Concepts and Meanings of a Formal Domain

这个项目将对高中阶段理解代数所涉及的知识和过程进行科学分析。这项研究将使用三种主要方法：一种方法是概念成长分析，这是一种在认知发展研究中卓有成效的方法。第二种方法使用数学课程中的任务和相关任务，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方法来构建学生的知识和认知过程的模型。这项研究将侧重于学生对变量和函数概念的理解，以及这种理解与他们对代数符号表达式的知识之间的关系。关于概念发展的研究将研究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，其中块的最终位置取决于几个因素，另一个是液体从一个气缸转移到另一个气缸的能力，其中液体的最终高度取决于其他几个因素。以前的研究表明，学生在学习代数之前对这些系统中的函数有很大的理解，这项研究将记录学生在学习相关形式数学时对函数理解的增加。对符号表示理解的研究将研究学生对公式和图形的含义以及它们之间关系的理解。这些研究中使用的任务将包括课程中包含的问题，以及更多旨在挖掘学生理解的特定方面的开放式任务。计算机建模研究将利用实证研究的结果，就学生在理解概念时为执行课程中的任务和其他推理任务而获得的特定知识，以及这种理解的变化和发展方式，提出明确的假设。增加对代数中概念理解的科学知识将有助于我们理解概念发展的领域和分析理解符号的含义。以前关于概念增长的研究已经研究了非正式的知识领域，如分类类别和生物过程。这项研究将通过研究代数来扩展这些分析，代数是一个具有形式结构的领域。以前对符号理解的研究大多集中在普通语言上，而对代数形式系统的理解将为理解符号表征的意义提供新的见解。这些结果也将有助于改进学校的代数教学和其他对数学理解很重要的培训。

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.