Concrete Pictorial Abstract (CPA): A Maths Teaching Guide

What is a Concrete Pictorial Abstract approach?

Concrete, Pictorial, Abstract (CPA) is an effective method for teaching that offers a sustainable and deep understanding of maths to the students. Often marked as the concrete, representational, abstract framework, CPA was first proposed by the American psychologist Jerome Bruner.

For a comprehensive exploration of this approach in practice, see our mastery-based maths teaching guide.

CPA Activity Planner

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Key Takeaways

CPA fosters a deep, sustainable understanding of mathematical concepts by aligning with cognitive development. Jerome Bruner's theory of instruction, particularly his work on enactive, iconic, and symbolic modes of representation, underpins the CPA approach (Bruner, 1966). This structured progression allows pupils to build robust mental models, moving from hands-on experience to abstract thought, ensuring a profound grasp of mathematical principles rather than rote memorisation.
The concrete stage is crucial for developing foundational conceptual understanding in mathematics. Utilising physical manipulatives, as advocated by researchers like Zoltan Dienes, provides pupils with tangible experiences that bridge the gap between abstract mathematical ideas and their real-world applications (Dienes, 1960). This hands-on engagement is vital for constructing initial mental models and reducing cognitive load when encountering new concepts.
Effective CPA implementation systematically guides pupils towards relational understanding, not just instrumental knowledge. The deliberate progression through concrete, pictorial, and abstract stages is designed to cultivate relational understanding, where pupils comprehend *why* mathematical procedures work, rather than merely *how* to apply them (Skemp, 1976). This deep conceptual understanding is essential for problem-solving, transfer of learning, and long-term mathematical proficiency.
Successful CPA implementation necessitates a structured, iterative progression with ongoing assessment to ensure readiness at each stage. The "Check readiness" and "Check mastery" points highlighted in the CPA framework are crucial for effective teaching, ensuring pupils have fully grasped concepts at one stage before moving to the next (Bruner, 1966). This systematic approach prevents gaps in understanding, allowing for targeted intervention and fostering a secure foundation for subsequent learning.

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Feature	Concrete Stage	Pictorial Stage	Abstract Stage
Best For	Initial concept introduction and hands-on exploration	Bridging physical understanding to symbolic thinking	Advanced problem-solving and mathematical reasoning
Key Strength	Multi-sensory learning through physical manipulation	Visual representation without physical constraints	Efficient symbolic manipulation and generalisation
Limitation	Requires physical materials and storage space	May oversimplify complex relationships	Too abstract for early learners
Age Range	Early years to elementary (ages 3-10)	Elementary to middle school (ages 6-14)	Upper elementary onwards (ages 10+)

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Flow diagram showing CPA approach: concrete objects lead to pictorial representations, then to abstract symbols — Flow diagram: Concrete Pictorial Abstract (CPA) Teaching Approach

CPA approach is a crucial strategy to teach maths for mastery in Singapore. Through our work with schools, we have seen first hand how physical experiences can shape thinking. Many of the concepts children encounter within the curriculum are too abstract to fully understand during their early exposure. Learners need to be able to explore the problem using multi sensory approaches.

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A sequential process infographic showing the Concrete-Pictorial-Abstract (CPA) math learning model, including readiness and mastery checks between each stage. — CPA Learning Process

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The concrete pictorial abstract (CPA) approach is a widely used method to teach mathematics that begins with real-world objects and ends with abstract concepts. This approach emphasises conceptual understanding and helps students develop mathematical thinking by using a combination of real objects, block models, pictorial models, and bar and part-whole models.

The CPA approach is effective in helping students understand mathematical concepts at a deeper level. By starting with concrete objects and gradually moving towards abstract concepts, students develop a strong understanding of the underlying concepts of mathematics. This helps to build a solid foundation for future learning and problem-solving.

Furthermore, the CPA approach helps to promote student engagement and interest in mathematics. By using real-world examples and visual aids, the CPA approach helps to make mathematics more meaningful and relevant to students. This can help to increase motivation and interest in the subject.

The first stage of the CPA approach involves using real objects to help students understand mathematical concepts. For example, when teaching addition, teachers can use actual objects such as apples or pencils to help students count and add them up. This helps students develop a concrete understanding of the concept of addition, rather than just memorizing the procedure. This approach also supports students' working memory by providing physical anchors for abstract ideas.

In the second stage, block models are used to represent real-world objects. Blocks of different colors or shapes can be used to help students visualise mathematical problems. For example, blocks can be used to demonstrate the relationship between multiplication and area. This helps students to see the connection between mathematical concepts and the real world. Teachers often use questioning techniques during this stage to deepen understanding.

The third stage involves using pictorial models to represent mathematical problems. Pictorial models help students visualise mathematical concepts without the use of physical objects. For example, a pictorial model can be created to represent a fraction, which helps students understand the concept of parts of a whole. This visual representation serves as crucial scaffolding between concrete and abstract thinking.

The final stage of the CPA approach involves using bar and part-whole models to represent abstract mathematical concepts. Bar models are used to represent the relationship between two quantities, while part-whole models are used to represent the relationship between a part and a whole. For example, bar models can be used to teach fractions, and part-whole models can be used to teach the concept of percentages. Teachers can use this stage for assessment for learning to gauge student understanding.

Our research has also shown us that this teaching approach shouldn't be limited to just Maths. The CPA method develops crucial thinking skills that transfer across subjects, and teachers can provide effective feedback at each stage to support learning.across all areas of learning.

Research by Hattie and Timperley (2007) identifies effective feedback as having one of the highest effect sizes in education, while Dylan Wiliam (2011) provides practical frameworks for embedding formative assessment.

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The CPA approach offers numerous educational benefits that enhance both teaching effectiveness and student learning outcomes.

There are several benefits to using the Concrete Pictorial Abstract (CPA) approach in the classroom. Some of these benefits include:

Improved Conceptual Understanding: The CPA approach helps students develop a deeper understanding of mathematical concepts beginning with tangible items and progressively moving towards abstract concepts.
Increased Engagement: The use of real-world examples and visual aids helps to make mathematics more meaningful and relevant to students, increasing their engagement and interest in the subject.
Enhanced Problem-Solving Skills: By developing a strong understanding of mathematical concepts, students are better equipped to solve complex problems.
Greater Confidence: As students develop a deeper understanding of mathematical concepts, they become more confident in their ability to succeed in mathematics.
Supports Diverse Learning Styles: The CPA approach offer multiple ways for students to engage by incorporating hands-on activities, visual aids, and abstract concepts.

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Successfully implementing the CPA approach in the classroom requires careful planning, appropriate resources, and a structured progression through each stage.

Here are some tips for implementing the Concrete Pictorial Abstract approach in the classroom:

Start with Concrete Objects: Begin by using real-world objects to help students understand mathematical concepts.
Use Visual Aids: Incorporate visual aids such as block models, pictorial models, and bar and part-whole models to help students visualise mathematical problems.
Provide Hands-On Activities: Offer hands-on activities that allow students to manipulate objects and explore mathematical concepts.
Encourage Discussion: Facilitate discussions about mathematical concepts to help students develop a deeper understanding of the subject.
Provide Opportunities for Practise: Give students ample opportunities to practise mathematical problems using the CPA approach.
Adapt to Student Needs: Be flexible and adapt the CPA approach to meet the specific needs of your students.

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Overcoming Common CPA Implementation Challenges

One of the most significant challenges teachers face when implementing CPA approaches is the temptation to rush through the concrete and pictorial stages to reach abstract mathematics more quickly. This pressure, often driven by curriculum demands or assessment schedules, can undermine the entire approach. Students need sufficient time at each stage to develop deep conceptual understanding before progressing. Research by Richard Skemp demonstrates that relational understanding, built through meaningful engagement with concrete and visual representations, creates more strong mathematical knowledge than instrumental understanding gained through premature abstraction.

Another common pitfall involves selecting inappropriate concrete materials that confuse rather than clarify mathematical concepts. For instance, using colourful counters for place value work may inadvertently focus attention on colour rather than mathematical structure. Teachers should choose materials that highlight the specific mathematical relationships being taught whilst minimising cognitive distractions. Similarly, when creating pictorial representations, clarity and mathematical accuracy must take precedence over aesthetic appeal.

Successful CPA implementation requires careful assessment of when students are ready to transition between stages. Teachers should look for evidence that students can manipulate concrete materials confidently and explain their thinking before moving to pictorial work. Regular formative assessment through questioning and observation ensures that each student progresses at an appropriate pace, maintaining conceptual understanding throughout their mathematical process.

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Adapting CPA for Different Age Groups

The effectiveness of CPA approaches fundamentally depends on age-appropriate adaptation, as cognitive development significantly influences how students process mathematical concepts. Early years practitioners should emphasise extended concrete manipulation, allowing children ample time to explore physical materials before introducing pictorial representations. Jerome Bruner's developmental theory supports this approach, suggesting that younger learners require substantial hands-on experience to build secure mathematical foundations.

Secondary educators face different challenges, as older students may resist concrete materials they perceive as childish. However, research by John Sweller demonstrates that even adolescent learners benefit from carefully selected manipulatives when encountering unfamiliar concepts such as algebraic expressions or geometric proofs. The key lies in presenting concrete materials as sophisticated tools for problem-solving rather than elementary aids.

Successful age-specific implementation requires teachers to adjust the pace and complexity of transitions between CPA stages. Primary students might spend weeks moving between concrete and pictorial work, whilst secondary learners may progress more rapidly through stages whilst still requiring visual supports. Consider students' prior mathematical experiences and confidence levels when determining appropriate concrete materials and the timing of abstract introduction.

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Conclusion

The Concrete Pictorial Abstract (CPA) approach is a highly effective method for teaching mathematics and other subjects. Through initial实物 examples followed by moving towards abstract concepts, teachers can help students develop a deeper understanding of the underlying principles of the subject matter. This approach promotes student engagement, enhances problem-solving skills, and creates greater confidence in learning.

The CPA approach is not just a teaching strategy; it is a philosophy that emphasises the importance of building a strong foundation of conceptual understanding. When implemented effectively, the CPA approach can transform the way students learn and perceive mathematics, making it more accessible, meaningful, and enjoyable.

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The transformative power of CPA extends beyond individual lessons to reshape entire mathematical journeys. Students who experience consistent concrete pictorial abstract teaching develop strong mental models that support transfer of learning across topics and year groups. For instance, a child who explores fractions through physical manipulatives, visual fraction walls, and symbolic notation will more readily grasp percentage concepts later, recognising the underlying connections between these related areas of mathematical understanding.

Successful classroom implementation requires commitment to the process, particularly during the concrete and pictorial phases where students might initially appear to progress more slowly. However, this investment in foundational understanding pays dividends when students demonstrate greater independence and confidence in abstract mathematical thinking. Teachers should resist the temptation to rush towards symbolic representation, instead allowing sufficient time for deep conceptual learning at each stage.

The ripple effects of effective CPA teaching extend throughout the school community, creating a shared language for mathematical discussion and developing

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Frequently Asked Questions

What is the Concrete Pictorial Abstract approach?

The Concrete Pictorial Abstract (CPA) approach is a method for teaching that helps students develop a deep understanding of mathematical concepts. It involves three stages: using real objects (Concrete), using visual representations (Pictorial), and using symbolic notation (Abstract).

How do I implement the CPA approach in the classroom?

To implement CPA, start with concrete materials for initial concept introduction, then move to pictorial representations to bridge understanding, and finally use abstract symbols for advanced problem-solving. Ensure you check readiness at each stage and provide opportunities for mastery.

What are the benefits of using CPA in teaching?

CPA promotes multi-sensory learning, supports conceptual understanding, and enhances problem-solving skills. It also increases student engagement by making abstract concepts more relatable and meaningful.

What are common mistakes when using CPA?

Common mistakes include skipping stages, not checking readiness before moving on, and over-reliance on abstract symbols too early. It's important to ensure smooth transitions between each stage.

How do I know if CPA is working?

To determine if CPA is effective, observe students' ability to apply concepts in different contexts, their engagement levels, and their mastery of abstract symbols. Regular assessments and feedback can also help assess progress.

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Written by the Structural Learning Research Team

Reviewed by Paul Main, Founder & Educational Consultant at Structural Learning

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Concrete Pictorial Abstract (CPA): A Maths Teaching Guide

Paul Main