Concrete pictorial abstract
A teaching sequence that moves pupils from physical objects to drawings to abstract notation. The dominant approach to early maths teaching in Singapore-influenced curricula.
- Subject
- Maths / Science
- Key stage
- EYFS, KS1, KS2, KS3
CPA was developed in Singapore in the 1980s, drawing on Jerome Bruner’s earlier theory of enactive, iconic, and symbolic representations. UK primary maths reform in the last decade has put CPA at the centre of mastery teaching.
What it is
A three-stage teaching sequence. The concrete stage uses physical objects that pupils can move and arrange. The pictorial stage uses drawings or diagrams that represent the objects. The abstract stage uses symbols, equations, and formal notation.
Pupils move through the stages, but not in a one-way march. Strong CPA teaching brings learners back to the pictorial or concrete stages when new variations appear.
Why it works
The sequence builds intuition before formalism. By the time pupils meet the abstract notation, they have already manipulated the underlying quantities physically and seen them drawn. The symbols then anchor to existing mental images, which is exactly what schema theory and dual coding theory predict will help.
The NCETM treats CPA as one of the five big ideas in teaching for mastery. The EEF’s mathematics guidance reports recommend it for KS2 and the start of KS3.
How to use it
Start with manipulatives. Cubes, counters, base-ten blocks, fraction strips. Have pupils talk through what they are doing. Move to drawings, often bar models or part-whole diagrams. Then introduce the symbolic notation alongside the drawing, not in place of it.
Avoid the temptation to rush to the abstract stage. Pupils who skip ahead often appear fluent until the problem changes.
When not to use it
For content that has no concrete analogue, the sequence breaks down. Some topics start at the abstract stage and have to stay there. The principle of grounding new symbols in familiar representations still applies, even when the representation is itself a diagram.
Related Chalk tools
Worksheet Creator, Concrete Examples, and Visual Keywords all support the pictorial stage of CPA.
Evidence
Bruner's enactive-iconic-symbolic theory is well established. The Singapore-style CPA sequence has applied evidence from primary mathematics reform, anchored by the NCETM and the EEF maths guidance reports, although controlled comparison with other sequences is thin.
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Related concepts
Questions teachers ask
Is CPA only for primary maths?
How long should pupils stay at the concrete stage?
What is the difference between the pictorial and abstract stages?
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