Mathematics is an intellectually demanding, challenging, and exciting content area and, for some students, a content area upon which a future career can be built. The child's informal mathematical future begins with the parent but formally begins the moment she or he walks through the schoolhouse door. Teachers of mathematics at every grade level, starting at kindergarten, must understand mathematics conceptually in addition to the algorithms that they learned in their formal schooling. This conceptual understanding is developed through multiple representations, such as physical objects, drawings, charts, graphs, and symbols. At the middle school level, the primary challenge is to continue the conceptual understanding through multiple representations when offering advanced content to the mathematically talented student. A secondary challenge is to allow that student the ...

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