Concrete-Pictorial-Abstract (CPA): Guide to Building Mathematical Understanding and Supporting SEN

Mathematics can feel abstract and disconnected from reality for many pupils. Numbers, symbols, and operations often exist in a world that seems far removed from their everyday experiences. This is where the Concrete-Pictorial-Abstract (CPA) approach becomes invaluable, it provides a bridge between the tangible world pupils know and the abstract mathematical concepts they need to master.

What is the CPA Approach?

The CPA approach is a three-stage teaching method that moves pupils progressively from hands-on experiences to abstract mathematical thinking. Developed by Jerome Bruner and refined by educational researchers, this approach has become increasingly prominent in UK schools, particularly following the success of Singapore’s mathematics programmes and the recommendations in various Ofsted reports.

This approach recognises that learning happens most effectively when pupils can build understanding through multiple representations of the same concept. Think of it as building a house: you need a solid foundation (concrete), strong walls (pictorial), and a completed structure (abstract). Each stage supports the next, creating a comprehensive understanding that pupils can rely on throughout their mathematical journey.

The CPA approach aligns particularly well with the National Curriculum’s emphasis on mathematical reasoning and problem-solving, supporting pupils in developing fluency, reasoning, and problem-solving skills across all key stages.

The Three Stages Explained

Stage 1: Concrete

In this initial stage, pupils work with physical objects and manipulatives to explore mathematical concepts. They might use Numicon pieces to understand number bonds, Cuisenaire rods to explore fractions, or place value counters to grasp the base-ten system. This hands-on experience allows pupils to see, touch, and manipulate mathematical ideas.

The concrete stage is particularly powerful because it engages multiple senses and connects abstract concepts to physical reality. When a pupil physically combines two groups of objects to understand addition, they’re not just memorising a procedure, they’re experiencing what addition actually means.

Stage 2: Pictorial

Once pupils have developed a solid concrete understanding, they transition to working with visual representations. This might include drawings, diagrams, bar models, or other visual representations that capture the mathematical concept without requiring physical manipulation.

The pictorial stage serves as a crucial bridge between concrete and abstract thinking. Pupils learn to see mathematical relationships through visual patterns and representations. A pupil might draw circles to represent groups in a multiplication problem or use number lines to visualise addition and subtraction.

Stage 3: Abstract

Finally, pupils work with numbers, symbols, and mathematical notation, the traditional language of mathematics. At this stage, they can manipulate mathematical symbols confidently because they understand what those symbols represent in concrete and pictorial terms.

Pupils who have progressed through all three stages don’t just know that 3 + 4 = 7; they understand what that equation means in terms of combining physical objects and can visualise the process mentally.

Supporting SEN Through CPA

The CPA approach is particularly valuable for supporting pupils with Special Educational Needs, offering multiple pathways to mathematical understanding that can accommodate diverse learning differences.

Benefits for Different SEN Categories

Pupils with Learning Difficulties: The concrete stage provides essential scaffolding for pupils who struggle with abstract concepts. Physical manipulatives allow them to experience mathematical relationships in a tangible way, building confidence and understanding gradually.

Pupils with Autism Spectrum Conditions: The structured, predictable progression of CPA can provide the consistency many pupils with autism need. The visual and concrete elements help make abstract concepts more accessible and less anxiety-provoking.

Pupils with ADHD: The hands-on nature of the concrete stage can help maintain engagement for pupils who struggle with attention. The movement and manipulation involved in using concrete materials can support their learning whilst keeping them focused.

Pupils with Dyslexia: Visual and concrete representations can bypass some of the language processing difficulties that might otherwise impede mathematical understanding. The multi-sensory approach supports different learning pathways.

Pupils with Working Memory Difficulties: The CPA approach reduces cognitive load by breaking complex concepts into manageable stages. Pupils can rely on concrete and pictorial supports to supplement their working memory.

Practical SEN Strategies

When implementing CPA for SEN pupils, consider providing additional time at each stage, particularly the concrete phase. Some pupils may need extended periods working with manipulatives before they’re ready to move to pictorial representations.

Use consistent manipulatives and visual representations to build familiarity and reduce cognitive load. Create personal visual dictionaries or symbol banks that pupils can refer to throughout their mathematical journey.

Consider peer support and collaborative learning, where pupils can learn from each other’s approaches to using concrete materials and creating pictorial representations.

Implementing CPA in Your UK Classroom

Aligning with National Curriculum Requirements

The CPA approach supports all three aims of the National Curriculum for mathematics: fluency, reasoning, and problem-solving. The concrete stage builds fluency through repeated experience, the pictorial stage develops reasoning through visual representation, and the abstract stage enables sophisticated problem-solving.

When planning lessons, consider how each stage supports the specific objectives you’re addressing. For instance, when teaching fractions in Year 3, pupils might use fraction walls (concrete), draw fraction diagrams (pictorial), and work with fraction notation (abstract).

Assessment and Progression

Use the CPA stages as a framework for assessment. Can pupils explain a concept using manipulatives? Can they draw or diagram their thinking? Can they work with abstract symbols whilst still connecting back to concrete meaning?

Look for pupils who can move fluidly between stages, this flexibility indicates deep understanding. A pupil who can solve 15 – 8 using place value counters, draw the process using a number line, and explain their thinking using formal notation demonstrates mastery across all three levels.

For pupils with SEN, assessment might focus more heavily on concrete and pictorial understanding, recognising that these stages represent significant mathematical achievement in their own right.

Practical Classroom Management

Start with a core set of manipulatives that align with your school’s mathematics scheme. Popular choices in UK schools include Numicon, place value counters, Cuisenaire rods, and fraction tiles. Establish clear routines for distributing, using, and storing these materials.

Consider creating mathematics toolkits for each pupil, particularly those with SEN, containing their preferred manipulatives and visual supports. This personalised approach can reduce anxiety and support independence.

Evidence Base and Research

Research from the Education Endowment Foundation and various studies on mathematics mastery approaches consistently show that CPA implementation leads to improved outcomes, particularly for lower-attaining pupils and those with SEN.

The approach has been successfully implemented in thousands of UK schools through various mathematics mastery programmes, with Ofsted noting particular benefits for pupil engagement and mathematical reasoning.

Studies specifically focusing on SEN pupils show that the multi-sensory nature of CPA can significantly improve mathematical understanding and confidence, with many pupils making accelerated progress when previously they had struggled with traditional abstract approaches.

Common Challenges and Solutions

Time and Curriculum Pressure

The CPA approach may seem time-intensive, but it actually supports deeper learning that reduces the need for reteaching. Pupils who understand concepts thoroughly from the beginning can apply their knowledge more flexibly and require less remediation.

Resource Constraints

Start small with versatile manipulatives that can support multiple mathematical concepts. Many effective concrete materials can be made or sourced cheaply, counting bears, place value counters, and simple fraction strips provide excellent value for money.

Differentiation Across Ability Ranges

Use the CPA stages flexibly to support differentiation. Higher-attaining pupils might move quickly through concrete and pictorial stages, whilst pupils with SEN might spend extended time building solid foundations through hands-on experiences.

Supporting Teaching Assistants

Ensure that teaching assistants understand the CPA approach and can support pupils effectively at each stage. Provide training on how to use manipulatives purposefully and how to guide pupils through the progression from concrete to abstract thinking.

Beyond Primary Mathematics

Whilst the CPA approach is most commonly associated with primary mathematics, its principles apply across all key stages. Secondary pupils learning algebra can benefit from concrete models using algebra tiles, visual representations of equations, and gradual progression to abstract notation.

The key is maintaining the connection between concrete meaning and abstract symbols, regardless of the mathematical level. When pupils understand that algebra is about relationships between quantities, not just symbol manipulation, they become more confident and capable mathematical thinkers.

Making CPA Sustainable and Inclusive

Implementing CPA effectively requires patience and commitment, but the results are worth the effort. Start with one mathematical concept and gradually expand your use of the approach. Pay attention to how pupils respond and adjust your methods based on their needs and reactions.

Remember that effective mathematics teaching isn’t about covering material quickly, it’s about building understanding that lasts. The CPA approach provides a framework for creating that lasting understanding, one concrete experience at a time.

For pupils with SEN, this approach can be transformational, providing access to mathematical concepts that might otherwise remain out of reach. By honouring different learning styles and needs, CPA creates more inclusive mathematics classrooms where all pupils can experience success.

The goal isn’t just to teach mathematical procedures, but to help pupils become mathematical thinkers who can approach problems with confidence, creativity, and deep understanding. When pupils have solid foundations built through concrete experiences, they’re prepared for whatever mathematical challenges they’ll encounter in their future education and beyond.

Frequently Asked Questions (FAQ)

What is the CPA approach in mathematics?

The Concrete-Pictorial-Abstract (CPA) approach is a progressive teaching method that helps pupils understand mathematical concepts. It starts with hands-on experiences using physical objects (concrete), moves to visual representations like diagrams and models (pictorial), and finally introduces mathematical symbols and notation (abstract).

How does the CPA approach support pupils with SEN?

The CPA approach is highly effective for pupils with SEN because it provides multiple entry points to learning. The concrete stage offers a tangible, multi-sensory foundation that is crucial for pupils with learning difficulties, autism, or ADHD. The progressive structure reduces cognitive load and helps build confidence before introducing more challenging abstract concepts.

Can the CPA approach be used in secondary schools?

Yes, the CPA approach is applicable across all key stages, including secondary education. While the manipulatives might change (e.g., using algebra tiles instead of counting bears), the principle of building understanding from concrete to abstract remains the same. It is particularly useful for introducing complex topics like algebra and geometry.

How do I know when a pupil is ready to move to the next stage?

Pupils are ready to move to the next stage when they can confidently explain their thinking using the current representation. A flexible understanding is key; a pupil should be able to move back and forth between stages, for example, by using a pictorial model to explain an abstract equation. Assessment should focus on their ability to connect the different representations.

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