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Explore the Concrete Pictorial Abstract (CPA) approach, a method that bridges hands-on learning with conceptual maths mastery.
What is the Concrete Pictorial Abstract Method?
The Concrete Pictorial Abstract (CPA) approach is a highly effective method to teaching maths that helps pupils build and deepen their understanding of mathematical concepts. The CPA approach involves three stages: Concrete, Pictorial, and Abstract. In the Concrete stage, learners use real-world objects, like tomatoes, toy cars, or building blocks, to explore maths concepts. Handling everyday items gives them a practical way to make sense of new ideas. The Pictorial stage moves learners from physical objects to visual models, such as diagrams, number lines, or bar models. These images help bridge hands-on learning and abstract thinking. Finally, in the Abstract stage, pupils use mathematical symbols and equations alone. By this point, they’re able to solve problems confidently using just numbers and notation.
The CPA method began with the pioneering work of psychologist Jerome Bruner, who identified the value of guiding learners through a careful sequence: starting with real objects, moving to pictorial representations, then progressing to abstract ideas. This structured approach is now embedded in educational practice around the world. In Singapore, the CPA framework has played a key role in achieving outstanding results in international maths assessments. Its success illustrates how this evidence-based approach aids children in primary schools across the UK in developing a deeper and more secure understanding of mathematics.
How the CPA Approach Supports Maths Mastery
The CPA approach supports maths mastery by helping pupils build a deeper, more lasting understanding of mathematical concepts. Progressing through the Concrete, Pictorial, and Abstract stages means learners first handle tangible objects, then move to visual representations, and finally work with mathematical symbols. This sequence encourages pupils to develop robust problem-solving and critical thinking skills by making connections between real-life contexts and abstract ideas.
Throughout each stage, pupils practise visualising and manipulating mathematical concepts, which not only boosts their comprehension but also reinforces long-term retention. The gradual, structured progression gives every learner the chance to experience success at each step, strengthening their confidence in maths. By allowing learners to move forward at a manageable pace, the CPA method helps prevent the frustration and anxiety often linked to new or challenging material. This nurturing approach is particularly effective for building resilience and a positive maths mindset.
Confidence is further enhanced as learners are empowered to try flexible strategies, make sense of mistakes, and transfer their learning between different types of problems. For more practical ideas on how to nurture confidence and develop maths fluency in the classroom, explore our article on strategies for building maths fluency.
Implementing CPA in the Modern Classroom
Bringing the CPA approach to life in today’s classrooms means embedding practical activities, visual supports, and structured reasoning into every maths lesson. Teachers can use a blend of physical manipulatives, visual tools, and educational technology to enrich each stage of the CPA process and make mathematical concepts accessible to all pupils.
Introducing the CPA method can sometimes present a few hurdles, but reliable solutions are well within reach. For example, making sure every pupil has access to essential resources like manipulatives and visual aids can take some planning. Schools may find success by investing in a range of teaching materials that appeal to different learning preferences, ensuring all pupils can fully participate.
Supporting teachers with ongoing professional development is key to delivering consistent, high-quality CPA lessons. Tailored training helps educators feel confident as they guide pupils through the Concrete, Pictorial, and Abstract stages, using carefully chosen resources to meet every learner’s needs. With this approach, every child can thrive on their maths journey, developing a deep understanding and long-lasting confidence. It’s natural for some teachers to feel a little unsure about moving away from familiar routines. Offering ongoing training and friendly support can help build confidence with the CPA approach, making it easier to experience the positive difference it brings to pupils’ understanding and enthusiasm for maths.
Using Sumdog to Support CPA Teaching
Sumdog is an excellent tool to support the CPA approach in maths education. Its interactive games and activities provide an engaging way for pupils to practise mathematical concepts at their own pace. Sumdog's visual aids and personalised learning paths align with the CPA stages, making it easier for teachers to integrate the approach into their lessons.
Although Sumdog does not support the Concrete stage directly, as it is an online tool, its visual aids help pupils transition from hands-on activities to the Pictorial stage. As learners progress, they can tackle more abstract problems through tailored question sets that adapt to their learning needs. Sumdog's platform also offers detailed reports and insights, enabling teachers to monitor progress and adjust instruction accordingly.
Building Confident Mathematicians with the CPA Approach and Sumdog
The CPA approach lays a strong foundation for maths mastery by helping every pupil build deep understanding, confidence, and essential problem-solving skills. By combining hands-on learning, visual supports, and abstract reasoning, this method gives all learners the tools they need to succeed—now and in the future.
If you’re looking to embed CPA effectively in your classroom, Sumdog is here to help. With engaging activities, adaptable learning paths, and real-time insights, Sumdog complements the CPA approach. Explore how Sumdog can make maths more accessible and enjoyable for your pupils, and take the next step towards confident, resilient mathematicians.
References
Leong, Y. H., Ho, W. K., & Cheng, L. P. (2015). Concrete-Pictorial-Abstract: Surveying its origins and charting its future. The Mathematics Educator, 16(1), 1-18. Available at: http://math.nie.edu.sg/ame/matheduc/tme/tmeV16_1/TME16_1.pdf (Accessed: 29 September 2025)