Plenary Presentation at the Commission Internationale pour l’Étude et l’Amélioration de l’Ensignement des Mathématiques, Toulouse, France, July 1994. A Versatile Theory of Visualisation and Symbolisation in Mathematics David Tall Mathematics Education Research Centre, University of Warwick COVENTRY CV4 7AL, UK This presentation will consider the roles of visualisation and symbolisation in the cognitive growth of mathematics. Visualisation plays a fundamental role throughout, both in the global overview afforded by visual diagrams and the processing of symbolism through the ability to scan written symbols and shift attention to different aspects at will. Reference will be made to the author’s development of a visual computer approach to mathematics (the theory of generic organisers) and to the subtle role of symbols standing for both process and concept (the notion of procept ). This will be placed in an extension of the theory of Bruner in which each mode of mental representation has its distinct form of objects, actions and proof. Modes of mental representation Almost thirty years ago, when current graphic computer environments were not even a dream, Bruner (1966) distinguished three different modes of mental representation – the sensori-motor, the iconic and the symbolic. In his essay “Patterns of Growth” he wrote: What does it mean to translate experience into a model of the world. Let me suggest there are probably three ways in which human beings accomplish this feat. The first is through action. […] There is a second system of representation that depends upon visual or other sensory organization and upon the use of summarizing images. […] We have come to talk about the first form of representation as enactive, the second is iconic. […] Finally, there is a representation in words or language. Its hallmark is that it is symbolic in nature. Bruner, 1966, pp. 10–11. Bruner considered that these representations grew in sequence in the individual, first enactive, then iconic and finally a symbolic representation, where the latter may have a power of its own which then depended less on the first two. Furthermore he hypothesised that “any idea or problem or body of knowledge can be presented in a form simple enough so that any particular learner can understand it in recognizable form” (ibid. p. 44). Although this claim is far-reaching, it has proved possible to appear to explain certain sophisticated logical ideas in, say, a simple visual form. However, different representations use different forms of knowledge which has advantages and hidden difficulties. In figure 1 the bold lines show Bruner’s proposed sequence of growth, together with other connections between the systems. For instance, the symbolic system passes written communication to the enactive system for writing and typing, the iconic system passes its drawing actions to the enactive system and there are many links between iconic and symbolic in mathematics, for example between symbolic functions and graphs. (2) Iconic visual & other summarising images (1) Enactive representation through action (3) Symbolic language & mathematical symbols Figure 1: Bruner’s three modes of representation