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The dynamics of cognition: An ACTR model of cognitive arithmetic
 Kognitionswissenschaft
, 1998
"... not be interpreted as representing the official policies, either expressed or implied, of the ONR or the U.S. government. Keywords: ACTR, cognitive arithmetic, Bayesian learning, activation spreading, dynamical systems, parameter analysis, power law, machine learning, hybrid systems. Cognitive arit ..."
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Cited by 29 (11 self)
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not be interpreted as representing the official policies, either expressed or implied, of the ONR or the U.S. government. Keywords: ACTR, cognitive arithmetic, Bayesian learning, activation spreading, dynamical systems, parameter analysis, power law, machine learning, hybrid systems. Cognitive arithmetic, the study of the mental representation of numbers and arithmetic facts and the processes that create, access and manipulate them, offers a unique window into human cognition. Unlike traditional Artificial Intelligence (AI) tasks, cognitive arithmetic is trivial for computers but requires years of formal training for humans to master. Understanding the basic assumptions of the human cognitive system which make such a simple and wellunderstood task so challenging might in turn help us understand how humans perform other, more complex tasks and engineer systems to emulate them. The wealth of psychological data on every aspect of human performance of arithmetic makes precise computational modeling of the detailed error
Cognitive addition: Strategy choice and speedofprocessing differences in gifted, normal, and mathematically disabled children
 Developmental Psychology
, 1991
"... Sixty young and 60 elderly adults completed a pencilandpaper addition test and solved 40 computerpresented simple addition problems. Strategies and problem solution times were recorded on a trialbytrial basis and were classified in accordance with the distributions of associations model of stra ..."
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Cited by 25 (17 self)
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Sixty young and 60 elderly adults completed a pencilandpaper addition test and solved 40 computerpresented simple addition problems. Strategies and problem solution times were recorded on a trialbytrial basis and were classified in accordance with the distributions of associations model of strategy choices. The elderly group showed a performance advantage on the ability measure and for the developmental maturity of the mix of problemsolving strategies, but the young group showed an advantage for overall solution times. A componential analysis of the overall solution times for memory retrieval trials, however, showed no reliable age difference for rate of retrieving addition facts from longterm memory but did suggest that the elderly adults might have been slower than the younger adults for rate of encoding digits and verbally producing an answer. Overall results are interpreted within the context of the strategy choice model. The first purpose of this study was to extend into adulthood normative information on the development of problemsolving strategies in addition. The second purpose was to compare the performance of young and elderly adults on a relatively wellunderstood cognitive task: the mental solution of simple addition
A componential analysis of an early learning deficit in mathematics
 Journal of Experimental Child Psychology
, 1990
"... This study was designed to assess strategy choice and informationprocessing differences in normal and mathematically disabled first and second grade children. Twentythree normal and 29 learning disabled (LD) children solved 40 computerpresented simple addition problems. Strategies, and their assoc ..."
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Cited by 24 (14 self)
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This study was designed to assess strategy choice and informationprocessing differences in normal and mathematically disabled first and second grade children. Twentythree normal and 29 learning disabled (LD) children solved 40 computerpresented simple addition problems. Strategies, and their associated solution times, used in problem solving were recorded on a trialbytrial basis and each was classified in accordance with the distributions of associations model of strategy choices. Based on performance in a remedial education course, as indexed by achievement test scores, the LD sample was reclassified into an LDimproved group and an LDnochange group. No substantive differences comparing the normal and LDimproved groups occurred in the distribution of strategy choices, strategy characteristics (e.g., error rates), or rate of information processing. The performance characteristics of the LDnochange group, as compared to the two remaining groups, included frequent counting and memory retrieval errors, frequent use of an immature computational strategy, poor strategy choices, and a variable rate of information processing. These performance
A componential model for mental addition
 Journal of Experimental Psychology: Learning, Memory, and Cognition
, 1989
"... A componential model capable of representing simple and complex forms of mental addition was proposed and then tested by using chronometric techniques. A sample of 23 undergraduate students responded to 800 addition problems in a truefalse reaction time paradigm. The 800 problems comprised 200 prob ..."
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Cited by 10 (6 self)
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A componential model capable of representing simple and complex forms of mental addition was proposed and then tested by using chronometric techniques. A sample of 23 undergraduate students responded to 800 addition problems in a truefalse reaction time paradigm. The 800 problems comprised 200 problems of each of four types: two singledigit addends, one singleand one doubledigit addend, two doubledigit addends, and three singledigit addends. The results revealed that the columnwise product of addends, a structural variable consistent with a memory network retrieval process, was the best predictor of mental addition for each of the four types of problem. Importantly, the componential model allowed estimation of effects of several other structural variables, e.g., carrying to the next column and speed of encoding of digits. High levels of explained variance verified the power of the model to represent the reaction time data, and the stability of estimates across types of problem implied consistent component use by subjects. Implications for research on mental addition are discussed. Over the past 20 years, several types of models for mental addition have been proposedfor example, models hypothesizing that analog (Restle, 1970), counting (Groen & Parkman,
The problemsize effect in mental addition: Developmental and crossnational trends
 Mathematical Cognition
, 1996
"... Across two experiments, the magnitude of the problemsize effect in mental addition was examined for kindergarten and elementary school children, as well as adults, from mainland China and the United States. In North American samples, the problemsize effect represents the finding that arithmetic pr ..."
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Cited by 6 (1 self)
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Across two experiments, the magnitude of the problemsize effect in mental addition was examined for kindergarten and elementary school children, as well as adults, from mainland China and the United States. In North American samples, the problemsize effect represents the finding that arithmetic problems consisting of largervalued numbers (e.g. 8+7) take longer to solve and are more error prone than are problems consisting of smallervalued numbers (e.g. 2+3). This standard finding was found for the kindergarten, elementary school, and adult samples from the United States. For the Chinese children, the problem size effect was evident in kindergarten and at the beginning of first grade. However, the effect had disappeared at the end of first grade and had reversed (i.e. largervalued addition problems were solved more quickly than smallervalued problems) by the end of third grade. However, the standard problemsize effect “reappeared ” for the Chinese adults. The results are interpreted in terms of theoretical models of the nature of the memory representation for arithmetic facts and in terms of the mechanisms that govern the development of these representations. In the nearly 25 years since Groen and Parkman’s (1972) seminal study of the mental processes underlying the solution of simple addition problems, cognitive arithmetic has emerged as a vibrant area of research. Scientists in this area have mapped the cognitive processes and neurological correlates that govern the mental solution of simple and complex arithmetic problems and have extended these basic findings to more applied issues, such as
Multiplication Number Facts: Modeling Human Performance With Connectionist Networks
"... INTERNAL REPRESENTATION 8 3 Comprehension Arabic Numeral Comprehension Verbal Numeral Arabic Numeral Verbal Numeral 24 CALCULATION MECHANISMS Numeral Comprehension Mechanisms Numeral Production Mechanisms Production Production Four TwentyThree Times Eight Figure 8. General model of numerical proc ..."
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Cited by 2 (0 self)
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INTERNAL REPRESENTATION 8 3 Comprehension Arabic Numeral Comprehension Verbal Numeral Arabic Numeral Verbal Numeral 24 CALCULATION MECHANISMS Numeral Comprehension Mechanisms Numeral Production Mechanisms Production Production Four TwentyThree Times Eight Figure 8. General model of numerical processing first proposed by McCloskey, Caramazza, and Basili (1985). order fashion. During this activation update the bidirectional connections enable the answer units to affect hidden units, as well as themselves. Retrieval process. To retrieve an answer to a multiplication problem, the network is first initialized by setting the activation level of the problem units to the problem representation and that of the hidden and answer units to 0. During the retrieval process the activation of the hidden and answer units is modified, but the problem units are clamped (i.e., remain unchanged and always supply input). Simulated annealing is then used. This term is adapted from a process of heating and...
Befunde des Forschungsgebietes auf der Grundlage eines
"... untersuchen die mentale Repräsentation von Zahlen und arithmetischen Fakten sowie die kognitiven Prozesse die diese generieren, abrufen und manipulieren. Das Spannungsfeld zwischen der scheinbar einfachen formalen Struktur dieses Aufgabenbereichs und den Schwierigkeiten, die Kinder bei seiner Bewält ..."
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untersuchen die mentale Repräsentation von Zahlen und arithmetischen Fakten sowie die kognitiven Prozesse die diese generieren, abrufen und manipulieren. Das Spannungsfeld zwischen der scheinbar einfachen formalen Struktur dieses Aufgabenbereichs und den Schwierigkeiten, die Kinder bei seiner Bewältigung haben, stellt einen einzigartigen Zugang zum Studium kognitiver Prozesse dar. Der vorliegende