Using Technology in a Differential Equations Course: Lessons Learned Implementing a New Paradigm

Historically, a first course in Ordinary Differential Equations (ODEs) has been taught as a “methods course. ” Namely, different types of differential equations are trotted out and the method of solution for each class of ODEs is presented. The student is then left to “master the method” by way of lengthy algebraic manipulations, a good dose of calculus, and of course, getting all the pieces of the new method done in the right order. All in all, it can be a daunting task for students and it usually does little to pique the interest or curiosity of most students. This is the old paradigm. In an attempt to make the course more interesting and less daunting, a new paradigm was adopted—a paradigm that incorporates technology as a core component of the course, rather than just an “add on. ” This article examines various issues related to implementing this new paradigm as well as some lessons learned along the way. 1 Background Differential Equations is a four credit hour course in the mathematics curriculum at Benedictine University. Students usually take it after taking multivariate calculus and before (or along with) Linear Algebra. Typically, about 25% of the students are mathematics majors and about 75% are majoring in the natural sciences or engineering. The former group of students takes the course because it’s required for all mathematics majors and the latter group because it’s a cognate requirement for their major or because they are working on a minor in mathematics. Differential Equations is designated as a Writing Intensive (WI) course. This will be discussed in more detail later in the paper. Since Differential Equations is a transition course from lower level (more computationally oriented) courses to upper level (more theoretically oriented) courses, there are two main learning objectives. The first is to impart knowledge and understanding of the course content. This includes the methods required to obtain solutions for various classes of ODEs or a system of ODEs and how to analyze an ODE in order to make a qualitative statement about the solution without knowing what it is, e.g. use the phase portrait to discuss the stability of equilibrium solutions. The second is to foster growth in CODEE Journal http://www.codee. org/ each student toward the next level of mathematical thinking. This involves things like modeling a physical phenomenon mathematically, analyzing a model, bringing theoretical questions to bear on a model and using theorems to answer these questions, and learning how to think more abstractly about mathematics. The idea is to strike the right balance between the theoretical and the applied nature of the material and to use technology in some meaningful way to help the students learn the mathematics. Therein lies the challenge! 2 Technology Platforms The main technology platform used in the course is Maple, a computer algebra system. Other pieces of technology used include 1. the Desire2Learn (D2L) course management system, 2. PowerPoint slides, which come with the text and are uploaded to D2L for ease of access, 3. Maple files, which also come with the text and are related to HW problems or textbook examples, 4. the web, in the form of links to related web sites (these are posted to D2L); websites used for in-class demos, e.g. YouTube videos on bungee jumping; and the student companion website, which comes with the text and has many additional resources for the students, like chapter summaries and interesting projects that go beyond the course content. All in all, Differential Equations is a technology rich course. Before discussing howMaple is used in the course, a brief summary of the four software packages used over the past 21 years will first be given. The first piece of software used (1993) was MODELS. This software is user-friendly, has some very nice graphing and animating capabilities (it graphs in 2D and 3D), and allows the user to specify which numerical solver is used. The pro for MODELS was that it allows for easy creation and modification of lab explorations. The con was that you had to create the labs ex nihilo. The second piece of software used (1998) was Interactive Differential Equations (IDE). This software runs in a web browser and comes with a workbook that has lab experiments for students to work through and interactive illustrations that are useful for classroom demonstrations and discussions. The pro for IDE was that it came with pre-made lab experiments—a huge time saver! The con was the lab experiments couldn’t be (easily) modified, so the students were doing the same ones over and over. Although IDE had no glaring deficiencies, when ODE Architect hit the streets (around 1999), the IDE software was retired because ODE Architect did everything IDE did . and more. For instance, the videos that come with the ODE Architect software are great “attention grabbers” at the beginning of class since they show students how differential equations can be used to model problems/phenomena in the real-world. The multimedia piece of ODE Architect is useful for in-class demonstrations and to “prime the pump” for explorations in the computer lab. ODE Architect also has a handy interface with Windows Word Pad by way of the Report Writer tool on the Toolbar. This allows students