Preservice teachers' abilities and learning processes involving the composition of definitions of quadrilaterals.

摘要

In the following qualitative study I have used a combination of pre- and post- surveys and a conjecture driven teaching experiment to further explore pre-service elementary teachers' abilities and learning processes involving the composition of definitions. My first research question explored how pre-service elementary teachers defined types of quadrilaterals before enrolling in a college level geometry content course and what these definitions reveal about their concept of mathematical definitions in general. The main findings suggest that students have much stronger conceptions of squares, parallelograms, and rectangles than they do for kites, trapezoids, and rhombi, as evidenced by greater overall success when defining them. These students may not fully understand the form and purpose of a mathematical definition as evidenced by their limited success writing definitions which contain necessary, sufficient, and minimal conditions, and the frequency with which they write definitions based on typical examples rather than classes of figures. Definitions which were complete enough to be classified were fairly evenly split between representing inclusive hierarchical relationships between shape types and exclusive partitional relationships. These findings present three different profiles of what knowledge a student may bring into a geometry content course.;My second research question focused on how pre-service elementary teachers' thinking evolved in the context of tasks which emphasize the relationship between mathematical definitions of quadrilaterals and the sets of objects to which they correspond. It specifically explored how they learn about necessary, sufficient, and minimal sets of properties and how they learn about hierarchical and partitional definitions. The main findings of this question are: (1) Using a sorting of sample shapes helps students to gradually improve their ability to define and to think on more sophisticated levels. (2) Asking specific questions to help students assess their own definitions focusing on the way the definition functions helped them to recognize the importance of necessary, sufficient, and minimal conditions and to apply these standards when defining in the future. (3) Students working to understand a hierarchical relationship between two shape types must first recognize hierarchies among the properties of each shape.