Concrete pictorial abstract approaches in the classroom

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.  

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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.

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 visualize 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 visualize 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.

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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 visualize 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 Practice: Give students ample opportunities to practice 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 robust 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 journey.

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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 robust 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 collaborative learning environments. When students can articulate their thinking using concrete materials and visual representations, peer-to-peer explanation becomes more meaningful, and mathematical discourse flourishes naturally in the classroom.

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Further Reading

CPA approach research

Mathematics representations

Bruner enactive-iconic-symbolic

For teachers looking to examine deeper into the Concrete Pictorial Abstract approach, here are some relevant research papers:

• Bruner, J. S. (1966). *Toward a theory of instruction*. Cambridge, MA: Harvard University Press.
• Anghileri, J. (2000). Discussion, exposition and practice 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.