The Relationship Between Accuracy of Numerical Magnitude Comparisons and Children’s Arithmetic Ability: A Study in Iranian Primary School Children

The relationship between children’s accuracy during numerical magnitude comparisons and arithmetic ability has been investigated by many researchers. Contradictory results have been reported from these studies due to the use of many different tasks and indices to determine the accuracy of numerical magnitude comparisons. In the light of this inconsistency among measurement techniques, the present study aimed to investigate this relationship among Iranian second grade children (n = 113) using a pre-established test (known as the Numeracy Screener) to measure numerical magnitude comparison accuracy. The results revealed that both the symbolic and non-symbolic items of the Numeracy Screener significantly correlated with arithmetic ability. However, after controlling for the effect of working memory, processing speed, and long-term memory, only performance on symbolic items accounted for the unique variances in children’s arithmetic ability. Furthermore, while working memory uniquely contributed to arithmetic ability in one-and two-digit arithmetic problem solving, processing speed uniquely explained only the variance in single-digit arithmetic skills and long-term memory did not contribute to any significant additional variance for one-digit or two-digit arithmetic problem solving.

A possible explanation of the contrasting results reporting the relationship between numerical knowledge and arithmetic ability is that no commonly accepted, identical version of the magnitude comparison task exists (De However, it has been shown that these are not interchangeable constructs and reliable tools must assess the same construct for cross-comparison (Inglis & Gilmore, 2014;Lindskog, Winman, Juslin, & Poom, 2013). Inglis and Gilmore (2014) investigated the psychometric properties of (and interrelation between) these different indices.
They found that numerical distance or ratio effect has poor reliability with retesting in the same group and has no relationship with either Weber fractions or accuracy. Moreover, the researchers showed that Weber fractions have lower retest reliability than accuracy (Inglis & Gilmore, 2014). In response to these findings, the researchers recommended indexing acuity on an individual's numerical comparison by reporting just their accuracy figures (Inglis & Gilmore, 2014).
An identical test intended to use just the accuracy index to measure the acuity during the comparison of magnitudes is the "Numeracy Screener" (Nosworthy et al., 2013). In this test, participants are required to compare pairs of both symbolic (56 digit pairs) and non-symbolic (pairs of dot arrays) magnitudes ranging from 1-9. The screener controls the visual characteristics of dot arrays by ensuring that half of them have equal area, while the other half have equal perimeter. Display time is controlled by giving participants a consistent time period (2 minutes) to complete the test (Nosworthy et al., 2013).
After demonstrating the validity of the Numeracy Screener by finding age-related improvements in the accuracy of numerical comparisons, the researchers (Nosworthy et al., 2013) tried to explore whether the test is capable of explaining variability in children's arithmetic achievement. They found that the scores on both symbolic and non-symbolic items of this basic assessment tool significantly correlated with the scores on arithmetic and calculation skills, but only performance on the symbolic items accounts for the unique variance in arithmetic skills after controlling for the effect of some general cognitive abilities such as working memory, intelligence and reading ability.

General Cognitive Factors and Their Relationship to Arithmetic Ability
In addition to numerical magnitude comparison processing as a domain-specific numerical ability, other domaingeneral cognitive factors might explain individual differences in children's arithmetic abilities. Working memory, processing speed, and long-term memory are three cognitive abilities that may have a significant impact on individual differences in the arithmetic abilities of different children (Alloway & Alloway, 2010;Berg, 2008;Bull & Johnston, 1997;Dulaney, Vasilyeva, & O'Dwyer, 2015;Geary, 1993;Raghubar, Barnes, & Hecht, 2010). Before clarifying the plausible role of these cognitive factors in children's arithmetic abilities, the development of arithmetic achievement in primary school-aged children should be briefly explained.
In the process of solving single-digit arithmetic problems, children pass through simpler to more advanced strategies. First, they learn to use a procedure in which they count all of the numbers, a procedure of obtaining a sum in which both addends are counted to find the solution. Then, they learn to use "counting on" (or min) procedures, in which they state the value of the larger addend before counting the smaller one. In both strategies, children may use their fingers for counting or count aloud. Finally, frequent use of counting strategies leads to the formation of long-term memory association between problems and their answers, eventually resulting in direct retrieval of answers from long-term memory when solving simple arithmetic problems (Geary, 1994;Siegler & Shrager, 1984). As mentioned earlier, numerical magnitude comparison skills may affect the ability of children to make decisions on the larger addend when using a min strategy (De Smedt et al., 2009). Furthermore, working memory, processing speed, and long-term memory are three types of cognitive ability that may play a central role as children progress through the different procedures used to solve arithmetic problems. Working memory generally refers to a limited-capacity information-processing resource engaged in two key processes, the storage of information and the simultaneous processing of the same or other information (Berg, 2008).
It is assumed that, in solving arithmetic problems, children use their working memory capacity to store relevant information (both addends) and simultaneously process that information using sum or min procedures. Furthermore, for the construction of an association between problem and answer in long-term memory, both addends (first and second number) and the final answer must be simultaneously active in working memory (Geary, 1993). Thus, children with higher working-memory capacity may solve simple arithmetic problems more efficiently than children with lower working-memory capacity. In concert with this assumption, several researchers examining the relationship between working memory and arithmetic ability have found that working memory uniquely explains individual differences in children's arithmetic ability in the presence of other number-specific or general cognitive factors (e.g., De Smedt et al., 2009;Nosworthy et al., 2013;Rasmussen & Bisanz, 2005; Van der Ven, Van der Maas, Straatemeier, & Jansen, 2013).
Processing speed refers to the cognitive ability to execute simple cognitive tasks rapidly and efficiently (Case, 1985, as cited in Brown, 1986. During sum or min counting procedures to solve arithmetic problems, processing speed may facilitate counting speed to successfully pair problems and their answers in available working memory before decay. More efficient and faster pairing of problems and their answers in limited working-memory capacity results in more established long-term memory representations. In contrast, lower processing speed may increase the time necessary for deriving counted answers and for pairing a problem with its answer in working memory; hence, working memory may decay before the pairing is produced, impeding the development of an association in long-term memory (Geary, 1993). Bull and Johnston (1997) initially examined the role of processing speed in the arithmetic ability of children. They reported that processing speed uniquely contributed to arithmetic ability when other cognitive skills (including working memory and reading ability) also provided unique contributions.
Given that arithmetic performance transparently depends on automatic retrieval of answers from long-term memory, long-term memory itself may be a plausible determinant of children's arithmetic performance (Fuchs et al., 2006). However, given the role of working memory and processing speed in the retrieval of information from long-term memory, it is unknown to what degree arithmetic performance can be attributed to long-term memory rather than working memory and processing speed (Fuchs et al., 2006;Swanson & Beebe-Frankenberger, 2004).

Present Study
In the present study, we examine the relationship between both the symbolic and the non-symbolic tasks of the Numeracy Screener and arithmetic ability in Iranian second grade children. Another goal is to find the extent to which these tasks explain the variability in both single-digit and two-digits arithmetic problem solving scores given the presence of contributions from working memory, processing speed, and long-term memory. With regard to the literature, we hypothesize that both symbolic and non-symbolic numerical magnitude comparisons are significantly correlated with arithmetic ability. Furthermore, we expect that symbolic (but not non-symbolic) magnitude comparisons uniquely explain the variability in arithmetic scores in the presence of working memory, processing speed, and long-term memory. Tavakoli 571

Participants
For the initial sample, the subject group consisted of typically developing second-grade boys (N = 113) from four different classrooms. Using Raven's Standard Progressive Test (Raven & Raven, 2003), the IQ level of the children ranged from 95 to 125. Every student in the sample was born in 2007. Their vision was normal or corrected to normal; none of them had developmental disabilities. After obtaining written informed parental consent, each subject was assessed individually during a 45-60 minute session in a quiet room located inside the school. All assessments were conducted in one session at the end of the academic year, when the students would have been taught all mathematical topics that were part of the curriculum for that year.

Measures Numerical Magnitude Comparisons
Numeracy Screener was used to measure both symbolic and non-symbolic numerical magnitude comparisons.
The Numeracy Screener (Nosworthy et al., 2013) is a quick written tool to assess children's ability to compare different numerical magnitudes ranging from 1-9. The magnitudes were presented in both symbolic (56 digit pairs) and non-symbolic (56 pairs of dot arrays) formats. In the non-symbolic form, dot stimuli were controlled for area and density; in both symbolic and non-symbolic formats, each numerical magnitude was counterbalanced taking into account the side of presentation. During the assessment section, participants were given one minute to cross out the larger of the two dot arrays in the non-symbol comparisons and one minute to cross out the larger of the two Arabic digits in the symbol comparisons. Before starting the assessment children were instructed to complete three sample items with the examiner and practice nine items on their own to ensure their understanding of the task. Furthermore, during the instruction given for the dot array comparisons, they were told not to count the dots, but rather to just try and estimate them. When the examiner ensured that the child understand the rule, the main part of the test was started. In addition, during the test, the child was reminded not to count the dots, but just try to estimate them as quickly as possible. The order in which children did the symbolic and non-symbolic representation comparisons was varied so that half of the students started with the Arabic digit comparison and the other half started with the dot array comparison. With administration of the Numeracy Screener, two scores were provided for each participant. The first score was the number of correct answers in the non-symbolic task (labeled as the non-symbolic comparison score); the second was the number of correct answers for the symbolic task (labeled as the symbolic comparison score).
Since two-digit arithmetic tasks may require more abstract reasoning, further attentional resources, and basic arithmetic skills, any time pressure may negatively affect child's performance accuracy. Thus, unlike the math fluency problems, there was no time limitation in these tasks. The score on the tests was the number of correct answers for each task.

Working Memory
"Counting Span" is a well-known task to measure working-memory capacity. This task was originally presented by Case, Kurland, and Goldberg (1982) and was adapted by Keenan (1998). This task was used with some other modifications to adjust it to the ability of the participants. During the warm-up phase of the task, participants were told that they would be playing a memory game in which they had to count all of the red dots on a card and then remember the number of red dots. Each card had between two and six red dots and a number of distractor dots in blue and green. The task had six levels; each level had two trials. In the first level, the participant was presented one card at a time, told to count the red dots, and then asked how many red dots were on the card after the card was turned over. In the second level, the participant was shown one card and told to count its red dots. Subsequently, the card was turned over and another card was presented. The same procedure was repeated for the second card. Then, the participant was asked how many red dots were on the first card and the second card. In levels three through six, the procedure was exactly the same, but one more card was added to each level. The procedures terminated when the participant incorrectly answered twice at any level. The score was the number of correct answers across all levels, with a maximum of 12.

Processing Speed
The "WJ-III Visual Matching Task" (WJ is for "Woodcock-Johnson") was used to measure processing speed (Woodcock, McGrew, Mather, & Schrank, 2003). In this task, the participants were required to locate and circle identical numbers in rows of six numbers. They had three minutes to complete 60 rows presented on two A4-sized pages. The participant's score was the number of correctly circled matching numbers in each row. The test developer reported that the internal consistency of the task is .91.

Long-Term Memory
To measure semantic long-term memory, we used Retrieval Fluency, described in Woodcock and Johnson (1989).
In this task, the participants were asked to recall related items divided between two categories (animals and fruits) for 1 minute per category. Credit was earned for each correct, non-duplicated answer.

Results
The means and standard deviations (along with skewness and kurtosis) of all variables were presented in Table   1.  As is shown in Table 1, all variables had skewness and kurtosis between -1 and +1. Bivariate correlations were conducted between the raw scores of all variables to determine their interrelationships. The results are presented in Table 2. Based on Cohen (1988), most variables had significant moderate to large correlations with each other (Table 2).
With regard to the relationships between aptitude on numerical magnitude comparisons and arithmetic ability, the higher correlations were between symbolic magnitude comparisons and both single-digit addition and subtraction (.42 and. 44, respectively) and both two-digit addition and subtraction (.34 and .36, respectively). However, smaller correlations existed between non-symbolic magnitude comparison and all arithmetic tasks (.22, .25, .19, and .21). Regarding correlations between general cognitive abilities and arithmetic skills, single-digit addition and subtraction had significant correlations with long-term memory, working memory, and processing speed, while two-digit addition and subtraction were significantly correlated with working memory and processing speed, but not with long-term memory.
Since comparisons of both symbolic and non-symbolic magnitudes were moderately related to all arithmetic skills, in order to get more information about the unique contribution of each magnitude comparison to both math fluency and calculation skills, four linear regression analyses were conducted, two for single-digit addition and subtraction and two for two-digit addition and subtraction. The major focus of these analyses was to carefully examine the independence of each numerical magnitude comparison to predict single-digit and two digits addition and subtraction skills while controlling for the effect of working memory, processing speed, and long-term memory. For this purpose, all of the variables were entered in one step. The results were presented in Table 3 and Table 4.  Table 3 shows that the first linear regression analysis with single-digit addition as the dependent variable was significant (F (5, 113) = 9.563, p < .001, R 2 = .31). In this model, performance on working memory, processing speed, and symbolic comparison explained the significant and unique variance in single digit addition. Conversely, non-symbolic magnitude comparison and long-term memory did not uniquely explain the variance in that. The second linear regression analysis, which used single-digit subtraction as a dependent variable, was also significant (F (5, 113) = 6.211, p < .001, R 2 = .23). In this model, similar to first model, symbolic comparison, working memory, and processing speed had significant and unique contributions to single-digit subtraction. However, nonsymbolic comparison and long-term memory did not account for significant and unique contributions to (or variances in) single-digit subtraction skill.
Europe's Journal of Psychology 2016, Vol. 12(4), 567-583 doi:10.5964/ejop.v12i4.1175 Table 4 shows that the third linear regression analysis with two-digit addition as the dependent variable was significant (F (5, 113) = 5.322, p < .001, R 2 = .22). In this model, performance on working memory, and symbolic comparison explained the significant and unique variance in two-digit addition. Conversely, non-symbolic magnitude comparison, processing speed, and long-term memory did not uniquely explain the variance in that. The forth linear regression analysis, which used two-digit subtraction as a dependent variable, was also significant (F (5, 113) = 6.321, p < .001, R 2 = .24). In this model, similar to third model, symbolic comparison and working memory had significant and unique contributions two-digit subtraction. However, non-symbolic comparison, processing speed, and long-term memory did not account for significant and unique contributions to (or variances in) twodigit subtraction skill.

Discussion and Conclusion
The present study explored whether performance on the symbolic and non-symbolic items of a Numeracy Screener is related to individual differences in the arithmetic ability of second grade children in Iran. The arithmetic skills measured included single-digit addition and subtraction tasks and series of two-digit addition and subtraction problems. Furthermore, we intended to determine whether magnitude comparison tasks significantly explain variances of single-digit and two-digit addition and subtraction skills over some general cognitive factors including working memory, processing speed, and long-term memory.
We found that participants' scores on both symbolic and non-symbolic items were significantly correlated with scores on single-digit and two-digit arithmetic. Participants who scored highly on symbolic or non-symbolic magnitude comparison tasks on the Numeracy Screener also tended to receive high scores on single-digit and twodigit arithmetic. Nevertheless, the results of our regression analyses demonstrated that only participants' scores on the symbolic items of the Numeracy Screener uniquely explain the variances in both single-digit and two-digit addition and subtraction skills. These findings are closely comparable with the results reported by Nosworthy and colleagues (2013). These researchers initially used the Numeracy Screener to measure performance on numerical magnitude comparisons and determine its relation to arithmetic ability. In addition, similar results have also been reported by researchers from studies that used other forms of magnitude comparison tasks (e.g., Holloway & Ansari, 2009;Kolkman et al., 2013;Lonnemann et al., 2011;Mundy & Gilmore, 2009;Sasanguie et al., 2012;Vanbinst et al., 2015). For example, in a recent study Vanbinst et al. (2015) tried to explore the relationship between numerical magnitude comparison ability and simple arithmetic ability in first-and second-grade children using a computerized version of the magnitude comparison task. The researchers (Vanbinst et al., 2015) used accuracy and reaction time as indices to measure both symbolic and non-symbolic magnitude comparison. Their results showed that symbolic (but not non-symbolic) numerical magnitude comparisons uniquely explain the variances in arithmetic ability among primary school children in the first grade.
Our data also support the commonly held assumption that symbolic magnitudes are acquired through a process in which the mapping symbolic system is mapped onto a previously existing non-symbolic system (Mundy & Gilmore, 2009). The significant correlation between our symbolic and non-symbolic comparison tasks, as well as the significant relationships between both magnitude comparisons and arithmetic ability, suggests that similar developmental trajectories may underlie the processing and representation of both symbolic and non-symbolic numerical magnitudes. However, the unique contribution of symbolic magnitude comparison to arithmetic ability may suggest that symbolic magnitude comparisons cover the variances explained by non-symbolic representations. Accuracy of Numerical Magnitude Comparisons and Children's Arithmetic Ability 576 The results of our regression analyses also revealed that working memory uniquely explains the variance in both single-digit and two-digit arithmetic, in the presence of the unique contribution of symbolic magnitude comparison.
The central role of working memory in developing children's arithmetic ability have been reported by several studies (e.g., Dulaney et al., 2015;Raghubar et al., 2010;Van der Ven et al., 2013). It has been assumed that working-memory capacity provides a limited work space for children to use sum or min strategies when solving simple digit arithmetic problems, consequently leading to the development of associations between problems and their answers in long-term memory (Geary, 1993). Thus, higher working-memory capacity leads to more accurate performance while making calculations and more long-term memory representations. In addition, attentional resource of working memory can assist children as they attempt to perform accurate two-digit calculations. In concert with this assumption and in line with previous studies, our findings support the vital role of working-memory capacity in children's arithmetic achievement.
To elaborate further, the regression analyses revealed that processing speed uniquely contributed to single-digit and two-digit addition and subtraction, but not to two-digit arithmetic. It has been assumed that, in solving arithmetic problems, processing speed may facilitate counting speed to successfully pair problems and their answers in available working memory capacity before decay begins. More efficient and faster pairing of problems and their answers in limited working memory capacity result in more established long-term memory representations (Geary, 1993). Bull and Johnston (1997) initially examined the role of processing speed in children's simple digit arithmetic abilities. They reported that processing speed uniquely contributed to arithmetic ability in the presence of unique contributions of other cognitive skills. Furthermore, the unique contribution of processing speed to simple-digit arithmetic abilities has also been reported by other researchers (e.g., Durand et al., 2005;Fuchs et al., 2006;Kail & Hall, 1999;Swanson & Beebe-Frankenberger, 2004). In line with Bull and Johnston (1997), the present study also supports the unique contribution of processing speed to simple-digit arithmetic problem solving. In addition, both the single-digit arithmetic test and the processing speed task (visual matching) were paper-and-pencil, timerestricted tasks. Thus, the rate of children's sensory-motor performance may underpin their execution speed in both tasks. However, unlike single-digit arithmetic, participants had an unlimited amount of time to complete our two-digit arithmetic test. This may clarify the reason for the insignificant contribution of processing speed to calculation. In addition, compared with simple (single-digit) arithmetic, solving problems involving two-digit arithmetic may require more abstract reasoning, further attentional resources, and basic arithmetic skills. Thus, it might be expected that the variances in two-digit arithmetic are explained to a greater degree by fluid intelligence, working memory, or basic arithmetic abilities rather than processing speed.
In line with Swanson and Beebe-Frankenberger (2004) and Fuchs et al. (2006), our regression analyses revealed that long-term memory did not uniquely contribute to variances in single-digit or two-digit arithmetic. These findings are in contrast with the assumption that long-term memory itself may be a plausible determinant of arithmetic performance in children. Given the role of working memory in the retrieval of information from long-term memory, it is plausible that arithmetic performance can be attributed to working memory rather than semantic long-term memory.
In conclusion, the current findings demonstrated that significant relationships exist between the performances of second-grade primary school children in Iran on the Numeracy Screener items as well as on both measured arithmetic abilities (single-digit and two-digit arithmetic). In addition, it was found that only the symbolic comparison portion of the Numeracy Screener accounted for unique variances in both single-digit and two-digit arithmetic.
These results are in line with previous research in which the Numeracy Screener was initially used (Nosworthy  et al., 2013) and highlights the applicability of the test in educational setting. Quickly and easily assessing the numerical magnitude processing abilities of students using such a simple and easily available written test can help educators focus on these essential skills during math instruction in the classroom. In particular, this may assist educators in developing countries where there is no available advanced computerized equipment in many school settings. Furthermore, the significant relationship shown between non-symbolic items of the Numeracy Screener and both single-digit and two-digit arithmetic skill replicates the findings of Nosworthy and colleagues (2013)  The current findings demonstrated that a combination of numerical magnitude comparison tasks assessed via a Numeracy Screener, memory, and processing speed explain 30% and 22% of the variances in math fluency and calculation skill, respectively. The remaining variances might be explained by other domain-specific factors such as number line estimation, subitizing ability, counting knowledge, or other domain-general cognitive abilities such as attentive behavior, inhibition, task switching, language, and intelligence. Further research is recommended to determine the relationship between the Numeracy Screener, the other factors summed up above, and arithmetic ability in primary school children, especially in other developing countries.