University students’ school mathematics understanding and its growth in their learning of collegiate mathematics: Through the lens of a transformative transition framework
Open Access
- Author:
- Lee, Younhee
- Graduate Program:
- Curriculum and Instruction
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 16, 2018
- Committee Members:
- Mary Kathleen Heid, Dissertation Advisor/Co-Advisor
- Keywords:
- Mathematical connections
Double discontinuity
Characteristics/qualities of mathematical understanding
Qualitative levels of mathematical understanding
Growth in mathematical understanding
Transformative transition
Teaching interview
Factorization
Polynomial equation
Unique factorization theorem
Prospective secondary mathematics teachers
Mathematics-intensive majors - Abstract:
- In the context of addressing Klein’s (1908/1924) double discontinuity, the goal of this study was to address the question of “how might university students come to see school mathematics from an advanced viewpoint in their learning of collegiate mathematics?” The research design involved developing a categorical framework of transformative transition, which delineate four different ways in which a qualitative leap might take place in one’s existing understandings as the individual encounters a new construct. The four categories of transformative transition are extending, unifying, strengthening, and deepening. The framework built on Piaget and Garcia’s triad and the APOS theory. A set of teaching interviews [TIs] was designed to provide a setting for participants’ building on what they had previously known about factorization and polynomial equations in order to construct the unique factorization theorem for polynomials. Data collection included conducting a total of 40 interviews with six mathematics-intensive majors. Five of the six participants were observed to enrich their prior understandings of factorization and polynomial equations through qualitative advancement in levels of extending, unifying, strengthening, and deepening in the course of interviews. Some participants initially demonstrated assumptions and norms that seemingly had been exercised and established in their school mathematics (e.g., separating polynomial factorization from number factorization, thinking factorization always results in a product of lesser degree polynomials). These assumptions and norms appeared to be reexamined by the participants from a different angle during the TIs, and as a result, their prior understandings seemed to be reconstructed and interrelated to form an enhanced, coherent understanding. This study reports some features of the interview context that appeared to have supported and underlain participants’ transformative transitions. Also, when participants did not seem to make transformative transitions, individual tendencies that seemed to act as obstacles to making transformative transitions were identified. A major contribution of this study is an empirical elaboration of the categorical framework, which I hope could provide mathematics educators with ways to think about how to interpret and organize their experiences of university students’ learning and how to support university students to see school mathematics from an advanced viewpoint.