by Deborah Peart, Grade 2 Lead
Many people have an aversion to word problems. They cringe at the mention of them. In elementary classrooms, teachers often report that this is what their students struggle with most. When word problems show up in math class, even students who enjoy reading will release a sigh and let their shoulders droop. “Do words even belong in math class?” they wonder. The answer is yes, they do! But students need guidance in how to make sense of story problems because in many classrooms they are taught to compartmentalize their learning in math class. While students are often encouraged to integrate social studies and language arts, mathematics is more frequently taught in isolation. In order for students to see math in the world around them, we must consider all the ways in which we can bring mathematics to life through stories.
For the first 20 years of my teaching career, my expertise was in literacy: phonics, word study, and reading comprehension. Literacy was my favorite block of the day and the focus of my graduate work. Ironically, it was the elective I took in elementary mathematics at the end of my grad program that lit a spark in me. That professor answered my questions, and helped me connect mathematics to the world in ways no one else had in the past. I was inspired. As a result, I shifted my studies and professional pursuits towards mathematics.
For the past decade, I have immersed myself in the study of elementary mathematics. In my work on the IM K–5 Math curriculum writing team I have re-engaged my literacy background to think deeply about how our youngest learners engage with mathematics as emergent, novice, and decoding readers. When word problems show up in the early grades, how can we assure that all students have access to the content?
How can we support “sense-making” of stories in math class?
The mission of Illustrative Mathematics is to create a world where learners know, use, and enjoy mathematics. By using stories to help students see math in the world around them and recognize the ways in which using math is a part of their daily lives, word problems can become an enjoyable part of math learning. This starts with calling word problems “story problems” in the early grades. From there, other supports embedded in the curriculum include:
- providing relevant contexts and images with which students can engage
- supporting reading comprehension with routines and instructional practices, like Act it Out and Three Reads
- encouraging students to use visual representations to support sense-making
- inviting students to write their own math stories and ask questions that can be answered by them
Provide Relevant Contexts
During the process of writing this curriculum, the K–5 team had many conversations inspired by articles and books that we had read on the topic of culturally relevant pedagogy and practices. From these conversations we had time for reflection and made decisions about prioritizing contexts that were inviting to all students. Sometimes the context is an introduction to another culture or experience, while at other times the context is relevant to the students for other reasons. Launching an activity by establishing the context with pictures, and students sharing their own experiences, is an important part of this process. With unfamiliar contexts, it is important for students to have an introduction to them, notice and wonder about them, and ask questions that will help them gain access to the math content. Having a shared understanding of the context is the first criteria for supporting sense-making of math stories.
|In this second grade example, the teacher establishes the context in a launch. There is time for students to share personal connections or ask questions. When students solve problems with a partner or independently throughout the lesson, they have images to support their ideas and a realistic context in which measuring ribbon makes sense.
|These girls from India are wearing saree dresses. Sarees are usually worn by women and girls and made by wrapping 5–7 meters of fabric in a special way. Many sarees are made from brightly colored silk, which is a soft fabric. Sometimes when sarees get too small or are worn out, they are cut into strips to make saree ribbon. Priya and her friends are planning to make saree silk ribbon necklaces. They are solving problems to make sure they get the measurements correct.
|Priya had a ribbon that was 44 in. long. She cut off 18 in. How long is Priya’s ribbon now?
Support Comprehension with Routines
In the early grades, independent reading skills vary greatly. In order to support comprehension of story problems, we offer several instructional routines and supports.
Act it Out (K–2)
In this routine, students are presented with a story and a picture to help establish the setting. Once the story has been read aloud, the students notice and wonder about the picture and share with their partners what they believe the story is about. After the story is read a second time, students have the opportunity to act it out. By the end of the routine, after students have shared their interpretations of the story with a partner or group, there is a class representation made of the story including expressions. With this routine, students connect language to mathematical representations and approach problems from a place of understanding.
Three Reads (K–5)
Before students begin solving problems, it is critical that they know what they are being asked to do. Math Language Routine 6: Three Reads supports reading comprehension, sense-making, and meta awareness of mathematical language. In this routine, students are supported in reading a math story 3 times, as the title suggests. Each time there is a particular focus. The first read is done without a question presented to allow students to consider what is happening in the story. This relieves the pressure of rushing to find the solution and creates space for a conversation about the situation. The second read focuses on mathematizing the story by posing a question about the things that can be counted or measured in the story. It is not until the third read that the question or prompt is revealed, and students discuss possible methods for finding a solution.
|In this grade 1 example, students hear the story and engage with their classmates in discussions about what is happening in the story before working to find a solution. Once they are working on this problem independently, they are familiar with the story and have heard several different strategies for solving. They are equipped to make decisions about the tools and representations they will use to share their thinking and offer solutions, including an equation.
Kiran has some fish in his fish tank. He has 4 red fish and 5 blue fish.
How many fish does he have in all? Show your thinking using drawings, numbers, or words. Write an equation to represent the problem.
4 + 5 = 9
Students need to understand what the story is about and what they are being asked to find a solution for in order to avoid “number plucking.” If students are given the support they need with comprehension, they can approach problem solving from a position of understanding and confidence. If students relate to the context and understand the actions of the story, they can connect mathematical ideas and representations. When students have positive experiences with story problems, they will soon be inspired to write their own, and consider the mathematical questions they can pose.
Encourage Visual Representations
Starting in kindergarten, students use math tools to model the mathematics in story problems. Using 10-frames and connecting cubes, students represent addition and subtraction long before writing expressions or equations.
By grade 1, in addition to using concrete models like connecting cubes, students begin to include discrete mathematical drawings to represent people or objects and actions in story problems. These representations set the foundation for the introduction of the tape diagram in second grade.
In grade 2, students are introduced to the tape diagram as a representation designed to help them make sense of story problems. While students are not required to use tape diagrams, they use them to make connections to the mathematical ideas being presented. By analyzing the structure of the tape diagram, considering what the question mark represents or how the labels reflect the details of the story, students can visualize the actions and make decisions about their strategies for solving the problem.
Students learn to use tape diagrams in part-part-whole situations, as well as compare situations and determine which best represents the story. The use of this visual representation also allows students the space to deepen their understanding of the relationship between addition and subtraction, as they use their knowledge of number relationships to choose methods that make sense to them for solving. To emphasize that these diagrams are for making sense of stories and not for finding the solutions, several activities in the curriculum involve matching diagrams to stories without the cognitive load of solving a problem.
This is an example of how the context can connect to mathematical ideas and representations. As students consider measuring and cutting ribbon, they see examples of the tape diagram, a continuous representation which supports the idea of number as length, being used to illustrate the action of having a length that is cut into parts. The labels and placement of the question mark help students think about how they will find the difference.
Priya had a ribbon that was 44 in. long. She cut off 18 in. How long is Priya’s ribbon now?
Andre drew this diagram to help him think about the problem.
The “?” represents the ribbon Priya has left. The dotted line tells us how much ribbon is being cut off: 18 in.
44 – 18 = 26
Priya’s ribbon is 26 inches long.
Tape diagrams are a powerful tool because they can be used to represent all four operations, additive and multiplicative comparisons, fractions, decimals, and percents. It is a representation with longevity, as it is used not only in the elementary grades, but throughout middle school.
Let’s invite students to enjoy math stories. In reading class, students engage with stories and relish in the fact that there is a problem to be solved. In these stories, the reader takes comfort in knowing that the problem will be solved if they just keep reading. Once students feel connected to and understand the actions of a math story, they will have the courage and confidence to solve problems on their own.
Math stories are often presented as words on a page with some unrelatable problem to solve and unanswered questions. It’s no wonder some students are intimidated. If we want students to transfer the reading strategies and skills they acquire during the literacy block, here are a few recommendations:
- Provide relevant contexts and images with which students can engage.
- Support reading comprehension with routines and instructional practices.
- Encourage students to use visual representations to support sense-making.
- Invite students to write their own math stories and ask questions that can be answered by them.