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## [Confab] Circle-Square

The following problem has obsessed me since I first heard about it several months ago from a workshop participant in Boston. I believe it originates from The Stanford Mathematics Problem Book, though I’ve seen it elsewhere in other forms.

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

Here’s why I’m obsessed. In the first place, the task involves a lot of important mathematics:

1. making sense of precise mathematical language,
2. connecting the verbal representation to a geometric representation,
3. reasoning quantitatively by estimating a guess at the answer,
4. reasoning abstractly by assigning a variable to a changing quantity in the problem,
5. constructing an algebraic model using that variable and the formulas for the area of a square and a circle,
6. performing operations on that model to find a solution,
7. validating that solution, ensuring that it doesn’t conflict with your estimation from #3.

Great math. But here’s the interesting part. Students won’t do any of it if they can’t get past #1. If the language knocks them down (and we know how often it does) we’ll never know if they could perform the other tasks.

What can you do with this? How can you improve the task?

I’m going to update this post periodically over the next few days with the following:

• two resources I’ve created that may be helpful,
• commentary from some very smart math educators on the original problem and those resources.

Help us out. Come check back in.

Previous Confab

The Desmos team asked you what other Function Carnival rides you’d like to see. You suggested a bunch, and the Desmos team came through.

Man did you guys came to play. Loads of commentary. I’ve read it all and tried to summarize, condense, and respond. Here are your big questions as I’ve read them:

• Is learning to translate mathematical language the goal here? Or can we exclude that goal?
• What role can an animation play here? Do we want students to create an animation?
• What kinds of scaffolds can make this task accessible without making it a mindless walk from step to step? On the other end, how can we extend this task meaningfully?

There was an important disagreement on our mission here, also:

Mr. K takes one side:

It took me about 3-4 minutes to solve – the math isn’t the hard part. The hard part is making it accessible to students.

Gerry Rising takes the other …

If we want students to solve challenging exercises, we should not seek out ways to make the exercises easier; rather, we should seek ways to encourage the students to come up with their own means of addressing them in their pristine form.

… along with Garth:

Put it to the kids to make it interesting.

I’ll point out that making a task “accessible” (Mr. K’s word) is different than making it “easier” (Gerry’s). Indeed, some of the proposed revisions make the task harder and more accessible simultaneously.

I’ll ask Gerry and Garth also to consider that their philosophy of task design gives teachers license to throw any task at students, however lousy, and expect them to find some way to enjoy it. This seems to me like it’s letting teachers take the easy way out.

Lots of you jumped straight to creating a Geogebra / Desmos / Sketchpad / Etoys animation. (Looking at Diana Bonney, John Golden, Dan Anderson, Stephen Thomas, Angelo L., Dave, Max Ray here.) I’ve done the same. But very few of these appleteers have articulated how those interactives should be used in the classroom, though. Do you just give it to your students on computers? To what end? Do you have them create the applet?

Stephen Thomas asks two important questions here:

1. How easy is using [Geogebra, Desmos, Etoys, Scratch] for kids to construct their own models?
2. When would you want (and not want) the kids to construct their own models?

My own Geogebra applet required lots of knowledge of Geogebra that may be useful in general but which certainly wasn’t germane to the solution of the original task. It adds “constructions with a straightedge and compass” to the list of prerequisites also, which doesn’t strike me as an obviously good decision.

Lots of people have changed the wording of the problem, replacing the mathematical abstractions of points and line segments to rope (Eddi, Angelo L) and ribbon (Lisa Lunney Borden) and fencing (Howard Phillips).

This makes the context less abstract, yes, but the student’s work remains largely the same: students assign variables to a changing quantity on the line segment, then construct an algebraic model, and then solve it. The same is true for some suggestions (though not all) of giving the students actual rope or ribbon or wire.

So I’m interested now in suggestions that change the students’ work.

Kenneth Tilton proposes a “stack” of scaffolding questions:

1. If the length of AB is 1, what is the length of AP?
2. What is the ratio of AP to PB?
3. Given Ps, the perimeter of a square, what is the area of the square?
4. Given Pc, the perimeter of a circle, what is the area of a circle?
5. How would you express “the two areas are equal” algebraically?

The trouble with scaffolds arises when a) they do important thinking for students, and b) when they morsel the task to such a degree it becomes tasteless. Tilton may have dodged both of those troubles. I don’t know.

David Taub lets students choose a point to start with. Choosing is new work.

Mr K asks students to start by correcting a wrong answer. Correcting is new work.

It seems to me that a simple model of the problem, (picture of a string) with a failed attempt (string cut into two equal parts) should be enough to pique the kids “I can do better” mode. Providing actual string with only one chance to cut raises the stakes above it being a guessing game.

I think more important would be to start with some “random” points and some concrete numbers and see what happens.

Max Ray builds fluency in mathematical language into the end of the problem:

So I would have my students solve the problem as a rope-cutting problem. Then I would invent or find a mathematical pen-pal and have them try to pose the rope problem to them.

If our mathematical language is as efficient and precise as we like to believe, its appeal should be more evident to the students at the end of the task than if we put it on them at the start of the task.

Gerry Rising offers us an extension question, which we could call “Circle-Triangle.” I’d propose “Circle-Circle,” also, and more generally “Circle-Polygon.” What happens to the ratio on the line as the number of sides of the regular polygon increases? (h/t David Taub.)

On their own blogs:

• Justin Lanier offers a redesign that starts with a general case and then becomes more precise. I’m curious about his rationale for that move.
• Jim Doherty runs the task with his Calculus BC students and reports the results.
• Mike Lawler gives us video of his son working through the problem.

2014 Feb 26. Some of my own resources.

Here’s one way this problem could begin:

1. Show this video. Ask students to tell each other what’s happening. What’s controlling how the square and circle change?
2. Then show this video. Ask students to write down and share their best guess where they are equal.

The problem could then proceed with students calculating whether or not they were right, formulating an algebraic model, solving it, checking their answer against their guess, generalizing their solution, and communicating the original problem in formal mathematical language.

Mr. K has already anticipated my redesign and raised some concerns, all fair. My intent here is more to provoke and less to settle anything.

I’m going to link up this video also without commentary.

2014 Feb 27. Other smart people.

I asked some people to weigh in on this redesign. I showed the following people the original task and the videos I created later.

• Jason Dyer, math teacher and author of the great math education blog Number Warrior.
• Keith Devlin, mathematician at Stanford University.
• Two sharp curriculum designers on the ISDDE mailing list, whose comments I’m reproducing with permission.

Here’s video of a conversation I had with Jason where he processes and redesigns the original version of the task in realtime. It’s long, but worth your time.

I immediately drew a simple sketch – divide the interval, fold a square from one segment, wrap a circle from the other, and then dive straight into the algebraic formulas for the areas to yield the quadratic. I was hoping that the quadratic or its solution (by the formula) would give me a clue about some neat geometric solution, but both looked a mess. No reason to assume there is a neat solution. The square has a rational area, the circle irrational, relative to the break point.

So in the end I just computed. I got an answer but no insight. I guess that reveals something of a mathematician’s meta cognitive arsenal. You can compute without insight, so when you don’t have initial insight, do the computation and see if that leads to any insight.

In the case of the obviously similar golden ratio construction, the analogous initial computation does lead to insight, because the equation is so simple, and you see the wonderful relationship between the roots

So in one case, computation just gives you a number, in the other it yields deep understanding.

Off the ISDDE mailing list, Freudenthal Institute curriculum designer Peter Boon had some useful comments on the use of interactives and videos:

I would like to investigate the possibility of giving students tools that enable them to create those videos or something similar themselves. As a designer of technology-rich materials I often betray myself by keeping the nice math (necessary for constructing these interactive animations) for myself and leaving student with only the play button or sliders. I can imagine logo-like tools that enable students to create something like this and by doing so play with the concept variable as tools (and actually create a need for these tools).

Leslie Dietiker (Boston University) describes how you can make an inaccessible task more accessible by giving students more work to do (more interesting work, that is) rather than less:

If the need for the task is not to generate a quadratic but rather challenge students to analyze a situation, quantify with variables, and apply geometric reasoning with given constraints, then I’m pretty certain that my students would appreciate a problem of cutting and reforming wire for the sake of doing exactly that …

I disagree with people who are saying that this problem as written is inherently bad or artificial. As an undergrad math major, a big part of the learning for me was figuring out that statements worded like this problem were very precise formulations of fundamental insights — insights that often had tangible models or visualizations at their core.

I remember lectures about knots, paper folding, determinants, and crazy algebras that the lecturers took the time to connect to interesting physical situations, or even silly but understandable situations about ants taking random walks on a picnic blanket. For a moment I even entertained the idea of graduate work in mathematics, because I realized that math was actually a pretty neat dance between thinking intuitively and thinking precisely.

Terrence Tao writes about that continuum here.

tl;dr version: Translating this problem from precise to intuitive and intuitive to precise, is part of the real work that research mathematicians (and their college students) do, and not something we should always keep from our students. It’s a skill we should help them hone.

2014 Mar 4. As usual Tim Erickson got here first.

## [Fake World] Limited Theories of Engagement

Let’s just call them “theories of engagement” for now. Every teacher has them, these generalized ideas about what engages students in challenging mathematics. Here’s the theory of engagement I’m trying to pick on in this series:

This theory says, “For math to be engaging, it needs to be real. The fake stuff isn’t engaging. The real stuff is.” This theory argues that the engagingness of the task is directly related to its realness.

This is a limited, incomplete theory of engagement. There are loads of “real” tasks that students find boring. (You can find them in your textbook under the heading “Applications.”) There are loads of “fake” tasks that students enjoy. For instance:

No context whatsoever in any of them. Perhaps the relationship actually looks more like this:

I’m being a little glib here but not a lot. Seriously, none of those tasks are “real-world” in the sense that we commonly use the term and yet they captivate people of all ages all around the world. Why? According to this theory of engagement, that shouldn’t happen.

Here are fake-world math tasks that students enjoy:

My point is that your theory of engagement might be limiting you. It might be leading you towards boring real-world tasks and away from engaging fake-world tasks.

We need a stronger theory of engagement than “real = fun / fake = boring.”

Homework Time!

Choose one:

• Write about a fake-world math task you personally enjoy. What makes it enjoyable for you? What can we learn from it?
• Write about an element that seems common to those enjoyable fake-world tasks above.

## Teaching With Three-Act Tasks: Act Three & Sequel

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 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?”

Post-Game Analysis

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.)

What’s Missing

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.

Homework

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.

## Teaching With Three-Act Tasks: Act Two

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.

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.

## [LOA] Hypothesis #5: Bet On The Ladder, Not On Context

#5: Kids care less about context — “real world” problems — than they do about problems that start at the bottom of the ladder. “Real world” is a risky bet.

Real World

Here is a “real world” problem:

The caterers Ms. Smith wants for her wedding will cost \$12 an adult for dinner and \$8 a child. Ms. Smith’s dad would like to keep the dinner budget under \$2,000. Ms. Smith would like to invite at least 150 guests to her wedding. How many children and adults can Ms. Smith invite to her wedding while staying within budget?

There is nothing to predict. Nothing to compare. The important information has already been abstracted. The question has been fully defined. The problem, as a whole, has been stretched tight and nailed to a board. The student’s only task is to represent the important information symbolically and then apply some operations to that representation.

And so hands go up around the room. The students attached to those hands say, “I don’t know where to start.” The task has hoisted them up to a middle rung on the ladder of abstraction and left their feet dangling in the air. Students are frustrated and disengaged in spite of the “realness” of the task.

Fake World

Meanwhile here is a “fake world” problem:

1. What are the new percents? Write down a guess.
2. Which quantities change?
3. Which quantities stay the same?
4. What names could we give to the quantities that are changing?

These questions include students in the process of abstraction. Each student guesses the new percents and is consequently a little more interested in an answer. Students aren’t just asked to accept someone else’s arbitrary abstraction [pdf] of the context. They get to make their own arbitrary abstraction of the context. (Why ABCD? Why not WXYZ?) All of these tasks prepare them to work at higher levels of abstraction later.

Solution

My preference is a combination of the two — a context that is real to students and a task that lets them participate in the abstraction of that context.

But I can’t tell you how many conversations I’ve had with teachers (veteran and new) and publishers (big and small) who tell me the fix for material that students don’t like is to drape some kind of context around the same tasks. Rather than expanding and enriching their tasks to include the entire ladder of abstraction, they insert iPads or basketballs or Justin Bieber or whatever they perceive interests students.

Real-world math is a risky bet. Bet on the bottom of the ladder. Here are some of those bets:

1. With the wedding task above, the teacher can ask students to pick any combination of children and adults they think will work. Any combination. 100 kids and 50 adults? Fine. Now tell me how much it costs. We’re all invested for a moment in a problem of our own choosing. Then we assemble student work side-by-side and notice that we’re all doing the same kind of calculations. Then we say, “All your work looks the same. What’s happening every time?” The students are participating in the symbolic abstraction.
2. Louise Wilson is using the images and videos on 101questions to give students practice just asking questions about a context. Asking questions is the assignment. Getting answers isn’t.
3. Andrew Stadel is giving his students daily practice with estimation, another task at the bottom of the ladder.

We ask our students to work most often at the top of the ladder and the result is a pervasive impression that a successful math student is a student who can memorize formulas and implement them quickly and correctly. Those are, of course, great and useful skills, but mathematicians also prize the ability to ask good questions, make good estimations, and create strong abstractions. These are skills where other students may excel. There is unrewarded excellence in our math classrooms because we have defined excellence narrowly as being good at abstract skills. You can only find (and then reward) that excellence by betting on the bottom of the ladder of abstraction.