Yesterday, I participated in a learning round.  I visited several k-5 classrooms with two other teachers.  We looked for evidence of modeling with mathematics.  In third grade, we found an opportunity to observe what it looks like when a model pushes students to the edge of their understanding. Incidentally, we found ourselves on the edge too.

When we walked into third grade, there was a quite hum of math talk and it took us a minute to find the teacher.  She was on the floor with a small group.   Several other students were working independently at their desks.  A group of boys caught my attention across the room. They were gathered around a large piece of poster paper having an intense conversation about something.

Tina, a Kindergarten teacher, and I went over to the boys to get a closer look at the poster.

“Do you guys mind if we watch you do math?”

“No.  We are working on this problem.”

(Thank you k-5mathteachingresources for sharing this problem with us!)

As Tina and I pulled up chairs and read the problem, the boys went back to their Math. Joe was writing on the number line and narrating his thinking as he counted the groups of partial pounds (2/3) of potatoes.

(replication of student work)

Bobby was coloring in groups of 2/3 on the circles that were drawn on another section of the poster.

I asked, “How do you know how many people there are?”

“We read the story. There are 6 people at the dinner. We are figuring out how many pounds of potatoes she needs.”

“What do you have so far?”

“Each person gets 2/3 of a pound so we think she need 4 pounds of potatoes.”

(He pointed to the groups of two thirds on his number line as he counted.”

“Wait a minute,” his partner said.  “I think we need to change something.”

“What do you mean?” he asked.

“Something doesn’t seem right about our circles.”

I asked Bobby to tell us some more about what he was thinking.

“Well. We already have 4 pounds of potatoes, but we aren’t done showing all the people. We still have to show potatoes for 2 more people. We need to draw some more circles.”

Mike thought about it. “We can’t draw any more circles or we will have more than 4 pounds. Each circle is a pound. Our number line says we need 4 pounds.”

I asked, “How do you know you have 4 pounds?”

They showed me again on the number line. “Why do you need two models?”

“Mrs. T said we needed to create two different models.”

I decided to push more, “Where are the pounds in the circles?

“See,” (they pointed to each circle) “1, 2, 3, 4”

“Oh!”  Bobby said.  “We need to move that (points to 1/3 in second circle) over there. (points to the blank space in the first circle.”

One of the boys reached for a marker and motioned as if he was going to scratch out all the circles and start again.

“Wait!” I said.

My colleague and I came into the classroom looking for students who might be “checking to see if the model makes sense within the context of the situation and changing the model when necessary.”  These students realized that their model didn’t make sense.  Their solution was to abandon the model.

Welcome to the edge.

What are these boys trying to understand? What are they confused about? What would you do next?  Post your thoughts.