Representations that create informative visual displays are powerful tools for communicating mathematical concepts. The National Council of Teachers of Mathematics encourages the use of manipulatives (NCTM 2000). Manipulative materials are often used to present initial representations of basic numerical principles to young children, and it is through these early developmental experiences that children frequently receive their first introduction to formal mathematics. Manipulative displays are well suited to serve as proxies for real-world problems, taking on the role of representing quantities. Teachers intuitively assemble manipulative displays to construct such representations, often attaching language to scaffold the real-world connections they are trying to portray. Many children prosper from the interaction; however, others do not. For the latter, confusion can result when the arrangement of the display is not clearly connected to the concept being taught (Garcia-Mila, Marti, and Teberosky 2004).
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