Preservice Mathematics Teachers’ Understanding of Quadrilateral Properties and Hierarchical Relationships

Understanding Students’ Learning Obstacles in Geometric Transformations and Their Implications for Didactical Design

Geometric transformations are one of the essential topics in mathematical learning, as they support students’ spatial reasoning, spatial visualization, and understanding of geometric relationships. Despite its importance, many students still have difficulties in interpreting transformation concepts beyond routine procedural exercises, particularly problems that are presented in unfamiliar or contextual situations. Therefore, this study aimed to explore students' learning obstacles in learning geometric transformations and examine their implications for the development of didactical design. Qualitative approach with a didactical design research framework was employed using a phenomenological perspective. Participants involved were 36 ninth-grade students and one mathematics teacher from a junior high school in West Bandung Regency, Indonesia. Research data were collected through a written test on mathematical spatial ability consisting of four contextual problems, a semi-structured interview, and a praxeological analysis of a textbook. The written-test instrument was validated by two mathematics education experts and one experienced teacher before implementation. The data were analyzed using the Miles and Huberman interactive model, involving data reduction, display, and conclusion drawing, to categorize students’ difficulties into epistemological, ontogenic, and didactical obstacles. The research revealed that students experienced interconnected categories of learning obstacles. Epistemological obstacles appeared in students’ inability to concept generalization, misconceptions of the transformations’ properties, and dependence of procedural examples. Ontogenic obstacles related to limited spatial relations ability, weak understanding of Cartesian coordinates, and difficult to interpret contextual problems. Lastly, didactical obstacles emerged from teacher-centered instructions, limited use of interactive visual media or technology, and emphasis on procedural completion compared to conceptual exploration. These obstacles interacted with one another and influenced students' understanding of geometric transformations.  The study implies that effective didactical design should integrate conceptual understanding, spatial visualization, multiple representations, contextual learning, and dynamic technology to facilitate meaningful learning of geometric transformations. Keywords: learning obstacles, geometric transformation, epistemology, ontogenic, didactic.

Reconceptualising The Geometry Gap: Theoretical Insights From Van Hiele And Constructivist Perspectives On Triangle Theorem Instruction In South Africa

Geometry forms a fundamental component of secondary school mathematics because it supports the development of spatial reasoning, logical thinking, and deductive proof skills. However, persistent learner difficulties in understanding geometric concepts, particularly triangle theorems, remain a significant concern in many education systems. This conceptual paper examines the factors contributing to the geometry gap in South African schools, with a particular focus on the teaching and learning of triangles and their associated theorems. The paper adopts a conceptual research approach that synthesises recent literature and theoretical perspectives in mathematics education, including the van Hiele theory of geometric thinking, constructivist learning theory and socio-cultural perspectives on learning. Through an integrative analysis of recent scholarship, the paper identifies structural, pedagogical and cognitive factors that collectively influence learners’ geometry learning outcomes. The study further proposes a conceptual framework explaining how these dimensions interact to produce persistent gaps in learners’ geometric reasoning and problem-solving abilities. The interconnected nature of these challenges offers theoretically grounded insights that may inform curriculum development, teacher professional development and instructional innovation aimed at improving geometry education. The findings contribute to ongoing debates on mathematics education equity and provide a foundation for future empirical research on strategies for strengthening geometry instruction in South African schools.

Foster Student Understanding of Parallelism Based on the Toulmin Model and Pirie-Kieren Theory

Students’ understanding of the definition and classification of quadrilaterals serves as a crucial foundation in geometry learning because it allows students to make connections between geometric figures based on their essential properties. Previous studies have primarily examined the structure of arguments or levels of geometric thinking, leaving the dynamics of the development of understanding during the argumentation process inadequately explored. This study aims to how collective argumentation fosters the development of junior high school students’ understanding of the definition and classification of quadrilaterals through collective argumentation. The research framework integrates Toulmin’s argumentation model to analyze the structure of arguments with Pirie and Kieren’s theory of the development of understanding to identify how it fosters trajectories of understanding. This study employs a qualitative approach involving a mathematics teacher and eighth-grade students selected through purposive sampling. Data were collected through classroom recordings, student work samples, observations, field notes, and interviews. The results indicate that the development of understanding progresses through the layers of primitive knowing, image making, image having, property noticing, formalising, and observing, accompanied by the process of folding back, while the characteristics of structuring were not fully and explicitly identified in the data. The integration of these two theories shows that the Toulmin model reveals the construction of arguments, while the Pirie-Kieren theory explains the development of understanding that underlies those arguments. These findings emphasize that collective argumentation fosters a mechanism that drives the reorganization of knowledge and strengthens students’ conceptual understanding of the definition and classification of quadrilaterals.

Exploring the nature of mathematical connections in pre-university students’ triangle similarity tasks

Triangle similarity is a mathematical concept with substantial conceptual richness; however, its teaching typically emphasizes procedural approaches over conceptual understanding, leading to a disconnect between concepts and procedures. In this context, the present study aimed to identify the mathematical connections established by a group of Mexican pre-university students while solving tasks related to triangle similarity. A conceptual framework based on the notion of mathematical connections and their typologies was adopted. The study followed a qualitative methodology with a descriptive scope, using a case study approach. Data were collected through task-based interviews. Three tasks were designed and completed by eight students, and the resulting data were analyzed using thematic analysis. The findings revealed six types of mathematical connections: procedural, feature-based, different representations, meaning, implication, and interconceptual; the part–whole connection was not observed. This absence suggests that students tend to focus on isolated properties and procedures rather than recognizing triangle similarity as part of broader mathematical structures. The results suggest the need to design instructional tasks that foster both conceptual understanding and procedural accuracy.

FOSTERING UNDERSTANDING OF INFINITE GEOMETRIC SERIES THROUGH A VISUAL AREA MODEL: A DIDACTIC DESIGN BASED ON APOS THEORY

Geomaniguras: A Manipulative Resource for Supporting Conceptual Understanding in Geometry Education

Manipulative materials are widely considered valuable resources in geometry education because they may support visualization, spatial reasoning, and conceptual understanding when implemented within appropriate instructional contexts. However, many existing resources focus on isolated concepts and provide limited opportunities for connecting geometric relationships through transformation and decomposition. This study presents Geomaniguras, a manipulative material designed for upper primary and lower secondary education. The research adopts a design and development approach focused on pedagogical design and expert-based validation rather than measuring student learning outcomes. The study combined prototype construction, expert evaluation, and iterative refinement. Twelve specialists in mathematics education assessed the material using a structured validation guide addressing pedagogical usefulness, conceptual coherence, usability, visual design, and classroom applicability. Results indicate that experts perceived Geomaniguras as a potentially valuable resource for exploring polygon classification, area relationships, geometric decomposition, circumference, and the Pythagorean theorem. The findings provide preliminary evidence regarding the pedagogical plausibility and classroom applicability of the material, although no direct conclusions can yet be drawn about instructional effectiveness. The originality of the proposal lies in its integrated conceptual framework, which connects multiple geometric learning experiences within a single manipulative system.