All About The Concrete Representational Abstract (CRA) Method

ALI Staff | Published  December 06, 2022 | Updated May 17, 2024

Think back to your earliest memories of math class...

Do you remember using blue graphing blocks or cut-outs of circles and dots? What about using pattern blocks?

You probably didn’t know it then, but you were learning through a specific stage of the CRA modelthe concrete stage.

When it comes to building the foundations in mathpractice makes perfect. However, traditional rote learning doesn’t work for most students, especially when it comes to understanding new and complex math concepts. It's even more challenging for those students trying to get back on track in math. The CRA model offers a differentiated approach to understanding new information.

Let’s take a closer look at CRAwhat it is, how to incorporate it into your classroom, and what it offers you and your students.


An image showing math manipulatives and formulas, showcasing various ways to represent the same math problem.


An image of a block quote stating the benefits the Concrete Representational Abstract approach has on test scores and learning


What is the CRA model?

The acronym CRA stands for Concrete, Representational, Abstract and is an instructional framework for teaching math. The CRA method provides the best opportunity for students to master content as they progress through the three stages.

CRA focuses on developing a deep understanding of a concept and allowing students to see patterns and relationships through the different ways they interact with it. There is always a situation where CRA can be used. CRA can be used at grade levels to teach any topic.  

Any configuration, whole class, small group, or individual, can benefit from using the CRA framework. 


An image defining what the concrete representational abstract approach is.


Concrete - Concrete Representational Abstract (CRA) Method

This is the “doing” stage. It starts with teachers modeling with concrete materials, followed by students working independently using their physical objects (manipulatives such as coins, graphing blocks, buttons, etc.).

Interacting with concrete objects leads to more retention.

Students make the connection with the concrete materials when, later, they practice the same skills at the abstract stage. Even middle and high school students benefit from using concrete materials to introduce concepts, for example, using concrete models of equations to see what has to be done to keep both sides equal.  

The concrete stage incorporates many different learning styles. The obvious one is kinesthetic or tactile. Students are physically manipulating materials to help them learn. While using concrete materials, students can see what is happening, so visual learning is also activated.  

Students working with manipulatives in groups are talking to one another about what is happening and what they are doing. Auditory learners are allowed to learn from the conversations and what is being discussed during this time that manipulatives are being used.  

An example of this would be a teacher modeling the counting and organizing of buttons. The teacher would model counting aloud and physically organizing the buttons.


Concrete in the concrete representational abstract method


Representational - Concrete Representational Abstract (CRA) Method

This is the “seeing” stage in which visual representations replace physical objects (e.g., buttons). Students will draw, tally, or use stamps to replace the buttons used in the concrete stage.

Students can see the problem model actualized differently, adding a layer of understanding through their drawings.

An example of this stage would be a teacher modeling the recording of the addition or subtraction of buttons on a number line– tracking the number of buttons when added or subtracted.


An image showing the representational portion of the concrete representational abstract approach



This is the “symbol” stage. This is the stage in which numbers and symbols are used instead of manipulatives or pictures.   In the abstract stage, students solve problems using only numbers and symbols.

Most students will need to write and solve the problem on paper, at least at first, but will eventually be able to solve problems mentally.  

Students must be given plenty of opportunities at each stage to practice and master the needed skills. Students must also be allowed to go back and forth between stages when needed.

CRA instruction provides students with many positive, permanent outcomes. Students have a better conceptual understanding and do not have to guess and ask the “WHY?” during the abstract stage.  

For example, a teacher would model importing information from the previous stages into an algebraic formula, showing written numbers to recreate the addition or subtraction of buttons.


the abstract part of the concrete representational abstract method


How to Implement the cra model in math

  • Start with modeling how to use concrete manipulatives. Students must be taught the rules and procedures for handling manipulatives.  
  • When students are ready, the teacher will need to make explicit and even demonstrate the connection between the concrete object and the visual model and allow students to practice this connection.  
  • Similarly, teachers must make explicit and demonstrate the connection between the visual model and the abstract number or symbol. Students need the opportunity to practice numerous times making the connection between representational and abstract.
  • Students need to be shown the connection between all three stages. Concrete can also be connected directly to the abstract. Teachers have to explain and demonstrate how all three are connected.  
  • Students should be able to move freely between concrete, representational, and abstract models. This autonomy ensures students will be more comfortable and engaged and provides the opportunity for them to demonstrate their understanding of how they are related. 


No physical math manipulatives? No problem! See how we use Virtual Manipulatives in STEMscopes Math:



An image displaying different aspects of CRA in a 5th grade activity from STEMscopes Math

Benefits of Using the CRA model in Math

Here are just some of the benefits of using the tried-and-true strategy of CRA in your classroom:

  • Increases student engagement and student buy-in
  • Appeals to multiple learning styles (kinesthetic, tactile, visual, auditory)
  • Allows for differentiation, student-choice, and peer-learning
  • Demonstrates the relationships between concepts through different instructional
  • Provides an understanding of core concepts in Math

Keep in mind that the three stages don't need to be followed in a linear order. Two or more models can be used simultaneously, which creates the chance to make connections between the stages.

Often times students and teachers want to move past the concrete state because they think of it as "too easy". Teachers, especially those in higher grade levels, might feel that this stage is more useful for elementary students. They may also find difficulty in gathering materials and may not be 100% convinced of their usefulness for instruction. This is not true.

All students benefit from the concrete stage. Some students find the concrete stage boring because they “get it” and want to move on. This is the perfect opportunity for those students to engage in peer learning and help struggling students master the concept.

Using the CRA framework goes beyond simply mastering standards and helps students make meaningful connections between concrete, real-world phenomena and abstract mathematics.

CRA addresses multiple learning styles and utilizes multisensory input to get students a true understanding of math concepts. It also supports the development of mathematical skills in students of all learning types at all levels.

Using CRA in your classroom will improve student learning and contribute to the accumulation of knowledge and skills.

See how STEMscopes Math helps students explore complex math concepts through a scaffolded approach using the CRA approach

An image with a magnifying glass over the 3 components of CRA



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