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SAMPLE ARTICLE  WINTER 2014
AN INTEGRATIVE APPROACH TO TEACHING MATHEMATICS, COMPUTER SCIENCE,
AND PHYSICS WITH MATLAB
Lukasz Pruski and
Jane Friedman
Department of
Mathematics and Computer Science
University of San
Diego
5998 Alcala Park
San Diego,
California 92110
lukaszpruski@gmail.com,
janef@sandiego.edu
1.
Introduction
The purpose of this article is to report on an innovative course for
firstsemester college freshmen, which integrates calculusbased
physics with computer programming through the lens of mathematical
modeling. We describe various aspects of the course so that it could
serve as a model for similar courses at other institutions.
Literature on education shows that connections foster better
learning and that learning in context is more effective than
compartmentalized learning. NCTM’s document Principles and
Standards for School Mathematics [6, pp. 6566]
unequivocally states that “school mathematics experiences at all
levels should include opportunities to learn about mathematics by
working on problems arising in contexts outside of mathematics”.
Research has shown that integrating physics and mathematics provides
pedagogical benefits. For instance, studies show that understanding
of calculus concepts helps in learning physics (for example, [5])
and that understanding of physics concepts enhances learning
calculus (for instance, [4]). In [10], researchers report that
carefully aligning highschool physics and calculus courses and
making explicit connections between the courses led to deeper
understanding of calculus concepts.
The
authors of this paper have been teaching for many years and they
have long been aware of dangers of “compartmentalization” in
education. Even within the field of mathematics compartmentalization
produces students who enter college full of misconceptions and who
are unable to use the mathematics effectively [1, p. 652]. The
majority of students perceive subjects as separate and disjoint
entities (even subjects as close as mathematics and physics), and
are thus prevented from taking advantage of “crossfertilization”
between the disciplines, where understanding of one subject feeds
off knowing more and more about other subjects. Freshmen students
usually do not have many opportunities to see the connections
between mathematics, physics, and computer science. At the
University of San Diego (USD), like at many other institutions,
mathematics majors are required to take Computer Programming I, but
it is just a single course and if any connections to mathematics (or
physics) appear, they are ad hoc rather than integral to the course
whose primary audience is computer science majors. Mathematics is
often referred to as the language of physics, but the curricula of
the two subjects are usually not coordinated, so that physics majors
often learn the underlying mathematics much later than needed. We
have often witnessed physics majors telling us in calculus courses
“Now what we were doing in physics makes sense!” as well as
mathematics majors saying “I wish I had learned that before I took
my physics course!”
Various experiments with integrating teaching mathematics and
physics at the undergraduate (freshmen) level aimed at addressing
the need for connections and context have been undertaken. For
example, interdisciplinary courses that integrate calculus and
physics are described in [2, 3]. Our project at USD has been
motivated by the same concerns but there are important differences.
We add introductory computer science to the mix of mathematics and
physics. The course we have developed does not replace a calculus
course (students take the second or the third calculus course
simultaneously). Nor does it replace our introductory computer
science course. It does however include enough of the content of
USD’s calculusbased Introduction to Mechanics and Wave Motion to
satisfy the Core requirement in physical sciences and to serve as a
prerequisite to the Thermodynamics, Electricity, and Magnetism
course required for physics majors. Furthermore, our course contains
a significant MATLAB component, and using MATLAB becomes the “glue”
that helps integrate the mathematics, physics, and programming
threads.
MATLAB is an excellent language for teaching a course of this
nature. Not only is MATLAB a good teaching tool, it is also a
powerful research tool for many scientists and mathematicians, so
introducing students to MATLAB helps them get acquainted with
software they may later use in research. Many institutions have site
licenses for MATLAB and the student edition is available at
relatively low cost. However, it would also be possible to teach
this course using other programming environments, such as Maple
or Mathematica.
2.
CONTEXT
Faculty in the mathematics, physics and computer science programs at
USD, recognizing the links between these fields and the benefits of
students understanding the relationships between them, wrote an NSF
grant entitled “Attracting Students to Mathematics, Computer Science
and Physics at USD” [7]. The main objective of the project is to
increase the number of academically strong and diverse USD students
graduating with a major in mathematics, computer science, or physics
and with an appreciation of the connections between these fields to
meet the high need for this expertise in our country. The course
described here was inspired by the desire to create an integrated
academic experience for these students as first semester freshmen.
This course and an interdisciplinary senior capstone course bracket
the students’ experience at USD. Two cohorts, each of 9 freshmen
students called “SSTEM scholars” (the first S for scholarships),
took the course in the fall semesters of 2010 and 2011. The students
in each cohort receive a scholarship from NSF. They are cohoused,
extensively mentored, and participate in various seminars and social
events.
3. GENERAL
CHARACTERISTICS OF THE COURSE
The title of the course is “Modeling Mechanics: Math, MATLAB n’More”,
so chosen because of an alliterated acronym. This is a fourunit
class, meeting four times a week over 14 weeks (for the total of 55
meetings). Originally, one meeting per week was scheduled as a
MATLAB lab; however the distinction between a "classroom meeting"
and a "lab meeting" disappeared toward the end of the course. The
first time the course was offered 11 students were enrolled, 9 of
whom were the SSTEM scholars. The second time it was offered 13
students were enrolled, again with 9 SSTEM scholars.
The
following four fundamental ideas guided the development of the
course:
·
Seamless integration of the three subject areas; the course has been
designed neither as a mathematics course with components of physics
and CS, nor as a physics or CS course with components of the two
other fields.
·
Grounding of the course topics in – from the physics side –
intuitively understandable phenomena of simple mechanics, i.e.,
kinematics and dynamics of motion and mechanical vibrations and in –
from the mathematics side – an approach based on modeling using
differential equations.
·
Exposing the students to the basic tenets of computational science –
building mathematical models and using computer programming to
describe, solve, and visualize problems of science (in this case
physics).
·
Extensive use of the MATLAB environment not only for building and
solving mathematical models of phenomena of physics but also to
introduce some of the basic concepts of computer programming, such
as sequence, selection, iteration, and recursion.
Two additional
integrative principles guided the pedagogical development of the
course:
·
Integration of symbolic methods of solving differential equations
with basic numerical methods.
·
Integration of a lecture component (involving all three areas:
mathematics, computer science, and physics) with the lab component (MATLAB).
The course achieved the full integration of the three threads in the
second half of the semester. The physics topics matched the
differential equations topics, and the material was synchronized
with coverage of MATLAB topics. The course culminated with a final
integrative experience: the students worked on a MATLAB programming
project that required using numerical methods for solving
secondorder differential equations to model either a physical
problem of damped mechanical vibrations with forcing or of a
nonlinear pendulum with forcing, and to visualize the behavior of
the solution.
3.1 Students'
Preparation
None
of the students had any serious prior exposure to differential
equations. The physics preparation ranged from virtually
nonexistent to a solid highschool physics course. Several students
had some experience with computer programming, but others had none.
None of the students had any exposure to a mathematical programming
environment such as MATLAB.
In
the first offering of the course two students did not have prior
calculus exposure (they were taking Calculus I concurrently with the
course). This caused some problems with integration of the three
threads in the first part of the course. For this reason, calculus
is now a prerequisite, and all students who took the course the
second time had AP credit (mostly ABlevel; and BClevel in some
cases), which allowed a much smoother integration of threads.
3.2 The Textbook
A
standard physics textbook, Serway and Jewett, Principles of
Physics: A CalculusBased Text, Fourth Edition [11] was used in
the course. This text matched the desired level of coverage of
physics topics quite well. The problems in the text obviously
covered only the purely “physics thread”, without any connections to
differential equations or computer programming. The coverage of
mathematics and programming components of the course was supported
with handouts authored by the instructor and with publicly available
webbased sources. The problems reflecting the integrative aspects
of the course were prepared by the instructor. Here is an example of
a problem that illustrates the restatement of a standard projectile
motion problem of physics (Newton’s laws in two dimensions) in a way
that connects with the topic of the secondorder linear differential
equations concurrently studied in class:
State the
initial value problem that describes the motion of a ball thrown
with an initial velocity of
at an angle a from
a platform
located h feet above the ground. Solve the initial value problem and
find how far from the point of the toss the ball will hit the
ground.
Further examples
of assignments that reflect the integration of all three threads of
the course, i.e., programming of computational aspect of
calculusbased solution to a physics situation are included in
Section 4.
3.3 Course Topics
Two
diagrams (Figure 1 and Figure 2) show the flow of topics covered
during the course and merging of the course threads. The leftmost
column shows mathematics topics, the next – physics topics, the
rightmost one – programming and MATLAB topics. The rounded edge
rectangles represent integrated mathphysics topics, and the
“3Drectangles” represent topics that integrate all three threads of
the course.
4. INTEGRATIVE
ASPECTS OF THE COURSE
The
course was developed in the belief that through using mathematical
models of physical phenomena programmed in MATLAB, students will not
only develop a deeper understanding of physics, but also a deeper
understanding of mathematical concepts not explicitly taught in the
course. The philosophy was that understanding physics deepens
understanding of mathematics and understanding mathematics deepens
understanding of physics and that understanding of both physics and
mathematics concepts are enhanced by modeling and programming in the
MATLAB environment.
4.1 Integration
of Mathematics, Physics, and
Computational Science Components
From
the point of view of content, the integration of mathematics,
physics, and computational components in the course was relatively
easy to design (and worked out well in practice). The topics of
mechanics yield easily to an interdisciplinary approach. The
following is a problem taken from inclass Test #3 (Fall 2010) that
illustrates the integration:
A mass of 100
g attached to a spring moves horizontally without any resistance.
The spring is such that it stretches by 50 cm under the weight of a
1 kg mass. Suppose at the initial instant of time the mass attached
to the spring is stretched by 10 cm and given an initial velocity of
5 cm/s in the positive direction. Consider Euler's numerical method
with step h = 0.1 (it would not be good enough in practice, but this
is just a test). Find the position and speed of the mass at time t =
0.2 if the initial time is t = 0.
Figure 1: Flow of topics in the course and merging of the course
threads – Part 1.
Figure 2: Flow of topics in the course and merging of the course
threads – Part 2.
The
integration of mathematics and physics components was present during
each class meeting and in most assignments. The most elegant example
of this integration was the complete mathematical proof that the
solutions of a damped vibrating system always decay with time. Some
students were visibly excited when the proof was completed on the
board, which strengthened the connections between the two subjects.
4.2
Integration of the MATLAB Component with
the Theoretical Material
The
course attained full integration of the three subjects in the second
half of the semester. For several students this was their first
experience with computer programming, and following the syntax of a
programming language, and adhering to the language semantics proved
quite daunting for these students. For this reason, the students
needed about six weeks to gain some familiarity with MATLAB, before
they could use it effectively as a modeling tool.
The
class met three days a week in a traditional classroom enhanced with
audiovisual technology whereas one day of the week the class met in
a stateoftheart computer classroom. Thus, during the first half
of the course, there were three theory class meetings per week and
one MATLAB class meeting a week. During the second half of the
course the distinction faded and the instructor used MATLAB for
demonstrations during “lecture meetings” and worked with students on
theory during “lab meetings”. In the second edition of the course
the distinction was even less pronounced.
MATLAB provides a complete environment for teaching introductory
computer programming. In this course, only about 10 contact hours
(nominally, as the course threads were integrated and intertwined)
were dedicated to teaching programming, and not all topics taught in
a standard one semester introduction to computer programming course
could be covered. However, all students learned a basic set of
programming constructs and concepts, i.e., sequence, selection,
iteration, recursion, and procedural abstraction, which are some of
the most important constructs necessary for helping students develop
their algorithmic thinking. These are also some of the more
difficult topics taught in an introductory programming course. The
students were able to use these constructs and concepts when
programming numerical solutions of differential equations models of
phenomena of mechanics, with the use of Euler or RungeKutta
methods, thus demonstrating their understanding of basic programming
concepts.
4.3
Integration of Symbolic and Numerical Approaches to
Solving Differential Equations
MATLAB provides dedicated procedures for solving differential
equations (the ode( ) family of functions). The students had
to first learn to write the code for numerical solutions. A
foundation was built by studying the basic Euler method, which
provides an easytograsp introduction to numerical methods. The
examples that student teams worked with in class were carefully
designed to raise issues of convergence and stability of the method.
The students also had an opportunity to observe the influence of the
selection of step size on the accuracy of the method. Next, the
students learned and programmed the RungeKutta method. This ensured
that the students were not using the ode( ) routines as a
black box and that they understood the details of the method. Only
then did they use the packaged MATLAB functions. The use of two
different methods allowed us to study the role of complexity in
numerical computations. The subsequent use of MATLAB routines
reinforced the notions of procedural abstraction.
4.4 Integrative
Programming Project
The
course culminated in a smallscale final summative experience for
the students. They were assigned a programming project, in which
they were to write a MATLAB system of functions that solved
differential equations describing a model involving oscillatory
behavior. They were also required to program an elementary animation
which depicted the behavior.
In the first implementation of the course, the mechanical model was
a massspringdamperforcing function system, and the students had
to illustrate the effects of change of parameter values (mass,
damping and spring constants, forcing amplitude and frequency) on
the behavior of the oscillatory system, by showing the motion of a
dot representing the vibrating mass. The following is the statement
of the assignment (from the students’ handout):
Write a MATLAB
function that uses the RungeKutta method to solve a forced damped
vibrations initial value problem. In addition to the time span and
the step, the function takes as parameters the damping coefficient,
the spring constant, the amplitude and frequency of a cosinusoidal
forcing function (with no phase shift), as well as the initial
displacement and speed of the mass (mass is assumed to be 1) and
returns the vector of positions of the mass.
Write a MATLAB
script that calls the function, obtains the vector of positions and
produces an animation of the motion of the mass. For purposes of
graphics you may assume the position of the mass will be limited to
the interval [5, 5].
In
the second implementation of the course, the model was a nonlinear
pendulum (without smallangle simplification), with external
periodic forcing applied to the mass. The students were told to
solve the differential equations of the system and provide a simple
animation. The statement of the assignment included the following:
Write a MATLAB
function that solves and animates the general case of pendulum
motion with external forcing of the form
. The function should take the following
parameters (in the following order): length, the initial amplitude
(angle), the initial angular speed, the constants
, , , time span, step, and a flag (0 or 1) that
indicates the method to be used. The function should return the
vectors of time points, values of angle, and values of angular
velocity. The first method (flag 0) uses your own RungeKutta code,
and the other (flag 1) uses MATLAB's ode45( ) routine.
5.
DID IT ALL WORK?
In informal terms, the course was a success. The students (two
cohorts combined) received the usual spectrum of grades: 5 A’s, 10
B’s, 7 C’s and one each of D and F, which is slightly higher than
the usual grade distribution in our department. The results of the
final exam and of the programming project showed that the students,
for the most part, learned the material.
As
was mentioned earlier, all students in the second cohort had
previously taken calculus in high school. This allowed us to merge
the three threads of the course, i.e., physics, differential
equations, and MATLABbased computer simulations, much earlier. As
might be expected, because of experiences gained, the second
implementation ran more smoothly. The pace of the course was a
little slower at the beginning, which clearly helped the students,
some of whom had the usual freshmen adjustment issues.
Several students from the first cohort reported that the course
helped them in other courses they were taking at USD. For example,
the exposure to the concept of an algorithm helped in computer
science courses, the exposure to mechanics context helped in
learning calculus, and the experience with differential
equationsbased models helped a student in a more advanced physics
course. The instructor of the course, who has extensive experience
with teaching an upper division Differential Equations course,
provided further anecdotal evidence. Class discussions, quizzes and
exams showed that the students in the course grasped the meaning of
differential equations models significantly better than more
experienced students in upper division Differential Equation courses
taught by the instructor.
Students' interpretational difficulties with differential equations
in modeling contexts are well known, and some have been documented
in the literature (for example, [9]). One of the typical
difficulties for students learning to model mechanical vibrations
with secondorder differential equations is the correct
interpretation of the three terms of the basic equation (inertia,
resistive force, and restoring force). The student sample was too
small in this course for any statistically significant conclusions,
but qualitative observations, based on responses to quiz and
examination questions indicate that the students were able to
interpret these terms with greater understanding than students in
the standard differential equations course taught by the same
instructor.
One
of the strongest positives of the course was the use of MATLAB. This
powerful tool aided the students’ understanding of mathematics and
mathematical modeling and it helped them learn the basics of
computer programming and computational science. It clearly improved
the students’ ability to visualize the time behavior of mathematical
models of physical situations. We attribute the success to the fact
that in the differential equations courses we taught in the past the
computer environments were used solely as solving and visualizing
tools whereas in Modeling Mechanics the students were required to
program in MATLAB language thus forming a stronger, feedbackrich
relationship with the computer environment.
6.
Lessons learned, ISSUES TO
RESOLVE,
and future plans
Prior student experience with calculus was clearly needed. The
second time the course was taught, all students had had some form of
calculus in high school. This allowed elimination of about five
class periods that were spent for reviewing differential and
integral calculus during the first implementation of the course. For
example, introduction to numerical methods was covered during the 34^{th}
class period in the Fall of 2010, and during the 28^{th}
period in the Fall of 2011. This allowed more time to focus on
integration of the threads of the course through solving additional
class problems, both in the lecture environment and in the MATLAB
programming environment. More time was also available to present
more difficult aspects of the material, such as forced vibrations.
Another, rather unsurprising observation from having twice taught
the course is that the students benefited the most from the class
meetings where the mathematical behaviors of the physical models
were demonstrated in real time via computer visualizations. For
instance, the topic of resonance has the potential of strengthening
the connections between students' internal representations of
physics phenomena and their representations of mathematical models
(differential equations). Another such topic is the dissipation of
energy in damped vibrations, which manifests itself in decreasing
amplitude of vibrations. The next time the course is taught, we plan
to spend several additional class meetings devoted to detailed,
indepth, integrated sessions utilizing MATLAB visualizations.
Another issue that still needs to be resolved is whether to include
wave motion in the course. Both times the course was taught,
coverage of wave motion took about five class meetings. The
advantages of including the topic are that it broadens the students'
perspective on the connections between physics and mathematics and
that it makes the course similar to the usual firstsemester
calculusbased physics course, so that the course may serve as a
prerequisite to further physics courses. Particularly attractive is
the fact that the coverage of wave motion allows a gentle
introduction to functions of two variables (one spatial and one
temporal) and thus provides a glimpse into partial differential
equations. On the other hand, without covering wave motion, there is
more coherence in the course material, and the flow of topics is
more natural. Leaving out wave motion would pave the way for
spending significantly more time on the core topics.
7.
CONCLUSIONS
Perhaps the most important result of running the Modeling Mechanics
course was providing a model for an interdisciplinary course that
integrates several STEM disciplines for freshmen students. At USD a
new category of freshmen courses was created – INST
(Interdisciplinary Studies). The course may serve as a stimulus
(and, hopefully, also a model) for other institutions that may want
to develop a course of a similar nature. It is too early to
confidently claim an unconditional success, yet the indications are
all positive. We are planning to make the course a regular component
of the USD freshmen year curriculum. Many universities offer smaller
freshmen seminars, which are often interdisciplinary in nature.
Institutions offering such courses include liberal arts colleges and
large public and private universities (examples of such institutions
include the University of Oregon, Connecticut College, and Yale
University). A course such as the one described here, would make an
ideal freshman seminar.
Our
job as teachers is to educate students both as thinkers and as
future productive members of society. Challenging integrative
experiences, such as this course, help develop students' critical
thinking skills and enhance their conceptual understanding of
standard mathematical and physical concepts. They introduce students
to the power of using computer technology to explore models. We
believe courses like this one provide both intellectual development
and attainment of specific skills. We believe this is a model worth
exploring further as the faculty continues to work to provide
meaningful educational experiences for their students.
Acknowledgment:
This
work has been partially supported by NSF grant 0965940.
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