Math 3-6

Concrete-Representational-Abstract (CRA)

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Overview

CRA is a sequential instructional approach during which students move from working with concrete materials to creating representational drawings to using abstract symbols. Using this concrete-representational-abstract sequence helps students develop the thorough mental representations that are foundational for conceptual understanding.

Use It in the Classroom

Watch how CRA can be used to teach difficult math concepts. By using algebra tiles, students can connect concrete objects to abstract algebraic expressions.

  • When teaching a math concept within Algebraic Thinking, such as algebraic expressions or factoring quadratics, teachers can first model with concrete objects like algebra tiles. Then, when students demonstrate mastery with these objects, teachers can work with students to draw the algebra tiles that represent the algebraic expressions. Finally, teachers can introduce the concept with only numbers and math symbols to target abstract understanding. This same process also works to develop Number Sense and Operations.

    • Design It into Your Product

    Videos are chosen as examples of strategies in action. These choices are not endorsements of the products or evidence of use of research to develop the feature.

    Learn how ST Math uses CRA to build mathematical understanding. All math concepts are first presented visually in games requiring digital manipulation of objects. As students progress, the math symbols are slowly introduced until a student sees the full equation represented in the game they are playing.

  • Products can use CRA to sequence and present their lessons using digital features such as virtual manipulatives and drawing tools. These elements can support Speed of Processing and moving concepts from Short-term to Long-term Memory.
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    Resources

    Below are additional examples, research, and professional development. These resources are possible representations of this strategy, not endorsements.

    CRA Instructional Approach

    Informational guide on CRA

    CRA Intervention Template

    Description, procedures, and critical components of implementing CRA

    CRA Video

    Walk through of CRA with a fraction lesson

    Math Interventions

    Subtopic that includes research and evaluations on various interventions to improve children's math skills on Digital Promise's Research Map

    Sequence of Instruction

    Detailed explanation of CRA and how to implement it

    Student Confidence in Math

    The importance of embracing errors as learning, emphasizing reasoning, and relating math to student strengths from Digital Promise's Ask a Researcher

    Factors Supported by this Strategy

    Mathematics
    Algebraic Thinking
    Arithmetic Fact Retrieval
    Math Communication
    Mathematical Flexibility
    Number Sense
    Operations
    Proportional Reasoning
    Cognition
    Long-term Memory
    Sensory Integration
    Short-term Memory
    Spatial Skills
    Speed of Processing
    Working Memory
    Social and Emotional Learning
    Emotion
    Self-regulation
    Student Background
    Adverse Experiences
    Math Learning Environment
    Physical Well-being
    Safety
    Sleep

    More Instructional Approaches Strategies

    Cognitively Demanding Tasks

    Providing math tasks with high cognitive demand conveys high expectations for all students by challenging them to engage in higher-order thinking.

    Direct Instruction: Math Language

    Knowing the language of math is critical because students must use this language to understand math concepts and determine calculations needed.

    Direct Instruction: Patterning

    Thinking of and about patterns encourages learners to look for and understand the rules and relationships that are critical components of mathematical reasoning.

    Direct Instruction: Problem Structures

    Teaching students to recognize common problem structures helps them transfer solution methods from familiar to unfamiliar problems.

    Direct Instruction: Problem-solving Strategies

    Discussing strategies for solving mathematics problems after initially letting students attempt to problem solve on their own helps them understand how to organize their mathematical thinking and intentionally tackle problems.

    Family Engagement

    Students are more likely to come to school when families feel like a valued part of the community.

    Guided Inquiry

    In guided inquiry, teachers help students use their own language for constructing knowledge by active listening and questioning.

    Math Centers

    Math centers support learner interests and promote the development of more complex math skills and social interactions.

    Mindfulness Activities

    Through short but regular mindfulness activities, students develop their awareness and ability to focus.

    Multimodal Instruction

    Instruction in multiple formats allows students to activate different cognitive skills to understand and remember the steps they are to take in their math work.

    Real-world Math

    When teachers connect math to the students' world, students see how math is relevant and applicable to their daily lives.

    Untimed Tests

    Untimed tests provide students the opportunity to flexibly and productively work with numbers, further developing their problem-solving abilities.

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