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Edited by: Xinlin Zhou, Beijing Normal University, China

Reviewed by: Melissa M. Kibbe, Boston University, United States; Maciej Haman, University of Warsaw, Poland

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.

The approximate number system (ANS) is widely considered to be a foundation for the acquisition of uniquely human symbolic numerical capabilities. However, the mechanism by which the ANS may support symbolic number representations and mathematical thought remains poorly understood. In the present study, we investigated two pathways by which the ANS may influence early math abilities: variability in the acuity of the ANS representations, and children’s’ ability to manipulate ANS representations. We assessed the relation between 4-year-old children’s performance on a non-symbolic numerical comparison task, a non-symbolic approximate addition task, and a standardized symbolic math assessment. Our results indicate that ANS acuity and ANS manipulability each contribute unique variance to preschooler’s early math achievement, and this result holds after controlling for both IQ and executive functions. These findings suggest that there are multiple routes by which the ANS influences math achievement. Therefore, interventions that target both the precision and manipulability of the ANS may prove to be more beneficial for improving symbolic math skills compared to interventions that target only one of these factors.

Math ability when a child first enters schooling is the strongest predictor of later math and overall academic achievement (

The ANS is frequently hypothesized to be a cognitive foundation for symbolic math abilities. Lending support to this view is the finding that the acuity of the ANS, typically measured by an individual’s ability to compare two arrays of dots, correlates with symbolic math achievement throughout the lifespan (see

Although many studies have focused on the link between ANS acuity and math achievement, relatively less attention has been paid to children’s ability to manipulate approximate numerical quantities. Beyond simply representing quantities, the ANS enables infants (

However, though previous work suggests that the precision and manipulability of the ANS both contribute to symbolic math achievement, it is currently unknown whether these are separable factors. In other words, do children with more precise ANS representations necessarily also more adept at manipulating approximate quantities in arithmetic operations? If this is the case, then we would expect ANS manipulability to mediate the relation between ANS acuity and symbolic math achievement. Alternatively, if ANS acuity and manipulability are distinct, we would expect both factors to contribute unique variance to children’s early symbolic math performance.

In the present research, we explicitly tested how ANS acuity and manipulability each contribute to symbolic math achievement in preschool-aged children. We focused on preschool-aged children because they have not yet started formal schooling, so they have not yet been exposed to formal symbolic math education. Thus, we could assess how different aspects of children’s intuitive sense of number relate to their symbolic math proficiency. Children were tested with a non-symbolic numerical comparison task to assess ANS acuity, a non-symbolic approximate addition task to assess ANS manipulability, and a standardized symbolic math test. In addition, children performed a general IQ test and a subset of children performed an executive functions task^{1} in order to control for domain-general factors that also contribute to math achievement.

One hundred and seventy children participated in this experiment (mean age: 4.59 years, range: 4.48–4.90 years; 89 female). Of these, 145 children completed the non-symbolic numerical comparison, non-symbolic addition, symbolic math, and IQ assessments, and 75 of those children additionally completed the executive functions task. Twenty-five children did not complete one or more of the primary tasks of interest and were therefore excluded from all analyses. Participants were recruited as part of a larger longitudinal studying tracking the development of numerical cognition from infancy into the preschool years. Data was collected between October 2011 and July 2015, and data collection was stopped when the lab moved to a new institution out of state.

Children were tested in two separate sessions each lasting less than 1 h. During the first visit, children completed the symbolic math assessment, one session of the non-symbolic number comparison task, and the executive functions task. During the second visit, children completed the IQ assessment, a second session of the non-symbolic number comparison task, and a non-symbolic approximate arithmetic task. All children were tested individually in a quiet room, and the order of the tasks within each session was counterbalanced across participants. At each visit, parents gave written consent to a protocol approved by the local Institutional Review Board. Parents were compensated monetarily and children received a small toy.

On each trial, a touchscreen computer displayed two squares (8 cm × 9.5 cm) containing arrays of dots. Children were instructed to touch the square that contained more dots and to make this choice without counting. Arrays contained between 4 and 14 dots, and the numerical ratio between the arrays was 1:2, 2:3, 3:4, or 6:7. To control for non-numerical perceptual cues, the parameters of the arrays varied such that the smaller and larger numerical array each had the larger cumulative surface area on 50% of trials. All of the dots within a single array were homogenous in element size and color, and the color of each array varied randomly from trial to trial. Differential audio-visual feedback was provided after each trial, and children received a small sticker for each correct response to keep them engaged. Children performed practice trials until they made three consecutive correct responses or completed a maximum of ten trials. Children were tested with 60 trials in each session for a total of 120 trials at each time point. Each child’s ANS acuity was estimated using a psychophysical modeling technique (e.g.,

This task was adapted from

Schematic of the approximate arithmetic task.

The Day/Night task (

Children’s mathematical ability was assessed with the Test of Early Mathematics Ability (TEMA-3) (

Descriptive statistics and a correlation table for all measures of interest can be found in Tables

Descriptive statistics for all variables of interest.

Task | Measure | Mean |
---|---|---|

Non-symbolic numerical comparison | Accuracy (% correct) | 79.45 (8.32) |

Weber fraction ( |
0.31 (0.15) | |

TEMA-3 (math achievement test) | Standardized score | 111.57 (12.99) |

RIAS (IQ test) | Standardized score | 128.94 (16.15) |

Day/Night (executive functions task) | Efficiency score | 0.51 (0.26) |

Approximate arithmetic | Accuracy (% correct) | 77.27 (13.63) |

Correlation matrix of Pearson r values for all variables of interest.

w | Symbolic math | IQ | Approximate addition | Executive functions | |
---|---|---|---|---|---|

w | - | -0.27 | -0.16 | -0.22 | -0.16 |

Symbolic math | -0.27 | - | 0.42 | 0.32 | 0.33 |

IQ | -0.16 | 0.42 | - | 0.21 | 0.18 |

Approximate addition | -0.22 | 0.32 | 0.21 | - | 0.40 |

Executive functions | -0.16 | 0.33 | 0.18 | 0.40 | - |

First we performed planned paired

In the first series of analyses, we used multiple regression models to investigate the unique variance contributed by each of our measures of interest (Table _{w}_{ApproxAdd}_{IQ}_{w}_{ApproxAdd}_{IQ}_{EF}

Regression models predicting symbolic math achievement.

Model 1 | Model 2 | |||
---|---|---|---|---|

^{2} |
0.247 | 0.383 | ||

145 | 75 | |||

_{Adjusted} |
_{Adjusted} |
|||

ANS acuity | -0.179 | 0.018 | -0.239 | 0.014 |

ANS manipulability | 0.237 | 0.002 | 0.267 | 0.011 |

IQ | 0.325 | 0.332 | <0.001 | |

Executive functions | – | – | 0.196 | 0.054 |

Scatterplots illustrating the relation between

Next we used structural equation modeling to determine whether the relation between ANS acuity and symbolic math achievement in mediated by ANS manipulability (Figure

Mediation models assessing whether

We also tested whether executive functions mediate the relation between approximate arithmetic performance and symbolic math. This model indicated that both the direct effect (c′ = 13.71,

The goal of the present research was to investigate the mechanisms by which approximate number representations contribute to preschoolers’ emerging symbolic math capabilities. Consistent with previous studies, we found that individual differences in the precision of the ANS are related to symbolic math achievement in preschool-aged children (e.g.,

The majority of studies relating the ANS to symbolic math have focused on individual differences in the acuity of approximate number representations. However, the present results suggest that the manipulability of these representations is a second mechanism by which the ANS influences symbolic math. Although both non-symbolic numerical comparison and approximate arithmetic tasks require representing approximate numerical quantities, approximate addition additionally requires the manipulation of those quantities. Previous studies in infants, young children, and monkeys, all of whom have no understanding of symbolic arithmetic, demonstrate that the ANS supports arithmetic operations (

Because approximate addition requires mental manipulation, it likely places a greater demand on executive functions, including working memory and updating, compared to non-symbolic numerical comparison. Given the well documented link between executive functions and math achievement in children (e.g.,

However, executive functions are a multifaceted construct (

In contrast to a previous finding (

A limitation of these data is that our approximate addition task only used numerosities between 1 and 8, which means that many of the numerosities fall within the subitizing range. However, the presence of ratio effects for both accuracy and reaction times suggests that children were not relying on subitizing to solve the addition problems. In addition, due to the speed of the addition animation, it is unlikely that children were counting the items or using a symbolic labeling strategy, and such strategies were actively discouraged. Previous work in human adults (

The ANS endows young children with a robust sense of quantity prior to beginning formal mathematics instruction. Although many studies have provided evidence for a correlation between the fidelity of the ANS and symbolic math achievement, there remain key open questions concerning the mechanisms underlying this relation. In the present study, we provide evidence that the acuity and manipulability of the ANS have separable influences on preschoolers’ early symbolic math proficiency. In particular, the influence of ANS manipulability may stem from its ability to support arithmetic operations. The shared demand for manipulating quantities may form a conceptual bridge between non-symbolic and symbolic arithmetic. Our findings therefore suggest a nuanced relation between approximate number representations and symbolic math achievement in which multiple features of the ANS contribute to the emergence of symbolic math ability in young children. In light of these results, interventions designed to target one or both of these pathways may be differentially beneficial for children depending on their level of symbolic number knowledge and mathematical proficiency.

The study and protocol were reviewed and approved by the Duke University’s Institutional Review Board. Written informed consent was obtained from the guardians of all participants and assent was obtained from all participants.

AS and EB conceived and planned the experiment. AS and RT collected the data. AS performed the analyses. All authors discussed the results and contributed to the final manuscript.

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.

The executive functions task was added to the battery after a 2013 paper (