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All Journal Progressive of Cognitive and Ability International Journal of Applied Learning and Research in Algebra
Kashaf Arzoo
Government College University, Faisalabad
Education Mathematics Other

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Students' Mathematical Understanding According to Kilpatrick and Findell: The Number Pattern Generalisation Process of Junior Secondary School Students Syafruddin Kaliky; Sufyani Prabawanto; Muhammad Hafeez; Kashaf Arzoo
International Journal of Applied Learning and Research in Algebra Vol. 3 No. 2 (2026)
Publisher : EDUPEDIA Publisher

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.56855/algebra.v3i2.2478

Abstract

Purpose – This study investigates how junior secondary school students generalise number patterns through Kilpatrick and Findell’s framework of mathematical understanding, focusing on the interaction among conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Methodology – A qualitative descriptive design was conducted with 30 Year 8 students at an Islamic junior secondary school in Ambon, Indonesia. Screening tests, informal interviews, and repeated testing were used to identify students who consistently demonstrated indicators of mathematical understanding. Two students, coded WNF and FF, were selected for in-depth analysis. NVivo 12 Plus supported the mapping and visualisation of their reasoning processes, while data were analysed using the Miles and Huberman model of data collection, reduction, display, and conclusion drawing. Findings – Both students progressed from recursive reasoning based on differences between consecutive terms to correspondence-based symbolic generalisation. They expressed the nth term using arithmetic and quadratic forms. This transition reflected interconnected development in conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning. Novelty – The study reveals how mathematical understanding operates within the transition from recursive to explicit generalisation, rather than treating pattern generalisation solely as a procedural outcome. Significance – The findings contribute to international research on early algebra by identifying a transferable reasoning trajectory from recursive pattern recognition to symbolic generalisation. This trajectory provides a basis for designing mathematics instruction that supports algebraic reasoning across diverse junior secondary educational contexts.  
Mathematical Misconceptions in Contextual Geometry Problems: A Multiple-Case Study of Reflective and Impulsive Students Yusmarni Yusmarni; Muhammad Hafeez; Kashaf Arzoo
Progressive of Cognitive and Ability Vol. 5 No. 4 (2026)
Publisher : EDUPEDIA Publisher

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.56855/jpr.v5i4.2508

Abstract

Purpose – This study investigates mathematical misconceptions among students with reflective and impulsive cognitive styles when solving contextual geometry problems, focusing on how students interpret geometric structures, select mathematical formulas, and execute problem-solving procedures. Methodology – A qualitative descriptive study with a multiple-case analytical approach was conducted among Grade 9 students at a junior secondary school in Jambi, Indonesia. Of 23 students classified using the Matching Familiar Figures Test (MFFT), six were purposively selected, comprising three reflective and three impulsive students. Data were collected through a mathematics diagnostic test, semi-structured interviews, and supporting documentation. Within-case and cross-case analyses were conducted using data condensation, evidence displays, and interpretive verification. Findings – All six participants demonstrated a shared conceptual misconception by applying the total surface area formula for a complete cone to an open-bottom birthday hat. Reflective students performed calculations consistent with their selected formulas, whereas two impulsive students exhibited additional arithmetic or measurement-unit errors. The findings revealed a common difficulty in translating contextual geometric structures into appropriate mathematical models, regardless of cognitive-style classification. Novelty – This study distinguishes shared conceptual misconceptions from individual variations in mathematical representation and execution through an integrated analysis of cognitive styles, written solutions, and students' explanations. Significance – The findings inform mathematics teachers, curriculum developers, and researchers in designing diagnostic assessments and instructional strategies that strengthen geometric interpretation, conceptual understanding, and mathematical modeling.
Co-Authors Muhammad Hafeez Sufyani Prabawanto, Sufyani Syafruddin Kaliky Yusmarni Yusmarni