What eye-movements tell us about ratios and spatial proportions

Complex computer interfaces often provide quantitative information which the user must integrate. For example a scroll bar gives information about position in a document. The user can adjust the scroll bar to maintain an appropriate ratio of the portion of the document displayed relative to the documents position. Users are often given icons to select, does the icon's proportionality influence their choice? The task of identifying two shapes that are geometrically similar, though different in size, is a proportional one, but also one in which spatial processing may be important. Our research examines ways in which spatial presentations of the data can facilitate the process of integrating quantitative information.Mathematically a proportion is defined as a four-term relation among quantities: A is to B as C is to D. In essence, the task of proportional reasoning is to compare the relation between one pair of terms (or quantities) with the relation between the other pair. One of the simplest examples of proportional reasoning occurs in the identification of shapes. Even young children can tell whether two shapes are equally proportioned, in terms of the ratios of their heights to their widths for instance—even if they are very different in overall size.In effect, in this task they are identifying a proportional equivalence between the ratio relating the width and height of the smaller shape and the ratio relating the width and height of the larger one. Although psychological tests of proportional reasoning typically control for perceptual, as our shape comparison example illustrates there is nothing in principle that makes a perceptually based judgment of proportionality mathematically unlike other proportionality judgments. What is important is that non-relational bases for task performance be controlled. Thus, for instance, matching of the fatness of two shapes that are the same height might not involve a consideration of ratios, since comparisons based on the horizontal dimension alone would be sufficient to determine whether the shapes were the same or different. Differences in overall magnitude are thus an essential element in any assessment of proportional judgments.In order to learn more about the visual processes underlying the apprehension of spatial-configurational ratios, we performed an experiment which recorded adults' eye movements during the performance of a shape comparison task. We found that observers do not need to fixate on both shapes in order to make a proportional comparison between them, suggesting that the perception of height: width ratios does not depend on foveal vision. At the same time, it is clear that scanning patterns were affected by characteristics of the scenes. The variation in looking at left vs. right sides of the scenes as a function of whether the correct (fatter) stimulus was on the left or the right is particularly intriguing. It appears that the smaller stimulus could be processed with relatively few fixations, but when it was the correct choice viewers verified its fatness with more extended viewing before responding. An interesting implication of this finding is that visual objects can be too big to be processed with optimal efficiency. Whereas it is often assumed that bigger is better, at least when what viewers need to discern is the overall shape of an object a large size may actually impede rapid apprehension by requiring more extensive scanning. Providing adequate spacing between visual elements may therefore be a more effective means of facilitating processing than enlarging the sizes of those objects (beyond some point). These findings support the idea that, when overall size is not too great, the proportional relations that characterize the shape of an object can be rapidly apprehended, often without directly fixating it. This supports the conjecture with which we began, that spatial configurations can be a powerful way of presenting information about the relations between two quantitative measures.