Clip 3/9: Interpreting Lesson Part 1C
In this clip, Mia Buljan works with her students on a number talk, using a mental math approach to multiplying a two-digit number and a one-digit number. She encourages her students to identify regularity among the various approaches identified, and guides them toward identifying the commutative property.
MIA BULJAN: I had collected the answers “69,” “70,” “56,” and “65” for the answers to the problem I presented (5 x14). Enmy is the third student I call on to share her strategy. More than half my students used her strategy to solve the problem. Sometimes I feel like our education system is so obsessed with the “basic facts” that we miss opportunities to develop our students’ conceptual understandings. Just because 5 x 14 and 14 x 5 make the same product does not mean that they are the same thing. I want to focus their [the students’] discussion on the math property that lets us switch our factors around (commutative), and I want to make sure that I’m modeling the repeated addition and skip counting for counting by fives so they can visually see how the two things (5 x 14 and 14 x 5) can be compared and contrasted.
Because so many of my students used this strategy, I wanted to focus on the discussion and the practices of being precise and critiquing the arguments of others.
In order to help the students situate their understanding of the commutative property, I am using the Sociomathematical Norm of continuing to ask questions until it makes sense. I first ask how she knows she can do 14 groups of 5 instead of 5 groups of 14, and it takes the whole group to articulate the idea that “it can be switched” or turned around, which I name as commutative. We immediately connect this word to how the commutative property works with addition and how it doesn’t work with subtraction.
When the discussion turns toward why we use the commutative property, I set it in the context of CCMP 2 [Common Core State Standards for Mathematics Standard for Mathematical Practice 2, Reason abstractly and quantitatively]. In other words, once we understand the problem, and contextualize it, we can step out of the context and use any number of strategies to solve for an answer before we step back into the context and see if our answer makes sense. Even though 14 groups of 5 doesn’t match the context we set with Marlene’s visualization of five equal groups of 14, we can articulate, with precision, what properties of the mathematics let us solve it this way.
I ask Enmy about how she knows when to stop counting by fives, and while I’m recording the skip counting on the poster we discuss how she kept track of the number of fives she had used on her fingers. It may seem odd that I appear to be forcing Enmy to count by fives in front of the whole class when she is struggling with it. However, it’s a normal part of the class culture to think about our math out loud, to want to prove and understand; so it’s not so odd for my students to struggle in front of each other. We embrace and applaud that struggle as part of learning. By having her count the fives out loud, I’m asking the rest of the class to listen to Enmy and see if they understand her and if they agree with her. Everybody heard the mistake, including Enmy, when we slowed down and listened to her count. The support comes after we identify the problem, when we all count by fives chorally.
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