Visual Representations in Mathematics¶

Strategy · All strategies
Evidence · 4 claims (3 for, 1 mixed) · 15 studies (5 causal, 5 review, 3 quant-synthesis, 1 associational, 1 theoretical), q1–q4 · 2 of 15 report an effect size

Description¶

This strategy teaches learners to construct and interpret visual representations — schematic diagrams, number lines, strip/bar models, graphs, and graphic organizers — that accurately depict the quantities and relationships in a mathematics problem or concept. The visual serves as an intermediate bridge between the concrete situation and abstract symbols, externalizing problem structure so it can be inspected, discussed, and revised. Critically, the benefit comes from schematic representations that map mathematical relationships, not from decorative pictures.

Design Implications¶

Visual representations reduce working memory demands by holding intermediate quantities and relationships externally Chunking reduces working memory load. [+M], and schematic diagrams in particular predict stronger problem-solving performance than pictorial ones Hegarty & Kozhevnikov found schematic (not pictorial) representation use correlates with success. [+M]. The strategy aligns with Dual Coding Theory: pairing verbal and visual codes creates two retrieval routes. However, visuals are not automatically helpful — learners must be taught how to construct them, and poorly matched visuals can add load rather than reduce it.

Context¶

Requirements¶

• Explicit instruction in specific representation types (e.g., strip diagrams for part–whole relations, number lines for magnitude) with teacher modeling of how to build them
• Practice constructing representations, not just interpreting pre-drawn ones
• Attention to accuracy: a diagram that misrepresents the relationships produces wrong solutions with high confidence
• Prompts for self-explanation [+M] connecting the visual to the symbolic expression

Constraints¶

• Learners with weak spatial skills or poor diagram-construction skills may produce disorganized or inaccurate visuals that harm performance [-M]
• Decorative or photorealistic images that do not map mathematical structure can distract and depress learning, consistent with coherence findings in Cognitive Load Theory [-M]
• For learners with high prior knowledge, forced use of diagrams can be redundant and slow problem solving, consistent with the expertise reversal effect [~M]
• Over-reliance on one representation type can block flexibility; learners need to know when a diagram helps, not just how to draw one [-W]

Implementation Variability¶

• Schema-based instruction: teach a small set of problem-type diagrams (e.g., part–whole, compare, change) and match problems to schemas — the approach with the strongest evidence base for word problems
• Concrete–pictorial–abstract (CPA): sequence from manipulatives to drawings to symbols, as in Singapore Math
• Student-generated vs. provided visuals: provided diagrams scaffold interpretation; generated diagrams build deeper processing but require more support
• Multiple contrasting representations: showing the same relationship as a strip diagram, number line, and equation supports abstraction Multiple contrasting cases support abstraction. [+M]

Target Learners¶

• Students with mathematics learning disabilities or difficulties, who benefit most from explicit representation instruction van Garderen & Montague found students with LD produce less effective representations without instruction. [+M]
• Novices encountering a new concept or problem type, for whom the visual externalizes structure
• Less beneficial — and sometimes counterproductive — for advanced learners who already reason symbolically [~M]

Target Learning Goals¶

• Conceptual understanding: making quantities and relationships visible (fractions as magnitudes, ratios as strip comparisons)
• Word-problem solving: translating text into a solvable structure
• Concrete-to-abstract transition: linking manipulatives, diagrams, and symbols
• Mathematical communication: using diagrams to explain and justify solutions

Instructions¶

1. Select a representation type matched to the target structure (number line for magnitude, strip diagram for part–whole, graph for covariation).
2. Model construction with Think-Aloud reasoning, showing how quantities are extracted from the problem text.
3. Have students construct their own representations with Practice on a sequence of problems, fading teacher support as accuracy grows (Scaffolding).
4. Prompt students to explain how the diagram maps to the equation, leveraging Self-Explanation [+M].
5. Compare multiple representations of the same problem to build flexibility and support abstraction Multiple contrasting cases support abstraction. [+M].

Related Strategies¶

• Worked Examples — worked examples can embed completed diagrams, letting learners study expert representation choices
• Concrete-Representational-Abstract — the broader sequencing framework this strategy fits within
• Analogies — a diagram functions as a visual analogy between an unfamiliar problem and a known structure
• Külppe's recommendations: formal exercises in thinking, more abstract thinking, and practice strengthening determining tendencies

Examples¶

• Singapore Math / Math in Focus — strip (bar) models are the central representation for word problems across grades K–8; students draw bars to represent known and unknown quantities before writing equations.
• Khan Academy — fraction and arithmetic exercises use interactive number lines and area models that students manipulate before symbolic work.
• Schema-Based Instruction (Jitendra) — published intervention programs teach students to classify word problems into schema types and complete the corresponding diagram; see the IES Practice Guide on RTI in mathematics for recommendations.
• Desmos — graphing as a dynamic visual representation linking equations to covarying quantities in algebra classrooms.

Key Sources¶

• Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689. doi:10.1037/0022-0663.91.4.684
• van Garderen, D., & Montague, M. (2003). Visual–spatial representation, mathematical problem solving, and students of varying abilities. Learning Disability Quarterly, 26(4), 246–262.
• Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060). National Center for Education Evaluation and Regional Assistance. https://ies.ed.gov/ncee/wwc/PracticeGuide/2
• Booth, R. D. L., & Thomas, M. O. J. (2000). Visualization in mathematics learning: Arithmetic problem-solving and student difficulties. Journal of Mathematical Behavior, 18(2), 169–190.
• Paivio, A. (1990). Mental representations: A dual coding approach. Oxford University Press. doi:10.1093/acprof:oso/9780195066661.001.0001