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.
