Article
Addition and Subtraction: A Cognitive Perspective
The development of addition and subtraction problem solving skills 01/1982;

 "For the purposes of this study, only responses to addition or subtraction problems were recorded. Therefore, the task was reduced to 20 items: 6 change AWPs, 6 compare AWPs, 6 equalize AWPs, and 2 combine AWPs, based on the classification of Carpenter and Moser (1983; Cronbach's alpha reliability value for the 20 items is .95, and when only change, compare, and equalize problems are used this value is .94). "
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ABSTRACT: Arithmetic word problem (AWP) solving is a highly demanding task for children with learning disabilities (LD) since verbal and mathematical information have to be integrated. This study examines specifically how syntactic awareness (SA), the ability to manage the grammatical structures of language, affects AWP solving. Three groups of children in elementary education were formed: children with arithmetic learning disabilities (ALD), children with reading learning disabilities (RLD), and children with comorbid arithmetic and reading learning disabilities (ARLD). Mediation analysis confirmed that SA was a mediator variable for both groups of children with reading disabilities when solving AWPs, but not for children in the ALD group. All groups performed below the control group in the problem solving task. When SA was controlled for, semantic structure and position of the unknown set were variables that affected both groups with ALD. Specifically, children with ALD only were more affected by the place of the unknown set. 
 "The assertion that, for instance, the sum of two and four more is equal to four and two more still involves a unary interpretation of addition. Weaver [26] calls such an assertion "pseudocommutativity" as it does not describe the property of an operation. Though such statements as 2 + 4= 6 and 4+ 2 = 6 are mathematically equivalent, psychologically they imply different meanings – even for adults. "
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ABSTRACT: Based on research on expertise a person can be said to possess integrated conceptual knowledge when she/he is able to spontaneously identify task relevant information in order to solve a problem efficiently. Despite the lack of instruction or explicit cueing, the person should be able to recognize which shortcut strategy can be applied  even when the task context differs from the one in which procedural knowledge about the shortcut was originally acquired. For mental arithmetic, first signs of such adaptive flexibility should develop already in primary school. The current study introduces a paperandpencilbased as well as an eyetrackingbased approach to unobtrusively measure how students spot and apply (known) shortcut options in mental arithmetic. We investigated the development and the relation of the spontaneous use of two strategies derived from the mathematical concept of commutativity. Children from grade 2 to grade 7 and university students solved threeaddends addition problems, which are rarely used in class. Some problems allowed the use of either of two commutativitybased shortcut strategies. Results suggest that from grade three onwards both of the shortcuts were used spontaneously and application of one shortcut correlated positively with application of the other. Rate of spontaneous usage was substantial but smaller than in an instructed variant. Eyetracking data suggested similar fixation patterns for spontaneous an instructed shortcut application. The data are consistent with the development of an integrated concept of the mathematical principle so that it can be spontaneously applied in different contexts and strategies. 
 "Regarding numerical skills, children under the age of two can make value judgments by recognizing small discrete quantities [55]–[58], and larger numerosities, albeit in an imprecise way [59]–[63]. At around the age of 2, they are able to count to about six and detect a violation of counting [64]–[67], but children cannot master the same counting principles as adults before six years of age, i.e. the sequence of number words, the oneone correspondence between objects and words, and the cardinal principle [68], [69]. By the age of 5 or 6, they solve verbal calculation problems requiring arithmetic skills [70]–[72], although younger children can already predict the outcomes of simple additions and subtractions [73]–[76]. "
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ABSTRACT: To investigate the rise of economic abilities during development we studied children aged between 3 and 10 in an exchange situation requiring them to calculate their investment based on different offers. One experimenter gave back a reward twice the amount given by the children, and a second always gave back the same quantity regardless of the amount received. To maximize payoffs children had to invest a maximal amount with the first, and a minimal amount with the second. About one third of the 5yearolds and most 7 and 10yearolds were able to adjust their investment according to the partner, while all 3yearolds failed. Such performances should be related to the rise of cognitive and social skills after 4 years.
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