ABSTRACTTHE VAN HIELE THEORY THROUGH THE DISCURSIVE LENS:PROSPECTIVE TEACHERS' GEOMETRIC DISCOURSES BySasha WangOver the past decade, there has been an increasing trend in the mathematics education research community to study students' reasoning in the teaching and learning of mathematics, and to examine issues emphasizing the use of vocabulary, terminology, and words in the mathematics classroom. In response, this study investigates changes in prospective elementary teachers' levels of geometric thinking, and the development of their geometric discourses in the classification of quadrilaterals. In Sfard's (2008) Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing, she introduces her commognitive framework, a systematic approach to analyzing the discursive features of mathematical thinking, including word use, visual mediators, routines, and endorsed narratives. To examine thinking about geometry, this study connects Sfard's analytic framework to another, namely the van Hiele theory (see van Hiele, 1959/1985). The van Hiele theory describes the development of students' five levels of thinking in geometry. Levels 1 to 5 are described as visual, descriptive, theoretical, formal logic and rigor, respectively. This study used the van Hiele Geometry Test from the Cognitive Development and Achievement in Secondary School Geometry (CDASSG) project (Usiskin, 1982) as the pretest and posttest to determine prospective elementary school teachers' van Hiele levels. This study also produces, on the basis of theoretical understandings and of empirical data, a detailed model, namely, the Development of Geometric Discourse. This model translates the van Hiele levels into discursive stages of geometric discourses with respect to word use, visual mediators, routines, and endorsed narratives. This study reveals discursive similarities and differences in participants' geometric discourses at the same van Hiele level, as well as changes in geometric discourse as a result of changes in levels of geometric thinking. The study also investigates the usefulness of a discursive framework in providing "rich descriptions" of participants' thinking processes.
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