What is The Concrete Pictorial Abstract (CPA) Approach And How To Use It In Maths
The Concrete Pictorial Abstract approach helps learners understand abstract concepts first through concrete resources and then pictorial representations before moving on to abstract symbols.
It is now an essential tool in teaching maths at KS1 and KS2, so here we explain in more depth 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 specifically be on how the CPA approach can be used to support the teaching and learning of the four written calculation methods.
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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. Millions of pounds have been injected into the Teaching for Mastery programme, a programme involving thousands of schools across the country.
Before 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 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.
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Watch Skye in actionThe CPA maths model in Teaching for Mastery
To find the origins of the mastery maths approach, we need to go further back in time and look much closer to home.
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, 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 scaffolding through effective representations and maths manipulatives.
He found that when pupils used the CPA approach as part of their mathematics education, they could build on each stage towards a greater mathematical understanding of the concepts being learned. In turn, this led to learners internalising information and knowledge to a greater degree.
Many teachers mistakenly believe mastery, and specifically the CPA approach, to have been a method imported from Singapore.
However, the Singapore Maths curriculum is 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 can solve problems without having to memorise.
When teaching reading to young children, we accept that children need to see the word 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 them. This applies equally to mathematics teaching at KS1 or KS2.
What is a ‘Concrete’ representation in the CPA approach?
In the concrete stage, children use physical manipulatives or practical equipment to understand new mathematical concepts and model problems. These physical resources, 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.
The abstract nature of maths can confuse children, but through the use of concrete materials, they can ‘see’ and make sense of what is actually happening.
Previously, there was the misconception that concrete resources are only for learners who find maths difficult. 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.
What is a ‘Pictorial’ representation in the CPA approach?
Once children are confident with a concept using concrete resources, they progress to the pictorial stage. During the pictorial stage, children use visual representations if the concrete resources. They might draw pictorial representations or quick sketches of the concrete resources they used, such as blocks or dienes. This pictorial stage helps learners connect the physical resources to the more abstract representations.
Some teachers choose to leave this stage out, but pictorial recording is key to ensuring that children can link a concrete resource and abstract notation. Without it, children can find 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
What is an ‘Abstract’ representation in the CPA approach?
Once children have a secure understanding of a maths concept through concrete resources and visual images, they can move on to the abstract stage. In the abstract stage, learners use symbols and numbers to solve problems. They do not use concrete objects or physical representations to help them. To apply the abstract stage effectively, children need to be confident with the concrete and pictorial stages.
For the most effective learning, children need to constantly go back and forth between each of the stages. This ensures concepts are reinforced and understood.
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 pictorial, then 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.
When concrete resources, pictorial representations and abstract recordings are all used within the same activity, it ensures pupils 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. Some are free to download and use.
Using ‘The Four Operations’ to model how CPA works
In the following section, I look at the ‘four operations’ and how the CPA approach can be used when teaching them. In particular, I examine how the 3 parts of the CPA approach should intertwine rather than be taught as 3 separate things.
As this blog is to share ideas rather than say how the calculation methods should be taught, I cover the four operations briefly.
It’s important to consider your school’s Calculation Policy 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 with 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.
Initially, children complete calculations where the units do not total 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 record them pictorially, again recording the digits to ensure links are constantly made between the concrete, pictorial and abstract stages.
The next step is for children to use more formal mathematical equipment. Dienes base ten should be introduced alongside the straws, to enable children to see what is the same and 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.
Alongside the concrete resources, children should record the numbers on the baseboard with 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.
Introduce these in the same way as the other resources, with children using a baseboard without regrouping initially, then progressing to calculations which do involve regrouping.
As with the other equipment, give children the opportunity to record the digits alongside the concrete resources and to progress to recording pictorially once they are secure. You can use place value counters 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.
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.
Introduce these alongside the straws so pupils make connections between the two resource types. Note digits down alongside the concrete resources, and once secure in their understanding, children can record the Dienes pictorially, to ensure link building between the concrete and abstract.
Once secure with the value of the digits using Dienes, children progress to using place value counters.
Again, the counters enable children to work concretely with larger numbers and bridge the gap from Dienes to the abstract. Along with the counters, children should record the digits and 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 when teaching multiplication, as it helps children 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.
As with addition and subtraction, children should record the digits alongside the concrete apparatus, and record pictorially once they are confident with the concrete resources.
Children can 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, they must understand what is meant by 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.
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 help them understand that 2-digit numbers can be partitioned and divided by both sharing and grouping. This can be through 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 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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