Published in Proceedings of PME25, Utrecht, July, 2001, 65–72. Relationships between embodied objects and symbolic procepts: an explanatory theory of success and failure in mathematics Eddie Gray and David Tall Mathematics Education Research Centre University of Warwick, UK <E.M.Gray@warwick.ac.uk>, <David.Tall@warwick.ac.uk> 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 process- object 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. Introduction The theory presented here builds on work that has developed steadily over the last two decades (Gray, 1991; Tall & Thomas, 1991; Gray & Tall, 1994; Tall, 1985, 1995). But it is not a simple restatement of earlier theories. A simple switch of viewpoint is seen to reveal powerful insight into very different ways in which individuals construct mathematical concepts. To gain insight into this viewpoint, we consider the situation in which embodied objects are perceived by and acted upon by individuals. (The precise nature of embodied objects will be discussed in more detail shortly, but essentially they begin with human perceptions using the fundamental senses, and become more mentally based through reflection and discussion over time.) Our viewpoint then compares the developing embodied meanings of the objects and their configurations with other mental constructions relating processes and concepts through the use of symbolism. Our purpose is to compare the meaning embodied in the objects and their configurations on the one hand with process-object abstraction on the other. We seek a theory with the power of both explanation and prediction of the varied nature of cognitive development throughout mathematics. We require a viewpoint that is theoretically sound yet has a simple and practical meaning relevant to the spectrum of practitioners from teachers of young children to university mathematicians.