Concrete-Representational-Abstract
Category: Mathematics
Grade Level: All
- What is the purpose of Concrete-Representational-Abstract (CRA) sequence of instruction in mathematics?
The purpose of using Concrete-Representational-Abstract (CRA) sequence of instruction is to ensure students attain a thorough understanding of the mathematical concepts they are learning. This three-step instructional method allows students to develop familiarity with concrete mathematical examples, which is later linked to abstract mathematical concepts. The use of representationals to facilitate transfer from concrete to abstract level of thinking reinforces students’ understanding. This sequence of instruction also acts as a model for students in future problem solving (Special Connections, 2005).
- With whom can they be used?
CRA sequence of instruction can be used atall grade levels, from elementary to secondary schools. However, appropriate selection of concrete objects is necessary for each grade and their respective mathematical topics. This method of instruction can be used on students of all abilities, including students with learning disabilities.
- What is the format of this technique?
CRA sequence of instruction incorporates three interlocking and graduated stages to reinforce students’ understanding of mathematical concepts.
This method moves from the use of concrete objects to introduce mathematical models, to the use of pictorial representations to further consolidate understanding, and finally to the introduction of abstract mathematical symbols to reach conceptual mastery.
- What teaching procedures should be used with CRA sequence of instruction?
- Concrete Stage: select materials suitable to the mathematical concept and grade of students
Teacher selects appropriate concrete objects to model mathematical concepts under interest. For example, concrete objects suitable for elementary level math (e.g., toothpicks for counting) may be different from those suitable for secondary level math (e.g., geometric bars for geometry).
Provide students with many opportunities for practice and only move on to the next stage of instruction when students demonstrate mastery at the concrete level.
- Representational: use of pictorials to reinforce understanding
Teacher uses the pictures that represent objects used in concrete stage of instruction to model the mathematical relationships (semi-concrete). Examples include the use of tallies, dots, or circles.
The repetitive drawing of more abstract shapes to model previously learned concrete examples reinforces students’ understanding of these mathematical concepts (Special Connections, 2005).
Provide students with many opportunities for practice at this stage and only move on to the next stage of instruction when students demonstrate mastery at the representational level.
- Abstract: Use of mathematical symbols to model concepts
Teacher uses abstract mathematical symbols and numbers to teach the mathematical concept of interest.
Provide students with many opportunities for practice using only numbers and symbols. After students show mastery at this level, teacher should ensure students stay at the level of abstract understanding with periodic monitoring and practice opportunities.
Students with learning disabilities may have difficulty with abstract conceptualizations; there are two strategies to aid conceptualization:
Re-teach concrete level modeling
If students have not achieved concrete understanding, teacher should re-teach the mathematical concepts using concrete objects and explicitly connect the concrete materials to abstract symbols.
Elicit student verbalization
If students havemastered the concrete stage, encourage them to verbalize their actions and explain their understanding of the mathematical concepts. This allows the teacher to gauge their level of understanding and modify instruction accordingly.
- In what types of settings should CRA sequence of Instruction be used?
CRA sequence of instruction can be used in classroom settings, group settings, and individually (Hauser, 2000). For students with learning disabilities, this method of instruction is effective by helping them make explicit connections between concrete and abstract levels of understanding through a graduated and organized structure (Math VIDS, n.d.).
- To what extent has research shown CRA sequence of instruction to be useful?
Research has demonstrated that students instructed under the CRA method develop more comprehensive understanding of mathematical concepts, motivation, and better application to other life situations (Hauser, 2000). For example, Flores found that students struggling in mathematics performed better under the CRA condition when learning subtraction with regrouping and had higher performance maintenance compared to the control group. Strickland and Maccini (2012) also found that CRA sequence of instruction was effective for students with learning disabilities in learning to multiply linear expressions, and participants also generalized this material to solving novel problems. Therefore, research has shown CRA sequence of instruction to be an effective instruction method for students with learning disabilities or struggling with mathematical concepts.
References
Flores, M. M. (2009). Using the Concrete–Representational–Abstract Sequence to Teach Subtraction With Regrouping to Students at Risk for Failure.Remedial and Special Education.
Hauser , Jane. (2010). Concrete-representational-abstract instructional approach.The Access Centre. Retrieved October 10th, 2014:
Math VIDS. (n.d.). Concrete-representational-abstract sequence of instruction. Retrieved October 10th, 2014:
Special Connections, (2005). Concrete-to-representational-to abstract (C-R-A) instruction. Retrieved October 9th, 2014:
Strickland, T. K., & Maccini, P. (2012). Effects of Concrete–Representational–Abstract Integration on the Ability of Students With Learning Disabilities to Multiply Linear Expressions Within Area Problems.Remedial and Special Education, 0741932512441712.
Math Resources
Math VIDS:
A multi-media website used to aid teachers in math instruction. This website is especially valuable for teaching students struggling with math.
Website:
Prepared by: Shiming Huang