Concreteness Fading¶

Principle · All principles
Evidence · 27 claims (12 for, 13 mixed, 2 against) · 29 studies (14 causal, 4 quant-synthesis, 4 review, 4 theoretical, 2 qualitative, 1 associational), q1–q4 · 5 of 29 report an effect size · 23 claims rest on one study

Conditional relationship¶

This page owns the decision of in what order concrete and abstract representations of one idea are given. When the target is an abstract symbol system or formal structure (an equation and the equal sign, a fraction, a group property, a permutation formula, a physical law written as an equation), and learners do not yet read the notation as meaning anything, beginning with a concrete model of the structure (objects to handle, or a familiar situation), moving to an iconic version of the same model (a drawing, a diagram, a schematic), and ending with the abstract notation, with each step explicitly connected to the one before, is expected to give better learning and transfer than starting with the notation or staying with the concrete model.

The relationship is conditional on four things, each with how firmly the wiki's claims hold it:

The order and the ending, not the concrete material. The one direct test recorded here, McNeil and Fyfe (2012) as reported by Antonides and Battista (2022), had undergraduates learning abelian group properties in a fading, a generic (abstract only) or a concrete condition; the fading group did better than both. It is reported second-hand with no effect size. The review behind the principle (Fyfe et al. 2014) argues the benefit is more than the sum of concrete and abstract materials taken separately.
The explicit link between representations. Concrete material helps only when learners are helped to connect it to the symbol (Fyfe et al. 2014, as recorded on the manipulatives claim). An object that stays an object does not carry the mathematics.
What the concrete stage adds over symbols alone. Across 55 studies from kindergarten to college, mathematics taught with manipulatives beat abstract-symbols-only instruction by a small-to-moderate amount, most on retention and least on problem solving and transfer (Carbonneau et al. 2013). That is the first stage against no first stage, not the fading sequence against staying concrete.
Who the learner is. The review's mechanism, interpreting opaque symbols through well-understood objects, presumes the symbol is still opaque. For a learner who already reads the notation, the concrete stage is expected to be redundant, by analogy with the expertise reversal [~W] found for other kinds of guidance; no claim here tests that for concrete representations.

How it sits beside its neighbours. It is not one of the multimedia learning principles: those decide how words and pictures are combined at one moment; this decides the order of representations across a sequence. It is a fading of the representation, where scaffolding and fading and worked examples fade support or solution steps while the representation stays the same; the two can run together but are different relationships, and the evidence for one is not evidence for the other. Its goal is the one transfer of learning owns, reaching a structure that is not tied to one surface; transfer of learning gets there by varying surfaces and comparing them, this by stripping one surface away in steps. Sequencing cites the same experiment as one case of ordering instruction.

Default design, while the dose is untested¶

The claims establish the direction (concrete, then iconic, then abstract, linked) and that concrete material with no link to the symbol does not teach the concept. They set no stage length, no number of sessions and no criterion for moving on. Everything below that is a time, a count or a rule is a labelled proposal.

Check the condition. Is there an abstract notation or formal structure the learner must end up using, and do they not yet read it as meaning anything? Ask them to say what the symbol means, or to solve one item in notation and explain it. If they can, start at step 5. (Untested proposal.)
Choose one concrete model whose structure is the structure of the idea: a balance for an equation, cube towers for permutations, fraction strips whose lengths are the part–whole relation. Keep it plain: no decorative detail that is not part of the structure. Basis: manipulatives require connection to concept [+M]; decorative detail as a cost is carried from decorative illustrations [-M].
Concrete stage, about 10–20 minutes in the first session. Learners act on the model and say what they did. From the start, name the parts with the words the notation will use ("this side", "equal", "one ten") and, near the end, write the matching symbol beside what they did. Basis for the linking: manipulatives require connection to concept [+M]; the timing is an untested proposal.
Iconic stage, in the same or the next session, about 10–15 minutes. The same situations drawn: a sketch of the balance, a diagram of the towers. Learners draw or complete them, then write the symbol under each. Ask what each mark in the notation corresponds to in the drawing. Basis: the three stages follow the cube-tower teaching experiment [+W]; the timing is an untested proposal.
Abstract stage, from the second or third session. Problems in notation only. When a learner errs, point back to the drawing or the object ("what would the balance do?") rather than restating the rule. Basis: the review's argument that the concrete stage leaves images usable when symbols lose meaning, fading builds memorable images [+W].
Make the concrete stage impractical rather than forbidden. Raise the size of the problems (more cubes, larger numbers) until handling the objects is slower than writing the symbol, so the learner chooses the notation. Basis: the fading move in the cube-tower teaching experiment [+W] (see the strategy increase task size to force fading).
Moving on. Move a learner from one stage to the next when they can do two new items at the current stage and say which part of the earlier representation each part corresponds to; move back a stage after two errors in a row on the same structure. (Untested proposal.)
Across a unit of three to five sessions: sessions 1–2 carry the three stages for the first idea; later sessions start each new, related idea at the iconic stage, and start at the concrete stage again only for an idea with a new structure. (Untested proposal.)
Judge it on the abstract notation and on transfer items without the objects, not on how well learners handle the model. Basis: fading condition outperformed generic and concrete conditions [+W], which measured learning and transfer.

Fitting the design to a situation¶

Three facts change the decision most. Ask for them when the brief omits them: does the learner already read the notation? (ask what the symbol means to them, or for one item solved and explained); is the goal the abstract structure and transfer, or fluent use of the concrete situation itself? (an electrician wiring a circuit may need the situation more than the equation); and what can learners handle, and for how long? (objects, a screen, paper only; minutes per session).

If the brief says…	Then change…	Basis
Learners are novices who cannot yet say what the symbols mean	Use all three stages, with the links in steps 3–4	Fading condition outperformed generic and concrete conditions [+W]; Manipulatives require connection to concept [+M]
Learners are advanced and already read the notation (`high-prior-knowledge`)	Start at the abstract stage; keep a drawing ready to point back to on errors	Expertise reversal [~M], carried from guidance formats to concrete representations, untested for them here
Young children (early primary, kindergarten)	Keep the concrete stage longer, across several sessions, with the teacher's words naming the relation as children act	Manipulatives improve math learning [+M] (kindergarten to college); the longer stage is an untested proposal
Adults or undergraduates learning a formal structure	Use a short concrete stage (one session or less) and move to iconic quickly	Fading condition outperformed generic and concrete conditions [+W] (undergraduates); the length is untested
The goal is retention of a procedure more than transfer	The concrete stage still helps; expect most of the gain on retention, less on problem solving and transfer	Manipulatives improve math learning [+M]
Self-paced online, no teacher to make the links	Build the links into the material: each drawing shown beside its object and its symbol, with a prompt to say which part matches which	untested proposal; the need for the link rests on Manipulatives require connection to concept [+M]
System-paced video or slides	Show the object, then the same scene redrawn as a diagram, then the symbol, with a pause and a question at each change; do not cut straight from object to equation	untested proposal
Phone-only, or no physical materials	Use virtual objects learners move on screen for the concrete stage	Physical experience enhances science learning [~M] (virtual and physical manipulation were equally effective for undergraduates' heat and temperature concepts)
VR or a rich simulation	Keep the scene plain enough that the structure stands out, and still end with the notation; a vivid scene is not the abstract stage	untested proposal; cost of irrelevant detail carried from Decorative illustrations [-M]
Print only	Use the iconic stage as the first stage (a drawing learners mark or cut out), then the notation	untested proposal
A single session	Run all three stages in one session at about 10 minutes each, and end on two items in notation	untested proposal
A fixed short time budget (10 minutes a day)	One stage per day across three days, opening each day with one item from the previous stage	untested proposal
One teacher for many learners	Do the concrete stage as a demonstration with a few learners acting it out, the iconic stage individually on paper; in discussion, take a learner's symbolic answer back down to the drawing or the object before moving on	Unfolding knowledge elements produces deeper conceptual learning opportunities [+W] (opportunities in the discourse, not measured learning)
Low literacy, or learning in a second language	Keep the concrete and iconic stages, which need little text; add the spoken and written word for each part as it is linked to the symbol	untested proposal
Learners with a visual impairment	Keep the concrete stage tactile; make the iconic stage a raised-line or tactile diagram before braille or spoken notation	untested proposal
Learners with a hearing impairment	No change to the stages; give the linking talk in captions or sign beside the object	untested proposal
The brief's own manipulatives, curriculum materials, routine or rubric	Keep them as given; add only the iconic step and the spoken links if they are missing	untested proposal
The brief's own assessment	Run it as given; add no test it did not ask for. If it tests only with the objects present, say that it cannot show the abstract stage was reached	untested proposal
Private or personal material	Not applicable to the order of representations; no change	untested proposal
A whole task kept whole (an authentic problem, a project)	Keep the task whole; apply the stages to the one formal idea inside it the task needs	untested proposal
High stakes (a qualifying exam in notation)	Keep the stages short and spend the saved time on practice in the notation the exam uses; the exam judges the abstract stage only	untested proposal
A concrete and an abstract version of the same animation are on offer, with no sequence	Expect no difference between them by themselves; the choice that matters is the order and the links	Abstract and concrete animation do not differ [~M]

Observation, state and explanation¶

These rows are proposals, untested with learners as diagnostics.

Response	Plausible explanations	Observation that could change the interpretation
Learner handles the objects fluently but cannot write or read the symbol	The link was never made; the model's structure does not match the notation's	Ask them to point to what each symbol stands for in the model; check whether the model has a part for every symbol.
Learner solves items in notation but cannot say what any step means	They skipped to a memorised rule; the abstract stage came before the links were secure	Ask them to draw the item; if they cannot, go back to the iconic stage.
Learner keeps reaching for the objects on large problems	The concrete stage has not been made impractical; the notation is not yet trusted	Raise problem size; ask them to predict with symbols and check with objects.
Learner transfers to a problem with the same objects but not to one without	Knowledge is tied to the model's surface	Give a second, different concrete model of the same structure and ask what the two share.

Evidence and qualifications¶

A conference paper reports that, in McNeil and Fyfe (2012), undergraduates taught with concreteness fading outperformed those taught in generic or concrete conditions [+W]: the only direct test of the full sequence against both alternatives recorded here, McNeil and Fyfe (2012), reported second-hand in a conference paper, with no effect size.
Manipulatives require connection to concept [+M]: Fyfe et al.'s (2014) systematic review rejects the concrete-versus-abstract choice and argues for concrete material explicitly and gradually faded to the symbol. No effect size.
Concreteness fading provides advantages beyond the sum of the benefits of concrete and abstract materials [+W]: the same review's central argument.
Four mechanisms the review proposes, none tested in the text read: interpreting opaque symbols through well-understood objects [+W], embodied grounding of abstract thinking [+W], a store of memorable images for when symbols lose meaning [+W], and stripping away extraneous concrete properties [+W].
Manipulatives improve math learning [+M]: Carbonneau et al.'s (2013) meta-analysis of 55 studies (N = 7,237, kindergarten to college); manipulatives over abstract symbols alone, small to moderate overall, moderate to large on retention, small on problem solving and transfer, moderated by how the instruction was done. No pooled d was recorded. It tests the concrete stage against none, not fading against staying concrete.
A cube-tower instructional sequence instantiated concreteness fading [+W] and students transferred cube-based permutation reasoning to tasks without cubes [+W]: two preservice teachers moved from building towers, to imagining them, to multiplicative symbols, and later used the reasoning without cubes. Two learners, one-on-one, no comparison.
Physical experience enhances science learning [~M]: a brief physical experience of angular momentum raised college physics quiz scores (Kontra et al. 2015), but physical and virtual manipulation were equally effective for heat and temperature (Zacharia & Olympiou 2011). The concrete stage need not be physical.
Abstract and concrete animation representations do not differ [~M]: in a middle-school electricity quasi-experiment, a concrete versus an abstract animation, given alone, made no detectable difference under any of five text conditions (equivalence not tested). Picking one representation is not the relationship this page states.
Expertise reversal [~M]: guidance that helps novices loses value or reverses as expertise grows, shown for other formats; carried here to the concrete stage.

Taken together: in the claims read here, the case for the full concrete–iconic–abstract order rests on one experiment reported second-hand and on a systematic review's argument; the case that a concrete first stage helps over symbols alone rests on a meta-analysis; and the case that the link between representations is what matters rests on the review. No claim in the wiki tests stage length, the iconic stage on its own, or the sequence with children against staying concrete.

The evidence by question¶

Does moving from concrete to iconic to abstract, with explicit links, beat starting abstract or staying concrete? Manipulatives require connection to concept [+M]
Does a concrete stage help at all over abstract symbols alone? Manipulatives improve math learning [+M]
By what route is concreteness fading thought to work? Concreteness fading helps learners interpret opaque abstract symbols [+W]

Further evidence, not yet read against this model¶

Hands-on learning improves achievement [+M]: the same Carbonneau et al. (2013) meta-analysis as the manipulatives claim, under a broader title.
Dale argued learning becomes more meaningful when abstract learning and concrete experience are related, not that more realism is better [+W]: the older version of the argument; see Cone of Experience.
The signaling effect holds only for abstract animation representations [~M]: signalling helped with an abstract animation (d = .35) but not with a concrete one; a hint that abstract representations need more cueing.
Self-explanation improves abstract transfer of logical rules [+W]: explaining concrete problems raised transfer to abstract ones; a candidate for the linking prompts in steps 3–4.
Comparing cases side by side improves learning and transfer [~M]: two concrete instances compared as a route to the abstraction, beside fading one instance; see Analogical Reasoning.
Spatial training improves math performance [~M]: transfer was larger when training used concrete manipulatives than computerised training.
Using more representations of mathematical ideas is associated with increased student math achievement [~W]: correlational, from classroom observation.
After an early algebra intervention, kindergarteners giving a relational definition of the equal sign went from one to nine [~W]: a single-group intervention that used balances and cubes for equations; it does not isolate the order.
Learners learned most about those principles that were faded [~W]: fading of worked solution steps, a different relationship; listed so it is not mistaken for evidence on fading representations.

Objective and learner-valued goal¶

The designer's objective is that learners can use the notation as meaning something, and use it on problems where no objects are at hand. Ask separately what the learner wants: a learner who wants to pass a symbolic test next week may see the concrete stage as childish or slow, and needs to be told what it is for and how short it will be; a learner whose goal is the concrete situation itself (measuring, building, wiring) may not need to reach the abstract stage at all.

What would revise this model?¶

The expectation should weaken if direct comparisons with comparable learners and equal time find that the fading sequence does no better than starting abstract on a delayed test in notation, or no better than staying concrete on transfer items without the objects. It should be narrowed if the benefit appears only for learners who cannot yet read the notation, or only in mathematics. It should be extended by tests of stage length, of the iconic stage on its own, and of self-paced versions where the links are built into the material.

Claims¶

Scaffolding and Fading — fading of support, not of representation
Worked Examples — fading of solution steps, often combined with this
Transfer of Learning — the goal this serves, reached by varying surfaces rather than stripping one
Analogical Reasoning — a concrete model as an analogy whose mapping must be made explicit
Sequencing — the ordering decision this is one case of
Dual Coding — words and a visual at one moment, beside an order of representations across sessions
Multimedia Learning — not an umbrella for this page; it combines words and pictures, this orders representations
Cognitive Load Management — why a plain model and an explicit link spare novices the search

Examples¶

Concrete-Representational-Abstract (CRA) — the classroom strategy that carries this sequence in mathematics.
Increase task size to force fading — the cube-tower fading move.
Concrete Modeling and Fading — the building blocks.
Provide Manipulatives and Using Manipulatives — the concrete stage on its own.
Seven-Lesson Kindergarten Equal Sign Sequence — a design using balances and cubes for equations.
Prior Knowledge — the learner variable that decides where to start.
DragonBox Algebra 5+ App
Five key representations for introductory statistics: verbal descriptions, visualizations, word equations, GLM notation, and R code

Key Sources¶

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
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
Antonides, J., & Battista, M. T. (2022). Towards an Elaboration of Concreteness Fading: Reflections on a Constructivist Teaching Experiment. Proceedings of the forty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. https://eric.ed.gov/?id=ED630441
Kontra, C., Lyons, D. J., Fischer, S. M., & Beilock, S. L. (2015). Physical Experience Enhances Science Learning. Psychological Science, 26(6), 737–749. doi:10.1177/0956797615569355
Zacharia, Z. C., & Olympiou, G. (2011). Physical versus virtual manipulative experimentation in physics learning. Learning and Instruction, 21(3), 317–331. doi:10.1016/j.learninstruc.2010.03.001
Kutbay, E., & Akpınar, Y. (2020). Investigating modality, redundancy and signaling principles with abstract and concrete representation. International Journal of Education in Mathematics, Science and Technology, 8(2), 131–145. https://eric.ed.gov/?id=EJ1255527
Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38(1), 23–31. doi:10.1207/s15326985ep3801_4
Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2013). Learning through case comparisons: A meta-analytic review. Educational Psychologist, 48(2), 87–113. doi:10.1080/00461520.2013.775712
Seels, B. (1997). The Relationship of Media and ISD Theory: The Unrealized Promise of Dale's Cone of Experience. https://eric.ed.gov/?id=ED409869

Page metadata

Field	Value
`type`	`principle`
`id`	`concreteness-fading`
`title`	`Concreteness Fading`
`description`	When learners must come to understand an abstract symbol system or formal structure (an equation, a group property, a permutation formula), starting with a concrete model of it, moving through an iconic picture of the same model, and ending with the abstract notation, each step explicitly linked to the last, is expected to produce better learning and transfer than starting abstract or staying concrete; one experiment, reported second-hand, tests the full sequence here, a meta-analysis supports concrete materials over symbols alone, and the rest is a review's argument and one teaching experiment.
`status`	`review`
`canonical`	`True`
`generated`	by `claude/unspecified` on `2026-10-14`
`trust`	unverified — no one has confirmed this page's content
`sources`	9 — `fyfe-et-al-2014`, `carbonneau-et-al-2013`, `antonides-battista-2022`, `kontra-et-al-2015`, `zacharia-olympiou-2011`, `kutbay-akpinar-2020`, `kalyuga-et-al-2003`, `alfieri-et-al-2013`, `seels-1997`

Frontmatter is the page's machine-readable record — see the Schema & Guide.