What is the Concrete-Representational-Abstract (CRA) Approach in Math?
- Serena Readhead
- May 27
- 4 min read
Updated: 5 days ago
Helping Students Build Deep Mathematical Understanding
Table of Contents
When I was in second grade, I learned how to subtract multi-digit numbers the same way many students do: memorize the steps and follow the procedure. If the top number was smaller than the bottom number, I "borrowed" from the digit next door. I crossed out numbers, wrote new ones above them, and somehow arrived at the correct answer.
The problem was that I had no idea why I was doing any of it.
Years later, as an elementary education major, sitting in my math methods course, my professor pulled out base-ten blocks and modeled subtraction in a way I had never seen before. For the first time, I understood what "borrowing" actually meant. I wasn't borrowing at all--I was regrouping. I was trading one ten for ten ones because the value stayed the same. What had once felt like a mysterious rule suddenly made perfect sense.
And that left me wondering: What would my math experience in school have been like if I had received this kind of instruction? How would it have affected my confidence as a mathematician? How many concepts had I learned to perform without ever truly understanding?
As an educator, math became far more exciting once I began teaching with concrete tools like base-ten blocks and fraction strips. Again and again, I watched students experience the same realization I had in that college classroom. Concepts that once felt like a collection of arbitrary rules suddenly made sense. Students weren't just memorizing procedures—they were building understanding.
And that understanding matters. When students understand the "why" behind the mathematics, they are better equipped to solve unfamiliar problems, make connections between concepts, and approach challenging tasks with confidence. The goal of math instruction isn't simply to help students arrive at the correct answer. It's to help them develop a deep understanding that will support their learning for years to come.
Concrete-Representational-Abstract Framework (CRA)
The Concrete-Representational-Abstract (CRA) framework, sometimes called Concrete-Pictorial-Abstract (CPA), is an instructional approach that helps students develop conceptual understanding by moving from hands-on experiences to visual models and finally to mathematical symbols and procedures.
Rather than starting with procedures and memorizing rules, the CRA framework builds understanding step by step. Each stage serves as a bridge to the next, helping students connect what they can see and touch to increasingly abstract mathematical ideas.
Concrete
In the concrete stage, students use physical objects, or manipulatives, to explore mathematical concepts. These tools allow students to model their thinking and make abstract ideas visible.
Examples of concrete tools include:
Base-ten blocks
Counters
Connecting cubes
Pattern blocks
Number tiles
For example, a student learning subtraction with regrouping might physically exchange one tens rod for ten unit cubes (ones cubes). Rather than memorizing a borrowing procedure, the student can see and experience what is happening.
Representational
In the representational stage, students move from physical objects to drawings, diagrams, and visual models. They begin creating pictures that represent the manipulatives they previously used.
Examples of representational models include:
Tape diagrams
Arrays
Number lines
Bar models
Area models
Drawings of base-ten blocks
Continuing the subtraction example, students might draw tens and ones on paper and show the regrouping process through their sketches. These visual representations help students transition away from manipulatives while still providing a concrete reference for their thinking.
Abstract
In the abstract stage, students work with mathematical symbols, numbers, and equations without relying on physical objects or drawings.
Examples include:
Standard algorithms
Number sentences
Equations
Variables and expressions
At this stage, students solve problems such as 42 - 18 using a standard equation with numbers and symbols. Because they have already built understanding through concrete and representational experiences, the procedures have meaning rather than being a series of steps to memorize.
Why Does the CRA (or CPA) Framework Matter?
When we introduce our students to abstract procedures too early, they may learn how to perform a skill without understanding why it works. The CRA framework helps students build mental models that support long-term understanding.
The goal is not for students to remain in the concrete stage forever. In fact, one of the most common misconceptions about the CRA framework is that students should always rely on manipulatives or drawings to solve problems. I once taught at a school where administrators discouraged teachers from encouraging fourth graders to memorize their multiplication facts. Instead, students were expected to draw an array for every basic fact to show their work. While arrays are a powerful tool for building conceptual understanding, requiring students to draw them for every fact is neither practical nor helpful once they have developed a solid understanding of how multiplication works.
Manipulatives and drawings are essential tools for building understanding, and we should not rush students through these stages. However, they are not the end goal. Once students understand a concept, they should gradually move toward more efficient strategies and abstract reasoning. The purpose of the CRA framework is to provide the support students need so they can eventually think and reason abstractly with confidence.
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