Practise Math at Home With the Concrete, Representational, Abstract Model (CRA)

Math Fluency

Fluency is the goal of math education, just as it is with reading and writing; your child will need to solve problems automatically and accurately as they move through progressively challenging concepts and algorithms over the years. This is true for students with and without learning disabilities.

Old-school rote learning drills the “procedures” into us, but math fluency requires both procedural and conceptual knowledge. Students don’t just need accurate skills; they need to understand when and how to employ them. In this way, math is its own sort of ‘language.’

One method of building fluency for students who are at-risk or who have a disability is the concrete-representational-abstract sequence of instruction, also known as the concrete-pictorial-abstract model.

What is the Concrete-Representational-Abstract Model?

The Concrete-Representational-Abstract (CRA) model is a multisensory, exploratory approach to teaching mathematical concepts and skills. The CRA model is based on research into child cognitive development by Jerome Bruner. Bruner’s constructivist theory suggests learning is most effective when it progresses through three stages:

  • Enactive representation (action-based)
  • Iconic representation (image-based)
  • Symbolic representation (language-based)

This progression holds true for learners of all abilities and ages, from early childhood to adulthood.

The Concrete Stage

This is the “doing” stage, where your child uses concrete objects to model problems. In the concrete stage, the teacher or parent models the concept using concrete materials such as blocks, coloured chips, or other manipulatives.

The Representational Stage

The representational, or pictorial, stage uses representations of the objects to model problems. When a child is ready for this level, the teacher or parent demonstrates transferring concrete concepts to semi-concrete pictures of circles, dots or tallies to represent the former concrete objects. 

The Abstract Stage

This is the “symbolic” stage, where abstract symbols and numbers alone are used to model problems. The parent or teacher uses numbers and operation symbols (+, × ,÷, –) to indicate addition, subtraction, multiplication, or division.

As the child explores each stage, the teacher or parent’s role is to provide support and feedback. This unintrusive supervisory approach was termed “scaffolding” by Bruner.

How to Use the CRA Model at Home

The CRA model has been proven to help children of all ages move toward better understanding of math and algebraic concepts.

It’s a misconception that manipulatives are only for young children or children facing obstacles to their learning. Practical math strategies often involve the manipulation of real-life objects, and life calls for the use of math in practical, hands-on situations.

As a parent, you can help your child to take a step back through the CRA model when you notice a lack of understanding. Here are a few examples of how that might happen:

  • If your child is struggling with arithmetic or algebra, model the representational stage to help bridge the gap. Draw a simple picture or use a tally to model the problem, then encourage your child to experiment with pictorial representations while they continue with the assigned problems.
  • If your child is struggling with word problems, model the concrete stage to create conceptual understanding. Use blocks or other household objects to tangibly demonstrate the math problem and stand by while your child explores the concrete concepts inherent in each problem.
  • If your child is beginning to work with representational or abstract math problems, have concrete manipulatives at the ready. Demonstrate how to use the objects in order to understand a problem, then supervise while your child ues the manipulatives to transfer understanding to the pictures or symbols in the assigned math problems

Try not to rush your child to move from concrete to representational, or from representational to abstract stages. Encouraging them to move backward between the stages as needed will help them be more independent when they encounter a new math task.

Spending plenty of time in the concrete and representational stages is key to the conceptual knowledge your child needs for math success. Focus on depth of understanding instead of completing a certain number of problems and you’ll help your child gain real math knowledge. 


Do you use concrete manipulatives and representations in your math practice at home? Share your experience below!

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