Cognitive Discontinuity at the Boundaries of Mathematical Systems and Pedagogical Reconstruction: Philosophical Reflections and Model Innovation from "Points and Segments" to "Zero Vectors"

Abstract There is a significant ontological gap in the transition from junior to senior high school mathematics. Junior high plane geometry strictly defines that "a point is not a line segment," while the senior high vector system stipulates that "the zero vector can be represented by a point" and that "the zero vector is parallel to any vector." This definitional contradiction undermines the transitivity of parallel and perpendicular relations, causing severe cognitive conflicts among students. From the perspectives of mathematics philosophy and pedagogy, this paper analyzes the root of this logical contradiction. Based on the innovative value of "boundary cases," this paper proposes a revision of textbook definitions and introduces the "infinitesimal directional chess piece" model as a concrete teaching tool for the zero vector, aiming to bridge the cognitive gap between elementary and advanced mathematics. 摘要 中学数学教育在初高中衔接阶段存在明显的本体论断层。初中平面几何严格定义“点不是线段”,而高中向量体系却规定“零向量可用点表示”且“零向量与任意向量平行”。这种定义上的矛盾破坏了平行与垂直关系的传递性,导致学生产生严重的认知冲突。本文从数学哲学与教育学视角出发,剖析了这一逻辑矛盾的根源。基于“边界情形”的创新价值,本文提出了教材定义的修正方案,并引入“无限小带方向棋子”模型作为零向量的具象化教学工具,旨在弥合初等数学与高等数学之间的认知鸿沟。