A componential model for mental addition

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 true-false reaction time paradigm. The 800 problems comprised 200 problems of each of four types: two single-digit addends, one single- and one double-digit addend, two double-digit addends, and three single-digit 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.     

Individual differences in cognitive arithmetic

Unities in the processes involved in solving arithmetic problems of varying operations have been suggested by studies that have used both factor-analytic and information-processing methods. We designed the present study to investigate the convergence of mental processes assessed by paper-and-pencil measures defining the Numerical Facility factor and component processes for cognitive arithmetic identified by using chronometric techniques. A sample of 100 undergraduate students responded to 320 arithmetic problems in a true-false reaction-time (RT) verification paradigm and were administered a battery of ability measures spanning Numerical Facility, Perceptual Speed, and Spatial Relations factors. The 320 cognitive arithmetic problems comprised 80 problems of each of four types: simple addition, complex addition, simple multiplication, and complex multiplication. The information-processing results indicated that regression models that included a structural variable consistent with memory network retrieval of arithmetic facts were the best predictors of RT to each of the four types of arithmetic problems. The results also verified the effects of other elementary processes that are involved in the mental solving of arithmetic problems, including encoding of single digits and carrying to the next column for complex problems. The relation between process components and ability measures was examined by means of structural equation modeling. The final structural model revealed a strong direct relation between a factor subsuming efficiency of retrieval of arithmetic facts and of executing the carry operation and the traditional Numerical Facility factor. Furthermore, a moderate direct relation between a factor subsuming speed of encoding digits and decision and response times and the traditional Perceptual Speed factor was also found. No relation between structural variables representing cognitive arithmetic component processes and ability measures spanning the Spatial Relations factor was found. Results of the structural modeling support the conclusion that information retrieval from a network of arithmetic facts and execution of the carry operation are elementary component processes involved uniquely in the mental solving of arithmetic problems. Furthermore, individual differences in the speed of executing these two elementary component processes appear to underlie individual differences on ability measures that traditionally span the Numerical Facility factor.(ABSTRACT TRUNCATED AT 400 WORDS)     

Mental Arithmetic: A Componential Analysis of Speed-of-Processing Across Monolingual,Weak Bilingual,and Strong Bilingual Adults

Two experiments compared rates of solving simple and complex addition and multiplication problems in groups of speakers of French or English in Experiment 1 (n = 35) and Spanish or English in Experiment 2 (n = 84). Subjects were divided into groups of English unilinguals, weak bilinguals, and strong bilinguals according to their performance on a naming task. In both experiments, simple problems consisted of two single-digit numbers. At least three single-digit numbers were used for complex problems in Experiment 1 and double-digit numbers in Experiment 2. Mean solution times, particularly for complex problems, were lowest for the monolingual group, followed in turn by the weak bilingual and strong bilingual groups, but these differences were not statistically reliable in either experiment. In Experiment 2, however, componential analyses of solution times indicated that strong bilingual subjects were slower at executing the carry operation when solving complex problems, relative to the two remaining groups. Results were interpreted in terms of the relationship between bilingualism and the representation and processing of numerical information.     

Computational and reasoning abilities in arithmetic: Cross-generational change in China and the United States

A Chinese advantage over Americans was found for economically relevant computational and reasoning abilities in arithmetic for groups of 6th- and 12th-grade students matched or equated on general intelligence. No cross-national difference for computational or reasoning abilities was found for samples of older (60- to 80-year-old) Chinese and American adults equated on general intelligence. The pattern of change in arithmetical competencies across cohorts suggests that the Chinese advantage in 6th and 12th grade is due to a cross-generational decline in competencies in the United States and a cross-generational improvement in China.     

Numerical and arithmetical cognition: a longitudinal study of process and concept deficits in children with learning disability

Based on the stability and level of performance on standard achievement tests in first and second grade (mean age in first grade = 82 months), children with IQ scores in the low-average to high-average range were classified as learning disabled (LD) in mathematics (MD), reading (RD), or both (MD/RD). These children (n = 42), a group of children who showed variable achievement test performance across grades (n = 16), and a control group of academically normal peers (n = 35) were administered a series of experimental and psychometric tasks. The tasks assessed number comprehension and production skills, counting knowledge, arithmetic skills, working memory, the ease of activation of phonetic representations of words and numbers, and spatial abilities. The children with variable achievement test performance did not differ from the academically normal children in any cognitive domain, whereas the children in the LD groups showed specific patterns of cognitive deficit, above and beyond the influence of IQ. Discussion focuses on the similarities and differences across the groups of LD children.     

Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability

Groups of first-grade (mean age = 82 months), third-grade (mean age = 107 months), and fifth-grade (mean age = 131 months) children with a learning disability in mathematics (MD, n = 58) and their normally achieving peers (n = 91) were administered tasks that assessed their knowledge of counting principles, working memory, and the strategies used to solve simple (4+3) and complex (16+8) addition problems. In all grades, the children with MD showed a working memory deficit, and in first grade, the children with MD used less sophisticated strategies and committed more errors while solving simple and complex addition problems. The group differences in strategy usage and accuracy were related, in part, to the group difference in working memory and to group and individual differences in counting knowledge. Across grade-level and group, the switch from simple to complex addition problems resulted in a shift in the mix of problem-solving strategies. Individual differences in the strategy mix and in the strategy shift were related, in part, to individual differences in working memory capacity and counting knowledge.     

Numerical and arithmetical cognition: patterns of functions and deficits in children at risk for a mathematical disability.

Based on performance on standard achievement tests, first-grade children (mean age = 82 months) with IQ scores in the low-average to high-average range were classified as at risk for a learning disability (LD) in mathematics, reading, or both. These at-risk children (n = 55) and a control group of academically normal peers (n = 35) were administered experimental tasks that assessed number comprehension and production skills, counting knowledge, arithmetic skills, working memory, and ease of retrieving information from long-term memory. Different patterns of intact cognitive functions and deficits were found for children in the different at-risk groups. As a set, performance on the experimental tasks accounted for roughly 50% and 10% of the group differences in mathematics and reading achievement, respectively, above and beyond the influence of IQ. Performance on the experimental tasks thus provides insights into the cognitive deficits underlying different forms of LD, as well as into the sources of individual differences in academic achievement.     

Trade-offs and individual differences in evolved traits

Comments on Evolutionary psychology: Controversies, questions, prospects, and limitations (see record 2010-02208-001) by Confer et al. We applaud Confer et al.'s (February-March 2010) clarifications of the many misconceptions surrounding the use of evolutionary analyses in psychology. As they noted, such misunderstandings are common and result in a curious tendency of some of our colleagues to criticize evolutionary psychology without a firm understanding of evolution itself. Confer et al. also did an admirable job acknowledging current unresolved issues among evolutionary psychologists (e.g., the relative importance of group selection on humans). The above said, we disagree with their view that a current limitation of evolutionary psychology is its inability to explain phenomena "that appear to reduce an individual's reproductive success, and cannot be explained by mismatches with, or hijacking of, our psychological mechanisms by modern-day novel inputs" (Confer et al., 2010, p. 122). Mismatches between modern environments and environments of evolutionary adaptedness are only one set of explanations for seemingly maladaptive traits (Nesse, 2005). Another set involves evolutionary trade-offs.