Error Analysis

While scrambled solutions focus on the order of operations, another powerhouse technique from cognitive science focuses on the accuracy of those operations: Error Analysis (sometimes called "What Went Wrong?").

If scrambled solutions are about building a logical skeleton, Error Analysis is about developing the "mathematical immune system." In this activity, students are given a fully solved problem that contains exactly one intentional mistake. Their job is not to solve the problem, but to find the error, fix it, and explain why the original "mathematician" made that choice.

Many students view math through a lens of "fragile perfection"—if they make one mistake, the whole endeavor is a failure. This creates high anxiety. Error Analysis flips the script by making the mistake the object of study rather than a personal failing.

From a brain-based perspective, this technique triggers comparative thinking. To find an error, a student must mentally run the correct procedure alongside the flawed one. This dual-processing strengthens their understanding of the "boundary conditions" of a rule—knowing not just what to do, but what not to do and why.  The error chosen should be a high-frequency misconception.  For instance, many students when doing the distributive property, forget to distribute the outside term across both inside terms.  Students for a problem like 3(x + 5) will say it equals 3x + 5, not 3x + 15.

One suggestion is to create the "math autopsy which is a wonderful collaborative activity for small groups.  Begin by giving each group a "Case File" (a worksheet) featuring a character—let’s call him "Messy Marvin"—who has consistently gotten the wrong answer.  Students must use a red pen to circle the exact line where Marvin made his mistake.  In a dedicated column, students must rewrite the problem correctly and write a "note to Marvin" explaining the rule he forgot. This forces the use of mathematical vocabulary (e.g., "Marvin, you forgot to use the Inverse Property...").

In a digital environment, Error Analysis can be made highly interactive, so the second method is called "Spot the Bot".  Use the Desmos Activity Builder to show a pre-animated solution. Students can use the "Sketch" tool to circle the error directly on the screen. Or you can  present a solved problem with four different potential "fixes." Students vote on which fix actually addresses the root cause of the error. Or you could  give students a solution generated by an AI that contains a subtle logical hallucination. Have them "peer review" the AI's work.

Error Analysis is the perfect companion to Scrambled Solutions. While you use scrambled solutions during the Bridge Phase to build logic, you use Error Analysis during the Refinement Phase (the end of a lesson or the start of the next day).

It is especially effective as a "Do Now" or "Bell Ringer." By starting class with a "broken" problem, you immediately engage the students' critical thinking. It signals that the classroom is a safe place to discuss mistakes, and it prepares their brains to be on the lookout for those same pitfalls in their own work.

Experts in any field—whether they are surgeons, engineers, or mathematicians—are defined by their ability to self-correct. By intentionally bringing errors into the light, we move students away from "answer-getting" and toward "sense-making." When a student can explain why a mistake happened, they are no longer just following a recipe; they are becoming the chef.

Teaching Error Analysis in Math.

 Many math teachers do not take time to have students analyze errors to learn more.  I know most of my students would rather not look at any problems to see what they've done incorrectly.

Learning to look at errors and analyze them is an important skill. There are several reasons to teach students to perform error analysis.  First, this promotes higher level thinking skills because it requires students to create, analyze, and proving hypothesis, thoughts, and ideas.

Secondly, it helps students connect the conceptual with the steps needed to solve a problem.  When students focus on the process, they often loose sight of the concept associated with the problem.  When they are able to find the mistake in the process and explain the error, they are showing they understand the concept.  Finally, it helps students prepare to take tests with multiple choice questions.  If they find an answer that is not one of the choices, they can review their math to find the errors.

There are some activities one can include in class to help students learn to analyze to find errors.  It is important to use these activities to help students learn to analyze their work for mistakes because most do not know how to do it.  I know from personal experience most of my student's analysis before they learn to do it usually consist of "I didn't know how to do the problem" or "I have no idea".    In addition, learning to analyze mistakes helps students develop their ability to communicate their thinking.

No matter what the error is in the problem, students need to rework the problem correctly so they get the expected answer.  It is not enough just to identify the error but they still need to do the problem correctly.  Both parts are needed for students to complete the error analysis.

One way is to assign a problem to students to work.  Then collect the solutions and check them quickly to find the ones that were done incorrectly.  Choose on of those to have students look and see if they can figure out what was done incorrectly and what needs to be done to do it correctly.  The other choice is to do a problem incorrectly and have students identify the error and how to correct it.  Some textbooks now have problems in each section in which allow students to practice their error analysis.

Create a gallery walk with several problems done incorrectly on them, spaced around the the classroom.  Divide the students up into groups of two or three and have them go through and find the error on each sheet.  You can have them write down their solutions on the sheets or on a separate sheet of paper.  By putting students into small groups, they need to discuss the error and it improves their communication skills.

It is suggested that students stop erasing mistakes as they do the problem.  Instead, have them circle the mistake and explain what it was before continuing to the end of the problem.  This teaches students to avoid making the same mistake in the future and it shows students that making mistakes is a part of the process.  It helps students accept that making mistakes is fine but you need to use those mistakes to learn to do the problems correctly.  Furthermore, students have a way of helping themselves assess their understanding.

My next entry will be about using centers in middle school or high school math because one center can be on analyzing errors.  Let me know what you think, I'd love to hear.  Have a great day.