A cross-sectional investigation of the development of the function concept

Applying Covariational Reasoning While Modeling Dynamic Events: A Framework and a Study

“…Informed by these studies and our research from the last several years (Carlson, 1998;Carlson, Jacobs, & Larsen, 2001) we define covariational reasoning to be the cognitive activities involved in coordinating two varying quantities while attending to the ways in which they change in relation to each other. We concur with Saldanha's and Thompson's (1998) view that images of covariation are developmental.…”

“…Research suggests that undergraduate students are entering the university with weak understandings of functions, and entry level university courses do little to address this deficiency (Monk, 1992;Monk & Nemirovsky, 1994;Thompson, 1994a;Carlson, 1998). Recent investigations of college students' understandings of functions have documented that even academically talented undergraduate students have difficulty modeling functional relationships of situations involving the rate of change of one variable as it continuously varies in a dependency relationship with another variable (Monk & Nemirovsky, 1994;Thompson, 1994a;Carlson, 1998). Research has also shown that this ability is essential for interpreting models of dynamic events (Kaput, 1994;Rasmussen, 2000) and is foundational for understanding major concepts of calculus (Kaput, 1994;Thompson, 1994a;Cottrill, 1996;Zandieh, 2000) and differential equations (Rasmussen, 2000).…”

“…In recent years our understanding of ways in which college students interpret and represent dynamic function situations has been informed by considerable research (Monk, 1992;Sierpinska, 1992;Kaput, 1994;Nemirovsky, 1996;Thompson, 1994b;Carlson, 1998). In examining the thinking of calculus students when attempting to interpret the changing nature of "rate of change" for intervals of a function's domain, several studies (Monk, 1992;Monk & Nemirovsky, 1994;Thompson, 1994a;Nemirovsky, 1996;Carlson, 1998) have revealed that this ability is slow to develop, with specific problems reported in students' ability to interpret graphical function information.…”

“…In examining the thinking of calculus students when attempting to interpret the changing nature of "rate of change" for intervals of a function's domain, several studies (Monk, 1992;Monk & Nemirovsky, 1994;Thompson, 1994a;Nemirovsky, 1996;Carlson, 1998) have revealed that this ability is slow to develop, with specific problems reported in students' ability to interpret graphical function information. Studies by Monk (1992) and Kaput (1992) have noted that calculus students show a strong tendency to become distracted by the changing shape of the graph, and in general, do not appear to view a graph of a function as a means of defining a covarying relationship between two variables.…”

Covariation between variables in a modelling process: The ACODESA (collaborative learning, scientific debate and self-reflection) method

“…Bloch, 2003;Breidenbach et al, 1992;Carlson, 1998;Harel & Dubinsky, 1992;Hitt, 1994;Mesa, 2004;Trigueros & Martínez-Planell, 2010),…”