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By analyzing longitudinal data from the start to the end of primary education, we aimed to investigate whether symbolic numerical magnitude processing at the start of primary education predicted arithmetic at the end, and whether arithmetic at the start of primary education predicted later symbolic numerical magnitude processing skills at the end. In the first grade (start) and sixth grade (end) of primary education, the same group of children’s symbolic numerical magnitude processing skills and arithmetic competence were assessed. We were particularly interested in exploring the direction of the association between symbolic numerical magnitude processing and arithmetic and observed that this association was bi-directional across primary education. Symbolic numerical magnitude processing skills in first grade predicted arithmetic in sixth grade; but also the reversed direction turned out significant: Early arithmetic predicted later symbolic numerical magnitude processing skills. Both directions remained significant after controlling for motor speed and nonverbal reasoning. Critically, when controlling for auto-regressive effects of prior abilities, the symbolic comparison-arithmetic association was no longer significant, the reversed direction became marginally significant. This suggests that children’s arithmetic development across primary education to some extent strengthens their ability to process the numerical meaning of Arabic digits.

Studies on the prediction of arithmetic indicate that variability in symbolic numerical magnitude processing skills (

Prior studies showed large individual differences in arithmetic in early (i.e.,

Previous longitudinal studies only considered one direction, i.e. from symbolic numerical magnitude processing to arithmetic, but they did not consider the reversed direction. Could learning arithmetic in primary education in turn affect children’s acquisition of symbolic numerical magnitude processing skills? It might be that when solving additions like 3 + 4 children indirectly train their symbolic numerical magnitude processing skills, as they learn that the numerical magnitude of the corresponding solution 7 will be larger than the numerical magnitude of each of these numbers separately. So children learn that 7 is larger than 3, but also larger than 4. Practicing arithmetic might therefore refine and strengthen someone’s numerical magnitude representations of numbers.

Interestingly, studies in the reading research field have shown bi-directional associations between the core underlying cognitive correlate of reading, i.e. phonological processing, and reading ability (

The present study further elaborated on the 3-year longitudinal data reported by Vanbinst and colleagues in studies focusing on domain-specific and domain-general cognitive correlates of children’s acquisition of arithmetic solving strategies (

As an additional strict control, we also investigated whether the long-term association between symbolic numerical magnitude processing skills at the start of primary education and arithmetic at the end of primary education, remained significant when the autoregressive effect of prior competence in arithmetic was taken into account. For the reversed direction, we controlled whether the long-term association between arithmetic at the start of primary education and symbolic numerical magnitude processing at the end of primary education remained significant after considering the autoregressive effect of prior symbolic numerical magnitude processing skills. By taking into account children’s prior competencies in arithmetic as well as symbolic numerical magnitude processing, we can more carefully explore the strength of the long-term associations between these measures.

Participants came from a longitudinal research project of which earlier findings were reported (see _{age} = 6 years and 2 months, _{age} = 11 years and 2 months,

Materials were paper-and-pencil-tasks and computerized tasks designed with the E-prime 1.0 software (

To individually assess symbolic numerical magnitude processing at the start of primary education, a classic computerized comparison task was used. During this task, children had to compare two simultaneously presented Arabic digits, displayed on either side of a 15-inch computer screen. They had to indicate the larger of two Arabic digits by pressing a key on the side of the larger one. Stimuli comprised all combinations of digits 1 to 9, yielding 72 trials. The position of the largest digit was counterbalanced. Each trial was initiated by the experimenter and started with a central 200ms fixation point, followed by a blank of 800ms. Stimuli appeared 1000ms after trial initiation, and remained visible until response. The computer registered the answers as well as the response times from stimulus onset. To familiarize children with the key assignments, three practice trials were presented.

We used a paper-and-pencil task that was recently developed by

The group-based paper-and-pencil comparison task contains 60 pairs of digits between 1 and 9, presented in 4 columns of 15 pairs (Verdana font, size 12). Participants are instructed to cross out the largest digit of each presented pair, and were given 30 seconds to solve as many items as possible. The theoretical maximum score was 60. Speed and accuracy are combined into one index score. To ensure that children understood the instruction, the task started with four practice trials.

Arithmetic was evaluated with the Tempo Test Arithmetic (TTA) (

To control for children’s general speed, a motor choice task was individually administrated. Two figures, of which one was filled in white, were displayed simultaneously on a computer screen. One displayed on the left, one displayed on the right. Participants had to press, as fast as possible, the key corresponding to the side on which the filled figure was presented (Left

A measure of nonverbal reasoning was included as a control measure, and was assessed with the Raven’s Standard Progressive Matrices (

All tasks were administered at the participant’s own school. At the start of primary education, all participants individually completed the computerized version of the symbolic comparison task as well as the motor choice task in a quiet room (February 2011). Raven’s Matrices, which was a group-based test, was also assessed at this point in time. The TTA was collectively administered at the start (October 2011) and at the end (October 2016) of primary education. Simultaneously with the assessment of the TTA at the end of primary education, participants collectively completed the paper-and-pencil task of the symbolic comparison task.

Descriptive statistics of measures collected at the start and at the end of primary education are presented in

Variables under study | Minimum | Maximum | ||
---|---|---|---|---|

Start of primary education | ||||

Symbolic accuracy (% correct) | 91.27 | 4.66 | 78.00 | 100.00 |

Symbolic speed (ms)ᵃ | 1126.53 | 232.13 | 616.82 | 1765.74 |

TTA addition | 13.36 | 2.71 | 8 | 19 |

TTA subtraction | 12.76 | 3.60 | 5 | 20 |

Motor choice accuracy (% correct) | 97.41 | 4.09 | 85.00 | 100.00 |

Motor choice speed (ms) | 622.30 | 107.20 | 451.40 | 892.2 |

Nonverbal reasoning | 107.20 | 14.27 | 80.00 | 141.00 |

End of primary education | ||||

Symbolic comparisonᵇ | 36.43 | 5.61 | 27.00 | 51.00 |

TTA addition | 26.61 | 4.15 | 19.00 | 35.00 |

TTA subtraction | 24.18 | 4.26 | 16.00 | 31.00 |

To examine long-term associations across primary education, Pearson correlation coefficients as well as Bayes factors (_{10}_{10}_{10}_{10}_{10}_{10}_{10}

For the computerized task of symbolic numerical magnitude processing, we calculated a score that combined response time and accuracy into one index by dividing an individual’s mean response time by his/her mean accuracy (e.g.,

Both directions of the long-term association between symbolic numerical magnitude processing and arithmetic were tested.

Variable | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1. Symbolic comparison Start | ||||||

Pearson's |
- | |||||

_{10} |
- | |||||

2. Symbolic comparison End | ||||||

Pearson's |
.479*** | - | ||||

_{10} |
131.480 | - | ||||

3. Arithmetic Start | ||||||

Pearson's |
.645*** | .461*** | - | |||

_{10} |
158193.710 | 76.530 | - | |||

4. Arithmetic End | ||||||

Pearson's |
.421** | .449*** | .563*** | - | ||

_{10} |
24.870 | 58.880 | 2903.860 | - | ||

5. Motor speed | ||||||

Pearson's |
.505*** | .368** | .363** | .221 | - | |

_{10} |
268.100 | 6.480 | 5.830 | 0.600 | - | |

6. Nonverbal reasoning | ||||||

Pearson's |
-.088 | -.159 | -.183 | -.062 | .100 | - |

_{10} |
0.206 | 0.324 | 0.400 | 0.186 | 0.220 | - |

_{10}

_{10}

_{10}

_{10}

_{10}

_{10}

*

We conducted a series of strict control regression analyses to carefully unpack the associations observed in _{inclusion}

Model, Predictor | _{inclusion} |
|||
---|---|---|---|---|

Arithmetic |
||||

Model 1: ^{2} |
||||

Motor speed | - .0003 | -0.003 | .998 | 0.329 |

Nonverbal reasoning | -.041 | -0.319 | .751 | 0.324 |

Symbolic comparison |
-.430 | -2.887 | .006 | 19.196 |

Model 2: ^{2} |
||||

Motor speed | .013 | 0.093 | .926 | 0.266 |

Nonverbal reasoning | .025 | 0.208 | .836 | 0.271 |

Autoregressor → Arithmetic |
.496 | 3.037 | .004 | 55.183 |

Symbolic comparison |
-.110 | 0.635 | .528 | 0.340 |

Symbolic comparison |
||||

Model 3: ^{2} |
||||

Motor speed | -.242 | -1.922 | .060 | 1.378 |

Nonverbal reasoning | .270 | 2.253 | .029 | 2.115 |

Arithmetic |
.425 | 3.323 | .002 | 25.552 |

Model 4: ^{2} |
||||

Motor speed | -.178 | -1.313 | .195 | 0.826 |

Nonverbal reasoning | .258 | 2.161 | .036 | 1.660 |

Autoregressor → Symbolic comparison |
-.214 | -1.260 | .214 | 1.785 |

Arithmetic |
.302 | 1.891 | .065 | 2.183 |

_{inclusion}

On top of this, we also evaluated whether

Similarly, we tested whether

By analyzing longitudinal data from the start to the end of primary education, we aimed to investigate whether symbolic numerical magnitude processing skills at the start of primary education predicted arithmetic at the end, and whether early arithmetic predicted future symbolic numerical magnitude processing skills. The present study extended the findings of prior longitudinal studies (e.g.,

By using a 6-year longitudinal design, this study demonstrates how variability in the ability to process symbolic numerical magnitudes at the beginning of primary education was correlated with individual differences in arithmetic competence at the end of primary education. First graders with proficient symbolic numerical magnitude processing skills outperformed their fellow students with difficulties in processing symbolic numerical magnitudes on a timed arithmetic task when they were in the sixth grade. In class, children are directly and frequently instructed to solve arithmetic, and we found that competence in arithmetic at the beginning of primary education was correlated with later symbolic numerical magnitude processing skills at the end of primary education. Each direction of this long-term association between symbolic numerical magnitude processing and arithmetic remained significant after controlling for motor speed and nonverbal reasoning.

We also controlled for autoregressive effects and tested whether symbolic numerical magnitude processing skills at the start of primary education predicted children’s future arithmetic ability while controlling for their prior arithmetic ability. When this autoregressive effect was taken into account, the well-known and previously studied direction of the association, i.e. that symbolic numerical magnitude processing predicts later arithmetic, was no longer significant. The reversed direction of this long-term association from arithmetic to later symbolic numerical magnitude processing became marginally significant after the autoregressive effect of initial symbolic numerical magnitude processing skills was taken into account. Bayesian statistics provided anecdotal evidence for the assumption that learning arithmetic strengthens children’s later symbolic numerical magnitude processing skills on top of their initial symbolic numerical magnitude processing skills. These results highlight that across primary education children continue to expand their competence in arithmetic, which in turn seems to affect children’s acquisition of symbolic numerical magnitude processing skills over time. As has been observed in the field of reading (

This longitudinal study was a first attempt to explore the potentially bi-directional nature of the association between symbolic numerical magnitude processing and arithmetic over developmental time. The effects that we observed were small, but so was our sample size. Replicating this study by using a larger number of participants might be interesting. It would also have been more powerful to use the same tasks to measure symbolic numerical magnitude processing at each point in time, as this might have affected the results. Our correlational data illustrated that the long-term association between symbolic numerical magnitude processing and arithmetic runs in both directions, but the evidence that supports this bi-directionality entirely changed after controlling for autoregressive effects. Such strict control analyses are rarely applied in the field of numerical cognition, but are obviously crucial. As a result of taking into account autoregressive effects, no evidence was found for the frequently suggested direction, i.e. from symbolic numerical magnitude processing to later arithmetic. Anecdotal evidence was found for the reversed direction, i.e. from arithmetic to symbolic numerical magnitude processing. These findings might be specific to the extended period of 6 years that we covered. Different association patterns might be found when exploring the reciprocity between symbolic numerical magnitude processing and arithmetic across the first few years of primary education. Future research on this topic seems therefore needed.

In addition to these longitudinal but correlational approaches, it might also be interesting to test these directions more experimentally. It has already been demonstrated that reading intervention enhances children’s phonological processing skills together with their competence in reading (

Evidence of bi-directionality would have important implications for the understanding of the cognitive mechanisms underlying dyscalculia. More specifically, it has been consistently shown that children with dyscalculia, who have difficulties in learning to calculate, have deficits in symbolic numerical magnitude processing (

We would like to thank all participating kindergartners and their teachers as well as their parents for participating to this scientific study.

The authors have declared that no competing interests exist.