Spatial knowledge and geometric representation in a child blind from birth

On the biology and politics of cognitive sex differences

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.

Brain differences, anthropological stories, and educational implications

Geary's anthropological assumptions are questioned as he uses literature influenced by sociobiology to back the claim that Female humans do not engage in environmental orientation. Yet, female gatherers or migrators do need and use spatial skills. Geary's exploration of gender differences in math skills is speculation that hasharmful, ungrounded, and misleading educational implications and applications, particularly in light of research on the status of gender equity in education.

The twain shall meet: Uniting the analysis of sex differences and within-sex variation

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.

Sex differences and evolutionary by-products

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.

Between-sex differences are often averaging artifacts

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.

Able youths and achievement tests

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.

Differences in male and female cognitive abilities: Sexual selection or division of labor?

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.

Arithmetic and old lace

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.

Sex differences in mathematical abllity: Genes, environment, and evolution

Geary proposes a sociobiological hypothesis of how (and why) 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.

Mary has more: Sex differences, autism, coherence, and theory of mind

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.

Is there a comparative psychology of implicit mathematical knowledge?

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.

On an evolutionary model of sex differences in mathematics: Do the data support the theory?

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 are far from understanding sex-related differences in spatial-mathematical abilities despite the theory of sexual selection

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.

Do gender differences in spatial skills mediate gender differences in mathematics among high-ability students?

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.

Genetic influences on sex differences in outstanding mathematical reasoning ability

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 (OMRA) 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.

Sexual-selection accounts of human characteristics: Just So Stories or scientific hypotheses?

We evaluate three of Geary's claims, finding that (1) there is little evidence for sex differences in object- vs. person-orientation; (2) sex differences in competition, even if biologically caused, lead to sex differences in mathematics only given a certain style of teaching; and (3) sex differences in mental rotation, though real, are not well explained in a sociobiological framework or by the proximate biological variables assumed by Geary.

A critic with a different 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.

Omissions relevant to gender-linked mathematical abilities

Analyses of bodies of data usually omit some relevant studies. Geary omits (1) some studies looking at functional correlates of basic biological data, (2) studies of developmental implications for functioning, and (3) the recent achievement of acceleration of cognitive development.

Resources dimorphism sexual selection and mathematics achievement

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.

The logic of the sociobiological model Geary-style

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.

Some problematic links between hunting and geometry

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.

Crossmodal changes in the somatosensory vibrissa/barrel system of visually deprived animals

Cats deprived of vision from birth adapt remarkably well to their situation and show little behavioral impairment. They seem to compensate for their lack of vision by relying more on their auditory and tactile senses. We report here that the facial vibrissae, which are most important for tactile orientation in many animals, show supernormal growth in both cats and mice that have been deprived of vision from birth. Furthermore, the whisker representation in the somatosensory cortical barrel field shows a concomitant enlargement in binocularly enucleated mice: individual barrels are expanded in size by up to one-third. The increased use of the vibrissae in visually deprived animals may stimulate both their own growth and, via activation of the respective neural pathways, the expansion of their central representation.

Acquisition of Route and Survey Knowledge in the Absence of Vision

The ability of sighted, blindfolded individuals to navigate while walking was assessed in two types of tasks, one requiring knowledge of a route that previously had been navigated and another requiring more complex spatial inference or computation. A computerized measurement system monitored spatial position. The route tasks included maintenance of a heading, distance and turn reproduction and estimation, and turn production. The inferential task required completion of a multisegment pathway by returning directly to the origin. pathways were replicated at two different scales. Measures for the route-knowledge tasks indicated a substantial ability to navigate in the absence of visual cues. Route reproduction performance was particularly accurate despite intrinsic veering tendencies. A substantial increase in error was observed in the pattern-completion task. Errors in pathway completion increased with pathway complexity and were quite similar in the two scales. Correlational data suggested that performance on different route-knowledge tasks reflected differing underlying representations. The completion task led to a high correlation between absolute turn and distance error but had minimal correlations with the route tasks. The data suggest that a survey representation with some degree of scale independence was constructed for use in the pathway completion task.

Mirror reversal by blind subjects in cutaneous perception and motor production of letters and numbers

A letter drawn on the forehead is often perceived as its mirror reversal. Similarly, people produce mirror reversals when asked to write a letter on the underside of a table at which they are sitting. We varied the orientation and position of the stimulated or inscribed surface, and found several examples of these two mirror-reversal phenomena in blind subjects (who had no, or very little, visual experience), as well as in sighted subjects. Furthermore, the mirror-reversal rate of the two groups of subjects as a function of orientation and position of the surface was highly correlated. Thus, the mirror-reversal phenomena should be attributed to a nonvisual, spatial cognitive scheme that is perhaps shared by sighted and blind subjects.

Understanding maps as symbols: the development of map concepts in children

We expect that many readers encountered this article with the beliefs that maps are highly specialized devices primarily used for wayfinding; that they represent the spatial world in a single, correct form; that they are readily transparent; and that their sole contribution to psychology is their role in externalizing environmental cognition. By discussing the myriad functions and forms of maps, by highlighting their symbolic nature, and by considering some of the misconceptions about maps, we have attempted to demonstrate the value of maps for addressing a wide range of developmental questions. Our review of past research literature suggests that research conducted within individual disciplines has both strengths and limitations. Work in the psychological tradition is characterized by attention to important subject characteristics and to carefully described and implemented research designs, procedures, coding, and analyses. At the same time, this work reveals, at best, highly restricted views about maps, and at worst, fundamental misconceptions about maps. Work in the geographic and environmental traditions, in contrast, samples a broader range of map forms and functions, but it suffers from inattention to procedural details that makes the conclusions less compelling than they might otherwise be. A conventional wisdom is emerging from the work in both traditions: That children's map understanding occurs extremely early and extremely easily. The limitations of both research traditions, however, suggest the need for caution in accepting this view. Developmental and cartographic theories provide a compelling reason to reexamine the early and easy view and suggest the need for alternative conceptual and empirical approaches. We have argued that future work should integrate the traditions of psychology and geography. Illustrative data from an interdisciplinary program of research were presented. We described work demonstrating the gradual and difficult process of mastering the representational and geometric correspondences that link the map to its referent in the world. Our data suggest that there are significant achievements in map conceptualization (the understanding of the concept of a map), map identification (understanding the formal components of a map), and map utilization (the ability to use maps). Our data support the view that maps are not transparent and that children's abilities to understand, use, and create maps are linked to their developing representational and spatial skills. In concluding, we should acknowledge that we have deliberately pushed interpretations about understanding maps as symbolic representations to the extreme. The reason for this strategy is simple: We believe that work on maps--both in the public schools and in academia--is assumed to be an expendable and irrelevant luxury.(ABSTRACT TRUNCATED AT 400 WORDS)

Styles of mental imagery in blind adults

Two tests of mental imagery were administered to six blind and four sighted subjects. One test, involving pictorial imagery, required subjects to imagine a checkerboard of which only certain squares, specified by the experimenter, were filled in; the other test was similar but emphasized nonpictorial imagery, requiring subjects to imagine a three-dimensional assemblage of small cubes. In both cases, the subject's task was to name a common object which the imaged pattern resembled. It was found that the greater the proportion of life for which each of the blind subjects had been without sight, the less was his or her pictorial score, expressed as a proportion of total imagery score. This suggests that the nature of mental imagery slowly changes following the loss of sight. Relative facility with the two types of imagery also varied from one subject to another within the sighted group; hand movements used by some of them as an aid to imagery may have been a factor in producing this variation.

Gerbils in space: performance on the 17-arm radial maze

In Experiment 1 six hungry gerbils received six trials per day on a 17-arm radial maze. During each trial the subjects were allowed to choose freely among the arms, each of which contained a food pellet, until each arm had been visited once or until eight minutes had elapsed. An error was recorded when the subject entered a previously visited arm. The gerbils quickly learned not to re-enter previously visited arms and generally made errors on fewer than 15% of entries, performance comparable to that of the rat and superior to that of other species tested in the radial arm maze. The intertrial-interval duration did not affect accuracy of arm choices during acquisition but did influence asymptotic accuracy. Accuracy did not change systematically over the six trials. A high proportion of arm entries were to nearby arms. Errors occurred most often towards the end of a trial. Odor cues were not important. When the number of trials per day was reduced from six to one, accuracy deteriorated slightly. In Experiment 2 neither the transposition of extramaze cues nor the placement of the maze in a different room had large disruptive effects on accuracy. In Experiment 3 the addition of three explicit intramaze brightness cues aided accuracy, perhaps by permitting the subjects to decompose the large maze into three smaller mazes, although there was no direct evidence that this was the case. Implications of a number of these results for models of spatial maze performance were discussed.