Understanding and Teaching the Mathematical Concepts of Functions and Their Graphs

这个项目将进行三个相互关联的研发研究，以理解和教授笛卡尔平面上的函数及其图形的数学概念。这些研究(1)阐述了具体材料和直观概念在学生发展对函数和图形的概念理解中的作用；(2)开发和检验了计算机软件和课程，以加深对这些概念的理解；(3)阐述了互动在学习中的作用，表现了通过计算机辅导系统进行学习的一些局限性。研究内容如下。(1)“基于仪器的”学习与在纯符号环境中学习的比较。这些研究人员计划阐述(A)关于物理对象的直觉在帮助学生掌握其数学表征方面的作用，(B)在这一领域中强健理解的概念基础，以及(C)概念习得的机制。(2)审查创新课程：(A)根据学生群体的多样性选择学校，重点是吸纳有风险的学生和少数族裔学生；(B)在项目实验室中从同一学校的学生中抽取样本。这些调查将探索每一种环境对发生在其中的学习类型的影响，并看看从实验室到课堂可以做出什么类型的推断。这项工作是与交互式数学项目(IMP)合作进行的。IMP的特许是开发与新的加州数学“框架”一致的试点课程。这些单元具有广泛分布的潜力。(3)人机辅导策略与计算机辅导策略比较。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，而实际上辅导是由一个人远程完成的。对“现场”辅导和“远程”辅导的比较将提供对以下方面的见解：(A)人类导师使用的依赖于个人互动的策略，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.