Students’ Understanding Of Sequence And Series As Applied In Electrical Engineering

Across all engineering fields, upper-level engineering courses often build upon a strong mathematical foundation. As such, a critical component of understanding how students learn engineering concepts is studying how students apply their mathematical background to the engineering domain. Studying how students apply mathematical knowledge in engineering courses allows us to identify challenges, pitfalls, and common misconceptions. Through an NSFfunded study, we are addressing the broad goal of developing a better understanding of how students transfer mathematical knowledge to concepts in engineering. In this component of the study, we focus on electrical engineering students’ application of concepts in sequence and series to a junior-level signals and systems course. A strong understanding of sequence and series is fundamental to the study of discrete-time signals and systems. A discrete-time system response takes the form of a sequence, and the characteristics of this sequence and of the associated series dictate system properties such as causality, stability, memory, and finite vs. infinite impulse response. Additionally, sequence and series are the basis of discrete-time Fourier series and transforms, which provide the primary tool for frequency-domain signals and systems analysis. We have selected and analyzed five group problems based on their connection to significant concepts in sequence and series. The results of our analysis indicate that students encounter challenges in associating mathematical expressions with physical realities, providing logical justifications for their conclusions, and manipulating multiple representations of series. These results have direct application to instructional design, since the design of assessment items and problems can be informed by students' interpretations of items. Background and Motivation All facets of engineering require students to transfer mathematical knowledge from introductory mathematics courses into engineering courses. Electrical engineering is a mathematically intensive discipline, and the subfield of signals and systems has a particularly strong mathematical basis including applications for Fourier analysis, Laplace transforms and advanced calculus. In this work, we focus on students in an introductory discrete-time signals and systems course. As part of a larger NSF-funded study through which we aim to better understand how students transfer mathematical knowledge to concepts in engineering, we analyze student work in the discrete-time signals and systems course in an effort to characterize challenges students face in transferring knowledge of sequence and series to the study of discrete-time signals and systems. While the mathematics education community has studied students' understanding of limits and of the convergence and divergence of series, these studies have not addressed the link to engineering applications. In addition, there are few studies about students’ understanding of periodicity 1 that is foundational to understanding signals and systems. In our broader work, we aim to study the depth of both the procedural and conceptual elements of students' understanding as it applies to mathematics in electrical engineering. Procedural knowledge has been described as the understanding of rules and algorithms for mathematics P ge 14092.2 where conceptual knowledge describes understanding of the relationships and connections between mathematical ideas or skills. 2,3 In this paper, we focus on the application of sequence and series in the context of discrete-time signals and systems. At a procedural level, determination of series limits arises in computation of the system energy, Fourier transform, and convolution sum. At a conceptual level, the students need to understand the connections between the mathematical techniques and the associated physical phenomena they represent in addition to connecting mathematical ideas broadly. Given that little research has been performed studying students’ use of sequence and series in an engineering context, we focus our work on the following research question: What are the procedural and conceptual difficulties students encounter in applying sequence and series knowledge to signals and systems content? Through qualitative analysis of students' in-class work, we have identified two areas of interest and drawn conclusions addressing students' approach, challenges, and misconceptions.