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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 each level of the curriculum:
| COURSE |
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|---|---|
| LESSON |
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| ACTIVITY |
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This interwoven approach is grounded in four design principles that promote mathematical language use and development:
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.
Strengthen opportunities and structures 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).
Strengthen opportunities and structures for constructive mathematical conversations (partners, groups, and whole class). Conversations are back-and-forth interactions, with multiple turns that build up ideas about math. Conversations act as scaffolds for students to develop mathematical language because they provide 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.
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 are motivated to 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 that 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.
The Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. The MLRs emphasize the use of language that is meaningful and purposeful, and isn’t only about getting answers. The routines 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 and incorporate these flexible MLRs across the lessons in each unit to support students at all stages of language development in improving their use of English and disciplinary language.
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 Guide for the lessons in which they appear.
Each unit also includes optional, suggested MLRs that facilitate access to the language demands of a lesson or an activity. These are described in either the Launch or the Activity Synthesis of the activities in which they appear. 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.
The MLRs facilitate attention to student language in ways that support in-the-moment teacher-, peer-, and self-assessments. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also the questions they ask to clarify their understanding of others’ ideas.
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 routines helps to maintain students’ engagement in mathematical discourse and ensures that the struggle remains productive. Decide which optional MLRs to use and when to use them, based on students’ individual needs. For a description of each of the eight MLRs used in the curriculum, see the Instructional Routines section of this Course Guide.
Sentence frames support students’ 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 |
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| Explain |
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| Justify |
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| Generalize |
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| Critique |
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| Compare and Contrast |
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| Represent |
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| Interpret |
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