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 and asked whether a directional derivative would be positive or negative, James made a wavy hand gesture and stated “It would depend on the way the temperature’s changing on the curve.”

The Linear Approximation Task In order to explore linear approximation in multivariable calculus, the students were given a physical surface representing a two-variable function with one point on the surface identified with a blue dot. The students were given the following task: A. The surface represents the density of gold (in grams per cubic mile) beneath the ground. You own a small mine located at the blue dot. Estimate the density of gold at your mine and measure how the density of gold changes in the north and east directions.

B. You want to buy one of three mines which are for sale; their locations (relative to yours) are given below. Estimate the density of gold at each mine using only your previous measurements. Mine A: 1.2 Miles North Mine B: 1.2 Miles North and 0.8 Miles East

Figure 1: Linear Approximation Task The students were able to apply their prior knowledge about partial derivatives to quickly answer Part A of the task finding that the height of the surface at the blue dot was 3.5, the rate in the north direction was 0.28 and the rate in the east direction was -1.1. When beginning Part B the students were quickly able to incorporate the fact that changes in the density will be proportional to changes in the north direction in order to approximate the density at Mine A.

James: Oh yeah, right, so we’re at 3.5 right. So they’re 1.2 north. And the rate of change is - 1.1 per inch. Willy: No that’s the east direction, north is 0.28 James: Alright, [a rate of] 0.28, that means that gets slightly taller. Willy: Or is it 0.28 [the rate of change] times 1.2 [the change in distance]? Mo: That is exactly what it is. This is all we need to do. If it says north, we times it with dz/dy if it says east we times it with dz/dx.

As the students attempted to approximate the density at Mine B, they needed to consider changes in both the north and east directions simultaneously.

Willy: But what about, like, [Mines] B and C where it moves both north and east? Mo: You can add them? James: But look – you have to multiply the rate of change by the direction and add that to our mine. You get what I’m saying? Because it’s 1.2 inches north, so you have to multiply that by the rate of change, and you have to add that to the mine, to see the height at that mine. Because it’s moving .25 grams per mile to the fourth in that direction. So we multiply that by 1.2 and then add that to the 3.5 to see where their height, quote unquote, would be. Mo: I see what you’re saying, yes... Willy: Yeah, that makes sense.

In the above excerpt we see that the additive property of linearity for changes in the dependent variable came naturally for the students. The students offered varied justifications for the linearity of their solutions when asked specifically why they believed it was appropriate to add the two components together. In the quotation below Willy argues that adding these changes together is similar to adding together vectors in three dimensional space.

Willy: So, say this is the original point (indicating the blue dot on the surface). Then when we went east it decreased a bit, and when we went north it increased a bit. So we are basically adding them. So, it’s basically vectors. Like, you have 3i + 4j and you are basically adding them up, something like that… It’s like, think of that parallelogram thing we learned. We’re getting the resultant vector from the north and east. So, we are adding them up, basically.

As seen in the above excerpt, Willy has made connections between this activity and his prior knowledge of vector addition, which incorporates the key properties of linearity, addition and scalar multiplication. It is not clear whether he is recognizing that the partial derivatives can be represented as vectors on a tangent plane or whether he is just acknowledging that the additive behavior and directionality seen in vectors is similar to the approximation calculation. Mo subsequently embraced the vector-style reasoning:

Mo: They’re not vectors but they behave like vectors. (He draws a rectangle with vectors as the edges.) If you want to get to this point you have to do this plus this. So I guess it’s like, they act like vectors, but they’re not really vectors.

James constructed a justification from a different point of view.

James: When you multiply them out ... you’re left with the change in z … you get the same unit as this one. And the same thing goes for here so you can add them all up.

He confirmed that in this context the units became the same for each term. While not a complete justification for linearity, it demonstrates its plausibility, as the inverse scenario would rule it out. After successfully approximating the density of gold at Mine B the students quickly applied the same principles to approximate the density at Mine C. Following this task they were able to work together to create a generalized formula for linear approximations at any point in the domain. Furthermore, upon returning for the second session of the interviews the students immediately applied the same additive approximation scheme when given a similar task.

Directional Derivatives Immediately after the development of their linearity schema for linear approximations, the students were asked if, given the partial derivatives of a two-variable function, they could evaluate the derivative in another specific direction. Their initial response is in the excerpt below:

Interviewer: Let’s make this vector more precise. Let’s make it 1i [plus] 2j. Could we figure out what the rate of change is in that direction? Willy: 2 over 1. Mo: It’s actually the magnitude. Willy: Its 2 over 1. James: No it’s not the magnitude. The magnitude is the length of the… Willy: Remember Pythagoras theorem, James: It’s the length of the steepest point.

In the above excerpt we see the students attempting to connect this problem to several prior experiences in mathematics, but significantly they have not connected this problem with the just completed linear approximation activity, and have not invoked any part of their linear combination schema. During the second interview session the students were once again tasked with finding the directional rate of change. The function had partial derivatives fx = 0.41 and fy = -0.19, and they were given the direction of 3.5i + 1.25j.

James: What if we add both rates, shouldn’t it give you that rate? Mo: Actually… yes. James: Yeah, it should because it’s going to give you the same points… Interviewer: So, tell me what you’re going to add? James: Wait let me see if it makes sense first? [cross talk] To get from point A to point B, you just add them. Interviewer: So where did the .22 come from? James: I just added the rates.

Here we see the students begin to explore incorporating addition into their problem solution; however, they are adding the rates and not the changes in the values. Thus we see that from their linearity schema they have utilized addition but not scalar multiplication. A short while later they recognize that they need to multiply the rate by the change in distance, but they still have not connected this process to the linear approximation schema developed earlier. This observation is finally made in the following excerpt.

Mo: Ok, I get it. So this rate is not for this distance, it is for anywhere. This is how much it is changing for a unit distance. So we need to multiply by this distance. James: Wait, yo, it’s what we did originally. It is. Mo: Yeah, I think the rate is this (writes 3.5 * 0.41 + -0.19 * 1.25). 3.5 times what was the rate, 0.41, times the distance, plus, again the rate, -.19, time the distance 1.25, and you’re going to divide it by the square root of it, to get this distance (writes square root of 3.52 + 1.252). That’s it. (does a victory fist pump)

When asked why they needed to divide in the above expression, the students responded:

Mo: I had the rate in this direction (indicates the x-direction), but I had to multiply it by the distance. But, since I don’t want the rate times distance in this direction (indicates the direction of the directional derivative) I had to divide by the distance. James: Yeah, to get the unit vector of unit rate.

Immediately following this excerpt the students extended their result and wrote a generalized formula for the directional derivative as a linear combination.

Discussion & Conclusion Over the course of the two interviews, we saw a progression of the students’ schema for linearity and its connectedness to linear approximation and directional derivatives. Initially the students did not display evidence of a connected linearity schema, arguing that there should be no relationship between rates of change in different directions. However, engaging with the approximation task prompted the students to introduce linear combinations into their linear approximation schema. Their construction of a correct procedure and answer for linear approximation represented a restructuring of their schema by schema induction. They were able to generalize their result, indicating an act of tuning. During discussion the next day on linear approximation they each comfortably utilized linearity in the same fashion, indicating that their schema had been strengthened. They offered varied justifications for implementing linearity. Two students had a justification for linearity which was context-free (vector addition) and one was more context dependent (adding like units). The ability to justify the use of linearity in appropriate contexts is a significant development since prior research shows that students often apply linearity and its properties to mathematical scenarios where it is inappropriate (De Bock et al., 2007). In spite of this schema development, the students were not able to evoke linearity once the students changed tasks to determine the value of a directional rate of change. This indicates that, at that time, there was no connection between their schemas for directional derivative and for either linearity or linear approximation. Instead they went about re-creating the linearity schema within the context of directional derivatives. Finally, after the linearity property was re-created by the students, James exclaimed “Wait, yo, it’s what we did originally!” This appears to be the moment that James recognized that he could use his linearity schema from the linear approximation task and adapt it in order to reason about directional derivatives. The other group members made the same realization and quickly incorporated linearity into their problem solution for directional derivatives. Their resulting schemas thus had connections between linearity, linear approximation and directional derivative. Given their inability to calculate the last two previously (in multivariable calculus), this indicates a significant restructuring of these schemas. It is significant that the expression of linearity within the context of linear approximation did not immediately lead to the use of the schema when finding directional derivatives. This is reasonable, as the former deals with total change, whereas the latter involves a rate, and an additional division must occur. Indeed, in the construction of the directional derivative expression, this division is the final step the students took. It is notable that it was, in fact, the recognized connection between linear approximations and directional rates of change that allowed the students to complete their formulation of the directional derivative. Many major textbooks (e.g. Stewart, 2012) do not make the connection between these topics explicit. This development points to several areas of possible future research. Is this connection between linear approximation and directional differentiation commonly observed among students? How does the instructional sequence of linear approximation followed by directional derivative compare to other alternative instructional sequences? How does the choice to contextualize the linear approximation task impact a student’s development of a linearity schema? Finally, do the observations reported here generalize to large student populations? This study has contributed to the body of research in multivariable calculus by observing how three students invoked their linearity schemas at varying levels of robustness while investigating linear approximation and directional derivative, by analyzing the connections they made, and has suggested new lines of inquiry as well.

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