Understanding and Teaching the Mathematical Concepts of Functions and Their Graphs

本项目将对笛卡尔平面上的函数及其图的数学概念的理解和教学进行三个相关的研发研究。这些研究(1)阐述具体材料和直观概念在学生发展对函数和图形的概念性理解中的作用；(2)开发和检验基于计算机的软件和课程，以发展对这些概念的深刻理解；(3)阐述互动在学习中的作用。通过基于计算机的辅导系统的窄带宽来表征学习的一些局限性。研究如下：(1)“基于装置”的学习与纯符号环境下的学习的比较。这些研究者计划详细阐述(a)关于物理对象的直觉在帮助学生掌握数学表征方面的作用，(b)在这一领域中稳健理解的概念基础，以及(c)概念习得的机制。(2)对创新课程(a)在学生群体多样化的学校中进行选择，重点是包括有风险的和少数民族的学生，以及(b)在项目实验室中与来自同一学校的学生样本进行测试。这些调查将探讨每种环境对其中发生的学习类型的影响，并看看从实验室到教室可以做出什么样的推断。这项工作是与互动数学项目（IMP）合作进行的。IMP被特许开发与新的加州数学“框架”一致的试点课程。这些单位有广泛销售的潜力。(3)人工辅导与计算机辅导策略的比较。使用名为OZ的基于计算机的界面，项目人员将在函数和图形上模拟基于计算机的辅导，而实际上辅导是由远距离的人完成的。“现场”和“远程”辅导的比较将提供以下方面的见解：(a)依赖于个人互动的人类导师使用的策略，以及(b)有效的机器辅导需要克服的各种限制。

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.