That external representations have a pivotal role in problem solving is well understood in cognitive science. It is argued that the external representations used for conceptual learning are of comparable significance. The ontological, structural and functional properties of a representation in itself may substantially determine what is learnt and how easily it is done. Poor representations may create obstacles that exacerbate conceptual difficulties, whilst effective representations will promote coherent and well integrated networks of concepts for target domains. Six characteristics of effective representations derived from previous empirical work are considered in this paper. They are: (1) the integration of the levels of abstraction in the domain; (2) the provision of a globally homogenous but locally heterogeneous representation of concepts; (3) the integration of different perspectives or interpretations of the domain; (4) the provision of malleable expressions in the representation; (5) the use of compact procedures and (6) uniform procedures for manipulating the representation. Here the characteristics are used as guidelines for the design of a novel representation for a complex mathematical domain, specifically probability theory. The new representational system, Probability Space (PS) diagrams, uses geometric and spatial constraints to encode the laws of probability in such a way that each instantiation of a diagram (drawing) isomorphically represents the structure of a particular problem situation. PS diagrams are members of a class of diagrammatic representations called Law Encoding Diagrams (LEDs), which have been shown to support problem solving and to promote learning in science. Specific criteria for the design of effective LEDs were also used in the development of PS diagrams. PS diagrams and traditional algebra based approaches to learning probability theory are contrasted by presenting ideal solutions to a selection of difficult and counter intuitive problems. Predictions are made concerning how PS diagrams should ameliorate conceptual difficulties and thus promote learning, in contrast to the traditional approach.
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