Sets, subsets, and the empty set: students' constructions and mathematical conventions

This study investigates students’ understanding of the basic concepts of introductory set theory: set, set element, cardinality, subset, and the empty set. The data was collected from a group of preservice elementary school teachers by means of written assessment, clinical interviews, and students’ participation in a computer-based project. The project included experimentation with basic set concepts in an open computerbased environment with mathematical computer language ISETL. A constructivist-oriented framework was used in analyzing the data. The results reveal complexities in students’ understanding, especially when set elements involved are sets themselves. Special attention is given to the description of students’ difficulties with the concept of the empty set. The primary concern of this study is with learners’ understanding of mathematical concepts. Specifically, the learners are preservice elementary school teachers and the concepts are underlying sets. Within the growing body of literature on research in undergraduate mathematics education, including research on teacher education and concept formation, we found only one reference—Baxter (1994)—with an explicit focus on the concepts related to set theory. Baxter presented an “action research” with a goal for enhancing teaching and learning via improving the design of learner-centered materials. “Clearly,” stated the editors in their comments on Baxter’s paper, “it is a beginning for the process of building general and theoretically

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