Concrete Representational Abstract (CRA) is a three step instructional approach that has been found to be highly effective in teaching math concepts.  The first step is called the concrete stage.  It is known as the “doing” stage and involves physically manipulating objects to solve a math problem.  The representational (semi-concrete) stage is the next step.  It is known as the “seeing” stage and involves using images to represent objects to solve a math problem.  The final step in this approach is called the abstract stage.  It is known as the “symbolic” stage and involves using only numbers and symbols to solve a math problem.  CRA is a gradual systematic approach.  Each stage builds on to the previous stage and therefore must be taught in sequence.  This approach is most commonly used in elementary grades, but can be found in some middle and high school classrooms.

 

Steps

  1. Teach the math concept using manipulatives (concrete level).
  2. Allow ample opportunities for students to practice the concept using various manipulatives.
  3. Make sure students understand the concept at the concrete level before moving on to the representational level.
  4. Introduce pictures to represent objects (representational level).  Model the concept.
  5. Provide plenty of time for students to practice the concept using drawn or virtual images.
  6. Check student understanding.  Do not move to the abstract if students haven’t mastered the representational level.
  7. Teach students the math concept using only numbers and symbols (abstract level).  Model the concept.
  8. Provide plenty of opportunities for students to practice using only numbers and symbols.
  9. Check student understanding.  If students are struggling, go back to the concrete and representational levels.
  10. Once the concept is mastered at the abstract level, periodically bring back the concept for students to practice and keep their skills fresh.
  • Remember that modeling the concept and providing lots of opportunities to practice is extremely important at all three levels.  Also, do not rush through the levels.  Students need time to make connections and build on what they already know.  Give them time to process the information before moving on to the next level.
Concrete Manipulative Examples:
  • colored chips
  • beans
  • unifix cubes
  • candy (ex. Skittles)
  • popsicle sticks
  • fraction blocks
  • fraction pizzas / cakes
Representational Examples:
  • tally marks
  • dots
  • circles
  • pictures of objects

 

Benefits

  • Provides students with a structured way to learn math concepts
  • Students are able to build a better connection when moving through the levels of understanding from concrete to abstract
  • Makes learning accessible to all learners (including those with math learning disabilities)
  • Taught explicitly using a multi-sensory approach
  • Follows Universal Design for Learning guidelines
  • Research has proven that this method is effective
  • Able to use across grade levels, from early elementary through high school
  • Aligned with NCTM standards
  • Helps students learn concepts before learning rules
  • Can be used in small groups or entire class

 

Negatives

  • Not commonly used past upper elementary grades (though it should)

 

Video Examples

 

 

MathVIDS also has some excellent video clips of math being taught using the CRA method (Concrete, Representational, and Abstract).  They are located under the Explicit Teaching Modeling section.

 

Multiple Intelligences / Learning Styles

  • Visual-Spatial: Helps students to think in pictures and create a mental image to retain concepts.
  • Verbal-Linguistic: Helps students to organize words in math problems in a way that makes sense.
  • Bodily-Kinesthetic: Students learn through hands-on activities.
  • Logical-Mathematical: Students use logic to organize information, classify and categorize, make connections and build relationships.

 

Universal Design for Learning

  • Multiple ways to teach math concepts
  • Multiple means of representation offered through the use of various manipulative items, visual images, and technology (SmartBoard, computer games/software, video, etc.)
  • Allows options for how students learn and express their understanding of a math concept (assessment example: Use SmartBoard clickers to ease student anxiety when having them give answers to math problems.  This will in turn increase student engagement and participation.)
  • Flexible methods for engaging students (able to incorporate student interests and use real life examples)
  • Accessible to all students regardless of ability level
  • Allows for accommodations to be made
  • Learning is active

 

According to the research cited by Terry Anstrom (n.d.), “students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations.”  Research shows that using the CRA approach is very effective for students who have a learning disability in math (Anstrom, n.d.).  Students are more apt to gain and retain an understanding of math concepts when they are taught using CRA (Anstrom, n.d.).

I think using the CRA approach is definitely the way to go.  It’s too bad that isn’t used much past the elementary grades.  I struggled with learning Algebra and Geometry in high school.  Being taught the concepts in a concrete way would have made all the difference for me.

 

 

References

Anstrom, T. (n.d.). Supporting students in mathematics through the use of manipulatives. Washington, DC: Center for Implementing Technology in Education. Retrieved April 14, 2012. From http://www.cited.org/library/resourcedocs/Supporting%20Students%20in%20Mathematics%20Through%20the%20Use%20of%20Manipulatives.pdf

Bender, W. (2009). Differentiating math instruction: Strategies that work for K-8 classrooms. Thousand Oaks: Corwin Press.

Sousa, D. (2008). How the brain learns mathematics. Thousand Oaks: Corwin Press.

The Access Center: Improving Outcomes for All Students K-8. (n.d.). Concrete-Representational-Abstract instructional approach. Retrieved April 14, 2012. From http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp#top


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