IEEE TRANSACTIONS ON EDUCATION, VOL. 35, NO. 4, NOVEMBER 1992 383

Signals and Linear Systems: A Teaching Approach Based on Learning Styles Concepts

Duane C. Hanselman, Senior Member, IEEE

Abstract-The exclusive teaching of continuous-time concepts before discrete-time concepts in undergraduate signals and linear systems is advocated when circuit analysis is a prerequisite. The reasons for this approach are based on learning styles concepts of inductive/deductive organization, sequentiaVgloba1 understanding, and sensory/intuitive perception. This approach is promoted because it: 1) supports inductive progression of material; 2) addresses the needs of students who learn through sensing; and 3) supports both global and sequential learners. Arguments are made that the presentation of continuous-time concepts first promotes the deepest understanding of the material in the largest number of students.

I. INTRODUCTION VER since discrete-time (DT) concepts have found an E application in engineering, there has been an ongoing

debate as to how undergraduate signals and linear systems should be taught. Some texts promote the exclusive teaching of continuous-time (CT) concepts first, followed by DT con- cepts [1]-[6]. Other texts present alternating chapters of each [7]-[ lo], while still others present both concepts concurrently [11]-[16], and one promotes the teaching of DT concepts before CT concepts [17]. The arguments for each of these approaches are many and varied, as is discovered by reading the prefaces of these texts or by reading a recent article by Strum and Kirk [HI.

In this work, it is argued that exclusive teaching of CT concepts followed by DT concepts is the best approach for most students. This opinion, supported by research on learn- ing and teaching styles [19]-[21], is based on viewing the material from the student’s perspective. This is in contrast to approaches that commonly center around the ideas of less rigorous mathematics and/or the importance of the parallels and similarities between CT and DT concepts [7]-[18].

A multitude of arguments can be made for this or any approach to the teaching of signals and linear systems. Rather than attempt an exhaustive elaboration of all the reasons for the approach promoted here, this work illustrates only those that are clearly supported by learning and teaching styles concepts. Furthermore, in agreement with most texts, this work assumes that one or more circuit analysis courses precede the introduction of signals and linear systems material.

Manuscript received January 1990; revised March 1991. The author is with the Department of Electrical and Computer Engineering,

IEEE Log Number 9203268. University of Maine, Orono, ME 04469-0107.

11. REASON ONE: INDUCTIVE PROGRESSION OF MATERIAL The foremost reason for teaching CT concepts first is

that CT concepts are a natural generalization of the specific material students have spent one to two semesters studying in circuit analysis. This movement from specific to general concepts supports the natural human learning style of induction [19]. Inductive learning is the process that starts from specifics (circuit analysis) and progresses to generalities (CT system concepts). When CT concepts are taught first and examples from circuit analysis are used to support concepts as they come up, students learn new material that is connected to some- thing they already understand. These connections reinforce the material, stimulate motivation, and foster student confidence.

Contrary to inductive learning is deductive teaching, which is the natural teaching style [19]. It is usually easier to present material from the top down rather than from the bottom up. That is, it is easier to start with some general concept or principle, then see how it applies to the real world. For example, the Fourier, Laplace, and Z transforms are commonly introduced in texts by simply giving the defining equations, moving on the transform properties, and then finally to applications. In spite of the ease with which material can be presented in this manner, deductive teaching hinders inductive learning. Of the ways in which signals and linear systems can be taught, only a CT first approach fully supports inductive teaching and therefore inductive learning. While DT concepts themselves can be taught inductively, doing so disregards the wealth of knowledge the student has already acquired about CT concepts through circuit analysis. The exclusive teaching of CT concepts first supports inductive learning that encompasses material learned in previous courses. This approach provides a bridge that helps the student more fully take in and process the material presented. A DT first approach offers no analogous bridge.

The frequency domain is perhaps the most important con- cept to be introduced in signals and linear systems. Possibly because of how powerful this concept is, students often have difficulty gaining a productive conceptual understanding of the Fourier and Laplace transforms. This difficulty is minimized in the CT first approach by using examples from circuit analysis, and relating phasor analysis to the Fourier transform and complex frequency analysis to the Laplace transform. With this basis, the subsequent teaching of DT frequency domain concepts is more readily understood by presenting the DT Fourier and Z transforms as inductive extensions of the Fourier and Laplace transforms, respectively. By doing so, the important DT concepts of aliasing and the limita-

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384 IEEE TRANSACTIONS ON EDUCATION, VOL. 35, NO. 4, NOVEMBER 1992

tions involved in signal reconstruction can be more readily understood.

A DT first approach does not foster this conceptual under- standing of the Fourier and Laplace transforms. When DT concepts are taught first, the Z transform and DT Fourier transform are commonly introduced deductively in an abstract manner, and any connection to the student’s past circuit analysis experience or any other experience is difficult, if not impossible, to communicate effectively. Moreover, in a DT first approach, it is difficult to exploit the fact that most DT signals encountered in engineering result from the sampling of CT signals. When CT concepts are taught first, this fact can be used to relate important DT concepts back to analogous CT concepts, and thus provide a bridge that can help the student gain a more productive conceptual understanding of DT concepts.

111. REASON TWO: SUPPORT OF SENSORS

The second reason for a CT first approach is that it supports the sensing learning style, which is exhibited by most engi- neering students [20]. Sensing refers to the way in which one perceives the world. Sensors perceive through their senses- by observing, by gathering data, by experimenting. The other means of perception is through intuition. Intuitors perceive through their unconscious-by speculation, by imagination.

The introductory laboratory course that is typically coreq- uisite with circuit analysis facilitates the needs of sensors. In the laboratory, sensors are nourished. They touch components, they build circuits, they observe first-hand the very concepts being taught in circuit analysis. Examples of CT concepts that are readily reinforced experimentally include the frequency content of signals, the frequency response of a system, the concept of superposition, and the interrelationships among system representations. To the extent that this experience carries over to the classroom setting, a CT first approach supports students who perceive through sensing.

DT concepts are much more difficult to demonstrate with hands-on examples. The difference equations describing a savings account or the Newton algorithm for finding the square root of a number are typical first examples of DT systems. While these and other DT systems can be simulated on a computer, they lack the concreteness of a physical circuit containing resistors, capacitors, and inductors. This lack of concreteness forms a possible learning obstacle for sensors that is difficult to overcome. Without the feedback provided by experimentation, the results of simulations alone are difficult for most sensors to accept without reservations. By teaching CT concepts first, these reservations can be minimized by relating DT concepts back to analogous CT concepts that were reinforced by experimentation.

Iv. REASON THREE: SUPPORT OF SEQUENTIAL AND GLOBAL LEARNERS

The last reason for a CT first approach is that it best supports both global and sequential learning styles. The terms “global” and “sequential” refer to how students internalize concepts [ 191. Sequential learners are comfortable with the

sequential nature of college teaching; they readily understand material presented in a logical and orderly fashion. Global learners, on the other hand, often struggle with the material until a complete conceptual picture appears to them. After this picture appears, global learners commonly find even involved problems relatively straightforward to solve.

A. Sequential Learners

Any approach to the teaching of signals and linear systems can work well for sequential learners, provided concepts are organized logically. A common difficulty for sequential learners, however, is that it is often only with substantial hindsight that they form a general picture of the concepts presented. For example, a sequential learner may be very good at performing the common steps of finding transforms, manipulating algebra, and finding partial fraction expansion coefficients, but will not understand the underlying process except that it leads to the right answer if performed diligently. Because of this difficulty, following an exclusive CT or DT first approach is helpful to sequential learners. Mixing CT and DT approaches together unnecessarily adds to the complexity of the material for students being introduced to it for the first time. Both sequential and global learners most easily internalize and put into perspective the similarities, differences, and subtleties between CT and DT concepts only after a thorough understanding of one approach or the other.

Of the CT and DT first approaches, the DT first ap- proach is often promoted as the better approach because the mathematical complexity and abstraction involved are less demanding than those required to learn CT concepts [18]. Missing from this argument is the fact that a student’s depth of understanding is not linked solely to the ease with which material can be manipulated or described. Students, especially sequential learners, may find it easier to perform typical DT mathematical operations and therefore make fewer algebra errors and perform better on tests. At the same time, however, they may not develop as deep an understanding as they would with a CT first approach because of Reasons One and Two.

While greater abstraction is required to understand CT con- cepts, students are usually familiar with this abstraction after taking two years of college calculus, where the concepts of limits, differentiation, and integration are rigorously covered. This, however, is not true for the abstraction required to un- derstand DT concepts such as sequences, difference equations, and the convergence of series, which are given little if any time in calculus. Although some use of these concepts may be used to introduce integration and differentiation, seldom are DT concepts covered for the sake of their importance alone. As a result, the CT first approach is a better approach for sequential learners.

B. Global Learners

Global learners often present the most difficult teaching challenge since they are commonly lost until an overall con- ceptual picture is apparent to them. Once again, an exclusive CT or DT first approach is helpful in this situation. By

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385 HANSELMAN: SIGNALS AND LINEAR SYSTEMS

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completing CT or DT concepts first before moving on to the other, global learners are exposed to a complete set of concepts somewhere near the middle of a semester, thereby giving them a natural place to synergize the various concepts covered. Since global learners can do well with either approach, it makes sense to follow the CT first approach because of Reasons One and Two discussed earlier.

In addition, the use of diagrams illustrating how various course concepts are related is particularly helpful to global learners. An example of a global diagram that has proven helpful to my students is shown in Fig. 1. By distributing diagrams such as this at the beginning of the semester, global learners begin to establish their global perspective immediately, thereby minimizing the amount of time spent struggling to form that perspective.

V. CONCLUSION There are many valid arguments for all popular approaches

to the teaching of signals and linear systems. As with most things, no one approach is perfect. The best that can be hoped for is one approach that will foster the deepest understanding of the material in the largest number of students. The applica- tion of research in learning and teaching styles indicates that the complete presentation of CT concepts first can achieve this goal, especially if the ensuing coverage of DT concepts is related back to analogous CT concepts.

REFERENCES

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386 IEEE TRANSACTIONS ON EDUCATION, VOL. 35, NO. 4, NOVEMBER 1992

[2] A. D. Poularikas and S. Seely, Signals and Systems, 2nd Ed. Boston, M A : PWS, 1991.

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Duane C. Hanselman (S’75-M’7&M’SS-SM’91), photograph and biography not available at the time of publication.

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