Graduate Thesis Or Dissertation
 

College algebra students' understanding and algebraic thinking and reasoning with functions

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  • The purpose of study was to investigate college algebra students' understanding of function concepts. In addition, their solution strategies and algebraic thinking and reasoning were explored. Twenty-four volunteer students from one college algebra recitation class participated in the study to access their understanding of functions. Five out of 24 volunteer students were selected to participate in problem-solving interview sessions to provide a rich description of their understandings of functions and their algebraic thinking and reasoning. A Function Understanding Questionnaire was administered to gather these college students' understandings of functions after they completed the college algebra course. The questionnaire consisted of four questions asking students to identify their understanding of: (1) the definition of function, (2) multiple representations of functions, (3) the use of functions in doing mathematics, and (4) the use of functions in the real-world situations. Formal interviews prior to, during, and after instruction on functions with the five students were conducted, and their work on homework problems, quizzes and tests were explored to clarify these college students' understanding of functions and to explore their solution strategies and algebraic thinking and reasoning while solving problems. Overall, instruction supported students' understanding of functions. The students' definitions of a function improved toward a more formal definition. In addition, students had a better understanding of multiple representations, function transformations, and the application of functions to new mathematical situations and to real-world situations after completing the course. Algebraic reasoning includes the ability to approach and solve mathematical problems in multiple ways. The students in this study were able to use different methods to solve mathematical problems when they were encouraged to do so. However, the instruction did not encourage this activity. Perhaps for this reason, their algebraic thinking and reasoning abilities did not seem to progress as much. In concert with the recommendation of the several mathematics education organizations, more research needs to deal with the development of algebraic thinking and reasoning. In addition, research involving the communication of mathematical ideas and connection of mathematical understanding, thinking, and reasoning to other mathematics disciplines, to different subject areas, and to real-world situations are recommended.
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