Clip 1/9: lesson part 1a
In this number talk, Mia Buljan engages her third-grade learners with a one-digit-by-two-digit multiplication problem — 5 x 14. She comments that her students have shown that while some numbers are easier for them to decompose (for example, tens and fives), other numbers have proven more difficult for them. In particular, with some numbers, her students have more difficulty mentally identifying “friendly” numbers to use to help solve the problem. She invites students to share the different solutions they found and then defend their thinking.
Buljan says: “This lesson is a ‘number talk’ lesson designed for learners to apply what they had been learning about multiplication. A _number talk_ is a process in which students are presented with a computation problem that they solve using mental math. I try to pick numbers strategically, hoping to elicit certain strategies [from my students]. In this case, I knew that my struggling class would be competent at counting by fives, meaning there was a strong likelihood that they would be successful with finding an answer using the commutative property of multiplication. I also picked a two-digit number (14) because my students had been developing ideas around breaking numbers apart to multiply them while they were doing direct modeling to solve word problems.
“_Direct modeling_ is part of Thomas Carpenter’s work from his book, _Children’s Mathematics: Cognitively Guided Instruction_. In the research project that underpins the book, he and his colleagues talk about problem types that are easier to model directly because students use tools (blocks or other manipulatives) to build the numbers in the order that they [the numbers] appear in the story problem, and then [students] model the action in the story by combining the quantities, separating them, and/or comparing them.
“During problem solving sessions, my students had become very competent at building equal groups of two-digit numbers as TEN and some EXTRAS using place value blocks. This model naturally led them into collecting the tens first, and counting them, then collecting up the extras and counting them. They then combined both quantities for a final total. By doing this, they essentially ‘created’ the distributive property for multiplying across addition, and I wanted to see if they could apply this idea in real time without the blocks.
“In order to promote discourse with young children, I use a combination of focusing on the Sociomathematical Norms (based on the work of Erna Yackel and Paul Cobb) and using strategies that push students toward more productive classroom discourse (_Classroom Discussions in Math_ by Nancy Anderson, Suzanne Chapin, Cathy O’Connor, and others).
“There are five tenets of Sociomathematical Norms that I rely on during number talks: 1) Errors are gifts — they promote discussion and learning, 2) The answer is important — but it is _not_ the only math, 3) Keep asking questions until it makes sense, 4) Think with language and use language to think, and 5) Use multiple strategies and multiple representations.
“In _Classroom Discussions_, they describe four steps a teacher can take to move a group toward more productive classroom discussions: 1) Help individual students to clarify and share their own thoughts (‘What are you saying?’), then 2) Help students to orient [to] the thinking of others (‘What are others saying?’). The next steps are 3) to help students deepen their thinking and 4) engage with the reasoning of others).”
MIA BULJAN: I had collected the answers “69,” “70,” “56,” and “65” for the answers to the problem I presented (5 x 14). Marlene is the first student I called on and her strategy was to visualize 5 groups with 14 in each group, then use skip counting to total the ones (5 groups of 4) and then the tens (5 groups of 10) using skip counting for both. She then adds the total for the tens to the total for the ones.
Although we know this as the distributive property of multiplication, I chose to name it for them as “decomposing.” I want them to connect this strategy to all the other ways that they have been breaking apart numbers to make them [the numbers] friendlier to work with. They [the students] have plenty of time to learn that this [property] has a special name in multiplication. I’m more worried that they anchor the concept first in what they already know about number sense and how numbers work.
A number talk is such a natural place for students to practice communicating their ideas and testing their conjectures around how the math works. When she [Marlene] tells us that she “counted the ones,” I actually had assumed that she meant the “ones” that were in the tens place of each 14 that she had visualized. I notice I am using one of the simplest (and one of the most effective) teaching techniques for encouraging discourse and trying to make sense of Marlene’s strategy: try to get students to say a second sentence. Following up with, “Tell us how you did that” or “What did that look like” is so simple and powerful.
In terms of the Sociomathematical Norms, I see that I am using her words (“I pictured 5 groups with 14 in each group”) to create a quick picture of 5 circles with the number 14 in each circle that we can refer to throughout the number talk. This picture representation helps support the numbers she is using to calculate the product.
For the first two steps of promoting discourse (clarify their own thoughts and orient on the thoughts of others), I’m relying heavily on restating and revoicing. When restating Marlene’s words, I’m making sure everybody heard her and checking for understanding. Revoicing allows me to use Marlene’s words to give her ideas “handles” so that other students can pick it [Marlene’s idea] up and use it. Revoicing means I can use her idea but then highlight terms or strategies that I want everyone else to hone in on. This is a particularly important strategy when trying to encourage very young students to engage in productive discourse; they don’t always recognize how profound and mathematically powerful their ideas are; it’s up the to the teacher to recognize and name what the students are doing and describing, as they [students] are developing the language and concepts to take on that responsibility for themselves.
There is a really clear example of encouraging students to engage with the thinking of others ([from Classroom Discussions], Step 4 for productive class discussions) where I ask Marlene if she is done after skip counting by fours. She says, “No.” I restate, “Marlene says she is not done,” and then I open it up to the rest of the class to orient on her thinking by asking, “What else does Marlene need to do?” Chase then uses the picture representation to explain that Marlene still needs to count all the tens.
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