## 1. Introduction

The representational epistemic (REEP) approach is being developed as a method for the analysis and design of complex representations and visual displays. It has been used to design novel diagrams to support demanding task domains involving large quantities of information, including examination timetabling (Cheng, Barone, Cowling, & Ahmadi, 2002), personnel rostering (Cheng & Barone, 2004), and manufacturing production and scheduling (Cheng & Barone, 2007). Novel diagrammatic systems for learning in conceptually challenging topics in science have also been invented, including electricity (Cheng, 2002) and kinematics (Cheng, 1999). The cognitive benefits of the novel representations have been successfully demonstrated in the laboratory (Cheng, 2002) and authentic instructional contexts (Cheng & Shipstone, 2003). For example, after 120 min of instruction with the AVOW diagrams for electricity young adult participants with little prior knowledge were able to solve problems that are challenging for students who have completed conventional courses of instruction on the topic (Cheng, 2002).

Generalizing over these studies, the central tenet of the REEP approach is that the *fundamental conceptual structure* of a target knowledge domain should be directly encoded in the structure of the representational system. Fundamental conceptual structure refers to the principal invariants, regularities, symmetries, constraints, and laws that essentially make the domain what it is, rather than some other domain. It is claimed that when the fundamental conceptual structure of a domain is directly encoded, the representational system is likely to have *semantic transparency* and *plastic generativity* (Cheng, 2002; Cheng et al., 2002). Semantic transparency concerns the availability of the conceptual content of the domain; how easily concepts can be accessed through the representational system. Plastic generativity concerns the ease of manipulating the components of a representational system to generate meaningful expressions during reasoning and problem solving. Obviously, a representation that enables its users to readily comprehend the meaning of its expressions is a desirable goal for design, as is a representation that allows meaningful statements, and only meaningful statements, to be simply and quickly derived without error. The REEP approach proposes four different design principles (Cheng & Barone, 2007), which will be enumerated and discussed in the third section of the paper. These *epistemic* principles consider how the fundamental sets of concepts that constitute a domain should each be encoded in different representational schemes and how those schemes should be coherently interrelated. Representational schemes are things such as coordinate systems, hierarchical trees, syntactic notations, spatial configurations, and geometric relations.

The primary goal of this paper is to provide further support for the claim that encoding the fundamental conceptual structure of a domain directly in the structure of the design of a representational system will yield an effective representation with semantic transparency and plastic generativity. The REEP approach is used to recodify the conceptually challenging domain of probability theory with the design of a novel diagrammatic system—probability space (PS) diagrams. The creation of PS diagrams demonstrates how to analyze the fundamental conceptual structure of a domain and how the epistemic design principles may be applied. An experiment is also reported to evaluate the relative benefits of PS diagrams and the conventional approach to learning about probability.

The REEP approach differs from other techniques in terms of its assumption about what should be the basis for analysis and design. Various approaches consider that the structure of task activities should be the focus (e.g., Endsley, Bolté, & Jones, 2003; Vincente, 1996). They provide methods for identifying the hierarchy of goals for particular classes of tasks and give guidelines for the design of displays that make information needed to support those goals readily apparent. In contrast, the REEP approach concentrates on the fundamental conceptual structure of the target domain and claims that representations design on this “higher” level may support a range of tasks in a domain, although not necessary as well as a bespoke display specially created for a particular problem. Many approaches focus on the information dimensions of the target domain and attempt to map those types of information identified to visual properties or representational formats (e.g., Card, MacKinlay, & Shneiderman, 1999; Engelhardt, 2002; Zhang, 1996). In the REEP approach, matching surface-level informational dimensions to graphical properties is a secondary concern, because the larger scale representational structures that it creates to directly encode fundamental conceptual structures provides stringent constraints on permissible lower level mappings of types of information to graphical properties.

Probability theory provides an interesting test case for the REEP approach for five reasons. (a) It has a rich conceptual structure with a variety of underpinning laws that are applied to diverse situations. (b) It combines two domains of knowledge: (i) set theory and combinatorics; (ii) the theory of chance or stochastics. (c) There are alternative Bayesian and Frequentist interpretations of probability, and alternative measures of quantities of chance in terms of probabilities and odds. (d) It is imaginable that the conventional representations for the topic constitute an effective encoding of the domain, because eminent mathematicians have worked on the notations for over three centuries. Hence, creating a better representation will be an achievement for the REEP approach. (e) The counterintuitive and paradoxical nature of the domain has been well documented in the literature (e.g., Austin, 1974; Falk,1992; Fischbein & Schnarch, 1997; Garfield & Ahlgren, 1988; Kahneman, Slovic, & Tversky, 1982; Shafir, 1994; Shaughnessy, 1992; Shimojo & Ichikawa, 1989), and approaches to support reasoning and instruction, including innovations with visual models, are yet to make an impact on the majority students (Armstrong, 1981; Cosmides & Tooby, 1996; Dahlke & Fakler, 1981; Gigerenzer & Hoffrage, 1995; Ichikawa, 1989; Shaughnessy, 1992).

The first of the following five sections will examine the conceptual structure of the domain as commonly portrayed in current courses on probability. The second section considers the design of PS diagrams by initially analyzing the conceptual structure of the domain and specifies how the conceptual structure is encoded in the new representation using the epistemic design principles. The third section is a theoretical comparison of the conventional approach and probability space (PS) diagrams in terms of their semantic transparency and plastic generativity. An experiment is then presented that demonstrates some of the advantages of the PS diagrams. The final discussion section draws out some of the wider implications of the central thesis that effective representations should encode fundamental conceptual structures.