It always feels good when we ask a question and we get a correct answer in response. It feels good when our students do well. But sometimes their “success” fools us. We might think of correct answers as an example of learning. But along the way I’ve learned that I had it all wrong. Mostly, when my students gave a correct answer they recalled something that they already knew. And so, I have found that their wrong answers were the best answers for learning. The wrong answers signal misunderstanding and misconception. The wrong answers are indicative of flawed or incomplete reasoning. The wrong answers signal that we have work to do – that there is learning to be had.
I’ve learned that I can expose and use mistakes for learning. I’ve learned there are a few things I can do to make that happen. I’ve learned that some of the most critical moves happen before any answer is offered. Here are some of the things I’ve learned.
Select Tasks that Mine for Misconceptions
In recent years, we’ve heard much about task quality and task selection. We know it matters. And it makes sense. The better the task, the better our chances of uncovering unfinished learning or flawed reasoning. Consider the tasks below. Which solicits thinking? Which provides insight into thinking? Which simply pursues an answer?
The left task is open for students to interpret and explain. It can provide insight into their understanding of part-to-part ratios, part-to-whole ratios, shading (A and D are both 2:3), and equivalent ratios. The task on the right has an answer. It will tell us if they can find a ratio. And often we find quite a few of these lower-level tasks on a single sheet for students to complete. But make no mistake, the quantity of low-level tasks or questions doesn’t overcome the lack of quality.
Do the Task
This might sound odd. But it is as simple as it sounds. Selecting a task should be followed with doing the task ourselves. Doing it reinforces our understanding of it, a strategy that might be used to solve it, and how one might justify their reasoning. Doing the task might uncover our bias for a certain solution or strategy. It can uncover what we might inadvertently lead students to believe as the best approach. Doing the task, positions us for the next part of the planning process.
Anticipate What Students Might Do
Doing the task leads us to anticipating. Knowing how we might solve a problem can help us anticipate what some of our students might do. But we also need to anticipate other solution paths and representations. This is also an opportunity for us to consider what misconceptions or flawed strategies our students might have. Anticipating what students might do or the errors they might make enables us to respond intentionally rather than randomly.
Anticipate with Others (Collaborative Planning)
As noted, doing the task ourselves helps us think about what students might do. It can also provide insight into bias or preference for solutions or strategies. This is where collaborative planning comes into play. Talking with others about how they solved a problem helps us consider approaches that we might not have considered on our own. It also provides access to others’ experiences with student approaches and common misconceptions. The exchange of these ideas is what collaborative planning truly can be.
Leverage Misconceptions and Mistakes Instead of Saying They Are OK
“It’s ok to make a mistake.” We have all said it. But what do we do after a student makes a mistake? I have declared that it’s ok myself. Then, one day David exclaimed, “but you don’t use them.” He essentially called me out on the old notion of it’s not what you say, it’s what you do. In other words, I said mistakes were ok. But I never used them FOR learning in the classroom. I didn’t leverage them to move students forward. I didn’t help students think about their errors and why their ideas didn’t work mathematically. I didn’t let them explore why certain approaches or solutions don’t work.
Erase the Eraser
It’s hard to imagine a mathematics classroom without pencils… and erasers. It’s easy to see our students erasing their mistakes. It’s easy to see some erasing so much that their paper is almost translucent. There are so many lost opportunities when the mistakes are erased. I discovered long ago the value in not erasing my mistakes. Without erasures, the written record remains. Strategies, calculations, and errors aren’t rubbed away. We can see our flawed reasoning. We can see why a solution wasn’t found. So, it may be wise to unleash pens in mathematics or at least bite the erasers off our pencils.
So, it turns out I had some wrong answers about my teaching all along. But I have learned. I’ve learned that mistakes, misunderstandings, and misconceptions matter. I’ve learned that there is a lot that I need to do to take advantage of them. I’ve learned that a good bit of that is done before the lesson ever happens. I’ve learned that planning is invaluable but it’s not what I was trained to believe it to be. I’ve learned that valuing mistakes means using and discussing mistakes. I’ve learned that we can transform mathematics instruction from the pursuit of answers to the pursuit of learning. I’ve learned that we can move students from doing math to knowing math. I’ve made mistakes. I’ve learned.