The Role of Paradox in Argumentation and Concept Transformation in a Community of Mathematical Inquiry : A Dialectical Analysis

This study is an inquiry into (1) the role of paradox as an organizing structure for students\u27 argumentation, and a trigger for and a mediator in cognitive change; and (2) the nature and character of conceptual transformation through a process of argumentation in a community of mathematical inquiry. Its theoretical approach is based on Vygotsky\u27s theory of social cognition, which in turn is grounded in the broader framework of Hegelian and Marxist dialectical theories, and systems theory. The study spans several areas of scholarship and research: (1) the use of paradox-resolution as an organizing activity for classroom discussion; (2) students\u27 reasoning in general and reasoning with paradoxes in particular; (3) the structure and dynamics of collaborative argumentation; (4) the process of mathematical concept-building and transformation and how it evolves in argumentation; and (5) all of the above embedded in a pedagogical context characterized by communal dialogical inquiry. It is primarily a qualitative study with a quantitative component. The primary data consist of 19 transcripts of classroom discussions in a community of inquiry format in which 21 fifth grade students participated over a period of one academic year. The study found that different types of paradoxes provided for different structures of argumentation, and consequently supported different types of learning, associated with first- and second-order cognitive change. Further, it was concluded that paradox fostered integrated reasoning—a multidimensional pattern which incorporates formal, informal, interpersonal, and philosophical reasoning—in the process of collective argumentation. In addition, the study performed a quantitative analysis of the development of argumentation structure and its transformation over the course of the discussions. The changes registered in argumentation patterns were interpreted as an outcome of a dialectical interplay between the macro- and the micro-structure of argumentation, where the quantitative changes registered on a micro-level resulted in qualitative shifts on the macro-level of the argumentation structure. Finally, the study explores mechanisms of conceptual transformation in a dialogical group setting, and concludes that the process follows a dialectical model, which the author develops and illustrates through an analysis of a group discussion of the concept of infinity