What is the CPA approach in maths?

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 and also in Numberblocks.

CONCRETE

The tangible, ‘handling’ stage uses physical objects to solve problems. For example: There are 4 buttons in the pot. Stanley has 2 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 nonrelated 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

The next step is for learners to 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

By Jemma Z Smith

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 concepts they find more difficult. This is particularly true when working with fractions. It enables learners to see how much of a ‘whole one’ has been shaded in and they can visualise the problem more clearly.

ABSTRACT

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

Brooke has 10 sweets and Millie has 12 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; for example ‘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.

THIS IS SO DIFFERENT FROM WHEN I LEARNT MATHS: WHAT’S THE POINT? Education methods have changed a lot since we were at school. For many adults today, classroom teaching involved having methods drummed into us that we didn’t necessarily understand and which were promptly forgotten. Could you explain why we ‘put a zero’ in the ones column when multiplying by a tens number? Why we ‘knock on the next number’s door and borrow a 1’ when exchanging in column subtraction? How long division really works?

Using physical ‘concrete’ counters show us the mechanics behind these methods and gives children an understanding of what maths is, rather than rote learning for an exam and forgetting the learning when moving on to the next concept. The learning will stick for longer and be more beneficial to them as they progress to the next stage of their education. With CPA, maths is seen by teachers as a house or a pyramid. Without a strong understanding of the foundations, more complex topics such as algebra or percentages will be a struggle.

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