Instructor-Student Applications for Vector Line Integrals: Symbolic Scaffolding and Personalized Visualization as Cognitive Reward

Abstract

This paper presents a paired Instructor-Student application for computing 3D vector line integrals that integrates the SymPy symbolic engine with the Manim procedural animation library. To address the lack of specialized authoring tools for vector calculus problem sets, the Instructor module provides a high-level interface for defining v ector fi eld co mponents and parameterizing integration paths. It empowers educators to construct custom task libraries through a rigorous authoring pipeline: each variant undergoes mathematical pre-computation to ensure solvability, followed by the generation of a 3D animation for visual validation before the complete problem-video pair is committed to the database. Developed to address the challenges of autonomous assessment in distance education, the Student module enforces step-by-step validation via a conditional Scaffolding Lock mechanism, effectively preventing error accumulation. The core pedagogical novelty lies in integrating symbolic verification with the concept of cognitive reward: after a successful step-bystep solution, the student receives a personalized 3D animation of the vector field and path from their own task. This visualization serves not merely as a correctness token, but as a cognitive reward that reveals the spatial-geometric properties of the specific problem - information that students rarely receive under traditional teaching methods. Functional testing demonstrated the system's operational stability and computational efficiency, laying the groundwork for transitioning from pilot studies to comprehensive pedagogical evaluation.