Using the CRA Framework in Elementary Math

The CRA (Concrete Representational Abstract) framework is a model that many math teachers use to engage students in conceptual understanding and procedural learning. Using the model, students typically start with concrete tools (e.g., math manipulatives), move into semi-concrete or pictorial representations (e.g., drawings, digital models), and work toward abstract (e.g., algorithmic) applications of math concepts.

Using the CRA framework can help to build a student-centered elementary math classroom grounded in evidence-based practices. Too often, students rush to learn standard algorithms without opportunities to learn math in the context of concrete and pictorial models.

The CRA framework allows teachers to differentiate instruction based on student needs, understandings, and misconceptions. While it’s widely known and used in math instruction, we focus on how to best integrate the framework in an elementary setting to prioritize concrete, hands-on learning.

Utilizing CRA: Introducing Math Tool kits

We’ve enjoyed working together to combine our respective expertise in the theory of CRA (Tori) and its implementation in practice (Michaela). Here, Michaela shares stories from a first-person perspective about using CRA in her classroom in meaningful ways, which Tori builds upon from a research-based lens.

In my (Michaela’s) first two years of teaching, my math block focused heavily on whole group instruction and lacked hands-on experiences. While this approach worked for some, there were many drawbacks. I was uncertain of students’ abilities and found myself having difficulty differentiating instruction. After some reflection, I revamped my math block.

I began by creating math tool kits for my students—large zipper bags that contain math manipulatives (e.g., base-10 blocks, coins, pattern blocks, red and yellow counters) for daily use. The purpose was to give students constant access to math manipulatives for more efficient transitions, while also supporting the need for concrete representations and hands-on learning in math. 

I introduced the kits on the first day of school, explaining to students that these tools would support their learning. I explicitly modeled how they should use the materials. Since introducing the kits, I’ve seen tremendous growth. Unencumbered access to math manipulatives means that using math tools has become a part of their daily instructional routine, and their comprehension has deepened as a result; many students have transitioned to representational and abstract models with a greater understanding of the meaning behind these representations, building their independence and confidence in applying math skills to various contexts. 

Restructuring Instruction

With the addition of math tool kits, my approach to Tier 1 instructional delivery has shifted to a small group model. Students engage in four 20-minute rotations, including direct small group instruction from me, game-based learning, math centers, and independent practice. 

Since each student has access to their own tool kit, they’re able to utilize math manipulatives during each rotation. This change has allowed me to gain a deeper understanding of my students’ strengths and areas of needs and to support their understanding of math concepts through the use of concrete models and pictorial representations. It’s also allowed them to see me modeling concrete and pictorial representations up close.

Students now get the individualized attention they need and engage more fully in their learning, often through collaborating with peers. They practice using models more consistently and can receive immediate teacher feedback. Ultimately, they’re better at making connections between specific math concepts and abstract representations, drawing from prior experiences with building concrete and pictorial models in multiple contexts. 

Aligning Tools with Instruction

Math tool kits provide multiple entry points into the CRA framework and support students’ conceptual understandings of grade-level content. When considering how to best align math tools with this instructional approach, it’s important to take into account the specific math concepts you’re teaching. In other words, identify specific math tools or pictorial representations that will best lend themselves to related math concepts. 

For example, if your instruction focuses on place value, base-10 blocks or place value mats may best support this concept. In contrast, if the instructional focus is fractions, then fraction bars, Cuisenaire rods, number lines, or pattern blocks may be more appropriate. If the concept is more skill-based, such as addition and subtraction operations, then base-10 blocks, 100s charts with chips, Unifix cubes, number lines, or two-colored counters may work best, depending on the value of the numbers and complexity of the operations. 

It’s also important to align expectations for students’ concrete and pictorial representations with grade-level standards. For example, if elementary students are expected to identify or represent fractions using area and set models, they need to have hands-on opportunities to build these specific models and to understand the fractional components of different model types. 

Math tool kits, paired with teacher modeling, allow students to select math manipulatives for specific concepts, building conceptual understanding. By giving students a finite amount of tools to pick from, you can reasonably ensure that available tools support mathematical concepts and allow student choice, without overwhelming students. 

This element of structured choice enables students to determine how best to represent their thinking, whether through concrete models, pictorial representations, or abstract and algorithmic thinking. In order to maximize the effectiveness of the CRA framework, it is important for teachers to offer multiple opportunities for students to develop, reinforce, and apply conceptual understandings and procedural knowledge through the use of math tools, digital resources, and student-centered representations. 

Achieving Differentiation

When used with purpose and intentionality, the CRA framework can support diverse learners in various instructional contexts. By aligning current mathematical practices with the CRA framework, you provide students with ongoing opportunities to develop conceptual understandings across multiple representations, often created with the use of concrete tools. 

As teachers, we have found that it’s important to carefully consider which readily accessible mathematical tools can best support students’ learning and understanding during instruction, guided practice, and their independent application of skills. Based on the CRA continuum, students need consistent practice with concrete tools and pictorial models/representations to foster conceptual understanding before moving into the abstract or algorithmic process of mathematics. This means tool kits are just the start—a catalyst, of sorts, for other forms of mathematical engagement.