What is the CPA approach in maths?

By Jemma Z Smith

CPA (concrete, pictorial, abstract) is a maths teaching approach that has grown in popularity in recent years. It involves using tangible objects for children to add, subtract, multiply or divide. They then progress to using pictorial representations of the objects and ultimately, abstract symbols. This way of learning is one of the foundations of Singaporean maths, now being adopted in the UK.

Concrete

The tangible, ‘handling’ stage uses physical objects to solve problems. For example:

There are four buttons in the pot. Stanley has two buttons in his hand. How many buttons are there altogether?

In this problem, the children might first count out actual buttons before moving on to non-related objects such as counters to represent the buttons. Of course, almost anything can be used to help a child master this stage from pencils to plastic pots, to counting beans and coins. You don’t need to spend money on concrete resources to support learning at home.

Pictorial

Next, learners move on to making their own representations. Returning to the example in the paragraph above, the buttons could be drawn as four circles on one part of the paper and two more on another part, representing the pot and the hand. Children can then count each small circle and get the same answer as they would have done with the buttons.

Building or drawing a model makes it easier for children to grasp more difficult concepts, particularly when working with fractions. It enables learners to see how much of a ‘whole one’ has been shaded in and visualise the problem more clearly.

Abstract

This is where children use abstract symbols to solve maths problems.

Brooke has ten sweets and Millie has twelve sweets. How many sweets do they have altogether?

Children at the abstract stage would be able to solve the problem by writing it out as 10 + 12 = 22.

Before moving onto this stage, children need a secure understanding of mathematical vocabulary; eg ‘altogether’, ‘add’ and ‘plus’ all mean that the numbers are increasing.

Although CPA has three distinct stages, it is always useful to revisit the previous stages to ensure that learners have a full grasp of each concept.

Jemma Z Smith is Director of The Education Hotel (www.educationhotel.co.uk)

What if my Year 6 child still uses props for maths?

Should you be concerned? The answer is that the use of concrete and pictorial resources is not only for very young children or children who find new concepts tricky.

While running a SATs booster class, I used the bar model, a way of visualising addition, subtraction, multiplication and division using blocks, with a group of Year 6 children. They had to work out how much of one type of liquid was used in a ratio word problem. The lightbulb moments started to happen: what a privilege to witness!

Once learners understand the mechanics of problems, they can apply their knowledge from the concrete and pictorial stages into the abstract final answer and, if they are really secure, even teach it to someone else as well, demonstrating true mastery of parents shouldn’t be concerned if their children are They are using them to their advantage and building a solid base of understanding that will see them through into later years.