Concepts and Meanings of Formal Domain

形式域的概念和含义

• 批准号: 8711342
• 负责人: Alan Schoenfeld
• 金额: $ 15.11万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-09-01 至 1991-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8711342&HistoricalAwards=false
• 关键词: Concepts; Meanings; Formal; Domain

This project will develop a scientific analysis of the knowledge and processes involved in understanding algebra at the high- school level. The research will use three main approaches: One approach is analysis of conceptual growth, a method that has been used productively in the study of cognitive development. The second approach uses tasks taken from the mathematics curriculum and related tasks designed to show what students are able to do and what knowledge they have that enables them to do it. The third approach uses methods of artificial intelligence to construct models of students' knowledge and cognitive processes. The research will focus on students' understanding of the concepts of variables and functions and how this understanding relates to their knowledge of the symbolic expressions of algebra. The research on conceptual growth will study students' ability to reason about two physical systems involving functional relations, a winch in which the final position of a block depends on several factors, and a transfer of liquid from one cylinder to another, where the final height of liquid depends on several other factors. Previous research has shown that students have significant understanding of functions in these systems before they study algebra, and this research will document the increases in students' understanding as they study relevant formal mathematics. The research on understanding symbolic representations will study students' understanding of the meanings of formulas and graphs and their relations. Tasks used in these studies will include problems that are included in the curriculum, as well as more open-ended tasks designed to tap specific aspects of students' understanding. The research on computer modelling will use the results of the empirical studies to develop definite hypotheses about specific knowledge that students acquire in order to perform tasks in the curriculum and other reasoning tasks when they understand the concepts, and about the ways in which that understanding changes and grows. Increased scientific knowledge about the understanding of concepts in algebra will contribute to our understanding of the domains of conceptual growth and the analysis of understanding the meanings of symbols. Previous research on conceptual growth has studied informal domains of knowledge, such as taxonomic categories and biological processes. This research will extend those analyses by studying algebra, a domain with a formal structure. Most previous studies of symbolic understanding have focused on ordinary language, and the study of understanding the formal system of algebra will provide new insights into ways that meanings of symbolic representations are understood. Results will also be useful in the improvement of school instruction in algebra and for other training in which mathematical understanding is important.

本计画将发展一套科学分析高中代数知识与过程的方法。该研究将使用三种主要方法：一种方法是对概念增长的分析，这种方法在认知发展的研究中已经得到了有效的应用。第二种方法使用数学课程中的任务和相关任务来展示学生能够做什么以及他们拥有哪些知识使他们能够做到这一点。第三种方法是利用人工智能方法构建学生的知识和认知过程模型。研究将集中在学生对变量和函数概念的理解，以及这种理解如何与他们对代数符号表达的知识联系起来。概念成长的研究将学习学生对两个涉及函数关系的物理系统的推理能力，一个是绞车，其中一个块的最终位置取决于几个因素，另一个是液体从一个圆柱体转移到另一个圆柱体，其中液体的最终高度取决于几个其他因素。先前的研究表明，学生在学习代数之前对这些系统中的函数有了重要的理解，而这项研究将记录学生在学习相关形式数学时对这些系统的理解的增加。理解符号表征的研究将研究学生对公式和图形的意义及其关系的理解。这些研究中使用的任务将包括课程中包含的问题，以及旨在挖掘学生特定理解方面的更多开放式任务。计算机建模的研究将利用实证研究的结果，对学生在理解概念时为了执行课程任务和其他推理任务而获得的特定知识提出明确的假设。以及这种理解是如何变化和发展的。对代数概念理解的科学知识的增加将有助于我们对概念增长领域的理解和对理解符号意义的分析。以往关于概念成长的研究主要集中在非正式的知识领域，如分类范畴和生物过程。本研究将通过研究代数（一个具有形式结构的领域）来扩展这些分析。以往对符号理解的研究大多集中在日常语言上，而对代数形式系统的理解研究将为理解符号表示的意义提供新的见解。结果也将有助于提高学校代数教学和其他训练，其中数学理解是重要的。