Radian riddles: semiotic challenges in students’ understanding of the semantic role of <i>π</i> in trigonometry

This study investigates how high school students understand the semantic role of π in trigonometry, using the lens of semiotic representational systems and task-based clinical interviews. Specifically, it examines conceptual problems and ambiguities in students’ initial understanding of π and other real numbers as sine and cosine arguments, immediately after school trigonometry, following the convention for representing radians with π notation and degrees without it. Results reveal that students associated π in different representations with multiple real values (e.g. associating π with 180 or –1, and 2π with 1 or 0). These results highlight the vital role of understanding semiotic conventions (e.g. representing radians with π) as a customary choice among equally-valid-alternatives (e.g., with or without π) for recognising invariant mathematical actions behind signs-and-symbols representing mathematical entities. Based on the findings, we propose a model for coherent understanding of the set of real numbers as domains of sinusoidal functions.