I wrote last week about the measuring-polygons activity we did to begin our investigation into pi, and hinted that many exciting things happened in class as a result.  Well, here’s one of them – the More Cheese Means Less Cheese Conundrum.

I may be wrong, but I think that actually measuring the shapes (and the What Can You Do With This? work we did several weeks ago) got the kids thinking about measuring ratios in the world around them.  Because in the middle of our work with the polygons, two girls in one of my classes said they had a burning question they’d like to ask everyone.  I love it when kids have burning questions – whether related to math or not – so up they went to the board to share.

Their question was: You know how Swiss cheese has holes?  Well, when you make a block of Swiss cheese bigger, the holes get bigger, right?  So, when you make a block of Swiss cheese bigger, do you actually have less cheese?

The answer to this question was obviously no.  That wouldn’t make any sense, so long as you don’t mean inflate the holes by “make a block of Swiss cheese bigger.”  But, if you’re a sixth grader and your practical understanding of proportion isn’t so strong, this question is baffling.  The holes get bigger – and more holes must mean less cheese, right?

When we put the question to a vote, the class was about 3 to 1 in favor of more cheese means less cheese.  And as the more cheese = less cheese faction shared their arguments (mostly that more cheese means more holes, and more holes means less cheese), the ratio shifted to about 6 to 1.  In a 14-person class, that makes the more cheese means more cheese faction pretty tiny – but they were very vocal for their position.  Debate was heated – but also without much evidence or solid logic either way.

This seemed to me like a very important question to resolve.  I couldn’t have my students going into a grocery store and buying a smaller amount of Swiss cheese thinking that they’d actually be getting more cheese.  But, class was over.  So, instead of having more polygon-measuring homework, I asked them to come up with a compelling argument in favor of their cheese position.  Before I sent them on their way, however, we discussed what could be important to include in a compelling argument.  Evidence, they said.  So, how will you get evidence?  The responses ranged from actually buying and measuring cheese to making an accurate diagram.

By the following class, only one student remained in favor of more cheese = less cheese.  But I couldn’t have asked for a better student to be pro more cheese = less cheese.

This particular student is very good at actively learning.  And by this, I mean he often has misguided ideas, but values solid logic very highly.  When he has an idea, he always has a logical argument for it that he believes to be strong.  And he will stick to his idea and logical argument until you give him a really good reason that the opposite is true.  So, I knew that his classmates would have to really know their stuff to bring him around.

Oddly, our single hold-out’s argument rested on proportion.  The holes must increase as the block of cheese increases, he argued, because the proportion the cheese taken up by the holes must remain constant.  So, the holes increase in volume as the entire cheese increases.  Meaning that the amount of solid cheese must decrease.

The other students practically fought to give an argument that would persuade him otherwise.  Some came to the board and drew models of what would happen with more cheese.  He didn’t buy it – how could you draw a really accurate model on the board?  As he doubted their accuracy, his classmates began to realize that they’d need some numbers to bring him around.  So, one student suggested making a mathematical model.

With numbers at her command, she demonstrated that, given the assumption that the larger piece of cheese is a scaled-up version of the smaller piece of cheese, the holes-to-cheese ratio must stay the same.  If the whole cheese doubles in size, the holes double, but so does the cheese.

The argument was convincing!  Those who preferred picture arguments altered their pictures to fit her mathematical model.  We talked for the rest of the period about whether her mathematical model would work regardless of the holes-to-cheese proportion and how scaling of geometric objects alters and doesn’t alter measurements you make of them.

We’d been working with that last question for almost a week with the polygons and proportions, and we had already talked about how ratios of lengths don’t change with scaling.  So, I could look at this incident – in which they didn’t take that knowledge and immediately apply correctly it to the new context – and be disappointed with my teaching and their learning.  Maybe I could judge myself as more successful if they’d immediately seen the connection between this question and the polygons.  Yes, there are things I can learn about my polygon lesson from the cheese experience – we should spend more time with scaling, and I should probably incorporate scaling into other work we do.

But, in my opinion, at least, learning math isn’t climbing a ladder or building a wall.  Just because something has been “taught” doesn’t mean you’ve topped that rung and you’ll never see it again, or that the brick has been mortared into place and completely dried, forming a foundation for future learning.  Learning math is more like playing a game of Entanglement.  (Play it, if you haven’t.  It’s fun!)  New tiles are laid, connections are made, but that doesn’t mean that you won’t use old tiles in new ways or pass through old connections – intentionally or entirely by accident.  My kids used their knowledge of ratio gained from the polygon problems to learn about cheese, whether intentionally or not.  And now, hopefully, it’s a little more tightly woven into their mesh of mathematical knowledge.

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