Unlocking Learners’ Thinking Using the Mathematical Language Routines

By IM Certified Facilitators, Cheryl Fricchione and Rachel Rundstrom

Consider this:
A child whose primary language isn’t English does not participate in the classroom to the degree that their English-speaking peers do. Parent-teacher conferences are held with family members who are eager to hear what the child is learning in class, the skills that they have in math class, and where the teacher wants them to grow more across the year. Everyone at the table cares deeply about the child’s success, yet language is hindering communication for the student, the teacher, and the student’s family. 

If it doesn’t already sound familiar to you yet, it soon will. According to the National Education Association, by 2025, 25 percent of students in classrooms across the country will be English learners, or ELs.

Often, perceptions of ELs in the classroom are built on the assumption that language barriers limit the students’ access to grade-level content. Many teachers frontload vocabulary work for ELs, asking students to memorize lists of mathematical vocabulary devoid of context, believing that this is a necessary prerequisite for ELs to construct meaning or communicate their ideas. However, according to Judit N. Moschkovich in Language and Mathematics: Multiple Perspectives and Directions for Research, ELs “can participate in discussions where they grapple with important mathematical content,” even as they are acquiring English.

A Third Grade Classroom Meets Fractions

In February 2021, a grade 3 classroom returned to school from remote learning. Due to the COVID-19 pandemic, the students hadn’t been in school with peers for eleven months, and they were in the middle of their first lesson from IM K–5 Math on fractions: Unit 5, Lesson 1, Activity 2: Fold and Name.

The teacher launched the activity by giving each student 4 paper rectangles.Then she invited the students into the activity. “We are going to fold these rectangles to partition them. Fold each rectangle into 3, 6, 4, or 8 equal parts. Draw lines where you folded to partition the rectangles. Be prepared to share how you folded your rectangles.”

Students began working independently. After 5 minutes, the teacher asked students to work in pairs to share how they folded their rectangles. As students worked, the teacher used the Math Language Routine Collect and Display (MLR2), described below, to record mathematical words and ideas the students expressed. Then she invited students to share their approaches to the task.

Student A, an EL in the class, raised his hand and described what he did:

“I took the paper and I folded it down and put the corners like this” (showing how he aligned the corners and edges of the rectangle). “Then I did this” (lining the corners up again, and folding it in half again).

Student B, a native English-speaker, then raised her hand and said, “Yes. I folded it in half like he did. Then I folded it in half again to make four equal parts.”

More students shared, refining their language as they heard from their peers. The teacher, aware that Student A had begun the conversation by modeling his understanding using the paper rectangle, encouraged more. During the synthesis, she asked him, “How did you use the partitions of four equal parts to partition eight equal parts?”

This time, Student A responded with more excitement. “I folded the rectangle in half like this to make two parts, and then in half again to get four parts. Then I folded the four parts in half again to get eight equal parts.”

How did the teacher position Student A?
The teacher saw that Student A mathematically understood the idea of partitioning and that his approach to folding yielded equal parts. She noted that he was using halves to partition fourths. Being an EL did not mean that Student A lacked understanding that, if shared, would benefit the class. The teacher recognized this and took the time to amplify his voice. By using the Math Language Routine Collect and Display (MLR2), she allowed him to become more comfortable with the language needed to describe the math that he was doing. His peers became a resource for him as they shared their strategies using mathematical terms, and his mathematical thinking was reinforced. In the end, she positioned him so that he could demonstrate his command over his mathematical understanding and the language that he was developing. It was exciting to see how much he contributed and the opportunities helped him grow more confident.

How can we all recreate opportunities like these in our classrooms?
Moskovitch suggests three guiding instructional principles for teachers that view language as a resource, rather than an obstacle, and focus on student achievement, rather than just on learning English.

Principle 1: Focus on listening to students’ mathematical reasoning, not on their accuracy in using language.

Principle 2: Provide opportunities for students to use language as part of the mathematical processes, not as single words or definitions.

Principle 3: Provide opportunities for students to engage in the complexity of mathematical language, including multiple representations (objects, pictures, words, symbols, tables, graphs), modes (oral, written, receptive, productive), kinds of talk (exploratory, expository), and audiences (teachers and peers).

The eight Mathematics Language Routines, or MLRs, that are built into the math curriculum provide a way for teachers to put these principles into action. The MLRs support all students as they move from using informal language to using more formal mathematical vocabulary, but, as the Course Guide states, they are “particularly well-suited to meet the needs of linguistically and culturally diverse students who are simultaneously learning mathematics while acquiring English”. The following examples highlight this.

Collect and Display
The purpose of Collect and Display (MLR2) is to preserve the diverse, fleeting, and often informal language students use as they engage in mathematical tasks. The teacher listens for and charts the words students use during partner, small-group, or whole-class discussions to describe an image or solve a problem using written words, diagrams, and pictures and posts them for later reference.

In doing the work of this MLR, teachers and students also engage in all three of Moskovitch’s principles. In this Grade 5 activity, Unit 3, Lesson 13, Activity 1: Paper Strips, students divide whole numbers by three different unit fractions using the context of strips of paper. This helpful diagram leads to students recognizing important relationships between the divisor, dividend, and quotient even without explicitly using these words. As students compare and contrast the three problems, the teacher listens for the authentic language they use to describe division.

Compare and Connect
The purpose of Compare and Connect (MLR7) is to give students the opportunity to explore mathematical ideas by comparing the approaches and representations of others to their own. The teacher and students work together to create a visual display that they can use to find connections between the strategies, with the goal of deepening their understanding of the mathematical concept. They work to develop their understanding of the mathematical concept as well as their mathematical language.

In doing the work of this MLR, teachers and students engage in all three of Moskovitch’s principles. In this Grade 1 activity, Unit 3, Lesson 24, Activity 2: Find the Number That Makes Each Equation True, students share different methods for solving 15 – 12. As they share, teachers listen to their reasoning, students explain their approaches, and they engage in the complexity of the mathematical language of subtraction.

Discussion Supports
The purpose of Discussion Supports (MLR8) is to support rich and inclusive discussions in math classrooms. Rather than the same routine over and over, the Discussion Supports are a collection of pedagogical strategies and teacher moves that can be combined to support discourse during almost any activity. One of the strategies is the use of sentence frames.

Sentence frames support all three of Moskovitch’s principles by providing students with a structure to communicate. The sentence frames included in the curriculum are open-ended so that they amplify language rather than constrain it. Some are generic, such as, “First, I _____ because…” and “I noticed _____ so I…” Others support the specific content and language functions of particular lessons such as, “This card shows an even number because…” and “This card shows an odd number because…” from Unit 8, Lesson 3, Activity 2: Card Sort: Even or Odd in Grade 2.

If we focus on how students are able to use informal language to express their understanding rather than the use of accurate and technical vocabulary, we are more likely to recognize the contributions that they bring to the classroom community. 2025 is less than 4 years away. In order to make sure all of our students are able to know, use, and enjoy mathematics, we need to open the door for students to use whatever language helps them to make sense of the mathematics.

Next Steps

We shared a glimpse into one of our classrooms. Now it’s your turn! What opportunities will you create for your students to use informal language? How will you honor their language and position them as competent contributors in your math classroom?

Share your experiences and success stories with us on Twitter using #LearnwithIM.

 

 

Rachel Rundstrom
IM Certified Facilitator

Rachel Rundstrom began her career as a grade 4 teacher in New York City. After four years, Rachel moved to Connecticut and was a founding grade 4 teacher at a charter school in New Haven. In this role, Rachel’s determination to advance the thinking of all students grew. She then became a teacher at a magnet school in Hartford, CT.  There, Rachel realized her love of professionally developing and supporting teachers. She currently works as an Elementary Math Specialist in Hartford, CT. Rachel’s favorite part of her job is supporting teachers in building math classrooms that are grounded in the belief that all students can learn. When Rachel isn’t diving into math, she enjoys the outdoors, traveling, and spending time with her husband and their 6-year-old twin girls.

Cheryl Fricchione
IM Certified Facilitator

Cheryl Fricchione began her career in education as a computer teacher in Perth Amboy, NJ, but quickly realized that math was her true love. A former middle school math teacher and elementary math coach, Cheryl is now a Math Content Lead who supports instructional improvement with school system partners throughout the country. After earning her MS in Mathematics Learning and Teaching, Cheryl was part of an NSF-funded grant aimed at improving math teachers’ ability to formatively assess students’ understanding and provide appropriate and valuable feedback. She believes that when students become active doers rather than passive consumers, the greatest learning gains can be realized. A fun fact about Cheryl is that she appeared on TV with The Fonz during the Statue of Liberty’s Centennial celebration.