How Do We Respond to Student Mistakes and Correct Answers?

Imagine a discussion during math class. The teacher poses a problem; the class solves it mentally and shares their thinking with a partner. During the discussion that follows, a student offers an incorrect answer, and their reasoning is faulty. What do you do? Do you let them know they’ve made a mistake? Do you ignore them and ask for someone else’s idea? Do you try to explain or provide a counterargument? Conversely, if a student offers the correct answer, then what do you do? Knowing how to respond to students on the spot, in real time, is no easy task. It requires us to listen carefully to students and understand their thinking, whether it makes sense and is mathematically correct or not. 

Our Words Are Important

While it’s beneficial for our students to get lots of opportunities to talk during math class, as teachers, we tend to do a lot of talking. And, what we say and how we respond to students really matters. Everything we say, and even the things that we don’t say, communicate a message to students that tells them whether it is okay to proceed. Our words can determine their success or their failure. It’s a big responsibility!

We want our students to feel comfortable and daring during math time. We want them to take a risk and propose an idea that might be “out there” but makes people think. We want them to feel safe and trust that, no matter what they say, we will still think they’re smart. 

How Do We Respond to Student Mistakes?

Let’s look at a scenario where the teacher has posed the problem 48 x 3 during a number talk. The students are given time to solve the problem mentally, then they share their thinking with a partner before the teacher elicits a few of their ideas. One student explains, “I rounded 48 to 50 to make it easier, then I multiplied 50 x 3 to get 150.” After listening to her student, the teacher must decide how to respond. 

Following are 5 different types of possible responses that a teacher might provide: denying, ignoring, providing a counterargument, and validating and probing (Bresser & Fargason, Becoming Scientists, 2014). For each, you’ll see an example of an actual teacher response, and the potential effects that their words have on student learning. 

Denying

Teacher Response: “Well, no, not really. Does anyone else have an idea?”

Potential Effect These Words Have on Student Learning:  The student who contributed feels embarrassed      at having given an incorrect answer, which halts his thinking about his strategy. The next time a question is asked, this student will think twice before contributing and may feel anxious about being incorrect again. The students know that you hold the correct answer and may be reluctant to speak unless they are sure they know it.

Ignoring

Teacher Response: “Okay. How about someone else?”

Potential Effect These Words Have on Student Learning:  The student who contributed the answer feels that his idea was not good enough to merit a response. This may affect future participation and can contribute to an unhealthy disposition about math. He no longer feels that he is a smart mathematician.

Providing a Counterargument

Teacher Response: “You added 2 to the 48, so you must take away something at the end.”

Potential Effect These Words Have on Student Learning:  This response may prompt some more thinking from the students since they will want to figure out the answer to your counterargument. However, some students may think their contributions are meaningless since you seem to have the right answer.

Validating and Probing

Teacher Response: “Hmm…that’s an interesting strategy. So, are you saying that to make the problem easier, you rounded 48 to 50? I wonder how that might change the answer.”

Potential Effect These Words Have on Student Learning:  The student thinks that his contribution was helpful and may lead the class to discover a new strategy. The rest of the class begins to think about the idea since you have validated it. Students may come up with questions or counterarguments on their own. “Wait a minute, if you add two at the beginning, don’t you have to subtract it at the end?” The discussion progresses, and other students join in to solve the problem.

Summarizing the Scenario

In the scenario above, the student who shared his strategy had a wonderful idea. He knew that rounding to a friendly number would make the multiplication easier; he just forgot to compensate for adding the 2 to the 48. The teacher’s response had the power to either shut learning down or validate an idea, encourage further thinking and participation, and help students see that mistakes are keys to learning. The point is not to ignore mistakes or misconceptions but to take advantage of them and use them as material to work with, fueling productive arguments that further learning and making space for self-correction. 

Anticipating student responses while planning a math lesson is key to helping teachers adequately respond to students’ ideas during a lesson. For example, when I choose a problem like 48 x 3 to pose to students, I brainstorm possible strategies while planning. Will students round to 50 then compensate? Will they multiply 40 x 3 then 8 x 3 and finally add the partial products? Will they add 48 three times? And what errors or misconceptions might they make or hold? No matter what grade level or math content, I go through this exercise before teaching, and it really helps when I have to make decisions about how to respond to students on the spot. 

How Do We Respond to Correct Answers?

Just as we need to respond in productive ways when students make mistakes, we also must respond to and question students’ correct answers. Let’s look at different ways teachers might respond to a student who correctly solves 48 x 3 by using the compensation strategy mentioned in the previous scenario. 

Confirming

Teacher Response: “Yes! You are correct. The answer is 144.”

Potential Effect These Words Have on Student Learning:  The student who shared her thinking feels successful. The rest of the class may be disappointed that they didn’t have a chance to answer since they also were correct. The class stops thinking about the problem and waits for the teacher to pose another one.

Confirming and asking for more

Teacher Response: “Yes! You are right. Tell us why you rounded to 50 and then subtracted 6 at the end?”

Potential Effect These Words Have on Student Learning:  Although the teacher asks for an explanation from the student, the rest of the class may not listen attentively to the explanation because the teacher has already confirmed that the answer was correct.

Probing

Teacher Response: “Can you explain why you chose to subtract 6 from 150 at the end?” Or the teacher might say to the class, “Talk with a partner about why the student subtracted 6 from 150.”

Potential Effect These Words Have on Student Learning:  Notice that the teacher did not confirm whether the student got the correct answer. Rather, they asked the student to defend her thinking, providing everyone an opportunity to learn about the compensation strategy. 

Summarizing the Scenario

Even when students give a correct answer, it’s important to probe and question, teasing out the details of a strategy, making the student’s thinking transparent so all can benefit. 

Using Teacher Talk Moves is another way the teacher can get more mileage out of a math discussion. For example, the teacher might ask the student, “Can you say more about that?” Or, the teacher might ask, “Can someone repeat what Juan said in their own words?” Asking students to restate someone else’s idea helps them “orient to the thinking of others” and encourages active listening. And finally, the teacher might ask students to apply their own reasoning to someone else’s idea by asking, “Do you agree or disagree, and why?” This encourages everyone to evaluate what was said and provide further reasoning for its accuracy or inaccuracy.

Let the Students Wrestle with the Answers

I remember facilitating a math discussion with a group of second graders. They had just worked on the problem 33-4. One student gave thirty-one as his answer, making the classic error of simply inverting the problem in the ones column (from 3-4 to 4-3) rather than decomposing a ten. This isn’t a surprise because they’d probably been told that you can’t subtract a larger digit from a smaller one so they just flipped the digits. 

I suppressed my impulse to explain his error and instead honored his thinking by probing his reasoning and then jotting down his answer on the board before eliciting other ideas from students. As the discussion unfolded, other answers and strategies emerged, the most popular being, “Just start with 33 and count back 4!” The class (including the student whose answer was 31) eventually came to a consensus that the result was 29. 

Guiding Principles

Despite the benefits of using mistakes and correct responses as learning sites, about a quarter of teachers still correct students without requiring them to explain their math reasoning. Nearly half respond to students’ errors by explaining and justifying the correct steps. And only about a third use student-centered discussion as a strategy for responding to mistakes and correct answers (Sage Journals, 2023, Vol. 2, issue 2). These findings are not surprising since teachers face time constraints that push them to keep lessons moving and default to procedural interventions that textbooks often model and encourage. The challenges are real, but there’s still a better way worth trying.

My responses to student errors and correct answers have changed over time. Rather than asking myself, “How am I going to fix it?,” I hand more of the thinking over to the students. I now view mistakes as opportunities to expand students’ horizons. I see students as the answer keepers. I try to talk less and listen and inquire more. But most importantly, I want students to feel safe and trust that no matter what they say, I will still think they’re smart. These are the guiding principles that help me be more responsive when students share their thoughts. We hope you try them out!

A special thanks to friend and colleague Sharon Fargason for brainstorming ideas with me for this post!