Relationships between Possible Predictor Variables and Arithmetic

After exploring the relationship between the comparison tasks the next step was to investigate the relationships between the comparison measures and arithmetic

achievement. Arithmetic was defined as a measure that assesses basic number skills (e.g. identifying and writing numbers and counting) and basic calculations involving addition, subtraction, multiplication and division. Two measures of arithmetic were chosen for this study: the first was a standardised test that assesses children’s ability to complete numerical operations (WIAT: Numerical Operations). The measure begins with identifying and writing numbers, counting items and then on to addition, subtraction and

multiplication problems; these progress from simple one digit sums to two digit sums (see Appendix 1 for a list of the items). On this measure the children were under no time pressure to complete the items. The second arithmetic measure was one minute addition, which requires children to complete simple one and two digit addition problems. This test

was a speeded measure and therefore assesses fluency at completing the sums. The relationship between the comparison tasks and these two arithmetic tests were analysed separately, this allows for the investigation of whether any relationships found are related to arithmetic achievement in general (speeded and untimed), or perhaps just arithmetic fluency. The best fitting comparison task CFA was taken (magnitude comparison and letter comparison) and a further CFA was conducted including arithmetic and the control

measures that might influence the relationship between children’s magnitude comparison ability and arithmetic achievement (age, verbal ability, nonverbal ability, number

identification). There were significant correlations between both the WIAT and speeded addition measures and both comparison latent variables, even with the other variables controlled for (age, verbal ability, and nonverbal ability). This finding provides support for previous studies who found a significant relationship between arithmetic achievement and performance on magnitude comparison tasks (e.g. symbolic: Durand et al. 2005;

nonsymbolic: Libertus et al. 2011; both: Mundy & Gilmore, 2009).

Due to the large sample size it was more likely that any associations would be significant, therefore it is important to look at the strength of the associations. Taking WIAT arithmetic achievement, the correlations with both the magnitude and letter comparison tasks were moderate in strength (the association between arithmetic and magnitude comparison was slightly stronger). There were also significant relationships between children’s arithmetic achievement and age, vocabulary, nonverbal ability, and number identification. The weakest association was between arithmetic and age; this is possibly due to the focussed age range of the children in the study. An interesting finding was that the strongest association was with number identification, which warranted further

investigation. For speeded addition, there were again moderate associations with both comparison tasks but they were very similar in strength. The association with number identification was this time similar in strength to the comparison factors. As moderate relationships were found (which were stronger than some found by previous studies) this merited the investigation of the predictive relationships of the measures.

Correlations only inform us about the strength of the linear relationship between the measures and do not mean that an improvement on one measure will result in an improvement in the other (i.e. predict individual differences). Therefore to explore these relationships further a full path model was run to investigate which, if any, of the measures would predict significant variance in children’s arithmetic achievement. Overall it was found that the variables explained 54% of variance in children’s WIAT achievement and 40% of

the variance in children’s addition fluency scores. Magnitude comparison was found to be a significant predictor of children’s arithmetic achievement (both untimed and speeded), this provides support for previous studies (e.g. symbolic: Durand et al., 2005; nonsymbolic: Libertus et al., 2011) and extends previous findings as the current study included controls for a variety of other factors. For example, the comparison process itself was controlled for due to the inclusion of the letter comparison task, children’s general cognitive ability (both nonverbal ability and vocabulary) and age were also included in the CFA. However, these results are in contrast to previous studies that found that individual differences on symbolic but not nonsymbolic comparison tasks predicted variance in children’s arithmetic scores (e.g. Holloway & Ansari, 2009; Sasanguie, De Smedt et al., 2012), and studies that found significant differences between children with arithmetic scores in the average range and those described as having a mathematics learning disorder on symbolic but not

nonsymbolic comparison tasks (e.g. De Smedt & Gilmore, 2011). Although letter

comparison was found to be related to children’s arithmetic achievement it did not predict significant variance in their arithmetic scores, suggesting that whilst digits and letters have an ordinal structure, it is the action of comparing the magnitude information that is represented by digits that predicts variance in children’s arithmetic skill (alongside the comparison of nonsymbolic numerosities).

Although the inclusion of a measure of children’s knowledge and familiarity of numbers (number identification) was initially included to act as a control measure it was found to be a significant predictor of variance in children’s arithmetic scores (alongside magnitude comparison). This may be intuitive that children who have greater knowledge of symbolic numbers are also more proficient on tests which use the symbolic number system; however, there is conflicting evidence on this in the wider literature. This finding that familiarity with symbolic numbers is important for arithmetic achievement is consistent with existing literature with both typically developing children and children with

mathematical difficulties (e.g. De Smedt & Gilmore, 2011; Gilmore et al., 2010; Landerl et al., 2004; Lembke & Foegen, 2009; Mazzocco & Thompson, 2005). Nevertheless, it is in contrast to some evidence from typically developing children that performance on number recognition (reading) tasks is not related to arithmetic achievement (De Smedt et al., 2009; Soltész, Szűcs & Szűcs, 2010; however see Gilmore et al., 2010). The number identification measure used in the present study required children to match a verbal number word to its visual Arabic form, ignoring the distractor items that included common errors that young children make; this task is therefore more complex than asking children to simply read

numbers that they are highly familiar with (i.e. 1 to 10). The task is potentially assessing children’s knowledge of the symbolic system, knowledge of what the symbol represents (i.e. mapping the symbol to the magnitude it represents) and/or knowledge of place value. All of these skills could be important for arithmetic achievement. Children who understand place value may also be better at checking their answers on the arithmetic task and correcting any errors. This predictive relationship observed between number knowledge and arithmetic could be analogous to the relationship between letter knowledge and reading ability.

Neither age nor general ability (nonverbal ability or vocabulary) were predictors of variance in children’s WIAT arithmetic scores. However, age was a significant predictor of variance in children’s performance on the speeded arithmetic task. The finding that differences in the age of the children did not predict individual differences in untimed arithmetic achievement is not surprising due to the small age range of the children. All children were from the same year group at school so will have received a similar degree of formal numeracy teaching. The predictive relationship found between age and children’s performance on the speeded arithmetic measure could therefore reflect speed of

processing differences between the youngest and oldest children (e.g. Kail, 1991; Kiselev, Espy & Sheffield, 2009).