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Performance across different areas of mathematical cognition in children with learning difficulties

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Abstract

The performance of 210 2nd graders in different areas of mathematical cognition was examined. Children were divided into 4 achievement groups: children with difficulties in mathematics but not in reading (MD-only), children with difficulties in both mathematics and reading (MD/RD), children with difficulties in reading but not in mathematics, and children with normal achievement. Although both MD groups performed worse than normally achieving groups in most areas of mathematical cognition, the MD-only group showed an advantage over the MD/RD group in exact calculation of arithmetic combinations and in problem solving. The 2 groups did not differ in approximate arithmetic and understanding of place value and written computation. Children with MD-only seem to be superior to children with MD/RD in areas that may be mediated by language but not in ones that rely on numerical magnitudes, visuospatial processing, and automaticity. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
... Although DSM and ICD suggest to measure number sense for MD identification, it is unclear how each sub-ability (e.g., quantity processing) differs between people with and without MD. Also, most studies have focused only on abilities which were measured by using accuracy scores (e.g., number of solved items) while response time scores are clearly under researched and not discussed in terms of their validity to differentiate between people with and without MD (e.g., Hanich et al., 2001;Mammarella et al., 2013). ...
... For those with MD, additional time constraints can complicate tasks thus negatively affecting their performance compared to people without MD. However, studies about MD rarely analyze response time scores (e.g., Hanich et al., 2001;Mammarella et al., 2013). Consequently, it is unclear if and how response time scores can be used for MD identification and whether the effect sizes are similar regardless of scoring type. ...
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Math difficulties (MD) manifest across various domain-specific and domain-general abilities. However, the existing cognitive profile of MD is incomplete and thus not applicable in typical settings such as schools or clinics. So far, no review has applied inclusion criteria according to DSM or ICD, summarized domain-specific abilities or examined the validity of response time scores for MD identification. Based upon stringent clinical criteria, the current meta-analysis included 34 studies which compared cognitive performances of a group with MD (n = 680) and a group without MD (n = 1565). Criteria according to DSM and ICD were applied to identify MD (percentile rank ≤ 16, age range 8–12 years, no comorbidities/low IQ). Effect sizes for 22 abilities were estimated and separated by their level and type of scoring (AC = accuracy, RT = response time). A cognitive profile of MD was identified, characterized by distinct weaknesses in: (a) computation (calculation [AC], fact retrieval [AC]), (b) number sense (quantity processing [AC], quantity-number linking [RT], numerical relations [AC]), and (c) visual-spatial short-term storage [AC]. No particular strength was found. Severity of MD, group differences in reading performance and IQ did not significantly moderate the results. Further analyses revealed that (a) effects are larger when dealing with numbers or number words than with quantities, (b) MD is not accompanied by any weakness in abilities typically assigned to reading, and (c) weaknesses in visual-spatial short-term storage emphasize the notion that number and space are interlinked. The need for high-quality studies investigating domain-general abilities is discussed.
... The existence of Tibetan-Chinese-Mathematical Language interpretation chain produces dual obstacles, possibly also affecting comprehension errors to occur among students with MLD in Tibet. In addition, problems in fact retrieval and comprehension in mathematical word problems can be affected by cognitive deficits in representing and retrieving information from phonetic and semantic memory (Hanich et al., 2001). Transformation errors were also reported as one of the most frequent error types in this study. ...
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A notably high percentage (33%) of students with mathematics learning difficulties (MLD) has been reported in Tibet, which can partly be explained by the current state of mathematics teaching. Less known is what kind of errors these students are making in different types of mathematical tasks. This study investigated the types and frequency of errors made in mathematics tasks by Tibetan seventh-grade students (n = 30) with MLD, as well as whether gender and school type had any effects on these errors. The novel Mathematics Error Pattern Identification Test (MEPIT) was used to identify the following eight different error types: visual-spatial, comprehension, transformation, relevance, fact, procedural, measurement, and presentation. Students identified as having MLD completed the MEPIT in two sessions due to its length (64 items). Regression analysis and t-tests were used. The most frequent error types were fact and comprehension errors. Compared to the boys in the study, the girls seemed to be more vulnerable to fact and relevance errors. The students in the rural school made more comprehension errors compared to the students in the urban school. Our exploratory study calls for further research on assessing errors that students with MLD make in mathematical tasks. MEPIT was shown to be a promising assessment tool for Tibetan teachers to identify errors that students with MLD may make in mathematics tasks.
... Thus, each child's score was the total number of correct trials (maximum possible score was 10; α = .85 at Time 1 and .82 at Time 2). 1 The Which N has __? task was a multiple-choice adaptation of the digit correspondence task (e.g., Hanich et al., 2001;Kamii, 1989). Children were presented with three written numerals arranged in a horizontal line (e.g., 2, 20, and 10). ...
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Place value concepts were measured longitudinally from kindergarten (2017) to first grade (2018) in a diverse sample (n = 279; Mage = 5.76 years, SD = 0.55; 135 females; 41% Black, 38% White, 8% Asian, 12% Latino). Children completed three syntactic tasks that required an explicit understanding of base-10 symbols and three approximate tasks that could be completed without this explicit understanding. Approximate performance was significantly better in both age groups. A factor analysis confirmed that syntactic and approximate tasks tapped separate latent variables in kindergarten, but not in first grade. Path analyses indicated that only kindergarten approximate performance predicted overall first-grade place value understanding. These findings suggest that explicit understanding of base-10 principles develops from implicit, partial knowledge of multidigit numbers.
... [3][4][5] Such struggle is usually manifested by high reaction times (RT), low accuracy, and the use of immature strategies, such as finger counting, while solving arithmetic operations. 3,[6][7][8] It is still unclear how basic number systems contribute to the difficulties observed in MD. Some numerical cognition models propose that complex mathematics skills build on primitive systems dedicated to processing nonsymbolic numerical magnitudes. ...
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It is still debated if the main deficit in mathematical difficulties (MD) is nonsymbolic or symbolic numerical magnitude processing. Objectives: In the present study, our main goal was to investigate nonsymbolic and symbolic numerical magnitude processing in MD and the relationship between these abilities and arithmetic. Methods: The Brazilian school-age children with MD completed a nonsymbolic and a symbolic numerical magnitude comparison task and an arithmetic task. We compared their performance with a group of children with typical achievement (TA) and investigated the association between numerical magnitude processing and arithmetic with a series of regression analyses. Results: Results indicated that children with MD had low performance in the nonsymbolic numerical magnitude comparison task. Performance in both nonsymbolic and symbolic numerical magnitude comparison tasks predicted arithmetic abilities in children with TA, but not in children with MD. Conclusions: These results indicate that children with MD have difficulties in nonsymbolic numerical magnitude processing, and do not engage basic numerical magnitude representations to solve arithmetic.
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Examining how informal knowledge systems change after formal instruction is imperative to understanding learning processes and conceptual development and to implementing effective educational practices. We used network analyses to determine how the organization of informal knowledge about multidigit numbers in kindergartners (N = 279; mean age = 5.76 years, SD = 0.55; 135 females) supports and is transformed by a year of in-school formal instruction. The results show that in kindergarten, piecemeal knowledge about the surface properties of reading and writing multidigit numbers and the use of base-10 units to determine large quantities are strongly associated with each other and connected in a stringlike manner to other emerging skills. After a year of instruction, each skill becomes connected to the "hub" abilities of reading and writing multidigit numbers, which also become strongly connected to more advanced knowledge of base-10 principles. These findings provide new insights into how partial knowledge provides the backbone on which explicit principles are learned.
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We replicated a study of a kindergarten mathematics intervention, ROOTS, delivered in the context of a research-based core program. We randomly assigned 62 classrooms to treatment (ROOTS) or a business-as-usual control. All classrooms implemented the research-based core program (Early Learning in Mathematics). Participants included 163 treatment students and 145 control students nested within classrooms. Key differences between the current replication study and the original study included geographical region, instructional context, and student initial skill. In contrast to the significant positive effects (Hedges’s g values of .30 to .38) found in the original study, no significant differences were found between the treatment and control conditions in this study (Hedges’s g values of –.09 to .12) Pretest skills did not moderate treatment effects. We discuss these results’ implications for replication research and evaluating intervention efficacy.
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Students with learning disabilities in mathematics often struggle with the underlying concepts of multidigit addition and subtraction. To help students build a conceptual understanding of these computations, teachers can utilize evidence-based practices such as the concrete, semi-concrete, abstract frameworks and the use of multiple, visual representations. This column presents five key strategies that incorporate evidence-based practices and teach whole number operations that rely on place value understanding.
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Bu çalışmada kaynaştırma/bütünleştirme uygulamalarının yürütüldüğü okul öncesi sınıflarda eğitim gören özel gereksinimi olan ve olmayan öğrencilerin matematik performanslarının öğretmen görüşüne dayalı olarak belirlenmesi amaçlanmıştır. Bu amaçtan hareketle öncelikle geçerliği ve güvenilirliği ispatlanmış bir ölçme aracının alanyazına kazandırılması, okul öncesi sınıflardaki öğrencilerin matematik performans düzeylerinin belirlenmesi ve matematik performans düzeylerinde farklılık oluşturan değişkenlerin ortaya konulması hedeflenmiştir. Ölçek geliştirme aşamaları dikkate alınarak hazırlanan çalışmada betimsel tarama modeli kullanılmıştır. Çalışma grubu amaçsal örnekleme yöntemi ile belirlenmiştir. Çalışmanın amaçları doğrultusunda öncelikle madde havuzu oluşturulmuş, uzman görüşü alınarak ölçeğin pilot uygulama formu oluşturulmuştur. Daha sonra geçerliğe ve güvenirliğe ilişkin analizlerin yapılması ve ölçme aracının nihai formunun oluşturulması için bağımsız anaokullarında hizmet veren 194 öğretmenden 970 öğrenciye ilişkin veri toplanmıştır. SPSS paket programına aktarılan veriler üzerinde normal dağılıma ilişkin varsayımlar incelenmiş ve uygun istatistiksel işlemler kullanılarak analizler yapılmıştır. Çalışmanın ilk amacı doğrultusunda öncelikle, geliştirilen üç alt formdan oluşan Matematik Performansı Değerlendirme Aracı'nın (MAPEDA 36-72 Ay) kapsam ve yapı geçerliği analizleri ile geçerlik; Cronbach alfa, iki yarı test, alt-üst %27 ve madde toplam korelasyonu analizleri ile güvenirlik analizleri gerçekleştirilmiştir. Yapı geçerliği analizi için yapılan AFA ile ölçeğin faktör yapıları belirlenmiş; 5 faktörlü 27 maddelik MAPEDA (36-48 Ay), 5 faktörlü 34 maddelik MAPEDA (49-60 Ay) ve 5 faktörlü 34 maddelik MAPEDA (61-72 Ay) formları elde edilmiştir. MAPEDA'nın (36-72 Ay) açıklanan toplam varyans değerleri; MAPEDA (36-48 Ay) formu için %66, MAPEDA (49-60 Ay) formu için %71 ve MAPEDA (61-72 Ay) formu için %73'tür. DFA ile test edilen uyum değerlerinin (RCI, CFI, NFI, NNFI, IFI, RF, RMSEA, RMR, SRMR, GFI, AGFI) her bir alt form için mükemmel uyum gösterdiği görülmüştür. MAPEDA (36-72 Ay) için yapılan güvenirlik analizleri sonucunda Cronbach α güvenirlik değerleri; MAPEDA (36-48 Ay) formu için .98, MAPEDA (49-60 Ay) formu için .99 ve MAPEDA (61-72 Ay) formu için .99, iki yarı test güvenirliği r değerleri; MAPEDA (36-48 Ay) formu için .97, MAPEDA (49-60 Ay) formu için .96 ve MAPEDA (61-72 Ay) formu için .96 bulunmuştur. MAPEDA'nın (36-72 Ay) her bir alt formu alt-üst %27 güvenirlik p değeri .05 düzeyinde anlamlı bulunurken madde toplam korelasyon değerlerinin alanyazında belirtilen kriterleri sağladığı tespit edilmiştir. Çalışmanın ikinci amacı doğrultusunda okul öncesi kaynaştırma/bütünleştirme uygulamalarının yürütüldüğü sınıflarda eğitim gören öğrencilerin matematik performans düzeyleri MAPEDA (36-72 Ay) ile değerlendirilmiş ve öğrencilerin çoğunun (%51) çok düşük ve düşük matematik performans düzeyine sahip oldukları belirlenmiştir. Araştırmanın son amacına yönelik kaynaştırma/bütünleştirme uygulamaları yürütülen okul öncesi sınıflarında eğitim gören öğrencilerin matematik performans düzeyleri; aileye (aile gelir düzeyi, anne baba eğitim düzeyi, ailedeki çocuk sayısı), öğrenciye (cinsiyet, eğitim alma süresi, tanı durumu), öğretmene (cinsiyet, yaş, mesleki kıdem) ve sınıfa (sınıf mevcudu, tanılı öğrenci sayısı, yardımcı/gölge öğretmen durumu) yönelik değişkenlere göre değerlendirilmiştir. Analiz sonuçlarına göre; aile gelir durumu, öğrenci cinsiyeti, öğrencinin tanı durumu ve yardımcı/gölge öğretmen durumunun MAPEDA'nın (36-72 Ay) her bir alt formunda öğrencilerin matematik performans düzeyleri üzerinde anlamlı farklılıklara yol açtığı tespit edilmiştir.
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Examined the performance of 42 middle- and 42 low-income kindergarten children on arithmetic calculations presented in a nonverbal format as well as in 3 different verbal formats. On the nonverbal task, the child was shown an initial set of disks, which was then hidden with a cover. The set was transformed by adding or removing disks. After the transformation, the child's task was to construct an array of disks that contained the same number of disks as in the final hidden set. A significant interaction between income level and task format was obtained. Although middle-income children performed better than low-income children on each of the verbal calculation tasks, the 2 income groups did not differ in performance on the nonverbal calculation task. The findings suggest that the nonverbal task format is less sensitive to socioeconomic variation than are the verbal task formats. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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A substantial body of data on young children's counting is now available. Portions of these data can be explained by social context interpretations, such as those advanced by Elbers, but other parts of the data cannot be. The most urgent need is for models that illuminate how primitive conceptual understanding of cardinality and ordinality contributes to learning of counting procedures, and how experience with the counting procedures, in turn, enriches children's conceptual understanding.
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The research reported in this paper was designed in a cross-sectional perspective to examine differences between mathematically disabled children (MD children) and mathematically normal children (MN children). Examined was the pattern of development that unfolds when the children move up through primary school, as reflected in their level of performance, in the discrepancy between their performance on simple number-fact problems compared with simple word problems, as well as in their use of task-specific strategies, identified as material, verbal, and mental strategies. The MN children's performance gradually improved from the second to the sixth grade and showed progression from strategies based on use of material through verbal strategies to mental strategies. The MD children's performance showed a course of development that had almost peaked in Grade 2, and their problem solving reflected inflexible use of a narrow register of different task-specific strategies characterised by functional efficiency dependent on problem type. The MD children's consistent use of material strategies was in accordance with the view that these children had both fact-retrieval problems and working-memory problems, and it indicated absence of an adequate domain-specific knowledge base of task-specific strategies. The findings highlight the MD children's need for mathematics instruction to move from focus on computation to strategy-learning activities.
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We traced the emerging relations between children's understanding of multidigit numbers and their computational skill and investigated how instruction influenced these relations. We followed about 70 children over the first 3 years of school while they were learning about place value and multidigit addition and subtraction in 2 different instructional environments. By interviewing the students several times each year, we found that understanding and skill were closely related on tasks for which students had not yet received instruction as well as on more difficult tasks even after instruction. Students appeared to apply specific understandings to invent new procedures and modify old ones. The alternative instruction, which encouraged students to develop their own procedures and to make sense of procedures presented by others, appeared to facilitate higher levels of understanding and closer connections between understanding and skill.
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The aim of the study is to investigate the informal and formal mathematical knowledge of children suffering from "mathematics difficulty" (MD). The research involves comparisons among three groups: fourth-grade children performing poorly in mathematics but normal in intelligence; fourth-grade peers matched for intelligence but experiencing no apparent difficulties in mathematics; and a randomly selected group of third graders. These children were individually presented with a large number of tasks designed to measure key mathematical concepts and skills. The findings suggest that: (1) MD children are not seriously deficient in key informal mathematical concepts and skills; (2) MD children seem to have elementary concepts of base ten notation but experience difficulty in related enumeration skills, particularly when large numbers are involved; (3) MD children's calculational errors often result from common error strategies; (4) MD children display severe difficulty in recalling common addition facts; and (5) in the area of problem solving, MD children are capable of "insightful" solutions and can solve simple forms of word problems, but experience difficulty with complex word problems. MD children are in many respects similar to normal, younger peers; an hypothesis of "essential cognitive normality" is advanced. The only and dramatic exception occurs in the area of number facts. While clinical experience corroborates this finding, its explanation is not evident.
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Knowledge of addition combinations has long been thought to facilitate the learning of subtraction combinations (e.g., 8 - 5 = ? can be answered by thinking 5 + ? = 8). Indeed, it follows from Siegler's (1987) model that an associative facilitating effect should make the correct answer the most common response to a subtraction combination, even in the earliest phase of mental-subtraction development. Children in the initial or the early phase of development were examined in 2 studies. Study 1 involved 25 kindergartners and 15 first graders in a gifted program; Study 2 involved 21 first graders in a regular program. Participants were presented with pairs of items, such as 4 + 5 = 9 and 9 - 4 = ?, and asked if the first item helped them to answer the second. Many participants, particularly the less developmentally advanced ones, did not recognize they could use a related addition equation to determine a difference. Study 2 participants were also administered a subtraction timed test. Contrary to Siegler's model, developmentally less advanced children responded with the correct difference relatively infrequently on nearly all items, and even developmentally advanced children did so on more difficult items. The results of both studies are consistent with earlier findings that suggested the complementary relation between addition and subtraction is not obvious to children. They further indicate that an understanding of the complementary relation is not an all-or-nothing phenomenon. It often develops first with subtraction combinations related to the addition doubles, apparently because such addition combinations are memorized relatively early. Ready facility with related addition combinations may make it more likely that children will connect their knowledge of subtraction to their existing intuitive knowledge of part-whole relations. This process may also account for why Study 2 participants were able to master subtraction complements without computational practice. Methodological and educational implications are discussed.
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Provides a longitudinal assessment of skill development in addition for 26 normal and 12 mathematically disabled 1st- or 2nd-grade children. At the first time of measurement, the children solved 40 simple addition problems. 10 mo later, all Ss were readministered the addition task and a measure of working memory resources. Across times of measurement, the normal group showed increased reliance on memory retrieval and decreased reliance on counting to solve the addition problems, as well as an increase in speed of counting and retrieving addition facts from long-term memory. The math-disabled group showed no reliable change in the mix of problem-solving strategies or in the rate of executing the counting or memory retrieval strategies. Finally, reliable differences, favoring the normal group, were found for the index of working memory resources. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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This study provides follow-up data on the development of calculation abilities in middle- and low-income children after formal instruction in first grade. Two conventional verbal calculation tasks (story problems, number-fact problems) and one nonverbal calculation task were used. Before formal instruction, middle-income kindergarten children performed better than low-income kindergarten children on both verbal calculation task, but the two income groups did not differ in performance on the nonverbal calculation tasks (Jordan, Huttenlocher, & Levine, 1992). After formal instruction in first grade, there still were no income group differences on the nonverbal calculation tasks. Moreover, there no longer were income group differences on number-fact problems. This finding was associated with the development of more effective calculation strategies among the low-income children. However, on story problems low-income children still performed more poorly than middle-income children. The findings show that even after formal instruction low-income children have difficulties with certain verbal arithmetic tasks.
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This paper examines the effects of quantitative literacy on the likelihood of employment among young adults in the United States. The data set used is the 1985 Young Adult Literacy Assessment Survey. This survey of persons 21 to 25 years old makes available scores achieved by individuals sampled on a test measuring proficiency in the application of arithmetic skills to practical problems encountered every day. We use these scores as one of a set of variables in a probit model explaining the probability of a person being fully employed. It is found that quantitative literacy skills are a major factor raising the likelihood of full-time employment. Furthermore, low quantitative literacy appears to be critical in explaining the lower probability of employment of young Black Americans relative to Whites.