Errors in Transitions Between Algebraic and Other Representations: A Qualitative Study of 7th and 8th Grade Students

Mathematical understanding is closely related to how concepts are represented and how learners coordinate different forms of representation. Within this context, multiple representations constitute a central component of mathematics learning by enabling students to construct and communicate meaning. This study aims to examine 7th and 8th grade students' ability to transition between multiple representations (e.g., graphical, tabular, and verbal forms), focusing specifically on the errors made in transitions from algebraic representations to other forms and vice versa. This study was designed as a qualitative case study. The study group consisted of middle school students, and data were collected through open-ended tasks requiring students to translate between different representations. The analysis was guided by a multiple representation framework based on Lesh (1979) and Janvier (1987), and the data were analyzed using content analysis. The findings indicate that students experienced the greatest difficulty in transitions from graphical to algebraic representations. Many students failed to interpret key graphical features such as slope and intercept and translate them into algebraic expressions. While transitions between tabular and algebraic representations showed similar error patterns across grade levels, differences emerged in algebraic–graphical and algebraic–verbal transitions. The most common error was performing numerical calculations without constructing algebraic expressions. In many cases, students focused on computing specific values rather than generalizing symbolically. Overall, the results highlight significant difficulties in representational transitions and emphasize the need for instructional practices that explicitly support representational fluency and conceptual understanding.