Graduation Term

Summer 2013

Degree Name

Doctor of Philosophy (PhD)

Department

Department of Mathematics

Committee Chair

David Barker

Committee Co-Chair

Edward Mooney

Committee Member

Amee Adkins

Committee Member

Saad El-Zanati

Committee Member

Craig Cullen

Abstract

The difficulty in observing, recording, and examining internal representations has been well documented (Goldin & Shteingold, 2001). However, the important role that these internal representations play in the learning and understanding of mathematical concepts has been noted (Yackel, 2000). This study sought to develop a framework for examining the internalization patterns of school-aged children, gain a deeper understanding of how these internal representations are developed and used, and search for internalization patterns associated with successful generalizations.

Semi-structured interviews were conducted with six sixth-grade students and six tenth-grade students. During these interviews the students were asked to solve a series of varying algebraic tasks. The data gathered through these interviews were used to extend and adapt an existing internalization framework developed in a previous study involving third-grade students (Barker, Cross, Witkowski, & Fisher, 2010). This framework consisted of representation categories created to describe the external and internal representations used by the students. The remaining analysis consisted of using the new K-12 Internalization Framework to identify and define patterns that described how students were making use of their internalization process. These patterns were based on when, if ever, the student used an external representation, and when, if ever, the student used an internal representation. While searching for patterns associated with success, I decided to only consider tasks where the student was being challenged appropriately. Because the number of tasks considered for this phase of analysis was small, I was unable to satisfactorily determine internalization patterns associated with success. The findings from this study suggest that it is important to allow students to use small cases to build deep and meaningful internal representations that can be reproduced and extended when appropriate. Future research calls for more information regarding other grade levels to strengthen the internalization framework used in this study.

Access Type

Dissertation-Open Access

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Included in

Mathematics Commons

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