HALVES, PIECES, AND TWOTHS: CONSTRUCTING AND USING REPRESENTATIONAL CONTEXTS IN TEACHING FRACTIONS
Deborah Loewenberg Ball Michigan State University
Learning to teach mathematics for understanding is not easy. First, teaching itself is complex. Second, many teachers' traditional experiences with and orientations to mathematics and its pedagogy are additional hindrances. This chapter examines the territory of practice and reviews some of what we know about those who would traverse it -- prospective and experienced elementary teachers. In analyzing practice, the author focuses on one major aspect of teacher thinking in helping students learn about fractions: the construction of instructional representations. Considerations entailed are analyzed and the use of representations in the classroom is explored. The term representational context is used to call attention to the interactions and discourse constructed in a classroom around a particular representation. The author provides a window on her own teaching practice in order to highlight the complexity inherent in the joint construction with students of fruitful representational contexts. The article continues with a discussion of prospective and experienced teachers' knowledge, dispositions, and patterns of thinking relative to representing mathematics for teaching. The author argues that attempts to help teachers develop their practice in the direction of teaching mathematics for understanding requires a deep respect for the complexity of such teaching and depends on taking teachers seriously as learners.
Current discourse about the desirable ends of mathematics teaching and learning centers on the development of mathematical understanding and