B REVE CONCLUSIÓN : CON LA MIRADA PUESTA EN EL SIGLO

There is a need to move beyond the question of “whether” to “why” significant correlations are repeatedly reported between mathematical and spatial constructs. Using evidence from longitudinal studies, Bailey (2017) presented a convincing argument for a causal effect of spatial skills on mathematics in the pre-school years. However, these findings are not definitive and cannot easily be translated to older children. Understanding the causal relationship and underlying explanations for spatial-mathematical associations, is key to integrating spatial learning into the mathematics classroom and to developing successful classroom interventions (Clements & Sarama, 2004; Hawes, Tepylo & Moss, 2015; Mix & Cheng, 2012). Findings on spatial-mathematical relations do not support a simple linear coupling between spatial and mathematical cognition (Fias & Bonato, 2018). Instead, it has been proposed that several different explanations underpin spatial-mathematical associations, depending on the mathematical and spatial sub-domains assessed (Fias & Bonato, 2018). Historically the Mental Number Line, or the idea that numbers are represented spatially in the brain, was proposed to explain observed associations between spatial and mathematical constructs (Barsalou, 2008; Lakoff & Núñez,

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2000). The Spatial-Numerical Association of Response Codes (SNARC) effect, thought to reflect the presence of the Mental Number Line, has been demonstrated in several studies. For example, individuals are faster to respond to small numbers with their left hand and larger numbers with their right hand, suggesting that small numbers are spatially represented to the left and larger numbers are represented to the right in the brain (Dehaene et al., 1993). Findings from arithmetic-based studies show that individuals typically overestimate addition results (right-side-of-space bias) and underestimate subtraction results (left-side-of-space bias) (Fischer & Shaki, 2014; Werner & Raab, 2014). In a similar way arithmetic performance is also influenced by the spatial presentation of equations and numbers (Fisher, Borchert, & Bassok, 2011; Landy & Goldstone, 2007; McNeil & Alibali, 2004). However, accepting the Mental Number Line as the driver of all spatial-mathematical relations is inconsistent with the differential associations observed between certain spatial and mathematical sub- domains, reported by Mix et al. (2016; 2017) among others. Instead, it has been suggested that not all associations between spatial and mathematical tasks can be explained in the same way, and a range of other explanations have subsequently been proposed as theoretical accounts for specific spatial-mathematical relations. First, it has been proposed that extrinsic-static spatial tasks, particularly spatial scaling tasks, rely on intensive quantification skills, or proportional reasoning (Newcombe, Möhring, & Frick, 2018). Magnitude can be encoded using two different quantification systems, an extensive system (using absolute amounts) or an intensive system (using proportions or ratios). Accurate spatial scaling between two different sized spaces requires the intensive coding strategy, with proportional mapping of relative, not absolute, distances. In mathematics, similar proportional mapping between extensive discrete representations of numbers to continuous intensive representations, is required for number line estimation and reasoning about formal fractions (Möhring, Newcombe, Levine, & Frick, 2016; Rouder & Geary, 2014). Theoretically, ANS tasks may also require proportional reasoning to facilitate ordinal comparisons, while performance on some geometry, area and distance tasks also rely on proportional and not absolute judgements (Barth & Paladino, 2011; Dehaene, Piazza, Pinel, & Cohen, 2003). In support of this, at 4 to 5 years significant correlations

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between spatial scaling performance and proportional reasoning performance (identification of the strength of flavour of different combinations of cherry juice and water) have been reported (Möhring, Newcombe, & Frick, 2015). Taken together, extrinsic-static spatial task performance is expected to correlate with mathematics tasks that rely on intensive quantity processing or proportional reasoning.

Second, for intrinsic-dynamic and extrinsic-dynamic spatial tasks, active processing including mental visualisation and manipulation of objects in space is thought to be required for successful task completion (Lourenco, Cheung, & Aulet, 2018; Mix et al., 2016). It is postulated that the generation of mental models allows individuals to visualise not only individual components of problems but also the relations between parts (Lourenco et al., 2018). Theoretically, in mathematics, individuals may use mental visualisations to represent and solve complex mathematical word problems, e.g., by keeping terms together and structuring order of operations tasks, or to represent and organise complex mathematical relationships such as multi-digit numbers (Huttenlocher, Jordan, & Levine, 1994; Laski et al., 2013; Thompson et al., 2013). Mental visualisations may also be used to ground abstract concepts. For example, in missing term problems of the format 4 + __ = 5, individuals may use visualisations of blocks or other concrete objects to balance the equation presented (Lourenco et al., 2018). Dynamic spatial tasks are thus expected to correlate with mathematical tasks requiring the mental manipulation or organisation of numbers. Third, intrinsic-static spatial tasks are reliant on form perception; the ability to distinguish shapes from a more complex background or to break more complex pictures into parts (Mix et al., 2016). Form perception is theoretically useful for spatial tasks such as map reading and figure drawing (Newcombe & Shipley, 2015), and for mathematics tasks such as distinguishing symbols such as + and × symbols, interpreting charts and graphs, and accurately completing multistep calculations which require an understanding of the spatial relations between symbols (Landy & Goldstone, 2007; 2010; Mix et al., 2016). As such, intrinsic-static spatial skills are predicted to relate to mathematics tasks that require identification and use of symbols or visual aids.

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Proposing theoretical explanations for associations between spatial and mathematical skills is further complicated by the role of developmental age. As outlined in the previous section, the relationship between spatial skills and mathematics appears to show sensitivity to developmental age. One explanation for this is that the role of spatial skills as predictors of mathematics may be greater for novel tasks compared to practiced, automatic mathematics skills (Ackerman, 1988; Uttal & Cohen, 2012; Young, Levine, & Mix, 2018). Spatial skills may provide scaffolding when students are faced with new mathematics material but may play a reduced role as mathematics skills become increasingly automatic or familiar (Mix et al., 2016). Alternatively, age-based differences in spatial-mathematical associations may be due to variations in the mathematical content that children are exposed to across school years (Mix et al., 2016). For example, in the early school years spatial scaling may be associated with number line estimation skills, but scaling performance is unlikely to correlate with performance on complex equations, to which children are exposed to in later school years.

In this section, it has been outlined that not all associations between spatial and mathematical tasks can be explained in the same way and a range of explanations have been proposed as theoretical accounts for specific spatial-mathematical relations. However, it is noteworthy that these accounts are based on theoretical predictions and, to date, there is limited evidence exploring the specificity of spatial- mathematical associations in primary school children in the context of these mechanistic accounts. Furthermore, this list of possible underlying mechanisms put forward to explain spatial-mathematical associations is not exhaustive and there may be additional explanations for other spatial-mathematical relations that have yet to be identified. Further research is needed to corroborate and refine the proposed explanations for spatial-mathematical relations, which considers the specificity and developmental sensitivity of these associations.