one of these things...

I remember, in high school, when a teacher was talking about series and sequences of numbers. She presented a series, such as 3, 6, 9* and asked what the next number is. Of course, she was expecting us to answer 12. I noted that, if you say that the sequence is defined by a polynomial expression, you can justify anything as the next number by picking the right coefficients. 3, 6, 9, 75? There's a third-degree polynomial that fits it. And, by the way, there are infinitely man fourth degree polynomials that fit as well.** 

I used to have a friend named Angry Bob. On more than one occasion he called me, annoyed about a sample Mensa test (or somesuch) he saw in a magazine. He didn't score in the genius range, and wanted to plead his case. Specifically, there would be four pictures, and a question of which one doesn't belong. It was easy enough to figure out what the quiz-writers intended, but Angry saw it differently. He pleaded his case. I explained that the quiz was looking for a different answer. He'd plead again, arguing that his choice was just as legitimate***.This went on until I said that, yeah, he's right, he deserves credit. We would go through several questions this way until he had enough questions right to score in the "genius" range.

I was reminded of the above vignettes as we listened to Christopher Danielson at the National Museum of Mathematics (Mo Math). Blair, Sharon and I (along with one of Sharon's friends and her family) went to see Danielson's presentation, "Which One Doesn't Belong?" which was part of the museum's "Family Fridays" program.

Danielson is an educator of educators. That is, his "thing" is helping people figure out how to teach math to kids. His websites are talkingmathwithkids.com and christopherdanielson.wordpress.com.

The main part of the program consisted of Danielson showing slides with pictures of four objects, and asking for the particpants' thoughts about which object didn't belong. But instead of focusing on what is the one right answer, he focused on the fact that any answer could be right. Consider, for example, the picture above.

• The shape in the lower left doesn't belong because it's the only one that's convex.
• The shape in the lower right doesn't belong because it's the only one with multiple lines of symmetry.
• The shape in the upper left doesn't belong because it's the only shape whose boundary has no straight lines.
• The shape in the upper right doesn't belong because it's the only shape whose boundary consists entirely of straight lines.

This wasn't as advanced a topic as Francis Su's cake-cutting talk (which I write about here). And to a degree we were out of place; Sharon and her friend were older than most of the kids there. But it was interesting nonetheless. Danielson was trying to introduce the youngsters to flexible mathematical thinking. And it was vindication for that former high schooler in me who bristled at an inflexible teacher. I only hope that Angry Bob, wherever he is, would feel vindicated as well.

*I don't remember what the actual numbers were, so I'll use these for the purposes of the narrative.

**Yeah, I was a smart-ass in high school.

***In all fairness, he was right.