Concrete-Representational-Abstract (CRA)¶

Strategy · All strategies
Evidence · 7 claims (4 for, 2 mixed, 1 against) · 23 studies (10 causal, 7 review, 5 quant-synthesis, 1 theoretical), q1–q4 · 5 of 23 report an effect size · 1 claim rests on one study

Description¶

CRA is an instructional approach that uses concrete materials, visual representations, and abstract symbols to teach math concepts. Learners first manipulate concrete objects to understand a concept, then draw pictures representing the objects, and finally use abstract numbers and symbols. As instructors move through the sequence, abstract symbols should be used in conjunction with concrete materials and drawings — not after them — to promote association between symbolic notation and the underlying quantities. The approach descends from Bruner's enactive–iconic–symbolic sequence and is a staple of special education mathematics instruction.

Design Implications¶

CRA manages intrinsic cognitive load by giving abstract notation a referent learners can see and touch, reducing the risk that symbols are manipulated procedurally without meaning Overloaded working memory degrades learning outcomes. [+M]. Pairing representations with symbols also exploits dual channels of processing Dual coding of verbal and visual information improves recall. [+M]. However, concrete materials are not automatically helpful: poorly chosen manipulatives can distract from the target structure rather than illuminate it Manipulatives support learning only when they make the target structure salient. [~M].

Context¶

Requirements¶

• Concrete materials matched to the concept (e.g., base ten blocks for place value, fraction tiles for equivalence), drawing materials, and explicit teaching methods
• Instructor modeling and description at each level, with ample practice and feedback before transitioning
• Deliberate overlap between levels: abstract symbols introduced alongside, not after, concrete and pictorial work
• A mastery criterion at each level before fading to the next
• Manipulatives or concrete materials that genuinely model the target structure (not merely decorative objects)
• Explicit mapping language at each transition ("the three counters are the same as the 3 in the equation")
• A plan for fading: each phase is temporary, with the abstract phase as the destination
• Consistent representations across phases so learners can map one onto the other
• Manipulatives or concrete materials that map transparently onto the target structure (not richly decorative objects)
• Deliberate bridging: each representational step must be explicitly linked to the one before it
• A plan for fading to abstract symbols within a defined timeframe, not treating concrete work as an end state
• Concrete materials that genuinely model the target mathematical structure (place value, proportionality), not merely decorate it
• A parallel representational form (drawings, diagrams, number lines) that maps one-to-one onto the concrete materials
• Explicit connection language and side-by-side presentation so learners see the same problem in multiple forms
• A planned transition point to abstract notation, with continued access to representations as a fallback (Fading)
• Concrete materials that genuinely map onto the target structure (e.g., base-ten blocks whose place-value grouping mirrors the algorithm)
• Explicit, teacher-led linking language at each transition ("two tens is written 20 because…")
• A plan for fading: criteria for moving from concrete to representational to abstract, not a fixed timetable

Constraints¶

• Transitions that move too quickly leave learners with symbols but no underlying concept; learners with weak fact fluency or memory difficulties may stall at the abstract level
• Concrete materials that are themselves complex (e.g., money for teaching decimals) can add extraneous load Overloaded working memory degrades learning outcomes. [-M]
• For learners with strong prior knowledge of a concept, the concrete phase is redundant and slows progress Guidance and support become less effective as learner expertise increases. [~M]
• Requires more instructional time than symbol-first teaching; impractical as a universal default for every topic
• Manipulatives left unconnected to symbols can become a distraction; learners may play with materials without extracting the intended structure Concrete materials can increase extraneous load when their perceptual richness is not tied to the target concept. [-M]
• Staying too long in the concrete phase delays fluency with symbols and can produce dependence on physical supports
• Highly realistic or feature-rich concrete materials can divert attention from the deep structure to surface features Perceptually rich manipulatives can reduce transfer compared with bland ones. [~M]
• Highly realistic or feature-rich concrete materials can draw attention to irrelevant surface features and depress transfer [-M] — learners encode the object, not the structure
• Keeping learners at the concrete phase too long can impede development of abstract reasoning; the sequence is a bridge, not a destination [~S]
• Poorly designed manipulatives that do not map one-to-one onto the symbolic system (e.g., non-proportional materials for place value) produce confusion rather than insight [-M]
• For learners with strong prior knowledge of the symbols, starting concrete is redundant and slows progress [~M]
• Concrete materials that are perceptually rich but structurally misleading (e.g., brightly colored, distracting manipulatives) can divert attention from the target concept Decorative illustrations do not improve learning and can reduce it. [-M]
• Prolonged reliance on concrete objects without fading can bind understanding to specific materials and block transfer to symbols [~M]
• Poorly chosen manipulatives (e.g., using the same color for ones and tens) can actively model the wrong structure [-M]
• Poorly designed manipulatives (intrusive, novel, or richly decorated objects) can become the focus of attention and increase cognitive load rather than reduce it [~M]
• Concrete knowledge does not automatically transfer to abstract notation; without explicit bridging, learners may perform well with materials but fail on symbols [-M]
• Extended concrete-only instruction can promote procedural manipulation of objects without conceptual insight [~M]
• Less effective for learners who already reason symbolically; starting concrete can be redundant and slow [~M]

Implementation Variability¶

• Gradual vs. simultaneous: classic CRA fades each level out before the next; a simultaneous variant presents all three representations side by side throughout
• Virtual manipulatives: apps such as Brainingcamp and The Math Learning Center's Number Frames replicate concrete materials digitally with comparable outcomes in several studies
• CRA for algebra: Witzel's CRA sequence uses concrete cups and counters for integers and equation solving, extending the approach beyond arithmetic
• CRA-I (with explicit integration lessons between phases) — used in mathematics interventions for students with learning disabilities
• Partial CRA — starting at the representational phase when learners already have concrete experience with the concept
• Virtual manipulatives — digital versions (e.g., Brainingcamp, Didax) that can scaffold the concrete phase at scale
• Simultaneous linking — showing concrete, pictorial, and symbolic forms side by side rather than in strict sequence, useful when time is limited
• Concreteness fading: start concrete and fade to abstract within a single lesson sequence (Fyfe et al., 2014)
• Simultaneous linking: show concrete, pictorial, and symbolic forms side by side rather than strictly sequentially
• Virtual manipulatives: digital versions (e.g., Brainingcamp, Didax virtual manipulatives) trade tactile experience for scalability and clean fading
• CRA with explicit instruction: common in special education, pairing the sequence with scripted teacher demonstration and student verbalization
• Gradual vs. simultaneous: fade stage by stage, or present all three representations side by side from the start for learners who can handle the load
• Virtual manipulatives: digital tools (e.g., Brainingcamp, Didax virtual manipulatives) can substitute for physical objects, with similar effects in meta-analytic comparisons [~M]
• Domain extension: CRA originated in mathematics but the logic applies to science (physical models → diagrams → formal notation) and programming (physical artifacts → flowcharts → code)
• CRA-I (CRA with explicit instruction): teacher demonstration at each stage followed by guided practice — the most researched variant for students with learning disabilities
• Gradual release within stages: full demonstration → guided → independent at each representation level
• Simultaneous linking: showing concrete, drawn, and symbolic forms side by side rather than strictly sequentially [~M]
• CSA (Concrete–Semiconcrete–Abstract): an older variant where "semiconcrete" uses pictures of real objects rather than schematic drawings

Target Learners¶

• Learners with mathematics learning difficulties and disabilities — the strongest evidence base CRA instruction improves math outcomes for students with learning disabilities. [+M]
• Young learners encountering a concept for the first time, for whom symbols alone carry no meaning
• Less beneficial for advanced learners who already operate symbolically Guidance and support become less effective as learner expertise increases. [~M]
• Novices in a symbol-heavy domain (typically early mathematics, but also science notation and music) Concrete representations reduce working-memory demands for learners without existing symbolic schemas. [+M]
• Learners with learning disabilities or limited prior knowledge, for whom CRA-based interventions show consistent gains in computation and word-problem performance CRA sequences improve arithmetic outcomes for students with mathematics difficulties. [+M]
• Novices encountering a new symbolic system for the first time, especially in elementary mathematics [+M]
• Less beneficial for advanced learners who already operate symbolically; starting concrete wastes time and can reduce engagement [~M]
• Elementary and middle-grades students encountering a mathematical concept for the first time [+M]
• Learners with learning disabilities or working memory constraints, for whom CRA is one of the better-evidenced mathematics interventions [+S]
• Students with learning disabilities or mathematics difficulties, for whom CRA has the strongest evidence base [~S]
• Novices encountering a new symbolic system for the first time Reducing extraneous cognitive load during instruction improves learning outcomes. [+M]
• Younger learners whose prior knowledge of the domain is thin and who benefit from Activation of informal, embodied understanding
• Less beneficial for advanced learners, who can often begin at the representational or abstract stage [~M]

Target Learning Goals¶

• Conceptual understanding: why a procedure works, not just how
• Procedural fluency grounded in meaning, supporting retention and transfer
• Representational flexibility: connecting symbols, pictures, and quantities
• Conceptual understanding of abstract symbol systems (place value, fractions, algebraic variables)
• Procedural fluency grounded in meaning rather than memorized steps
• Representational flexibility — the ability to translate among concrete, pictorial, and symbolic forms
• Conceptual foundations for symbolic procedures (place value, fractions, algebraic expressions)
• Transfer from manipulable situations to abstract notation
• Flexible representation use — the ability to move among concrete, pictorial, and symbolic forms
• Conceptual understanding: grasping why an algorithm or symbolic procedure works
• Procedural fluency grounded in meaning rather than rote steps
• Representational flexibility: moving fluently among concrete, pictorial, and symbolic forms
• Conceptual foundations for procedural skills: place value, regrouping, fraction operations, integer operations, early algebra
• Representational flexibility: mapping between physical, pictorial, and symbolic forms
• Procedural fluency with symbols, once the abstract stage is reached and practiced

Instructions¶

1. Activate relevant prior knowledge and preteach prerequisite skills (Direct Instruction, Supportive Information)
2. Concrete phase: model the concept with physical materials while narrating the mathematics; learners manipulate and explain (Practice)
3. Representational phase: replace objects with drawings or diagrams of the same objects, explicitly linking each mark to the material it stands for
4. Abstract phase: introduce symbols alongside the drawings, then fade the supports (Procedural Information, Simple-to-Complex Sequencing)
5. Provide immediate corrective feedback at each level and assess performance against a mastery criterion before advancing

Related Strategies¶

• Cognitively Guided Instruction (CGI for Math) — shares CRA's commitment to grounding symbols in learners' own models of quantities
• Explicit Teaching — CRA is typically delivered through explicit modeling, guided practice, and independent practice
• Manipulative-Based Instruction — the concrete phase generalizes beyond mathematics to science and other domains
• Worked Examples — CRA phases can be delivered as worked examples with representations shown step by step
• Manipulative-Based Instruction — the concrete phase in isolation; CRA adds the critical transitions
• Use CRA-based small-group math tutoring with scripts, manipulatives, and computer fact practice as secondary prevention
• Concreteness fading — the general principle CRA instantiates; research supports fading across mathematics and science domains
• Multiple representations — CRA is a specific, ordered case of teaching with linked representations
• Multiple Representations — the broader principle of linking forms; CRA is a temporal sequencing of it
• Scaffolding — the stage sequence is a scaffold that must be faded as symbolic competence grows

Examples¶

• Place value: learners build 342 with base ten blocks, then draw flats/longs/units, then write the numeral — with the numeral introduced while blocks are still visible
• Fraction equivalence: fraction tiles compared physically, then bar diagrams, then the symbolic procedure for finding common denominators
• Witzel's CRA algebra curriculum: solving x + 3 = 5 with cups and counters before pictorial sketches and finally symbolic manipulation
• Number Frames / Number Pieces apps — free virtual manipulatives supporting the concrete and representational phases digitally
• Cognitively Guided Instruction — builds arithmetic understanding from children's informal, concrete modeling strategies toward abstract number sentences
• Concrete-Representational-Abstract instruction for fractions — Butler, Miller, Crehan, Babbitt & Pierce (2003) compared CRA and representational-only treatments for teaching fraction concepts to secondary students with learning disabilities; both improved, CRA strongest for subtraction
• National Library of Virtual Manipulatives — free virtual manipulatives (base-ten blocks, algebra tiles) supporting the concrete-to-representational transition in digital contexts
• Singapore Math / Math in Focus — published curriculum built on the concrete–pictorial–abstract sequence, using bar models as the representational bridge to algebraic reasoning
• Witzel's CRA algebra curriculum — algebra tiles used to teach linear equation solving to middle school students with math difficulties, moving from tile manipulation to drawn area models to symbolic steps.
• Butler et al. fraction intervention — fraction tiles and circles to teach equivalent fractions and fraction addition, with measured gains over representational-only instruction.
• Bridges in Mathematics — a published K–5 curriculum that systematically sequences hands-on models, visual models, and symbolic notation across units.
• Numicon — structured number tiles used in UK primary classrooms as the concrete stage of early arithmetic instruction.

Key Sources¶

• Bouck, E. C., Satsangi, R., & Park, J. (2018). The concrete–representational–abstract approach for students with learning disabilities: An evidence-based practice synthesis. Remedial and Special Education, 39(4), 211-228. doi:10.1177/0741932517721712
• Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.
• Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380–400. doi:10.1037/a0031084
• Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching early algebra to students with mathematics learning difficulties. Learning Disability Quarterly, 26(4), 265–280.
• Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research & Practice, 18(2), 99–111. doi:10.1111/1540-5826.00066
• Bouck, E. C., Satsangi, R., & Park, J. (2018). The concrete–representational–abstract approach for students with learning disabilities: An evidence-based practice synthesis. Remedial and Special Education, 39(4), 211–228. doi:10.1177/0741932517721712
• Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9–25. doi:10.1007/s10648-014-9249-3
• McNeil, N. M., & Fyfe, E. R. (2012). "Concreteness fading" promotes transfer of mathematical knowledge. Learning and Instruction, 22(6), 449–455. doi:10.1016/j.learninstruc.2012.05.001
• Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings. Learning Disabilities: A Contemporary Journal, 3(2), 49–60.