#### Abstract

INVESTIGATION OF THE PROCESS OF INSTRUMENTAL GENESIS OF 11TH GRADE STUDENTS ON THE GRAPHS OF QUADRATIC FUNCTIONS

One of the basic concepts of mathematics education is the concept of function. Learning processes in mathematics education have a complex structure. Instrumental theory is a theoretical approach that provides the analysis of these learning processes, taking into account the complexity. While the studies in the literature examine the action schemes created by the use of technological tools, this study differs in that it examines the use of the roots of a quadratic equation as a tool. The aim of this study is to examine the instrumental action schemes that appear in the graphs of the quadratic functions by using the roots of the quadratic equation with one unknown as a tool. The implementation of the research was carried out with eleventh-grade students of a public high school in the central district of Düzce province in the spring term of the 2021-2022 academic year. Participants were determined using the easily accessible sampling method. In the research, qualitative research method was adopted and case study design was used. A worksheet consisting of five questions containing the graph of quadratic functions and semi-structured interview technique were used as data collection tools. The worksheet was applied to 52 students and semi-structured interviews were conducted with six of the students who performed the worksheet application The data obtained from the worksheets and semi-structured interviews were first subjected to descriptive analysis. As a result of the descriptive analysis, six categories were created based on the answers given by the participants to the five questions on the worksheet. These categories are correct answer, incorrect answer, incomplete answer, irrelevant/indifferent answer, inconsistent information/formula, and blank. In order to analyze the solution of the questions, the instrumented action schemes of the students were tried to be determined by using the content analysis method. As a result of the findings obtained by the analysis of the worksheets, it was determined that the roots of the quadratic equation were used by the students in the solution process of the questions in the codes under the categories of correct answer, incorrect answer, incomplete answer and inconsistent information/formula. It was determined that the instrumented action schemes presented in the graphs of the quadratic functions were similar for all participants. As a result of the analysis of the data obtained from the semi-structured interviews, it was determined that asking the questions in different styles and forms did not change the instrumented action schemes used in the solutions.

Keywords
Roots of the equation, Quadratic functions, Instrumental genesis, Instrumented action schemes

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