Making Things Explicit

I teach students to do polynomial division using the box method, which I originally learned about from Anna Vance. I’ve found students typically really enjoy the box method once they get the hang of it, and it reinforces their understanding of multiplication at the same time. But the initial stages of teaching the topic have always been hard and result in a lot of student confusion and frustration. Here are a series of puzzles I designed to try to improve the transition for students:

For those curious, here is a doc with the images and a template for making more, and I create the smudges by uploading images to Google Keep.

Pretty clever, right? I found students work their way through the puzzles without too much guidance, applying what they know about multiplication and gradually working up to problems where they’re doing polynomial division. Then, I say “hey, you don’t even realize it, but you’re doing polynomial division! Isn’t that cool!” Then I give them some problems for practice, and they’ve magically learned a new topic.

Only it didn’t work out so smoothly. Students could work through the puzzles, but the transition from puzzles to a division problem stumped them. The connection was totally unclear. Some students given a polynomial division problem just multiplied the two polynomials and built bad habits before I realized what was happening. I ended up having to completely reteach the topic, and to work through some well-justified frustration along the way.

So I took the same puzzles, but used a different approach. After each problem, we paused and summarized what was happening. What patterns are we seeing, and why are they there? Why are like terms on diagonals? How is that useful? Which expressions are being multiplied? What is the corresponding division sentence that must be true? How would you write that division sentence? How does that reasoning apply to this next problem? What’s a different way to write this problem?

Then, on the transition to polynomial division problems without the scaffolding of the puzzle, I modeled what the first steps of the setup look like. Suddenly life was great, polynomial division was easy, and the lesson breezed by.

I think there’s a really important bit of learning in the difference between these two approaches. I love finding ways to reframe content as a puzzle. It gets students curious and it reflects the discipline of mathematics. But there’s also something significant happening cognitively when students see a topic as a puzzle, rather than starting a lesson with SWBAT polynomial division. Starting with an explanation creates all kinds of potential confusion. Students bring in their associations with polynomials, with division, with negative experiences learning a new and abstract concept. Framing the topic as a puzzle helps students to zoom in on the nitty-gritty, get some muscle memory with the basic elements of the procedure, activate background knowledge, and build confidence. But that transition from a fun puzzle to the abstractions I want students to take with them needs to be explicit in the lesson; if I leave those connections to chance, students’ learning will never get linked to the contexts I want them to apply it in, living in their minds as a fun but isolated puzzle.

A similar topic came up at PCMI last summer. We were talking about Smudged Math, and one participant shared frustration that it often seems like teachers can find a million ways to help students to do algebra but feel like they’re not doing algebra, without ever actually linking that learning to algebra. I think that’s dead on. It’s often fashionable in math teaching to talk about how students figured something out entirely on their own, or to tell students they’ve been applying a particular concept without realizing it. I see these as fantastic pedagogical opportunities, but only when we make clear exactly how what students are doing connects to new and more abstract math we also want them to do. As teachers we see the bigger picture, but students, and any novices learning a topic, struggle to step back and see the forest for the trees, getting lost in the specifics of a particular question or task. I think the crucial pedagogical move here is to make those connections explicit for students rather than leaving them to chance. Otherwise, it’s just clever packaging for a lesson students enjoy but don’t learn anything they can apply in the future.

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