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ability has generally been supported, the relationship between nonsymbolic comparison and arithmetic is less clear. As noted previously, researchers generally adopt one of two methods when including nonsymbolic comparison tasks in their research. Either to present items in the range of 1 to 9 and manipulate the effect of distance to match a symbolic version that is also being administered (Holloway & Ansari, 2009; Mundy & Gilmore, 2009; Sasanguie, De Smedt et al., 2012; Vanbinst et al., 2012), or use larger numerosities and manipulate the ratio difference between them (Bonny & Lourenco, 2013; Fuhs & McNeil, 2013; Gilmore et al., 2013; Inglis et al., 2011; Libertus et al., 2011; 2013). When varying the ratio between numerosities, the analysis tends to focus on accuracy scores and often a Weber fraction is calculated (to represent the acuity of the approximate number system), whereas when smaller numerosities are used the analysis is performed on reaction time data (and sometimes explores accuracy data). The aim of this next section will be to give an overview of all of these measures.

Holloway and Ansari (2009) found that the size of the nonsymbolic distance effect did not explain variance in children’s arithmetic scores (calculation or fluency subtests from the WJ-III) and this was the case whether using reaction time or accuracy measures (after controlling for age, symbolic comparison processing speed, symbolic comparison distance effect and reading). The only significant relationship between nonsymbolic comparison performance (calculated using overall accuracy, distance effect on accuracy scores, mean comparison RT, distance effect on RTs) and arithmetic, was between mean comparison RT and mathematics fluency scores, however this was weak in strength (r = -.21). They proposed that as a relationship with arithmetic achievement was found only with the symbolic comparison task, that this reflects the ability to access numerical magnitude information from symbols.

Mundy and Gilmore (2009) found no relationship between children’s mathematics scores and the nonsymbolic distance effect on response times but there was a relationship with overall accuracy (r = .35, p = .05). Accuracy performance on the task did predict

variance in children’s mathematics scores when it was entered first into the regression model (12.2%) but not over and above performance on the symbolic version of the task. Therefore the relationship between nonsymbolic comparison performance and

mathematics may be mediated by performance on symbolic comparison tasks. Similar results were observed in two further studies that administered both symbolic and nonsymbolic comparison tasks. While relationships were found between mathematics ability and symbolic comparison performance, no corresponding relationships were found with nonsymbolic comparison performance (De Smedt & Gilmore, 2011; Sasanguie, De Smedt et al., 2012; Vanbinst et al., 2012).

The nonsymbolic tasks used in the above studies include numerosities ranging from 1 to 9. Often this is because a symbolic version is also presented and so that the two tasks are analogous. It was highlighted earlier that the action of enumerating small range numerosities (1 to 3) is different to that of larger range numerosities (4 and beyond) (Landerl et al., 2004; Mandler & Shebo, 1982). Individuals are able to enumerate small numbers of items fast and accurately (Kaufman et al., 1949), whilst when presented with larger arrays the response is based on counting (Mandler & Shebo, 1982). Therefore it is possible that the processes involved when comparing small numbers of items differs to that when comparing larger number of items (Revkin, Piazza, Izard, Cohen & Dehaene, 2008).

An alternative way to estimate the precision with which numerical magnitudes are represented is to calculate a Weber fraction. Libertus et al. (2011) investigated whether ANS acuity in young children who had received little formal instruction in mathematics would be related to their arithmetic achievement. Children were recruited from preschools and had an average age of 4 years 2 months (range = 2 to 6 years old). The comparison task involved comparing two arrays (one blue, one yellow) with the number of dots in each ranging from 4 to 15, and the ratio between the two being 1:2, 2:3, 3:4, or 6:7. Both children’s accuracy and reaction times were recorded enabling three measures of performance: accuracy as percent correct, accuracy as the Weber fraction, and reaction time (this included times from both correct and incorrect comparisons). To assess children’s arithmetic achievement the Test of Early Mathematics Ability (TEMA-3; Ginsburg &

Baroody, 2003) was administered which includes tasks such as reading numbers, counting, and addition and subtraction calculation problems. Children who were more accurate and faster on the comparison task were found to have higher arithmetic scores (accuracy (% correct) r = .42; Weber fraction r = -.26; RT r = -.28), moreover, all three measures were

found to predict variance in arithmetic with age and vocabulary knowledge controlled (RT and accuracy: RT = 5%, accuracy = 13% of variance explained; RT and Weber fraction: RT = 8%, Weber fraction = 6% of variance explained).

In the same sample of children re-tested on the same measures, these significant associations were found to remain six months later (Libertus et al., 2013). Again, children who were more accurate and faster on the comparison task were found to have higher arithmetic scores (accuracy (% correct) r = .52; Weber fraction r = - .42; RT r = -.36) and all three measures were found to be significant predictors (ranging from 4 to 18% of variance explained).

Even when using the same arithmetic outcome measure as previous studies (TEMA- 3) and children of a similar age range (3 to 5 years old), the findings are not always

consistent. Fuhs and McNeil (2013) found no relationship between accuracy performance (percentage correct) and arithmetic achievement. The nonsymbolic comparison task used did differ to those used in the majority of studies: it was not presented to children on a computer and a larger range of numerosities, with some within the subitizing range (1 to 30), were presented. This will be explored more in Chapter 4 but does highlight the fact that the relationship between performance on nonsymbolic comparison tasks and arithmetic may depend on the stimuli, presentation or the data analysis of the task.

To explore whether the relationship between number acuity and arithmetic was the same in children (mean age = 8 years 6 months) and adults, Inglis et al. (2011)

calculated a Weber fraction for both groups and assessed arithmetic using the calculation subtest from the WJ-III. After controlling for nonverbal ability and age, it was found that there was a significant relationship between the size of the Weber fraction and arithmetic in children (r = -.55) but not in adults. The adults completed additional subtests (including mathematics fluency and more applied problems) from the WJ-III but there were no significant associations with any of these measures (all correlations were weak in strength). The comparison task completed by the adults included larger numerosities (9 to 70) than the task presented to the children (5 to 22), but with similar ratios, to make the task more difficult and to avoid ceiling effects. This therefore led to a wide range in the size of the Weber fraction in both groups. The authors speculate that the approximate number system does play a role in the early development of the understanding of number but that after mastering these early skills other factors, such as strategy choice or working memory capacity, may then lead to individual differences in arithmetic ability. What is therefore

needed is a longitudinal study that explores both the concurrent and longitudinal predictive relationships between number acuity and arithmetic, whilst also assessing other possible factors that could lead to individual differences.

In contrast, Lyons and Beilock (2011) did find a significant relationship between individual differences in the size of the Weber fraction and (complex) mental arithmetic problems in a group of adults (r = -.34). However, this was mediated by individuals’ symbolic number ordering ability (deciding whether or not triads of Arabic digits were in increasing order). This finding is similar to that of Mundy and Gilmore (2009) who found that children’s accuracy on the nonsymbolic comparison task was related to their arithmetic achievement but did not predict variance in arithmetic scores once symbolic comparison ability was controlled.

1.5.1.3. Differences in the relationship with arithmetic between symbolic and