Collaborative Research: Tutoring to Apply Conceptual Understanding of Ordinary Differencial Equations: Analysis for Cognitive and Computable Patterns

微分方程是对动态现象建模有用的数学描述。诸如描述流体流动、疾病传播和股票期权价格等问题，往往借助于微分方程来解决。当科学和工程专业的学生学习微分方程的技巧时，他们很难从技术性的、象征性的数学操作转向更定性的、概念性的理解。本项目探讨如何指导学生进行这样的概念理解。它将研究专家导师使用的技术和教学知识，并尝试将这些技术传授给经验不足的同行导师。它将开发新的测试来衡量学生在概念层面上理解微分方程的程度，并研究各种辅导方法的效果。它还将研究如何构建计算机辅助辅导微分方程的软件，在这个软件中，计算机可以跟踪对话，并向没有经验的导师建议一些专家方法。

Differential equations are mathematical descriptions useful for modeling dynamic phenomena. Problems such as describing fluid flow, the spread of a disease, and stock option prices are often solved by appealing to differential equations. When students of science and engineering learn the techniques of differential equations it can be difficult for them to move from technical, symbolic mathematical manipulations to a more qualitative, conceptual understanding. This project investigates how to tutor students in such conceptual understanding. It will study the techniques and pedagogical knowledge employed by expert tutors and try to teach these techniques to less experienced peer tutors. It will develop new tests to measure how well students comprehend differential equations on a conceptual level and study the efficacy of various tutorial methods. And it will investigate how to build software for computer-assisted tutoring of differential equations, where a computer follows the conversation and suggests some of the expert methods to the inexperienced tutor.