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WCER Working Paper No. 2010-9

Enhancing Middle School Students' Representational Fluency:
A Classroom-Based Study
Mitchell J. Nathan , Martha W. Alibali, Kate Masarik, Ana C. Stephens, and Kenneth R. Koedinger
July 2010, 29 pp.

ABSTRACT:

The present study examined middle school students’ (N = 82) representational fluency, the ability to reason with and between multiple representations, using tabular, graphical, verbal, and symbolic representations of linear and nonlinear relations. Preinstructional data replicated the complexity-based trade-off among representations: Students showed greater success using verbal representations when solving lower complexity (linear) problems, but greater success using equations when solving higher complexity (nonlinear) problems. Students then had 9 weeks of algebra instruction using either Connected Mathematics (CM), a widely adopted reform curriculum, or Bridging Instruction (BI), a novel curriculum taught by the same teacher that drew on students’ mathematical preconceptions and invented solution strategies. Both instructional approaches improved students’ abilities to solve problems using linear equations. But BI students showed larger improvements overall, with significant gains using equations and word expressions for linear and nonlinear functions. Translation between representations improved for both treatments but was far greater and broader for BI students, who showed gains in generating tables and graphs from all the input representations, as well as in translating between word expressions and symbols.

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Keywords: Early Algebra; Problem Solving; Representations; Mathematics Education; Bridging Instruction