Teaching with High Standards

Building a Math House of Knowledge

A Highly Effective Approach to Teaching Mathematics 

Mastering Math Concepts: The Concrete-Representational-Abstract (CRA) Approach

Educators are always searching for the most effective way to help students master math concepts. In today’s classrooms, meeting the diverse needs of all learners in one lesson can be a challenge. The CRA approach — moving from Concrete to Representational to Abstract methods — offers a structured path that gives students time to deeply understand concepts before advancing. By building this progression into your math instruction, you can make a difference in how students grasp and master each concept.

Three Important steps to Building a "Math House" of Knowledge

Step 1: Concrete Method — Building the Foundation

Before building a house, the foundation must be laid. In math, this foundation is built by using concrete manipulatives, which offer a range of benefits. They promote student engagement and provide hands-on opportunities for students to construct ideas that support deeper conceptual understanding. Manipulatives help students develop cognitive models, giving them a tangible way to explore the math concepts they’re learning.

Concrete Manipulative Examples:

• Base Ten Blocks
• Unifix Cubes
• Fraction Bars
• Colored Tiles
• Candy (M&M, Skittles, etc.)

Step 2: Representational Approach — Constructing the Walls

While hands-on exploration is critical, students must also transition to representational thinking to succeed in formal assessments and real-world applications. At this stage, students begin drawing models or creating visual representations to demonstrate what they’ve learned with manipulatives. These drawings or diagrams help students verify their understanding and reinforce the concepts they built in the concrete stage. As they refine these cognitive ideas, they move closer to working with abstract representations.

Representational Examples: 

• Pictures or Drawings
• Tally Marks
• Place Value Charts
• Hundredths Grids

Step 3: Abstract Method — Finishing the Roof

By the time students reach the abstract level, they’re ready to express concepts using numbers and symbols alone. Students who can demonstrate abstract understanding and articulate the reasoning behind their methods have truly mastered the concept. This is the point at which they can apply standard algorithms and show they understand not just how to get an answer, but why the process works.

Abstract Examples:

• Standard Algorithms

Benefits of the CRA Approach

• Provides a structured progression that supports understanding.
• Builds conceptual understanding by moving from concrete to abstract thinking.
• Allows for differentiated instruction tailored to students' levels.
• Utilizes a multi-sensory approach for diverse learning needs.
• Aligns with Common Core State Standards.
• Helps students learn underlying concepts, not just procedural rules.

By building a strong foundation, adding layers of representational understanding, and achieving abstract mastery, the CRA approach offers a transformative pathway to math success for all learners.

Planning:

Planning for these lessons can be extremely time consuming. Once planned, you still need to find the materials needed to implement the plan you created.  Wouldn’t it be nice to find a unit that already incorporates these ideas in a progression of lessons ready for student use? Check out my link to find already made resources to use with students to build this mathematical house of knowledge.
https://www.teacherspayteachers.com/Store/Teaching-With-High-Standards