BookNook Insights

Math Education Insights: Boosting Student Performance with Concrete and Semi-Concrete Representations




Are Virtual Manipulatives as Effective as Tactile Manipulatives in Math Education?

SERIES: Part 1

This article is the first of a two-part series discussing the impact of physical and virtual math manipulatives on student performance.

Subscribe to the BookNook Insights blog for Part 2 (next week) to read more about virtual vs. physical manipulatives. 

In light of recent national math scores, it’s clear that innovative and effective teaching methods are more important than ever. One approach that has consistently proven its worth is the use of concrete and semi-concrete representations. These methods not only make abstract mathematical concepts more tangible but also significantly enhance student engagement and comprehension.

Understanding Concrete Representations

Concrete representations, often called physical manipulatives, involve using physical objects to illustrate mathematical concepts. These are tangible items that students can manipulate to better grasp abstract ideas. Examples include counting blocks, base-ten blocks, fraction circles, and coins.

Examples of Physical Manipulatives

Store-bought Math Manipulatives
  • Counting Blocks: Used for basic arithmetic operations like addition and subtraction.
  • Fraction Circles: Help in understanding fractions by visually demonstrating parts of a whole.
  • Base-Ten Blocks: Aid in teaching place value and arithmetic operations.
Everyday Objects
  • Legos: These versatile building blocks can be used to teach concepts such as fractions, addition, subtraction, and even multiplication and division.
  • Food Items: Items like pizza slices, candy pieces, or fruit segments can make fractions and ratios more tangible and relatable.
  • Buttons and Beads: Great for counting, sorting, and pattern recognition activities.
  • Bottle Caps: Can be used for counting, sorting, and grouping exercises.
  • Cardboard Cutouts: Shapes cut from cardboard can represent different geometric figures, fractions, or algebraic concepts.
  • Egg Cartons: Useful for demonstrating arrays, multiplication, and division.
Interactive and Sensory Tools
  • Sand and Water Tables: Drawing shapes and numbers in sand or using water to measure volumes can make learning sensory and engaging.
  • Clay and Playdough: These materials can be molded into different shapes to represent geometric figures, fractions, or even solve algebraic problems visually.
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Benefits of Concrete Representations

Concrete representations are especially beneficial in early math education. They provide a solid foundation by allowing students to physically interact with the objects, thus reinforcing their understanding through tactile experiences. This hands-on approach makes learning more engaging and helps in retaining concepts longer.
  • Engagement: Using unexpected items can capture students' interest and make learning fun.
  • Relatability: Everyday objects make math more relatable and less intimidating.
  • Creativity: Encourages creative thinking and problem-solving.
  • Accessibility: Many of these items are readily available and cost-effective, making them accessible to all educators.


Exploring Semi-Concrete Representations

Semi-concrete representations, also known as pictorial representations, involve using drawings or visual aids to represent physical objects. These visual tools act as a bridge between concrete manipulatives and abstract symbols.
As students progress in their learning journey, they transition from using physical objects to visualizing these objects through drawings or diagrams. This gradual shift helps them move towards abstract thinking.

Examples of Visual Aids

  1. Drawing Pictures: Students can draw objects like circles or squares to represent numbers or quantities.
  2. Number Lines: Used to visually represent addition, subtraction, and other operations.
  3. Bar Models: Helpful in solving word problems by visually breaking down the information.

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Virtual Manipulatives: A Modern Blend

Virtual manipulatives are digital tools that mimic the functionality of physical manipulatives. They can be interacted with using a stylus, mouse, or touchscreen, providing a blend of concrete and semi-concrete experiences. Early research suggests both physical and virtual manipulatives have a positive impact on student performance. That being said, the most benefit was seen when the two were used together. 

Technology-Enhanced Tools

Virtual manipulatives bridge the gap between concrete and semi-concrete representations by combining the interactive nature of physical objects with the convenience of digital tools. Examples include digital counting blocks, fraction circles, and base-ten blocks available in educational apps or software. Additional examples include: 

  1. Augmented Reality (AR): AR apps like GeoGebra AR can overlay digital information on physical objects, making math concepts interactive and immersive.
  2. 3D Printing: Custom manipulatives can be created to fit specific educational needs, such as geometric shapes, complex solids, or customized fraction pieces. This example is unique in that it blends digital technology with more traditional tools. 

Benefits of Digital Tools

  1. Interactivity: Allows for direct manipulation similar to physical manipulatives.
  2. Accessibility: Easily accessible on various devices, making learning flexible and scalable.
  3. Engagement: Interactive features keep students engaged and motivated.


Practical Applications for Physical and Virtual Manipulatives

Concrete and semi-concrete representations have a wide range of practical applications in various educational settings. These tools are invaluable in early childhood education, special education, and math tutoring and intervention programs. By providing tangible and visual ways to understand abstract concepts, manipulatives play a crucial role in enhancing mathematical comprehension and student engagement across diverse learning environments.

Early Childhood Education

Teachers use manipulatives like blocks and counters in kindergarten and early elementary grades to introduce basic arithmetic concepts. These tools help young learners visualize and understand math problems better.

Special Education

Students with learning differences benefit significantly from concrete and semi-concrete tools, both physical and virtual. These representations make abstract concepts more accessible, catering to their unique learning needs.

Math Tutoring and Intervention Programs

High-impact tutoring programs, such as those offered by BookNook, leverage manipulatives to provide tailored learning experiences. These tools are integral in helping students catch up and excel in math.

Impact on Student Engagement and Understanding

Using manipulatives enhances student engagement by making learning interactive and fun. It also improves understanding by providing multiple ways to approach and solve problems.

Recommendations for Educators

  1. Early Stages: Introduce concrete manipulatives when students are first learning new concepts.
  2. Transition Gradually: Move from concrete to semi-concrete representations as students gain confidence.
  3. Incorporate Digital Tools: Use virtual manipulatives to provide a versatile learning experience.

Importance of Transitioning

Transitioning from concrete to semi-concrete to abstract stages helps students build a robust understanding of mathematical concepts. This scaffolded approach ensures that they are well-prepared for more complex math problems. It also promotes cognitive development by gradually encouraging students to think more abstractly and independently. This progression mirrors the natural learning process and supports the development of critical thinking skills essential for success in math and other academic areas.

Connecting to Pedagogical Concepts

The use of concrete and semi-concrete representations in math education not only enhances student engagement and understanding but also aligns seamlessly with various pedagogical approaches. By integrating these methods, educators can create a more effective and inclusive learning environment that caters to diverse learning styles and reinforces key mathematical concepts.

Relation to Scaffolding and Differentiated Instruction

The use of manipulatives aligns with scaffolding techniques by gradually increasing the complexity of tasks. Differentiated instruction is also supported, as manipulatives can be tailored to meet the diverse needs of students.

Connection to Constructivist Learning Theory

Constructivist learning theory emphasizes the importance of students building their own understanding through hands-on experiences. Manipulatives provide these experiences, making abstract concepts more concrete.

Integrating Manipulatives with Other Teaching Methods

Combining manipulatives with traditional teaching methods can create a comprehensive learning environment. This integration helps reinforce concepts and caters to different learning styles.


Concrete and semi-concrete representations play a vital role in math education. They help students develop a deep understanding of mathematical concepts through hands-on and visual experiences. Educators are encouraged to incorporate these methods into their teaching practices to enhance student engagement and comprehension. 

As digital tools continue to evolve, the future of math education looks promising with the integration of innovative manipulatives.

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