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Edited by: Maciej Haman, University of Warsaw, Poland

Reviewed by: Christine Schiltz, University of Luxembourg, Luxembourg; Mojtaba Soltanlou, Eberhard Karls Universität Tübingen, Germany

^{†}These authors have contributed equally to this work.

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) and the copyright owner(s) 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.

Mental calculation is thought to be tightly related to visuospatial abilities. One of the strongest evidence for this link is the widely replicated operational momentum (OM) effect: the tendency to overestimate the result of additions and to underestimate the result of subtractions. Although the OM effect has been found in both infants and adults, no study has directly investigated its developmental trajectory until now. However, to fully understand the cognitive mechanisms lying at the core of the OM effect it is important to investigate its developmental dynamics. In the present study, we investigated the development of the OM effect in a group of 162 children from 8 to 12 years old. Participants had to select among five response alternatives the correct result of approximate addition and subtraction problems. Response alternatives were simultaneously presented on the screen at different locations. While no effect was observed for the youngest age group, children aged 9 and older showed a clear OM effect. Interestingly, the OM effect monotonically increased with age. The increase of the OM effect was accompanied by an increase in overall accuracy. That is, while younger children made more and non-systematic errors, older children made less but systematic errors. This monotonous increase of the OM effect with age is not predicted by the compression account (i.e., linear calculation performed on a compressed code). The attentional shift account, however, provides a possible explanation of these results based on the functional relationship between visuospatial attention and mental calculation and on the influence of formal schooling. We propose that the acquisition of arithmetical skills could reinforce the systematic reliance on the spatial mental number line and attentional mechanisms that control the displacement along this metric. Our results provide a step in the understanding of the mechanisms underlying approximate calculation and an important empirical constraint for current accounts on the origin of the OM effect.

Adults and children (

A prerequisite to perform approximate mental calculation is the capacity to estimate and manipulate numerical quantities, which is a phylogenetically ancient cognitive tool that humans share with other animals (

Approximate calculation also follows the Weber–Fechner law (

It has been proposed that mental calculation is grounded in neural circuits that originally evolved for processing visuospatial information (

The attentional shift account has been developed to explain the OM effect in adults. Therefore, no predictions or hypotheses were proposed regarding how the attentional shifts on the MNL that accompany addition and subtraction emerge and whether they undergo substantial changes during development. Here, we propose that formal schooling (i.e., acquiring arithmetical skills) could reinforce (or even contribute to develop) the idea that addition is related with shifts toward larger numbers and subtraction toward smaller numbers. Namely, although mental calculation might be implemented as an attentional shift on the MNL before formal schooling, repeated exposition to spatial-numerical associations (e.g., the number line) might consolidate a systematic movement direction during the acquisition of arithmetical skills. Moreover, the systematic association between operations and results (i.e., when adding, the result is always larger than both operands; when subtracting, the result is always smaller than the first operand), that children are exposed to, could boost the attentional shift on the MNL. The influence of the attentional shift in the estimation of the result might increase with age and in turn a larger and more systematic bias would emerge. Therefore, one may predict an increasing OM effect during childhood. Moreover, it is worth noting that the co-opting of visuospatial attention during mental calculation seems to increase with age. In fact, significant functional changes associated with the neural activity elicited by symbolic arithmetic problem-solving have been found between 2nd and 3rd graders, that is 7–9 years old children (

The ^{5.9} ≈ 26 × 14 ≈ 364. However, the actual approximate addition performed by the system is much more accurate (see for example

What developmental trajectory of the OM effect is expected according to the compression account? This account focuses on the logarithmic compression of the MNL. A large body of evidence suggests that the representational metric of the MNL shifts from a logarithmic to a linear scale during childhood (

The sample and the tasks analyzed in the present paper were administered to children as part of a larger study conducted in Brazil (for a more precise description of this larger study see

One hundred seventy-two children from first to sixth grade were recruited from private and public schools in Brazil. Ten children were not able to perform non-symbolic numerical tasks, as shown by the fact that they failed to perform a non-symbolic number comparison task (this task is not reported here, for a more detailed description of this task see ^{2}< 0.2) in the estimation of the Weber fraction, and thus were excluded from the study. These ten children were also not included in the present analyses. The final sample consisted of 162 children (66 boys, 96 girls) between 8 and 12 years of age (mean = 9.7 years,

All children performed above the 25th percentile in the spelling (mean = 110.08,

In this task children were asked to estimate and report verbally the numerosity of a set of dots visually presented on a computer screen. Dots were displayed in black within a white circle, which was presented against a black background. The following numerosities were presented: 10, 16, 24, 32, 48, 56, or 64 dots. Each numerosity was presented five times (in a different configuration), resulting in a total of 35 trials. The same numerosity never appeared in consecutive trials. Each trial started with a fixation point (i.e., a white cross at the center of the screen) presented for 500 ms, followed by the onset of the set of dots which remained on the screen until spacebar was pressed or for up to 1000 ms. During the presentation of the dots, as soon as the child responded, the examiner, who was seated next to the child, pressed the spacebar on the keyboard and typed the child’s answer. The next trial started after an intertrial interval of 700 ms, which consisted of a black screen. Dots were displayed on the screen for up to 1000 ms only to prevent counting. To prevent the use of non-numerical features, total dot area was held constant across the trials and thus it could not be used as a clue to estimate the different numerosities. The average dot-size of the dots was selected so that the total area remained constant, but the dot-size of each dot could vary with a normal distribution with the mean selected to provide constant area across the trials. Therefore, while the average dot-size covaried negatively with numerosity, the dot-size of the single dots could not be used as a cue to evaluate the numerosity of the set. To avoid memorization effects due to the repetition of a specific numerosity, on each trial, the stimuli were randomly chosen from a set of 10 precomputed images with the given numerosity. To exclude extreme responses, the normalized mean estimated value was calculated for each child and each of the seven presented numerosities, then responses ±3 SD from the mean estimated value were considered outliers and excluded from the analysis (3.5% of the trials). Children’s number acuity was measured in term of individual mean coefficient of variation (i.e., separately for each numerosity, the ratio of standard deviation and mean chosen value).

This task has been adapted from ^{i/3}), where

Operands, correct outcome (C) and deviant (D) outcomes presented in the non-symbolic arithmetic problems.

Operands | Correct results and deviant proposed outcomes |
|||||||
---|---|---|---|---|---|---|---|---|

1/2.5 | 1/1.8 | 1/1.4 | 1 | 1.4 | 1.8 | 2.5 | ||

5 | 5 | 4 | 5 | 7 | 10 | 14 | 18 | 25 |

6 | 4 | 4 | 5 | 7 | 10 | 14 | 18 | 25 |

8 | 8 | 6 | 9 | 12 | 16 | 22 | 29 | 40 |

10 | 6 | 6 | 9 | 12 | 16 | 22 | 29 | 40 |

13 | 13 | 10 | 14 | 19 | 26 | 35 | 48 | 65 |

18 | 8 | 10 | 14 | 19 | 26 | 35 | 48 | 65 |

20 | 20 | 16 | 22 | 29 | 40 | 54 | 74 | 100 |

26 | 14 | 16 | 22 | 29 | 40 | 54 | 74 | 100 |

16 | 6 | 4 | 5 | 7 | 10 | 14 | 18 | 25 |

20 | 10 | 4 | 5 | 7 | 10 | 14 | 18 | 25 |

24 | 8 | 6 | 9 | 12 | 16 | 22 | 29 | 40 |

32 | 16 | 6 | 9 | 12 | 16 | 22 | 29 | 40 |

40 | 14 | 10 | 14 | 19 | 26 | 35 | 48 | 65 |

52 | 26 | 10 | 14 | 19 | 26 | 35 | 48 | 65 |

62 | 22 | 16 | 22 | 29 | 40 | 54 | 74 | 100 |

80 | 40 | 16 | 22 | 29 | 40 | 54 | 74 | 100 |

Low | D | D | D | C | D | |||

High | D | C | D | D | D |

To provide a child-friendly paradigm, problems were embedded in a story of a monkey having a box of balls (

Trial sequence of the non-symbolic approximate calculation task. The example shows the screenshots from a non-symbolic addition trial. During the response period, the five response alternatives were presented in a circle-like shape around the center of the screen (i.e., green star) with two boxes on the left of the screen, two on the right, and one on the top.

All analyses were performed using R-project software (

The results of all the ANOVAs performed on the tasks are reported in the Appendix B (Supplementary Table

The first analysis aims to evaluate the performance of children in the non-symbolic number estimation task. Mean chosen numerosity and CV were analyzed with a repeated measure ANOVA with displayed numerosity (i.e., 10, 16, 24, 32, 48, 56, and 64 dots) as within-subject factor and age (i.e., 8 to 12 years old) as between-subject factor. Mean chosen numerosities significantly increased with displayed numerosity [^{2} = 0.47]. However, as shown in _{rm}(971) = -0.57, 95% CI = [-0.61, -0.53],

On the basis of the assumption that mental numerosity representation is subjected to the Weber–Fechner law, the CV should not covary with displayed numerosity (i.e., the CV should be constant across numerosities). As shown in _{rm}(971) = 0.16, 95% CI = [0.10, 0.22],

To account for putative effects of inflated variance due to small number of trials in each displayed numerosity, we repeated these analyses using the

In each trial, the set of five proposed alternatives was sampled from either the lower range of responses (alternatives from 1 to 5, see ^{2} = 0.16] and the age × range × response category interaction [^{2} = 0.01] were significant. Moreover, the four-way interaction showed a tendency toward significance [^{2} < 0.01]. The tendency of the four-way interaction and

Mean (arcsine-transformed) percentage of choice across the response category (

For addition, the main effect of response category was significant [^{2} = 0.06]. Moreover, the age × response category [^{2} = 0.03], the range × response category interaction [^{2} = 0.43] and the three-way interaction [^{2} = 0.03] were significant (

For subtraction, only the main effect of response category [^{2} = 0.07] and the age × response category interaction [^{2} = 0.03] were significant, whereas neither the range × response category interaction [

In order to evaluate children’s performance in approximate addition and subtraction, mean chosen response and standard deviation were analyzed with a repeated-measure ANOVA with correct outcome (i.e., 10, 16, 26, and 40) and operation (i.e., addition vs. subtraction) as within-subject factors and age (i.e., 8–12 years old) as between-subject factor. For mean chosen response, the main effect of correct outcome was significant [

Standard deviation significantly increased with correct outcome [

To investigate whether children’s mental numerosity representation follows Weber–Fechner law, a third ANOVA was performed on CV with correct outcome and operation as within-subject factors and age as between-subject factor. The main effect of correct outcome was significant [_{rm}(485) = 0.005, 95% CI = [-0.08, 0.09], _{rm}(485) = -0.17, 95% CI = [-0.25, -0.08],

To investigate the developmental trajectory of the OM effect, the mean response bias was analyzed with a repeated-measure ANOVA with operation as within-subject factor and age as between-subject factor. Response bias was calculated as the mean difference between the logarithm of the chosen response and the logarithm of the correct outcome. Response bias was significantly different between addition (-0.0004, ^{1}, from no effect for younger children to a strong effect for older children (see

Mean response bias (i.e., difference between the logarithm of the chosen response and the logarithm of the correct outcome) as a function of age and operation (addition in black, subtraction in gray dashed). Error bars represent the standard error of the mean. The horizontal dotted line represents no bias.

Age group | Addition |
Subtraction |
df | Cohen’s _{z} |
Hedges’ _{av} |
|||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | Mean | |||||||||

8 | 24 | –0.020 | 0.057 | –0.028 | 0.094 | 0.4 | 23 | >0.1 | 0.08 | 0.10 |

9 | 54 | –0.012 | 0.041 | – |
0.075 | 3.61 | 53 | 0.005 | 0.49 | 0.71 |

10 | 50 | 0.005 | 0.048 | – |
0.093 | 4.55 | 49 | <0.001 | 0.64 | 0.94 |

11 | 20 | 0.019 | 0.058 | – |
0.052 | 4.52 | 19 | 0.002 | 1.01 | 1.46 |

12 | 14 | 0.029 | 0.045 | – |
0.073 | 5.04 | 13 | 0.002 | 1.35 | 2.04 |

_{z}and Hedges’ g

_{av}) refers to

In Appendix A, we report an additional set of analyses that by and large confirms these findings.

This study aimed to investigate the developmental trajectory of the OM effect in children aged from 8 to 12 years old and to assess whether the current accounts are able to predict these age-related changes. Concerning the non-symbolic estimation task, consistent with previous research (_{rm} = 0.16). Moreover, both mean estimated values and standard deviation increased with displayed numerosity. This suggests that children’s performance was by and large well captured by Weber–Fechner law, even if the CV was not perfectly linear across the entire numerical range. In line with previous findings that suggest that the Weber fraction decreases with age (

In the approximate addition task, the distribution of responses clearly peaked around the correct outcome showing that children were able to solve these problems. The response distribution for subtraction problems, however, showed a different pattern. The distribution was flat for younger children (8 years old, see

How well do the current accounts predict the developmental-related changes of the OM effect? The

In line with the recycling theory (

Although the attentional shift account is consistent with our results, a more complex picture emerges if the results from previous studies are taken into account. In fact, the inverse OM effect found in 6/7 years old children (

The presence of a standard OM effect in infants (

This study has some limitations. First, children’s performance in subtraction was low compared to addition. The higher difficulty to estimate the result of approximate subtraction could be due to the use of different strategies to perform the two operations. To better understand how children perform approximate calculation, future research should further investigate this difference in performance. Second, despite the fairly large sample, 6/7 years old children were not included, that is the age group that showed the inverse OM effect. Future studies should include a larger age range in order to confirm the inverse OM effect and to further investigate the development of this effect. Third, we did not include any task to measure visuospatial attention. Future studies should investigate whether there is a correlation between the developmental trajectories of visuospatial attention and of the OM effect. Finally, the effect of education is also accompanied by the maturation of neural network that supports mental calculation. In the analysis we focused on age, future research, however, should also disentangle the influence of age (neural maturation) and grade (education) on the OM effect. These two independent factors could make distinct contribution at various stages of development.

To sum up, we provided a novel finding on the developmental trajectory of the OM effect in children from 8 to 12 years old. The OM effect monotonically increases with age. This developmental pattern is inconsistent with the compression account. On the other hand, the attentional shift account provides a possible explanation of these results based on the functional relationship between visuospatial attention and mental calculation and on the effect of the acquisition of arithmetical skills during formal schooling. The attentional shift account leads to new predictions about a correlation between visuospatial processing and mental calculation which can be addressed in future studies. Our results provide an important empirical constraint to further explore the origin of the OM effect.

This study was carried out in accordance with the recommendations of ethics review board of the Federal University of Minas Gerais, Brazil (COEP–UFMG) with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. Informed written consent was obtained from the parents and oral consent from the children. The protocol was approved by the ethics review board of the Federal University of Minas Gerais, Brazil (COEP–UFMG).

PP-C, VH, GW, and AK designed the research. PP-C performed the research. DD and PP-C analyzed the data. DD drafted the manuscript. DD, PP-C, AK, VH, and GW contributed to write and revise the paper.

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 acknowledge support by the German Research Foundation (DFG) and the Open Access Publication Fund of Humboldt-Universität zu Berlin.

The Supplementary Material for this article can be found online at:

Since the sample size is unequal in the different age groups, we also performed two Spearman’s correlation analyses between mean response bias and age (in months), separately for addition and subtraction. For addition, there was significant positive correlation [