Abstract
One of the foundational issues in education is pertained to teacher education and the integration of technology to better support scholars, educators and teachers to meet the needs of a 21st century classroom. This research focuses on two powerful mathematical ideas - variance and invariance - and the way they play a role in teachers' conceptualization of and reasoning about mathematical structures in a Dynamic Geometry Environment (DGE). In this exploratory study, I investigated conceptualization and reasoning through the lens of three key elements of variance and invariance: (a) discerning invariant properties; (b) interpretation; and (c) continuous variation of an object versus a set of examples. I developed a conceptual framework that is based on Piaget's work and several aspects of Gestalt theory to explain, describe and emphasize how individuals might conceptualize and perceive mathematical structures in a dynamic way using technology such as DGE. I designed four activities using the iPad app version of the Geometer's Sketchpad® that is called the Sketchpad® Explorer to provide teachers with opportunities to explore mathematical structures, in terms of variance and invariance, and to share their perception, thinking and reasoning about these objects. The primary data were collected from a set of two 45-minutes task-based interviews with each teacher. All interviews were videotaped and transcribed verbatim. The focus of my analysis was on descriptions in which participants described aspects of the DGE object related to the three elements of variance and invariance. My research contributions attempt to deepen our theoretical and practical understanding of these three elements as they pertain to the development of rich and advanced conceptualization and reasoning in the context of structures, DGE and invariance. My work offers possible implications to teacher education programs; illuminates theoretical aspects of key elements underlying invariance and variance; and informs future research in this domain. I found four important categories that invariant properties were related to: (a) shape, (b) measurement, (c) location, and (d) calculation. I concluded that these categories are essential components to the development of rich and deep conceptualization and reasoning of mathematical structures, in terms of variance and invariance, in a DGE. I also found that the teachers in this study tend not to communicate invariant properties unless they were specifically asked to consider invariance. I found throughout this work interpretations that were described and shared through operations and actions; and some of these interpretations indicate that the status of an invariant property can change to include a generalization of a given situation. Lastly, I found that in working with variance and invariance the perspective of generating examples was extremely more dominant than looking at an object under continuous variation. This finding raises concerns about conceptualization and reasoning as being mainly associated with one perspective - a set of examples - that is more static in nature than dynamic. It is concluded that future research should examine how invariance, structures, and DGE are presented in curriculum frameworks, especially when considering teaching and learning of mathematics at the high school level.