I was asked last week to teach a guest lesson for 3rd graders on fractions using Everyday Math. The book had some nice lessons, including one that involved building pattern block

designs and then figuring out what fraction of them are triangles, hexagons, etc. A sweet little lesson, no complaints (though a caveat for those who use pattern blocks to talk about area: begin by using only the pattern blocks shown here. The squares and tan rhombuses have incomparable areas.)

Before I’m ready to bring this in, I’m plagued by a question: why am I asking students to answer this question? By which I mean, why should they care?

The question comes down to motivation, as it so often does, and I give myself a hard time. It doesn’t strike me as fair to ask students to do arbitrary activities, even if there’s some idea I want them to get out of it. There has to be some reason for us to want to understand, be it internal (some fundamental curiosity about the math itself, i.e. pure math) or external (some reason the math is useful for helping us make sense of another issue we care about, i.e. applied math).

Now my background is in pure math, and I tend to lean toward pure motivation. In this case, though after long discussions with Katherine and after mulling over different possibilities, I ended up launching the lesson in the following way.

I chose two volunteers and explained to the class that they lived in a world where pattern blocks were wealth. I gave the first student 1 hexagon and 2 trapezoids, and the second 2 hexagons and 4 trapezoids. Then I explained that due to a national emergency, the government needed all the trapezoids of their citizens, and I took all of their trapezoids.

Every kid immediately felt the immediacy of the question, because it’s a question kids deal with constantly: *was this fair*?

There was a spirited debate right away: some students declared the tax unfair because one student had to give up twice as many trapezoids as the other; others said it was fair because they had each given up half of their wealth. Other proposals were put forward, defended, disposed of. Eventually I told them that I didn’t have the answer to what was fair, but that it came down to an issue of what “one” equals (was it a trapezoid? Or was it the amount of wealth you start with?), and to talk to each other for real we would need to understand how fractions work.

Then I asked some questions about fractions and launched into the book’s lesson; they built shapes from pattern blocks, and missed a huge opportunity.

The lesson went fine, so it wasn’t a calamity, but in retrospect, I had given myself a gift and then dropped it. Instead of asking:

“What fraction of this shape does a trapezoid represent?” as the book suggest, I could have said:

Our classroom government has decided to tax you in the amount of all your trapezoids.

And then the question becomes not just “what fraction of your shape is made from trapezoids?” but a whole host of questions that circle the overwhelming question: is this fair? Are all the students in the class treated fairly? Would there have been a fairer way to collect a tax to get the same total amount of pattern block area from the classroom? Who is worst off from the tax? Who is best off? How do you compare these things?

There’s a real discussion there, and more so, it’s one that echos the national discussion of the moment. Kids can get in and say something real here, and understanding fractions and comparing them is what enables them to make powerful arguments. The need here is clear, and I bet the understanding of fractions that results would be much more substantial and long lasting. After all, you don’t need the “right” answer because your teacher says so; you need it to convince your classmates of something you care about. In fact, you need them to make sense of whether what’s happening is really fair or not.

Ah, well. There will always be missed opportunities. I’ll get this one next time. I will say that in terms of the original intention—motivating the students to learn about fractions using the pattern blocks—the class went great, with students describing the fraction of their shape that was a particular shape (or, for those need a greater challenge, color) beautifully. The lesson also offered a spectrum of difficulty: for those who needed more practice with easier shapes, they had it; for those who wanted a real challenge, they could get that for themselves as well.

## Comments 3

Great post and amazing lesson. I was particularly struck by your line that read, “there has to be some reason for us to want to understand it, be it internal or external.” This is something I have been thinking about a lot lately also. An excerpt from “In Search of Understanding: The Case for Constructivist Classrooms” reads, “students’ fundamental quest is discrepancy resolution.” The idea, as I interpret it, is that our role as teachers is to provide students with questions/situations that create a challenge to their existing mental framework. You clearly did this with your lesson here but it your lesson ALSO seems to be situated in what many might call an applied context. So, I was left wondering…does it need to be internal OR external or is it ALWAYS internal and sometimes we do a good job of situating that in an external scenario? Thanks for making me think.

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Thanks, Bryan. This is a really interesting point you raise. Some part of what I wrote needs a clarification of the language, I think. The kids always need to have some internal motivation, by which I mean something inside themselves that wants to know, to proceed, to work. That’s in contrast to the intrinsic/extrinsic of whether the draw of the lesson comes from a pure math aesthetic vs. an applied, real-world motive.

Either way, the students have to have some motivation to proceed, and at the end of the day, that’s a response from inside of them to whatever I provide.

I totally agree about presenting students with challenges to their existing mental framework. That’s one of my favorite parts of the job. Did you ever listen to that radiolab about the kids who had to turn the Furby’s upside down? There was a very interesting confusion about whether the toy was alive or not.

I wouldn’t be too hard on yourself for the missed opportunity. It shows us mistakes happen to everyone and lead the way to real understanding, as you have told us.