# 5th graders who can’t see equivalence

So I’ve uncovered another math misconception, and I’m fairly baffled.

I’m pretty sure most teachers write blog posts to share something they’ve mastered, something they’ve learned, something on which they consider themselves an expert. This isn’t one of those posts.

I’m admitting that I’ve got a lot to learn, even after 28 years of teaching math to kids. The more I learn about teaching math, the more I believe that I DON’T know about teaching math.

A 5th grader looked at these two fraction models today, and could not figure out where the yellow portion was shown in the red shaded model. I asked him to find the yellow part in the model on the top and outline it. He first drew a line around three sides of a 1/6 piece on the upper model. He didn’t even know what it meant to outline something, let alone find 1/3 in a model of sixths.

The students were trying to solve this problem: 1/3 + 1/6 =

My confused gentleman was looking at another student’s model of the problem, which looked like the red example above.

So how did I get him to see the 1/3 hidden in the model of sixths? I did some improv teaching and grabbed two pieces of half papers. I partitioned them like the models above. I shaded the yellow part on one paper with a dark sharpie, so that the shading would show through the other paper. Then I placed the sixths model over the top of the thirds so he could see the fraction lined up. He then said, “OH!” and was able to return to the model on the board and outline the correct portion.

Wow! It’s amazing what you learn about kids when you force them to explain things thoroughly.

Am I at a place where I can fully understand how kids think? No way! Do I know how to use progressive instruction to develop spatial sense in all my students, so they don’t lack the skills to identify equal portions? No way!

I’m going to need a few more lifetimes to master this job we call teaching.