The Three Acts Of A Mathematical Story

Some writing that went on to define much of my work in math education.

I’m taking a break in July from social media and this newsletter in order to complete a couple of heavier writing projects. Instead of letting this newsletter go completely fallow, I have decided to pull out several published and perhaps unpublished pieces from years ago and offer some brief commentary on how they hold up and don’t.

One of the major identity crises of my young adulthood was the fact that I had an abundance of interest in screenwriting, directing, and storytelling and very little talent for it. Meanwhile, you could flip that relationship for math teaching. I graduated university with a math degree, headed into math teaching, and fretted about this constantly in my early twenties.

Long after writing this article, I’d come to read Willingham and other cognitive scientists on “the privileged status of story,” but early in my career it was a tremendous joy to realize that math teaching and storytelling were not, in fact, separate disciplines.

I think my description here of “story” as a pedagogical framework wobbles quite a bit in “Act Two,” where I seem pretty under-opinionated about how students should learn a new skill. (“Just let ‘em get it from Khan Academy or something!”) But my conviction that the introduction of a new mathematical skill should engage a student’s sensory experience and intuition, my conviction that the conclusion of learning a new mathematical skill should result in a feeling of power and resolution for the learner, I think has held up pretty well for me over the years, and directed the majority of my professional work.

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Act One

Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.

With Jaws your first act looks something like this:

The visual is clear. The camera is in focus. It isn’t bobbing around so much that you can’t get your bearings on the scene. There aren’t any words. And it’s visceral. It strikes you right in the terror bone.

With math, your first act looks something like this:

The visual is clear. The camera is locked to a tripod and focused. No words are necessary. I’m not saying anyone is going to shell out ten dollars on date night to do this math problem but you have a visceral reaction to the image. It strikes you right in the curiosity bone.

Leave no one out of your first act. Your first act should impose as few demands on the students as possible – either of language or of math. It should ask for little and offer a lot.

Act Two

The protagonist/student overcomes obstacles, looks for resources, and develops new tools.

Before he resolves his largest conflict, Luke Skywalker resolves a lot of smaller ones – find a pilot, find a ship, find the princess, get the Death Star plans back to the Rebellion, etc. He builds a team. He develops new skills.

So it is with your second act. What resources will your students need before they can resolve their conflict? The height of the basketball hoop? The distance to the three-point line? The diameter of a basketball?

What tools do they have already? What tools can you help them develop? They’ll need quadratics, for instance. Help them with that. [Quite a lot of hand waving right there! -DM]

Act Three

Resolve the conflict and set up a sequel/extension.

The third act pays off on the hard work of act two and the motivation of act one. Here’s act three of Star Wars.

That’s a resolution right there. Imagine, though, that Luke fired his last shot and instead of watching the Death Star explode, we cut to a scene inside the Rebellion control room. No explosion. Just one of the commanders explaining that “the mission was a success.”

That what it’s like for students to encounter the resolution of their conflict in the back of the teacher’s edition of the textbook.

If we’ve successfully motivated our students in the first act, the payoff in the third act needs to meet their expectations. Something like this:

Now, remember Vader spinning off into the distance, hurtling off to set the stage for The Empire Strikes Back. You need to be Vader. Make sure you have extension problems (sequels, right?) ready for students as they finish.

Conclusion

Many math teachers take act two as their job description. Hit the board, offer students three worked examples and twenty practice problems. As the ALEKS algorithm gets better and Bill Gates throws more gold bricks at Sal Khan and more people flip their classrooms, though, it’s clear to me that the second act isn’t our job anymore. [Huge miss here from me. I’m skeptical about the efficacy of all three of those solutions. - DM] Not the biggest part of it, anyway. You are only one of many people your students can access as they look for resources and tools. Going forward, the value you bring to your math classroom increasingly will be tied up in the first and third acts of mathematical storytelling, your ability to motivate the second act and then pay off on that hard work.