
05807765
j
2010f.00773
Caglayan, G\"unhan
Olive, John
Eighth grade students' representations of linear equations based on a cups and tiles model.
Educ. Stud. Math. 74, No. 2, 143162 (2010).
2010
Springer Netherlands, Dordrecht
EN
H33
C33
D23
linear equations
cognitive dissonance
cognitive operation
manipulatives
models and modeling
quantitative reasoning
quantitative unit conservation
quantitative unit coordination
referential relationship
representation
classroom research
lower secondary
doi:10.1007/s106490109231z
Summary: 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, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glaserfeld (Ed.), Radical constructivism in mathematics education (pp. 5374). The Netherlands: Kluwer Academic Publishers; Kaput, J. J., Blanton, M. L., \& Moreno, L. (2008). Algebra from a symbolization point of view. In J. J. Kaput, D. W. Carraher, \& M. L. Blanton (Eds.), Algebra in the early grades (pp. 1955). New York: LEA \& NCTM). 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 representationsdrawn pictures and the algebraic equation) experienced by students and teacher.