@MISC{Wong04theoreticalconsiderations, author = {Khoon Yoong Wong}, title = {Theoretical Considerations}, year = {2004} }

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Abstract

Mathematics consists of abstract ideas. They are communicated through the use of external representations. The external representations must embody the key properties of the respective ideas. Since a particular mode of representation cannot embody an abstract idea completely, it is necessary to use more than one representation. Even mathematicians use different modes of mathematical thinking, including visual and kinetic imagery and less frequently, mental words or algebraic symbols (Hadamard, 1945). In recent years, the principle of multiple representations has attracted much attention among mathematics educators. For example, the Representation Standard of the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000) states that instructional programs should enable all students to “select, apply, and translate among mathematical representations to solve problems ” (p. 67). An instructional issue is how many modes of representations need to be used at various levels of mathematics instruction. Four decades ago, Bruner (1964) proposed three modes of representation, namely the enactive, iconic, and symbolic, to be introduced in that order when teaching mathematics. These three seem to be the minimum number of modes to use. Two decades later, Lesh, Post and Behr (1987) stressed that a student who “understands ” a mathematical idea “can (1) recognize the idea embedded in a variety of qualitatively different representational systems, (2) flexibly manipulate the idea within given representational systems,