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# Reframing Mistakes in the Math Classroom

One of the greatest challenges of teaching (and what makes it so exhausting) is responding to student ideas in real time. Student contributions can be complex, nuanced, incomplete, incorrect, or hard to comprehend. We do our best to plan for the lesson by anticipating student responses, but the great work of teaching is responding to those comments, questions, and contributions that we didn’t plan for, and leveraging them for the learning of the whole class.

One of the first things we listen for is whether the answer is wrong or right. But wrong and right are very simplistic terms when it comes to analyzing and evaluating a student response. Was the mistake due to a misunderstanding, a conceptual error, or a computational error? How can we leverage the mistake as a tool for fostering critical thinking in our classrooms? The first step is to truly understand the student’s response and to invite the input of other students rather than just evaluating it as correct or incorrect. Here are a few of our go-to responses:

• Tell me more.

• What makes you say that?

• ______ just said something interesting. Let’s explore it a little more.

• Hmmm. You bring up something interesting. What do others think about that?

Allowing a student to explain his or her thinking instead of just saying “No, that’s not right.” develops student agency and sharpens their skills of justification.

The most important thing is that you can not just question students when their answers are wrong, you have to do this with correct answers as well! Asking students to explain their reasoning, justify their thinking, clarify their argument, or opening up someone’s idea to the class for further discussion provides valuable opportunities for sense making, critical thinking, and consolidation of ideas, whether the given answer was correct or incorrect.

## What's the value of a mistake?

We need to get students to see (and sometimes even convince ourselves) of the value of errors. We live in an educational landscape that punishes errors at every turn (-1 for forgetting units), which has instilled fear in our students and an unwillingness to take academic risks. Students are less likely to share incomplete thoughts, and they miss out on some of the productive struggle of group work by frequently asking the teacher if their answer is right. I've had to train myself not to answer these questions. Instead, I generally stick to the “I don’t know. You tell me,” smile-and-walk-away approach.

Why should we shift our classroom culture around making mistakes?

• To acknowledge the fact that mistakes are valuable! They help us determine what a student understands and doesn’t understand and provide an opportunity to point out and discuss a common misconception, so students can avoid it in the future.

• To reinforce the mindset that failure is not fatal

• To encourage risk taking and increase participation

• To redefine a valuable contribution as anything that moves our collective thinking forward; we can learn from an incorrect answer just as much as from a correct answer

## My Favorite No

One of my all-time favorite instructional routines to foster a positive disposition towards mistakes is an activity called “My Favorite No.”

Give students a problem to work on, specifically one where you want to draw out a common misconception. Have students work individually and write their solution on a notecard. I do not require students to write their names on the notecard.

After about 5 minutes, collect the notecards and sort them into a “Yes” pile and a “No” pile based on if the solutions are correct or incorrect. Then choose one of the “No” cards and write up the student solution on the board (it’s important to re-write the student answer, we don’t want students to get distracted by trying to figure out who wrote it based on handwriting). Make sure you pick an incorrect solution that will garner good discussion or highlight an important misconception, not just one that has a computational error.

Then ask the class these two questions:

• What did this student do well?

• How could this solution be improved?

When you do this activity for the first or second time, there is a lot of framing that has to be done up front in order for this activity to be productive. I make a big deal about telling students I’m going to pick a card that showcases a really important mistake that will be beneficial to discuss for everyone in the class. I also tell students that it is completely irrelevant who wrote the solution, and that our goal is not to try to figure out whose notecard it is. I remind students that it is completely anonymous and that we are talking about the ideas, and not the person. I also emphasize that it is in fact my “favorite” no. “Ooooh, I found a really good one. I love this one. This is a great one.”

When framing the activity in this way, I find that students are very willing to engage and don’t get nervous or anxious about seeing their incorrect answer on the board.

Make sure you spend at least as much time, if not more, talking about what the student did well as you do talking about the mistakes. Even incorrect answers demonstrate a lot of good thinking, reasoning, and mathematical understanding. It is almost impossible to find a response that is completely devoid of strengths. As my favorite movie quotes, there is always “a certain rightness to your wrongness.”

## Conclusion

Explaining and evaluating ideas are a crucial component of critical thinking, and fostering critical thinking is one of the best tools for increasing equity in our classrooms. But this will only work in environments where students are not afraid to make mistakes. By reframing how we think about mistakes and getting students to see their value, we can greatly increase the opportunities for sense making, promote student agency, and prepare students for citizenship in 21st century society.