Math Language Routines: Discourse with a Purpose

By Dr. Kristen Taylor, IM Certified® Facilitator 

Math teachers can talk all day about math! We get super excited when we encounter someone else who enjoys these conversations as well.

But too many of our students don’t get to experience that joyful discourse, which limits their understanding of and positive experiences with math.

How would our students’ math learning and perception about math improve if they also had regular opportunities to talk about math? How might these opportunities help us create a world where learners know, use, and enjoy mathematics?

Let’s explore the Math Language Routines, which are embedded in IM K–12 Math.

What are math language routines?

Stanford Graduate School of Education’s Understanding Language and the Stanford Center for Assessment, Learning, and Equity (SCALE) designed eight Math Language Routines (MLRs) to help teachers “amplify, assess, and develop students’ language” in math class. These routines are structured, but adaptable to the content and classroom context. The routines “emphasize the use of language that is meaningful and purposeful, not inauthentic or simply answer-based . . . [and] can and should be used to support all students learning mathematics.” (from Understanding Language/SCALE’s Principles for the Design of Mathematics Curricula: Promoting Language and Content Development) There are four specific design principles used to entice students to discuss the concepts and ensure that the discussions provide opportunities for students to learn something new. The Math Language Routines also provide opportunities for students to engage in the Common Core’s Standards for Mathematical Practice.

As students engage in math language routines, MLRs help students to advance their thinking in the following ways: 

  • OPTIMIZING OUTPUT as students describe their own math thinking to others orally and visually or writing as physical, graphical, pictorial, or symbolic representations.
  • SUPPORT SENSE-MAKING as students scaffold their thinking and amplify language so students can make their own meaning between concepts and representations.
  • CULTIVATE CONVERSATION by providing opportunities to simultaneously make meaning, communicate that meaning, and refine the way content understandings are communicated.
  • MAXIMIZE META-AWARENESS as students organize their own experiences, ideas, and learning for themselves through reflection of the experience of discussing their math ideas, reasoning, and language.

This doesn’t just mean that teachers tell students to talk about math and they do. We know that if we simply open the space and tell students to talk, much of the discussion will veer off-topic. Math Language Routines help teachers create a structure in which students can bring those four design principles to life.

Let’s see what opportunities each MLR provides for students to practice exploring their math thinking and the MLRs’ connections to the Standards for Mathematical Practice.

MLR 1: Stronger and Clearer Each Time – Students write a response to a math problem, then verbally share their response with a partner to get feedback from the listener to improve the response, and revise their original written response based on the feedback they received.

Students learn to both listen to their own thoughts and the thoughts of another with the purpose of strengthening their “first draft” thinking. This MLR strengthens student communication of their thinking. They make their explanations clearer through repeated “reviews” and revisions, which allow them to practice giving, receiving, and applying constructive feedback, and develop listening skills. This MLR helps students communicate their mathematical reasoning with greater clarity and focus.

(This connects to Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them, 3: Construct viable arguments and critique the reasoning of others, and 6: Attend to precision.

As students verbalize and review in writing the strategies they used and justification of those strategies, students have the opportunity to get feedback which can be used to critique the precision of their solution strategy and answer.)

MLR 2: Collect and Display – Students access their own and others mathematical ideas as the teacher scribes the language, strategies, and concepts students use during partner, small group, or whole-class discussions using written words, diagrams, and pictures.

By creating stable representations of student thoughts, the teacher extends the involvement of students in the math classroom. Student contributions are displayed for everyone to see. During that discussion as well as future conversations, students refer to these written ideas to improve the clarity and focus of their own communication. They can use the vocabulary, reference previous relationships between concepts, and identify new connections. This MLR helps make student thinking visible and promotes student ownership over mathematical ideas.

(Standards for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others, and 6: Attend to precision.

As students view the thinking of their classmates from the representation, they can address any differences they see between their representation and their classmates’ representation in type and kind, further pushing students to find the most efficient or precise representation to solve the problem.)

MLR 3: Critique, Correct, and Clarify – Students rewrite a math response from an example that is incorrect, incomplete, or otherwise ambiguous.

In addition to providing students with a space to further engage in math, students employ a “growth mindset” and consider revisions to their work in order to strengthen their communication. This MLR helps students “try on” others’ math thinking in order to improve it.

(Standards for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

Similar to doing error analysis, students have the opportunity to engage with a strategy for solving that might be different from theirs in order to find ways to improve the situation representation, address and clarify misconceptions about the math, and “borrow” parts of the strategy that were relevant to add to their toolbox.)

MLR 4: Information Gap – In a group, each student has different parts of the math context and they work together to piece together that information orally and/or visually to bridge the gap between the parameters of a math situation and a question to solve a math problem.

As students work to make sense of individual pieces of information about a math context, they also identify what information they need in order to answer the math question. This creates an environment where students work together as they consider which strategies are the most efficient and what information they need to know to solve the problem. This MLR helps students explore math contexts with a purpose.

(Standards for Mathematical Practice 2: Reason abstractly and quantitatively, and 4: Model with mathematics.

As students imagine the kinds of information they need to know and explain why they need to know that information, they are building an understanding of the problem and possible solutions related to the parameters of the problem and the kinds of data they need to solve for an answer.)

MLR 5: Co-Craft Questions and Problems – Students take a provided math “answer” to create a problem that could result in that answer or use a given context to create a problem that can be answered using that context.

With this MLR, students use their creativity to explore a math context as they iterate their understanding in order to identify questions that can be asked and answered by the context. As students share their understandings with others, they also uncover the importance of slowing down to explore and think before they try to answer a question, which results in students understanding the math more deeply. This MLR helps students define math contexts.

(Standards for Mathematical Practice 4: Model with mathematics.

By creating their own questions and problems, students learn to visualize the parameters of the problem as they consider what kind of answers are possible with the information the problem provides.)

MLR 6: Three Reads – Students are guided to read the problem three separate times with three separate purposes with quick discussions between each read.

When students slow down to consider how they can use the information in a problem, they notice things about their understanding of the problem context and their own math understanding that they wouldn’t have noticed before. Giving students a purpose to read helps to focus students to identify (1) the context of the mathematical situation, (2) relationships that exist within that context, and (3) strategies that can be used to answer the question using the problem context. This MLR helps students think metacognitively about their math problem solving.

(Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them.

As students set a purpose for reading each of the three times, students practice and see the importance of understanding the problem context and using the information they know from the problem in order to identify a strategy for solving that relates directly to the problem context and available information.)

MLR 7: Compare and Connect – Students identify, compare, and contrast their own understandings with other students’ mathematical approaches, representations, concepts, examples, and language.

As students encounter other’s math thinking, this MLR opens the floor for students to consider how they see and talk about math. This MLR specifically helps students interact with each other’s thinking.

(Standards for Mathematical Practice 4: Model with mathematics, 6: Attend to precision, and 7: Look for and make use of structure.

As students compare (find differences) and connect (find similarities) between their strategy for solving and another classmate’s strategy for solving, students mine their understanding of the math concept in relation to the structure and steps of another student’s understanding of the math. This allows both students to attend to precision because to be understood by another, they have to be precise with their explanation.)

MLR 8: Discussion Supports – Students access strategies in multi-modal ways, like visuals, aural, animation/movement, and symbols as modes, to help students better make sense of complex language, ideas, and classroom communication.

Enhancing access to the mathematical context through these multi-modal supports helps students develop understanding while also continuing to work on their mathematical communication. This MLR helps students deepen their mathematical understandings, and find ways to communicate their understandings in different ways.

(Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them, 3: Construct viable arguments and critique the reasoning of others, 5: Use appropriate tools strategically, and 6: Attend to precision.

Using discussion supports helps students to see the reasoning of others as well as identify which tools to solve are most appropriate. In using these supports, students can be successful with in-depth or multistep problems with support from teachers and classmates.)

Next Steps

Consider what your students need to encounter math more deeply and in collaboration with others. More heads are better than one!

Looking for more support for distance learning? Check out IM’s collection of resources.