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This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Symbolic numerical magnitude processing skills are assumed to be fundamental to arithmetic learning. It is, however, still an open question whether better arithmetic skills are reflected in symbolic numerical magnitude processing skills. To address this issue, Chinese and German third graders were compared regarding their performance in arithmetic tasks and in a symbolic numerical magnitude comparison task. Chinese children performed better in the arithmetic tasks and were faster in deciding which one of two Arabic numbers was numerically larger. The group difference in symbolic numerical magnitude processing was fully mediated by the performance in arithmetic tasks. We assume that a higher degree of familiarity with arithmetic in Chinese compared to German children leads to a higher speed of retrieving symbolic numerical magnitude knowledge.

According to the recently proposed “integrative theory of numerical development", numerical magnitude processing skills are at the core of numerical development and individual differences regarding these skills are assumed to be related to individual differences in arithmetic proficiency and math performance (

Recent meta-analyses revealed a significant association between non-symbolic numerical magnitude processing skills and math performance (

Cross-national assessments of mathematical achievement have repeatedly demonstrated that Chinese children outperform their non-Chinese peers at various ages (e.g.,

To further explore the association between arithmetic skills and symbolic number magnitude processing skills, we tested Chinese and German third graders. In their 3rd year of elementary school, children typically possess basic arithmetic skills. If better arithmetic skills are reflected in symbolic numerical magnitude processing skills, a superior Chinese performance should not only exist for arithmetic skills but also for symbolic numerical magnitude processing skills. Moreover, if arithmetic skills shape symbolic number magnitude processing skills, a performance difference between Chinese and German children in symbolic numerical magnitude processing should be mediated by arithmetic skills. As a performance difference between Chinese and German children in the arithmetic tasks as well as in the symbolic numerical magnitude comparison task might be due to the fact that Chinese number words can be verbalized more quickly than German number words (e.g.,

The German sample consisted of 33 third graders (18 female, mean age 9.1, range 8–10 years) recruited from a public primary school in Mühlheim am Main (Germany). The Chinese sample was the one described by

All children started with the symbolic numerical magnitude comparison task, then proceeded to the arithmetic tasks, and finally worked on the task assessing speed of number pronunciation. All tasks were carried out individually.

In the symbolic numerical magnitude comparison task, two single-digit Arabic numbers were presented on a screen. The two stimuli were arranged in a horizontal fashion. Children had to indicate the side with the larger numerical magnitude by using the left index finger when it was larger on the left hand side and by using the right index finger when it was larger on the right hand side. Responses were given by pressing the ‘S’ and ‘L’ keys on a notebook keyboard. Comparison pairs varied along four numerical distances (see ^{®} software (Neurobehavioral Systems, Inc.). Black-colored Arabic digits were presented in Times 60-point font on a 17^{′′} color screen against a white background. A target stimulus was presented until the response was given but only up to a maximum duration of 4000 milliseconds (ms), and was followed by a black screen for 700 ms. If no response was given, a trial was classified as erroneous. Correct responses were used for computing mean RT. Response times below 200 ms were excluded from further analysis as well as responses outside an interval of ±3 standard deviations around the individual mean. Trimming resulted in 1.5% of response exclusions for Chinese participants and in 1.3% of response exclusions for German participants. A reciprocal transformation (dividing 1 by each score) was carried out on mean RT to yield more normally distributed data (the Shapiro–Wilk test revealed that the distribution was not significantly different from a normal distribution after transformation, for Chinese participants

Comparison pairs for the different numerical distances.

Distance |
|||
---|---|---|---|

1–2 | 1–3 | 2–5 | 1–5 |

2–3 | 2–4 | 3–6 | 2–6 |

4–5 | 3–5 | ||

5–6 | 4–6 |

The arithmetic tasks consisted of nine blocks of ten problems each (see

Children received two sheets of paper, each listing 60 Arabic digits. Stimuli were arranged in six rows of ten items and presented in Times New Roman 48-point font. Children were instructed to correctly name the items as quickly as possible and to proceed from left to right, starting at the top row and continuing to the bottom row. The first sheet contained the numbers 1–3 and the second one the numbers 4–6 with no consecutive identical stimuli. Response time was measured using a stopwatch from a start signal until the child named the last stimulus. The mean response time of both sheets was used to estimate speed of number pronunciation. To yield more normally distributed data, a reciprocal transformation (dividing 1 by each score) was carried out on mean response time (the Shapiro–Wilk test revealed that the distribution was not significantly different from a normal distribution after transformation, for Chinese participants

By using two-sample

To assess effects of the distance between the two to-be-compared Arabic digits in the symbolic numerical magnitude comparison task, we looked for linear trends based on reciprocal RT separately for Chinese and German participants. ER was low in the symbolic numerical magnitude comparison task and it did not significantly differ between groups (see

Comparison of Chinese and German children (paired-sample

Chinese children |
German children |
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Age | 111 | 4.13 | 0.73 | 109 | 5.94 | 1.03 | |

RT comparison^{∗} |
656 | 115.12 | 20.35 | 812 | 161.71 | 28.15 | |

ER comparison | 2.86 | 2.05 | 0.36 | 2.21 | 2.24 | 0.39 | |

Arithmetic | 78 | 7.70 | 1.36 | 37 | 13.64 | 2.37 | |

Speed of number pronunciation^{∗} |
28 | 6.68 | 1.18 | 37 | 7.68 | 1.34 |

^{∗}p-value based on analysis of reciprocal reaction/response times.

In order to test whether a possible performance difference between Chinese and German children in the symbolic numerical magnitude comparison task was mediated by arithmetic skills, we used mediation analyses. On the one hand, mediation analysis allows to investigate direct associations, used in this study to examine the relation between the factor group (Chinese vs. German) and individual performance in the symbolic numerical magnitude comparison task, while holding constant the performance in the arithmetic tasks. On the other hand, mediation analysis provides estimates of the statistical significance of indirect associations, used in this study to evaluate whether arithmetic skills mediate the association between the factor group and symbolic numerical magnitude processing skills. A second mediation model was tested to check the opposite direction of influence, i.e., to examine whether a possible performance difference between Chinese and German participants in the arithmetic tasks was mediated by the performance in the symbolic numerical magnitude comparison task. The mediation models were tested using the INDIRECT macro in SPSS (

Moreover, Pearson correlation coefficients (before and after correction for attenuation) were employed to verify associations between arithmetic skills and reciprocal RT in the symbolic numerical magnitude comparison task as well as between arithmetic skills and reciprocal speed of number pronunciation, separately in both groups. The respective correlation coefficients of both groups were compared directly using the Fisher

While Chinese and German children did not significantly differ with regard to age (

Reaction times in the symbolic numerical magnitude comparison task increased as the numerical distance between the two to-be-compared Arabic digits decreased: significant linear trends were found for Chinese [_{p}^{2} = 0.71] and for German children [_{p}^{2} = 0.77; see

The first mediation model revealed that the group difference in reciprocal RT in the symbolic numerical magnitude comparison task was no longer significant after controlling for arithmetic skills [direct effect = 0.0000,

For Chinese participants, reciprocal RT in the symbolic numerical magnitude comparison task was marginally correlated with performance in the arithmetic tasks (

We compared Chinese and German third graders regarding their performance in arithmetic tasks and in a symbolic numerical magnitude comparison task. Chinese children showed better performance in the arithmetic tasks, corresponding to previous findings (e.g.,

Mediation analysis revealed that the group difference in symbolic numerical magnitude processing was fully mediated by the performance in the arithmetic tasks. After controlling for arithmetic performance, the difference between Chinese and German children’s performance in the symbolic numerical magnitude comparison task was no longer significant. The difference between Chinese and German children in arithmetic was partially mediated by symbolic numerical magnitude processing skills. Indeed, the group difference in arithmetic performance was significantly mediated by the performance in the symbolic numerical magnitude comparison task but it was still significant after controlling for the performance in the symbolic numerical magnitude comparison task. Hence, while the group difference in arithmetic performance was only partially mediated by symbolic numerical magnitude processing skills, the group difference in symbolic numerical magnitude processing was fully mediated by the performance in the arithmetic tasks. The influence of arithmetic skills on symbolic numerical magnitude processing skills accordingly seems to be higher than the opposite direction of influence, at least in children who have already developed basic arithmetic skills. These findings might be seen as evidence for the notion that arithmetic skills shape symbolic numerical magnitude processing skills. Based on the assumptions that (a) children’s familiarity and fluency of manipulating symbolic numbers serves as the crucial link between symbolic numerical magnitude processing and arithmetic skills (

In accordance with previous findings, RT in the symbolic numerical magnitude comparison task correlated with arithmetic skills in German children (see e.g.,

It is important to note that the cross-sectional design of the current study does not offer means of assessing cause. Based on the different results of the two mediation models, we assume that a higher degree of familiarity and fluency with arithmetic in Chinese compared to German third graders causes a higher speed of retrieving symbolic numerical magnitude knowledge. To substantiate this notion, however, longitudinal studies are needed. The assessment of both the development of symbolic numerical magnitude processing skills and the development of arithmetic skills in Chinese and German children over time would lead to a better understanding of the interrelationship between these skills. Moreover, it would be possible to examine whether the direction of influence changes in the course of development and determine to what extent the developmental trajectories are culture-specific.

Another limitation of our study is that the two groups under study might have differed with respect to other factors that may account for the group differences in symbolic numerical magnitude processing and in arithmetic skills, but were not assessed in this study. For example, general cognitive abilities of Chinese and German children were not assessed. Instead of controlling for general cognitive abilities, we used a domain-specific control task, allowing us to rule out that our findings can be explained by between-group differences in the speed of number pronunciation. It can, however, not be ruled out that our findings are due to between-group differences in general intellectual abilities. Nonetheless, findings from previous studies do not support this notion but demonstrated that proficiency in comparing symbolic numbers is not related to children’s intellectual abilities (

To conclude, results from our study revealed that differences in arithmetic performance between Chinese and German children are accompanied by differences in processing of symbolic numerical magnitude. Chinese third graders did not only show a higher fluency in solving arithmetic tasks but were also able to process symbolic numerical magnitude information at a faster pace than their German peers. The group difference in symbolic numerical magnitude processing was fully mediated by the performance in arithmetic tasks, suggesting that arithmetic skills shape symbolic numerical magnitude processing skills. We assume that a higher frequency of exposure to arithmetic leads to a higher degree of familiarity with arithmetic in Chinese compared to German children, in turn leading to a higher speed of retrieving symbolic numerical magnitude knowledge.

JL, JL, MH, and SL substantially contributed to the conception and design of the work, the acquisition, analysis, and interpretation of data for the work. JL, JL, MH, and SL substantially contributed to drafting the work and revising it critically for important intellectual content. JL, JL, MH, and SL substantially contributed to final approval of the version to be published. JL, JL, MH, and SL agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank all the participating children, Guopeng Chen (East China Normal University, Shanghai), Gerd Lüer, Uta Lass, Markus Reitt (Georg-August-University, Göttingen), and Song Yan (Jacobs-University, Bremen) for their support.