The role of reflective abstraction in the conceptualization of infinity and infinite processes

James Alan Petty, Purdue University

Abstract

The purpose of this study is to explore and elaborate on the role that reflective abstraction plays in the individual's construction of knowledge about infinity and infinite processes. Particular emphasis is given to how individuals reflect upon their own conceptions of infinity and infinite processes, the structures which are formed, and the progressively higher levels of conceptual development displayed. In order to clarify the role of reflective abstraction, a four stage model is proposed involving the role of infinity in the individual's understanding of limiting processes and is used in conjunction with a model describing six levels of increasingly abstract reflective activity. This results in focusing upon both the activity used by the individual, as well as, the resulting conceptual structures developed. The explanation of the role of reflective abstraction in this process is guided by a Cognitive constructivist theoretical framework following Piaget. This framework assumes that as problematic tasks are encountered individuals will reconstruct or reorganize their conceptual structures in the process of solving these tasks. Four students are interviewed over an eight week period as they attempt to resolve problematic situations involving infinity and infinite processes. The tasks for this interviewing format are designed to create conflict within the conceptual structures of each individual. As each student attempts to resolve his/her conflict, the interviewing format provides an opportunity for observing how these conceptual structures are utilized and subsequently altered. These tasks are also designed with enough similarity as to provide an opportunity to observe how these altered conceptual structures influence the solution activity of each individual. Four detailed student case studies are provided based on written and verbal responses in conjunction with transcribed video tapes. Based upon these case studies, the solution activity produced increasingly abstract levels of reflective thinking which resulted in students being able to predict their potential solution activity. The complexity of the students' conceptual structures were altered as they began to use their own conceptual structures as objects of reflection. The importance of this finding suggests that by generating increasingly higher levels of abstract reflective thinking, students will increase their understanding of limits and infinite processes.

Degree

Ph.D.

Advisors

Harel, Purdue University.

Subject Area

Mathematics education|Cognitive therapy|Educational psychology

COinS