False Dichotomies

I’ve been thinking about Ben Blum-Smith’s most recent post, about Bowen and Darryl’s strategies for facilitating problem sets at PCMI. It brought up an idea that I’ve been exploring recently; that I have a tendency to think about teaching in terms of false dichotomies, focusing too much on polarized ideas and less on the middle ground.

One of the mathematical teaching practices from Principles to Actions is to “facilitate meaningful mathematical discourse”. The 5 Practices book has been influential for me in how I work to facilitate discourse, and Bowen and Darryl use those principles well in their discussions. Most of my effort around discourse in my class has been to facilitate 5 Practices-style discussions. But Ben’s post was a good reminder of other strategies that Bowen and Darryl use, in particular around norm-setting and group composition that facilitate discourse within groups. Kids can talk to each other about math in small groups, in partners as a break in a larger discussion, in writing, or to the full group. That discourse can be more formal, as in times when they have a chance to prepare their ideas, or off-the-cuff. And I’m not sure that any one of these forms of discourse is fundamentally superior — they’re all tools that serve different pedagogical purposes, and are useful at different times.

A second idea that I have been using regularly in my class is visible random groupings. I love using this handy spreadsheet to create random groups for students. Sometimes I regroup more than once in a class to mix things up (helpful for my 90 minute blocks). I think there is a great deal of value on random groups to level the playing field among students and reduce perceptions of status differences. But I also think I have become too focused on visible random groupings and ignored the potential of more deliberate groupings. Bowen and Darryl present an alternate framework, thinking about students as either “speed demons” or “katamari” — whether they rush through the work, or take the time to learn from all that problems have to offer.  Ensuring each group has more katamari than speed demons is an alternate approach with a ton of potential, and is relatively easy to implement. I’m not sure how to articulate the places in my teaching where one method of grouping will be more effective, but I’m sure that they both have value, and that I can work to incorporate both into my teaching.

I’m not trying to make an argument for or against discussion or grouping, or say anything profound on either topic. I just want to reflect on what seems to be a heuristic that I need to be aware of — the tendency to see only the black and the white and ignore the gray in the middle. I think this is exacerbated by the volume of rhetoric arguing for one side or the other in many of these issues, oversimplifying complexity and distracting from the most effective pedagogy that is enormously context-dependent, and often in the middle.

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