Evidence for spatial mathematical relations

1.4.1.1 Adult and adolescent studies

Spatial ability has been identified as a reliable predictor of STEM outcomes in many large-scale longitudinal studies (N > 500), following both normative and intellectually gifted populations through adolescence and adulthood (Shea, Lubinski, & Benbow, 2001; Wai et al., 2009). Talent Search participants are young people from the United States who qualify for special educational programmes due to high performance on college entrance exams at a young age (Wai et al., 2009). Even after controlling for quantitative and verbal skills, longitudinal studies of Talent Search participants have reported significant correlations between high spatial ability scores (intrinsic- dynamic spatial skills) at 13 years and later STEM outcomes (Shea et al., 2001). The STEM outcomes measured included: a preference for mathematics as a high school subject at 18 years, achievement of undergraduate and graduate degrees in STEM measured at 23 years, and future careers in STEM domains relative to careers in the humanities measured at 33 years (Shea et al., 2001). Similar findings have been reported in studies of non-gifted students. It has been reported that those who pursue STEM careers and complete STEM degrees at both undergraduate and masters level have higher spatial ability scores at 13 years (Wai et al., 2009). The spatial ability measure used in these studies was a composite of performance across a range of spatial tasks, predominantly targeting intrinsic-dynamic spatial skills. This pattern of associations between spatial thinking and STEM outcomes in adults is mirrored in cross-sectional studies. Spatial ability has been implicated as an

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important factor in undergraduate science success, medicine, dentistry, and engineering performance (Hegarty, 2014; Hegarty, Keehner, Cohen, Montello, & Lippa, 2007; Hegarty, Keehner, Khooshabeh, & Montello, 2009; Uttal, Miller, & Newcombe, 2013). Mental rotation skills (intrinsic-dynamic spatial skills) have been associated with undergraduate students’ abilities to translate organic chemistry diagrams (Stull, Hegarty, Dixon, & Stieff, 2012). For physics, spatial visualisation skills (intrinsic-dynamic sub-domain) are significantly correlated with mechanics problem solving (Kozhevnikov & Thornton, 2006) while for engineering, mental rotation skills (intrinsic-dynamic sub-domain) are significantly associated with an individual’s efficiency in learning to use computer aided design software (Sorby & Baartmans, 2000).

More specifically for mathematics, mental rotation skills (intrinsic-dynamic sub- domain) has been associated with mathematical performance in adults using number line estimation and magnitude comparison tasks (Thompson, Nuerk, Moeller, & Cohen Kadosh, 2013). Similarly, in adolescents intrinsic-dynamic spatial skills are significantly correlated with mental arithmetic and problem solving at 15 to 16 years (Reuhkala, 2001), geometry performance at 13 years (Delgado & Prieto, 2004), and mathematical word problems at 12 years (Hegarty & Kozhevnikov, 1999). Neuroimaging findings suggest that these spatial-mathematical associations may be attributable to shared processing requirements for spatial and mathematical tasks. There is evidence that overlapping circuits in the parietal lobe are activated in the completion of both number, and spatial tasks (Cutini, Scarpa, Scatturin, Dell’Acqua, & Zorzi, 2014; Hubbard, Piazza, Pinel, & Dehaene, 2005; Winter, Matlock, Shaki, & Fischer, 2015).

Overall, the evidence from longitudinal, cross-sectional and neuroimaging studies of adults and adolescents supports the existence of associations between spatial thinking and STEM domains; in particular the mathematics domain. However, as outlined in sections 1.2 and 1.3, both spatial and mathematical skills undergo significant development in childhood (before 13 years). Therefore, it is important to establish whether spatial-mathematical relations are present at all stages of development, or whether they emerge when mature performance levels are

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reached. The next section reviews the evidence for spatial-mathematical associations in childhood populations.

1.4.1.2 Longitudinal studies in childhood populations

Longitudinal studies in childhood populations have also measured the associations between spatial skills and mathematics. These studies predominantly focus on the predictive role of pre-school spatial skills. Verdine et al. (2014) reported that spatial skills at 3 years, assessed using the Test of Spatial Assembly (TOSA), a measure of intrinsic-dynamic spatial ability, predicted a significant 27% of the variation in mathematical problem solving, measured using the Wechsler Individual Achievement Test (WIAT) at 4 years. Similarly, a preliminary report from Farmer et al. (2013) indicated that spatial performance on the TOSA at 3 years is significantly correlated with a combined mathematics measure, at 5 years. Wolfgang, Stannard, and Jones (2001) demonstrated that spatial play in the pre-school years, in particular adaptiveness and integration in block play, is associated with mathematics achievement at 12 years. However, these results should be interpreted cautiously as interpretation of free block play is subjective and subject to errors. Furthermore, block play does not exclusively measure spatial thinking as it is influenced by a range of cognitive skills including attention and executive functions (Wolfgang et al., 2001). In a study of primary school children, Gunderson, Ramirez, Beilock, and Levine (2012) reported that performance on the Thurstone Mental Rotation Task (intrinsic-dynamic sub-domain) at 7 years, predicted improvement in number line estimation 6 months later. Gunderson et al. (2012) extended these results to show that performance on the CMTT, also a measure of intrinsic-dynamic spatial skills, at 5 years was predictive of approximate symbolic calculation at 8 years. These results were found to be mediated by number line estimation scores at 6 years.

As seen for adult studies, a majority of longitudinal studies that have explored spatial- mathematical associations in children, measure spatial skills in the intrinsic-dynamic spatial sub-domain. However, there is also some evidence that these associations hold for other spatial sub-domains and mathematics. This suggests that the association between spatial ability and mathematics competence is wide-ranging.

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Performance on a spatial relations task, which required input from both intrinsic- static and intrinsic-dynamic spatial sub-domains, was found at 3 years to be a significant predictor of arithmetic at 10 years (Zhang et al., 2014). Similarly, a composite measure of spatial skills, assessing both intrinsic-static and intrinsic- dynamic sub-domains at 7 years, significantly predicted mathematics achievement at 10 years (Carr et al., 2017). Casey et al. (2015) reported that spatial skills in girls, assessed using a composite measure generated from block design (intrinsic-dynamic spatial sub-domain) and mental transformation tasks (intrinsic-static and intrinsic- dynamic spatial sub-domains), at 7 years were a significant predictor of mathematics reasoning at 11 years. Longitudinal studies of primary school students have also reported correlations between visuospatial skills, including visual perception and motor integration at 6 years, and mathematics achievement at 9 years. However, these findings were confounded by the visual and motor demands of the tasks used (Lachance & Mazzocco, 2006; Mazzocco & Myers, 2003).

Overall, there is evidence that spatial abilities in the pre-school years, particularly intrinsic-dynamic spatial skills, are associated with later mathematics performance. In older children there is evidence that general spatial abilities in the primary school years are associated with later mathematics outcomes at 9 to 11 years. However, because most studies of primary school aged children use spatial composite scores, it is unclear which spatial sub-domains drive associations between spatial and mathematical performance in middle childhood.

1.4.1.3 Cross-sectional studies in childhood populations

Further insights into spatial-mathematical relations can be obtained from cross- sectional studies in primary school populations (from 5 to 10 years). Significant correlations have been reported between mental rotation (an intrinsic-dynamic spatial skill) and both calculation and arithmetic in children aged 6 to 8 years (Cheng & Mix, 2014; Hawes et al., 2015). For other intrinsic spatial tasks including disembedding (an intrinsic-static spatial skill) and spatial visualisation (an intrinsic- dynamic spatial skill), performance has been associated with a range of mathematics achievement measures at 10 and 11.5 years respectively (.37 < r < .42 ) (Tosto et al.,

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2014). Performance on mental rotation and disembedding tasks (intrinsic-static and intrinsic-dynamic sub-domains) and VSWM, was also identified as a significant predictor of standardised mathematics achievement (measured using the WIAT) at 8 to 10 years (Simms et al., 2016). In contrast, Carr, Steiner, Kyser, and Biddlecomb (2008) reported no significant association between mental rotation (an intrinsic- dynamic spatial skill) and standardised mathematics performance at 7 years.

Mix et al. (2016; 2017) have completed the most extensive cross-sectional research to date on spatial and mathematical thinking in the primary school years. In both initial exploratory factor analysis (EFA) (2016) and follow-up CFA (2017) studies, Mix

et al. found that, although spatial and mathematics tasks are highly correlated, they

form distinct factors (Mix et al., 2016; 2017). By comparing children of differing ages on the same spatial and mathematics tasks, Mix et al. (2016; 2017) have provided important preliminary evidence that there are distinct relations between individual spatial sub-domains and specific aspects of mathematics performance, and that these relations vary with age. More specifically, mental rotation (an intrinsic-dynamic spatial skill) was a significant predictor of mathematics (a general mathematics factor derived from performance on a range of mathematics measures) at 6 years only, while VSWM was a significant predictor at 11 years only. VSWM was measured using a spatial location memory task. No spatial predictors were identified for mathematics at 9 years. These findings suggest that associations between spatial thinking and mathematics in the primary school years may not be limited to the intrinsic-dynamic spatial domain. However, of note, some cross-factor loadings were not replicated across both the EFA and CFA studies. These inconsistencies suggest that there is instability of cross-factor loadings across different populations, which weakens the generalisability of the results. Thus, the findings should be interpreted cautiously (Mix et al., 2016; 2017).

In summary, current literature supports the organisation of spatial and mathematics domains as two distinct factors, with some cross-factor loadings. Cross-sectional studies provide evidence that different sub-domains of spatial thinking and different aspects of mathematics are differentially associated. That is, not all spatial and mathematics skills are associated to the same degree. Furthermore, there is evidence

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that the relationship between spatial and mathematical skills changes with development. Associations between some spatial and mathematics skills are present at specific developmental stages only. However, no known study investigates the role of different spatial sub-domains for mathematics, at different developmental ages in primary school.

These findings across both longitudinal and cross-sectional studies in children highlight a need to further elucidate the specificity of spatial-mathematical relationships across different tasks and skills. In particular, there is limited research on spatial-mathematical relations across the primary school years. Elucidating these relations in primary school children is important as there is evidence that the relationship between spatial skills and mathematics is sensitive to developmental age.