From the perspective of evolutionary theory, we believe it makes more sense to view the sex differences in spatial cognition as being an evolutionary by-product of selection for optimal rates of fetal development. Geary does not convince us that his proposed selective factors operated with “sufficient precision, economy, and efficiency.” Moreover, the archaeological evidence does not support his proposed evolutionary scenario.

The target article draws on evolutionary theory to formulate a biosocial model of sex differences in quantitative abilities. Unfortunately, the data do not support some of the crucial hypotheses. The male advantage in geometry is not appreciably greater than the male advantagi in algebra, and the greater male variability in mathematics cited by Gear is not cross-culturally invariant.

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...) from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

The male advantage in certain mathematical domains contributes to the difference in the numbers of males and females that enter math-intensive occupations, which in turn contributes to the sex difference in earnings. Understanding the nature and development of the sex difference in mathematical abilities is accordingly of social as well as scientific concern. A more complete understanding of the biological contributions to these differences can guide research on educational techniques that might someday produce more equal educational outcomes in mathematics and (...) other academic domains. (shrink)

I have provided evidence that Geary's model does not explain male dominance in spatial abilities by sexual selection. The current literature concerning the relations of nonverbal IQ to testosterone, hand preference, and right- and left-hand skill, as well as the organizing effects of testosterone on cerebral lateralization during the perinatal period, does not support Geary's arguments.

In Darwinian terminology, “sexual selection” refers to purely reproductive competition and is conceptually distinct from natural selection as it affects reproduction generally. As natural selection may favor the evolution of sexual dimorphism by virtue of the division of labor between males and females, this possibility needs to be taken very seriously.

To the extent that Geary's theory concerning biologically primary and secondary behaviors depends on factor analytic methods and findings, it is woefully weak. Factors have been mistakenly called primary mental abilities, but the adjective “primary” represents reification of a mathematical dimension defined by correlations. Fleshing out a factor beyond its mathematical properties requires much additional quantitative experimental and correlational research that goes far beyond mere factoring.

Geary suggests that implicit mathematical principles exist across human cultures and transcend sex differences. Is such knowledge present in animals as well, and is it sufficient to account for performance in all species, including our own? I attempt to trace the implications of Gearys target article for comparative psychology, questioning the exclusion of “subitizing” in describing human mathematical performance, and asking whether human researchers function as cultural agents with animals, elevating their implicit knowledge to secondary domains of numerical performance.

This commentary focuses on one of the many issues raised in Geary's target article: the importance of gender differences in spatial ability to gender differences in mathematics. I argue that the evidence for the central role of spatial ability in mathematical ability, or in gender differences in it, is tenuous at best.

Geary's project faces the severe methodological difficulty of tracing the biological effects of gender on mathematical ability in a system that is massively open. Two methodological stratagems he uses are considered. The first is that pancultural sex differences are biological in nature, which is dubious in the domain of mathematics, since it is completely culture-bound. The second is that sociosexual differences are partly caused by biosexual differences, which renders his thesis unfalsifiable and empirically empty.

The principles of sexual selection were used as an organizing framework for interpreting cross-national patterns of sex differences in mathematical abilities. Cross-national studies suggest that there are no sex differences in biologically primary mathematical abilities, that is, for those mathematical abilities that are found in all cultures as well as in nonhuman primates, and show moderate heritability estimates. Sex differences in several biologically secondary mathematical domains are found throughout the industrialized world. In particular, males consistently outperform females in the solving (...) of mathematical word problems and geometry. Sexual selection and any associated proximate mechanisms influence these sex differences in mathematical performance indirectly. First, sexual selection resulted in greater elaboration in males than in females of the neurocognitive systems that support navigation in three-dimensional space. Knowledge implicit in these systems reflects an understanding of basic Euclidean geometry, and may thus be one source of the male advantage in geometry. Males also use more readily than females these spatial systems in problem-solving situations, which provides them with an advantage in solving word problems and geometry. In addition, sex differences in social styles and interests, which also appear to be related in part to sexual selection, result in sex differences in engagement iii mathematics-related activities, thus further increasing the male advantage in certain mathematical domains. A model that integrates these biological influences with sociocultural influences on the sex differences in mathematical performance is presented in this article. (shrink)

Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio (8 vs. 16 but not 8 vs. 12). The capacities to represent approximate numerosity found in adult animals and humans evidently develop in human infants prior to language and symbolic counting.

Male superiority in mathematical ability (along with female superiority in verbal fluency) may reflect the operation of an X-Y homologous gene (the right-shift-factor) influencing the relative rates of development of the cerebral hemispheres. Alleles at the locus on the Y chromosome will be selected at a later mean age than alleles on the X, and only by females.

The ratio-bias (RB) phenomenon refers to the perceived likelihood of a low-probability event as greater when it is presented in the form of larger (e.g. 10-in-100) rather than smaller (e.g. 1-in-10) numbers. According to cognitive-experiential self-theory (CEST), the RB effect in a game of chance in a win condition, in which drawing a red jellybean is rewarded, can be accounted for by two facets of concrete thinking, the greater comprehension (at the intuitive-experiential level) of single numbers than of ratios, and (...) of smaller than of larger numbers. In a lose condition, in which drawing a red jellybean is punished, the assumption of a third facet of concrete thinking, the ''affirmative-representation principle'', is necessary, as many participants reverse their focus of attention from the undesirable red to the desirable white jellybeans. Results supported the CEST explanation of the RB effect by demonstrating a predicted negative linear relation between the magnitude of the RB effect and the magnitude of the probability-ratios in the win condition and a positive linear relation in the lose condition. Support was also found for the associative principle of experiential processing. (shrink)

The aim of this paper is to demonstrate that empirically delimited structures of mind are also differentiable by means of systematic logical analysis. In the sake of this aim, the paper first summarizes Demetriou's theory of cognitive organization and growth. This theory assumes that the mind is a multistructural entity that develops across three fronts: the processing system that constrains processing potentials, a set of specialized structural systems (SSSs) that guide processing within different reality and knowledge domains, and a hypecognitive (...) system that monitors and controls the functioning of all other systems. In the second part the paper focuses on the SSSs, which are the target of our logical analysis, and it summarizes a series of empirical studies demonstrating their autonomous operation. The third part develops the logical proof showing that each SSS involves a kernel element that cannot be reduced to standard logic or to any other SSS. The implications of this analysis for the general theory of knowledge and cognitive development are discussed in the concluding part of the paper. (shrink)

Geary's is the traditional view of the sexes. Yet each part of his argument – the move from sex differences in spatial ability and social preferences to a sex difference in mathematical ability, the claim that the former are biologically primary, and the sociobiological explanation of these differences – requires considerable further work. The notion of a biologically secondary ability is itself problematic.

Spatial visualization as a key variable in sex-related differences in mathematical problem solving and spatial aspects of geometry is traced to the 1960s. More recent relevant data are presented. The variability debate is traced to the latter part of the nineteenth century and an explanation for it is suggested. An idea is presented for further research to clarify sex-related brain laterality differences in solving spatial problems.

Geary's model is a worthy effort, but ambiguous on important issues. It ignores differential resource allocation, although this follows directly from sexual selection via differential parental investment. Dimorphism in primary traits is arbitrarily attributed to sexual selection via intramale competition, rather than direct evolutionary pressures. Dubious predictions are made about the consequences of raising mathematics achievement.

Criticism of sex differences in mathematical ability and sex roles in sociobiology and the pernicious influence of these ideas on education.

Geary proposes a sociobiological hypothesis of how sex differences in math and spatial skills might have jointly arisen. His distinction between primary and secondary math skills is noteworthy, and in some ways analogous to the closed versus open systems postulated to exist for language. In this commentary issues concerning how genes might affect complex cognitive skills, the interpretation of heritability estimates, and prior research abilites are discussed.

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event (...) depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS. (shrink)

‘Number sense’ is a short‐hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain‐specific, biologically‐determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including language; the existence of a homology (...) between the animal, infant, and human adult abilities for number processing; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher–level cultural devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution. (shrink)

Research from ethology and evolutionary biology indicates the following about the evolution of reasoning capacity. First, solving problems of social competition and cooperation have direct impact on survival rates and reproductive success. Second, the social structure that evolved from this pressure is the dominance hierarchy. Third, primates that live in large groups with complex dominance hierarchies also show greater neocortical development, and concomitantly greater cognitive capacity. These facts suggest that the necessity of reasoning effectively about dominance hierarchies left an indelible (...) mark on primate reasoning architectures, including that of humans. In order to survive in a dominance hierarchy, an individual must be capable of (a) making rank discriminations, (b) recognizing what is forbidden and what is permitted based one's rank, and (c) deciding whether to engage in or refriin from activities that will allow one to move up in rank. The first problem is closely tied to the capacity for transitive reasoning, while the second and third are intimately related to the capacity for deontic reasoning. I argue that the human capacity for these types of reasoning have evolutionary roots that reach deeper into our ancestral past than the emergence of the hominid line, and the operation of these evolutionarily primitive reasoning systems can be seen in the development of human reasoning and domain-specific effects in adult reasoning. (shrink)

It is commonly supposed that evolutionary explanations of cognitive phenomena involve the assumption that the capacities to be explained are both innate and modular. This is understandable: independent selection of a trait requires that it be both heritable and largely decoupled from other `nearby' traits. Cognitive capacities realized as innate modules would certainly satisfy these contraints. A viable evolutionary cognitive psychology, however, requires neither extreme nativism nor modularity, though it is consistent with both. In this paper, we seek to show (...) that rather weak assumptions about innateness and modularity are consistent with evolutionary explanations of cognitive capacities. Evolutionary pressures can affect the degree to which the development of a capacity is canalized by biasing acquisition/ learning in ways that favor development of concepts and capacities that proved adaptive to an organism's ancestors. q 1999 Elsevier Science B.V. All rights reserved. (shrink)

We challenge the notion that differences in spatial ability are the best or only explanation for observed sex differences in mathematical word problems. We suggest two ideas from the study of autism: sex differences in theory of mind and in central coherence.

Geary is highly selective in his use of the literature on gender differences. His assumption of consistent female inferiority in mathematics is not necessarily supported by the facts.

Cognitive developmental evidence is sometimes conscripted to support ''naturalized epistemology'' arguments to the effect that a general epistemic stance leads children to build theory-like accounts of underlying properties of kinds. A review of the evidence suggests that what prompts conceptual acquisition is not a general epistemic stance but a series of category-specific intuitive principles that constitute an evolved ''natural metaphysics''. This consists in a system of categories and category-specific inferential processes founded on definite biases in prototype formation. Evidence for this (...) system provides a better understanding of the limited ''plasticity'' of ontological commitments as well as a computationally plausible account of their initial state, avoiding ambiguities about innateness. This may provide a starting point for a ''naturalized epistemology'' that takes into account evolved properties of human conceptual structures. (shrink)

The central problem in Geary's theory is how differences are conceptualized. Recent research has shown that between-sex differences on certain tasks are a consequence of averaging within sex differences. A mixture distribution models between-sex differences on several tasks well and does not appear congenial to a sexual-selection perspective.

We evaluate three of Geary's claims, finding that there is little evidence for sex differences in object- vs. person-orientation; sex differences in competition, even if biologically caused, lead to sex differences in mathematics only given a certain style of teaching; and sex differences in mental rotation, though real, are not well explained in a sociobiological framework or by the proximate biological variables assumed by Geary.