The latest post at Educating Grace is about breaking down classroom cultures of “fishing for or steering people towards the right answer, treating wrong answers as dangerous, only valuing people who give right answers.”  My comment got so long that Blogger wouldn’t accept it — so I’m posting here instead.

Grace starts by posting a short video clip of a PD session with math teachers, focusing on a moment where a math teacher tries to come up with a non-standard algorithm but ends up getting the wrong answer.  You should go watch it now.

I actually found it hard to watch.  I felt uncomfortable with the responding teacher’s growing embarrassment, as well as with the vocal performance of her embarrassment.  In the moment, I interpreted the stage-whispers she shared with a seat-mate as a way of letting the rest of the room know that she knew, of course she knew why it was wrong.  Which goes back to Grace’s point — we are in a culture where it would be shameful not to know. Mistakes require some gesture of face-saving by the mistake-maker to prove that she knows, and probably some gesture of chiding from bystanders to prove that they also know — lest they be dragged into the shame of not-knowing by association.  If it was uncomfortable for me to watch, it makes me think of how squirmy it must make students…

1.  I wanted to spend less time unpacking the idea when it was first mentioned, not more. Maybe because she loses face more for every minute she continues to make the mistake?  But maybe also because asking someone to repeat their point is (in the generic classroom of my imagination) often a cue that the teacher wants you to say something else.

So I was imagining myself writing down her process as soon as she said it, and collecting more.  Sometimes I have had success undermining the cult of correctness by putting some distance between the speaker and the strategy.  After there are 5 or 6 strategies on the board, especially if they don’t all match, I can go back and ask students to think about the pros and cons of each one.

2. Another strategy that sometimes helps me is getting people to pool their answers in small groups, and report back as a group.  This doesn’t solve the problem — they will still tend to correct each other, argue, and be mortified if their solution is different from their group-mates’ — but it means the loss of face happens in front of fewer people, where it might be more manageable.

Sometimes I explicitly help students practise recording all the strategies from their group and reporting all of them — I encourage them to discuss the differences without trying to convince the others.  Their default strategy for listening is often “decide whether it’s right or wrong”, so just telling them to stop doing that doesn’t work as well as giving them something else to do: “try to figure out why a reasonable person might think that.”

3. Another thing I don’t do as often as I would like is asking people to record all the strategies that they think work, and then strategies that look plausible but don’t work.  Recording *all* of them on the board, asking people not to say which ones are which, helps break down the assumption that all things written on boards are automatically true.  This is somewhat inspired by Kelly O’Shea’s “Mistake Game“.

When we are looking through strategies that don’t work, I go back to the class with “why might a reasonable person think this.”  With teachers, we might be able to deflect attention away from ourselves by asking, “This is a tempting strategy that a student could easily use.  Why might a student think this?  If they did, what could help them sort it out?”

A related approach is, “what question is this the right answer to?”  Where one commenter on the original post found something good about the strategy’s algebra, I’m finding something good about the strategy’s heuristics.  I’m not thinking about what you should actually multiply the denominator by.  I’m thinking, “that would be a good strategy if we were maintaining the same speed and trying to figure out how far we got in 1.5 hours (27 miles).” In this question we’re maintaining the same distance, and asking how fast, not how far…. but it’s still an example of using the previous problem to solve a new one.

In the debrief, I found myself wanting to talk about, how easy it is to answer a different question than we meant to, especially if we’re trying to do things in a new way.  This must happen to students all the time — they likely have some experience with speed and distance, and some comfortable ways of thinking about them.  We’re asking them to think about familiar things in unfamiliar ways, and that’s going to be disorienting.  It points to the idea that we have to be careful in our assessments — just because someone gives that kind of answer, it doesn’t mean they don’t understand speed and distance.  In fact, it might be an indicator of a new layer of connectedness in our thinking — similar to what Brian Frank refers to as “U-shaped development.”

The fact that the responding teacher was deliberately trying to come up with a non-standard algorithm shows intellectual courage and autonomy, traits I want to encourage in my students.  What helped her develop that courage?  How could we help our students develop it?  I’d be curious to hear the answers from the teachers in the PD session.