David H. Allsopp, Ph.D. University of South Florida [email protected]

Effective Mathematics Teaching Practices:

Explicit CRA Practices – Making Mathematics Transparent

David H. Allsopp, Ph.D.University of South Florida

[email protected]

“Explicit” Means To Provide Students…

ACCESSACCESS

ACCESS ACCESS

ACCESS

to the target mathematics concept.

Providing Access

“Students can’t hit what they can’t see...”

Student

Mathematics Concept

Make mathematics concepts accessible to your students by...

Student Mathematics Concept

Authentic ContextsAuthentic ContextsVisualsVisualsLanguage ExperiencesLanguage ExperiencesTeach Problem Solving StrategiesTeach Problem Solving StrategiesMultiple Opportunities to Apply UnderstandingsMultiple Opportunities to Apply UnderstandingsData-based Decision-makingData-based Decision-making

Concrete-to-Representational-to-Abstract Concrete-to-Representational-to-Abstract ExperiencesExperiences

What is CRA?What is CRA?

• It is not a “natural” process for some students

• Systematically teaching mathematics through a Concrete-to-Representational-to-Abstract Sequence

• Concrete Level - materials that students can manipulate to represent mathematical concepts and to problem solve.

• Representational Level - teaching drawing strategies to represent mathematical concepts and to problem solve.

• Abstract Level - representing mathematical concepts and problem solving using numbers and mathematical symbols without the use of concrete materials and drawings.

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Systematic CRA Teaching Process

Systematic CRA Teaching Process

Identify learner

objectives

Provide guided

practice & independent

practice

Provide advance

organizer

Provide modelsEvaluate

learning

Make instructional decisions

Monitor Progress

Specific Correctiv

e Feedback

Specific Positive

Reinforce-ment

1717

What is CRA?What is CRA?

Numbers and other mathematical symbols should be used at all three

levels and should be explicitly associated with the concrete materials and drawings that

represent them.

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Example of Explicit CRA SequenceExample of Explicit CRA Sequence

• Let’s examine the CRA sequence for the following algebraic expression:

4x = 8

Concrete Level: Start with Concrete Experiences

• Use simple (discrete) materials to represent the abstract numbers and symbols.

• Model how to manipulate the materials in ways to problem solve.

Representational Level: Teach Strategies for Drawing Representations of the Same Concept or Problem Solving Situation

• Model how to draw a Model how to draw a representation of the representation of the concept/problem concept/problem solving situation.solving situation.

• Model how to Model how to manipulate the manipulate the drawings in order to drawings in order to do the mathematics do the mathematics involved.involved.

4x = 8

X=2

Abstract Level: Gradually Fade Use of Drawings So Students Do Mathematics Using Numbers & Symbols Only

Concrete

Representational

Abstract4x = 8

x = 2

4x = 8

x = 2

4x = 8

x = 2

CRA LevelsCRA Levels

Other Examples in Resource Packet

CRA Examples pp. 65-68

Part 2: Effectively Implementing Explicit CRA Instruction for Struggling

Learners

Part 2: Effectively Implementing Explicit CRA Instruction for Struggling

Learners

• Important Considerations

• Effective Explicit CRA Instructional Practices

18-1918-19

Important Considerations for Struggling Learners

Important Considerations for Struggling Learners

Use a variety of appropriate concrete materials.

Use appropriate drawings.

Use appropriate strategies for helping students transition from concrete to abstract levels of understanding.

Allow students to reach mastery at each level of understanding before moving to the next level.

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Appropriate Use of Concrete Materials

Accurate representation of concept

Discrete objects

Attributes more “accessible”

Can be “manipulated” more easily

Use a variety over time to stimulate generalization

Manipulative Examples: (More Abstract)Manipulative Examples: (More Abstract)

•Potential benefits?

•Potential barriers?

More About Concrete MaterialsMore About Concrete Materials

• Discrete vs. Continuous

• Proportional vs. Non-Proportional

• Linked vs. Non-linked

Drawing Examples (see MathVIDS website)Drawing Examples (see MathVIDS website)

Other Drawing Examples (see MathVIDS website)Other Drawing Examples (see MathVIDS website)

ActivityIndividually or with a group…

Draw solutions for one or more of the following operations:

14 + 5 = 3 x 6 = 15 ÷ 3 = 7 – 4 =

Individually or with a group…

Draw solutions for one or more of the following operations:

½ x ¼ = 1 ¼ + ¼ = -8 + 7 = 3x = 24

Periodically “Move Down” Levels to Reinforce Conceptual Understandings

Gained At Prior Levels

Periodically “Move Down” Levels to Reinforce Conceptual Understandings

Gained At Prior LevelsMaintenance Activities

Periodic opportunities for students to revisit previously mastered abstract level knowledge and skills through 5-10 minute activities.

Teacher prepares a response prompt that engages students in thinking about/describe one or more key features of the previously mastered concept or skill.

Students use concrete materials and/or drawings to emphasize important features of the previously mastered concept or skill.

Purpose is to help students maintain what they have previously mastered AND to further enhance conceptual understanding.

The link below will take you to the MathVIDS Teaching Plans site where you can read about this practice activity:http://coe.jmu.edu/mathvids2/plans/cflud/C_plan4.html

Reflection Activity

Individually or with a group…

Write and/or share your observations from reviewing the information on Effective CRA Instructional Practices

How Does Systematic CRA Instruction Address Needs of

Struggling Learners?

How Does Systematic CRA Instruction Address Needs of

Struggling Learners?

?

By Providing StudentsMultisensory ACCESS

To The Distinctive Features of Mathematical Concepts &

Processes

Student Mathematics Concept

How Does Systematic CRA Instruction Address Needs of

Struggling Learners?

CRA Assessment and Teaching In Action - Multimedia ModelsCRA Assessment and Teaching In Action - Multimedia Models

http://fcit.usf.edu/mathvids/index.html

CRA Assessment and Teaching Resources on the MathVIDS website:

Mathematics Dynamic Assessment - Click “Instructional Strategies - Complete List of Strategies”

CRA Instruction Sequence - Click “Instructional Strategies - Complete List of Strategies”

Explicit Teacher Modeling at CRA Levels - Click “Instructional Strategies - Complete List of Strategies”

Teaching Plans for Selected Mathematics Concepts/Standards at CRA Levels that integrate research supported instructional practices - Click “Teaching Plans”

From the MathVIDS Professional Development Website:

CRA Provides Students A Tangible Foundation for Conceptual Understanding of Abstract Concepts & Processes

Through Concrete & Representational Learning Experiences

That Are Purposeful;

That Are Systematic in Implementation;

That Are Explicitly Associated with the Abstract;

That Provide Multiple Opportunities To Respond;

Where Students Use Language To Describe What They Understand.

How Does Systematic CRA Instruction Address Needs of Struggling Learners?

CRA Instruction & Related Instructional Practices Research

Support

CRA Instruction & Related Instructional Practices Research

Support

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