UC Merced Proceedings of the Annual Meeting of the Cognitive Science Society

UC Merced Proceedings of the Annual Meeting of the Cognitive Science Society Title Modeling Spatial Ability in Mental Rotation and Paper-Folding Permalink https://escholarship.org/uc/item/4jc23853 Journal Proceedings of the Annual Meeting of the Cognitive Science Society, 35(35) ISSN 1069-7977 Authors Lovett, Andrew Forbus, Kenneth Publication Date 2013 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Modeling Spatial Ability in Mental Rotation and Paper-Folding Andrew Lovett ([email protected]) Kenneth Forbus ([email protected]) Qualitative Reasoning Group, Northwestern University, 2133 Sheridan Road Evanston, IL 60208 USA Abstract perform each task consistently. This analysis allows us to address a longstanding debate about the effects of shape Spatial ability tests like mental rotation and paper-folding provide strong predictions of an individual’s achievement in complexity on mental rotation. It also provides hypotheses science and engineering. What cognitive skills are involved in about the skills supporting fast, efficient mental rotation, them? We use a computational model to analyze these tasks, and thus the skills underlying spatial ability. asking how much information must be processed to perform We begin with background on mental rotation and the them. The models demonstrate that in some cases stimuli can question of shape complexity. We show how paper-folding be vastly simplified, resulting in consistent performance appears to violate many researchers’ conclusions, as it regardless of stimulus complexity. The ability to produce a scaled-down representation of a complex stimulus may be a involves simple shapes but requires great deliberation and key skill underlying high spatial ability. effort. We next present our computational model, which builds on previous cognitive models of perception, Keywords: spatial ability; mental rotation; paper-folding; comparison, and visual problem-solving (Falkenhainer, cognitive modeling. Forbus, & Gentner, 1989; Lovett & Forbus, 2011). We apply the model to the two tasks, determining the amount of Introduction information that must be carried through the There is strong evidence linking spatial ability to academic transformations, and showing why paper-folding is a more achievement. Children who perform well on spatial ability difficult task. Finally, we discuss the results and consider tests are more likely to study STEM disciplines (Science, the ramifications for spatial ability in general. Technology, Engineering, and Mathematics) and to go into a STEM profession (Shea, Lubinski, & Benbow, 2001; Wai, Background Lubinski, & Benbow, 2009). This effect holds even when controlling for verbal and mathematical ability, suggesting Mental Rotation that spatial ability is an independent component of intelligence. If we are to improve STEM achievement, it is Mental rotation is frequently used to evaluate spatial ability critical that we better understand the skills that compose (Vandenberg & Kuse, 1978). Typically the distractors—the spatial ability and how they can be taught. shapes that aren’t a valid rotation—are mirror reflections. Traditionally, spatial ability has been evaluated using When they are presented sequentially, there is often a cue tasks such as mental rotation and paper-folding. In mental indicating what the orientation of the second shape will be rotation (Figure 1A, 1B), individuals are shown two shapes (e.g., Cooper & Podogny, 1976; Figure 1B). A common and asked whether a rotation of one shape could produce the finding across task variations is that the response time is other. In paper-folding, they are shown a line-drawing of proportional to the angle of rotation between the shapes. paper and asked to imagine the results of unfolding (Figure That is, response times increase linearly with angular 1C) or folding up (Figure 1D) the paper. Both tasks appear distance. This finding has led to the claim that people use a to measure spatial visualization, the ability to manipulate mental space, analogous to the physical space, and that they mental representations of images (McGee, 1979). There is rotate their representation through this space just as an evidence that the tasks are linked, with training on one object might rotate physically (Shepard & Cooper, 1982). improving performance on the other (Wright et al., 2008). One common question concerns how shapes are rotated However, many questions remain about what skills enable through mental space. Are they rotated piecemeal, with one people to perform them quickly and accurately. part rotated at a time, or are they rotated holistically, with Here, we study the mental rotation and paper-folding every part rotated together (Bethell-Fox & Shepard, 1988)? tasks using a computational model. The model operates These two possibilities produce different predictions about directly on 2D line drawings (sketches), automatically how shape complexity interacts with rotation speed. If generating representations, transforming them, and shapes are rotated piecemeal, then people should rotate evaluating the results of the transformation. We use the complex shapes more slowly, because there are more parts model to analyze the tasks, asking how much information to rotate. If shapes are rotated holistically, then shape must be encoded and carried through the transformations to complexity may not affect rotation speed. 930 A) B) C) D) Figure 1. Mental rotation (A, B) and paper-folding (C, D) tasks (A: Shepard & Metzler, 1971; B: Cooper & Podogny, 1976; C: Ekstrom et al., 1976; D: Shepard & Feng, 1972). The results on shape complexity provide evidence for people should rotate shapes more slowly, particularly when both piecemeal and holistic rotation. Overall, it appears that the shapes are complex. There is evidence supporting both rotation speed depends more on other factors, such as the predictions (1: Cooper & Podgorny, 1976; 2: Folk & Luce, familiarity of the objects (Bethell-Fox & Shepard, 1985; 1987). Yuille & Steiger, 1982), the similarity of the distractors (Folk & Luce, 1987), and the strategy and overall ability of Paper-Folding the participant (Yuille & Steiger, 1982; Heil & Jansen- In contrast with mental rotation, paper-folding has seen Osmann, 2008). These findings suggest that dealing with relatively little study. This is surprising, given that it is also shape complexity may itself be a spatial skill. Skilled often used to evaluate spatial ability (Ekstrom et al., 1976). participants may apply heuristics to simplify shapes for Here, we focus on a version of paper-folding that emerged rapid rotation. However, when these heuristics fail, they are at about the same time as mental rotation (Shepard & Feng, reduced to rotating one piece at a time, which is slower. 1972). While this version is used less frequently in spatial One straightforward heuristic for simplifying a shape is to ability evaluations, there is direct evidence linking it to ignore parts of it. When Yuille and Steiger (1982) told mental rotation (Wright et al., 2008). participants they could complete a mental rotation task Figure 1D shows an example. The letters have been added using only the top halves of the shapes, participants rotated for illustrative purposes and are not part of the stimulus. In the shapes more quickly. Alternatively, participants might this task, participants are shown six connected squares, utilize scalable representations (Schultheis & Barkowsky, representing the surfaces of a cube that has been unfolded. 2011) that support dynamic variation of detail based on task Two edges are highlighted by arrows, and one square is demands. Both the degree and the type of detail may vary. grayed out, indicating it is the base of the cube. Participants For example, while we can imagine both the locations and are asked whether the highlighted edges would align if the orientations of objects in space, a task might require squares were folded back into a cube. considering only one of these. In this paper, we use the term Unlike mental rotation, this task requires a sequence of spatial smoothing for any process that removes spatial rotations. For example, Figure 1D requires three rotations. detail, producing a simpler representation. One solution (Figure 2) would be: 1) Rotate squares A, B, Participants may smooth out the details in complex and C up, so that they stick out from the plane. 2) Rotate shapes, producing representations with equal complexity to squares B and C down to make the top surface of the cube. those of simpler shapes. However, when the distractors are 3) Rotate square C farther down, making the front surface of particularly similar to the base shapes, participants may the cube. At this point, the two arrows align perfectly. require additional detail, and so they may use more complex Surprisingly, even though each of these three rotations representations that are more difficult to rotate. seems simple, they appear to be piecemeal rotations. This hypothesis leads immediately to two predictions: 1) Participants’ response times are not a function of the When similarity of distractors is kept constant and relatively number of rotations performed, but of the number of times low, people should rotate shapes at the same rate regardless every square is rotated. In this case, three squares are rotated of shape complexity. 2) As distractors become more similar, (Figure 2B), then two squares (2C), then one square (2D), A) B) C) D) Figure 2. Possible solution for Figure 1D. 931 so

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