Reading and Understanding Written Math Problems¶

Strategy · All strategies
Evidence · 3 claims (2 for, 1 mixed) · 9 studies (3 review, 2 causal, 2 quant-synthesis, 1 associational, 1 theoretical), q1–q4 · 1 of 9 report an effect size

Description¶

Word problems require learners to read text, build a situation model of the quantities and relationships described, and only then select and execute a mathematical operation. This strategy treats the reading and comprehension phase as an explicit object of instruction: the teacher models translating words into pictures, diagrams, and number sentences, teaches the vocabulary and syntax of mathematical language, and has students verbalize and critique each other's interpretations. It is especially valuable for English language learners, for whom the linguistic demands of word problems can mask mathematical competence.

Design Implications¶

Word-problem failure is often a comprehension failure rather than an arithmetic failure: learners who build a proper situation model choose correct operations even when computation is harder, while keyword-based strategies (e.g., "altogether means add") produce systematic errors on inconsistent problems Hegarty et al. showed successful solvers build situation models rather than translate keywords directly into operations. [+S]. Instruction should therefore emphasize representing the problem structure — through drawings, bar models, or schemas — before any computation, and should have students explain their interpretations aloud, since verbalizing the translation from text to model strengthens conceptual understanding Self-explanation improves conceptual understanding. [+M].

Context¶

Requirements¶

• Explicit instruction in the vocabulary and syntax of mathematical language (comparative terms, relational phrases, implied quantities) Accessible Vocabulary & Syntax
• A modeling routine in which the teacher demonstrates the translation from words → picture/diagram → number sentence, making reasoning visible
• Hands-on or visual materials so learners can act out the situation before formalizing it
• Daily structured practice with a consistent comprehension routine (Practice)
• Repeated readings of the problem text to free attention for the mathematical structure

Constraints¶

• Keyword-matching instruction (teaching learners to map "shared," "total," "left" directly to operations) actively harms performance on problems where language and operation diverge [-S] — it substitutes a shallow translation heuristic for comprehension
• Heavy linguistic load can consume working memory entirely for ELLs, leaving no capacity for the mathematics; unmodified full-length problems can overload novices Chunking reduces working memory load. [~M]
• Requires sustained planning and materials; a one-off lesson does not build the schema library that makes the strategy effective
• If modeling dominates, learners may copy the teacher's representation without learning to construct their own; representations should be faded over time

Implementation Variability¶

• Schema-based instruction: classify problems into a small set of structures (change, group, compare, combine) and teach a representation template per schema
• Cognitively Guided Instruction: use knowledge of problem-type difficulty to sequence problems and build on learners' informal strategies
• Numberless word problems: present the problem first with quantities blanked, forcing attention on relationships before numbers
• Repeated reading plus retell: learners reread the problem and restate it in their own words before solving

Target Learners¶

• English language learners in math classrooms, who face compounded linguistic and mathematical demands [+M]
• Novices who rely on keyword strategies and fail on multi-step or inconsistent problems
• Struggling readers whose decoding effort competes with situation-model construction Automatic word recognition frees resources for comprehension. [+M]
• Less necessary for learners with strong mathematical reading fluency, for whom explicit translation routines may be redundant

Target Learning Goals¶

• Comprehension of mathematical text: extracting quantities, units, and relationships
• Schema formation: recognizing underlying problem structures independent of surface story
• Translation skill: moving flexibly among words, pictures, and number sentences
• Mathematical discourse: explaining and critiquing interpretations orally

Instructions¶

1. Pre-teach key vocabulary and relational language in context, not in isolation (Accessible Vocabulary & Syntax)
2. Read the problem aloud while students follow; reread and have students restate the situation in their own words before any math
3. Model the translation chain — words → picture or diagram → number sentence — thinking aloud at each step (Practice follows the model)
4. Have students construct their own representations, then explain their thinking to a partner and correct each other's interpretations
5. Provide daily brief problem-solving practice with hands-on materials, fading supports as interpretation becomes fluent
6. Use learners' enthusiasm and oral explanations as a bridge into reading fluency work with math texts

Related Strategies¶

• Activating Prior Knowledge — connecting the problem's story to familiar situations supports the situation model
• Annotating — marking quantities and relationships in the problem text is the individual form of this routine

Examples¶

• Cognitively Guided Instruction (CGI) — Teachers sequence addition/subtraction and multiplication/division problem types by difficulty and elicit children's informal solution strategies before formal notation (CGI for Math).
• Schema-based instruction in special education — Learners learn a four-step routine (read, paraphrase, represent with a diagram, solve) applied across problem sets sharing one structure.
• Singapore bar models — Students draw unit-bar diagrams to represent comparison and part-whole relationships before writing equations, making the text's relational language visible.

Key Sources¶

• Kintsch, W., & Greeno, J. G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92(1), 109–129. doi:10.1037/0033-295X.92.1.109
• Cummins, D. D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role of understanding in solving word problems. Cognitive and Instruction, 5(1), 1–38. doi:10.1016/0010-0285(88)90011-4
• Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87(1), 18–32. doi:10.1037/0022-0663.87.1.18
• Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's mathematics: Cognitively Guided Instruction. Heinemann.
• Jitendra, A. K., et al. (2007). A comparison of single and multiple strategy instruction on third-grade students' mathematical problem solving. Journal of Educational Psychology, 99(1), 115–127. doi:10.1037/0022-0663.99.1.115