Elsevier

Cognition

Volume 194, January 2020, 104104
Cognition

The knowledge of the preceding number reveals a mature understanding of the number sequence

https://doi.org/10.1016/j.cognition.2019.104104Get rights and content

Highlights

  • CP-knowers master the successor (n+1) but not the predecessor (n-1) knowledge.

  • Not all CP-knowers display a linear mapping between the ANS and the counting list.

  • Predecessor knowledge and visual order of numbers related to number comparison skills.

  • The knowledge of the number sequence relates to symbolic numerical magnitude.

Abstract

There is an ongoing debate concerning how numbers acquire numerical meaning. On the one hand, it has been argued that symbols acquire meaning via a mapping to external numerosities as represented by the approximate number system (ANS). On the other hand, it has been proposed that the initial mapping of small numerosities to the corresponding number words and the knowledge of the properties of counting list, especially the order relation between symbols, lead to the understanding of the exact numerical magnitude associated with numerical symbols. In the present study, we directly compared these two hypotheses in a group of preschool children who could proficiently count (most of the children were cardinal principle knowers). We used a numerosity estimation task to assess whether children have created a mapping between the ANS and the counting list (i.e., ANS-to-word mapping). Children also completed a direction task to assess their knowledge of the directional property of the counting list. That is, adding one item to a set leads to he next number word in the sequence (i.e., successor knowledge) whereas removing one item leads to the preceding number word (i.e., predecessor knowledge). Similarly, we used a visual order task to assess the knowledge that successive and preceding numbers occupy specific spatial positions on the visual number line (i.e., preceding: [?], [13], [14]; successive: [12], [13], [?]). Finally, children's performance in comparing the magnitude of number words and Arabic numbers indexed the knowledge of exact symbolic numerical magnitude. Approximately half of the children in our sample have created a mapping between the ANS and the counting list. Most of the children mastered the successor knowledge whereas few of them could master the predecessor knowledge. Children revealed a strong tendency to respond with the successive number in the counting list even when an item was removed from a set or the name of the preceding number on the number line was asked. Crucially, we found evidence that both the mastering of the predecessor knowledge and the ability to name the preceding number in the number line relate to the performance in number comparison tasks. Conversely, there was moderate/anecdotal evidence for a relation between the ANS-to-word mapping and number comparison skills. Non-rote access to the number sequence relates to knowledge of the exact magnitude associated with numerical symbols, beyond the mastering of the cardinality principle and domain-general factors.

Introduction

One of the fascinating questions in the development of early numerical skills is understanding how children learn the numerical meaning of number words and Arabic digits, a process known as symbol-grounding (Harnad, 1990; Leibovich & Ansari, 2016). Children learn to enumerate objects in their environment by respecting three counting principles: each item in a set must be associated with only one number word (one-to-one correspondence), the counting list must be recited in the established order (stable-order principle), and the last recited number word denotes the numerosity of the set (cardinality principle; Gelman & Gallistel, 1978). The development of counting skills is widely assessed using the Give-a-number task (GaN task), in which children are presented with a large set of items (e.g., 15 tokens) and asked to give different numerical quantities. Children sequentially learn the meaning of number words from one to four and correctly give them when requested. The limit of four corresponds to the number of objects children can keep in visual short-term memory (Feigenson, Dehaene, & Spelke, 2004; Knops, Piazza, Sengupta, Eger, & Melcher, 2014; Piazza, 2010; Piazza, Fumarola, Chinello, & Melcher, 2011). At this stage, children are defined as subset-knowers because their cardinal knowledge is still limited to the first number words (Le Corre & Carey, 2007; Sarnecka & Carey, 2008). Then, children crucially understand that next counted object corresponds to the next number word in the counting list (i.e., successor function; Carey, 2004). Children become cardinal-principle knowers (CP-knowers), and they can correctly count both small and large numerical sets. The acquisition of the cardinality principle represents a crucial milestone in the development of early numerical skills: CP-knowers have understood how counting represents numbers (Gelman & Gallistel, 1978; Sarnecka, 2015). An early mastering of the cardinality principle enables children to understand the numerical meaning of numbers and constitutes an advantage for acquiring later mathematical skills (Chu, vanMarle, Rouder, & Geary, 2018; Geary et al., 2018).

The present study further explores the symbolic numerical knowledge in preschool children who were (mainly) classified as CP-knowers. In particular, we examined preschoolers’ understanding of the exact numerical magnitude that is associated with number words and Arabic numbers and its relation with other numerical skills.

The mapping between numerical quantities and the counting list is usually assessed by asking individuals to estimate the number of items in briefly presented visual sets (Izard & Dehaene, 2008; Reeve, Reynolds, Humberstone, & Butterworth, 2012). The quick presentation of the target numerosities prevents the use of serial counting and allows to assess a mapping that is not the byproduct of a procedure. CP-knowers can precisely estimate small numerosities (<3-4 items), an effect called subitizing, which depends on the parallel individuation of distinct items in the set via the Object Tracking System (OTS; Kaufman et al., 1949; Le Corre, 2014; Odic, Le Corre, & Halberda, 2015; Revkin, Piazza, Izard, Cohen, & Dehaene, 2008; Sella, Lucangeli, & Zorzi, 2018; Trick & Pylyshyn, 1994). Large numerical quantities, instead, are processed via the Approximate Number System (ANS), in which each numerosity is represented as a Gaussian curve of activation whose width increases with numerical magnitude (Feigenson et al., 2004; Piazza, 2010). Only some CP-knowers, called mappers (i.e., CP-mappers), have established a linear mapping between the ANS and the counting list (i.e., ANS-to-word mapping) whereas some CP-knowers still lack such mapping (i.e., CP-non-mappers). Linearity is usually operationalised as a positive slope when regressing the estimates on target numerosities (Crollen, Castronovo, & Seron, 2011; Izard & Dehaene, 2008). The presence of a linear relation between estimates and target numerosities marks the understanding of the later-greater principle, that is, later number words in the counting list correspond to large numerical quantities (Le Corre & Carey, 2007; Le Corre, 2014). Children who master the cardinality principle should know that the late number words in the counting list represent large numerosities. However, this is not the case. Children acquire the later-greater principle after the cardinality principle (Le Corre & Carey, 2007). Accordingly, only CP-mappers provide estimates that increase with large numerical quantities whereas CP-non-mappers usually provide the same estimate (e.g., “five”) for large target numerosities (Odic et al., 2015). Crucially, CP-mappers have a better understanding of the symbolic system as indexed by their superior performance in comparing number words (Le Corre, 2014; but see Sella, Lucangeli, & Zorzi, 2018).

Other studies have suggested that the ANS-to-word mapping (and more generally the ANS) has a marginal role in the development of symbolic numerical skills (Carey, 2001; Lyons & Beilock, 2011; Lyons, Ansari, & Beilock, 2012; Lyons, Price, Vaessen, Blomert, & Ansari, 2014). Conversely, number words and Arabic digits acquire their numerical meaning when children learn their reciprocal relation, in a symbol-to-symbol association (Reynvoet & Sasanguie, 2016). In this vein, the knowledge and the experience in accessing the numerical sequence play a crucial role in the symbol-grounding. The numerical sequence can be represented in its verbal and visuospatial format, respectively, the counting list (i.e., “one”, “two”, “three”, “four”, …) and the number line (i.e., 1-2-3-4-…).

Children early memorise the counting list and can recite it forward. However, this does not imply that children have grasped the directional property of the counting list, that is, adding one item to a set leads to the next number word (i.e., n+1) whereas removing one leads to the previous number word (i.e., n-1). Sarnecka and Carey (2008) developed the unit task, whereby children watched the experimenter adding one or two items (n+1 and n+2) to a box already containing four or five items. Children had to say the number of items in the box after the manipulation by choosing between the n+1 and n+2 answer. Similarly, in the direction task, children were presented with two sets both containing five objects, then one object was moved to one set to the other and children were requested to indicate which set had now four or six objects. CP-knowers performed the unit and direction task above the chance level as they grasp the directional property of the counting list whereas subset-knowers displayed poor performance. However, not all CP-knowers could adequately perform the two tasks (see also Davidson, Eng, & Barner, 2012). Surprisingly, all CP-knowers can proficiently add up items to a set when performing the GaN task, but not all of them can perform the same adding-one transformation in the unit task. This discrepancy might emerge from the fact some CP-knowers are implementing a rote behaviour in the GaN task without really grasping the meaning of the manipulation. In this light, the implementation of a task capable of assessing the knowledge of the successive and preceding number across the counting list without requiring any motor routine would be a perfect tool to measure the knowledge of the directional property of the counting list in preschool children. Sella and colleagues (Sella, Lucangeli, Cohen Kadosh, & Zorzi, in press) showed children an opaque box and told them that inside there were some felt strawberries. Then, a strawberry was either added or removed from the box, and children had to tell the number of strawberries in the box after the manipulation. The starting number of strawberries in the box varied from two to eight to explore the ability to perform n+1 and n-1 transformations in the entire numerical interval from one to nine. CP-knowers displayed an accurate performance when one item was added (n+1) to the box as an index of their mastering of the successor knowledge. Conversely, their accuracy in performing the n-1 transformation was high with small starting numbers and decreased with large starting numbers. CP-knowers display an immature mastering of the predecessor knowledge, which is successfully applied to the first, but not the later, number words in the counting list. The performance in direction task correlated with the exact symbolic numerical knowledge, as measured using a number words comparison task, even when controlling for cardinality knowledge and memory capacity (Sella et al., in press).

The knowledge of the relation between numbers can also be assessed by measuring children’s familiarity with the number line, as done in the number-to-position task (Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010; Siegler & Opfer, 2003). In the task, children mark the position of target numbers onto a visual line entailing a numerical interval, usually from 1 to 10 for young children. Some CP-knowers, called spatial-mappers, display a linear ordering of target digits on the visual line whereas other CP-knowers, called non-spatial-mappers, tend to place all the target digits in the middle of the line or at the start or end points (Sella, Berteletti, Lucangeli, & Zorzi, 2017). CP-spatial-mappers can reproduce the spatial arrangement of numbers and demonstrate better knowledge of the exact symbolic magnitude, as measured by a higher accuracy in a digits comparison task compared to CP-non-spatial-mappers (Sella et al., 2017). To further investigate the spatial mapping of numbers in preschool children, Sella and colleagues have developed a computerised task to separately assess direction, ordinality and accuracy of the spatial mapping (DOS task; Sella et al., 2018b). In the DOS task, a digit (e.g., 2) is presented in the middle of a horizontal line and the child is requested to place an additional digit (e.g., 1) in two possible locations, one on the left side and one the right side compared to the centred digit. The positioning of the target digit immediately determines the direction of the mapping, whether the child prefers to map numbers with increasing magnitude from left-to-right (e.g., 1–2) or right-to-left (e.g., 2-1). Then, another digit (e.g., 3) is presented, and the child is requested to place it. The positioning of the last digit reveals whether children respect the spatial order of numbers (e.g., 1-2-3) or not (e.g., 1-3-2) while also measuring the precision of the mapping, especially whether children respect the equidistance between digits (e.g., 1-2-3 vs 1-2---3). Preschool children mainly mapped numbers from left-to-right (according to the canonical direction in Western society), and the accuracy in ordering digits decreased with large triplets (e.g., 7-8-9) whereas the precision of mapping remained stable across triplets. Crucially, only spatial order related to the knowledge of the magnitude associated with digits, as measured in a digit comparison task (Sella et al., 2018b; Sella et al., in press). Children who can place digits in the correct spatial order within the number line can also indicate the exact numerical magnitude they represent.

Taken together, the results of the above-mentioned studies suggest that children’s exact symbolic numerical knowledge might emerge from the conceptual understanding of the directional property of the counting list and the knowledge of the spatial ordinal relation between numbers within the number line. The former represents the understanding that adding or removing items from a set is respectively associated with moving forward and backwards on the counting list. The latter indicates the knowledge of the position each number occupies within the number line. The counting list and the number line are both structures, whereby the ordinal position of a number can inform about its relative magnitude compared to the other numbers in the sequence.

In the present study, we explored the ANS-to-word mapping and the knowledge of number sequence and how these components relate to the understanding of the magnitude associated with number words and Arabic numbers. On the one hand, the cardinality principle and the ANS-to-word mapping represent a symbol-quantity relation, in which numerical quantities are mapped to symbolic numbers. On the other hand, the knowledge of the number sequence, as indexed by the directional property of the counting list and the spatial arrangement of numbers, reflect a symbol-symbol relation as numbers acquire a numerical meaning in relation to each other.

We selected a group of preschool children, who were more likely to be proficient counters (CP-knowers), to explore the knowledge of the symbolic system beyond the acquisition of the cardinality principle. We also ensured that all preschoolers in our sample knew the meaning of “numerically more” and could recite the counting list sufficiently high to perform the numerosity estimation task. Children performed a numerosity estimation task to assess the linearity of their ANS-to-word mapping. We used a direction task to measure the knowledge of the successive (n+1) and preceding number (n-1) in the counting list (Sella et al., in press). Children were presented with a box containing a starting number of tennis table balls, then one ball was added or removed, and children had to indicate the number of balls inside the box after the transformation. Crucially, this task does not require the implementation of rote behaviour, as the serial counting in the Give-a-number task. Moreover, we used both small and large starting numerosities, which could differentiate the knowledge of successive and preceding numbers within and outside the limit of the working memory capacity (i.e, subitizing limit in the OTS). When the box only contains 2 balls, children could perform the n+1 and n-1 transformation by tracking the number of items whereas, children had to rely on their knowledge of the directional property when the box contained a larger number of balls. We assessed the knowledge of the number line using a visual order task, whereby children saw three squares, two of them containing consecutive numbers: one in the central and one in the right (e.g., [], [2], [3]) or left square (e.g., [1], [2], []). Children had to say the number that goes in the empty square. Compared to previously used tasks, the visual order task resembles the transformations of direction task by separately assessing what number comes after (+1) and before (-1) in the number line. Moreover, it does not require any fine motor skills, as in the number-to-position and DOS task, and the effect of lexical knowledge was removed by reading all the numbers to children. Children chose the larger between two number words and Arabic numbers to assess their knowledge of the numerical magnitude of numbers. The direct request to compare the numerical magnitude of symbols and the fact that the response cannot be achieved by implementing a behavioural routine, as in the GaN task, make the comparison tasks the perfect tools to evaluate the knowledge of the exact magnitude associated with numerical symbols. In the number comparison task, error rates and reaction times increase as a function of the numerical distance (i.e., distance effect) and the size of the numbers (i.e., size effect; Moyer & Landauer, 1967). These effects relate to a general ratio effect as the representation of numbers overlaps with increasing numerical magnitude, as in the ANS, which is supposed to be the underlying mechanism of both non-symbolic and symbolic number comparison (Dehaene, 1992; Piazza, 2010), even though alternative models for number comparison performance have been proposed (Krajcsi, Lengyel, & Kojouharova, 2018; Verguts, Fias, & Stevens, 2005; Zorzi & Butterworth, 1999). Moreover, the speed in comparing Arabic digits has been repeatedly associated with arithmetic skills (see Schneider et al., 2016, for a review) and has been found reduced in children with developmental dyscalculia (Rousselle & Noël, 2007), which makes the exploration of the same ability in preschool children relevant, even though based on accuracy rather than response time.

Most of the studies have focused the investigation of early numerical skills on the range from 1 to 10 in preschool children. Nevertheless, it should be considered that preschool children, especially CP-knowers, should in principle correctly associate numerical quantities with the corresponding number words in their counting list, which extends above ten. In this light, preschool children might already have preliminary knowledge of the numerical magnitude represented by numbers above ten (e.g., Gilmore, McCarthy, & Spelke, 2007). Therefore, we chose to present numbers below and above ten to have a better description of children’s numerical knowledge of the broader numerical sequence. We used the same target numerosities and numbers across tasks to ensure perfect comparability of the performances. Finally, we assessed verbal and visuospatial memory as domain-general control measures.

We expect some of the children in our sample to display a linear mapping between the ANS and the counting list (Le Corre & Carey, 2007; Le Corre, 2014; Sella et al., 2018a). Children should display better performance in the n+1 trials of the direction task compared to the n-1 as children are more familiar with the process of adding up items as in the GaN task compared to remove items from a set (Sella et al., in press). We expect the knowledge of the spatial arrangement of numbers to decrease with large triplets (Sella et al., 2018b). Finally, we specifically examine the relative contribution of the ANS-to-word mapping, the knowledge of the counting list and the number line on number comparison skills. Such direct contrast should inform whether the knowledge of the magnitude associated with number words and Arabic numbers relates to children’s knowledge of the numerical sequence or the ANS-to-word mapping.

Section snippets

Participants

One hundred and four preschool children from five schools located in northeastern Italy took part in the study after parents, or legal guardians gave their informed consent. Parents or legal guardians also filled in a questionnaire regarding the family background and demographic information (e.g., nationality, parents’ education). We excluded those children who: failed to correctly choose the larger set in the dots comparison task at least in ten out of twelve trials (n = 5); failed to enumerate

Results

We ran the statistical analyses using the free software R (R Core Team, 2016) along with the BayesFactor package (Morey & Rouder, 2015) using default priors for Bayesian analyses. We reported Bayes factors (BF10) expressing the probability of the data given H1 relative to H0 (i.e., values larger than 1 are in favour of H1 whereas values smaller than 1 are in favour of H0; Wagenmakers et al., 2017, 2016). When comparing regression models, we reported the Bayes factors (BF) as the ratio of BFs10

Discussion

A proficient counting represents the first step in a long process that leads children to a mature understanding of the numerical meaning of number words and Arabic numbers. In the present study, we explored how preschool children performed a variety of tasks, which specifically assessed the mastering of different numerical concepts beyond the cardinality principle. To this aim, we ensured that all children involved in the study could proficiently count, knew sufficiently well the numerical

Conclusions

Despite the mastering of the cardinality principle, only some of the children in our sample have established a mapping between the ANS and the counting list. The direction task revealed a discrepancy between the mastering of the successor and predecessor knowledge. Children frequently said the successive number in the counting list even when one item was removed. The tendency to name the successive number might have inflated children’s mastering of the successor knowledge and, more generally,

Acknowledgements

The authors wish to thank the children and their parents for participating in the present study, as well as Giulia Alfiero and Sara Zennaro for their help in collecting data.

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