The nature of students' understanding of quadratic functions
Metcalf, Rebecca Colleen
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This case study presents a detailed description of the nature of students' understanding of quadratic functions in an undergraduate pre-calculus course in a developmental mathematics program. Three questions served to guide the study: What is the nature of students' understanding of algebraic and graphical representations of quadratics? What is the nature of students' understanding of the connections between these two representations and solutions of quadratics? To what extent, if any, do students' previous sources of knowledge and understanding about quadratics influence their understanding? Three students are described in detail. Using a "think out loud" protocol, a series of open-ended tasks administered over three interviews allowed the participants to be observed repeatedly in situations where they thought about quadratic concepts and their multiple representations. One participant confidently performed several procedures, but showed limited relational understanding of the concepts. Neither of the other two participants was proficient with procedures, and each had distinct pieces of relational understanding that were not interconnected. Neither of the participants showed much flexibility in moving between the representations, they also exhibited difficulties with communication. The tasks used in this study proved to be successful instruments for data collection. They also served as occasions for learning by the participants. Use of problems similar to these may give students opportunities to form a better understanding of the concepts of quadratics. Further research on students' understanding of these mathematical concepts and how the teaching of this topic can be improved is recommended.