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On the epistemological model of fractions arithmetic: Learning the praxeologies from classroom works

This study, utilizing the Anthropological Theory of the Didactic to analyze sixth-grade classroom data, argues that understanding fractions as quantities of units and recognizing arithmetic operations as manifestations of a common organization are essential for developing a coherent and unified understanding of fraction arithmetic.

Original authors: Sani Sahara, Tatsuya Mizoguchi, Nadya Syifa Utami, Yurniwati Yurniwati, Novi Andri Nurcahyono, Mira Amelia Amri, Wily Wandari

Published 2026-09-15

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Original authors: Sani Sahara, Tatsuya Mizoguchi, Nadya Syifa Utami, Yurniwati Yurniwati, Novi Andri Nurcahyono, Mira Amelia Amri, Wily Wandari

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Mathematics in school often feels like a collection of disconnected rules: here is how you add, there is how you multiply, and over here is a special trick for dividing. For many students, fractions represent the point where this collection of rules breaks down. They are taught to see a fraction as a slice of a pie or a part of a whole, but when asked to perform calculations, they must suddenly switch to a different set of instructions that seem to have no logical connection to the picture they were just looking at. This disconnect creates a persistent barrier in learning, leaving students to memorize procedures without understanding the underlying logic. The question researchers ask is not just why students struggle, but whether the way schools teach fractions is actually the source of the problem. If the foundation is built on isolated symbols rather than a single, coherent idea, the entire structure becomes unstable.

A team of researchers in Indonesia and Japan set out to test a different approach in a sixth-grade classroom. Instead of treating fractions as a new, confusing topic to be memorized, they introduced the students to the concept of a "unit fraction" as the single building block for all fraction arithmetic. In this model, a fraction is not a pair of numbers separated by a line, but a specific quantity of identical pieces. The researchers, working with twenty-one students over seven sessions, guided the class through a process where they physically cut ribbons, arranged them on a number line, and counted these pieces to discover the rules of addition, subtraction, multiplication, and division for themselves. The goal was to see if students could reconstruct the entire system of fraction math from this single, unified idea, rather than learning a series of separate tricks.

The study began by asking students to cut a strip of paper into equal parts. At first, the students treated the numbers as symbols, writing down "one-fourth" as a static label. However, as they physically manipulated the strips, their understanding shifted. They began to see the fraction not as a label, but as a count of pieces. A piece cut from a strip divided into four became a "fourth," and two of those pieces became "two fourths." This simple change in perspective allowed the students to treat fractions like whole numbers. When they added two fourths and one fourth, they simply counted the pieces to get three fourths. The rule was no longer a mysterious instruction to "add the top numbers and keep the bottom numbers"; it was a natural consequence of counting identical units.

The real test came when the students faced fractions with different denominators, such as adding one-half to one-fourth. In a traditional classroom, this often triggers a rote application of a formula. Here, the students encountered a genuine problem: they could not add a half and a fourth because the pieces were different sizes. One student compared it to trying to add a pencil and a marker, noting that you cannot combine them until they are the same kind of object. This realization drove the class to invent their own solution. They discovered that to add the fractions, they had to break the larger piece down until it matched the smaller one. By cutting the half into two fourths, they created a common unit that allowed the addition to proceed. The students had effectively rediscovered the concept of a common denominator, not as a rule to memorize, but as a necessary step to make the pieces compatible.

This logic extended to multiplication and division as well. When asked to multiply one-half by one-third, the students did not multiply the top numbers and bottom numbers blindly. Instead, they visualized taking one-third of a half, which resulted in a new, smaller piece. They saw that multiplication was a process of scaling, creating a new unit size. Similarly, division was understood as a measurement problem: how many of these smaller pieces fit inside the larger one? By counting how many one-sixths fit into a half, the students arrived at the answer of three. Throughout these sessions, the students moved from manipulating physical ribbons to using a number line, and finally to working with abstract symbols. The physical objects served as a bridge, allowing the students to validate their ideas before the teacher formalized the rules.

The researchers found that by anchoring the entire system in the idea of a unit fraction, the students were able to treat addition, subtraction, multiplication, and division as different actions performed on the same type of object. The confusion that usually arises from seeing fractions as separate rules vanished because every operation was just a way of handling these units. The study suggests that the difficulties students face with fractions often stem from the way the subject is introduced in schools, where fractions are presented as disconnected procedures rather than a unified system of quantities. The findings indicate that when students are given the opportunity to build the system from the ground up using a consistent unit, they can develop a coherent understanding that connects whole numbers and fractions seamlessly.

The study concludes that the fraction unit is a crucial mathematical object that is often missing from standard instruction. Without this central concept, the rules of fraction arithmetic appear arbitrary and ad hoc. By focusing on the unit, the researchers showed that the four basic operations can be understood as different manifestations of a single arithmetic organization. The students in this classroom did not just learn how to calculate; they learned why the calculations work. Their journey from cutting paper to solving complex problems demonstrates that when the underlying logic is made visible, the barrier to understanding fractions can be removed, allowing students to see the mathematics as a logical, connected whole rather than a list of confusing exceptions.

Technical Summary: On the Epistemological Model of Fractions Arithmetic

Problem Statement
The paper addresses the persistent difficulties students face in learning fraction arithmetic, attributing these challenges not merely to cognitive limitations but to the "incoherence" of how fractions are taught in school mathematics. Drawing on the work of Wu (2002, 2011) and others, the authors argue that school curricula often present fractions as a collection of disconnected definitions (part-whole, quotient, ratio, operator) and isolated procedural rules (e.g., "invert and multiply") rather than as a logical extension of whole-number arithmetic. This fragmentation leads to the "whole-number bias," where students treat numerators and denominators as independent integers rather than as a single numerical quantity. Furthermore, the transition from natural numbers ( $\mathbb{N}$ ) to rational numbers ( $\mathbb{Q}$ ) is often treated as a new domain rather than a continuous numerical system, resulting in ad hoc procedures lacking mathematical justification. The study seeks to investigate how an epistemological model based on unit fractions as the primary mathematical object can reorganize fraction arithmetic into a coherent system within an Indonesian classroom context.

Methodology
The study employs a qualitative inquiry grounded in the Anthropological Theory of the Didactic (ATD) and the Theory of Didactical Situations (TDS).

Participants and Setting: The research involved 21 sixth-grade students in a regular Indonesian classroom over a seven-session instructional sequence spanning two months. The researcher also served as the classroom teacher, acting as an interventionist to manage the evolution of didactic situations.
Theoretical Framework: The study utilizes the Reference Epistemological Model (REM) proposed by Author et al. (2025), which posits unit fractions ( $\hat{u}$ ) as the dominant praxeology. In this model, fractions are defined as quantities of unit units, and arithmetic operations are viewed as actions on these units.
Data Collection: Data included video recordings of all sessions, field notes, transcripts of discussions, and students' written productions.
Analytical Approach: The analysis focused on the evolution of praxeologies (the combination of tasks $T$ , techniques $\tau$ , technology $\theta$ , and theory $\Theta$ ). The study tracked the progression through TDS phases: action, formulation, validation, and institutionalization. The goal was to observe how students reconstructed their understanding of fractions through interaction with a mathematical milieu (ribbons, number lines) and how the teacher facilitated the transition from situated knowledge to institutional knowledge.

Key Contributions and Results
The study demonstrates that when fraction arithmetic is reorganized around the concept of the unit fraction, students can reconstruct arithmetic operations as coherent actions rather than isolated algorithms.

Reconstruction of the Unit: In Sessions 1 and 2, students moved from viewing fractions as static symbolic expressions ( $\frac{k}{n}$ ) or parts of a whole to understanding them as quantities of unit fractions ( $k\hat{u}$ ). Through partitioning ribbons and mapping them to a number line, students validated that the denominator defines the unit ( $\hat{u} = \frac{1}{n}$ ) and the numerator counts the quantity of those units. This shift allowed for the natural emergence of equivalent fractions and improper fractions as extensions of counting units beyond one whole.
Unification of Arithmetic Operations: In Sessions 3 through 6, the study showed that the four arithmetic operations could be understood as different manifestations of a single organizational structure based on units:
- Addition/Subtraction: Performed only on identical units. When units differed (e.g., $\frac{1}{2} + \frac{1}{4}$ ), students were motivated to find a common unit (reduction) to make the operation possible, analogous to adding pencils to pencils.
- Multiplication: Reconceptualized as scaling. Multiplying fractions (e.g., $\frac{1}{3} \times \frac{1}{2}$ ) was understood as generating a new, smaller unit ( $\frac{1}{6}$ ) through the interaction of two units.
- Division: Reconceptualized as measurement. Division (e.g., $\frac{1}{2} \div \frac{1}{6}$ ) was framed as determining how many units of the divisor fit into the dividend.
Institutionalization: In Session 7, the study documented the process of decontextualization, depersonalization, and detemporalization. The students' situated knowledge, derived from physical manipulation and classroom discourse, was transformed into abstract, symbolic mathematical rules (e.g., $\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$ ). The paper argues that this transition is essential for knowledge to become part of the mathematical institution.

Significance and Claims
The paper claims that the fraction unit is a critical mathematical object often absent in conventional instruction. Its absence contributes to the ad hoc nature of fraction procedures and the disconnect between whole-number and fraction arithmetic.

Epistemological Coherence: By establishing the unit fraction as the primary object, the study provides a unified framework where addition, subtraction, multiplication, and division are not independent rules but logical consequences of operating on quantities of units.
Reframing the Whole-Number Bias: The authors suggest that the "whole-number bias" may not solely stem from students' prior knowledge but from the institutional presentation of fractions as two separate integers. By framing fractions as counts of units, whole-number knowledge (counting, iterating) becomes a productive resource rather than an interference.
Didactic Implications: The study validates the utility of combining TDS and ATD to analyze conceptual development. It suggests that a coherent technological organization (centered on units, equivalence, and reduction) is central to developing deep fraction understanding, offering a counter-narrative to curricula that emphasize isolated procedural fluency.

The authors conclude modestly that while their findings support an integrated, unified approach to rational numbers, the study was limited to a specific institutional context (Indonesian Grade 6) and did not explicitly examine higher levels of co-determinacy such as national curriculum or textbook constraints, which remain avenues for future research.

Technical Summary: On the Epistemological Model of Fractions Arithmetic

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