C-R-A Approach

What is It?

• A 3 step instructional strategy• Each step builds off of the other• Used to explain concept of the problem before

executing the problem• Steps

• Concrete “doing” Stage• Representational “seeing” Stage• Abstract “symbolic” Stage

Concrete–Representational–Abstract (CRA)

As students move through the three stages of learning, hands-onactivities address the needs of auditory, visual, and kinestheticlearners.• The Concrete (Manipulative) stage consists of actual contact with

objects. Students look at, touch, and move objects to understand mathematical concepts.

• In the Representational (Pictorial) stage, students bridge the gap between the concrete and the abstract with strategies such as drawing pictures and talking about math.

• In the Abstract (Numbers/Signs) stage, real objects and pictures are connected to the abstract numerals and signs of arithmetic and algebra. Students gain understanding of abstract concepts from experiences in the concrete and representational stages.

Let’s Move through each stage together….

Understanding the Math

• The “Why” behind the math- students need to know why and be able to explain why!

• Develop the concept then teach the procedure that goes along with that.

Concrete Representational Abstract

Core Image of Mathematics

“one”“two”

“three”

+ -

1 2 3

x =

Quantity Concrete display

of concept

Symbols Simply record keeping!

Mathematical Structure Discussion

of the concrete

V. Faulkner and DPI Task Force adapted from Griffin

12

CRA Sorting Activity

• In your group, sort the activities by Concrete, Representational and Abstract

CWT Connection

• 1d. Identify key math instructional phases. Concrete presentation of material Representational presentation of material Abstract presentation of material

CWT Operational Definition• Concrete- The teacher is modeling the mathematical concept with concrete materials

(e.g., red and yellow chips, cubes, base-ten blocks, pattern blocks, fraction bars, geometric figures, algebra tiles, Hands On Equations, measuring devices, etc) and/or the students are using the manipulative objects or a touchable visual or a physical representation to display and solve math problems. This is the “doing” stage using concrete objects to model problems.

• Representational- The teacher is transforming the concrete models into a representational (semi-concrete) level, which may involve drawing pictures or using pictorial representations. The students could be using circles, dots, tallies, drawings, tables, graphs, stamps, etc. to represent the math. The teacher and/or students could also be involved in a discussion of the math to make sense of the concept (connecting the concrete objects to the abstract symbols). This is the “seeing” stage using representations of the objects to model problems.

• Abstract- The teacher is modeling the mathematics concept at a symbolic level, using only numbers, notations, and mathematical symbols to represent the mathematics. The students are using symbols (numbers/mathematical symbols) to solve the problems. This is the “symbolic” stage using abstract symbols to model problems.

Table Talk

• Think about ways that you could incorporate this into your lessons and assessments.

Concrete Representational Abstract

Writing & Talking in Math

My definition of a good teacher has changed from “one who explains things so well that students understand” to “one who gets students to explain things so well they can be understood”.

~Steven C. Reinhart

Writing Across the Curriculum

• The importance• Open Ended Questions• ACE Rubric

ACE Rubric

Subject A C EMath Answer the

problemCompute your work

Explain how you got your answer: reasonable?