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Dr Nicole Fonger, University of Wisconsin-Madison
Student's understanding of functions is central to school algebra in which two perspectives are valued: a change and variation perspective, and a structural or symbolic perspective. An oft-cited goal of algebra curriculum and instruction is to cultivate students' abilities to create, interpret, and connect numeric, graphic symbolic, and verbal representation of functions. Research in this domain often focuses on students' representational performances to the detriment of understanding what representations mean to students and how this representational activity is supported or constrained by students' conceptions of mathematical ideas. In this talk I will present a research study in which I explore the nature of students' conceptions of functions, the ways in which these understandings support their ability to symbolize quadratic function rules, and the meanings students make of these rules. I analyzed middle school students' problem solving activity during a 15-day small group teaching experiment (n=6) emphasizing quadratic growth and covariation of quantities. Results indicate four modes of reasoning supportive of students' symbolization of quadratic function rules: (a) correspondence, (b) variation and correspondence, (c) covariation, and (d) flexible covariation and correspondence. For students with strong covariational conceptions of functions, their symbolizations of quadratic "correspondence" rules may indeed represent covariational relationships between quantities. I will discuss implications for research on learning, and the design of tasks and instruction to support students' representational fluency and quantitative reasoning
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