| The purpose of this study was to examine prospective secondary mathematics teachers' understandings while exploring non-routine geometric problems with a dynamic geometry tool. Another purpose was to characterize the subjects' conceptions of dynamic geometry representations and to describe how those understandings influenced their mathematical exploration.;Five prospective secondary mathematics teachers participated in the study. The research design was a case study that incorporated some aspects of the teaching experiment, namely the clinical interview and the teaching episode. Four classroom observations supplemented the two interviews and seven teaching episodes were conducted with each subject. All classroom observations, interviews, and teaching episodes were audiotaped and videotaped. Two subjects were selected as case studies. Verbatim transcripts of the interviews and teaching episodes, students' coursework related to dynamic geometry tool use, students' journal entries, and the researcher's notes were analyzed from 15 hours with each of these subjects. The researcher synthesized the Teaching for Understanding (TfU) framework (Wiske, 1998) and the van Hiele levels that served as the theoretical framework for coding and analyzing data.;Both subjects were characterized at the apprentice level within the TfU knowledge dimension and at the novice level for the TfU method dimension, which corresponds to the abstract/relational level within the van Hiele model. This characterization suggests that the subjects were able to reason abstractly and realize the importance of proof in mathematics, but they exhibited a reliance on empirical verification. Although the subjects were characterized at the same levels, they differed in their willingness to continue their exploration beyond the most obvious features of the dynamic representation for the geometry problem.;Analysis of the data also supports the existence of stages of knowing with a dynamic geometry tool. The stages were the intuitive stage, the participatory stage, and the anticipatory stage. It appeared that the students were operating at a participatory stage of knowing in which they underutilized the dynamic capabilities of the tool, partly due to their lack of understanding of functional dependence within the dynamic geometry tool environment. |
