Mapping students’ mathematical representation trajectories across five habits of mind performance levels in similarity and congruence problems

Abstract

This study aims to map students’ trajectories of mathematical representation across five Habits of Mind performance levels in solving similarity and congruence problems. Unlike previous descriptive studies, which generally examine the relationship between mathematical representation ability and Habits of Mind, this study emphasizes how students’ pictorial, symbolic, and verbal representations develop, intersect, and differ across performance levels. A descriptive qualitative approach was employed involving 30 ninth-grade junior high school students selected through purposive sampling. The data were collected using a mathematical representation ability test, a Habits of Mind questionnaire, and semi-structured interviews. The findings reveal five representation trajectories. At the beginner level, students’ representations were fragmented and lacked meaningful connections to the problem. At the limited level, representations began to appear but remained partial, inconsistent, and procedurally weak. At the developing level, students showed emerging ability to connect pictorial, symbolic, and verbal forms, although conceptual errors still occurred. At the proficient level, students demonstrated more structured symbolic and pictorial representations, but their written verbal explanations still required refinement. At the superior level, students were able to coordinate representations flexibly, systematically, and meaningfully. A notable finding is that representation development does not always progress linearly; some students at the developing level showed representational characteristics close to the proficient level. Therefore, this study contributes a five-level trajectory map that explains the dynamics of students’ mathematical representation ability through the lens of Habits of Mind performance.