Understanding and Teaching the Mathematical Concepts of Functions and Their Graphs

本项目将进行三项相互关联的研发研究， 理解和教授函数的数学概念， 它们在笛卡尔平面上的图形。 这些研究（1）阐述了 具体材料和直观概念在学生发展中的作用 函数和图形的概念理解（2）发展和 研究基于计算机的软件和课程， 这些概念的理解和（3）阐述的作用 学习中的相互作用，表现出学习的一些局限性 通过基于计算机的辅导系统的窄带宽。 的 研究如下。 (1)“基于仪器”的学习与 纯粹的象征性环境。 这些研究人员计划详细说明（a） 关于物理对象的直觉在帮助学生理解中的作用 掌握它们的数学表示，（B）概念 在这一领域的坚实理解的基础，以及（c） 概念获得的机制。 (2)（a）在选定的学校中审查创新课程， 学生群体的多样性，重点是 包括处于风险中的学生和少数民族学生，以及（B）在项目中 实验室从同一所学校的学生中抽取样本。 这些 调查将探讨每种设置对各种 学习其中发生的事情，看看有什么样的推断， 从实验室到教室都可以做到。 这项工作是在 互动数学项目（IMP） IMP是 特许开发与《公约》相一致的试点课程， 新的加州数学“框架”。 这些单位有潜力 广泛分布。 (3)比较以人为基础的辅导策略和以计算机为基础的辅导策略。 使用 一个名为OZ的基于计算机的界面，项目人员将模拟 基于计算机的函数和图形辅导，当辅导是在 一个人在远处做的事情。 “live”和“at a”的比较 远程”辅导将提供洞察（a）使用的策略， 依赖于个人互动的人类导师，以及（B） 需要克服的有效的机器辅导的constants。

This project will conduct three interrelated R & D studies on understanding and teaching the mathematical concepts of functions and their graphs in the Cartesian plane. These studies (1) elaborate the role of concrete materials and intuitive notions in students'developing conceptual understandings of functions and graphs' (2) develop and examine computer-based softare and curricula for developing deep understandings of these concepts' and (3) elaborate the role of interactions in learning, characterizing some limitations of learning via the narrow bandwidth of computer-based tutorizl systems. The studies are as follows. (1) A comparison of "apparatus-based" learning with learning in a purely symbolic environment. These investigators plan to elaborate (a) the role of intuitions about pysical objects in helping studends come to grips with their mathematical representations, (b) the conceptual underpinnings of robust understanding in this domain, and (c) the mechanisms of concept acquisition. (2) The examination of innovative curricula (a) in schools chosen for the diversity of their student populations, with an emphasis on including at risk and minority students, and (b) in the project laboratory with a sample of students from the same schools. These investigations will explore the effects of each setting on the kinds of learning that take place in it, and to see what kinds of extrapolations can be made from the laboratory to the classroom. This work is in collaboration with the Interactive Mathematics Project (IMP). IMP is chartered with the development of pilot curricula consistent with the new California Mathematics "Framework". The units have the potential for wide distribution. (3) A comparison of human and computer-based tutoring strategis. Using a computer-based interface named OZ, the project staff will simulate computer-based tutoring on functions and graphs, when the tutoring is in fact done by a person at a distance. Comparisons of "live" and "at a distance" tutoring will provide insights into (a) the strategies used by human tutors that depend on personal interactions, and (b) the kinds of constaints that need to be overcome for effective machine tutoring.