On the relation between the Give-a-Number task and Working Memory Capacity in young children

Abstract Although in the literature there is a lively debate on the factors that affect performance on the Give-a-Number task and its development, the role of working memory remains largely under-investigated. In this preregistered exploratory study, connected with the larger cross-cultural project “ManyNumbers” (Abreu-Mendoza et al., 2024), we examine the relation between working memory capacity (WMC) and the Give-a-Number task, and the quantitative patterns of WMC development that might underpin Give-a-Number performance. The data will be analyzed both within languages and pooled over languages. Three specific predictions will be tested: (a) The transition from non-knower to subset-knower is associated with a WMC increase; (b) The transition from subset-knower to cardinality principle (CP)-knower or counter is associated with an additional WMC increase; (c) Consistent with neo-Piagetian developmental theories (cfr. Morra et al., 2008), and in particular using Case’s (1985) definitions of WMC stages, the minimum WMC required for achieving the stages of subset-knower and counter are three interrelational schemes and one dimensional scheme, respectively. Furthermore, we explore whether transitions from a subset-knower level to the following subset-knower level are also associated with increases in WMC. Because language ability and inhibitory control could also predict Give-a-Number performance, we shall also test by means of regression analyses whether WMC accounts for performance improvement in the Give-a-Number above and beyond inhibition and vocabulary, or whether it interacts with them. Introduction The effect of early numeracy skills on math achievement in childhood and adolescence is well documented (e.g., Watts et al., 2014). The Give-a-Number task (Schaffer et al., 1974; Wynn, 1990; Sella et al., 2021; Marchand et al., 2022) is a widely adopted test to assess young children’s knowledge of number words meaning, and it is predictive of later math cognition (e.g., of first graders’ strategic skills in addition problems; Chu et al., 2018). Different versions of the Give-a-Number task exist. On each trial of the most common versions, the child is required to pick a specified number of objects from a larger set and subsequently count them to check that they have picked the correct number. However, the cognitive abilities and processes that underlie children’s performance on this task are not yet fully understood. A large, international, multi-lab project called ManyNumbers (Abreu-Mendoza et al., 2024) is currently in progress to investigate children’s early number knowledge, including the Give-a-Number as one of the main tasks. The present registered report presents an “exploratory project” related to the larger ManyNumbers project, aiming to examine the relation between performance on the Give-a-Number task and working memory capacity development. Although in the literature there is a lively debate on the factors that may affect Give-a-Number performance and its development, it appears that the role of working memory has not yet been adequately investigated, except for some correlational evidence reported by Gordon et al. (2021) and Li et al. (2024). Some studies highlighted that the early acquisition of mathematical abilities is linked to domain-general processes that enable children to properly use and manipulate their knowledge (e.g., Passolunghi & Lanfranchi, 2012; Traverso et al., 2021). Among the various domain-general processes found to be associated with or predictive of math achievement, working memory capacity (WMC) appears to be particularly important (e.g., Morosini et al., 2025; Morra et al., 2019; Traverso et al., 2021) because, as suggested by neo-Piagetian theories (e.g., Case, 1992; Pascual-Leone & Johnson, 2021; see also Morra et al., 2008), a certain size of WMC is required for the child’s construction of novel, complex cognitive structures that mark a developmental advance (including, for instance, the concepts that mark advances in the child’s comprehension of number). In line with this theoretical approach, Morra et al. (2019) examined the development of the natural number “central conceptual structure” (CCS; see Case, 1992) starting from the preschool years. According to Case (1992), central conceptual structures are networks of declarative and procedural knowledge (i.e., figurative and operative schemes, following Piagetian terminology), which organize the individuals’ conceptual knowledge in a broad domain and also affect their understanding of other, related domains. Case (1992) specifically considered three CCS: narrative, quantitative, and spatial. Central conceptual structures undergo development in childhood and adolescence; regarding in particular the quantitative CCS, it would go through a first stage based on natural numbers and a second stage that includes rational number knowledge – each stage divided in a few substages or levels. The development of a CCS obviously depends much on experience and learning opportunities; however, the degree of complexity of concepts and their systematicity would be heavily constrained by maturation of WMC, because a child could hardly acquire a concept whose acquisition requires integrating more information than his/her working memory can hold and manage (Case, 1992; Case & Okamoto, 1996). Consistent with Case’s theory, Morra et al. (2019) found that children’s understanding of whole number advances by one developmental level per additional unit of WMC. However, the studies carried out by Case and colleagues (e.g., Griffin, 2009; Okamoto & Case, 1996) and followed up recently by Morra et al. (2019) have considered the development of the whole number conceptual structure starting from a point where children (approximately 4 years old) can count. The relation between WMC and very early number knowledge (such as the early learning of number words and counting), from infancy to the preschool years, to the best of our knowledge is definitely under-investigated. In particular, the Give-a-Number task assesses the acquisition of the first number-words meaning and the transition to counting. In line with the literature on kindergartners and schoolchildren briefly reviewed in the foregoing paragraphs, we hypothesize that already in very young children WMC could play an important role in the early learning of number words and counting. For this reason, we propose to investigate the relationship between a battery of different WMC tasks and children’s performance in the Give-a-Number task. Children’s knowledge of number words’ meaning, cardinality and counting is often analysed in terms of a sequence of stages or developmental steps (see Wynn, 1990; 1992; Wege et al., 2024). Children would acquire first the meaning of “one”, then “two”, and later “three”; children at these stages are labelled as one-knowers, two-knowers, and three-knowers. According to Wynn (1990, 1992), acquisition of the meaning of “four” is followed very soon after by the acquisition of the cardinality principle and counting – although a few studies have reported the finding of some children who know also the meaning of “five” even though they do not yet use counting (see Sella et al., 2021; Wege et al., 2024; Krajcsi & Reynvoet, 2024 for reviews). We speculate that acquiring the meaning of the first number words could depend on coordinating several pieces of information: the word itself (e.g., “two”), a mental image of a set (e.g., a small group of toy fish, if the request is “two fish”), and an object-tracking procedure in which each object is individually indexed (which would enable the child to represent a set of two). It is often claimed (e.g., Carey, 2004) that on average it takes a few months for a one-knower to become a two-knower, a few more months to become a three-knower, and yet a few additional months to become a four-knower or a counter. The available evidence on this temporal pattern is not fully consistent. We have identified several studies that report the mean age or the age range of children at each stage (Almoammer et al., 2013; Ceylan & Aslan, 2018; Le Corre et al., 2006, 2016; Meyer et al., 2020; Nikoloska, 2009; Rousselle & Vossius, 2021; Sarnecka & Carey, 2008; Sarnecka et al., 2015; Sarnecka & Lee, 2009; Shusterman et al., 2016; Slusser et al., 2013; Spaepen et al., 2018; Wynn, 1990, 1992), some of which (Rousselle & Vossius, 2021; Shusterman et al., 2016; Wynn, 1992) report also on longitudinal data. Most of these studies seem to indicate that progress from non-knower to counter through various subset-knower stages takes extended time, i.e. one year or more, although some of them seem to indicate little age difference from one- to three-knowers, or from two- to four-knowers. These inconsistencies could be due to random variation (also because some studies did not use large samples) or to subtle differences across studies (e.g., the participants’ language, the numbers tested, the titrated or non-titrated procedure, the form of the questions, and other methodological aspects discussed by Wege et al., 2024 and Abreu-Mendoza et al., 2024). Overall, however, it seems that transition from one stage to the next typically requires about three or four months on average. The transitions from non-knower to subset-knower and from subset-knower to counter are particularly important. Becoming a subset knower represents the beginning of a child’s acquisition of symbolic number, and becoming a counter is another major qualitative leap, because it entails understanding the cardinality principle and paves the way to mental representation of exact, large numbers (e.g., Le Corre et al., 2006; Sarnecka et al., 2015). Different accounts of these transitions have been proposed in the literature. According to a nativist hypothesis, innate principles would guide the child’s acquisition of the meaning of number words and the counting procedure

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