MASL Research • Capstone 2026 • GradBot Deep Note

Academic Literacy in Mathematics for English Learners

Judit N. Moschkovich • Journal of Mathematical Behavior, 40 (2015) 43–62
📄 View Original Article (PDF)

Moschkovich, Judit N. "Academic Literacy in Mathematics for English Learners." Journal of Mathematical Behavior, vol. 40, Elsevier, Dec. 2015, pp. 43–62. DOI: 10.1016/j.jmathb.2015.01.005.

About the Original Article's Tone

This is a peer-reviewed theoretical framework paper published in the Journal of Mathematical Behavior — which means the intended audience is mathematics education researchers and graduate students who already fluent in sociocultural learning theory. It's not a practitioner guide. Moschkovich is talking to people who already know what Vygotsky, Lave and Wenger, and Gee mean when they show up in a bibliography.

It uses:

The vibe: This feels like a very well-organized conference keynote written down. Big framework, careful distinctions, one extended example to anchor everything. Think "respected scholar making the case for a paradigm shift" rather than "someone teaching you how to run a lesson."

What it glosses over: Moschkovich argues powerfully for the integrated ALM framework and against deficit views, but the paper stays almost entirely at the theoretical and descriptive level. There's no discussion of how you actually train teachers to see ALM in action, how you assess it without replicating the deficit move she critiques, or what happens when the demands of standards-based accountability run directly counter to the participation-based vision she describes. The classroom example is rich but was selected for illustrative purposes — she explicitly says it wasn't chosen for representativeness, which a careful reader should note.

Visual Metaphor

Imagine this:

A few centuries of undisturbed growth. Old-growth forest doesn't look like a plantation — there's no single row of trees doing one job. There are canopy giants, mid-story trees, fallen logs decomposing into soil, root networks laced through everything, fungi feeding nutrients between species, birds whose foraging creates the gaps that let seedlings reach light. Remove any one layer and the whole system shifts. The diversity is not decoration. It's the mechanism.

A student speaking mathematics in this ecosystem is not transmitting vocabulary. She's participating in the whole thing — the concepts, the practices, the discourse — all at once, even when she's using informal language, even when her English is imperfect. Precision lives in the network, not in any one word.

What This Is Really About

Here's the move schools make constantly with English Learners in math class: strip the language down. Vocabulary lists. Word walls. Sentence starters like "I know the answer is ___ because ___." The thinking goes: students need language before they can do math, so let's isolate the language piece and teach it first, then add the math back in. It seems logical. It's also, Moschkovich argues, fundamentally backwards — and actively harmful.

Moschkovich's Academic Literacy in Mathematics (ALM) framework says that language instruction and mathematics instruction are not separable. Strip the mathematical practices out and you've removed the very context that gives mathematical language its meaning. Strip the mathematical discourse out and you've cut students off from the social processes through which mathematical understanding is built. What you're left with isn't a simpler version of mathematics. It's a distorted version that teaches neither language nor mathematics well.

The ALM Framework: Three Intertwined Components

Moschkovich defines Academic Literacy in Mathematics as the integration of three components that cannot be separated in instruction or assessment without damage to all three:

1. Mathematical Proficiency — Moschkovich draws on Kilpatrick et al. (2001)'s five intertwined strands:

  1. Conceptual understanding — comprehension of mathematical concepts, operations, and relations; knowing what a result means and why a procedure works
  2. Procedural fluency — skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
  3. Strategic competence — ability to formulate, represent, and solve novel mathematical problems (not just routine exercises)
  4. Adaptive reasoning — logical thought, reflection, explanation, and justification
  5. Productive disposition — a habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with belief in one's own efficacy

Schools routinely collapse "mathematical proficiency" into only #2 — procedural fluency with arithmetic. Moschkovich argues all five strands are essential, and that ELs are especially harmed when instruction is reduced to computation drills.

2. Mathematical Practices — Moschkovich draws on NCTM Standards and the Common Core State Standards (CCSS) for Mathematical Practice, which describe eight practices that mathematics educators should develop in students:

  1. Make sense of problems and persevere in solving them
  2. Reason abstractly and quantitatively
  3. Construct viable arguments and critique the reasoning of others
  4. Model with mathematics
  5. Use appropriate tools strategically
  6. Attend to precision
  7. Look for and make use of structure
  8. Look for and express regularity in repeated reasoning

Mathematical practices are culturally organized, semiotic activities. They involve social membership — participating in the way mathematicians (and students in reform classrooms) work together. And critically: many of them are discursive. You can't practice "construct viable arguments and critique the reasoning of others" in silence.

3. Mathematical Discourse — This is the component most often reduced to "vocabulary" in school practice, and Moschkovich spends the most time correcting this reduction. Mathematical discourse is not principally about formal vocabulary. It involves:

The "I went by twos" classroom transcript is her proof of concept: two bilingual students and a teacher have an extended mathematical discussion using exclusively informal language. Nobody says "scale," "ratio," or "unitizing." But the discussion involves real conceptual content (unitizing — the cognitive assignment of units to segments on an axis), real mathematical practices (constructing arguments, attending to precision at the discourse level), and real mathematical discourse (coordinating utterances with views of inscriptions). Strip the vocabulary test on "scale" and you've measured nothing of what actually happened in this room.

The Vocabulary Trap: Why "Academic Language" Usually Gets It Wrong

Moschkovich makes a targeted distinction between "academic language" (the phrase most schools and curricula use) and "Academic Literacy in Mathematics." The problem with "academic language" in most implementations is that it reduces mathematical communication to words: teach students the formal terms, assume that fixes the language problem, move on. This approach has three specific failures:

Her example: CCSS Practice #6, "Attend to precision." Schools often implement this as "use the correct formal vocabulary." But precision in mathematics is a discourse-level phenomenon, not a word-level one. The claim "multiplication makes bigger" is imprecise not because the word "bigger" is informal, but because the claim doesn't specify when it applies. The precise version — "multiplication makes the result bigger, only when you multiply by a positive number greater than 1" — uses equally informal words. Precision lives in the claim structure, not in individual word choice. Requiring formal vocabulary in the name of precision is actually teaching students a wrong model of what mathematical precision means.

The Deficit Critique: What It Means to Focus on "Progress"

Moschkovich is consistently, carefully critical of deficit-oriented frameworks. A sociocultural perspective, she argues, shifts the focus "to the potential for progress in what learners say and do, not on learner deficiencies or misconceptions." This is not just a philosophical preference. It has direct instructional consequences.

When you frame ELs as language-deficient, you position their imperfect English as a problem that has to be fixed before real mathematical work can happen. But research (including Moschkovich's own longitudinal work) documents repeatedly that ELs can participate in complex mathematical discussions while still learning English — using gestures, objects, home language, informal phrases, all of which carry real mathematical content. A student who uses informal language to make a mathematically correct claim is not showing a deficiency. She is demonstrating exactly the situated, hybrid, multi-resource character that the ALM framework predicts as normal.

The practical implication: instruction for ELs should not lower cognitive demand in the name of language access. Reducing tasks to computation or vocabulary matching actually takes mathematics away from students who most need access to all five strands of proficiency.

What the Classroom Transcript Actually Shows

Carlos and David are eighth-grade bilingual students comparing two coordinate graphs they drew for homework. Their graphs look different even though they plotted the same data. The discussion: why? They're using the phrase "I went by twos / fives / ones" — a phrase native to this particular classroom — to describe the scale intervals on their axes. The teacher uses the same phrase but with a different meaning. Three participants, one phrase, three distinct situated meanings. The discussion lasts several minutes, involves mathematical argument, conceptual reasoning about unitizing, and negotiation of competing claims. Not one formal mathematical term appears until near the end, when the teacher says "two and a half."

This is Moschkovich's evidence that mathematical discourse is inherently hybrid and situated — drawing simultaneously on everyday register and academic register, coordinating utterances with focus on inscriptions (the actual graphs), and constructing meaning socially rather than retrieving it from a dictionary.

The Big Picture: ALM for All Students, Not Just ELs

Here's the move MASL inherits from this paper: Moschkovich argues that the ALM framework matters for all mathematics learners, but is essential for ELs. This matters for MASL's audience claim. If algebraic notation language is part of mathematical discourse (which ALM says it is), and mathematical discourse is part of mathematical literacy for everyone (which ALM says it is), then MASL isn't "ESL support for math class." It's mathematical literacy instruction that happens to be especially visible as a need for students who don't yet have the conventional spoken forms — which, as the Baseline Language Assessment demonstrates, includes most students, not just multilingual ones.

Key Vocabulary

Academic Literacy in Mathematics (ALM)
Simply: The whole package of what it means to really do mathematics — not just knowing facts or speaking correctly, but participating in the ways mathematicians think, talk, and make meaning together.
Moschkovich's proposed framework defining mathematical competence as the integrated, simultaneous engagement of three components: mathematical proficiency, mathematical practices, and mathematical discourse. The term "literacy" is chosen specifically to shift focus away from language-as-words toward participation in practices and discourses. Neither component functions properly when isolated from the other two.
CCSS (Common Core State Standards) for Mathematical Practice
Simply: The eight behaviors mathematically expert people actually do — like making sense of problems, constructing arguments, and attending to precision — as distinct from just knowing content.
The Common Core State Standards Initiative's list of eight practices describing "varieties of expertise that mathematics educators at all levels should develop in their students." Moschkovich uses these as one anchor for describing mathematical practices, noting that they overlap with NCTM Standards and the NRC's proficiency strands, and that they connect mathematics content to process in ways that foreground language and discourse.
Communicative Competence
Simply: Not just knowing the grammar rules of a language, but knowing when, how, and with whom to use it appropriately — which turns out to be most of what communication actually is.
Hymes' (1972) concept, used by Moschkovich to define mathematical discourse as "the communicative competence necessary and sufficient for competent participation in mathematical practices." Distinguishes between grammatical knowledge and the social, contextual knowledge required to actually communicate in a community — positioning mathematical discourse as a form of sociocultural competence, not merely linguistic fluency.
Deficit View
Simply: The framing that sees what students are missing rather than what they bring — leading to instruction that fills holes instead of building on existing capacity.
A perspective in educational research and practice that describes students — particularly ELs — primarily in terms of what they lack: missing vocabulary, insufficient English fluency, gaps in academic language. Moschkovich argues against deficit views and for a sociocultural alternative that focuses on "the potential for progress in what learners say and do." The deficit view has documented consequences: it leads to lower cognitive demand, reduced access to mathematical practices, and instruction that treats language remediation as prerequisite to mathematical learning.
English Learners (ELs)
Simply: Students who are still developing proficiency in English — though Moschkovich consistently argues the research on ELs is relevant to bilingual and multilingual learners more broadly.
Moschkovich's chosen label for students learning English, acknowledging that "there are many labels used to refer to students who are learning English." She notes that research on ELs may also be relevant to bilingual and multilingual mathematics learners, and to students learning a language of instruction other than English — a framing that explicitly extends her claims beyond a narrowly defined population.
Hybrid Resources
Simply: The full mix of everything students bring to a mathematical conversation — home language, informal phrases, gestures, drawings, formal terms, and everything in between — all used simultaneously as resources.
Mathematical discourse draws on hybrid resources — "multiple resources: modes of communication, symbol systems, as well as both everyday and academic registers" — rather than relying exclusively on formal academic language. Building on Gutiérrez et al.'s (1999, 2001) concept of hybridity, Moschkovich argues that the everyday register is not a stepping stone to be replaced by academic language, but a legitimate resource that functions alongside it in actual mathematical activity.
Mathematical Discourse
Simply: Everything involved in talking and thinking mathematically — not just knowing the vocabulary, but participating in the patterns of argument, justification, representation, and meaning-making that define mathematical communities.
Moschkovich's use of "discourse" (rather than "language") signals a shift from words to participation. Mathematical discourse includes communicative competence for mathematical practices; involves other symbolic systems and artifacts; is embedded in mathematical practices; and has meanings that are situated and develop through participation. Academic mathematical discourse is characterized by general modes of argument: precision, brevity, logical coherence, abstracting, and generalizing — but these are discourse-level properties, not word-level ones.
Mathematical Practices
Simply: The things mathematicians actually do — not the content they know, but the habits of mind, argument, and reasoning they use to do mathematics — and which students must be apprenticed into.
"Culturally organized" activities (Scribner, 1984) that involve symbol systems and are related conceptually to other mathematical activities. Mathematical practices are not purely cognitive — they are also social, cultural, and semiotic. They arise from communities and mark membership in communities. Both general practices (CCSS Standards) and practices specific to particular topics or concepts are included. Critically for ELs: many valued mathematical practices are inherently discursive, meaning language is constitutive of the practice, not just a vehicle for reporting on it.
Mathematical Proficiency
Simply: The full picture of what it means to be good at mathematics — five strands working together, not just fast and accurate computation.
Kilpatrick et al.'s (2001) framework of five intertwined strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Moschkovich argues that ELs are frequently given access only to procedural fluency (arithmetic computation) under the assumption that language barriers preclude the other four strands — a decision that both underestimates students and harms their mathematical development.
Situated Meanings
Simply: Words and phrases mean what they mean because of context — the specific classroom history, the artifact being looked at, the moment in the conversation — not because a dictionary says so.
Meanings for mathematical utterances are "situated locally in the ecology of each classroom, in the history and interactions that precede a discussion." The classroom transcript illustrates this: "I went by twos" has three different situated meanings for three participants, none of which are accessible from the words alone without knowing what inscription each participant is looking at and how they're viewing it. Situated meanings are negotiated through interaction, coordinated with focus of attention on inscriptions, and may shift for the same individual at different times.
Unitizing
Simply: Deciding what "one chunk" is — whether you're counting individual cans or six-packs, individual grid squares or labeled intervals — and then consistently treating that chunk as your unit.
Lamon's (1994, 1996) concept: "the cognitive assignment of a unit of measurement to a given quantity; it refers to the size chunk one constructs in terms of which to think about a given commodity." In the classroom transcript, the teacher's explanation of the two different scales requires students to recognize that "two and a half" is the actual unit value of each grid segment on Carlos's graph — a conceptual demand that the informal phrase "I went by fives" obscures. Moschkovich uses unitizing to show that seemingly procedural discussions (about axis labels) can involve genuine conceptual understanding.

🎯 MASL Connection

This Study Supports:

Design Implications:

Evidence Strength for MASL:

Strong for theoretical framing; limited for effect-size claims. Moschkovich's evidence base is observational and case-study based — the paper analyzes one 90-second classroom transcript selected for illustrative purposes, explicitly not for representativeness. This is appropriate for theoretical framework development, but cannot support claims about effect sizes or efficacy. The ALM framework is widely cited, deeply influential, and published in a peer-reviewed mathematics education journal (Journal of Mathematical Behavior). It provides the strongest available theoretical grounding for MASL's core claim that notation-language instruction is mathematics instruction. However, for empirical support of MASL's design choices, Ke & Newton (2024), Barbieri et al. (2023), and Mercer & Sams (2006) carry the quantitative weight that Moschkovich cannot.

Connections to MASL Framework (click to expand)
  • MASL Trio (Math / We Say / Meaning cards): The three-card structure maps directly onto ALM's three components: Math card = mathematical practices and proficiency; We Say card = mathematical discourse; Meaning card = conceptual understanding (the meaning-making function of mathematical discourse). The inseparability of the three cards in the sort mirrors ALM's inseparability of the three components in mathematical activity.
  • Sentence frames: ALM supports frames that structure participation in mathematical discourse practices — justifying, explaining, attending to precision at the discourse level. It would critique frames that simply substitute formal vocabulary for informal vocabulary without connecting to practices. MASL's frames ("because the negative sign stays with ___") are warranted; rote definition frames would not be.
  • Irregular forms instruction: ALM's emphasis on the situated and hybrid nature of mathematical meanings supports MASL's "Mathematical Language Irregulars" concept — these are notation conventions that cannot be inferred from general rules, that exist as community norms, and that must be explicitly taught as discourse practices (not vocabulary). "x squared" vs. "x to the second power" is a discourse-level convention, not a word-level fact.
  • Scaffolding fading: ALM's participation framework is compatible with Renkl's fading principle: scaffolded participation is appropriate at early stages, but full scaffolding permanently creates dependency rather than genuine membership in mathematical discourse communities. ALM would support the fading plan as moving toward genuine participation rather than sustained scaffolded approximation.
  • Cross-connections: Sfard (2007/2008) commognition framework is the closest theoretical relative — both argue thinking IS communicating in mathematics. Arcavi (1994) symbol sense provides the specific cognitive-linguistic construct that ALM grounds at the sociocultural level. Zwiers et al. (2017) Math Language Routines operationalize ALM principles into structured classroom activities. Grapin (2023) from TESL_5621 extends the equity framing to ask whether access-oriented interventions (like MASL) can exist simultaneously with equity-transformation goals.

💬 Key Quotes

Copy-paste ready quotes for papers, discussions, and reflections.

"The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse."
p. 43 Thesis
Why this quote: The foundational statement of the ALM framework — the one to cite when introducing the framework in any MASL context.
"Separating language from mathematical thinking and practices can have dire consequences for English Learners. First, such a separation can make ELs seem more deficient than they might actually be, since they may not be able to express their mathematical ideas through language, but may still be engaged in correct mathematical thinking."
p. 44 Challenge
Why this quote: The strongest statement of the stakes — and the diagnostic problem the Baseline Language Assessment must navigate. Students can do mathematics correctly while lacking the conventional spoken forms. The BLA must measure the discourse gap without mislabeling the mathematical competence.
"Mathematical discourse is more than language — it involves other symbolic systems as well as artifacts, discourse is embedded in mathematical practices, and meanings are situated and develop through participation in mathematical practices."
p. 48 Definition
Why this quote: The clearest statement of what ALM means by "discourse" vs. everyday uses of "language" — essential for distinguishing MASL from vocabulary instruction.
"The sociocultural perspective used here expands academic literacy in mathematics beyond simplified views of language as words. Simplified views of academic language focus on words, assume that meanings are static and given by definitions, separate language from mathematical knowledge and practices, and limit mathematical discourse to formal language."
p. 44 Foundational
Why this quote: A four-part critique of the "vocabulary approach" that MASL needs to explicitly differentiate itself from — this is the thing MASL is not.
"What makes a discussion mathematical is not the use of formal mathematical words, but mathematical concepts, which can sometimes be expressed using informal words and phrases, and mathematical practices, such as justifying a claim, which are not at the word level."
p. 56 Provocative Claim
Why this quote: Directly unsettles the word-wall and vocabulary-first approach. Mathematical quality lives in the practices and concepts, not in the lexicon.
"Instruction should not emphasize low-level language skills but, instead, provide students opportunities to actively communicate about their mathematical solutions, ideas, and reasoning."
p. 58 Practical
Why this quote: The core instructional recommendation — directly supports MASL's insistence that notation language work must be embedded in mathematical reasoning, not treated as a standalone language drill.
"Although these three components are important for all mathematics learners, it is essential that mathematics instruction for ELs include and maintain a simultaneous focus on all three components."
p. 43 Foundational
Why this quote: The "all students, but essential for ELs" framing that grounds MASL's universal-plus-targeted audience claim — notation language instruction matters for everyone and is critical for multilingual learners.

📚 References & Further Reading

📚 References & Further Reading
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press.
Foundational

What it is: The National Research Council's landmark synthesis of mathematics education research, introducing the five-strand model of mathematical proficiency. Tone: Dense but accessible committee report. Why it matters: The five-strand framework is the backbone of Moschkovich's Component 1; if you cite ALM, someone will ask you to explain the proficiency model. Buzz: Thousands of citations; standard reference in every math ed paper since 2001. Verdict: Skim the five-strand chapter (Ch. 4) — it's exactly what Moschkovich describes and worth having read directly.

Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. National Governors Association.
Foundational

What it is: The CCSS-M document, including the eight Standards for Mathematical Practice that Moschkovich cites as her source for mathematical practices. Tone: Policy document — flat and list-based. Why it matters: If you're working in any US secondary school context, these eight practices frame what "doing mathematics" officially means. Buzz: The most-cited K-12 curriculum document in US education since 2010. Verdict: Read the eight practices (first 2 pages of the Standards for Mathematical Practice section) and no more. The rest is content standards.

Gee, J. P. (1999). An Introduction to Discourse Analysis: Theory and Method. Routledge.
Advanced

What it is: Gee's foundational text on Discourse analysis (capital D), from which Moschkovich draws her analysis questions and the broader definition of Discourse as involving not just language but images, equations, gestures, and social positions. Tone: Theoretical, accessible for linguists but dense for math educators. Why it matters: Provides the intellectual framework that lets Moschkovich call oral discussion a "literacy practice." Buzz: Highly cited across literacy studies, education research, and applied linguistics. Verdict: For researchers only unless you want to understand why Moschkovich capitalizes "Discourse."

Halliday, M. A. K. (1978). Sociolinguistic aspects of mathematical education. In M. Halliday (Ed.), The Social Interpretation of Language and Meaning (pp. 194–204). University Park Press.
Foundational

What it is: Halliday's description of the "mathematics register" — the set of meanings, structures, and vocabulary specific to mathematics as a discourse community. Tone: Dense sociolinguistics. Why it matters: The foundational text for understanding why mathematical language is treated as a register rather than just vocabulary — directly relevant to MASL's claim about algebraic notation as a register with its own conventions. Buzz: Foundational; cited in nearly every paper on mathematical language. Verdict: Worth finding and reading the relevant pages — it's short and gives you the intellectual lineage of the "mathematics register" concept.

Lave, J., & Wenger, E. (1991). Situated Learning: Legitimate Peripheral Participation. Cambridge University Press.
Foundational

What it is: The book that introduced "communities of practice" and "legitimate peripheral participation" to education research — the theoretical backbone of participation-based views of learning. Tone: Accessible theoretical text. Why it matters: Moschkovich's use of "participation in practices" as the unit of learning is grounded here. Buzz: One of the most-cited education research texts of the last 40 years. Verdict: Worth reading Ch. 1-2 if you want to understand why sociocultural theorists frame learning as participation rather than acquisition.

Moschkovich, J. N. (2008). "I went by twos, he went by one": Multiple interpretations of inscriptions as resources for mathematical discussions. Journal of the Learning Sciences, 17(4), 551–587.
Must Read

What it is: The earlier, longer analysis of the same classroom transcript analyzed in this 2015 paper — the "I went by twos" discussion. Tone: Accessible research article with full transcript. Why it matters: If you want to understand the transcript analysis in depth — including the full analysis of each participant's meanings and how the discussion resolved — this is the source. The 2015 paper only has space for excerpts. Buzz: Widely cited in mathematics education and language-in-mathematics research. Verdict: Read it alongside this paper — it fills in everything the 2015 version abbreviates.

Sfard, A. (2008). Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing. Cambridge University Press.
Must Read

What it is: Sfard's commognition framework — the argument that mathematical thinking and mathematical discourse are not separate phenomena but two aspects of the same thing. Tone: Demanding but rewarding theoretical text. Why it matters: The closest theoretical relative to ALM; where Moschkovich grounds ALM in community of practice theory, Sfard grounds the same inseparability thesis in cognitive science. MASL needs both. Buzz: Highly cited and influential in mathematics education and learning sciences. Verdict: Essential for MASL — read at least the introduction and Chapter 1.

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. Stanford University.
Must Read

What it is: The Math Language Routines (MLR) framework — eight structured routines for integrating language and content development in mathematics, used in the Illustrative Mathematics curriculum. Tone: Practitioner-accessible framework document. Why it matters: This is the practitioner operationalization of what Moschkovich describes theoretically — ALM principles translated into classroom structures. MASL activities live in the same intellectual ecosystem. Buzz: Widely adopted by math curriculum developers after IM curriculum adoption. Verdict: Required reading for MASL — this is your connection to classroom implementation.

🧐 Test Your Understanding

Six conceptual questions — test your understanding of ALM, not your memorization of definitions.

1. Moschkovich argues that separating "academic language" from mathematical practices and proficiency is dangerous. What is the most direct consequence she identifies for English Learners?

2. In the "I went by twos" transcript, Carlos, David, and the teacher are all using the same phrase with different meanings. What does this illustrate about mathematical discourse?

3. Moschkovich says "Attending to Precision" (CCSS Practice #6) should not be interpreted as "using the perfect word." What does she argue precision actually means in mathematical practice?

4. Moschkovich argues that focusing instruction for ELs on vocabulary is counterproductive. Which of these correctly captures her argument about what vocabulary-focused instruction actually does to student learning?

5. For MASL's Baseline Language Assessment, ALM raises a tension: Moschkovich argues against deficit views, but a diagnostic instrument necessarily measures what students can't do yet. How does ALM's framework actually help resolve this tension?

6. ALM was developed primarily through research with bilingual and multilingual students. What is the most important question to ask before extending ALM's claims universally — to all students regardless of language background?

🤔 Match the Concepts

Drag each term or framework element to its correct description. Cards are shuffled on load.

Terms & Concepts

Mathematical Proficiency (5 strands)
Mathematical Practices
Mathematical Discourse
Situated Meanings
Hybrid Resources
Attending to Precision (CCSS #6)
Deficit View
Unitizing
ALM vs. "Academic Language"
Vocabulary-Focused Instruction

What It Means / Does

Conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition — five strands that must all be developed together
Culturally organized, semiotic activities — ways of reasoning, arguing, and symbolizing that are social and discursive, not just cognitive
Communicative competence for mathematical participation; involves multiple modes, symbol systems, and registers; meanings are constructed through use, not given by definitions
Meanings grounded in the local history of a classroom, the artifact being examined, and the moment of interaction — not fixed in words themselves
Drawing simultaneously on everyday register, home language, formal vocabulary, gestures, and objects — the multi-resource character of real mathematical communication
A discourse-level property of mathematical claims, not a word-level property — means specifying when a claim applies and when it doesn't, not using formal vocabulary
Framing students by what they lack rather than what they bring; causes ELs' mathematical competence to be underestimated by treating discourse gaps as mathematical incapacity
The cognitive assignment of a unit to a given quantity — deciding the "chunk size" for counting or measuring, as in labeling tick marks on a graph axis
ALM includes practices and discourse as inseparable from proficiency; "academic language" reduces literacy to vocabulary or grammar features divorced from mathematical activity
Limits student access to all five strands of mathematical proficiency by removing the practices and concepts that give mathematical language its meaning

Reflect

  1. You opened with the image of an old-growth forest — a system where the complexity is the mechanism, not decoration. Now that you've worked through ALM: what does "removing one layer" actually look like in a mathematics classroom? What layer gets removed first, and what does the system look like after it's gone?
  2. Moschkovich argues against deficit views — but MASL's Baseline Language Assessment literally measures what students can't say yet. Write out the full response to someone who says "the BLA is just a deficit-framing tool with better PR." What does ALM actually let you say back, and where does the argument get genuinely hard?
  3. ALM says discourse is mathematics — participating in the discourse communities of mathematics is not how you report on mathematical thinking, it's how mathematical thinking happens. So: is a student who solves every algebra problem correctly but never speaks, writes, or gestures during class actually doing mathematics, by Moschkovich's definition? If not, what are they doing? And does your answer change how you should teach them?
  4. Moschkovich chose the "I went by twos" example specifically because it illustrates ALM in action — but she also explicitly says she didn't choose it for representativeness. Bilingual eighth-graders in a reform classroom with 90 minutes per session and a teacher who allows open discussion: how many algebra classrooms in the US look like this? What happens to the ALM argument in a 45-minute period with 32 students, no collaborative norms established, and a pacing guide due next week?
  5. ALM was developed from research with bilingual and multilingual students in mathematics classrooms. When Moschkovich argues it applies to all students, she's making a generalization move that is strategically useful for MASL (everyone needs notation-language instruction) but potentially politically risky. Name one thing that the universal framing makes more possible, and one thing it makes less visible.
  6. Moschkovich says that focusing on formal vocabulary alone "limits the linguistic resources teachers can use to teach mathematics." Think about that from the teacher side: if you've been taught to manage discussions by correcting informal language into formal vocabulary, what does ALM ask you to unlearn? Is that realistic to ask of a first-year teacher managing a classroom full of students with varying language profiles?