A Lesson in Discussion in Mathematics

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It wasn't exactly where I had anticipated directing the discussion yesterday. And as it turns out, that was not only a moment of revelation, it was a glimpse into good things that can happen to mathematical discussions.Have you seen a problem that is something like this one?

31 students are going on a field trip. They travel in cars holding 4 students and a driver. How many cars will they need?

With second language learners, I expected that we would have a discussion about what should be done with the remaining students.  We never actually got to that.

At the Summary point in our lesson, I asked volunteers to explain their thinking and computation for the problem. Three student volunteers stepped up to the document camera and explained their thinking: student one divided 31 by 3; student two divided by 4 and student three divided by 5.  Which one was correct?

The rest of the students kept turning around to me to see which of the three students had the right - as in which student had the one correct solution.  I've been working on this area of my teaching for a while now, and fortunately I did not take the bait.

Because had I stepped into the discussion as "teacher as the holder of all things correct" , I would have missed one of the all-time great moments of teaching -- the time when the students follow all those discussion norms we've worked on and have a debate about which student had the most logical interpretation of this word problem.  I wish I had filmed it!

What did these fourth grade mathematicians decide? Although student one's  interpretation made sense to him, there was general agreement "4 students and a driver"  did not mean divide by 3. Student two pointed out that bus drivers don't go inside on field trips, so neither would car drivers; if the students were fourth graders, divide by 4. And student three? Student three is steadfastly holding the position that if the students were high school aged, one of them might be able to drive; therefore, divide by five.

And my question - what to do with the remaining students? Well, we'll work on that one on another day.