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Predictors of Early Numeracy: Is There a Place for Mistakes when Learning about Number?

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Title: Predictors of Early Numeracy: Is There a Place for Mistakes when Learning about Number?
Language: English
Authors: Muldoon, Kevin P., Lewis, Charlie, Berridge, Damon
Source: British Journal of Developmental Psychology. Nov 2007 25(4):543-558.
Availability: British Psychological Society. St Andrews House, 48 Princess Road East, Leicester, LE1 7DR, UK. Tel: +44-116-254-9568; Fax: +44-116-227-1314; e-mail: enquiry@bps.org.uk; Web site: http://www.bpsjournals.co.uk
Peer Reviewed: Y
Physical Description: PDF
Page Count: 16
Publication Date: 2007
Document Type: Journal Articles
Reports - Research
Education Level: Early Childhood Education
Descriptors: Numeracy, Young Children, Number Concepts, Developmental Psychology, Mathematics Skills, Statistical Analysis, Error Patterns, Arithmetic, Numbers, Predictor Variables
DOI: 10.1348/026151007X174501
ISSN: 0261-510X
Abstract: It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy-two 5-year-olds were given a battery of nine tasks on each of three sessions (at 3-monthly intervals). Tasks measured procedural proficiency, conceptual understanding (using an error-detection paradigm) and the ability to compare sets using number knowledge. Procedural skills remained fairly stable over the 6-month period, and preceded children's ability to detect another's violations to those procedures. Regression analysis revealed that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinal numbers for set comparisons. We suggest that although children's conceptual understanding of well-rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.
Abstractor: As Provided
Entry Date: 2011
Accession Number: EJ940091
Database: ERIC
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  Value: <anid>AN0027563598;99e01nov.07;2019May29.11:49;v2.2.500</anid> <title id="AN0027563598-1">Predictors of early numeracy: Is there a place for mistakes when learning about number? </title> <p>It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy‐two 5‐year‐olds were given a battery of nine tasks on each of three sessions (at 3‐monthly intervals). Tasks measured procedural proficiency, conceptual understanding (using an error‐detection paradigm) and the ability to compare sets using number knowledge. Procedural skills remained fairly stable over the 6‐month period, and preceded children's ability to detect another's violations to those procedures. Regression analysis revealed that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinal numbers for set comparisons. We suggest that although children's conceptual understanding of well‐rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.</p> <p>Counting and sharing are two of the most important early number skills. They provide the foundation for mathematical understanding. However, it is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. Not only does it take a long time for children to master these procedures – up to 2 years in the case of counting ([<reflink idref="bib34" id="ref1">34</reflink>]) – but they still have a long way to go if they are to build on them in the classroom. A great deal of research has gone into testing children's procedural competence, and the contexts in which they do or do not apply their skills, but it is still unclear how developments in procedural ability inform early mathematical understanding. Similarly, although an association between first graders' conceptual knowledge of counting principles and arithmetical skill development was found some years ago ([<reflink idref="bib15" id="ref2">15</reflink>]), more recent results have proved less conclusive ([<reflink idref="bib16" id="ref3">16</reflink>]).</p> <p>Notwithstanding these mixed findings on later arithmetical progress, we set out to assess the relative impacts of these two components – procedural mastery and conceptual understanding – on a fundamentally important step in children's numerical knowledge; that is, the significance of cardinal numbers when judging quantitative relationships between multiple sets.</p> <p>Consider counting. Accurate counting demands one‐to‐one correspondence between number words and the items being counted. If one‐to‐one correspondence is maintained, then the last number word in the sequence – the cardinal number – tells us how many there are (assuming the count starts with 'one' and adheres to the conventional number order thereafter). Although early counting behaviour often displays sensitivity to this form of one‐to‐one correspondence, preschoolers are often less sure about the meaning of the cardinal number. It is often not clear whether children who successfully answer 'How many?' questions use the principle of cardinality, or simply apply a 'last‐word' rule, which entails either repeating ('one, two, three ... three!') or emphasizing ('one, two, THREE!') the final number‐tag. Although an association between last‐word responding and counting proficiency has been found ([<reflink idref="bib11" id="ref4">11</reflink>]), this cannot be taken as evidence that counting proficiency inevitably yields a grasp of cardinality because such social conventions can be rote‐learned without insight into their numerical significance.</p> <p>Such uncertainty over children's understanding of what they are doing when they count has informed long‐running – and ongoing – debates concerning the order in which children master procedural counting and appreciate the significance of the underlying principles. The 'principles‐first' view (e.g. [<reflink idref="bib17" id="ref5">17</reflink>]) argues that an innate sensitivity to numerical principles guides later counting, whereas the 'skills‐first' view (e.g. [<reflink idref="bib14" id="ref6">14</reflink>]) contends that counting proficiency precedes a grasp of those same principles. A more complex 'mutual development' view ([<reflink idref="bib2" id="ref7">2</reflink>]) posits the case that some counting skills may be learned mechanically and that other counting skills and principles may be constructed in tandem. To date, there is a persuasive body of evidence to suggest that children learn how to count first, and then gradually learn which aspects of counting are important and which are not (e.g. [<reflink idref="bib4" id="ref8">4</reflink>]; [<reflink idref="bib14" id="ref9">14</reflink>]; [<reflink idref="bib20" id="ref10">20</reflink>]; [<reflink idref="bib21" id="ref11">21</reflink>]; [<reflink idref="bib30" id="ref12">30</reflink>]; [<reflink idref="bib31" id="ref13">31</reflink>]). For example, [<reflink idref="bib8" id="ref14">8</reflink>] found an association between children's developing mastery of the counting procedure and their ability to discriminate between the order‐irrelevance principle (i.e. changing the order in which items are counted does not affect cardinality) and addition and subtraction (which do affect cardinality). Although procedural counting is typically in place by the age of 6, conceptual understanding of that procedure is still undergoing change ([<reflink idref="bib21" id="ref15">21</reflink>]).</p> <p>Accordingly, while being able to count up to 10, and then through the teens up to 20 and beyond, are landmarks in early numeracy, such achievements do not necessarily demand insight into what the purpose of counting is ([<reflink idref="bib19" id="ref16">19</reflink>]). Furthermore, for counting skills to develop beyond simple enumeration, it is essential to learn another important role that cardinal numbers serve; that is, as a means of comparing two or more sets of items. When we count two sets of items, it is possible to determine whether there are as many items in one set as another by comparing cardinals. Recognizing this is a key advance in the development of numeracy, but demands a qualitative shift in children's representation of number words. This is because the logic of item‐to‐item correspondence between two sets does not automatically follow from the logic of word‐to‐item correspondence when counting a single set. There is nothing in counting a single set, and deriving a cardinal number, that necessarily ties that cardinal value with another. The connection can be made, however, because there <emph>is</emph> a logico‐mathematical relationship between the two. Equivalent sets have items that are in one‐to‐one correspondence, such that every item in set A can be mapped onto an item in set B, and vice versa. Once both sets have been counted, there is also correspondence in the assigned number words (i.e. each number word assigned to set A will also have been assigned to set B, and vice versa).</p> <p>[<reflink idref="bib29" id="ref17">29</reflink>] highlighted the fragility of children's grasp of the relationship between item‐to‐item correspondence and number in his test of conservation. Children are also prone to rely on length cues when comparing <emph>static sets</emph> even where guidelines make the correspondence between those sets explicit ([<reflink idref="bib5" id="ref18">5</reflink>]), and after both sets have been counted ([<reflink idref="bib6" id="ref19">6</reflink>]). Cowan's findings reveal the problems that young counters have in reasoning about the relationship between counting, cardinal numbers and numerical correspondence. It takes otherwise proficient counters some time to learn that cardinal numbers define numerical relationships, including the realization that same number word implies one‐to‐one corresponding sets.</p> <p>It is perhaps unsurprising, therefore, that they have difficulty in extending their knowledge of sets in one‐to‐one correspondence to number words. What makes this striking, however, is the fact that children can display great sensitivity to the significance of item‐to‐item correspondence between sets in the second of the two early‐learned procedures; that is, sharing. Sharing proficiency emerges in the preschool years, and is thus largely contemporaneous with learning how to count ([<reflink idref="bib28" id="ref20">28</reflink>]). The development of sharing knowledge is similar to that of counting, in that procedural mastery appears fairly stable between preschool and school years ([<reflink idref="bib23" id="ref21">23</reflink>]), but outstrips awareness of what sharing achieves. Using a 'same/different' task, [<reflink idref="bib7" id="ref22">7</reflink>] found that 3‐year‐olds understood that distributing items using a 'one‐for‐you, one‐for‐me' rule results in sets that are the same, particularly if shares were small (i.e. less than four items) and visible. They were not tested on whether they knew how many items each set contained, however, and young children experience difficulty mapping number words on to their judgments of equivalence. [<reflink idref="bib13" id="ref23">13</reflink>] asked 4‐year‐olds to share out sweets between dolls, then counted aloud one set for children, and asked how many sweets there were in the other. Even though most children had shared out successfully, none would infer the number of sweets in the second set (every child started to count). It seems that having learned how to establish equivalent sets by maintaining item‐to‐item correspondence using a 'one‐for‐you, one‐for‐me' strategy, it takes children some time to recognize that the resulting sets must have the same cardinal number. Up to the age of 8, children often insist that counting is necessary to reach a conclusion about the equality of evenly distributed shares ([<reflink idref="bib9" id="ref24">9</reflink>]; [<reflink idref="bib13" id="ref25">13</reflink>]). There is insufficient longitudinal data available to explore how this crucial skill develops and what predicts such cognitive change.</p> <p>[<reflink idref="bib24" id="ref26">24</reflink>] has found that children's counting accuracy correlates with the recognition of numerical equivalence on set‐comparison tasks, particularly where there are perceptual differences (e.g. in size, colour, etc.) between the sets being compared. The ability to maintain one‐to‐one correspondence between words and items when counting appears to offer a developmental advance. However, it is less clear how the ability to count single sets accurately might help children gain the conceptual insight that sharing evenly produces sets with the same cardinal. It is intuitively appealing to suspect that the ability to maintain item‐to‐item correspondence between shares is a more likely predictor of this ability.</p> <p>In order to explore the development of children's early numeracy, we probed their understanding of the two procedures of counting and sharing during their first school year. Specifically, we examined children's understanding using three simple number tests. We examined counting in two ways. On one test, we counted two rows of wooden blocks to see whether children would compare them on the basis of their respective cardinal numbers. On a separate test, we looked to see whether they would count similar rows spontaneously in order to compare them if we did not do the counting for them. To examine sharing, we got children to share items out before asking them to count the number of items in one set and infer the number of items the second set contained.</p> <p>We had two types of independent variables; some measured procedural knowledge and others conceptual knowledge by way of an error‐detection paradigm. For procedural knowledge, we assessed children's ability to count accurately sets of 10 and 20 items and share out fairly sets of between 10 and 14 items (i.e. to produce equivalent sets of 5 or 7 items, respectively) by adhering to an item‐for‐item form of distribution. To assess conceptual understanding of these procedures, we asked children to judge the procedural proficiency of a puppet. For counting, they were asked to watch closely as a puppet counted to see if he made any mistakes. If they spotted an error, they were probed about the numerical consequences of the miscount. Similarly, they were also invited to monitor the puppet's ability to share items out in a fair manner (i.e. by maintaining item‐to‐item correspondence).</p> <p>We included these two types of measure – procedural and conceptual – for three important reasons. First, it allowed us to assess children's conceptual knowledge independently of their procedural knowledge ([<reflink idref="bib18" id="ref27">18</reflink>]). Secondly, as the study was longitudinal, it allowed us to identify which type of knowledge showed the greatest development over a 6‐month period, and whether any of these developments were key to progress on the three number tests. Thirdly, the inclusion of error‐detection tests informed the design in another important way.</p> <p>Efforts have been made to interpret reliably what children's ability to detect miscounts tells us about their conceptual understanding. Error‐detection tasks have been used to examine children's sensitivity to core counting principles independently of procedural performance (e.g. [<reflink idref="bib18" id="ref28">18</reflink>]). According to Gelman, children sometimes make errors when they count (because they have difficulty coordinating the tasks of pointing to objects, partitioning one object from another and saying the number‐tags), but an innate grasp of counting principles helps them to monitor and correct any errors they may make. [<reflink idref="bib34" id="ref29">34</reflink>] asked children how many items there were following correct counts and miscounts, and found that some 3‐year‐olds could identify and discriminate appropriately between valid and invalid cardinals on the grounds of procedural accuracy.</p> <p>[<reflink idref="bib12" id="ref30">12</reflink>] have gone further in highlighting the utility of error‐detection tasks as probes for children's <emph>understanding</emph> of miscounts. A key development in early numeracy, they argue, is when children shift from disregarding miscounts as functionally irrelevant to where they recognize that miscounts have logical consequences that can be corrected. They found 5‐year‐olds who not only knew that the last word in a miscounted set was invalid, but also how the true cardinal was related to the type of miscount (e.g. that adding 'one' was necessary if an item had been missed). The ability to categorize counts and miscounts as either valid or invalid instances of counting and to recognize <emph>why</emph> miscounts are invalid, is particularly revealing, as it tells us that they know <emph>how</emph> counting delineates quantity, something that procedural mastery cannot. Freeman <emph>et al.</emph> concluded that once children grasp that a single set has one – and only one – cardinal number, and that true cardinality rests on maintaining word‐to‐item correspondence, they are in a position to learn from miscounts.</p> <p>Support for this last claim has come from [<reflink idref="bib25" id="ref31">25</reflink>] discovery that the ability to reason about miscounts has a striking effect on children's emerging conceptualization of what is achieved by counting. Children who were able to detect and reason about miscounts were more likely to count when asked to compare two sets, and to create equivalent sets, than children who, although procedurally proficient, either ignored or failed to detect another's miscounts.</p> <p>It is possible that similar patterns of procedural competence, conceptual understanding and problem solving are to be found in the context of sharing skills. Inferences made following sharing are similarly bound by a principle of correspondence. Accurate sharing rests on item‐to‐item correspondence, and only if this is maintained are sets of equal size. Accordingly, children who are sensitive to mis‐sharing might also be in a position to learn from errors. What is not yet known is whether, or how, this type of insight about counting or sharing procedures is associated with developments in early numeracy.</p> <p>In this paper, we examine the possibility that a grasp of the importance of errors in numerical procedures makes a unique contribution to children's representations of number. We examined children's knowledge of counting and sharing from three perspectives; that is, their procedural proficiency, their conceptual understanding of those procedures and the relevance of each procedure to simple set‐equivalence problems. The solution to all three set‐equivalence problems demanded recognition of the relationship between counting (word‐to‐item correspondence) and item‐to‐item correspondence. The aim of the study was to identify whether any of our measures of numerical understanding (counting or sharing, procedural or conceptual) was able to predict children's success on these problems.</p> <hd id="AN0027563598-2">Method</hd> <p></p> <hd id="AN0027563598-3">Participants</hd> <p>Seventy‐two children (27 girls1) from three schools in a predominantly White, urban population took part. None of the children had any identified learning difficulties and were aged between 50 and 62 months (<emph>M</emph> = 54.3, <emph>SD</emph> = 3.0) at the first time of testing. The tests were administered at 3‐month intervals, with the first session being conducted in November, 2 months after children had started school.</p> <hd id="AN0027563598-4">Materials and procedure</hd> <p>The study comprised a total of nine tasks. The experimenter and child were seated at a small table in a quiet corner of the classroom, and each child received the battery of tasks in the same fixed order at each time of testing. For each task, counterbalancing of all variables was observed as far as possible.</p> <hd id="AN0027563598-5">Sharing and set‐inferences</hd> <p>Two shoeboxes resembling an ice cream van and a small shop were placed on the table. The roof of the shop had a small slot through which items could be 'posted', whereas the ice cream van had no roof. Toy ice creams (either 10, 12 or 14) were placed in front of the child who was asked to share them out evenly (Task 1 – 'Sharing') between the van and the shop. Once the child had distributed ice creams, the experimenter asked, 'Did you share them out properly?' Any child who said 'No' was invited to share them out again until he or she was happy that the sharing had been completed satisfactorily, and was asked 'How many ice creams does the ice cream van have to sell?' followed by 'How many ice creams does the shop have to sell?' (Task 2 – 'Set inference'). If any child miscounted the ice creams in the van, they were asked to 'Count them again carefully'. Each child received two trials with different numbers of ice creams.</p> <hd id="AN0027563598-6">Pre‐ and post‐count comparisons</hd> <p>Children were shown two sets (one blue and one green) of wooden blocks (each block 3×3 cm). The experimenter introduced a toy puppet hedgehog called Harry, explaining that they were going to see how good he was at making fair sets. There were four trials. On the first two trials (Task 3 – 'Pre‐count comparison'), the child closed their eyes while Harry made two parallel rows of blocks, after which the child was asked 'Have you got more blocks than me, have we both got the same amount of blocks, or have I got more than you?' On the third and fourth trials (Task 4 – 'Post‐count comparison'), the child was first asked to count the number of blocks in his/her row (until the row had been counted accurately if any miscounts occurred), and then Harry counted the number of blocks in the experimenter's row, followed by the same question about equivalence.</p> <p>There were two types of array. Equal number/unequal length (EN/UL) arrays contained equal numbers of blocks, but one of the rows was longer than the other. In unequal number/equal length (UN/EL) arrays, one row contained one block more than the other, even though the rows were of the same length.</p> <hd id="AN0027563598-7">Counting</hd> <p>Pennies were put in a line, with 1 cm spacing. The child was asked, 'How many pennies are there?' and gently urged 'Go on, you count them' if there was any hesitation. There were two trials: on the first, the row contained 10 pennies (Task 5 – 'Count 10') and on the second, there were 20 pennies (Task 6 – 'Count 20'). If any child miscounted on either trial, they were invited to count again carefully in order to be sure that the miscount was a true procedural error rather than the result of (e.g.) overeagerness. If a child failed to count accurately in two attempts, they were scored as zero for that set size.</p> <hd id="AN0027563598-8">Detect and reason miscounts</hd> <p>The child was told that Harry was just learning to count and wanted to practice. The child was told to watch Harry carefully because sometimes he got it right (i.e. counted correctly) and counted OK, but sometimes he got it wrong and his counting was not OK. A row of between seven and nine blocks was placed on the table and the puppet was made to count aloud. Four trials were randomized, two with correct counts and two with miscounts (once by skipping an item and once by double‐counting an item). The child was scored on whether they judged the correct counts as being 'OK' and the miscounts as 'not OK' (Task 7 – 'Detect miscount'). After each of the puppet's miscounts, the child was asked 'How many does Harry think there are?' followed by the reality control question 'How many are there really?' (Task 8 – 'Reason miscount'). The child had to answer both questions correctly to pass Task 8.</p> <hd id="AN0027563598-9">Detect mis‐shares</hd> <p>The child was told that their job was to watch Harry carefully as he shared out (either six or eight) ice creams (the number of ice creams was smaller than that used in Task 2 to minimize the chances of children's attention wandering from what Harry was doing if it took too long) to see if he shared properly. On two trials the ice creams were distributed in an item‐to‐item manner, and on two the item‐to‐item correspondence was violated by repeatedly placing two ice creams in the shop and only one in the van. The trials were presented in pairs, with one correct trial and one incorrect trial in each pair. Scoring was done on the basis that a child had to judge the correct procedure as 'OK' and the mis‐share as 'not OK' in each pair of trials, giving a possible maximum score of 2.</p> <p>Children were not asked to explain why correct sharing produces equivalent sets or why incorrect sharing produces unequal sets for two reasons. First, the battery of tasks was substantial as it was and we did not want to jeopardize children's willingness to participate. Secondly, the results of an earlier study ([<reflink idref="bib26" id="ref32">26</reflink>]) showed that, on sharing tasks, young children do not benefit from being asked how they knew why a judgment was correct or incorrect. This may be due to the fact that unlike set‐comparisons where either (or both) visible, spatial cues or spoken, numerical cues to quantity are salient, sharing is typically a silent procedure where set‐correspondence is temporally determined, and there is no equivalent to the verbalization of cardinality.</p> <hd id="AN0027563598-10">Results</hd> <p>As anticipated, more than half the children were able to count and share out small sets on Session 1, and almost all were procedurally proficient 6 months later (Table 1 shows the number of children scoring zero, one and two out of two2 for Tasks 1–9 over Sessions 1–3).</p> <p>Graph</p> <p>Furthermore, over half the children were also sensitive to the puppet's miscounts and mis‐shares at Session 1, although most had, at best, a fragile grasp of the consequences of miscounting (as measured by the 'Reason miscount' task). However, 6 months later, at Session 3, nearly 50% appeared to understand that miscounting implied that the true cardinal was not the same as the last number‐tag used.</p> <p>Of the set‐equivalence problems, children were most successful on the 'Set inference' task, showing considerable improvement over the three sessions. But even children who knew that cardinal numbers denoted equivalent shares did not necessarily count spontaneously on the 'Pre‐count comparison' task, even though most were proficient counters. This might simply indicate a production deficiency (the failure to select a particular strategy rather than the absence of that ability; see [<reflink idref="bib10" id="ref33">10</reflink>]), although evidence against this is the fact that children still had difficulty trusting cardinal numbers over length on the 'Post‐count comparison' task.</p> <p>Separate analyses were carried out on two of the three set‐equivalence tasks ('Set‐inference' and 'Pre‐count comparison'). We looked particularly closely at children's performance on these tasks, as they would tell us the most about children's grasp of the relationship between the two forms of correspondence we were investigating. To infer a corresponding set, children had first to understand that sharing items evenly (by maintaining item‐to‐item correspondence) produces sets of the same size. Equally important to success on 'Set inference' was a grasp of cardinality (the result of word‐to‐item correspondence); children had to know that a cardinal number from one set could be extended to another set known to be in item‐to‐item correspondence.</p> <p>Similarly, we were confident that children who counted spontaneously on the 'Pre‐count comparison' task knew that counting establishes a cardinal representation of a set, and that comparing two cardinals is a good way of assessing the degree of correspondence between the respective sets. Success on the 'Post‐count comparison' task was not quite as stringent; that is, children had to choose between two salient cues, but did not have to display evidence that they incorporated that knowledge into their problem‐solving strategy. Nonetheless, the importance of relying on cardinals where they were given is addressed separately (see below).</p> <p>We focused on children who, on either the first or second session, displayed no evidence that they had the target skills we were interested in (i.e. children who scored 0/2). Regression models were constructed to compare the odds of making a gain (i.e. moving from a score of 0 to a score of 1 or 2) to the odds of making no gain (i.e. staying with a score of 0) on each of these measures (see below for more details). In other words, scores of 1 or 2 on Sessions 2 or 3 where the previous session had yielded a score of 0 were our two dependent measures. The aim was to construct a model that identified the key areas of knowledge – counting vs. sharing, procedural mastery vs. error‐detection – associated with gains on the target skills. Time was treated as a fixed variable with two levels representing the difference between Sessions 1 and 2 (Time 2) and between Sessions 2 and 3 (Time 3).</p> <p>Before reporting our findings, it is important to detail the statistical analyses we used. With the exception of the procedural counting tasks ('Counting 10' and 'Counting 20'), children were given a score of 0, 1 or 2 for each task on each session. Responses with only three possible values, together with small sample sizes and a hypothesis regarding change (thereby implicitly breaching the principle that cell variation remains homogeneous) negated the possibility of assuming an underlying normal distribution. Having only two response categories would have allowed us to use a binary logistic model. However, in our data, each response variable comprised three categories (score of 0, 1 or 2), hence we used a generalization of binary logistic regression, called the multinomial logistic model. Such models extend the classic normal random effects model to data (counts, binary scores and ordinal values), where the assumption of normality for the response is inappropriate, and are a special case of a generalized linear mixed model (see [<reflink idref="bib1" id="ref34">1</reflink>]; [<reflink idref="bib22" id="ref35">22</reflink>]).</p> <p>By conditioning on a score of 0 at time (<emph>t</emph>−1), where <emph>t</emph> denotes Session 2 or Session 3, the relative risks (<reflink idref="bib1" id="ref36">1</reflink>) and (<reflink idref="bib2" id="ref37">2</reflink>) may be expressed in terms of transition probabilities as follows:</p> <p></p> <ulist> <item> Prob (score of 1 at time <emph>t</emph>, given score of 0 at time (<emph>t</emph>−1))/Prob (score of 0 at time <emph>t</emph>/given score of 0 at time (<emph>t</emph>−1)) and</item> <p></p> <item> Prob (score of 2 at time <emph>t</emph>, given score of 0 at time (<emph>t</emph>−1))/Prob (score of 0 at time <emph>t</emph>/given score of 0 at time (<emph>t</emph>−1)).</item> </ulist> <p>This conditional multinomial model allows us to assess the extent to which the explanatory variables and time (treated as a factor with two levels), observed at time <emph>t</emph> (Session 2 or 3), affect the probability of a shift in score (from 0 to either 1 or 2) from (<emph>t</emph>−1) (i.e. Session 1 or 2, respectively).</p> <p>For example, a RRR greater than 1 (see Table 2) indicates that the chance of a child – who got a score of 0 at time (<emph>t</emph>−1) – getting a score of 1 or 2 at time <emph>t</emph>, relative to the chance of getting a score of 0 at time <emph>t</emph>, is greater if the child has detected a sharing error at time <emph>t</emph>, as opposed to not detecting a sharing error at time <emph>t</emph> (where <emph>t</emph> is either the second or third session).</p> <p>Graph</p> <p>It should be noted that by adding interactions between each of the explanatory variables and time to this model, we were able to test whether the main effect of each explanatory variable changed significantly over the three sessions. However, none of the interaction terms proved significant; those main effects found to be significant (see Table 2) remained consistent over the 6‐month period.</p> <hd id="AN0027563598-11">Set inference</hd> <p>Figure 1 shows the developmental trend of children's performance on the 'Set inference' task relative to a number of independent variables.</p> <p>Graph: 1 Relative levels of performance for 'Set inference', 'Count 10 ', 'Sharing' and 'Detect mis‐share' over Sessions 1–3</p> <p>As expected, many children were capable of sharing out items perfectly and counting the resulting shares accurately, on Session 1, but failed the 'Set inference' test. However, success on the 'Set inference' test increased by 100% between Sessions 1 and 3. We carried out a regression analysis to identify which, if any, of our independent variables should be included in a model of change. Performance on 'Detect mis‐share' predicted children moving from a score of 0 to a score of 1, and moving from 0 to 2, and was the biggest contributor to the model (see Table 2). Sensitivity to another's sharing accuracy – rather than their own procedural mastery of sharing – increased the likelihood that children had learned that sharing produces equivalent sets. 'Reason miscount' also predicted improvement from 0 to 1.</p> <p>In contrast, 'Detect miscount' was significantly but negatively correlated with learning; this was unexpected, as we had anticipated that error‐detection skills would reflect greater understanding than procedural skills alone. There are two possible explanations for this finding. First, detecting miscounts has no positive association with a conceptual understanding of the numerical significance of sharing, because the former concerns a single set whereas the latter concerns the relationship between multiple sets. Secondly, it is conceivable that the miscount detection task does not tap into conceptual understanding of counting at all; if a child's counting is little more than a well‐rehearsed routine with no numerical significance, then detecting a violation to that routine will not necessarily indicate conceptual understanding of the numerical significance of counting. Of course, this objection to the meaning of miscount detection could equally apply to detecting mis‐shares, but this ability was positively and significantly associated with success on the set‐inference problems. We can speculate that there is a key difference between the monitoring of words‐to‐items and the more temporally demanding monitoring of one‐to‐one sharing that results in differing levels of conceptual awareness of the meaning of the two procedures.</p> <p>Due to their statistical significance, all three explanatory variables were retained in a parsimonious model (see Table 2[a]), and account for nearly 40% of the variance in the set inference data (χ<sups>2</sups>(<reflink idref="bib6" id="ref38">6</reflink>, _I_N_i__SP_obs_sp_=75)=57.52, <emph>p</emph><.001)3. The regression analyses allow us to make inferences about the patterns of change. Procedural skills remained fairly stable over the 6‐month period (i.e. over Sessions 1 to 3) and, as we anticipated, preceded children's ability to detect another's violations to those procedures. Figure 1 shows this clearly with respect to 'Sharing' when compared with 'Detect mis‐share'. By Session 3, children's ability to detect the puppet's mis‐share matches their procedural ability, but only the conceptual task predicts their understanding that sets created through sharing have the same cardinal. The absence of any time interaction confirms that there was no significant difference between the number of children improving between Sessions 1 and 2 and between 2 and 3 (22 and 17 children, respectively).</p> <hd id="AN0027563598-12">Pre‐count comparison</hd> <p>Figure 2 shows the developmental trend of children's performance on the 'Pre‐count comparison' test relative to a number of independent variables. There were only two significant predictors of improvement on this task. By far, the biggest was 'Detect mis‐share'. It is worth noting that its procedural counterpart – 'Sharing' – was, again, not itself a significant predictor.</p> <p>Graph: 2 Relative levels of performance for 'Pre‐count comparison', 'Count 10', 'Miscount detection', 'Detect mis‐share' and 'Reason miscount' tasks over Sessions 1–3</p> <p>This finding seems particularly interesting. Although we had anticipated that a conceptual task would predict children's spontaneous use of a counting strategy, we had expected this to be 'Detect miscount' or 'Reason miscount', on the basis that we would credit children who passed these tests with a firmer understanding of counting than children who's counting‐related skills were restricted to procedural competence. However, this is to miss the fact that our counting tasks ('Count 10/20' and 'Detect/Reason miscount') only tapped knowledge of the principles underlying the enumeration of a <emph>single</emph> set ([<reflink idref="bib17" id="ref39">17</reflink>]). Our sharing measures, on the other hand, tapped children's knowledge of item‐to‐item correspondence between two sets. We can speculate that the latter informs children's recognition of the relationship between number words and set‐correspondence in a way that single‐set enumeration does not.</p> <p>Time did exert a significant effect here; more children showed improvement between Sessions 2 and 3 than between 1 and 2 (25 vs. 5 children, respectively). A parsimonious model that includes 'Detect Mis‐share' and 'Time' accounts for nearly 15% of the variance (see Table 2[b]), again a highly significant proportion of variation in the data; (χ<sups>2</sups>(<reflink idref="bib4" id="ref40">4</reflink>, _I_N_i__SP_obs_sp_=124)=25.71, <emph>p</emph><.001). However, we decided to carry out an additional analysis on these data by constructing a regression model that included performance on 'Post‐count comparison'. We suspected that children who recognized the importance of cardinals over spatial cues when those cardinals were provided would be most likely to succeed on a task where spontaneous counting was the outcome measure, and this was the case. Adding 'Post‐count comparison' to the parsimonious model accounted for an additional 30% of the total variance (<emph>R</emph><sups>2</sups>=.45; χ<sups>2</sups>(<reflink idref="bib1" id="ref41">1</reflink>, _I_N_i__SP_obs_sp_)=76.60, <emph>p</emph><.001).</p> <p>A chi‐squared analysis was conducted to confirm the association between 'Post‐count comparison' and 'Pre‐count comparison' tasks. The positive relationship between these two variables was significant on Session 1 (χ<sups>2</sups>(<reflink idref="bib1" id="ref42">1</reflink>, _I_N_i_ = 72)=14.88, <emph>p</emph><.001), Session 2 (χ<sups>2</sups>(<reflink idref="bib1" id="ref43">1</reflink>, _I_N_i_ = 72)=16.52, <emph>p</emph><.001) and Session 3 (χ<sups>2</sups>(<reflink idref="bib1" id="ref44">1</reflink>, _I_N_i_ = 72)=34.06, <emph>p</emph><.001). An inspection of cross‐tabulation cells revealed that no children counted spontaneously on the 'Pre‐count comparison' task unless they based their judgment on cardinals rather than length on the 'Post‐count comparison' task. This prompted us to examine which counting skills – procedural vs. conceptual – predicted whether children relied on cardinals rather than length. A regression analysis (not shown) revealed only one predictor: success on the 'Reason miscount' task (χ<sups>2</sups>(<reflink idref="bib2" id="ref45">2</reflink>, _I_N_i__SP_obs_sp_=42)=9.17, <emph>p</emph><.05) accounted for 11% of the variance. Children are more likely to trust cardinal numbers over misleading, but salient, spatial cues to number if they have a grasp of the consequences of miscounting. In‐turn, these children are more likely to use counting spontaneously to compare sets even when spatial cues are evident.</p> <p>Together, these findings suggest that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinality for numerical comparisons, when compared with children who, although they can count and share accurately themselves, appear to have only an implicit grasp of the principles underlying successful counting and the constraints on sharing evenly.</p> <hd id="AN0027563598-13">Discussion</hd> <p>In order to view counting as a way of comparing sets, and sharing as establishing equivalent sets, children's mental representations of these two procedures must be qualitatively different from those representations that are sufficient to simply reproduce the procedure ([<reflink idref="bib3" id="ref46">3</reflink>]). Specifically, they need to recognize that counting and sharing are not ends in themselves, but means to different ends. Why should error‐detection tasks be associated with children's recognition of counting and sharing as means of comparing multiple quantities?</p> <p>Counting and sharing might be initially rote‐learned, with the numerical significance of these routines still to be discovered. Our data support this suggestion, as children's ability to detect another's procedural errors lagged behind their own procedural mastery. [<reflink idref="bib33" id="ref47">33</reflink>] has argued that conceptual knowledge of counting, and the application of counting as an arithmetical tool, must be linked. Supporting evidence for this claim comes from our finding that children's ability to reason about miscounts predicted whether they would trust cardinal numbers over misleading spatial cues when comparing sets. These children knew the impact that a miscount had on the cardinal number for a set, because they understood that the puppet would think there were a different number of items than there was in reality.</p> <p>Their ability to reason about miscounts highlights what [<reflink idref="bib12" id="ref48">12</reflink>] refer to as children's 'functional organization' of [<reflink idref="bib17" id="ref49">17</reflink>] how‐to‐count principles, and they go on to refer to this ability as a 'theory of error'. In this model of number development, a theory of error signifies that a child has advanced from conceptualizing miscounts as things to be ignored to recognizing that miscounts have logical consequences. We expected children who had this ability to show greater gains on the set inference and set comparison tasks than other children who did not exhibit similar knowledge. However, even these more demanding counting‐related skills failed to predict children's spontaneous use of a counting strategy to compare sets when those sets were not counted beforehand. In contrast, the ability to detect the puppet's mis‐shares did. Why was the ability to detect mis‐shares more strongly associated with the ability to solve set‐equivalence problem solving than comparable counting ability?</p> <p>[<reflink idref="bib28" id="ref50">28</reflink>] found only a small, non‐significant, association between counting and sharing proficiency. A key difference between these two procedures is that they are founded on different forms of one‐to‐one correspondence. Counting demands correspondence between words and items when enumerating a single set. A child has to maintain temporal coordination between saying the next word in the sequence and (in this case) pointing to the next object to be counted, until all objects have been assigned a word. The final word summarizes the items in that set alone. The demands of sharing are quite different. Here, a child has to monitor the distribution of items to different recipients. If sharing is performed accurately, the achievement is a specific relationship between two sets. Children who can share accurately may not be aware of this if sharing is rote‐learned. The ability to detect mis‐shares suggests an understanding that one‐to‐one correspondence is the objective of sharing, and that violations to the procedure jeopardize this objective. It may be that sensitivity to mis‐shares informs children's appreciation of correspondence <emph>between</emph> sets in a way that detecting miscounts does not.</p> <p>Counting and sharing skills come together when children count items as they are sharing them out (e.g. 'One‐one, two‐two, three‐three, and so on). We did not record whether children did this, but it is conceivable that children self‐scaffold the integration of number understanding and sharing proficiency in this manner. Counting‐while‐sharing has the potential to act as a conceptual bridge between the cardinality of a single set and the way cardinal numbers define the numerical relationship between two sets. It is possible that some children were counting 'silently' as they shared out items in the present study. Contrasting the development of children who are trained to count as they share, with a control group who are not encouraged to count, would test whether the effects we observed are influenced by children verbally maintaining numeral‐for‐numeral correspondence as they share or whether the physical distribution alone is sufficient to confer a benefit.</p> <p>An association between age‐related improvements in sharing and the ability to infer corresponding shares has recently been found ([<reflink idref="bib27" id="ref51">27</reflink>]), and the present findings add to this developmental picture. We can infer that children who display sensitivity to another's sharing mistakes are attuned to the goal of sharing, namely the maintenance of item‐to‐item correspondence. It might be too much to claim that this reveals a greater understanding than procedural mastery alone, were it not for the finding that this skill predicted children's ability to judge numerical equivalence between the resulting shares using cardinal numbers, and the recognition that correspondence between sets can be judged via counting.</p> <p>Thus, the error‐detection paradigm illuminates two important functions. First, it is a means of more accurately assessing children's grasp of how underlying principles determine what procedures like counting and sharing achieve. Secondly, error‐detection performance is much more than a marker of conceptual knowledge. Others have recognized the utility of error‐detection skills in revealing nascent appreciation of numerical principles (e.g. [<reflink idref="bib18" id="ref52">18</reflink>]) and insight into those principles ([<reflink idref="bib12" id="ref53">12</reflink>]), but identifying the impact of error‐detection ability on number development has received less attention. Our data are instructive with regard to error‐detection tasks as learning mechanisms.</p> <p>What are the implications for classroom practice? We began by arguing that although being able to count accurately is a landmark in early numeracy, children still have a great deal to learn about the meaning of number words. In particular, they must learn how the words they use to denote the size of a single set also determine the numerical relationship between two or more sets; this is the basis for formative mathematical development and is a skill typically targeted in school curricula (e.g. numeracy targets for Key Stage 1 [Year 1] in the UK, outlined by the Department for Education and Skills). We believe that the error‐detection paradigm has a place in the classroom. The demands of teaching can result in an emphasis on performance or procedures at the expense of conceptual understanding. This is likely to produce the gains in procedural mastery we observed (i.e. procedural counting reaching ceiling before the age of 6), but our results highlight the potential of error‐detection performance to predict the more advanced type of numerical thinking that teachers want to foster. Educational studies have found that 7‐year‐olds identified as having arithmetical deficiencies are relatively poor at identifying violations to the counting principles ([<reflink idref="bib15" id="ref54">15</reflink>]), and that a central feature of slow mathematical development is an inflexible conceptualization of the counting procedure. The present findings suggest that an explicit sensitivity to the rules governing counting and sharing is associated with a child breaking free from this type of inflexible thinking.</p> <p>Accordingly, we end by recommending the inclusion of activities where children are asked to focus on another's procedural performance, in particular, where children have to monitor another's sharing for mistakes. However, a causal account is required before such a recommendation can be accepted. [<reflink idref="bib32" id="ref55">32</reflink>] found that asking children to explain judgments increases their conceptual understanding of balance‐scale problems. We believe it would be possible to incorporate a miscount/mis‐share‐detection paradigm into a training study to examine a number of possible influences on children's conceptual grasp of the counting and sharing procedures. It might be that mere exposure to procedural errors, or giving children feedback on their judgments about another's procedural accuracy, is sufficient to promote children's interest in the principles that underlie counting and sharing. Alternatively, gains in conceptual understanding might require prompts for children to reflect on – by way of providing an explanation for – the significance of procedural accuracy. Notwithstanding these questions still to be answered, we suggest that although children's conceptual understanding of well‐rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.</p> <hd id="AN0027563598-14">Acknowledgements</hd> <p>We would like to thank the staff and children of Cathedral Catholic Primary School and Scotforth Church of England Primary School, Lancaster, and Great Wood Primary School, Morecambe for their participation and enthusiasm for this project. The first‐named author also wishes to thank the Economic and Social Research Council, UK, for funding the doctoral position during which research was conducted.</p> <ref id="AN0027563598-15"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref34" type="bt">1</bibl> <bibtext> <sups>3</sups> <emph>N</emph> <sups>obs</sups> refers to the number of observations on which the analysis is based (e.g. 75 = the number of times a score rose from 0 to either 1 or 2).</bibtext> </blist> <blist> <bibl id="bib2" idref="ref7" type="bt">2</bibl> <bibtext> <sups>2</sups>Except for the counting proficiency tasks – Tasks 6 and 7 – which were scored as either 0 or 1.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref46" type="bt">3</bibl> <bibtext> <sups>1</sups>The ratio of boys to girls is significantly different (P<.05). However, this reflects the composition of the three classes of children taking part.</bibtext> </blist> </ref> <ref id="AN0027563598-16"> <title> References </title> <blist> <bibtext> Agresti A., Caffo B. Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures American Statistician 54 280 – 288 2000.</bibtext> </blist> <blist> <bibtext> Baroody A. The development of preschooler's counting skills and principles Bideaud J., Meljac C., Fischer J.‐P Pathways to number: Children's developing numerical abilities. 99 – 126. Hillsdale, NJ: Lawrence Erlbaum and Associates 1992.</bibtext> </blist> <blist> <bibtext> Bialystok E. Symbolic representation of letters and numbers Cognitive Development 7 301 – 316 1992.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref8" type="bt">4</bibl> <bibtext> Briars D., Siegler R. S. 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  Data: Predictors of Early Numeracy: Is There a Place for Mistakes when Learning about Number?
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  Data: British Psychological Society. St Andrews House, 48 Princess Road East, Leicester, LE1 7DR, UK. Tel: +44-116-254-9568; Fax: +44-116-227-1314; e-mail: enquiry@bps.org.uk; Web site: http://www.bpsjournals.co.uk
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  Data: <searchLink fieldCode="DE" term="%22Numeracy%22">Numeracy</searchLink><br /><searchLink fieldCode="DE" term="%22Young+Children%22">Young Children</searchLink><br /><searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Developmental+Psychology%22">Developmental Psychology</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Analysis%22">Statistical Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Error+Patterns%22">Error Patterns</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic%22">Arithmetic</searchLink><br /><searchLink fieldCode="DE" term="%22Numbers%22">Numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Predictor+Variables%22">Predictor Variables</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1348/026151007X174501
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0261-510X
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy-two 5-year-olds were given a battery of nine tasks on each of three sessions (at 3-monthly intervals). Tasks measured procedural proficiency, conceptual understanding (using an error-detection paradigm) and the ability to compare sets using number knowledge. Procedural skills remained fairly stable over the 6-month period, and preceded children's ability to detect another's violations to those procedures. Regression analysis revealed that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinal numbers for set comparisons. We suggest that although children's conceptual understanding of well-rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2011
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ940091
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ940091
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1348/026151007X174501
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 16
        StartPage: 543
    Subjects:
      – SubjectFull: Numeracy
        Type: general
      – SubjectFull: Young Children
        Type: general
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Developmental Psychology
        Type: general
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Statistical Analysis
        Type: general
      – SubjectFull: Error Patterns
        Type: general
      – SubjectFull: Arithmetic
        Type: general
      – SubjectFull: Numbers
        Type: general
      – SubjectFull: Predictor Variables
        Type: general
    Titles:
      – TitleFull: Predictors of Early Numeracy: Is There a Place for Mistakes when Learning about Number?
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Muldoon, Kevin P.
      – PersonEntity:
          Name:
            NameFull: Lewis, Charlie
      – PersonEntity:
          Name:
            NameFull: Berridge, Damon
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 11
              Type: published
              Y: 2007
          Identifiers:
            – Type: issn-print
              Value: 0261-510X
          Numbering:
            – Type: volume
              Value: 25
            – Type: issue
              Value: 4
          Titles:
            – TitleFull: British Journal of Developmental Psychology
              Type: main
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