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Research Based Math Interventions - Concrete Representational Abstract Approach

If you know anything about the alphabet soup that special education can sometimes be, then you've likely heard of acronyms like RTI (response to intervention) and SRBI (scientific research based interventions).  What do these two acronyms mean to school staff?  Basically, that there needs to be a research base to the interventions they use to help struggling students.  What this means to parents is that research based interventions are likely being tried to help their children every day.  Speaking of research based interventions, or in this case research based math interventions, one that has shown some promise is the Concrete Representational Abstract Approach.

Yep, that is a pretty scary name.  But if you want to improve fraction skills, algebra, place value, and more, the Concrete Representational Abstract Approach is one you'll want to become familiar with.  So read on to find out what this math intervention is, and what the research says about it.

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What is the Concrete Representational Abstract Approach?

The Concrete Representational Abstract Instructional Approach (CRA) is a research based math intervention that is broken down into stages by its name.  These stages are:

Concrete: Most that run in educational circles realize that multimodal instruction can help students learn.  In other words, adding words to tactile representations or visuals (tactile, visual, auditory) helps.  In CRA, the concrete stage refers to the use of concrete materials (ex.- blocks, marbles, chips) to introduce mathematical concepts.

Representational: In the representational stage, the teacher moves things to a higher level by transitioning instruction from the concrete level to a less concrete platform.  In other words, students might go from using blocks to using dots, picturer, or some other type of drawing.

Abstract: Math is about symbols, right?  Well, in this stage instruction moves to the symbols and numbers we've become so accustomed to in math.  You've seen them before- +, -, %, etc.

An Example of the Concrete Representational Abstract Approach


Goal: Learn about reading and writing fractions.

Concrete: The teacher might offer students 10 marbles.  Five of them are red, five are yellow.  He/she might ask the group to count the total number of marbles, then how many are red, how many are yellow.

Representational:  The teacher might discuss the fact that the total number of marbles is 10.  Since that is the total number of marbles, it represents the whole thing or the whole in the following fractional representation.

part
whole

Abstract: The teacher might then ask students to write 10 where the word "whole" is in the aformentioned representation.

From there, the teacher does not necessarily need to go back to the concrete stage.  Rather, they might transition back to the representational stage to ask their students, "how many marbles are yellow?" When they say five, the teacher might conclude for them that the yellow marbles represent part of the whole.  The instruction would then once again enter the abstract phase, where the students would likely write down the number 5 where the word "part" is in the aformentioned representation, leaving the fraction as 5/10.

The Research Supporting the Use of Concrete Representational Abstract Instructional Intervention as a Research Based Math Approach

Drum roll, please.  After all, we've promised you that the CRA is a research based math intervention.  So check out some of the online research that supports its use by following the links below.

A Learning Disabilities Research article tells the story of comparing a CRA method of teaching initial place value to an abstract-only one. Learning-disabled elementary and middle school students using the CRA method performed significantly better on initial place value assessments following.

Nine math disabled students aged 7-11 were effectively taught abstract level problems after seven lessons in a CRA sequence.

Sixth and seventh grade learning disabled and/or students at risk for algebra difficulties learned algebraic transformation equations better when offered CRA instruction as opposed to traditional mathematics instruction.

Further References

K8 Access Center

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