Concrete, Representational, Abstract (CRA): A Powerful Strategy for Building Mathematical Understanding

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When students struggle in mathematics, it's often because they've been asked to work with abstract symbols before developing a conceptual understanding of what those symbols represent. 

One instructional framework that can help bridge this gap is the Concrete, Representational, Abstract (CRA) approach.

CRA is an evidence-based teaching strategy that moves students from hands-on experiences to visual representations and finally to symbolic notation. The goal is not simply memorization, but deep mathematical understanding. Research suggests that moving between concrete materials, visual models, and abstract symbols helps students build stronger mental representations of mathematical concepts. 

What is the CRA Approach?

CRA stands for:

Concrete

Students manipulate physical objects.

Representational

Students create drawings, models, or pictures.

Abstract

Students solve problems using numbers and symbols.

Many teachers think CRA is a strict sequence:

Concrete → Representational → Abstract

However, effective instruction often involves moving back and forth between all three stages as students deepen their understanding. 

Stage 1: Concrete

At the concrete stage, students physically interact with mathematical ideas.

Examples include:

• Linking cubes
• Two-color counters
• Ten frames
• Base ten blocks
• Fraction strips
• Place value disks

For example, when solving:

8 + 5

Students might build eight cubes, build five cubes, combine them, and count the total.

This stage allows students to:

✓ Build conceptual understanding

✓ Explain their reasoning

✓ Develop mathematical language

✓ Reduce cognitive load

Many students receiving intervention benefit from spending additional time in the concrete stage before moving to symbolic procedures. 

Classroom Tip

Ask students:

"What do you notice?"

"What changed?"

"How do the manipulatives show the math?"

Foam Two-Color Counters

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Concrete learning experiences help students build meaning before introducing symbols.

Stage 2: Representational

Once students demonstrate understanding with manipulatives, they begin representing their thinking through drawings and models.

Examples include:

  • Sketching ten frames
  • Drawing cubes
  • Open number lines
  • Number bonds
  • Bar models
  • Tape diagrams

For our example:

8 + 5

Students might draw:

○○○○○○○○

○○○○○

Then count thirteen objects.

Representational thinking allows students to visualize mathematics without needing physical materials every time.

This stage serves as an important bridge between concrete experiences and abstract notation. 

Stage 3: Abstract

The final stage focuses on symbolic notation.

Students solve problems using numbers, equations, and algorithms.

Examples include:

13 + 7 = ___

54 – 28 = ___

3 × 6 = ___

At this stage, students should be able to connect symbols back to concrete and visual models.

One misconception about CRA is that students leave manipulatives behind forever.

Strong mathematicians frequently move back and forth between all three stages when encountering new concepts or challenging problems. 

CRA in Action

Let's look at subtraction.

Concrete

Put 9 counters on a ten frame.

Remove 7 counters.

Count what remains.

Representational

Draw 9 circles on a piece of paper of white board.

Cross out 7 circles.

Count the remaining circles.

Abstract

9 − 4 = 5

Students who experience all three stages tend to develop stronger conceptual understanding than students who are taught procedures alone. 

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CRA in Action: Building Number Sense

One activity I enjoy using during the representational stage is having students match different representations of a number. Students might build the quantity with counters, identify it on a ten frame, and then connect it to a numeral or equation.

Resources that provide multiple representations of numbers can help students make meaningful connections between concrete models and abstract symbols.

Looking for additional number representation practice? Check out my Representing Numbers 0–5 Cards resource in my TPT store.

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Many combination activities naturally support the CRA framework because students can physically model quantities before transitioning to visual representations and equations.

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Why CRA Matters in Math Intervention

Many students receiving intervention have experienced years of procedural instruction.

They may know steps to follow but lack a true understanding of what those steps mean.

CRA helps students:

  • Develop number sense
  • Build confidence
  • Improve retention
  • Explain mathematical thinking
  • Strengthen problem-solving skills
  • Connect ideas across mathematical concepts

I've found CRA especially useful when teaching:

    ✓ Place value

    ✓ Addition strategies

    ✓ Subtraction

    ✓ Multiplication

    ✓ Fractions

    ✓ Decimals

Common Mistakes Teachers Make with CRA

Moving Too Quickly

Students need time to explore and discuss mathematical ideas.

Skipping the Representational Stage

Pictures serve as an essential bridge between concrete materials and abstract symbols.

Treating CRA as a One-Way Path

Students should continue using manipulatives and visual models whenever needed.

Even older students benefit from concrete experiences when learning new concepts. 

Linking cubes remain one of the most versatile manipulatives for implementing CRA in elementary classrooms.

MathLink Cubes

Concrete learning experiences help students build meaning before introducing symbols.

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Final Thoughts

CRA is much more than a teaching strategy.

It's a framework that helps students build lasting mathematical understanding.

When students can manipulate, visualize, and symbolize mathematical concepts, they develop the flexibility and confidence needed for long-term success.

The next time a student struggles with a concept, consider asking:

"Do they need more practice with the abstract, or do they need another opportunity to experience the mathematics concretely?"

Sometimes the most powerful intervention is simply giving students the chance to make mathematics visible.

Disclosure: As an Amazon Associate, I earn from qualifying purchases. This comes at no additional cost to you.

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