Concepts and Meanings of Formal Domain

该项目将对知识进行科学分析， 和理解代数的过程 学校水平。 研究将采用三种主要方法： 方法是概念增长的分析，这种方法已经被 在认知发展研究中卓有成效地使用。 的 第二种方法使用数学课程中的任务 以及相关的任务，旨在展示学生能够做什么 以及他们所拥有的知识使他们能够做到这一点。 第三种方法使用人工智能的方法， 建构学生的知识与认知过程模型。 研究将侧重于学生对 变量和函数的概念以及这种理解 与他们对象征性表达的知识有关， 代数 概念发展的研究主要是研究学生的概念发展， 推理两个物理系统的能力， 关系，一个绞车，其中一个块的最终位置取决于 几个因素，和液体从一个缸转移到 另一个，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 之前对这些系统中的功能有深入的了解 他们学习代数，这项研究将记录 在学生学习相关的正式课程时， 数学 符号理解研究 代表将研究学生的理解 公式和图形的含义及其关系。 使用的任务 在这些研究将包括问题，包括在 课程，以及更多的开放式任务，旨在挖掘 学生理解的具体方面。 研究 计算机模型将使用实证研究的结果 发展关于特定知识的明确假设， 学生获得，以便执行课程中的任务， 其他推理任务，当他们理解的概念， 这种理解的变化和发展的方式。 增加科学知识，了解 代数中的概念将有助于我们理解 概念发展领域与理解分析 象征的意义。 关于概念增长的研究 研究了非正式的知识领域，如分类学， 分类和生物过程。 这项研究将扩大 这些分析通过学习代数，一个域与一个正式的 结构 以往大多数关于符号理解的研究 专注于普通语言，以及对理解 代数的形式系统将提供新的见解， 理解符号表示的含义。 结果 也将有助于改善学校教学， 代数和其他数学训练 理解是重要的。

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.