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Concrete-Representational-Abstract Instruction. Objectives. At the end of this training you will be able to describe the components of concrete-representational-abstract (CRA) instruction create a lesson supported by the CRA sequence relate the benefits of CRA.

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Concrete representational abstract instruction

Concrete-Representational-AbstractInstruction

2010 Region 3 Education Service Center / Texas A&M University


Objectives
Objectives

At the end of this training you will be able to

  • describe the components of concrete-representational-abstract (CRA) instruction

  • create a lesson supported by the CRA sequence

  • relate the benefits of CRA

2010 Region 3 Education Service Center / Texas A&M University


Components of effective math instruction
Components of Effective Math Instruction

Explicit strategy instruction using a step-by-step approach

Mastery at each step

Effective teaching components (teacher modeling, guided practice, independent practice, immediate corrective positive feedback)

A wide range of examples and nonexamples

Clear, concise instruction with definitions of potentially confusing terms and skills

Cumulative review

2010 Region 3 Education Service Center / Texas A&M University


Cra instructional sequence
CRA Instructional Sequence

CRA is a three-part instructional strategy

Each subsequent part builds on previous instruction

The sequence promotes student learning and retention

2010 Region 3 Education Service Center / Texas A&M University


Research evidence
Research Evidence

  • Teaching equivalent fraction concepts and procedures (Butler et al., 2003)

    • Fraction circles and beans

    • Coloring portion of items to make fraction

    • Abstract symbols and algorithm

  • Teaching multiplication facts and related word problems (Morin and Miller, 1998)

    • Paper plates and wooden blocks

    • Pictures of containers and items

    • Numbers

2010 Region 3 Education Service Center / Texas A&M University


Cra sequence
CRA Sequence

2010 Region 3 Education Service Center / Texas A&M University


Cra principles
CRA Principles

The CRA sequence should be very explicit

and systematic, with multiple opportunities

for both guided practice and independent

practice.

2010 Region 3 Education Service Center / Texas A&M University


Definition of concrete
Definition of Concrete

The use of manipulatives or models that represent the concept being studied is critical.

It is the “doing” step of instruction.

Students must be fluent in manipulation of the concrete materials and must be able to explain the connection to the skill.

Howell and Barnhart (1992) identified six steps in teaching from concrete to representational levels.

2010 Region 3 Education Service Center / Texas A&M University


Stage i concrete
Stage I: Concrete

Step 1: Free exploration

Step 2: Purposeful exploration

Step 3: Number cards

Step 4: Number sentence

Step 5: Mathematical word problems

Step 6: Verbal explanation

(Howell and Barnhart, 1992)

2010 Region 3 Education Service Center / Texas A&M University


Step 2 purposeful exploration
Step 2: Purposeful Exploration

2010 Region 3 Education Service Center / Texas A&M University


Step 3 number cards
Step 3: Number Cards

6

+

3

=

9

2010 Region 3 Education Service Center / Texas A&M University


Step 4 number sentence
Step 4: Number Sentence

3 stars + 4 stars = 7 stars

2010 Region 3 Education Service Center / Texas A&M University


Step 5 mathematical word problems
Step 5: Mathematical Word Problems

There are three green buttons in a box. There are also two yellow buttons in the box. How many buttons are in the box altogether?

3

2

+

=

5

2010 Region 3 Education Service Center / Texas A&M University


Step 6 verbal explanation
Step 6: Verbal Explanation

Students should verbalize what they did to solve the problem and why.

Students should be taught to mastery level for each step before moving to the next step.

2010 Region 3 Education Service Center / Texas A&M University


Activity
Activity

Activity #1

  • Working with a partner, teach students to solve a subtraction problem by using manipulatives.

  • Keep the steps in mind:

    Step 1: Free exploration

    Step 2: Purposeful exploration

    Step 3: Number cards

    Step 4: Number sentence

    Step 5: Mathematical word problems

    Step 6: Verbal explanation

2010 Region 3 Education Service Center / Texas A&M University


Guidelines for using manipulatives
Guidelines for Using Manipulatives

Handout #1

  • Select manipulatives that clearly illustrate the concept.

  • Use a variety of manipulatives that illustrate the concept.

  • Provides verbal explanations while incorporating the manipulatives.

  • Provide multiple opportunities for guided practice.

  • Encourage the use of manipulatives and strategies across settings.

  • Plan a graduated instructional sequence that will allow transition from concrete to symbolic representation.(Maccini and Gagnon, 2000)

2010 Region 3 Education Service Center / Texas A&M University


Definition of representational
Definition of Representational

Replacing concrete manipulative objects with pictures or drawings

The “seeing” step of instruction

Division into a semiconcrete level and a semiabstract level

Six steps in the representational stage (Howell and Barnhart, 1992)

2010 Region 3 Education Service Center / Texas A&M University


Stages and levels
Stages and Levels

Abstract stage

Semiconcrete level

Semiabstract level

Representational stage

Concrete stage

(Howell and Barnhart, 1992)

2010 Region 3 Education Service Center / Texas A&M University


Stage 2 representational
Stage 2: Representational

(Howell and Barnhart, 1992)

Step 1: Picture representation cards

Step 2: Number sentences (semiconcrete)

Step 3: Mathematical word problems

Step 4: Graphic representation

Step 5: Number sentences (semiabstract)

Step 6: Mathematical word problems

2010 Region 3 Education Service Center / Texas A&M University


Step 1 picture representation cards
Step 1: Picture Representation Cards

2010 Region 3 Education Service Center / Texas A&M University


Activity for step 1
Activity for Step 1

Activity #2

2010 Region 3 Education Service Center / Texas A&M University


Step 2 number sentences semiconcrete
Step 2: Number Sentences (Semiconcrete)

Activity #3

2010 Region 3 Education Service Center / Texas A&M University


Step 3 mathematical word problems
Step 3: Mathematical Word Problems

Activity #4

Number

sentence: 3 + 6 = 9

2010 Region 3 Education Service Center / Texas A&M University


Step 4 graphic representation
Step 4: Graphic Representation

Activity #5

2010 Region 3 Education Service Center / Texas A&M University


Step 5 number sentences semiabstract
Step 5: Number Sentences (Semiabstract)

3

+

4

=

7

2010 Region 3 Education Service Center / Texas A&M University


Step 6 mathematical word problems
Step 6: Mathematical Word Problems

4

+

12

=

16

Jackie has $4 allowance left. Her mom gives her $12 more today. How much allowance does she have altogether?

2010 Region 3 Education Service Center / Texas A&M University


Definition of abstract
Definition of Abstract

The “symbolic” step of instruction

Written mathematical numbers are used to represent concepts or skills

Thinking strategy to solve word problems, with five questions to ask (Howell and Barnhart, 1992)

2010 Region 3 Education Service Center / Texas A&M University


Stage 3 abstract
Stage 3: Abstract

(Howell and Barnhart, 1992)

Five questions used as procedural prompts that help students become independent problem solvers:

1. What is the question?

2. What are the numbers in the problem?

3. What do I need to do with the numbers?

4. What is the answer?

5. How can I check the answer?

2010 Region 3 Education Service Center / Texas A&M University


Hint words
Hint Words

Handout #2

  • Preteach vocabulary

    • Addition—add, in all, sum, together, total

    • Subtraction—subtract, more, less, difference

    • Multiplication—multiply, time, by, each, of

    • Division—divide, group, how many times

  • Tips

    • Follow CRA sequence, and teach to mastery for each stage before moving to the next

    • Use explicit and systematic instruction

    • Provide many opportunities to respond

    • Use real-life contexts

2010 Region 3 Education Service Center / Texas A&M University


Step 1 what is the question
Step 1: What Is the Question?

How many more?

Adam has 12 marbles, and Jack has 6 marbles. How many more marbles does Adam have?

2010 Region 3 Education Service Center / Texas A&M University


Step 2 what are the numbers
Step 2: What Are the Numbers?

How many more?

Adam 12

Jack 6

Adam has 12 marbles, Jack has 6 marbles. How many more marbles does Adam have?

2010 Region 3 Education Service Center / Texas A&M University


Step 3 what do i need to do with the numbers
Step 3: What Do I Need to Do with the Numbers?

How many more?

Adam 12 12

Jack 6 – 6

2010 Region 3 Education Service Center / Texas A&M University

Adam has 12 marbles, Jack has 6 marbles. How many more marbles does Adam have?


Step 4 what is the answer
Step 4: What Is the Answer?

How many more?

Adam 12 12

Jack 6 – 6

6

Adam has 12 marbles, and Jack has 6 marbles. How many more marbles does Adam have?

2010 Region 3 Education Service Center / Texas A&M University


Step 5 how can i check the answer
Step 5: How Can I Check the Answer?

How many more?

Adam 12 //// / ////// 12

Jack 6 //// / – 6

////// 6

2010 Region 3 Education Service Center / Texas A&M University

Adam has 12 marbles, and Jack has 6 marbles. How many more marbles does Adam have?


Activity1
Activity

Activity #7

Work with a partner. Plan to teach students to multiple four by six with the CRA sequence.

  • Identify what manipulatives to use

  • Make up a word problem

  • Identify what activities to use for each step in each stage

  • Discuss what other strategies you will use to teach the skill

2010 Region 3 Education Service Center / Texas A&M University


Closure
Closure

Take out your Change of Practice Plan. Think about what you learned in this module, and relate it to your classroom. Write down some ideas of what you want to start using in your classroom.

2010 Region 3 Education Service Center / Texas A&M University