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Discovering the Linearity in Directional Derivatives and Linear Approximation

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Discovering the Linearity in Directional Derivatives and Linear Approximation Discovering the Linearity in Directional Derivatives and Linear Approximation Brian Fisher Jason Samuels Lubbock Christian University City University of New York - BMCC Linear functions of more than one variable exhibit the property that changes in the dependent variable are linear combinations of changes in the independent variables. Although multivariable calculus makes frequent use of this linearity condition, it is not known how students reason about linearity within this context. This report addresses this question by analyzing how three students incorporate linearity into their schemas for linear approximation and directional derivative. The students in this report showed a progression in their understanding from not using linearity within their reasoning to incorporating linearity into first their scheme for linear approximation and finally into their scheme for directional derivative. The results indicate that the context of linear approximation was useful for developing concepts of linearity and aiding their development of the concept of directional derivative. Keywords: Multivariable Calculus, Linearity, Schema Theory, Directional Derivative Introduction & Literature Review There is a recent surge of interest in student learning in multivariable calculus, which is a crucial course for all STEM majors (PCAST, 2012). Although there is a rich body of research investigating students’ understanding of rate of change (Johnson, 2013; Lobato & Siebert, 2002; Stump, 2001; Teuscher & Reys, 2010) in general and the concept of derivative (Zandieh, 2000; Park, 2011; Samuels, 2012; Orton, 1983) in particular, there is limited research on how students interpret these concepts in multivariate settings. Recent studies have indicated that students may struggle to generalize the concept of slope from two to three dimensions (McGee, Conner, & Rugg, 2011) and that by the end of a multivariable calculus class few students have developed a meaningful conceptualization of total derivative (Trigueros Gaisman, Martinez-Planell & McGee, 2018). Additionally, research in physics education shows that students having completed multivariable calculus struggle to appropriately apply partial differentiation in physics contexts (Thompson, Bucy & Mountcastle, 2006). However, there is evidence that the use of physical manipulatives may support students’ conceptions of rate in multivariable calculus (Samuels & Fisher, 2018; McGee, Moore-Russo & Martinez-Planell, 2015). Linear functions of one variable exhibit the important property that changes in the dependent variable are always proportional to changes in the independent variable. Calculus takes advantage of this fact when using a linear approximation to estimate nearby values of a function with the equation Δy ≈ f’(x)·Δx. Using the language of differentials this property can be summarized with the equation dy = f’(x) dx. However, in multivariable calculus the concept of linearity takes on the additional property that changes in the independent variables are additive when determining the change in the dependent variable. As a linear approximation for functions of two variables, we have that Δz ≈ fx·Δx + fy·Δy; the corresponding differential property is dz = fx dx + fy dy. At a given point, these expressions become linear combinations. It is similar for the directional derivative: where v = (Δx, Δy), Dvf = fx·Δx/|v| + fy·Δy/|v|. Although there is significant research on student understanding of linearity within the context of one variable functions (e.g. Ellis, 2007; Greenes, C., Chang, K. & Ben-Chaim, D., 2007; Moschkovich, J., Schoenfeld, A. & Arcavi, A., 1993) and an emerging body of research on linearity within linear algebra (Wawro & Plaxco, 2013; Wawro, Rasmussen, Zandieh, Sweeney & Larson, 2012), there is an absence of research on how students experience linearity in multivariable calculus. This study adds to the literature by exploring student conceptions of linearity in this context. In particular, we seek to answer the question: what conceptions of the role of linear combinations do students form in the context of linear approximation and directional derivative for multivariable functions? Theoretical Framework This report aims to analyze student understanding through the lens of schema theory. Schema theory has a long history of development with many significant influences (e.g. Bartlett, 1932; Piaget, 1926; Anderson, 1984) whose models of cognition subtly differ from one another. For this reason there are multiple definitions of the word schema throughout the literature. For the purposes of this study we will define a schema as an internal framework used to guide encoding, organization and retrieval of information (Stein & Trabasso, 1982). In this way a schema characterizes the relations among its components (Anderson et al., 1978). From our perspective, schemas are functional in the sense that they are continuously undergoing change (Iran-Nejad & Winslerin, 2000), reshaping themselves as the individual undergoes new experiences and reflects upon past experiences. This reshaping can occur in three ways: accretion, in which new facts are assimilated into the existing knowledge structure, tuning, in which the knowledge structure is slightly modified without changing relationships, and restructuring, in which new knowledge structures are created. There are four types of tuning: refining accuracy, generalizing, exemplifying, and creating an archetype. There are two types of restructuring: patterned generation, in which an old schema is modified into a new schema, and schema induction, in which a recurrent relationship among schemas is retained as a new schema. The latter is the most difficult and rare form of learning (Rumelhart & Norman, 1978). Methodology The data for this report were obtained from semi-structured task-based interviews with three students working together as one group as they encountered the ideas of multivariable linear approximation and directional derivative for the first time. The students were enrolled in a multivariable calculus course incorporating physical manipulatives using the Raising Calculus to the Surface materials (Wangberg & Johnson, 2013). The interviews took place in two separate sessions. The first session consisted of open-ended questions and tasks designed to elicit their prior understanding of rates of change in single and multivariable calculus followed by a series of activities designed to explore the ideas of linear approximation and directional derivative. The second session revisited the linear approximation and directional derivative tasks to assess further changes in the way the students viewed these concepts. The sessions were video recorded and analyzed. During the analysis the authors identified instances in which the students actively described or utilized schemas which incorporated aspects of rate of change or linearity. These instances were then analysed from the perspective of schema theory in order to identify the pattern of connections evoked by the students related to rate of change and linearity. These patterns were then analyzed over the duration of the interviews to determine significant changes within the students’ schemas as a result of their explorations during the task. Results Students’ Prior Knowledge In response to the open-ended questions prior to the linear approximation task, each student exhibited a robust understanding of rates of change in single variable calculus. Their initial schemas included a description of derivatives as measurable rates of change in geometric and contextual situations arrived at through a limiting process. The students were then able to extend these ideas to a two-variable setting by adding an element of directionality to their mental framework. They were able to evoke this rate of change schema in order to measure partial derivatives in multiple settings: on a three-dimensional physical representation of a surface, in the applied context of a heated plate, and on a contour map. At this point in the interview the students’ primary use of directionality was to reduce a three dimensional problem to a problem of only two dimensions by looking at the traces of the surface on the coordinate planes. This is described below with the first evidence that the students were also considering directions other than those along the coordinate axes. Interviewer: You mentioned earlier that the idea of derivative is connected to the idea of tangent line. Is there any sort of similar idea that holds in multivariable calculus? Willy: When you’re dealing with more dimensions, kind of like a plane, which plane the rate of change is happening, the xz-plane or the yz-plane. Mo: You have to specify a direction. Interviewer: You were nodding, was your description of plane similar to [Mo]’s description of a direction? (Willy nods) What sort plane are you thinking about? Willy: There are infinite amounts of planes (gestures vertical planes in many directions). So you have to specify which direction the tangent line is in. As we see in the above excerpt, the students were able to consider rates of change in many directions; however, prior to the linear approximation tasks they did not demonstrate an ability to measure or calculate rates of change along directions other than the coordinate directions. When asked if there is a relationship between the rates of change in different directions, Willy responded, “No… I don’t think there is any relation between one slope and another.” Similarly, when given two partial derivatives
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