Document Type
Article
Publication Title
ACMS Journal and Proceedings
Abstract
Jacques Hadamard describes a four-stage epistemological process: preparation, incubation, illumination, and verification. Hadamard developed this scheme based upon Henri Poincaré’s reflections on the process that led to his discovery of Fuchsian functions (i.e., automorphic functions) which he described in his wonderful paper La Genese de la Creation Mathematique (Mathematical Discovery in English). Philosopher and theologian Bernard Lonergan builds on the writings of Aquinas to develop his theory of insight which elaborates the transition from incubation to illumination. Insight and illumination provide an analytical lens for transitional moments as students learn ever more abstract mathematical concepts. We offer exemplars of these transitions at the elementary, middle, high, undergraduate, and graduate school levels. We provide connections to theorists in mathematics education such as Jean Piaget (stages of development of understanding functions), Pierre van Hiele and Dina van Hiele-Geldof (levels of understanding geometry), and Dietmar Küchemann (levels of understanding letters or variables). We also provide pedagogical strategies which align with Christian faith commitments and promote growth in understanding.
First Page
73
Last Page
83
Publication Date
5-2025
Recommended Citation
Klanderman, David; Klanderman, Sarah; and Turner, James, "Insight and Illumination in Mathematical Learning: Exemplars of Transitions to Understanding in the Classroom from Elementary to Graduate School" (2025). University Faculty Publications and Creative Works. 950.
https://digitalcommons.calvin.edu/calvin_facultypubs/950