Teaching Math to Multilingual Learners
Overview
Multilingual learners (MLs) — also called English Language Learners (ELLs) or Emergent Bilinguals — face a compounded challenge in math classrooms: they must simultaneously acquire English and learn mathematics, two cognitively demanding tasks. Research consistently shows that language barriers, not mathematical ability, account for a significant share of ML performance gaps on standardized assessments.
MLs use more than one language to make meaning of mathematics. Teachers should position multilingualism as an asset — not a deficit — and design tasks that leverage bilingualism as a tool to negotiate mathematical meaning.
Simplify language, not content. The goal is to change the language barrier, not the mathematics. Word problems should be rewritten for linguistic clarity while maintaining cognitive demand.
Language Proficiency: BICS vs. CALP
A critical distinction underlies almost all ML math instruction research:
| BICS | CALP | |
|---|---|---|
| Full name | Basic Interpersonal Communication Skills | Cognitive Academic Language Proficiency |
| What it is | Everyday, informal, context-rich communication | Abstract, decontextualized academic language |
| Time to acquire | 1–2 years | 4–7 years |
| Math relevance | Classroom conversation, peer discussion | Word problems, proofs, formal explanations |
1. Language as Barrier vs. Language as Asset
The dominant research consensus has shifted away from treating ML students' home languages as deficits to overcome, toward treating multilingualism as a cognitive and mathematical asset.
Second language learners' performance in mathematics is significantly affected by academic language features, independent of their mathematical knowledge. Language barriers account for performance gaps even when underlying math understanding is equal.
Never reduce the cognitive demand of a task. Provide language scaffolds instead. Lowering mathematical expectations because of language proficiency conflates two separate skills and denies students access to grade-level mathematics.
Key instructional implications
- Maintain rigorous, standards-aligned tasks — add time and language support, not simpler math
- Allow multiple modes of demonstrating understanding: verbal, written, drawn, gestured
- Create classroom norms that expect and value mathematical communication from all students
- Provide additional instruction time — not remedial content, but linguistic scaffolding around the same content
2. The Word Problem Problem
Word problems are the primary site of language-based difficulty in math for MLs. Research identifies several specific linguistic features that compound difficulty for students still developing academic English.
Passive Voice
Passive voice obscures agency — the reader cannot easily determine who performs the action. For an ML student parsing "The triangle was divided into three parts", the math question and the actor are both hidden behind the grammatical construction. Active voice makes the mathematical action explicit.
| ❌ Passive — avoid | ✅ Active — prefer |
|---|---|
| "The rectangle was divided equally." | "Divide the rectangle into equal parts." |
| "A value was found by multiplying." | "Multiply to find the value." |
| "How many times has the base been used?" | "How many times does the base repeat?" |
| "The equation is solved by isolating x." | "Isolate x to solve the equation." |
| "The answer can be found below." | "Find the answer below." |
Words with Dual Meanings (Polysemy)
Many common English words carry different mathematical meanings than their everyday meanings. For MLs, the everyday meaning is typically learned first — causing confusion when the same word means something different in math.
math: average of a set
math: divisibility by 2
math: 3-dimensional space
math: organized data
math: 90° angle
math: also n²
math: angle or temperature unit
math: initial investment amount
math: charge on a loan
math: non-terminating, non-repeating decimal
math: measuring tool
math: fraction > 1
Sentence Complexity
Greater linguistic complexity increases the difficulty of math items for ELLs compared to non-ELLs of equivalent math proficiency. The linguistic demands of a word problem function as a second test — of language — layered on top of the mathematical test.
Specific structures that compound difficulty for MLs:
- Subordinate clauses — "If the train leaves at 3, and the distance is 200 miles, which is greater than..." creates multiple parsing tasks simultaneously
- Embedded phrases — "The value, which represents the total cost of 4 items each priced at $3, is..." buries the mathematical relationship inside a noun phrase
- Modal verbs — "should", "would", "could", "might" create ambiguity about whether action is required or hypothetical
- Conditional structures — "If... then..." requires understanding of hypothetical framing that depends on cultural/linguistic familiarity
- Vague pronouns — "it", "they", "this" without clear antecedents force students to hold and track multiple referents
- Dense noun phrases — multiple modifiers stacked before a noun: "the first positive even integer less than 10" requires parsing 5 constraints
3. Vocabulary Instruction in Mathematics
Vocabulary instruction for MLs in math must address two categories of words: math-specific academic terms (exponent, coefficient, theorem) and everyday words that appear in word problems with potentially unfamiliar meanings (purchase, per, equally, remaining).
Core strategies
- Pre-teach before the problem — introduce critical vocabulary before students encounter it in a task, not after confusion arises
- Concept first, word second — establish the mathematical concept through visuals or manipulatives, then attach the vocabulary label to something already understood
- Pair academic terms with plain-language definitions inline — "Write the exponent — the small raised number — in the box"
- Visible word banks — maintain a class word wall or reference sheet with current-unit vocabulary; anchor discussion to it
- Address polysemous words explicitly — flag when a familiar word carries a mathematical meaning different from its everyday meaning
Cognates (especially for Spanish speakers)
30–40% of English words have a related Spanish word (cognate). Direct instruction in recognizing cognates can accelerate vocabulary acquisition — but research shows that simply having cognates available is not enough. Students need explicit instruction in recognizing and using them.
| English | Spanish cognate | English | Spanish cognate |
|---|---|---|---|
| Fraction | Fracción | Algebra | Álgebra |
| Triangle | Triángulo | Operation | Operación |
| Rectangle | Rectángulo | Multiplication | Multiplicación |
| Circle | Círculo | Division | División |
| Geometry | Geometría | Equation | Ecuación |
4. Visual and Multimodal Approaches
Visual representations reduce dependence on language for communicating mathematical structure. Research from Cisco found that students given text + visuals learn better than students given text alone — and this effect is amplified for language learners.
Research-supported visual supports
- Ratio tables and coordinate planes — make proportional relationships visible without requiring language to parse
- Fraction models (area models, number lines) — give concrete referents for abstract fraction vocabulary
- Algebra tiles and base-ten blocks — make place value and algebraic structure physically manipulable
- Anchor charts / word walls — persistent visual vocabulary reference during instruction
- Color coding — consistent semantic color assignment across activities reduces the cognitive load of parsing text
- Gestures and demonstrations — convey the same information as words through a non-linguistic channel
- Think-alouds — making teacher reasoning visible models how to process a problem, not just what answer to produce
5. Mathematical Notation as a Second Language
For students learning English as a second language, mathematical notation functions as a third language. The language of mathematics is not English — it is a specialized system with its own grammar, vocabulary, and conventions that even native English speakers must explicitly learn.
Mathematical language consists of two layers: (1) a substrate of natural language using technical terms and conventions peculiar to mathematical discourse, and (2) a highly specialized symbolic notation system. MLs must acquire both simultaneously while also acquiring English.
Notation conventions vary across countries
Many ML students arrive with prior math education in a different country where notation conventions differ. These differences create confusion independent of language:
- Decimal separators: 3.14 (US/UK) vs. 3,14 (many European and Latin American countries)
- Thousands separators: 1,000,000 (US) vs. 1.000.000 (many others) vs. 1 000 000 (spaces, ISO standard)
- Division symbol: ÷ (US/UK) vs. : (many European countries)
- Measurement systems: Imperial (US) vs. metric (most of the world)
- Long division layout: visual format varies significantly by country
6. Mathematical Language Routines (MLRs)
Stanford's Understanding Language team developed eight Mathematical Language Routines — structured, adaptable formats for amplifying, assessing, and developing students' language in math class. They are grounded in the interdependence of language learning and content learning.
The 8 Mathematical Language Routines
Students draft, share, get feedback, and revise their mathematical explanation through multiple iterations.
Teacher captures students' language during discussion and displays it publicly, making emerging language patterns visible for the class.
Students evaluate and improve mathematical statements that contain errors, ambiguity, or incomplete reasoning.
Students must ask questions to get missing information needed to solve a problem, creating authentic communication necessity.
Students develop their own inquiry questions about a mathematical context before solving, building investment and language simultaneously.
Close-reading protocol for word problems — three structured reads with distinct language goals each time. (See below.)
Students analyze relationships between different mathematical representations, requiring precise language to describe similarities and differences.
Structured conversation protocols (think time, partner talk, sentence frames) that scaffold full participation in whole-class mathematical discourse.
The Three-Read Protocol (MLR 6 in detail)
The Three-Read Protocol directly addresses the compounded challenges of word problems for MLs. Each read has a distinct, bounded language goal:
"What is happening in this problem?"
Read aloud; use choral reading. Build context understanding without any math processing. Use visuals and graphic organizers for key vocabulary. The goal is to ensure every student understands the situation before encountering any numbers.
"What quantities are present, and what do they represent?"
Identify all numbers and their mathematical meaning. What is being counted, measured, or compared? Students work with peers; sentence stems support peer discussion. No solving yet.
"What do I need to find, and how will I approach it?"
Students formulate their mathematical approach and discuss with partners before solving. This separates understanding from execution.
Sentence Frames and Sentence Stems
Sentence frames and stems scaffold academic language production — they model correct syntax and content-area vocabulary while giving students a supported entry point into mathematical discourse.
Language scaffolds are beneficial for all learners, with amplified benefits for multilingual learners and students with language-based learning differences. Sentence frames do not reduce rigor — they reduce the language production barrier so cognitive effort can focus on mathematics.
Example frames for math discourse
| Purpose | Frame / stem |
|---|---|
| Explaining a strategy | "The strategy I used was ___ because ___" |
| Agreeing with elaboration | "I agree with ___ and I want to add ___" |
| Disagreeing with reason | "I disagree because ___" |
| Making a claim | "My answer is ___ because ___" |
| Asking for clarification | "Can you explain why you ___?" |
| Connecting representations | "This method is similar to ___ because ___" |
| Restating a problem | "The problem asks me to ___" |
| Describing a pattern | "I notice that when ___, then ___" |
7. Translanguaging
Translanguaging recognizes that multilingual people do not switch cleanly between separate language systems — they draw from a single integrated linguistic repertoire that includes features of all languages they know. Treating this as a classroom asset (rather than suppressing home language use) has measurable benefits for math learning.
Students use their home language to make better sense of course content, leading to better performance without negative effects on English acquisition. (Celic & Seltzer, 2013, as cited in CPM Educational Program)
Practical translanguaging strategies
- Ask students to share the name of mathematical concepts in their home language — builds cognate awareness and validates home language knowledge
- Place students in same-home-language groups for initial problem discussion, then bridge to English for whole-class sharing
- Allow journal entries and scratch work in any language
- Use bilingual word walls where possible
- Have students explain a concept in their home language first, then in English — explanation in L1 consolidates understanding; English translation is then a language task, not a math task
8. WIDA Framework for Mathematics
WIDA's ELD Standards Framework provides language expectations specifically for mathematics — pointing to how students use language to meet grade-level content standards across six proficiency levels.
Three dimensions of language (WIDA)
| Dimension | What it covers | Math example |
|---|---|---|
| Word / phrase | Individual vocabulary and short expressions | "exponent", "to the power of", "squared" |
| Sentence | Grammar, syntax, sentence structure | "Multiply the base by itself n times" vs. "The base is multiplied n times" |
| Discourse | Extended communication — paragraphs, explanations, arguments | Explaining a multi-step solution strategy verbally or in writing |