High-school students’ errors and misconceptions regarding absolute value and related interpretations: an empirical study

Students’ difficulties with absolute value are widely reported in mathematics education research and often persist into upper secondary and tertiary education. This study investigates and categorizes high-school students’ errors and misconceptions across different aspects of absolute value, including numerical, algebraic and functional interpretations. Data were collected through a questionnaire comprising open-ended and closed-ended questions administered to 432 high-school students, followed by 15 semi-structured interviews. The findings reveal systematic patterns of errors and misconceptions, particularly in the transition from numerical to algebraic contexts, the perception of absolute value as a ‘positive-making machine’, confusion between domain and range and difficulties with equations, inequalities and logical operators. These errors are interpreted within a constructivist theoretical framework, drawing on the distinction between concept image and concept definition, as well as on didactic and epistemological obstacles. Rather than being viewed as isolated mistakes, students’ errors are understood as coherent cognitive structures shaped by prior learning experiences and instructional practices. Based on these findings, specific instructional proposals are suggested, which make systematic use of realistic contexts, dynamic geometry software and multiple representations. These interventions aim to facilitate conceptual change and bridge the gap between procedural fluency and a deeper structural understanding of absolute value.