Embodied Numeracy: Making Sense of Mathematics Through the Linear Abacus®

Genovieve Grouios

Have you ever watched your child dutifully follow the steps of a math procedure, only to realise they couldn't explain why it works or apply it in a different context? Perhaps you've experienced that moment when your child correctly solved a problem but couldn't make sense of the answer? These experiences reveal a common disconnect in mathematics education—where procedures are often taught without enough attention to meaning and understanding.

This disconnect isn't merely academic—it shapes how children see themselves as mathematical thinkers. Many adults still carry the scars of mathematics education that emphasised memorisation over meaning, leading to the all-too-common refrain: "I'm just not a math person." The Linear Abacus® offers a different path—one where mathematical understanding grows naturally through meaningful physical experiences.

Unlike traditional approaches that rush to symbolic representation, the Linear Abacus® honors a fundamental truth about how we learn: our understanding begins in our bodies. When children physically manipulate the beads on the Linear Abacus®, they're not just illustrating abstract concepts—they're actually building the neural foundations for those concepts. Mathematics becomes something they can see, touch, and make sense of, not just symbols to manipulate according to mysterious rules.

Through the games in the Linear Abacus® Games Book, children develop a unique understanding of numbers as "names of places in an order that answers 'how many?'"—an approach that unifies the cardinal (how many) and ordinal (which position) aspects of number. This seemingly simple insight unlocks a coherent view of mathematics as a meaningful system rather than a collection of disconnected procedures.

The Philosophy Behind the Linear Abacus®

At the heart of the Linear Abacus® approach lies a powerful vision of mathematics education that challenges conventional practices. While traditional approaches often rush to abstract symbolism, the Linear Abacus® recognises that deep mathematical understanding must be built on a foundation of embodied experience.

Sense-Making: The Heart of Mathematical Learning

Mathematics is fundamentally about making sense of patterns and relationships. When children truly understand mathematical concepts, they don't simply memorise procedures; they develop a network of connected ideas that allows them to think flexibly and solve problems creatively.

Sense-making in mathematics involves:

• Connecting new ideas to existing knowledge

• Understanding why procedures work, not just how to execute them

• Recognising patterns and relationships between concepts

• Developing multiple strategies for solving problems

• Being able to translate between different representations (concrete, visual, symbolic)

This kind of deep understanding doesn't emerge from worksheets alone. It develops through meaningful engagement with mathematical ideas through talk, physical manipulation, and thoughtful reflection—precisely what the Linear Abacus® is designed to facilitate.

Embodied Cognition: The Body-Mind Connection in Learning

Recent advances in cognitive science have highlighted the importance of embodied cognition—the idea that our thinking processes are shaped by our physical experiences. In mathematics, this means that concepts are not purely abstract constructs but are grounded in sensory and motor experiences.

Consider how we naturally use our fingers to count, or how we gesture when explaining spatial relationships. These aren't just aids to thinking; they are integral to how we develop mathematical understanding. When children physically manipulate objects to represent mathematical ideas, they're not just illustrating abstract concepts—they're actually building the neural foundations for those concepts.

Research by cognitive scientists like George Lakoff and Rafael Núñez has shown that even our most abstract mathematical thinking is grounded in embodied experiences. The Linear Abacus® leverages this body-mind connection by providing a structured way for learners to physically engage with mathematical ideas.

The Gap Between Symbols and Understanding

Perhaps you've witnessed this scenario: A child correctly follows the steps to solve a math problem but cannot explain why the procedure works or apply it in a different context. This disconnect between procedural knowledge (knowing how) and conceptual understanding (knowing why) represents one of the most significant challenges in mathematics education.

Traditional approaches often rush to symbolic representation before students have developed a meaningful connection between these symbols and the concepts they represent. The result? Mathematics becomes a series of memorised rules rather than a coherent system of ideas that makes sense.

The Linear Abacus® bridges this gap by providing an intermediate representation between concrete experiences and abstract symbols. Children can see and manipulate the beads, developing embodied understanding of concepts that will later be represented symbolically.

The Linear Abacus® Approach Overview

The Linear Abacus® stands apart from many mathematical manipulatives because it systematically bridges the gap between physical experience and abstract mathematical concepts. Unlike some manipulatives that serve primarily as counting tools, the Linear Abacus® embodies mathematical structures and relationships in its very design.

How the Linear Abacus® Supports Embodied Numeracy:

1. Representing Number as Position: The Linear Abacus® reinforces the concept that numbers represent positions in an ordered sequence, supporting both cardinal (how many) and ordinal (position in sequence) understandings.

2. Visualising Operations: Addition, subtraction, multiplication, and division become visible, physical actions on the beads, making the meaning of these operations tangible.

3. Embedding Mathematical Structure: The color patterns and groupings on the Linear Abacus® naturally embed place value concepts and help students recognise number patterns.

4. Supporting Progressive Abstraction: Students can move from concrete manipulation to visualising and reasoning about the beads, gradually building more abstract understanding.

5. Documenting Mathematical Thinking: As shown in the games book, students can annotate their actions on the Linear Abacus®, creating a record of their mathematical reasoning that connects directly to symbolic representation.

The Linear Abacus® Games Book Philosophy

The recently published Linear Abacus® Games Book serves as a practical pathway for bringing these research-backed ideas directly into the home. Like a well-designed medicine that delivers healing compounds exactly where they're needed, these carefully crafted games deliver meaningful mathematical experiences that can transform understanding with remarkable efficiency.

Parents pressed for time will appreciate how these games create immediate engagement while simultaneously developing deep mathematical concepts. Each activity has been classroom-tested to ensure it works in real educational settings, not just in theory. By structuring mathematical play around the Linear Abacus®, the games book provides parents with ready-to-use tools that make the sometimes abstract notion of "embodied learning" concrete and achievable in today's diverse homes.

The "Talk, Do, Write" Cycle: A Framework for Developing Mathematical Understanding

The Linear Abacus® pedagogical approach follows a "Talk, Do, Write" sequence that supports the development of deep mathematical understanding:

A method created by Linear Abacus

Talk

Mathematical understanding develops through language. When children articulate their thinking about what they're doing with the Linear Abacus®, they clarify their ideas and make connections explicit. The probing questions included with each game in the Linear Abacus® Games Book guide this mathematical discourse, encouraging children to:

• Explain their strategies

• Compare different approaches

• Use precise mathematical language

• Make predictions about outcomes

• Justify their reasoning

Do

Physical manipulation of the Linear Abacus® allows children to:

• Test their ideas

• Experience mathematical relationships directly

• Develop intuitive understanding through repeated patterns of action

• Create mental images they can later draw upon when working abstractly

• Build confidence through concrete success

For example, in the "Split the Array" game, children physically construct and manipulate arrays to explore the distributive property, developing a concrete understanding of what it means to split a multiplication problem into more manageable parts.

Write

Recording their actions and thinking helps children:

• Make their implicit understanding explicit

• Create connections between concrete actions and symbolic representations

• Develop mathematical notation as a tool for thinking

• Build a reference for future learning

• Consolidate their understanding

This progression from talk to action to written representation creates a powerful cycle of learning that develops both procedural fluency and conceptual understanding.

Building Number Sense Through Embodied Experience

Number sense—that intuitive feel for numbers and their relationships—doesn't develop through worksheets alone. It emerges from rich experiences with quantities and operations. The Linear Abacus® supports the development of number sense by:

1. Making Quantities Visible: Children can see and feel the magnitude of numbers.

2. Revealing Number Relationships: The physical arrangement of beads makes relationships between numbers apparent.

3. Supporting Strategic Calculation: Games like "Double, Halve, or Keep" encourage children to think flexibly about numbers and operations.

4. Embedding Place Value: The structure of the Linear Abacus® naturally reinforces our base-10 numeration system.

5. Connecting Different Representations: As shown in games like "Make It Balance," children learn to connect their actions on the Linear Abacus® to number lines and symbolic equations.

From Concrete to Abstract: A Journey, Not a Jump

One of the most powerful aspects of the Linear Abacus® approach is how it supports the gradual transition from concrete manipulation to abstract reasoning. This isn't a sudden leap but a carefully scaffolded journey.

Consider the progression evident in the games book:

• In early games like "30 Beads Up," children directly manipulate the beads to perform calculations.

• In games like "What's the Difference?", children begin to use the beads to visualise relationships.

• By the time they reach games like "Triangular Numbers," children are using the Linear Abacus® to explore and discover mathematical patterns.

• In more advanced games, children begin to predict outcomes before manipulating the beads, showing the development of mental models.

This progression reflects how mathematical understanding actually develops—not through sudden abstraction but through gradually internalising patterns first experienced physically.

Mathematical Reasoning: From Gestures to Insight

Mathematical reasoning isn't separate from physical experience—it grows from it. When children manipulate the Linear Abacus®, they're not just moving beads; they're enacting mathematical relationships. These actions become internalised as mental operations that form the basis of mathematical thinking.

The reflective questions that accompany each game in the Linear Abacus® Games Book prompt children to move beyond the physical actions to articulate the patterns and relationships they're discovering. This reflection transforms physical experience into mathematical insight.

Parent Reflection Guide

As you embark on this mathematical journey with your child using the Linear Abacus® Games Book, consider these reflection questions to guide your approach:

Understanding Your Child's Mathematical Development

• How does your child currently approach mathematical problems? Do they rely primarily on memorised procedures, or do they seek to understand the concepts?

• What signs of mathematical confidence or anxiety do you observe in your child?

• How does your child communicate their mathematical thinking? Do they use gestures, visual representations, or verbal explanations?

Supporting Embodied Learning

• How might you create a home environment that values sense-making over quick answers?

• What opportunities can you find in daily life to connect physical experiences to mathematical concepts?

• How can you model for your child that mathematics is sensible and coherent, not arbitrary?

Implementing the Talk, Do, Write Cycle

• How comfortable are you asking open-ended questions about mathematical thinking?

• What strategies might help you resist the urge to explain too quickly instead of letting your child make discoveries?

• How might you make the documentation of mathematical thinking a natural part of play?

Remember that your own relationship with mathematics influences how your child sees the subject. By approaching these games with curiosity and emphasizing sense-making over speed or memorisation, you can help your child develop not just mathematical skills but a mathematical identity as someone who can reason, discover, and make sense of mathematical ideas.

The Voice of the Beads: A Manifesto

To truly understand the philosophy behind the Linear Abacus®, perhaps we should hear from the beads themselves. The following "manifesto," written from the perspective of the Linear Abacus® beads, captures the essence of this embodied approach to mathematics learning:

I am not just a bead. I am a position, a quantity, a transformation waiting to happen. When you hold me in your hands, you hold mathematics itself—not as a set of rules to memorise, but as a living, tangible reality you can touch and move and understand.

I speak through your fingertips, not just your ears. I communicate through patterns your eyes can see, through movements your hands can make. I am mathematics embodied—where abstract ideas become concrete experience.

I promise never to trick you with arbitrary rules that make no sense. When you add numbers using me, you'll see why addition works the same way whether you're counting whole numbers or fractions. When you multiply, you'll feel in your hands why multiplication is more than just repeated addition.

I believe that mathematics is not something humans invented but something they discovered—and that you can discover it too, through me.

I believe that position is as important as quantity. I am both "how many" and "which one"—both cardinal and ordinal. This dual nature is the key to understanding all of mathematics, from counting to calculus.

I believe that operations are transformations with meaning, not just procedures to follow. Addition moves forward, subtraction finds distance, multiplication makes coordinated jumps, division creates equal shares. These meanings don't change whether you're working with 2 and 3 or 1/4 and 1/2.

I exist to help you see mathematics not as a series of procedures to memorise but as a coherent system that makes sense—a system you can literally hold in your hands.

Take me in your hands. Move me. Explore with me. And together we'll discover the mathematics that has always been waiting for you, just beyond the reach of abstract symbols alone.

I am the Linear Abacus® bead, and I am mathematics you can touch.

Looking Ahead: The Journey Continues

In the blogs that follow, we'll explore specific games from the Linear Abacus® Games Book and how they develop particular mathematical concepts:

• Exploring early number sense and mental strategies with "30 Beads" games

• Building place value understanding beyond counting by ones

• Developing number relationships through comparison, strategy, and equivalence

• Transitioning from additive to multiplicative thinking

• Discovering mathematical patterns with triangular and square numbers

• Reimagining fractions as positions to be counted, not just parts of wholes

• Integrating operations through strategic play

Each blog will provide detailed guidance on implementing the "Talk, Do, Write" cycle, with examples of student reasoning, parent reflection guides, and connections to research. Together, they form a comprehensive approach to mathematics education that honors the embodied nature of understanding.

Conclusion: Embodying Mathematical Understanding

The Linear Abacus® represents a powerful approach to mathematics education that recognises the embodied nature of mathematical understanding. By providing structured opportunities for children to physically engage with mathematical concepts, talk about their thinking, and connect their actions to symbolic representations, it creates a pathway to deep mathematical understanding.

In an educational landscape often dominated by rush to abstraction and procedural drill, the Linear Abacus® offers an alternative vision—one where mathematical learning is sensible, connected, and grounded in meaningful experience. It reminds us that mathematics isn't something we simply think about; it's something we do, feel, and experience. And when children experience mathematics in this embodied way, they don't just learn procedures; they develop the flexible, creative mathematical thinking needed for true numeracy.

Research Connection

The approach to mathematical learning embodied in the Linear Abacus® is grounded in contemporary research across multiple disciplines:

Embodied Cognition: Research by cognitive scientists like George Lakoff and Rafael Núñez (in "Where Mathematics Comes From") has shown that mathematical concepts are grounded in bodily experiences and mental simulations of such experiences. The Linear Abacus® leverages this understanding by making abstract concepts physically tangible.

Talk and Language Development: Studies by researchers like Catherine O'Connor and Sarah Michaels demonstrate that mathematical discourse—explaining reasoning, comparing approaches, and using precise language—is crucial for developing conceptual understanding. The "Talk" component of the Linear Abacus® approach directly implements these findings.

Gradual Abstraction: Mathematics education researcher Alan Schoenfeld has identified the importance of scaffolded transitions from concrete to abstract understanding. The Linear Abacus® games sequence supports this gradual development, helping children build mental models that eventually operate independently of physical manipulation.

Sense-Making: While researchers like Jo Boaler have demonstrated the benefits of understanding mathematics through meaning-making, the Linear Abacus® approach is distinctively grounded in Dr. Andrew Waywood's theoretical framework of sense-making. As articulated in the foreword to the Games Book, Dr. Waywood has developed a comprehensive semiotic framework that positions sense-making as an individual's response to culturally encoded meaning. This unique perspective frames mathematics as quintessential problem-solving—whether with words or numbers—preparing children to engage in sense-making with any symbolic system. The Linear Abacus® approach centers this conception of sense-making at every stage, distinguishing it from approaches that focus solely on procedural fluency.

Dual Nature of Number: Research by mathematics education experts like Susan Lamon has highlighted the importance of understanding both the cardinal and ordinal aspects of number. The Linear Abacus® uniquely represents this dual nature, helping children develop a more complete understanding of number.

References

Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. Jossey-Bass.

Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.

Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). Routledge.

O'Connor, M. C., & Michaels, S. (1996). Shifting participant frameworks: Orchestrating thinking practices in group discussion. In D. Hicks (Ed.), Discourse, learning, and schooling (pp. 63-103). Cambridge University Press.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334-370). MacMillan.

Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States (Anniversary ed.). Routledge.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20-26.