Mathematics instruction is a complex process that attempts to make abstract concepts tangible, difficult ideas understandable, and multifaceted problems solvable. Visual representations bring research-based options, tools, and alternatives to bear in meeting the instructional challenge of mathematics education (Gersten et al., 2008).
Visual representations, broadly defined, can include manipulatives, pictures, number lines, and graphs of functions and relationships. “Representation approaches to solving mathematical problems include pictorial (e.g., diagramming); concrete (e.g., manipulatives); verbal (linguistic training); and mapping instruction (schema-based)” (Xin & Jitendra, 1999, p. 211). Research has explored the ways in which visual representations can be used in solving story problems (Walker & Poteet, 1989); learning basic math skills such as addition, subtraction, multiplication, and division (Manalo, Bunnell, & Stillman, 2000); and mastering fractions (Butler, Miller, Crehan, Babbitt, & Pierce, 2003) and algebra (Witzel, Mercer, & Miller, 2003).
Concrete-Representational-Abstract (CRA) techniques are probably the most common example of mathematics instruction incorporating visual representations. The CRA technique actually refers to a simple concept that has proven to be a very effective method of teaching math to students with disabilities (Butler et al., 2003; Morin & Miller, 1998). CRA is a three-part instructional strategy in which the teacher first uses concrete materials (such as colored chips, base-ten blocks, geometric figures, pattern blocks, or unifix cubes) to model the mathematical concept to be learned, then demonstrates the concept in representational terms (such as drawing pictures), and finally in abstract or symbolic terms (such as numbers, notation, or mathematical symbols).
During a fraction lesson using CRA techniques, for example, the teacher might first show the students plastic pie pieces, and explain that, when the circle is split into 4 pieces, each of those pieces is ¼ of the whole, and when a circle is split into 8 pieces, each piece is ⅛ of the whole. After seeing the teacher demonstrate fraction concepts using concrete manipulatives, students would then be given plastic circles split into equal pieces and asked what portion of the whole one section of that circle would be. By holding the objects in their hands and working with them concretely, students are actually building a mental image of the reality being explored physically.
After introducing the concept of fractions with concrete manipulatives, the teacher would model the concept in representational terms, either by drawing pictures or by giving students a worksheet of unfilled-in circles split into different fractions and asking students to shade in segments to show the fraction of the circle the teacher names.
In the final stage of the CRA technique, the teacher demonstrates how fractions are written using abstract terms such as numbers and symbols (e.g., ¼ or ½). The teacher would explain what the numerator and denominator are and allow students to practice writing different fractions
on their own.
As the Access Center (2004) points out, CRA works well with individual students, in small groups, and with an entire class. It’s also appropriate at both the elementary and secondary levels. The National Council of Teachers of Mathematics (NCTM) recommends that, when using CRA, teachers make sure that students understand what has been taught at each step before moving instruction to the next stage (Berkas & Pattison, 2007). In some cases, students may need to continue using manipulatives in the representational and abstract stages, as a way of demonstrating understanding.