Special Issue of PRIMUS: Modeling Approach to Teaching Differential Equations

1. Universal Introduction to Teaching with Modeling

We believe conversations about incorporating inductive reasoning through modeling in teaching differential equations will be helpful to all. Furthermore, we seek to enliven the study of applied mathematics in this pivotal course for STEM majors, including especially those students who will teach mathematics to the next generation of students in our K–12 system.

As stated by the distinguished professor of history, Charles G. Sellers (UC Berkeley) [2] across disciplines, real problems motivate the process of learning:

The notion that students must first be given facts and then at some distant time in the future will ‘think’ about them is both a cover-up and a perversion of pedagogy… . One does not collect facts he does not need, hang on to them, and then stumble across the propitious moments to use them. One is first perplexed by a problem and then makes use of the facts to achieve a solution.

Mathematics is driven by the desire to better understand reality. Give elementary school students a collection of buttons and ask them to describe what they see. They categorize and sort them, on color, size, shape, mass, etc. Then they invent, on the spot, the notion of histogram to render their narrative. This is what happens when modeling comes first; mathematics is created, used, and often better understood by students. So it is also in differential equations.

The modeling-first approach closely parallels the underlying principles found in problem-based learning, inductive learning, and inquiry-based learning, which suggest that students learn best by doing and retain best when they construct their own paradigms [1, 6, 8, 9, 11]. Additionally, putting the mathematics into the context of real-world problems makes the subject meaningful, applicable, interesting, and powerful in the mind of the students. This can improve student attitudes about mathematics, resulting in increased curiosity, persistence, and perceived usefulness [10, 12]. Moreover, such an approach can enhance transferability of the mathematical knowledge, as it is based in reality along with vivid imagery.

In a seminal study on inductive teaching and learning in the premier engineering education journal, the American Society for Engineering Education’s (ASEE) Journal of Engineering Education, the authors conclude that “… inductive methods are consistently found to be at least equal to, and in general more effective than, traditional deductive methods for achieving a broad range of learning outcomes.” [9, p. 1]

We present both broad coverage and specific-filled material for reader consideration and trust the authors offer evidence of the value of modeling in teaching differential equations.

2. The Teaching and Learning of Differential Equations: Change is Afoot

As teachers of mathematics we developed a curiosity for what mathematics is and what makes it work, as well as a desire to address the issue of how mathematics can solve real problems. The topic of differential equations, that is the study of change, offers our students an entry level to apply their mathematics in real situations. Moreover, connecting to cognate disciplines that apply differential equations can stimulate students to want to learn more mathematics, even as we introduce them to that mathematics.

With the wide availability of technology and computational methods, students can now consider phenomena with real data and render potentially meaningful analyses. This practical approach is especially valued in the engineering community [7]. Moreover, in the 2015 MAA CUPM Curriculum Guide: Course Reports - Differential Equations there is strong support for including modeling and technology in differential equations courses [4]. Also, from the same CUPM report, under Technology and the Mathematics Curriculum, there is strong encouragement for the values of technology in many aspects of coursework: exploration, computation, communication, assessment, and motivation [5].

There is strong evidence and support for the use of modeling to motivate student learning in mathematics [3, 4]. Several movements, such as COMAP, CODEE, and SIMODE, are reaching out to the broader community that is interested in incorporating modeling in teaching. The Consortium for Mathematics and its Applications (COMAP) at www.comap.com offers general modeling support through its journal, The UMAP Journal, its text and video series, and is many modeling competitions. The Community of Ordinary Differential Equations Educators (CODEE) with its revitalized journal (CODEE Journal at http://scholarship.claremont.edu/codee/) and SIMIODE through its supportive community at www.simiode.org all offer materials to support the teaching of differential equations. SIMIODE focuses on using modeling and technology upfront and throughout the learning process. In this free and open-access community, colleagues can explore, communicate, collaborate, publish, teach, contribute, archive, and more. On registering, teachers can gain access to double-blind, peer-reviewed materials with instructor comments on pedagogy for use in class.

Teachers are doing exciting things with technology to collect data in support of modeling, and are reaching out to other disciplines to find areas of applications of mathematics. The topic of differential equations is a natural bridge for such collaboration. Students practice sound modeling principles as part of learning differential equations, often to motivate the study of the subject. Most importantly, students have a contextual environment in which to communicate their mathematics and its application through projects, reality comparisons, data analyses, parameter estimation, and prediction. Part II of this special issue will highlight many rich activities to use modeling in teaching differential equations courses in situational efforts which teachers can bring to their classroom.

3. Special Issue of PRIMUS devoted to a Modeling First Approach to Differential Equations

Part I of this two-volume special issue on the Modeling Approach to Teaching Differential Equations is devoted to broader issues surrounding the use of modeling in differential equations along with its accompanying support issues, whereas Part II will offer specific examples of activities which use modeling to motivate and teach differential equations.

The first three papers in this issue offer guidance on how to have students discover techniques that are useful in better understanding the solutions in differential equations. The techniques range from creating bifurcation diagrams to developing solutions of second-order equations. In the opening paper of this issue, “Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram,” by Chris Rasmussen, Justin Dunmyre, Nicholas Fortune, and Karen Keene, the approach of realistic mathematics education is used to illustrate how students discover relevant mathematics. Next, James S. Sochacki, Roger Thelwell, and Anthony Tongen develop students’ intuitions on the nature of solutions to a second-order equation for various forcing functions in “Forced Differential Equations: Problems to Impact Intuition.” The paper, “A Sinusoidal Twist with Exponential Influences” by Satyanand Singh, illustrates guiding students through the rigorous analysis necessary to determine the distance traversed by the mass of an under-damped oscillating spring mass system with sinusoidal displacement that results in a nice, closed-form expression.

The fourth paper, “Inquiry-Based Modeling of Population Dynamics with Logistic Differential and Difference Equations,” by Celil Ekici and Chris Plyley, demonstrates how to bring forth inquiry-based learning when building and comparing difference and differential equation models of logistic growth while combining numerical and graphical methods.

Wandi Ding, Ryan Florida, Jeffery Summers, Puran Nepal, and Ben Burton, in the paper, “Experience and Lessons Learned from Using SIMIODE Modeling Scenarios,” offer a narrative of how a research faculty member moves to incorporating modeling in a differential equations course with rich illustrations.

The introduction of ways to use technology and develop open source materials to enhance students understanding of material discussed in class is the focus of the paper, “Creating Dynamic Documents with R and Python as a Computational and Visualization Tool for Teaching Differential Equation,” by Boyan Kostadinov, Johann Thiel, and Satyanand Singh, while the following paper, “Echoes of the Instructor’s Reasoning: Exemplars of Modeling for Homework,” by Jennifer A. Czocher, Jenna Tague, and Greg Baker offers a formal study on the use of pencast technology to enhance student understanding of modeling.

The final paper of Part I of this special issue, “Make the Eigenvalue Problem Resonate with Our Students,” by Jeffrey A. Anderson and Michael V. McCusker, transitions to a specific illustration of self-designed and built laboratory equipment used to collect data on a spring-coupled pair of pendula. Students analyze the data and system and start to make connections with appropriate linear algebra techniques.

Part II of this special issue will offer more illustrations of classroom activities and specific projects to illustrate how modeling can enhance and support the teaching and learning of differential equations. We share the essential terms to whet the appetite of the reader: weighing fog on first day of class, modeling transport of leaves in a stream, modeling drone homing flight, using United States census data, collecting data on rolling dog treat ball, modeling particles in a box with one sticky wall, and using pharmacokinetic models to motivate student learning.

References

Acknowledgements

We thank all the authors for their creation in writing, their enthusiastic sharing of their ideas, and their willingness to revise and improve their presentation. We value the fellow guest editors and referee’s contributions of their time, intellect, and effort in giving constructive feedback for improved revisions of the papers. Finally, we appreciate the meticulous guidance, encouragement, and, above all, patience of the Editor-in-Chief Matt Boelkins.

Additional information

BIOGRAPHICAL SKETCHES

Chris McCarthy, Mathematics, Borough of Manhattan Community College, New York NY has conducted research in both pure and applied mathematics, especially in mathematical biology, and in the mathematics of adsorption. He enjoys including students in his research projects. This has led to a number of his students making presentations at national conferences, collaborative publications, and his research receiving continuing funding from CUNY. He has reviewed, edited, and written materials for SIMIODE.

Ellen Swanson, Mathematics, Centre College, Danville KY, is an applied mathematician who has organized contributed paper sessions at MAA and SIAM national meetings. She enjoys modeling with her students both in the classroom and through undergraduate research. She strives to make students more aware that mathematics can be used to better understand all aspects of the world around us from social phenomena to biological processes to physical behaviors.

Brian Winkel founded and edited the following journals: Cryptologia (30 years), Collegiate Microcomputer (11 years), and PRIMUS (20 years) and has organized many conferences and activities which engage in faculty development and this particular area of modeling in differential equations coursework. He is the Director of SIMIODE, Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations, where a community of learners and teachers is offering support of a modeling first approach to teaching differential equations. SIMIODE is currently supported by a 3-year National Science Foundation grant.