The beliefs that students and teachers hold about mathematics have been well documented in the research literature in recent years (e.g., Cooney 1985; Frank 1988, 1990; Garofalo 1989a, 1989b; Schoenfeld 1987; Thompson 1984, 1985, 1988). The research has shown that some beliefs are quite salient across various populations. These commonly held beliefs include the following (Frank 1988):
Denise A. Spangler
Allyson Hallman-Thrasher and Denise A. Spangler
We share ideas for preparing for and enacting high-cognitive demand tasks in ways that support students in articulating and justifying their ideas. We offer strategies for developing and posing several types of purposeful questions: (1) eliciting thinking, (2) generating ideas, (3) clarifying explanations, and (4) justifying claims.
Denise A. Spangler and Allyson Hallman-Thrasher
We describe an activity designed to help preservice elementary school teachers develop mathematical knowledge for teaching in the domain of facilitating mathematical discussions. The activity involved preservice teachers writing task dialogues, imaginary conversations between a child and teacher about a problem-solving task, in which they practice responding to correct, partially correct, and incorrect student responses. Preservice teachers then implemented these same tasks with children in a field experience setting. We describe 2 different iterations of the activity and field experience in detail as well as the insights into preservice teacher knowledge each iteration afforded us.