Effective Mathematics Teaching Practices:
Explicit CRA Practices – Making Mathematics Transparent
David H. Allsopp, Ph.D.University of South Florida
“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
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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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