What Is The Concrete Representational Abstract (CRA) Model In Math?
The concrete representational abstract approach is now an essential tool for teaching math in elementary school. Here we explain what the CRA model is, why it works, where teachers still get it wrong, and how you can use it in your own math lessons.
In this blog, we focus throughout on one thing: using the CRA approach to teach the four written calculation methods.
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Download Free Now!What is the Concrete Representational Abstract (CRA) approach?
CRA stands for concrete, representational, abstract. It is a three-stage teaching sequence that moves students from handling physical objects, to drawing what they handled, to working with numbers and symbols alone. You will also see it called the CRA model, the CRA method, the CRA framework or the CRA instructional sequence. Outside the US it is usually called concrete pictorial abstract, or CPA.
The three stages of the CRA model
- Concrete – students solve the problem using objects they can pick up, such as base ten blocks, counters or craft sticks.
- Representational – students draw the objects instead of handling them. This stage is also called semi-concrete or pictorial.
- Abstract – students solve the same problem using numerals and symbols alone.
What is the Concrete Representational Abstract approach in Math?
The concrete representational abstract (CRA) model is a system of learning that uses physical and visual aids to build a student’s understanding of abstract topics.
You introduce a new mathematical concept through concrete resources, such as fruit, base ten blocks or fraction bars. Once students solve problems comfortably with physical aids, you give them problems with pictures, usually visual representations of the same objects.
Then you ask them to solve problems where only the abstract remains: numbers or other symbols. Building these steps across a lesson helps students understand the relationship between numbers and the real world, and it secures their grasp of the concept they are learning.
Origins of Concrete Representational Abstract Approach
The origins of the CRA approach take us back to the teaching methods of the 1960s, when American psychologist Jerome Bruner proposed it as a way of scaffolding learning.
Bruner believed the abstract nature of learning, which is especially true in math, to be a “mystery” to many students. It therefore needs scaffolding through effective representations and math manipulatives.
He found that students using the CRA approach built on each stage towards a greater mathematical understanding of the concepts they were learning. That, in turn, led them to internalize information and knowledge to a far greater degree.
Many teachers mistakenly believe the CRA approach arrived from Singapore.
In fact, the Singapore math curriculum draws heavily on a combination of Bruner’s ideas about learning and the recommendations of the 1982 Cockcroft Report, an English inspectorate report which argued that computational skills should be related to practical situations and applied to problems.
Why use the Concrete Representational Abstract approach in Math?
Students achieve a much deeper understanding when they do not have to resort to rote learning and can solve problems without memorizing.
When we teach reading to young students, we accept that they need to have seen what a word means to understand it. Putting together the letters c-a-t would be meaningless and abstract to a child who had never seen a cat or a picture of one.
People often don’t think this way about math. Yet to students, many mathematical concepts are equally meaningless without a concrete model or a picture to go with them. The same applies to mathematics teaching right through elementary school.
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What is a ‘Concrete’ representation in the CRA approach?
In the concrete stage you introduce new concepts through physical objects that students can handle. Sometimes called math manipulatives, these range from everyday household items such as straws or dice to specific mathematical resources such as base ten blocks and ten frames.
Handling a concrete model lets students “see” what is actually happening. The abstract nature of math confuses many students, and physical materials make the mathematics visible.
Teachers once assumed that concrete resources were only for students who find math difficult. In fact you can use them in a great variety of ways at every level. Every student benefits from practical materials, because understanding then goes beyond learning a procedure.
Practical materials also promote reasoning and discussion, so students can articulate and explain a concept. Watching them work shows you where the misconceptions sit and how deep the understanding really goes.
How can we portray the “representational” part of the CRA approach?
Once students feel confident with a concept using concrete resources, they move on to drawing visual representations or quick sketches of the objects. They are no longer manipulating physical materials, but they still get the visual support those materials provide.
Some teachers skip the representational stage, also known as the semi-concrete stage. That is a mistake. It is the stage that lets students link a concrete resource to abstract notation, and without it many students struggle to visualize a problem at all.
The bar model is one of the most common methods in the representational stage, and you will see it most often in multi-step problem solving.
Read more: What is a bar model
What is an ‘Abstract’ representation in the CRA approach?
Once children have a secure understanding of the concept through concrete resources and visual images, they can move on to the abstract stage. Here, children are using abstract symbols to model problems – usually numerals. To access this stage effectively, children need access to the previous two stages.
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.
Teaching through the Concrete Representational Abstract method in elementary school
A common misconception with the CRA model is that you teach the concrete, then the representational, 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.
When concrete resources, visual representations and abstract recordings are all used within the same activity, it ensures students are able to make strong links between each stage.
If you’re concerned about differentiating effectively using the CRA approach, have a look at the Standards for Mathematical Practice or our differentiation strategies guide for ideas to get you started.
Using ‘The Four Operations’ to model how CRA works
In the following section, we will be looking at the ‘four operations’ and how the CRA approach can be used at different stages of teaching them. In particular, we will examine how the 3 parts of the CRA 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, we will only cover the four operations briefly.
Teaching addition using the Concrete Representational 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 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 representationally, again recording the digits alongside to ensure links are constantly being made between the concrete, representational and abstract levels.
The next step is for children to progress to using more formal mathematical equipment. Base ten blocks should be introduced alongside the straws, to enable children to see what is the same and what is different. As confidence grows using the base ten blocks, 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 be recording the numbers on the baseboard, and again have the opportunity to record representational representations.
Once children are completely secure with the value of digits and the base ten nature of our number system, base ten blocks can be replaced with place value counters. These help children as they progress towards the abstract, as unlike the base ten blocks, they are all the same size.
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 representationally 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 Interventions In Upper Elementary
Teaching subtraction using the Concrete Representational 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 representationally, again recording the digits alongside.
As with addition, children should eventually progress to using formal mathematical equipment, such as base ten blocks.
These should be introduced alongside the straws so students 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 base ten blocks representationally, to ensure links are built between the concrete and abstract.
Once secure with the value of the digits using base ten blocks, children progress to using place value counters.
Again, the counters enable children to work concretely with larger numbers, as well as bridge the gap from the use of base ten blocks to the abstract. Along with the counters, children should be recording the digits and they should have the opportunity to record representationally once confident with the method using concrete resources.
Read also: How To Teach Subtraction For Interventions In Upper Elementary
Teaching multiplication using the Concrete Representational 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.
Students 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 be recording the digits alongside the concrete apparatus, and recording representationally 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.
Once children are confident using the counters, they can again record them representationally, 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 Interventions in Elementary
Teaching division using the Concrete Representational Abstract (CRA) Approach
As children work towards the standard algorithm for division (long 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. Students 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 as bundles of ten and individual straws, or base ten blocks), children can record them representationally, before progressing to more formal short division.
As children work towards understanding long division to find the quotient, 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 base ten 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 base ten blocks.
Read also: How To Teach Students Division For Interventions In Upper Elementary
The Concrete Representational Abstract Approach is here to stay
It may have taken many years for CRA to reach the level of popularity it has today, but it is definitely here to stay.
I have seen firsthand 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 math that I never had as a child.
Read more:
- What Is Part Whole Model?
- What Is Mean In Math?
- Math Problems For 1st Graders
- 2nd Grade Math Problems
- Math Problems For 3rd Graders
- 4th Grade Math Problems
- Addition Worksheets: 1st -8th Grade
- Division Worksheets for 3rd-8th Graders
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The content in this article was originally written by assistant headteacher Emma Johnson and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.
