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We investigated the associations between young children’s domain-general executive functioning (EF) skills and domain-specific spontaneous focusing on number (SFON) tendencies and their performance on an approximate number system (ANS) task, paying particular attention to variations in associations across different trial types with either congruent or incongruent non-numerical continuous visual cues. We found that children’s EF skills were strongly related to their performance on ANS task trials in which continuous visual cues were incongruent with numerosity. Novel to the current study, we found that children’s SFON tendencies were specifically related to their performance on ANS task trials in which continuous visual cues were congruent with numerosity. Children’s performance on ANS task trials in which children can use both congruent numerical and non-numerical continuous visual cues to approximate large quantities may be related to their unprompted tendency to focus on number in their early environment when there are not salient distractors present. On the other hand, children’s performance on incongruent ANS trials may be less a function of number-specific knowledge but more of children’s domain-general ability to inhibit salient but conflicting or irrelevant stimuli. Importantly, these effects held even when accounting for global math achievement and children’s cardinality knowledge. Overall, results support the consideration of both domain-specific and domain-general cognitive factors in developmental models of children’s early ability to attend to numerosity and provide a possible means for reconciling previous conflicting research findings.

According to the core knowledge perspective, the approximate number system (ANS) is an innate system that facilitates large number approximations and comparisons in humans and other species (

In the current study, we tested the association between children’s ANS task performance and their domain-general executive functioning (EF) skills, domain-specific spontaneous focusing on number (SFON) tendencies, and early math skills (global math achievement and cardinality knowledge). Importantly, we paid specific attention to these associations with children’s performance on ANS tasks across different tasks types that have congruent (e.g., numerically “more” covers a

Theorists describe the ANS as an innate core number system that allows individuals to estimate and compare large quantities without counting, improves over development, and is ratio-dependent wherein individuals can discriminate sets up to a certain ratio threshold (

There is some empirical support for a link between children’s ANS acuity as assessed by the non-symbolic number comparison task and their early math skills (e.g.,

Early math skills assessments used in correlational analyses with ANS tasks are often global achievement measures, making it difficult to specify associations between ANS task performance and specific math skills.

Associations between children’s ANS task performance and math skills also may be due at least in part to shared variance with children’s executive functioning (EF) skills (e.g.,

One limitation of many studies including EF skills is that a single task has often been used to assess inhibitory control. Although inhibitory control and the other two sub-skills comprising EF skills (cognitive flexibility and working memory) are most often conceptualized as three interrelated but distinct skills in adulthood (

The association between children’s domain-general EF skills and the ANS task performance may also be much more specific to children’s cognitive skills related to attending to number even when there are not conflicting attentional demands in a task. Children’s ability to naturally attend to numerical information in their environment without being instructed to do so, or their spontaneous focusing on numerosity (SFON) tendencies, have been found to predict later symbolic math achievement (e.g.,

Non-symbolic number comparison tasks used to assess ANS acuity are confounded by overlap between numerical and non-numerical visual features. Various controls have been used to try to account for the influence of continuous non-numerical properties of object sets (e.g., surface area) on children’s ANS task performance, resulting in continuous visual properties that are either congruent or incongruent with numerosity ratios (e.g.,

Leibovich and colleagues (

Given prior conflicting findings on the link between young children’s ANS task performance and their early math skills, we designed a study with ANS task performance as the outcome of analyses rather than the predictor of math achievement to better identify which domain-general and domain-specific cognitive and math skills are associated with children’s ANS task performance. We examined the associations between children’s performance on an ANS task and their domain-general EF skills and domain-specific SFON tendencies as well as specific math skills, including math achievement and cardinality knowledge. We addressed three research questions:

What are the relative associations between domain-general EF skills and domain-specific SFON tendencies and children’s performance on an ANS task?

Do these associations vary depending on the relations between numerosity and continuous surface area features of dot arrays across different ANS task trial types?

Do these associations hold when accounting for math skills (math achievement globally and cardinality knowledge) that have been found to be linked to ANS task performance in prior studies?

We were specifically interested in ANS task performance across three different conditions used in prior work (

This study was approved by a university human subjects ethics committee and adhered to the US Federal Policy for the Protection of Human Subjects. Data were collected from 119 preschool-aged children (_{age} = 55.91 months; _{age} = 9.86 months; 74 females) across two study sites to overcome limitations of using homogenous populations in past studies and to assess the sensitivity of the findings in a sample with varying demographics. There were 57 children (_{age} = 60.75 months; _{age} = 10.15 months; 35 females) recruited from middle-to-high-SES backgrounds attending a private university-based childcare center in the northeastern United States at the first study site. The majority of children at this study site were White (63%), 12.3% were biracial or other, and 24.6% of parents declined to report their child’s ethnicity. There were 62 children (_{age} = 51.36 months; _{age} = 7.20 months; 39 females) recruited from local childcare centers in the Midwestern United States at the second site. Precise racial and ethnic identity information was not available at this study site. Compared to the first study site, participants at the second study site were younger and from more diverse socioeconomic backgrounds (44% of children attended childcare centers primarily serving children who qualified for tuition assistance).

We administered a language assessment to children to serve as a control for general cognitive ability. At the first study site, we administered the Peabody Picture Vocabulary Test – 4 (PPVT-4;

We administered two assessments of executive functioning skills. The first task was the Day/Night task (

The second task we administered was the Dimensional Change Card Sort (DCCS;

We assessed children’s SFON tendencies using the Set-Matching Task (

There were 16 trials presented in four blocks, each with a different configuration of objects placed in the bowls: homogenous, mixed color, mixed shape, and heterogeneous. Within each of the blocks, the child was asked to ‘copy’ trials in which objects in set sizes of 1 – 4 were presented. The order of object sets (1 – 4 objects) and the order of blocks administered were pseudo-random in their presentation. There were two possible orders of object set size presentation, and each object set size order had four possible orders of block type presented. We tested for order effects and did not find that order of trial type presentation had an effect on children’s outcomes. Though the original Negen and Sarnecka task included a set of trials wherein children had to reconstruct the sets once they have been covered, we chose not to include these trials as they required working memory and we did not want to conflate our assessments of SFON tendencies and EF skills. Children scored a 1 on each trial if they matched the correct number of items, with total SFON number-matching accuracy scores ranging from 0 – 16. We computed the percentage of correct trials for use in analyses. Note that we also conducted a test of sensitivity by re-running our analyses using only the first four trials of the task given the use of only 3 – 4 trials in several other SFON tasks in the literature. As reported in the results section, our findings were consistent across these two scoring methods. The Cronbach’s alpha for SFON number-matching accuracy scores on items 1 – 16 was .89.

We used a paper version of this task for ease of administration and created the trials based on prior work with young children (

We administered Applied Problems from the standardized Woodcock-Johnson III Achievement Battery (WJ-III;

Children completed an adapted Give-A-Number task (

We administered the Corsi Blocks task (

All children completed assessments with a trained experimenter in a quiet area of their preschool. Assessments were administered in a fixed order across two sessions with the SFON Set-Matching Task always administered first so that children were not primed to think about number prior to the task.

Descriptive statistics for the full sample (

Variable | Min. | Max. | Skewness | ||
---|---|---|---|---|---|

General Cognitive Ability Covariate | 73.00 | 150.00 | 114.67 | 13.97 | 0.03 |

Day/Night Total Scores | 0.00 | 16.00 | 11.01 | 5.05 | -0.96 |

DCCS Post-Switch Total Scores | 0.00 | 18.00 | 9.84 | 5.27 | -0.77 |

SFON Number-Matching Accuracy | 0.06 | 1.00 | 0.91 | 0.17 | -2.83 |

ANS Accuracy | 0.42 | 1.00 | 0.73 | 0.14 | -0.07 |

ANS Mean Area Equal Accuracy | 0.33 | 1.00 | 0.76 | 0.17 | -0.47 |

ANS Total Area Equal Accuracy | 0.33 | 1.00 | 0.72 | 0.17 | -0.25 |

ANS Inverse Accuracy | 0.25 | 1.00 | 0.71 | 0.18 | -0.19 |

To establish that the ANS task performance yielded a ratio effect, we compared children’s performance on larger ratios (1:2 and 2:3) to their performance on smaller ratios (3:4 and 4:5) using a paired-samples t-test and found that children’s mean accuracy was significantly better on larger ratios (

Correlations are included in

Variable | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1. Age | – | ||||||||

2. General Cognitive Ability Covariate | .04 | – | |||||||

3. Day/Night Total | .44** | .23* | – | ||||||

4. DCCS Post-Switch Total | .37** | .26** | .50** | – | |||||

5. EF Composite | .47** | .29** | .87** | .87** | – | ||||

6. SFON Number-Matching Accuracy | .43** | .26** | .48** | .47** | .55** | – | |||

7. ANS Accuracy | .48** | .23* | .45** | .48** | .54** | .50** | – | ||

8. ANS Mean Area Equal Accuracy | .29** | .27** | .37** | .40** | .45** | .47** | .86** | – | |

9. ANS Total Area Equal Accuracy | .50** | .17^{†} |
.32** | .41** | .42** | .38** | .85** | .66** | – |

10. ANS Inverse Accuracy | .42** | .15 | .45** | .42** | .50** | .31** | .83** | .55** | .53** |

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Prior to using mixed models to examine results by ANS trial type, we first analyzed the data collapsing across ANS trial types to compare results to other studies that commonly use this method. We examined associations between children’s overall performance on the ANS task and their EF skills and SFON tendencies. We included sample as a dichotomous covariate given that the data were collected at two sites. The results of this analysis are presented in

Variable | β | ||
---|---|---|---|

Sample | 0.18 | 1.13 | .260 |

Age | 0.27 | 3.28 | .001 |

General Cognitive Ability Covariate | 0.11 | 1.59 | .114 |

EF Composite | 0.25 | 3.14 | .002 |

SFON Number-Matching Accuracy | 0.12 | 1.50 | .136 |

However, this regression analyses did not allow us to examine associations by ANS task trial type. To examine the relative associations between domain-general EF skills and domain-specific SFON tendencies and children’s ANS task performance by trial type, we utilized mixed models in SPSS v. 24 as children’s scores on the different trial types (type_{t}) were nested within each individual (children_{i}) (see

_{ti}= γ

_{00}+ γ

_{10}*Inverse

_{ti}+ γ

_{20}*Total_Area

_{ti}+ γ

_{01}*Age

_{0i}+ γ

_{02}*Sample

_{0i}+ γ

_{03}*Language

_{0i}+ γ

_{04}*EF

_{0i}+ γ

_{05}*SFON

_{0i}+ γ

_{14}*Inverse

_{ti}*EF

_{0i}+ γ

_{15}*Inverse

_{ti}*SFON

_{0i}+ γ

_{24}*Total_Area

_{ti}*EF

_{0i}+ γ

_{25}*Total_Area

_{ti}*SFON

_{0i}+ μ

_{0i}+ r

_{ti}

The outcome variable was children’s percentage correct on the ANS task trials, and we used two dummy codes (γ_{10} and γ_{20}) at the within-subjects level of our model to compare children’s performance on our three trial types with the _{01}), sample (γ_{02}), the general cognitive ability covariate (γ_{03}), EF composite scores (γ_{04}), and SFON number-matching accuracy scores (γ_{05}). To test if the associations between children’s ANS task performance and their EF skills and SFON tendencies varied by ANS task trial type, we entered interactions between our dummy-coded ANS task trial type comparison variables and both EF skills (γ_{14} and γ_{24}) and SFON number-matching accuracy scores (γ_{15} and γ_{25}). All variables were entered as fixed effects. To compute standardized estimates, all variables in this model were z-scored prior to being entered into the model. Results are presented in

Parameter | Estimate | ||
---|---|---|---|

Intercept | 0.08 | 0.12 | .472 |

Dummy-coded ANS 1: ANS |
-0.34 | 0.08 | .000 |

Dummy-coded ANS 2: ANS |
-0.24 | 0.08 | .004 |

Sample | 0.18 | 0.16 | .260 |

Age | 0.27 | 0.08 | .001 |

General Cognitive Ability Covariate | 0.11 | 0.07 | .114 |

EF Composite | 0.16 | 0.10 | .111 |

SFON Number-Matching Accuracy | 0.24 | 0.10 | .014 |

Dummy-coded ANS 1*EF Composite | 0.24 | 0.10 | .017 |

Dummy-coded ANS 2*EF Composite | 0.04 | 0.10 | .684 |

Dummy-coded ANS 1*SFON | -0.26 | 0.10 | .010 |

Dummy-coded ANS 2*SFON | -0.11 | 0.10 | .280 |

To probe the interactions, we used an online interactive computational tool to estimate simple intercepts and slopes (

Associations between ANS Task Performance and EF Skills by ANS trial type.

We also found a significant interaction between ANS task performance and children’s SFON number-matching accuracy only for the ANS

Associations between ANS Task Performance and SFON Number-matching accuracy by ANS trial type.

One might argue that using 16 trials in the Set-Matching Task, as compared to 3 – 4 trials used in most of Hannula and colleagues’ tasks (e.g.,

We performed a second test of sensitivity to examine the possibility that our findings could be driven primarily by children who scored high on the assessment given the skewness of the distribution. To explore this possibility, we removed all children (

For a subsample of children (study site two from the previous analyses;

Descriptive statistics for the subsample for which we had additional measures are presented in

Variable | Min. | Max. | Skewness | ||
---|---|---|---|---|---|

General Cognitive Ability Covariate | 87.00 | 134.00 | 109.76 | 10.16 | 0.16 |

Day/Night Total Scores | 0.00 | 16.00 | 9.73 | 5.70 | -0.58 |

DCCS Post-Switch Total Scores | 0.00 | 18.00 | 8.92 | 5.84 | -0.39 |

SFON Number Matching Accuracy | 0.06 | 1.00 | 0.87 | 0.21 | -2.23 |

ANS Accuracy | 0.42 | 0.94 | 0.70 | 0.14 | -0.23 |

ANS Mean Area Equal Accuracy | 0.33 | 1.00 | 0.75 | 0.17 | -0.22 |

ANS Total Area Equal Accuracy | 0.33 | 1.00 | 0.69 | 0.16 | -0.38 |

ANS Inverse Accuracy | 0.25 | 1.00 | 0.68 | 0.17 | -0.42 |

Additional Measures | |||||

Applied Problems Standard Score | 75.00 | 139.00 | 109.18 | 12.95 | -0.09 |

Give-A-Number Knower Level | 0.00 | 6.00 | 4.51 | 1.97 | -0.85 |

Corsi Backward Span | 0.00 | 5.00 | 1.70 | 1.46 | 0.16 |

Variable | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1. Age | – | |||||||||||

2. General Cognitive Ability Covariate | -.10 | – | ||||||||||

3. Day/Night Total | .51** | .20 | – | |||||||||

4. DCCS Post-Switch Total | .50** | .27* | .61** | – | ||||||||

5. Corsi Backward Span | .33* | .12 | .28* | .29* | – | |||||||

6. EF Composite | .58** | .26* | .81** | .82** | .68** | – | ||||||

7. SFON Number-Matching Accuracy | .40** | .29** | .48** | .52** | .24^{†} |
.54** | – | |||||

8. ANS Accuracy | .49** | .13 | .48** | .58** | .42** | .64** | .46** | – | ||||

9. ANS Mean Area Equal Accuracy | .31* | .18 | .35** | .51** | .30* | .50** | .48** | .86** | – | |||

10. ANS Total Area Equal Accuracy | .49** | .11 | .38** | .48** | .34** | .52** | .37** | .85** | .71** | – | ||

11. ANS Inverse Accuracy | .38** | .04 | .44** | .42** | .38** | .54** | .27* | .72** | .37** | .36** | – | |

12. Applied Problems Standard Score | -.07 | .47** | .22^{†} |
.40** | .18 | .35** | .30** | .28* | .27* | .16 | .23^{†} |
– |

13. Give-A-Number Knower Level | .18 | .27** | .24^{†} |
.35** | .12 | .31** | .42** | .28* | .24^{†} |
.23^{†} |
.21 | .41** |

^{†}

We replicated prior analyses with the subsample with additional measures to test the generalizability of findings when controlling for other measures commonly associated with ANS task performance. When collapsing across ANS task trial types, we found that children’s EF skills were closely associated with their ANS task performance even when controlling for children’s cardinality knowledge and their math achievement (see Table 7).

Variable | β | ||
---|---|---|---|

Age | 0.18 | 1.33 | .190 |

Language Covariate | -0.04 | -0.37 | .713 |

EF Composite | 0.40 | 2.71 | .009 |

SFON Number-Matching Accuracy | 0.15 | 1.15 | .255 |

Applied Problems Standard Score | 0.12 | 0.95 | .345 |

Give-A-Number Knower Level | 0.03 | 0.28 | .782 |

Next, we replicated our prior mixed model analyses (Equation 1) but added in children’s math achievement (γ_{06}) and their cardinality knowledge (γ_{07}) as additional between-subjects predictors and included the working memory measure in the EF composite. Results are presented in

Parameter | |||
---|---|---|---|

Intercept | 0.23 | 0.11 | .035 |

Dummy-coded ANS 1: ANS |
-0.41 | 0.13 | .002 |

Dummy-coded ANS 2: ANS |
-0.36 | 0.13 | .006 |

Language Covariate | -0.05 | 0.09 | .713 |

Age | 0.15 | 0.12 | .191 |

EF Composite | 0.22 | 0.15 | .142 |

SFON Number-Matching Accuracy | 0.27 | 0.13 | .041 |

Give-A-Number Knower Level | 0.03 | 0.09 | .782 |

Applied Problems Standard Score | 0.10 | 0.11 | .345 |

Dummy-coded ANS 1*EF Composite | 0.23 | 0.15 | .128 |

Dummy-coded ANS 2*EF Composite | 0.09 | 0.15 | .601 |

Dummy-coded ANS 1*SFON | -0.32 | 0.15 | .035 |

Dummy-coded ANS 2*SFON | -0.15 | 0.15 | .297 |

To probe the interactions, we used the same online computational tool to estimate simple intercepts and slopes (

Associations between ANS Task Performance and EF Skills by ANS Trial Type controlling for math knowledge.

Associations between ANS Task Performance and SFON by ANS Trial Type controlling for math knowledge.

We examined the associations between young children’s domain-general and domain-specific cognitive and math skills and their ANS task performance. We found support for our prediction that the pattern of associations among domain-general and domain-specific cognitive skills and ANS task performance would vary by ANS task trial type. Specifically, children’s domain-specific SFON tendencies were significantly associated with their performance on ANS task trials in which the numerosity ratio of the object sets was positively correlated with the total surface area ratio of the object sets (

When examining associations between overall ANS task performance and children’s EF skills and SFON tendencies, we found that children’s EF skills were most closely associated with their overall performance on the ANS task even when math achievement and children’s counting skills were taken into account. Importantly, however, results of overall ANS task performance alone do not provide the entire picture of how domain-general and domain-specific cognitive skills relate to performance on ANS task trials that vary in their control of continuous visual features of the stimuli. As others have argued (

Children’s domain-specific cognitive skills, or SFON tendencies, were associated with their performance only on the

As expected, children’s EF skills were specifically associated with their performance on

Interestingly, we did not find evidence of a significant association between children’s cardinality knowledge and ANS task performance across surface area control types. There are several possible explanations of these findings. First, it may simply be that because many children in the sample were cardinal principle knowers, there may have been less variation in the Give-A-Number task that could be associated with children’s ANS task performance. The findings of this study have implications for children who for the most part were in their last year of preschool, which precludes us from making conclusions about how these associations may operate in much younger children. Replicating this study in a younger sample would help address this open question. Second, prior studies reporting a link between children’s cardinality knowledge and their ANS task performance may have been capturing at least in part an association that could be explained by children’s more general attention to number or SFON tendencies given that children’s SFON tendencies are significantly related to both their cardinality knowledge (

These results provide unique insights about what skills the ANS task may be measuring that can be useful in thinking about conflicting theoretical and empirical work in the literature concerning links between ANS acuity and children’s math achievement. The recently proposed sense of magnitude hypothesis has been criticized as limited by not focusing enough on the interactions among learning mechanisms that may be at work in children’s ANS task performance, including not only domain-general EF skills but domain-specific attention to number and number knowledge (

These results also provide practical takeaways for the design of early math assessments as well as early instruction for children who may be at risk for difficulties in math achievement. Children’s performance on an ANS task was closely related to their domain-general and domain-specific cognitive skills depending on the stimulus properties of objects sets that are compared, more so than to their performance on math-specific skills. Also, by simply examining the correlation tables, one can see that the correlations between children’s domain-general and domain-specific cognitive skills and their math achievement were stronger than the correlations between ANS task performance (both overall and by trial type) and their math achievement. Taken together, these results suggest that there may be limited utility in using the ANS task specifically as it is currently designed to uniquely predict young children’s math achievement when other predictors are more strongly directly linked to their performance. Rather, targeting children’s EF and SFON skills in early math activities by providing guided practice with numerical and non-numerical magnitudes in contexts with and without significant distractions could potentially be a promising avenue for future intervention research.

There are several limitations to acknowledge in this study. First, we chose to focus on surface area as the non-numerical continuous visual feature to vary systematically across ANS trials because it has been argued that this feature is one that is particularly salient to young children. But this does not rule out the possibility that other features we could have varied such as convex hull may also reveal similar patterns of associations with EF and math skills. The current study also only utilized one measure of children’s SFON tendencies. Recent research has suggested that different approaches to measuring children’s SFON can lead to different patterns of results (e.g., tasks that require a physical response vs. ones that require verbal responses;

The results of this study support the hypothesis that young children’s performance on an ANS task is related to both their domain-general and domain-specific cognitive skills. Associations with children’s SFON tendencies were specific to

Many thanks to the research assistants, children, families, and schools involved in this project in both Dayton, OH and Durham, NH.