Math Disagreements are Valuable!

My favorite moment in math class is when I take a poll about possible answers to a math problem and one student stands by their answer while others all pick the same one. This brave student may be incorrect, but they have a reason and a strategy they’ve used to get their answer and they are willing to disagree with their classmates. So the rest of us watch as the student shows us his or her reasoning behind their confidence in their answer. If this student has swayed his or her classmates, they change their answer. Others stand by their original solution, telling their reasoning and strategy. It’s compelling actually.

An article in the May 2011 issue of Teaching Children Mathematics states that disagreement discussions in math can “provide students with the impetus to think deeply about mathematics in an effort to make sense of a situation. The discourse that surrounds the disagreement allows students to organize their thoughts, formulate arguments, consider other students’ positions, and communicate their positions to their classmates.” (Barlow and McCrory pg. 531)

For me these discussions provide insight into student thinking, and how they are understanding concepts. It also reveals their misconceptions. Recently several students disagreed over which was larger 4/5 or 5/4. Many students agreed that 4/5 was bigger. During their discussion of why it was the larger number, I discovered they carried the misconception that the bigger the denominator, the bigger the number. I pulled out our fraction pieces and had the students build the two fractions. They discovered that 5/4 is the larger number. They reminded each other that we discussed this when we first started fractions using the analogy that I shared with them—splitting the candy bar between three people and having 1/3 gives you a bigger piece than sharing it between five people and getting 1/5.

Thankfully our students work in a community that allows for them to feel safe to disagree and are encouraged to state their ideas during these discourses. Throughout the school we value risk taking, open discussion and differing ideas, and not just at math time. The ability to disagree well is taught and modeled by the teachers and staff to students in the Rocks and Rivers class all the way to the 8th grade.

The next day the students had a fractions assessment. The students faced the same challenge they had the day before, but it was with different numbers. The question, which is larger 2/3 or 3/2, spurred less discussion this time. The number of students who had struggled with the misconception about denominators the day before lessened and understanding was evident. Sometimes it doesn’t happen this quickly, and more conversations must occur. Thankfully we have the time and place to do so.