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, 39 Mathematical Language Routines Handbook for Algebra 1
Introduction to Mathematical Language Routines.....................................................................................2
Mathematical Language Routines................................................................................................................ 4
Navigating Support for English Language Learners...................................................................................5
MLR 1 Stronger and Clearer Each Time........................................................................................................ 6
MLR 1 Stronger and Clearer Each Time: How It Happens..........................................................................7
MLR 1 Stronger and Clearer Each Time: Graphic Organizer......................................................................8
MLR 2 Collect and Display............................................................................................................................ 10
MLR 2 Collect and Display: How It Happens.............................................................................................. 10
MLR 2 Collect and Display: Graphic Organizer.......................................................................................... 11
MLR 3 Clarify, Critique, Correct................................................................................................................... 12
MLR 3 Clarify, Critique, Correct: How It Happens.....................................................................................13
MLR 3 Clarify, Critique, Correct: Graphic Organizer.................................................................................14
MLR 4 Information Gap................................................................................................................................ 16
MLR 4 Information Gap: How It Happens.................................................................................................. 17
MLR 4 Information Gap: Graphic Organizer.............................................................................................. 18
MLR 5 Co-Craft Questions............................................................................................................................ 20
MLR 5 Co-Craft Questions: How It Happens.............................................................................................. 21
MLR 5 Co-Craft Questions: Graphic Organizer.......................................................................................... 22
MLR 5 Co-Craft Questions Guide................................................................................................................. 24
MLR 6 Three Reads....................................................................................................................................... 25
MLR 6 Three Reads: How It Happens.......................................................................................................... 26
MLR 6 Three Reads: Graphic Organizer..................................................................................................... 27
MLR 6 Three Reads: Guide........................................................................................................................... 28
MLR 7 Compare and Connect...................................................................................................................... 29
MLR 7 Compare and Connect: How It Happens........................................................................................30
MLR 7 Compare and Connect: Graphic Organizer....................................................................................31
MLR 7 Compare and Connect: Guide.......................................................................................................... 32
Openstax CC BY NC SA
MLR for Algebra 1, brought to you by OpenStax 1
,MLR 8 Discussion Supports.......................................................................................................................... 33
MLR 8 Discussion Supports: How It Happens............................................................................................ 34
Algebra 1 and MLR Crosswalk..................................................................................................................... 36
References..................................................................................................................................................... 42
Introduction to Mathematical Language Routines
Language development can be built into teachers’ instructional practice and students’ classroom
experience by intentionally designing materials, teacher commitments, administrative support and
professional development.
Theory of Action
Mathematical Language Routines are supported by the Theory of Action, which is grounded in four
key concepts:
Interdependence of Learning
Mathematical understanding and language competence develop interdependently. Deep
conceptual learning is gained through language. Ideas take shape through words, texts,
illustrations, conversations, debates, examples, etc. Teachers, peers, and texts serve as language
resources for learning.
Scaffolding Routines
Scaffolding provides temporary supports that foster student autonomy. Language learners can
engage deeply with mathematical ideas under instructional conditions. Mathematical language
development occurs when students use language to make meaning and engage with challenging
problems beyond their ability to solve independently, requiring interaction with peers.
Instructional Responsiveness
Instruction supports learning when teachers respond to students’ verbal and written work. Eliciting
student thinking through language allows teachers and students to respond formatively to the
language students generate. Formative peer and teacher feedback creates opportunities for
revision and refinement of content understanding and language.
Student Agency
Students are agents in their own mathematical and linguistic sense-making. Mathematical
language proficiency is developed through the process of actively exploring and learning
mathematics. Language is action. In the very “doing” of math, students have naturally occurring
opportunities to need, learn, and notice mathematical ways of making sense and talking about
ideas and the world. These experiences support learners to expand their existing language toolkits.
Openstax CC BY NC SA
MLR for Algebra 1, brought to you by OpenStax 2
, Design Principles
The framework for supporting emerging bilingual students in this curriculum includes four design
principles for promoting mathematical language use and development. These four principles are
guides for curriculum development, planning and execution of instruction, and the structure and
organization of interactive opportunities for students. They also serve as guides for observation,
analysis, and reflection on student language and learning. The design principles motivate the use
of mathematical language routines.
Support Sense-Making
Teachers can make language more accessible for students by amplifying rather than simplifying
speech or text. Amplifying means anticipating where students might need language support to
understand concepts or mathematical terms, and providing multiple ways to access them.
Optimize Output
Linguistic output is when students communicate their ideas to others in oral, written, or visual
formats. All students benefit from repeated, strategically optimized, and structured opportunities
to articulate mathematical ideas into linguistic expression.
Cultivate Conversation
Conversations are back and forth interactions with multiple turns that build up ideas about math.
They scaffold students developing mathematical language because conversations provide
opportunities to simultaneously make meaning, communicate that meaning, and refine how
content understandings are communicated.
Maximize Meta-Awareness
Language is a tool that not only allows students to communicate their math understanding to
others, but also to organize their own experiences, ideas, and learning for themselves. Meta-
awareness is consciously thinking about one's own thought processes or language use. It can be
strengthened when teachers ask students to explain to each other the strategies they used to solve
a challenging problem.
Openstax CC BY NC SA
MLR for Algebra 1, brought to you by OpenStax 3
MLR for Algebra 1, brought to you by OpenStax 0
, 39 Mathematical Language Routines Handbook for Algebra 1
Introduction to Mathematical Language Routines.....................................................................................2
Mathematical Language Routines................................................................................................................ 4
Navigating Support for English Language Learners...................................................................................5
MLR 1 Stronger and Clearer Each Time........................................................................................................ 6
MLR 1 Stronger and Clearer Each Time: How It Happens..........................................................................7
MLR 1 Stronger and Clearer Each Time: Graphic Organizer......................................................................8
MLR 2 Collect and Display............................................................................................................................ 10
MLR 2 Collect and Display: How It Happens.............................................................................................. 10
MLR 2 Collect and Display: Graphic Organizer.......................................................................................... 11
MLR 3 Clarify, Critique, Correct................................................................................................................... 12
MLR 3 Clarify, Critique, Correct: How It Happens.....................................................................................13
MLR 3 Clarify, Critique, Correct: Graphic Organizer.................................................................................14
MLR 4 Information Gap................................................................................................................................ 16
MLR 4 Information Gap: How It Happens.................................................................................................. 17
MLR 4 Information Gap: Graphic Organizer.............................................................................................. 18
MLR 5 Co-Craft Questions............................................................................................................................ 20
MLR 5 Co-Craft Questions: How It Happens.............................................................................................. 21
MLR 5 Co-Craft Questions: Graphic Organizer.......................................................................................... 22
MLR 5 Co-Craft Questions Guide................................................................................................................. 24
MLR 6 Three Reads....................................................................................................................................... 25
MLR 6 Three Reads: How It Happens.......................................................................................................... 26
MLR 6 Three Reads: Graphic Organizer..................................................................................................... 27
MLR 6 Three Reads: Guide........................................................................................................................... 28
MLR 7 Compare and Connect...................................................................................................................... 29
MLR 7 Compare and Connect: How It Happens........................................................................................30
MLR 7 Compare and Connect: Graphic Organizer....................................................................................31
MLR 7 Compare and Connect: Guide.......................................................................................................... 32
Openstax CC BY NC SA
MLR for Algebra 1, brought to you by OpenStax 1
,MLR 8 Discussion Supports.......................................................................................................................... 33
MLR 8 Discussion Supports: How It Happens............................................................................................ 34
Algebra 1 and MLR Crosswalk..................................................................................................................... 36
References..................................................................................................................................................... 42
Introduction to Mathematical Language Routines
Language development can be built into teachers’ instructional practice and students’ classroom
experience by intentionally designing materials, teacher commitments, administrative support and
professional development.
Theory of Action
Mathematical Language Routines are supported by the Theory of Action, which is grounded in four
key concepts:
Interdependence of Learning
Mathematical understanding and language competence develop interdependently. Deep
conceptual learning is gained through language. Ideas take shape through words, texts,
illustrations, conversations, debates, examples, etc. Teachers, peers, and texts serve as language
resources for learning.
Scaffolding Routines
Scaffolding provides temporary supports that foster student autonomy. Language learners can
engage deeply with mathematical ideas under instructional conditions. Mathematical language
development occurs when students use language to make meaning and engage with challenging
problems beyond their ability to solve independently, requiring interaction with peers.
Instructional Responsiveness
Instruction supports learning when teachers respond to students’ verbal and written work. Eliciting
student thinking through language allows teachers and students to respond formatively to the
language students generate. Formative peer and teacher feedback creates opportunities for
revision and refinement of content understanding and language.
Student Agency
Students are agents in their own mathematical and linguistic sense-making. Mathematical
language proficiency is developed through the process of actively exploring and learning
mathematics. Language is action. In the very “doing” of math, students have naturally occurring
opportunities to need, learn, and notice mathematical ways of making sense and talking about
ideas and the world. These experiences support learners to expand their existing language toolkits.
Openstax CC BY NC SA
MLR for Algebra 1, brought to you by OpenStax 2
, Design Principles
The framework for supporting emerging bilingual students in this curriculum includes four design
principles for promoting mathematical language use and development. These four principles are
guides for curriculum development, planning and execution of instruction, and the structure and
organization of interactive opportunities for students. They also serve as guides for observation,
analysis, and reflection on student language and learning. The design principles motivate the use
of mathematical language routines.
Support Sense-Making
Teachers can make language more accessible for students by amplifying rather than simplifying
speech or text. Amplifying means anticipating where students might need language support to
understand concepts or mathematical terms, and providing multiple ways to access them.
Optimize Output
Linguistic output is when students communicate their ideas to others in oral, written, or visual
formats. All students benefit from repeated, strategically optimized, and structured opportunities
to articulate mathematical ideas into linguistic expression.
Cultivate Conversation
Conversations are back and forth interactions with multiple turns that build up ideas about math.
They scaffold students developing mathematical language because conversations provide
opportunities to simultaneously make meaning, communicate that meaning, and refine how
content understandings are communicated.
Maximize Meta-Awareness
Language is a tool that not only allows students to communicate their math understanding to
others, but also to organize their own experiences, ideas, and learning for themselves. Meta-
awareness is consciously thinking about one's own thought processes or language use. It can be
strengthened when teachers ask students to explain to each other the strategies they used to solve
a challenging problem.
Openstax CC BY NC SA
MLR for Algebra 1, brought to you by OpenStax 3
