## Wednesday, 15 July 2015

### Which One Doesn't Belong?

I really like the Which One Doesn't Belong? idea, that Christopher Danielson has made such a good book of! It's becoming a phenomenon, with a twitter account and a great website created by Mary Bourassa devoted to assembling the growing body of WODBs.

Here's one of mine:
Something's niggling me about them, and I must just get it clear in my head. I wondered if I'd got it wrong, especially when John Golden asked
Perhaps I should be making my differences more different?

What do I mean? Well take this one, by Barb Seaton:
It works like a good WODB - you can find a reason for any of the four. Take the top right one: you could say, "It's the only brown one. The rest are white." How would that be, if they were all different colours? Like this:
Could you still say, "The top right one - because it's dark brown"? I've been - in my slow way- pondering the difference between these two cases.

In the first one there's a binary difference: there's the brown dog and considering the category of colour the others are all the same - white. In the second case, as far as colour is concerned, you could pick on any of them and say it's different because it's such and such a colour and the others aren't. It's not binary in quite the same way: there are four values for the category of colour.

Christopher Danielson goes for both kinds of difference in his original example:
He says (my notes in brackets):
• The bottom left shape doesn’t belong because it’s not shaded in. (Binary)
• The top left shape doesn’t belong because it only has three sides, while the others have four. (Binary)
• The top right doesn’t belong because it is the only square. (Not binary - there are three shapes. He could have said, "It has right angles." Somehow that feels a little more binary as right angle- not right angle is such a major distinction with angles.)
• The bottom right doesn’t belong because it’s the only one resting on a side. (Binary)
I'm kind of pleased that even this one has a not binary example in it. It seems to open up the possibilities a bit, to relax the whole thing. Of course, binary is satisfying.

I only did one lesson on Christopher Danielson's book. It was worthwhile, but I didn't pay attention to the distinction I'm making, and how the students were relating to it.

Just recently, Dani  Ruiz Aguilera has posted a good pattern-block example:

I wonder if he had in mind the not-binary category of order of rotational symmetry? Certainly there are lots of other features you could pick as well, some of them binary.

I'm intrigued by these shapes and see lots of interesting things in them. Like, that you can transform the top right one into the others with a bit of internal rotation:
I wonder, what is the proper formal language for this binary-not binary distinction in differences?
Does it matter? Do you have a preference?