“Tens and ones didn’t work so we had to try a new way.”

Even as early as second grade, we are working on breaking down seemingly fixed mindsets about math.

During the first few weeks of math workshop, when we would share a story problem, friends would immediately ask, “Is it joining or separating?” “Are we adding or subtracting?”

We challenged them: “Can you model the story with math?”

The first Common Core Standards for Mathematical Practice is “Make sense of problems and persevere in solving them.” When our second graders quickly jumped to, “It’s adding, yeah, it’s definitely adding,” they depended on a single operation and joined the numbers together without thinking about the problem. Often, in a part-unknown problem, friends would join the whole and the part, revealing a misunderstanding of the problem.

For the past week, we challenged that by trying out comparing problems. As always, students first unpacked the learning target, discussing what modeling is and how they would show their model in writing.DSCF2579

Friends were stuck.

“Tawanda gave Lucy more pens?”

“So Lucy and Tawanda shared the pens.”

“I think Lucy lost some pens.”

This is where modeling becomes critical. It was clear that students were having difficulty making sense of the problem as we discussed it at the carpet. Then, working at their desks, they tried out different models. Some acted out the problem. “You’re Lucy; I’m Tawanda.” Others grabbed publishing pens and started counting out piles of pens. “These are Lucy’s. These are Tawanda’s.” Using concrete models helps students make sense of the problem.

Escarlet and Sandra used unifix cubes to represent the publishing pens.DSCF2580First, they set up Lucy and Tawanda’s pens. Instead of counting out the cubes by ones, Sandra counted out sticks of ten by ten and then counted on the ones. “I wanted them in tens and ones becuse it was easier.” She explained. Promise observed, “It’s faster to use use stacks of tens. Counting by ones would use all of your time. Instead you can count 10, 20, 30, 40…”

But it was tricky for them to compare them with the cubes set up as tens and ones. This set-up did not show how many more pens Tawanda had.
DSCF2581

“So then we lined them up,” Sandra explained. To make a clear, visual model, they lined up the cubes. It was immediately apparent that Tawanda truly had more publishing pens.DSCF2583

“They lined them up side-by-side,” Lucinda recapped. Immediately, students began pointing. “That’s Lucy’s, and that’s Tawanda’s. That’s where she has more!”DSCF2585Escarlet pointed out, “This is where they are the same. And this is where Tawanda has more.” This model made it clear. They were not joining. They were not separating. They were simply comparing the groups and finding out where the two groups are the same and where they are different. DSCF2587They added labels to show where both Tawanda and Lucy have 25 and where Tawanda has more. Escarlet broke off the “more.” Together, they then counted the “more.”

This type of problem solving requires perseverance. It requires friends to be open to trying a new model when their favorite strategy does not work. It requires them to take a step back from comfortable operations and to think critically about the problem. It requires them to be flexible and to try out different models.

This week friends are pushing themselves to focus on the second learning target–taking their concrete models and representing them with pictures and words on paper.

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