Two studies evaluated the effectiveness of Euler circles in aiding participants in drawing conclusions to deductive reasoning problems. The problems were the ones that typically cause reasoner the most difficulty because their prior beliefs about conclusions interfere with their judgments of deductive validity. The use Euler circles reliably contributed to reasoners' inability to solve the problems. This pattern was shown for both young, university students and elderly retired people.

The process of member validation requires researchers to present their findings back to the communities that have been studied to gain their appraisal of the work. By depicting subject matter that ranges from the physical to the conceptual, diagrams provide a valuable alternative to the written documents traditionally used in member validation. This paper reports on a study in which diagram-based member validation was used to assess the accuracy and acceptability of the researchers

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Canonical correlation analysis is a type of multivariate linear statistical analysis. In a study of crew interaction with the automatic flight control system of the Boeing 757/767 aircraft, we observed 60 flights and recorded every change in the aircraft control modes, as well as every observable change in the operational environment. The complete dataset consisted of 1665 such snapshots, each characterized by values on 75 variables. To quantify the relationships between the state of the operating environment and pilots‚ actions and responses, we used canonical correlation because of its unique suitability for finding multiple patterns in large datasets. Traditionally, the results of canonical correlation analysis are presented by means of numerical tables, which are not conducive to recognizing multidimensional patterns in the data. Such patterns are extremely important in characterizing the most important environmental conditions and their effects, and in revealing deviations (outliers) indicative of operational errors. We created a sun-ray-like diagram where all the independent variables are on the right side of the circle, and all the dependent variables are on the left -- a heliograph. Alexander's theory describes 15 heuristic properties that help create wholeness in a design, and which can be extended to the problem of data integration. We applied this theory to guide the design of our heliograph, as well as to interpret its strengths as a data-rich display. Our ongoing work aims to extend this theory to deal with problems of information integration and packing of large amounts of data for visualization.

The Propositional Statechart formalism, described in Dunn-Davies et al. (2005), is a specialised form of David Harel's Statechart formalism (Harel, 1987) specifically tailored to the design and description of interaction protocols. Propositional Statecharts harness the intuitive nature of the Statechart formalism, and add precise action semantics to enable protocols to be defined unambiguously. Here we descibe an extension of the Propositional Statechart formalism that enables it to represent interaction protocols in a modular fashion. We also show how this extension of the Propositional Statechart formalism can be applied in practice to provide an unambiguous description of a recursive protocol.

There is undoubtedly something like a 'grammar of graphics'. Various syntactic principles can be identified in graphics of different types, and the nature of visual representation allows for visual nesting and recursion. We propose a limited set of possible 'building blocks' for constructing graphic spaces, and a limited set of possible syntactic functions of graphic objects. Based on these ingredients, and the rules for their combination, the syntactic structure of any visual representation can be drawn as a hierarchically nested tree. We claim that the presented visual syntax applies to all types of visual representations.

The construction of graphs is underdetermined by the mathematical information that they are intended to display. The present research explores the possibility that they are constrained by the social information that they depict. Several lines of research suggest that social categories are androcentric and render males the default gender. As English language users parse spatial information from left-to-right and top-to bottom, and graphs are conventionally defined to be encoded in those ways, we hypothesized an androcentric preference for constructing graphs depicting gender differences that positioned data about males above or to the left of data about females.

Fifty-four British undergraduates were presented with written prompts to draw vertical bar graphs representing differences between two sub-groups within four separate categories, including gender. Participants'graphs positioned the typical entity to the left of the atypical entity by a ratio of 3:1, and males' data first rather than females'data first by a ratio of 3:1. These results suggest that males may be positioned first in graphs of gender differences because they are deemed more typical of generic human categories than women are.

The second study was a content analysis of psychologists' written reports of gender differences. 388 articles reporting gender differences were systematically sampled from four APA journals over the period from 1965 to 2004. Twenty of these articles (from the journal Developmental Psychology) included both parents and children as study participants. Here, children were treated as the principal participants and gender differences among parents were analyzed separately. On average, graphs and tables within the article positioned males' data first rather than females' data first by a ratio of 3:1. This finding was not moderated by year of publication or author gender, but varied somewhat between journals. In contrast, within the 20 articles reporting gender differences between parents, four times as many graphs positioned mothers first as fathers first.

Both relatively naive undergraduates and relatively sophisticated scientists show a preference to graph males before females more than the reverse, except when females are more typical of the overarching category, mirroring established effects of androcentrism on verbal explanations of gender differences. These studies are the first to examine how androcentrism affects spatial cognition, they raise the hypothesis that typicality determines preferences for order of information in graphs, and they call for future research on the rhetorical functions of scientific representations of group differences.

In contrast to the visuospatial properties of traditional static diagrams, the temporal properties of the newly emerging phenomenon of animated diagrams tend to closely reflect attributes of the referent subject matter. However, the educational strengths of static diagrams are intimately connected with their extensive manipulation of the referent's visuospatial properties. This paper explores possibilities for extending this tradition of manipulation to the temporal properties of animated diagrams as a way of improving their educational effectiveness, particularly when complex dynamic subject matter is represented. A pilot study is reported that examined the effect of manipulating the playing speed of a Newton's Cradle animation and the number of times the animation was viewed. The preliminary results obtained suggest that such manipulations affect the type and level of information that is extracted.

Hierarchical diagrams are common in both everyday and scientific contexts. For example, a hierarchy can be used to represent relations among members of the animal kingdom or the search space at a given point in a chess game. Cladograms, a type of hierarchical branching structure, are one of the most important tools that contemporary biologists use to reason about evolutionary relationships. These diagrams depict the distribution of characters (i.e., physical, molecular, and behavioral characteristics) among taxa. They are hypotheses about nested sets of taxa that are supported by shared evolutionary novelties. Cladograms can be represented in both a tree form and a ladder form. In the present study, we investigated students' understanding of the hierarchical relations in these different cladogram formats. We compared college students who had more and less prior coursework in biology. Both groups found ladders more difficult to understand and interpret than trees. This was especially true for the low knowledge students. The nature of subjects' errors suggests that the difficulty in understanding ladders may reflect both perceptual and conceptual principles --- the Gestalt principle of good continuation and a misleading analogy to real-world ladders, respectively. We discuss both the psychological and educational implications of our work.

We outline an approach to diagrammatic reasoning (more specifically, geometric reasoning) which is completely model-theoretic. Geometric diagrams are represented as relational structures. From our abstract logical point of view, a geometric proof is seen as a series of relational structures approaching a limit. Mathematically, we capture these ideas in terms of productive direct systems and their direct limits.