Making Maths Real

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When I start working with a new child in maths, I often find myself explaining to parents that I will use a range of tools that their child can touch and move to help teach concepts.  This is regardless of the age of the child.  I prepare the child and the adult to use materials they maybe used in younger classes and I find that once prepared, the parents accept this and the children enjoy it.  But why do I use this approach?

Well, this approach does have a name, CPA Approach.  This stands for Concrete, Pictorial and Abstract Approach.  It is helpful for teaching all ages and stages maths, even adults!  It is about giving learners the tools to solve the problems in front of them.  It links previous learning to their new abstract learning.  But, what does it really mean?

Lets start by looking at the stages.


This is the stage where children are “doing”.  In traditional teaching methods the teacher will demonstrate the problem and the child is expected to follow the process to complete follow up questions themselves.  In CPA the learning is brought to life.  For example, if children are learning to add they might have 2 dinosaurs whilst I have another 2.  The child can then count the number of dinosaurs altogether, doing the sum 2+2.  From this we can move towards using cubes, glass counters or other such materials.



This is the “seeing” stage.  Drawing can help children learn more complex theories such as division or multiplication.  It also helps them make the connection between this and the concrete stage.  So we might have 3 groups of dinosaurs with 4 dinosaurs in each group.  This equation is 3×4.




Finally, we have the abstract or “symbolic” stage.  Children here use abstract symbols to solve problems.  It is essential that children understand and are competent at the concrete and pictorial stages before we progress to this stage.  This is when we start using the mathematical symbols.  So instead of solving the problem of I have 2 dinosaurs and you have 3 dinosaurs, children will solve the equation 2+3=



Although there are 3 distinct stages I will move through them in a cyclical fashion, reinforcing earlier learning and using it to extend to new learning.  I will use a range of tools for each stage and ensure it is available for the child at all lessons.  This helps children develop a range of strategies for solving problems.

Difference Between Tutoring and School

Often in school the concrete materials are removed by the time the child reaches primary 3 or 4.  This is because they can be considered a distraction or babyish.  Even if available, children may be hesitant to use them as they do not want teased by their classmates.  However, this can lead to children being forced to encounter abstract concepts too early leading to children missing out a key step in their mathematical understanding.

Whilst children can not use concrete materials forever, I firmly believe this method helps build understanding and in turn helps the child develop confidence.  It helps children clarify their mathematical thinking and most importantly builds a strong foundation for future learning.