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