An Exploration of the Confusion Between Concept and Formalization Amongst the Community of Teachers in Physics

Smigiel and Simon

INSA Strasbourg, Archives Poincaré, University of Strasbourg

This study focuses on the links between concepts in physics and the mathematical formalisms that translate them. A physics concept ought to be explored from an epistemological disciplinary perspective, one that shouldn’t be confused with the formalization process that aims at translating it. The notion of divergence of a vector field can be used to highlight the confusions that might exist between concept and formalization. Using an internet survey, an important proportion of French professors of higher education were asked to give the definition of the divergence of a vector field. 80% of the answers defined that term as the sum of the partial derivatives of the components of the field in relation to the corresponding coordinates. The paper shows how Maxwell and Heaviside have clarified this concept and how they have shown that an intrinsic definition based on vector analysis leads to the correct articulation between former concepts and new ones. By defining divergence as the limit of the electric flux per unit volume through a closed surface when the volume tends towards zero, the introduced concept takes root in previous knowledge whose limits were highlighted; it helps in pursuing the initial reflection and hence in making more sense. The poll showed surprisingly that this definition rarely appears. This article shows that much work on Science teaching combined with History of Science may improve teaching efficiency despite the great amount of results that the discipline has already achieved.