Middle school students' sense-making of algebraic symbols and construction of mathematical concepts using symbols
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The purpose of this study was to investigate how students constructed meaning for algebraic symbols and mathematical concepts with symbols in relation to narrative, tabular and graphical representations. This study focused on four seventh-grade students who were beginning to learn algebra with the mathematical context of representing changing situations with two variables in relation to each other. Kaput’s (1991) referential relationship model guided the present study as the major theory in combination with the other complementary theories such as the Structure of Observed Learning Outcomes (SOLO) taxonomy (Biggs & Collis, 1982) and the procept model (Tall et al., 2001) to investigate students’ referential relationships. This study was conducted within the activities of an ongoing project, Coordinating Students’ and Teacher’s Algebraic Reasoning (CoSTAR), funded by the National Science thFoundation. The four participating students were selected from Ms. Moseley’s 7 grade class, who participated in one of the case studies of the CoSTAR project. Data were collected in the * The present study was supported by the National Science Foundation under Grant #0231879. The opinions, findings, and recommendations expressed in this study are those of the author and do not necessarily reflect the views of the National Science Foundation. form of videotaped interviews with pairs of students based on classroom activities. The method for analysis of videotaped data was informed by iterative videotape analysis. The results of the data analysis mainly explained the process and the nature of students’ referential relationships according to the three bi-directional ways of referential relationships centered around algebraic notations: algebraic ” narrative, algebraic ” tabular, and algebraic ” graphical. Going from algebraic to the other forms of representation appeared more difficult than the other way around. Also, students’ conceptions of variables and rates in the referential relationships played an explanatory role. The conclusion of this study raised relevant issues for understanding students’ referential relationships. The issues included students’ appreciation of representing changing situations in various forms of representation in mathematics, their understanding of algebraic equations as an abstract form of representation, and their conceptions of variables and rates in the referential relationships. These issues also suggest instructional implications for teachers and mathematics educators to help enhance students’ understanding. Some implications for future research were also discussed.