A cognitive systems analysis of engineering students' mathematical reasoning in signals and systems.

摘要

This study was designed to investigate student understanding of fundamental concepts in the engineering course, signals and systems. The aim of this study was threefold: (1) to identify faulty reasonings students invoke in their study of signals and systems; (2) to identify the reasoning resources that explain the origin of student faulty reasonings; and (3) to identify consistencies in students' invocation of reasoning resources across different signals and systems topics. Fifty-one undergraduate students majoring in aerospace engineering at the Massachusetts Institute of Technology participated in this study. Seven oral problems were designed to test student understanding of central topics in the study of continuous-time linear, time-invariant systems. Participants were divided into seven cohorts and each cohort was interviewed on a different problem. Interview transcriptions were analyzed based from a complex systems perspective to identify the knowledge elements of reasoning resources that characterize student reasoning in signals and systems.; Results relate to student understanding of four central signals and systems topics: linearity and time-invariance, convolution, the Laplace transform, and Bounded-Input-Bounded-Output stability. Findings indicate that student faulty reasonings can be ascribed to an intuitive knowledge base of reasoning resources that they inappropriately invoke in their response to a signals and systems problem, and that are cued by external features of the problem. Two reasoning resources---interval matching and symmetry---were identified as having special explanatory power in accounting for student faulty reasoning about different signals and systems topics. Also, a consistency in student reasoning was identified and ascribed to systematicity in the invocation of interval matching.; This study provides an understanding of the nature of naive knowledge in signals and systems that can be used to explain and predict student reasoning in signals and systems, and in other mathematics and engineering disciplines. In particular, the study suggests the need for further investigations on the role of interval matching and symmetry in naive reasoning in science and mathematics. The findings inform the development of instruction in signals and systems, and yield important implications for the continued development of a signals and systems concept inventory.