Adapted with permission from work done by Understanding Language at Stanford University. For the original paper, Principles for the Design of Mathematics Curricula: Promoting Language and Content Development, please visit https://ul.stanford.edu/resource/principles-design-mathematics-curricula.

In a problem-based mathematics classroom, sense-making and language are interwoven. Mathematics classrooms are language-rich—and therefore language-demanding—learning environments for every student. The linguistic demands of doing mathematics include reading, writing, speaking, listening, conversing, and representing (Aguirre & Bunch, 2012). Students are expected to say or write mathematical explanations, state assumptions, make conjectures, construct mathematical arguments, and listen and respond to the ideas of others. In an effort to advance the mathematics and the language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. 

To support students who are learning English in their development of language, this curriculum includes instruction devoted to advancing language development alongside mathematics learning, and fostering language-rich environments in which there is space for all students to participate.

This table reflects the attention and support for language development at different levels of the curriculum:

COURSE	Foundation of curriculum: theory of action and design principles that drive a continuous focus on language development Student glossary of terms
LESSON	Language goals, embedded in learning goals, describe the language demands of the lesson Definitions of new glossary terms
ACTIVITY	Strategies to support access for English learners, based on the language demands of the activity Math language routines

This interwoven approach is grounded in four design principles that promote mathematical language use and development:

Principle 1: Support sense-making

Scaffold tasks and amplify language so students can make their own meaning. Students need multiple opportunities to talk about their mathematical thinking, negotiate meaning with others, and collaboratively solve problems. Provide targeted guidance. Make language more accessible by amplifying rather than simplifying speech or text. Simplifying includes avoiding the use of challenging words or phrases. Amplifying means anticipating where students might need support in understanding concepts or mathematical terms, and providing multiple ways to access them.

Principle 2: Optimize Output

Strengthen opportunities for students to describe their mathematical thinking to others, orally, visually, and in writing. All students benefit from repeated, strategically optimized, and supported opportunities to articulate mathematical ideas into linguistic expression, to communicate their ideas to others. Opportunities for students to produce output are strategically optimized for both (a) important concepts of the unit or course, and (b) important disciplinary language functions (for example, explaining reasoning, critiquing the reasoning of others, making generalizations, and comparing approaches and representations).

Principle 3: Cultivate Conversation

Strengthen opportunities for constructive mathematical conversations. Conversations are back-and-forth interactions with multiple turns that build up ideas about math. Conversations act as scaffolds for students developing mathematical language because they offer opportunities to simultaneously make meaning, communicate that meaning (Mercer & Howe, 2012; Zwiers, 2011), and refine the way content understandings are communicated. During effective discussions, students pose and answer questions, clarify what is asked and what is happening in a problem, build common understandings, and share experiences relevant to the topic. Foster meaningful conversations, using activities and routines as opportunities to build a classroom culture that motivates and values students’ efforts to communicate.

Principle 4: Maximize Meta-awareness

Strengthen the meta-connections and distinctions between mathematical ideas, reasoning, and language. Meta-awareness, consciously thinking about one's own thought processes or language use, develops when students consider how to improve their communication and reasoning about mathematical concepts. When students use language in ways that are purposeful and meaningful to themselves, in their efforts to understand—and to be understood by—each other, they attend to ways in which language can be both clarified and clarifying. Students learning English benefit from an awareness of how language choices are related to the purpose of the task and the intended audience, especially if oral or written work is required. Both metacognitive and metalinguistic awareness are powerful tools to help students self-regulate their academic learning and language acquisition.

These design principles and related mathematical language routines ensure language development is an integral part of planning and delivering instruction. Moreover, they work together to guide teachers to amplify the important language that students are expected to know and use in each unit.
 

Mathematical Language Routines

Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. The MLRs included in this curriculum were selected because they simultaneously support students’ learning of mathematical practices, content, and language. They are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while concurrently acquiring English. Adapt these flexible routines to support students at all stages of language development in improving their use of English and disciplinary language.

MLRs, included in select activities of each unit, offer all students explicit opportunities to develop mathematical and academic language proficiency. These “embedded” MLRs are described in the Teacher Guide for the lessons in which they appear.

Use the optional, suggested MLRs to support access and language development for English learners, based on the language demands students will encounter. They are described in the Activity Narrative, under the heading Access for English Learners. Use the suggested MLRs and language strategies, as appropriate, to provide students with access to an activity, without reducing the mathematical demands of the task. When using these supports, take into account the language demands of the specific activity—and the language needed to engage the content more broadly—in relation to students’ current ways of using language to communicate ideas as well as their English language proficiency. Using these supports can help maintain students’ engagement in mathematical discourse and ensure that the struggle remains productive. Use the MLRs, as needed, and phase them out as students develop understanding and fluency with the English language.

MLR1 Stronger and Clearer Each Time

This routine offers a structured and interactive opportunity for students to revise and refine their ideas and their verbal and written output (Zwiers, 2014). This routine provides a purpose for students’ conversation through the use of a discussion-worthy and iteration-worthy prompt.
How It Happens:
Response – First Draft. Students think and write individually about a thought-provoking question or prompt. (2–3 minutes)
Structured Partner Meetings. A structured pairing strategy facilitates students meeting with 2 or 3 different partners. During each meeting, students take turns at playing the roles of the speaker, who shares their ideas and writing, and the listener, who asks the speaker clarifying questions and gives feedback. (1–2 minutes per meeting)
Response – Final Draft. Students write a final draft that is stronger (showing evidence of incorporating or addressing new ideas, examples, and reasoning about mathematical concepts) and clearer (showing evidence of refinement in language and precision). When finished, students compare their first and second drafts. (2–3 minutes)

MLR2 Collect and Display

The intent of this routine is to stabilize the varied and fleeting language in use during mathematical work, in order for students’ own output to become a reference in developing mathematical language. Organize, revoice, or explicitly connect to other terms in a display that all students can refer to, build on, or make connections with during future discussion or writing. Throughout the course of a unit (and beyond), reference the displayed language as a model, update and revise the display as students’ language changes, and make bridges between prior student language and new disciplinary language (Zwiers et al., 2017).
How It Happens:
Collect: Circulate and listen to students talk during partner, small-group, or whole-class discussion. Jot down the words and phrases students use, together with helpful sketches or diagrams. Capture a variety of uses of language, both formal and informal, to connect to the lesson content goals.
Display: Organize the collected output in a display to refer back to during whole-class discussions throughout the unit. Encourage students to suggest revisions, updates, and connections to add as they develop new mathematical ideas and new ways of communicating, over time.

MLR3 Critique, Correct, Clarify

This routine invites students to analyze a piece of mathematical writing that is not their own, and then to improve the work by correcting errors and clarifying meaning. Demonstrate how to effectively and respectfully critique the work of others, with meta-think-alouds, and press for details when necessary. More than just error analysis, this routine purposefully engages students in considering both the author’s mathematical thinking as well as the features of their communication. 
How It Happens:
Original Statement: Create or curate a written mathematical statement that intentionally includes conceptual (or common) errors in mathematical thinking as well as ambiguities in language. (1–2 minutes)
Partner Discussion: Students discuss the original statement in pairs. Ask guiding questions, such as: “What do you think the author means?” “Is anything unclear?” or “Are there any reasoning errors?” (2–3 minutes)
Improved Statement: Students revise the original to create an improved statement, resolving any mathematical errors or misconceptions, and clarifying ambiguous language. (3–5 minutes)

MLR4 Information Gap

This routine, also called Info Gap, cultivates conversation by creating an authentic need for students to communicate (Gibbons, 2002). Students share ideas and information in order to bridge a gap and accomplish a task that they could not have done alone. Demonstrate how to ask for and share information, how to justify a request for information, and how to clarify and elaborate on information.
How It Happens:
Problem/Data Cards: Students work together as partners. Partner A is given a card with a problem to solve, and Partner B has the information needed to solve the problem on a Data Card. Neither student should read or show their card to their partner.
Bridging the Gap: Partner A determines what information is needed, and asks Partner B for that specific information. Partner B should not share information unless Partner A specifically asks, and justifies the need for, the information.
Solving the Problem: Partner A shares the Problem Card, and both students solve the problem independently. Partner B shares the Data Card, and students compare their strategies and solutions.

MLR5 Co-Craft Questions

This routine invites students to get familiar with a context before feeling pressure to produce answers. Students produce the language of mathematical questions themselves, and analyze how to use different mathematical forms and symbols to represent different situations. 
How It Happens:
Hook: Present the context. Display a problem stem, a graph, a video, an image, or a list of interesting facts. Optional: Students keep books or devices closed.
Students Write Questions: Ask students: “What mathematical questions can you ask about this (situation)?” Although it is preferable that they write these questions, if students still are developing their writing skills, they can state their questions orally or discuss them with a partner. The questions should be answerable by doing math and could be about the situation, information that is unknown, or assumptions that students think are important. (1–2 minutes)
Students Compare Questions: Students compare, with a partner or a small group, the questions they generated (1–2 minutes), before sharing questions with the whole class. Questions are displayed for all to see, and the whole class may discuss what the questions have in common, how they are different, the language of mathematical questions, and so on.
Actual Question(s) Revealed: Students continue with the task as designed. If time allows, students also may select from the list of student-generated questions.

MLR6 Three Reads

Use this routine to ensure that students know what they are asked to do, create opportunities for students to reflect on the ways mathematical questions are presented, and equip students with tools used to actively make sense of mathematical situations and information (Kelemanik, Lucenta, & Creighton, 2016). This routine supports reading comprehension, sense-making, and meta-awareness of mathematical language.
How It Happens:
In this routine, students are supported in reading and interpreting, three times, a mathematical text, a situation, a diagram, or a graph, each time with a particular focus. At times, withhold the intended question or main prompt until the third read so that students can concentrate on making sense of what is happening before rushing to find a solution or method.
First Read: “What is this situation about?” After a shared reading, students describe the situation or context. This is the time to identify and resolve any challenges with non-mathematical vocabulary. (1 minute)
Second Read: “What can be counted or measured?” After the second read, students list all quantities in the situation that are countable or measurable. Examples: “number of people in a room” rather than “people,” “number of blocks remaining” instead of “blocks.” Record the quantities as a reference to use when solving the problem after the third read. (3–5 minutes)
Third Read: “What are different ways or strategies we can use to solve this problem?” Students discuss possible strategies. They may find it helpful to create diagrams to represent the relationships among quantities identified in the second read, or to represent the situation with a picture (Asturias, 2012). (1–2 minutes)

MLR7 Compare and Connect

This routine fosters students’ meta-awareness, leveraging the powerful mix of disciplinary representations available in mathematics as a resource for language development. In this routine, students make sense of mathematical strategies other than their own by relating and connecting other approaches to their own. Prompt students to reflect on, and verbally respond to, these comparisons (for example, by exploring why or when to approach, represent, or say a mathematical problem a certain way, or by identifying and explaining correspondences between different mathematical representations or methods).
How It Happens:
Students Display Their Work: Students are given a problem they can approach and solve, using multiple strategies, or a situation they can model, using multiple representations. Students prepare a display of their work, paying attention to the language and the details they include to allow others to make sense of their approach and their reasoning.
Compare: Students investigate each other's work, pointing out important mathematical features and making comparisons. These comparisons should focus on the typical structures, purposes, and affordances of the different approaches or representations: what worked well in this or that approach, or what is especially clear in this or that representation.
Connect: Students identify correspondences in how specific mathematical relationships, operations, quantities, or values appear in each approach or representation. During the discussion, amplify language students use to communicate about mathematical features that are important for solving the problem or modeling the situation. Call attention to the similarities and the differences between the ways those features appear.

MLR8 Discussion Supports

Instructional moves and strategies support inclusive discussions about mathematical ideas, representations, contexts, and strategies (Chapin, O’Connor, & Anderson, 2009). Combine and use these instructional moves and strategies to support discussion during almost any activity. These include multimodal strategies for helping students make sense of complex language, ideas, and classroom communication. Over time, students may begin using these strategies themselves to prompt each other to engage more deeply in discussions.
Examples of Possible Strategies:

• Revoice students’ ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
• Press for details in their explanations by requesting that students challenge or elaborate on an idea, or give an example.
• Show central concepts multi-modally by using different types of sensory inputs: act out scenarios or invite students to do so, show videos or images, use gestures, and talk about the context of what is happening.
• Use choral response or repetition to give all students opportunities to produce verbal output, and to support the transfer of new vocabulary to long-term memory.
• Think aloud by talking through thinking about a mathematical concept while solving a related problem or working on a task.
• Provide prompts or structures to support discussion when students work as partners or in small groups.
• Give students time to make sure that everyone in the group can explain or justify each step or part of the problem. Then make sure to vary who is called on to represent the work of the group, so students get accustomed to preparing each other to fill that role.

Sentence Frames

Sentence frames can support student language production by providing a structure to communicate about a topic. Helpful sentence frames are open ended to amplify language production rather than constrain it. The table shows examples of generic sentence frames that can support common disciplinary language functions across a variety of content topics. Some of the lessons in these materials include suggestions of additional sentence frames that could support the specific content and language functions of that lesson.

Language Function	Sample Sentence Frames and Question Starters
Describe	I notice that . . . I wonder if . . . The next time I _____, I will . . .
Explain	First, I _____ because . . . Then I . . . I noticed _____ so I . . .
Justify	I know _____ because . . . I heard you say . . . I agree, because . . . I disagree, because . . .
Compare and Contrast	_____ and _____ are the same/alike because . . . _____ and _____ are different because . . . _____ reminds me of _____ because . . .
Question	Can you say more about . . .? Why did you . . .? What does this _____ mean?