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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

Alexandre Muniz:

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

Evan Weinberg:

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.

Jason Dyer:

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).

Last June, Stanford history education professor Sam Wineburg went to Umeå University in Sweden to accept an honorary doctorate. He had prepared remarks on his recent Howard Zinn critique [pdf] but instead chose to analyze his complicated sense of relevance in the age of Twitter.

It's fascinating introspection on what counts as scholarship, how status is awarded, and how we define academic relevance in the 21st century. It's particularly interesting given that he's a tenured professor at an elite university. He's throwing stones from inside the glass house, basically. On a personal level, having felt forced to maintain something of a firewall between my Internet advocacy and my work at Stanford, it was cathartic to hear that one of my professors isn't just aware of Twitter but understands that it complicates and enriches his professional existence.

There are plenty of interesting moments throughout the 20-minute talk. Here are a few I wanted to transcribe and collect:

On publishing:

After I received tenure … I had an opportunity to reflect upon the way that my own values had become changed by the culture of the university in which I was rewarded for publishing in the most prestigious journals, whether or not those journals had any effect on anyone else except for the small number of people reading those journals.

A foundation supervisor asked him about his theory of change. His response:

Our theory of change is that we will produce materials that are better than those that are commercially bought and the commercial companies will start to see that people are downloading our materials for free and they will start to copy what we are doing and imitate what we are doing.

On the waning relevance of universities:

Our worry is that, as we continue to produce only our refereed journal articles, we are not understanding the profound changes in how information is disseminated in modern society. The university, particularly professional schools that are supposed to be producing knowledge for practitioners, are being left behind. We are becoming less and less relevant to the people who most need our knowledge — teachers and students and principals and decisionmakers in the field.

He's on Twitter. I'll encourage him (and anyone else feeling his angst) to look to Jon Becker, Cedar Riener, Sara Goldrick-Rab, and Bruce Baker as academics doing interesting advocacy on Twitter.

2013 Feb 19. Here is another disclosure from Prof. Wineburg, a brief interview on his struggle with depression.

Andrew Leonard:

All we need is one superb remedial algebra course that can be effectively delivered online and, theoretically, the demand for a zillion remedial algebra courses taught at a zillion community colleges suddenly drops off a cliff.

This hypothetical drives me up the wall, oblivious as it is to all the very interesting things that can happen in a brick-and-mortar classroom that can't yet happen on the Internet.

The Internet is like a round pipe. Lecture videos and machine-scored exercises are like round pegs. They pass easily from one end of the pipe to the other.

But there are square and triangular pegs: student-student and teacher-student relationships, arguments, open problems, performance tasks, projects, modeling, and rich assessments. These pegs, right now, do not flow through that round pipe well at all.

So I'm aggravated by the hypothetical and, especially, its seductive allure to money-men and policy-makers.

But it also energizes me. It makes our job rather clear, doesn't it?

Promote the hell out of the square and triangular pegs.

Push them into the plain view of anybody who'd love to believe math education isn't anything more than a set of round pegs ready for a trip down the round pipe.


Michael Pershan:

I'm going to commit to finding things that are intellectually taxing that are central to my teaching. It's going to require experimentation to find the right combination, but I think this search itself constitutes a sort of hard practice.

Evan Weinberg:

I need to be a lot more aware of the level of my own excitement around activity in comparison to that of the students. I showed one of the shortened videos at the end of the previous class and asked what questions they really wanted to know. They all said they wanted to know where the bird would land, but in all honesty, I think they were being charitable. They didn’t really care that much.

There you have two bloggers who are open and honest about their classroom misadventures. They don't just say to themselves, "Well, if critical feedback comes through my blog, I'm sure I'll be better off for it." They're bloggers who actively seek out that criticism. That isn't easy to do, but a career is way too short to let the Internet's vast store of criticism and insight go to waste.

Alan November, in a workshop with teachers in Hong Kong where I found myself kind of randomly today:

  • Give your students two sites that offer two competing versions of the truth. Have them determine which one is true.
  • Assign one student each day the role of official researcher. Henceforth, whenever a question arises, the researcher answers it, not the teacher. Students eventually start asking more researchable questions more often.
  • Come into class with your own research question. Tell the class you need someone on the planet who knows more than you do about it. Have them find that person.

I like the list. How adaptable are these items to a mathematics classroom?

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