By Kathryn Garforth, Graduate Student, Faculty of Education, University of British Columbia and Linda Siegel, PhD, University of British Columbia
The goal of mathematics instruction is for lessons to occur in a step-by-step manner, allowing the learner to move from needing concrete manipulatives to solve a problem to a point where they are able to think abstractly through the steps to solve a problem (Miller & Mercer, 1993).
Mathematics can be a challenging topic for students who have learning disabilities (LDs), especially as the concepts and instructional methods become more abstract. Prior literature reviews have found using direct and explicit instruction for students with LDs in mathematics to have strong effect sizes (e.g., Baker, Gersten & Dae-Sik, 2002; Gersten, Chard, Jayanthi, Baker, Morphy & Flojo, 2009; Zheng, Flynn & Swanson, 2013).
Description of the CRA Strategy
The Concrete – Representational – Abstract (CRA; also known as Concrete – Semiconcrete – Abstract) instructional strategy combines effective components of both behaviourist (direct instruction) and constructivist (discovery-learning) practices (Sealander, Johnson, Lockwood & Medina, 2012; Mercer & Miller, 1992). This strategy is especially effective when used to teach individuals with LDs across grade levels and in many different topic areas in mathematics (Witzel, Riccomini & Schneider, 2008).
CRA uses demonstration, modeling, guided practice followed by independent practice and immediate feedback which are aspects commonly found in direct instruction. CRA also includes discovery-learning strategies involving representation to help students’ transition between conceptual knowledge and procedural knowledge (Sealander, Johnson, Lockwood & Medina, 2012).
CRA is a sequential three level strategy promoting overall conceptual understanding, procedural accuracy and fluency by employing multisensory instructional techniques when introducing the new concepts. Each level builds on the concepts taught previously (Witzel, Riccomini & Schneider, 2008).
The three stages
During the concrete stage of instruction, three-dimensional objects are employed so students can use the manipulatives to assist while they are learning the new concept (Miller & Kaffar, 2011). The use of manipulatives increases the number of sensory inputs a student uses while learning the new concept, which improves the chances for a student to remember the procedural steps, needed to solve the problem (Witzel, 2005).
In the representational stage of instruction, students are taught to use two-dimensional drawings (instead of the manipulatives from the concrete stage) to represent the same concepts.
In the abstract stage, students are taught how to translate the two-dimensional drawings into the conventional mathematics notation to solve the problem (Miller & Kaffar, 2011).
The manipulations in the concrete and representational stages allow students to rationalize the conceptual mathematical procedures into logical steps and understandable definitions (Witzel, Riccomini, & Schneider, 2008). When students encounter a difficult mathematical problem, they are able to construct pictorial representations to assist in find the solution (Witzel, 2005).
It is important to remember CRA is an interconnected instructional sequence where each lesson relates to the previous lesson(s). These explicit connections between lessons and stages are crucial in order for students to learn the targeted skill as well as comprehend the associated concepts(Witzel, Riccomini & Schneider, 2008). Students with LDs generally need explicit guidance and support when learning to new concepts and skills across settings (Witzel, Riccomini & Schneider, 2008).
Sealander, Johnson, Lockwood and Medina (2012) suggest each stage should consist of three lessons. Miller, Mercer and Dillon (1992) note that each lesson should follow the same format. At the beginning of a lesson, students should be given a graphic organizer. The teacher should demonstrate the new skill and have the students model the process. Through guided practice students try some problems and receive feedback on the process. Finally students independently practice the new skill(s).
Creating a CRA Instructional Sequence
Witzel, Riccomini and Schneider (2008) developed an acronym teachers can use to assist them in creating their own CRA instructional sequence. CRAMATH outlines seven steps teachers can use to create a mathematical unit:
- Choose the math topic to be taught.
- Review procedures to solve the problem.
- Adjust the steps to eliminate notation or calculation tricks.
- Match the abstract steps with an appropriate concrete manipulative.
- Arrange concrete and representational lessons.
- Teach each concrete, representational and abstract lesson to student mastery.
- Help students generalize what they learn through word problems. (p. 273.).
LD@school has gathered ideas from a number of resources to help outline how the Concrete-Representational-Abstract approach can be used to investigate adding and subtracting integers. Educators can build these ideas into their own lesson plans to suit the needs of their students.
Summary of Evidence
Numerous studies have shown the CRA instructional strategy to be effective for students both with LDs and those who are low achieving across grade levels and within topic areas in mathematics such as:
- basic facts and place value (Bryant et al., 2008; Miller & Mercer, 1993),
- addition, subtraction, multiplication and division (Mancl, Miller, & Kennedy, 2008; Miller & Kaffar, 2011; Miller & Mercer, 1993; Sealander, Johnson, Lockwood & Medina, 2012),
- word problems (Hutchinson, 1993; Maccini & Hughes 2000),
- fractions (Butler, Miller, Chrehan, Babbitt & Pierce, 2003; Jordan Miller & Mercer, 1999; Misquitta, 2011), and
- algebra (Witzel, 2005; Witzel, Mercer & Miller, 2003).
CRA has been effective in inclusive, small group and one-on-one settings (e.g., Bryant et al., 2008; Sealander, Johnson, Lockwood & Medina, 2012; Witzel, 2005).
When students are taught using CRA, they have been able to generalize and maintain the progress they made during the CRA intervention period (e.g., Bryant et al., 2008; Sealander, Johnson, Lockwood & Medina, 2012; Witzel, 2005).
Related Resources on the LD@school Website
Research-Based Education Strategies and Methods: Concrete Representational Abstract (CRA). Click here to access a short article about Research-Based Education Strategies and Methods.
MathVIDS: Concrete-Representational-Abstract Sequence of Instruction. Click here to access an article about Concrete-Representational-Abstract Sequence of Instruction
Strategies for Teaching Elementary Mathematics: Concrete-Representational-Abstract Instructional Approach.Click here to access this article
Baker, S., Gersten, R., & Dae-Sik, L. (2002). A synthesis of empirical research on teaching mathematics to low achieving students. Elementary School Journal, 103, 51-73.
Bryant, D. P., Bryant, B. R., Gersten, R. M., Scammacca, N. N., Funk, C. Winter, A., Shih, M., & Pool, C. (2008). The effects of tier 2 intervention on the mathematics perfomance of first-grade students who are at risk for mathematics difficulties. Learning Disability Quarterly, 31 47-63.
Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research & Practice, 18(2), 99-111.
Gersten, R., Chard, D.J., Jayanthi, M.J., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79, 1202-1242. doi 10.3102/0034654309334431
Hutchinson, N.L. (1993). Students with disabilities and mathematics education reform – Let the dialogue begin. Remedial and Special Education, 14(6), 20-23.
Jordan, L., Miller, M. D., & Mercer, C. D. (1999). The effects of concrete to semi-concrete to abstract instruction in the acquisition and retention of fraction concepts and skills. Learning Disabilities: A Multidisciplinary Journal, 9, 115-122.
Maccini, P., & Hughes, C.A. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research and Practice, 15(1), 10-21.
Mercer, C.D., & Miller, S. P. (1992). Teaching students with learning problems in math to achieve, understand and apply basic math facts. Remedial and Special Education, 13, 19-35.
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Zheng, X., Flynn, L.J., & Swanson, H. L. (2013). Experimental intervention studies on word problem solving and math disabilities: A selective analysis of the literature. Learning Disabilities Quarterly, 36, 97-111. doi: 10.1177/073194871244427