We are two weeks into our math workshop. Our first focus is on the addition and subtraction Common Core standard, “Represent and solve problems involving addition and subtraction. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.”
With two-digit addition and subtraction problems as our context, we have narrowed on efforts on developing structures that support the Standards for Mathematical Practice, especially using appropriate tools strategically, making sense of problems and persevering in solving them, and modeling with mathematics.
At the beginning of our math workshop, we unpack our learning targets, ensuring that all friends understand the goal for the work that day. From there, we read the math task–the story problem–together. Students unpack the problem together. They share what they know, (She had 49 pieces of popcorn and she ate 27 pieces) what they want to know, (How much popcorn is left in the bag?), and how they might get started (I’m going to grab 49 unifix cubes.) They create a larger story. “She went to the movie theatre to see a movie. She bought a bag of popcorn and it had 49 piece. When she sat down, she started eating 27 pieces. But she got full so she saved some for later in the car.” We spend all of four minutes on this part and with an, “Off you go!” friends grab stacks of unifix cubes, pencils, and paper and head to tables to create their own models of the problem.
The word modeling is key here. We avoid language like, “find the answer,” or “solve the problem.” Rather, we encourage students to model what happened in the story. For some, that means acting out the problem with a friend and modeling it with concrete objects like the popcorn, itself. For others, that means using unifix cubes to represent the popcorn. Some friends put the unifix cubes into stacks of tens to represent the numbers in the problem. Still others draw pictoral representations of tens and ones.
After about 15 to 20 minutes of grappling, students join together at the carpet to debrief their strategies. Often, multiple strategies are shared. This week, Ashley counted out 49 cubes one by one, explaining with a wave of her hand, “This is the popcorn that Genesis bought.” She then counted out 27 cubes from the pile and said, “And this is the popcorn that she ate.” Pointing to the now smaller pile, “And this is what she didn’t eat. Now we have to count it.”
Georgia shared her strategy for a second problem. “I put them in tens and ones with the same color because I could count them easier. 10, 20, 30, 31, 32, 33, 34, 35, 36, 37, 38. I counted by tens because I know how to count by tens. Twos or fives would take longer. So when I was at the [unifix cube] bin, I grabbed smaller stacks [of cubes] to make stacks of ten. Then I labeled them with sticky notes so I could keep myself organized. That’s how I set it up.”Angel, too, stacked hers in tens and ones, but mixed up her colors. “It was faster. I wanted to get the cubes as fast as I could so I could start my math.” After comparing the strategies, friends concluded that using sticky notes and a single color for each group were helpful modeling strategies. Angel had a good point, though; it was faster to just grab whichever cubes and start solving right away.
“Let’s store them in stacks with the same color,” Nathanael suggested, “Then we can grab them quickly. So when you put them away, stack them in the same color and gently put them in the bin so the stacks don’t break.” He modeled.
Teaching math this way is laborious and doesn’t always go the way we planned. But it is also rich and rigorous as students own their learning and construct their own understanding as we go. And always, it is a process.