Children’s understandings of counting: Detection of errors and pseudoerrors by kindergarten and primary school children

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Abstract

In this study, the development of comprehension of essential and nonessential aspects of counting is examined in children ranging from 5 to 8 years of age. Essential aspects, such as logical rules, and nonessential aspects, including conventional rules, were studied. To address this, we created a computer program in which children watched counting errors (abstraction and order irrelevance errors) and pseudoerrors (with and without cardinal value errors) occurring during a detection task. The children judged whether the characters had counted the items correctly and were asked to justify their responses. In general, our data show that performance improved substantially with age in terms of both error and pseudoerror detection; furthermore, performance was better with regard to errors than to pseudoerrors as well as on pseudoerror tasks with cardinal values versus those without cardinal values. In addition, the children’s justifications, for both the errors and pseudoerrors, made possible the identification of conventional rules underlying the incorrect responses. A particularly relevant trend was that children seem to progressively ignore these rules as they grow older. Nevertheless, this process does not end at 8 years of age given that the conventional rules of temporal and spatial adjacency were present in their judgments and were primarily responsible for the incorrect responses.

Highlights

► A computer program to assess knowledge of counting: conventional and logical rules. ► Error and pseudoerror trials presented with a detection task. ► Error trials were easier than pseudoerror trials. ► Knowledge of logical rules does not promote the understanding of conventional rules. ► Children’s justifications show that they gave priority to some conventional rules.

Introduction

Counting is one of the first numerical skills that children acquire, and the publication of Gelman and Gallistel’s (1978) The Child’s Understanding of Number was a milestone in research on this skill. The authors’ proposal challenged the Piagetian view of the number. Whereas Piaget and Szeminska (1941) suggested that the concept of number was based on the synthesis of seriation and classification, for Gelman and Gallistel it was based on counting skill. This skill was best understood as a complex cognitive skill based on five principles: (a) the one-to-one principle (every item in a display must be tagged once and only once); (b) the stable order principle (the tags, regardless of their nature, must be ordered in a stable list of unique tags); (c) the cardinal principle (the last tag used in a count represents not only the last item in the array but also the cardinality of the set); (d) the abstraction principle (any collection of discrete items can be counted, thereby making it possible to establish its cardinality); and (e) the order irrelevance principle (as long as the other rules are maintained, the order of the count is immaterial because the cardinal values remain the same). The first three principles, called the how-to-count principles, form the conceptual structure of the number and define the rules for proceeding with valid counting. Acquisition of the abstraction and order irrelevance principles grants “permission” to introduce counting variations, whereas the count remains correct (e.g., Sophian, 1998).

One of the tasks particularly well-suited to studying children’s understandings of counting is the detection task (where children must indicate whether a puppet has counted correctly or not) because it frees them from performance demands and allows researchers to freely manipulate the counting scenarios. In this study, we expected that the detection task would allow us to establish whether children can distinguish essential counting aspects, which are governed by logical rules, from nonessential counting aspects, which are governed by conventional rules.

When learning to count, children acquire logical rules, which involve understanding the principles underlying counting, and conventional rules, which involve recommendations that depend on the social context (e.g., customs, common school practices). Thus, the violation of a logical rule (e.g., labeling a single element with two number words) leads to an incorrect count. The same result does not necessarily occur when conventional rules are violated; for instance, a common practice in Western culture consists of counting the elements of a row from left to right, but this rule may be violated (counting from right to left) without necessarily yielding an incorrect count. Consequently, because conventional rules refer to nonessential aspects of counting, they are modifiable (any change, in this sense, leaves the counting logic intact) and so are optional, whereas logical rules are unchangeable and obligatory (Laupa, 2000).

The detection task involves determining whether a puppet has counted appropriately. During the task, children watch both correct and incorrect counts as well as pseudoerrors (correct though unconventional counts). Correct counts are governed by logical and conventional rules, such as the adjacency rule (counting the elements without skipping forward or backward). Incorrect counts occur when the puppet violates the logical rules. Nevertheless, some incorrect counts also fail to fulfill one or more of the conventional rules. For instance, when the puppet skips an element of the row, it violates not only the logical rule of one-to-one correspondence but also the conventional rule of adjacency. Pseudoerrors, in contrast, respect the logical counting rules but not the conventional rules. For example, when the puppet counts all elements of a row nonconsecutively, the conventional adjacency rule is broken but the logical counting rules are not broken.

In empirical studies, when incorrect performances were assessed, 3- to 5-year-old children demonstrated very high achievement on items based on the how-to-count principles. Gelman and Meck (1983) found that 3- and 4-year-olds correctly detected errors of one-to-one correspondence (when an item was skipped or counted twice) in 75% of the trials. Briars and Siegler (1984) obtained similar rates of success with 3- and 4-year-olds, who detected 71.7% of the omitted and repeated object errors. Gelman and Meck also observed that 3- and 4-year-olds detected 92% of the erroneous cardinality trials and that 3- to 5-year-olds correctly identified stable order errors in 89.7% of the trials. LeFevre et al., 2006, Kamawar et al., 2010 reported that rejections of incorrect counts ranged from 82% to 85% in 5- to 10-year-olds. Regarding the abstraction and order irrelevance principles, information about children’s capacities to detect such errors is unavailable because research has focused on the principles’ usefulness in relation to pseudoerrors.

The situation is more complex in the case of pseudoerrors. There are seven types of pseudoerrors described in the literature: (a) counting from the middle, (b) counting alternate colors, (c) right to left counting, (d) beginning counting at any element, (e) double pointing, (f) skipping an element in the middle and counting it at the end, and (g) counting with nonstandard tags (e.g., Briars and Siegler, 1984, Geary et al., 2004, Gelman and Meck, 1983, Kamawar et al., 2010, LeFevre et al., 2006). In addition to other theoretical and methodological differences (e.g., participants’ ages, requesting justifications of the responses, quantities used, instructions given), none of the aforementioned investigations used all seven of the pseudoerrors and the results found are divergent.

Whereas Gelman and Meck (1983) found that 96% of the 3- and 4-year-olds participating considered the pseudoerrors of counting from the middle and counting alternate colors to be correct, Briars and Siegler (1984) reported that only 65% of the 3-year-olds and 35% of the 4-year-olds in their study considered the counting to be correct. Moreover, in this work, only 47% of the 5-year-olds rated the puppet’s performance, which included these pseudoerrors, as valid. Subsequent investigations did not resolve the differences in the results in terms of knowledge of counting and differentiation of logical and conventional rules. Specifically, Gelman and Meck (1986) ratified and extended the data from 1983 (∼90% correct responses at 4 years of age and 93% correct responses at 5 years of age), but the studies by Frye, Braisby, Lowe, Maroudas, and Nicholls (1989) (found 52.5% correct trials using 4-year-olds), LeFevre and colleagues (2006) (reported 53.5% correct trials using 5- and 6-year-olds), and Kamawar and colleagues (2010) (found 50.4% correct trials using the same ages) ratified and extended the findings of Briars and Siegler (1984). The latter works seem to confirm Briars and Siegler’s affirmation that the process of differentiating essential and nonessential characteristics of counting starts at 3 years of age and does not end at 5 years. Thus, some authors have sought to extend this analysis to children in the primary grades (e.g., Geary et al., 1992, Geary et al., 2000, Kamawar et al., 2010, LeFevre et al., 2006, Saxe et al., 1989). Nevertheless, the data from these investigations are not conclusive. LeFevre et al., 2006, Kamawar et al., 2010 reported that children in the primary grades did not reach the percentages of success indicated by Gelman and Meck even at 10 years of age (70.3% success rate reported by Kamawar et al.). However, studies led by Geary yielded success rates of approximately 74% in 7- and 8-year-olds and 89% in 10- and 11-year-olds (Geary et al., 1992, Geary et al., 2000, Geary et al., 2004). Saxe and colleagues (1989) found that only 11-year-olds accepted (as correct) the counts with nonstandard tags, although 8-year-olds did begin to accept them in certain contexts.

The research carried out to date shows that the elaboration of counting is prolonged beyond kindergarten. It also demonstrates that logical and conventional rules coexist during the acquisition and development of counting. Thus, it can be assumed that they play different roles throughout the process. In the beginning, this coexistence could improve children’s knowledge of counting, easing the application of the logical rules (e.g., the adjacency rule could impede the omission or repetition of one or more items in a row). However, it could also hinder progress if children do not establish appropriate distinctions among the rules, such as prioritizing logical rules over conventional rules, or if they cannot apply the rules flexibly.

The current research had three goals. The first was to identify the age at which children switch from relying on conventional (arbitrary or unnecessary) counting rules to logical (essential, unchangeable, and obligatory) counting rules as the basis of their judgments regarding counting. Second, the study sought to determine children’s capacities to distinguish logical and conventional rules in their judgments of errors and pseudoerrors. Finally, the research evaluated whether stating the cardinality (the number of items in the set) affects children’s performances in trials containing pseudoerrors.

For these purposes, we presented a detection task (containing errors and pseudoerrors), using a semistructured interview, to children of various ages. With regard to the error trials, one of the novelties of this work is that it incorporated errors of the abstraction and order irrelevance principles that violate logical and conventional rules. The hope was that the participants’ verbal reports would provide a degree of insight into the underlying thought processes, allowing us to differentiate the children who reject the error trials because they violate a logical rule from those who reject them because either a logical rule and a conventional rule or only a conventional rule has been broken. As for pseudoerrors, most of the trials were created ad hoc for this work to cover as many conventional rules as possible. In half of the cases, the cardinality response was included to determine whether the introduction of the cardinal number of the set emphasized the functional aspect of counting. The justifications given by the children in the individual semistructured interviews may reveal the conventional rules that they use to reject pseudoerrors.

Taking into account these goals, and in accordance with the above-mentioned research, we expected that children’s performances related to pseudoerrors would improve with age, such that the conventional rules would be less frequently applied in unconventional counting situations, especially among 7- and 8-year-olds. Second, we expected that the children’s performances would be better with error trials than with pseudoerror trials. In addition, we predicted that improved performance on pseudoerror trials would be more evident in trials with the cardinality response because the cardinal value would encourage the children to note that deviation from conventional counting is not important. In other words, the presence of the cardinal number could generate increased acceptance of correct (but unconventional) counts because it would relegate the procedure by which one had determined the cardinality of a given set of elements to a lesser role.

Section snippets

Participants

In this work, the participants were 74 children distributed into three groups: (a) 25 kindergartners (11 boys and 14 girls), with ages ranging from 64 to 75 months (M = 70 months); (b) 24 children in first grade (12 boys and 12 girls), with ages ranging from 75 to 86 months (M = 81 months); and (c) 25 children from second grade (14 boys and 11 girls), with ages ranging from 87 to 101 months (M = 93 months). According to information provided by school staff, the participants were of lower middle

Results

A 3 (Grade Level: kindergarten, first grade, or second grade of primary education) × 4 (Detection Task: abstraction errors, order irrelevance errors, pseudoerrors with cardinal value, or pseudoerrors without cardinal value) mixed analysis of variance (ANOVA), with the detection task as the repeated measure, was conducted on the arcsine transformed proportions of correct responses (Table 2). Prior to the analysis, an arcsine transformation was applied to the proportions of correct responses to

Discussion

There is a tendency to think that a child who recites numerals knows how to count and is prepared to begin learning arithmetic concepts and skills, such as addition and subtraction. Learning to count, however, is a long and difficult process that does not end with the mere recitation of counting words and the recognition of their graphs. Knowing how to count involves not only implementing a set of logical rules but also ignoring conventional rules that gradually become unnecessary. This process

Acknowledgment

This investigation was funded by the Autonomous Community of Madrid (CCG10-UCM/HUM-4698). The authors wish to thank David Bjorklund and the anonymous reviewers for providing many helpful comments.

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