An Innovative Modeling Approach at the Freshman/Sophomore Level

这是一个创新项目，为不同文化背景的学生在低年级开设互动建模课程。本课程将侧重于使用基于差分和微分方程（系统）的示例的建模过程。课程的第一部分将介绍连续模型所需的基本术语和背景：导数的概念，如变化率和切线的斜率。有了这个背景，学生可以推导出模型方程，并使用Mathematica创建的斜率场和相平面图定性分析解决方案。该课程将在Macintosh实验室教授，使用Joy of Mathematica作为用户友好的菜单驱动的数学界面。 在课程的第二部分，学生将从事生物学，化学，经济学和物理学的应用。来自这些学科的合作者将确定基于差分和微分方程的“标准模型”，并从他们的专业领域中找到有趣的应用。这些模型将作为交互式Mathematica文本实现;因此，将创建一个模型库供学生探索。学生将在海报会议上展示他们的期末项目。 拟议的培训班强调视觉内容以促进理解，这将使该培训班能够作为低年级的培训班提供。该课程将针对数学背景较弱的学生，他们不会参加传统建模课程所需的高级课程，科学专业的学生准备在其学科中建模应用，以及未来的教师准备基于NCTM标准的课程。 学生应该离开课程的建模过程中有一个坚实的理解，提高批判性思维和写作能力，并欣赏数学建模跨学科的重要性。这是我们的希望，他们将不那么害怕数学，并认为数学作为一个组成部分，为他们了解世界。

This is an innovative project establishing an interactive modeling course at the lower division level for culturally diverse students. The course will focus on the modeling process using examples based on (systems of) difference and differential equations. The first part of the course will introduce basic terminology and background necessary for continuous models: the notions of derivative as rate of change and as slope of the tangent line. With this background students can derive model equations and qualitatively analyze the solution using slope field and phase plane graphs created by Mathematica. The course will be taught in the Macintosh laboratory, using Joy of Mathematica as a user-friendly, menu-driven interface to Mathematics. In the second part of the course, students will work on applications from biology, chemistry, economics and physics. Collaborators from these disciplines will identify "standard models" based on difference and differential equations with interesting applications from their area of expertise. These models will be implemented as interactive Mathematica texts; a library of models will thus be created for student exploration. Students will present their final projects at a poster session. The emphasis of the proposed course on visual components for understanding will make it possible to offer the course as a lower division course. The course will be aimed at students with weak mathematics background who would not take the advanced courses needed for a traditional modeling course, science majors in preparation for modeling applications in their disciplines, and prospective teachers in preparation of a curriculum based on the NCTM Standards. Students should leave the course with a solid understanding of the modeling process, improved critical thinking and writing skills, and an appreciation of the importance of mathematics for modeling across the disciplines. It is our hope that they will be less intimidated by mathematics, and view mathematics as an integral part for their understanding of the world.