Relationship of Covariational Reasoning on College Algebra Students' Interpretation of Function Notation

In this study, I examined links between College Algebra students' covariational reasoning and their conception of general function notation (y = f(x)). I investigated the following research questions: How might students' conceptions of function impact their conceptions of function notation? How might covariational reasoning related to function impact students' conceptions of function notation? How do students conceive of general function notation ( y = f(x))? I posit three levels of students' conceptions of function notation: "function notation as label," "function notation as convention," and "function notation as a relationship between variables," and draw connections to students' engagement in quantitative, variational, and covariational reasoning, as well as their employment of a correspondence approach to function. For this study, I report three cases of students, Jack, Dave, and Lisa, who demonstrated different conceptions of function notation and different forms of variational and covariational reasoning. These students were enrolled in a College Algebra course at a public university in a large US city. I conducted a sequence of four task-based clinical interviews with the first interview serving as the Pre interview and the last interview serving as the Post interview. I analyzed the data using Wolcott's (1994) constructs of "Description," "Analysis," and "Interpretation." I used constant comparative analysis (Corbin & Strauss, 2008) to detect any differences in reasoning from the Pre interview to the Post interview. I found a link between students' engagement in covariational reasoning and their conception of function notation: Students engaging in early levels of covariational reasoning could conceive of function notation as a relationship between variables. Furthermore, students' conceptions of the definition of function mitigated their conceptions of function notation. In addition, when engaging with different kinds of tasks, they demonstrated different conceptions of function and function notation, and engaged in different forms of covariational reasoning. To promote students' conceptions of function and general function notation (y = f(x)) expressing an invariant relationship between quantities, researchers/teachers should leverage technology-rich tasks incorporating two different graphs that represent the same relationship, and tasks providing opportunities for students to make sense of others' claims about graphs. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]