How Concreteness Fading Works | From Concrete Examples to Abstraction

eduKateSG Learning Node Series · 0022

Concrete examples give abstract ideas somewhere to stand. But if the learner never leaves the concrete example, the idea may never learn to travel.

A child understands equivalence with a balance scale. A science student understands current through water-flow analogies. A mathematics learner sees fractions through pizza slices. These representations are useful because they borrow meaning from familiar experience.

Then comes the harder job: remove the pizza, the water and the physical balance without removing the relationship they helped reveal.

Concreteness fading is an instructional approach that deliberately moves from concrete representations toward increasingly idealised or symbolic ones while making the underlying structure explicit.

Quick Read: Concrete → Intermediate → Idealised

Fyfe, McNeil, Son and Goldstone’s review of concreteness fading in mathematics and science describes a progression designed to combine the benefits of concrete materials with the generality of abstract representations. Later work by Fyfe and Nathan sharpened the idea as a three-step progression from a concrete representation to a generic or idealised representation of the same concept.

Start with something the learner can see or manipulate. Then remove irrelevant surface detail while preserving the invariant relationship.

The theory is attractive because it solves a real tension. Concrete representations can make ideas meaningful, memorable and embodied. Abstract representations are better for generalisation, compression and transfer across contexts.

Why Concrete Examples Help

Abstract notation is efficient only after it has meaning.

The symbol “=” is extraordinarily compact, but a novice may treat it as a command to calculate rather than a statement of equivalence. A physical balance can make the relationship visible: change one side and balance is lost unless the other side changes correspondingly.

Concrete representations can activate prior knowledge, provide perceptual support, create memorable images and reduce the opacity of symbols.

That is why How Concrete Examples Work remains an important adjacent owner.

Why Concrete Examples Can Also Trap Learning

Concrete representations come with extra features.

A pizza has cheese, slices, shape and cultural meaning. A balance has physical arms and weights. A money example carries prices and currency. These features can help memory, but they can also become part of what the learner thinks the concept is.

If fractions are always pizza, the learner may struggle when fractions appear as ratios, probabilities or algebraic expressions. If equality is always a balance picture, symbolic equations may feel like a different topic.

Concreteness fading tries to preserve the deep structure while progressively stripping away surface dependence.

The Three-Representation Bridge

A useful sequence often contains three representational states.

Concrete: physical objects, rich images, familiar stories or realistic situations.
Intermediate: simplified diagrams, schematic pictures, number lines or stripped-down icons.
Idealised: symbols, equations, graphs, formal notation or general verbal principles.

The middle representation matters because it acts as a bridge. Jump directly from manipulatives to symbolic notation and the learner may fail to see that both representations encode the same relationship.

Fading Must Be Explicit

Simply showing three representations in sequence does not guarantee that learners connect them.

Teachers should point to correspondences.

“This physical block represents one unit. In the drawing, the rectangle plays the same role. In the equation, the variable now stands for that quantity.”

The learner must see what changed and what did not.

What Should Fade?

Fade features that are helpful for initial interpretation but unnecessary for the general concept.

Decorative realism.
Context-specific labels.
Physical manipulatives once their relationships are understood.
Colour cues once categories are stable.
Extra spatial detail not required by the formal model.
Teacher-provided mappings once learners can generate them.

Do not fade the invariant.

Concreteness Fading in Mathematics

Mathematics is a natural domain because symbols compress relationships so aggressively.

For equivalence, learners might begin with a physical balance, move to a diagram of equal groups, and then work with equations. For place value, physical base-ten blocks can become drawn bundles and then positional notation. For fractions, partitioned objects can become bar models and then symbolic fractions.

The instructional goal is not permanent dependence on manipulatives. It is to let meaning survive their disappearance.

Continue through the Mathematics Learning Hub.

Concreteness Fading in Science

Science often moves from observable objects to models that represent invisible systems.

A learner may begin with a physical magnet and coil, move to a field-line representation, and then work with symbolic relationships such as Faraday’s law. Or begin with a real circuit, move to a circuit diagram, then reason from equations.

Recent physics research is important because it shows that concreteness fading is not universally superior. Studies of electromagnetic induction have found equivalent or mixed outcomes when comparing concrete-to-abstract sequences with reversed or simultaneous arrangements.

That is a useful boundary condition: the principle should be treated as an instructional hypothesis to be fitted to learner and domain, not a ritual sequence.

Continue through the Science Learning Hub.

Concreteness Fading in English

English uses fewer physical manipulatives, but the same representational movement exists.

A concrete argument may begin with a familiar school scenario. The learner identifies claim, evidence and reasoning in that specific case. The same architecture is then shown in a schematic paragraph map. Finally, the learner uses the abstract structure on an unfamiliar topic.

A narrative lesson may begin with a photograph, move to a scene map, then ask the learner to construct a scene without visual support.

The concrete case opens the door. The abstract structure lets the learner leave the room.

Concreteness Fading and Embodied Learning

Concrete representations are sometimes physical actions rather than objects.

A learner may physically walk a number line, gesture a rotation or manipulate tiles before moving to diagrams and symbols. The body provides the first representation.

But the final goal is not permanent movement. The learner should eventually carry the structure internally.

See How Embodied Learning Works.

The Transfer Test

The key question is whether the learner can recognise the same relationship in a new surface form.

If a child understands equivalence only when weights sit on a balance, the concrete representation has not yet released the concept. If the learner can solve a symbolic equation and explain how the equation preserves the same relationship, fading has succeeded more deeply.

Transfer is the audit.

Why Simultaneous Representations Can Sometimes Help

Fading is sequential, but some learners benefit from seeing representations side by side.

Place the physical object, diagram and symbol together. Ask the learner to map corresponding parts. Then remove one representation and reconstruct it from the others.

This can make the bridge more explicit than a sequence where the earlier representation disappears before the relationship is secure.

Research comparing sequential fading with simultaneous representation in physics reinforces the broader lesson: representational design depends on what the learner already knows and what structural mapping the task requires.

The Reverse Route Can Sometimes Work

Some studies have found that abstract-to-concrete sequences can equal or outperform concrete-to-abstract sequences in particular domains.

That should not be treated as a contradiction that destroys the whole idea. It tells us that “concrete” is not automatically easier and “abstract” is not automatically harder.

A realistic picture can contain more visual detail than a clean schematic. A symbolic representation can sometimes make the important relation clearer because irrelevant detail has already been removed.

The real design problem is structural accessibility, not a simplistic ladder from easy objects to difficult symbols.

How Fading Fails

The concrete example is memorable but the invariant is never named.
Representations change without explicit mapping.
The sequence fades too quickly.
The concrete form contains distracting detail.
The abstract form is introduced before the learner understands what it represents.
The concrete context becomes the concept itself.
The learner never meets varied examples, so transfer remains untested.

A Practical Design Protocol

Choose the invariant: identify the relationship the learner must preserve.
Choose the concrete anchor: use a familiar or manipulable representation that expresses that invariant clearly.
Build a bridge: create a schematic or intermediate representation.
Introduce the formal representation: symbols, equations, graphs or general rules.
Map explicitly: show which parts correspond.
Remove support: ask the learner to reconstruct missing representations.
Vary the surface: use new contexts.
Test transfer: present a case where no concrete cue remains.

Concreteness Fading and the First Weak Link

When transfer fails, ask where the bridge broke.

Did the learner misunderstand the concrete representation? Did the learner fail to map the intermediate diagram to the object? Did the symbol lose its connection to meaning? Did the learner overlearn one surface feature?

Repair the failed mapping rather than simply adding more abstract exercises.

A Parent Example

Suppose a child understands fractions with cake slices but struggles with 3/4 + 1/8.

Do not stay forever with cake. Move from cake to a clean bar model. Then map the bar model to equivalent fractional notation. Ask the child to explain why the denominator changes and what remains invariant.

The parent’s job is to help the meaning migrate into the notation.

A Tutor Example

For algebra, start with a balance metaphor only if it clarifies equality. Then quickly move to a schematic balance, then equations. Ask the learner to identify the same legal move in each representation.

Once the learner can explain the invariant—whatever operation is performed to one side must preserve equality—remove the metaphor.

The scaffold has done its job.

The Deep Principle: Meaning Must Survive Compression

Abstract representations are powerful because they compress.

An equation can represent infinitely many concrete situations. A graph can summarise a relationship across a range of values. A symbol can stand for an entire conceptual category.

But compression is useful only if the learner can still recover meaning from the compressed form.

Concreteness fading is therefore not about abandoning reality for symbols. It is about building symbols that still know where they came from.

Use This Tomorrow

Take one concept currently taught with a concrete example. Create a three-step sequence: concrete, schematic, symbolic. Ask the learner at each transition what changed and what stayed the same. Then test the concept in a new context with no original surface cues.

Research and Further Reading

eduKateSG Learning Node Series · 0022. Previous: 0021 — How Completion Problems Work.

How Concreteness Fading Works | Move From Objects and Stories to Symbols Without Losing the Structure