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Alumna shares the "write" way to overcome math misconceptions

Kimberle Kembitzky

Kimberle Kembitzky (PhD '09, Mathematics Education) is in her 14th year as a mathematics teacher at Hilliard Davidson High School, Hilliard, Ohio, and her 23rd year of teaching math. At the invitation of her former advisor, Professor Doug Owens, Teaching and Learning, she explained to a class of graduate students how to use ERNIeS, an error analysis process she designed for use with high school mathematics students. She is both a Master Teacher and National Board Certified Teacher.

As a successful math teacher, I continually ask the question, "How can I help students confront their own math misunderstandings?"

During my doctoral coursework, I examined the different types of reflective writing advocated by mathematics researchers to support learning. I knew, from my experience and the research, that students' faulty mathematics logic, once acquired, is extremely difficult to correct.

In fact, research shows that students who appear to overcome a misconception after instruction often return to it only a short time later.

To address this issue, I used knowledge gleaned from my PhD work to modify an intervention instrument that I had been using called ERNIeS, which stands for Error Analysis. I first created the process because I believed that if students were cognitively aware of why they made their errors and communicated the reason in writing, they were more likely to overcome their misconceptions permanently.

Over the years, I had used the process with my students, refining it based on their feedback and results. Once I became a doctoral candidate, I realized the perfect dissertation research lay before me. I could validate the effectiveness of the ERNIeS process by testing its success in helping geometry students.

My study was among the few to look at students' use of written analysis to reflect on their own misconceptions in mathematics, especially geometry. The results show clear benefits to students of using the ERNIeS process compared to students who did not use it.

I also learned that the ERNIeS process specifically helps students struggling with thinking spatially -- a skill that can influence success in geometry class. Spatial ability level alone did not influence the accuracy with which the error was analyzed. However, the higher the spatial ability level, the less likely the student was to repeat the misconception.

The ERror aNalysIS (ERNIeS) process: How it works

The ERNIeS process calls for students to:

Rewrite the original problem (including any diagrams) that they missed on a quiz.
Rework the problem, correcting any error made in the original attempt. They may get assistance from others.
Determine if the error was minor (careless procedural) or major (conceptual, e.g., didn't use the right process at all or were not sure where to start the problem).
Describe, in their own words, why they think they missed the problem. Think carefully and explain as best they can. Consider what they were thinking when they did the problem the first time and discuss what they didn't understand then but do understand now.
Turn in the ERNIe with their quizzes.

Students and teachers benefit

In my study questionnaire, students said the ERNIeS process:

Improved their comprehension of the concept
Enhanced their metacognitive abilities concerning their errors
Was complementary to different learning styles
Prevented the repetition of the misconception
Was an excellent study tool

High math achievers and motivated learners benefit the most because they're naturally reflective and analytical. But students with challenges can also benefit from using the ERNIeS process.

For instance, students with low math achievement who are good at writing enjoy the process. Students who struggle with writing can talk through their ERNIeS with the teacher. Use of manipulatives can help them explain their ideas.

Teachers benefit from ERNIeS because in reading the analysis, they see students' thinking, This is especially helpful since they can't have a one-to-one discussion with every student.

As our student population continues to grow, it is not uncommon that math teachers can have almost 150 students. This is how they can have more one-on-one time with their students.

Implementing ERNIeS for the first time

From experience, I know that timing of feedback to students is important. If they address their misconceptions when they first occur, they are less likely to repeat them later.

I suggest fitting the ERNIeS process into a class master plan. Even with my experience, I don't use ERNIeS all the time. It's time consuming, and feedback must be provided to students quickly.

If using the process for the first time, I recommend setting a modest goal: perhaps two or three times a semester. The process is best used on early quizzes, as it allows teachers to catch student misconceptions early. Students have a better chance of correcting their misunderstandings permanently.

I provide students with model analyses so they can see what a good one looks like. I also recommend getting everyone in the school on board, such as intervention specialists. The more people who help with the process, the more likely students are to correct their misconceptions.

When grading math writing, I use a rubric. I recommend consulting with the school English or language arts department to create such a tool. As a math teacher, I'm looking for correct math facts and logical explanations, The English faculty can offer guidelines about how to grade other categories such as organization of the paper, audience and purpose, and presentation, spelling and grammar.

More math writing ideas and tips

Writing about math helps improve students' ability to communicate mathematically. I offer students a variety of fun, creative writing opportunities to stimulate their thinking and engage their related interests.

Here is a sampling of math writing activities from my list:

Write a haiku about a math vocabulary word.
Write a math autobiography at the beginning of the year so teachers can understand students' backgrounds.
Create a children's book explaining math concepts.
Collect other people's incorrectly worked problems and write an ERNIe on them, referred to as fuzzy math review.
Write a brief reflection at the end of class on what was learned that day in class. What question did they end with? What question would they ask tomorrow? This prompts questions from those hesitant to ask.

To sum up students' feelings about the ERNIeS process, I recall one student's response to the study questionnaire: "I would do it in another math class. And I'm thinking about doing it in other classes in general because being reflective is never a bad thing."

Read Kembitzky's full dissertation on OhioLink ETD.

The Ohio State University

www.osu.edu