Concrete Pictorial Abstract (CPA): A Maths Teaching Guide
The CPA approach explained: concrete manipulatives, pictorial representations, and abstract symbols. Based on Bruner's model with worked examples across primary maths.
Concrete Pictorial Abstract (CPA): A Maths Teaching Guide describes a way to teach maths. Learners first handle objects, then show the same idea with diagrams. Finally, they use numbers, symbols and formal notation. The approach draws on Bruner’s theory of representation (Bruner, 1960), but in class it works best when teachers connect the three forms instead of treating them as a fixed ladder.
For example, a Year 4 class learning fractions might fold paper strips and shade matching bar models. They might then write 3/4 + 1/4 = 1 and discuss how each representation shows the same relationship. This matters because well-chosen concrete and pictorial models can secure mathematical meaning. Poorly linked resources, however, can add cognitive load or leave learners dependent on apparatus.
Concrete Pictorial Abstract Approach
Concrete, Pictorial, Abstract (CPA) is an effective teaching method. It helps learners build a lasting and deep understanding of maths. CPA is also known as the concrete, representational, abstract framework. It is linked to Jerome Bruner’s work on enactive, iconic and symbolic representation, and was later developed and used in Singapore mathematics education.
Mastery learning in mathematics is detailed in our guide. It gives educators practical classroom strategies. We cite research by Bloom (1968), Carroll (1963) and Guskey (1997). This resource aids effective learner progress.
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Key Takeaways
- CPA builds 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 learners 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 important for developing foundational conceptual understanding in mathematics. Utilising physical manipulatives, as advocated by researchers like Zoltan Dienes, provides learners 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 learners towards relational understanding, not just instrumental knowledge. The deliberate progression through concrete, pictorial, and abstract stages is designed to cultivate relational understanding, where learners 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 important for effective teaching, ensuring learners 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 developing a secure foundation for subsequent learning.
| 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+) |
CPA approach is a important 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.
The CPA approach teaches maths using objects first, then abstract ideas. This helps learners understand concepts (Bruner, 1966). They develop mathematical thought with objects and models (Skemp, 1976; Piaget, 1954).
Research consistently finds the CPA approach helps learners grasp maths concepts. Start with objects, then move to abstract ideas for strong understanding (Bruner, 1966; Leong, Ho & Cheng, 2015). This builds a firm base for later learning and problem solving.
Research shows that CPA boosts learner maths engagement. It uses visuals and real examples, making maths more relevant. This method helps learners to find maths more interesting. This increases their motivation.
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CPA starts with real objects, helping learners grasp maths concepts. For instance, use apples to teach addition, (Bruner, 1966). This solidifies understanding over simple memorisation. It also aids working memory (Baddeley, 2000) by anchoring abstract concepts.
Block models use real objects (Bruner, 1966). Learners use coloured blocks to picture maths (Dienes, 1960). Blocks link multiplication and area, connecting concepts to life. Questioning builds learner understanding (Vygotsky, 1978).
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Use pictorial models to show maths problems. These models help learners see maths ideas, not objects. For example, use pictures for fractions; learners grasp parts of a whole (Bruner, 1966). Visuals support the move from concrete to abstract thought (Piaget, 1936).
Abstract maths uses bar and part-whole models (Bruner, 1966). Bar models show relationships between two amounts. Part-whole models show part and whole links, a hallmark of the Singapore Maths approach (Leong, Ho & Cheng, 2015). Use this stage to check learner understanding (Black & Wiliam, 1998).
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Our research has also shown us that this teaching approach shouldn't be limited to just Maths. The CPA method develops important thinking skills that transfer across subjects, and teachers can provide effective feedback at each stage to support learning.across all areas of learning.
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Feedback can make a clear difference to learner outcomes (Hattie & Timperley, 2007). Dylan Wiliam (2011) shares practical ways to use formative assessment in classrooms.
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Concrete Pictorial Abstract (CPA): A Maths Teaching Guide
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The CPA approach offers numerous educational benefits that enhance both teaching effectiveness and learner learning outcomes.
There are several benefits to using the Concrete Pictorial Abstract (CPA) approach in the classroom. Some of these benefits include: Use it as a starting point for professional discussion: identify the learner's current need, record evidence from more than one lesson, and agree the next classroom adjustment with the SENCO or family.
- Improved Conceptual Understanding: The CPA approach helps learners 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 learners, increasing their engagement and interest in the subject.
- Enhanced Problem-Solving Skills: By developing a strong understanding of mathematical concepts, learners are better equipped to solve complex problems.
- Greater Confidence: As learners 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 learners to engage by incorporating hands-on activities, visual aids, and abstract concepts.
- Start with Concrete Objects: Begin by using real-world objects to help learners understand mathematical concepts.
- Use Visual Aids: Incorporate visual aids such as block models, pictorial models, and bar and part-whole models to help learners visualise mathematical problems.
- Provide Hands-On Activities: Offer hands-on activities that allow learners to manipulate objects and explore mathematical concepts.
- Encourage Discussion: support discussions about mathematical concepts to help learners develop a deeper understanding of the subject.
- Provide Opportunities for Practise: Give learners ample opportunities to practise mathematical problems using the CPA approach.
- Adapt to Learner Needs: Be flexible and adapt the CPA approach to meet the specific needs of your learners.
- Learners apply theory through classroom tasks.
- Abstract ideas are scaffolded into usable lesson moves.
- Challenge stays in the task, not in decoding instructions.
- Bruner, J. S. (1966). *Towards a theory of instruction*. Cambridge, MA: Harvard University Press.
- Anghileri, J. (2000). Discussion, exposition and practise in primary mathematics classrooms. *Mathematics Education Research Journal, 12*(3), 179-193.
- Wong, N. Y., & Lee, P. Y. (2009). The Model Method: Singapore Children's Tool for Representing and Solving Algebraic Word Problems. *Journal of Mathematics Education at Teachers College, 1*(1), 32-40.
- McKendree, J., Small, C., Stenning, K., & Conlon, T. (2002). The role of representation in understanding and problem solving: theoretical and practical issues. *Educational Psychology, 22*(5), 551-566.
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: Use it as a starting point for professional discussion: identify the learner's current need, record evidence from more than one lesson, and agree the next classroom adjustment with the SENCO or family.
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Concrete Pictorial Abstract (CPA): A Maths Teaching Guide
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Overcoming Common CPA Implementation Challenges
Teachers often rush the concrete and pictorial stages of CPA (Richard Skemp). Learners need enough time to understand concepts fully. Rushing to abstract maths undermines the approach. Relational understanding, built with objects and visuals, creates stronger maths knowledge (Richard Skemp).
Poor resources can confuse learners and weaken maths understanding. For example, coloured counters can draw attention away from place value structure (Uttal et al., 2009). Teachers should choose materials that show maths relationships clearly (McNeil & Jarvin, 2007). Clear, accurate pictures work better than images that only look attractive.
Teachers check if learners are ready to progress through CPA stages. Do learners show confidence with materials and explain concepts clearly? Questioning acts as formative assessment to check learner understanding. This supports each learner's progress (Bruner, 1966; Skemp, 1976; Piaget, 1952).
Adapting CPA for Different Age Groups
CPA effectiveness relies on adapting to age (Bruner). Early years teachers should stress concrete tools. Give learners time to explore materials before pictures. This builds strong maths foundations (Bruner). Use it as a starting point for professional discussion: identify the learner's current need, record evidence from more than one lesson, and agree the next classroom adjustment with the SENCO or family.
Manipulatives can support older learners as well as younger ones, particularly when learners are still novices at a concept; they can help learners grasp algebra and geometry concepts. Frame them as problem-solving aids.
Teachers must adjust CPA pace for each age group. Younger learners might need weeks on concrete and pictorial tasks. Older learners progress faster but still need visuals (Bruner, 1966). Consider learners' past maths and confidence (Skemp, 1976) for choosing materials and abstract concepts (Piaget, 1936).
Next Steps for Classroom Practice
The Concrete Pictorial Abstract (CPA) approach helps learners understand maths. Teachers use real examples before abstract ideas. This gives learners a deeper grasp of the topic. CPA improves problem-solving and builds learner confidence (Jerome Bruner, 1966).
Researchers like Bruner (1966) show CPA builds conceptual understanding. It changes how learners perceive maths, making it more accessible. Skemp (1976) also supports this approach, increasing enjoyment.
CPA methods reshape maths learning over time. Learners build firm mental models using concrete, pictorial, abstract steps. This supports transfer, as Bransford et al. (2000) found. For example, fractions taught visually help learners understand percentages later (Bruner, 1966).
Research by Bruner (1966) shows concrete learning helps learners later. Give learners time with pictures and objects, as suggested by Clements and Sarama (2009). Avoid rushing to abstract maths too quickly. Learners gain confidence if they deeply understand each stage, according to Skemp (1976).
CPA helps learners build a deeper understanding of maths ideas (Bruner, 1966). When learners grasp these ideas, they can use their knowledge to solve new and harder problems (Skemp, 1976). It also builds problem-solving skills that can transfer across subjects (Boaler, 1998).
Over time, CPA can support a classroom culture where learners explore maths, take risks, and talk about ideas with peers and teachers (Vygotsky, 1978). In short, CPA builds maths understanding (Bruner, 1966). Learners apply knowledge to new problems when they understand the concepts (Skemp, 1976).
Problem-solving skills can transfer across subjects (Boaler, 1998). Learners can also explore ideas and discuss maths with more confidence (Vygotsky, 1978).
Frequently Asked Questions
Concrete Pictorial Abstract in Maths
Bruner (1966) described the Concrete Pictorial Abstract (CPA) approach. This method helps learners understand maths concepts well. Learners use objects (Concrete), then pictures (Pictorial). Finally, they use symbols (Abstract) once the concept is secure.
Using CPA in the Classroom
This approach helps learners build strong maths foundations (Bruner, 1966). First, use physical objects to introduce new ideas. Then use drawings, so learners link concrete examples to abstract ideas (Kilpatrick et al., 2001).
Finally, use symbols for more advanced problem-solving (Skemp, 1971). Check that learners are ready before you move on.
What are the benefits of using CPA in teaching?
Bruner (1966) and Piaget (1954) found the Concrete Pictorial Abstract approach aids learning. Skemp (1976) showed CPA boosts learners' problem-solving skills. This method also makes learning more interesting and useful.
What are common mistakes when using CPA?
These errors hinder learners' understanding (Piaget, 1936). Check learners' readiness before you progress to the next stage (Bruner, 1966). Too much focus on abstract symbols can confuse learners (Vygotsky, 1978). Smooth transitions between stages are vital for success (Ausubel, 1968).
How do I know if CPA is working?
Evidence for CPA's impact comes from learners applying concepts elsewhere. Engagement and symbol mastery also show success. Use regular checks and feedback to track learner progress (Bruner, 1966; Skemp, 1976; Piaget, 1954).
Free Resource Pack
Multi-sensory and varied resources are available now. This free pack includes posters and desk cards for learners. CPD materials are also provided for staff. Use it as a starting point for professional discussion: identify the learner's current need, record evidence from more than one lesson, and agree the next classroom adjustment with the SENCO or family.
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Limitations and Critiques
CPA is useful, but it is not a complete theory of mathematics teaching. One criticism is that the approach is often presented as a simple ladder from concrete to pictorial to abstract. Fyfe et al. (2014) argue instead for concreteness fading, where representations overlap and teachers make the links explicit. If learners only proceed through the stages, they may miss the structure that connects objects, diagrams and symbols.
A second limit concerns manipulatives. Carbonneau et al. (2013) found that physical resources can improve learning. However, the effect depends on age, task design and the level of teacher guidance. Apparatus can add extraneous cognitive load when learners focus on colour, shape or play instead of the mathematical relationship.
There are also cultural and method limits. CPA is often linked to Singapore mathematics, but its success also depends on curriculum order, teacher knowledge and system conditions. Evidence from one setting may not transfer straight into another school system without careful adaptation.
In UK classrooms, Ofsted (2023) warns that teachers need to use representations consistently across year groups. If not, learners may meet disconnected models that do not build understanding over time.
Finally, CPA can create barriers for some neurodivergent learners. Learners with dyscalculia, working memory difficulties or aphantasia may find it hard to move from objects to mental images. They may need verbal reasoning, gesture, structured diagrams or digital models alongside pictorial work.
Even with these limits, CPA remains useful. It works best when teachers use it flexibly, ask careful questions and keep the focus on mathematical meaning.
References
Bruner, J. (1960). The process of education.
Karpicke, J. (2008). The critical importance of retrieval for learning.
Vygotsky, L. (1978). Mind in society: The development of higher psychological processes.
Further Reading
Bruner's (1966) work explores learning through action, imagery, and symbols. Skemp (1976) discusses understanding maths using relational and instrumental approaches. Vygotsky (1978) highlights social interaction's role in learner knowledge construction. These papers offer insights into the Concrete Pictorial Abstract method.
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