In our last post, we talked about how culturally responsive math educators pose contexts that serve as either windows or mirrors for students. “A mirror,” Emily Style says, “is a story that reflects your own culture and helps you build your identity. A window is a resource that offers a view into someone else’s experience.”
In this post, we’ll explore how children’s literature can provide story contexts that serve as both mirrors and windows for our students and help make abstract concepts like measurement division relatable and accessible for those in grades 3-6.
Launching with Literature
I love it when authors highlight characters who are strong math role models, especially when the characters are girls or children of color, two groups that are historically underrepresented in math contexts.. That’s one of the reasons I like Annabelle Swift, Kindergartner by Amy Schwartz (Orchard Books, 1988). The
other reason is that the context is one that all students can relate to.
You can share the story using video here, or you can purchase the book for your classroom library if your budget allows.
The story opens with Annabelle facing her first day of kindergarten. Armed with (poor) advice from her older sister, she seems to say all the wrong things. After the kindergartners turn in six cents each for milk (yes, milk was cheap back in the ‘80s), they begin to count how much they have altogether. The other children give up at around ten cents, but Annabelle counts all the way to $1.08, impressing everyone in the room. The teacher appoints Annabelle milk monitor, and she goes to the school kitchen to collect milk cartons.
When Annabelle gets to the cafeteria, she walks up to “a big lady wearing a scary red hairnet” and hands her an envelope with the $1.08 and waits for the lady to give her a tray full of little milk cartons and straws. This is where I pause and ask the students to think about how many milk cartons were on the tray.
Teaching with a Problem-Solving Focus
Providing students with opportunities to make sense of math story problems gives them a sense of agency and is one of the important Common Core Math Practices. When problem solving, students must understand the context, figure out what numbers to use, and then decide which operation (s) they’ll use on those numbers. Finally, they must evaluate the answer they get and determine whether it makes sense. So, before revealing the answer in the book, I handed a blank piece of paper to each of the fourth graders I was working with and sent them to work, interested in what they would come up with.
The students solved the milk problem in a variety of ways. Nichelle started with 6 cents and kept adding 6s until the total came to one dollar and eight cents. She then added the number of 6s she used to get a total of 18.
Rolf started at $1.08 and kept subtracting 6 cents until he got to zero (see below).
Cassie wrote me a letter, explaining how she did the computation mentally. “Dear Rusty, I multiplied 6 and 10. I got 60, so I did 60 x 2 (to get from 10 to 20 milks) and got 120. I knew that was too big, so I did 120-108 = 12. I got 12 and that is two milks. Now I have eighteen milks.” (See her letter below).
Two Types of Division
The Annabelle Swift problem is an example of quotative or “measurement division.” This type of division can be viewed as the action of separating items into groups when the number of items in each group is known. This type of division is called measurement division because the quantity in each group is already measured. In Annabelle’s case, we know that each group (milk carton) cost 6 cents. What we don’t know is how many groups, or milk cartons, there will be. That’s what the fourth graders had to figure out.
Partitive division is different from measurement division in that the number of groups is known, but the amount in each group is what you must figure out. For example, let’s say that Annabelle knew that there were 18 students in her class. She needs to get 18 milk cartons (the number of groups) from the cafeteria. She collects the $1.08, but what she doesn’t know is how much each milk costs (the amount in each group). This video from San Francisco Unified School District Math does a good job of explaining the difference between partitive and measurement division as they explain division of fractions. PBS offers an additional video explanation of the two types of division using cuisenaire rods.
What’s important is that students benefit from solving both types of division problems. But in school, what students typically get is a steady diet of partitive or sharing problems such as, “I have a piece of licorice 10 inches long and I want to share it with a friend. How much of the licorice will each of us get? I can use a model to show what that would look like:
With measurement division, the question might be, “How many 2-inch bites of licorice will there be in a 10-inch piece of licorice?” I can use a model to show what that would look like:
Early experience with measurement division will help students when they learn how to divide fractions in 5th and 6th grade.
Annabelle Swift for Fifth and Sixth Graders
Older kids love hearing about Annabelle’s adventures on her first day of kindergarten, especially since many have younger brothers and sisters. With older students, I read through the book, and like with the fourth graders, I pause before finishing and ask the students to solve the problem mentally. This makes for a good number talk.
Next, I pose a different problem related to the Annabelle story. “What if,” I tell the students, “Annabelle’s teacher buys 6 pints of ice cream for a class party. If he serves each student 1/3 of a pint of ice cream, how many students can he serve?” In other words, how many ⅓ servings are in 6 pints of ice cream? This problem models measurement division because we know how many servings for each student (or group), but we don’t yet know how many students (or groups) we can serve with this much ice cream.
I asked students to solve the problem and show their thinking using words, equations, and diagrams. Below is one example of a fifth grader’s work showing how they figured out how many guests or students Anabelle’s teacher could serve. Notice how the diagram and written explanation helps make the student’s thinking transparent.
Divide and Ride: Another Context for Measurement Division
In Divide and Ride by Stuart J. Murphy (Murphy, 1997), eleven friends go to a carnival and must divide into groups for each ride. The Dare-Devil ride seats 2 people in each seat, so the eleven friends must figure out how many seats they will use and whether there will be anyone leftover. The Satellite Wheel holds 3 people per chair, so when the eleven friends get on this ride, they have another measurement division problem to solve. Each ride at the carnival has seats that fit different numbers of people: 2, 3, and 4.
Divide and Ride is a great way to provide a mirror that reflects third and fourth graders' experiences, or a window for students who haven’t yet been to a carnival or amusement park. The book also serves as a relatable way to introduce students to measurement division in preparation for dividing fractions in fifth and sixth grades. Read about how a teacher uses Divide and Ride to introduce a division lesson here.
What books can you recommend that serve as contexts for measurement division?