Date of Award

Winter 1995

Project Type

Dissertation

Program or Major

Mathematics Education

Degree Name

Doctor of Philosophy

First Advisor

Karen J Graham

Abstract

Mathematics educators are realizing the impact that technology is having on the way mathematical functions can be represented and manipulated. The increased use of graphing technology in the classroom is paralleled by an increased emphasis on the role of the graphical representation of a function to solve problems. These changes together with a recognition of the significance and complexity of developing a rich understanding of the graphical representations of polynomial functions are the motivation behind this research.

The study was designed to explore students' conceptual understandings of the graphs of polynomial functions. Guided by a constructivist approach to conceptual change, the investigations were primarily directed towards determining how the students' ability to interpret the graphs of polynomial functions of degree greater than two depends and builds on their understandings of the graphs of linear and quadratic functions.

Based on three case studies of students within an Algebra II class in a large high school in Northern New England, the inquiry was qualitative in nature. Clinical interviews were used to probe into the students' developing understandings of the graphs of linear, quadratic, and cubic functions. Based on findings from the clinical interviews, teaching episodes were designed in an attempt to enhance connections between the classes of polynomial functions. Data used in the analysis came from multiple sources: videotapes of the classroom instruction, collected student work, videotaped clinical interviews and teaching episodes, student journal entries, and field notes.

Results indicate that the students in the study exhibited links between their understandings of the graph of a cubic function and their understandings of the graphs of linear and quadratic functions. These connections, however, were often hindered by the classroom instructional emphases and activities as well as by conventional notation and terminology.

The focus of the teaching episodes was on enhancing the connections between the classes of polynomial functions by building polynomial functions from products of linear expressions. This method did foster connections between the classes of polynomial functions, as well as connections between the algebraic and graphical representations.

Suggested modifications to the curriculum, implications for pedagogy, and avenues for future research are given.

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