Using Manipulatives¶

Strategy · All strategies
Evidence · 5 claims (5 for) · 14 studies (5 causal, 3 quant-synthesis, 3 review, 3 associational), q2–q4 · 3 of 14 report an effect size

Description¶

Manipulatives are physical objects — counters, base-10 blocks, fraction strips, pattern blocks, geoboards, algebra tiles — that learners handle to model and reason about mathematical concepts. They serve to introduce, practice, or remediate ideas across number, algebra, geometry, measurement, and data strands. Effective use requires making the manipulative's mathematical structure transparent, connecting it explicitly to symbols and language, and fading it as learners internalize the concept.

Design Implications¶

Manipulatives work as symbols for mathematical ideas, not as self-interpreting objects: learners must understand the mapping between object and concept for handling to produce learning [~M]. Meta-analytic evidence shows moderate positive effects on retention when instruction with manipulatives is combined with traditional instruction, but effects shrink or reverse when manipulatives replace rather than supplement other instruction [~S]. The central design task is the bridge from concrete to abstract — manipulatives paired with self-explanation and explicit symbol connection outperform free exploration.

Context¶

Requirements¶

• Manipulatives whose structure actually maps onto the target concept (base-10 blocks for place value, not any countable object)
• Instructor guidance on how the object represents the mathematics, plus familiarization time so tool handling does not consume working memory
• Explicit connection to symbols, drawings, and mathematical language during and after use
• A plan for fading: moving from concrete objects to drawings to symbolic notation (Fading)

Constraints¶

• Unstructured exploration without guidance produces little learning; learners may play with the objects without extracting the mathematics [-M]
• Perceptually rich manipulatives (colorful, realistic objects) can distract from the mathematical structure and impair transfer compared with bland ones [-M]
• Learners can treat the manipulative as the answer rather than a model — relying on it as a crutch instead of internalizing procedures [-M]
• Benefits diminish and can reverse for learners with sufficient prior knowledge, consistent with the expertise-reversal effect [~M]
• If the mapping between object and concept is opaque (e.g., using a single counter for "ten"), the manipulative misleads rather than supports [-M]

Implementation Variability¶

• Concrete–Representational–Abstract (CRA) sequencing: manipulatives first, then drawings, then symbols — the most common structured adaptation
• Virtual manipulatives: digital versions (e.g., Brainingcamp, The Math Learning Center apps) offer repeatability and annotation; evidence suggests comparable effects to physical objects when the pedagogy is the same [~W]
• Comparing multiple representations: using two different manipulatives for the same concept supports abstraction Multiple contrasting cases support abstraction. [+M]
• Learner-created materials: fraction strips students cut themselves can deepen the object–concept mapping

Target Learners¶

• Elementary and middle school learners encountering a concept for the first time, especially where the concept is not observable in everyday objects (place value, fractions, negative numbers)
• Learners with weak prior knowledge who need an external model to reduce working-memory load Chunking reduces working memory load. [+M]
• Less beneficial — and potentially redundant — for learners who already reason fluently with symbols [~M]

Target Learning Goals¶

• Conceptual understanding of abstract structures (place value, equivalence, area)
• Procedural grounding: understanding why regrouping or common denominators work before executing them symbolically
• Mathematical argumentation: learners can use manipulatives to justify and demonstrate claims
• Early spatial and number sense Block play predicts math achievement. [+W]

Instructions¶

1. Select a manipulative whose structure transparently maps to the target concept; verify learners can operate it fluently before instruction begins.
2. Model the mapping explicitly — name what each object, position, or action represents (Demonstration).
3. Have learners solve problems with the manipulative while explaining what each action means mathematically Self-explanation improves conceptual understanding. [+S].
4. Connect each manipulation to symbolic notation in the same lesson — record alongside, not after.
5. Fade support: move from objects to drawings to symbols as accuracy grows (Fading) Fading support promotes transfer of responsibility. [+M].
6. Assess transfer with symbolic problems the manipulative did not directly model.

Related Strategies¶

• Worked Examples — a teacher's worked demonstration with a manipulative gives learners a model of the object–concept mapping before independent use
• Self-Explanation — prompts learners to articulate the mapping between object and symbol, the step most often skipped
• Comparing Cases — contrasting two manipulative representations of the same concept supports abstraction

Examples¶

• Base-10 blocks for regrouping: learners trade ten units for one "long" while recording the symbolic exchange in the same step, then solve subtraction problems with blocks removed.
• Fraction strips for equivalence: students fold and compare paper strips to establish 1/2 = 2/4, then generalize to a number line.
• The Math Learning Center free virtual manipulative apps (Number Frames, Number Pieces) used in a CRA sequence.
• Cognitively Guided Instruction classrooms, where children's own counting strategies with counters drive instruction.

Key Sources¶

• 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
• Sowell, E. J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20(5), 498–505. doi:10.5951/jresematheduc.20.5.0498
• Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18(1), 37-54. doi:10.1016/s0193-3973(97)90013-7
• McNeil, N. M., & Jarvin, L. (2007). When theories don't add up: Disentangling the manipulatives debate. Theory Into Practice, 46(4), 309–316. doi:10.1080/00405840701593899
• Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.