Investigations in Conceptual Understanding of Polynomial Functions and the Impact of Mathematical Belief Systems on Achievement in an Accelerated Summer Program for Gifted Students

Abstract

This study investigated the achievement of gifted students on mathematics problems that were designed to assess both conceptual and procedural knowledge of polynomial functions, and it attempted to determine the impact of the students? mathematical belief systems on this achievement. The students were enrolled in a three-week Algebra II course at a summer program for gifted mathematics students. Data sources were belief scales, in-class examinations, and in-depth interviews. Qualitative and quantitative analyses indicated that the students were able to make a variety of connections among concepts related to polynomials and functions, and they easily applied their mathematical knowledge to real world phenomena. The participants suffered, however, from several misconceptions relating to the understanding of the roles of the independent and dependent variables in functions. They also struggled with the concept of symmetry and how it relates to polynomial functions. Statistical analyses suggested that belief systems were correlated with achievement, but the conclusions from this study were ambiguous since the correlations were unexpectedly negative. Through its identification of potential conceptual difficulties that gifted students may encounter in their learning of polynomial functions, this study suggested specific topics that teachers of gifted students should consider when planning their instructional activities.