Summary and Research Aims

It has been proposed that even young children have a system for representing numerical magnitude nonverbally (approximate number system), that the symbolic number system may map onto this, and therefore forms the basis for which formal arithmetic knowledge may build upon. Measures which tap the acuity of this system have been designed - magnitude comparison tasks, these simply involve choosing the larger

numerosity and those that include nonsymbolic rather symbolic stimuli can be presented to young children. Multiple indices have been used to quantify performance (e.g. accuracy, reaction time, distance/ratio effect, Weber fraction). Over development, performance on these tasks improves and the effect of the distance between numerosities on comparison ability decreases, as does the size of the Weber fraction. This has been taken to suggest that the acuity of representations within the approximate system become more precise (whether this is due to less compression or less variability is unknown and also difficult to entangle).

While the change in the distance (and ratio) effect and size of the Weber fraction has been investigated in different age groups the majority of these studies are cross- sectional, i.e. include different children. What is missing from the existing literature is the development of this in the same children over the initial period of formal schooling, i.e. longitudinal data. As many different ways to investigate the effect of distance and to estimate the acuity of the ANS have been used with differing findings, it will be important to address this issue within the construction of the tasks and the analysis of the data.

The importance of this approximate number system on arithmetic development has therefore become an emerging area of interest. There is consensus that some aspect of symbolic comparison ability is related to individual differences in arithmetic achievement in different age groups of children both concurrently (e.g. Durand et al., 2005; Holloway & Ansari, 2009; Mundy & Gilmore, 2009) and longitudinally (e.g. De Smedt et al., 2009; Sasanguie, Van den Bussche & Reynvoet, 2012; Sasanguie et al., 2013). Whether this is accuracy, speed, or the effect of distance on the task has varied along with the arithmetic or mathematics measure used to represent this ability.

The findings investigating the relationship between measures of nonsymbolic comparison and arithmetic are less clear. Halberda and colleagues (Halberda et al., 2008; Libertus et al., 2011; Libertus et al., 2013; Mazzocco et al., 2011b) consistently find a relationship between the two skills both concurrently and longitudinally, in different age

groups, using different presentations of the task, and when also controlling for other cognitive skills. Whereas less consistent results are found by other groups (e.g. Holloway & Ansari, 2009; Mundy & Gilmore, 2009; Sasanguie et al., 2013). There are many differences between the methods used which could be possibilities for these inconsistencies, for example the way the comparison task is presented, the number of items being compared, the method of analysing the data, and the arithmetic measure used to represent numerical skill. This warrants further investigation within the same sample of children, for example by completing nonsymbolic comparison tasks with these different manipulations and by calculating multiple indices of performance.

A more systematic approach should therefore be taken when investigating the relationship between both symbolic and nonsymbolic comparison and arithmetic achievement, to enable further understanding of this association. Further research is needed with a longitudinal design to investigate the relationship of magnitude comparison ability with arithmetic to expand the existing literature. Whilst studies incorporating a longitudinal design have been conducted it will be important to include measures of both arithmetic and magnitude comparison at all time points, this will enable the investigation of the bidirectional relationship between the two. Children’s early arithmetic skill should also be controlled in the analysis which means that it is the growth in arithmetic knowledge that is being investigated. When prior arithmetic skill is not controlled any relationships found between magnitude comparison and later arithmetic may be reflecting the association between arithmetic ability at the two time points, rather than a specific relationship between the two different measures (magnitude comparison and arithmetic). At an early time point arithmetic ability and magnitude comparison ability may be sharing variance, therefore by controlling for prior differences in arithmetic skill you are removing the variance due to early arithmetic ability.

The differing findings of previous research may be due to the different arithmetic and mathematics measures used. Many of the measures include items that could be described as assessing basic understanding of number rather than calculation abilities and some of the items may overlap with the processes involved in the magnitude comparison tasks (e.g. identifying larger numbers, ordering numbers). Therefore it is important that future research also includes measures that avoid this possible overlap. Another suggestion would be to include both timed and untimed measures to investigate whether there is a specific relationship with arithmetic fluency (speed at which simple problems are solved or retrieved from memory) or with arithmetic ability in general (i.e. also more difficult

calculation problems). In addition the relationship between magnitude comparison and arithmetic may differ with different operations (addition, subtraction, multiplication). Therefore to extend this research area an examination of the relationships between magnitude comparison and these different numerical operations assessed separately must be carried out as well as with overall calculation ability.

An additional point to note is that in some studies the age range of the children spans two or three years or grades. To help clarify the relationships a more concentrated age range should be used, with the measures of magnitude comparison administered as early as possible before children have received a large amount of formal schooling in numeracy.

As highlighted earlier, there are other domains that will be important for children’s arithmetic development (i.e. working memory, counting, attention), as well as

environmental influences, for example teaching and social economic status. To include every possible predictor of children’s arithmetic development is beyond the scope of this thesis; however, children’s verbal and nonverbal ability will be estimated to account for their general cognitive ability, this will allow for the investigation of whether any relationships found are due to individual differences in this ability.

The main aims of this thesis are therefore:

 To assess the relationship between symbolic and nonsymbolic comparison tasks and whether they are tapping the same underlying construct.

 To investigate performance on these magnitude comparison tasks over development, within the same sample of children (i.e. longitudinally).  To explore the concurrent relationships between numerical processing and

arithmetic achievement, while controlling for children’s general cognitive ability, in a large representative sample.

 To extend this longitudinally by also exploring the predictive value of these measures to later arithmetic achievement over an extended period of time (one and two years).

 In order to address some of the previous aims a large number of children will need to be recruited and the most efficient way to complete this will be to use a group testing design. Therefore in order to verify the findings, a subgroup of children will

also need to complete individually presented measures, more akin to those frequently used in the literature.