Continuity: From Intuition to Formalization – A Comprehensive Study

Abstract

This study examines the difficulties students face in understanding the concept of continuity and its connection to key theorems such as the Fundamental Theorem of Algebra. The initial hypothesis suggests that the challenge lies not in the intuitive notion of continuity, but in its mathematical formalization, which is defined through limits within a topological framework. Although students approach problems continuously and intuitively, many struggle to translate their geometric reasoning into formal algebraic representations. To support this transition, the “Robot Trajectories” activity was designed using a Problem-Based Learning (PBL) approach. In this task, 79 students, organized into 29 teams, analyzed different paths a robot could take to reach a target point. The trajectories, represented as piecewise polynomial functions, required students to identify the optimal route and link it to the corresponding function rule, often considering obstacles along the way. While most teams relied on geometric intuition to approach the problem, only a few succeeded in achieving a solid algebraic formalization. These results highlight the gap between intuitive understanding and mathematical formalization, emphasizing the need for support for this transition in engineering calculus education.