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Cross-Notation Knowledge of Rational Numbers Predicts Fraction Arithmetic

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Title: Cross-Notation Knowledge of Rational Numbers Predicts Fraction Arithmetic
Language: English
Authors: Boby Ho-Hong Ching (ORCID 0000-0002-3526-7704), Xiang Yu Li, Tiffany Ting Chen
Source: British Journal of Educational Psychology. 2024 94(3):717-737.
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 21
Publication Date: 2024
Document Type: Journal Articles
Reports - Research
Descriptors: Number Concepts, Fractions, Arithmetic, Young Children, Prior Learning, Learning Processes, Mathematics Skills, Predictor Variables, Foreign Countries, Cognitive Ability, Intelligence, Computation, Recall (Psychology), Beginning Reading, Phonological Awareness, Addition, Subtraction
Geographic Terms: China
DOI: 10.1111/bjep.12674
ISSN: 0007-0998
2044-8279
Abstract: Background: Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered. Aims: To examine the extent to which children's earlier knowledge of cross-notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally. Sample: Three hundred and fifty-four Chinese children (Mage = 112.1 months). Methods: During the first wave of assessment, a range of cognitive abilities of children were measured, including within-notation fraction and decimal magnitude comparisons, whole-number arithmetic fluency, non-verbal intelligence, attentive behaviours, counting recall, word-level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division. Results and Conclusions: Multiple linear regressions showed that within-notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole-number arithmetic were controlled. Cross-notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within-notation fraction and decimal magnitude knowledge and other covariates. However, within-notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross-notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross-notation knowledge in the assessments of children's mathematics abilities and teaching.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1434514
Database: ERIC
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  Value: <anid>AN0178883000;6kx01sep.24;2024Aug09.05:58;v2.2.500</anid> <title id="AN0178883000-1">Cross‐notation knowledge of rational numbers predicts fraction arithmetic </title> <p>Background: Recent research showed that cross‐notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered. Aims: To examine the extent to which children's earlier knowledge of cross‐notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally. Sample: Three hundred and fifty‐four Chinese children (Mage = 112.1 months). Methods: During the first wave of assessment, a range of cognitive abilities of children were measured, including within‐notation fraction and decimal magnitude comparisons, whole‐number arithmetic fluency, non‐verbal intelligence, attentive behaviours, counting recall, word‐level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division. Results and Conclusions: Multiple linear regressions showed that within‐notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole‐number arithmetic were controlled. Cross‐notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within‐notation fraction and decimal magnitude knowledge and other covariates. However, within‐notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross‐notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross‐notation knowledge in the assessments of children's mathematics abilities and teaching.</p> <p>Keywords: arithmetic; cross‐notation knowledge; decimals; fractions; integrated theory of numerical development; magnitude knowledge</p> <p>The objective of this study was to examine the predictive role of cross‐notation knowledge of fractions and decimals on children's fraction performance longitudinally. Previous research indicates that proficiency in rational numbers is a strong predictor of mathematical achievement (Booth et al., [<reflink idref="bib8" id="ref1">8</reflink>]; Siegler et al., [<reflink idref="bib62" id="ref2">62</reflink>]). However, in most educational settings, children typically acquire knowledge of whole numbers before they learn about rational numbers. This sequential acquisition may hinder their progress in learning rational numbers because the properties of whole numbers do not necessarily apply to other types of numbers (Ni & Zhou, [<reflink idref="bib48" id="ref3">48</reflink>]; Vamvakoussi et al., [<reflink idref="bib72" id="ref4">72</reflink>]; Van Hoof, Janssen, et al., [<reflink idref="bib74" id="ref5">74</reflink>]; Van Hoof, Vandewalle et al., [<reflink idref="bib75" id="ref6">75</reflink>]). One particular challenge in learning fractions is that they possess multiple meanings, such as magnitudes, parts of a whole, and ratios (Behr et al., [<reflink idref="bib6" id="ref7">6</reflink>]). These meanings are based on multiplicative relationships rather than additive ones (Gray et al., [<reflink idref="bib32" id="ref8">32</reflink>]; Lamon, [<reflink idref="bib39" id="ref9">39</reflink>]; Nunes & Bryant, [<reflink idref="bib49" id="ref10">49</reflink>]). Thus, understanding fractions presents a complex learning task for children.</p> <p>Research has consistently shown that a significant number of children struggle with fraction arithmetic, as evidenced by various studies (e.g., Hansen et al., [<reflink idref="bib34" id="ref11">34</reflink>]; Siegler & Pyke, [<reflink idref="bib66" id="ref12">66</reflink>]). Common errors in arithmetic include mistakenly adding the numerators and denominators when adding fractions (e.g., 4/7 + 5/8 = 9/15) and forgetting to multiply the common denominator when multiplying fractions (e.g., 1/7 × 3/7 = 3/7), among others. Given the challenges students face in learning rational numbers, it is crucial to investigate the factors that contribute to success in rational number arithmetic. Such insights can inform the development of improved curriculum designs and more effective instructional strategies. Building upon the integrated theory of numerical development (Siegler et al., [<reflink idref="bib67" id="ref13">67</reflink>]), the present study examined the predictive role of cross‐notation knowledge of fraction and decimal magnitudes in children's fraction arithmetic performance.</p> <hd id="AN0178883000-2">MAGNITUDE UNDERSTANDING OF RATIONAL NUMBERS</hd> <p>The integrated theory of numerical development (Siegler et al., [<reflink idref="bib67" id="ref14">67</reflink>]) posits that numerical development encompasses the understanding that all real numbers possess magnitudes that can be positioned and ordered on number lines. According to this theory, magnitude understanding plays an important role in the acquisition of arithmetic skills. Consistent with this perspective, accurate estimations of magnitudes for whole numbers have been found to be crucial across various domains of mathematical development. These domains include counting (e.g., Whyte & Bull, [<reflink idref="bib78" id="ref15">78</reflink>]), whole‐number arithmetic (e.g., Booth & Siegler, [<reflink idref="bib9" id="ref16">9</reflink>]), memory for and categorization of numbers (e.g., Thompson & Siegler, [<reflink idref="bib69" id="ref17">69</reflink>]), and overall mathematical achievement (e.g., Halberda et al., [<reflink idref="bib33" id="ref18">33</reflink>]).</p> <p>According to this theory, one implication is that difficulties in rational number arithmetic may arise from challenges in comprehending the magnitudes associated with rational numbers. It has been suggested that knowledge of rational number magnitudes contributes to arithmetic performance by enhancing the estimation of arithmetic results. This, in turn, helps individuals identify and reject implausible answers and facilitates the accurate retrieval of correct solutions (Booth & Siegler, [<reflink idref="bib9" id="ref19">9</reflink>]). Indeed, research has consistently shown that a better understanding of fraction magnitude knowledge is linked to improved performance in arithmetic, both for fractions (Bailey et al., [<reflink idref="bib4" id="ref20">4</reflink>]; Siegler et al., [<reflink idref="bib67" id="ref21">67</reflink>]; Siegler & Pyke, [<reflink idref="bib66" id="ref22">66</reflink>]; Torbeyns et al., [<reflink idref="bib71" id="ref23">71</reflink>]) and decimals (Rittle‐Johnson & Koedinger, [<reflink idref="bib57" id="ref24">57</reflink>]). Experimental studies have provided evidence that a causal relation exists between fraction magnitude understanding and proficiency in fraction arithmetic (Dyson et al., [<reflink idref="bib27" id="ref25">27</reflink>]; Fuchs et al., [<reflink idref="bib29" id="ref26">29</reflink>]).</p> <p>Despite its importance, research has shown that students, up to at least eighth grade, often struggle with understanding equivalence within a notation, specifically in fractions and decimals (e.g., Behr et al., [<reflink idref="bib5" id="ref27">5</reflink>]; Braithwaite & Siegler, [<reflink idref="bib12" id="ref28">12</reflink>]; Fyfe & Brown, [<reflink idref="bib30" id="ref29">30</reflink>]; Knuth et al., [<reflink idref="bib38" id="ref30">38</reflink>]; Matthews et al., [<reflink idref="bib43" id="ref31">43</reflink>]). For example, many eighth‐grade students did not recognize that the product of multiplying a fraction by A/A (where A is a nonzero whole number) results in the same quantity as the original fraction. As a result, they may perceive equivalent fractions with larger numerators and denominators (e.g., 8/10 vs. 4/5) as larger fractions. While there was improvement in accurately judging fraction equivalence from fourth to eighth grade, approximately one‐third of the students in the United States still demonstrated significant inaccuracies in this area even by eighth grade (Braithwaite & Siegler, [<reflink idref="bib12" id="ref32">12</reflink>]). Similar findings have been observed in different countries (Christou, [<reflink idref="bib24" id="ref33">24</reflink>]; McMullen & Van Hoof, [<reflink idref="bib46" id="ref34">46</reflink>]; Ni & Zhou, [<reflink idref="bib48" id="ref35">48</reflink>]; Obersteiner et al., [<reflink idref="bib50" id="ref36">50</reflink>]; Vamvakoussi & Vosniadou, [<reflink idref="bib73" id="ref37">73</reflink>]; Van Hoof, Janssen, et al., [<reflink idref="bib74" id="ref38">74</reflink>]; Van Hoof, Vandewalle et al., [<reflink idref="bib75" id="ref39">75</reflink>]).</p> <p>Evidence also suggests that many students aged 9–11 struggle with understanding decimal equivalence. For example, they failed to recognize that 0.5 is equal to 0.50 or 0.500 (Durkin & Rittle‐Johnson, [<reflink idref="bib26" id="ref40">26</reflink>]). These students often overlook the significance of zeros in the tenth place, leading to misconceptions such as claiming that 0.07 is equal to 0.7 or believing that adding a zero at the end of a decimal makes the number larger (e.g., 0.70 > 0.7). These findings highlight the common challenges students face in understanding equivalence within notations, including fractions and decimals, and the need for targeted instructional support to address these misconceptions.</p> <hd id="AN0178883000-3">CROSS‐NOTATION KNOWLEDGE OF FRACTIONS AND DECIMALS MAGNITUDES</hd> <p>Vergnaud ([<reflink idref="bib76" id="ref41">76</reflink>]) argues that a mathematical concept can be represented in different ways, providing different perspectives on the same underlying concepts. According to this view, the ability to connect different representations of rational numbers, such as fractions and decimals, can facilitate students' performance in fraction arithmetic. The integrated theory of numerical development (Siegler et al., [<reflink idref="bib67" id="ref42">67</reflink>]) also emphasizes the importance of integrating numerical knowledge. It suggests that numerical development involves the recognition that many properties that hold true for whole numbers do not necessarily apply to all types of numbers. To succeed in mathematics, children need to transition from initially understanding numbers based on the prominent characteristics of whole numbers to later recognizing the distinctions between the defining properties of all real numbers and those that only apply to specific types of numbers. This theory explains why knowledge of fractions is important for mathematical achievement. It provides an opportunity for children to realize that certain properties of whole numbers are not universally applicable to all numbers. Therefore, integrating and differentiating between whole number and rational number knowledge is essential for the numerical development of children.</p> <p>In a recent study by Braithwaite et al. ([<reflink idref="bib11" id="ref43">11</reflink>]), a complementary aspect of the integration process has been emphasized – an integration of magnitude knowledge within rational numbers. Based on their arguments, it is not enough to understand fractions and decimals separately (referred to as <emph>within‐notation knowledge</emph>), such as determining which is larger between 5/8 and 4/3 in a fraction comparison task or between 0.36 and 0.5 in a decimal comparison task. It is equally important to understand the relation between these notations (known as <emph>cross‐notation knowledge</emph>), for example, determining which is larger between 2/3 and 0.8. Braithwaite et al. ([<reflink idref="bib11" id="ref44">11</reflink>]) postulate that integrating knowledge of fractions and decimals plays an important role in refining individuals' understanding of fractions, decimals, and rational numbers as a whole.</p> <p>It has been suggested that the ultimate goal for representing numerical magnitudes is to have an integrated mental number line that incorporates both within‐notation and cross‐notation relations. This integrated mental number line allows for precise estimation of individual numbers as well as accurate comparisons and calculations involving numbers expressed in different notations (Braithwaite et al., [<reflink idref="bib11" id="ref45">11</reflink>]; Schiller & Siegler, [<reflink idref="bib60" id="ref46">60</reflink>]). For instance, when asked to estimate the sum of "4/9 + 4/13," employing either a within‐notation approximate translation (such as 1/2 + 1/3) or a cross‐notation approximate translation (like 0.5 + 0.3) would likely aid in providing a reasonable answer.</p> <p>The use of cross‐notation approximation could also potentially help prevent students from making errors influenced by a bias towards whole numbers in estimation. For example, estimating 12/13 + 7/8 as 19 (the sum of the numerators) or 21 (the sum of the denominators) (Carpenter et al., [<reflink idref="bib13" id="ref47">13</reflink>]). Similarly, comparing 20/27 and 24/49 might initially appear challenging, but employing either a within‐notation approximation (such as 3/4 vs. 1/2) or a cross‐notation approximation (like 75% vs. 50%) is likely to facilitate the comparison (Schiller & Siegler, [<reflink idref="bib60" id="ref48">60</reflink>]). Indeed, some 6th and 8th‐grade students relied on translating fractions to decimals or percentages when asked to estimate the magnitudes of fractions on number lines (Siegler et al., [<reflink idref="bib67" id="ref49">67</reflink>]). Rather than directly placing the fractions on the number line, these students tended to convert the fractions into decimals or percentages before making their estimations. Taken together, an integrated mental number line with both types of relations may enable a comprehensive understanding of numerical magnitudes and facilitate individuals' performance in various mathematical tasks.</p> <p>Previous research has employed tasks that assess participants' abilities to convert between fractions and decimals and engage in cross‐notation comparison or ordering (Binzak & Hubbard, [<reflink idref="bib7" id="ref50">7</reflink>]; McMullen et al., [<reflink idref="bib45" id="ref51">45</reflink>]; Van Hoof, Janssen, et al., [<reflink idref="bib74" id="ref52">74</reflink>]; Van Hoof, Vandewalle et al., [<reflink idref="bib75" id="ref53">75</reflink>]). However, these studies did not directly investigate the connections between cross‐notation knowledge and other aspects of rational number understanding. More recently, Park and Esposito ([<reflink idref="bib51" id="ref54">51</reflink>]) conducted a study involving children aged 10 to 12. They found that children who demonstrated a conceptual understanding of fractions and decimals as quantities across different notations had higher levels of mathematical achievement. This evidence suggests that recognizing the relation between fractions and decimals across different notations can contribute to overall mathematical proficiency in children.</p> <p>In the domain of rational numbers specifically, Mazzocco and Devlin ([<reflink idref="bib44" id="ref55">44</reflink>]) showed that difficulties in relating numerical representations across different notations were linked to a higher risk of encountering general difficulties in understanding rational numbers among middle school students. Braithwaite et al. ([<reflink idref="bib11" id="ref56">11</reflink>]) examined the relation between cross‐notation magnitude knowledge and performance in fractional and decimal arithmetic among children in grades 4–6. It was found that individual differences in cross‐notation magnitude knowledge made unique contributions to the variance in both fraction and decimal arithmetic performance. Importantly, these contributions remained significant even after controlling for within‐notation fraction and decimal magnitude knowledge. This suggests that cross‐notation knowledge provides benefits beyond those conferred by within‐notation knowledge alone. The study also revealed that fraction magnitude knowledge did not predict fraction arithmetic more strongly than decimal arithmetic, and vice versa for decimal magnitude knowledge. These findings indicate that the associations between fraction and decimal magnitude knowledge and fraction and decimal arithmetic were not specific to a particular notation. Instead, knowledge of each notation contributed to the successful performance of arithmetic tasks involving the other notation as well.</p> <p>Consistent with previous evidence, Schiller and Siegler ([<reflink idref="bib60" id="ref57">60</reflink>]) also showed that an integrated number sense, which encompasses the skilful representation and comparison of magnitudes within and across different notations, was predictive of overall mathematical proficiency and specific mathematical outcomes, including the estimation of fraction sums. The study conducted by Hurst and Cordes ([<reflink idref="bib36" id="ref58">36</reflink>]) revealed that children's ability to compare fractions and decimals significantly predicted their pre‐algebra knowledge. This finding held true even after accounting for their performance in rational number arithmetic and their grade level. The results suggest that the skill of comparing fractions and decimals has implications for a broader mathematical understanding beyond just rational number arithmetic.</p> <p>Despite its importance, research suggests that a significant number of middle and high school students in the United States and several other countries lack a strong understanding of cross‐notation magnitudes. For instance, in Greece, only 30% of 8th‐grade students were found to fluently transition between fractions and decimals, such as performing operations like 1/10*45 = 0.1*45. In the same study, only 8% of students accurately translated between percentages and other notations, like comprehending that 10% of 45 is equivalent to 1/10 of 45, which is also equal to 0.1 of 45 (Lemonidis & Pilianidis, [<reflink idref="bib40" id="ref59">40</reflink>]). Even in high school, students often hold misconceptions about the relations between fractions and decimals. For example, some students claimed that only decimals, not fractions, exist between other decimals, and vice versa, that only fractions, not decimals, exist between other fractions (Vamvakoussi & Vosniadou, [<reflink idref="bib73" id="ref60">73</reflink>]). Another study showed that many middle school students struggled to correctly order fractions and decimals when presented with numbers in both notations and asked to rank them from least to greatest. They often made errors, such as considering all fractions to be larger than any decimals or vice versa (Mazzocco & Devlin, [<reflink idref="bib44" id="ref61">44</reflink>]). These findings suggest that despite years of exposure to rational numbers, a significant number of students have not fully integrated their understanding of different notations into a unified mental model. The lack of an integrated understanding of rational number notations indicates a need for targeted instruction and support to promote a more comprehensive grasp of these concepts.</p> <hd id="AN0178883000-4">OVERVIEW AND HYPOTHESES</hd> <p>While previous research has demonstrated an association between cross‐notation magnitude knowledge and fraction arithmetic in children, studies with a longitudinal design are lacking. Bradley and Bryant ([<reflink idref="bib10" id="ref62">10</reflink>]) argue that both longitudinal and intervention studies are important for establishing a cause‐and‐effect relation between variables. Intervention studies are particularly useful in determining the causal connection between specific skills and mathematical achievement. By employing this type of study design, we can investigate whether training in cross‐notation magnitude knowledge, for instance, leads to an enhancement of these skills and fraction arithmetic. However, prior to implementing an intervention study, it is necessary to identify factors that significantly influence children's fraction arithmetic. Longitudinal studies provide a valuable opportunity to examine the chronological sequence of events. Understanding whether a predictor precedes fraction arithmetic is essential, as it serves as a prerequisite for determining causal relation between variables. Statistical techniques, such as multiple regression analysis, can be employed to identify the direction and strength of associations between a predictor and fraction arithmetic. These techniques can also allow for comparing the unique contributions of each predictor to the variation in the outcome. Thus, a longitudinal study is regarded as an important initial step in the development of an intervention.</p> <p>However, the majority of research in the area of interest of this study has utilized a cross‐sectional research design. This cross‐sectional evidence has provided a snapshot of the association between cross‐notation magnitude knowledge and children's fraction arithmetic. However, it does not capture how well cross‐notation magnitude knowledge predicts later performance in fraction arithmetic beyond a range of cognitive factors at an earlier time point. Thus, we employed a longitudinal design spanning one year to address this limitation and sought to provide a rigorous examination of the unique contributions of cross‐notation knowledge to fraction arithmetic by incorporating several covariates.</p> <p>Specifically, the study considered the potential interplay between fluency in whole‐number arithmetic and domain‐general cognitive resources in supporting fraction arithmetic performance. One potential cognitive covariate is working memory, which plays a role in various higher cognitive tasks that require the simultaneous representation and manipulation of information (Baddeley & Hitch, [<reflink idref="bib3" id="ref63">3</reflink>]). According to Baddeley's theoretical model ([<reflink idref="bib2" id="ref64">2</reflink>]), working memory consists of four components: the phonological loop, visuospatial sketchpad, central executive, and episodic buffer. The phonological loop is responsible for storing and rehearsing auditory information, while the visuospatial sketchpad performs a similar function for visuospatial information. The central executive enables concurrent processing and storage of information, and the episodic buffer facilitates interaction between the central executive and long‐term memory.</p> <p>Working memory has been argued to be important for mathematics competence across various developmental stages, from initial skill acquisition to the integration of new information with existing mathematical knowledge. Previous studies have shown that working memory makes a unique contribution to quantitative reasoning and rational number arithmetic performance, even after controlling for other cognitive factors (Ching, [<reflink idref="bib15" id="ref65">15</reflink>]; Jordan et al., [<reflink idref="bib37" id="ref66">37</reflink>]). Non‐verbal intelligence, language and literacy skills, and attentive behaviour have also been found to be related to success in rational number arithmetic (Bailey et al., [<reflink idref="bib4" id="ref67">4</reflink>]; Jordan et al., [<reflink idref="bib37" id="ref68">37</reflink>]; Siegler et al., [<reflink idref="bib62" id="ref69">62</reflink>]; Vukovic et al., [<reflink idref="bib77" id="ref70">77</reflink>]). These cognitive resources may further support the development of fraction arithmetic skills. By considering these covariates, the study aimed to provide a comprehensive analysis of the unique contributions of cross‐notation knowledge to fraction arithmetic performance while accounting for the influence of other cognitive factors.</p> <p>Simple arithmetic knowledge, particularly whole‐number arithmetic, has long been recognized as a crucial foundation for learning more complex mathematical skills (Pillay et al., [<reflink idref="bib54" id="ref71">54</reflink>]). The relation between whole‐number arithmetic and fraction arithmetic is noteworthy because the calculation processes often involve manipulation of whole numbers. Proficiency in whole‐number arithmetic can also reduce the demands on domain‐general cognitive resources, allowing these resources to be allocated more efficiently to support the processing of fraction and decimal arithmetic problems. Children who demonstrate fluency in whole‐number arithmetic have an advantage when it comes to fraction arithmetic. They can allocate more cognitive resources to the selection and implementation of appropriate fraction procedures.</p> <p>In contrast, children with weaker proficiency in whole‐number arithmetic may experience a disadvantage. They must allocate more cognitive resources to calculating whole‐number arithmetic, which can come at the expense of the cognitive processes necessary for selecting and executing fraction computation‐related procedures. By having a solid foundation in whole‐number arithmetic, children can free up cognitive resources to handle the unique challenges of fraction arithmetic, such as understanding the concept of equivalence, comparing and ordering fractions, or performing operations with fractions. Therefore, the proficiency in whole‐number arithmetic may support the development of fraction arithmetic skills. Given the potential interplay between whole‐number arithmetic, domain‐general cognitive resources, and cross‐notation knowledge, it is indeed crucial to consider these factors simultaneously when examining the contributions to success in fraction arithmetic.</p> <p>Based on the existing literature, the study had two hypotheses. The first hypothesis was that within‐notation fraction and decimal magnitude knowledge would longitudinally predict fraction arithmetic performance, even after controlling for the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole‐number arithmetic. This hypothesis suggests that an individual's understanding of fraction and decimal magnitudes within their respective notations would have a significant impact on their ability to perform fraction arithmetic tasks over time.</p> <p>The second hypothesis was that cross‐notation magnitude knowledge would make independent contributions to fraction arithmetic performance longitudinally, beyond the influence of within‐notation fraction and decimal magnitude knowledge and other covariates. This hypothesis suggests that the ability to understand and compare magnitudes across different notations (fractions and decimals) would have a unique effect on fraction arithmetic performance, even when accounting for other cognitive factors and whole‐number arithmetic proficiency. By testing these hypotheses, the study aimed to provide a comprehensive understanding of the distinct contributions of within‐notation and cross‐notation knowledge to the development of fraction arithmetic skills, while considering the influence of other cognitive factors and whole‐number arithmetic proficiency.</p> <hd id="AN0178883000-5">METHOD</hd> <p></p> <hd id="AN0178883000-6">Procedure and participants</hd> <p>This study obtained ethical approval from the institutional research ethics committee at the researchers' university. Participants were recruited through personal contacts with local primary schools in the Guangdong‐Hong Kong‐Macao Greater Bay Area. Initially, six schools were contacted, and three of them agreed to participate in the study. The children, their parents, and school principals were provided with letters informing them about the study, and active consent was obtained from all participants. No monetary remuneration was provided for participation.</p> <p>In the first wave of assessment, a total of 375 children participated in the study. However, data from 21 children could not be included in the analyses due to either (a) providing random responses or (b) failing to complete all the measures. Therefore, the final sample for analysis consisted of 354 Chinese children who completed all the measures in both waves of assessments without any missing data. Among the participants, 173 identified as male (48.9%) and 181 identified as female. Prior to conducting the study, an a‐priori power analysis using G*Power software indicated that a sample size of 114 participants would be necessary to achieve a power of 0.80, with an alpha level of 0.05, to detect a medium effect size (<emph>f</emph><sups><emph>2</emph></sups> = 0.15). Therefore, the sample size of 354 participants in this study was considered sufficient to meet the power requirements for the analysis.</p> <p>During the first wave of assessment, all participants were in Grade 4, with a mean age of 112.1 months (<emph>SD</emph><subs>age</subs> = 2.17). The participants' school teachers reported that they had normal intelligence for their age and did not have any diagnosed learning difficulties such as dyslexia, specific language impairments, or emotional or behavioural problems. To estimate socioeconomic status, the highest educational levels attained by the children's mothers were used as a proxy measure. Students in the upper grades of a primary school in the Guangdong‐Hong Kong‐Macao Greater Bay Area are expected to acquire an understanding of concepts related to fractions and decimals. They should also be able to perform the four arithmetic operations, involving whole numbers, fractions, and decimals.</p> <p>During the first wave of data collection, children provided verbal assent and participated individually in a quiet classroom within their primary schools. The assessment session at Time 1 (Grade 4) lasted approximately 15–20 min and included various measures. The children were administered tests to assess non‐verbal intelligence using Raven's Standard Progressive Matrices. Working memory was evaluated through a counting recall task. Word‐level reading fluency was measured using a Chinese word recognition task, and phonological awareness was assessed with tasks focusing on onset and tone awareness. These measures were administered individually.</p> <p>Paper‐and‐pencil measures were used to evaluate magnitude knowledge, both within‐notation (comparing magnitudes of fractions and decimals within the same notation) and cross‐notation (comparing magnitudes of fractions and decimals across different notations). Paper‐and‐pencil assessment was also conducted for whole‐number arithmetic. These measures were administered in a group setting, with approximately 30 participants per group. The presentation of tasks was counterbalanced, meaning that the measures appeared in three different orders to control for any order effects. During Time 1, school teachers also completed a questionnaire on the children's attentive behaviour, providing additional information on this aspect. At Time 2, which was 12 months later, each child participated in a group setting where the measure of fraction arithmetic was administered.</p> <hd id="AN0178883000-7">Measures</hd> <p></p> <hd id="AN0178883000-8">Fraction arithmetic</hd> <p>The fraction arithmetic measure consisted of two tasks: one focused on fraction addition and subtraction, and the other on fraction multiplication and division. The addition and subtraction measure included a total of 30 items involving 15 addition items and 15 subtraction items. Ten of the items (5 addition‐ and 5 subtraction‐items) involved fractions with an identical denominators, while 20 (10 addition‐ and 10 subtraction‐items) involved fractions with different denominators. Similarly, in the fraction multiplication and division task, the children were presented with 30 problems that included 15 multiplication items and 15 division items. Ten of the items (5 multiplication‐ and 5 division‐items) involved fractions with an identical denominators, while 20 (10 multiplication‐ and 10 division‐items) involved fractions with different denominators. The children were awarded one point for each correct answer. The maximum possible score for each task was 30. The internal consistencies of both measures were assessed and found to be good. The coefficient alpha (<emph>α</emph>) for the addition and subtraction task was.85, indicating a high level of internal consistency. Similarly, the coefficient alpha for the multiplication and division task was.82, also indicating good internal consistency.</p> <hd id="AN0178883000-9">Within‐ and cross‐notation knowledge of fractions and decimals</hd> <p>The magnitude knowledge assessment consisted of three tasks: within‐notation fraction magnitude comparison, within‐notation decimal magnitude comparison, and cross‐notation magnitude comparison between fractions and decimals. In the within‐notation fraction magnitude comparison task, participants were presented with pairs of fractions (e.g., 5/8, 4/3) and were asked to circle the larger fraction. Similarly, in the within‐notation decimal magnitude comparison task, pairs of decimals (e.g., 0.36, 0.5) were presented, and participants were asked to circle the larger decimal. If the numbers had the same magnitude, participants were instructed to circle both numbers. Each task included 15 items, resulting in a total of 15 fraction items and 15 decimal items.</p> <p>The cross‐notation magnitude comparison task involved comparing magnitudes between fractions and decimals. Each problem in this task required participants to compare one fraction and one decimal (e.g., 1/8, 0.8) and circle the number with the larger magnitude. Again, participants received one point for each correct answer. The cross‐notation task also consisted of 15 items. The internal consistencies of all three tasks were assessed and found to be good. The within‐notation fraction magnitude comparison task had a coefficient alpha (<emph>α</emph>) of.87, indicating high internal consistency. Similarly, both the within‐notation decimal magnitude comparison task (<emph>α</emph> = .86) and the cross‐notation magnitude comparison task (<emph>α</emph> = .82) had good internal consistency.</p> <hd id="AN0178883000-10">Whole‐number arithmetic fluency</hd> <p>The whole‐number arithmetic measure included four operations: single‐digit addition, single‐digit subtraction, single‐digit multiplication, and single‐digit division. For single‐digit addition, the combinations of addends ranged from 0 to 9. Similarly, for single‐digit subtraction, the subtrahends ranged from 0 to 9. In the case of single‐digit multiplication, the multiplicands and multipliers also ranged from 0 to 9. Finally, for single‐digit division, the dividends and divisors ranged from 0 to 9. Participants were asked to complete as many of the 60 problems as possible within a one‐minute time limit. They were required to compute the correct answers for each problem and were awarded one point for each correct answer. The maximum possible score for this whole‐number arithmetic task was 60.</p> <hd id="AN0178883000-11">Non‐verbal intelligence</hd> <p>The non‐verbal intelligence measure used in the study was based on three subtests of Raven's Standard Progressive Matrices (Raven et al., [<reflink idref="bib55" id="ref72">55</reflink>]). Each subtest consisted of 12 items. In each item, participants were presented with a target matrix that had a missing piece. They were then provided with 6 to 8 options from which they had to select the best figure to complete the target matrix. For each item, participants received one point for a correct answer. Therefore, the maximum possible score for each subtest was 12, and the maximum possible score for all three subtests combined was 36.</p> <hd id="AN0178883000-12">Working memory</hd> <p>The Counting Recall subtest, which was based on the Working Memory Test Battery for Children developed by Pickering and Gathercole ([<reflink idref="bib53" id="ref73">53</reflink>]), was employed in this study for testing children's working memory. This working memory measure has also been used in prior studies conducted with Chinese children (Ching, [<reflink idref="bib14" id="ref74">14</reflink>]; Ching et al., [<reflink idref="bib19" id="ref75">19</reflink>]). In the Counting Recall task, children were presented with sets of four, five, six, or seven dots, with each dot on a separate card. They were instructed to count the number of dots on each card and then recall the number of dots on each card. The task consisted of six trials at span levels ranging from one to six. At each span level, the number of items to be recalled increased by one. For example, at span level one, children had to recall the number of dots on a single card. At span level two, they had to recall the numbers on two cards, and so on. The test was terminated when a child failed three trials of the same length. Two practice items were given to familiarize children with the task, and no feedback was provided during the test. Each correct answer was awarded one point. The maximum possible score for the Counting Recall task was 36.</p> <hd id="AN0178883000-13">Attentive behaviour</hd> <p>In this study, the criteria for attention‐deficit/hyperactivity disorder (ADHD) from the Diagnostic and Statistical Manual of Mental Disorders (DSM‐5; American Psychiatric Association, [<reflink idref="bib1" id="ref76">1</reflink>]) were used to assess the presence of ADHD symptoms in the participating children. Specifically, the head teacher of each child was asked to rate the extent to which the child met certain criteria for ADHD. The ratings were made using a 7‐point scale, ranging from 1 (<emph>very frequent occurrence</emph>) to 7 (<emph>very infrequent occurrence</emph>). The ratings were collected for two domains: inattention (items 1–9) and hyperactivity and impulsivity (items 10–18). To calculate a summary score for each domain, the mean score across all items within the domain was computed. Higher scores on this measure indicated higher levels of attention. The internal consistency of the measure was assessed and found to be high, with a coefficient alpha (<emph>α</emph>) of.94.</p> <hd id="AN0178883000-14">Word‐level literacy</hd> <p>A Chinese word recognition task was used as a proxy measure of literacy. This task was adapted from a similar task used in previous research (Ching & Nunes, [<reflink idref="bib21" id="ref77">21</reflink>], [<reflink idref="bib22" id="ref78">22</reflink>]). In the Chinese word recognition task, children were presented with 30 written two‐character Chinese words and were asked to read them aloud. For each correctly pronounced word, participants received one point. Therefore, the maximum possible score for this task was 30. The task aimed to assess participants' ability to recognize and pronounce Chinese words, serving as an indirect measure of their literacy skills in Chinese. The internal consistency of the task was found to be high, with a coefficient alpha (<emph>α</emph>) of.87.</p> <hd id="AN0178883000-15">Phonological awareness</hd> <p>The phonological awareness measure used in the study consisted of two tasks: onset awareness and tone awareness. These tasks were adapted from previous research (Ching et al., [<reflink idref="bib16" id="ref79">16</reflink>]; Ching & Nunes, [<reflink idref="bib20" id="ref80">20</reflink>]). For the onset awareness task, children were orally presented with three monosyllabic words accompanied by pictures. Each word shared the same tone, but the rhymes were different. However, only two of the words had the same onset. The children's task was to identify which two words had the same onset by pointing to the corresponding picture. Each correct answer was awarded one point. The maximum possible score for this task was 12. Similarly, for the tone awareness task, children were orally presented with three monosyllabic words accompanied by pictures. This time, the three words shared the same rhyme, but the onsets were different. Only two of the words had the same tone. The children were asked to identify which two words had the same tone by pointing to the corresponding picture. Again, each correct answer was given one point, with a maximum possible score of 12 for this task. The scores from the onset awareness and tone awareness tasks were added together, resulting in a combined score representing phonological awareness. Higher scores on this combined measure indicated higher levels of phonological awareness. The internal consistency of the measure, as assessed by coefficient alpha (<emph>α</emph>), was reported to be.92 for the combined phonological awareness measure.</p> <hd id="AN0178883000-16">RESULTS</hd> <p></p> <hd id="AN0178883000-17">Preliminary analyses and bivariate correlations</hd> <p>In our preliminary analyses, we examined whether demographic characteristics (specifically children's sex and maternal educational levels) had an impact on children's performance on each task. Independent t‐tests were conducted for children's sex, and one‐way analysis of variance (ANOVA) was performed for maternal educational levels. Each task, including fraction arithmetic and the cognitive predictors, was analysed separately. The results indicated that none of the cognitive variables under study significantly differed based on participants' gender or maternal educational levels (<emph>p</emph>s > .05). This suggests that children's performance on the tasks was not significantly influenced by these demographic factors.</p> <p>Next, we explored whether there was evidence of clustered observations within school units. Since the participants were recruited from three different schools, the intraclass correlation coefficient (ICC) was computed to examine the clustering effect on the outcome variables. The ICCs were reported to be.008 for fraction addition and subtraction, and.015 for fraction multiplication and division. These values indicate that the clustering effect within the school units was lower than the conventional cut‐off of.05, which suggests substantial clustering (Heck et al., [<reflink idref="bib35" id="ref81">35</reflink>]). Based on these findings, we decided to combine the data from all three schools for subsequent analyses. This decision was made because the observed clustering effect was not substantial, and it was deemed appropriate to pool the data together to increase the sample size and statistical power.</p> <p>In Table 1, the bivariate correlations among all the cognitive predictors are presented. Different patterns of correlations for the two specific tasks, namely fraction addition and subtraction and fraction multiplication and division, were observed. Higher performance in fraction addition and subtraction was correlated with better performance in several cognitive predictors. These predictors included within‐ and cross‐notation magnitude knowledge (knowledge of magnitudes within and across different notations), whole‐number arithmetic fluency, non‐verbal intelligence, attentive behaviours, counting recall, word‐level reading fluency, and phonological awareness. By contrast, we found a more limited set of significant correlations for the tasks of fraction multiplication and division. Specifically, performance in fraction multiplication and division was significantly related to cross‐notation magnitude knowledge, whole‐number arithmetic fluency, attentive behaviours, and counting recall.</p> <p>1 TABLE Bivariate correlations among cognitive variables (N  = 354).</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left">1</th><th align="left">2</th><th align="left">3</th><th align="left">4</th><th align="left">5</th><th align="left">6</th><th align="left">7</th><th align="left">8</th><th align="left">9</th><th align="left">10</th><th align="left">11</th></tr></thead><tbody valign="top"><tr><td align="left">1. Fraction addition and subtraction</td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">2. Fraction multiplication and division</td><td align="left">.407<xref ref-type="fn" rid="tfn2" /></td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">3. Cross‐notation magnitude</td><td align="left">.332<xref ref-type="fn" rid="tfn2" /></td><td align="left">.181<xref ref-type="fn" rid="tfn2" /></td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">4. Fraction magnitude</td><td align="left">.326<xref ref-type="fn" rid="tfn2" /></td><td align="left">0.076</td><td align="left">.178<xref ref-type="fn" rid="tfn2" /></td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">5. Decimal magnitude</td><td align="left">.307<xref ref-type="fn" rid="tfn2" /></td><td align="left">0.081</td><td align="left">.437<xref ref-type="fn" rid="tfn2" /></td><td align="left">.228<xref ref-type="fn" rid="tfn2" /></td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">6. Whole‐number arithmetic</td><td align="left">.340<xref ref-type="fn" rid="tfn2" /></td><td align="left">.161<xref ref-type="fn" rid="tfn2" /></td><td align="left">.166<xref ref-type="fn" rid="tfn2" /></td><td align="left">.293<xref ref-type="fn" rid="tfn2" /></td><td align="left">.163<xref ref-type="fn" rid="tfn2" /></td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">7. Non‐verbal intelligence</td><td align="left">.236<xref ref-type="fn" rid="tfn2" /></td><td align="left">0.021</td><td align="left">0.084</td><td align="left">.209<xref ref-type="fn" rid="tfn2" /></td><td align="left">0.062</td><td align="left">.134<xref ref-type="fn" rid="tfn1" /></td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">8. Attentive behaviours</td><td align="left">.314<xref ref-type="fn" rid="tfn2" /></td><td align="left">.174<xref ref-type="fn" rid="tfn2" /></td><td align="left">.141<xref ref-type="fn" rid="tfn2" /></td><td align="left">.278<xref ref-type="fn" rid="tfn2" /></td><td align="left">.190<xref ref-type="fn" rid="tfn2" /></td><td align="left">.283<xref ref-type="fn" rid="tfn2" /></td><td align="left">.137<xref ref-type="fn" rid="tfn2" /></td><td align="left">1</td><td align="left" /><td align="left" /><td align="left" /></tr><tr><td align="left">9. Counting recall</td><td align="left">.376<xref ref-type="fn" rid="tfn2" /></td><td align="left">.345<xref ref-type="fn" rid="tfn2" /></td><td align="left">.171<xref ref-type="fn" rid="tfn2" /></td><td align="left">.149<xref ref-type="fn" rid="tfn2" /></td><td align="left">.149<xref ref-type="fn" rid="tfn2" /></td><td align="left">.274<xref ref-type="fn" rid="tfn2" /></td><td align="left">.161<xref ref-type="fn" rid="tfn2" /></td><td align="left">.283<xref ref-type="fn" rid="tfn2" /></td><td align="left">1</td><td align="left" /><td align="left" /></tr><tr><td align="left">10. Word‐level reading</td><td align="left">.122<xref ref-type="fn" rid="tfn1" /></td><td align="left">0.078</td><td align="left">0.063</td><td align="left">.164<xref ref-type="fn" rid="tfn2" /></td><td align="left">0.081</td><td align="left">0.039</td><td align="left">.110<xref ref-type="fn" rid="tfn1" /></td><td align="left">0.032</td><td align="left">.113<xref ref-type="fn" rid="tfn1" /></td><td align="left">1</td><td align="left" /></tr><tr><td align="left">11. Phonological awareness</td><td align="left">.112<xref ref-type="fn" rid="tfn1" /></td><td align="left">0.02</td><td align="left">0.026</td><td align="left">0.082</td><td align="left">0.084</td><td align="left">0.094</td><td align="left">0.084</td><td align="left">.133<xref ref-type="fn" rid="tfn1" /></td><td align="left">0.038</td><td align="left">0.078</td><td align="left">1</td></tr><tr><td align="left">Mean</td><td align="left">23.83</td><td align="left">20.23</td><td align="left">9.14</td><td align="left">11.03</td><td align="left">10.87</td><td align="left">42.21</td><td align="left">29.63</td><td align="left">3.96</td><td align="left">26.34</td><td align="left">26.68</td><td align="left">22.10</td></tr><tr><td align="left">SD</td><td align="left">3.52</td><td align="left">3.57</td><td align="left">2.35</td><td align="left">2.12</td><td align="left">2.15</td><td align="left">6.15</td><td align="left">2.79</td><td align="left">1.29</td><td align="left">2.69</td><td align="left">2.02</td><td align="left">1.55</td></tr><tr><td align="left">Theoretical range of scores</td><td align="left">0–30</td><td align="left">0–30</td><td align="left">0–15</td><td align="left">0–15</td><td align="left">0–15</td><td align="left">0–60</td><td align="left">0–36</td><td align="left">1–7</td><td align="left">0–32</td><td align="left">0–30</td><td align="left">0–24</td></tr><tr><td align="left">Observed range of scores</td><td align="left">16–30</td><td align="left">13–29</td><td align="left">2–15</td><td align="left">6–15</td><td align="left">7–15</td><td align="left">26–53</td><td align="left">24–36</td><td align="left">1.11–6.11</td><td align="left">22–32</td><td align="left">22–30</td><td align="left">19–24</td></tr></tbody></table> </ephtml> </p> <p>1 * Correlation is significant at the 0.05 level (2‐tailed).</p> <p>2 ** Correlation is significant at the 0.01 level (2‐tailed).</p> <hd id="AN0178883000-18">Relative contributions of within‐ and cross‐notation magnitude knowledge</hd> <p>We employed multiple linear regression analyses to test our hypotheses regarding the longitudinal prediction of fraction arithmetic performance. Two specific hypotheses were examined: (a) Within‐notation fraction and decimal magnitude knowledge predicted fraction arithmetic longitudinally, even after controlling for the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole‐number arithmetic. (b) Cross‐notation magnitude knowledge made independent contributions to fraction arithmetic performance longitudinally, beyond the influence of within‐notation fraction and decimal magnitude knowledge and other covariates.</p> <p>The results for fraction addition and subtraction are presented in Table 2. In Step 2, within‐notation fraction magnitude knowledge (standardized <emph>β</emph> = .142, <emph>p</emph> = .004), within‐notation decimal magnitude knowledge (standardized <emph>β</emph> = .181, <emph>p <</emph> .001), whole‐number arithmetic fluency (standardized <emph>β</emph> = .155, <emph>p</emph> = .002), counting recall (standardized <emph>β</emph> = .231, <emph>p <</emph> .001), attentive behaviours (standardized <emph>β</emph> = .109, <emph>p</emph> = .028), and non‐verbal intelligence (standardized <emph>β</emph> = .116, <emph>p</emph> = .013), were significant predictors. The predictions of word‐level reading fluency and phonological awareness became non‐significant when other cognitive predictors were present (<emph>ps ></emph> .05). This model accounted for 31.1% of variance. Consistent with our first hypothesis, within‐notation fraction and decimal magnitude knowledge made independent contributions (additional variance explained = 5.5%) to fraction addition and subtraction.</p> <p>2 TABLE Relative contributions of cognitive predictors to fraction addition and subtraction (N  = 354).</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left" /><th align="left">Unstandardized coefficients</th><th align="left">Standardized</th><th align="left"><italic>t</italic></th><th align="left">Sig.</th></tr><tr><th align="left" /><th align="left" /><th align="left"><italic>β</italic></th><th align="left">SE</th><th align="left"><italic>β</italic></th><th align="left" /><th align="left" /></tr></thead><tbody valign="top"><tr><td align="left">Step 1</td><td align="left">Non‐verbal intelligence</td><td align="char" char=".">.174</td><td align="char" char=".">0.060</td><td align="char" char=".">.137</td><td align="char" char=".">2.886</td><td align="char" char=".">.004</td></tr><tr><td align="left" /><td align="left">Attentive behaviours</td><td align="char" char=".">.440</td><td align="char" char=".">0.137</td><td align="char" char=".">.161</td><td align="char" char=".">3.213</td><td align="char" char=".">.001</td></tr><tr><td align="left" /><td align="left">Counting recall</td><td align="char" char=".">.320</td><td align="char" char=".">0.065</td><td align="char" char=".">.244</td><td align="char" char=".">4.883</td><td align="char" char="."><.001</td></tr><tr><td align="left" /><td align="left">Word‐level reading</td><td align="char" char=".">.110</td><td align="char" char=".">0.082</td><td align="char" char=".">.063</td><td align="char" char=".">1.342</td><td align="char" char=".">.180</td></tr><tr><td align="left" /><td align="left">Phonological loop</td><td align="char" char=".">.104</td><td align="char" char=".">0.107</td><td align="char" char=".">.045</td><td align="char" char=".">0.967</td><td align="char" char=".">.334</td></tr><tr><td align="left" /><td align="left">Whole‐number arithmetic</td><td align="char" char=".">.116</td><td align="char" char=".">0.028</td><td align="char" char=".">.203</td><td align="char" char=".">4.084</td><td align="char" char="."><.001</td></tr><tr><td align="left">Step 2</td><td align="left">Non‐verbal intelligence</td><td align="char" char=".">.147</td><td align="char" char=".">0.059</td><td align="char" char=".">.116</td><td align="char" char=".">2.503</td><td align="char" char=".">.013</td></tr><tr><td align="left" /><td align="left">Attentive behaviours</td><td align="char" char=".">.299</td><td align="char" char=".">0.135</td><td align="char" char=".">.109</td><td align="char" char=".">2.211</td><td align="char" char=".">.028</td></tr><tr><td align="left" /><td align="left">Counting recall</td><td align="char" char=".">.302</td><td align="char" char=".">0.063</td><td align="char" char=".">.231</td><td align="char" char=".">4.770</td><td align="char" char="."><.001</td></tr><tr><td align="left" /><td align="left">Word‐level reading</td><td align="char" char=".">.058</td><td align="char" char=".">0.080</td><td align="char" char=".">.033</td><td align="char" char=".">0.724</td><td align="char" char=".">.469</td></tr><tr><td align="left" /><td align="left">Phonological loop</td><td align="char" char=".">.079</td><td align="char" char=".">0.104</td><td align="char" char=".">.035</td><td align="char" char=".">0.761</td><td align="char" char=".">.447</td></tr><tr><td align="left" /><td align="left">Whole‐number arithmetic</td><td align="char" char=".">.089</td><td align="char" char=".">0.028</td><td align="char" char=".">.155</td><td align="char" char=".">3.151</td><td align="char" char=".">.002</td></tr><tr><td align="left" /><td align="left">Fraction magnitude</td><td align="char" char=".">.236</td><td align="char" char=".">0.082</td><td align="char" char=".">.142</td><td align="char" char=".">2.864</td><td align="char" char=".">.004</td></tr><tr><td align="left" /><td align="left">Decimal magnitude</td><td align="char" char=".">.297</td><td align="char" char=".">0.077</td><td align="char" char=".">.181</td><td align="char" char=".">3.878</td><td align="char" char="."><.001</td></tr><tr><td align="left">Step 3</td><td align="left">Non‐verbal intelligence</td><td align="char" char=".">.141</td><td align="char" char=".">0.058</td><td align="char" char=".">.111</td><td align="char" char=".">2.435</td><td align="char" char=".">.015</td></tr><tr><td align="left" /><td align="left">Attentive behaviours</td><td align="char" char=".">.294</td><td align="char" char=".">0.133</td><td align="char" char=".">.107</td><td align="char" char=".">2.208</td><td align="char" char=".">.028</td></tr><tr><td align="left" /><td align="left">Counting recall</td><td align="char" char=".">.285</td><td align="char" char=".">0.063</td><td align="char" char=".">.217</td><td align="char" char=".">4.544</td><td align="char" char="."><.001</td></tr><tr><td align="left" /><td align="left">Word‐level reading</td><td align="char" char=".">.055</td><td align="char" char=".">0.079</td><td align="char" char=".">.031</td><td align="char" char=".">0.698</td><td align="char" char=".">.486</td></tr><tr><td align="left" /><td align="left">Phonological loop</td><td align="char" char=".">.088</td><td align="char" char=".">0.102</td><td align="char" char=".">.039</td><td align="char" char=".">0.867</td><td align="char" char=".">.386</td></tr><tr><td align="left" /><td align="left">Whole‐number arithmetic</td><td align="char" char=".">.083</td><td align="char" char=".">0.028</td><td align="char" char=".">.145</td><td align="char" char=".">2.985</td><td align="char" char=".">.003</td></tr><tr><td align="left" /><td align="left">Fraction magnitude</td><td align="char" char=".">.222</td><td align="char" char=".">0.081</td><td align="char" char=".">.134</td><td align="char" char=".">2.736</td><td align="char" char=".">.007</td></tr><tr><td align="left" /><td align="left">Decimal magnitude</td><td align="char" char=".">.184</td><td align="char" char=".">0.082</td><td align="char" char=".">.112</td><td align="char" char=".">2.245</td><td align="char" char=".">.025</td></tr><tr><td align="left" /><td align="left">Cross‐notation magnitude</td><td align="char" char=".">.256</td><td align="char" char=".">0.074</td><td align="char" char=".">.171</td><td align="char" char=".">3.448</td><td align="char" char=".">.001</td></tr></tbody></table> </ephtml> </p> <p>Adding cross‐notation magnitude knowledge to the model contributed to explaining an additional 2.3% of the variance of the outcome variable. In the final model, cross‐notation fraction magnitude knowledge (standardized <emph>β</emph> = 171, <emph>p</emph> = .001), within‐notation fraction magnitude knowledge (standardized <emph>β</emph> = .134, <emph>p</emph> = .007), within‐notation decimal magnitude knowledge (standardized <emph>β</emph> = .112, <emph>p</emph> = .025), whole‐number arithmetic fluency (standardized <emph>β</emph> = .145, <emph>p</emph> = .003), counting recall (standardized <emph>β</emph> = .217, <emph>p <</emph> .001), attentive behaviours (standardized <emph>β</emph> = .107, <emph>p</emph> = .028), and non‐verbal intelligence (standardized <emph>β</emph> = .111, <emph>p</emph> = .015), were significant predictors. Consistent with our second hypothesis, cross‐notation magnitude knowledge of fractions and decimals was uniquely predictive of children's performance in fraction addition and subtraction.</p> <p>The results for fraction multiplication and division are different (Table 3). In Step 2, within‐notation fraction and decimal magnitude knowledge did not significantly predict fraction multiplication and division. The only significant predictor at this step was counting recall (standardized <emph>β</emph> = .309, <emph>p <</emph> .001). This model accounted for only 13.2% of the variance in total. The non‐significant predictions of within‐notation fraction and decimal magnitude knowledge were not consistent with our first hypothesis. Adding cross‐notation magnitude knowledge to the model contributed to explaining an additional 1.4% of the variance of the outcome variable, which was statistically significant (<emph>p</emph> = .018). In the final model, only cross‐notation magnitude knowledge (standardized <emph>β</emph> = .134, <emph>p</emph> = .018) and counting recall (standardized <emph>β</emph> = .299, <emph>p <</emph> .001) were significant predictors. Consistent with our second hypothesis, cross‐notation magnitude knowledge of fractions and decimals was uniquely predictive of children's performance in fraction multiplication and division.</p> <p>3 TABLE Relative contributions of cognitive predictors to fraction multiplication and division (N  = 354).</p> <p> <ephtml> <table><thead valign="bottom"><tr><th align="left" /><th align="left" /><th align="left">Unstandardized coefficients</th><th align="left">Standardized</th><th align="left">t</th><th align="left">Sig.</th></tr><tr><th align="left" /><th align="left" /><th align="left"><italic>β</italic></th><th align="left">SE</th><th align="left"><italic>β</italic></th><th align="left" /><th align="left" /></tr></thead><tbody valign="top"><tr><td align="left">Step 1</td><td align="left">Non‐verbal intelligence</td><td align="char" char=".">−.066</td><td align="char" char=".">0.066</td><td align="char" char=".">−.051</td><td align="char" char=".">−1.001</td><td align="char" char=".">.317</td></tr><tr><td align="left" /><td align="left">Attentive behaviours</td><td align="char" char=".">.211</td><td align="char" char=".">0.15</td><td align="char" char=".">.076</td><td align="char" char=".">1.407</td><td align="char" char=".">.16</td></tr><tr><td align="left" /><td align="left">Counting recall</td><td align="char" char=".">.412</td><td align="char" char=".">0.072</td><td align="char" char=".">.31</td><td align="char" char=".">5.754</td><td align="char" char="."><.001</td></tr><tr><td align="left" /><td align="left">Word‐level reading</td><td align="char" char=".">.078</td><td align="char" char=".">0.09</td><td align="char" char=".">.044</td><td align="char" char=".">0.874</td><td align="char" char=".">.383</td></tr><tr><td align="left" /><td align="left">Phonological loop</td><td align="char" char=".">−.016</td><td align="char" char=".">0.117</td><td align="char" char=".">−.007</td><td align="char" char=".">−0.137</td><td align="char" char=".">.891</td></tr><tr><td align="left" /><td align="left">Whole‐number arithmetic</td><td align="char" char=".">.035</td><td align="char" char=".">0.031</td><td align="char" char=".">.06</td><td align="char" char=".">1.121</td><td align="char" char=".">.263</td></tr><tr><td align="left">Step 2</td><td align="left">Non‐verbal intelligence</td><td align="char" char=".">−.065</td><td align="char" char=".">0.067</td><td align="char" char=".">−.05</td><td align="char" char=".">−0.969</td><td align="char" char=".">.333</td></tr><tr><td align="left" /><td align="left">Attentive behaviours</td><td align="char" char=".">.211</td><td align="char" char=".">0.154</td><td align="char" char=".">.076</td><td align="char" char=".">1.368</td><td align="char" char=".">.172</td></tr><tr><td align="left" /><td align="left">Counting recall</td><td align="char" char=".">.411</td><td align="char" char=".">0.072</td><td align="char" char=".">.309</td><td align="char" char=".">5.705</td><td align="char" char="."><.001</td></tr><tr><td align="left" /><td align="left">Word‐level reading</td><td align="char" char=".">.079</td><td align="char" char=".">0.091</td><td align="char" char=".">.045</td><td align="char" char=".">0.867</td><td align="char" char=".">.386</td></tr><tr><td align="left" /><td align="left">Phonological loop</td><td align="char" char=".">−.017</td><td align="char" char=".">0.118</td><td align="char" char=".">−.007</td><td align="char" char=".">−0.146</td><td align="char" char=".">.884</td></tr><tr><td align="left" /><td align="left">Whole‐number arithmetic</td><td align="char" char=".">.035</td><td align="char" char=".">0.032</td><td align="char" char=".">.061</td><td align="char" char=".">1.098</td><td align="char" char=".">.273</td></tr><tr><td align="left" /><td align="left">Fraction magnitude</td><td align="char" char=".">−.013</td><td align="char" char=".">0.094</td><td align="char" char=".">−.008</td><td align="char" char=".">−0.137</td><td align="char" char=".">.891</td></tr><tr><td align="left" /><td align="left">Decimal magnitude</td><td align="char" char=".">.02</td><td align="char" char=".">0.087</td><td align="char" char=".">.012</td><td align="char" char=".">0.232</td><td align="char" char=".">.817</td></tr><tr><td align="left">Step 3</td><td align="left">Non‐verbal intelligence</td><td align="char" char=".">−.07</td><td align="char" char=".">0.066</td><td align="char" char=".">−.054</td><td align="char" char=".">−1.048</td><td align="char" char=".">.295</td></tr><tr><td align="left" /><td align="left">Attentive behaviours</td><td align="char" char=".">.206</td><td align="char" char=".">0.153</td><td align="char" char=".">.074</td><td align="char" char=".">1.351</td><td align="char" char=".">.178</td></tr><tr><td align="left" /><td align="left">Counting recall</td><td align="char" char=".">.397</td><td align="char" char=".">0.072</td><td align="char" char=".">.299</td><td align="char" char=".">5.528</td><td align="char" char="."><.001</td></tr><tr><td align="left" /><td align="left">Word‐level reading</td><td align="char" char=".">.077</td><td align="char" char=".">0.09</td><td align="char" char=".">.043</td><td align="char" char=".">0.847</td><td align="char" char=".">.398</td></tr><tr><td align="left" /><td align="left">Phonological loop</td><td align="char" char=".">−.01</td><td align="char" char=".">0.117</td><td align="char" char=".">−.004</td><td align="char" char=".">−0.081</td><td align="char" char=".">.935</td></tr><tr><td align="left" /><td align="left">Whole‐number arithmetic</td><td align="char" char=".">.031</td><td align="char" char=".">0.032</td><td align="char" char=".">.053</td><td align="char" char=".">0.959</td><td align="char" char=".">.338</td></tr><tr><td align="left" /><td align="left">Fraction magnitude</td><td align="char" char=".">−.024</td><td align="char" char=".">0.093</td><td align="char" char=".">−.014</td><td align="char" char=".">−0.255</td><td align="char" char=".">.799</td></tr><tr><td align="left" /><td align="left">Decimal magnitude</td><td align="char" char=".">−.069</td><td align="char" char=".">0.094</td><td align="char" char=".">−.041</td><td align="char" char=".">−0.731</td><td align="char" char=".">.465</td></tr><tr><td align="left" /><td align="left">Cross‐notation magnitude</td><td align="char" char=".">.203</td><td align="char" char=".">0.085</td><td align="char" char=".">.134</td><td align="char" char=".">2.381</td><td align="char" char=".">.018</td></tr></tbody></table> </ephtml> </p> <hd id="AN0178883000-19">DISCUSSION</hd> <p>The purpose of this study was to investigate the predictive power of cross‐notation knowledge of fractions and decimals on children's fraction arithmetic performance. Our first hypothesis was that within‐notation fraction and decimal magnitude knowledge would predict children's performance in fraction addition and subtraction over time. Consistent with the hypothesis, within‐notation fraction and decimal magnitude knowledge significantly predicted children's arithmetic performance in fraction addition and subtraction longitudinally. These predictions remained significant even after controlling for the effects of working memory, nonverbal intelligence, phonological awareness, word‐level reading, attentive behaviour, and whole‐number arithmetic. It appears that within‐notation magnitude knowledge has a unique and robust influence on children's performance in fraction addition and subtraction tasks.</p> <hd id="AN0178883000-20">Contributions of within‐notation knowledge to fraction addition and subtraction</hd> <p>The integrated theory of numerical development (Siegler et al., [<reflink idref="bib67" id="ref82">67</reflink>]) suggests that magnitude knowledge is important for children's mathematical learning. Previous research has consistently demonstrated that a better understanding of fraction magnitude knowledge is associated with improved performance in arithmetic tasks involving fractions (Bailey et al., [<reflink idref="bib4" id="ref83">4</reflink>]; Siegler et al., [<reflink idref="bib67" id="ref84">67</reflink>]; Siegler & Pyke, [<reflink idref="bib66" id="ref85">66</reflink>]; Torbeyns et al., [<reflink idref="bib71" id="ref86">71</reflink>]) and decimals (Rittle‐Johnson & Koedinger, [<reflink idref="bib57" id="ref87">57</reflink>]). Experimental studies have further supported this line of research by showing that enhancing fraction magnitude understanding can lead to higher levels of proficiency in fraction arithmetic (Dyson et al., [<reflink idref="bib27" id="ref88">27</reflink>]; Fuchs et al., [<reflink idref="bib29" id="ref89">29</reflink>]). These findings highlight the importance of magnitude knowledge, particularly in the context of fractions, for developing competence in arithmetic.</p> <p>While it may seem intuitive that within‐notation magnitude knowledge would be specifically related to arithmetic within the same notation system, the current findings do not support this notion. The results demonstrated that both fraction and decimal magnitude knowledge made an independent and significant contribution to children's performance in fraction addition and subtraction. These findings suggest that children's knowledge of rational numbers may not be compartmentalized by notation, with distinct compartments for magnitude knowledge and arithmetic within each notation. Instead, there appears to be a more interconnected and interdependent relation between magnitude knowledge and arithmetic across different notation systems.</p> <p>One possible explanation for the observed results is that within‐notation measures of magnitude knowledge capture a general understanding of rational number magnitudes that is not specific to any particular notation. This general understanding may be related to proficiency in arithmetic with both fractions and decimals. It may also involve concepts and principles that are applicable to both fractions and decimals, such as representing numerical magnitudes as positions on a number line or the principle that the sum of positive numbers is greater than either addend. Overall, the current findings align with the integrated theory of numerical development and add to the existing literature demonstrating the importance of magnitude knowledge within the realm of rational numbers for predicting and facilitating children's performance in fraction addition and subtraction.</p> <hd id="AN0178883000-21">Contributions of between‐notation knowledge to fraction addition and subtraction</hd> <p>Our second hypothesis was that cross‐notation magnitude knowledge would independently contribute to fraction arithmetic performance, beyond the influence of within‐notation magnitude knowledge and other variables. The rationale behind this hypothesis was based on the idea that successful numerical development involves the ability to represent numbers flexibly, irrespective of the notation used. Integrating information across notations is considered to require more advanced knowledge and skills compared to integrating within a single notation. Therefore, we expected that cross‐notation understanding would play a role in explaining variance in fraction arithmetic that goes beyond what within‐notation understanding can account for.</p> <p>Consistent with our hypothesis, the current study showed that cross‐notation magnitude knowledge made independent contributions to arithmetic performance in fraction addition and subtraction, even when accounting for within‐notation magnitude knowledge and other covariates. These findings support the theory that the connections between different representations of mathematical concepts, such as fractions and decimals, should be considered when learning mathematics. In particular, while understanding the magnitudes of fractions and decimals is important, recognizing the relations <emph>between</emph> these notations represents a further step towards a more comprehensive understanding of rational numbers. The evidence highlights the value of multiple representations of rational numbers in developing a nuanced understanding of these concepts. Taken together, while the integration of different types of numbers (e.g., whole numbers and rational numbers) is crucial for children's numerical development, the integration of different notations within the same type of numbers is also important.</p> <p>The results entail several educational implications. First, research by Siegler and Oppenzato ([<reflink idref="bib65" id="ref90">65</reflink>]) suggests that there is a scarcity of problems in textbooks that involve multiple notations. This lack of balance in the representation of different notations in textbooks and the problems assigned by teachers has been found to have an impact on the learning of various mathematics topics (Siegler et al., [<reflink idref="bib63" id="ref91">63</reflink>]; Siegler & Tian, [<reflink idref="bib68" id="ref92">68</reflink>]). Thus, textbooks should not only present each rational number notation in isolation, but also emphasize their interconnectedness. Second, numerous assessments of children's understanding of rational number magnitudes have primarily relied on tasks that involve comparing magnitudes within a specific notation system. These tasks include comparing the magnitudes of fractions (Fazio et al., [<reflink idref="bib28" id="ref93">28</reflink>]; Gabriel et al., [<reflink idref="bib31" id="ref94">31</reflink>]; Meert et al., [<reflink idref="bib47" id="ref95">47</reflink>]), estimating the position of fractions on a number line (Booth et al., [<reflink idref="bib8" id="ref96">8</reflink>]; Resnick et al., [<reflink idref="bib56" id="ref97">56</reflink>]), comparing the magnitudes of decimals (DeWolf et al., [<reflink idref="bib25" id="ref98">25</reflink>]; Roell et al., [<reflink idref="bib59" id="ref99">59</reflink>]), and estimating the position of decimals on a number line (Durkin & Rittle‐Johnson, [<reflink idref="bib26" id="ref100">26</reflink>]; Rittle‐Johnson et al., [<reflink idref="bib58" id="ref101">58</reflink>]). However, the current findings suggest that focusing solely on tasks within a single notation system may overlook important individual differences. Therefore, assessments of children's understanding of rational number magnitudes should also include tasks that involve multiple notation systems.</p> <p>Third, teachers may also encourage students to conceptualize fractions and decimals as related entities. However, it is important to note that the current study does not provide experimental evidence of causality. Therefore, it is not possible to confidently provide specific instructional recommendations based solely on these findings. It is worth mentioning a previous study by Malone et al. ([<reflink idref="bib42" id="ref102">42</reflink>]), which showed that an integrated intervention that focused on teaching fractions, decimals, and the connections between them did not lead to better performance in fraction arithmetic compared to a fractions‐only intervention. This highlights the need for more experimental research to examine whether cross‐notation magnitude knowledge can be improved and determine whether improved cross‐notation knowledge contributes to enhanced performance in fraction arithmetic. Thus, while the findings suggest the importance of considering cross‐notation magnitude knowledge in assessments of rational number understanding, further experimental research is needed to establish causal relations and determine the efficacy of interventions targeting cross‐notation knowledge in improving children's fraction arithmetic abilities.</p> <hd id="AN0178883000-22">Magnitude knowledge and fraction multiplication and division</hd> <p>Whereas cross‐notation magnitude knowledge emerged as a unique predictor of fraction multiplication and division, within‐notation fraction and decimal magnitude knowledge did not show significant correlations with fraction multiplication and division. These findings are consistent with a combined analysis of data in a previous study (Braithwaite et al., [<reflink idref="bib11" id="ref103">11</reflink>]), which showed that fraction magnitude knowledge uniquely predicted accuracy in fraction addition and subtraction tasks, but not in fraction multiplication and division tasks. Similarly, decimal magnitude knowledge was found to predict addition and subtraction accuracy more strongly than multiplication and division accuracy. One possible explanation for these results is that students are more inclined to consider numerical magnitudes in the context of addition and subtraction compared to multiplication and division. This tendency might arise because addition and subtraction operations inherently involve the composition and decomposition of magnitudes, which makes the connection between magnitudes and these operations more transparent. Supporting this explanation, it has been observed that most middle school children have a solid understanding that adding positive fractions or decimals results in a larger magnitude, while subtracting them results in a smaller magnitude. However, their understanding of how multiplication or division by fractions or decimals affects numerical magnitudes is often less developed or accurate (Lortie‐Forgues & Siegler, [<reflink idref="bib41" id="ref104">41</reflink>]; Siegler & Lortie‐Forgues, [<reflink idref="bib64" id="ref105">64</reflink>]). Thus, children's intuitive understanding of magnitudes may be more readily applied to addition and subtraction tasks, where the effects on magnitude are clearer. By contrast, multiplication and division tasks may require more explicit and systematic reasoning about the impact of these operations on numerical magnitudes, which is a more challenging aspect of rational number understanding.</p> <p>Cross‐notation magnitude knowledge made a small but unique contribution (it only explained an additional 1.4% of variance) to fraction multiplication and division. It remains an open question as to why recognizing and connecting magnitudes across different notations may provide some benefit in fraction multiplication and division. One possibility is that converting between fractions and decimals often involves applying arithmetic operations, such as division or multiplication. Individuals need to know how to divide the numerator by the denominator to convert a fraction into a decimal or multiply the decimal by an appropriate power of 10 to convert it into a fraction. Therefore, the competence to convert between fractions and decimals may be related to their general proficiency in doing multiplication and division.</p> <p>However, it is important to acknowledge that the contribution of cross‐notation magnitude knowledge to fraction multiplication and division was small, while other factors, such as multiplicative reasoning may play more substantial roles in predicting performance in these specific operations (Ching & Kong, [<reflink idref="bib17" id="ref106">17</reflink>]; Ching & Wu, [<reflink idref="bib23" id="ref107">23</reflink>]). The ability to reason multiplicatively and understand the relational aspects of fractions may be crucial for fraction multiplication and division (Behr et al., [<reflink idref="bib6" id="ref108">6</reflink>]; DeWolf et al., [<reflink idref="bib25" id="ref109">25</reflink>]; Piaget, [<reflink idref="bib52" id="ref110">52</reflink>]; Thompson & Saldanha, [<reflink idref="bib70" id="ref111">70</reflink>]). Multiplicative reasoning involves understanding the relations between the numerator and denominator of a fraction and how changes in one affect the other, while the ability to reason about quantities may contribute to a solid understanding of magnitudes (Ching & Kong, [<reflink idref="bib18" id="ref112">18</reflink>]). For example, recognizing that 3/9 is equivalent to 1/3 because both the numerator and denominator are divisible by 3 demonstrates an understanding of the multiplicative relation between the parts of a fraction. Further, a mastery of relational concepts is important for fraction multiplication and division. For instance, the understanding that unit fractions are multiplicatively related to the whole, such as 1/4 × 4 = 1, contributes to the ability to recognize and apply these relations to solve fraction multiplication and division problems. The concept of the multiplicative inverse may also be relevant, where the reciprocal of a fraction, such as the inverse of 2/3 being 3/2, allows for solving division problems through inverse multiplication. The hypothesis that relational reasoning about fractions, encompassing concepts, such as fraction equivalence, inverse/reciprocal relations, and ratio relations (DeWolf et al., [<reflink idref="bib25" id="ref113">25</reflink>]), plays a central role in fraction arithmetic awaits further investigation. It is plausible that this type of reasoning is more directly related to fraction multiplication and division than magnitude knowledge alone. Future research can explore the specific role and interactions between relational reasoning and magnitude knowledge in different types of fraction arithmetic tasks to provide a more comprehensive understanding of the cognitive processes involved.</p> <hd id="AN0178883000-23">Limitations and future directions</hd> <p>The current study had several strengths, including its longitudinal design and the inclusion of a wide range of covariates. However, there are several important caveats that should be considered when interpreting the findings. First, the study was conducted within a specific cultural context with a specific demographic group (e.g., children in the Guangdong‐Hong Kong‐Macao Greater Bay Area). It is essential to recognize that cultural and contextual factors can influence mathematical development. Therefore, caution should be exercised when generalizing the results to other cultural settings. Replication studies in different cultural contexts are needed to ascertain the generalizability of the findings. Second, the study focused solely on magnitude comparison tasks and did not include number line estimation tasks. Although magnitude comparison tasks and number line estimation tasks are related, they capture different aspects of numerical magnitude understanding. Future research could incorporate number line estimation tasks to examine the relative contributions of different variables to arithmetic performance. However, it is worth noting that previous research suggests that magnitude comparison tasks have greater predictive utility for subsequent mathematical performance compared to number line estimation tasks (Schneider et al., [<reflink idref="bib61" id="ref114">61</reflink>], for a review).</p> <p>Third, the study only examined children's magnitude understanding of two specific kinds of notations for rational numbers (i.e., fractions and decimals), but the understanding of other notations for rational numbers may also warrant investigation in the future. For example, percentages are another representation of rational numbers that are commonly encountered in real‐world contexts. It would be valuable for future research to explore children's magnitude understanding of other notations, such as percentages, and investigate whether cross‐notation knowledge between fractions (decimals) and percentages also contributes uniquely to arithmetic competence. Taken together, while the current study had strengths such as its longitudinal design and inclusion of covariates, it is important to consider the limitations. Replication studies in diverse cultural contexts, incorporating different types of magnitude tasks, and exploring additional notations of rational numbers would provide a more comprehensive understanding of the role of magnitude knowledge in arithmetic competence.</p> <hd id="AN0178883000-24">CONCLUSION</hd> <p>In conclusion, the current study contributes to our understanding of the role of cross‐notation magnitude knowledge in children's fraction arithmetic. The findings support the theoretical perspective that successful mathematics learning involves the integration of mathematical knowledge, including the connections between different notations within the same number type. The study highlights the importance of considering cross‐notation knowledge in assessments of numerical magnitude. Future research and educational practices may benefit from incorporating cross‐notation tasks in assessments of magnitude knowledge to gain a more comprehensive understanding of children's mathematical abilities. Taken together, while cross‐notation magnitude knowledge of rational numbers was found to be a significant predictor of fraction arithmetic, there is a need for additional research to explore other factors that may be more strongly related to children's arithmetic performance in the multiplicative domain of fractions.</p> <hd id="AN0178883000-25">AUTHOR CONTRIBUTIONS</hd> <p> <bold>Boby Ho‐Hong CHING:</bold> Conceptualization; methodology; data curation; supervision; formal analysis; project administration; resources; validation; investigation; writing – original draft; writing – review and editing. <bold>Xiang Yu LI:</bold> Conceptualization; methodology; project administration; formal analysis. <bold>Tiffany Ting CHEN:</bold> Conceptualization; methodology; project administration; formal analysis.</p> <hd id="AN0178883000-26">CONFLICT OF INTEREST STATEMENT</hd> <p>The authors declare that there is no conflict of interest.</p> <hd id="AN0178883000-27">DATA AVAILABILITY STATEMENT</hd> <p>The data that support the findings of this study are available from the corresponding author upon reasonable request</p> <hd id="AN0178883000-28">INFORMED CONSENT</hd> <p>Informed consent was obtained from each participant included in the study.</p> <ref id="AN0178883000-29"> <title> REFERENCES </title> <blist> <bibl id="bib1" idref="ref76" type="bt">1</bibl> <bibtext> American Psychiatric Association. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Cross-Notation Knowledge of Rational Numbers Predicts Fraction Arithmetic
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Boby+Ho-Hong+Ching%22">Boby Ho-Hong Ching</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-3526-7704">0000-0002-3526-7704</externalLink>)<br /><searchLink fieldCode="AR" term="%22Xiang+Yu+Li%22">Xiang Yu Li</searchLink><br /><searchLink fieldCode="AR" term="%22Tiffany+Ting+Chen%22">Tiffany Ting Chen</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22British+Journal+of+Educational+Psychology%22"><i>British Journal of Educational Psychology</i></searchLink>. 2024 94(3):717-737.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 21
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2024
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Fractions%22">Fractions</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic%22">Arithmetic</searchLink><br /><searchLink fieldCode="DE" term="%22Young+Children%22">Young Children</searchLink><br /><searchLink fieldCode="DE" term="%22Prior+Learning%22">Prior Learning</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Processes%22">Learning Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Predictor+Variables%22">Predictor Variables</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Cognitive+Ability%22">Cognitive Ability</searchLink><br /><searchLink fieldCode="DE" term="%22Intelligence%22">Intelligence</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Recall+%28Psychology%29%22">Recall (Psychology)</searchLink><br /><searchLink fieldCode="DE" term="%22Beginning+Reading%22">Beginning Reading</searchLink><br /><searchLink fieldCode="DE" term="%22Phonological+Awareness%22">Phonological Awareness</searchLink><br /><searchLink fieldCode="DE" term="%22Addition%22">Addition</searchLink><br /><searchLink fieldCode="DE" term="%22Subtraction%22">Subtraction</searchLink>
– Name: Subject
  Label: Geographic Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22China%22">China</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1111/bjep.12674
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0007-0998<br />2044-8279
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Background: Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered. Aims: To examine the extent to which children's earlier knowledge of cross-notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally. Sample: Three hundred and fifty-four Chinese children (Mage = 112.1 months). Methods: During the first wave of assessment, a range of cognitive abilities of children were measured, including within-notation fraction and decimal magnitude comparisons, whole-number arithmetic fluency, non-verbal intelligence, attentive behaviours, counting recall, word-level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division. Results and Conclusions: Multiple linear regressions showed that within-notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole-number arithmetic were controlled. Cross-notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within-notation fraction and decimal magnitude knowledge and other covariates. However, within-notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross-notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross-notation knowledge in the assessments of children's mathematics abilities and teaching.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2024
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1434514
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1434514
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1111/bjep.12674
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 21
        StartPage: 717
    Subjects:
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Fractions
        Type: general
      – SubjectFull: Arithmetic
        Type: general
      – SubjectFull: Young Children
        Type: general
      – SubjectFull: Prior Learning
        Type: general
      – SubjectFull: Learning Processes
        Type: general
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Predictor Variables
        Type: general
      – SubjectFull: Foreign Countries
        Type: general
      – SubjectFull: Cognitive Ability
        Type: general
      – SubjectFull: Intelligence
        Type: general
      – SubjectFull: Computation
        Type: general
      – SubjectFull: Recall (Psychology)
        Type: general
      – SubjectFull: Beginning Reading
        Type: general
      – SubjectFull: Phonological Awareness
        Type: general
      – SubjectFull: Addition
        Type: general
      – SubjectFull: Subtraction
        Type: general
      – SubjectFull: China
        Type: general
    Titles:
      – TitleFull: Cross-Notation Knowledge of Rational Numbers Predicts Fraction Arithmetic
        Type: main
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          Name:
            NameFull: Boby Ho-Hong Ching
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            NameFull: Xiang Yu Li
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            NameFull: Tiffany Ting Chen
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          Dates:
            – D: 01
              M: 09
              Type: published
              Y: 2024
          Identifiers:
            – Type: issn-print
              Value: 0007-0998
            – Type: issn-electronic
              Value: 2044-8279
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            – Type: volume
              Value: 94
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              Value: 3
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            – TitleFull: British Journal of Educational Psychology
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