Students' Use and Coordination of Symbolic and Number-Line Representations in Fraction Comparison: A Qualitative Multiple-Case Study

Miftahul Jannah; Rustanto Rahardi; Dahliatul Hasanah

doi:10.56741/IISTR.ijlree.002408

Authors

Miftahul Jannah Universitas Negeri Malang
Rustanto Rahardi Universitas Negeri Malang https://orcid.org/0000-0001-8974-840X
Dahliatul Hasanah
dahliatul.hasanah.fmipa@um.ac.id
Universitas Negeri Malang https://orcid.org/0000-0002-6606-3341

DOI:

https://doi.org/10.56741/IISTR.ijlree.002408

Keywords:

Conceptual Coordination, Final Warrant for Correctness, Fraction Comparison, Mathematical Representations, Number Line

Abstract

Using more than one representation in fraction comparison does not, by itself, indicate that students have coordinated the mathematical relationships across representations. This study aimed to analyze the use and functions of symbolic and number-line representations, shifts and coordination across representations, and the final basis on which students accepted their answers as correct when comparing fractions. A qualitative multiple-case study design was employed. Twenty-four seventh-grade students completed an open-ended task comparing and . Based on variation in written representational patterns, four focal cases (S7, S11, S17, and S24) were purposively selected and participated in two rounds of task-based interviews. Data were analyzed using a hybrid deductive–inductive thematic analysis integrating written work, interview data, and cross-case comparison. The findings showed that the number line served different functions across cases, including initial solution search, visualization, explanation, and verification, whereas symbolic procedures were used to obtain, clarify, and justify results. Conversion, cross-representational consistency, and conceptual coordination also showed distinct patterns: evidence of coordination was partial for S7, could not be established for S11, was stronger within the task context for S17, and could not be assessed across representations for S24. For S7, S11, and S17, symbolic procedures carried greater justificatory weight than the number line, whereas for S24, the symbolic procedure was the only observable procedural basis. These findings highlight the importance of distinguishing representational form from function, conversion and cross-representational consistency from conceptual coordination, and representational function from justificatory function when analyzing students’ reasoning in fraction-comparison tasks.

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Author Biographies

Miftahul Jannah, Universitas Negeri Malang

is a master’s student in Mathematics Education at Universitas Negeri Malang, Indonesia. Her academic interests include mathematical representation, mathematical cognition, problem solving, and mathematics education. Her current work focuses on how students use and coordinate multiple representations when reasoning about mathematical concepts, particularly fractions. (email: miftahul.jannah.2503118@students.um.ac.id).

Rustanto Rahardi, Universitas Negeri Malang

is a lecturer in the Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Malang, Indonesia. His academic activities involve teaching and research in mathematics and mathematics education. He is also engaged in scholarly publication and academic supervision within the department. (email: rustanto.rahardi.fmipa@um.ac.id).

Dahliatul Hasanah, Universitas Negeri Malang

is a lecturer in the Department of Mathematics, Universitas Negeri Malang, Indonesia. Her areas of expertise include analysis, applied mathematics, and mathematics education. She is actively involved in research, academic publication, and international scholarly activities, and has served as Chair of The International Conference on Mathematics and Its Applications. (email: dahliatul.hasanah.fmipa@um.ac.id).

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