The VIBE Framework: A Student-Centered Approach to Teaching Knot Theory in Secondary Mathematics

TL;DR

The paper introduces the VIBE framework, a student-centered pedagogical approach for teaching knot theory in secondary math, emphasizing visualization, inquiry, collaboration, and context to enhance engagement and understanding.

Contribution

It presents a novel, constructivist-based framework for integrating knot theory into secondary education through structured activities and interdisciplinary connections.

Findings

Enhanced spatial reasoning and engagement in students.

Effective integration of knot theory into secondary curricula.

Visual and inquiry-based activities foster deep understanding.

Abstract

Knot theory, a visual and intuitive branch of topology, offers a unique opportunity to introduce advanced mathematical thinking in secondary education. Despite its accessibility and cross-disciplinary relevance, it remains largely absent from standard curricula. This paper proposes the {\it VIBE framework}, a student-centered approach, structured around four pedagogical pillars: Visual, Inquiry-based, Braided (collaborative), and Embedded (contextualized) learning. Rooted in constructivist theory, VIBE supports cognitive development, spatial reasoning, and mathematical engagement across diverse learners. We present a sequence of low-threshold, high-ceiling activities designed to develop core topological concepts while fostering creativity and exploration. Through qualitative heatmaps, clustering visualizations, and classroom snapshots, we demonstrate how knot theory can be transformed into a powerful medium for inquiry and interdisciplinary connection. We believe that the VIBE framework provides a structured yet adaptable approach that supports the integration of deep, meaningful mathematical experiences into secondary education.