A few months ago, I swung by Justin Reich’s classroom and showed his undergrad some math mistakes. (Read about it here or here.) In planning the session, I practically begged Justin to let me use a class set of mistakes instead of individual pieces of interesting mistakes. Here’s what I wrote:
Your first question in the protocol is “Look at three problems on the board. Predict all of the mistakes that students might make.” I love this question. But I feel lingering guilt about how mathmistakes.org
usually responds to this sort of question with a single
example of student work. This has felt problematic in some of the conversations that I’ve had surrounding student work, because someone might be entirely right in their predictions in a way that isn’t affirmed by the chosen work. I worry that this feeling of “gotcha” sometimes kills discussion around student work since the initial predictions aren’t entirely engaged.
My proposal is to tweak the protocol a bit. Instead of showing kids what a single student actually did, what about showing them a class set of responses? Then we can better check our initial predictions and ask a whole host of other interesting questions. (e.g. What patterns do you see? Do you think you can tell how these kids were taught? Why do kids tend to make this mistake?)
I ended up using a class set of fraction-comparison work
for Justin’s class. My experience cemented my opinion: if you want to talk about teaching or student thinking, it’s gotta be the class set. Why?
Reason #1: You can’t be right or wrong with a prediction about one piece of student work.
This is what I mentioned in my email to Justin, but I want to expand on it here. A great way to learn something is to make a prediction, and then check it against reality. But knowledge about student thinking is most powerful either in the aggregate or in the very specific. Meaning, to know something about student thinking is either to know something about how kids, in the aggregate, often think about something, or it’s knowledge about how this kid, right here thinks about it.
This is a long way of saying that checking a prediction against one wacky error is inevitably a bit of a letdown:
- OK people! Here’s this math problem. What thinking do you predict you’d see from kids here?”
- You collect predictions
- “OK here’s this piece of student work. Were you right?”
- Umm. Yes? No?
It feels unfair to me to ask — as I have in the past — for people to invest themselves with a prediction that I can’t honor with a realistic response. Using a class set better respects other people’s predictions.
Reason #2: You can’t really make connections and form generalizations about student thinking from one example of student work.
When I worked with Justin’s students, I saw them making connections across different students. They noticed the prevalence of the area model in the student work, and that led to interesting observations, connections and questions on their part. Wouldn’t have happened if I just showed them the craziest one.
Reason #3: Math Mistakes discussions sometimes devolve into “Well this kid needs one-on-one tutoring/special attention/I would pull her aside and ask…” The class context nudges us away from that.
Look back on the comments from the first few years of this site, and you’ll see this line appearing over and over again. This imagines a context where you’ve got one student having a hard time, while the class as a whole mostly gets the math. This allows participants in a math mistakes discussion to shift the responsibility onto the individual student.
That’s a fine context to imagine, but offering a class set of math mistakes offers a much richer context for conversation. Here is a class, in the middle of learning something. On the whole, they know some things and struggle with others. What are those things? What could we do next?
Reason #4: It’s more authentic to the actual work of teaching.
Finally, the class set more closely resembles the actual work of teaching, so participants get to practice an aspect of that actual work. We can sort and categorize student work
. We can talk about groups of students, and develop language to describe different sorts of students and different sorts of struggles.
I haven’t posted very many class sets on this site, though I’d like to do more. Feel free to continue submitting individual pieces of student work, but you’ll get a special high five if you send me a set of a class’ work ready to post.
Here are the class sets that I’ve posted so far: