Listening to Learn is an exciting K–5 digital interview tool from highly respected educators Marilyn Burns and Lynne Zolli. Listening To Learn one-on-one assessments help teachers learn how their students reason numerically—the essential information they need to plan instruction. This blog comes from Marilyn Burns Math, where Marilyn regularly posts blogs, videos, and podcasts to help teachers help students learn math. Marilyn tweets at @mburnsmath.
Interviewing students—asking them to do math without paper and pencil—has pushed me to learn more about how they reason numerically than I ever did before in my years of teaching. When focusing on division in our Listening to Learn interview tool, one question we ask students to solve is 100 ÷ 3. We included that problem in the interview for several reasons.
Meet Zaynab and Mallika
Zaynab was a fifth grader, here interviewed by Mallika Scott. I’d spent a good deal of time in Zaynab’s class both when she was a fourth grader and then the next year when this interview was done. She was a thoughtful student and a competent math thinker. (I’ve worked with Mallika on a variety of projects, including Do The Math and the Math Reasoning Inventory. She now teaches math methods courses at California State University, Fullerton.)
Before presenting 100 ÷ 3, Mallika asked Zaynab to solve 32 ÷ 5. For this question, Zaynab thought for a moment and then answered, “Six remainder 2.” She explained, “Because I knew that 6 times 5 equals 30 and there would be 2 remainder.”
Zaynab solved 53 ÷ 5 just as easily, using 5 x 10 to reason.
From these responses, we learn that Zaynab was able to derive answers from known facts (here 6 x 5 and 5 x 10), solve division problems without contexts that had remainders, and apply the inverse relationship between multiplication and division (which she regularly did to solve division problems both in and out of contexts).
However, when Mallika presented the problem 100 ÷ 3, Zaynab’s response was decidedly different. Take a look.
Reflecting on Zaynab Solving 100 ÷ 3
To solve 100 ÷ 3, Zaynab first added 25 + 25 + 25 and commented that “each of them got 25.” (It wasn’t until later that I realized what she meant by that comment. I explain below.) Zaynab knew she had 25 left from the 100 to deal with and she began to work on dividing 25 by 3. Mallika watched Zaynab think quietly about 25 ÷ 3 for 30 seconds and then gently asked, “Can you say what you’re thinking out loud?” Zaynab explained how she was using multiplication, multiplying 3 x 3, then 3 x 4, and continuing up to 3 x 8 = 24. Then she said, “All the classes would get . . ..” Here Zaynab paused to add 25 + 8 and then concluded that “all of them would get 33 and there would be a remainder of 1.”
What I learned from Zaynab:
About Sharing vs Grouping Division Problems
The context of the Origami paper problems led Zaynab to think about 100 ÷ 3 as a sharing division problem―sharing 100 pieces of paper among three classes. She first gave each class 25 pieces and then figured out how to share the remaining 25 pieces. A grouping division problem , still using Origami papers, might be: I have 100 pieces of Origami paper and I want to give each student three pieces. How many students will get a packet of three Origami papers? When we introduce students to division, we provide them experiences with both sharing and grouping situations. The answer is numerically the same for both problems―33 with a remainder of 1―but students’ solution strategies might differ. Students benefit from experiences with solving problems that involve both interpretations of division.
Using Zaynab’s Thinking for Class Instruction
It might be interesting to use Zaynab’s reasoning for a class lesson. One of the values of video clips like this one is how they can help spark other students’ thinking. For example, imagine showing your class this video clip and then asking the students to discuss how Zaynab reasoned. This would be appropriate only after the students had grappled with the problem themselves. (You can read a previous blog I wrote about engaging Zaynab’s class in thinking about how Jonah, a student from another school, solved 100 ÷ 3 and got the answer of 33 ¼.)
Representing Zaynab’s Solution Strategy
When I think about what Zaynab did to divide 100 by 3, here’s how I might mathematically represent her thinking:
100 = 75 + 25
75 ÷ 3 = 25
25 ÷ 3 = 8 remainder 1
25 + 8 = 33
100 ÷ 3 = 33 remainder 1
Another way to represent Zaynab’s thinking, that shows how her reasoning relates to the Distributive Property and uses the kind of representation that students learn in middle school:
100 ÷ 3 = (75 ÷ 3) + (25 ÷ 3)
Listening to Learn Labs―Interactive Professional Learning:
Watching the video clip multiple times was incredibly helpful for me to understand Zaynab’s reasoning. Also, I talked about the clip with several colleagues. These two experiences―watching the clip multiple times and talking with colleagues―is what I found to be an extraordinary professional learning opportunity for me.
Watching video clips, reflecting on them, and talking with colleagues is an opportunity we provide in our three categories of Listening to Learn Labs. In our Reasoning Labs, we focus on how students solve specific interview questions. In our Strategy Labs, we unpack the numerical reasoning strategies that students need for solving problems mentally. In our Interview Labs, we address questions that relate to interviews more generally, questions like When is wait time important and useful during interviews?, What if a student self-corrects and changes an answer?, When is it helpful to ask additional probing questions during an interview? and more.
Next Steps Suggestions for Instruction:
At the end of all of our Listening to Learn Labs, we provide next-steps suggestions. Here are two instructional ideas.
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