Manipulatives¶

Strategy · All strategies
Evidence · 3 claims (2 for, 1 mixed) · 9 studies (3 review, 2 causal, 2 quant-synthesis, 1 associational, 1 theoretical), q2–q4 · 1 of 9 report an effect size

Description¶

Manipulatives are physical or tangible objects — blocks, counters, maps, models — that learners handle to make abstract concepts concrete. The strategy works by grounding symbolic or verbal content in perceptual, motor experience, giving learners something manipulable to reason with before or alongside formal notation. It is carried out by selecting objects that map transparently onto the target concept, structuring guided interaction with them, and explicitly connecting the objects to the abstract representation they stand for.

Design Implications¶

Manipulatives leverage embodied and dual-coding pathways: pairing a concept with a tangible referent gives learners a second, more intuitive representation to reason from Dual Coding Theory. Their benefit is real but conditional — a meta-analysis of mathematics instruction found concrete manipulatives outperform abstract instruction alone at moderate effect sizes, but only when the mapping between object and concept is made explicit Carbonneau et al. meta-analysis of concrete manipulatives. [+M]. Manipulatives are not self-teaching: without teacher guidance connecting "what you did with the blocks" to "what the symbols mean," learners often treat the activity as play and fail to build the intended schema [~M].

Context¶

Requirements¶

• Readily available manipulatives that map transparently onto the target concept (one-to-one correspondence between object and idea)
• A teacher who actively supports and guides their use, not just distributes them
• Explicit linking of concrete actions to abstract representations — verbalizing and labeling during use (Provide Guidance)
• Follow-up Practice that moves learners toward the symbolic form of the concept

Constraints¶

• Some students might not have the cognitive awareness to benefit from manipulatives — learners must recognize that the object stands for something else; very young or low-prior-knowledge learners may treat the object as the concept itself [~M]
• Overly attractive or feature-rich objects can distract from the target concept; learners manipulate the surface features instead of the structure [~M]
• Prolonged concrete-only instruction can impede transition to abstract reasoning; the concrete phase should be faded, not permanent [~M]
• Ineffective when the manipulative's structure does not actually mirror the concept's structure (e.g., using arbitrary objects for place value)

Implementation Variability¶

• Concrete → Representational → Abstract (CRA) sequencing: start with objects, move to drawings, then symbols — the most validated adaptation
• Virtual manipulatives: digital equivalents (e.g., PhET simulations) offer repeatability and scalability, with roughly comparable effects to physical objects [+W]
• Learner choice: allow learners to select manipulatives that work for them; extend beyond numeracy to history artifacts, maps and globes for geography, physical models in science
• Self-explanation prompts: asking learners to explain what the object shows deepens the object–concept link Self-explanation improves conceptual understanding. [+M]

Target Learners¶

• Learners who face speed of processing challenges — tangible objects reduce the working-memory demands of holding abstract information while reasoning Chunking reduces working memory load. [+M]
• Novices encountering an abstract concept for the first time, who lack the prior knowledge to reason symbolically
• Less beneficial for learners with strong abstract understanding of the concept, for whom concrete materials add extraneous steps Worked-example guidance becomes less effective as learner expertise increases. [~M]

Target Learning Goals¶

• Conceptual comprehension: building an intuitive model of an abstract structure
• Retention: concrete referents provide memorable retrieval cues
• Bridging to symbolic fluency: preparing learners for formal notation and procedures

Instructions¶

1. Select a manipulative whose structure maps directly onto the target concept; avoid decorative or high-distraction objects.
2. Introduce the object with Provide Guidance — model how the object represents the concept, naming the correspondence explicitly.
3. Have learners actively manipulate the object to solve problems or explore relationships (Practice).
4. Prompt learners to explain what the object shows and why (Self-Explanation or Think-Aloud).
5. Fade toward representational drawings and then abstract symbols, connecting each stage back to the object.

Related Strategies¶

• Worked Examples — a demonstration with manipulatives shows the object–concept mapping before learners manipulate independently
• Analogies — manipulatives function as physical analogies; both rely on learners mapping structure from a familiar surface to an unfamiliar target
• Multiple Representations — concrete objects are one representation among several; coordinating them builds flexible understanding
• Külppe's recommendations: formal exercises in thinking, more abstract thinking, and practice strengthening determining tendencies

Related Elements¶

• Practice — manipulation must be followed by practice that consolidates the concept
• Provide Guidance — unguided object play rarely produces the intended learning
• Analogies — the cognitive mechanism underlying why a concrete object can stand in for an abstract idea

Patterns That Use This Strategy¶

• Cognitively Guided Instruction (CGI for Math) — children's concrete modeling of word problems drives instruction
• Cognitive Apprenticeship — physical artifacts anchor modeling and coaching in observable activity

Examples¶

• Base-ten blocks for place value — the canonical mathematics use; effective when teachers explicitly link trades of tens to column regrouping in written algorithms.
• PhET Interactive Simulations — virtual manipulatives for physics and math (e.g., circuit construction, fractions) with guided exploration prompts.
• Maps and globes in geography — handling physical maps and globes makes spatial relationships and scale concrete before map-reading conventions are taught.
• Artifact handling in history — physical or replica objects ground abstract historical concepts in tangible evidence.
• Resource: Speed of Processing micro-credential, Friday Institute at NC State.

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
• 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
• Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3, pp. 41–95). Lawrence Erlbaum.
• Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.
• Paivio, A. (1990). Mental representations: A dual coding approach. Oxford University Press. doi:10.1093/acprof:oso/9780195066661.001.0001