Modeling Abstract Numeric Relations Using Concrete Notations

Abstract relational reasoning is a core component of human thinking. While abstract relations are understood using a wide variety of methods, the formal algebraic equation is among the most powerful and general mechanisms for representing relational statements. It has often been assumed that the means by which expressions represent relations are purely semantic, and are encoded in an abstract syntax that governs the use of notation without regard to the details of its physical structure (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). In contrast, we propose an image of equation construction that highlights the role of concrete physical relations in mediating the interpretation of equations. In this account, construction processes involve a structural alignment across representation systems. Alignment biases reasoners toward the selection of representations that maintain the concrete structure of source representations. We demonstrate that this approach accounts naturally for a variety of previously reported phenomena in equation construction, and correctly predicts several new phenomena.relational reasoning is a core component of human thinking. While abstract relations are understood using a wide variety of methods, the formal algebraic equation is among the most powerful and general mechanisms for representing relational statements. It has often been assumed that the means by which expressions represent relations are purely semantic, and are encoded in an abstract syntax that governs the use of notation without regard to the details of its physical structure (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). In contrast, we propose an image of equation construction that highlights the role of concrete physical relations in mediating the interpretation of equations. In this account, construction processes involve a structural alignment across representation systems. Alignment biases reasoners toward the selection of representations that maintain the concrete structure of source representations. We demonstrate that this approach accounts naturally for a variety of previously reported phenomena in equation construction, and correctly predicts several new phenomena.