Understanding Osculating Circle and Sphere: Modeling with Mathematica

Abstract

This research paper investigates how a digital tool Mathematica supports students’ comprehension of mathematical concepts, specifically the osculating circle and osculating sphere in differential geometry. This research is grounded on Technological Pedagogical Content Knowledge (TPACK) framework, to answer a research question “how do students understand the dynamics of the osculating circle and osculating sphere using mathematical simulation?”  I employed a qualitative phenomenography methodology, a branch of phenomenology among thirteen students from the Central Department of Education (CDED), Tribhuvan University. I gathered data through semi‑structured interviews, direct observations, and recordings of students’ interactions with Mathematica‑based dynamic and interactive simulations. I analyzed the data through generation of transcriptions, meaning condensation, categorization of structural aspects, and generation of thematic outcomes reflecting on students’ understanding of osculating concepts and processes. The study found that Mathematica-coded dynamic visualization enhanced students’ conceptualization of osculating process linking formal definitions with mental images. Particularly, it helped in visualizing how the osculating circle and sphere evolve along a curve. The simulations helped bridge the gaps between concept-definition and concept-image. The study suggests that digital pedagogy aligned with the TPACK framework can promote students’ conceptual understanding of advanced mathematics.