Concrete–Pictorial–Abstract (CPA) describes a progression in which a learner can first work with tangible quantities or manipulatives, then represent the same mathematical relationship visually, and finally express it using abstract symbols and notation.
The power of CPA does not come from completing three compulsory stages for every question. It comes from maintaining the same mathematical structure across different representations until the learner can reason independently.
Concrete
Concrete experiences can include counters, blocks, strips, measuring tools or other manipulatives. The learner can act on the quantity and observe what changes and what remains invariant.
Pictorial
Diagrams, number lines, arrays, bar models, tables and graphs preserve the relationship without requiring the physical object. This is often where a learner begins to see general structure.
Abstract
Numbers, variables, operators, equations and formal notation compress the same relationship. Abstract representation becomes powerful only when the symbols still carry meaning for the learner.
CPA is not a one-way staircase
Strong learners move between representations. An algebra student can return to a graph; a fraction student can return to a strip model; a probability student can build a table. Moving back to a representation is not failure—it can be a deliberate diagnostic move.
Common implementation mistakes
- Using manipulatives without connecting them to mathematical language.
- Letting pictures become decorative rather than structural.
- Moving to symbols before the relationship is understood.
- Keeping visual scaffolds long after the learner no longer needs them.
- Teaching one representation as the only correct representation.
A support-fading rule
If the learner can explain the relationship, choose an appropriate representation and solve a nearby unseen problem, support can be reduced. If success disappears as soon as the picture is removed, the representation has not yet become internalised knowledge.
HMH’s current Math in Focus programme explicitly describes a concrete-pictorial-abstract approach using manipulatives, diagrams, models and abstract symbols. Singapore’s broader curriculum framework similarly emphasises conceptual understanding, visualisation, tools and problem solving.
Sources reviewed 26 September 2026: Singapore MOE Primary Mathematics curriculum and current Math in Focus programme materials.
World Mathematics route: return to the World Mathematics Atlas to connect Singapore Math methods with school levels, international curricula, examinations, competitions and university Mathematics.
CPA is a representation progression, not a three-box ritual
Concrete means mathematically chosen objects
Counters, cubes, fraction strips, place-value discs and other manipulatives should expose the relationship being learned. Random objects do not become good teaching merely because they are physical.
The concrete representation must preserve structure
Ten loose counters and one ten-stick both represent ten, but they make different place-value relationships visible. Choose materials that support the intended concept.
Pictorial representation compresses the physical model
Drawings, number lines, arrays, bar models and diagrams retain important relationships while removing some physical detail. This is the bridge toward abstraction.
Abstract symbols are efficient because meaning was built earlier
An equation such as 3×4=12 is compact. The learner should still be able to connect it to equal groups, an array or another valid representation.
Movement is not strictly one-way
A student working symbolically may return to a diagram when a new problem is difficult. Representation flexibility is a strength, not evidence of failure.
Do not force all three stages into every question
Once a concept is secure, unnecessary manipulatives can increase cognitive load. Use the least support that preserves understanding.
Language belongs inside CPA
Ask what each object, drawing and symbol represents. The learner should explain correspondences rather than merely imitate the teacher’s movements.
Fractions show why CPA matters
Fraction strips can establish equal parts and equivalence; diagrams can compare fractions; symbols then support operations. Skipping meaning makes later algorithms brittle.
Algebra can also use CPA principles
Tiles, balance models and bar diagrams can make equality and unknown quantities visible before symbolic transformations become fluent.
Common implementation failure
Teachers sometimes present concrete, then a picture, then symbols in rapid succession without checking whether the learner connected them. The stages become three demonstrations instead of one concept.
Assessment should test translation
Give a symbol and ask for a model; give a model and ask for an equation; give a story and ask for both. Flexible translation is stronger evidence than completing one representation repeatedly.
Transfer principle
The durable idea is representation design: make structure visible, gradually compress it and retain the ability to unpack symbols when reasoning becomes difficult.