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Transcoding is the process of translating between spoken and written numbers, and it is correlated with other mathematical skills. In the present study, we investigated the link between French number writing of 49 students in the third grade (aged 7-9 years) and their language skills. Transcoding in French is of particular interest because the spoken number language system does not completely correspond to that of the written digits (e.g., quatre-vingt-dix [four-twenty-ten] and 90). We hypothesised that the complex linguistic structure of spoken numbers in French would be challenging for students who are learning to transcode. First and second French-language learners' accuracy and errors were recorded during a writing task of 3- to 7-digit numbers. Children also completed linguistic tests (e.g., receptive vocabulary, receptive syntax). Results showed that first- and second-language learners did not differ in their transcoding accuracy. Number size, decade complexity of stimulus number words in French (i.e., numbers containing a complex decade, operationalized as a number between soixante-dix, 70, and quatre-vingt-dix-neuf, 99), and receptive vocabulary predicted children's French transcoding skills. Students were more likely to produce errors (e.g., 68 or 6018 for 78) when they transcoded complex decade numbers compared with simple decade numbers. When an error was made on the complex decade portion of a number, it was likely a lexical error. In conclusion, third graders, both first- and second-language learners, found complex decade numbers challenging and their performance was related to their general vocabulary skills.

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This article synthesizes findings from an international virtual conference, funded by the National Science Foundation (NSF), focused on the home mathematics environment (HME). In light of inconsistencies and gaps in research investigating relations between the HME and children's outcomes, the purpose of the conference was to discuss actionable steps and considerations for future work. The conference was composed of international researchers with a wide range of expertise and backgrounds. Presentations and discussions during the conference centered broadly on the need to better operationalize and measure the HME as a construct - focusing on issues related to child, family, and community factors, country and cultural factors, and the cognitive and affective characteristics of caregivers and children. Results of the conference and a subsequent writing workshop include a synthesis of core questions and key considerations for the field of research on the HME. Findings highlight the need for the field at large to use multi-method measurement approaches to capture nuances in the HME, and to do so with increased international and interdisciplinary collaboration, open science practices, and communication among scholars.

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In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 + 3 = 8), subtraction (e.g., 5 - 3 = 2), and multiplication (e.g., 5 × 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All families were English-speaking, and parent education levels ranged from high school to postgraduate, with a median of community college. In Grade 2, children completed general cognitive tasks (i.e., receptive vocabulary, working memory, nonverbal reasoning, and inhibitory control). In both grades, children completed single-digit addition and complementary subtraction problems. In Grade 3, they completed single-digit multiplication problems and measures of applied mathematics, specifically, word-problem solving, algebra, and measurement. We found that addition and subtraction were reciprocally related (controlling for cognitive skills). Subtraction fluency predicted multiplication in Grade 3, whereas addition fluency did not. In Grade 3, both subtraction and multiplication fluency were predictors of applied mathematics, with multiplication partially mediating the relation between subtraction and applied mathematics performance. These findings support the view that learning arithmetic associations is a hierarchical process. As students practice each new skill, individual differences reflect the integration of the novel component into the developing associative network. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

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In the current study, we adopted the Pathways to Mathematics model of LeFevre et al. (2010). In this model, there are three cognitive domains--labeled as the quantitative, linguistic, and working memory pathways--that make unique contributions to children's mathematical development. We attempted to refine the quantitative pathway by combining children's (N=141 in Grades 2 and 3) subitizing, counting, and symbolic magnitude comparison skills using principal components analysis. The quantitative pathway was examined in relation to dependent numerical measures (backward counting, arithmetic fluency, calculation, and number system knowledge) and a dependent reading measure, while simultaneously accounting for linguistic and working memory skills. Analyses controlled for processing speed, parental education, and gender. We hypothesized that the quantitative, linguistic, and working memory pathways would account for unique variance in the numerical outcomes; this was the case for backward counting and arithmetic fluency. However, only the quantitative and linguistic pathways (not working memory) accounted for unique variance in calculation and number system knowledge. Not surprisingly, only the linguistic pathway accounted for unique variance in the reading measure. These findings suggest that the relative contributions of quantitative, linguistic, and working memory skills vary depending on the specific cognitive task.

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The purpose of this study was to propose and test a model of children's home numeracy experience based on Sénéchal and LeFevre's home literacy model (Child Development, 73 (2002) 445-460). Parents of 183 children starting kindergarten in the fall (median child age=58 months) completed an early home learning experiences questionnaire. Most of the children whose parents completed the questionnaire were recruited for numeracy and literacy testing 1 year later (along with 32 children from the inner city). Confirmatory factor analyses were used to reduce survey items, and hierarchical regression analyses were used to predict the relation among parents' attitudes, academic expectations for their children, reports of formal and informal numeracy, and literacy home practices on children's test scores. Parental reports of formal home numeracy practices (e.g., practicing simple sums) predicted children's symbolic number system knowledge, whereas reports of informal exposure to games with numerical content (measured indirectly through parents' knowledge of children's games) predicted children's non-symbolic arithmetic, as did numeracy attitudes (e.g., parents' enjoyment of numeracy). The home literacy results replicated past findings; parental reports of formal literacy practices (e.g., helping their children to read words) predicted children's word reading, whereas reports of informal experiences (i.e., frequency of shared reading measured indirectly through parents' storybook knowledge) predicted children's vocabulary. These findings support a multifaceted model of children's early numeracy environment, with different types of early home experiences (formal and informal) predicting different numeracy outcomes.

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Individuals who do well in mathematics and science also often have good spatial skills. However, the predictive direction of links between spatial abilities and mathematical learning has not been firmly established, especially for young children. In the present research, we addressed this issue using a sample from a longitudinal data set that spanned 4 years and which includes measures of mathematical performance and various cognitive skills, including spatial ability. Children were tested once in each of 4 years (Time 1, 2, 3, and 4). At Time 3 and 4, 101 children (in Grades 2, 3, or 4 at Time 3) completed mathematical measures including (a) a number line task (0-1000), (b) arithmetic, and (c) number system knowledge. Measures of spatial ability were collected at Time 1, 2, or 3. As expected, spatial ability was correlated with all of the mathematical measures at Time 3 and 4, and predicted growth in number line performance from Time 3 to Time 4. However, spatial ability did not predict growth in either arithmetic or in number system knowledge. Path analyses were used to test whether number line performance at Time 3 was predictive of arithmetic and number system knowledge at Time 4 or whether the reverse patterns were dominant. Contrary to the prediction that the number line is an important causal construct that facilitates learning arithmetic, no evidence was found that number line performance predicted growth in calculation more than calculation predicted number line growth. However, number system knowledge at Time 3 was predictive of number line performance at Time 4, independently of spatial ability. These results provide useful information about which aspects of growth in mathematical performance are (and are not) related to spatial ability and clarify the relations between number line performance and measures of arithmetic and number system knowledge.

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We examined the role of executive attention, which encompasses the common aspects of executive function and executive working memory, in children's acquisition of two aspects of mathematical skill: (a) knowledge of the number system (e.g., place value) and of arithmetic procedures (e.g., multi-digit addition) and (b) arithmetic fluency (i.e., speed of solutions to simple equations such as 3+4 and 8-5). Children in Grades 2 and 3 (N=157) completed executive attention and mathematical tasks. They repeated the mathematical tasks 1 year later. We used structural equation modeling to examine the relations between executive attention and (a) concurrent measures of mathematical knowledge and arithmetic fluency and (b) growth in performance on these measures 1 year later. Executive attention was concurrently predictive of both knowledge and fluency but predicted growth in performance only for fluency. A composite language measure predicted growth in knowledge from Grade 2 to Grade 3. The results support an important role for executive attention in children's acquisition of novel procedures and the development of automatic access to arithmetic facts.

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A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.

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Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality-that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children's conceptual knowledge of counting.

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The development of conceptual and procedural knowledge about counting was explored for children in kindergarten, Grade 1, and Grade 2 (N = 255). Conceptual knowledge was assessed by asking children to make judgments about three types of counts modeled by an animated frog: standard (correct) left-to-right counts, incorrect counts, and unusual counts. On incorrect counts, the frog violated the word-object correspondence principle. On unusual counts, the frog violated a conventional but inessential feature of counting, for example, starting in the middle of the array of objects. Procedural knowledge was assessed using speed and accuracy in counting objects. The patterns of change for procedural knowledge and conceptual knowledge were different. Counting speed and accuracy (procedural knowledge) improved with grade. In contrast, there was a curvilinear relation between conceptual knowledge and grade that was further moderated by children's numeration skills (as measured by a standardized test); the most skilled children gradually increased their acceptance of unusual counts over grade, whereas the least skilled children decreased their acceptance of these counts. These results have implications for studying conceptual and procedural knowledge about mathematics.

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