“Find the area enclosed by...” Parceling an especially robust model of reasoning among first-year students

Abstract

Mathematics education research has been aware that calculus students can draw on single definite integrals as a model to compute areas (SImA), without minding whether the function changes its sign in the assigned interval. In this study, I take conceptual and empirical steps to understand this phenomenon in more depth. Building on Fischbein’s theory, I conceptualize area as a figural concept and associate its calculation with intra-mathematical modelling routes that pass through the network of figures and integrals. To characterize these routes, I analyzed video clips that students in a large first-year service course in mathematics submitted as part of their coursework. This led to the construction of an analytical cycle explicating the models that students generated when implementing SImA. To appreciate how widespread SImA can be, I use the responses of nearly twelve thousand students in final multiple-choice exams in the same course over a decade. This analysis found that nearly thirty percent of the students chose answers that are consistent with this model of reasoning; in some exams the SImA-options were selected more frequently than the correct answers. Drawing on these findings, I make several teaching-oriented comments about the relations between areas and definite integrals.