I’ve been tutoring some struggling mathematicians after school a couple days a week. This last week, we were looking at comparing fractions with different numerators and denominators. I showed the students region models that represented 2/3 and 3/4.
Just about every one of the 10 kids looked at the model and said that 2/3 was greater than 3/4. I was dumbfounded. How could the kids look at these two models, where one was clearly bigger than the other, and say the smaller one was larger?
I tried to get inside the heads of my students. Where was the misconception? Then it hit me. I had been showing numerous examples of unit fractions and pounding into the kids’ heads how 1/4 was smaller than 1/3, 1/10 smaller than 1/5, etc.
I did my best to drive home the point that the larger denominator actually indicated the smaller fraction. Then, we switched to comparing fractions like 3/4 and 2/3. Eureka! The kids were looking at the individual pieces, not the entire shaded region! When I asked which was bigger, they said 2/3 was greater than 3/4 because 1/3 is greater than 1/4!
Lesson learned. When kids seem illogical and out-of-touch, there’s a reason. Our job as educators is to pinpoint these misconceptions and help students make sense of the world around them.
For a really cool visual math resource, check out this fraction model on the NCTM website, Illuminations.
Have you discovered any math misconceptions with your students that you could share?