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Concrete – Representational – Abstract:

Updated: Aug 12, 2018

An Instructional Strategy for Math

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).

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). 

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

Concrete: 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).

Representational: 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.

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).

Abstract: 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).