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Showcase August 2013: Twisting space: Are rigid and Non-rigid mental transformations separate spatial skills?

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Twisting space: Are rigid and Non-rigid mental transformations separate spatial skills?

Kinnari Atit1, Thomas F. Shipley1, and Basil Tikoff2

1 Temple University and 2 University of Wisconsin-Madison


For the archival version of this research, and the preferred citation, see:

  • Atit, K., Shipley, T. F., & Tikoff, B. (2013). Twisting space: Are rigid and non-rigid mental transformations separate spatial skills? Cognitive Processing: Spatial Learning and Reasoning Processes, 14(2), 163-173.[DOI]

What is spatial thinking? What are spatial skills?

Spatial thinking ranges from the planning of real-time movements and actions to the abstract visualization of multi-dimensional space. Tasks such as strategically packing your car before a long road trip requires you to think spatially. It involves visualizing and manipulating a broad class of spatial information, the relations between different locations, configurations, shapes, and objects and how they change over time (Newcombe & Shipley, in press). The ability to represent and transform spatial relations makes up one’s spatial skills (Carroll, 1993). Much research in cognitive science has focused on trying to characterize the skills that make up spatial thinking and on trying to define the distinct categories. Although most researchers agree that spatial thinking is important and that it is not composed of just one skill, (e.g. Guildford & Lacey, 1947; McGee, 1979; Thurstone & Thurstone, 1941), there is surprisingly little agreement about the details of what makes up this skill set (Caplan, MacPherson, & Tobin, 1985).

Scientists have taken many different approaches to try and characterize the different types of spatial thinking that exist, such as identifying commonalities in the cognitive mechanisms (e.g. Maccoby & Jacklin, 1974), and identifying clusters in skill using statistical methods (e.g. Ekstrom, French, Harman, & Derman, 1976; Linn & Petersen, 1985). Though a number of attempts have been made to define this set of skills, the types of events used by cognitive scientists to study spatial thinking have mostly involved rigid mental transformations (i.e. transformations within which the distance between two points on the object is preserved, such as in mental rotation) (e.g. Hegarty & Waller, 2004; Pani, Chariker, Dawson, & Johnson, 2005; Shepard & Metzler, 1971). Yet, humans can visualize events that involve complex non-rigid changes (i.e. transformations within which the distance between two points on the object is not preserved, such as in mental bending). For example, we can readily imagine the hood of a car before and after a collision. As a result of the collision, the hood underwent a non-rigid transformation, where the distances among points on the hood changed as the metal crumpled.

Recently, Chatterjee (2008) proposed two broad dichotomies of spatial thinking based on research in psychology, linguistics, and neuroscience, suggesting that humans process intrinsic (an object’s shapes and part-based representations) and extrinsic (relations among objects and between objects) spatial properties differently, and static (stationary) and dynamic (moving) spatial properties differently. This framework seems to classify all spatial skills into four broad categories that involve the four types of spatial relations: intrinsic static, intrinsic dynamic, extrinsic static, and extrinsic dynamic.

Non-rigid Mental Transformations & Our Experiment

In cognitive science, researchers have recognized that humans can visualize non-rigid transformations (e.g. Gibson, Owsley, & Johnston, 1978; Gibson & Spelke, 1983; Gibson & Walker, 1984), and that the brain processes complex transformations (e.g. Shepard, 2001; Shepard & Cooper, 1986; Shepard & Metzler, 1971), but research on how humans visualize complex non-rigid transformations is scant. Aside from a handful of cases, such as Resnick and Shipley’s (2013) work on mental breaking (discontinuous changes) (see January 2010 Showcase), research about the mental visualization of non-rigid transformations mainly consists of research on mental folding. To study the simulation of non-rigid events with continuous changes (i.e. bending), we developed the Non-rigid Bending test, which required participants to visualize a continuous non-rigid transformation applied to an array of objects by asking simple spatial questions about the position of two forms on a bent transparent sheet of plastic (see Figure 1). Participants were to judge the relative position of the forms when the sheet was unbent.

                                          Figure 1            Figure 1b
Figure 1. Examples of stimuli from the Non-rigid Bending Test. Figures were taken from Atit, Shipley, and Tikoff (2013).

In this experiment, to see if the skills needed to visualize rigid events are separable from those needed to visualize non-rigid events, and to see if skill in reasoning about one type of event predicted skill on others, in addition to the Non-rigid Bending test (a test of continuous non-rigid mental transformations) we asked undergraduate participants to complete three more tests of mental transformations –the Paper Folding test and the Mental Brittle Transformation test (two tests of non-rigid mental transformation with local rigid transformations), and the Mental Rotation test (a test of rigid mental transformation).

Results & Conclusions

Results from our study showed that performance on the Non-rigid Bending test was predicted by performance on measures of other non-rigid transformations (the Paper Folding test and the Mental Brittle Transformation test), but not rigid transformations (the Mental Rotation test). Interestingly, performance on the Paper Folding test and the Mental Brittle Transformation test were both related to performance on the Mental Rotation test and the Non-rigid Bending test, but breaking and folding were not strongly correlated with each other. Combining the findings of this study with existing distinctions made in spatial thinking (e.g. Chatterjee, 2008; Resnick & Shipley, 2013; Shepard & Cooper, 1986), we propose that the skills used for rigid mental transformations are distinct from those used for non-rigid mental transformations, and skills used to manipulate intrinsic spatial properties are distinct from those used for extrinsic spatial properties. These relationships are illustrated in Figure 2.

Figure 2Figure 2. This Venn diagram illustrates the relationships among the four spatial skills measures. In the diagram, the overlap between circles suggests a shared set of spatial skills. No overlap between circles indicates possibly no significant overlap in skills. We hypothesize that the vertical axis represents a continuum of non-rigid to rigid transformations, and that the horizontal axis represents the continuum of extrinsic to intrinsic spatial skills. This figure was taken from Atit, Shipley, and Tikoff (2013).

By considering the nature of natural events (e.g. a car crash) and developing tools that can measure skills in reasoning about a broad array of mental transformations, we have identified a new type of spatial reasoning, and achieved some new insights regarding previous findings on the breakdown of spatial ability. The motivation for this study and the context of our interpretations reflect an interest in broadening the scope of spatial cognition research to consider complex spatial problems and the information humans use to solve them. Studying the skills required to visualize transformations present in the environment should help assure that what is learned in the laboratory can inform cognitive science about the many spatial problems humans reason about in their lives. This environment-driven approach to delineating spatial problems can also provide insight into the types of skills that should be developed and fostered in the classroom. Since high spatial ability has been found to predict success in science disciplines (Shea, Lubinski, & Benbow, 2001), identifying and categorizing the spatial information in spatial problems could point to new approaches for curriculum development to help train students’ spatial thinking skills, develop their general problem-solving skills, and enhance learning in the sciences.


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