What is The Concrete Pictorial Abstract (CPA) Approach And How To Use It In Maths

The Concrete Pictorial Abstract approach is now an essential tool in teaching maths at KS1 and KS2, so here we explain what it is, why its use is so widespread, what misconceptions there may be around using concrete resources throughout a child’s primary maths education, and how best to use the CPA approach yourself in your KS1 and KS2 maths lessons.

The maths curriculum is far too broad to cover in one blog, so the focus here will be on specifically how the CPA approach can be used to support the teaching and learning of the four written calculation methods.

The CPA approach explained
The CPA approach illustrated by one of our Third Space Learning resources

What is the Concrete Pictorial Abstract in Maths?

The Concrete Pictorial Abstract (CPA) approach is a system of learning that uses physical and visual aids to build a child’s understanding of abstract topics.

Pupils are introduced to a new mathematical concept through the use of concrete resources (e.g. fruit, Dienes blocks etc). When they are comfortable solving problems with physical aids, they are given problems with pictures – usually pictorial representations of the concrete objects they were using.

Then they are asked to solve problems where they only have the abstract i.e. numbers or other symbols. Building these steps across a lesson can help pupils better understand the relationship between numbers and the real world, and therefore helps secure their understanding of the mathematical concept they are learning.

Origins of Concrete Pictorial Abstract Approach

Anyone working in primary mathematics education can’t fail to have noticed that the word ‘maths’ is rarely heard these days without a mention of the term ‘mastery’ alongside it. 

This is no surprise, with ‘mastery’ being the Government’s flagship policy for improving mathematics and with millions of pounds being injected into the Teaching for Mastery programme; a programme involving thousands of schools across the country.

Prior to 2015, the term ‘mastery’ was rarely used. With the constant references to high achieving Asian-style Maths from East Asian countries including Singapore and Shanghai (and the much publicised Shanghai Teacher Exchange Programme), a teacher could be forgiven for believing ‘teaching for mastery’ to be something which was imported directly from these countries.. 

The fact that the CPA approach is a key component in maths teaching in these countries only added to the misconception. 

The CPA maths model in Teaching for Mastery

To find the origins of the mastery maths approach, we need to go much further back in time and look much closer to home.  

Teaching for Mastery diagram
How CPA fits into the teaching for mastery, as explained by NCETM.

The concept of ‘mastery’ was first proposed in 1968 by Benjamin Bloom. At this time the phrase ‘learning for mastery’ was used instead. Bloom believed students must achieve mastery in prerequisite knowledge before moving forward to learn subsequent information. 

Bloom suggested that if learners don’t get something the first time, then they should be taught again and in different ways until they do.

Jerome Bruner and Concrete Pictorial Abstract

Looking more specifically at the origins of the CPA approach, we again need to go back to the teaching methods of the 1960s, when American psychologist Jerome Bruner proposed this approach as a means of scaffolding learning.  

He believed the abstract nature of learning (which is especially true in maths) to be a ‘mystery’ to many children. It therefore needs to be scaffolded by the use of effective representations and maths manipulatives

He found that when pupils used the CPA approach as part of their mathematics education, they were able to build on each stage towards a greater mathematical understanding of the concepts being learned, which in turn led to information and knowledge being internalised to a greater degree.

Many teachers mistakenly believe mastery, and specifically the CPA approach, to have been a method imported from Singapore. 

In actual fact, the Singapore Maths curriculum has been heavily influenced by a combination of Bruner’s ideas about learning and recommendations from the 1982 Cockcroft Report (a report by the HMI in England, which suggested that computational skills should be related to practical situations and applied to problems). 

To get a better handle on the concept of maths mastery as a whole, take a look at our Ultimate Maths Mastery guide.

Why use the Concrete Pictorial Abstract approach in Maths?

Pupils achieve a much deeper understanding if they don’t have to resort to rote learning and are able to solve problems without having to memorise. 

When teaching reading to young children, we accept that children need to have seen what the word is to understand it. Putting together the letters c- a- t would be meaningless and abstract if children had no idea what a cat was or had never seen a picture. 

People often don’t think of this when it comes to maths, but to children many mathematical concepts can be equally meaningless without a concrete resource or picture to go with it. This applies equally to mathematics teaching at KS1 or at KS2. 

What is a ‘Concrete’ representation in the CPA approach?

As part of the CPA approach, new concepts are introduced through the use of physical objects or practical equipment. These can be physically handled, enabling children to explore different mathematical concepts. These are sometimes referred to as maths manipulatives and can include ordinary household items such as straws or dice, or specific mathematical resources such as dienes or Numicon

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The abstract nature of maths can be confusing for children, but through the use of concrete materials they are able to ‘see’ and make sense of what is actually happening. 

Previously, there has been the misconception that concrete resources are only for learners who find maths difficult. In fact concrete resources can be used in a great variety of ways at every level. All children, regardless of ability, benefit from the use of practical resources in ensuring understanding goes beyond the learning of a procedure. 

Practical resources promote reasoning and discussion, enabling children to articulate and explain a concept. Teachers are also able to observe the children to gain a greater understanding of where misconceptions lie and to establish the depth of their understanding. 

An explanation of the concrete part of the CPA approach
Tangible objects help children to represent numbers by observing the concrete method.

What is a ‘Pictorial’ representation in the CPA approach?

Once children are confident with a concept using concrete resources, they progress to drawing pictorial representations or quick sketches of the objects. By doing this, they are no longer manipulating the physical resources, but still benefit  from the visual support the resources provide. 

Some teachers choose to leave this stage out, but pictorial recording is key to ensuring that children can make the link between a concrete resource and abstract notation. Without it, children can find actually visualising a problem difficult.

One of the most common methods of representing the pictorial stage is through the bar model which is often used in more complex multi step problem solving.

Read more: What is a bar model

An explanation of the pictorial part of the CPA approach
Visualisation through pictures is the next step as we see here.

What is an ‘Abstract’ representation in the CPA approach?

Once children have a secure understanding of the concept through the use of concrete resources and visual images, they are then able to move on to the abstract stage. Here, children are using abstract symbols to model problems – usually numerals. To be able to access this stage effectively, children need access to the previous two stages alongside it. 

For the most effective learning to take place, children need to constantly go back and forth between each of the stages. This ensures concepts are reinforced and understood.

An example of the abstract part of the CPA approach
Typically written maths problems like this will now be more approachable.

How to teach using the Concrete Pictorial Abstract method at primary school

A common misconception with this CPA model is that you teach the concrete, then the pictorial and finally the abstract. But all stages should be taught simultaneously whenever a new concept is introduced and when the teacher wants to build further on the concept. 

Bar model used in a TSL lesson
The bar model in action in a Third Space Learning 1-to-1 lesson.

When concrete resources, pictorial representations and abstract recordings are all used within the same activity, it ensures pupils are able to make strong links between each stage. 

If you’re concerned about differentiating effectively using the CPA approach, have a look at our differentiation strategies guide for ideas to get you started. Or if you’re short on time, our White Rose Maths aligned lesson slides incorporate the CPA approach into them and some are free to download and use.

Using ‘The Four Operations’ to model how CPA works

Diagram of the 'four operations' in mathematics
‘The four operations’ we approach in primary level maths.

In the following section I will be looking at the ‘four operations’ and how the CPA approach can be used at different stages of teaching them. In particular, I will examine how the 3 parts of the CPA approach should be intertwined rather than taught as 3 separate things.  

As this blog is to share ideas rather than say how the calculation methods should be taught, I am only going to cover the four operations briefly.

It’s important to take your school’s Calculation Policy into account when determining how the CPA approach can work best for you.

Teaching addition using the Concrete Pictorial Abstract approach

In the early stages of learning column addition, it is helpful for children to use familiar objects. For example, straws or lollipop sticks can be bundled into groups of ten and used individually to represent the tens and ones. 

Once children are familiar making 2-digit numbers using these resources, they can set the resources out on a baseboard to represent the two numbers in a column addition calculation.  

Lollipop sticks used to shows 10's and 1's
The concrete method sees lollipop sticks used by a teacher to differentiate between digits.

Initially children complete calculations where the units do not add to more than 9, before progressing to calculations involving exchanging/ regrouping. Alongside the concrete resources, children can annotate the baseboard to show the digits being used, which helps to build a link towards the abstract formal method.

Once children are confident using the concrete resources they can then record them pictorially, again recording the digits alongside to ensure links are constantly being made between the concrete, pictorial and abstract stages.

The next step is for children to progress to using more formal mathematical equipment. Dienes base ten should be introduced alongside the straws, to enable children to see what is the same and what is different. As confidence grows using the Dienes, children can be introduced to the hundreds column for column addition, adding together 3-digit and 2-digit numbers

Example image of column addition using dienes
A basic example of pictorial stage of column addition with Dienes.

Alongside the concrete resources children should be recording the numbers on the baseboard, and again have the opportunity to record pictorial representations.

Once children are completely secure with the value of digits and the base ten nature of our number system, Dienes equipment can be replaced with place value counters. These help children as they progress towards the abstract, as unlike the dienes they are all the same size. 

column addition example image
An example of counters as the intermediary between the concrete and abstract steps.

These should be introduced in the same way as the other resources, with children making use of a baseboard without regrouping initially, then progressing to calculations which do involve regrouping.

As with the other equipment, children should have the opportunity to record the digits alongside the concrete resources and to progress to recording pictorially once they are secure. The place value counters can be used to introduce children to larger numbers, calculating column addition involving the thousands and then the ten thousands column.

Read also: How To Teach Addition For KS2 Interventions In Year 5 and Year 6

Teaching subtraction using the Concrete Pictorial Abstract

The method for teaching column subtraction is very similar to the method for column addition. Children should start by using  familiar objects (such as straws) to make the 2-digit numbers, set out on a baseboard as column subtraction.

example image of column subtraction
The same lollipop sticks can now be used for subtraction.

To begin with, ensure the ones being subtracted don’t exceed those in the first number. Once children are confident with this concept, they can progress to calculations which require exchanging. As with addition, the digits should be recorded alongside the concrete resources to ensure links are being built between the concrete and abstract. 

Once secure with using the concrete resources, children should have the opportunity to record pictorially, again recording the digits alongside.

As with addition, children should eventually progress to using formal mathematical equipment, such as Dienes. 

example image of column subtraction using dienes
An example of Dienes illustrating subtraction.

These should be introduced alongside the straws so pupils will make the link between the two resource types. Digits are noted down alongside the concrete resources and once secure in their understanding children can record the Dienes pictorially, to ensure links are built between the concrete and abstract.

Once secure with the value of the digits using Dienes, children progress to using place value counters.

Column subtraction using place counters
Tackling subtraction using place value counters.

Again, the counters enable children to work concretely with larger numbers, as well as bridging the gap from the use of Dienes to the abstract. Along with the counters, children should be recording the digits and they should have the opportunity to record pictorially once confident with the method using concrete resources.

Read also: How to Teach Subtraction for KS2 Interventions in Year 5 and Year 6

Teaching multiplication using the Concrete Pictorial Abstract approach

The grid method is an important step in the teaching of multiplication, as it helps children to understand the concept of partitioning to multiply each digit separately. 

Pupils can begin by drawing out the grid and representing the number being multiplied concretely. For example, 23 x 3 can be shown using straws, setting out 2 tens and 3 ones three times. This way, children can actually see what is happening when they multiply the tens and the ones. 

multiplying straws grid method
Consistent object choices through the operations can assist a child’s learning process.

As with addition and subtraction, children should be recording the digits alongside the concrete apparatus, and recording pictorially once they are confident with the concrete resources.

Children are then able to progress to representing the numbers in a grid, using place value counters.

The video above is a great example of how this might be done.

Once children are confident using the counters, they can again record them pictorially, ensuring they are writing the digits alongside both the concrete apparatus and the visual representations.

Progressing to the expanded method and then the short method of column multiplication is much easier for children if these are introduced alongside the grid method, to enable them to see the link.

Read also: How to Teach Multiplication for KS2 Interventions in Year 5 and Year 6

Teaching division using the Concrete Pictorial Abstract (CPA) Approach

As children work towards the formal written method for division, it is important they understand what is meant by both division as grouping and division as sharing. 

Concrete resources are invaluable for representing this concept. Pupils need to understand how numbers can be partitioned and that each digit can be divided by both grouping and sharing. 

Partitioning with straws
In this example, the straws will be shared out between the stuffed toys.

Once confident using concrete resources (such bundles of ten and individual straws, or Dienes blocks), children can record them pictorially, before progressing to more formal short division.

As children work towards understanding short division (also known as the bus stop method), concrete resources can be used to help them understand that 2-digit numbers can be partitioned and divided by both sharing and grouping. This can be through the use of bundles of ten straws and individual straws or dienes blocks to represent the tens and ones. 

As with the other operations, it’s important that children  are recording the digits alongside the concrete resources and are having the opportunity to draw visual representations. As children grow in confidence and once they are ready to progress to larger numbers, place value counters can replace the dienes. 

Read also: How to Teach Division for KS2 Interventions in Year 5 and Year 6

The Concrete Pictorial Abstract Approach is here to stay

It may have taken many years for CPA to reach the level of popularity it has today, but it is definitely here to stay. I’m not one to jump on the bandwagon when it comes to the latest teaching ‘fad’, however this has been one I’ve been happy to jump on. 

I have seen first-hand how successful it can be when children have the opportunity to work in this way and I love the fact that children are now starting to have the conceptual understanding in maths that I never had as a child.

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