“I just counted by tens!” Or how we generated more efficient strategies.

During our inverted math workshop, students are given ample time to problem solve independently or with a partner.  When I was in first and second grade, my teachers modeled how to solve an addition or subtraction problem, and then I practiced in the same exact way on multiple problems. Now, our goal is that students generate their own problem-solving strategies so that they can have deep, solid, conceptual understandings of how to add and subtract.  And so, during our inverted math workshop, we pose a complex story problem to students.  At the carpet, they visualize the problem, discuss the problem with a partner, and then discuss it whole group.

They independently select from a variety of tools like unifix cubes, number lines, and hundred charts to help them solve the problems.  At their desks, they model the story problem using either concrete tools like unfix cubes or by drawing pictures. Some students have begun to model story problems using series of number sentences.IMG_9418During the first few weeks of school, the numbers in the problems ranged from zero to 35. Many students were counting by ones to add and subtract. For example, some counted out 23 cubes, and then counted one-by-one to take away 11. On paper, they drew 23 lines and then crossed out 11 lines, one at a time.

This week, we expanded the range to include numbers through 60.  During our first debrief, friends reflected on the new challenges.  “There were so many that my cubes fell off the table!” Laila shared. “I kept losing track of my counting,” Dani reflected.  We guessed that once we start working with three-digit numbers, that counting by ones would not be the best strategy.  And so we began hunting for more efficient strategies.

During our second debrief, Marlon came to the carpet with a strategy that he was convinced would be more efficient. He took on the problem, “There were 29 friends in the commons.  13 were kindergartners.  How many were first graders?”

“The unifix cubes are in stacks of ten,” he explained, “and so I didn’t break them apart.  I kept them together. I just counted by tens! I did 10, 20, 30.  Then I took one off of the ten to make nine for 29.”IMG_9415 “The problem didn’t say that the kindergartners went away, so I didn’t take away.  I separated them.  I made the 13 kindergartners. Then I knew that the others were the first graders.”IMG_9416 “So I counted them. 10, 11, 12, 13, 14, 15, 16.” [Yes, we caught the mistake with the ones after these pictures were taken.]IMG_9417As a class, we charted Marlon’s work.  We’ve spent time talking about how just getting an answer isn’t enough when you are creating your own strategy.  Instead, in the words of Amy, “You need to prove it!” And so we drew Marlon’s cubes in green.  We labeled the counting with numbers and the groups with words.  Finally, in our own words, we explained the process that Marlon went through so that someone else could look at the chart and follow Marlon’s strategy.

Throughout the year, students will engage in this type of work in math.  They will persevere through challenging problems, trying out multiple strategies before landing on one that works.



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