^{*}

^{a}

^{b}

^{c}

While the approximate number system (ANS) has been shown to represent relations between numerosities starting in infancy, little is known about whether parallel individuation – a system dedicated to representing objects in small collections – can also be used to represent numerical relations between collections. To test this, we asked preschoolers between the ages of 2 ½ and 4 ½ to compare two arrays of figures that either included exclusively small numerosities (< 4) or exclusively large numerosities (> 4). The ratios of the comparisons were the same in both small and large numerosity conditions. Experiment 1 used a between-subject design, with different groups of preschoolers comparing small and large numerosities, and found that small numerosities are easier to compare than large ones. Experiment 2 replicated this finding with a wider range of set sizes. Experiment 3 further replicated the small-large difference in a within-subject design. We also report tentative evidence that non- and 1-knowers perform better on comparing small numerosities than large numerosities. These results suggest that preschoolers can use parallel individuation to compare numerosities, possibly prior to the onset of number word learning, and thus support previous proposals that there are numerical operations defined over parallel individuation (e.g., Feigenson & Carey, 2003;

According to structuralism, an influential theory of the nature of mathematical objects, the natural numbers are nothing but a system of relations (e.g.,

“If the only sense in which the brain represents number is that there is a sensory/perceptual mapping from numerosity to brain states (the activities of detectors for specific numerosities), which make possible simple numerical discriminations, then the brain’s representation of number is a representation in name only. Only if the brain brings combinatorial processes to bear on the neural entities that represent numerosities may we say that the brain represents number in an interesting sense of the term representation.” (p. 159).

One of the possible developmental roots of our representations of relations between numbers is the Approximate Number System (ANS). The ANS represents number approximately – i.e., it does not discriminate all pairs of numerosities with equal precision (

However, multiple studies have shown that, in various contexts, infants and adults often do not use the ANS to represent the numerosity of collections when they include 4 or fewer objects. Instead, they use a distinct system whose capacity is limited to collections of up to 3 or 4 objects (

Some have proposed that, in addition to representing objects, parallel individuation supports numerical operations (

If parallel individuation does indeed support numerical operations, one might expect that the system could be used to compare distinct collections of objects on the basis of their numerosity. Some studies have shown that infants can use parallel individuation to compare collections. However, these studies found that infants’ comparisons were based on physical attributes of the collections (e.g., their total physical size) rather than on their numerosity (

To test whether parallel individuation can support numerical comparisons, a study must include numerosities that are small enough to be compared with parallel individuation. But this is not enough, since, in principle, participants can also use the ANS to compare small numerosities (

Pairs of collections that straddle the boundary between parallel individuation and the ANS must be avoided because the pattern of performance to be expected in such situations is not well known. Unfortunately, the boundary between the two systems is somewhat unclear. Some studies suggest that parallel individuation can hold up to 4 objects, but many others suggest that it cannot hold more than 3 (e.g.,

Previous studies with preschoolers have shown that children can compare two collections on the basis of number, but these studies do not meet the conditions outlined above (

To our knowledge, only two studies met all of the demands laid out above. One of these studies tested adults (

The present study aims to contribute to research on the development of parallel individuation as a system for numerical reasoning by applying a numerical comparison task with the right design features to preschoolers, namely two- to four-year-olds. Since, other than one study with inconclusive results (

A total of 99 2½ - to 4½-year-olds participated in this study. Fifty of them were tested on comparisons of small collections only (average age of 3 years 7 months; range: 2 years 7 months – 4 years 7 months; 22 males), and 49 were tested on comparisons of large collections only (average age of 3 years 7 months; range: 2 years 6 months – 4 years 7 months; 30 males). All of the children were recruited in Southwestern Ontario, Canada, and were predominantly monolingual speakers of English. An additional two children were excluded for always choosing the same side.

Children were tested on a numerical comparison task and on Give-

The numerical comparison task always started with an experimenter introducing two puppets to the children – a frog and a duck. Then, children were shown a picture of a rectangle and asked to name it, e.g., “Do you know what this is?” If children failed to provide a label, the experimenter suggested one (e.g., block, rectangle), and encouraged children to repeat it. After familiarization with the stimuli, the experimenter showed children two pictures of blocks, placed one in front of each of the two puppets, and said, “Froggie has some [blocks], duckie has some [blocks], who has more [blocks]?” This instruction was repeated for each trial.

The numerical comparison task had two between-subject conditions that differed on the range of numerosities tested: small (< 4) and large (> 4). In each condition, collections that differed by a 1:3 and 2:3 ratio were shown. Children in the small numerosity condition were asked to compare 1 vs. 3 (4 trials) and 2 vs. 3 (4 trials). Children in the large numerosity condition were asked to compare 6 vs. 18 (4 trials) and 6 vs. 9 (4 trials). All collections consisted of red rectangles printed on letter-sized paper. The rectangles in each individual collection were all of the same physical size. However, the relation between the physical size of individual rectangles, their total perimeter and area, and numerosity varied across trials (see

In the small numerosity condition, for size-number congruent trials, the cumulative surface area of the collections ranged from 1.5 cm^{2} to 12 cm^{2}, and their cumulative perimeter ranged from 5.32 cm to 30 cm. For size-number incongruent trials, the cumulative surface area was always 6 cm^{2}, and cumulative perimeter was 18 cm. In the large numerosity condition, for size-number congruent trials, the cumulative surface area of the collections ranged from 18 cm^{2} to 72 cm^{2}, and their cumulative perimeter ranged from 31.92 cm to 144 cm. For size-number incongruent trials, cumulative surface area ranged from 18 cm^{2} to 36 cm^{2}, and cumulative perimeter ranged from 54 cm to 108 cm.

Pairs of collections of rectangles were presented in one of two item orders. The orderings of correct side (left or right), ratio (1:3 or 2:3), numerosity pair (small condition: 2 vs. 3, 1 vs. 3; large condition: 6 vs. 9, 6 vs. 18), and trial type (size-number congruent or size-number incongruent) were randomized across orders. No two consecutive trials were of the same type or pair of numerosity. No feedback was given.

A size-number congruent trial in the small numerosity condition in Experiment 1.

A size-number incongruent trial in the small numerosity condition in Experiment 1.

A size-number congruent trial in the large numerosity condition in Experiment 1.

A size-number incongruent trial in the large numerosity condition in Experiment 1.

The purpose of this task was to assess children’s number word knowledge. Children were first introduced to a puppet, a tub of 10 fish, and a plate. They were then told that the puppet wanted to eat some fish, and the experimenter asked the child to give the puppet

Children were called ‘

Across all three experiments, whenever multiple comparisons were performed, the alpha level was adjusted using the Holm-Bonferroni method.

The number of children and the mean age in each knower-level group are presented in

Knower-Level | Small |
Large |
||||
---|---|---|---|---|---|---|

Mean Age | Range | Mean Age | Range | |||

Non-knowers | 8 | 3;1 | 2;9 - 3;8 | 7 | 3;5 | 3;2 - 3;10 |

1-knowers | 12 | 3;5 | 2;7 - 4;0 | 10 | 3;0 | 2;6 - 3;7 |

2-knowers | 10 | 3;5 | 2;8 - 3;11 | 12 | 3;9 | 3;2 - 4;7 |

3-knowers | 12 | 3;10 | 3;1 - 4;7 | 11 | 3;8 | 2;11 - 4;2 |

CP-knowers | 8 | 3;11 | 3;2 - 4;7 | 9 | 3;11 | 2;11 - 4;7 |

Our first analysis found no effect of order or gender (all

We included 2-way interaction terms involving Number Range (Number Range x Congruence, Number Range x Ratio, Number Range x Knower-Level, and Number Range x Age) because we were primarily interested in the effects of that factor. In particular, the model included Number Range (small vs. large), Size-Number Congruence (congruent vs. incongruent), Ratio (1:3 vs. 2:3), Knower-Level (non- and 1-knowers vs. 2- and 3-knowers vs. CP-knowers; CP-knowers as the reference category) and centered Age as fixed effects, with by-subject random intercept. This model did not increase the fit over a model with main effects only, χ^{2}^{i} A main-effects-only model revealed a main effect of Number Range, β = -.87,

Overall proportion correct on small and large numerosity comparisons on congruent trials (top panel) and incongruent trials (bottom panel), separated by knower-levels in Experiment 1. Violin plots are used to depict the distribution and probability density of the data. Error bars represent standard errors of the mean.

Finally, we asked whether the ability to use parallel individuation to compare numerosities is available even at the earliest stages of number word learning. To test this, we analyzed whether children with minimal number word knowledge – non-knowers and 1-knowers – performed significantly above chance on size-number incongruent trials for small numerosities, and whether their performance for small numerosities was better than that for large numerosities. We combined 2 vs. 3 and 1 vs. 3 comparisons and found that non-knowers and 1-knowers as a group performed significantly above chance on small comparisons, ^{ii} We also found that they performed significantly better on comparisons of small numerosities (1 vs. 3 and 2 vs. 3) than of large numerosities (6 vs. 18 and 6 vs. 9; ^{iii} These results suggest that parallel individuation can support numerical comparisons even in children with little or no knowledge of number word meanings. They also suggest that children’s performance on the numerical comparisons task cannot be explained by counting, because these children have not acquired the cardinal principle, and thus cannot have used counting to solve the task (

Two- to four-year-old children were tested on a non-verbal numerical comparison task in one of two conditions: comparing collections of three or fewer objects or collections of six or more objects. The ratios of the numerosities in both the small and large numerosity comparisons were the same. Despite that, children did not perform equally well on all comparisons. Rather, they were more accurate on small than on large comparisons. This difference also held in children at the earliest stages of number word learning – i.e., non-knowers and 1-knowers. These children performed significantly above chance when they compared small numerosities but not when they compared large ones. Given that the number range effect was observed in children at the earliest stages of number word learning and that these children performed above chance on comparisons of small numerosities even when area and numerosity were not congruent, our results also suggest that the development of the ability to use parallel individuation to compare numerosities does not depend on number word learning.

Two aspects of preschoolers’ performance on large numerosity comparisons warrant discussion. First, children at the earliest stages of number word learning (i.e., non- and 1-knowers), and to some extent, 2- and 3-knowers, performed poorly on comparisons of large numerosities (see

Second, none of the effects involving ratio were significant. Since this suggests that children were not more accurate on the large comparisons with a large ratio (6 vs. 18) than on the large comparisons with a smaller ratio (6 vs. 9), it may seem to pose problems for our suggestion that children used the ANS to compare large numerosities. That is, one may be concerned that our task was not appropriately designed to test our question of interest because it may not have engaged the ANS. We believe that this concern is unwarranted for two reasons. First, three other studies of non-verbal numerical comparisons in preschoolers with designs similar to ours also failed to find differences of accuracy between ratios similar to ours – i.e., 1:2 and 2:3 (

There is no other plausible explanation of children’s performance on large comparisons in our study. The only other strategies that are available to

Another possible concern is that children controlled how long the collections were presented to them, so that, consequently, they could have used counting instead of parallel individuation and/or the ANS to compare numerosities. Since small collections are easier to count than large ones, this could explain why children were more accurate on small comparisons. However, the fact that children who did not know the cardinal principle – i.e., non-knowers and 1-knowers – showed this pattern of results makes this very unlikely. Nevertheless, in Experiment 3, we address this concern by asking children to compare the numerosities of collections that are presented too quickly to be counted.

Finally, it is possible that the better performance on comparisons of small numerosities was driven by better performance on the 1 vs. 3 comparisons only. By physical necessity, whenever one of the choices in a comparison is one object, the other choice is always the right answer. Thus, comparisons where one of the choices is a single object may be easier than all other comparisons. That is, one may predict that the effect of Number Range is specific to comparisons with a 1:3 ratio. Nevertheless, the lack of an interaction between Number Range and Ratio has ruled out this possibility.

There remain two alternatives that cannot be ruled out directly by the results of Experiment 1. First, it may be that children performed better on comparisons of small numerosities because the correct answer in this condition was always the same – i.e., 3 – whereas the correct answer in the large numerosity condition alternated between 9 and 18. Second, it could be that children were, in fact, relying on the ANS for all comparisons but that they performed more poorly on large comparisons because these comparisons make greater demands on non-numerical aspects of processing. For example, larger collections necessarily require children to divide their attention over more objects than small collections, and this could cause a decrement in performance. Although possible in theory, this “processing demands” alternative is unlikely to be the right explanation. On this alternative, the difficulty of comparisons should increase continuously as a function of the absolute size of numerosities. Contrary to this prediction, previous research provides strong evidence that when infants (

Like Experiment 1, Experiment 2 included one group of children who compared small numerosities only and another group who compared large numerosities only. Unlike Experiment 1, Experiment 2 only included size-number incongruent trials. This feature was changed to increase the number of trials that test whether children use distinct systems to compare collections on the basis of numerosity.

Experiment 2 had two main goals. First, we sought to replicate the effect of Number Range observed in Experiment 1. Second, we aimed to address the alternative explanations raised in Experiment 1. To address the processing demands alternative the large comparisons included a wider range of pairs of numerosities of the same ratio. To ensure that the number of trials was reasonable for young preschoolers, we only included comparisons with a ratio of 2:3, namely, 6 vs. 9, 10 vs. 15 and 12 vs. 18. Evidence that children’s accuracy decreases as numerosity increases would support this alternative. On the other hand, evidence that (1) children perform better when they compare small numerosities than when they compare large numerosities at the same ratio,

We also tested whether the children in Experiment 1 who compared small numerosities were more accurate than those who compared large ones because the correct answer was always the same numerosity in the former condition but not in the latter. To test this, the small numerosity condition of Experiment 2 included comparisons of 1 vs. 2 and 3 vs. 4 in addition to comparisons of 2 vs. 3. Evidence that children were nonetheless more accurate on 2 vs. 3 than on the large comparisons would show that this alternative is false.

A total of 86 2 ½ - to 4 ½-year-olds participated in this study. There were 45 children in the small numerosity condition, with an average age of 3 years 5 months (range: 2 years 6 months – 4 years 5 months; 24 males), and 41 children in the large numerosity condition, with an average age of 3 years 5 months (range: 2 years 6 months – 4 years 9 months; 21 males). A majority of the children were recruited in Southwestern Ontario, Canada (

The stimuli and procedure were similar to Experiment 1. All children completed the numerical comparison task before the Give-

In the small numerosity condition, the cumulative surface area ranged from 6 to 8 cm^{2} and cumulative perimeter ranged from 18 to 24 cm. In the large numerosity condition, cumulative surface area ranged from 18 to 36 cm^{2}, and cumulative perimeter ranged from 54 cm to 108 cm.

The number of children and the mean age in each knower-level group are presented in ^{iv}

Age was significantly correlated with children’s knower-level, Pearson’s

Knower-Level | Small |
Large |
||||
---|---|---|---|---|---|---|

Mean Age | Range | Mean Age | Range | |||

Non-knowers | 3 | 3;0 | 2;10 - 3;3 | 3 | 2;10 | 2;8 - 3;0 |

1-knowers | 8 | 3;0 | 2;6 - 3;10 | 12 | 3;1 | 2;7 - 4;9 |

2-knowers | 10 | 3;2 | 2;7 - 4;5 | 12 | 3;3 | 2;6 - 4;0 |

3-knowers | 9 | 3;5 | 2;10 - 4;2 | 5 | 3;8 | 2;8 - 4;3 |

CP-knowers | 14 | 3;10 | 3;3 - 4;5 | 9 | 4;1 | 3;4 - 4;7 |

Preliminary analyses showed that children recruited in Canada and in the US performed similarly on the numerical comparison task,

First, we asked if the effect of Number Range in Experiment 1 could be replicated. We constructed a linear regression using overall proportion correct on comparisons that differed by a 2:3 ratio as the dependent variable, and Number Range (small vs. large), Knower-Level (non- and 1-knowers, 2- and 3-knowers, and CP-knowers; CP-knowers as the reference category), and centered Age as independent variables. We found that adding the 3-way interaction or the 2-way interactions did not improve the fit of the regression model. We thus used a main-effects-only model in our final analysis. Results revealed a main effect of Number Range, β = -.13,

Overall proportion correct on small and large numerosity comparisons on incongruent trials, separated by knower-levels in Experiment 2. Only 2:3 ratio comparisons were included. Violin plots are used to depict the distribution and probability density of the data. Error bars represent standard errors of the mean.

We also asked whether children succeeded on 2 vs. 3 because they somehow defaulted to choosing 3 or to avoiding 2 as the correct answer without doing the comparison. If children used the first strategy, they should have always chosen the wrong collection on comparisons of 3 vs. 4. If they used the second, they should have always chosen the wrong collection on comparisons of 1 vs. 2. This was not so. Rather, children performed significantly above chance on 1 vs. 2 (

To test the processing demands alternative, we compared the large numerosity comparisons to each other. No significant differences were found (all

Proportion correct on small and each of the large numerosity comparisons in Experiment 2. Error bars represent standard error of the mean.

Finally, we asked whether children’s ability to recruit parallel individuation to compare small numerosities is available even in children at the earliest stages of number word learning – i.e., non-knowers and 1-knowers. We found that they performed significantly above chance on small comparisons,

Experiment 2 replicates several of the most important results of Experiment 1. First, it shows that children were more accurate on comparisons of small than large numerosities. We also found that non- and 1-knowers were significantly more accurate on small than on large comparisons. Since these children did not know the cardinal principle, it cannot be that this small-large difference was due to counting. In addition, counting is a poor explanation of the fact that children were more accurate on 2 vs. 3 than on the large comparisons, but were equally accurate on all the latter comparisons. Indeed, if children had used counting to compare numerosities, one might have expected their performance to decrease gradually as numerosity increased.

Experiment 2 also rules out a number of alternative explanations of the fact that children were more accurate on comparisons of small numerosities. First, given that the numerosity of the correct answer varied in both the small and the large comparison conditions, the results of Experiment 2 show that greater accuracy on small comparisons in Experiment 1 was not due to the fact that the correct answer was always the same in the small numerosity condition but varied in the large numerosity condition. It also shows that children did not simply default to always choosing the collection of 3 or to avoiding the collection of 2 on the 2 vs. 3 comparison. Second, Experiment 2 provides evidence against what we called the processing alternative – i.e., they show that, contrary to the prediction of this alternative, children were more accurate on 2 vs. 3 than on comparisons of large numerosities of the same ratio, despite the fact that the large numerosities varied in size from 6 vs. 9 to 12 vs. 18.

The use of a between-subject design in the previous experiments leaves open the question of whether the effect of Number Range can be observed in the same children on the same task. To address this, we adopted a within-subject design, with the same group of children comparing exclusively small collections (2 vs. 3)

Experiment 3 made two further changes. First, although the fact that children who did not know the cardinal principle (e.g., non-knowers and subset-knowers) showed better performance on small collections than large collections suggests that this difference is not due to counting, we sought convergent evidence against this possibility by controlling the presentation time of the stimuli. That is, to prevent children from counting, the pairs of collections to be compared were presented for 2 seconds only (see also ^{v} Thus, to compare collections of 2 to collections of 3 by counting both of them, children would need at least 3.5 seconds, which exceeds the presentation time used in Experiment 3. Second, to ensure that our task engaged the ANS, we tested whether we could obtain a ratio effect in the large number range by including pairs of numerosities that differed by a 5:6 ratio, in addition to pairs that differed by a 2:3 ratio. Evidence that children perform more poorly on the 5:6 than on the 2:3 comparisons of large numerosities would confirm that our task taps the ANS.

As in Experiment 2, we only included size-number incongruent stimuli. We also included comparisons of 3 vs. 4 in the small numerosity block to prevent children from using the strategy of always choosing 3 without doing the comparison, and to match the small and large numerosity blocks in terms of number of types of comparisons (both blocks have two types).

A total of 50 2 ½ - to 4 ½-year-olds participated in this study, with an average age of 3 years 5 months (range: 2 years 3 months – 4 years 8 months; 21 males, 29 females). An additional 18 children were tested and removed for providing no response (

The procedure was similar to Experiment 2. All children completed the numerical comparison task before the Give-

The numerical comparison task always started with an experimenter introducing two stuffed animals - a blue bear sitting on the left side of a laptop computer and a red bear sitting on the right side. To ensure that children could identify the bears by their color, she asked children to point to the red bear and the blue bear. All children correctly identified the bears. Then, the experimenter said they were going to play a game with some blocks on the computer, and that their job was to indicate which bear had more blocks. The experimenter explained that the blue bear has blue blocks (while gesturing to the left of the computer) and that the red bear has red blocks (while gesturing to the right). Children were told that the blocks would go away very quickly and thus they would have to look at the screen very carefully.

On each trial, the experimenter asked, “Who has more blocks?” right before a white asterisk on a grey background appeared on the screen for 1500 ms. Then, an array of blue blocks and an array of red blocks appeared simultaneously for 2000 ms. Each array of blocks was presented on a 10 cm x 6 cm white background. The arrays were 4.3 cm apart from each other. The overall background of the screen was grey (see

A 2:3 ratio comparison in the small numerosity condition in Experiment 3 (size-number incongruent).

A 2:3 ratio comparison in the large numerosity condition in Experiment 3 (size-number incongruent).

The study began with four familiarization trials, which included two collections of 1 vs. 3 and two collections of 10 vs. 30, presented in alternating order (1 vs. 3, 10 vs. 30, 1 vs. 3, 10 vs. 30). If a child failed to respond on the first trial, the experimenter repeated the instructions. If the child continued to provide no response, the experimenter terminated the study session and thanked the child for participating.

After the familiarization trials, the small and large comparisons were presented in blocks that were counterbalanced across children: half of the children completed the small comparisons block first and the other half completed the large comparisons block first. In the small comparisons block, children were asked to compare 2 vs. 3 (3 trials) and 3 vs. 4 (3 trials). In the large comparisons block, they were asked to compare 6 vs. 9 (3 trials) and 15 vs. 18 (3 trials). Stimuli for the 2 vs. 3, 3 vs. 4, and 6 vs. 9 comparisons were chosen from a subset of the stimuli used in Experiment 2. Each collection was digitized to ensure that the two collections in each pair of comparison had an equal number of pixels. Stimuli for the 15 vs. 18 comparisons were created in the same way as number-size incongruent stimuli in Experiment 1, and were then digitized as images.

Size and number were incongruent in all comparisons. Pairs of collections of rectangles were presented in one of two item orders. For one order, the first comparison was at a 2:3 ratio in both small and large comparisons blocks, and for the other order, the first comparison was at a 3:4 ratio in the small comparisons block and a 5:6 ratio in the large comparisons block. The correct side (left or right) was counterbalanced in each item order. The trials within each block were randomized such that no two consecutive trials were of the same ratio of comparison.

The number of children and the mean age in each knower-level group are presented in ^{vi} Age was significantly correlated with children’s knower-level, Pearson’s

Knower-Level | Mean Age | Range | |
---|---|---|---|

Non-knowers | 6 | 2;9 | 2;4 - 3;3 |

1-knowers | 9 | 2;8 | 2;3 - 3;0 |

2-knowers | 8 | 3;7 | 2;8 - 4;2 |

3-knowers | 5 | 3;9 | 3;6 - 4;0 |

CP-knowers | 22 | 3;11 | 2;10 - 4;8 |

Preliminary analyses revealed no effect of item order, block order, or gender, and thus, we collapsed across these variables in subsequent analyses (all

Children performed significantly above chance on the practice trials,

To examine the effect of number range, we constructed a logistic mixed-effects model predicting children’s correct responses on 2:3 ratio comparisons (2 vs. 3 and 6 vs. 9) with Number Range (small vs. large), centered Age in months, and Knower-Level (non- and 1-knowers, 2- and 3-knowers vs. CP-knowers) as fixed factors, with by-subject random slopes for Number Range. We also included 2-way interaction terms involving Number Range (Number Range x Age, Number Range x Knower-Level). This model did not increase the fit over a model with main effects, χ^{2}

Overall proportion correct on small and large numerosity comparisons on incongruent trials, separated by knower-levels in Experiment 3. Only 2:3 ratio comparisons were included. Violin plots are used to depict the distribution and probability density of the data. Error bars represent standard errors of the mean.

Next, we asked if there was a ratio effect in the large number range. To test this, we selected children who answered at least two out of three 6 vs. 9 comparisons correctly and asked if they performed worse on the 15 vs. 18 comparisons (_{age}_{av} = 1.30 [.44, 1.36]^{vii}, indicating that the comparison with a 2:3 ratio comparison was easier than the one with a 5:6 ratio. Nevertheless, children who could compare 6 vs. 9 were no better at comparing 10 vs. 30 (_{av} = 0.16 [-.51, .27].

We also asked whether children succeeded on 2 vs. 3 because they somehow defaulted to choosing 3 without doing the comparison. Against this explanation, children performed significantly above chance on 3 vs. 4 (

Unlike Experiments 1 and 2, Experiment 3 did not show a main effect of knower-level – CP-knowers (^{2}

Finally, we asked whether children who were at the earliest stages of number word learning used parallel individuation to compare small numerosities. Contrary to what we found in Experiments 1 and 2, we found that non-knowers and 1-knowers did not perform significantly above chance on 2 vs. 3 (_{av}

Although non-knowers and 1-knowers performed better on small comparisons than large ones in all three experiments, the mean difference was significant in only two of the three experiments. We thus sought to gather cumulative evidence on whether non- and 1-knowers can recruit parallel individuation to compare small collections by conducting a mini meta-analysis (see

Experiment 3 made two important methodological changes in an attempt to replicate and expand previous findings. We adopted a within-subject design such that the same children compared pairs of small and pairs of large numerosities, and we prevented counting by limiting presentation time to 2 seconds. Despite these changes, we replicated the finding that 2 vs. 3 comparisons were easier than comparisons of large numerosities with an equal ratio. This provides strong evidence that preschoolers can use parallel individuation to compare small numerosities.

The lack of a number range effect at the 1:3 ratio may raise doubts about whether the small-large difference reflects the use of distinct systems for each number range. However, to assess whether children can recruit parallel individuation (instead of the approximate number system) to compare small numerosities, one need not show that small numerosity comparisons are different from large numerosity comparisons

Finally, we found that CP-knowers were better at comparing numerosities non-verbally than subset-knowers and non-knowers. However, the difference between CP-knowers and subset- and non-knowers observed in Experiment 3 was smaller than in Experiments 1 and 2.

Unlike what was found in Experiments 1 and 2, we found that children who were at the earliest stages of number word learning did not show a significant number range effect. However, a mini meta-analysis summarizing across all three experiments suggests that the difference between small and large numerosity comparisons is likely a reliable effect for non- and 1-knowers.

In three different experiments, we find that preschoolers between the ages of two-and-a-half and four-and-a-half – were modestly, but reliably more accurate in comparing collections of 2 and 3 than pairs of numerosities larger than 5 with the same ratio. This difference was found regardless of whether performance on comparisons of small numerosities was compared to performance on large numerosities across different children or within the same ones, and regardless of whether the presentation time was child-controlled or limited to prevent counting. This suggests that it is a highly robust result.

On our view, the best explanation of this difference is that, in this task, children use distinct systems to compare small and large numerosities: parallel individuation for 1 to 3 and the ANS for more than 5. Various aspects of our results support this conclusion. We know that children based their choices on the numerosities of the collections because the evidence for this explanation was found on comparisons where size and numerosity were incongruent – i.e., where children could not base their decision on the total area or the total perimeter of the collections because these were equated, and where they could not base it on the individual object sizes because these were larger in the collection with the smaller numerosity. Our results cannot be explained by counting because children were more accurate on small than on large comparisons even if (1) they did not understand the cardinal principle and thus could not count to compare numerosities and (2) they could not count any of the collections because they were presented too quickly to allow them to do so. Moreover, it is unlikely that children were more accurate on comparisons of small than large numerosities because they could name the former but not the latter. Indeed, in two out of three experiments, we found the same pattern of results in children who could not name the small numerosities – namely non-knowers and 1-knowers. Our results cannot be explained by any hypothesis that predicts that accuracy decreases gradually as numerosity increases – i.e., while children were more accurate on comparisons of small numerosities (2 vs. 3) than on any of the comparisons of large numerosities, they were equally accurate on comparisons of a relatively wide range of large numerosities (6 vs. 9, 10 vs. 15, and 12 vs. 18). Finally, Experiment 3 provided direct evidence that our task did engage the ANS when numerosities were large – i.e., children performed more poorly on comparisons of 15 vs. 18 than on comparisons of 6 vs. 9, suggesting that their accuracy on these comparisons was controlled by the ratio of the numerosities and not by the absolute difference between them. This leaves the view that children used parallel individuation for small comparisons and the ANS for large ones as the only plausible explanation of our results.

To be clear, we are not claiming that the ANS is never used to represent small numerosities. Indeed, some studies have shown that, in some contexts, infants (

Across all three experiments, we found that children who had not learned the meaning of any number word beyond “one” (i.e., non- and 1-knowers) demonstrated better performance on small numerosity comparisons than on large numerosity comparisons. Although the difference between small and large numerosity comparisons was significant in only two of the three experiments, a mini meta-analysis using effect sizes suggests that the number range effect in this group of children is reliable. Thus, although the difference between performance on small and large comparisons may be small for this group, we take it to suggest that it is likely that the capacity of parallel individuation to support numerical comparisons is available prior to the acquisition of number word meanings.

Interestingly, studies of comparison of small collections in infants suggest that it might not be available in the form observed here from the beginning of life. On the one hand,

In two of the three experiments, we found that CP-knowers were significantly better at numerical comparisons than subset-knowers. In the last experiment, we did not find a significant effect of acquiring the cardinal principle on average accuracy, but we found that CP-knowers were more likely to answer at least 4 out of 6 trials correctly than subset-knowers and non-knowers. Two other studies of numerical comparisons where the relative size of the elements in the collections conflicts with their relative numerosity (what we called “size-number incongruent” comparisons) show similar results (

Why is acquiring the cardinal principle related to an improvement in children’s performance on non-verbal numerical comparisons? On one view, the acquisition of the cardinal principle causes the representations of numerosity created by the ANS to become more precise (

In light of these results, Negen and Sarnecka argue that it is not the case that the acquisition of the cardinal principle provokes changes in the precision of the ANS. Rather, they suggest that (1) subset-knowers still do not clearly differentiate the continuous meaning of “more” from its discrete meaning (as in “more chocolate” vs. “more candy bars”), and that (2) CP-knowers are more likely than subset-knowers to spontaneously pay more attention to numerosity than to area. Our data provide some evidence against the first part of Negen and Sarnecka’s hypothesis. We found that in Experiments 1 and 2, children who had not acquired the cardinal principle (i.e., non-, 1-, 2-, and 3-knowers) performed above chance on comparisons of small numerosities (^{viii}

Unlike the ANS, the symbols created with parallel individuation are not symbols for cardinalities; they are symbols for objects. For example, when infants use parallel individuation to represent two toy cars hidden in a box, they do not represent the contents of the box as

The present study is the first to provide evidence that the ability to use parallel individuation to compare distinct collections on the basis of numerosity is available at least by age 2 and that the development of this ability likely does not depend on number word learning. Following Carey and colleagues (

We thank all the children and families for participating, and preschools and teachers for their help with this project. Without them, this research would not be possible. We also want to thank Miranda Sollychin and Alexandra Butti for their assistance with data collection. We are grateful to Susan Carey and Anna Shusterman for allowing the first author to use their lab resources to recruit participants for Experiment 2 and Experiment 3, respectively. We thank two anonymous reviewers for providing helpful feedback on this manuscript.

We fit all mixed effects models using the lme4 package in R (

As a stronger test of whether the capacity to use parallel individuation to compare numerosities depends on number word learning, we also analyzed non-knowers on their own – i.e., without 1-knowers. We found that the non-knowers performed significantly above chance on these comparisons,

All effect sizes (Cohen’s d) and their 95% confidence interval for independent samples t-tests were computed using the compute.es package in R (_{av}) were computed using

Across both small and large numerosity conditions, we found 1 4-knower and he was classified as a 3-knower in the analyses. One child in the small numerosity comparison did not complete the Give-

We found one 4-knower. She was grouped with the 3-knowers in the analyses.

We reported Cohen’s _{av}

We conducted post-hoc analyses to further explore this proposal. Despite a lack of an interaction between Size-Number Congruence and Knower-level, using data from Experiment 1, we found that for large comparisons, subset-knowers were at chance on size-number incongruent trials (

The authors have declared that no competing interests exist.