Deaf children's use of cognitive strategies in simple arithmetic problems

Research shows that deaf children have inferior achievement in simple arithmetic compared to their hearing agemates. The reported study investigates whether the reasons for deaf children's poor achievement can be sought in their strategy development. As this is a central issue, the strategies used by deaf children when solving symbolic addition and subtraction problems are identified, classified and compared to findings from earlier research, involving both deaf and hearing children. The effect of Sign Language in strategy invention and use is the main concern in this study. One result from the present study is that structural aspects of Sign Language counting may influence deaf children's thinking in a way that does not lead to a developed conceptual knowledge base, but instead to refined procedural competence. This is a development in simple arithmetic that is compatible with that of less able hearing children. The counting procedures used by the deaf children involve both oral counting and Sign Language counting. Due to the small sample size, and the shortcomings of the research design, the results are more suggestive than conclusive. Thus, further studies are needed in this area.This revised version was published online in September 2005 with corrections to the Cover Date.     

Distinguishing adaptive from routine expertise with rational number arithmetic

Adaptive expertise is a valued, but under-examined, feature of students' mathematical development (e.g. Hatano & Oura, 2012). The present study investigates the nature of adaptive expertise with rational number arithmetic. We therefore examined 394 7th and 8th graders’ rational number knowledge using both variable-centered and person-centered approaches. Performance on a measure of adaptive expertise with rational number arithmetic, the arithmetic sentence production task, appeared to be distinct from more routine features of performance. Even among the top 45% of students, all of whom had strong routine procedural and conceptual knowledge, students varied greatly in their performance the arithmetic sentence production task. Strong performance on this measure also predicted later algebra knowledge. The findings suggest that it is possible to distinguish adaptive expertise from routine expertise with rational numbers and that this distinction is important to consider in research on mathematical development.     

Manipulatives and number sentences in computer aided arithmetic word problem solving

In this study we investigated the relative contribution of two main components often used in the instruction of arithmetic word problem solving to first-grade children and children with learning problems, external representation with manipulatives and formal mathematical representation with number sentences. Four computer aided treatments were developed along these dimensions. Furthermore, an assessment control group was formed.It was shown that performance improved significantly after treatment in comparison to the performance of the assessment control group. Comparison of the different training conditions showed that a number sentence treatment was the most effective training component. However, this effect was only present with the relatively high competent children and not with the relatively low performing children.     

错题探讨引出等差数列的一个性质

先看下面的两个题 :题 1　一个等差数列共有 2ｎ 1项 ,其中奇数项之和为 3 0 5 ,偶数项之和为 2 76,试求第ｎ 1项。(见华东师范大学数学系编《数学学习丛书》高中代数 (二 )第 71页 3 0题 )题 2　已知数列ａ1,ａ2 ,…ａ2ｎ 1为等差数列 ,设Ｐ=ａ1 ａ3 … ａ2ｎ 1,Ｑ =ａ2 ａ4 … ａ2ｎ,求Ｐ/Ｑ的值。 (见华东师范大学出版社出版、上海市中、小学通用教材 ,高中数学二年级第二学期《数学习题册》第 2 8页第 1 2 (1 )题 )两个题的略解如下 :解　对于题 1 ,设题给的等差数列的各项依次为ａ1,ａ2 ,… ,ａ2ｎ 1,公差为ｄ ,依题意有 :…     

Early number and arithmetic performance of Ecuadorian 4-5-year-olds

This study aimed at (a) constructing a reliable and valid test to assess Ecuadorian 4–5-year olds’ number and arithmetic skills; (b) providing empirical data on Ecuadorian 4–5-year olds’ number and arithmetic skills; and (c) confronting these children’s actual performances with the performances expected by national experts in this domain. We administered the Test of Early Number and Arithmetic (TENA), developed on the basis of the Ecuadorian mathematics standards to 86 Preschoolers and 127 Kindergartners and asked 10 experts to evaluate TENA’s validity and predict children’s performances on it. Results supported the overall reliability and validity of the TENA. Furthermore, we observed differences in number and arithmetic competencies between and within Preschoolers and Kindergartners, but not between boys and girls. Finally, experts overestimated children’s performances on the test. The scientific and practical implications of these results are discussed.     

Adaptive number knowledge and its relation to arithmetic and pre-algebra knowledge

Traditionally measured skills with arithmetic are not related to later algebra success at levels that would be expected given the close conceptual relation between arithmetic and algebra. However, adaptivity with arithmetic may be one aspect of arithmetic competences that can account for additional variation in algebra attainment. With this in mind, the present study aims to present evidence for the existence and relevance of a newly acknowledged component of adaptivity with arithmetic, namely, adaptive number knowledge. In particular, we aim to examine whether there are substantial individual differences in adaptive number knowledge and to what extent these differences are related to arithmetic and pre-algebra skills and knowledge. Adaptive number knowledge is defined as the well-connected knowledge of numerical characteristics and relations. A large sample of 1065 Finnish late primary school students completed measures of adaptive number knowledge, arithmetic conceptual knowledge, and arithmetic fluency. Three months later they completed a measure of pre-algebra skills. Substantial individual differences in adaptive number knowledge were identified using latent profile analysis. The identified profiles were related to concurrent arithmetic skills and knowledge. As well, adaptive number knowledge was found to predict later pre-algebra skills, even after taking into account arithmetic conceptual knowledge and arithmetic fluency. These results suggest that adaptive number knowledge is a relevant component of mathematical development, and may help account for disparities in algebra development.     

An analysis of diverging approaches to simple arithmetic: Preference and its consequences

Earlier research by the author indicated that many below average attainers do not remember number facts and use alternative strategies to obtain solutions to basic arithmetical problems. These alternatives were frequently seen as the best way of finding a solution.This paper considers the relationship between the various strategies used by mixed ability children aged 7 to 12. An analysis of alternatives suggests that the selection is not underpinned by regression through the learning sequence, but by regression dominated by the child's preference for certain strategies over others. Through the evaluation of a hierarchy of preferences, divergence between the strategies available to the less able and the more able child is revealed. The alternative strategies used are based either on counting — procedural strategies, or on the use of selected known knowledge — deductive strategies. Above average children have both available as alternatives; evidence of deduction is rare amongst below average children. The more able child appears to build up a growing body of known facts from which new known facts are deduced. Less able children — relying mainly on procedural strategies — do not appear to have this feedback loop available to them.The paper contends that, for some children, procedural methods do not encourage the need to remember; the procedure provides security. On the other hand, deductive methods initially enhance the ability to remember other basic facts and eventually help children make extensive use of facts that are known to remove the need to remember new ones. More able children appear to be doing a qualitatively different sort of mathematics than the less able.     

Developmental changes in the contributions of phonological processing skills to Chinese character reading and arithmetic

This study examined the associations of phonological processing skills with reading and arithmetic ability in Chinese kindergartners (Mage = 5.56 years), third graders (Mage = 9.72 years), and fifth graders (Mage = 11.75 years) (N = 413) of Han descent. The results showed that phonological awareness and rapid automatized naming (RAN) showed stronger relations than phonological memory with reading and arithmetic across grades. Furthermore, the associations of phonological awareness and RAN with reading were much stronger in kindergartners than in primary school children, whereas their relationships with arithmetic remained stable across grades. Among phonological skills, phonological awareness has a unique influence on arithmetic that is independent of Chinese character reading in third-graders and kindergartners. In contrast, RAN uniquely explained the variation in arithmetic skills in fifth graders when reading was statistically controlled for. These findings have important implications for understanding the co-development of reading and arithmetic across grades and raise the possibility of training in phonological awareness and/or RAN to help children at risk for learning disabilities.     

Pictorial representations of simple arithmetic problems are not always helpful: a cognitive load perspective

This study examined immigrant parents’ involvement in early years mathematics learning, focusing on learning of multiplication in in- and out-of-school settings. Ethnographic interviews and workshops were conducted in an urban city in Japan, to examine out-of-school practices of immigrant families. Drawing from sociocultural theory of learning and the concept of appropriation (Wertsch, 1998), the role of power and identity was examined in relation to children’s appropriation of an informal multiplication method that was taught by their parents. An intergenerational analysis, between immigrant parents and their children, revealed heterogeneous perspectives towards appropriation. Immigrant parents in this study framed their involvement in their children’s early years mathematics learning in relation to their positional identities and the pressures to conform to the mainstream practices of their host country. During their early years of schooling, students in this study were already aware of academic tracking in the school and were aware of what was believed to be legitimate in school mathematics learning. The significance of diversifying mathematics curriculum and pedagogy was discussed to affirm the knowledge and identities of immigrant students and families.     

Choice of operation in verbal arithmetic problems: the effects of number size,problem structure and context

12-and 13-year-olds were tested with two types of tasks to test their understanding of applications of the multiplication and division of positive numbers: (i) writing down calculations required to solve verbal problems, and (ii) making up stories to fit given calculations. Selected pupils were interviewed to investigate further the thinking processes involved. The results indicate (a) the pervasive nature of certain numerical misconceptions, (b) the effects of structural differences among the items; particularly whether multiplication can be conceived as repeated addition or not, and whether division has the structure of partition, quotition or rate, (c) specific effects of context attributable to such aspects as relative familiarity, and (d) various interactions between these three sets of factors.With the collaboration of Joanna Rigg and Malcolm Swan.     

Pathways to arithmetic: The role of visual-spatial and language skills in written arithmetic,arithmetic word problems,and nonsymbolic arithmetic

This study set out to develop and test a pathway model of the relations between general cognitive skills, specifically visual-spatial and spoken and written language skills, and competence in three forms of arithmetic that vary in modes of number representation. A total of 88 Chinese 4-year-olds participated and were tested first in kindergarten second grade (K2) and then in kindergarten third grade (K3). Language skills, including phonological, morphological, and visual-orthographic skills, and visual-spatial skills were measured at K2, and arithmetic outcomes, including written arithmetic, word problems, and nonsymbolic arithmetic, at K3. The results generally supported our model. Specifically, visual-spatial skills contributed to the prediction of all three types of arithmetic outcomes. Morphological skills predicted word problems, whereas phonological skills predicted written arithmetic. Finally, visual-orthographic skills contributed to both written and nonsymbolic arithmetic. These findings underscore the importance of delineating the specificity of cognitive processes in learning diverse forms of arithmetic.     

An epidemiological study of number processing and mental calculation in Greek schoolchildren

The aim of this study was to validate and standardize an instrument for the diagnosis of developmental dyscalculia (mathematics disorder) in a Greek population and to obtain relevant epidemiological data. We used the Neuropsychological Test Battery for Number Processing and Calculation in Children (NUCALC) in a community sample of 240 students ages 7 to 11 years from urban and rural schools. There were no differences between genders in arithmetical performance; however, the effects of grade and socioeconomic level were significant. Prevalence was higher in the rural than in the urban area. A cross-cultural comparison of the Greek data with those obtained with the same instrument in other countries in schoolchildren of the same age was performed.     

多项式数列和的简捷方法

给出了求关于自然数k的m次多项式数列f(k)=α0k^m α1k^m-1 … αm-1k αm=∑i=0^m αik^m-i的前n项和∑k=1 m f(k)的简单递推公式，而无需应用Bernoulli数，推广了文[1]、[2]、[3]的结论。     

Problem solving in arithmetic

Growing numbers of children face conditions that severely limit their ability to learn and develop in good health. As a consequence, both P-12 schooling and higher education will need to modify their practices to help these children reach their full potential. Family-centered integrated services systems, in conjunction with interprofessional preparation, are proposed as being important steps in meeting the needs of disadvantaged children and families. Suggestions for implementation of family-centered programming are offered, and issues that need consideration at the postsecondary level in order to pursue interprofessional collaboration are discussed. Collaboration is essential to the process.     

The role of inhibition in overcoming arithmetic natural number bias in the Chinese context: Evidence from behavioral and ERP experiments

The natural number bias (NNB) in arithmetic operations refers to the application of natural number properties to reasoning about rational numbers. Previous studies found the NNB interferes with students’ problem-solving. However, few studies have examined it in the Chinese context or the underlying mechanism by which it can be overcome. Addressing these gaps, in Experiments 1a (n = 31) and 1b (n = 30), we found that Chinese students demonstrate the NNB despite linguistic differences between Chinese and western languages. Experiment 2 (n = 38) adopted a negative priming paradigm and found that inhibitory control was necessary to overcome the NNB. Experiment 3 (n = 34) employed the event-related potential technique; we observed increased P2 amplitude when students solved congruent problems, and increased N2 and decreased P3 amplitude when they solved incongruent problems. These results indicated that the NNB is rooted in intuitive thinking, and overcoming this bias relies on inhibition.     

等差数列前n项和公式的又一形式及应用

由等差数列的通项公式不难推出如下性质 :若{an}是等差数列 ,am、an、ap、aq 分别是该数列的第m、n、p、q项 ,且m n =p q,则am an=ap aq。又显然 ,1 n =k (n 1 -k) ,故由上述性质可知 :a1 an=ak an 1-k,k∈N ,且k≤n将这一结果代入等差数列前n项和公式中 ,便有Sn=n(a1 an)2 =n(ak an 1-k)2 。等差数列前n项和的这一形式 ,具有非常好的解题功能。下面略举数例说明之。例 1　 ( 1 995年全国高考题 )　等差数列 {an}、{bn}的前n项和分别为Sn 与Tn,若 SnTn=2n3n 1 ,则limn→∞anbn等于 (　　 )(A) 1　　 (B) 6/ 3　　 (C) 23　　 (…     

Foundation of basic arithmetic

This is the second of a series of articles, in which we demonstrate the utility of the place value system of representing numbers.     

Foundation of basic arithmetic

This is the first of a series of articles, in which we shall briefly outline the early evolution of numerals as we currently know it.