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

Abstract

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.