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Evaluation of Engineering & Mathematics Majors' Riemann Paper ID #14461 Evaluation of Engineering & Mathematics Majors’ Riemann Integral Defini- tion Knowledge by Using APOS Theory Dr. Emre Tokgoz, Quinnipiac University Emre Tokgoz is currently an Assistant Professor of Industrial Engineering at Quinnipiac University. He completed a Ph.D. in Mathematics and a Ph.D. in Industrial and Systems Engineering at the University of Oklahoma. His pedagogical research interest includes technology and calculus education of STEM majors. He worked on an IRB approved pedagogical study to observe undergraduate and graduate mathe- matics and engineering students’ calculus and technology knowledge in 2011. His other research interests include nonlinear optimization, financial engineering, facility allocation problem, vehicle routing prob- lem, solar energy systems, machine learning, system design, network analysis, inventory systems, and Riemannian geometry. c American Society for Engineering Education, 2016 Evaluation of Engineering & Mathematics Majors' Riemann Integral Definition Knowledge by Using APOS Theory In this study senior undergraduate and graduate mathematics and engineering students’ conceptual knowledge of Riemann’s definite integral definition is observed by using APOS (Action-Process- Object-Schema) theory. Seventeen participants of this study were either enrolled or recently completed (i.e. 1 week after the course completion) a Numerical Methods or Analysis course at a large Midwest university during a particular semester in the United States. Each participant was asked to complete a questionnaire consisting of calculus concept questions and interviewed for further investigation of the written responses to the questionnaire. The research question is designed to understand students’ ability to apply Riemann’s limit-sum definition to calculate the definite integral of a specific function. Qualitative (participants’ interview responses) and quantitative (statistics used after applying APOS theory) results are presented in this work by using the written questionnaire and video recorded interview responses. Participants are asked to calculate the definite integral of the function f(x) = x 2 on the interval [1, 2] by using the limit definition of Riemann integral. Missing conceptual knowledge of the participants in calculus are observed when they were incapable of determining the solution to the problem. Key Words: Riemann integral, functions, derivative, triad classification, APOS theory. Introduction Riemann integral is an important concept in calculus that is often used by engineering and mathematics majors during their undergraduate and graduate studies. Given a continuous function f on an interval [a, b], the Riemann integral (for definite integral) of f on the given interval can be determined by using the limit of sums: b n ∫ )( dxxf = ∑ (af + ∆xi )∆x. lim ∞→ a n i=1 This definition will be called the limit definition of Riemann integral throughout this work. This definition of Riemann integral is taught at early stages of calculus education, therefore Riemann sum approximation needs to be known by the Numerical Methods/Analysis students to be able to solve a question related to the Riemann integral’s limit definition. ___________________________________________________________________________ Special thanks to Drs. Deborah A. Trytten and Gizem S. Aydin for their valuable discussions and input during the preparation of the IRB approved form. This definition involves knowledge of concepts such as algebra, functions, limit, and summation rules. Integral calculations by using limit concept can be challenging and a mistake throughout the calculations can result in a misleading path towards finding the solution. In this work, the goal is to observe senior undergraduate and graduate mathematics and engineering students' ability to apply the limit definition of Riemann integral to calculate the integral of f(x)=x2 on the interval [1, 2]. Methodology Seventeen participants of this study are asked to complete a questionnaire with a follow up interview to explain their written questionnaire responses. The questionnaire questions covered concepts such as functions, limits of functions, function derivatives, Riemann integral, power series of functions, and programming preferences of the participants. The participants of this study are engineering and mathematics undergraduate and graduate students who were either enrolled or recently completed a numerical methods or analysis course in a particular semester at a large Midwest university in the United States. The participants completed a series of pre-requisite calculus courses in which the questionnaire concepts are covered. Post-interview results are designed to have a better understanding of the pre-interview (i.e. written) responses of the participants. The responses to the Riemann integral question are evaluated by considering the concepts that take place in the solution of the research question. The data collected in this work is expected to help understanding the missing conceptual knowledge of STEM majors during the application of the limit definition of Riemann integral. The collected written and interview response data is evaluated by using the Action-Process-Object-Schema (APOS) theory of Asiala, Brown, DeVries, Dubinsky, Mathews, and Thomas (1996) and the triad classification taking place in schema development. Relevant Literature In this section theories used for evaluation of the research question are explained. APOS theory and triad classification will be used to observe students’ ability to apply Riemann integral definition to the research question. The pedagogical literature on determining the Riemann integral of functions by using paper-pencil solution is limited. Asiala et al. (1996) pointed out the difficulty of writing a code to find the integral of functions and asked the participating students to write a code to approximate the integral by sampling points. Thompson (1994) states ...We must think of integration as the culmination of a limiting process, but at the same time consider that process, applied over an interval of variable length, as producing a correspondence... and invites to do research on determining the integral of functions: ...A curricular and instructional emphasis in algebra and pre-calculus on having students develop images of arithmetic operations in analytically-defined functions as operations on functions would seem to prepare them for a deeper understanding of this aspect of the calculus. At the same time, a conception of operations in expressions as operating on numbers and not on functions would seem to be an obstacle to understanding the derivative and integral as linear operators. These are empirically testable hypothesis; I would welcome research on them... Thompson (1994) observed senior mathematics undergraduate and graduate students’ weak rate of change concept knowledge resulted in weak understanding of the integration concept. The first derivative knowledge of the students appeared to be the major problem in answering the research question of Thompson (1994). Schema Development Clark, Cordero, Cottrill, Czarnocha, DeVries, St. John, Tolias, and Vidakovic (1997) used the stages of the triad classification; Intra, Inter and Trans to investigate how first year calculus students construct the concept of chain rule. Their attempt to use the APOS theory resulted in insufficiency by itself therefore they included the schema development idea of Piaget et al. (1989). Clark et al. (1997) used triad classification after realization of not being able to apply the APOS theory. Similar to Clark et al. (1997) APOS theory appears to be inappropriate for evaluating the research question in this work because students' responses didn't reflect a proper setting to apply the APOS theory; therefore, participating students' responses are analyzed by using the schema development idea. The Triad classification in this setting is as follows: • Intra Stage: Students classified in this category if they didn't know how to start solving the problem algebraically. This categorization includes a) Students who started solving the problem by writing the summation terms, b) Suggested to solve the problem by approximating it (i.e. Choose a particular value of n.) • Inter Stage: Students in this stage knew how to solve the problem but made a mistake either during the summation or limit calculations. • Trans Stage: Students were able to apply the definition and successfully found the correct answer. The following terms are excluded from the triad classification because this information is provided to the students. n (nn + )1 n (nn + 2)(1 n + )1 ∑i = and ∑i2 = i=1 2 i=1 6 Action-Process-Object-Schema (APOS) Theory By relying on Piaget`s study of functions in 1977 (Piaget et al. 1977), Action-Process-Object idea in mathematics education for the undergraduate curriculum was initiated by Breidenbach, Dubinsky, Hawks and Nichols in 1992 who studied students’ conceptual view of the function in their research. In 1996, Asiala, et al. applied APOS theory to understand students’ function knowledge and explained this theory as the combined knowledge of a student in a specific subject based on Piaget’s philosophy. Dubinsky and McDonald (2001) explained the components of the APOS theory as follows: An action is a transformation of objects perceived by the individual as essentially external and as requiring, either explicitly or from memory, step-by-step instructions on how to perform the operation... When an action is repeated and the
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