This wasn’t a dream. The MidSchoolMath conference organizers had proposed the idea months ago. “Why don’t you do some actual teaching instead of just talking about teaching?” basically. They’d find the kids. I was game.

But what kind of math should we do together? I needed math with two properties:

• The math should involve the real world in some way, by request of the organizers.
• The math should ask students to think at different levels of formality, in concrete and abstract ways. Because these students would be working in front of hundreds of math teachers, I wanted to increase the likelihood they’d all find a comfortable access point somewhere in the math.

So we worked through a Graphing Stories vignette. We watched Adam Poetzel climb a playground structure and slide down it.

I asked the students to tell each other, and then me, some quantities in the video that were changing and some that were unchanging. I asked them to describe in words Adam’s height above the ground over time. Then I asked them to trace that relationship with their finger in the air. Only then did I ask them to graph it.

I asked the students to “take a couple of minutes and create a first draft.” The rest of this post is about that teaching move.

I want to report that asking students for a “first draft” had a number of really positive effects on me, and I think on us.

First, for me, I became less evaluative. I wasn’t looking for a correct graph. That isn’t the point of a rough draft. I was trying to interpret the sense students were making of the situation at an early stage.

Second, I wasn’t worried about finding a really precise graph so we (meaning the class, the audience, and I) could feel successful. I wanted to find a really interesting graph so we could enjoy a conversation about mathematics. I could feel a lot of my usual preoccupations melt away.

After a few minutes, I asked a pair of students if I could share their graph with everybody. I’m hesitant to speculate about students I don’t know, but my guess is that they were more willing to share their work because we had explicitly labeled it “a first draft.”

I asked other students to tell that pair “three aspects of their graph that you appreciate” and later to offer them “three questions or three pieces of advice for their next draft.”

• I like how they show he took longer to go up than come down.
• I like how they show he reached the bottom of the slide a little before the video ended.
• I think they should show that he sped up on the slide.
• Etc.

If you’ve ever participated in a writing workshop, you know that workshopping one author’s rough draft benefits everyone’s rough draft. We offered advice to two students, but every student had the opportunity to make use of that advice as well.

And then I gave everybody time for a second and final draft. Our pair of students produced this:

Notice here that correctness is a continuous variable, not a discrete one. It wasn’t as though some students had correct graphs and others had incorrect ones. (A discrete variable.) Rather, our goal was to become more correct, which is to say more observant and more precise through our drafting. (A continuous variable.)

And then the question hit me pretty hard:

Why should I limit “rough-draft talk” (as Amanda Jansen calls it — paywalled article; free video) to experiences where students are learning in front of hundreds of math teachers?

My students were likely anxious doing math in front of that audience. Naming their work a first draft, and then a second draft, seemed to ease that anxiety. But students feel anxious in math class all the time! That’s reason enough to find ways to explicitly name student work a rough draft.

That question now cascades onto my curriculum and my instruction.

How should I transform my instruction to see the benefits of “rough-draft talk”?

If I ask for a first draft but don’t make time for a second draft, students will know I really wanted a final draft.

If I ask for a first draft, I need to make sure I’m looking for work that is interesting, that will advance all of our work, rather than work that is formally correct.

How should I transform my curriculum to see the benefits of “rough-draft talk”?

“Create a first draft!” isn’t some kind of spell I can cast over just any kind of mathematical work and see student anxiety diminish and find students workshopping their thinking in productive ways.

Summative exams? Exercises? Problems with a single, correct numerical answer? I don’t think so.

What kind of mathematical work lends itself to creating and sharing rough drafts? My reflex answer is, “Well, it’s gotta be rich, low-floor-high-ceiling tasks,” the sprawling kind of experience you have time for only once every few weeks. However I suspect it’s possible to convert much more concise classroom experiences into opportunities for rough-draft talk.

To fully wrestle my question to the ground, how would you convert each of these questions to an opportunity for rough-draft talk, to a situation where you could plausibly say, “take a couple of minutes for a first draft,” then center a conversation on one of those drafts, then use that conversation to advance all of our drafts.

I think the questions each have to change.

Geometry

Arithmetic

Algebra

[photo by Devin Rossiter]