PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 9, 020110 (2013) Student understanding of Taylor series expansions in statistical mechanics Trevor I. Smith,1 John R. Thompson,2,3 and Donald B. Mountcastle2 1Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA 2Department of Physics and Astronomy, University of Maine, Orono, Maine 04469, USA 3Maine Center for Research in STEM Education, University of Maine, Orono, Maine 04469, USA (Received 2 March 2012; revised manuscript received 23 May 2013; published 8 August 2013) One goal of physics instruction is to have students learn to make physical meaning of specific mathematical expressions, concepts, and procedures in different physical settings. As part of research investigating student learning in statistical physics, we are developing curriculum materials that guide students through a derivation of the Boltzmann factor using a Taylor series expansion of entropy. Using results from written surveys, classroom observations, and both individual think-aloud and teaching interviews, we present evidence that many students can recognize and interpret series expansions, but they often lack fluency in creating and using a Taylor series appropriately, despite previous exposures in both calculus and physics courses. DOI: 10.1103/PhysRevSTPER.9.020110 PACS numbers: 01.40.Fk, 05.20.Ày, 02.30.Lt, 02.30.Mv students’ use of Taylor series expansion to derive and I. INTRODUCTION understand the Boltzmann factor as a component of proba- Over the past several decades, much work has been done bility in the canonical ensemble. While this is a narrow investigating ways in which students understand and learn topic, we feel that it serves as an excellent example of physics (see Refs. [1,2]). Some of this work has studied student knowledge transfer from mathematics and previous student understanding in particular content areas including physics courses to more advanced topics. This topic is also classical mechanics [3–6], electrodynamics [7–11], optics particularly interesting because an understanding of Taylor [12,13], thermal physics [14–24], and quantum mechanics series expansion is, in fact, necessary to understand the [25–27]. Other studies have focused on students’ abilities physical implications of the final mathematical expression: to transfer knowledge and understanding between ProbðEÞ¼ZÀ1 exp½E=kT, where Z is the canonical domains: either between subfields of physics [28,29]or partition function. In this way, the mathematical tool across disciplines such as mathematics [30–33]. Most (Taylor series) is not only a means to an end but a vital studies have been conducted with introductory students, component to understanding the physics of the canonical but researchers have become increasingly interested in ensemble. intermediate and upper-division undergraduate popula- We have previously reported various student difficulties tions [8,11,18–22,25,34–44]. These populations have the using the Boltzmann factor in appropriate contexts as well potential to be particularly fascinating as they provide a as efforts to address these difficulties [21]. Our main effort glimpse of ‘‘journeyman physicists’’ on their way toward has been the creation of a tutorial that leads students expertise [45]. through a derivation of the Boltzmann factor and the We are presently involved in an ongoing multi- canonical partition function [50]. As part of the derivation, institutional investigation into students’ understanding of students need to produce a Taylor series expansion of topics related to thermal physics. We are primarily inter- entropy as a function of energy. ested in identifying specific difficulties that upper-division The Taylor series expansion of a function fðxÞ centered students have with topics in and related to thermal physics at a given value, x ¼ a, is a power series in which each and addressing these difficulties via guided-inquiry work- coefficient is related to a derivative of fðxÞ with respect to sheet activities (i.e., tutorials) as supplements to and/or x. The generic form of the Taylor series of fðxÞ centered replacements for lecture-based instruction [46]. This at x ¼ a is work has led to additional investigations of student use (and understanding) of various mathematical techniques df 1 d2f fðxÞ¼fðaÞþ ðx À aÞþ ðx À aÞ2 þÁÁ within thermal physics classes [17,22,43,44]. In this paper dx 2 dx2 we report preliminary results of an investigation into a a 1 X 1 dnf ¼ ðx À aÞn: n (1) n¼0 n! dx a Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- bution of this work must maintain attribution to the author(s) and The series is often truncated by choosing a finite upper the published article’s title, journal citation, and DOI. limit for the summation based on the specific context. 1554-9178=13=9(2)=020110(16) 020110-1 Published by the American Physical Society SMITH, THOMPSON, AND MOUNTCASTLE PHYS. REV. ST PHYS. EDUC. RES. 9, 020110 (2013) The Maclaurin series is a special case, generated by setting understanding. One difficulty observed throughout the a ¼ 0 in Eq. (1). interview was the student’s apparent lack of understanding The Taylor series expansion is used ubiquitously of the importance of the center point [in their case a ¼ 0 in throughout physics to help solve problems in a tractable Eq. (1)]. The student was prone to examine the behavior of way. It is the mathematical root of several well-known the Taylor series approximation as the argument of the formulas across physics, ranging from one-dimensional, function got further from, rather than closer to, the center constant acceleration kinematics, [61]. However, they did not ask the student to consider Taylor series in which the expansion point was nonzero xðtÞ¼x þ v ðt À t Þþ1aðt À t Þ2; 0 0 0 2 0 (2) (and would, therefore, appear explicitly in the series), so it is unclear what effect this complexity may have on student through electricity and magnetism and quantum understanding. mechanics. Given the previous work that has been done and our Other common uses of Taylor series expansions include focus on student understanding of thermal physics, we are numerical computations, evaluations of definite integrals interested in three main questions: and/or indeterminate limits, and approximations [52]. (1) How familiar are students with Taylor series expan- Approximations are particularly useful in physics at times sions (either in the context of thermal physics or in when a solution in its exact form is unnecessary or too other math or physics domains)? difficult to obtain; in situations where information is (2) To what extent can students graphically interpret a known about various derivatives of a function at a specific Taylor series? That is, when given a graph of a point, but nothing more is known about the function itself; function and a generic Taylor series expansion of or in situations in which one is investigating sufficiently that function, how well do students correctly con- small fluctuations about (or changes to) an average value. nect the algebraic elements of the series with their Here we are interested in students’ understanding of these graphical equivalents? uses in the context of expanding the entropy of a relatively (3) Can students generate the Taylor series expansion large thermodynamic system (a ‘‘thermal reservoir’’) as a of entropy as a function of energy that is used to function of the energy of a much smaller system with derive the Boltzmann factor? If not, what difficulties which it is in thermal equilibrium [53]. do they exhibit and how much guidance do they Relatively few studies exist within the mathematics need? education literature pertaining to students’ understanding and use of Taylor series, the majority of which focus Our research questions differ considerably from those on conceptions related to series convergence [54–62]. present in the mathematics literature. The commonality Moreover, most of these articles (similar to this one) across all three mathematical studies mentioned above present the results of studies in which the authors inves- is the comparison between graphical representations tigate Taylor series as part of a broader project and not as a main focus [59]. However, a dominant theme emerging [58,59,61]. Students (and experts in Martin’s study [59]) from many articles is the difficulty of synthesizing many were asked to compare the graph of a function to the graph previously learned calculus concepts to generate a robust of a truncated Taylor series approximating it and comment understanding of Taylor series. Martin examines the dif- on the validity of the approximation. We are interested in ferences between expert and novice approaches to tasks students’ abilities to connect the algebraic forms of indi- related to Taylor series convergence and reports that a vidual terms in the Taylor series to specific features of a ‘‘graphical understanding of Taylor series may be the graph of the function (e.g., the slope at a given point) rather single most notable effect separating novices from experts than their abilities to compare two graphs. We are also [63].’’ Habre also focuses on graphical representations and interested in student use of Taylor series in a specific found that visual reasoning of Taylor series convergence physical context, that of the canonical ensemble. This may be possible even for students with poor mathematical physical context is the motivation for our study and guides backgrounds and who may not be able to reason about our research design and methods. convergence analytically [58]. To answer our research questions as completely as pos- Recently, Champney and Kuo presented a case study of sible we collected data using written surveys, classroom a single sophomore physics major using graphical images observations, and student interviews. Our objective was to to reason about series truncation and the usefulness of the identify and document specific difficulties that students resulting truncation in both a purely mathematical context displayed while engaging with Taylor series expansions.