Mathematics is a subject that unfolds like a story; one that moves from tangible experiences to symbolic reasoning. In elementary classrooms, students benefit from a structured progression that helps them build deep, conceptual understandings. The Concrete, Representational, Abstract (CRA) model is one such instructional framework that guides learners along this journey. The CRA model lays a strong foundation for mathematical thinking and problem solving by beginning with hands-on experiences, moving to visual representations, and finally arriving at abstract symbols.
For decades, educators and researchers have emphasized the importance of starting with what students can physically manipulate and see. When children interact with concrete objects, they develop an intuitive sense of mathematical concepts. Then, by transitioning to representational models, such as drawings, diagrams, or pictorial representations, students begin to form connections between what they see and the underlying mathematical concepts. Finally, moving into the abstract stage, learners are much more likely to apply standard mathematical symbols and notations with confidence. This gradual progression not only enhances understanding but also helps students become flexible thinkers who can navigate complex problems.
What Is the CRA Model?
The CRA model is a three-part instructional approach that supports students as they move from concrete experiences to abstract reasoning.
Concrete Stage
In the concrete stage, students work with physical objects, or manipulatives such as connecting cubes, counters, or fraction strips, to explore mathematical ideas through hands-on activities that make abstract concepts more accessible. The concrete stage is often referred to as the “doing” stage because students are physically experiencing the mathematics.
Representational Stage
Next, in the representational stage, also known as the pictorial or visual phase, students use drawings, diagrams, or visual models to represent the problem or concept. This approach translates their physical experiences into images, which serve as a bridge to more abstract language and symbols.
Abstract Stage
Finally, in the abstract stage, students work with symbols, numbers, and formal mathematical notation, applying the concepts they have built through the concrete and representational stages to solve problems and express their ideas in standard mathematical form. It is in the abstract stage that students attach social conventions, such as mathematical symbols and vocabulary, which they typically learn through explicit instruction, to the concepts they have already understood in the concrete and representational stages.
The CRA progression mirrors the natural way children learn. It respects the developmental process by ensuring that learners do not jump into abstract symbols before they are ready. Instead, students are guided through a continuum that reinforces understanding at every step.
The Origins and Research Behind the CRA Model
The roots of the CRA model date back to decades-old research focused on how people learn and develop. In the 1960s, psychologist Jerome Bruner introduced the idea that children learn in three ways: through hands-on experiences (concrete), through pictures or visual models (representational), and through symbols (abstract). His work helped us understand that learners first experience things directly, then form mental images, and finally move to abstract thinking.
Educators later used Bruner’s ideas to create a step-by-step approach for teaching math: students first work with physical materials, then create drawings or diagrams of what they learned, and finally express their understanding using formal mathematical symbols.
Later research confirmed that this three-stage approach is very effective for building deep math understanding. Studies by researchers like Carraher and others have shown that when students start with tangible objects, such as cubes, counters, or pattern blocks, they grasp basic math ideas more easily before moving on to abstract notation. This research indicates that the CRA model not only helps with understanding and memory but also boosts problem-solving skills and overall confidence in math.
Today, the CRA model is widely used in elementary classrooms as a powerful strategy that fits naturally with how children learn, supporting a diverse range of students to succeed in mathematics.
The Benefits of the CRA Approach
Implementing the CRA model in the classroom offers many benefits for both students and teachers. It helps students develop a deep understanding of math by starting with concrete materials that let them experience math in a hands-on way. This direct interaction lays the foundation for grasping core ideas instead of just memorizing steps. As students move to the representational stage, they create visual connections that strengthen their understanding and make the jump to abstract reasoning smoother.
The CRA model also boosts problem-solving skills by encouraging students to explore different ways of representing the same problem. Whether they are using physical objects, drawing models, or writing equations, students learn that there are multiple ways to approach and solve a math challenge. This flexibility encourages creative problem solving and critical thinking.
When students explain their ideas, such as describing how a set of manipulatives shows a number or how a drawing illustrates a pattern, they build strong communication skills. This process deepens their own understanding and makes classroom discussions richer, as peers learn from each other.
Additionally, by connecting familiar, concrete experiences to more challenging abstract ideas, the CRA model helps reduce math anxiety and can improve a child’s math identity. Students build confidence as they see that trying different approaches and making mistakes is a natural and valuable part of learning. This supportive progression makes math more accessible for everyone.
Finally, the CRA model is great for differentiated instruction. Teachers can tailor activities at each stage to meet the varied needs of their students, offering extra hands-on exploration for those who need more time, while challenging advanced learners to explain the connection between the stages or tackle more complex problems.
Implementing CRA in the Classroom
Implementing the CRA model in your classroom doesn’t have to be overwhelming. With thoughtful planning and intentional practice, you can weave concrete, visual, and abstract activities into your daily math lessons.
Begin by selecting concrete manipulatives that directly relate to the concept you’re teaching. For example, when introducing basic addition or subtraction, you might use counters, ten frames, or connecting cubes, and when teaching fractions, offer fraction strips or Cuisenaire rods. Allow students ample time to explore and experiment with these tools so they begin to internalize key math ideas, recognize patterns, and even make their own conjectures.
Next, encourage your students to create drawings or diagrams that represent their experiences, connecting what they’ve learned with the physical materials to visual representations. Use these drawings as a bridge to help them understand and gradually connect with formal mathematical symbols.
A key to understanding the CRA model is that it isn’t about spending a few days on each phase in a set order; rather, all three components work together simultaneously to promote high engagement and deep understanding.
It’s also important to keep in mind that when using CRA, you may cover fewer problems, but you’ll go much deeper with each one; prioritize depth over breadth. The idea is not to think of CRA as a strict progression where concrete activities belong only in the early grades and abstract work only in the later grades; manipulatives, hands-on experiences, and drawings have a place in every classroom from kindergarten through high school.
However, it’s unrealistic to expect every lesson to incorporate all components of the CRA model right away. Instead, as you plan your lessons, consider these guiding questions:
- What manipulatives can help my students understand this concept?
- If I don’t have access to ideal tools, how can I creatively use what I do have? What hands-on experiences or real-world connections can I offer?
- Am I giving students enough time and space to draw and translate their concrete actions into visual models?
- And am I providing opportunities for them to work together and share their thinking?
Reflecting on these prompts and gradually incorporating CRA strategies will help you create a classroom environment that supports deep, meaningful learning in mathematics.
Creating a Supportive Environment for Mathematical Exploration
The success of the CRA model relies on a classroom culture that values exploration, risk-taking, and collaboration. To foster an environment where students feel safe to experiment and express their ideas, it is essential to celebrate multiple approaches. Remind students that there is no single “correct” way to solve a problem; every method is a step toward deeper understanding.
Encouraging peer discussion is also key; having students explain their thinking at each stage through pair-share activities, small-group work, or whole-class discussions can clarify misunderstandings and spark new insights.
In addition, providing consistent feedback and modeling the process are crucial steps. Offer timely and specific feedback that focuses on the process rather than just the final answer, and praise effort, creativity, and perseverance as students work through the CRA stages.
Demonstrate the CRA approach by thinking aloud as you work through a problem on the board, showing students how to move from a physical demonstration to a diagram and finally to a symbolic representation. When students know that their learning journey is supported and that mistakes are a natural part of the process, they are more likely to take risks and fully engage with challenging mathematical ideas.
Overcoming Challenges and Supporting Student Learning
Like any instructional approach, integrating the CRA model can present challenges. Some students might resist moving from a familiar, hands-on activity to a more abstract stage, while others who excel at memorizing rules and procedures may prefer to rely solely on abstract representations, not recognizing the value of the concrete and representational phases. To address these challenges, teachers can pace the instruction flexibly, allowing students who need more time to spend additional moments in the concrete or representational phases.
Reinforcing the connections between stages by using multiple representations is also essential. For instance, when solving a problem, displaying a solution that includes manipulatives, a drawing, and the corresponding equation can help students make these links.
Additionally, differentiating instruction by offering tiered activities or varied levels of challenge within each CRA stage ensures that every student, regardless of their current understanding, can engage meaningfully with the material.
Collaborating with colleagues to share successful strategies and resources can lead to new insights and help create a school-wide culture that embraces the CRA model. Anticipating potential hurdles and planning proactive interventions is a way to ensure that every student benefits from this scaffolded approach to learning mathematics.
Conclusion
The CRA model is more than just an instructional strategy, it’s a pathway that transforms how students engage with mathematics. It nurtures flexible problem solvers by building robust conceptual understanding that begins with hands-on experiences, moves through visual representations, and culminates in abstract symbols. Students not only learn how to perform mathematical operations but also develop a deeper appreciation for the “why” behind the procedures.
In today’s diverse classrooms, where every student brings unique strengths and challenges, the CRA model offers a way to meet learners where they are and guide them toward higher levels of mathematical thinking. When teachers intentionally embed concrete materials, visual models, and abstract reasoning into their lessons, they create an environment where curiosity is encouraged, mistakes are viewed as learning opportunities, and every student has the chance to succeed.
The CRA model provides a way for teachers to transform mathematics instruction from a series of disconnected steps into a coherent, meaningful conversation; a conversation that begins with the tangible, moves through the visual, and culminates in the abstract world of symbols. When students see the connections between these stages, they gain not only mathematical proficiency but also the confidence to explore, question, and grow as lifelong learners.
