Year of Graduation

2026

Document Type

Thesis

Degree Name

MALS: Interdisciplinary Studies

Directing Professor

Dr. Bryan Faulkner

Abstract

This thesis develops a vertically integrated curriculum in topology spanning advanced high school, undergraduate, and graduate levels of mathematical instruction. The study is motivated by the observation that topology possesses a relatively stable collection of structural concepts—including continuity, connectedness, compactness, separation properties, and topological equivalence—that persist across all stages of mathematical development.

The curriculum framework proposed here distinguishes between structural concepts and the technical machinery used to investigate them. Structural concepts are introduced early through visualization and conceptual exploration, while increasingly sophisticated methods are incorporated as students progress toward proof-based and research-oriented study. Separate curriculum models and mathematical foundations are developed for each educational level, with attention given to cognitive readiness, pedagogical methodology, and disciplinary authenticity.

Through comparative analysis, the thesis demonstrates that topology can serve as a coherent pathway from intuitive mathematical reasoning to advanced abstraction. The resulting model provides a framework for introducing higher mathematics in a manner that preserves conceptual continuity while supporting increasing levels of rigor and sophistication. More broadly, this work contributes to discussions of curriculum design in mathematics education by illustrating how advanced mathematical disciplines may be organized around enduring conceptual structures that support long-term intellectual development.

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