THE JOURNEY
OUR BEGINNINGS
How the Linear Abacus® started
A VISION TURNED INTO REALITY
Transforming Mathematics Education
At the heart of our journey is a simple yet powerful conversation—one that began during a lunch break between two passionate educators who shared a deep concern about mathematics education.
The Origin of Change
Dr. Andrew Waywood and Genovieve Grouios MEd, both lecturers with extensive experience in education, noticed a troubling pattern in mathematics education that demanded attention and action. Dr. Waywood, approaching the issue as a pedagogical theoretician, recognised that mathematical thinking isn't merely an academic subject but the essential currency of the modern world. His concern centered on how poorly equipped many students were to navigate an increasingly symbol-mediated reality where making sense of abstract systems is a fundamental requirement. For Andrew, the stakes extend far beyond classroom performance—this is about preparing children to participate fully in a STEM-driven society.
Genovieve Grouios, as an educational practitioner and reformer, observed this same problem through a different lens. In classrooms from Prep to Year 8 and among pre-service teachers, she witnessed the anxiety and disconnection many experienced with mathematics. Her focus became addressing the emotional and practical barriers preventing students from engaging meaningfully with mathematical concepts and developing confidence in their abilities.
A Collaborative Vision
When these complementary perspectives converged, they created a powerful partnership of theory and practice. Waywood's semiotic framework provided the theoretical foundation to understand how mathematical meaning is constructed, while Genovieve brought practical expertise in transforming classroom experiences. Together, they recognised that neither theoretical understanding nor practical reform alone would be sufficient—the solution required both.
Their collaboration represents a unique integration of visionary theory with compassionate practice—a partnership dedicated to transforming how mathematics is taught and learned, preparing students not just for tests but for thoughtful participation in a world increasingly organised through mathematical structures. Both believe that the greatest gift to give a learner is understanding.
The Linear Abacus®: Our Innovative Solution
Central to their approach is the concept of sense-making. Mathematics isn't about memorisation, but about understanding. The Linear Abacus® emerged from this philosophy—a practical, intuitive tool designed to transform mathematical learning from a source of anxiety to a journey of discovery.
More Than Just a Resource
What started as a conversation between two friends has grown into a proud Australian initiative committed to changing how we think about and teach mathematics. We're not just developing educational resources; we're building confidence, sparking curiosity, and opening doors to mathematical understanding for children and adults alike.
Our story continues, and we invite you to be part of this transformative educational journey.
OUR MISSION
To empower every child with the mathematical thinking needed for tomorrow's world by making Linear Abacus® practices central to education - where physical actions reveal how numbers model the structure of thought itself, creating learners who understand mathematics as a powerful way to make sense of their world.
The Linear Abacus® is a classroom tool used to help students learn and make meaning of mathematical concepts. It was designed to provide a consistent multiyear resource for teaching algebraic thinking as the basis for STEM subjects. The development and inspiration of the Linear Abacus® was motivated by the need to find simple but rich manipulatives that can be used easily in the developing world to teach number, measure, and arithmetic from prep through to grade 8.
For children, learning to manipulate materials to make meaning of mathematical concepts and to generate mathematical activity is a crucial step in their development. Meaning can be achieved as they learn to use and talk about the materials to build intuitive understanding of key mathematical ideas. The Linear Abacus® allows children to learn through their actions and gestures, through their observations, by constructing models, by translating mathematical ideas, or by demonstrating how numbers interact with their world.
For teachers, this tool provides a wholistic approach to the teaching of numeration and arithmetic, where language and number interact in framing concepts. The Linear Abacus® embodies all the arithmetical relationships covered in the primary and early secondary stages of schooling and can be used to teach arithmetic for understanding. This means that the teaching of arithmetic with the Linear Abacus® goes beyond rote learning procedures. Teachers can help students generate meaning by coordinating various communications in the classroom and three important types of activities:
1. modelling with materials,
2. solving word problems, and
3. performing calculations.
ABOUT THE LINEAR ABACUS®
Key Characteristics of the Linear Abacus®
The Linear Abacus® is a foundational model of number for young children. It is made up of multi-coloured cubic centimetre beads which are threaded with a double string to keep the beads in place. Excess string is provided to allow children to perform various calculations with ease.
The abacus string reflects the structure of the base 10 place value system. Each consecutive group of ten beads alternate colours, for example, ten yellow beads, ten blue beads and so forth. This pattern repeats 5 times over as there are a total of 100 beads on the string.
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The Linear Abacus® is multifunctional and since cubic centimetre beads are used, the full 100-bead abacus string is equivalent to the measure of 1 metre. This means that the abacus string can also be used in measurement to model length, area (as a process of covering), and volume (as a process of filling).
Class sets can be used for rich collaborative problem solving and modelling to build deep number and measure intuitions. For instance, children can build a square metre, explore the number of strings used to circle an oval, or determine ways to convert between metric units of length, area, and volume.
THE LINEAR ABACUS® CAN BE USED ACROSS DIFFERENT MATHEMATICAL AREAS SUCH AS NUMBER, ALGEBRA, & MEASUREMENT
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Counting using one-to-one correspondence to order an unordered collection. This is the first notion of a number.
Numeration- naming numbers using base 10 place value. The Linear Abacus® helps students expand numbers multiplicatively.
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Number operations and basic facts are explored through concrete and visual representations. Here a student discovered the connection between quotative division and multiplication. The Linear Abacus® helped them construct the number line and array model.
Area is explored through ‘covering’. Here a square metre is being constructed. Students discovered that 100 strings were required to cover the square metre. The Linear Abacus® helped them see the submultiples of the unit and convert between units.
Volume is explored through ‘filling’ boxes. The Linear Abacus® helps students understand the formula to find the volume of prisms i.e., base area x height.
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Growing Patterns made with pattern blocks are connected to the Linear Abacus®. This helps students describe the pattern and recognise how it changes in each step. In this example, they were exploring skip counting with fractions: 2/3, 4/3, 6/3...
THE DIFFERENT READINGS ON THE LINEAR ABACUS®
The abacus string can be read in different ways. Each of these interpretations are based on the idea that each bead is 1cm in length (as the beads are a cubic centimetre).
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An individual bead on the abacus string can be thought of as a count because each bead is an object in an order. The counts are represented as numerals on the bead face and are colour coded red. The figure below shows a count of 5 discrete things. The numeral 5 links to the fifth thing counted.
The basis of counting and additive thinking.
1 2 3 4 5
The basis of counting and additive thinking.
1 2 3 4 5
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A span or a measure (in the sense of a ruler) on the Linear Abacus® starts from the beginning of the abacus string to the end or boundary of one bead to the next. These numerals are colour coded purple. On the abacus string this is shown as an annotated arrow below the string. The figure below shows an informal measure of 5 units.
5 beads long
The basis of measuring and the foundation of multiplicative thinking.
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When two beads touch a point is found on the abacus string. Each of the points between the beads on the abacus string can be thought of as marks on a scale or a number line. These numerals are colour coded black and are marked as dots between the beads. The figure below shows the 5th centimetre mark on a scale.
5cm
The beginning understanding of scale and algebraic thinking with numbers.
WAYS TO USE THE LINEAR ABACUS®
In the classroom the process of expressing and developing an understanding of concepts is facilitated by the discourse between teachers and students, students with other students, and students with themselves. For instance, if a child is given a simple number sentence (SNS) such as 18÷12=1½ and they are able to build a model on the Linear Abacus®, interpret a word problem, or do a calculation, whilst connecting all three interpretations simultaneously, then it is safe to assume that they have understood a concept in arithmetic.
In this problem students are comparing 18 to 12 multiplicatively.
SNS
In the World
Jack has 18 marbles in a bag, and Jill has 12 marbles in a bag. How many times more marbles does Jack have than Jill?
On the Linear Abacus® Model
Start by locating Jack and Jill’s bead on the abacus string as both are things that can be counted.
Then ask yourself, “how many times does Jill’s total go into Jack’s total?” The arrows above the beads show the answer and dashed beads have been included in the diagram to show how many times Jill’s total goes into Jack’s total. This can also be performed with a second abacus string.
This concrete representation shows, one lot of “what Jill has” and another half of “what Jill has”.
Another way to interpret this is to let 12 marbles be one full bag. If 12 represents 1 bag, then 18 is one bag and 6 out of a second bag.
12 18
Jill has Jack has Bead Bank
Using a Calculation
The diagrams below include both the additive and multiplicative interpretations.
An additive interpretation
A multiplicative interpretation
This answers the word problem i.e., Jack has 1½ times as many marbles as Jill. 1½ is a multiplicative relation as it focuses on ‘how many times’ not how many.
For more ideas on how to use the Linear Abacus® when teaching numeration and arithmetic, you can purchase the Concise Instructional Manual for the Linear Abacus® which provides descriptions and detailed examples with annotations.
The manual could be used by a parent to support a child, for home schooling, by a teacher looking to innovate new approaches, or as a systematic review of key concepts.
