College Students' Understanding of the Domain and Range of Functions on Graphs
Cho, Young Doo
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The mathematical concept of function has been revisited and further developed with regularity since its introduction in ancient Babylonia (Kleiner, 1989). The difficulty of the concept of a function contributes to complications when students learn of functions and their graphs (Leinhardt, Zaslavsky, & Stein, 1990). To understand the concept of a function, students must understand the sub-concepts, such as correspondence, domain, and range. A function's domain and range are critical to understanding the graph of that function. Through a review of the literature, it is apparent that many researchers have studied students' concepts of functions. However, no study has focused on how students understand the graphical representation of a function's domain and range. In this research, I explored students' transitional conceptions (often referred to by those with different theoretical framings as "misconceptions") of the domain and range of a graphical representation of a function. The research questions are as follows: 1. Which conceptions and strategies are evident when students consider the domain and range of a graphical representation of a function? 2. How do students' use of strategies and their understanding of concepts and representations impact their understanding of the domain and range of a graphical representation of a function? The findings of this study exposed that many students have diverse transitional strategies and conceptions. Twenty-two strategies and conceptions were discovered. These were categorized as (a) projecting graphs to the x-axis or y-axis, (b) following or tracing graphs, (c) working with horizontal lines and discontinuous functions, and (d) representing with interval notation. Among the 22 strategies and conceptions, four are grouped as the fully developed strategies and the other four are the partially transitional strategies and conceptions, while the remaining 14 are the transitional strategies and conceptions. Among transitional strategies and conceptions, three strategies and conceptions were predominantly used by the majority of the interviewees: (a) difficulty with the notation in representing the range of horizontal lines, (b) belief that a horizontal line or segment of a line has no range, and (c) tracing or chasing the graph from left to right. In addition, most transitional strategies and conceptions stem from measuring the range of a graph, not the domain. This implies the need for more instructional focus on measuring the range as well as for additional study on the matter.