# ALGEBRA 1 STUDENTS' ABILITY TO RELATE THE DEFINITION OF A FUNCTION TO ITS REPRESENTATIONS

#### Abstract

One hundred high school Algebra students from a southern California school participated in this study to provide information on students’ ability to relate the definition of function to its representations. The goals of the study were (1) to explore the extent to which students are able to distinguish between representations of functions/non-functions, (2) to compare students’ ability to distinguish between familiar/unfamiliar representations of functions/non-functions, (3) to explore the extent to which students are able to apply the definition of function to verify function representations, and (4) to explore the extent to which students are able to provide an adequate definition of function. Data was collected from written responses on a math survey consisting of items that asked students to decide if given illustrations are representations of functions, to explain how the decision was made, and to supply the domain and range when applicable. The questions included seven types of illustrations: graphs, equations, ordered pairs, tables, statements, arrow diagrams, and arbitrary mappings. Findings indicated that students were more able to correctly identify familiar than unfamiliar function representations. The easiest representation for students to correctly identify was the graph of a linear function and the most difficult was the graph of a piecewise function. A conjecture as to why this occurred is that the formal definition of function is not often emphasized or referenced when function and its representations are introduced so students do not have a deep understanding of how the function definition is related to its representations. The explanation, domain, and range responses were sketchy. A conjecture as to why this occurred is that in general, students have difficulty expressing themselves orally and in writing or perhaps students had not learned about domain and range. A separate question asked students, “What is a function?” To this question, students provided a variety of responses. It is suggested that conducting further studies that include student interviews and participants from multiple teachers, would provide increased understanding of how students learn the concept of function and the extent to which they are able to relate it to the definition of function.