As classroom teachers, we know that our roles often go beyond conveying curriculum. Sometimes we teach seemingly unrelated lessons that tie back to our subject area. That happened to me a few years ago in my eighth-grade math class.
My student Jessica was walking into class with her heavy backpack on (names have been changed). When she went to take off her backpack to sit down, Billy was leaning over in her direction and got hit directly in the face with her backpack. Jessica said, “I’m sorry, I didn’t mean to do that. I wish you didn’t lean in.” Billy responded, “Well, it did. I wish you didn’t store your entire locker in your backpack,” and the conversation only escalated from there. Billy was angry, and Jessica was both confused and frustrated.
Jessica didn’t understand Billy’s reaction, and this gave me a chance to share with her some advice about a proper apology. Too often students think that saying “I’m sorry” is enough. I shared that nobody makes a mistake on purpose, and we all knew she didn’t mean to do it. But she needed to take ownership of her actions and not blame Billy for what was her fault. Later, she gave a much better apology, and the two of them reconciled.
A good apology leads to learning of both knowledge and behaviors. From a good apology, we learn how to be intentional with our actions in preventing future incidents in similar situations. Abstracting this framework for a good apology, we can improve learning in any context, especially math.
I like to connect math corrections with a good apology. Students will make mathematical mistakes in my class, but they won’t always have interpersonal conflicts. Still, learning how to handle these situations is just a good life skill that I like to pass on to help them down the road.
Learning from Our Mistakes
In math, we know the value of learning from mistakes. We can’t thank Jo Boaler (and Carol Dweck) enough for sharing the benefits of mistakes and how they contribute to learning. When error analysis and corrections are combined with the framework for a good apology, students are able to meticulously reflect on their current conception and iteratively progress to deepen their mathematical understanding.
I provide the following prompts for students to do corrections:
- What did you do (not what you didn’t do) that led to your mistake, and/or what were you thinking that led to your mistake?
- What should you have done?
- Redo the part of the problem where the mistake occurred (not the entire problem).
- Create and solve your own example of this part.
Each individual step is intentional to support student learning. The first part explicitly asks students to state what they did, not what they didn’t do. This is important because we want students to honor their current conception, not misconception.
Let’s look at an example. A common mistake a student might make when finding “3 squared” is getting the result of 6. When asking them to reflect on their mistake, I might get a response like “I didn’t get 9.” That’s the misconception. It’s deflecting and acknowledging the result they should have gotten but doesn’t address the root cause of the problem that stemmed from the current conception. This type of response might allow the mistake to occur in the future, similar to how superficial apologies lead to unresolved conflicts.
If the student were to honor their current conception, they would say, “I multiplied my base 3 with my exponent 2 and got 6. I should have known that the exponent represents two factors of my base being multiplied (3x3), which would have led to 9 as my result.”
The second response shows a reflection of the current conception with clear next steps. I would argue that the second response completed steps 1–3 of the corrections, and all they would need to do would be to create and solve their own example to show some sort of iterative transfer in their learning, something like “4 squared is 16.” In this response, the student takes ownership for what they did and what they should have done, leaving the error less likely to occur in the future. Conflict resolved.
In contemporary education, one of the hardest things for teachers to assess is learning. We assess what students know and do, but we don’t know where or when students learned it—if it was from family, a previous teacher, a tutor, or the internet. Structured corrections that allow students to reflect on their current conceptions and iteratively progress show learning in the moment. For students, learning becomes real. They appreciate being able to do something they previously hadn’t been able to do. It’s even better when they see this all happen in one class period.
Corrections, in alignment with a framework for a good apology, intentionally deepen student learning and make students better at handling conflict resolution. It’s a win-win. As educators, we need to create conditions that allow students to make and document their mistakes, and that provide the time for them to reflect on and value learning from such mistakes. When we do this, we cultivate the growth mindset in our students through concrete examples of their iterative growth.