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Conceived and designed the experiments: MM LF JH. Performed the experiments: MM LF JH. Analyzed the data: MM JH. Contributed reagents/materials/analysis tools: MM LF JH. Wrote the paper: MM LF JH.

The authors have declared that no competing interests exist.

The Approximate Number System (ANS) is a primitive mental system of nonverbal representations that supports an intuitive sense of number in human adults, children, infants, and other animal species. The numerical approximations produced by the ANS are characteristically imprecise and, in humans, this precision gradually improves from infancy to adulthood. Throughout development, wide ranging individual differences in ANS precision are evident within age groups. These individual differences have been linked to formal mathematics outcomes, based on concurrent, retrospective, or short-term longitudinal correlations observed during the school age years. However, it remains unknown whether this approximate number sense actually serves as a foundation for these school mathematics abilities. Here we show that ANS precision measured at preschool, prior to formal instruction in mathematics, selectively predicts performance on school mathematics at 6 years of age. In contrast, ANS precision does not predict non-numerical cognitive abilities. To our knowledge, these results provide the first evidence for early ANS precision, measured before the onset of formal education, predicting later mathematical abilities.

The Approximate Number System (ANS) is a mental system of magnitude representations that produces an intuitive “number sense” across species

Unlike numerical skills targeted by formal schooling, magnitude representations of the ANS are independent of symbolic representations such as numerals (e.g.,

What is the relationship between the intuitive number sense that is supported by the ANS and more formal mathematical abilities? Recent investigations of individual differences in ANS representations suggest corresponding differences in mathematical ability. Individual differences in the ANS have been measured in terms of differences in the precision of people's approximate number representations. When a person views an array of items (e.g., 105 berries) too quickly to count, an ANS representation (e.g., “approximately one hundred”) is activated. The noise surrounding this estimate is large for numbers of this magnitude because the degree of error in the ANS representation increases linearly as the number being represented increases. This leads to ratio-dependent performance on numerical discrimination tasks (such as judging which of two briefly presented arrays is more numerous), in accord with Weber's Law

Critically, individual differences in the ANS appear to be linked to mathematics ability. For example, we have shown that the precision of ninth graders' ANS representations retrospectively correlates with their standardized mathematics achievement scores obtained up to 8 years prior (i.e., at kindergarten)

What drives the reported relationship between ANS precision and math ability? Two competing hypotheses propose that the ANS either underlies, or is itself refined by, formal mathematical learning. If the ANS underlies formal mathematical ability in childhood, it may be a fruitful target for early instruction and intervention (as proposed by Wilson and colleagues

These two hypotheses are not mutually exclusive. The ANS and formal mathematical ability might support and refine one another, in both directions, across development. One way of determining the initial state of their relationship is to ask whether ANS skills measured prior to schooling (i.e., prior to differences in the quality of children's formal mathematics instruction) predict achievement levels attained after the onset of formal math instruction. To date, the few studies of numerical ability predicting mathematics achievement have not relied on pure measures of the ANS as predictors, and have been carried out with children already enrolled in school. These studies show that symbolic number skills such as verbal magnitude comparison predict later math achievement. For instance, counting, reading or writing numerals, and symbolic magnitude comparisons measured at kindergarten predict math achievement at Grade 2

Here we assessed whether ANS precision measured prior to entering school predicts school mathematics during or after kindergarten. We first measured children's ANS precision at 3 to 4 years of age, using a nonverbal, non-symbolic comparison task; we then measured the same children's mathematics abilities two years later. We found that children's ANS precision measured at preschool, prior to formal instruction in mathematics, selectively predicted their school mathematics performance at age 6.

The research procedures described below were completed in accordance with approval from the Institutional Review Board at the Johns Hopkins University. Written consent was obtained from parents of all participants prior to testing.

We tested 17 children (7 girls, 10 boys) who, at preschool, had participated in a cross-sectional study of ANS precision in 3 to 6 year olds

Time Point | Mean | Std. Deviation | Age Range |

Age at Preschool | 4; 2 | 0; 4.5 | 3; 5 to 4; 11 |

Age at Follow up | 6; 8 | 0; 4.2 | 6; 2 to 7; 5 |

Interval between assessments | 2; 6 | 0; 2.8 | 2; 0 to 2; 9 |

Recruitment was conducted via mail and telephone. Most of the participants were white (n = 14), and all 17 had parents of middle socioeconomic status who had completed at least some higher education.

At both preschool and follow up testing, children were seen individually for one session, in the research lab. The assessment administered when participants were in preschool was limited to one measure. At follow up testing, several measures were administered, in a fixed order.

We used performance on an ANS numerical discrimination task as our predictor measure of preschool ANS precision (described elsewhere in greater detail

We measured children's ANS precsion by having children judge whether Big Bird or Grover had more objects [e.g., crayons], with objects flashed too briefly to allow verbal counting.

Several controls ensured that children remained focused on the number of objects throughout the task – as opposed to other dimensions, such as object size. Displays were controlled either for average object size (area correlated trials) or summed continuous extent (area anti-correlated trials). For each ratio presented, on half of the trials the larger numerosity had more total surface area (area correlated trials), and on the other half of trials the smaller numerosity had more total surface area (area anticorrelated trials). Area anticorrelated trials equated the total summed perimeter of Big Bird's and Grover's objects and anticorrelated their total surface area, two dimensions of continuous extent to which infants have shown sensitivity

Sixty-six test trials followed. These were structured just like the simultaneous portion of the practice trials: Big Bird's and Grover's objects appeared simultaneously (synchronized to the phrase, “Who has more [crayons]?”) and remained visible for a fixed interval (either 1200 (n = 12) or 2500 ms (n = 5) depending on children's age at testing). After the objects disappeared, the images of Big Bird, Grover, and the empty background frames remained onscreen until children responded.

Each trial displayed numbers of objects drawn from a wide range of numerical ratios. Unequal numbers of trials of each ratio were presented, in order to focus on the more difficult ratios. Each child was tested with two trials per ratio bin for the ratios 1∶2, 2∶3, 3∶4, and 4∶5, with ten trials per ratio bin for the ratios 5∶6, 6∶7, and 7∶8, and with 14 trials per ratio bin for ratios 8∶9 and 9∶10, with the absolute number of objects in each array ranging from 1 to 14. (Trials of the same ratio could include different absolute numbers of objects (e.g., 5∶10 and 7∶14)). Across trials, the ratio, number of objects within each array, and object type varied randomly for each child, with the restriction that each child received the same number of trials from each ratio. A recorded voice provided positive or negative feedback after a child responded. The entire procedure lasted approximately 5 minutes.

Measures of ANS precision can be derived for groups or individual participants. By combining all subjects into a single group, accuracy on this ANS task as a function of ratio can be modeled psychophysically to determine the most difficult ratio that still results in accurate discrimination (i.e., the Weber fraction,

When children returned at 6 years of age, we used standardized tests to assess their mathematical and general cognitive abilities. First, we administered the Test of Early Mathematics Ability – Third Edition (TEMA-3

We used the Wechsler Abbreviated Scale of Intelligence (WASI

Finally, we administered three subtests of the Rapid Automatized Naming (RAN) test, a timed measure of lexical retrieval

We hypothesized that ANS precision would predict formal mathematics skills (TEMA-3), but not other aspects of cognitive performance. Hence, we predicted significant associations between ANS precision and the TEMA-3, but not the WASI subtests. Moreover, we hypothesized that ANS precision would be associated with only the Numbers subtest of the RAN.

We first evaluated performance on the ANS numerical discrimination task at preschool, in order to verify that the task engaged children's ANS. Collapsing across all trials, percent correct scores ranged from 43% to 82% (Mean = 61.09%, SD = 11.14), and were normally distributed. As anticipated, when examined across three levels of ratio size – small (8∶9 and 9∶10), intermediate (4∶5, 5∶6, 6∶7, and 7∶8), and large ratios (3∶4, 2∶3, and 1∶2) – the mean Percent Correct score increased with ratio size, consistent with Weber's Law (

Group mean values correspond to percent correct scores for three levels of ratio size: small, intermediate, and large. Error bars are 95% confidence intervals.

Measure | Mean | Std. Deviation | Range |

Preschool Numerical Discrimination (Percent Correct) | 61.09 | 11.14 | 43.27–82.25 |

TEMA – 3 (Standard Score) | 114.12 | 8.57 | 98–130 |

Vocabulary (T score) | 59.88 | 5.86 | 50–69 |

Block Design (T score) | 54.71 | 10.90 | 40–80 |

Matrix Reasoning (T score) | 63.18 | 13.45 | 37–79 |

RAN Color (RT in seconds) | 35.02 | 8.17 | 25.23–53.42 |

RAN Letter (RT in seconds) | 27.24 | 8.46 | 16.41–45.40 |

RAN Number (RT in seconds) | 24.06 | 6.30 | 15.32–37.52 |

Despite the overall improvement in performance accuracy as ratio size increased, four preschoolers performed quite poorly overall, scoring at or near chance on the ANS task (≈50%). Although it is possible that these children engaged a strategy independent of the ANS (e.g., they may have guessed), it is also possible that they were more numerically challenged than their peers, even when faced with arrays conforming to a 2∶1 ratio. As representatives of preschoolers with the least precise ANS skills, their inclusion in the subsequent analyses is important; however, if indeed these children were not engaging ANS supported skills, their exclusion is warranted. Since it is unclear which of these two explanations accounts for these children's poor performance, the subsequent sets of analyses were first conducted with the entire sample of 17 children, and then were repeated without those children performing at chance. Descriptive statistics for this latter subgroup are reported in

Measure | Mean | Std. Deviation | Range |

Age at Preschool, in Years, Months | 4; 2 | 0; 5 | 3; 5–4, 11 |

Age at Follow up, in years, months | 6; 8 | 0; 4 | 6; 2–7; 5 |

Preschool Numerical Discrimination (Percent Correct) | 65.10 | 9.42 | 55.54–82.25 |

TEMA – 3 (Standard Score) | 113.92 | 9.11 | 98–130 |

Vocabulary (T score) | 59.23 | 5.69 | 50–67 |

Block Design (T score) | 53.15 | 10.38 | 40–80 |

Matrix Reasoning (T score) | 62.69 | 14.06 | 37–79 |

RAN Color (RT in seconds) | 35.99 | 8.78 | 25.23–53.42 |

RAN Letter (RT in seconds) | 27.06 | 8.86 | 16.41–45.40 |

RAN Number (RT in seconds) | 22.50 | 5.93 | 15.32–37.52 |

We used three sets of linear regression models to address our primary research questions. In each case, ANS precision at 3- to 4-years of age, indexed as the total Percent Correct on the numerical discrimination task, was entered as the predictor variable. Stimulus display times for this task varied across participants as a function of age, so in all analyses we calculated residual scores to adjust for age and display times at preschool testing.

Our first question was whether ANS precision at preschool predicts school mathematics performance. We conducted a linear regression analysis to evaluate the prediction of TEMA-3 scores (adjusted for age and grade at follow up testing) from the total Percent Correct score on the preschool ANS numerical discrimination task (adjusted for age and display time at initial testing). The model was significant, with ANS precision accounting for 28% of the variance in TEMA-3 score, ^{2}^{2}

For all three measures, higher scores indicate better performance.

This pattern of findings held when the analyses were limited to the children whose ANS performance exceeded chance levels. For these 13 children, ANS precision at preschool (adjusted for age and display time at testing) accounted for 35% of the variance in TEMA-3 performance at age 6 years (adjusted for age and grade at testing), ^{2}

As aforementioned, although the traditional index of ANS precision in our work and that of others has been the Weber fraction score (^{2} = .208. Although this did not reach statistical significance,

The predictive value of early ANS precision on later school mathematics may reflect a specific relationship between intuitive and formal numerical tasks, or it could just reflect an association between earlier and later cognitive skills. To test this we asked whether ANS precision at preschool predicted scores on any of the WASI subtests. For each subtest we conducted a linear regression analysis with WASI subtest score as the outcome variable (adjusted for age and grade at testing), as predicted by the preschool ANS task percent correct score (adjusted for age and display time). ANS precision was not a significant predictor of Vocabulary at 6 years, ^{2}

As an additional test of specificity, we asked whether ANS precision at preschool (adjusted for age and display time) predicted response time on RAN Numbers, but not RAN Colors or RAN Letters, at primary school (controlling for age and grade at follow up testing). For each subtest we conducted a linear regression analysis with ANS precision at preschool as the predictor and RAN RT at primary school as the outcome. When RAN Colors or Letters subtest response times were included as the outcome variable, neither model was significant, ^{2}^{2}

For ANS performance, higher scores indicate better performance. For RAN response times (RT), higher scores indicate poorer performance.

This is the first study to show that ANS precision measured years prior to formal schooling predicts mathematics ability in primary school. This association is not explained by possible confounds of general full-scale IQ. It appears specific to mathematics, since no such association emerged for ANS precision and measures of expressive vocabulary (i.e., WASI), perceptual organization (i.e., Block Design, Matrix Reasoning), or non-numerical lexical retrieval (i.e., RAN Colors and Letters). Finally, the strength of the relationship we observed in this sample (^{2}^{2}

It is noteworthy that our findings emerged despite the relatively restricted range of average to above average TEMA-3 scores (98 to 130) obtained by our small study sample, particularly given the wider range of TEMA-2 scores reported in our earlier longitudinal study of 64 ninth graders (60 to 133 at Kindergarten, as reported elsewhere

Although here and elsewhere

Finally, influences on mathematical learning other than the ANS and other “number sense” skills range from motivational factors

If ANS skills influence mathematical ability, they may be important targets for early intervention or instruction and may even guide efforts to vary some aspects of mathematics instruction on the basis of individual students' foundational skills. Whether this proves to be the case depends on the nature of this association. Effective applications will require greater specification regarding the ways in which the ANS drives mathematical learning, whether its role is direct or indirect, whether its primary role is to support early symbolic instruction

The authors would like to thank Andrea Stevenson for her assistance with data collection.