Abstract
This study examines eighth grade students’ use of a representational metaphor (cups and tiles) for writing and solving equations in one unknown. Within this study, we focused on the obstacles and difficulties that students experienced when using this metaphor, with particular emphasis on the operations that can be meaningfully represented through this metaphor. We base our analysis within a framework of referential relationships of meanings (Kaput 1991; Kaput, Blanton, and Moreno, et al. 2008). Our data consist of videotaped classroom lessons, student interviews, and teacher interviews. Ongoing analyses of these data were conducted during the teaching sequence. A retrospective analysis, using constant comparison methodology, was then undertaken in order to generate a thematic analysis. Our results indicate that addition and (implied) multiplication operations only are the most meaningful with these representational models. Students also very naturally came up with a notation of their own in making sense of equations involving multiplication and addition. However, only one student was able to construct a “family of meanings” when negative quantities were involved. We conclude that quantitative unit coordination and conservation are necessary constructs for overcoming the cognitive dissonance (between the two representations—drawn pictures and the algebraic equation) experienced by students and teacher.
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Acknowledgments
The research reported in this article was supported by the National Science Foundation under Grant REC-0231879. The opinions expressed are our own and do not necessarily reflect the views of the National Science Foundation. We would like to thank Ms. Jennings and her students for allowing us to be part of their classroom.
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Caglayan, G., Olive, J. Eighth grade students’ representations of linear equations based on a cups and tiles model. Educ Stud Math 74, 143–162 (2010). https://doi.org/10.1007/s10649-010-9231-z
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DOI: https://doi.org/10.1007/s10649-010-9231-z