The Concrete-Pictorial-Abstract Approach
The CPA approach is a research-based framework for teaching mathematics that supports all learners, particularly those with SEND. Each concept is introduced in three progressive stages:
- Concrete: Pupils use physical objects (counters, Base 10, place value counters) to explore mathematical concepts hands-on.
- Pictorial: Concepts are represented visually through drawings, bar models, number lines and diagrams.
- Abstract: Pupils work with numbers and symbols once understanding is secure.
Many pupils with SEND need extended time at the concrete stage. Do not rush to abstract representations before understanding is secure.
Reducing Cognitive Load
Mathematical tasks place significant demands on working memory. Reduce cognitive load by:
- Breaking multi-step problems into separate, manageable stages
- Providing worked examples before independent practice
- Using consistent methods and layouts across the class
- Pre-teaching vocabulary before introducing new concepts
- Allowing calculators when calculation is not the learning objective
- Providing reference materials (number lines, times tables grids, formula sheets)
- Reducing the number of questions while maintaining challenge
- Using templates and writing frames for word problems
When pupils can focus on the mathematical concept rather than remembering procedures, learning deepens.
Visual Representations
Visual models make abstract concepts accessible:
- Bar models: Support problem-solving across number, fractions, ratio and algebra
- Part-whole models: Show how numbers can be partitioned and combined
- Number lines: Support counting, ordering, and understanding intervals
- Place value charts: Make the structure of number visible
- Arrays and area models: Show the structure of multiplication
- Fraction walls: Demonstrate equivalence and comparison
Display these models permanently in the classroom. Reference them consistently during teaching.
Manipulatives and Resources
Physical resources help pupils develop mathematical understanding:
- Base 10 equipment for place value
- Cuisenaire rods for fractions and algebra
- Place value counters for all operations
- Double-sided counters for part-whole
- Numicon for number sense
- Fraction tiles and circles
- Geoboards for shape and angles
- Interlocking cubes for volume
- Dice and spinners for probability
- Measuring equipment for practical maths
- 100 squares for pattern spotting
- Blank number lines for flexibility
Keep manipulatives accessible. Pupils should use them whenever they need concrete support, not just in early years.
Scaffolding Word Problems
Word problems present multiple barriers: reading demands, identifying relevant information, selecting appropriate operations, and organising working. Support pupils by:
- Reading problems aloud or providing audio versions
- Highlighting or underlining key information
- Crossing out irrelevant details in problems
- Using bar models to structure thinking
- Providing sentence starters for explanations
- Teaching question stems ("How many more?" means subtract)
- Reducing linguistic complexity while maintaining mathematical challenge
- Allowing drawing and annotation of problems
Alternative Recording Methods
Not all pupils can easily record their mathematical thinking in writing. Offer alternatives:
- Photographs of concrete models as evidence
- Voice recording to explain reasoning
- Printed or photocopied questions to reduce copying
- Partially completed working with gaps to fill
- Squared paper to support column alignment
- Technology-based recording (tablets, laptops)
- Scribed responses for some pupils
- Multiple-choice options alongside open responses
Focus assessment on mathematical understanding, not presentation.
Pre-Teaching and Overlearning
Some pupils benefit from seeing concepts before whole-class teaching:
- Introduce key vocabulary in advance with visual supports
- Pre-teach the concrete representation of new concepts
- Use short, focused pre-teaching sessions (10-15 minutes)
- Revisit and consolidate previous learning regularly
- Provide extra practice with varied examples
- Use low-stakes retrieval practice (quick quizzes, flashcards)
Overlearning helps secure basic facts and procedures, freeing up cognitive capacity for problem-solving.
Mathematical Language
Precise mathematical vocabulary is essential but can be a barrier. Support language development by:
- Displaying vocabulary with images and definitions
- Teaching multiple ways to express concepts (add, plus, sum, total)
- Using stem sentences to practise language ("___ is ___ more than ___")
- Encouraging pupils to rehearse explanations with a partner
- Accepting approximate language initially, then refining
- Creating word banks for different topics
Building Fluency
Fluency with number facts and procedures reduces cognitive load. Support fluency through:
- Short, regular practice sessions rather than long blocks
- Clear progression from concrete to abstract
- Teaching calculation strategies explicitly
- Using games and technology for engaging practice
- Celebrating incremental progress
- Allowing continued use of resources when needed
Fluency develops at different rates. Some pupils may need resources long-term while building automaticity.