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 to and respond to the ideas of others. In an effort to advance the mathematics and 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 fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.

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 with targeted guidance from the teacher. Teachers can 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 should be 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 provide opportunities to simultaneously make meaning, communicate that meaning, and refine the way content understandings are communicated. During effective discussions, students pose and answer questions, clarify what is being asked and what is happening in a problem, build common understandings, and share experiences relevant to the topic. Meaningful conversations depend on the teacher using activities and routines as opportunities to build a classroom culture that motivates and values 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 are using language in ways that are purposeful and meaningful for themselves, in their efforts to understand—and be understood by—each other, they are motivated to attend to ways in which language can be both clarified and clarifying. Students learning English benefit from being aware 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, described below, ensure language development is an integral part of planning and delivering instruction. Moreover, they work together to guide teachers to amplify the most 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 simultaneously acquiring English. These routines are flexible and can be adapted to support students at all stages of language development in using and improving their English and disciplinary language use. 

MLRs are included in select activities in each unit to provide all students with explicit opportunities to develop mathematical and academic language proficiency. These “embedded” MLRs are described in the teacher notes for the lessons in which they appear. 

Each lesson also includes optional, suggested MLRs that can be used 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.” Teachers can use the suggested MLRs and language strategies as appropriate to provide students with access to an activity without reducing the mathematical demand of the task. When using these supports, teachers should take into account the language demands of the specific activity and the language needed to engage the content more broadly, in relation to their students’ current ways of using language to communicate ideas as well as their students’ English language proficiency. Using these supports can help maintain student engagement in mathematical discourse and ensure that struggle remains productive. All of the supports are designed to be used as needed, and use should be faded out as students develop understanding and fluency with the English language. 

MLR1 Stronger and Clearer Each Time

The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine both their ideas and their verbal and written output (Zwiers, 2014). This routine provides a purpose for student conversation through the use of a discussion-worthy and iteration-worthy prompt.
How It Happens:

1. Response - First Draft. Students think and write individually about a thought-provoking question or prompt. (2–3 minutes)
2. Structured Pair Meetings. Students meet with 2–3 different partners. During each meeting, students take turns being the speaker, to share their ideas and writing, and the listener, who asks the speaker clarifying questions and gives feedback. (1–2 minutes each meeting)
3. Response - Final Draft. Students write a final draft. Final drafts should be: 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. After students are finished, their first and second drafts can be compared. (2–3 minutes)

MLR2 Collect and Display

Use this routine to capture a variety of students’ oral words and phrases into a stable, collective reference. This collected output can be organized, revoiced, or explicitly connected to other language 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), teachers can reference the displayed language as a model, update and revise the display as student language changes, and make bridges between prior student language and new disciplinary language (Dieckman, 2017). 
How It Happens:

1. Collect: Circulate and listen to student talk during paired, 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, that can be connected to the lesson content goals. 
2. Display: Organize the collected output on a visible display to refer back to during whole-class discussions throughout the unit. Refer back to the display regularly, and encourage students to suggest revisions, updates, and connections to be added as they develop new mathematical ideas and new ways of communicating, over time.

MLR3 Clarify, Critique, Correct

This routine invites students to analyze a piece of mathematical writing that is not their own, and then improve on the work by correcting errors and clarifying meaning. 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. Teachers can demonstrate how to effectively and respectfully critique the work of others with meta-think-alouds and press for details when necessary. 
How It Happens:

1. 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)
2. Discussion with Partner: Students discuss the original statement in pairs. “What do you think the author means?,” “Is anything unclear?,” or “Are there any reasoning errors?” (2–3 minutes)
3. 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 cultivates conversation by creating an authentic need for students to communicate (Gibbons, 2002). With an information gap, students need to share ideas and information in order to bridge a gap and accomplish something that they could not have done alone. Teachers should 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:

1. Problem/Data Cards: Students are paired into Partner A and Partner B. Partner A is given a card with a problem that must be solved, and Partner B has the information needed to solve it on a “data card.” Neither partner should read or show their card to their partner.
2. Bridging the Gap: Partner A determines what information they need, and asks Partner B for that specific information. Partner B should not share information unless Partner A specifically asks for it and justifies the need for the information. 
3. Solving the Problem: Partner A shares the problem card and both students solve the problem independently. The Data Card is revealed and students compare their strategies and solutions. 

MLR5 Co-craft Questions

This routine invites students to get familiar with a context before the pressure to produce answers. Students produce the language of mathematical questions themselves, and analyze how different mathematical forms and symbols can be used to represent different situations. 
How It Happens:

1. Hook: Present the context. Display a problem stem, graph, video, image, or list of interesting facts. Optional: Students keep books or devices closed. 
2. Students Write Questions: Ask, “What mathematical questions might be asked about this (situation)?” It is preferable that students write down these questions; however, if students are still developing their writing skills they can state their questions orally or discuss them with a partner. These should be questions that students think are answerable by doing math and could be questions about the situation, information that appears to be missing, and even about assumptions that they think are important. (1–2 minutes)
3. Students Compare Questions: Students compare the questions they generated with a partner or small group (1–2 minutes) before select questions are shared 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.
4. Actual Question(s) Revealed: Students continue with the task as designed. If time allows, students may also select from the list of student-generated questions.

MLR6 Three Reads

Use this routine to ensure that students know what they are being 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 a mathematical text, situation, diagram, or graph three times, each with a particular focus. Optional: At times, the intended question or main prompt may be intentionally withheld until the third read so that students can concentrate on making sense of what is happening before rushing to find a solution or method.

1. Read #1: “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 any non-mathematical vocabulary. (1 minute)
2. Read #2: “What can be counted or measured?” After the second read, students list all quantities, focusing on naming what is countable or measurable in the situation. 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)
3. Read #3: “What are different ways or strategies we can use to solve this problem?” Students discuss possible strategies. It may be helpful for students to create diagrams to represent the relationships among quantities identified in the second read, or to represent the situation with a picture (Asturias, 2014). (1–2 minutes)

MLR7 Compare and Connect

This routine fosters students’ meta-awareness as they identify, compare, and contrast different mathematical approaches and representations. Students are prompted to reflect on, and linguistically respond to, these comparisons; for example, exploring why or when one might do or say something a certain way, or by identifying and explaining correspondences between different mathematical representations or methods.
How It Happens:

1. Students Prepare Displays of Their Work: Students are given a problem that can be approached and solved using multiple strategies, or a situation that can be modeled using multiple representations. Students prepare a visual display of their work, paying attention to the language and details they include that will allow others to make sense of their approach and reasoning. 
2. Compare: Students investigate each other's work, pointing out important mathematical features, and making comparisons. 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. 
3. 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 differences between the ways those features appear.

MLR8 Discussion Supports

This routine includes a variety of instructional moves and strategies that support inclusive discussions about mathematical ideas, representations, contexts, and strategies (Chapin, O’Connor, & Anderson, 2009). This collection of strategies can be combined and used to support discussion during almost any activity. They 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:

• Revoice student 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 students’ explanations by requesting that students challenge an idea, 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 provide all students with 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 doing a task.
• Provide prompts or structures to support discussion when students work in pairs or 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, so as to amplify language production, not 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?

Adapted with permission from work done by Understanding Language at Stanford University. For the original paper, Guidance for Math Curricula Design and Development, please visit https://ell.stanford.edu/content/mathematics-resources-additional-resources.

References

• Aguirre, J. M. & Bunch, G. C. (2012). What’s language got to do with it?: Identifying language demands in mathematics instruction for English language learners. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs. (pp. 183–194). Reston, VA: National Council of Teachers of Mathematics.
• Chapin, S., O’Connor, C., & Anderson, N. (2009). Classroom discussions: Using math talk to help students learn, grades K–6 (2nd ed.). Sausalito, CA: Math Solutions Publications.
• Gibbons, P. (2002). Scaffolding language, scaffolding learning: Teaching second language learners in the mainstream classroom. Portsmouth, NH: Heinemann.
• Kelemanik, G, Lucenta, A & Creighton, S.J. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann.
• Zwiers, J. (2011). Academic conversations: Classroom talk that fosters critical thinking and content understandings. Portland, ME: Stenhouse.
• Zwiers, J. (2014). Building academic language: Meeting Common Core Standards across disciplines, grades 5–12 (2nd ed.). San Francisco, CA: Jossey-Bass.
• Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the design of mathematics curricula: Promoting language and content development. Retrieved from Stanford University, UL/SCALE website: https://ul.stanford.edu/resource/principles-design-mathematics-curricula