Wondering what a number string looks like in a classroom? Curious about the details of the routine? The following video features site co-founder Rachel Lambert teaching a group of 4th grade students at the Citizens of the World Charter School in Mar Vista, CA. These students have a wonderful classroom teacher, Hayley Roberts, who does number strings regularly with the students as part of a rigorous, inquiry-based mathematics curriculum. Hayley is off camera in Part 1 leading the students in a mathematics mindfulness exercise.

Students in this class had done lots of number strings with the open number line for addition and subtraction, as well as the array as a model for multiplication. On the previous day, the number string had been a multiplication doubling and halving string on the open number line. Today’s number string was designed to help students understand division on the open number line, focused on using equivalence as a strategy. Before you watch, you might want to anticipate how 4th graders might solve these problems, and how Rachel will represent strategies on the open number line.

Make sure to click on captions if that is useful. For info on how I finally figured out captions, go to a new post on my other blog mathematizing4all.com

2 x 50

4 x 100

100 ÷ 2

100 ÷ 4

200 ÷ 4

400 ÷ 8

800 ÷ 16

800/16

In the first segment, Rachel reminds the students of the strategies they used the day before (giving students credit), and gives the first (2 x 50) and second (4 x 25) problems.

Notice the way the classroom teacher leads her students to be mindful mathematically before they begin. This is a beautiful way to help students orient themselves to the mathematics.

In the second segment, Rachel takes more strategies for 4 x 25, and then presents the third problem, 100 ÷ 2.

In Part 3, Rachel begins by asking students if they can find an open number line that represents 100 ÷ 2. Throughout this segment, she asks students to visualize representations before she models them. This segment continues, moving fairly quickly through 100 ÷ 4 and 200 ÷ 4, before moving more slowly through 400 ÷ 8 and 800 ÷ 16. It ends with one student beginning to generalize a strategy for understanding the relationship between the dividend and the divisor in division.

In Part 3, Rachel also adds another support to the number string: context. She asks the students to imagine that the division problems represent a lottery winning beings shared by students equally. She wants this context to support them in in thinking about the relationships between the dividend, the divisor and the quotient. What happens when the amount of money in the lottery is doubled? What happens when the number of students sharing the money doubles?

Part 4 continues with another student sharing their thinking about that generalizable relationship. Like the first student, this generalization is supported by the context of the lottery.

Rachel then tries to wrap up the number string, but a student offers what they think will be the next problem: 1600 ÷ 32. Students love to predict the next problem in a number string! Although she is over her time limit, Rachel throws out what would have been the last problem if she had time: 800/16 as a fraction.

In the final part, Rachel asks students, not to solve this problem, but if they can think of any equivalent fractions for 800/16.

A couple of small points. Students use the sign for “agree” in American Sign Language to indicate that they used the same strategy as a classmate. Rachel stops the class several times for a turn and talk, particularly at moments in which they were unsure, or she was unsure how much students understood. The turn and talk gives students a chance to talk through their thinking, helping them make sense of still emerging understandings.

Finally, we present this number string not as an example of perfect practice, by any means, but as a sample of practice that can be analyzed. What did you notice? What are your questions?

We were watching this in our math for elementary teaching class, and we had some notice & wonders:

Notice and wonders about the teaching: when a student was explaining their answer the teacher would go the same pace as the student and would let them complete their thought rather than her finishing it for them. Ask the question in an inviting way. When students talked over one another she praised them for staying on the subject but also said it in a way that let them know they need to be quieter when another student is talking. When having students talk to each other she also joined the conversation. She gave a lot of time on how the students came about the answer instead on focusing on just the answer.

Student thinking: They felt comfortable sharing multiple methods while also getting to the same answer. Overall, the students were very excited about the number string. They wanted to share their answers. It’s exciting to watch students get excited about math. A lot of them would use previous strategies to solve for the current problem. They noticed the patterns such as doubling or halving.

We wonder what the next day of number strings would be. Would it start where they left off or use a different number string to get new strategies? Would you revisit what they previously learned? Maybe something like 60/10, 120/20, 30/5, 90/15?

Absolutely loved and connected to this artful practice of facilitating student learning with number strings. Wondering which SMPs and standards this activity addressed.