by Tessa Henry, Supervising Editor—Math Editorial
“You must never feel badly about making mistakes . . . as long as you take the trouble to learn from them. For you often learn more by being wrong for the right reasons than you do by being right for the wrong reasons.”
― Norton Juster, The Phantom Tollbooth
Think of your students who say, “I hate math.” Why is this? For some, the reason is the fear of making mistakes, of getting a wrong answer, especially on a test or in front of the class. There is no person in history who has never made a mistake, so why are we so afraid of it?
Try encouraging your students to flip the script, to embrace math because you can work hard and get a right answer. It might not be your first answer, but you can try again until you get there. Think about puzzles and brain teasers, like sudoku. You might struggle with them for a while, erase some stuff, or even start over. But when all the pieces fall into place, it can be a very satisfying accomplishment. In fact, the harder the puzzle, the more satisfying it is. Making mistakes along the way is just a part of the process. In the same way, math can be like a puzzle. Maybe you don’t really need to know the weight of 45 identical watermelons or how long a piece of ribbon is, but if you keep at it until you get the answer, you can know that it is right.
At Six Red Marbles, we think about mistakes not just in classrooms but also in how math programs are designed. The way mistakes are handled by teachers, learners, and even curricula can change outcomes.
Learning from Your Mistakes
This is an oversimplification, but there are two basic ways to learn something: You can learn from your mistakes, or you can learn from other people’s mistakes. Over time, people have developed a body of knowledge we call mathematics. Much of this was done through trial and error, experimentation, and deduction. Mathematicians have always challenged and refined each other’s work and continue to do so today. The fact that you are handed this information in a neat package is due to their hard work, making and fixing mistakes for thousands of years. This is how you learn from other people’s mistakes.
Of course, we still continue to make our own mistakes. And each mistake is an opportunity. For teachers, it is an opportunity to understand students’ thought processes. For students, it is an opportunity to gain a deeper understanding.
When students make mistakes, emphasize that it is part of the process of learning. Encourage them to consider why: Was it a careless error, a computational slip, or a conceptual misunderstanding? Analyzing and fixing mistakes helps students avoid repeating them while giving teachers insight into thought processes.
These classroom strategies also mirror how strong curricula are designed. At SRM, we build in opportunities for reflection and revision to help learners engage with mistakes productively.
Here are some ways to use mistakes as learning opportunities:
- Use a “correct and reflect” strategy for tests, where students correct one or two errors, briefly explain what they did wrong, and suggest how they could avoid this error next time.
- Encourage students to keep track of the types of mistakes they make so they can focus on these areas for future tests. Perhaps keep track of it in an “Oopsie Journal” or “Math Bloopers Log.”
- Have students brainstorm about types of errors and ways to avoid them. Compile the results in a classroom chart.
- Have students play a “two truths and a lie” type game, where each partner creates three versions of a solution to a problem. One is correct, and the other two have errors. Partners trade and try to find each other’s correct solution.
- Pair up students based on the types of mistakes they tend to make, to benefit both partners. For example, if one student understands what they are doing but makes mistakes because their work is messy, their partner could be a student who struggles with the concepts but shows their steps in an organized way.
- Reward students for catching your mistakes, if they can explain what you did wrong. The reward could be something like a sticker or an extra point on a test or assignment. You may have to make an occasional mistake on purpose to give them a chance at this.
Praising Effort
Praising the effort of a student who made a mistake can be tricky. If a student is focused on what they did wrong, “good try” can sound generic, insincere, or even sarcastic. Try to be genuine and specific about what they did well, and don’t just follow up with a critique every time.
Here are some examples you might use:
- I’m impressed with the way you kept at it when the problems got challenging.
- You did a great job of trying different strategies until you found one that worked for you.
- I like the way you show all your steps clearly so that it is easy to follow your work.
- It’s great how engaged you are in class, especially when you ask questions that other students might be too shy to ask.
- You’re very good at collaborating and supporting your classmates.
- Your hard work is really helping you improve; I know you’re going to do great if you keep at it!
Promoting a Growth Mindset
Students may think that because you are a teacher, math has always been easy for you. For many math educators, this wasn’t always the case. Your students may find it reassuring if you share some of your own difficulties in your mathematical journey. None of us were born knowing how to calculate percents or write a two-column proof. Even if you think you are “bad at math,” it is possible to improve.
The ideas that challenge a person most are often the ones that endure. What comes easily can vanish as swiftly as it appears, but the concepts that demand struggle—the ones wrestled with, questioned, and finally forced into clarity—are the ones that remain.
Everyone makes mistakes, even famous mathematicians. One example is Pierre de Fermat, who wrote a margin note to himself around 1637 about a theorem now known as Fermat’s Last Theorem. The note stated, “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” He never published this demonstration; it is assumed that he started to write it and found an error. The theorem was not proved until 1995, over 350 years later. In the meantime, there were many attempts and many mistakes.
As a teacher, you can also have a growth mindset. Look back over your tests, and see where most students made errors. Could the question have been unclear? Is there a gap in their collective understanding that needs to be addressed? If time allows, discuss some of these questions with the class.
In our work, we aim to design math materials that encourage this same mindset: content that’s rigorous but forgiving so students and teachers alike see mistakes as part of the journey.
Some Ways of Avoiding Errors
Although it is important to create a classroom environment that normalizes mistakes, it is also important for students to learn from these mistakes and avoid them in the future. Here are some strategies that can help.
For careless mistakes:
- Graph paper can help students write neatly and line up numbers properly.
- Solve equations vertically. Students usually make fewer mistakes and find this easier to understand. Of course, if a student is solving correctly and understands the process, either method is acceptable.
- When working on a computer, students may resist writing out their work and instead try to do computations in their heads. Provide scratch paper or encourage students to create a “digital scratch paper” document for showing their work.
- For paper tests, leave room for students to show their work with the problem, if possible. This avoids students making errors in copying their problem to scratch paper or copying the answer back to the test.
- Make sure students read the problem carefully, more than one time if needed to understand what is being asked. It may help to make a list of important information and restate the question. Try to avoid wording questions in a confusing way.
- Make sure that students do all the parts of any multistep problems.
For computational or procedural errors:
- Encourage students to check their work by solving the problem a different way or substituting their answer back into the problem.
- For problems involving a step-by-step process, students may benefit from writing out the steps in one place in their notebook to look at while they practice or using a mnemonic to remember the steps.
- Have students check for reasonableness by rounding to estimate an answer or making sure the solution of a real-world problem makes sense (really, 54,267 watermelons?).
- Suggest that students break up longer problems into simpler ones and check their work as they go.
- Allow students to check answers on a computer or calculator.
For conceptual errors:
- Leave time for students to try problems on their own and ask questions in class. They may think they understand from watching you do it but may struggle once they try it on their own.
- Make sure students are strong in any prerequisite skills that could be a barrier to understanding a topic. Review and practice if needed.
- Have students explain the process in their own words so that you can identify where their misunderstanding is.
- Encourage students to see a tutor or come to office hours, if possible.
- Students may benefit from a different explanation of a concept. Try a different approach or suggest online videos for another perspective.
- If students have access to AI, help them generate prompts that can deepen their understanding. For example, “How does long division work?” or “What are the steps for completing the square? Explain the purpose of each one.” Emphasize that they can ask for clarification on any parts they don’t understand or ask for a new explanation as many times as they want. Note that this technology is still being developed and some answers may contain errors; however, it can still be a useful tool.
Ultimately, mistakes aren’t setbacks; they’re stepping stones. As content creators, we believe every error is a chance to deepen understanding. That’s why Six Red Marbles prioritizes clarity, reflection, and accessibility in the math programs we help build. When students and teachers embrace mistakes as opportunities, everyone moves forward.
Learn more about Six Red Marbles’ math solutions here: https://www.sixredmarbles.com/k-12-math
Here are some additional resources you might want to check out:
- https://www.nwea.org/blog/2025/embrace-mistake-making-in-math/
- https://cognitivecardiomath.com/cognitive-cardio-blog/how-to-use-math-mistakes-as-a-teaching-tool/
- https://catlintucker.com/2020/02/error-analysis-station-math/
- https://www.tarheelstateteacher.com/blog/the-power-of-mistakes-in-learning-math
About the Author
Tessa Henry is a Supervising Editor for Six Red Marbles. She has been writing and editing math products from pre-K to calculus for over 25 years, with a particular fondness for geometry. Tessa enjoys spending her free time outdoors hiking, relaxing with a good book, and discovering new live music. She also has a creative side, often unwinding with crochet projects.