Love that distinction between “the answer” and “an answer,” Ann. Thanks for sharing that, and for the reference to Peter Johnston as well.

Love it.

It was a real ah-ha moment for the both of us. I’m also an ELA/writing teacher by training, so am all about workshop, revision, talk, and the like. I might suggest a good place to read more about this kind of talk are Peter Johnston’s books, Choice Words or Opening Minds. Both are short and incredibly powerful. He’s in ELA, but his writing about talk is useful for all disciplines in all kinds of classrooms.

I’m interested in adapting/inventing/testing ways to use public instruction that are cost-effective and time-efficient, and found this example to be inspiring on a number of levels.

Same! I know you’ve been working on that project for a long time, Kevin. If you’ve published anything on your work at NTC or elsewhere, please let me know.

Helpful caveat to my suggestion above, Aaron, and usefully illustrated.

Another approach I’ve tried in similar situations is to invite students to choose any three numbers before they know the problem we’ll be considering.

“Choose three numbers. Okay now which of them makes x(x+1) closest to 1. How will your results inform your next guess.”

There are signs in my classroom that say, “It’s o.k. not to get it the first time” Rough draft and final draft of work is a common concept that students need to realize applies to math.

I thought The presentation went great – especially I came up and bugged you right before you went on . . . sorry. ;-o

Whoa! Zack, “verbal ping-pong” is exactly what most of my classroom interactions feel like to me, and it’s one of the things I dislike most about teaching in front of a classroom instead of talking with a small group. I never saw this discomfort clearly until you pointed it out, and I never knew there were classroom setups designed to avoid it.

While I agree that this rewording might have the side effect of adding freedom to the problem, I think its most important effect is to lower the floor of the problem in a certain way. You could say it enables rough draft talk by turning a step into a ramp.

For me, the original wording creates a barrier I have to get over to get any reward from the problem. If I don’t have a solution, it feels like I’ve failed to achieve anything. I’m not going to hold up my failed attempts as rough drafts; I’m just going to throw them out.

The new wording invites me to just try something! I don’t have to find a solution; I just have to look for one. I can plug in random numbers, see that I didn’t find a solution, and still feel like I’m doing what the problem asked me to do. That’s a pretty low floor.

But this low floor slopes upward. When I’m plugging in numbers, I might notice that some guesses feel closer than others. If my first draft is a random guess, my second draft could be a better guess, my third draft could be a proposed rule for identifying good guesses before I plug them in, my fourth draft could be a refinement of that rule, and so on.

To illustrate the difference I see between adding freedom and “turning a step into a ramp,” consider this problem.

“Find the positive real number x for which x(x-1) = 1.”

I can add freedom by allowing more solutions.

“Find a real number x for which x(x-1) = 1.”

However, I think this leaves the problem just as inaccessible to rough draft talk.

I can “turn a step into a ramp” by asking for approximate solutions.

“By adjusting the real number x, make x(x-1) as close as you can to 1.”

Now every guess is a rough draft, and revising that draft might lead me to explore methods for approximating the solutions of polynomial equations.

I’d be interested to see examples where adding freedom is enough, by itself, to enable more rough draft talk. I also wonder if the heuristic would work the opposite way: if a rewording doesn’t add any freedom to the problem, is it unlikely to enable more rough draft talk?