# Math Made Visible: CRA Process

### Concrete, Representational and Abstract Models

## What is CRA?

Students make multiple representations while they solve problems, which progress from concrete to abstract as they make sense of a contextual problem (Gravemeijer et al., 2002; Stylianou, 2011).

## Concrete Stage: "Seeing" and "Touching" Math ProcessIn the concrete stage, the teacher begins instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, base-ten blocks, pattern blocks, fraction bars, and geometric figures). | ## Representative: Stage for Using Pictures for Math ProcessIn this stage, the teacher transforms the concrete model into a representational (semiconcrete) level, which may involve drawing pictures; using circles, dots, and/or tallies for counting. | ## Abstract: Using Symbols to Communicate Math Process At this stage, the teacher models the mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the number of circles or groups of circles. The teacher uses operation symbols (+, –, ) to indicate addition, multiplication, or division. |

## Concrete Stage: "Seeing" and "Touching" Math Process

In the concrete stage, the teacher begins instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, base-ten blocks, pattern blocks, fraction bars, and geometric figures).

## Representative: Stage for Using Pictures for Math Process

In this stage, the teacher transforms the concrete model into a representational (semiconcrete) level, which may involve drawing pictures; using circles, dots, and/or tallies for counting.

## Abstract: Using Symbols to Communicate Math Process

## CRA process and Instruction

Instruction at the representational level follows the same steps, but the manipulative objects are replaced by pictures and/or drawings. This serves as a transition from the use of pictures or drawings to the use of numbers only—that is, the abstract phase.

During this final phase (abstract), students use numbers only in completing a mathematical task, and instruction focuses on fluency.(Flores, 2009, p.185)

## CRA Process: Topic and Dates

## CRA Process for Addition, Subtraction, Multiplication and Division: One Day Session

Click to Register: BLaST Williamsport Office - March 24