Modelo de radiación

This chapter focused primarily on the nature and development of the ANS measured using magnitude comparison tasks. For this purpose, distance and ratio effects were first investigated followed by a detailed analysis of the structure of magnitude comparison tasks and its change over time using confirmatory factor analyses.

Overall, the comprehensive analyses of children’s performance on symbolic and nonsymbolic magnitude comparison tasks revealed three findings: First,

children’s performance on magnitude comparison tasks generally showed significant distance and ratio effects for both symbolic and nonsymbolic comparisons with better performance on the far trials than close confirming previous findings (Barth et al., 2003; Piazza et al., 2010; Xu and Spelke, 2000; Halberda et al., 2008; Gilmore et al. 2010; Libertus et al., 2011; Mazzocco et al., 2011; Halberda and Feigenson, 2008). There was a significant interaction for the symbolic distance effect across Times 1 and 2. Children performed significantly better on far trials than close trials at Time 2, but the distance effect for the symbolic task at Time 1 (numeric distance effect) was not significant. This may be due to the fact that some children had difficulties reading the Arabic numerals. It was noted that a third of the children made at least two mistakes in reading single digit Arabic numerals. If mastery of the single digit Arabic numerals is taken into account, a marginal distance effect can be

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found even in young children. As expected, no such limitation applied to performance on nonsymbolic comparison.

However, the findings for nonsymbolic comparison included two interactions: at Time 1 where the manipulation of square size affected far trials (easier items in general) and at Time 5 when square size influenced the performance of items in the close condition (harder condition). The findings suggest that

manipulating square size affects very young children differently than older children suggesting that exposure to the task as well as a developmental improvement in magnitude comparison skills boost children’s performance. These interactions may support the findings from Sekuler and Mierkiewicz (1977) that fourth and seventh graders slope of the function relating to judgement time to distance was comparable to adult performance, whereas the function of kindergarten and first grade children was much steeper. The authors concluded that there are no qualitative differences supported by the fact that the shape of the numerical difference effect was the same for the groups, but rather quantitative differences. According to the authors, a steeper slope may imply either that the representation of numerical magnitudes is

compressed in younger children, or that discriminal dispersion around the means are larger in young children, or a combination of the two.

However, Sekuler and Mierkiewicz (1977) used only symbolic comparison tasks. The nonsymbolic interaction may be more complicated. These interaction may be due to the perceptual advantage of fixed size stimuli over surface-area matched stimuli, or the requirement to supress incongruent stimuli in fixed size condition may affect children differently at different ages according to the difficulty of the

comparison (far versus close).

Second, the results revealed nonsymbolic ratio effects (2:3 > 3:4 > 5:6) across all time points. Children performed more accurately on ratios with a large difference (i.e. 2:3) than ratios with a small difference (5:6). Furthermore, manipulating the feature size impacts children’s performance on comparison tasks. Fixed size arrays were generally easier to discriminate for both, distance and ratio trials, than surface- area matched arrays suggesting that it is more difficult for children to ignore the prominent feature size in the surface-area matched condition where the array with fewer stimuli has bigger squares compared to many tiny squares. Previous studies have shown that performance on ANS measures increases with age, with adults

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discriminating numerosities outside of the ability of infants. Halberda and Feigenson (2008) identified the Weber fraction of ANS in three-, four-, five- and six-year-old children and adults using dot arrays ranging from 1 to 14 dots spanning the ratios from 1:2 through 9:10. They further controlled for object area similar to the current fixed size versus surface-area matched manipulation. The results showed an increase in performance over time with six year-olds performing like adults. The current study found a comparable increase in performance over time.

Third, the dynamic relation between symbolic and nonsymbolic magnitude comparison tasks changes over time. This change coincides with children’s entry to the formal school system. Symbolic and nonsymbolic comparison tasks loaded on separate factors at four to five years of age. Interestingly, the same pattern emerged at Time 3, with magnitude comparison tasks being represented by two separate underlying factors (symbolic and nonsymbolic) rather than one general comparison factor. However, the distinction between the factors is declining and it seems that children’s representation and processing of magnitude comparison tasks at the age of 5 years and 6 months (autumn term of Year One) is changing towards the general comparison ability construct. To further investigate this hypothesis, analyses of the subsequent two time points are crucial. If this hypothesis is true, a shift towards the single-factor model should occur. At Times 4 and 5, the single-factor model should be preferred meaning that magnitude comparison tasks load on one general

comparison factor and not two distinct factors (symbolic and nonsymbolic) confirming the developmental trend towards a general magnitude comparison factors.

Children’s pre-school representation of magnitude comparison tasks may best be described by two distinct underlying factors: symbolic and nonsymbolic

magnitude comparison (Libertus et al., 2011, Piazza, 2010; Piazza and Dehaene, 2004).

This distinction is vanishing slowly around Year One moving from two constructs towards one general comparison ability construct (see also Kolkman et al., 2012). It seems that the shift in the processing of magnitude comparison tasks may be complete by the end of Year One (6 years and 4 months of age). Questions remain on why this change in the representation and processing of the magnitude

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One possibility would be that children’s mastery of the Arabic numeral system and their understanding of magnitude in general may play a crucial role in the development of comparison tasks. Interestingly, the change appears around the time after children entered school and are formally trained in numeracy. Also, after five testing sessions, children were very familiar with the task and stimuli and this

exposure may further foster the change in processing. The findings suggest that at an early, pre-school age, processing of magnitude comparisons may load heavily on cognitive resources, and that children devise different strategies to solve symbolic and nonsymbolic magnitude tasks. Through exposure and formal training on numeracy the processes become more automatized and rely on broader general comparison abilities.

This hypothesis is supported by findings on high achievers on number reading task. These number wizards achieved the maximum score on the number reading task at Time 1 (four years of age). If using only data from the number wizards, then the two-factor and single-factor models do not significantly differ, suggesting, according to the principle of parsimony, that the latter is the better model and that the performance of number wizards can best be explained by one general magnitude comparison construct at Time 1. At Times 2 and 3 however, the two models do slightly differ favouring the two-factor model, though the significance alpha level was only .05. Nevertheless, these findings point to the fact that children’s understanding of numerals may be mediating the divide between symbolic and nonsymbolic magnitude comparison. The Times 3, 4 and 5 results are similar to the findings from the whole sample that symbolic and nonsymbolic magnitude

comparison form one general magnitude comparison construct. All in all, the findings indicate that children’s mastery of Arabic numerals plays a crucial part in the structure of the ANS. Once children have a complete understanding of the single digit numerals, then the symbolic and nonsymbolic comparison task are best

described by a one-factor model. These findings will inform future studies which should also account for children’s number recognition skills when examining the relationship between ANS and arithmetic.

In summary, the results clarify the little investigated structure of symbolic and nonsymbolic magnitude comparison. At pre-school age, ANS tasks show two distinguishable skills compared to the integration of the ANS skills into one general magnitude comparison structure. In view of these recent results, previous findings on

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the relationship between symbolic and nonsymbolic comparisons and their impact on arithmetic skills at school age should carefully be re-examined. The following

chapters will investigate the concurrent as well as longitudinal prediction of arithmetic focusing on the role of magnitude comparison.

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Chapter 4. Concurrent Prediction of Early Arithmetic across Time.