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

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.”

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.

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?”

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.

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?”

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

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.