modeling with fractions, revisited

In my last post, I described three boys attempting to model a problem about servings of chicken and potatoes.  My colleague and I were watching the boys wrestle with how to represent the potato servings with fraction circles.  At the end of the post, I “saved” the boys from crossing out their representation and starting all over.

I tend to consider myself a pretty decent math teacher.  I try not to say anything that a kid can say. I ask a lot of questions.  I use rich tasks that promote deep thinking.  I encourage my students to persevere and engage in productive struggle. And, yet, here I was rescuing these boys from their learning.  What, exactly, was I saving them from?  Maybe I thought I was saving them from starting over.

A mathy friend read my last post and she asked me, “why didn’t you let them cross it out?”

“because I….. Well, I thought I…..hmmmm.  Oh, man. Why DIDN’T I let them cross it out?”  I sighed as I realized that  I may have hijaked their learning. I saw those boys on the edge and I yanked them right back away from it.  Why did I do that?  Maybe I felt rushed.   I was super excited to watch them in the thick of making some really cool connections. Maybe old habits are hard to break. Did I ruin their learning?  Of course not.  Would it have been interesting to see what the boys would have done if I hadn’t said, “wait!”?  I think so.

It is really challenging to maintain the balance of providing and removing scaffolding for students.  How do we keep the struggle productive?  What makes a struggle productive? I have been thinking about this for awhile now.  Here is what I came up with:

A struggle is productive when learning is happening, connections are being made,  questions are being asked, and  doubt and uncertainty are being examined.

My colleague and I were on a learning round. We were “looking for” evidence of students checking to see if an answer makes sense within the context of a situation and changing a model when necessary.  In my last post,  I commented that I thought the boys were abandoning their model if they crossed it out, but I have changed my thinking. I don’t think they were abandoning their model. I think they wanted to change their model so it more accurately reflected the context of the situation.  They were sense making. They were struggling productively.

I love looking at student work. It is amazing.  I often walk away with more questions than answers, but that is okay.  I also walk away with a deeper appreciation for how my students are making sense of math. Below is the poster that the boys shared with their teacher and their classmates after they had finished solving the problem. The top portion is their work on the potatoes part of the problem.  The bottom portion is their work on the chicken part of the problem. They completed the bottom part after I left. I have been looking at their work for awhile.  I think it is beautiful.

My favorite parts are the “cross outs”.

modeling with fractions (twice)

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.