The Concrete-Pictorial-Abstract Approach - Building Deep Mathematical Understanding

Why do some students memorize procedures without understanding, while others develop deep mathematical thinking? The answer often lies in how concepts are introduced. The Concrete-Pictorial-Abstract (CPA) approach is a research-backed teaching method that builds genuine understanding by starting with physical objects, progressing to visual representations, and finally moving to abstract symbols.

What Is the CPA Approach?

The CPA approach is a three-stage learning sequence:

1. Concrete Stage - “Do It”

Students use physical objects (manipulatives) to explore mathematical concepts through hands-on experiences.

Examples:

• Counting blocks for addition
• Fraction circles for understanding parts of wholes
• Base-ten blocks for place value
• Measuring cups for capacity

2. Pictorial Stage - “See It”

Students use visual representations—drawings, diagrams, models—to represent the mathematical concepts they explored concretly.

Examples:

• Drawing circles to represent counting
• Sketching fraction diagrams
• Drawing base-ten blocks
• Illustrating word problems

3. Abstract Stage - “Symbolize It”

Students use mathematical symbols, notation, and formal algorithms to represent concepts they now understand deeply.

Examples:

• Writing equations: 3 + 5 = 8
• Using fraction notation: 3/4
• Recording multi-digit calculations
• Solving problems symbolically

Why CPA Works: The Science Behind It

Brain-based learning:

• Hands-on experiences create stronger neural connections
• Multiple representations reinforce understanding
• Physical-visual-abstract sequence mirrors natural cognitive development

Research shows:

• Students who learn through CPA have deeper conceptual understanding
• CPA learners can apply knowledge to new situations more effectively
• The approach reduces maths anxiety by making abstract concepts tangible
• Long-term retention improves significantly

The problem with jumping to abstract: When we start with symbols (3 + 5 = ?), many students memorize procedures without understanding what addition means. They know “the answer is 8” but don’t conceptualize combining quantities.

Implementing CPA in the Classroom

Concrete Stage: Choosing and Using Manipulatives

Effective manipulatives for different concepts:

Number sense & operations:

• Counters, blocks, buttons for counting
• Number lines (physical, floor-sized)
• Ten-frames for building number sense

Place value:

• Base-ten blocks (ones, tens, hundreds)
• Place value discs
• Bundling sticks

Fractions:

• Fraction circles and bars
• Pattern blocks
• Paper folding
• Food items (pizza, chocolate)

Geometry:

• 2D and 3D shapes
• Geoboards with rubber bands
• Tangrams

Measurement:

• Rulers, measuring tapes
• Scales and balances
• Containers for capacity
• Clocks (analog with moveable hands)

Money:

• Play coins and notes
• Cash register for role-play

How to use manipulatives effectively:

1. Introduce purposefully: Don’t just give out materials without context

   • “Today we’re using these blocks to explore what happens when we combine numbers”

2. Allow exploration time: Let students play briefly before structured use

   • Curiosity satisfied = focus on learning

3. Model explicitly: Show how manipulatives represent concepts

   • Think aloud: “I’m taking 3 blocks, and adding 5 more blocks…”

4. Connect to language: Narrate actions using mathematical vocabulary

   • “When we combine these groups, we call that addition”

5. Encourage representation: Have students show their thinking

   • “Can you use the blocks to show me your solution?”

Pictorial Stage: From Objects to Images

The bridge between concrete and abstract is crucial. Students transition from manipulating objects to drawing representations.

Drawing strategies:

Simple drawings (not artistic!):

• Circles or dots for counting
• Rectangles for blocks
• Lines for number lines
• Basic shapes for geometry

Example progression for addition (5 + 3):

1. Concrete: 5 red counters + 3 blue counters physically moved together
2. Pictorial: Draw 5 circles, draw 3 circles, count all circles
3. Abstract: Write 5 + 3 = 8

Bar models (Singapore Math approach): Visual rectangles representing quantities—extremely powerful for word problems.

Example: “Sarah has 12 apples. She gives 5 to Tom. How many does she have left?”

• Draw bar showing 12 (whole)
• Divide into two parts: 5 (given away) and ? (remaining)
• Visual makes subtraction obvious: 12 - 5 = 7

Benefits of pictorial stage:

• Develops visualization skills
• Makes thinking visible
• Supports problem-solving
• Provides assessment evidence
• Creates independence from physical materials

Abstract Stage: Symbols with Understanding

Only after concrete and pictorial understanding should students work with pure symbols.

Introducing abstract notation:

1. Connect to previous stages: Always link back

   • “Remember when we used blocks? And drew pictures? Now we can write it with numbers”

2. Maintain the connection: Keep concrete/pictorial available

   • Students can return to earlier stages if confused

3. Use mathematical language precisely:

   • “The plus sign means combine” (not just “add the numbers”)
   • “The equals sign means the same value on both sides” (not “the answer comes next”)

4. Emphasize meaning, not just procedure:

   • “Why does this algorithm work?” not just “Follow these steps”

CPA Across Mathematical Topics

Example 1: Teaching Multiplication

Concrete:

• Create equal groups with counters
• “Let’s make 4 groups with 3 counters in each”
• Count total: “How many counters altogether?”

Pictorial:

• Draw arrays: 4 rows of 3 dots
• Draw groups: 4 circles with 3 dots each
• Show on number line: jumping in 3s, four times

Abstract:

• Write equation: 4 × 3 = 12
• Understand what symbols mean: “4 groups of 3 equals 12”

Example 2: Teaching Fractions

Concrete:

• Fold paper into quarters
• Cut pizza/chocolate into parts
• Use fraction circles/bars

Pictorial:

• Draw circles divided into parts
• Shade portions to show fractions
• Create fraction walls showing equivalence

Abstract:

• Write fraction notation: 3/4
• Understand numerator/denominator
• Compute with fraction symbols

Example 3: Teaching Place Value

Concrete:

• Use base-ten blocks (ones, tens, hundreds)
• Bundle and unbundle straws
• Group objects into tens

Pictorial:

• Draw representations of base-ten blocks
• Use place value charts
• Show groupings visually

Abstract:

• Write multi-digit numbers
• Understand position determines value
• Perform standard algorithms

Common CPA Mistakes and Solutions

Mistake 1: Rushing through concrete stage

• Symptom: Students struggle when manipulatives removed
• Solution: Spend adequate time until genuine understanding emerges

Mistake 2: Skipping the pictorial stage

• Symptom: Huge gap between concrete and abstract
• Solution: Always include drawing/visual representation phase

Mistake 3: Never reaching abstract

• Symptom: Students dependent on manipulatives for simple tasks
• Solution: Deliberately transition to symbols once understanding is solid

Mistake 4: Using all stages every time

• Symptom: Lessons drag; students frustrated
• Solution: Once mastery achieved, students can work abstractly

Mistake 5: Poor quality manipulatives

• Symptom: Confusion rather than clarity
• Solution: Use clear, purposeful materials that match the concept

Age and Ability Considerations

Early years (Foundation - Year 2):

• Heavy emphasis on concrete experiences
• Lots of time at each stage
• Frequent return to concrete when needed

Middle years (Years 3-6):

• Begin with concrete for new concepts
• Pictorial stage becomes powerful problem-solving tool
• Increasing time at abstract level

Upper years (Years 7-10):

• Concrete still valuable for complex new concepts (algebra tiles, geometry models)
• Pictorial crucial for visualization and problem-solving
• Most time spent at abstract level

Important: Even older students benefit from concrete experiences with new, complex concepts. This isn’t “babyish”—it’s how our brains construct understanding.

Assessing Understanding at Each Stage

Concrete stage:

• Can student use manipulatives to demonstrate concept?
• Can they explain what they’re doing?
• Do they choose appropriate manipulatives independently?

Pictorial stage:

• Can student draw accurate representations?
• Do drawings show mathematical understanding?
• Can they explain their visual representations?

Abstract stage:

• Can student work with symbols accurately?
• Do they understand what symbols represent?
• Can they move flexibly between all three stages?

Warning sign: Student can perform abstract procedures but can’t explain with concrete/pictorial representations—indicates procedural knowledge without conceptual understanding.

Bringing CPA Into Your Teaching

For teachers new to CPA:

Week 1: Choose one concept to teach using full CPA approach

• Gather manipulatives
• Plan all three stages
• Notice student engagement and understanding

Week 2-4: Gradually extend CPA to more topics

• Build manipulative collection
• Develop pictorial strategies
• Create routines and expectations

Ongoing: Make CPA your default approach

• New concepts = start concrete
• Struggling students = return to concrete
• Always connect all three representations

The Bottom Line

The Concrete-Pictorial-Abstract approach isn’t just a teaching technique—it’s a philosophy that respects how mathematical understanding develops. When we rush to abstract symbols before students have concrete and visual understanding, we create a foundation of sand. When we build systematically through CPA, we create mathematical thinkers who truly understand.

The time invested in concrete and pictorial stages isn’t wasted—it’s essential. Students who build this foundation not only master current content more deeply, but develop thinking skills that serve them throughout their mathematical journey. That’s not slower learning—it’s deeper learning that ultimately proves faster and more lasting.