Relationships among AP calculus teachers' pedagogical content beliefs, classroom practice, and their students' achievement

Graduation date: 1997This study investigated associations among teachers' pedagogical content beliefs,\ud approaches to teaching, and their students' achievement in a high school Advanced\ud Placement (AP) Calculus setting. The three major research questions concerned: (a)\ud how well Al' calculus teachers' pedagogical beliefs about mathematics, curriculum, and\ud instruction aligned with a constuctivist point of view, (b) how AP teachers' pedagogical\ud content beliefs were reflected in their approaches to teaching and goals for instruction,\ud and (c) relationships among AP teachers' pedagogical content beliefs, approaches to\ud teaching, and their students' achievement.\ud Teachers' pedagogical content beliefs were categorized as cognitively based\ud (CB) or less cognitively based (LCB) using a belief questionnaire adapted from research\ud with first-grade teachers. Telephone interviews with nine CB and eight LCB teachers\ud served to provide additional insight into their beliefs and to gain information on how\ud teachers approach teaching differential calculus. A researcher-designed Differentiation\ud Test was administered to assess student achievement.\ud Interviews with nine CB teachers and eight LCB teachers revealed: (a) CB\ud teachers were more likely to believe the role of the teacher was that of a facilitator/guide\ud and the role of the student was to explore. LCB teachers were more likely to believe\ud their role as that of a knowledge base, and the role of the student was to learn from the\ud teacher. (b) CB teachers' self-reported classroom practices were found to be more\ud student-centered and conceptual in nature than LCB teachers. (c) CB teachers were\ud more likely to use word problems when introducing topics, emphasize student\ud involvement, have their students work in groups, emphasize visual approaches to topics,\ud and consider students' knowledge when planning instruction. LCB teachers were more\ud likely to present rules and theorems, work examples, and require students to memorize\ud rules of differentiation.\ud Students of CB teachers were found to have a better conceptual understanding\ud of differential calculus than students of LCB teachers. Students of CB teachers were\ud better able to interpret graphical information and to interpret information given in a table.\ud No differences were found in students' ability to work with symbolic information or to\ud use a graphing calculator in conjunction with problem solving