Spatial Organization and Presentation Mode in the Representation of Complex Data

Spatial Organization and Presentation Mode in the Representation of Complex Data David W. Braithwaite (baixiwei@gmail.com) Robert L. Goldstone (rgoldsto@indiana.edu) Department of Psychology, Indiana University, 1101 E. 10th Street Bloomington, IN 47405 USA Abstract External representations are more effective when spatial di- mensions are used to represent numeric variables. However, this principle may result in suboptimal representations when the number of numeric variables to be represented is large. To test this possibility, participants studied a set of graphs repre- senting a parametrized function under different parameter val- ues. The graphs were displayed either using a grid organiza- tion, with parameter values represented by spatial dimensions (horizontal and vertical position of the graphs), or juxtaposed in a single area, with parameter values represented by non-spa- tial dimensions (color and texture). Juxtaposed organization led to better learning. However, this advantage was eliminated when the graphs were presented successively rather than sim- ultaneously. The results suggest that juxtaposed organization can improve comprehension of complex data by facilitating comparison between parts of the data. Such organization may be preferable even if it precludes use of spatial dimensions for some numeric variables. Keywords: external representations; graphs; human factors; mathematics; education. Background External representations of information, such as pictures, di- agrams, and graphs, play an important role in human thought, serving as tools to facilitate learning, problem solving, and communication. The effectiveness of external representa- tions depends in large part on the degree to which they respect cognitive constraints on their design. A well-studied class of constraints known relates to the selection of representational dimensions – that is, the choice of which features or dimen- sions of the external representation will represent which fea- tures or dimensions of the information to be represented. For example, in a network diagram, degree of connectivity be- tween nodes could be represented by either spatial proximity or thickness of the lines connecting them (Tversky, Corter, Yu, Mason, & Nickerson, 2012), while in a graph of experi- mental results, different data points could be represented by either bars or points on lines, and so on. One general cognitive constraint on the selection of repre- sentational dimensions is congruence: external representa- tions are more effective when dimensions of the information are represented by similarly structured dimensions of the rep- resentation (Hegarty, 2011; Tversky, Morrison, & Betrancourt, 2002; Zhang, 1996). For example, in diagrams, categorical variables are best represented by categorical vis- ual dimensions such as inclusion in containing shapes, while continuous variables are best represented by continuous vis- ual dimensions such as proximity (Tversky et al., 2012). Ra- tio scale variables, including most numeric variables, are best represented by visual dimensions that also possess a ratio scale (Zhang, 1996). Spatial dimensions, e.g. horizontal or vertical position, fall into this category, while many non-spa- tial dimensions, e.g. line color and line texture, do not. As an example, it is preferable to represent numeric variables by position on the x-axis of a graph, a spatial dimension, than by the graph legend, typically corresponding to a non-spatial di- mension such as line color or texture (Shah & Carpenter, The constraints just discussed apply primarily to the selec- tion of representational dimensions for individual dimensions of the represented information. If the information to be rep- resented involves multiple dimensions, additional constraints may arise when these dimensions are considered together. Consequently, a selection of representational dimensions might be optimal from the perspective of each dimension considered separately, but suboptimal when considered as a whole. Thus, a complete understanding of cognitive con- straints on the selection of representational dimensions must include an understanding of constraints that apply specifi- cally to the selection of multiple dimensions. This type of constraint has received comparatively little attention in re- search to date. To make this issue more concrete, consider Figures 1A and 1B. These figures present a number of graph lines illustrative of the binomial distribution. The binomial distribution gives the probabilities of the different possible outcomes of a se- quence of independent binomial trials, here termed “repeti- tions,” where an outcome is defined as a certain proportion of repetitions resulting in “successes. ” The probability mass function over possible outcomes is determined by two param- eters, the number of repetitions n and the probability of suc- cess on a single repetition p. Figures 1A and 1B each show 9 graph lines for the probability mass functions correspond- ing to 9 different combinations of values of n and p. In both figures, the different possible outcomes are represented by the x-axis and the corresponding probabilities by the y-axis. However, the figures differ in their selection of visual dimen- sions to represent the parameters n and p. In Figure 1A, the graph lines are contained in separate graphs, which are ar- ranged in a grid, and the parameters are represented by spatial dimensions: each vertical position in the grid represents a dif- ferent number of repetitions n, and each horizontal position a different probability of success p. In Figure 1B, the graph lines are contained in a common plot area, and the parameters are represented by non-spatial dimensions: each line texture represents a different number of repetitions n, and each line color a different number of repetitions p. The graphs shown in Figures 1A and 1B all involve the same four variables: the proportions of repetitions resulting in success, the probabilities of each such proportion, and the