Zwiers, Jeff, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. "Principles for the Design of Mathematics Curricula: Promoting Language and Content Development." Understanding Language/Stanford Center for Assessment, Learning and Equity, Stanford University Graduate School of Education, 28 Feb. 2017, http://ell.stanford.edu/content/mathematics-resources-additional-resources.
About the Original Article's Tone
This is a practitioner design framework — published by Stanford's Understanding Language/SCALE center and aimed at mathematics curriculum developers, instructional coaches, and math teachers working with English learners. It is not a peer-reviewed journal article and does not present original data. It presents a framework and a toolkit. Think less "journal article" and more "design specification document written for thoughtful practitioners."
It uses:
Collaborative, inclusive "we" voice — the authors position themselves as partners with teachers, not authorities above them
Framework language with clear numbered structures — four design principles, eight routines, each with labeled examples
Moderate academic register — it cites Vygotsky and Halliday but translates their ideas into classroom-ready language
Instructional how-to format — many sections read like professional development materials (numbered steps, teacher moves, example scripts)
Equity framing throughout — "access," "scaffolding," "agency," and "language learners" recur as core commitments, not afterthoughts
The vibe: This reads like a well-designed professional development workshop distilled into a document — it has structure, purpose, and genuine pedagogical thinking behind it, without the methodological weight of an empirical study. Think "Stanford workshop handout for smart teachers" rather than "academic paper for academic audiences."
What it glosses over: The framework emerged from EL-focused research and contexts but is now deployed universally across all students, including native English speakers. That mainstreaming is barely mentioned. It also offers no empirical evidence that any specific routine produces language gains — the routines are design heuristics grounded in theory, not effect sizes. Finally, the relentless positivity of the document papers over the real implementation challenge: these routines require significant teacher skill, and "the curriculum can support it" is not the same as "teachers can do it without substantial training and coaching."
Visual Metaphor
Rain falls across a limestone plateau. It doesn't stay on the surface — it percolates down through cracks and layers, each stratum filtering, slowing, and routing the water differently. The rock doesn't create the water. But without the rock, the water runs off and is lost.
Deep underground, the water collects in an aquifer — pressurized, purified, patient. Months or years later, it emerges at a spring on the valley floor: clear, cold, reliable. The aquifer didn't manufacture anything. It gave the water time and structure to become what it already was.
MLRs are the permeable strata. Mathematical thinking is the rain.
What This Is Really About
Somewhere in the last fifty years of mathematics education, we convinced ourselves that language was optional — that if students could do the math, the talking about it was decoration. Zwiers and his Stanford colleagues have a different theory: the talking is the doing. Language is not how students report their mathematical understanding after the fact. Language is the medium through which mathematical understanding forms in the first place.
This document is the theoretical and practical foundation of the Math Language Routines (MLRs) that now appear in Illustrative Mathematics (IM) curriculum materials across the country. If you've seen "MLR6: Three Reads" in a lesson plan, this is the document that invented it and explains why.
The Theory of Action
The document's theory of action rests on four interconnected claims:
Mathematical understanding and language competence develop interdependently. You cannot truly separate "learning math" from "learning to talk about math." The ideas take shape through language — through words, debates, explanations, and written arguments. Vygotsky's insight that teachers and peers serve as language resources for learning isn't a philosophical abstraction here; it's a design constraint.
Students are agents in their own sense-making. The document explicitly challenges two persistent myths: (a) that mathematical language proficiency means using only formal vocabulary, and (b) that developing disciplinary language requires "time-out" mini-lessons separate from math content. Both assumptions are wrong, and both are costly — the first sets an unrealistic bar, the second wastes instructional time by treating language as a detour from math rather than a part of it.
Scaffolding provides temporary supports that foster independence. Students — including English learners at any proficiency level — can engage deeply with grade-level mathematical ideas when appropriately scaffolded. The key word is temporary: scaffolds are not accommodations. They're training wheels that come off. The error most classrooms make is either refusing to provide scaffolds (sink-or-swim immersion) or leaving them in place so long they become crutches.
Instruction works when teachers respond to student language. The routines create the conditions for teachers to actually hear what students are saying and doing, enabling real-time formative adjustment. The MLRs aren't just student activities; they're teacher listening devices.
The Four Design Principles
The document organizes its recommendations around four principles for language-responsive curriculum and instruction:
Design Principle 1: Support Sense-Making. Scaffold tasks and amplify language so students can make their own meaning. The key distinction here is amplifying vs. simplifying — simplifying means avoiding challenging language; amplifying means providing multiple pathways into it (visuals, manipulatives, analogies, think-alouds). Supported routines: MLR2 (Collect and Display), MLR6 (Three Reads), MLR8 (Discussion Supports).
Design Principle 2: Optimize Output. Strengthen opportunities for students to describe their mathematical thinking clearly — orally, visually, and in writing. Students need repeated, strategic, iterative practice making ideas stronger (better reasoning) and clearer (more precise language). Output means all forms of student expression; conversation is covered separately in Principle 3. Supported routines: MLR1 (Stronger and Clearer), MLR3 (Critique, Correct, Clarify), MLR4 (Info Gap), MLR5 (Co-Craft Questions), MLR7 (Compare and Connect).
Design Principle 3: Cultivate Conversation. Strengthen back-and-forth mathematical conversations in pairs, groups, and whole class. Conversations differ from output: they involve multiple turns, negotiated meaning, and genuine communicative purpose. The "information gap" is the engine — when students actually need each other's information to solve a problem, talk becomes authentic rather than performative. Supported routines: MLR1, MLR3, MLR4, MLR5, MLR7, MLR8.
Design Principle 4: Maximize Meta-Awareness. Strengthen "meta-" connections between mathematical ideas, reasoning, and language. This means both metacognitive (thinking about one's own thinking) and metalinguistic (reflecting on language choices and how they shape meaning). When students compare how they say something in their home language vs. English, or when they notice that "negative" and "minus" name different things with the same symbol — that's meta-awareness in action. Supported routines: MLR2, MLR3, MLR5, MLR6, MLR7, MLR8.
The Eight Math Language Routines (MLRs): Full Descriptions
The eight MLRs are the operational core of this framework. These are not optional enrichment activities; they are structured, adaptable formats for simultaneously learning mathematical practices, content, and language. Each one maps to one or more design principles.
MLR1: Stronger and Clearer Each Time. Students write an initial response individually, then share it through structured successive pairing — talking to two or three different partners in sequence, each time refining and strengthening their explanation before writing a final version. The goal is iterative revision: not just talking more, but producing demonstrably better mathematical communication with each pass. Two sub-variants: Successive Pair Shares (multiple rotating partners) and "Convince Yourself, a Friend, a Skeptic" (three audience-based iterations of the same argument). Supports: Principles 2 and 3.
MLR2: Collect and Display. While students work, the teacher circulates and collects — on a visible display — actual student words, phrases, diagrams, and representations. This "stabilizes the fleeting language" students use during group work, making it available as a shared reference point. The display evolves throughout a unit: teacher can add to it, revise it, and explicitly bridge student informal language to formal disciplinary terms. Supports: Principles 1 and 4.
MLR3: Critique, Correct, and Clarify. Students receive a piece of mathematical writing that is not their own — one containing an error, ambiguity, or gap — and improve it. The key is that the given response should be partially right and contain an ambiguous phrase or informal expression: not an obvious wrong answer, but a response that requires genuine analysis. Variant: Always-Sometimes-Never, where students evaluate whether mathematical statements are always, sometimes, or never true. Supports: Principles 2 and 4.
MLR4: Information Gap. Partners receive different pieces of necessary information. Partner A has the problem; Partner B has the data. Partner A must ask for specific information — and justify why they need it — before Partner B will share. Neither partner reads their card aloud or shows it. The information gap creates genuine communicative need: this is not "pretend to need each other," this is actually needing each other to solve the problem. Supports: Principle 3.
MLR5: Co-Craft Questions and Problems. Students receive a context or problem stem (without the question) and generate the mathematical questions themselves — then compare their questions to classmates' before the actual question is revealed. Variants include co-creating entirely new problems and co-creating situations from mathematical representations. The goal is getting students inside a context before they face production pressure. Supports: Principles 3 and 4.
MLR6: Three Reads. Students read a mathematical situation or problem three times, each with a distinct focus: (1) comprehension — what is happening? (2) mathematical structure — how is the language organizing the math? (3) solution methods — what are possible approaches? A variant uses a two-column graphic organizer (Values | Units) where students systematically annotate quantities and their meanings before writing expressions. Supports: Principles 1 and 4.
MLR7: Compare and Connect. Multiple student solution strategies, representations, or approaches are displayed simultaneously, and students are asked to identify what is similar and what is different — then focus on specific mathematical relationships that appear (or don't appear) across strategies. This is not just a gallery walk; students are pressed with specific questions: "Why does this approach use multiplication and that one doesn't? Where is the 10 in each approach?" Variant: Which One Doesn't Belong? — four mathematical objects, students decide which three share a category and which doesn't fit. Supports: Principle 4.
MLR8: Discussion Supports. A collection of multi-modal strategies for facilitating rich, inclusive whole-class and small-group discussion: revoicing, pressing for elaboration, multi-modal representation (gesture, video, acting out), choral response, think-alouds, and Numbered Heads Together (students count off in groups, teacher calls a random number, that student reports their group's reasoning). These supports are not a single routine but a toolkit of teacher moves that amplify any of the other seven MLRs. Supports: All four principles.
What This Challenges
The document quietly demolishes two professional habits that are deeply entrenched in math classrooms: the assumption that language instruction is the EL teacher's job (not the math teacher's), and the belief that mathematical correctness is sufficient evidence of mathematical understanding. Zwiers' framework says both are wrong. The math teacher IS a language teacher — always has been, whether or not they knew it. And a student who can get the right answer without being able to explain their reasoning is not fully demonstrating mathematical understanding; they're demonstrating one slice of it.
It also challenges the instinct to simplify: when a student is struggling, the temptation is to reduce language demand. But Zwiers argues the opposite — amplify, don't simplify. Provide more pathways in, not fewer words on the page.
The Big Picture
This document was written with English learners in mind but explicitly designed for all students. That move — universalizing what started as an EL-targeted intervention — is both a strength and a tension worth watching. The strength: if scaffolding language produces gains for ELs, it likely produces gains for everyone, and embedding it universally eliminates the stigma of "EL-only" support. The tension: when a framework designed to address racialized inequity gets universalized, it sometimes gets defanged — the equity edge softens as the practice gets absorbed into general "good teaching." Whether the MLRs retain their equity force as they become universal practice in IM classrooms is an open question this document doesn't address.
Key Vocabulary
Amplifying (vs. Simplifying)
Simply: Instead of turning down the difficulty dial, you give students more doors into the same room — visuals, examples, analogies — so they can find their own way in.
A pedagogical stance in which teachers anticipate where students may need language support and provide multiple access pathways (visuals, manipulatives, analogies, modeling) rather than reducing the linguistic complexity of the task or text. Zwiers explicitly contrasts this with simplifying, which avoids challenging language and produces watered-down mathematics. Amplifying maintains rigor while increasing accessibility.
Disciplinary Language
Simply: The specific way mathematicians talk, write, and argue — including vocabulary, sentence patterns, and conventions that aren't taught anywhere except by actually doing math.
The specialized register of a subject-matter community — in mathematics, this includes technical vocabulary, argumentation conventions, symbolic notation practices, and the implicit norms of how claims are made and justified. The document argues that disciplinary language develops through doing the discipline, not through separate language instruction; teachers are therefore always already teaching disciplinary language whether they acknowledge it or not.
Information Gap (as a pedagogical structure)
Simply: When students actually need each other's information to solve a problem — not pretend-need, but genuinely can't proceed without asking — talk becomes real rather than performative.
A deliberate instructional design feature in which two partners hold different, complementary pieces of necessary information, creating authentic communicative need. The information gap is the structural engine of MLR4 but also the underlying principle of why conversation in mathematics classrooms must be purposefully designed — students need a genuine reason to talk. Without an information gap, group work easily collapses into one student solving while others watch.
Language as Action (van Lier & Walqui)
Simply: Language isn't the receipt you print after you've bought something — it's the transaction itself. You don't figure out the math and then describe it; you figure it out by describing it.
A theoretical stance drawn from van Lier & Walqui (2012) in which language is understood as constitutive of disciplinary learning, not merely reportive. Students do not first understand a mathematical concept and then produce language about it; rather, they understand through the production of language. This framing underpins the entire document's argument that language instruction and content instruction cannot be productively separated.
Mathematical Language Routine (MLR)
Simply: A reusable classroom structure — like a move in chess — that reliably creates conditions where students talk about math in useful ways, as opposed to just getting an answer and going quiet.
A structured but adaptable format for amplifying, assessing, and developing students' mathematical language. MLRs emphasize language that is meaningful and purposeful rather than inauthentic or answer-based. There are eight MLRs in the framework (MLR1–MLR8), each designed to be usable across grade levels and embedded within existing lesson structures rather than added as supplementary activities.
Mathematical Register
Simply: The particular "flavor" of language that mathematics uses — not just vocabulary, but grammar, argumentation style, and conventions that are different from everyday speech even when using the same words.
The specialized linguistic variety used within mathematical discourse, characterized by specific vocabulary, clause structures, argumentation norms, and symbolic conventions. The document argues that developing students' command of the mathematical register is not an additional burden but is already embedded in the goal of producing powerful mathematical thinkers — you cannot fully think mathematically without some command of how mathematics communicates.
Meta-Awareness (Metacognitive / Metalinguistic)
Simply: Thinking about your own thinking, or noticing your own language choices — stepping outside the math for a moment to reflect on how you're doing the math or saying the math.
Consciously attending to one's own thought processes or language use. Metacognitive awareness involves reflecting on how one is approaching a problem ("What strategy am I using? Where am I stuck?"). Metalinguistic awareness involves reflecting on language itself ("How did I say that? How does that phrase compare to how I'd say it in Spanish?"). Zwiers' Principle 4 (Maximize Meta-Awareness) targets both dimensions and argues they are mutually reinforcing.
Scaffolding
Simply: Temporary help structures that let students do harder things than they could do alone — with the key word being "temporary." Scaffolds that never come off aren't scaffolds; they're crutches.
Temporary instructional supports that enable students to engage with content and language demands beyond their current independent ability. Scaffolding theory (Vygotsky, Walqui & van Lier) emphasizes that the supports must be designed to fade as learners internalize strategies — the goal is student independence, not permanent assistance. The document positions scaffolding as an equity imperative: students with emerging language can engage with rigorous content under appropriate scaffolded conditions.
🎯 MASL Connection
This Study Supports:
MASL's overall design language and curriculum positioning: MASL is implemented as supplementary materials for the Illustrative Mathematics (AccessIM) curriculum, which is built on Zwiers' MLR framework. Understanding what MLRs are and what they do is not background knowledge for MASL — it's the scaffolding MASL sits on top of. When MASL slides use sentence frames, structured partner talk, and critique routines, they are using Zwiers' vocabulary and Zwiers' design logic. The capstone argument must be positioned relative to this framework.
MASL's sentence frame design (all four activities): Zwiers' Principle 2 (Optimize Output) and Principle 3 (Cultivate Conversation) provide the theoretical justification for using sentence frames in Partner Card Sort, Language Frames, and Suggest Improvements activities. Zwiers specifically argues that students need "spiraled practice in making their ideas stronger with more robust reasoning and making their ideas clearer with more precise language" — which is exactly what MASL's sentence frame architecture targets. The Barko-Alva critique of sentence frames lands differently here: Zwiers' own MLR3 (Critique, Correct, Clarify) uses partial responses with ambiguous language as the vehicle, which is closer to MASL's Suggest Improvements activity than to the rote fill-in-the-blank Barko-Alva critiques.
MASL's MLR variant activities in the slide generator: MASL's slide generator produces MLR-tagged activity variants (MLR1, MLR3, MLR6, MLR7, MLR8 appear explicitly in MASL slide templates). Zwiers et al. is the source document that defines what those tags mean. Any MASL slide labeled "MLR6: Three Reads" is implementing a routine from this paper; any paper or presentation that references MASL's MLR alignment should cite Zwiers et al. (2017) for the routine definitions.
MASL's Baseline Language Assessment (diagnostic framing): Zwiers' discussion of formative assessment through language observation — teachers using student language as evidence for in-the-moment instructional adjustment — supports the rationale for MASL's baseline assessment. The 34-item diagnostic operationalizes what Zwiers describes as the teacher's need to observe students' developing disciplinary language. MASL makes this observation systematic and instrument-based rather than anecdotal.
Design Implications:
Position MASL as adding a notation layer MLRs don't address. Zwiers' routines target mathematical discourse broadly — arguing, explaining, comparing, questioning. What none of the eight MLRs address is the verbal production of mathematical notation specifically. MLR6 (Three Reads) asks students to comprehend a word problem; it does not ask them to produce the spoken name for the symbols in that problem. MASL occupies exactly this gap. When writing the capstone paper, this non-overlap is the strongest positioning argument available: MASL is additive, not redundant.
Distinguish MASL's sentence frames from Zwiers' output-support frames. Zwiers uses sentence frames as scaffolds for argumentation and explanation ("I think ___ because ___," "My claim is ___ and my evidence is ___"). MASL's frames target notation verbalization specifically ("I say this expression as ___ because ___," "The symbol ___ should be read as ___"). The design specification must be explicit: MASL frames foreground the spoken register of notation, which is a category of linguistic production that Zwiers' framework acknowledges exists (the "mathematical register") but does not operationalize as a specific instructional target.
Use Zwiers' "amplify, don't simplify" principle to defend notation instruction for all students. One likely critique of MASL will be "notation instruction is only for ELs." Zwiers explicitly rejects this kind of tracking logic: the MLR framework was designed for ELs but intentionally universalized. The same logic applies to MASL's notation layer — notation irregulars (x² vs. "x squared," f(x) vs. "f of x") are genuinely difficult for all students, not just ELs. Zwiers provides the framework for making this universal-access argument.
The MLR8 Discussion Supports toolkit maps directly to MASL's Partner Card Sort facilitation moves. Revoicing, pressing for detail, choral response, and Numbered Heads Together from MLR8 can all be adapted for use during Card Sort activities. MASL's slide generator should flag MLR8 as the discussion facilitation layer that wraps around Card Sort, Language Frames, and Suggest Improvements — not just for the activities that explicitly carry an MLR tag.
Evidence Strength for MASL:
Zwiers et al. (2017) is a design framework, not an empirical study. It provides design language, design logic, and design heuristics — but no effect sizes, no comparison groups, and no outcome data. This is the right tool for positioning and framing MASL within the field's shared vocabulary; it is not the right tool for causal claims about what notation instruction does. For MASL's capstone argument, Zwiers provides the "what are we building within" scaffolding; the causal claims come from Mercer & Sams (2006), Barbieri et al. (2023), Booth et al., and the other empirical sources. Citing Zwiers for empirical warrant would be a category error.
Connections to MASL Framework (click to expand)
MASL Trio (Math / We Say / Meaning cards): The "We Say" card is a direct operationalization of Zwiers' claim that the mathematical register must be taught explicitly. Zwiers argues students need to develop command of how mathematical ideas are communicated — the "We Say" card makes the spoken register visible as an object of study alongside the symbol and its meaning. This is Zwiers' theory made concrete in cardstock.
Sentence frames: Zwiers' Optimize Output principle and Cultivate Conversation principle both support sentence frames as scaffolds for mathematical discourse. MASL's frames narrow the target to notation-specific production; Zwiers' framework provides the broader theoretical umbrella. The Barko-Alva critique applies to frames that pre-specify both form AND content; MASL's frames pre-specify the linguistic structure but leave the mathematical content as the production target — aligning more closely with Zwiers' own MLR3 design than with rote fill-in-the-blank.
Irregular forms instruction: Zwiers discusses the mathematical register as including "tools and conventions" that students come to see as useful — but does not specifically identify suppletive/irregular spoken forms (x²="x squared") as a category requiring explicit instruction. MASL's Mathematical Language Irregulars framework is the extension Zwiers' theory implies but does not specify.
Scaffolding fading: Zwiers' scaffolding theory (temporary supports that foster independence) provides the theoretical warrant for MASL's fading plan in Language Frames (full example → partial → blank). Zwiers doesn't name Kalyuga's expertise reversal effect, but the principle — scaffolds that don't fade become barriers to fluency — is exactly his Principle 1 scaffolding commitment. Cite both Zwiers (design justification) and Kalyuga (expertise reversal, technical mechanism) in the capstone.
Cross-connections: Zwiers' framework is the curriculum architecture of AccessIM (Illustrative Mathematics for ELs); Mercer & Sams (2006) provides the empirical data for structured partner talk that Zwiers' Principle 3 assumes but doesn't measure; Barko-Alva & Chang-Bacon (2023) is the direct challenge to sentence frames that must be answered using Zwiers' own MLR3 design logic as a defense.
💬 Key Quotes
Copy-paste ready quotes for papers, discussions, and reflections.
"Mathematical understandings and language competence develop interdependently. Deep disciplinary learning is gained through language, as it is the primary medium of school instruction."
p. 3Thesis
Why this quote: The foundational claim of the entire document — the theoretical stake in the ground. This is the sentence that makes every MLR necessary.
"Teachers should make language more 'considerate' to students by amplifying rather than simplifying speech or text... Amplifying means anticipating where students might need support in understanding concepts or mathematical terms, and providing multiple ways to access those concepts and terms."
p. 6Practical
Why this quote: The amplify-vs.-simplify distinction is the most practically significant and most frequently cited move in the document — it reframes scaffolding from reduction to provision.
"A commitment to help students develop their own command of the 'mathematical register' is therefore not an additional burden on teachers, but already embedded in a commitment to supporting students to become powerful mathematical thinkers and 'do-ers'."
p. 4Challenge
Why this quote: Directly addresses the "language instruction is extra work" objection. Key citation for any MASL argument that notation language instruction is part of — not in addition to — math teaching.
"A 'math language routine' refers to a structured but adaptable format for amplifying, assessing, and developing students' language. The routines emphasize the use of language that is meaningful and purposeful, not inauthentic or simply answer-based."
p. 9Definition
Why this quote: The definitional anchor for what an MLR is — essential when referencing MLRs in any MASL presentation or paper that needs to explain the term to audiences unfamiliar with IM curriculum.
"Learners with emerging language – at any level – can engage deeply with central disciplinary ideas under specific instructional conditions."
p. 4Foundational
Why this quote: The equity claim in five words that matter — "at any level." This isn't "ELs can participate in simple tasks." This is ELs can engage with rigorous content when conditions are right. Critical for MASL's access argument.
"Systemic barriers for language learners persist not only in tasks and materials, but in educators' presentational language, expectations for peer interactions, and assessment practices."
p. 3Challenge
Why this quote: Expands the problem beyond "bad curriculum" to teacher practice and assessment design — the barriers are systemic, not textbook-deep. Directly relevant to the argument that MASL addresses one specific layer of systemic barrier.
"Students need repeated, strategic, iterative and supported opportunities to articulate complex mathematical ideas into words, sentences, and paragraphs... making their ideas stronger with more robust reasoning and examples, and making their ideas clearer with more precise language and visuals."
p. 7Practical
Why this quote: The production rationale — why output-focused routines like MLR1 and MLR3 exist. Directly applies to MASL's Language Frames and Suggest Improvements activities, which target exactly this kind of iterative precision-building.
📚 References
References & Further Reading (click to expand)
Walqui, A. & van Lier, L. (2010). Scaffolding the Academic Success of Adolescent English Language Learners: A Pedagogy of Promise. WestEd.
Foundational
What it is: The theoretical scaffolding framework that Zwiers draws on most heavily — particularly the amplify-vs-simplify distinction and the "language as action" stance. Tone: Academic but accessible; practitioner-researcher audience. Why it matters: If you want to understand why Zwiers designs the way he does, this is the upstream source. Buzz: Widely cited across EL and content-area language instruction literature. Verdict: Worth reading if you're writing seriously about scaffolding theory — Zwiers is applying this work, not originating it.
Moschkovich, J.N. (2013). Principles and guidelines for equitable mathematics teaching practices and materials for English Language Learners. Journal of Urban Mathematics Education, 6(1), 45–57.
Must Read
What it is: The companion framework from Moschkovich that situates mathematical language within academic literacy — "mathematics proficiency = practices + discourse + content inseparably." Tone: Dense but clearly argued; academic journal. Why it matters: Core MASL source (Academic Literacy in Mathematics framework); Zwiers and Moschkovich are working in the same theoretical space and citing each other. Verdict: Read this before writing anything that positions MASL within the broader mathematics education for ELs literature.
Mercer, N. & Howe, C. (2012). Explaining the dialogic processes of teaching and learning: The value and potential of sociocultural theory. Learning, Culture, and Social Interaction, 1(1), 12–21.
Worth Reading
What it is: Sociocultural theory applied to classroom dialogue — conversations as scaffolds for simultaneous meaning-making and language development. Tone: Academic. Why it matters: Zwiers cites this for Principle 3 (Cultivate Conversation); provides theoretical grounding for why structured partner talk works, not just that it works. Verdict: Worth reading to understand the theory behind MLR1–MLR4; skip if you're already fluent in sociocultural/Vygotskian theory.
Cazden, C. (2001). Classroom Discourse: The Language of Teaching and Learning. Heinemann.
Foundational
What it is: The classic text on classroom discourse structures — including the IRF (Initiation-Response-Feedback) pattern that Zwiers is largely working against. Tone: Academic but readable. Why it matters: Understanding what MLRs are designed to disrupt requires understanding what the default classroom discourse looks like; Cazden describes the default. Buzz: Thousands of citations; textbook reference. Verdict: Skim for the IRF concept and the discourse analysis framework; dense if read cover-to-cover.
Kelemanik, G., Lucenta, A. & Creighton, S.J. (2016). Routines for Reasoning: Fostering the Mathematical Practices in All Students. Heinemann.
Worth Reading
What it is: Practitioner text on math talk routines; Zwiers cites it for MLR6 (Three Reads) specifically. Tone: Practitioner-friendly; teacher audience. Why it matters: More detailed implementation guidance for Three Reads and similar routines than Zwiers' brief treatment; useful for classroom-level design work. Verdict: Worth reading if you need extended implementation guidance; skip if you're working at the theory level.
Aguirre, J.M. & Bunch, G.C. (2012). What's language got to do with it?: Identifying language demands in mathematics instruction for ELLs. In Celedón-Pattichis & Ramirez (Eds.), Beyond Good Teaching: Advancing Mathematics Education for ELLs. NCTM.
Worth Reading
What it is: Framework for identifying and planning around language demands in mathematics instruction. Tone: Practitioner-academic hybrid; NCTM audience. Why it matters: Zwiers opens by citing this — it establishes the scope of "language demands" as including reading, writing, speaking, listening, conversing, and representing. Verdict: Quick read; good framing for the breadth of language demands the MLR framework addresses.
Vygotsky, L.S. (1978). Mind in Society. Harvard University Press.
Foundational
What it is: The theoretical bedrock — Zone of Proximal Development, scaffolding as concept, and the social-mediated nature of cognitive development. Tone: Dense Soviet academic prose, translated. Why it matters: Everything in this document stands on Vygotsky's shoulders. You don't need to read all of it, but knowing ZPD and the social mediation argument will make the entire MLR framework make more sense. Verdict: Read Chapter 6 (the ZPD chapter) — 30 minutes, worth it.
🧠 Quiz — Test Your Understanding
Six conceptual questions. Not about recalling definitions — about understanding what's actually being argued.
1. According to Zwiers et al., the difference between "amplifying" and "simplifying" language is:
Correct. Amplifying means anticipating where students need support and providing multiple ways in — visuals, analogies, modeling, gesture — while keeping the mathematical complexity intact. Simplifying removes the challenge, which also removes the learning opportunity. The other options miss the key distinction: amplifying is about access routes, not about reducing difficulty. This matters for MASL because notation instruction should amplify students' ability to engage with symbolic language, not simplify the notation itself.
Not quite. Amplifying is not about word count, student level, or modality alone. Zwiers defines it specifically as providing multiple pathways into complex content while maintaining its rigor — as opposed to simplifying, which removes the complexity. Think of amplifying as giving students more doors into the same building, rather than replacing the building with a smaller one.
2. The document argues that developing students' command of the "mathematical register" is:
Correct. Zwiers' key move here is collapsing the separation between language instruction and content instruction: if you're committed to developing powerful mathematical thinkers, you're already committed to developing their command of how mathematics communicates. The other options all preserve the separation the document is explicitly arguing against. This framing is strategically important for MASL: notation language instruction isn't an add-on — it's part of the math teaching commitment.
The document explicitly argues the opposite of options A, B, and D. Zwiers' central move is arguing that mathematical language development is not separate from mathematical content development — it's embedded in it. A math teacher who is committed to developing powerful mathematical thinkers is already, whether they know it or not, a language teacher.
3. The "information gap" in MLR4 is important because:
Correct. The information gap creates authentic rather than performative conversation — students aren't talking because the teacher asked them to, they're talking because they genuinely cannot proceed without what their partner knows. Zwiers is explicit about this: "students need or want to share their thoughts (which are not the same), students have a reason or purpose in talking and listening to each other." The wrong options all describe secondary effects, not the structural driver. This is also why card sort activities without genuine information asymmetry can feel hollow — the social motivation isn't there.
The information gap isn't primarily about vocabulary, cognitive load, or role equity — it's about authentic communicative need. The key word is "genuine": when students actually cannot solve the problem alone, conversation is no longer a teacher-imposed ritual. It's a real tool for accomplishing something they actually need to do. This is the design principle behind MLR4 and it applies to any activity that wants to generate real rather than performed mathematical discussion.
4. Zwiers et al.'s "Principle 4: Maximize Meta-Awareness" targets which type of awareness?
Correct. Zwiers explicitly addresses both dimensions as "powerful tools to help students self-regulate their academic learning and language acquisition." Metacognitive (thinking about your own thinking: "What strategy am I using?") and metalinguistic (thinking about your own language: "How does 'minus' here compare to 'negative' there?") reinforce each other — and both are strengthened by the same routines (MLR2, MLR3, MLR5, MLR6, MLR7, MLR8). This both/and framing is directly relevant to MASL: notation verbalization requires metalinguistic awareness specifically, which is the least-developed of Zwiers' meta- dimensions in most math classrooms.
Zwiers is explicit: Principle 4 targets both metacognitive (thinking about how you're solving math) and metalinguistic (thinking about how you're talking about math) awareness — and explicitly argues they are mutually reinforcing. Option D is about correctness, which is orthogonal to meta-awareness. The distinction matters for MASL: standard math instruction develops metacognitive awareness; MASL is specifically targeting the metalinguistic layer that notation verbalization requires.
5. Which of the following best describes the relationship between scaffolding and student independence in Zwiers' framework?
Correct. Zwiers' entire scaffolding stance is built on the premise that the goal is student independence — not supported performance, but internalized capacity. Scaffolds that don't fade become barriers to fluency (the Kalyuga expertise reversal effect, though Zwiers doesn't name it). The document distinguishes this from the "sink-or-swim" view (no scaffolding, language learners just absorb through immersion) and from the "permanent accommodation" view (scaffolding as a crutch). This is directly relevant to MASL's Language Frames fading plan: full example → partial → blank.
Zwiers is clear that scaffolding is temporary — that's the whole point. A scaffold in construction is something you remove when the building can stand on its own. Permanent scaffolding isn't scaffolding; it's a prosthetic. The goal is always independent participation. This framing is important for defending MASL's fading plan against the concern that sentence frames become a crutch rather than a launching pad.
6. The MLR framework was designed specifically for English learners. Which tension does its universalization (application to all students) create?
Correct. The document explicitly universalizes: "while the framework can and should be used to support all students learning mathematics, it is particularly well-suited to meet the needs of linguistically and culturally diverse students." The tension is that mainstreaming equity-driven practices can defang their equity argument — when everything becomes "good teaching for all," the specific claim about racialized inequity and the targeted argument about ELs gets absorbed into general pedagogy. The framework itself doesn't address this tension. This is an important systemic critique: universalization increases adoption but may dilute the equity commitments that motivated the design. This applies to MASL too.
The tension isn't about implementation difficulty or effectiveness — the routines work for everyone, which is actually the argument for universalization. The deeper problem is what universalization does to the equity argument: when EL-targeted practices become general "good teaching," the specific claim about who has been systematically excluded and why tends to disappear. The practice gets adopted; the critique of why it was necessary gets dropped. This is a structural pattern worth naming explicitly in any work that universalizes EL-specific research.
🃏 Card Sort — Match the Concepts
Drag each MLR or principle from the left column to its matching description on the right. Shuffle resets the cards.
MLRs & Principles
MLR1: Stronger & Clearer
MLR2: Collect & Display
MLR3: Critique, Correct & Clarify
MLR4: Information Gap
MLR5: Co-Craft Questions
MLR6: Three Reads
MLR7: Compare & Connect
MLR8: Discussion Supports
Principle 2: Optimize Output
Amplify, Don't Simplify
What It Does
Students iterate through multiple partners, strengthening reasoning and clarifying language with each pass before writing a final response
Teacher captures student informal words, phrases, and diagrams on a visible display that evolves as a shared reference throughout the unit
Students receive someone else's partial or flawed mathematical argument and improve it by identifying errors and ambiguous language
Partners hold different pieces of necessary information; one must ask for what the other has — and justify why they need it — creating real communicative purpose
Students receive a context or problem stem without the question and generate their own mathematical questions before the actual task is revealed
Students read a mathematical situation three times with different foci: comprehension, then mathematical structure, then possible solution methods
Multiple solution strategies or representations are displayed together; students identify what is similar, what is different, and where specific quantities appear across approaches
A toolkit of teacher moves — revoicing, pressing for detail, choral response, Numbered Heads Together — that amplify any of the other seven MLRs
The design principle targeting repeated, strategic, iterative opportunities for students to describe mathematical thinking orally, visually, and in writing
Provide multiple access pathways (visuals, analogies, modeling) into complex mathematical language — maintain the rigor, multiply the doors
Reflect
You opened with the image of rain percolating through limestone strata — filtered, routed, and emerging as a spring. Now that you've worked through the document: what does the spring represent in your classroom that you haven't been deliberately building toward? Where is the water running off instead of being absorbed?
MLRs are now embedded in the Illustrative Mathematics curriculum and used in thousands of classrooms nationwide. But Zwiers et al. provide no empirical data on whether any specific MLR produces language gains. The routines are well-designed and theoretically grounded — but "theoretically grounded and well-designed" is not the same as "demonstrated to work." What would it take to know whether MLR1 specifically produces measurably better mathematical discourse, as opposed to just producing more of it?
MASL adds a notation-language layer on top of the MLR framework. Here's the uncomfortable question: if students are already working through MLR6 (Three Reads) and MLR3 (Critique, Correct, Clarify) in every IM lesson, are they already getting notation language practice implicitly? What does MASL add that couldn't emerge naturally from sustained MLR use — or does it add something qualitatively different?
Zwiers' framework originated in EL-focused research and was explicitly designed to address inequity in mathematics education for linguistically diverse students. It is now used universally. When you observe an MLR activity in a mainstream math classroom with no ELs, do you think the equity argument travels with the practice — or does mainstreaming quietly strip away the critique that made the practice necessary in the first place?
Zwiers argues teachers should "amplify, not simplify" — maintaining rigor while multiplying access points. But in your actual experience of math classrooms, which direction does the pressure run? When a student is struggling with a word problem, what is the instinctive move — and how does it compare to what the framework prescribes?
The document says meta-awareness is "strengthened when... teachers ask students to explain to each other the strategies they brought to bear." This is metacognitive. But when does a math class ever ask students to reflect explicitly on how they named a symbol, or why they said "f of x" rather than "f times x"? That's metalinguistic. How often does metalinguistic awareness get developed in math classrooms, as opposed to metacognitive? And if almost never — what does that absence cost?