A Modern Approach to Differential Equations Martha L. Abell and James P. Braselton At Georgia Southern University, we try to take a classic course in elementary ordinary differential equations and present it in a new light. On the surface, our course appears to have the same ingredients as other differential equations courses. However, upon closer inspection, we see that the course has some exciting features. We describe all standard methods in an easy-to-understand fashion and attempt to build a firm understanding of the methods. In addition, we point out the importance of observing solutions and analyzing their properties through the use of graphics. Although this does not seem to be an important issue, the student's ability to make a connection between a formula and a graph is imperative in understanding the intricacies of the study of differential equations. Above all, we attempt to incorporate technology in a meaningful way. we do this on several different levels. First, we encourage students to familiarize themselves with methods for solving first and higher order equations as well as systems of differential equations (both exactly and numerically) with technology. In doing this, students learn to become less concerned with a method of solution and more concerned with other aspects of the problem. In addition, students learn to spot any erroneous results given by technology when what they see on a computer screen does not match what they expect to see based on their experience and intuition. Second, we present In Touch With Technology problems so that students can explore topics on their own by making conjectures, drawing conclusions, and giving supporting evidence. For example, students may compare how a slight change in an initial condition affects solutions to linear and nonlinear equations. Third, we give students the opportunity to use their knowledge of differential equations and technology to solve some interesting applications projects called Differential Equations At Work. These projects, which include topics such as Controlling the Spread of Disease, the Mathematics of Finance, and Modeling the Motion of a Skier, encourage problem solving, communication, reasoning, and the use of mathematics in realistic situations. In order to complete these projects successfully, students work in groups and thoughtfully plan a course of action. They determine the skills that must be incorporated in answering specific questions, and they decide when to use technology and when to perform tasks by hand. In addition, they submit a written report to summarize their results. The completion of a Differential Equations at Work project is a worthwhile experience for the students in that they are asked to relate the material covered in the classroom to a problem they have never studied. In addition, they are required to call on knowledge from previous mathematics courses which helps to reinforce their overall understanding of mathematics. Although our course is not particularly a reform differential equations copurse in that it presents standard topics and methods in a familiar order, it modernizes the study of differential equations by calling on students to think about what they are accomplishing by solving a differential equation. Throughout our course we pose thought-provoking questions to the class as a whole. Sometimes, these questions encourage students to use technology to investigate a particular property of a solution. At other times, they encourage students to recall a related topic from a previous course. Still, at other times, they ask students to draw a conclusion by observing a graphic. In all cases, the student is asked to think and in doing so, go beyond merely writing down a formula. Our hope is that students eventually understand what a solution to a differential equation means, how they can use solutions to answer other questions, and why they can use differential equations beyond the classroom.