The case of Dan: Student construction of a functional situation through visual attributes

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

“…The new century offers educators a plethora of technologies (e.g., graphing calculators, geometry software, computer algebra systems, electronic laboratory probes, specialty software such as MathCars (Kaput, 1994)) and specially designed physical devices (e.g., Monk & Nemirovsky, 1994) for studying real-time dynamic events. Rich pedagogical opportunities abound for building on students' intuition about and experience with dynamically changing quantities.…”

“…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 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.…”

"You Have to Count the Squares": Applying Knowledge in Pieces to Learning Rectangular Area

“…Izsák (2000) analyzed processes of notation variation and mapping variation that eighth-grade students used when constructing algebraic models of linear motions in a physical device, and Roschelle (1998) introduced registrations to characterize similar knowledge when analyzing students' qualitative reasoning about Newtonian motion simulated by a computer microworld. Other researchers (Lobato, Ellis, & Muñoz, 2003;Lobato & Siebert, 2002;Meira, 1995;Monk & Nemirovsky, 1994;Moschkovich, 1998;Nemirovsky, 1994;Schoenfeld, Smith, & Arcavi, 1993) also reported instances in which learning to focus on and use features of graphs and tables-including y-intercepts, x-intercepts, and slopes-and attributes of physical objects for solving problems has been a significant accomplishment for students.…”

“…1 Growth and change consists of multiple, related processes including not only the construction of new knowledge elements, but also the coordination of diverse knowledge elements and the extension or constriction of conditions under which particular elements may be applied productively. Further research has applied the knowledge in pieces perspective to examine emerging competence in the domains of Newtonian mechanics (e.g., Roschelle, 1998;Sherin, 2001), intuitive epistemology (diSessa, Elby, & Hammer, 2003;Hammer & Elby, 2002), functions (e.g., Monk & Nemirovsky, 1994;Moschovich, 1998), fractions (Smith, 1995), and probability (Wagner, 2003). Most participants in these studies have been high school and college students.…”