Coordinating analytic and visual approaches: Math majors’ understanding of orthogonal Hermite polynomials in the inner product space [formula omitted] in a technology-assisted learning environment

Abstract

The purpose of this research study was to understand how linear algebra students in a university in the United States make sense of the orthogonal Hermite polynomials as vectors of the inner product space ℙnℝ in a Dynamic Geometry Software (DGS) – MATLAB facilitated learning environment. Math majors came up with a diversity of innovative and creative ways in which they coordinated visual and analytic approaches (Zazkis, Dubinsky, & Dautermann, 1996) for visualizing inner products of Hermite polynomials along with other notions inherent in the inner product space, such as Triangle Inequality, Pythagorean Theorem, Parallelogram Law, Orthogonality and Orthonormality, Coordinates Relative to an Orthonormal Basis. Research participants not only produced such creative inner product space visualizations of the Hermite polynomials with the induced improper integral inner product 〈f,g〉=∫−∞∞e−x2f(x)g(x)dx on the DGS, but they also verified their findings both analytically and visually in coordination. The paper concludes by offering pedagogical implications along with implications for mathematics teaching profession and recommendations for future research.