# Acquiring numerical equality

Véronique Izard, Valeria Giardino, Arnaud Viarouge

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Véronique Izard, Valeria Giardino, Arnaud Viarouge. Acquiring numerical equality. 2026. hal-05364342

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# Acquiring numerical equality

Véronique Izard¹, Valeria Giardino², Arnaud Viarouge³

¹ Université Paris Cité, CNRS, Integrative Neurosciences and Cognition Center, F-75006, Paris, France

² Institut Jean Nicod, UMR 8129, CNRS, F-75005, Paris, France

³ Université Paris Cité, CNRS, LaPsyDÉ, F-75005 Paris, France

*Note: This version predates copyediting and typesetting.*

## Abstract

We propose to analyze the notion of numerical equality as a complex, *Numerical Equality*, constituted of three facets (“same number” statements, numeral labels, and one-to-one pairings of collections) all mapping onto a same relation defined on collections, herein referred to as the *relation of equal numerosity*. With this framework in hand, we review experimental studies assessing whether children correctly map the three facets of Numerical Equality onto the relation of equal numerosity, and whether they understand that these three facets are interrelated. By and large, these studies indicate that children still struggle with various aspects of the Numerical Equality complex as they enter school – and provide no evidence that children represent the relation of equal numerosity itself before the age of 5 years. Our review also identifies significant gaps in the literature, leaving many aspects of children’s acquisition of numerical equality open to research.

## I. The Numerical Equality complex and its constituents

Imagine a child playing with an adult. The child will hear the adult use the word “number”, stating that some piles of toys have the “same number” of items, others not. Sometimes, adults will also use words like “two”, “three”, “five” to label piles of toys, or engage in a funny activity where they point
to each toy in turn while reciting such words in a fixed order. The words they use may or may not match across piles. Meanwhile, the child will also manipulate toys herself, trying to pair things that go together one-to-one, for example Barbies with hats, or eggs with egg cups. Attempts to do so may result in frustrating states of affairs, however: A Barbie may remain bareheaded, or some guests may have to eat their eggs without the proper tableware.

This description illustrates different facets of numerical equality, as it manifests itself in a young child's life. Importantly, the cases where two collections can be said to have the "same number" coincide with the cases where these collections should be labeled with matching numerals or their items can be paired one-to-one. In other words, "same number" statements, numeral labels, and one-to-one pairings all map onto the same relation defined over collections, which thus lies at the heart of numerical equality: we will refer to this relation as the *relation of equal numerosity*. The term "relation" should be understood in its strictest sense: a relation is a function that takes pairs of items as input and outputs one of two values, "related" or "unrelated". Accordingly, what we call the "relation of equal numerosity" is a black-box function that takes pairs of collections as input, and labels them either as "equal in numerosity" or "unequal in numerosity". The relation of equal numerosity and the three aforementioned facets – "same number" statements, numeral labels, one-to-one pairings between collections – together form a complex that we will name *Numerical Equality* (capitalized; see Figure 1).

In light of this framework, what does it mean for a child to understand numerical equality?

Ultimately, the child needs to be able to (A) represent the relation of equal numerosity (Figure 1, center), and (B) link this relation with the three facets of Numerical Equality (Figure 1, blue arrows). That is, ultimately they should be able to identify which collections can be said to have the "same number" of items, should be labeled with matching numerals, can be paired one-to-one. They also need to (C) understand the interrelations between "same number" statements, numeral labels and one-to-one pairings (Figure 1, black arrows). For instance, stating that two collections have the
"same number" of items implies that these collections should be labeled with matching numerals, and that their items can be paired one-to-one.

Before getting to this final level of understanding, however, children may go through developmental stages where they possess only snippets of this knowledge. Children at these earlier stages of development may interpret only a subset of the facets of numerical equality in terms of the relation of equal numerosity, and/or they may fail to recognize that these facets are related. Maybe too, there is a point in development where children cannot represent the relation of equal numerosity itself. Still, at this early stage they might be able to represent other relations of collections, and thus have access at least to some aspects of the relation of equal numerosity.

<figure><img src="image_2.png" /><figcaption>Figure 1: The Numerical Equality complex (red frame) is constituted of three facets ("same number" statements, numeral labels, one-to-one pairings between collections; black text) all mapping onto the same relation defined on collections, the relation of equal numerosity (blue text). The relation of equal numerosity is a "relation" in the strictest sense of the term: a black-box function that takes two collections as input and outputs one of two values, "equal in numerosity" or "unequal in numerosity".</figcaption></figure>
What could such a partial understanding of the relation of equal numerosity look like? To answer this question, we found inspiration in a philosophical paper [1], which implies that the relation of equal numerosity can be characterized via a list of principles describing how equality/inequality generalizes through transformations applied to collections. This approach will be described in Section II.

Accordingly, to assess whether children have an early (and potentially partial) understanding of the relation of equal numerosity, in Section III we will examine the experimental literature probing their understanding of this kind of principle, when applied to “same number” statements, numeral labels, or one-to-one pairings of collections. To complete this picture, in Section IV we will also review studies testing whether children understand the interrelations between “same number” statements, numeral labels and one-to-one pairings.

## II. Characterizing the relation of equal numerosity

As part of his search for logical foundations of arithmetic, Frege [2] demonstrated that the basic laws of arithmetic standardized by the Dedekind-Peano axioms can be derived from a single non-logical axiom, Hume's principle, which characterizes *equinumerosity* (a relation defined on sets) in terms of one-to-one correspondence: two sets are equinumerous if and only if they can be related by a bijection. The philosopher Heck [1] remarked that Frege's equinumerosity is too technical to possibly be accessible to lay people, let alone children engaged in learning about numbersª. They show that it is possible to construct (finite) arithmetic out of a more psychologically plausible primitive, which

<footnote>ª Frege defines cardinal numbers as *equivalence classes* under the relation of equinumerosity [see 2, §68]. Alternatively, in Husserl's view, equinumerosity (or one-to-one correspondence) is based on "sameness of number", which is obtained by counting. For Frege, this is unacceptable, since counting depends on a one-to-one correspondence: the correspondence between the numerals from 1 to n and the objects counted [3]. In their paper, Heck gets inspired from children's behavior and shows that, against Frege, the rules that govern the practice of counting can be mastered without first grasping the notion of one-to-one correspondence. However, they argue, cardinal numbers cannot be based on the practice of counting, since it is also possible to count without having a concept of cardinality. For these reasons, they introduce a notion that is less sophisticated than Frege's equinumerosity and which, they show, can serve as an alternative primitive for the concept of cardinal number.</footnote>
they call *Just As Many* (henceforth, JAM). Interestingly, Heck does not characterize JAM through
necessary and sufficient conditions, but through the following 3 principles describing some of its
properties:

1. "For any empty collections A and B, JAM holds between A and B."

2. "If JAM holds between collections A and B, and one item is adjoined to A and one item is adjoined to B, then JAM will continue to hold between A and B."

3. "If JAM holds between collections A and B, and one item is removed from A and one item is removed from B, then JAM will continue to hold between A and B."

Viewed as a pure relation on collections<sup>b</sup>, Heck's JAM coincides with our relation of equal numerosity: the same pairs of collections that are considered "just as many" by JAM are also considered "equal in numerosity" by the relation of equal numerosity. Heck's three principles above can thus form an implicit characterization of the relation of equal numerosity, just like they form an implicit characterization of JAM.

While originally written with the aim of revisiting the logical foundations of arithmetic, Heck's work
opens interesting perspectives for developmental psychology [for another discussion of the parallels
between Heck's proposal and developmental psychology, see 4]. First, Heck's analysis suggests an

<footnote><sup>b</sup> It is unclear from Heck's paper whether JAM should be considered as a pure relation on collections and can thus be fully equated with our relation of equal numerosity, or whether it incorporates aspects of what we describe as a facet of Numerical Equality: one-to-one pairings. For instance, Heck explicitly requires that children connect the phrase "just as many" to one-to-one pairings to be granted proper understanding of this phrase: "It seems obvious that someone could accept [principles 1-3] and still have no idea what to make of the claim that there are just as many children as cookies. What we need, then is some other unsophisticated answer to the question what "just as many" and "more" mean. (...) [S]uppose there are just as many children as cookies. (...) [F]rom the child's point of view, it need mean no more than that, if you start giving cookies to children, and you're careful not to give another cookie to anyone to whom you've already given one, then you won't run out, though you won't have any left. There are enough, that is to say, for everyone to have one, but not for anyone to have two." Importantly, irrespective of the connection between JAM and one-to-one pairings, Heck's three principles do not appeal to one-to-one pairings, making them a suitable characterization of the relation of equal numerosity.</footnote>
approach for investigating children's understanding of the relation of equal numerosity: if this
relation can be characterized via principles, then children's understanding of the relation can be
assessed via their understanding of such principles. Second, given that several principles are needed
to characterize the relation of equal numerosity, it may turn out that children know some but not all
of these principles. In this sense, they may possess only a partial knowledge of the relation of equal
numerosity.

At first view, Heck's analysis thus outlines a clear research program for experimentalists: children should be asked what they think of principles 1-3 above; if they agree that these principles are true, they can be granted knowledge of the relation of equal numerosity. Unfortunately though, designing such experimental studies raises a practical challenge. To turn Heck's principles into intelligible questions, experimentalists need to find a way to ask whether "the relation of equal numerosity holds". The most natural way to solve this issue is to rely on the facets of Numerical Equality: "Do these two collections have the same number? How many items do they have? Can they be paired one-to-one?". While asking such questions, studies do not assess directly whether children are able to represent the relation of equal numerosity. Rather, they test whether children correctly map the facets of Numerical Equality onto the relation of equal numerosity. Again, the logic from Heck's analysis can be borrowed to that aim: to assess whether children map a given facet onto the relation of equal numerosity, it is sufficient to test whether they believe that Heck's principles apply to this facet. Albeit indirectly, such studies can also shed light on children's knowledge of the relation of equal numerosity per se: if children map at least one of the facets tested onto the relation of equal numerosity, then it follows that they are able to represent this relation.
In the next section, we will thus review studies testing how children generalize “same number” statements, number words<sup>c</sup>, and one-to-one pairings as collections undergo transformations. As will soon become evident, no study has tested exactly Heck’s principles: equality between empty collections, equality continuing to hold after adjoining or removing an item to/from *both* collections. However, as we will show, tests of other types of transformations (e.g. adjoining or removing items to/from only one collection, altering the spatial configuration of items) provide precious information to analyze whether children map “same number” statements, number words, and one-one pairings onto the relation of equal numerosity, or onto a different relation.

### III. Do children map the three facets of Numerical Equality – “same number” statements, numeral labels, and one-to-one pairings – onto the relation of equal numerosity?

Research on children’s understanding of numerical equality originated in Piaget’s number conservation studies [9]. Tasks from this research tradition typically present children with two collections of objects arranged in lines and placed in visual one-to-one correspondence (e.g. eggs and egg cups). The experimenter then makes sure that the child agrees that “the two collections are the same”, that “they have the same number”, and/or that “there is one X for each Y”. Then, she changes the spatial position of the objects, making one line longer than the other, and asks the child whether the two collections are still “the same”, whether they still “have the same number”, “have one X for each Y”. At the age of 4-5 years, many children answer these questions negatively, pointing to the lengths of the lines to justify their answer: one of the lines is now longer and thus has more.

<footnote><sup>c</sup> We use “numeral label” as a general term referring to any kind of sign used to denote the numerosity of a collection. These signs can take a variety of forms, including spoken or written number words, written digits, body parts, tally marks, abacus configurations, and potentially more. In theory, experiments assessing children’s usage of any such numeral label, besides number words, would thus be relevant to our project. However, most studies have focused on number words, because they are the first type of numeral label learned by young children in western societies [for studies on written digits or abacus use, see 5–8].</footnote>
Taken at face value, these findings indicate that young children have not connected the phrase "same number" or one-to-one pairings with the relation of equal numerosity (top left and bottom blue arrows in Figure 1): They state that "same number" can cease to hold or that two collections can no longer be paired one-to-one after a mere displacement of their items. Later, however, conservation studies have been criticized for misleading children towards wrong answers, casting doubts about these conclusions. For instance, various authors argued that children mistook the experimenter's question as an invitation to look better and reevaluate their first judgment, or that the task places an undue burden on children's limited executive resources [e.g., 10-13]. Taking these caveats into account, it still remains that conservation studies fail to provide evidence that young children map any of the facets of Numerical Equality onto the relation of equal numerosity.

In the decades that followed, researchers turned their attention to children's use of counting and number words. This research identified a simple yet powerful test to diagnose children's understanding of number words: the "Give-a-Number" task, in which children are requested to provide a specific number of objects. Performance at Give-a-Number increases with age, with children progressing through two major stages [14–21]. Younger children are able to honor the experimenter's requests for just a few number words, but once the number requested becomes too large, they give large piles of items apparently at random. Asked to count the pile produced, they can do so but won't correct their response on this basis. These children are called "Subset-Knowers", because they appear to understand only a subset of the number words in their count list. Around the age of 3½, children's performance shifts: they spontaneously use counting, thus becoming able to produce the right numerosity for any number word in their count list. Because these children appear to know that the last word of a count represents the quantity of items counted (the "Cardinal Principle"), they are called "Cardinal Principle-Knowers" or "CP-Knowers" [for discussions of CP-Knowers' knowledge about counting and number, see also LeCorre, this volume, and Barner, this volume].
When asked to produce or to quantify collections, Subset-Knowers do not use number words in accordance with the relation of equal numerosity: by definition, they may sometimes associate collections of various numerosities with the same number word, or associate various number words with the same numerosity. Still, concluding from these results that Subset-Knowers do not interpret number words in line with the relation of equal numerosity might be hastened. Children's variable answers may simply be due to their difficulty finding the right number word (since they haven't identified counting as a good strategy to do so), not to their lack of understanding that number words should ideally be applied in accordance with the relation of equal numerosity.

New tasks were thus designed to directly test Subset-Knowers' understanding of the principles governing number word usage [15, 19], but these tasks yielded mixed results. In a first task [19, replicated in 22], an experimenter showed a box to children, and announced how many items had been placed in this box. The experimenter then applied a transformation (shaking the box, adjoining one item, removing one item) and gave children the choice between two number words to label the box's contents: the initial number word or a different number word. Subset-Knowers performed above chance, preferring the initial number word after a shaking event, and a different number word after all other transformations. Subset-knowers were however at chance in a second task [15, also replicated in 22]. In this task, the experimenter showed children two collections, and labeled one of them with a number word. She then applied a transformation to this collection (adjoining/removing one or several items, rearranging items), and asked the children to identify which collection possessed a given number of objects – either the same number word that had been used before the transformation event, or a different one. This second task may seem overly complex, especially for young children – explaining why they failed to answer correctly. Still, the interpretation of the findings from these two tasks remains debated. For instance, it has been argued that the box task overestimates children's knowledge, because the mere pragmatics of the situation leads them to pick the correct number word [22]. Moreover, Subset-Knowers succeed on the two-collection task when tested with small numbers, thus demonstrating that this task is actually within their reach [15].
It is thus unclear whether Subset-Knowers understand that number words should be used in
accordance with the relation of equal numerosity (top right blue arrow in Figure 1), again failing to
provide evidence that young children are able to represent this relation. However, young children
may fail in tasks involving number words simply because they haven't yet learned the meaning of
these words, not because they lack the relevant concepts. As a stronger test of Subset-Knower's
understanding of the relation of equal numerosity, we now turn to a study testing children in a non-
verbal context: When they manipulate collections, do children represent the relation of equal
numerosity? In particular, might they understand that one-to-one pairings track equal numerosity
(bottom blue arrow in Figure 1)?

To assess this question, Subset-Knowers were presented with a collection of finger puppets placed
on a collection of sticks, either perfectly paired one-to-one, or with one less puppet than sticks [23].
Then, the pairing was disassembled and a transformation was applied to the puppets or the sticks.
After this event, the children were instructed to place the puppets back on the sticks. Their searching
times for puppets were analyzed in order to assess whether they expected the one-to-one pairing
between puppets and sticks to remain as it initially was, or to change as a result of the
transformation applied.

The results show that Subset-Knowers do not interpret one-to-one pairings in line with the relation
of equal numerosity, but rather in line with a different relation on collections: the *relation of identity*.
The relation of identity holds between a collection and itself, independently of the spatial position of
the items comprising it. It is disrupted if novel items are introduced in the collection and/or some of
the items initially forming the collection are removed – including when some items are substituted
with other individuals. Accordingly, children expected one-to-one pairings to be preserved over
displacements applied to one of the collections. In contrast, when a transformation affecting the
identity of the items was applied (adjunction, removal, or substitution), they did not know whether
the one-to-one pairing would be preserved or altered.
In sum, studies using the three facets of Numerical Equality – “same number” statements, number words, one-to-one pairings – all failed to provide conclusive evidence that Subset-Knowers represent the relation of equal numerosity. These studies however establish that Subset-Knowers can represent another relation defined over collections, the relation of identity.

Could older children still rely on the relation of identity to interpret situations that ought to be understood in terms of equal numerosity? Strikingly, very few studies have attempted to contrast the relations of equal numerosity vs. identity, leaving the question largely open. For instance, to our knowledge research in the Piagetian tradition never applied substitutions - the crucial type of transformation to distinguish between the relation of identity and the relation of equal numerosity.

As one exception, one study found that 5-year-olds are not limited to the relation of identity in their interpretation of number words [24]. Participants to this study correctly appreciated the impact of a whole range of transformations on number words: they judged that a number word ceases to apply to a collection when an item is removed or adjoined, but applies again if a removed item is returned to the collection, or, crucially, if it is replaced by another item. Children applied these rules not only to number words they could count to, but also to number words out of their counting reach. At that age, thus, children appear to<sup>d</sup> (1) be able to represent the relation of equal numerosity, and to (2) understand that this relation determines the meaning of number words (top right blue arrow in Figure 1).

How do children arrive at this state of knowledge? As reported above, around the age of 3½ years, children become able to give any number of items in the Give-a-Number task, provided that the number requested is in their count list – they become “CP-Knowers”. Does children’s interpretation of number words align with the relation of equal numerosity at that moment? Are children able to

<footnote><sup>d</sup> The data in [24] is extremely clear: children succeeded at all the questions with over 75% accuracy. To understand why we nonetheless introduced a nuance in this conclusion, see Section V.</footnote>
represent the relation of equal numerosity innately, with the CP-Knower transition enabling them to
associate this relation with number words? Or, in reverse, do they acquire the relation of equal
numerosity from their practice of counting? These questions were addressed in a study conducted in
the Bolivian Amazon, with Tsimané children [25]. The Tsimané language has number words and a
counting list, but counting is not a very frequent activity in the Tsimané culture. As a result, children's
exposure to counting varies greatly, as well as the age at which they reach the CP-Knower stage in
the Give-a-Number task [18]. The authors designed a task to probe Tsimané children's understanding
of numerical equality. They presented children with two collections paired one-to-one, checking that
participants agreed that the collections had the “same quantity” (as translated from Tsimané by the
authors). Then, they brought the collections into piles to break the visual one-to-one pairing, and
applied a transformation to one of the collections (adjoining one item, subtracting one item,
substituting one item with a new item, removing one item and placing it back; plus the two control
transformations of stirring the pile and doubling the number of items). After this event, children
were asked whether the two collections still had the “same quantity”. To place it within the
framework of Figure 1, this task tests whether children interpret the Tsimané expression “same
quantity” and/or one-to-one pairings in line with the relation of equal numerosity (top left and
bottom blue arrows; a familiarization phase explicitly trained children to associate the phrase “same
quantity” with collections paired one-to-one).

The study brought two main findings. First, success at the numerical equality task and at Give-a-
Number (also administered to the same participants) was strongly related, such that CP-Knowers
reached higher levels of accuracy on numerical equality than Subset-Knowers, irrespective of age.
Thus, the children who understand counting (CP-Knowers by Give-a-Number) also tend to better
understand that the phrase “same quantity” and/or one-to-one pairings map onto the relation of
equal numerosity. Second, despite this correlation, there was no strict relation of implication
between the two achievements tested: A few Subset-Knowers were flawless on the numerical
equality task<sup>e</sup>, and reversely, some CP-Knowers made errors that could not be attributed to mere
distraction (for instance, they erred on the same transformations that Subset-Knowers tended to err
on, and did not make any errors on the two control transformations). These findings suggest that
understanding the relation of equal numerosity, just like understanding counting as a quantification
tool, is a learning achievement. They also indicate that, while counting can be a ladder on which to
build an understanding of numerical equality, it does not always lead to this knowledge.

IV. Do children understand the interrelations between the three facets of Numerical Equality –
“same number”, numeral labels, and one-to-one pairings?

We now turn to studies testing another aspect of the Numerical Equality complex: The interrelations
that exist between its three facets, i.e. “same number” statements, numeral labels, and one-to-one
pairings between collections (black arrows in Figure 1).

In line with the conclusion of the previous section, studies have revealed qualitative differences
between Subset-Knowers and CP-Knowers’ understanding of these interrelations. In a first study [26],
children were asked to reproduce the “number” of items in a target collection without counting. CP-
Knowers matched the exact numerosity of the target collection when they could rely on one-to-one
pairing as a strategy, while Subset-Knowers invariably produced approximate matches. This suggests
that CP-Knowers but not Subset-Knowers understand that one-to-one pairing produces collections
with the “same number” of items (bottom left black arrow in Figure 1). Relatedly, children’s
understanding of the relation between one-to-one pairings and number words (bottom right black
arrow in Figure 1) was assessed in several studies [19, 27, 28, LeCorre, this volume], all yielding

<footnote><sup>e</sup> In this volume, however, Barner [Barner, this volume] argues that the Subset-Knowers tested in this study may have actually reasoned on approximate quantities, some of them reaching ceiling performance only by chance. The Tsimané findings are thus compatible with the idea that children need to first reach the CP-Knower stage in Give-a-Number before they can interpret one-to-one pairings or “same quantity” statements in terms of the relation of equal numerosity.</footnote>
similar findings. For instance, Le et al. [27] presented vignettes where two groups of animals (rabbits, lions) entered a school while leaving their hats at the door, forming two distinct lines of hats.

Children were asked to estimate the (non-visible) number of animals (e.g. rabbits) using the (visible) hats, and the experimenters recorded whether they counted the hats belonging to the right animals - and only those hats. Only CP-Knowers succeeded, albeit with moderate levels of performance; Subset-Knowers were strictly at chance.

CP-Knowers thus start relating one-to-one pairings with “same number” and with number words: they identify one-to-one pairing as a good strategy to produce collections that have the “same number” of elements, and they understand that collections should be labeled with the same number word when they are perfectly paired one-to-one. In addition, preliminary findings reported in this volume [LeCorre, this volume] indicate that CP-Knowers can predict whether two collections can be paired one-to-one based on their number word labels.

Yet, other findings indicate that preschool children's understanding of the interrelations between one-to-one pairings and the other facets of Numerical Equality are far from complete: they do not understand that collections can be said to have the “same number” or should be associated with the same number words *only* when they can be paired one-to-one. Following up on their first finding, Le et al. [27] designed a second task where they showed CP-Knowers (aged 3-5 years) vignettes featuring just one kind of animal (rabbits) holding two types of items (e.g., hats and water bottles).

One type of item was perfectly paired one-to-one with the animals (i.e., each rabbit had a hat), while the pairing was off for the other type of item (i.e., one or several rabbits did not have a water bottle). These relations were emphasized to ensure that children had perfectly encoded the situation – and then the rabbits disappeared in a school building. As in the previous task, CP-Knowers spontaneously counted items to infer the number of (now invisible) rabbits, but when the distractors were off by just one item, their choices between hats and water bottles were at chance level. They only turned to the items paired one-to-one with animals when the number of distractors was markedly different.
At this age, CP-Knowers thus take number words to extend across collections that are *approximately* equal in numerosity (and they take counting as a method to arrive at those approximate numeral labels). Children's understanding of the interrelations between one-to-one pairings and number words is thus far from complete as they undergo the CP-Knower transition.

Based on findings from conservation tasks, these difficulties appear to continue throughout the preschool years. Indeed, if a child knows that two collections that can be paired one-to-one are *guaranteed* to have the “same number”, they should judge that two collections once paired one-to-one will still have the “same number” after their items are displaced. Yet, following a displacement, many 4- and 5-year-olds reply that the collection that now takes more space has more. Some children (classified as “intermediate” stage of knowledge by Piaget, typically 5 years of age) maintain at the same time that one collection has more and that the two collections can be paired one-to-one [9]. For 4- and 5-year-olds, one-to-one pairing may thus constitute just one piece of evidence to decide whether two collections have the “same number”, to be corroborated or invalidated by further evidence – for instance, visual evidence that two lines of items are equal or unequal in length. In line with this suggestion, kindergarteners (5-6 years) can oscillate between “Conserver” and “non-Conserver” answers depending on the size of the displacement operated by an experimenter [29]: they agree that two collections once paired one-to-one still have the “same number” for small displacements, but state that the collections are “different in number” for large displacements, as if weighting different bits of evidence varying in strength.

Unlike one-to-one pairings, children may realize early on that number words provide strong evidence to decide whether two collections have the “same number” of items (top black arrow in Figure 1). To our knowledge, no study has tested children's understanding of the interrelations between “same number” statements and number words directly. Such studies should test whether children judge that two collections that are labeled with the same number word should be said to have the “same number” of items, or reversely, that two collections said to have the “same number” of items should
be labeled with the same number word. Interestingly, though, when (nowadays) kindergarteners are
administered conservation tasks and asked to justify their answer, many of them attempt to count
the collections, as if counting constituted the ultimate proof to decide whether two collections have
the “same number” of items [Viarouge, personal observation]. On the basis of these observations, it
seems plausible that children integrate “same number” statements and number words quite early,
but that the connection with one-to-one pairings is made only much later.

V. Conclusion

We propose a new theoretical framework to gauge children’s understanding of numerical equality. In this framework, numerical equality is analyzed as a complex, “Numerical Equality”, with a central relation defined on collections, the relation of equal numerosity, interfaced with three facets: “same number” statements, numeral labels, and one-to-one pairings. Guided by this framework, we reviewed the experimental literature to answer three questions: First, when are children able to represent the relation of equal numerosity itself? Second, when do children start interpreting the facets of Numerical Equality in terms of this relation? Third, when do children understand that these three facets are interrelated, such that a statement made about one facet implies conclusions about the other two?

In order to assess how children interpret the three facets of Numerical Equality, we looked for
experiments testing how they think transformations affect “same number” statements, numeral
labels, and one-to-one pairings – an approach building on Heck [1]. As the literature currently stands,
no experiments have yielded any evidence that the relation of equal numerosity guides children’s
behavior before the age of 5 years, when it is reflected in their interpretation of number words.
There is, moreover, clear evidence that children do *not* interpret one-to-one pairings in terms of the
relation of equal numerosity at the age of 2½ years, raising the possibility that children may initially
lack the ability to represent this relation. In fact, many preschool-age children have not yet fully
integrated one-to-one pairings with the other constituents of Numerical Equality, failing for instance
to appreciate that two collections can be said to have the “same number” of items only when their
items can be paired one-to-one.

Numerical Equality thus seems to be first understood from the lens of verbal symbols, with one-to-
one pairings engrafted to the complex only later<sup>f</sup>. Moreover, access to a counting procedure appears
to play a key role in acquiring full-blown numerical equality. Indeed, US children start discovering the
links between one-to-one pairings and number words only *after* they have understood that number
words derive their meaning from counting (i.e., after the CP-Knower transition); and in a non-
industrialized society where learning number words and counting extends throughout childhood,
understanding one-to-one pairings and/or “same quantity” statements correlates with
understanding counting.

Our review also identifies many open questions and unresolved debates. How do children interpret
number words *before* the CP-Knower transition? As of now, relevant studies have yielded mixed
results, still in need of a reconciliation. It is thus possible that children have access to the relation of
equal numerosity much earlier than the age of 5 years currently identified in the literature, and
before they understand the numerical meaning of counting. When exactly do children come to
interpret “same number” statements and one-to-one pairings in line with the relation of equal
numerosity? When do they discover that the three facets of Numerical Equality coincide perfectly?
Are the constituents of the Numerical Equality complex learned in a piecemeal and protracted
manner, or do learnings cascade and accelerate once a few links are in place?

<footnote>f These findings are compatible with two interpretations. On one hand, children may acquire the relation of equal numerosity only as they learn the principles that govern number words usage, effectively creating a novel concept transcending their earlier resources. Alternatively, children may possess the relation of equal numerosity as part of their innate conceptual repertoire, but find it easier to discover that this relation applies to number words than to one-to-one pairings [for a related proposal see Barner, this volume].</footnote>
Besides these questions for experimenters, our review also raises pressing theoretical questions. Our approach was built on Heck [1], who identified three principles characterizing the relation of equal numerosity. Heck established that (a finite version of) Peano arithmetic can be derived from these three principles, given appropriate definitions of the arithmetical notions of zero and predecessor. Psychologists, however, have worked on different principles. How do the psychologists' principles relate to the relation of equal numerosity, and to the concept of natural number<sup>g</sup>? What other epistemological ingredients may be needed for a child to develop the concept of natural number out of these principles? In particular, one study found that 5-year-olds correctly appreciate the impact of adjunctions, removals and substitutions on number words [24]. Do these transformations uniquely characterize the relation of equal numerosity, as the transformations in Heck's principles do, or would they be compatible with other relations?

In conclusion, we urge the field to start thinking of numerical equality not as an indissociable whole, but as a constellation of abilities, resulting from separate learning achievements. We show that this approach helps make sense of seemingly contradictory results in the literature and raises novel questions. We hope that our framework will become a fruitful guide to future investigations of children's acquisition of numerical equality and of the concept of natural number.

## Acknowledgments

We thank Eric Snyder, Joonkoo Park, Daniel Sutherland and T.R. Virgil for thoughtful comments. For the purpose of open access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission.

<footnote><sup>g</sup> For instance, one study [23] introduced three principles on displacements, adjunctions/removals, and substitutions. The authors showed that these principles are *necessary* to characterize the relation of equal numerosity: if either of them is waived, the other two are compatible with different relations on collections. It is unclear, however, whether these principles are *sufficient* and uniquely describe the relation of equal numerosity.</footnote>
References

[1] R.G. Heck. Cardinality, Counting, and Equinumerosity. *Notre Dame Journal of Formal Logic*, 41(3):187–209. Jul. 2000. https://doi.org/10.1305/ndjfl/1038336841

[2] G. Frege. *The foundations of arithmetic: A logico-mathematical enquiry into the concept of number*. Northwestern University Press, Evanston, 1984/1980. J.L. Austin, translator.

[3] G. Frege. Review of E.G. Husserl, *Philosophie der Arithmetik*. I. In B. McGuinness, editor, *Collected Papers on Mathematics, Logic and Philosophy*. Basil Blackwell, Oxford, 1984.

[4] M. Pantsar and B. Assadian. *Where Does Cardinality Come From?* *Review of Philosophy and Psychology*, 16:429–451. Aug. 2024. https://doi.org/10.1007/s13164-024-00746-9

[5] L. Benoit, H. Lehalle, M. Molina, C. Tijus, and F. Jouen. Young children's mapping between arrays, number words, and digits. *Cognition*, 129(1):95–101. Oct. 2013. https://doi.org/10.1016/j.cognition.2013.06.005

[6] C. Jiménez Lira, M. Carver, H. Douglas, and J.-A. LeFevre. The integration of symbolic and non-symbolic representations of exact quantity in preschool children. *Cognition*, 166:382–397. Sep. 2017. https://doi.org/10.1016/j.cognition.2017.05.033

[7] D.C. Hyde, Y. Mou, I. Berteletti, E.S. Spelke, S. Dehaene, et al. Testing the role of symbols in preschool numeracy: An experimental computer-based intervention study. *PLoS ONE*, 16(11):e0259775. Nov. 2021. https://doi.org/10.1371/journal.pone.0259775

[8] D. Barner, G. Alvarez, J. Sullivan, N. Brooks, M. Srinivasan, et al. Learning Mathematics in a Visuospatial Format: A Randomized, Controlled Trial of Mental Abacus Instruction. *Child Development*, 87(4):1146–1158. Jul. 2016. https://doi.org/10.1111/cdev.12515

[9] J. Piaget. *The Child's Conception of Number*. Norton, 1965.

[10] R. Gelman. The nature and development of early number concepts. In H.W. Reese, editor, *Advances in Child Development*, volume 3, pages 115–167. Academic Press, New York, 1972.
[11] O. Houdé. Inhibition and cognitive development: object, number, categorization, and reasoning. *Cognitive Development*, 15(1):63–73. Jan. 2000. https://doi.org/10.1016/S0885-2014(00)00015-0

[12] J. McGarrigle and M. Donaldson. Conservation accidents. *Cognition*, 3:341–350. 1974.
https://doi.org/10.1016/0010-0277(74)90003-1

[13] J. Mehler and T.G. Bever. Cognitive Capacity of Very Young Children. *Science*, 158(3797):141–142. Oct. 1967. https://doi.org/10.1126/science.158.3797.141

[14] S. Carey. *The Origin of Concepts*. Oxford University Press, New York, NY, 2009.

[15] K.F. Condry and E.S. Spelke. The development of language and abstract concepts: the case of natural number. *Journal of Experimental Psychology: General*, 137(1):22–38. Feb. 2008.
https://doi.org/10.1037/0096-3445.137.1.22

[16] M. Le Corre, G. Van de Walle, E.M. Brannon, and S. Carey. Re-visiting the competence/performance debate in the acquisition of the counting principles. *Cognitive Psychology*, 52(2):130–69. Mar. 2006. https://doi.org/10.1016/j.cogpsych.2005.07.002

[17] E. Marchand, J.T. Lovelett, K. Kendro, and D. Barner. Assessing the knower-level framework: How reliable is the Give-a-Number task? *Cognition*, 222:104998. May 2022.
https://doi.org/10.1016/j.cognition.2021.104998

[18] S.T. Piantadosi, J. Jara-Ettinger, and E. Gibson. Children's learning of number words in an indigenous farming-foraging group. *Developmental Science*, 17(4):553–563. Jul. 2014.
https://doi.org/10.1111/desc.12078

[19] B.W. Sarnecka and S.A. Gelman. Six does not just mean a lot: preschoolers see number words as specific. *Cognition*, 92(3):329–52. Jul. 2004. https://doi.org/10.1016/j.cognition.2003.10.001

[20] K. Wynn. Children's understanding of counting. *Cognition*, 36(2):155–93. Aug. 1990.
https://doi.org/10.1016/0010-0277(90)90003-3

[21] K. Wynn. Children's acquisition of the number words and the counting system. *Cognitive Psychology*, 24(2):220–251. Apr. 1992. https://doi.org/10.1016/0010-0285(92)90008-P
[22] N. Brooks, J. Audet, and D. Barner. Pragmatic Inference, Not Semantic Competence, Guides 3-Year-Olds' Interpretation of Unknown Number Words. *Developmental Psychology*, 49(6):1066-1075. Jun. 2013. https://doi.org/10.1037/a0029384

[23] V. Izard, A. Streri, and E.S. Spelke. Toward exact number: Young children use one-to-one correspondence to measure set identity but not numerical equality. *Cognitive Psychology*, 72:27-53. Jun. 2014. https://doi.org/10.1016/j.cogpsych.2014.01.004

[24] J.S. Lipton and E.S. Spelke. Preschool children master the logic of number word meanings. *Cognition*, 98(3):B57-66. Jan. 2006. https://doi.org/10.1016/j.cognition.2004.09.013

[25] J. Jara-Ettinger, S. Piantadosi, E.S. Spelke, R. Levy, and E. Gibson. Mastery of the logic of natural numbers is not the result of mastery of counting: evidence from late counters. *Developmental Science*, 20(6):e12459. Nov. 2017. https://doi.org/10.1111/desc.12459

[26] R.M. Schneider, E. Brockbank, R. Feiman, and D. Barner. Counting and the ontogenetic origins of exact equality. *Cognition*, 218:104952. Jan. 2022. https://doi.org/10.1016/j.cognition.2021.104952

[27] K.N. Le, R.M. Schneider, and D. Barner. The development of cardinal extension: From counting to exact equality. *Developmental Psychology*, 61(6):1180-1195. Jan. 2025. https://doi.org/10.1037/dev0001922

[28] B.W. Sarnecka and C.E. Wright. The idea of an exact number: children's understanding of cardinality and equinumerosity. *Cognitive Science*, 37(8):1493-506. Dec. 2013. https://doi.org/10.1111/cogs.12043

[29] A. Viarouge, O. Houdé, and G. Borst. The progressive 6-year-old conserver: Numerical saliency and sensitivity as core mechanisms of numerical abstraction in a Piaget-like estimation task. *Cognition*, 190:137-142. Sep. 2019. https://doi.org/10.1016/j.cognition.2019.05.005