Things are always more complicated than I want them to be.
Here’s something that happens to me all the time. I introduce a concept through some activity or discussion. It seems like students understand it. I give them a few problems to check their understanding. Suddenly it’s a disaster, everyone is confused, and we have to circle back and clean up the mess. Now I’m all for seeing mistakes as learning opportunities. But too often students feel frustrated and that frustration leads to spiraling and entrenching negative feelings about math class. Definitely worth avoiding.
This fall I’m experimenting with diagnostic questions. I use them right before I have students try to apply a concept on their own. Here’s one I used in a class on graphing sine functions:
We had spent some time talking about how to find the period of sine functions. Which is a hard concept! And I thought they had it. Not so fast. I asked this question, and half the students answered B. It led to a great discussion. I had students chat with the person next to them, and most pairs reminded themselves of the formula for calculating period after talking with a partner. We talked briefly as a group and did another example together. I sent students off to practice feeling like I had done something productive, surfacing how students thought about period before letting them flail on their own.
One logistical note. I do what Dylan Wiliam recommends in his book Embedding Formative Assessment. There are lots of ways I could collect student answers, from clickers to cups to moving around the room. Dylan Wiliam’s thought is that students rarely forget to bring their fingers to class, and fingers don’t need an internet connection. One finger for A, two for B, three for C, and four for D. It’s been hard to get every student to raise their hand. I’m uneasy pressuring students to answer if they’re guessing, but I don’t want it to be too appealing to opt out either. Students only need to flash their answer for a moment; I try to reduce opportunities to look at each others’ answers and engage in social posturing.
So here’s my dilemma. When students are split between two answers, my next move as a teacher seems pretty straightforward: have students discuss, in pairs and then as a full class, which of those two answers makes more sense. But in a different class, working on writing exponential functions, I asked this diagnostic question:
This time, all but two students answered B, and the other two answered D. My instinct here was to pat myself on the back. Go me!
But what do I say to the class?
One option might be to say, “awesome, almost everyone got it right! Nice job!”
What message does that send to the two students who picked D?
Instead, I did the same thing as when the class was more evenly split. “I’m seeing some disagreement between B and D. Chat with the person next to you about which answer you think makes more sense.”
There’s a lot of complexity here. Coming in, my thinking was pretty straightforward. I wanted a better way to figure out whether I should move on, or if students needed more time as a whole class. I figured I should ask a quick question, and based on their answers decide whether to stop and discuss or move on.
But it will be pretty rare that every student gets a question right. And it always seems useful to take a moment and discuss a question like this. I now look at these more as discussion starters than diagnostic questions. The information I get about who answered what is definitely useful. But so is listening in on a quick partner discussion.
And even asking a quick question to gauge student thinking feels tricky. I like multiple choice here because it helps make the questions accessible and efficient. But trying to do it quickly can undermine the culture I want to create where speed isn’t the most important thing in math class. My goal is to figure out how students are thinking about one piece of a concept, and it feels hard to linger on a question for too long. But I really don’t want students to feel rushed — and it would probably be the same students every time who feel rushed, building a negative association with these types of questions.
I’ve found it useful to take a step back. One goal is to better understand what students know before they jump into independent work, to see if we need to spend more time talking as a full group. But an equally important goal is for students to avoid reinforcing negative narratives students might have about their ability as mathematicians. And there are all sorts of things here that are in tension with that goal. An implicit value on speed. Social risk in sharing answers so that every student can see. Comparing oneself to others. Surfacing ideas that might single out one student who feels like they are on the spot.
I think that the benefits outweigh the drawbacks here. There are also plenty of liabilities to throwing away diagnostic questions. And the issues above are ones I can manage through class culture. And that’s teaching. Something I thought would be simple actually has a lot more layers than I initially thought. And there’s a lot of useful stuff here. In both of these instances, I helped to avoid the phenomenon I wanted to avoid: sending students off for some independent practice when they still have very different conceptions of some mathematical idea. Avoiding that is worthwhile, but will take more nuance and subtlety than I first anticipated.