**Concrete/Representational(Pictorial) Abstract**

Representational/Pictorial

The "seeing" stage using representations (pictures) of the concrete objects to help model a problem. The teacher transforms the concrete level to representations (often called the semi-concrete level).

Examples: pictures using circles, dots, tallies, 2d images, etc.

(American Institutes for Research 2004 )

Concrete

The "doing" stage using concrete models to solve problems The teacher will begin by modeling the mathematical concepts with concrete materials.

Examples: place value blocks, red/white counters, algebra tiles, money, connecting cubes, tape measure/ruler.

(American Institutes for Research 2004 )

Sequencing

It is important to follow the CRA sequence and not skip from concrete to abstract. Students need to proceed through all three levels to demonstrate that they understand a visual for the concrete model. Often times that visual is what people us in their head when they are at the abstract level.

If the sequencing of CRA/CPA is not followed students performance on symbolic operations will simply be rote repetitions of memorized procedures. (Sousa 2008) Students will not have the understanding behind why they are doing something (for instance regrouping in subtraction or addition)

Abstract

The "symbolic" stage using abstract symbols to model a problem such as 3x4=12. In this stage students are able to create the equation, solve using an algorithm or provide strategy necessary in order to solve the problem.

(American Institutes for Research 2004 )

Essentially in this stage students will be applying the "rules" that they have been taught to solve concepts.

**What is CRA or CPA**

**This is a sequence of activities that transition from concrete to representation/pictorial to abstract. When the sequence of concrete, representational, abstract is followed, the students develop a mastery level of understanding of math concepts.**

Research

Witzel, Mercer, and Miller who wrote Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model, conducted a study among 6th and 7th graders who had difficulties in learning algebra. Student who learned how to solve algebra equations using the CRA/CPA method scored higher on post assessments and follow up assessments than the control peer group who received traditional instruction. Also those who had CRA/CPA instruction had fewer procedural errors when solving algebraic equations.

Misconceptions

At the secondary level, often instruction is at the abstract level for a few reasons:

-concrete and manipulatives were used as introduction in elementary school

-there isn't enough time to go through CRA

-students already have the concrete/visual understanding

Types of Concrete Materials

Discrete- those objects that are used to help count

Continuous-Those materials not used for counting but more for measurement

Example of how the CRA/CPA approach works

Eddie Lacey ran the ball for 12 yard pick up. On the next play Aaron Rodgers was sacked for a loss of 7 yards. What is the total yardage in that series?

Concrete- Red side of chip = positive and White= negative. Have students line up 12 red chips and below them 7 yellow. Cancel out until there are only on color chip left. There should be 5 (reds) so the answer is positive five.

Representational- Draw a row of 12 positive signs, with a row of 7 negative signs. Line up so a positive matches up with a negative. Cross out pairs of positive and negative until only one sign is left. (5 positive.)

Abstract: 12+-7=5

Is a student at the concrete, representational, or abstract cognitive thinking?

A student may be at different cognitive thinking levels for different skills. A student may be at the abstract level for integers, but may have to go back to the concrete level for fractions.

A student may also be able to demonstrate that they are at the abstract level, however they may have that understanding only through rote memorization but not the understanding of why something happens. Which is the thinking that students will be required to demonstrate on parts of the Smarter Balanced Assessment.

Again regrouping is the first example that comes to mind.

Why are we discussing a strategy called Concrete/Representational/Abstract (CRA)?

Many of our students come to us without the foundation needed to retain math concepts at their grade level. We see that there is a continual need to review fractions, decimals, integers, and fact fluency because the students have not yet mastered and retained the concepts. In order for students to continue on in math they need to have the number sense and previous topics mastered to build upon. Concrete/representational/abstract is one method to help our students master and retain these concepts.

Ideas for concrete materials

On the hand-out applied problems intervention strategy is a table of different concepts and the manipulatives that can be used for each of those concepts. If there are things that you feel you would be beneficial for your students, please let Jackie or Jennifer know.

Ideas of what we may already have.

How to Implement?

Meet with Jennifer to discuss groups of students that you have noticed that do not seem to have grasped a concept(s) (data taken from ALEKS/MOBY/ or classroom assessments). We can talk about what materials we can use with the students. Jennifer can begin the intervention (2-3 times a week in the math classroom) and then the teacher can take over. We can give a pre and post assessment to help us understand misconceptions and then address those misconceptions with the concrete and representational tools.

QUESTIONS??