An earlier version of the imageAnimations OSSLET has been used for two years in the first semester of the four semester mathematics core curriculum at the United States Military Academy. The course emphasizes applied mathematics through modeling and using effective problem solving strategies and modeling theory to solve complex and often ill-defined problems. The course exercises mathematical concepts while nurturing creativity, critical thinking, and learning through activities performed in disciplinary, interdisciplinary, and multidisciplinary settings. Special emphasis is placed on introducing calculus using continuous and discrete mathematics through applied settings. A variety of technological tools are explored to develop numerical, graphical, and analytical solutions that enhance understanding.
Course Objectives
The five course objectives below focus on the growth of the student.
Acquire a body of knowledge: The students are introduced, and in some cases reintroduced, to several important topics so that they are prepared to succeed in their future math, science, and engineering courses. The course is broken down into four blocks of instruction. These blocks, along with the specific goals for each of these blocks, are as follows:
Problem-solving, data analysis and modeling with functions.
Modeling with matrices, vectors, and linear systems.
Modeling with sequences -- that is, discrete dynamical systems.
Introduction to calculus and modeling with continuous dynamical systems.
Use technology appropriately and reflectively: The students' abilities to employ technology as an analytical tool and as an aid in the problem solving process are developed. The course utilizes the enhanced capabilities of technology to investigate possible solutions by determining numerical and graphical results. Technology allows students to adapt models to changes in variables, parameters and conditions. Specifically, students use a computer algebra system (Mathematica) and computer spreadsheets (EXCEL) to graph functions, model data with best fit functions, solve systems of linear equations, examine long-term behavior, and iterate difference equations.
Communicate effectively: The students have many opportunities to communicate their results both verbally and in writing in essay questions, a project write-up, and briefings on class work. First, however, students are expected to listen actively to a presentation by a student or instructor and then be able to verbally synthesize and summarize. Students begin to articulate the results of their work in board presentations. At the same time students begin to read a mathematics textbook on their own and either synthesize the material or "know what they don't know" in order to frame an appropriate question for class. Students receive guided instruction on the preparation of a technical report and begin to understand the process of articulating the problem solving process and its results. Students complete the course expected to know what "right" technical report looks like. Students are introduced to the Documentation of Written Work and are expected to document all assignments according to the procedures outlined in the document.
Become confident and competent problem-solvers: Modeling and problem solving abilities are developed through in-class experiences, homework exercises and a group project. These events require students to analyze real-world problems, make assumptions, model the system, solve the model, and then interpret the results. The events illustrate the direct applicability of the many tools and problems solving techniques discovered in the course.
Develop habits of mind: Students' reasoning power is improved by introducing different modes of thought: induction, deduction, algorithms, approximations, and implications. The students are also encouraged to learn how they best learn and to develop good study habits. Finally, the students are introduced to the importance of life long learning and to the characteristics of a life long learner.
The five goals outlined above are summarized in the self educational growth triangle shown below. As one moves from the bottom of the triangle to the top the level of difficulty becomes more complex for the student to grasp, and for the teacher to educate.
