*a/k/a Dave Major Goes Bananas*

**Shorter**: Dave Major and I are experimenting again with what math textbooks could look like on devices that are digital and networked. Our most recent experiment is Ice Cream Stand.

**Longer**: Last September, Kate posted this image to Twitter attached to the tweet, “Worst geometry problem ever: can’t be solved until after you solve it.”

Clever bit, right? Classic Kate.

We could print that out and have students use a compass and straightedge to construct the circumcenter (the point that’s equidistant from all three coffee shops). That’d be a fine summative assessment. Very “real world,” etc.

But if you’d like to use Kate’s tweet to motivate the *need* for the circumcenter, to give students a reason to care about the circumcenter, we’ll need to start much lower on the ladder of abstraction. We’ll need to throw out formal vocabulary and formal operations for a few minutes. We’ll need to start with intuition.

So we changed the domain from coffee to ice cream. We changed the environment from a roadway (a complicated space) to a park (an open space). And we gave students a few easy choices. “Which ice cream stand would you pick, given where you’re standing right now?”

Students see that they’re basically painting the field one dot at a time.

So we ask them to extend that metaphor and paint the entire field so that someone else can see which stand is the closest no matter where they are in the park.

This is a task that a lot of students can complete regardless of their mathematical knowledge. It’s expensive, but not impossible, to provide this task on paper. It’s impossible to do on paper what comes next.

We combine the entire class’ park paintings.

That’s a composite from three dozen people on Twitter.

Dave and I then asked students for some preliminary thoughts about how we could calculate the right painting. But that’s where we finished. The point is, students now want to know, “Who’s right? Who’s closest?” And what’s weird is that our intuition validates the math to a degree.

That is to say, you can see areas where Twitter agreed with itself. You can see areas where Twitter disagreed with itself. When you construct the circumcenter from the perpendicular bisectors, you’ll find that they overlay rather neatly on the areas of disagreement.

That’s the ladder of abstraction. It isn’t impossible to climb it with print-based tasks, but a digital networked device makes it a lot easier.

**Open Questions**

- Q: Where does this activity go next? We could add some expository text about the circumcenter. We could leave that to the teacher. We could calculate which student took the best guess in her painting of the field. A huge open question throughout these projects is, “What role does the teacher play here?”
- Q: Another huge, open question is, “What happens to the first student who runs through this activity?” Her composite painting is just her own painting. Dave and I are developing activities that exploit the network effect. They get better and more interesting when more students use them. So again: what happens to the first student through?

**BTW**. Dave Major wrote his own post about this project.

**Featured Comments**

The burning question I have after looking at this is, why is the average line a bit wrong? (Especially the blue/green line.)

The line of uncertainty shows where the intuitive power of the brain breaks down. This is where the power of mathematical tools can step in to hone in on a more precise answer. What strikes me here is that the mathematical tools don’t do that much better of a job.

If you allow the first student through to see the picture as it gets revised (via a reload button or some auto-update), I don’t see a terrible problem (except for the usual classroom dilemma of what you do with any student that finishes fast).

## 20 Comments

## Graeme Campbell

March 18, 2013 - 2:22 pm -Love the idea! I’ll volunteer now to help with coding/creating/trialing this future endeavour :)

Q1: The teacher should formalize the discussion. They should have the ability to pull up the completed drawings of each student and then discuss the meaning with the class. Afterwards, they’ll want to do some more concrete examples. Finally, I’d suggest that the textbook then allow students to fiddle with the Maps one – provide them with a ruler and a scale and see if that changes things. If they try the same approach they’ll get practice finding a circumcenter, but they can also notice that measuring the actual walking distances may change the answer slightly.

Q2: The first student through needs to have a method of reviewing his/her work as other students finish. In a class of 25 I don’t foresee there being a huge time difference between the first and second students. Once that hits, they both get immediate feedback with regards to how right they were – the map will be pretty obviously the same, or different.

Perhaps the students who finish can move some things around and upload a new “question” to a stored database for the class. The first couple of students can try and create good questions (and answers!) and other students, as they finish, can either review their work compared to the classes or try one of the challenge problems.

Love the thinking!

## Bob Lochel

March 18, 2013 - 3:47 pm -Dan, you keep out-doing yourself, and I found myself more engaged with this activity that the previous activeprompt attempts.

As for where to go, I’d like to see a reflective piece in the document, where students can communicate their struggles/ideas/generalizations. Something like Voicethread is what I am thinking….a tool to archive my thoughts in real time as I go through the activity, and come back to later.

I visited a class recently where a teacher introduced the concept of a secant line by simply showing one on the screen. The students then individually shared their own definition for what a secant line must be using a Google Form. Between all of the student ideas, a little of this, a little of that, there was a complete definition. And the power of having every student contribute something to the group was priceless.

## Alexandre Muniz

March 18, 2013 - 3:55 pm -The burning question I have after looking at this is, why is the average line a bit wrong? (Especially the blue/green line.) Is it just this particular random sample, or is there a cognitive bias in effect here? Direct overhead views are unusual in the real world, where we tend to see things in perspective. What angle do our brains think we’re looking at when we look at maps? Can we use this information to create maps that correct for our cognitive bias? (Assuming that’s what’s going on.) How would we have to transform the map to do so? Would the bias have been the same if we didn’t start with a map (of things we would have seen in perspective if we were standing at a spot at the bottom of the map) but we instead started with an abstract diagram? What do we do for people who ask questions that are interesting but derailing?

## Evan Weinberg

March 18, 2013 - 4:45 pm -The line of uncertainty shows where the intuitive power of the brain breaks down. This is where the power of mathematical tools can step in to hone in on a more precise answer. What strikes me here is that the mathematical tools don’t do that much better of a job.

In the case of the first few people participating, I would let the computer pick a number of random points and color them accordingly. It is a bit disingenuous to tell the user in this case that this is what other ‘people’ picked, but the end result of confirming/refuting one’s intuition is the same.

## Alex Eckert

March 18, 2013 - 7:55 pm -In college we’d go play Lasertag and I hated it. I’m red/green colorblind, and the vests for Lasertag were red for one team and green for the other. I’d end up shooting my own team half the time. My only suggestion is to think of us minorities and change the ice cream stands to blue, red, and yellow.

## Dave Major

March 18, 2013 - 11:10 pm -Putting the maths to one side (music grad. and all) I’m intrigued by the reaction to this one. Each previous experiment had been framed by a blog post – my stats say that about the number of people come via Dan’s blog than via direct links. I had a smattering of emails between floating this one and Dan writing it up asking “what’s next?”, “where is this going?” etc. That felt good. Here we aren’t framing the outcome – you all knew by page two of the squares task where we were going right?

I think there is some value in not having to sit under a section heading, or at least under one heading. How cool would a textbook be where you kicked off with this and then it, depending on your responses, followed up with different possible solutions. At the end all the students would be pulled back together to examine and evaluate what everyone did.

You know, like an ‘adaptive’ digital text, previously known as a ‘half-decent teacher’.

## Bowen Kerins

March 19, 2013 - 12:40 am -While the computerized implementation is cleaner, this activity can and has been done on paper … well, sort of: on transparency paper. Give each kid or small group a transparency to dot many dots, then combine the transparencies on an overhead projector.

It works really well, you still get overlapping regions. (The same sort of work can help students visualize the solution to an equation or inequality in the plane.)

For the computerized version I would love to see this sort of “overlapping” instead of the all-kids-combine page just flying in, because I might not believe that image came from everyone. With transparencies, I have to believe it, it’s right in front of me.

Sweet stuff. Now make a similar activity for the incenter ;)

## Patrick Honner

March 19, 2013 - 4:26 am -This is definitely a cool way to approach circumcenter. Here are a couple of possible extensions to this activity.

1) Zoom out to show a fourth ice cream stand. How will the regions change? Things can get a little complicated, but this leads to some very interesting ideas (like cyclic quadrilaterals).

2) If the owner of stand A wanted to open up a second shop, where should it be located in order to draw business away from B and C but not A?

## Jason Dyer

March 19, 2013 - 6:36 am -If you allow the first student through to see the picture as it gets revised (via a reload button or some auto-update), I don’t see a terrible problem (except for the usual classroom dilemma of what you do with any student that finishes fast).

Also: agree with Alex on the colorblind issue.

@Dave: I’ve been messing around with using nonlinear lesson design on a electronic lesson. I would recommend a Choose-Your-Own-Adventure style text prototype in Twine or something of the like.

## David Wees

March 19, 2013 - 10:16 am -The paintbrushes seem a bit thick to me, so I wonder what effect this has on how carefully people choose where to shade their regions.

It seems to me that there may be another way to ask this question (after asking people to paint the field) by dragging a central point around and automatically adjusting the painted regions. See this Geogebra file I created, for example: http://davidwees.com/geogebra/icecreamstand

I think we should ask important questions like “what is a good way of representing the optimal solution for each ice cream stand” with multiple representations so that our students can abstract beyond the limitations of whichever tool they use.

I love the task, and I’d like to extend it a bit further, and to add a step to the ladder of abstraction between the painting representation, and coming up with the important information in order to be able to find a solution.

## Michael Serra

March 19, 2013 - 1:55 pm -I like the activity of guessing, collecting all the guesses, and painting regions. However I feel it is more about regions (Voronoi diagrams) rather than circumcenter. This activity leads effortlessly to asking “What about four points?” Which has nothing to do with points of concurrency but definitely back to the key idea of a perpendicular bisector dividing a region into two regions so that any point in one region is always closer to the one endpoint of the segment in that region. Which is nicely extended to 3, 4, 5, … n points. Ten years ago I probably would have asked, given two points A and B (two post offices?), how would you divide the region into two regions so that any point in the region containing point A is closer to A than B and the same for point B. Then months later toss out the problems of three points (fire stations?) and then four points (Starbucks?) in the plane. (See Discovering Geometry 4th edition page 152 exercise 11, followed by page 165 exercise 13). Rather than scaffolding, now I see the greater value of jumping right to the more perplexing problem of three or four points and let them struggle.

Was the direct investigation of looking for the point in the park that was equally distant from all three (take your pick: entrances, water fountains, bathrooms, …) points, too boring? You’d have your Act 1 with students voting, compiling all votes and see who was closest and how close the aggregate voting gets to actual circumcenter. A nice follow up for the circumcenter would be to ask is the circumscribed circle the smallest disk that can cover any triangle?

Another direct investigation that would directly get at the incenter would be to ask “What is the largest disk that you can get that would fit in a triangular region and where would you place the center of the disk?

Again, the important teacher part after the class voting is to make sure the idea is gotten across that the “answer” is not correct because that is what the majority voted for.

## Brendan Murphy

March 20, 2013 - 3:38 am -My teachers will ask why not use three transparencies one with shops A and B, A and C and B and C. On each sheet color in the half of the park which you would go to which ice cream stand. Then stack them up? Identify the point where you could go to any ice cream stand.

Would this be better or worse? How would it effect adding a 4th ice cream stand?

## James Key

March 21, 2013 - 1:21 pm -@Michael: “I like the activity…however I feel it is more about regions (Voronoi diagrams) rather than circumcenter.”

I was thinking the same thing. At no time did anyone ask, “Which point in the park is the same distance from all 3 stands?”

@Dan: “But if you’d like to use Kate’s tweet to motivate the need for the circumcenter, to give students a reason to care about the circumcenter, we’ll need to start much lower on the ladder of abstraction.”

As Michael points, out this activity did not make me “need” the circumcenter. I found the activity interesting and fun, and I’m excited about what you and Dave are working on. But this just goes to show that we can have cool digital textbooks, with engaging prompts, compiled responses, etc etc — and *still* end up falling short *if we don’t ask the right questions.* The questions should feel logical and natural for the student.

Here is an easy remedy for your task. Ask students:

#1. Which points in the park are the same distance from stands A and B? Paint ’em.

#2. Which points…from stands B and C? Paint ’em.

#3. Which points…from all 3 stands? Paint.

Now the teacher follows up with a lesson on the perpendicular bisector, answering the question “How do we know the exact right answer?”

## Dan Meyer

March 21, 2013 - 2:19 pm -James Key:Chris Robinson asked the same thing. See, the circumcenter, on its own, is of no use to the person standing in the park. This lesson may head in that direction (all you’re seeing is a preface) or in any of the three directions you outlined earlier. But the circumcenter wouldn’t be an immediate concern to anybody but a math teacher.

## Chris

March 24, 2013 - 11:17 am -I would say that the abstraction needs to be formalized. Give students some (virtual) tools to make a geometric construction. Keep the interaction, and what others did in there. Check the geometric construction (which can be done by linking to a proof checker). I think the line that is ‘off’ shows that a formalization is necessary. So I would add some visual misconceptions.

BTW, I would say there is something pseudo-contextual about this task. Of course the circumcenter is of no use for someone standing in a park, but I doubt that geometric distance to a point is the main consideration for someone any way. Things like ‘where is my car?’, ‘where do I want to go after my ice cream’, ‘are people playing football over there or not’ are probably more important in determining what stand to go to.

## Michael Serra

March 24, 2013 - 12:29 pm -An interesting issue for discussion:

“there is something pseudo-contextual about this task.”

I would submit, MOST tasks presented in our math classes are pseudo-contextural and many of these task are so by necessity.

I’ve quoted Anthony Gardiner Professor of Mathematics at the University of Birmingham on Dan’s blog once before but here it is again:

“Good mathematical problems are necessarily artificial. In contrast, “realistic” problems tend to elicit “realistic” responses involving little or no mathematics. In mathematics teaching, what matters is not whether a problem is plausibly real or artificial, but whether it is such that pupils are prepared to enter into the spirit of the mental world it conjures up.”

–A. Gardiner

I don’t think any one believes that a person in the park will pull out a map of the park, locate the three ice cream stands on the map, construct the perpendicular bisectors of the segments connecting the three points to create the three Voronoi regions and then determine which region they are standing in to decide which ice cream stand to go to enjoy their cool reward. A realistic response is just what the voting is doing, students “eyeballing” their guesses. There is no mathematics but “eyeballing” is a reasonable problem solving approach to a realistic situation.

I don’t think we need to overly concern ourselves with how realistic a situation is that we are presenting to our students. A good teacher should be able to mix pseudo-realistic, fantasy, and puzzling situations to create curiosity or perplexity, “such that pupils are prepared to enter into the spirit of the mental world it conjures up.”

## Dan Meyer

March 24, 2013 - 1:48 pm -Chrisis a student of Freudenthal who believed very similarly to Gardiner (Freudenthal: “The fantasy world of fairy tales and even the formal world of mathematics can be very suitable contexts for a problem, as long as they are real in the student’s mind.”) which makes his objection all the more curious.## Fawn Nguyen

March 25, 2013 - 4:42 pm -Jennifer Wilson asked her students of a similar task at http://easingthehurrysyndrome.wordpress.com/2013/03/13/reflections-on-the-fire-hydrant/

She also mentioned extending this question to include a 4th building (or ice cream stand as in your example).

## Lauren

May 9, 2013 - 9:25 am -From a school lacking technology– You could do this with a class set of transparency sheets overlapping. No computer required.