Introductory Differential Equations Research Articles

Semiotic Aspects of Differential Equations: Analytical and Graphical Competency in the USA and Tunisia

In this article we present the results of an investigation into semiotic aspects of an introductory differential equations class from two countries. In order to perform a comparative analysis, students in the USA and Tunisia were given common questions on their midterm exams that examined analytical and graphical competencies. Our results indicate that across both classes, the ability of the students to answer analytical questions was much higher than their ability to answer corresponding graphical questions, especially in the USA, and that it cannot be assumed that the graphical properties of basic functions have been retained by students when entering the course. We contend that this may need to be addressed at the beginning of the semester, with inquiry-oriented pedagogies and an appropriate use of computer algebra systems offering ways to enhance proficiency in converting between the different registers.

Reinventing the Salty Tank Through Guided Inquiry

In this paper, we discuss mathematical modeling opportunities that can be included in an introductory Differential Equations course. In particular, we focus on the development of and extensions to the single salty tank model. Typically, salty tank models are included in course materials with matter-of-fact explanations. These explanations miss the opportunity for students to develop rate of change equations for themselves. In this paper, we highlight an open-ended single salty tank task that provides the foundation for more complicated salty tank models that we also detail herein. Furthermore, we discuss instructional implementation techniques that instructors can use in their classes to meet their specific learning goals.

Facing the Pandemic Together: Forming a Collaborative Research Group

This is an account of how a reading and writing project in an introductory differential equations course was transitioned to a professor-student research group collaborative project, in response to the global COVID-19 pandemic. Adapting on the fly to the ever-evolving pandemic, we collected data, estimated parameters in our models, and computed numerical solutions to SIR-based systems of differential equations. This is a description of what we did and how we found comfort in the project in this time of great uncertainty. The collaboration yielded successes and more questions than we had answers for, but the situation provided an opportunity of a lifetime for my students to engage in a real-world developing situation.

Phase portraits, Lyapunov functions, and projective geometry

We discuss two problems which grew out of an introductory differential equations class but were solved only later, each after having been put into a different context. First, how do you find a rather complicated Lyapunov function with your bare hands, without using a fully developed theory (while reconstructing the steps leading up to such a theory)? Second, how can you obtain a global picture of the phase-portrait of a dynamical system (thereby invoking ideas from projective geometry)? Since classroom experiences played an important part in the making of this paper, didactical aspects will also be discussed.

Critical stance within a community of inquiry in an advanced mathematics course for pre-service teachers

This paper reports a study of university students’ experiences of learning mathematics in an introductory differential equations course that takes an inquiry approach to learning and teaching. The students are on a trajectory leading toward a bachelor’s degree in mathematics to be followed by a year-long teacher education credential program. The analysis of students’ end of course presentations and written portfolios is framed with community of practice theory, complemented with analytic categories from community of inquiry, including notions of inquiry stance and critical alignment. Our social theorization of university mathematics education adds a complementary perspective to the more conventional cognitive and discursive analyses and is necessitated by the very nature of inquiry approaches to education. The analysis of students’ presentations and writing leads to exploring their development of the notion of critical stance, which the students are seen to take toward the mathematics, their learning, and approaches to teaching mathematics. We find that the notion of critical stance can be distilled into three components: awareness, self-evaluation, and agency.

Inquiry-oriented differential equations: a guided journey of learning

AbstractThis study conducted at the Lebanese American University in spring 2018 reports on an inquiry-oriented differential equations class where the course curriculum falls within the realm of Realistic Mathematics Education. The focus of the study is on some curriculum tasks and how they assist students reinvent successfully key mathematical notions covered in most introductory differential equations courses. While the results show remarkable cognitive gains in understanding and thinking, reinventing knowledge was demanding and in some cases required the intervention of the instructor to control and lead the discussion. Results also show that the lack of appropriate mathematical knowledge was sometimes an obstacle for the reinvention process, and the traditional view of mathematical learning a hindrance for accepting that an inquiry-oriented approach can fulfil the course outcomes.

Outcomes of a service teaching module on ODEs for physics students

ABSTRACTThis paper reports on the first part of a multiphase research project that seeks to identify and address the difficulties encountered by physics students when studying differential equations. Differential equations are used extensively by undergraduate physics students, particularly in the advanced modules of their degree. It is, therefore, necessary that students develop conceptual understanding of differential equations in addition to procedural skills. We have investigated the difficulties encountered by third-year students at Dublin City University in an introductory differential equations module. We developed a survey to identify these difficulties and administered it to students who had recently completed the module. We found that students’ mathematical ability in relation to procedural competence is an issue in their study of differential equations, but not as severe an issue as their conceptual understanding. Mathematical competence alone is insufficient if we expect our students to be able to recognize the need for differential equations in a physical context and to be able to set up, solve and interpret the solutions of such equations. We discuss the implications of these results for the next stages of the research project.

Numerical existence and stability of solutions to the distributed spruce budworm model

This paper presents the steady-state solutions and traveling wave solutions for a spatially distributed PDE version of the spruce budworm model. The ODE (undistributed) model has been used in practical scenarios to model the outbreaks of the spruce budworm in forest environments, alongside the study of concepts involving fixed points and bifurcations in introductory differential equations courses. This study represents the spread of an outbreak from one end of a forest to the other. Numerical simulations are conducted using spectral methods.

Why No Difference? A Controlled Flipped Classroom Study for an Introductory Differential Equations Course

Flipped classrooms have the potential to improve student learning and metacognitive skills as a result of increased time for active learning and group work and student control over pacing, when compared with traditional lecture-based courses. We are currently running a 4-year controlled study to examine the impact of flipping an Introductory Differential Equations course at Harvey Mudd College. In particular, we compare flipped instruction with an interactive lecture with elements of active learning rather than a traditional lecture. The first two years of this study showed no differences in learning, metacognitive, or affective gains between the control and flipped sections. We believe that contextual factors, such as a strong group-work culture at Harvey Mudd College, contribute to the similar performance of both sections. Additionally, to maintain a rigorous experimental design, we maintained identical content across the control and flipped section; relaxing this requirement in a non-study setting would allow us to take further advantage of educational opportunities afforded by flipping, and may therefore improve student learning.

Lacking a Formal Concept of Limit: Advanced Non-Mathematics Students’ Personal Concept Definitions

This mixed-methods study examines the conceptual understanding of limit among 22 undergraduate engineering students from two different sections of the same introductory differential equations course. The participants’ concepts of limit (concept images and personal concept definitions) were examined using written tasks followed by one-on-one investigative interviews. The findings suggest that, at the end of their mathematics coursework, many engineering students do not consider employing a formal definition of limit to solve limit-based mathematics problems. Furthermore, the findings suggest that most participants’ personal concept definitions of limit are both inoperable for solving limit problems and inconsistent with the formal definition of limit.

Teaching Introductory Differential Equations with ScicosLab

Perhaps the largest and most capable open source software for doing applied mathematics is the open source package Scicoslab. Widely used in Europe and Asia it is less well known in the United States. In this article we will explain how ScicosLab can be easily used to help teach differential equations and to also introduce students to a software package that they can take with them for the rest of their careers.

On the Inclusion of Difference Equation Problems and Z Transform Methods in Sophomore Differential Equation Classes

In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the “classical” methods for ODE's to discrete time problems arising in many applications.

An Engineering-Oriented Approach to the Introductory Differential Equations Course

The introductory differential equations course can be made more relevant to engineering students by including more of the engineering viewpoint, in which differential equations are regarded as systems with inputs and outputs. This can be done without sacrificing any of the usual topical coverage. This point of view is conducive to student exploration, which can be facilitated by software tools such as those available at http://dynsys.uml.edu/.

Advanced Mechanics of Solids

3R5. Advanced Mechanics of Solids. - OT Bruhns (Institut fur Mechanik, Ruhr-Univ Bochum, Universitatsstr 150, Bochum, D-44780, Germany). Springer-Verlag, New York. 2003. 206 pp. ISBN 3-540-43797-5. $49.95.Reviewed by P Puri (Dept of Math, Univ of New Orleans, 2000 Lakeshore Dr, New Orleans LA 70148).The author of this book is a well-known researcher in solid mechanics and has made some important contributions to the theory of elasticity and plasticity. In general, the material in this book is presented quite well and can be used as an introductory one-semester course in solid mechanics. Readers are expected to have the basic knowledge of ordinary and partial differential equations, mechanics, and some elementary knowledge of the theory of beams. The solved examples are well chosen and help the student understand the theory and procedure for solving problems given in the exercises. Useful exercises are given in the last section of every chapter, except Chapter 10. The reviewer likes the fact that answers to all exercises are given. The figures in the book are neat and clearly drawn. The book is not intended to be a comprehensive book on mechanics of solids like a recent book on elasticity by Hetnarski and Ignaczak. Chapter 1 presents the fundamental concepts, principles of linear continuum mechanics, and the derivation of the basic equations. A minor comment: Eq (1.84) is more similar to Eq (1.76) than to Eq (1.75). Chapter 2 contains the development of constitutive equations based on Hooke’s law. The expressions for strain and complementary energy are derived in Section 2. The three equilibrium equations in terms of displacements and the equivalent six Beltrami equations in terms of stresses are stated in Section 3. Section 4 discusses the influence of temperature. The rest of the sections are devoted to Hooke’s law in two dimensions, the strength criteria, and the principle of complementary virtual work. It appears to the reviewer that for the derivation of Eq (2.18) a reference to Eq (2.9) is more appropriate than Eq (2.6) and that it would be helpful to refer to Eq (1.53) for the derivation Eq (2.19). The theory of simple beams is developed in Chapter 3. Topics covered in this chapter are normal stresses, shearing stresses, shear center of thin-walled open sections, influence of distributed loads, and stresses in non-prismatic bars, and deflections of beams. In Chapter 4, the torsion of prismatic bars is explored. Various types of cross sections are considered in the first three sections, and the influence of restrained warping is discussed in Section 4. The next chapter is on curved beams. The four sections in this chapter analyze general static’s, beams with large curvature, beams with small curvature, and the deflection of curved beams. Reciprocity theorems, energy principles, theorems of Engesser and Castigliano, statically indeterminate systems, complementary and strain energy of the beams are explained in Chapter 6. Chapter 7 is concerned with two-dimensional problems of plane stress and plane strain, which are discussed in the first section. In the second section, the special case in which the body forces have potential is discussed. In the next section the equations of plane strain and plane stress are given in polar form. The theory of elastic plates and shells is presented in Chapter 8. General remarks are given in Section 1. Section 2 consists of the solved examples for discs. Discussions on thin plates, Kirchhoff’s theory, boundary conditions, axially symmetric bending of circular plates, and the elastic energy of plates are contained in Section 3. Membrane theory of shells of revolution is given in Section 4. The notion of stability, bifurcation with finite degrees of freedom, snap through buckling, and column buckling are the topics covered in Chapter 9. The principle of virtual work, Hamilton’s principle, Euler-Lagrange equations, and the Legrange multiplier method are given in Chapter 10. This is the only chapter, which has no solved examples or exercises. The next two chapters are concerned with vibration. The first two sections in Chapter 11 are on undamped and damped free vibrations. This material is generally covered in introductory differential equations. The third section consists of forced vibration with harmonic excitation, and the last section in this chapter is on isolators, which tend to dampen the vibrations. The last chapter covers the material on free and forced vibrations with several degrees of freedom. This chapter also includes answers to the exercises. Advanced Mechanics of Solids is recommended to people interested in learning the basics of solid mechanics and understanding the interaction of forces on solids. The book is particularly suitable for structural engineers.

DIFFERENCES IN LEARNING OUTCOMES: CALCULUS & MATHEMATICA VS. TRADITIONAL CALCULUS

ABSTRACT This study, conducted -at The Ohio State University, compared students from the calculus-reform course sequence Calculus & Mathematica with traditional students in two areas: conceptual and procedural understanding of calculus, and achievement in calculus-dependent courses. Task-based interviews were conducted with students from both groups after they had completed one of the two calculus sequences. These interviews were used to investigate understanding of calculus. An analysis of grades collected from the university database was conducted to investigate student achievement in calculus-dependent courses. Results from the interviews showed that the Calculus & Mathematica students were more likely to approach problems from a conceptual viewpoint of calculus knowledge, whereas the traditional students were more likely to approach problems procedurally. The Calculus & Mathematica group also demonstrated a more general understanding of the derivative and integral than the traditional group. Results from the analysis of grades show a significant difference in the introductory differential equations course favoring traditional students, and a significant difference in the first course of the calculus-based physics sequence favoring the Calculus & Mathematica students. Other significant differences, favoring the top third of Calculus & Mathematica students, were found in the introductory physics courses and in an introductory engineering mechanics course.

Physical Mechanics. Third Edition

The third edition of this book differs in a number of respects from the earlier ones. Though the title page does not now carry the sub-title—An Intermediate text-book for students of the physical sciences—the preface states that its fundamental purpose is to serve as the basis of an intermediate course for college and university students who have taken a substantial course in elementary general physics and college mathematics, through calculus and introductory differential equations.