Comprehension Of Visualized Ordered Sets

An Empirical Approach Based on the Theory of Knowledge Spaces

Christof Körner

Summary

In this study, human comprehension of visualized ordered sets was investigated. Ordered sets provide a non-numerical data structure, whose mathematical properties are very general. Therefore, they appear in many contexts in which ordered information is processed.

An outstanding feature of ordered sets is that they can be visualized by means of so-called upward drawings or Hasse diagrams. Such drawings can provide an easy access to the ordering information. Therefore, ordered sets and their visualizations play an important role in the (computer-aided) presentation of ordered information. There is a great interest in criteria for good drawings, i. e., for drawings which are easy to comprehend.

The visualization of ordered sets in upward drawings is not unique. In the mathematical and computer scientific literature, several mathematical properties referring to ordered sets are applied to upward drawings. This is done in order to provide criteria for what a good drawing should look like. The most prominent visual properties are planarity, slope and levels. The influence of these properties on comprehension and their adequate visualization in respective drawings were systematically investigated as well as the knowledge which must be available to a reader to guarantee proper comprehension.

In this study, mathematical properties of ordered sets and their drawings are reported, as far as they are relevant for the subsequent empirical investigations. Psychological assumptions on the cognitive processes involved in the course of comprehension of visualized ordered sets are considered. Starting from such assumptions, precise experimental hypotheses were derived. This was achieved by means of the theory of knowledge spaces, which provided the methodological framework of this study. The theory allowed for a systematic construction of the stimulus materials involving the visual properties. Moreover, it allowed for predictions regarding the efficiency of comprehension, as indicated by response latencies, and misconceptions evolving from partial knowledge. The predictions were tested in four closely interrelated experimental investigations.

In Investigations 0 and I, the influence of the visual properties on the course of information processing was investigated in case participants have complete knowledge on upward drawings. In an instruction and practice part, participants were provided with knowledge on upward drawings. In the main part of these investigations, eight different upward drawings were presented. They all were valid visualizations of the same ordered set. However, they were systematically constructed with respect to the visual properties. In addition, to each drawing, systematically constructed questions were posed which covered a broad range of knowledge on upward drawings.

As had been expected from the extensive practicing, the participants showed a high performance on this task in terms of correct answers. However, the efficiency of comprehension was highly dependent on the visual properties. The latencies for arriving at a correct answer were shown to systematically co-vary with the constructed upward drawings. These results suggest that an impairment of early (visual) processes by the visual properties of the drawings can be compensated by subsequent knowledge-based processes. They assure a proper comprehension; however, the efficiency of comprehension is affected in any case.

In Investigations II A and II B the consequences of partial knowledge on upward drawings were studied with respect to their interaction with visual properties on the one hand, and to possible misconceptions on the other hand. In an instruction and practice part, participants were provided with partial knowledge on upward drawings, which varied between the two investigations. In the main part of these investigations, the same drawings as in Investigations 0 and I were presented.

In the first stage of the main part, questions were posed which were simpler than in the previous two investigations. Participants' partial knowledge was sufficient to answer these questions. Accordingly, their performance in terms of correct answers was almost perfect. As predicted, no influence of visual properties on the efficiency of comprehension was found if participants had only little partial knowledge. However, an influence of visual properties was obtained for participants who were provided with more (partial) knowledge. Together with the findings on efficiency of comprehension obtained from the previous investigations, these results suggest an interaction between knowledge and visual properties. The more knowledge is available to (and required from) a reader of an upward drawing, the stronger is the effect of the visual properties on the efficiency of comprehension.

In the second stage of the main part, specifically designed questions were posed, in order to reveal possible misconceptions of the participants. A model was set up which yielded predictions on specific wrong answers to those questions. Participants of both investigations showed a substantial amount of wrong answers to the questions. The model could reasonably well account for these answers indicating that misconceptions actually evolved as a consequence of partial knowledge on upward drawings.

The consequences of the experimental findings are discussed and conclusions with respect to an adequate visualization of ordered sets are drawn.

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Last modification: Feb 12, 1999