However, studies with kindergarten through sixth grade children also show a great deal of overlap among a wide range of spatial skills Mix et al. Further, certain spatial skills, notably mental rotation and visuospatial working memory, have been found to cross-load onto a mathematical factor at particular grade levels. An important next step is to examine process models of spatial skills and how they are manifested or not on mathematical tasks, as illustrated below regarding mental rotation.

Mental rotation was first described based on the finding that time to simulate the rotation of an object was related to the angle through which the object was rotated Shepard and Metzler, Cognitive process models, supported by empirical studies, reveal that mental rotation actually involves multiple, non-obvious sub-components. Behavior is best fit by a model that involves carrying out small, successive, variable transformations, rather than a single rotation Provost and Heathcote, and empirical work suggests that individuals actually rotate just one part of the object rather than all parts of the whole object Xu and Franconeri, Further, modeling shows that the type of mental rotation problem influences the process that is engaged; when rotating complex stimuli, participants tend to be slower Bethell-Fox and Shepard, ; Shepard and Metzler, , which has been fit by computational models of mental rotation where task relevant features of the object are focused on and task irrelevant features are ignored Lovett and Schultheis, Participants also frequently err in problems with complex stimuli by selecting the mirror image of the correct choice that is rotated to the same degree as the correct choice e.

Relatedly participants tend to use a fast flipping transformation akin to matching features for simple, 2D stimuli, which models of mental rotation have taken this into account Kung and Hamm, ; Searle and Hamm, The varied components described by these models make clear that mental rotation is not a simple process, and that there are many steps needed to succeed at a mental rotation task.

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Each of these modeled components of mental rotation performance has a potential role to play in the observed relationship between mental rotation and various mathematical skills over the course of development. If spatial constructs are actually based on wide-ranging processes it opens up the hypothesis space to determine the source of connections between spatial and mathematical thinking. Rather than a simple connection between two monolithic skills, there are numerous possible connections based on the components of each, and possibly even multiple ways a spatial skill can act in a single math problem.

The work of figuring out which components are critical to the observed relation between spatial and mathematical skills, while daunting, is needed in order to unpack what otherwise are opaque connections. To take one example, Gunderson et al. Individual differences in mental rotation performance could have arisen as a difference in any of the components identified above: the ability to carry out rotations, to focus on relevant spatial information, or to carry out non-rotational stimulus matching.

Similarly, the number line estimation task, where participants are asked to determine the position of a number along a labeled line, could be decomposed into several components as well e. Any or all of these components might be the source of the connection between number line estimations and spatial skill see Figure 1. By designing studies that control for and model the components of both spatial and mathematical tasks, it should be possible to identify and understand the mechanisms that explain links between spatial and mathematical thinking. This approach compliments and enriches the work focused on looking at the latent structure of skills, while not dwelling on an explanation of any one task but focusing on explaining important connections between latent skills.

Figure 1. Potential connections among spatial and mathematical skill components. Meta-analyses provide strong evidence that training spatial skills in the laboratory result in significant improvements and transfer to other spatial skills Uttal et al.

### Richard W. J. Neufeld

However, evidence is more mixed about training spatial skills to improve mathematical skills e. Broader training regimes in and out of the classroom have helped to improve mathematics performance in multiple age groups e.

One finding substantiated by factor analyses and interventions is that spatial skills are more closely related to novel mathematical and scientific content than to STEM skills that are more familiar Stieff, ; Mix et al. Another set of findings suggests that providing students with a repertoire of spatial tools, such as gesture, rich spatial language, diagrams, and spatial analogies, Newcombe, ; Levine et al.

## Mathematical Psychology

Moreover, these tools, as well as 3-D manipulatives Mix, have been found to facilitate learning mathematical concepts e. An overarching principle to guide the use of spatial thinking and tools in education is that supporting spatial thinking and learning beginning early in life may result in improvements in mathematics understanding, based on the general connection between spatial and mathematical factors as well as evidence that training particular spatial skills shows some transfer to mathematics skills.

A promising avenue for future work is not just to support spatial thinking in general, but to show students how they can use this kind of thinking to solve particular kinds of mathematical problems Casey, In this review, we critically evaluate the contributions of the factor analytic method to identifying and elucidating the connection between spatial and mathematical thinking across development.

We highlighted a central gap in our knowledge—understanding the mechanisms connecting spatial and mathematical skills—which can be better addressed through targeted experimental studies that are informed by process models than by factor analytic studies. The findings that can emerge from this approach are important for increasing our basic understanding of why spatial and mathematical thinking are connected. They also hold promise for informing educational efforts to increase mathematical achievement by strengthening spatial thinking by training spatial skills, by encouraging the use of spatial tools, and by showing children how they can deploy these skills and tools to solve particular kind of mathematical problems.

CY wrote the original draft and led efforts to refine subsequent drafts for this article. All authors worked on a related chapter; SL and KM contributed substantially to the writing and editing of this article. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Aiken, L. Attitudes toward mathematics.

Armstrong, J.

### Edited by Jerome R. Busemeyer, Zheng Wang, James T. Townsend, and Ami Eidels

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## Psychological and Brain Sciences: Indiana University Bloomington

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