Relationships between embodied objects and symbolic procepts: an explanatory theory of success and failure in mathematics

In this paper we propose a theory of cognitive construction in mathematics that gives a unified explanation of the power and difficulty of cognitive development in a wide range of contexts. It is based on an analysis of how operations on embodied objects may be seen in two distinct ways: as embodied configurations given by the operations, and as refined symbolism that dually represents processes to do mathematics and concepts to think about it. An example is the embodied configuration of five fingers, the process of counting five and the concept of the number five. Another is the embodied notion of a locally straight curve, the process of differentiation and the concept of derivative. Our approach relates ideas in the embodied theory of Lakoff, van Hiele’s theory of developing sophistication in geometry, and the processobject theories of Dubinsky and Sfard. It not only offers the benefit of comparing strengths and weaknesses of a variety of differing theoretical positions, it also reveals subtle similarities between widely occurring difficulties in mathematical growth.

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