The concrete pictorial abstract (CPA) approach is a way to build children’s mathematical understanding through carefully structured progression, incorporating physical objects, visual representations and abstract symbols.
When implemented effectively, it creates a bridge between the tangible world and abstract mathematical concepts, allowing children to grasp the real-world relevance of maths and develop a solid foundation for more advanced concepts.
As a teacher, it can sometimes feel overwhelming to teach a new concept to a class with a wide variety of abilities, but CPA can really help, as in the example of teaching addition.
1. Start concrete
Begin by introducing physical objects children can manipulate with their hands, such as fruit, blocks or counters, showing them that maths is everywhere. During this “hands-on” stage, encourage children to combine groups of objects, this is the very beginning of developing an understanding of basic addition.
2. Move to pictorial
Once children are comfortable combining groups and finding a total with the concrete materials, it’s time to transition to visual representations. Encourage children to use drawings, diagrams or pictures representing the concrete objects. No art skills are required; the visual representation could be as simple as a dot, a dash or a squiggle, as long as the child is representing each individual object with a visual mark.
At this stage, remind the children of the previous stage and encourage them to solve similar problems using drawings or representations of those objects. A dry-wipe pen and mini whiteboard are ideal for this stage as they allow for drawings to be added and taken away fairly quickly.
3. Progress to abstract
Finally, it’s time to introduce abstract symbols like numbers and mathematical notation. At this stage, children should be ready to work with purely symbolic representations. Remind the children of what each number and symbol represents as they record their addition problems from the previous two stages in this abstract manner. It is essential to explicitly link these three stages so children can see how the abstract symbols relate to the pictures and concrete objects.
Making clear connections
The power of CPA lies in its flexibility. Don’t hesitate to revisit earlier stages if children struggle with abstraction; be prepared to move back and forth between stages as needed. The ultimate goal is to help every child build a robust, conceptual understanding of mathematics.
All children can benefit from this approach, don’t be tempted to reserve manipulatives for those children who are struggling. Encourage the high-achieving children to use concrete and pictorial methods to deepen their understanding and explain their reasoning.
When children face challenging problems, prompt them to model the problem using concrete objects, draw a pictorial representation and write the abstract mathematical equation.
You can also encourage children to explain their thinking using all three representations. You can do this by asking questions like: “Can you show me that using the blocks?”, “How would you draw a picture of what you just did?” or “Now, how can we write that as an equation?”
The power of CPA lies in using all three representations simultaneously, not as separate stages. Allowing children to make strong connections between concrete reality and abstract mathematics, by consistently implementing these strategies, you’ll create a classroom environment where children develop a deep, conceptual understanding of mathematics, rather than relying on rote memorisation.
Michelle Windridge is a lecturer in education at University College Birmingham, and a former assistant headteacher and EYFS aspiration leader
For the latest research, pedagogy and classroom advice, sign up for our weekly Teaching Essentials newsletter
