1. C-R-A Approach

2. 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

3. Concrete–Representational–Abstract (CRA)
As students move through the three stages of learning, hands-on
activities address the needs of auditory, visual, and kinesthetic
learners.
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.

4. Let’s Move through each stage together….

5. 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.
We can’t skip steps in the process
Concrete  Representational  Abstract

6. Core Image of Mathematics2 1
“one”
“two”
“three”
2 3
x =
Quantity
Concrete display
of concept
Mathematical
Structure
Discussion
of the concrete
Symbols
Simply record keeping!
V. Faulkner and DPI Task Force adapted from Griffin

7. CRA Sorting Activity
In your group, sort the activities by Concrete, Representational and Abstract
***This would need to be created at all schools or provided Jen had this activity
(Activity is posted on CRW wiki with this powerpoint presentation)

8. CWT Connection 1d. Identify key math instructional phases.
 Concrete presentation of material
 Representational presentation of material
 Abstract presentation of material

9. 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.

10. Table Talk
Think about ways that you could incorporate this into your lessons and assessments.
Concrete
Representational
Abstract

11. 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
Hard for our students to do if they’ve been used to answering multiple choice, yes/no, or fill in the blank questions.
Common Core demands evidence based writing in all subject areas (anchor stds. and math practices)

12. Writing Across the Curriculum
The importance
Open Ended Questions
ACE Rubric
District expectation is that writing will occur across the curriculum
-OEQs will continue to be a way that we assess student understanding
-ACE rubric can help us prepare our students for OEQs and more formalized assessments in math

13. ACE Rubric A C E Subject Math Answer the problem Compute your work
Explain how you got your answer: reasonable?