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While symbolic number processing is an important correlate for typical and low mathematics achievement, it remains to be determined whether children with high mathematics achievement also have excellent symbolic number processing abilities. We investigated this question in 64 children (aged 8 to 10), i.e., 32 children with persistent high achievement in mathematics (above the 90th percentile) and 32 average-achieving peers (between the 25th and 75th percentile). Children completed measures of symbolic number processing (comparison and order). We additionally investigated the roles of spatial visualization and working memory. High mathematics achievers were faster and more accurate in order processing compared to average achievers, but no differences were found in magnitude comparison. High mathematics achievers demonstrated better spatial visualization ability, while group differences in working memory were less clear. Spatial visualization ability was the only significant predictor of group membership. Our results therefore highlight the role of high spatial visualization ability in high mathematics achievement.

Large individual differences exist in children’s acquisition of mathematical abilities (e.g.,

In the current study, we therefore examined whether young children with high mathematics achievement excel in their basic ability to process the magnitude and order of symbolic numbers. We also examined the role of spatial ability and working memory — two domain-general cognitive abilities that have been linked with individual differences in mathematics achievement (

There are two ways of thinking about numbers: On the one hand, they describe magnitude (i.e., cardinality), and, on the other hand, they represent order (i.e., ordinality) (e.g.,

Numerical magnitude processing refers to people’s intuition about the quantity of numbers. In recent decades, there has been an increased interest in the role of numerical magnitude processing for explaining individual differences in mathematics achievement (

Another fundamental and distinct aspect of symbolic number processing is the processing of numerical order or ordinality. Numerical order refers to each number’s position in the counting sequence (e.g., number 7 comes before 8, but after 6) (

Symbolic numerical magnitude comparison and symbolic numerical order processing are correlated (

What explains the association between our understanding of number and mathematics achievement? Mathematics is considered a hierarchical subject, meaning that the concepts and skills, such as our understanding of number, that we acquire early in development are considered as foundational for learning more complex mathematical skills. An understanding of symbolic numbers and their relations is therefore assumed to positively contribute to the acquisition of advanced mathematics (

The widespread idea of a critical role of symbolic number processing for mathematical development, both in typical development and in dyscalculia, leads to the straightforward yet untested prediction that excellent mathematical abilities should coincide with excellent basic number processing skills. Surprisingly, the number processing abilities of individuals with high mathematics achievement have been very rarely investigated.

Against this background, the current study focused on the symbolic number processing abilities of children who demonstrate high mathematics achievement in primary school. In contrast to the previous studies that focused only on the magnitude component of number, we investigated if a high ability to process numerical magnitude and order are both characteristics of high mathematics achievement.

Domain-general cognitive factors, such as spatial ability (

Another domain-general cognitive factor that is thought to contribute to high mathematics achievement is working memory (

Given the above-mentioned research evidence, spatial visualization ability and working memory were included as variables of interest in the current study. Another reason for including them was that both variables have been linked to the development of symbolic number processing (

These associations between spatial visualization ability and working memory with symbolic number processing raise the question whether potential differences in symbolic number processing between high mathematics achieving children and average-achievers might be driven by domain-general differences in their spatial visualization ability and/or working memory (as mentioned by

The current study investigated the role of symbolic number processing in high mathematics achievement in children in third and fourth grade of primary school. This age was chosen because of the observed a shift in the middle grades (i.e., Grades 3 and 4) from cardinal to ordinal processing with regard to relative importance of symbolic skills to predict arithmetic performance from Grade 1 to Grade 6 (

The current study aimed to address three objectives. First, we investigated whether there were differences between high-achieving children in mathematics and average-achieving children in symbolic number processing. Based on the above-reviewed studies, we expected that the children who are high achieving in mathematics would outperform the average-achieving children on both measures of basic symbolic number processing. Second, we tested whether there were group differences in the spatial visualization ability and working memory. Based on the above-reviewed studies, we expected to replicate that the children with high mathematics achievement would outperform the average-achieving children on both domain-general cognitive abilities. Lastly, we assessed the unique contributions of both symbolic number processing and domain-general abilities to high mathematics achievement versus average mathematics achievement. We explored whether the symbolic number processing abilities would still be a significant contributing factor to high mathematics achievement, when we also considered group differences in domain-general cognitive abilities.

The participants were selected based on their persistent percentile ranking of the Flemish Student Monitoring System for mathematics or LVS-math (

All children in Grade 3 and 4 of the eight participating schools were given informed consent forms and 286 forms were returned. Fifty children were explicitly not allowed to participate by their parents. For those children who consented, we requested the schools to send us the most recent LVS data. For the children in Grade 3, we collected LVS-math data from the middle of Grade 2 (February) and the beginning of Grade 3 (September). For the children in Grade 4, we had LVS-math data from the middle of Grade 3 (February) and the beginning of Grade 4 (September). There were five to six months between the last LVS-math test and the start of the data collection.

To be included in the high mathematics achieving group, children had to score above the 90th percentile on the LVS-math on two consecutive time points that were at least 6 months apart. All high-achieving children for whom we received informed consent were included in our sample. Each high-achieving child was individually matched with a child from the same class that scored between the 25th and 75th percentile on the same LVS-math tests at the same two consecutive time points. Similar criteria have previously been used in research on high mathematics achievers (

The final sample consisted of 32 high-achieving children in mathematics and 32 average-achieving children in mathematics (^{2}(1) = 0.25,

Variable | High mathematics achievers | Average mathematics achievers | Cohen’s |
BF_{10}^{a} |
||
---|---|---|---|---|---|---|

Gender | 13 girls, 19 boys | 15 girls, 17 boys | – | – | – | – |

Age in months ( |
109.10 (6.51) | 109.16 (5.73) | -0.08 | .936 | -.02 | 0.30 |

Age range in months | 98 – 122 | 99 – 120 | – | – | – | – |

LVS range (Pc) | 91.50 – 99 | 25 – 75 | – | – | – | – |

^{a}The Bayes factor gives the relative support of the data for the alternative hypothesis (BF_{10}; evidence for group differences) compared to the null hypothesis (evidence for no group difference) (see section Analyses).

The two symbolic number processing tasks were computerized and developed with OpenSesame 3.1.9 Software (

To measure the magnitude aspect of number processing, we used a symbolic magnitude comparison task with Arabic digits. The child had to compare pairs of Arabic numerals: one displayed on the left side of the computer screen and another displayed on the right (

To measure the ordinal aspect of number processing, we used the numerical order task (see

We used the Block Design subtest of the WISC-III (

We measured the visual-spatial working memory with the backward Corsi block tapping task (

We measured verbal working memory with the backward digit span (

A motor speed task was included in order to control for the speed of response in answering on the keyboard (

Given that mathematical skills have been reported to be highly correlated with reading skills (e.g.,

As a control for children’s verbal ability, we used the vocabulary test of the WISC-III (

All testing was done individually in a quiet location at the children’s schools. Testing was done in two sessions of approximately 30 minutes each. All tasks were administered in the same order to all children. The first block consisted of the motor task, block design, OMT, and symbolic magnitude comparison. The second block consisted of the backward Corsi block tapping task, numerical order task, vocabulary test, and the backward digit span task.

Analyses were run in JASP (

Frequentist analyses were used as they are an often-used method to assess the statistical significance of results. Cohen’s _{01}) with the alternative hypothesis (BF_{10}), and thereby quantify the evidence in favor of one of these hypotheses (see _{10} = 20) indicates that the data are 20 times more likely under the alternative hypothesis than under the null hypothesis. Likewise, a Bayes factor of 0.5 (BF_{10}) indicates that the data are 20 times more likely under the null hypothesis than under the alternative hypothesis. By using Bayesian analyses, we could get a more fine-grained understanding of group differences or the lack thereof. A Bayes factor is a continuous measure of evidence, but there are some classification schemes that can be used for interpretation (e.g., _{10} = 1 indicates no evidence for either hypothesis, BF_{10} > 1 indicates anecdotal evidence, BF_{10} > 3 indicates moderate evidence, BF_{10} > 10 indicates strong evidence, BF_{10} > 30 indicates very strong evidence, and BF_{10} > 100 indicates decisive evidence for the alternative hypothesis. We used the default priors in JASP, as there was too little research available to determine informed priors.

Lastly, we ran a binary logistic regression analysis to determine which of the variables best discriminated among the two groups (i.e., high mathematics achievers versus average mathematics achievers) and to investigate if symbolic number processing abilities would still be a significant predictor of high mathematics achievement, when we also considered children’s domain-general cognitive abilities and other control variables.

The descriptive statistics for each group’s scores on symbolic number processing, spatial visualization ability, and working memory tasks are shown in

Variable | Max. | Reliability |
High mathematics achievers |
Average mathematics achievers |
||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Range | Skewness | Shapiro-Wilk | Shapiro-Wilk |
Range | Skewness | Shapiro-Wilk | Shapiro-Wilk |
|||||

Number processing | ||||||||||||

Symbolic comparison | ||||||||||||

Accuracy | 36 | .03^{a} |
35.06 (0.98) | 33-36 | -0.79 | .82 | < .001 | 34.81 (1.03) | 32-36 | -0.92 | .85 | < .001 |

Reaction time (ms) | – | .91^{b} |
712.40 (118.20) | 458.25-951.81 | -0.05 | .98 | .832 | 759.20 (138.50) | 532.72-1035.31 | 0.55 | .94 | .056 |

Numerical order | ||||||||||||

Accuracy | 24 | .55^{a} |
22.19 (1.93) | 14-24 | -2.69 | .73 | < .001 | 20.94 (1.90) | 16-24 | -0.54 | .73 | < .001 |

Reaction time (ms) | – | .84^{b} |
1706.80 (376.70) | 735.42-2659.05 | 0.45 | .94 | .070 | 1982.47 (658.80) | 1312.56-4044.25 | 1.50 | .53 | < .001 |

69 | .79^{a} |
48.03 (9.84) | 25-65 | -0.71 | .95 | .189 | 38.88 (10.52) | 20-58 | -0.05 | .97 | .415 | |

Visual-spatial working memory (Corsi) | 24 | .66^{a} |
11.22 (2.24) | 7-18 | 0.43 | .96 | .280 | 10.41 (2.03) | 7-16 | 0.10 | .96 | .249 |

Verbal working memory (Digit span) | 21 | .75^{a} |
8.44 (2.11) | 3-12 | -0.47 | .97 | .391 | 7.36 (1.91) | 4-12 | 0.34 | .95 | .170 |

^{a}Cronbach’s alpha. ^{b}Odd-even reliability using the average RT per participants for correct odd and even trials.

The results of the group comparisons on the symbolic number processing tasks and the domain-general cognitive measures are provided in

Variable | Cohen’s |
BF_{10} |
||
---|---|---|---|---|

Number processing | ||||

Symbolic comparison (Accuracy) | 1.28 | .210 | .23 | 0.40 |

Symbolic comparison (RT) | -1.78 | .085 | -.31 | 0.77 |

Numerical order (Accuracy) | 2.71 | .011 | .48 | 4.14 |

Numerical order (RT) | -2.65 | .013 | -.47 | 3.61 |

3.34 | .002 | .59 | 16.39 | |

Working memory | ||||

Visual-spatial working memory (Corsi) | 1.75 | .089 | .31 | 0.74 |

Verbal working memory (Digit span) | 2.22 | .034 | .39 | 1.59 |

The two groups differed significantly in spatial visualization ability, and the Bayes factor indicated that evidence for this difference was strong. There was no significant group difference for the backward Corsi block tapping task. However, a significant group difference in backward digit span was found, but the Bayes factor revealed that the evidence for this difference was only anecdotal.

We further verified if the two groups differed in terms of their motor speed, reading ability, and vocabulary (

Variable | High mathematics achievers |
Average mathematics achievers |
Cohen’s |
BF_{10} |
||||
---|---|---|---|---|---|---|---|---|

Motor RT (ms) | 517.10 | 97.86 | 524.00 | 85.03 | -0.36 | .722 | -0.06 | 0.20 |

One Minute Test | 60.84 | 12.78 | 55.13 | 13.14 | 1.94 | .062 | 0.34 | 0.99 |

Vocabulary | 34.00 | 6.25 | 30.81 | 1.90 | 2.43 | .021 | 0.43 | 2.37 |

To gain an understanding of the unique contributions of our investigated variables to high mathematics achievement, we conducted a binary logistic regression analysis with achievement group as the dependent variable (^{2}(10) = 23.76, ^{2}) and categorized 75% of the participants correctly in the two groups. Only spatial visualization ability emerged as a significant predictor and was associated with a higher likelihood of belonging to the high mathematics achieving group. More specifically, for each increase in the score on the Block Design test, the odds of belonging to the high mathematics achievement group increased by a factor of 1.076.

Variable | B | β | Exp(B) | ||
---|---|---|---|---|---|

Symbolic comparison (Accuracy) | 0.366 | 3.474 | .039 | .916 | 1.441 |

Symbolic comparison (RT) | -0.003 | 0.004 | -.354 | .502 | 0.997 |

Numerical order (Accuracy) | 3.529 | 2.577 | .546 | .171 | 34.094 |

Numerical order (RT) | -0.001 | 0.001 | -.640 | .162 | 0.999 |

Spatial visualization ability | 0.073 | 0.034 | .818 | .029 | 1.076 |

Verbal working memory | 0.219 | 0.177 | .453 | .215 | 1.245 |

Visual-spatial working memory | -0.205 | 0.185 | -.442 | .269 | 0.815 |

One Minute Reading Test | 0.031 | 0.030 | .410 | .302 | 1.032 |

Motor Reaction Time | 0.004 | 0.006 | .384 | .459 | 1.004 |

Vocabulary | 0.024 | 0.062 | .147 | .691 | 1.025 |

During the past decades, there have been major gains in our understanding of the cognitive characteristics of typically developing children, and of the cognitive factors associated with dyscalculia. Largely absent are studies that examine the cognitive correlates of children with high mathematics achievement. Previous studies in this research domain have mainly focused on adolescents and adults, with little attention to children. The role of spatial ability and working memory in high mathematics achievement has garnered some attention, but much less is known about the number processing abilities of individuals with high mathematics achievement. In the current study, we compared children that showed persistent high mathematics achievement to their persistent average-achieving peers. The main aim of this study was to examine whether children who demonstrate high mathematics achievement excel in their basic ability to process number. Of primary interest were the two components of basic symbolic number processing, namely magnitude and order, that have been found to play an important role in explaining mathematics achievement in typically developing children (e.g.,

Analyses with regard to the symbolic number processing tasks revealed small differences between the mathematics achievement groups for order (both accuracy and RT), but no differences for magnitude comparison. The latter result is somewhat surprising given the findings of

The current study observed group differences on the numerical order task, but these differences were small. A possible explanation for differences in numerical order, but not magnitude comparison, is the age-related shift in the predictive value of these two symbolic number processing abilities for later mathematics achievement (

The absence of strong evidence for group differences on the symbolic number processing ability tasks merits further comment. This finding does not preclude the possibility that increased symbolic number processing ability might be an important characteristic of high mathematics achievement. High achievers may have a heightened symbolic number processing ability earlier in development, but this may attenuate across development, when, for example, more complex number processing abilities start to become more important. Future studies should investigate whether symbolic magnitude and order processing might have a time limited role in high mathematics achievement in children, particularly at younger ages. Alternatively, future studies on symbolic number processing in children with high mathematics achievement should shift the focus to more complex numbers, such as fractions (e.g.,

It is plausible to assume that group differences in symbolic number processing ability might be observed when more fine-grained indices of the symbolic magnitude processing and order processing are considered, i.e., the canonical numerical distance effect and the reversed distance effect, respectively. Performance on a numerical magnitude comparison task is poorer (i.e., slower reaction times) for numbers that are closer together (e.g., 5 and 6) compared to numbers further apart (e.g., 2 and 7), which is the classic numerical distance effect (

Turning to the domain-general cognitive abilities, similar to

Findings were mixed regarding the role of working memory in high mathematics achievement. High mathematics achievers performed similar to controls on the Corsi block tapping task. Frequentist analyses showed a significant—yet small—group difference for digit span backwards, but the Bayes factor indicated that this evidence was only anecdotal. In all, we do not observe higher working memory capacity in children with high mathematics achievement, at least not at this age. Our results are inconsistent with findings on the contribution of working memory among children and adolescents of high mathematical achievement (e.g.,

The above-described findings demonstrate that visual-spatial working memory and spatial visualization ability might have different roles in high mathematics achievement, aligning with recent observations by

The inclusion of both symbolic number processing abilities as well as domain-general cognitive abilities in the present study allowed to examine their unique contribution to high mathematics achievement. This is crucial, as previous studies have revealed that these symbolic number processing abilities and domain-general cognitive abilities are highly related (

When evaluating the above findings, it should be kept in mind that they were based on a sample of children that were high achieving in mathematics, but not necessarily mathematically gifted (

Furthermore, a shortcoming of the current study was the small sample size, which might make it difficult to find small effects. Indeed, potential differences in symbolic processing might be very subtle between individuals with and without high mathematics achievement. It is therefore important to replicate this study with a larger sample size in order to address the role of symbolic number processing, spatial visualization ability, and working memory in high mathematics achievement more thoroughly.

It is clear that much remains to be learned about the cognitive characteristics of young children with high achievement in mathematics. In this study, we primarily focused on very basic symbolic number processing abilities. As mentioned earlier, it would be interesting to examine whether magnitude and order processing might play a more important role in high mathematics achievement earlier in development. Another interesting avenue for further research would be to examine mathematical domains that might really allow high achievers to show their mathematical potential, for example, domains such as flexibility in mathematical thinking identified by the Russian psychologist

Item | Stimulus |
Distance | Ratio | |
---|---|---|---|---|

Left | Right | |||

1 | 8 | 7 | .125 | |

3 | 6 | 3 | .5 | |

7 | 2 | 5 | .286 | |

8 | 9 | 1 | .89 | |

6 | 4 | 2 | .67 | |

7 | 5 | 2 | .714 | |

8 | 2 | 6 | .25 | |

9 | 6 | 3 | .67 | |

2 | 5 | 3 | .4 | |

3 | 1 | 2 | .33 | |

2 | 4 | 2 | .5 | |

7 | 8 | 1 | .875 | |

4 | 5 | 1 | .8 | |

7 | 1 | 6 | .143 | |

6 | 8 | 2 | .75 | |

2 | 3 | 1 | .67 | |

7 | 6 | 1 | .86 | |

9 | 3 | 6 | .33 | |

2 | 1 | 1 | .5 | |

5 | 6 | 1 | .83 | |

9 | 1 | 8 | .11 | |

3 | 5 | 2 | .6 | |

8 | 4 | 4 | .5 | |

7 | 9 | 2 | .78 | |

4 | 3 | 1 | .75 | |

5 | 9 | 4 | .56 | |

2 | 6 | 4 | .33 | |

8 | 3 | 5 | .375 | |

5 | 1 | 4 | .2 | |

9 | 2 | 7 | .22 | |

1 | 4 | 3 | .25 | |

3 | 7 | 4 | .43 | |

5 | 8 | 3 | .625 | |

7 | 4 | 3 | .57 | |

1 | 6 | 5 | .167 | |

9 | 4 | 5 | .44 |

Item | Order | Stimulus |
Distance | ||
---|---|---|---|---|---|

Left | Center | Right | |||

yes | 2 | 3 | 4 | 1 | |

no | 2 | 1 | 3 | 1 | |

no | 7 | 9 | 8 | 1 | |

yes | 3 | 4 | 5 | 1 | |

yes | 1 | 2 | 3 | 1 | |

no | 7 | 5 | 9 | 2 | |

yes | 5 | 6 | 7 | 1 | |

no | 6 | 2 | 4 | 2 | |

no | 6 | 4 | 5 | 1 | |

yes | 1 | 3 | 5 | 2 | |

yes | 6 | 7 | 8 | 1 | |

yes | 3 | 5 | 7 | 2 | |

no | 6 | 8 | 7 | 1 | |

no | 2 | 4 | 3 | 1 | |

yes | 5 | 7 | 9 | 2 | |

no | 8 | 4 | 6 | 2 | |

yes | 4 | 5 | 6 | 1 | |

yes | 4 | 6 | 8 | 2 | |

no | 5 | 3 | 7 | 2 | |

no | 4 | 5 | 3 | 1 | |

no | 6 | 7 | 5 | 1 | |

yes | 7 | 8 | 9 | 1 | |

no | 3 | 5 | 1 | 2 | |

yes | 2 | 4 | 6 | 2 |

We used a similar approach as _{10} = 1.394e+13. Next, we compared the NDE of the average and high achievers. While the size of the NDE was smaller for the high achievers (_{10} = 0.517.

We used a similar approach as _{10} = 193.95, showing the expected reverse distance effect. The RDE of the high achievers (_{10} = 1.00.

Lieven Verschaffel is an Associate Editor for the

MB: Conceptualization, methodology, formal analysis, writing – original draft, funding acquisition. EP: investigation, methodology, writing – review and editing. JT: Conceptualization, supervision, writing – review and editing, funding acquisition. LV: Conceptualization, supervision, writing – review and editing, funding acquisition. BDS: Conceptualization, supervision, writing – review and editing, funding acquisition, project administration.

The authors would like to thank all participating children and schools.

The Social and Societal Ethics Committee of the KU Leuven approved the study (G-2018 01 1100).

This work was supported by the Research Foundation - Flanders FWO (1124219N) and by the Research Fund KU Leuven (C16/16/001).