George M. Miller, Jr.
Nassau Community College

State University of New York


Thomas H. Sweeny

Nassau Community College

State University of New York



1967 - 2013, VOLUME 1, NO. 1


  Contained in the Winter 2014 Issue,

Volume 48, Number 1



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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 first-semester college freshmen, which integrates calculus-based 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. 65-66] 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 high-school 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  “cross-fertilization” 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 calculus-based 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.


             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 “S-STEM 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 co-housed, extensively mentored, and participate in various seminars and social events. 


             The title of the course is “Modeling Mechanics: Math, MATLAB n’More”, so chosen because of an alliterated acronym. This is a four-unit 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 S-STEM scholars. The second time it was offered 13 students were enrolled, again with 9  S-STEM 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 second-order differential equations to model either a physical problem of damped mechanical vibrations with forcing or of a non-linear 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 non-existent to a solid high-school 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 AB-level; and BC-level 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 Calculus-Based 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 web-based 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 second-order 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 calculus-based 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 math-physics topics, and the “3D-rectangles” represent topics that integrate all three threads 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 in-class 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.



Text Box: Pendulum motion



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 audio-visual technology whereas one day of the week the class met in a state-of-the-art 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 Runge-Kutta 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 easy-to-grasp 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 Runge-Kutta 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 small-scale 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 mass-spring-damper-forcing 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 Runge-Kutta 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 non-linear pendulum (without small-angle 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 Runge-Kutta code, and the other (flag 1) uses MATLAB's ode45( ) routine.


             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 MATLAB-based 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 equations-based 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 second-order 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, feedback-rich 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 34th class period in the Fall of 2010, and during the 28th 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, in-depth, 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 first-semester calculus-based 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.


            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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2. Joan Hundhausen and Richard Yeatts, “An experiment in integration: Calculus and physics for freshmen”, Journal of Engineering Education, Volume 84, Issue 4,  p. 369-374, DOI 10.1002/j.2168-9830.1995.tb00192.x (1995).

3. Martin Jackson and Andrew Rex, “A Course and Text Integrating Calculus and Physics”, 1999 Joint Mathematics Meetings, San Antonio, TX, 13-16 January, 1999.


4. Karen Marrongelle, “The role of physics in students’ conceptualizations of calculus concepts: implications of research on teaching practice”, 2nd International Conference on the Teaching of Mathematics (at the undergraduate level), Hersonissos, Crete, Greece, University of Crete, 1-6 July 2002. (http:// www.math.uoc.gr/~ictm2/Proceedings/pap153.pdf)

5. David Meltzer, “The relationship between mathematics preparation and conceptual learning gains in physics: A possible ‘hidden variable’ in diagnostic pretest scores”, American Journal of Physics, Volume 70, Issue 12, pp. 1259-1268,  ISSN: 0002-9505 (2002).

6. National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics, Reston, VA: NCTM, ISBN: 0873534808 (2000).

7. National Science Foundation Grant #0965940, Attracting Students to Mathematics, Computer Science and Physics at USD (2010).

8. Eric Page, “USD MCP: An Integrative Learning Environment for Mathematics, Computer Science and Physics”, Poster presented at Engaged STEM Learning: From Promising to Pervasive Practices, Miami, FL, 24-26 March 2011. (http:// www.aacu.org/meetings/STEM/11/resources.cfm)

9. David Rowland and Zlatko Jovanoski, “Student interpretation of the terms in first-order ordinary differential equations in modeling contexts”, International Journal of Mathematical Education in Science and Technology, Volume 35, Issue 4,  pp. 503-516, ISSN: 0020739X (2004).

10. Eileen Schwalbach and Debra Dosemagen, “Developing student understanding: Contextualizing calculus concepts”, School Science and Mathematics,  Volume 100, Issue 2, pp. 90-98, Online ISSN: 1949-8594 (2000).

11. Raymond Serway and John Jewett, Principles of Physics: A Calculus-Based Text, Fourth Edition, Brooks/Cole, ISBN: 9780534491437 (2006).