A few weeks ago I gave the following problem to my students:

I had them find the answer to the logic puzzle and then write a convincing “proof” of why their answer was the correct one. This was their first “proof” and the idea was to have them work in small groups to workshop their proof-writing skills. I put students in groups of 3 and had them read over each other’s proofs, talk about what worked well, what could be improved and then collaborate on a group write-up of the proof that incorporated the good things they were already doing with the suggestions they had made to each other. The whole activity went really well, if I do say so myself.

But this was my favorite part.

One student, Rosie, submitted this as her answer:

Her group members were adamant—that’s not what it means! It doesn’t work like that! One of them asked me to arbitrate. Who was *RIGHT?*

Well, as far as I know, there is no mathematically rigorous definition for the word “across”. So I said to him: “It looks like Rosie had a different assumption that you for what it means to be across the table. Is it that Rosie is *wrong*, or is it that the instructions aren’t clear enough?”

A thoughtful expression crossed the student’s face. “You’ve got me this time, Bree.”

I checked in with the group a few minutes later. They’d decided to define “across” as meaning along the diameter of the circular table[1]. Everyone was okay with this definition. More importantly no one felt like Rosie’s idea was inferior or that she wasn’t smart to have decided on a different definition earlier. They just needed to make a decision so that they were able to communicate effectively. And so they did.

[1] I think the basis for deciding this way was along the lines of “majority rules”. Not the most mathematically based reason, but oh well. I would have loved a discussion about how using the diameter limits the options for who is “across” from whom, but you can’t have everything.

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Interesting. Sounds well played on your part.

If you rotate the graphic 15 degrees, would Rosie be as likely to see E and C as

acrossfrom each other? Is her view an artifact, in other words, of the horizontal orientation of that “diagonal”?Good question. I think the rotation would likely change Rosie’s perception. Though I’m not sure. She was describing the picture in terms of a rectangular table. I’m guessing she has a rectangular table at home, where presumably she was working on this. The funny thing is, our lunchroom tables are all circular, so she’s had plenty of experience sitting at round tables before.

My MO is that, any time I can have students make a decision, I want to let them make it. I want them to feel a sense of autonomy in their experience with math. If they are always turning to an external arbiter, then their experience with math is one where someone is always telling them the “right way” to do things. And sometimes there isn’t a “right way.” I like to find those moments & those types of problems and make my students be the deciders.

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