“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 assigned interval. In this study, I take conceptual and empirical steps understand phenomenon more depth. Building Fischbein’s theory, conceptualize area figural concept associate calculation with intra-mathematical modelling routes pass through network of figures integrals. To characterize these routes, analyzed video clips large first-year service course mathematics submitted part their coursework. This led construction an analytical cycle explicating models generated when implementing SImA. appreciate how widespread SImA be, use responses nearly twelve thousand final multiple-choice exams same over decade. analysis found thirty percent chose answers are consistent reasoning; some SImA-options were selected frequently than correct answers. Drawing findings, make several teaching-oriented comments about relations between