Paper ID #25532

A Course in Differential Equations, Modeling, and Simulation for Engineer-ing Students

Prof. Scott W. Campbell, University of South Florida

Dr. Scott Campbell has been on the faculty of the Department of Chemical & Biomedical Engineering atthe University of South Florida since 1986. He currently serves as the department undergraduate advisor.Scott was a co-PI on an NSF STEP grant for the reform of the Engineering Calculus sequence at USF. Thisgrant required him to build relationships with engineering faculty of other departments and also facultyfrom the College of Arts and Sciences. Over the course of this grant, he advised over 500 individualcalculus students on their course projects. He was given an Outstanding Advising Award by USF and hasbeen the recipient of numerous teaching awards at the department, college, university (Jerome KrivanekDistinguished Teaching Award) and state (TIP award) levels. Scott is also a co-PI for a Helios-fundedMiddle School Residency Program for Science and Math (for which he teaches the capstone course) andis on the leadership committee for an NSF IUSE grant to transform STEM Education at USF. His researchis in the areas of solution thermodynamics and environmental monitoring and modeling.

Prof. Carlos A. Smith PhD, University of South Florida

Carlos A. Smith is a Professor Emeritus in the Department of Chemical & Biomedical Engineering at theUniversity of South Florida. His interests are in Process Control, Process Engineering, and EngineeringEducation. He is co-author of three editions of ”Principles and Practice of Automatic Process Control,”John Wiley, and two editions of ”A First Course ion Differential Equations, Modeling and Simulation,”CRC.

Silvia M. Calderon, Universidad de Los Andes, Venezuela

Professor, School of Chemical Engineering, University of Los Andes, Venezuela. Senior Research Fellow,University of Oulu, Finland. Ph.D in Chemical Engineering, University of South Florida, USA. M.Sc. inApplied Engineering Mathematics, University of Los Andes Venezuela.

c©American Society for Engineering Education, 2019

A Course in Differential Equations, Modeling and

Simulation for Engineering Students

Introduction

A course in differential equations generally is taken at a critical point in engineering

curricula – where a turn is made away from basic math and science courses towards

courses in which basic skills and knowledge are synthesized and applied. This raises the

question of whether the course should be a mathematics course, an engineering course, or

a hybrid. It has been argued [1], with supporting results, that the teaching of differential

equations through the modeling of physical and chemical phenomena is effective because

it allows students to overcome the cognitive obstacles more easily. On the other hand, it

has been reported [2] that there is no difference between students who took a math-based

differential equations course and those who took an engineering-based one, in regard to

their preparation for differential equations content in a later course.

The authors held that mathematics for engineering students is best learned in the context

of its applications, in line with articles [3], [4], [5] that have described different

approaches to increasing the engineering relevance of mathematics courses taken by

engineering students. With this belief, we developed a course Modeling and Analysis of

Engineering Systems that covers analytical solutions of ordinary differential equations,

how ordinary differential equations arise through modeling of physical systems, and how

ordinary differential equations are solved numerically, if needed. Our approach was to

motivate students to learn how to solve differential equations because interesting

practical problems required it.

Modeling and Analysis of Engineering Systems is offered at our university by the

College of Engineering and is an alternative to the standard Differential Equations course

offered by the Department of Mathematics. Engineering students may select either

course to meet degree requirements and all seven engineering disciplines within the

College of Engineering are represented in the student populations of both courses.

Below, we discuss the course content and structure, compare the content to that of the

Differential Equations course, and provide results of several assessments that compare

the two courses.

Content of the Modeling and Analysis Course

A little over half of class time is spent covering analytical solutions to ordinary

differential equations, with the remaining time spent on modeling physical systems and

obtaining numerical solutions through computer simulation. Specific topics covered are:

Analytical solutions of ordinary differential equations (approximately 8 weeks)

The objective for this component of the course is for students to learn analytical solution

methods for ordinary differential equations (ODEs) that commonly arise in engineering.

Methods covered include anti-differentiation, separation of variables, general solution to

first order linear ODEs, characteristic equations, undetermined coefficients, and Laplace

transforms. The significance of the roots of the characteristic equation for second order

equations to the qualitative response of the system (monotonic or oscillatory response,

and stable or unstable response) is stressed, as this is an important issue in engineering

and physics. When covering Laplace transforms we stress the concept of the transfer

function and relate it to the characteristic equation, and show how Laplace transforms are

useful in solving multiple coupled differential equations. Because most models

developed are in the form of first or second order differential equations, we purposely

analyze the response of these equations to typical forcing functions, and discuss terms

such as time constant, characteristic time, damping ratio, system gain, etc.

Modeling of physical systems (approximately 5.5 weeks)

The main objective of the modeling component of the course is to demonstrate how

mathematical modeling of a variety of physical systems of interest to engineers results in

differential equations. Applications include motion of bodies in a gravitational field

(with and without air resistance), translational and rotational mechanical systems, fluid

systems, thermal systems, and electrical systems. We continuously stress that the starting

point in modeling is the basic physical law that applies to the system. These include

Newton’s 2nd law for bodies on a gravitational field and for mechanical systems,

conservation of mass for fluid systems, conservation of energy for thermal systems and

Kirchhoff’s voltage and current laws for electrical systems.

Model development proceeds further by expressing terms in the basic law using

phenomenological relationships such as those for air resistance, for springs, damping, and

friction in mechanical systems, for valves in fluid systems, for convective heat transfer in

thermal systems, and for current-voltage relations of resistors, capacitors, and inductors

in electrical systems. We stress the relation of the physical concepts of natural response

and forced response to the respective mathematical concepts of homogeneous and

nonhomogeneous equations.

Computer solutions of models (approximately 1.5 weeks)

The objective of this component is for students to use commercial software (in our case,

Simulink) to solve single or coupled ordinary differential equations. Because Simulink (a

MATLAB tool kit) is very intuitive, students learn how to use it quite quickly, and after

three class meetings (and homework) they feel comfortable with it. In addition, Euler’s

method for the numerical solution of first and second order equations is covered in

lecture so that students have a feel for what the software is doing and understand why the

selection of step size is important.

Course Structure

Both authors teach the course but use different approaches. The text [6] used for the

course covers analytical solutions in chapters 1-5, applications in chapters 6-10 and

numerical solutions/computer simulation in chapter 11. One of us covers analytical

solutions first, followed by modeling applications for which either analytical solution or

computer simulation is employed, as appropriate. The other tends to cover analytical

solutions, modeling, and simulation simultaneously, with increasing complexity in the

differential equations as the semester proceeds. In either case, we seek for the students to

gain an understanding of how the three major topics relate to one another in an integrated

form that is greater than the sum of its parts.

We illustrate this last point with an example from the thermal systems component. A

metal pellet with mass M, heat capacity C and surface area A at an initial temperature

T(0) is dropped at t=0 into a small cooling bath containing a fluid with mass Mf and heat

capacity Cf, initially at a temperature Tf(0). A value for the convective heat transfer

coefficient h between the pellet and fluid is given. Students are asked to determine the

temperatures T of the pellet and Tf of the fluid as functions of time, ignoring any thermal

interactions between the cooling bath and surroundings. A diagram of the problem is

shown in Figure 1a.

Figure 1. Quenching of a pellet in a small bath (a) and in a large bath (b).

Previously, students have been exposed to the fundamentals of heat transfer to a lumped

parameter system through the basic notion of conservation of energy (rate of

accumulation of energy in the system = rate of energy entering – rate of energy leaving).

In addition, they have been exposed to phenomenological relationships representing the

rate of energy accumulation and the rates of energy entering (or leaving) by convection.

As a result, they are able to quickly write a mathematical model for the system as

follows:

Pellet: ))()((0)(

tTtThAdt

tdTMC f (1)

Fluid: 0))()(()(

tTtThAdt

tdTCM f

f

ff (2)

With the specification of initial conditions T(0) and Tf(0), this model is a completely

defined initial value problem to be solved for T(t) and Tf(t).

Solution of coupled differential equations by Laplace transforms is covered in the course

and students would probably use this technique if they were asked to solve the problem

analytically. However, in this exercise, students are asked to solve the problem

numerically using Simulink. After constructing a block diagram for equations (1) and (2)

and running a simulation, students obtain the result shown in Figure 2a.

Figure 2. Pellet and bath temperature vs time for a pellet quenched in a small bath

(a) and a much larger bath (b).

From their solution, students are able to observe that the fluid temperature increases as

the pellet temperature decreases and that the two temperatures eventually become equal

as the system reaches equilibrium. We believe this would be a useful exercise if it ended

here – but we take it further. Students are asked how they think the fluid temperature

would change if the bath was much larger, and are instructed to re-run their simulation

with a fluid mass 100 times larger than originally given, as suggested by Figure 1b. From

their results shown in Figure 2b, they are able to verify that the fluid temperature hardly

changes at all for this case.

In a separate exercise, students are asked to write the model as if the bath was large

enough that its temperature change could be neglected. This results in the model:

))((0)(

fTtThAdt

tdTMC (3)

Where Tf is a constant and there is a single dependent variable T(t). Students find the

analytical solution to this initial value problem either by separation of variables or using

the general solution of a first order linear equation, resulting in:

t

MC

hATTTT ff exp)( 0 (4)

Students then compare this analytical solution to the numerical solution shown in Figure

2b and find them to be in close agreement.

By now, students have solved two different (but related) problems using two different

techniques. Given the ease with which students were able to solve the more complicated

problem of the two, a natural question to ask is why anyone would bother solving the

simpler problem analytically. This leads to a discussion of what information is available

from the analytical solution versus what is available from the numerical solution, the

advantages of an analytical solution, and under what circumstances one would seek a

numerical solution. Finally, it leads to a point that is seldom appreciated when the three

main topics of this course are taught in a non-integrated manner - that the analytical

solution of a simpler case can serve as a limiting case check of the numerical solution of

a more complex case. As simulations become more and more complex, students are

encouraged to find ways to check their simulation results with limiting cases for which

the answer is known. This might involve a model simplification, as was applied here, or

comparing the simulation results at long time to the steady state predicted by the model.

By integrating model building, analytical solution and numerical simulation throughout

the coverage of the applied topics, our hope is that students will see these aspects less as

distinct topics and more as a set of interrelated tools.

Comparison of Modeling and Analysis to Differential Equations

Given that analytical solutions to differential equations are covered in a little more than

half of the semester, it is obvious that there must be some differences between the

Modeling and Analysis and Differential Equations courses. Topics covered in each

course are shown in Table 1. The Differential Equations content was extracted from a

syllabus provided by the Department of Mathematics.

Table 1. Comparison of course content for Modeling and Analysis (M&A) and

Differential Equations (DE)

Analytical methods M&A DE

First order ODE

Anti-differentiation X X

Separation of variables X X

Integrating factors X X

Exact equations X

Second order ODE

Homogeneous with constant coefficients X X

Nonhomogeneous - variation of parameters X

Nonhomogeneous - undetermined coefficients X X

Higher order differential equations X

Series solutions X

Matrix methods for linear systems X

Laplace transforms X X

Modeling

Motion of a falling body X X

Thermal systems X X

Free and forced vibrations X X

Fluid systems X

Coupled spring-mass systems X

Electrical circuits X

Numerical solution

by Euler's method X

by commercial software X

The comparison shown in Table 1 indicates that, indeed, the mathematics content in

Modeling and Analysis is lower than that of Differential Equations. In particular, series

solutions, matrix methods and higher (than 2nd) order equations are not covered in

Modeling and Analysis. Some modeling topics are covered in Differential Equations but

not as many, and at lower complexity, than in Modeling and Analysis. Numerical

solutions are not covered in Differential Equations. Both courses spend about the same

amount of time (2 to 3 weeks) on Laplace transforms.

Assessments

Because engineering students at our university can take either course, an opportunity

exists to compare those who took Modeling and Analysis to those who took Differential

Equations. In fact, such a comparison of these two courses was published [2] several

years ago by others not associated with either course. That assessment was performed in

a mechanical engineering course (Numerical Methods) for which a course in differential

equations was pre-requisite. It consisted of pre-and post-concept tests that each included

two questions on differential equations and a final exam that included four questions

related to differential equations. Details of the test items were not given other than they

were multiple choice.

This previously reported assessment occurred over several semesters between 2008 and

2010 and the student population consisted 42 students who took Modeling and Analysis

and 232 who took Differential Equations. The authors found no significant difference in

the performance of the two groups on the pre-tests, post-tests and final exam and were

unable to conclude that either group was more prepared than the other for differential

equations content in courses following their differential equations course.

In an attempt to probe more deeply whether the two courses have different effects on the

student experience, we provide results of additional comparisons and new assessments

below. Unless noted otherwise, statistical testing consists of two-sample, two-tailed t-

tests of the null hypothesis (no differences between means of comparison groups) at a

significance level of α = 0.05. The null hypothesis is rejected if p < α.

The authors have always considered the differential equations content of Modeling and

Analysis to be essential and the modeling component to be merely desirable. The first

question we wished to answer arose from our desire to ensure that the differential

equations content of the Modeling and Analysis course was covered at least as well as in

Differential Equations: Question 1. Is there any difference between students who took

Modeling and Analysis and those who took Differential Equations in regard to the ability

to solve differential equations in a subsequent course?

To answer this, we asked instructors of two different engineering courses to allow us to

give a short test to their students during the first week of class. During the spring of

2016, we gave the test to students taking Dynamics, a course primarily taken by students

majoring in Civil or Mechanical Engineering. In fall of 2016, we gave an identical test to

students taking Material and Energy Balances, which is populated exclusively by

Chemical Engineering students. No student was in both course offerings.

Students were not required to take the test and not all students participated. In addition,

neither Differential Equations nor Modeling and Analysis are pre-requisite for Dynamics

so we eliminated from the study any students who took our test but had not yet completed

one of these two courses. (Given our experience that students dislike exams, we found

this to be a surprisingly large number of students.) We also eliminated from the study

any students who took Differential Equations elsewhere, as the average number of

semesters between when they took the differential equations course and when they took

our test was much larger than the corresponding number of semesters for students who

took Differential Equations at our university. We will say more about this shortly.

This filtering resulted in 56 students who completed Differential Equations (DE) and 43

students who completed Modeling and Analysis (M&A). A comparison of characteristics

of these two groups is shown in Figure 3, which shows average university grade point

average (4.0 scale), average grade in the previous math course (DE or M&A, 4.0 scale)

and the average number of semesters since they took that math course. For students who

had taken several attempts at their math course, we considered only their most recent

completion of the course. Error bars represent 95% confidence intervals of the means.

Figure 3. Comparison of average course grades (in DE or M&A), average

university GPA and average number of semesters since the math course (DE or

M&A) was taken for students taking the test.

A t-test reveals no significant difference in average math course grade between students

who took Modeling and Analysis (M = 2.77, SD = 0.78) and those who took Differential

Equations (M = 2.92, SD = 0.87), t(97) = 0.93, p = 0.36. Likewise, there was no

significant difference in university GPA between students who took Modeling and

Analysis (M = 3.09, SD = 0.40) and those who took Differential Equations (M = 3.17, SD

= 0.44), t(97) = 0.88, p = 0.38. Finally, there was no significant difference in average

number of semesters since the math course was taken between students who took

Modeling and Analysis (M = 1.88, SD = 0.85) and those who took Differential Equations

(M = 1.85, SD = 1.09), t(97) = 0.22, p = 0.83.

2

023 2 50 ( ) with (0) 0 and 0t

d v dv dvv u t v

dt dt dt

For the test items, we chose problems of a type that engineering students would be likely

to encounter in any engineering discipline and that would be covered in detail in either

mathematics course. The test consisted of two items:

1. Obtain the analytical solution of )5(0002.0 Ttd

Td with T(0) = 25

2. Explain in as much detail as you can the different steps necessary to obtain the

analytical solution of

The first problem is solvable by several different methods, including separation of

variables and the general solution for a first order linear equation. Most students

(because of the presence of the unit step function u(t) in the forcing function) would use

Laplace transforms to solve the second problem but the methods of characteristic

equations and undetermined coefficients can be used as well.

Because we were sensitive to taking up too much class time and because the solution to

the second problem is somewhat time-consuming, students were asked to solve the first

problem but only to outline the solution to the second problem.

Student responses to each problem were scored on a five-point scale. The first item was

scored on correctness (zero credit for nothing correct, five points for fully correct and

intermediate points depending on number of errors and whether they were minor or

conceptual). The second item was scored based on identifying a proper method, and

providing the procedure to apply the method, with correctness and level of detail taken

into account.

After scoring, results were grouped by whether the student had taken Differential

Equations or Modeling and Analysis, then averaged within each group. Average scores

for Problem 1 and Problem 2 for the two groups of students are shown in Figure 4:

Figure 4. Performance of students on test items (1) and (2) for those who took

Modeling and Analysis (M&A) and for those who took Differential Equations (DE).

Leaving aside the observation that neither group scored as well as we would have hoped

on the test, we will examine whether the two groups performed differently. A t-test

reveals no significant difference in the ability to solve problem 1 between students who

took Modeling and Analysis (M = 2.28, SD = 1.83) and students who took Differential

Equations (M = 1.84, SD = 1.66), t(97) = 1.25, p = 0.21. However, there was a

significant difference in the ability to solve problem 2 between students who took

Modeling and Analysis (M = 2.16, SD = 1.77) and students who took Differential

Equations (M = 0.34, SD = 0.64), t(97) = 7.13, p <0.001.

Had students who took Differential Equations elsewhere (primarily in community

colleges) been included, the scores for the DE group would been lower for both

problems. However, the mean number of semesters between Differential Equations and

the taking of this test was 5.4 for students who took it elsewhere versus 1.8 for students

who took it at our university. Because the time since taking the math course is likely to

be an important variable, we excluded from the analysis students who transferred their

Differential Equations course from elsewhere.

According to these results, students who take Modeling and Analysis are better able to

solve differential equations in a later semester than students who take Differential

Equations. This conclusion is at odds with that of the previous study [2], which revealed

no significant differences between these two groups in their scores on pre-concept

questions in a subsequent course. This is possibly because the course used for testing

(Numerical methods) in that study came later in the curriculum than the Dynamics and

Material & Energy Balances courses used for testing here.

The results presented here are more consistent with the study of Czocher [1], who found

that students taking a differential equations course taught in a model-driven format

outperformed (on common final exam questions) students who took a more traditionally

taught course focused on analytical solutions. One way of reconciling all three studies is

to conclude that students taking a modeling-based course in differential equations are

better able to solve differential equations for a short time but the advantage is not

maintained indefinitely.

Because Modeling and Analysis has a mathematical modeling component, we were

curious as to whether this would prove advantageous to students in their later courses –

which led to our second question: Question 2. Do students who have passed Modeling

and Analysis perform better than their counterparts in courses that apply similar physical

models?

To address this question, we considered student performance in three courses

(Thermodynamics, Dynamics and Electrical Systems) that tie to the thermal systems,

mechanics, and circuits content covered in Modeling and Analysis. None of these three

courses requires either Modeling and Analysis or Differential Equations as a pre-

requisite. Results were extracted from a data set constructed in 2012, tracking student

performance between spring 2005 and fall 2011. The set contains student (first attempt)

grades in those three courses along with their grades in either Differential Equations or

Modeling and Analysis. The term that each course was taken is also part of the data set.

For each of the three engineering courses, the treatment group was comprised of students

taking the course who had previously passed Modeling and Analysis. The control group

would logically be anyone else taking the course, including students who had previously

passed Differential Equations, those who had not yet passed Differential Equations, and

those who had not yet passed Modeling and Analysis. However, this led to a disparity

both in overall academic performance and in academic maturity between the treatment

group and the control group. Because of this, we chose the control group to be students

taking the same engineering course who have previously passed Differential Equations.

For Dynamics, we created two groups of students: One which had passed Modeling and

Analysis (received a grade of C- or higher) prior to taking Dynamics and the other which

had passed Differential Equations prior to taking Dynamics. We then calculated the DFW

rate (percent of students who received a grade of D+ or less or withdrew from the course)

in Dynamics for both groups. This process was repeated for Thermodynamics and

Electrical Systems. Sample sizes (had passed M&A prior, had passed DE prior) were (88

,251), (172, 532) and (128, 377) for Dynamics, Electrical Systems and Thermodynamics,

respectively. Comparisons of the DFW rates are shown in Figure 5.

Figure 5. DFW rates for students who took the indicated course after passing

Modeling and Analysis or Differential Equations.

Chi-square tests for independence reveal no relation between previous math course and

DFW rate in a subsequent engineering course for any of the three courses Dynamics [χ2

(1, N = 339) = 0.06, p = 0.80], Electrical Systems [χ2 (1, N = 704) = 0.21, p = 0.65] or

Thermodynamics [χ2 (1, N = 505) = 2.38, p = 0.12]. Passing Modeling and Analysis

prior to taking Dynamics, Electrical Systems or Thermodynamics is not shown to give an

advantage over students who passed Differential Equations before taking these courses.

While performing these analyses, we noticed that about 40% of students taking any of

these three engineering courses were taking their math course (M&A or DE)

concurrently. This inspired us to ask a related question: Question 2b. Are students who

are taking Dynamics, Electrical Systems and/or Thermodynamics less likely to fail these

courses if they are taking and passing Modeling and Analysis concurrently? In other

words, does the mathematical modeling in Modeling and Analysis synergize with the

modeling covered in these engineering courses?

For Dynamics, we created two groups of students: One which was taking Modeling and

Analysis concurrently (and would ultimately pass it) and one which was taking

Differential Equations concurrently (and would ultimately pass it). The DWF rate in the

Dynamics course was then calculated for both groups and the process repeated for

Thermodynamics and Electrical Systems. Sample sizes (passing M&A, passing DE) were

(62 ,196), (128, 275) and (133, 254) for Dynamics, Electrical Systems and

Thermodynamics, respectively. Comparisons of the DFW rates are shown in Figure 6,

which appear to be quite different from those shown in Figure 5.

Figure 6. DFW rates for students who took the indicated course while taking (and

passing) Modeling and Analysis or Differential Equations.

For students taking and passing their math course (M&A or DE) concurrently with an

engineering course, chi-square tests for independence reveal that significant relations

exist between the DFW rate of the engineering course and which math course is being

taken, both for Electrical Systems [χ2 (1, N = 403) = 5.18, p = 0.02] and for

Thermodynamics [χ2 (1, N = 387) = 4.00, p = 0.045]. No such relation was found for

Dynamics [χ2 (1, N = 258) = 0.94, p = 0.33]. We conclude that students taking Electrical

Systems or Thermodynamics while taking and passing their math course have an

advantage if that math course is Modeling and Analysis rather than Differential

Equations.

The notion that a synergist effect would exist between Modeling and Analysis and other

engineering courses is not far-fetched. Students seeing (at the same time) similar models

in two different courses and in somewhat different contexts are forced to reconcile what

they may perceive to be inconsistencies, and in doing so probably gain a better

understanding of the material.

Given that the results for questions 1 and 2b indicate that Modeling and Analysis appears

to have beneficial immediate or short-term impact, we wondered whether there were any

longer-term impacts of the course on student retention and graduation. This suggested

our third question: Question 3: Are there differences in graduation rates and persistence

between students who completed Differential Equations and those who completed

Modeling and Analysis?

We addressed this question using the Civitas’s Illume Courses [7] platform at our

university. For any course in its database, Illume Courses uses historical data to report

average persistence and average graduation rate as a function of letter grade received in

the course. Students are considered to persist after a course if they register for a later

semester and to have graduated if they complete their degree requirements within six

years. The database for our institution contains historical course data from spring 2008 to

the present, so student start-terms from spring 2008 to spring 2013 were used to track six

year graduation rates. Illume Courses uses the most recent four years of data to calculate

persistence.

Modeling and Analysis was offered as an online course on several occasions. These

offerings were filtered out to obtain consistency with the Differential Equations course,

which is offered exclusively face-to-face. Students included in the sample were those

who were majoring in one of the seven engineering disciplines in our College and

included both transfer students and students who entered the university as freshmen. We

considered two different groups of students: those who took Differential Equations and

those who took Modeling and Analysis.

The average persistence rate for both courses was 97%. DFW rates were also identical at

16.9% and the average grade was nearly the same (2.9/4.0 for Differential Equations and

2.8/4.0 for Modeling and Analysis).The difference in graduation rate between the courses

is shown as a function of letter grade (for students who passed the courses) in Figure 7.

Figure 7 indicates that students who earn an A in Modeling and Analysis graduate at

higher rates than those who receive an A in Differential Equations but that the reverse is

true for students who earn a B or C. This is likely due to differences in grade

distributions for the two courses.

Figure 7. Six year graduation rates as a function of letter grade in math course for

students who took Modeling and Analysis (M&A) or Differential Equations (DE).

The percentage of students receiving a particular grade is shown for both courses in

Figure 8. Since the DFW rates are the same for both courses, the fraction of students

passing is the same. However, students earn fewer A grades in Modeling and Analysis

and more B and C grades. Because it is more difficult to earn an A in Modeling and

Analysis, it is likely that the “average A student” in M&A is a higher performer than his

or her counterpart in Differential Equations and would therefore be expected to have a

higher graduation rate.

Figure 8. Grade distributions of engineering students for Modeling and Analysis

(M&A) and for Differential Equations (DE).

Considering only students who pass (a grade of A, B or C) their math course, six-year

graduation rates are about the same (55% for DE and 53% of M&A) for both courses, as

shown in Figure 7. A chi-square test for independence reveals no relation between

graduation rate and which of the two math courses an engineering student takes, χ2 (1, N

= 1884) = 0.60, p = 0.44.

Finally, we were curious about student perceptions of the two courses, leading to a fourth

question: Question 4. Is there a difference in satisfaction between students who take

Differential Equations and those who take Modeling and Analysis?

To answer this question, we compiled the previous five years of student assessment

results (fall 2013 to summer 2018) for both courses. The instrument for student

assessment of instruction at our university consists of eight items, which are scored on a

five point Likert scale (5 = excellent, 1 = poor). Most of these items probe issues of

delivery (respect for students, availability outside of class, expression of expectations,

etc.) rather than content. For the comparison here, we selected the two assessment items

that relate most closely to content:

E2 Communication of Ideas and Information

E6 Stimulation of Interest in the Course

Average student ratings for these two items are shown in Figure 9.

Figure 9. Results from Student Assessment of Instruction for the years 2013-2018

for Modeling and Analysis (M&A) and for Differential Equations (DE).

Regarding communication of ideas, a t-test indicates a significant difference in student

satisfaction between those who took Modeling and Analysis (M = 3.97, SD = 1.25) and

those who took Differential Equations (M = 3.68, SD = 1.38), t(1,422) = 3.21, p = 0.001.

Likewise, for stimulation of interest, there is a significant difference in satisfaction

between those who took Modeling and Analysis (M = 3.99, SD = 1.30) and those who

took Differential Equations (M = 3.81, SD = 1.34), t(1,419) = 1.98, p = 0.048. It is noted

though that Modeling and Analysis is taken exclusively by engineering students while the

population of Differential Equations is, on the average, 75% engineering students, with

the remaining majoring in math or the sciences. The extent to which this difference

influences student assessment results is unknown.

With this caveat, students, on the average, are significantly (though not dramatically)

more satisfied with the Modeling and Analysis course than with the Differential

Equations course. Of course, few individuals are average, as exemplified by the two

student comments below that appeared in the spring 2018 assessment results for

Modeling and Analysis:

(1) This course has been my favorite so far due to the fact that it applies a lot of the math

that's introduced in prerequisite courses. Modeling engineering systems has allowed me

to understand why we actually need differential equations; I feel like I have an advantage

over someone who took a basic differential equations course.

(2) Should have taken Differential Equations.

Conclusions

We have described a course, Modeling and Analysis of Engineering Systems, in which

differential equations are placed in context using mathematical modeling. The course is

an alternative to the standard Differential Equations course offered at our institution,

which has a stronger emphasis on analytical solutions and a weaker emphasis on

application and numerical solutions.

Results reported here suggest that students who take Modeling and Analysis, for a

semester or two anyway, have the ability to solve differential equations in subsequent

courses to a higher degree than those who took Differential Equations. These results are

consistent with those reported by Czocher [1], who compared a modeling-based approach

to teaching differential equations to an approach focused on their analytical solutions.

The comparisons in that study were made at the end of the semester that the differential

equations course was taken rather than at the beginning of a later semester, as was done

here. However, both studies suggest that a modeling-based approach results in more

learning.

The modeling component of Modeling and Analysis does not appear to offer an

advantage to students who subsequently take Thermodynamics, Dynamics or Electrical

Systems. However, there appears to be a synergistic effect that students who are

currently passing Modeling and Analysis while taking either Electrical Systems or

Thermodynamics are successful in the engineering course at higher rates than those who

are currently passing Differential Equations while taking these courses.

There is no significant difference in student persistence or graduation rate between

Modeling and Analysis and Differential Equations, but student satisfaction is higher for

Modeling and Analysis.

Taken as a whole, these results suggest that there are immediate and short-term

advantages to students taking Modeling and Analysis instead of Differential Equations

but that the choice of math course has no long-term effect on persistence, graduation rate

or success in subsequent courses.

Acknowledgements

The authors are grateful to Dr. Gwendolyn Campbell for her suggestions in regard to the

statistical analyses used, and to Ms. Michelle King for preparing the data base used to

analyze performance of students in engineering courses.

References

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students in a traditionally taught differential equations course?", J. Math. Behav.

vol. 45, pp. 78–94, 2017.

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Exposition, June 23-26, 2013, Atlanta, GA. https://peer.asee.org/19456

[3] B. Howard, "Enough of this science and mathematics, let’s do some engineering

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Education. 29th Annual Frontiers in Education Conference, Nov 10-13, 1999, San

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