A Practical Framework for Helping Students Understand Math Concepts

Helping your students understand math can feel overwhelming, especially when you are balancing pacing, standards, and the wide range of learners in your classroom. One framework that consistently supported deeper understanding when I was in the classroom, and continues to support strong math instruction today, is Concrete–Representational–Abstract (CRA). While CRA is often described as a progression, it works best as a flexible framework that allows you to respond to student thinking. When used intentionally, CRA helps your students understand math by connecting hands-on experiences, visual models, and abstract symbols in meaningful ways.

Using Concrete Experiences to Help Students Understand Math

Concrete experiences give your students something tangible to think about when learning new math concepts. These experiences involve manipulatives or models that your students can touch, move, and explore. When your students understand math through concrete tools, they begin building meaning before symbols and numbers are introduced.

When I was in the classroom, I found that concrete experiences were especially helpful for introducing new concepts or when my students struggled to explain their thinking. Concrete tools allowed my students to slow down and make sense of what the math actually represented. These experiences created a foundation that made later abstract work more meaningful.

Concrete does not always mean physical objects alone. Digital manipulatives can also help your students understand math when they are used intentionally. Virtual base ten blocks, fraction strips, or algebra tiles can support sense-making. The key is that your students are actively modeling and reasoning. 

Focused concrete experiences can strengthen understanding without replacing core instruction. Using hands-on or digital manipulatives during the introduction, practice, or review of math concepts only helps our students build a stronger understanding of foundational math concepts.

Representational Models That Help Students Understand Math

Representational work, sometimes referred to as pictorial or semi-concrete, is the stage in which your students use drawings, diagrams, or visual models instead of physical objects. All of these terms describe the same middle phase of CRA. This stage plays a critical role in helping your students understand math by bridging hands-on experiences and abstract reasoning.

When I was teaching, representational work revealed far more about student understanding than pre-made visuals ever could. Student-created drawings, number lines, tape diagrams, or part-part-whole models showed how my students were thinking. These showed me a lot more than whether they arrived at the correct answer or not. These representations did not need to be neat or polished. What mattered was that they reflected student reasoning.

Representational work can look different from student to student. One of your students might draw a model while another uses manipulatives to solve the same problem. In partner work, one student might represent the math visually while the other models it concretely. This flexibility allows your students to understand math in ways that make sense to them and supports more effective math conversations.

Abstract Tools That Help Students Understand Math Concepts

Abstract work involves symbols, numbers, notation, and algorithms. This is where many traditional math tasks live. Abstract work is most effective when it builds on understanding developed through concrete and representational experiences. Your students will understand math more deeply when abstract symbols represent ideas they already know.

When I introduced algorithms in my classroom, I framed them as tools rather than shortcuts. Algorithms represented the most efficient way to do math once my students understood why the math worked. When my students memorized steps without understanding, gaps often surfaced later during problem-solving or explanations.

Moving Between Levels of the Framework 

Flexibility is the name of the game when it comes to our thinking around CRA. Students may move back and forth between the levels of support as they learn and practice math concepts. Similarly, abstract work does not signal the end of CRA. If your student can arrive at a correct answer but cannot explain how or why it works, that is often a cue to revisit a concrete or representational model. Moving back within the framework helps your students understand math concepts more fully and strengthens long-term learning.

Helping Students Understand Math Across Grade Levels

CRA is not limited to elementary classrooms. Students at all grade levels benefit from concrete, representational, and abstract thinking. This is extremely true as math concepts become more complex. Understanding math requires reasoning, not just computation.

Upper-grade students often benefit from tools such as area models for fraction operations. They'll also benefit from using algebra tiles for expressions and equations and number lines for integer reasoning. These representations help your students visualize relationships that might otherwise feel abstract or confusing.

When I worked with older students, these models often revealed misunderstandings that would not have surfaced through abstract work alone. Providing access to concrete and representational tools helped my students understand math concepts more deeply. They were also able to explain their thinking with greater confidence.

Using Technology to Help Students Understand Math

Technology can support CRA when used intentionally. Digital tools should help your students understand math by encouraging modeling, representation, and exploration. Tools focused solely on speed or quick answers often limit opportunities to make sense of the problem.

When I included technology in my math instruction, the focus was always on understanding rather than efficiency. I chose digital manipulatives because they allowed my students to move pieces, test ideas, and visualize relationships. 

When selecting technology, consider whether it supports student thinking and progression within the CRA framework. Used intentionally, digital tools can strengthen understanding. They provide another way for your students to engage meaningfully with math concepts.

Observing Student Thinking to Help Students Understand Math

One of the most important aspects of using CRA effectively is observing your students working. Watching how your students solve problems helps you decide which phase of the framework will best support understanding. CRA works best when instructional decisions are guided by evidence from student thinking.

When I noticed my students could compute accurately but struggled to explain their reasoning, it often signaled the need to revisit concrete or representational experiences. This was not a step backward. It was an intentional instructional move that helped my students understand math more deeply.

Sometimes we worry that spending time on models takes away from instruction. In reality, these moments often save time later. When your students truly understand math concepts, they make fewer errors, require less reteaching, and approach problem-solving with greater confidence. Sometimes you have to give time to get time.

Using Task Cards to Help Your Students Understand Math With CRA

My Multiplying Decimals by Whole Number Task Cards resource includes three different task card activities that support your students as they develop their understanding of multiplying decimals and whole numbers. I designed each activity to give your students multiple ways to make sense of the math before relying on an algorithm. This makes it easy to use within the Concrete-Representational-Abstract framework. 

The DINOmite Decimals activity focuses on building conceptual understanding. Your students need to determine how many groups to create, build the problem using math tools or visual models, write the expression, and then solve. This structure encourages your students to understand math by connecting the idea of groups to decimal values. The emphasis is on modeling and reasoning, not just computation.

The Stompin’ Decimals activity supports your students who are ready to move between representational and abstract thinking. Your students will identify multiplication sentences, draw or visualize models, and solve using strategies that fit their readiness level. This activity reinforces the meaning of multiplication while still allowing your students to show their thinking in their own way.

The Decimal BONEanza activity provides opportunities for more independent practice and review. Your students will be able to solve decimal multiplication problems with whole numbers and decimals. For this activity, there is a strong focus on place value reasoning. This activity works well when your students are transitioning into abstract work, but still need to revisit models or explanations to fully understand the math concepts.

You can choose between these tasks based on student readiness. You can even rotate activities during math centers, or use them to move your students back and forth within the CRA framework as needed. This flexibility supports a deeper understanding. It also allows your students to understand math in ways that make sense to them.

Helping Your Students Understand Math With Confidence

Helping your students understand math is not about rushing them through steps or sticking rigidly to one instructional path. It is about paying attention to how your students think and giving them the tools they need to make sense of ideas. When you view Concrete–Representational–Abstract as a flexible framework instead of a fixed sequence, you give yourself permission to respond to student needs in real time.

When your students have access to this framework, learning becomes more meaningful. They are better able to explain their thinking, apply strategies, and recover when they feel stuck. These learning moments build confidence and reduce frustration for both you and your students. 

Save for Later

Save this post to your favorite math Pinterest board as a reminder that CRA is not a checklist to complete or a sequence your students must master in order. It is a responsive framework that allows you to move back and forth based on student understanding.