post-Game Analysis

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.

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 your analysis to the second act of the task.

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.

Couldn't make it to NCTM this year? We're in the middle of recapping the conference over at MathRecap.com, one session per day, including:

• Forrest Clark on standards-based grading.
• Christopher Danielson on what students will and won't need for calculus in high school.
• Ryan Higgins on Socratic seminars in mathematics.
• Gary Rubinstein on the crossover between math and astronomy.
• Marc Garneau on visualization in math, and
• Mayim Bialik's opening keynote session.

Including handouts, slides, and other resources. More coming this week.

Lots of thanks to our recappers: Benjamin Graber, Raymond Johnson, Brandon Price, and Tanis Thiessen.

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.

Daniel Schneider, in a must-read piece:

I believe that standards-based grading, at its fundamental level, is only changing your gradebook so you grade individual standards. However, this change forces you to face realities about a traditional classroom that you can’t ignore and that you are forced to react to.

If this piece were only about the implementation of standards-based grading, it'd be indispensable. If you're thinking about making a constructive change to how you grade and treat your students, you should read the Schneider's how-to guide.

But it's also about changes Schneider made from year one to year two in that implementation, which makes it rarer and more valuable among all the SBG literature you can find.

But he also diagnoses how this one change to assessment then rolls along and affects every other aspect of his classroom. Curriculum, homework, relationships, the definition of math itself — nothing is spared. Assessment is only the first domino.

It's the best examination of the classroom as a thriving, codependent ecosystem I've read in a long while.