The CPA Approach
The Definitive Guide to Concrete-Pictorial-Abstract
A teaching method that supports all learners in building a solid understanding of mathematics concepts.
What is the CPA Approach?
In the pursuit of mathematical mastery, the Concrete-Pictorial-Abstract (CPA) approach stands as the primary instructional framework for schools in Canada, England, New Zealand, and Australia. While many view CPA as a simple three-step process, it is actually a sophisticated, interlaced learning system that bridges the gap between physical experience and conceptual understanding.
We at Maths — No Problem! integrate the CPA approach into every lesson to ensure students don't just memorize rules, but build a robust, relational understanding of how mathematics works.
Developed from the “modes of representation” researched by American psychologist Jerome Bruner, the CPA approach suggests that learning is most effective when it follows a progression from the tangible to the symbolic.
1. Concrete (The Enactive Mode)
The “doing” stage involves learning through physical action. Students interact with concrete manipulatives — such as base-ten blocks, counters, or multi-link cubes — to model mathematical problems.
→ The Goal: To build muscle memory and a tangible sense of quantity and structure.
→ Why it Works: It provides an accessible “low-floor” entry point for all learners, particularly those in the “Enactive” stage of cognitive development. In educational settings, this refers to the stage where hands-on activities are prevalent, e.g. building with blocks to understand balance or shapes.
2. Pictorial (The Iconic Mode)
The “seeing” stage uses mental images or drawings to represent concepts. This might involve bar models, number lines, or sketches of the manipulatives used in the concrete stage.
→ The Goal: To serve as a mental bridge between the physical object and the abstract symbol.
→ Why it Works: It allows students to “see” the math in their mind's eye, making it easier to visualize structures and relationships.
3. Abstract (The Symbolic Mode)
The “symbolic” stage uses abstract notation—numbers, letters, and operation signs—to represent mathematical ideas.
→ The Goal: To achieve fluency and efficiency in manipulating pure symbols.
→ Why it Works: Because the student has “done” the math and “seen” the math, the symbols (like 2 + 3 = 5) now carry deep, relational meaning rather than being arbitrary marks.
An interlaced learning system
A common “terminology conundrum” is the belief that CPA is separate from other pedagogical models, such as the Connective Model (linking symbols, language, and pictures). In reality, these theories are all interlaced and describe the same core learning journey.
→ CPA and Piaget: We use concrete tools to manage “productive struggle” or “disequilibrium.” If a student is stuck in the abstract, returning to the concrete allows them to “re-build” their mental schema.
→ CPA and Vygotsky: Within the Zone of Proximal Development (ZPD), pictorial representations often act as the “scaffold” provided by a teacher or a “More Knowledgeable Other” (MKO) to help a student reach the abstract level.
→ CPA and Dienes: We don't just use one manipulative; we use Variation. By showing “ten-ness” through blocks, then bead strings, then money (perceptual variability), we ensure students abstract the concept rather than the tool.
Why CPA is crucial for a super curriculum
When CPA is implemented within a coherent curriculum, it transforms the classroom into an inclusive environment where every child can succeed.
→ Relational understanding: It moves students away from “instrumental” rules (rules without reasons) and toward a “relational” map where they know what to do and why.
→ Depth over speed: For advanced learners, CPA provides the challenge of visualisation and communication. Proving a complex equation through a pictorial bar model requires more cognitive depth than simply calculating an answer quickly.
→ Reducing workload: When teachers use a coherent textbook series that has the CPA progression built-in, they can focus on “adaptive teaching” (responsive, in-the-moment diagnosis) rather than spending hours creating their own resources.
