Second-order covariation: enlarging the theoretical framework of covariational reasoning

Research in mathematics education has shown the need to explore new forms of covariational reasoning to conceptualize dynamic situations. Indeed, describing a dynamic phenomenon requires conceptualizing which quantities are varying, how they are varying and co-varying in relation to the other quantities, and also how the quantities may affect the behavior of the phenomenon itself. This paper introduces an enlarged theoretical framework for covariational reasoning. Among the three orders of covariation it includes, second-order covariation is then discussed more deeply by analyzing data from a 10th-grade class experiment on the conceptualization of the motion of a ball on an inclined plane with the support of digital tools. This study theoretically broadens the perspective on covariational reasoning by reviewing and framing coherently constructs already introduced in the literature; empirically, it elaborates on the characterization of students' second-order covariational reasoning and on the features of the adopted digital tools that supported it.