Problems in Learning

The three major content areas in the Signals and Systems courses are Signal properties, Fourier analysis, and system analysis (Laplace transform, and convolution) (Evans, Karam, West, & McClellan, 1993; Munson & Jones, 1999; Wage, Buck, Welch, & Wright, 2002). Some studies have suggested that the abstract nature and disconnection of these concepts from daily life could make them difficult to understand. Additionally, these concepts and their applications in the physical world are described through mathematics, which requires students to combine advanced mathematical concepts with their perception of physical systems (Nasr, Hall, & Garik, 2005). Consequently, a large part of this course deals with abstract mathematical constructs. A few studies have contended that these abstract mathematical constructs are difficult to visualize and comprehend (Shaffer, Hamaker, & Picone, 1998; Nasr, Hall, & Garik, 2005, 2007; Tsakalis et al., 2011). For conceptual understanding of any subject matter, students often need to know the usefulness of what they learn and want to be sure that the information they acquire is useful in daily life (Çetin, 2004). Nasr, Hall, and Garik (2005) have argued that the disjointed-from-everyday-life nature of concepts in Signals and Systems course content makes this course different from other courses in engineering, like Electronics and Circuit Analysis.

In addition, for better understanding of this course, sophisticated mathematical skills rather than just knowing formulas and carrying out fixed procedures to solve the problems are deemed necessary in most of the studies. There is sufficient anecdotal support that, these days, engineering students either lack the mathematical proficiency required to solve any problem in physics or engineering, or fail to apply their knowledge

of the mathematics to any physics or engineering context (Nasr, Hall, & Garik, 2005). Bruner (1962) argues that students can find difficulty in understanding mathematical concepts if they cannot understand them intuitively or be able to translate intuitive ideas into mathematics. Betz (1978) suggests that math anxiety in engineering students can influence their understanding of math-influenced engineering concepts. Moreover, in the university level engineering education, the gap between application-oriented expectations of students and theory-focused lectures is claimed to have a considerable effect on the motivation of students (Munz, Schumm, Wiesebrock, & Allgower, 2007).

There is a limited amount of work done in conceptual understanding of topics taught in Signals and Systems courses. I have divided this literature into three categories based on the particular course content investigated in each study. The categories are: i) Linear-time-invariant system analysis and convolution and, ii) mathematical concepts and thinking. The difficult concepts discussed in one category may not be uniquely attributed to the problems in learning that particular concept, but rather a combination of more than one category of difficult concepts.

2.1.2.1 Linear-Time-Invariant System Analysis and Convolution

The difficulties in learning system analysis and convolution identified in the previous literature are as follows:

i. Nasr, Hall, and Garik (2007) used the concept of DiSessa's p-prims (fundamental

knowledge structure) and coordination classes (large and complex knowledge structures composed of combination of p-prims) to explain the faulty cognitive resources underlying the mathematical reasonings of students attempting to learn

continuous time (CT) linear, time-invariant (LTI) electric circuits. They were specifically interested in finding reasons for conceptual problems in the context of LTI circuits, as the students in aerospace engineering program at MIT are taught Signals and Systems courses in this context. For this purpose, they interviewed 51 students enrolled in Signals and Systems course in the Department of Aeronautics at Massachusetts Institute of Technology, in 2002-2003. Their results suggested that the faulty reasonings of the students when engaging with topics related to superposition, convolution, and Laplace transform are mostly because of the inappropriate invocation of the interval matching readout strategy. In other words, they argued that the students employ the interval matching strategy in problems where its use is inappropriate. Readout strategies, as presented by DiSessa (1983, 2002) are part of a large complex knowledge system called coordination class, which is an integrated model of numerous smaller knowledge structures that result in an expert-like understanding of a certain scientific concept. Readout strategies constitute the ways in which a particular concept or a situation is observed or understood (DiSessa, 1983, 2002).

ii. In a follow-up study, Nasr, Hall, and Garik (2009) investigated naive reasoning of

aeronautical engineering students related to the concepts of linearity, time-invariance, and convolution, to provide a foundation for designing effective instructional materials for Signals and Systems courses. They suggested that their findings would help in designing a better pedagogy for this course as the knowledge of students' skills and pre-conceptions is necessary for effective pedagogical design (NBPTS, 2005). In addition to interval matching, symmetry invocation was also claimed to be a

commonly employed faulty naive reasoning. Symmetry invocation as defined by Nasr, Hall, and Garik (2009) is students' undue bias to apply symmetry properties to analyze all the systems including non-symmetric systems.

iii. Wage, Buck, and Hjalmarson (2006a) conducted semi-structured interviews with nine

students and they argued that in some instances the "connotations" of the daily use of the term "filter" limits the student's understanding of the concept of scaling factor in the concept of "filters as systems" taught in this course. According to them, students face difficulties in connecting the concept of a scaling factor to a filter as their perception of filters adheres to the everyday use of filter such as air filter, coffee filter, or spam filter. The difficulty in learning a new concept or term about which the students have prior familiarity in a different meaning is suggested in other studies and contexts in science education as well. These include Herman, Kaczmarczyk, Loui, and Zilles's (2008) study on computer science and computer engineering students' misconceptions in logic design concepts, and DiSessa, Gillespie, and Esterly's (2004) study on the K-12 students' concepts of force.

iv. Wage, Buck, and Wright (2004) have argued that students face difficulty in relating

the concepts of impulse response and complex frequency analysis (Laplace transform) to analyze a real system (Wage, Buck, & Wright, 2004).

2.1.2.2 Mathematical Concepts and Thinking

1. Wage, Buck, and Wright (2004) have used the Signals and Systems concept inventory (SSCI) (to be discussed in detail in section 2.2.1) to illustrate that a sound mathematical knowledge is helpful in understanding the concepts in Signals and

Systems course content. Although the reliability of an SSCI as an assessment instrument is not yet established, ever since its initial design in 2001 it has been widely used in over twelve schools for research in understanding problems encountered by students taking Signals and Systems courses (Wage, Buck, & Wright, 2004). Without the information about the reliability of an instrument, the consistency of the results of any research using that particular instrument remains questionable (Gilbert, 1989). However, in an interest to include all the discussion in the literature about possible obstacles in conceptual understanding of Signals and Systems course content, I am presenting the results of a study conducted using an SSCI (Wage, Buck, & Wright, 2004) that claims that the mathematical understanding of students contribute towards conceptual learning of concepts covered in Signals and Systems courses.

a) The study claimed a positive correlation of the gain in SSCI scores of the students with their grades in some prerequisite courses (calculus, differential equations, and circuits) within the curriculum of electrical and computer engineering.

b) Wage, Buck, and Wright (2004) argued, based on students' responses in SSCI pretests and posttests, about the presence of three persistent misconceptions in students. Firstly, they suggested that students incorrectly believed that the real impulse response corresponds only to systems with real poles and zeros. Secondly, they claimed that the students incorrectly thought that the multiplication in the time domain corresponds to multiplication in the frequency domain as well. Thirdly, they asserted that the students falsely believed that a frequency response with two resonant peaks have one pole in the left-half plane

and one in the right-half plane, that is, they mistakenly reverse the roles of the real and imaginary axes of the pole-zero plot.

c) A few studies have suggested that the students face difficulties in understanding the need and importance of transforms, which further confuse them to connect alternate shapes of the same signal in different domains (Wage, Buck, & Wright, 2004; Buck & Wage, 2005; Wage, Buck, & Hjalmarson, 2006a).

2. Nasr, Hall, and Garik (2009) have argued that the students find difficulties in doing convolution by graphical method. They suggest that while performing convolution by graphical method, students demonstrate problems in solving long integrals, multiplying two signals, putting appropriate limits, defining signals piece-wise, and flipping and shifting the signal. They further claimed that the difficulty in doing convolution was more significant when one of the two functions being convolved had any of these characteristics: (i) did not begin at t=0, (ii) was piece-wise, (iii) was non- causal, and (iv) had negative values over a certain interval of time (Nasr, Hall, & Garik, 2009).

3. Nelson, Hjalmarson, and Wage (2011) used two types of in-class assessments: group exercises and individual exams to observe students' understandings of Signals and Systems course content. They claimed that the mathematical areas where students exhibited significant gaps in their knowledge were i) definitions and/or evaluation of the conditions of causality and stability of a system, ii) mathematical representation of signals and systems as either a function or a graph, iii) different types of independent and dependent variables together in a function, and v) impulse response.