Conference Paper

Kavramsal Şipşak Sayılama Uygulamalarının Hesaplama Performansına Etkisi

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This study tested the hypothesis that subitizing ability may cause achievement differences in mathematics especially for students with mathematics learning disabilities. Students from 1st through 4th grade were applied to curriculum based math achievement tests (MAT). Based on MAT scores, they were dividied into four groups as Mathematics Learning Disorder (MLD) risks, low achievers (LA), typical achievers (TA), and high achievers (HA). All students were asked randomly and canonically arranged dot enumeration tasks with 3 through 9 dots. Median response times (MRT) were calculated for each task and plotted for each grade level and task types. There were virtually no differences in MRTs for number 3 and 4. On the other hand, the MLD risk group spent relatively more time on enumerating canonically arranged dots from 5 through 9. Results provided more support for the claim that rather than subitizing, numerosity coding mechanisms or the type of symbolic quantity manipulations is different in children with different mathematical achievements especially the lower group, the MLD risk group. Keywords: subitizing, numerosity coding, math achievement, symbolic manipulations of quantities
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Three pictures hang in front of a six month-old child. The first shows two dots, the others show one dot and three dots. The infant hears three drumbeats. Her eyes move to the picture with three dots. Young children spontaneously use the ability to recognize and discriminate small numbers of objects (Klein and Starkey 1988). But some elementary school children cannot immediately name the number of pips showing on the dice. What is this ability? When and how does it develop? Is it a special way of counting? Should we teach it? Subitizing: A Long History Subitizing is "instantly seeing how many." From a Latin word meaning suddenly, subitizing is the direct perceptual apprehension of the numerosity of a group. In the first half of the century, researchers believed that counting did not imply a true understanding of number but that subitizing did (e.g., Douglass [1925]). Many saw the role of subitizing as a developmental prerequisite to counting. Freeman (1912) suggested that whereas measurement focused on the whole and counting focused on the unit, only subitizing focused on the whole and the unit; therefore, subitizing underlay number ideas. Carper (1942) agreed that subitizing was a more accurate than counting and more effective in abstract situations. In the second half of the century, educators developed several models of subitizing and counting. They based some models on the same notion that subitizing was a more "basic" skill than counting (Klahr and Wallace 1976; Schaeffer, Eggleston, and Scott 1974). One reason was that children can subitize directly through interactions with the environment, without social interactions. Supporting this position, Fitzhugh (1978) found that some children could subitize sets of one or two but were not able to count them. None of these very young children, however, was able to count any sets that he or she could not subitize. She concluded that subitizing is a necessary precursor to counting. Certainly, research with infants suggests that young children possess and spontaneously use subitizing to represent the number contained in small sets and that subitizing emerges before counting (Klein and Starkey 1988). As logical as this position seems, counterarguments exist. In 1924, Beckmann found that younger children used counting rather than subitizing (cited in Solter [19761). Others agreed that children develop subitizing later, as a shortcut to counting (Beckwith and Restle 1966; Brownell 1928; Silverman and Rose 1980). In this view, subitizing is a form of rapid counting (Gelman and Gallistel 1978). Researchers still dispute the basis for subitizing ability, with patterns and attentional mechanisms the main explanations (Chi and Klahr 1975; Mandler and Shebo 1982; von Glaserfeld 1982). Lower animal species seem to have some perceptual number abilities, but only birds and primates also have shown the ability to connect a subitized number with a written mark or an auditory label (Davis and Perusse 1988).
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Thirty-one 8- and 9-year-old children selected for dyscalculia, reading difficulties or both, were compared to controls on a range of basic number processing tasks. Children with dyscalculia only had impaired performance on the tasks despite high-average performance on tests of IQ, vocabulary and working memory tasks. Children with reading disability were mildly impaired only on tasks that involved articulation, while children with both disorders showed a pattern of numerical disability similar to that of the dyscalculic group, with no special features consequent on their reading or language deficits. We conclude that dyscalculia is the result of specific disabilities in basic numerical processing, rather than the consequence of deficits in other cognitive abilities.
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What representations underlie the ability to think and reason about number? Whereas certain numerical concepts, such as the real numbers, are only ever represented by a subset of human adults, other numerical abilities are widespread and can be observed in adults, infants and other animal species. We review recent behavioral and neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core systems for representing number. Performance signatures common across development and across species implicate one system for representing large, approximate numerical magnitudes, and a second system for the precise representation of small numbers of individual objects. These systems account for our basic numerical intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.
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This paper aims to highlight the significance of a particular aspect of magnitude processing, namely counting and subitizing or the rapid enumeration of small sets of items, for learning. Emphasis is laid on the historical roots and the conceptual framework as well as on studies on pre-verbal and school-age children. Evidence of the potential value of this research for the assessment of children at risk of mathematical learning disabilities, is presented. Inherent to its nature, subitizing relies on rapid, preverbal analogue magnitude comparisons being triggered. We will highlight the differences with counting, and the implications of shortcomings in counting and subitizing in children with mathematical learning disabilities for the automaticity of number magnitude processing. Furthermore we especially look in this paper at the varying assessment paradigms which are used in research with different age groups, something which has received insufficient attention in the past. Finally, we outline the challenges for future research on mathematical learning disabilities.
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Kindergarten to 3(rd) grade mathematics achievement scores from a prospective study of mathematical development were subjected to latent growth trajectory analyses (n = 306). The four corresponding classes included children with mathematical learning disability (MLD, 6% of sample), and low (LA, 50%), typically (TA, 39%) and high (HA, 5%) achieving children. The groups were administered a battery of intelligence (IQ), working memory, and mathematical-cognition measures in 1(st) grade. The children with MLD had general deficits in working memory and IQ, and potentially more specific deficits on measures of number sense. The LA children did not have working memory or IQ deficits, but showed moderate deficits on these number sense measures and for addition fact retrieval. The distinguishing features of the HA children were a strong visuospatial working memory, a strong number sense, and frequent use of memory-based processes to solve addition problems. Implications for the early identification of children at risk for poor mathematics achievement are discussed.
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ABSTRACTThe ability of recognizing a number of brieflypresented items without actually counting is calledsubitizing (from lat. subito = suddenly). Adult subjectscan subitize 3 to 4 items. For greater numbers thesubjects begin a counting process relying on the visualmemory of the test pattern, which needs increasinglymore time as the number of items increases. Thedevelopment of accuracy and speed of subitizingand visual counting was measured for subjects up tothe age of 17 years. Furthermore, this study tests thehypothesis that children with difficulties in acquiringbasic arithmetic skills exhibit developmental deficits insubitizing and/or counting. The study does not intendto investigate theories on the nature of dyscalculiaeven though most test children can be classifiedas dyscalculic.Methods: Two-hundred-nineteen control subjectsand 156 test subjects with problems in arithmetic skillsin the age range of 7 to 17 years were given a visualcounting task in which 1 to 9 items were presentedfor 100 ms. The subjects had to press a digit key on anumerical keyboard to indicate the number of itemsthey had seen. Percentages of correct responses andresponse times were recorded.Results: The analysis shows systematic differencesbetween control and test children increasing with age.The percentage of test children performing belowthe 16-percentile of the age matched controls wasestimated to be between 40% and 78% (increasingwith age).Conclusions: We concluded that the deficit in abasic visual capacity may contribute to the problemsencountered by children with anomalies in acquiringbasic arithmetic skills.
Dyscalculia: Causes, identification, intervention and recognition. Paper presented at the Dyscalculia and Maths Learning Difficulties
  • B Butterworth
Butterworth, B. (2009). Dyscalculia: Causes, identification, intervention and recognition. Paper presented at the Dyscalculia and Maths Learning Difficulties, Holiday Inn, Bloomsbury (nr. Euston Station) London.