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Conceptual Power Series Knowledge of STEM Majors

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Conceptual Power Series Knowledge of STEM Majors Paper ID #21246 Conceptual Power Series Knowledge of STEM Majors Dr. Emre Tokgoz, Quinnipiac University Emre Tokgoz is currently the Director and an Assistant Professor of Industrial Engineering at Quinnipiac University. He completed a Ph.D. in Mathematics and another Ph.D. in Industrial and Systems Engineer- ing at the University of Oklahoma. His pedagogical research interest includes technology and calculus education of STEM majors. He worked on several IRB approved pedagogical studies to observe under- graduate and graduate mathematics and engineering students’ calculus and technology knowledge since 2011. His other research interests include nonlinear optimization, financial engineering, facility alloca- tion problem, vehicle routing problem, solar energy systems, machine learning, system design, network analysis, inventory systems, and Riemannian geometry. c American Society for Engineering Education, 2018 Mathematics & Engineering Majors’ Conceptual Cognition of Power Series Emre Tokgöz [email protected] Industrial Engineering, School of Engineering, Quinnipiac University, Hamden, CT, 08518 Taylor series expansion of functions has important applications in engineering, mathematics, physics, and computer science; therefore observing responses of graduate and senior undergraduate students to Taylor series questions appears to be the initial step for understanding students’ conceptual cognitive reasoning. These observations help to determine and develop a successful teaching methodology after weaknesses of the students are investigated. Pedagogical research on understanding mathematics and conceptual knowledge of physics majors’ power series was conducted in various studies ([1-10]); however, to the best of our knowledge, Taylor series knowledge of engineering majors was not investigated prior to this study. In this work, the ability of graduate and senior undergraduate engineering and mathematics majors responding to a set of power series questions are investigated. Written questionnaire responses of participating students and the follow-up interviews to have a better understanding of these written responses are analyzed qualitatively and quantitatively by using the Action- Process-Object-Schema (APOS) theory. Samples of the student responses to the written questionnaire and the transcribed interview data are displayed throughout this work. Written and oral interview data collected from participating STEM majors indicated a well-established knowledge of approximation, a poor knowledge of the meaning of center concept that takes place in the Taylor series expansion of functions, and a well-established knowledge of infinity as a part of infinite series concept. Key Words: Maclaurin series, Taylor series, power series, finite series, infinite series, APOS theory, schema Introduction Taylor series representation of functions attracted many mathematics and physics pedagogy researchers. In mathematics education, comparison of geometric (graphical) and algebraic representations of functions as well as comparison of approximate and exact representations of functions are considered for investigating students’ Taylor series knowledge. Series expansion of functions are also widely used in engineering and computer science applications. In this work, seventeen graduate and senior undergraduate students’ Taylor series knowledge will be investigated based on their responses to four Taylor series questions. The participants of this study completed a calculus course that covered power series of functions in addition to the completion of a Numerical Methods course that is offered either by the School of Computer Science or the Department of Mathematics at a large Midwest U.S. university. The responses of the participants are evaluated by using the Action-Process-Object- Schema (APOS) theory. Participants are given approximately 80 minutes to respond to the written questionnaire Special thanks to Drs. Deborah Tytten and Gizem Aydin for their support during the preparation of the IRB forms and structuring the study. that consisted of calculus questions and these participants are also interviewed for about 40 minutes. The calculus questions given in this questionnaire included function, derivative, integral, and power series questions. Applications of Taylor series expansions include approximations, numerical computations, evaluations of definite integrals, and indeterminate limits ([5]). Assuming f is a smooth (i.e. C∞) function, the Taylor series expansion of f centered at x=a is mathematically represented by the series (1) = + | − The Taylor series given in (1) is called the Maclaurin series of the function in the case when a=0. The truncation of the Taylor series by using a finite upper limit is the main idea of the smooth function approximation. Approximation of differentiable functions using Taylor series requires a good understanding of the difference between several terms of the series and the series itself. Students’ ability to explain the difference between the Taylor series approximation and the exact representation of smooth functions is the key point of algebraic understanding of the concept. This idea is investigated in physics and mathematics education by several researchers. The research on Taylor series expansion of functions in mathematics education is limited and mainly focuses on series convergence of functions ([6–14]). Most of these studies’ common theme is pertaining the calculus and series knowledge with a bigger scope than investigating the Taylor series knowledge of students. Students’ visual and non-visual reasoning of sequence and series convergences are investigated in [8] and [9] respectively. Relating graphical representation of Taylor series to the convergence of the series is observed to be possible even for students with poor mathematical backgrounds who may not have the ability to explain the convergence analytically ([9]). In [10] the geometric (i.e. graphical representation) understanding of the Taylor series is observed to be the distinguishing factor between the experts and novice approaches for the tasks related to the Taylor series convergence. This difference between the two groups is observed by investigating their knowledge on the truncated and actual Taylor series expansion of several functions. Another pedagogical research on infinite series is implemented in [15] in which sequence knowledge of students is also observed. Hardly any pedagogical research on engineering students’ conceptual power series knowledge is investigated prior to this study. The research on Taylor series expansion of functions in physics education is also limited and emphasized mainly on applications. In [4] students’ difficulties to connect the algebraic forms of individual terms in the Taylor series to the specific features of a graph of the function (e.g., the slope at a given point) are investigated and documented on the contrary to the pedagogical research implemented on Taylor series in mathematics education. The main aim of the research is to derive the Boltzmann factor using a Taylor series expansion of entropy. The findings in [4] also indicated many participating students’ familiarity with the Taylor series but their difficulty in its fluent use during the physical applications. Another result observed in [4] is the benefit of a pre-tutorial homework assignment for refreshing the memory of what exactly a Taylor series is and its use for modeling physical contexts. Action, Process, Object, and Schema (APOS) theory is a constructivist theory that deals with measuring learners’ successes and failures based on the mathematical instructions and mathematical backgrounds. One of the central ideas in APOS theory is to understand learners’ cognitive progress after a set of instructions and classify them to understand the success of the instructional design. An action is performing an activity based on external instructions. Cognitive development starts with the elementary “applications” as the actions that turn into processes when the learner manages to operate correlated variations. Process turns into object when new actions can be applied with the existing process. Schema is “more or less coherent collection of objects along with actions which the subject can perform on them” ([3]). Pedagogical research on APOS theory applications of functions’ series expansion is limited in the literature ([1]). The only research overlap on APOS theory and infinite series concepts’ is the report that describes a three-semester calculus course developed at Purdue University with support from the U.S. National Science Foundation. The design of the course was based on APOS theory and involved students writing and running programs in a mathematical programming language and making calculations on the computer using a symbolic computing system. The pedagogical strategy consisted of cooperative student group work in a computer lab where they are expected to develop mental structures by using the mathematics instructions. Students are confronted with problem situations designed to get them use the mental structures developed in the computer lab to construct their understanding of mathematical concepts in small groups. Assignments are intended to provide practice with standard calculus problems and reinforce their understandings ([2]). The literature on engineering and mathematics majors’ function series knowledge investigation prior to this study is very limited. In this work, the following four research questions are asked to the
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