If you were a “good” math student, your teacher may have filled your papers with a checkmark next to each correct answer. But handing back “perfect work” with a slew of checkmarks is a missed opportunity for math teachers. Every time we provide a student with feedback, we have a chance to send a message.
We can signal what we value and the strengths we see in the student’s work, providing insight into areas for growth and further learning. In math we value clarity and logic, creative solutions, perseverance, and curiosity. Used strategically, positive feedback can reinforce these cornerstones of the discipline. Although the examples I provide in this article are geared toward middle and high school math classes, positive feedback can be used at a variety of grade levels.
Reinforcing Effective Communication
Math teachers frequently ask students to show their work. When students give us insight into their thought processes by writing enough on the page, we can provide positive and specific comments in return. Our reinforcement helps students learn what specific aspects of their work were effective at clarifying their logic.
Here are some examples:
- Your explanation here helps me see how you got from one step to the next.
- The picture you drew helped me to understand how you’re thinking about this.
- When you defined the variable before using it, I was able to follow your reasoning.
- The sentence at the end helped me to see that this is your final answer.
All of these comments acknowledge the time that students put into explaining their reasoning. Providing positive reinforcement when students effectively convey their thought processes helps them develop into effective communicators of mathematics.
Recognizing a Mathematician’s Craft and Choices
Math teachers constantly remind their students that “there is more than one way to solve a problem.” This is true—the art of problem-solving allows students to discover elegant paths to a solution. Math talks have become popular because of their emphasis on multiple problem-solving techniques. The comments that we make on student work can reinforce the value of mathematical thinking.
You might write something like these:
- I didn’t think of this strategy!
- This is a clever implementation of factoring.
- This step reminds me of the example we looked at when _____.
- I like how you adapted the idea from _____ to this problem.
These comments will help students see their technique within the space of many pathways to a solution. When we reference the strategy that they used and contextualize it, we help them connect their problem-solving process to the content.
Celebrating Growth and Perseverance
Math is hard! Math teachers must find ways for students to grapple with the content and engage in productive struggle. If we give students credit for their progress and perseverance, we can encourage them to push through challenges the next time they arise.
Here are some examples of how you might emphasize growth and praise perseverance:
- Great job catching the mistake here.
- I can tell this was a long and messy computation. By keeping your work organized and sticking to your plan, you persevered.
- I noticed that you had trouble with this skill in the last unit, but you have mastered it now! Great job sticking with it.
Students don’t always notice their own growth. When we can point it out to them, they see that their hard work and struggle is worth it.
When students perform at a high level, positive comments can push their thinking beyond the standard content. With feedback, we can inspire their curiosity and encourage deeper thinking.
Here are some reflective comments you could make:
- What about this problem helped you realize that you needed this particular strategy?
- How did you check your work?
- What strategies did you use to keep track of all the steps needed to solve this problem?
- How will you remember the connection that you made here?
- What do you think would happen if the problem were slightly different? How would you have to adapt your approach?
All of these questions send the message that learning doesn’t stop at a perfect test score. We should continue to ask questions, pursue our curiosity, and find new and novel ways to apply what we’ve learned.
I hope that these ideas inspire you to provide comments beyond the simple check mark. We should be putting just as much care and attention into the feedback we provide to high-achieving students as what we give to students who need more practice.
While some may think that math teachers have it easy when it comes to grading, we shouldn’t take the easy way out. Writing a few comments might add a minute or two to the time it takes to mark a paper, but I assure you that the gains you will see in student confidence, motivation, creativity, and understanding will be worth it. By taking the time to provide meaningful, detailed, specific, and positive feedback, we provide all of our students the opportunity to grow.
Note: I wrote this piece after reading Alex Shevrin Venet’s “How to Give Positive Feedback on Student Writing,” because I was struck by how many of the guidelines that she provides are applicable to the math classroom. I recommend taking a look if you haven’t already!