A Framework for Research and Curriculum Development in Undergraduate Mathematics Educationjpeg

Abstract

Over the past several years, a community of researchers has been using and re ning a particular framework for research and curriculum development in undergraduate mathematics education. The purpose of this paper is to share the results of this work with the mathematics education community at large by describing the current version of the framework and giving some examples of its application. Our framework utilizes qualitative methods for research and is based on a very speci c theoretical perspective that is being developed through attempts to understand the ideas of Piaget concerning reective abstraction and reconstruct them in the context of college level mathematics. Our approach has three components. It begins with an initial theoretical analysis of what it means to understand a concept and how that understanding can be constructed by the learner. This leads to the design of an instructional treatment that focuses directly on trying to get students to make the constructions called for by the analysis. Implementation of instruction leads to the gathering of data, which is then analyzed in the context of the theoretical perspective. The researchers cycle through the three components and re ne both the theory and the instructional treatments as needed. In this report the authors present detailed descriptions of each of these components. In our discussion of theoretical analyses, we describe certain mental constructions for learning mathematics, including actions, processes, objects, and schemas, and the relationships among these constructions. Under instructional treatment, we describe the components of the ACE teaching cycle (activities, class discussion, and exercises), cooperative learning and the use of a mathematical programming language. Finally, we describe the methodology used in data collection and analysis. The paper concludes with a discussion of issues raised in the use of this framework, followed by an extensive bibliography.