Introduction

The Number System [1,2] is thought to be responsible for the basic processing that engages mental representations of numerical quantities. According to current theories, the Number System constitutes a domain-specific cognitive scaffolding on which high-level mathematical competence is assembled [3,4]. Therefore, the relationship between basic numerical capacities (BNC) and subsequently developed mathematics skills has been of increasing interest. Up to now, most studies have applied cross-sectional designs that are useful in evaluating the associations between variables but not in drawing conclusions on the causality of the associations [5,6]. This limitation can be overcome by a longitudinal approach, which may help to clarify the direction of causal relationships for typical development, and may also highlight the role of early impairment in later development. Most longitudinal studies have, however, focused on the relationship between arithmetic and cognitive abilities (e.g., [7]), or have focused on a narrow age range, specifically early grades which is the crucial period when children acquire the first symbolic numbers and learn counting principles [4,8–12]. Consequently, there is insufficient evidence on the specific and unique role of BNC in middle and later stages of mathematics learning in which more sophisticated knowledge is acquired.

As far as we know, two studies have focused on core indices of BNC and later mathematics skills with a longitudinal perspective. In the first one, the authors found that the numerical approximation ability of 14-year-old children correlated with the children’s past scores in standardized mathematics achievement tests, extending all the way back to kindergarten [13]. Moreover, this correlation remained significant when controlling for individual differences in a wide range of cognitive and performance factors. This finding strongly supports the domain-specific hypothesis on the role of BNC in mathematics attainment. However, owing to the retrospective design of this study, the direction of the causality is hard to define. Acuity in the mental representation of numerical magnitudes at age 14 might be the cause, but it can also be the consequence of mathematics proficiency (see [14] for a comprehensive analysis). In the second study, the authors reported the first longitudinal research that prospectively tracks children’s core numerical competencies and arithmetic development throughout all elementary school years [15]. The participants were grouped according to individual reaction time (RT) for numerical comparison and subitizing, as slow, medium and fast performers. The authors found that subgroup classification at 6 years of age predicted computation ability at 6 years, 9.5 years, and 10 years. In a separate analysis, they also found that the subgroups did not differ in processing speed or nonverbal reasoning and concluded that subitizing and number comparison do not tap general cognitive abilities but reflect individual differences that are specific to the domain of numbers.

Despite the relevance of these findings, this study failed to show evidence supporting the domain-specific hypothesis because the effects of the core BNC in the presence of the domain-general abilities were not extensively analyzed. However, this kind of stringent analysis is essential in testing whether the BNC measures retain their status as unique predictors. This point is critical because several studies have revealed the predictive power of the domain-general mechanisms (e.g., speed of information processing, working memory and logical reasoning) in solving arithmetic problems (e.g., [16–19]). Nevertheless, except for one [19] none of them have assessed these potential mechanisms simultaneously with one another and with domain-specific measures of BNC.

Given the limited evidence available and the need for further testing of the domain-specific hypothesis within a longitudinal approach, here we designed a one year follow-up study to evaluate whether measures of BNC are related to individual differences in more sophisticated mathematics skills at the end of elementary school after controlling for domain-general cognitive abilities. This design allowed us to evaluate the domain-specific role of BNC on mathematical learning above and beyond early stages of learning.

As a further step toward this goal, we have tried to address some methodological issues presented in previous developmental studies. Firstly, the multi-componential nature of mathematics was taken into account to design outcome measures. As a consequence, we can explore the relative contributions of BNC and domain-general mechanisms in relation to the nature of mathematics performance (e.g., [19]). Secondly, we considered the assessment of mathematics achievement with a time limit for performance. Under this condition, we can differentiate children who efficiently process numerical information from those who do not [20]. Moreover, previous reports revealed that BNC accounted for individual differences in performance with timed mathematics tests better than with untimed mathematics tests [8,21]. Thirdly, we focused on RT and accuracy trade-off analysis (efficiency) instead of response accuracy in BNC tasks because several developmental studies often needed to deal with ceiling effects from a certain age onward (e.g., [22]). Efficiency-based measures also allow us to inspect developments in the processing of numerosities and magnitudes and enable us to observe developments of standard effects of number processing (e.g., numerical distance effect) that are consistently reported in adults [21,23]. These effects could be more informative on how numbers and magnitudes are represented and processed in the cognitive system than the accuracy measures.

Considering these methodological issues, in the present research the measurement of BNC was focused on the size of well-known effects of basic numerical processing based on a RT-accuracy trade-off analysis and the measurement of mathematics outcomes relied on multi-component and speeded tests. This design could reveal a more fine-grained picture of the relationship between the distinctness of quantity representations and individual differences in performance of diverse components of mathematics knowledge in later elementary education.

At the beginning of the study we assessed BNC in 3^rd and 4^th graders using two RT-based tasks that required the understanding of numerosity and the ability to recognize and judge small numerosities (up to 9), which required minimal cognitive resources and low levels of formal mathematics achievement. We therefore used a simple number comparison, which would test whether children understood number magnitudes, and a simple enumeration task. Similar tasks had been extensively reported in previous developmental studies (see [24] for a review). As pointed out, we focused on the size of numerical effects elicited by these tasks because they provide indicators for the preciseness of representations of numerical magnitudes. Such is the case of the numerical distance effect: when adults and school-aged children compare numerical stimuli for their relative magnitude, they are faster and more accurate at making responses when the numerical distance separating two numbers is relatively large than when it is small [21,25]; and the subitizing and counting effects: when sets are enumerated and participants are faster and more accurate on set sizes of 1-3 or 4 than on set sizes of 5 or larger [8,15,25,26].

As stated above, the potential relationships between the numerical effects and mathematics achievement could be related to more general cognitive differences between individuals. Such general cognitive differences would presumably be related to both mathematical skills and other crucial cognitive abilities such as reading. We therefore also collected measures of the children’s nonverbal reasoning, processing speed and reading decoding to help assess the specificity of the relationships between BNC and mathematics achievement and to rule out that the possible relationships may be due to individual differences in more general cognitive processes, rather than to individual differences in the processing and representation of numerosities.

One year later, several components of mathematics achievement were evaluated using two timed tests. One test was based on the mathematics curriculum for the grade and the other on fluency in solving simple calculations. We focused on fluency because it has been identified as a core arithmetic skill [27]. Moreover, attaining calculation fluency is difficult for many children (e.g., [28]). In addition, reading fluency and reading comprehension were assessed as outcomes of reading achievement.

In summary, we had three aims. The first was to investigate whether the numerical effects are domain-specific predictors of mathematics development at the end of elementary school by exploring whether they explain additional variance of later mathematics fluency after controlling for the effects of other cognitive skills, focused on nonnumerical aspects. Our second aim was to address the same issues but applied to achievement in the mathematics curriculum that requires solutions to fluency in calculation. These analyses assess whether the relationship found for fluency generalize to mathematics content beyond fluency in calculation. As a third aim, the domain specificity of the numerical effects was examined by analyzing whether they contribute to the development of reading skills, such as decoding fluency and reading comprehension, after controlling for general cognitive skills and phonological processing.

Methods

Results

The means and standard deviations of all measures are shown in Table 1. Results of the tasks in T₁ and T₂ show considerable individual differences in the children’s predictors and outcomes.

To provide more detailed information on the children’s task performance for critical numerical predictors, a series of mixed-design analyses of variance (ANOVA) were performed for Dot Enumeration and Symbolic Comparison tasks.

The expectation that children would subitize quantities up to 3, and count for 4 to 8 was supported by EM data. EMs were analyzed in a 2 (grade: 3^rd, 4^th) x 8 (numerosities: 1 to 8) ANOVA. Because the assumption of sphericity was violated, the within-participants effect and interactions are reported using the Greenhouse–Geisser adjustment. Means and standard errors of the EMs by numerosity and by grade can be found in Table 2.

We found that the main effect of grade was nonsignificant but the main effect of numerosity was highly significant (F(3.49, 164.16) = 395.02, p<.001, power= .99). In addition, we did not find a significant Numerosity x Grade interaction. According to post hoc Bonferroni-corrected t tests (α= .05/120 = .0041) the children’s EMs on numerosities of one, two and three did not differ at either grade level. Differences were, however, significant for numerosities of four to eight. In previous studies the subitizing range had been variably defined, sometimes ranging to three [8,26,33,35,36] and sometimes to four [14,37]. As the current study involved children who might have a comparably restricted subitizing range, we define the subitizing range as the numerosities one to three. Numerosity four could not be clearly ascribed to either the subitizing or the counting range and was thus excluded from the statistical analysis. Furthermore, as previous studies (e.g., [37]) have demonstrated, end effects for the enumeration of the largest numerosity of a set, the 9-dot stimuli were also excluded from the statistical analysis and the counting range was determined as numerosities five to eight. Based on these results, the size of subitizing and counting effects were calculated to quantify individual differences as was described previously in the Method section.

Next, an ANOVA was conducted using EM for each numerical distance (three levels: distances 1–3) as the within-participants variable, and grade (two levels: 3^rd - 4^th) as the between-participants variable. In this case, the sphericity assumption was confirmed. Mean and standard error of the EM for each numerical distance by grade are shown in Table 2.

We found a significant main effect of distance, with the small distance having longer EM than the large distance ( F(2, 94) = 23.20, p < .001, power =.99 ). More specifically, Bonferroni corrected t tests (α= .05/3 = .16) demonstrated that the comparison with Distance 1 was more efficient than with distance 3; but EMs were nonsignificantly different for Distances 1 and 2. We also found a main effect of grade (F(1, 94) = 14.39, p < .001, power = .99), which reflects the finding that 4^th graders were significantly faster than 3^rd graders. Finally, we did not find a significant interaction between Grade and Distance. Based on these results, the size of NDE was calculated to quantify individual differences in symbolic comparisons as was described in the Method section.

Although we are testing directional hypotheses, significance values of all correlational analyses come from a two-tailed distribution. As shown in Table 3, a significant association occurred between all outcome measures (R: .38 to .69, p<.01, power: .78 to .99). Intellectual ability was significantly correlated with the outcome measures, indicating that children who were in general better in nonverbal reasoning also showed better performance in reading and mathematics (R: .28 to .65, p<.05, power: .50 to .99). However, the numerical effect indexes were not significantly correlated among themselves or with intellectual ability. Furthermore, the numerical effects were not related to any of the outcome measures, but the size of the subitizing effect was related to the mathematics fluency score (R= -.35, p<.05, Power= .71). Figure 1 shows this significant association. Note that the children who exhibited a larger subitizing effect (SE values close to zero) showed, one year later, higher scores in mathematics fluency. In contrast, efficiencies on lexical and phonological processing were significantly associated (R=.84, p<.001, Power= .99) and were also associated to reading outcome measures (R: -.37 to -.48, p<.01, power: .76 to .95).

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Figure 1. Scatterplot showing significant correlation between mathematics fluency and the size of the subitizing effect.

The solid line represents the linear regression for this relationship.

https://doi.org/10.1371/journal.pone.0079711.g001

Hierarchical regression analyses were conducted to investigate up to what extent the mechanisms involved in subitizing, counting and comparing numerical magnitudes explain unique variance in mathematics and reading achievement. Eight steps (see Table 4 and 5) were sequentially included in these analyses to determine whether the size of the numerical effects explained variance within two measures of mathematics outcome and two measures of reading outcome above and beyond general predictors: age (Step 1), nonverbal reasoning (Step 2), general processing speed (Step 3); and cognitive capacities that might also be reflected in other cognitive skills such as reading: lexical (Step 4) and phonological (step 5) processing. Predictor variables were entered in a fixed order to provide the most stringent test for the roles of variables predicting reading and mathematics skills after controlling for the effect of prior general cognitive skills. The constant order of entry was based on the outlined predictions and predictive correlations.

Mathematics Fluency and Mathematics CBM were defined as outcomes in separate analyses (power .82 to .84). In a model, the numerical predictors NDE, CE and SE were sequentially added in steps 6, 7 and 8 to determine whether individual differences in SE could explain variance over and above general predictors, NDE and CE. In a complementary set of analyses, steps 6, 7 and 8 were switched to determine the specific contribution of NDE and CE in explaining variance over and above general predictors and the other numerical predictors. The values of standardized beta coefficients, incremental R² and F obtained for all variables included in the models are shown in Table 4. Note that the total amount of the variance explained by general, reading and numerical factors in predicting Mathematics Fluency and Mathematic CBM were 47.5% and 61.4%, respectively. The hierarchical regression analyses showed that the size of SE was a domain-specific predictor of both mathematics outcomes assessed one year later. In fact, after controlling for other variables, the size of SE significantly predicted 7.1% of the variance observed in mathematics fluency and 9.5% of individual variability in the achievement in mathematics curriculum. The size of CE explained a marginally significant amount of unique variance in fluency (5.5%) and the size of NDE did not significantly contribute to explaining individual variability in the measures of later mathematics achievement. Furthermore, the reading predictors did not explain additional variance of later mathematics skills after controlling for the effects of general cognitive skills.

Complementary analyses were run to test the domain specificity of the numerical predictor variables. In this case, hierarchical regressions were run following the same procedure described above. Here, Reading Fluency and Reading Comprehension were defined as outcomes in separate prediction models (statistical power .43 and .25, respectively). As shown in Table 5, the general cognitive, numerical and reading predictors explained 43.2 % of the variance in Reading Fluency and 30% of the variance in Reading Comprehension. Note that the numerical predictors did not significantly contribute to explaining individual variability in reading skills. However, the efficiency in lexical decoding explained a significant amount of unique variance found for Reading Fluency and Reading Comprehension (8.2% and 19.8%, respectively). Interestingly, the efficiency in phonological processing did not significantly contribute to explaining individual variability in later reading achievement.

Discussion

The aim of this study was to explore the domain specificity of children’s individual differences in the size of several numerical effects (NDE, CE and SE) during intermediate grades of elementary school. We first examined the associations of the numerical effects with contemporaneous cognitive and linguistic variables, as well as with mathematics and reading skills, and then we tested, with hierarchical regression analyzes, whether numerical effects predict academic skills such as mathematics and reading one year later, after controlling for the effects of other predictor variables. The results showed that the size of SE in intermediate grades was a significant domain-specific predictor of mathematics fluency and also curricular mathematics achievement, but not reading skills, assessed at the end of elementary school. Furthermore, the size of CE also predicted fluency in calculation, although this association only approached significance.

This finding makes sense given that mathematics fluency and subitizing require quick recognition and combination of small magnitudes [19]. One possibility is that children having a strong foundation for subitizing and mapping the corresponding quantities are at an advantage in learning number facts. This explanation is in agreement with the point of view of Gallistel and Gelman [38] who consider that subitizing involves the use of a fast preverbal estimation of numerosities and the mapping from the resulting magnitudes to number words in order to rapidly generate the number words for small numerosities. On the other hand, the retrieval of the number facts is mediated via the inverse mappings from verbal and written numbers to preverbal magnitudes and the use of these magnitudes to find the appropriate cells in the tabular arrangements of the answers. This may explain the smaller subitizing range of children with mathematics disability found by Koontz and Berch [39]; it may also explain the common finding that children with mathematics disability have difficulties learning number facts [40]. It is noteworthy that the domain-specific relationship found for subitizing and fluency is generalized to other mathematical domains in which small numerosities are embedded. In line with this result, Fuchs and colleagues [19] found that precise representations of small quantities was uniquely predictive of mathematics fluency and word-problem tasks that change, combine, compare and equalize numerical relationships.

As highlighted in the Method section, one such common process used in both the enumeration task and the mathematics tasks, is the speed of processing. Thus, it could be argued that the predictive value of the SE in later mathematics skills is simply due to both kinds of tasks being speeded. However, by using each child’s RT to numerosity 1 as a baseline, we were able to calculate the SE as a ratio accounting for individual differences in RT. Moreover, our regression analysis demonstrated that the SE explained a significant amount of variance in mathematics outcomes after controlling individual differences in general processing speed.

Previous longitudinal studies have found that processing small numerosities is closely related to early mathematics learning [8,41,42], confirming that subitizing and counting constitute domain-specific foundational skills on which the formally acquired mathematics knowledge is built. As a novelty, the present research illustrates that mechanisms specialized in recognizing, representing and mentally manipulating small numerosities are not only “start –up” tools for the acquisition of mathematical competence during the first years of formal education, instead, they continue playing a domain-specific role on more sophisticated mathematics skills acquired in later grades, above and beyond phonological and lexical processing, nonverbal reasoning and general processing speed.

This finding therefore encourages more empirical work to clarify the extent to which the Small Number System and mathematics knowledge are related [14,43]. Further research into this relationship that would examine a broader developmental age range in a longitudinal approach could reveal how subitizing modulates numerical symbolic acquisition from early to more advanced stages of mathematics knowledge above and beyond domain-general cognitive and linguistic mechanisms. On the other hand, it is known that the size of the subitizing effect decreases over developmental time [15]. However, it is unclear if this reduction simply reflects developmental changes in domain-general speed of processing and whether it is specific to numerical compared with nonnumerical stimuli. To examine these open questions, children at different ages and adults could be tested with numerical and nonnumerical subitizing tasks controlling for a measure of processing speed.

Contrary to previous reports [21,44], we found that NDE did not account for a significant proportion of the variance for mathematics fluency and achievement in mathematics curriculum. Since these studies included children younger than those recruited for the present study, a possible explanation to this is that the predictive value of NDE decreases in the course of mathematics acquisition. Several authors [45,46] reported that the NDE decreases from kindergarten to fourth grade, with minor changes occurring thereafter. In fact, Holloway and Ansari [21] found a significant correlation between NDE and mathematical competence in 6 year-old children but this association was not significant in 8 year-olds.

Additionally, in our study word and pseudoword decoding tests also failed to explain additional variance of mathematics fluency and other mathematics skills above and beyond general cognitive and numerical mechanisms. However, there is much support for the relationship between phonological awareness and mathematics achievement (see [47]). The triple-code hypothesis [48] states that language itself is needed to construct concepts of exact numbers greater than four, and several authors [40,49,50] argue that phonological awareness reflects the ability to differentiate between meaningful segments of language and to manipulate them, and this should consequently facilitate the differentiation and manipulation of single words in the number word sequence. However, some previous reports are in line with our finding; for instance, Durand and colleagues [51] found that phoneme deletion was a unique predictor of individual differences in reading but did not predict subsequent arithmetic skills. Moreover, Jordan and colleagues [12] reported that although basic reading proficiency was a strong predictor of mathematics achievement, it did not predict mathematics achievement above and beyond BNC. In another study, Fuchs and colleagues [17] reported that phonological processing (measured by rapid digit naming, first sound matching, and last sound matching) was a unique determinant of fact fluency, but did not predict other aspects of mathematics performance (e.g., story problems). In a subsequent study [52], the authors reported similar results when phonological processing was measured by phonological decoding of pseudowords.

Just as interesting, besides the different contributions of SE and CE to each mathematics outcome after controlling for the effects of general cognitive skills, we found that the inclusion of domain-general predictors, in particular nonverbal reasoning, resulted in a 10% increase in R² for predicting fluency in calculation, compared to a substantial increase (46%) in the accounted variance in mathematics tasks based on curriculum. These findings lend support to the notion that fluency in calculation and curricular achievement may represent distinct domains of mathematical cognition. In line with this assumption, Fuchs and colleagues [53] found that children situated at the lower-end of the performance range for calculation tasks exhibited distinctive cognitive profiles compared with children at the lower-end of the performance range for verbal problem-solving; whereas Hart and colleagues [54] demonstrated that mathematics problem solving has different genetic and environmental influences compared with fluency in calculation.

In summary, in the present research, we find substantial evidence supporting that the domain-specific capacities, specifically subitizing and to a lower extent counting, are significantly related to more sophisticated mathematics skills acquired at the end of elementary school above and beyond domain-general abilities. This finding contrasts with proposals that the core numerical competencies measured by enumeration will bear little relationship to mathematics achievement. As practical implication, the present research supports the importance of training low-level numerical processing for enhancing mathematical competence in typically developed children even in later grades of elementary school.

Finally, certain considerations should be taken into account for interpreting the findings of the present study. Firstly, our sample size was modest. A small sample size makes it difficult to detect differences when following up significant multivariate interactions. However, power analyses were referred systematically in the Results Section to assist readers in evaluating whether power is a concern. This is particularly important as the conclusions are based partially on nonsignificant findings in certain relationships. Secondly, our measures of mathematics outcomes came from self-constructed tests without standardization. Although, this could be a partial limitation considering that the analysis was based on individual differences rather than on classifying children according to performance.