An Innovative Modeling Approach at the Freshman/Sophomore Level

这是一个创新的项目，为不同文化背景的学生在低年级开设互动建模课程。本课程将重点介绍基于差分和微分方程系统的建模过程。课程的第一部分将介绍连续模型的基本术语和必要的背景知识：导数作为变化率和切线斜率的概念。在此背景下，学生可以推导模型方程，并使用Mathematica生成的斜率场和相平面图定性分析解。该课程将在麦金塔实验室进行，使用Joy of Mathematica作为一个用户友好的菜单驱动的数学界面。在课程的第二部分，学生将学习生物学、化学、经济学和物理学的应用。来自这些学科的合作者将根据各自专业领域的有趣应用，确定基于差分和微分方程的“标准模型”。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。学生将在海报环节展示他们的期末项目。所建议的课程强调视觉组成部分以促进理解，这将使该课程有可能作为低年级课程提供。该课程将针对数学背景较弱的学生，这些学生不会参加传统建模课程所需的高级课程，为其学科中的建模应用做准备的科学专业学生，以及准备基于NCTM标准的课程的准教师。学生在结束课程时应该对建模过程有一个扎实的理解，提高批判性思维和写作技能，并认识到数学对跨学科建模的重要性。我们希望他们不会被数学吓倒，并将数学视为他们理解世界的一个组成部分。

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.