UNDERGRADUATE STUDENTS’ APPROACHES TO CONSTRUCTING MATHEMATICAL GENERALITIES
Open Access
- Author:
- Graysay, Duane Taylor
- Graduate Program:
- Curriculum and Instruction
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 08, 2016
- Committee Members:
- Mary Kathleen Heid, Dissertation Advisor/Co-Advisor
Mary Kathleen Heid, Committee Chair/Co-Chair
Glendon W. Blume, Committee Member
Rose Mary Zbiek, Committee Member
Richard Alan Carlson, Outside Member - Keywords:
- mathematics education
undergraduate mathematics
mathematical processes
advanced mathematical thinking
generalizing
mathematical generality - Abstract:
- This dissertation reports results of a study of the ways that mathematics-intending university students construct mathematical generalities, which are general statements associated with domains of mathematical objects. Such generalities are important elements of mathematical knowledge, and there is little research examining how such generalities are constructed outside of attention to the process of generalizing among K- 12 students. The goal of this research was to develop descriptions of the process of generality-constructing among University students in mathematics-intending majors to provide new insights into how individuals approach the process. Data consisted of responses from ten students to tasks of three types: Characterizing tasks, which request constructing a general property statement about a given domain; Populating tasks, which request constructing a domain of objects that satisfy a given general property; and Reconstructing tasks, which request revising a given generality to encompass a broader domain. Findings with respect to Characterizing type tasks are that individuals used superficial features to sort collections, peculiarized examples that did not fit the sorting criteria, and regularized collections by suggesting alternatives to peculiarized elements. Findings from Populating type tasks indicate that participants represented elements of the domain at various levels of generality, which had implications for the approaches through which they constructed domains to satisfy the given property statement. Findings from Reconstructing type tasks suggest some participants attempted to understand relationships between given information in order to scaffold construction of a iii iv generality by logical deduction. Others reduced such tasks to those of the Characterizing type or of the Populating type. Implications include potential learning goals for the development of generality- constructing and highlight the importance of attending to the ways in which curriculum provides opportunities for students to engage generality-constructing. Directions for further research include exploring ways of promoting the development of each approach.