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).”
3rd grade math - interpreting multiplication & division
Mia Buljan, Glassbrook Elementary School, Hayward Unified School District, Hayward, California
Lesson Part 1a
MIA BULJAN: We're going to do an equal groups problem and I'm going to ask you for the total. Okay? So here's how I'm going to write equal groups and I don't want you to write anything, I want you to think about what that would be in your head. How could you solve that without writing anything? And remember you're going to let me know by doing this, and if you have more than one answer, or one way, or 3 ways, or 4 ways, or 10 ways. Celine what was your answer?
STUDENT: Um, 70.
MIA BULJAN: Anybody agrees with her that it's 70.
MIA BULJAN: Did anybody get a different answer?
MIA BULJAN: Okay, anybody get a different answer?
MIA BULJAN: And another answer?
MIA BULJAN: And another answer? Everybody sees their answer up here right now?
MIA BULJAN: All right. Who wants to tell us how they got their answer? Marlene.
STUDENT: I put 5, 5 um, circles and then...and then I put 14 in each one.
MIA BULJAN: So you thought of this? What are these circles?
STUDENT: The 5.
MIA BULJAN: Oh the 5 the 5, so there's 1, 2, 3, 4, 5 circles and you put 14 in here?
MIA BULJAN: And then what you put in here?
MIA BULJAN: Uh-huh.
STUDENT: 14. 14. 14. 14.
MIA BULJAN: So if we were going to say this problem in words we would say 5 groups of 14. Who sees 5 groups of 14 in Marlene's picture that she visualized in her head? Okay.
STUDENT: Ms. B.
MIA BULJAN: Uh-huh.
STUDENT: You can just say 14*5 that would be easier then.
MIA BULJAN: Okay, hold on one second.
STUDENT: Yeah, that's what I did.
MIA BULJAN: Okay, hold on one second. So Marlene, tell us about what you did next.
STUDENT: Then I counted them all and...
MIA BULJAN: What's the first thing you counted?
STUDENT: I counted the ones.
MIA BULJAN: So tell me about that.
STUDENT: I counted 4 and then I got 8, 12, 16, 20.
MIA BULJAN: So she said, "I counted the ones," and I thought she meant she counted these ones. Everybody see those ones?
MIA BULJAN: But then she said, "I counted 4, 8, 12, 16, 20." Which ones is she talking about?
STUDENTS: The four ones.
MIA BULJAN: So this is a number that has some ones in it and then what is this over here? This is...
STUDENT: A 10.
MIA BULJAN: It has a 10 and some ones in it and every one of them is a 10 with some ones.
MIA BULJAN: A 10 with some ones, a 10 with some ones, and a ten with some ones.
MIA BULJAN: So what she did was she put all of these together first and she did it by skip counting, 4, 8, 12, 16, 20. All those fours came together as this number right here 20. Then what do you do next? Was she done?
MIA BULJAN: She's not done? How do you know she's not done Chase?
STUDENT: Because ...
MIA BULJAN: Thank you mija, you can leave it on my desk and then come sit down. So do it or not do it?
STUDENT: Do it.
MIA BULJAN: Okay, we'll talk about that in a second, sorry. Go ahead Chase. Is she done?
STUDENT: Uh, no. Because she hasn't counted the 10 because uh, 20...that's why she got 70 because she counted these tens.
MIA BULJAN: Ahhh.
STUDENT: And then she got 70 and then...
MIA BULJAN: So when she broke this apart into 10 and 4, she has to do the 4 part but she still has to do the 10 part also. So Marlene, what did that look like when you did the 10 part? Thank you Chase you can have a seat.
STUDENT: It looked like I counted 10, 20, 30, 40, and then I got 50.
MIA BULJAN: And your last one was 50. So this was 10, 20, 30, 40, 50. So now you have this 4 parts and the 10 parts, and what did you do?
STUDENT: And then I added 20 and 50.
MIA BULJAN: 20 and 50 is the same as...
MIA BULJAN: 70 all together. So everybody, anybody have a question for Marlene, or do you understand what she did?
MIA BULJAN: Okay. When she broke these apart, what Marlene did was she decomposed.
MIA BULJAN: This is Marlene's way. Did anybody else break them apart?
STUDENT: I did.
MIA BULJAN: You did, but you did it in a different way? Okay, we want to hear about this way. Hold on one second.
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.