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Taylor series are a wildly valuable tool in many professions, but students’ reasoning with this topic is vastly understudied. What sense do students make of Taylor series? Do they have any image at all for Taylor series and what they’re used for? The little research on students’ understanding of Taylor series speaks mostly to broad themes of characterizing expert/novice strategies (e.g. Martin, 2009), tendencies for reasoning with them (e.g. Alcock & Simpson, 2004 and 2005), grappling with formal definitions of convergence (e.g. Martin et al, 2011), or use of technology in instruction (e.g. Yerushalmy & Schwartz, 1999; Soto-Johnson, 1998). That is, most of these important studies on students’ use and understanding of Taylor series take a more global perspective, examining general themes, post hoc. But to develop a robust knowledge of the concept of Taylor series requires the synthesis of many previous calculus content topics, woven together and used appropriately, to form a more complete image. The question of how students synthesize their prior knowledge and arrive at their image has not been studied. That is, we have little idea about how students construct an understanding of Taylor series from less formal prior calculus notions, and how they attribute meaning to particular aspects of whatever representation of a Taylor series they espouse.
Moment-to-moment analyses about the construction of a mathematical topic are crucial not just to uncover ‘misconceptions’ that particular students have about the topic, but also to be able to put their responses to tasks about those topics in context. Habre (2009) discovered that even multiple exposures to the topic of Taylor series, at varying levels of mathematical sophistication, are often insufficient for even a broad comprehension of the material. Thus, knowing how students build their understanding can put into perspective some of the issues that persist around this topic.
Though it is not always students’ tendency to produce visual images for Taylor series tasks, many do (including the student in the case discussed in this preliminary report). Access to students’ visual images, supplemented by their descriptions and explanations, can provide 15TH Annual Conference on Research in Undergraduate Mathematics Education 369 additional insight into how they are constructing an understanding of topics such as Taylor series, as visualization is “a fundamental aspect to understanding students’ constructions of mathematical concepts” (Habre, 2009). Martin (2009) showed unsurprisingly that mathematicians were more fluent than novices in using graphical representations, both in their construction and interpretation, in the context of Taylor series. His dissertation made clear that “many students do not have a good visual image, if they have any visual image at all, of the convergence of Taylor series” (p. 288). Biza, Nardi, and Gonzales-Martin (2008) agree, citing an additional lack of useful imagery in textbooks chapters that students may use for reference. In our experience and works in progress, which align with Alcock and Simpson (2004), many students do in fact turn to visual images to explain and reason with Taylor series tasks. In fact, Alcock and Simpson (2005) also demonstrated that even “non-visualizers” may have a reliable graphical image, but tend to not call on it. So, in this paper, we endeavor to study, with a moment-to-moment analysis, the creation of one student’s graphical image that he chose to use to play out his reasoning with Taylor series approximation tasks.
Research Questions The strength of the case to be presented in this report is two-fold. As it is a detailed examination of the development of a student’s reasoning, as it plays out graphically, following this student’s process with a moment-to-moment analysis can allow for an examination of what he takes as calculus-based and physical evidence for claims he is making in his reasoning, and how those claims are manifested in his graphic. Second, and much more content-specific, Taylor series literature largely examines students’ (graphical, and other) reasoning or presentation at the completion of a problem, rather than as it is being built, negotiated from one moment to the next (exceptions include Martin et al, 2011, which focuses on formal definitions). Therefore, the
exploration of this case will speak to the following:
(1) How is additional evidence germane to a problem gathered and used to amend a visual image that serves to represent a particular concept for a student?
(2) In what ways are prior calculus concepts negotiated to construct and attribute meaning to a representation of Taylor series?
With the case presented here, these questions can only be addressed for one particular student, but can be used as a model both for future analyses, and to highlight ways in which calculusbased reasoning can (and does) influence students’ understanding of Taylor series.
Data Collection and Methods The study makes use of a particular 1.5-hour semi-structured interview with sophomore physics major Joe, who was participating in a larger, related study. Though it will not be discussed in this paper, the purpose of the larger study was to investigate students’ consistencies (and inconsistencies) in reasoning around a set of approximation tasks in calculus and physics contexts. The tasks discussed in this paper are the only two in the larger task set that elicit students’ thinking about approximation in the context of Taylor series (the text of which appear in Figs. 1 and 2). They are intentionally vague, and written to allow participants great freedom in what they choose to attend to as they respond. The interview was videotaped and transcribed for analysis. At the time of the interview, Joe had taken three semesters of calculus and two semesters of physics, and earned grades of “A” in all of them. He was identified by instructors as very competent in the subject matter. Upon completion of data collection for the larger study, Joe’s interview stood out for several reasons, of interest here are those related to his construction 370 15TH Annual Conference on Research in Undergraduate Mathematics Education of Taylor series images. An in-depth, microgenetic analysis of this interaction between the student and the calculus content at his disposal seemed a promising way to examine change in his notions of Taylor series approximation on a more fine-grained level than would be ascertained in other assessment situations (Calais, 2008).
displacement of the pendulum from vertical in radians. You want to calculate the period of oscillation for this pendulum. How big can the angle of displacement of the pendulum be before the equation for small oscillations isn’t a good approximation of the period?
€ Figure 2: Task 2 Results and Analysis.
The analysis that constitutes this preliminary report is ongoing, but an all-too-brief description of one data sample of an early transition in Joe’s thinking (below), while working on Task 1, will serve to highlight the nature of the transition points in the analysis that illuminate both how Joe uses his additional evidence to refine his image, and how that image represents the meaning of approximation with Taylor series (according to him). To carry out an analysis of this entire interview, it was broken into episodes during which Joe is appealing to a stable version of his visual image. Within each episode, it is then instructive to trace his thinking and evidence for his claims, both as he discusses them and as he amends his image based on those claims. When he abandons one image for another structurally different version, a new episode begins.
Data Sample. While working on Task 1, after drawing a graph of f(x)=arctan(x), Joe decides to draw a band around the horizontal asymptote of y= π /2 with two horizontal lines, y=1.47 and y=1.67 (see Fig. 3). Here referred to as “tolerance bands,” he emphasizes that it is reasonable to
be within roughly a 0.1-band on either side of π /2, stating:
€ 15TH Annual Conference on Research in Undergraduate Mathematics Education 371 € “You need to find where [the approximation] first enters the [tolerance band]. I think you can just assume it's a good approximation until then … And then once it enters the [tolerance band], you begin to encounter the possibility of it being a bad approximation, so then once it leaves that [tolerance band] you know that it’s become a bad approximation.” Joe’s language and drawing indicate that he believes the series approximation will look like the thicker line in Fig. 3, pointing out where it enters and exits the horizontal tolerance bands.
Though he explains in great detail why he believes this is a good strategy for determining when the approximation (a cubic) would represent a reasonable approximation for arctangent, and shows great skill in graphing and reasoning about end behavior, upon further reflection, Joe recognizes two problems with this representation. First, he recognizes that that it “starts outside the range” - That is, he notices that the point that the two graphs share (the origin) is outside of his band. He chooses to explain this away and not act on it, not recognizing the importance of the ‘center’ at x=0. However, Joe does act on a second problem – plugging the value of x=1,000 into the first two terms of the approximation, he realizes that that cubic should “go off to negative infinity.” This does not sync with his knowledge that arctangent will level off at π /2.
It is at this point that Joe shifts his thinking, uses a calculator to graph y = x − (1/3)x 3, and produces Fig. 4. Realizing with this new evidence that the approximation will never even reach his tolerance band, Joe’s attention is drawn to more local features such€ the maximum of the as cubic function. Noting that “arctangent is strictly increasing,” and that the cubic has a maximum, € Joe posits “[The cubic] is decreasing after a certain point, so once it passes that point you know it is rapidly becoming a bad approximation” While he had originally convinced himself thoroughly that tolerance bands around π /2 were appropriate, new evidence (both numerical and graphical) prompted Joe to, for the moment, abandon the idea of tolerance bands. No longer concerned with the asymptote, his focus shifts to the increasing/decreasing features of the two functions in question. That is, he gathered evidence € that caused an amendment in his graphic, momentarily foregoing end behavior to accommodate what he knows about a more local feature of the graph.
€ 372 15TH Annual Conference on Research in Undergraduate Mathematics Education Figure 5: tolerance banding around arctan Figure 6: tolerance banding around T0 Breakdown of example. This short description of one part of Joe’s work highlights the sorts of things that analysis of this case will attend to – namely evidence that is used and not used in amending the evolving visual image for Taylor series approximation, and the prior calculus concepts initiated, as well as how they are integrated into Joe’s image and understanding.
A complete treatment of the case of Joe, which will appear in a RUME Conference Report in 2012, serves to illuminate how he arrived at his final working visual images in Figs. 5 and 6, corresponding to Tasks 1 and 2. Upon inspection, it would appear that Joe may have a relatively robust understanding of what it means to approximate with Taylor series. Some of his language even supports this. For example, he later states “you're looking at the distance between [the functions] at any given value of θ,” which is the more normative way of examining error.
However, our analysis shows documentable, systematic gaps in his understanding that are not evident in examining his final products alone, and were not resolved in his construction of those figures. For example, by the end of the interview Joe still does not appreciate the role of ‘center,’ € he persists in attending to infinite behavior instead of local behavior even when the context changes, and more. Most importantly, we have a window into how earlier calculus concepts and understandings mediated the creation of those images, and served as evidence (to Joe) for the evolution of his image.
Continuing Analysis. The next steps in this research will be to complete the analysis on Joe’s episode, with more emphasis on the first research question. Most of the analysis to date (only a snapshot presented here) has concentrated on the ways that prior mathematics concepts were negotiated to assign meaning to the approximation image for Taylor series, but there are bigger picture issues to be dealt with in the continuing analysis. Namely, what types of claims merit revisions to the image vs. other claims that are discarded, explained away, or deemed less important? How is that additional evidence that eventually causes amendments in the visual image sought?
Contributions As Borgen and Manu (2002) emphasize, “an understanding of what images, both correct and incorrect, that students might construct is important if teachers are to help students work toward connected formalizations” (p. 164). Even better – knowing how students build those images provides additional perspectives for informing pedagogy around the topic of Taylor series.
Returning to Martin’s (2009) point, recognizing that graphical representations of Taylor series are one of the most significant factors in separating novices from experts, it is instructive to work 15TH Annual Conference on Research in Undergraduate Mathematics Education 373 on building students’ graphical images for such a topic. However, one cannot responsibly undertake that task without first exploring how students create that understanding for themselves.
ReferencesAlcock, L. & Simpson, A. (2004). Convergence of Sequences and Series: Interactions Between Visual Reasoning and the Learner’s Beliefs about their own Role. Educational Studies in Mathematics, 57, 1-32.
Alcock, L. & Simpson, A. (2005). Convergence of Sequences and Series 2: Interactions Between Non-visual Reasoning and the Learner’s Beliefs about their own Role. Educational Studies in Mathematics, 58, 77-100.
Alcock, L. & Simpson, A. (2009) Ideas from mathematics education: An introduction for mathematicians.
Borgen, K. & Manu, S. (2002). What do students really understand? Journal of Mathematical Behavior, 21, 151-165.