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:
- making sense of precise mathematical language,
- connecting the verbal representation to a geometric representation,
- reasoning quantitatively by estimating a guess at the answer,
- reasoning abstractly by assigning a variable to a changing quantity in the problem,
- constructing an algebraic model using that variable and the formulas for the area of a square and a circle,
- performing operations on that model to find a solution,
- 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:
- your thoughts,
- 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.
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:
- How easy is using [Geogebra, Desmos, Etoys, Scratch] for kids to construct their own models?
- 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:
- If the length of AB is 1, what is the length of AP?
- What is the ratio of AP to PB?
- Given Ps, the perimeter of a square, what is the area of the square?
- Given Pc, the perimeter of a circle, what is the area of a circle?
- 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.
Here's one way this problem could begin:
- Show this video. Ask students to tell each other what's happening. What's controlling how the square and circle change?
- 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.
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.
Keith Devlin had the following to say about the original task:
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 …
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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.