Cognitive and brain systems underlying early mathematical development.

We review current debate regarding the core competencies that support early mathematics learning, focusing on the contributions of the inherent system for representing approximate magnitudes, and domain-general systems that facilitate learning across academic domains. The latter include the executive control system that enables explicit processing of quantitative symbols, such as Arabic numerals, and the logical problem-solving abilities (intelligence) that facilitate learning the relations among numerals. The neural systems that underlie these abilities, as related to mathematical learning, are also discussed, albeit briefly. We place the contributions of inherent quantitative abilities and domain-general mechanisms in an evolutionary context and provide some discussion as to how they interact during the learning of evolutionarily novel mathematics.

Differential contribution of specific working memory components to mathematics achievement in 2nd and 3rd graders.

The contribution of the three core components of working memory (WM) to the development of mathematical skills in young children is poorly understood. The relation between specific WM components and Numerical Operations, which emphasize computation and fact retrieval, and Mathematical Reasoning, which emphasizes verbal problem solving abilities in 48 2nd and 50 3rd graders was assessed using standardized WM and mathematical achievement measures. For 2nd graders, the central executive and phonological components predicted Mathematical Reasoning skills; whereas the visuo-spatial component predicted both Mathematical Reasoning and Numerical Operations skills in 3rd graders. This pattern suggests that the central executive and phonological loop facilitate performance during early stages of mathematical learning whereas visuo-spatial representations play an increasingly important role during later stages. We propose that these changes reflect a shift from prefrontal to parietal cortical functions during mathematical skill acquisition. Implications for learning and individual differences are discussed.

Negative priming from activation of counting and addition knowledge.

In two experiments, subjects were presented with digit pairs (e.g., 32) and asked to respond to the rightmost number. Negative priming, that is, slowed processing, was evident when the rightmost number was a counting-string (e.g., 43 following 12) or addition-sum (e.g., 65 following 32) associate of the number pair from the preceding trial. The studies are the first to demonstrate negative priming with counting and arithmetical memory representations and suggest the obligatory activation of these representations with the presentation of number pairs. The results are also consistent with the view that negative priming often occurs at the semantic level.

When in doubt, do it both ways: brain evidence of the simultaneous activation of conflicting motor responses in a spatial stroop task.

Response competition is often considered an important contributor to the delayed reaction to stimuli for which physical and semantic information are in conflict ("Stroop" effect). Response competition implies that brain areas associated with correct and incorrect responses (e.g., left and right motor cortices) should be simultaneously activated in conflict conditions. However, there is at present little direct evidence of this phenomenon, in part because of the paucity of brain imaging techniques that can independently monitor the time course of activation of adjacent brain areas, such as the motor areas. In the present study, we show that the event-related optical signal (EROS) can provide these types of data. The results confirm the prediction that conflict trials elicit simultaneous activation of both motor cortices, whereas nonconflict trials elicit brain activity only in the contralateral motor cortex. These data support a parallel view of the human information processing system.

Estrogens and relationship jealousy.

The relation between sex hormones and responses to partner infidelity was explored in two studies reported here. The first confirmed the standard sex difference in relationship jealousy, that males (n=133) are relatively more distressed by a partner's sexual infidelity and females (n=159) by a partner's emotional infidelity. The study also revealed that females using hormone-based birth control (n=61) tended more toward sexual jealousy than did other females, and reported more intense affective responses to partner infidelity (n=77). In study two, 47 females were assessed four times across one month. Patterns of response to partner infidelity did not vary by week of menstrual cycle, but significant relations between salivary estradiol level and jealousy responses were obtained during the time of rising and high fertility risk. The implications, at least for females, are that any evolved psychological, affective, or behavioral dispositions regarding reproduction-related relationships are potentially moderated by estradiol, and that the use of synthetic hormones may disrupt this relation.

Sex differences in spatial cognition, computational fluency, and arithmetical reasoning.

Alternative explanations for the male advantage in arithmetical reasoning, as measured by the ability to solve complex word problems, include a male advantage in spatial cognition and a male advantage in computational fluency. The current study was designed to test these competing hypotheses. To this end, 113 male and 123 female undergraduates were administered arithmetical computations and arithmetical reasoning tests, along with an IQ test and a test of spatial cognition. There was no sex difference on the IQ test, but males showed significantly higher mean scores on the arithmetical computations, arithmetical reasoning, and spatial cognition measures. A series of structural equation models indicated that individual differences in arithmetical reasoning were related to individual differences in IQ, spatial abilities, and computational fluency. Moreover, the results suggested that the male advantage in arithmetical reasoning is mediated by the male advantages in both computational fluency and spatial cognition.

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.

Evolutionary developmental psychology.

Evolutionary developmental psychology is the study of the genetic and ecological mechanisms that govern the development of social and cognitive competencies common to all human beings and the epigenetic (gene-environment interactions) processes that adapt these competencies to local conditions. The basic assumptions and domains of this emerging field, as related to human life history and social and cognitive development, are outlined, as are implications for issues of importance in contemporary society.

Evolution and proximate expression of human paternal investment.

In more than 95% of mammalian species, males provide little direct investment in the well-being of their offspring. Humans are one notable exception to this pattern and, to date, the factors that contributed to the evolution and the proximate expression of human paternal care are unexplained (T. H. Clutton-Brock, 1989). The nature, extent, and influence of human paternal investment on the physical and social well-being of children are reviewed in light of the social and ecological factors that are associated with paternal investment in other species. On the basis of this review, discussion of the evolution and proximate expression of human paternal investment is provided.

From infancy to adulthood: the development of numerical abilities.

Developmental and school-related changes in basic number, counting, and arithmetic skills from infancy to old age are reviewed. Nearly all of the quantitative competencies that emerge during infancy and the preschool years appear to reflect the operation of a biological primary, or inherent, cognitive system, and appear to be universal in their expression and development. In contrast, most of the basic quantitative competencies acquired in school and that are of importance in industrial societies do not have a direct inherent foundation. As a result, the development of these secondary quantitative abilities varies considerably with educational practices and can, and often does, vary from one country or generation to the next. Variability in the development of secondary quantitative abilities greatly complicates the study of the relation between pathological (e.g., dyscalculia due to stroke) and age-related processes and these abilities.

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.

Functional organization of the human mind: implications for behavioral genetics research.

It is proposed that the fundamental goal of human behavior is to organize and control the social, biological, and physical environments that support reproduction and survival and that the human mind is organized as a system of social, biological, and physical modules that support attempts to seek such control. More precisely, modules are systems of motivational, emotional, cognitive, and behavioral processes that provide the interface between this fundamental goal and the environments within which this goal must be achieved. An outline of this functional systems approach to the human mind is provided and the implications of this perspective for behavioral genetics research are discussed.

Numerical cognition: age-related differences in the speed of executing biologically primary and biologically secondary processes.

Groups of younger and older adults were administered numerical and arithmetical tasks that varied in the extent to which they assess evolved versus culturally specific forms of cognition, termed biologically primary and biologically secondary abilities, respectively. Componential analyses of solution times suggested that younger adults are faster than older adults in the execution of biologically primary processes. For biologically secondary competencies, a pattern of no age-related differences or an advantage for older adults in speed of processing was found. The results are consistent with the view that there has been a cross-generational decline in arithmetical competencies in the United States and are discussed in terms of models of age-related change in cognitive performance.

Comparison of the big-five factor structure across samples of Chinese and American adults.

We compared the factor structure of Goldberg's (1992) 50-item Bipolar Rating Scale (50-BRS) in samples of Chinese (n = 198) and American (n = 303) students. Results confirmed the hypothesized five-factor pattern for the U.S. sample, and a simultaneous multisample confirmatory factor analysis showed that the same five-factor pattern fit the item covariances in the Chinese sample. High levels of internal consistency were found within each sample, and a high degree of congruency of corresponding item factor loadings was obtained across samples. Overall, results supported the potential utility of the Five-Factor Model and the 50-BRS for assessing personality dimensions in Chinese culture.

Development of arithmetical competencies in Chinese and American children: influence of age, language, and schooling.

The arithmetical competencies of more than 200 Chinese or American kindergarten, first-, second-, or third-grade children were assessed toward the beginning and toward the end of the U.S. school year. All children were administered a paper-and-pencil test of addition skills, a digit span measure, and an addition strategy assessment. The addition strategy assessment provided information on the types of strategies the children used to solve simple addition problems as well as information on the speed and accuracy of their strategy use. Information on the number of math instruction periods across times of measurement was also obtained for each of the first-, second-, and third-grade children. The pattern of arithmetical development across the academic year and across the Chinese and American children suggests that a mix of cultural and maturational factors influence the emergence of early arithmetical competencies and that the Chinese advantage in early mathematical development is related to a combination of language- and school-related factors.

Reflections of evolution and culture in children's cognition. Implications for mathematical development and instruction.

An evolution-based framework for understanding biological and cultural influences on children's cognitive and academic development is presented. The utility of this framework is illustrated within the mathematical domain and serves as a foundation for examining current approaches to educational reform in the United States. Within this framework, there are two general classes of cognitive ability, biologically primary and biologically secondary. Biologically primary cognitive abilities appear to have evolved largely by means of natural or sexual selection. Biologically secondary cognitive abilities reflect the co-optation of primary abilities for purposes other than the original evolution-based function and appear to develop only in specific cultural contexts. A distinction between these classes of ability has important implications for understanding children's cognitive development and achievement.

Mathematical disabilities: cognitive, neuropsychological, and genetic components.

Cognitive, neuropsychological, and genetic correlates of mathematical achievement and mathematical disability (MD) are reviewed in an attempt to identify the core deficits underlying MD. Three types of distinct cognitive, neuropsychological, or cognitive and neuropsychological deficits associated with MD are identified. The first deficit is manifested by difficulties in the representation or retrieval of arithmetic facts from semantic memory. The second type of deficit is manifested by problems in the execution of arithmetical procedures. The third type involves problems in the visuospatial representation of numerical information. Potential cognitive, neuropsychological, and genetic factors contributing to these deficits, and the relationship between MD and reading disabilities, are discussed. Finally, suggestions for the subtyping of mathematical disorders are offered.

Simple and complex mental subtraction: strategy choice and speed-of-processing differences in younger and older adults.

Thirty-six younger adults (10 male, 26 female; ages 18 to 38 years) and 36 older adults (14 male, 22 female; ages 61 to 80 years) completed simple and complex paper-and-pencil subtraction tests and solved a series of simple and complex computer-presented subtraction problems. For the computer task, strategies and solution times were recorded on a trial-by-trial basis. Older Ss used a developmentally more mature mix of problem-solving strategies to solve both simple and complex subtraction problems. Analyses of component scores derived from the solution times suggest that the older Ss are slower at number encoding and number production but faster at executing the borrow procedure. In contrast, groups did not appear to differ in the speed of subtraction fact retrieval. Results from a computational simulation are consistent with the interpretation that older adults' advantage for strategy choices and for the speed of executing the borrow procedure might result from more practice solving subtraction problems.

Effects of practice on component processes in complex mental addition.

Two experiments examined the effects of task practice on the speed of executing the component processes underlying the mental solution of complex addition problems. Componential analyses of Ss' response times in Experiment 1 demonstrated that the component process of carrying was reliably affected by amount of task practice. In contrast, the component processes of encoding single digits and of retrieving correct columnar answers from long-term memory appeared not to have been affected by amount of task practice. Computational feasibility checks indicated that the specificity of the practice effects could be explained by 2 distinct learning mechanisms: strengthening and composition. Results of Experiments 2A and 2B favor a composition explanation. We conclude that the differential practice effects in Experiment 1 are probably due to differential composition of component processes underlying complex mental addition.

Counting knowledge and skill in cognitive addition: a comparison of normal and mathematically disabled children.

The relationship between counting knowledge and computational skills (i.e., skill at counting to solve addition problems) was assessed for groups of first-grade normal and mathematically disabled (MD) children. Twenty-four normal and 13 MD children were administered a series of counting tasks and solved 40 computer-administered addition problems. For the addition task, problem-solving strategies were recorded on a trial-by-trial basis. Performance on the counting tasks suggested that the MD children were developmentally delayed in the understanding of essential and unessential features of counting and were relatively unskilled in the detection of certain forms of counting error. On the addition task, the MD children committed many more computational errors and tended to use developmentally immature counting procedures. The immature counting knowledge of the MD children, combined with their relatively poor skills at detecting counting errors, appeared to underlie their poor computational skills on the addition task. Suggestions for future research are presented.