We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

Spatial and mathematical abilities may be “sex-limited” traits. A sex-limited trait has the same determinants of variation within the sexes, but the genetic or environmental effects would be differentially expressed in males and females. New advances in structural equation modeling allow means and variation to be estimated simultaneously. When these statistical methods are combined with a genetically informative research design, it should be possible to demonstrate that the genes influencing spatial and mathematical abilities are sex-limited in their expression. This approach (...) would give an empirical confirmation of Geary's evolutionary speculations. (shrink)

Geary's emphasis on hunting ignores the possible importance of other human activities, such as scavenging and gathering, in the evolution of spatial abilities. In addition, there is little evidence that links spatial abilities and math skills. Furthermore, such links have little practical importance given the small size of most differences and girls' superior performance in mathematics classrooms.

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.

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.

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)

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.

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.

Based on Geary's theory, intelligence may determine which males utilize innate spatial knowledge to inform their mathematical solutions. This may explain why math gender differences occur mainly with higher abilities. In support, we found that mental rotation ability served as a mediator of gender differences on the math Scholastic Assessment Test for two high-ability samples. Our research suggests, however, that environment and biology interact to influence mental rotation abilities.

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)

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.

Sexual selection provides an adequate partial explanation for the difference in means between the distributions, but fails to explain the difference in variance, that is, the overrepresentation of both boys with outstanding mathematical reasoning ability and boys with mental retardation. Other genetic factors are probably at work. While spatial ability is correlated with OMRA, so are other cognitive abilities. OMRA is not reducible to spatial ability; hence selection for navigational skill is unlikely to be the only mechanism by which males (...) have gained an advantage in OMRA. (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.

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'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.

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.

Although Geary's partitioning of mathematical abilities into those that are biologically primary and secondary is an advance over most sociobiological theories of cognitive sex differences, it remains untestable and ignores the spatial nature of women's traditional work. An alternative model based on underlying cognitive processes offers other advantages.

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.

Analyses of bodies of data usually omit some relevant studies. Geary omits some studies looking at functional correlates of basic biological data, studies of developmental implications for functioning, and the recent achievement of acceleration of cognitive development.

Rips et al.'s analyses have boosted the plausibility of proposals that the human mind embodies some critical properties of natural numbers. I suggest that such proposals can be further evaluated by infant studies, neuropsychological data, and evolution-based considerations, and additionally, that Rips et al.'s model may need to be modified in order to more completely reflect infants' quantitative abilities.

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)

Achievement test differences between boys and girls and between young men and young women, mostly favoring males, extend far beyond mathematics. Such pervasive differences, illustrated here, may require an explanatory theory broader than Geary's.

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.

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.

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.

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.

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.

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.

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

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

Cognitive Science, Volume 46, Issue 2, February 2022.