Diagnosing Errors

Overview

Diagnosing Errors helps students learn from mistakes by analyzing incorrect solutions, discussing misconceptions, and explaining corrections. Instead of being told what went wrong, they collaboratively identify and articulate the source of the error, and explain how to correct it.

Through this practice you will be able to:

• Deepen student understanding through error identification and correction.
• Promote critical thinking via discussion of misconceptions.
• Create a safe space to learn from mistakes.
• Build student confidence and precision in problem solving.

Materials

Day Before Prep (~50 min)

1. Select a Task (5-15 minutes): Select 1–3 problems related to a current or recent topic that represent the key objectives of the unit or standard.
2. Solve the Problems Yourself (5-15 minutes): Solve the problem to ensure that you are familiar with the correct answer and are prepared to guide students in a discussion. As you work through it, ask yourself:
   • What strategies might students use?
   • What concepts or steps could be confusing?
   • Where might students get stuck or offer unexpected ideas?
3. Collect Samples (10-15 minutes): Create or collect samples of incorrect solutions. Remove student names if using real work. Identify the misconceptions that are present in the work. Ideally, these mistakes will be related to common conceptual misunderstandings that students are likely to encounter again.
4. Review Supporting Docs (5-10 minutes): Prepare guiding questions or sentence stems to guide students through the discovery process, encouraging them to share their thinking and provide evidence (e.g., “I think the student thought…”, “The mistake might have happened because…”).
5. Group Students (5-10 minutes): Consider grouping students strategically to support productive conversation.

Conducting the Practice (30-45 min)

1. Introduction (5 minutes): Explain to students that today’s goal is to understand how and why mistakes happen, not just how to get the right answer, and see how errors provide opportunities for learning. Here is sample language:

   “Today we're going to take a look at some common math mistakes. I want you to know right from the start…mistakes are nothing to be embarrassed about. In fact, they’re one of the best ways we learn. By looking closely at what went wrong and why, we can understand the math more deeply and become stronger problem solvers. So let’s work together, share ideas, and learn from these mistakes as a team.”

2. Present Errors (5-10 minutes): Show the sample problem(s) with errors. Keep the student work anonymous.

3. Small Group Discussion (10-15 minutes): In groups of 2–4, students analyze the error using guiding questions:

   • What do you think the student was trying to do?
   • Where did the mistake occur?
   • Why do you think that mistake was made?
   • How would you correct it? Students can also use the Noticings & Wonderings Protocol to individually think through the task before collaborating with their peers.

4. Whole-Class Share-Out (5-10 minutes): Invite groups to share their analysis. Ask follow-up questions to deepen reasoning. Share any additional explanations to help make sure that students have an accurate understanding of the error, how and why it was incorrect, and how to correct it.

5. Reflection (5 minutes): Students write individually about what they learned, how the mistake could be avoided, or what they will do differently in future problems.

What to Look For

• Students identifying not just what is wrong, but why the error occurred
• Use of precise mathematical vocabulary during group discussions
• Students building on each other’s ideas and respectfully disagreeing
• Increased participation, especially from students who may be quieter or typically reluctant to contribute
• Connections made between the error and the underlying concept

Pitfalls to Avoid

• Focusing only on the correct answer: The power of this routine lies in understanding the reasoning, not just fixing mistakes.
• Shaming or identifying students connected to the errors: Keep examples anonymous to maintain psychological safety.
• Rushing through the discussion: Allow enough time for students to talk through their ideas thoroughly.
• Choosing overly complex or trivial errors: Focus on mistakes that reveal conceptual misunderstandings students are likely to encounter again.
• Not providing discussion scaffolds: Some students need help getting started with analyzing errors. Be sure to provide sentence stems or guiding prompts as needed.

Additional Inspiration

Teaching & Learning Mathematics Through Error Analysis

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This practice was developed in collaboration with WestEd.