Geary, D.C. (1996). Sexual selection and sex differences in mathematical abilities. Behavioral and Brain Sciences 19 (2): 229-284.
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SEXUAL SELECTION AND SEX DIFFERENCES IN MATHEMATICAL ABILITIES

David C. Geary


Department of Psychology
210 McAlester Hall
University of Missouri at Columbia
Columbia, MO 65211
[email protected]

Keywords

sexual selection, sex differences, mathematical ability, spatial ability, sex-role stereotypes, social style

Abstract

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 pan-culturally, in nonhuman primates, and show moderate heritability estimates. Sex differences in several biologically secondary mathematical domains (i.e., those that emerge primarily in school) are found throughout the industrialized world. In particular, males consistently outperform females in the solving of mathematical word problems and in geometry. Sexual selection and any associated proximate mechanisms (e.g., sex hormones) appear to influence these sex differences in mathematical performance indirectly. First, sexual selection appears to have resulted in the greater elaboration of the neurocognitive systems that support navigation in 3-dimensional space in males than in females. Knowledge implicit in these systems appears to reflect an understanding of basic Euclidean geometry, and thus appears to be one source of the male advantage in geometry. Males also co-opt these spatial systems in problem-solving situations more readily than females, which provides males with an advantage in word problems and geometry. Moreover, sex differences in social styles and interests, which also appear to be related, in part, to sexual selection, result in sex differences in engagement in mathematics-related activities, which further increases 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. 


Introduction

The principles of sexual selection provide a useful theoretical framework for understanding human sex differences in certain social and sexual behaviors (Buss 1995), as well as sex differences in certain cognitive domains (Gaulin 1992; Kimura & Hampson 1994). In this article, sexual selection provides the primary theoretical context for examining the cross-national pattern of sex differences in mathematical abilities. The argument is not that complex mathematical abilities, or any associated sex differences, have been directly shaped by evolutionary pressures, but rather that sexual selection appears to have directly shaped the social and cognitive styles of males and females, which, in turn, influence mathematical development and performance and contribute to sex differences in certain mathematical domains.

In order to fully consider the potential influence of evolutionary pressures on mathematical abilities, a general framework for making inferences about the relative degree of biological and cultural influences on cognition is needed (Geary 1995). This is because children's mathematical development occurs primarily in school settings, and, as a result, the assessment of mathematical performance necessarily reflects some cultural influences. In fact, the administration of all psychometric tests, such as the Scholastic Achievement Test (SAT, formerly the Scholastic Aptitude Test) necessarily reflects culturally-taught skills, such as reading. Although performance on these measures must to a large degree reflect schooling and other sociocultural influences, this does not preclude more primary biological influences on psychometric test performance. In the first section below, a framework for making inferences about forms of cognition that are largely influenced by biological factors and forms of cognition that are more culturally-specific is presented; these respective forms of cognition are called biologically-primary and biologically-secondary (Geary 1995; Rozin 1976).

On the basis of this framework, a systematic assessment of sex differences in mathematical abilities requires a consideration of whether the differences in question are evident for biologically- primary or biologically-secondary mathematical domains, which are defined in Section 2. If sexual selection were directly related to sex differences in mathematical abilities, then any such sex difference should be most evident for biologically-primary mathematical domains, those least affected by sociocultural influences (e.g., schooling). In contrast, if there are no sex differences in biologically-primary mathematical domains but consistent sex differences in secondary mathematical domains, then there are two general potential sources of these sex differences. The first and most obvious is a difference in the schooling of boys and girls, given that school is the primary cultural context within which secondary mathematical abilities appear to emerge (Geary 1995). The second and less obvious source of sex differences in secondary mathematical domains is the secondary effects of sexual selection on the cognitive and social styles of boys and girls. Section 3 presents a framework for considering any such secondary influences of sexual selection on sex differences in mathematical abilities.

The present article is not the first to argue that cognitive sex differences in general and sex differences in mathematics in particular have biological origins. Indeed, there are many theoretical reviews in the literature on the potential biological influences on cognitive sex differences (e.g., Benbow, 1988; McGee 1979; McGuinness & Pribram 1979; Ounsted & Taylor 1972; Sherman 1967), and many well reasoned arguments that any such differences are largely the result of the differential socialization of boys and girls (e.g., Fennema & Sherman 1977; Sherman 1980). Benbow, for instance, presented evidence suggesting that the greater number of males than females at the upper end of the distribution of SAT scores reflects, at least in part, a sex difference in the functional organization of the left- and right-hemisphere. McGee argued that a sex difference in certain forms of spatial cognition has biological origins and contributes to sex differences in certain mathematical areas. Sherman (1980, 1981), in contrast, presented evidence suggesting that the sex difference in mathematics was primarily related to the greater mathematical confidence of boys than girls. On this view, the greater mathematical confidence of boys results in a sex difference in mathematical activities (e.g., course taking), favoring boys, and ultimately a male advantage in mathematical performance. In fact, the emergence of any complex cognitive skill almost certainly reflects an interaction between biologically based differences in the types of activities that boys and girls prefer to engage in and the environments made available to them by parents and by peers (McGuinness & Pribram 1979; Scarr & McCarthy 1983).

The present article builds on and extends previous theoretical treatments of the source and nature of sex differences in mathematics in several ways. First, as noted above, potential biological influences on mathematical sex differences are considered explicitly within the context of sexual selection. In particular, sexual selection is used as a theoretical context for examining sex differences in those cognitive and social domains that appear to influence mathematical development. As an example, it is argued in section 3.1 that sexual selection has resulted in a sex difference in the social preferences of males and females, with males being more object oriented and females being more person oriented, which, in turn, influences math-intensive career choices and mathematics course taking. Although the position that the sex difference in person/object orientations influences the sex difference in mathematical development is not novel (Chipman & Thomas 1985; Thorndike 1911; McGuinness & Pribram 1979), the present article contributes to this literature by attempting to place the potential relation between person/object orientations and sex differences in mathematical abilities in the wider context of human evolution.

Second, the article provides a theoretical framework for examining how evolved cognitive abilities might be manifested in evolutionarily novel contexts (such as schools). In the section below, it is argued that there are at least two ways in which biologically primary forms of cognition, such as certain spatial abilities, can be expressed in biologically secondary domains, such as complex mathematics. By more fully articulating the potential relation between primary spatial abilities and secondary mathematical abilities, this framework expands on McGee's (1979) position that the sex difference in certain mathematical areas is secondary to more primary sex differences in certain spatial abilities. Finally, the review of sex differences in mathematical abilities is more comprehensive in many ways than other treatments of this issue (e.g., Benbow 1988; Hyde, Fennema & Lamon 1990). Specifically, the review of mathematical sex differences provides a greater emphasis on cross-national patterns than have most previous reviews and includes a review of potential sex differences in very early numerical competencies, those that appear to be biologically primary.

1. Evolution, culture and cognition

In some respects, all forms of cognition are supported by neurocognitive systems that have evolved to serve some function or functions related to reproduction or survival. These basic neurocognitive systems appear to be found in human beings throughout the world, and appear to support the emergence of species-typical cognitive domains, such as language (Pinker & Bloom 1990; Witelson 1987). However, some forms of cognition, such as reading, emerge in some cultures and not others. This pattern suggests that while the emergence of some domains of cognition are driven largely by biological influences, other domains emerge only with specialized cultural practices and institutions, such as schools, that are designed to facilitate the acquisition of these cognitive skills in children (Geary 1995). Thus, when assessing the source of group or individual differences in cognitive abilities, it seems necessary to consider whether the ability in question is part of a species-typical biologically-primary cognitive domain, or whether the ability in question is culturally-specific, and therefore biologically-secondary.

Although biologically-secondary abilities appear to emerge only in specific cultural contexts, at least for large segments of any given population, they must perforce be supported by neurocognitive systems that have evolved to support primary abilities. Indeed, these culture-specific abilities might involve the co-optation of biologically-primary neurocognitive systems or access to knowledge implicit in these systems for purposes other than the original evolution-based function (S. J. Gould & Vrba 1982; Rozin 1976). The basic premise is that in terms of children's cognitive development, the interface between culture and biology involves the co-optation of highly specialized neurocognitive systems to meet culturally-relevant goals. In this section, a basic framework for distinguishing between primary and secondary abilities is presented (a more complete presentation can be found in Geary 1995).

1.1 Biologically-primary and biologically-secondary abilities. One way to organize our understanding of both biologically-primary and biologically-secondary forms of cognition is to hierarchically organize them into domains, abilities, and neurocognitive systems. Domains, such as language or arithmetic, represent constellations of more specialized abilities, such as language comprehension or counting. Individual abilities, in turn, are supported by neurocognitive systems, and might consist of three types of cognitive competencies; goal structures, procedural skills, and conceptual knowledge (Gelman 1993; Siegler & Crowley 1994). The goal of counting, for instance, is to determine the number of items in a set of objects. Counting is achieved by means of procedures, such as the act of pointing to each object as it is counted. Pointing helps the child to keep track of which items have been counted and which items still need to be counted (Gelman & Gallistel 1978). Counting behavior, in turn, is constrained by conceptual knowledge (or skeletal principles for the initial emergence of primary domains), so that, for instance, each object is pointed at or counted only once.

Although there are similarities in the cognitive competencies associated with primary and secondary abilities (Siegler & Crowley 1994), there also appears to be a number of important differences, two of which are relevant to the current discussion. First, inherent in the neurocognitive systems that support primary abilities is a system of skeletal principles (Gelman 1990). Skeletal principles provide the scaffolding upon which goal structures and procedural and conceptual competencies emerge. For instance, one skeletal principle that appears to be associated with biologically-primary counting abilities is "one-one correspondence" (Gelman & Gallistel 1978). Here, as noted above, the counting behavior of human children and even the common chimpanzee (Pan troglodytes) is constrained by an implicit understanding that each item in an enumerated set must be tagged once and only once (Boysen & Berntson 1989; Starkey 1992). Second, it appears, at least initially, that the knowledge that is associated with primary domains is implicit. That is, the behavior of children appears to be constrained by skeletal principles, but children cannot articulate these principles (Geary 1995; Gelman 1993).

While the initial structures for the cognitive competencies that might be associated with primary abilities appear to be inherent, the goal structures as well as procedural and conceptual competencies for secondary abilities are likely to be induced or learned from other people (e.g., teachers) and might emerge, at least in part, from primary abilities. For the latter, there appear to be two possibilities. As noted above, the first involves the co- optation of the neurocognitive systems that support primary abilities. Second, knowledge that is implicit in the skeletal principles of primary abilities can be made explicit and used in ways unrelated to the evolution of these principles (Rozin 1976).

For a germane example, consider the possibility that the development of geometry as a formal discipline involved, at least in part, the co-optation of the neurocognitive systems that have evolved to support navigation in the three-dimensional physical universe and access to the associated implicit knowledge. Arguably all terrestrial species, even invertebrates (e.g., insects), have cognitive systems that enable navigation in three-dimensional space (Gallistel 1990; J. L. Gould 1986; Landau et al. 1981; Shepard 1994). Cheng and Gallistel (1984), for instance, showed that laboratory rats appear to develop a "Euclidean representation of space for navigational purposes" (p. 420), and, as a result, are sensitive to changes in basic representation, such as shape, and metric, such as angle. Implicit in the functioning of the associated neurocognitive systems is a basic understanding of geometric relationships amongst objects in the physical universe. So, for example, even the behavior of the common honey bee (Apis mellifera) reflects an implicit understanding that the fastest way to move from one location to another is to fly in a straight line (J. L. Gould 1986).

Even though an implicit understanding of geometric relationships appears to be a feature of the neurocognitive systems that support habitat representation and navigation, this does not mean that individuals have an explicit understanding of the formal principles of Euclidean geometry. Rather, the development of geometry as a formal discipline might have been initially based on early geometer's access to the knowledge that is implicit in the systems that support habitat navigation. In keeping with this position, in the development of formal geometry, Euclid apparently "started with what he thought were self-evident truths and then proceeded to prove all the rest by logic" (West et al. 1982; p. 220). The implicit understanding, or "self-evident truth," that the fastest way to get from one place to another is to go "as the crow flies," was made explicit in the formal Euclidean postulate, "a line can be drawn from any point to any point" (In Euclidean geometry, a line is a straight line)" (West et al 1982; p. 221). The former appears to represent implicit, biologically-primary knowledge (i.e., a skeletal principle) associated with the neurocognitive systems that support habitat navigation, whereas the latter represents the explicit formalization of this knowledge as part of the formal discipline of geometry.

Moreover, although the neurocognitive systems that support habitat navigation appear to have evolved in order to enable movement in the physical universe (Shepard, 1994), they can also be co-opted or used for many other purposes. For an example of what I would call cognitive co-optation, consider the use of spatial representations to aid in the solving of arithmetic word problems. Lewis and Mayer (1987) showed that word problems that involve the relative comparison of two quantities are especially difficult to solve. For instance, consider the following compare problem from Geary (1994): "Amy has two candies. She has one candy less than Mary. How many candies does Mary have?" The solution of this problem requires only simple addition, that is, 2+1. However, many adults and children often subtract rather than add to solve this type of problem; the keyword "less" appears to prompt subtraction rather than addition. Moreover, the structure of the second sentence (i.e., Mary is the object rather than the subject of the sentence) also leads individuals to conclude that Amy has more candy than Mary. Lewis (1989) showed that one way to reduce the frequency of errors that are common with these types of relational statements is to diagram (i.e., spatially represent) the relative quantities in the statements (in this example the number of Amy's and Mary's candies).

It is very unlikely that the evolution of spatial abilities was in any way related to the solving of mathematical word problems. Nevertheless, spatial representations of mathematical relationships are used, that is co-opted, by some people to aid in the solving of such problems (Johnson 1984). More important, there appear to be differences in the ease with which these systems can be used for their apparent evolution-based functions and co-opted tasks. The use of spatial systems for moving about in one's surroundings or developing cognitive maps of one's surroundings appears to occur more or less automatically (Landau et al. 1981). However, most people need to be taught, typically in school, how to use spatial representations to solve, for instance, mathematical word problems (Lewis 1989). In short, the formal step-by-step procedures that can be used to spatially represent mathematical relationships are secondary with respect to the evolution of spatial cognition--these systems did not evolve for this purpose but nevertheless can be used for this purpose. In other words, the practices that occur within some cultural institutions, such as schools, can, in a sense, create cognitive skills that otherwise would not emerge. Any such culture-based skill must perforce be built upon more primary forms of cognition (Geary 1995).

In general, I am arguing that the neurocognitive systems that have more likely evolved to support movement in the three- dimensional physical universe (Gaulin 1992; Shepard 1994) can be adapted by human beings for purposes other than the original evolution-based function (Rozin 1976). With regard to mathematics, there are two resulting predictions. The first is that the relationship between spatial and mathematical abilities should be rather selective. Specifically, the prediction is that measures of 3-dimensional spatial abilities should be more strongly related to abilities in the domain of Euclidean geometry (e.g., an intuitive understanding of basic geometric relationships and graphing relationships in Euclidean space) than to other forms of mathematics (e.g., solving algebraic expressions). The prediction is for tests of 3-dimensional spatial abilities because the neurocognitive systems that appear to support habitat navigation evolved in the 3-dimensional physical universe (Shepard 1994). Second, in addition to Euclidean geometry, a relationship between spatial abilities and mathematical information that can be represented spatially is predicted. Example of such areas would include the solving of word problems, tables, graphs, etc.

From this perspective, a thorough assessment of sex differences in mathematical abilities should be based on a consideration of whether the abilities in question are likely to be biologically primary or biologically secondary. For secondary domains, we should consider whether any differences that might emerge could possibly involve the co-optation of biologically primary cognitive systems or access to knowledge implicit in these systems. Thus, as noted earlier, the sections below provide a brief overview of potential primary and secondary mathematical domains. Moreover, given the potential for co-optation, potential biologically-primary sex differences in cognitive and social styles that might influence mathematical development are considered in Section 3.

2. Potential primary and secondary mathematical abilities

The following subsections present a brief overview of potential biologically-primary and biologically-secondary mathematical domains.

2.1 Biologically-primary mathematical abilities. There is some evidence for the pan-cultural existence of a biologically-primary numerical domain which consists of at least four numerical abilities; numerosity, ordinality, counting, and simple arithmetic (see Geary 1995). A brief description of these abilities is provided in Table 1.

Numerosity represents the ability to quickly determine the quantity of a set of 3 to 4 items without the use of counting or estimating. The ability to quickly and accurately make numerosity judgments is evident in human infants in the first week of life, as well as in the laboratory rat, an African grey parrot (Psittacus erithacus), and the common chimpanzee (Pan troglodytes) (Antell & Keating 1983; Boysen & Berntson 1989; Davis & Memmott 1982; Pepperberg 1987). Moreover, numerosity judgments appear to be based on an abstract representation of quantity rather than on modality-specific processes, as these judgments can be made by human infants for auditory and visual information (Starkey et al. 1983; 1990). In support of this view is the finding that certain cells in the parietal-occipital cortex of the cat are selectively responsive to small quantities, whether the quantities are presented in the visual, auditory, or tactile modalities (Thompson et al. 1970).

A sensitivity to ordinal relationships, for example, that 3 is more than 2 and 2 is more than 1, is evident in 18-month-old infants (Cooper 1984; Strauss & Curtis 1984). Moreover, well- controlled studies have shown that nonhuman primates are able to make very precise ordinal judgments (Boysen 1993; Washburn & Rumbaugh 1991). For instance, after learning the quantity associated with specific Arabic numbers, a rhesus monkey (Macaca mulatta) named Abel could correctly choose the larger of two Arabic numbers more than 88% of the time (Washburn & Rumbaugh 1991). More important, Abel could choose the larger of two previously unpaired numbers more than 70% of the time.

Counting appears to be a pan-cultural human activity that is, at least initially, supported by a set of skeletal principles, or implicit knowledge, before children learn to use number words (Crump 1990; Geary 1994; Gelman & Gallistel 1978; Saxe 1982; Starkey 1992; Zaslavsky 1973). As noted earlier, one basic principle that appears to constrain counting behavior is one-one correspondence (Gelman & Gallistel 1978). Implicit knowledge of this skeletal principle is reflected in the act of counting when each item is tagged (e.g., with a number word) or pointed to once and only once (Gelman & Gallistel 1978). Some human infants as young as 18-months-of-age are able to use some form of tag in order to determine the numerosity of sets of up to three items (Starkey 1992), as can the common chimpanzee (Boysen 1993; Rumbaugh & Washburn 1993).

Other research suggests that 5-month-old infants are aware of the effects that the addition and subtraction of one item has on the quantity of a small set of items (Simon, Hespos & Rochat in press; Wynn 1992). Similar results have been reported for 18- month-olds (Starkey 1992) and for the common chimpanzee (Boysen & Berntson 1989). Moreover, these competencies in simple arithmetic appear to be qualitatively similar in the chimpanzee and human infants and young children (Gallistel & Gelman 1992). Preschool children appear to be able to add quantities up to and including three items using some form of preverbal counting, while at least some chimpanzees appear to be able to add items up to and including four items also by means of preverbal counting (Boysen & Berntson 1989; Starkey 1992).

Finally, psychometric and behavioral genetic studies support the argument that some numerical and arithmetical skills are biologically primary and cluster together. In the psychometric literature, human cognitive abilities are hierarchically organized. At the top are abilities that span many or all more specific lower- order ability domains (Cattell 1963; Gustafsson 1984; Horn & Cattell 1966; Spearman 1927; Thurstone & Thurstone 1941; Vernon 1965). These general abilities are often subsumed under the terms general intelligence (g) or fluid and crystallized intelligence and appear to represent cognitive skills, such as speed of processing or working memory, that support performance in many cognitive domains (e.g., Kyllonen & Christal 1990; Vernon 1983). From an evolutionary perspective, g might index, at least in part, individual differences in the ability to co-opt primary abilities, that is, use these abilities for purposes unrelated to their evolutionary functions (e.g., adapting primary abilities for school learning; Geary 1995). Either way, of particular importance to this discussion is the emergence of lower-order numerical and mathematical factors, or ability domains.

For five-year-olds, tests that assess number knowledge, memory for numbers, as well as basic counting and arithmetic skills cluster together and define a Numerical Facility factor (Osborne & Lindsey 1967). In fact, the Numerical Facility factor is one of the most stable lower-order factors ever identified through decades of psychometric research (e.g., Coombs 1941; Thurstone 1938; Thurstone & Thurstone 1941), and has been found throughout the lifespan, as well as with studies of American, Chinese, and Filipino students (Geary 1994; Guthrie 1963; Vandenberg 1959). Even Spearman, an ardent supporter of the view that individual differences in human abilities are best explained by g stated that basic arithmetic skills "have much in common over and above...g" (Spearman 1927; p. 251). Behavioral genetic studies of tests that define the Numerical Facility factor have yielded heritability estimates of about .5, suggesting that roughly 1/2 of the variability in at least some components of arithmetical ability are due to genetic differences across people (Vandenberg, 1962; 1966).

2.2. Biologically-secondary mathematical abilities. The argument that certain features of counting, number, and arithmetic are biologically primary should not be taken to mean that all numerical and arithmetical abilities are biologically primary. In fact, there are many features of counting and arithmetic that are probably biologically secondary. These features include skills and knowledge taught by parents (e.g., the names of number words), concepts that are induced by children during the act of counting (e.g., that counted objects are usually tagged from left to right), and skills that are formally taught in school (e.g., the base-10 system, trading, fractions, multiplication, exponents, etc.) (Briars & Siegler 1984; Fuson 1988; Geary 1994; Ginsburg et al. 1981). Moreover, it is likely that most features of complex mathematical domains, such as algebra, geometry, and calculus are biologically- secondary, given that the associated abilities only emerge with formal education. Of particular relevance to the goal of this article is the development of geometric and mathematical problem-solving abilities, because cross-national studies consistently find sex differences in these areas (see Section 4).

Even though I have argued that basic geometric knowledge might emerge from the neurocognitive systems that support habitat navigation, many formal geometric skills, as taught in high school in the United States, are very likely to represent secondary cognitive abilities. This is because the knowledge that appears to be implicit in the neurocognitive systems that support navigation is limited and imprecise (Gallistel 1990). Thus, while access to this knowledge might facilitate the learning of basic geometric principles, many other features of geometry probably need to be explicitly taught, such as geometric proofs, that the sum of the angles in a triangle is 180 degrees, etc.

Mathematical problem-solving abilities are typically assessed by the solving of arithmetical and algebraic word problems. The solving of mathematical word problems requires the ability to spatially represent mathematical relations, as described earlier, as well as the ability to translate word problems into appropriate equations, and an understanding of how and when to use mathematical equations (Mayer 1985; Schoenfeld 1985). Psychometric studies suggest that mathematical problem solving, or mathematical reasoning (as psychometric researchers call it), is a biologically-secondary cognitive domain. Although a lower-order Mathematical Reasoning factor has been identified in many psychometric studies (Dye & Very 1968; Thurstone 1938), this factor does not emerge in all samples, not even in some samples of college students (e.g., Guthrie 1963). In fact, a distinct Mathematical Reasoning factor is consistently found only with groups of older adolescents (i.e., end of high school or early college) who have taken a lot of mathematics courses (e.g., Very 1967). Thus, unlike the Numerical Facility factor which has emerged in nearly all studies that have included arithmetic tests, a Mathematical Reasoning factor emerges only in samples with prolonged mathematical instruction, suggesting that many individuals do not easily acquire the competencies that are associated with the solving of arithmetical and algebraic word problems.

The fact that a Mathematical Reasoning factor does not emerge until adolescence does not of course preclude biological influences on the acquisition of the associated skills. Indeed, factor analytic studies suggest that mathematical reasoning abilities initially emerge from biologically-primary mathematical abilities and general reasoning abilities, both of which are partly heritable (Vandenberg 1966). However, by the end of high school, the abilities subsumed by the Numerical Facility and Mathematical Reasoning factors are largely independent (Geary 1994). The point is that the abilities that are subsumed by the Mathematical Reasoning factor only appear to emerge with sustained mathematical instruction, and are therefore more likely to represent secondary abilities rather than primary abilities. In other words, it appears that for most individuals direct instruction (e.g., teaching the use of diagrams to solve word problems; Lewis 1989) is necessary for the co-optation of primary abilities and the eventual emergence of a coordinated system of secondary mathematical abilities.

3. Evolution of sex differences

One potential ultimate source of sex differences in social behavior and cognition is sexual selection (Darwin 1859, 1871; Buss 1995; Gaulin 1992). For most mammalian species, sexual selection is thought to operate through intramale competition and female choice of mating partners. For the present argument, sexual selection is broadly defined to include characteristics that directly influence the outcome intramale competition, such as physical strength, as well as less direct influences on the outcome of any such competition. On this view, the factors that might have influenced the evolution of sex differences in certain forms of spatial cognition, for instance, are included under the broader definition of sexual selection. Similarly, some influences on the sexual division of labor might also be considered under the broader definition of sexual selection. Those social and cognitive characteristics that might contribute to any sex difference in the level of investment in offspring would be examples of germane features of the sexual division of labor.

Indeed, in a seminal paper, Trivers (1972) argued that sex differences in the level of parental investment in offspring "governs the operation of sexual selection" (p. 141), and, as such, is the ultimate cause of any associated sex differences. From this perspective, the sex that invests the least in offspring will show less discriminant mating, and, relative to the higher investing sex, show more intrasexual competition over access members of the opposite sex, be physically larger, and have higher mortality rates, among other things. The higher-investing sex, in contrast, is expected to be much more discriminating in terms of choosing sexual partners, with discrimination focusing on a potential partner's physical characteristics or behaviors that might benefit future offspring. In most mammalian species, particularly polygynous species, males invest less than females in offspring, and are physically larger, show more intrasexual competition, and have higher mortality rates than females (Daly & Wilson 1983). For these species, the sex difference in investment in offspring is likely to reflect the internal gestation of offspring, which perforce makes the initial female investment larger than the initial male investment (see Trivers 1972).

There are a number of findings that indicate that human males and females differ on the above dimensions (e.g., sexual behavior, physical size), suggesting that sexual selection has quite likely influenced the physical, social, and cognitive evolution of males and females. For instance, although most human mating systems are functionally monogamous, most preliterate societies are polygynous, that is, a few males have more than one wife, and some males have no wife (Symons 1979). In keeping with the view that human males are potentially polygynous (if only in fantasy for most), are consistent sex differences in sexual behavior. In a large-scale meta-analysis, Oliver and Hyde (1993) reported that across cohorts (i.e., historical periods) males reported greater sexual activity and more permissive attitudes towards casual sex than females. Moreover, human males are, on average, physically larger than human females, although the magnitude of this difference appears to have decreased over the course of human evolution (Frayer & Wolpoff 1985), more frequently engage in extreme forms of intra- and inter-sexual aggression (e.g., homicide; Daly & Wilson 1988; Wilson & Daly 1985), have higher mortality rates, and invest less in their offspring, on average, than human females (Furstenberg & Nord 1985; Stillion 1985). The overall pattern for humans is consistent with Trivers' (1972) model, as well as with the pattern of sex differences found in most (i.e., polygynous) mammalian species (Daly & Wilson 1983).

Any related sex differences are likely to be directly associated with differences in the reproductive strategies of males and females and, as noted above, to any associated division of labor (Buss & Schmitt 1993; Eibl-Eibesfeldt 1989; Frayer & Wolpoff 1985; Ghiglieri 1987; Ruff 1987). We cannot of course recreate human evolution in order to directly test the effects of sexual selection on the psychological development of males and females. However, indirect evidence can be accrued. First, sex hormones are likely to be an important proximate mechanism for the development of any sex differences associated with sexual selection. Thus, social behaviors and cognitive abilities that are sensitive to fluctuations in hormonal levels are good candidates for primary attributes that have been shaped by sexual selection. Second, sex differences in social behaviors and cognitive abilities that are relatively insensitive to historical and cultural changes also need to be considered as potentially related to sexual selection.

Third, it is possible that sex differences in early play patterns influence later sex differences in cognitive abilities (Serbin & Connor 1979). If sex differences in these cognitive abilities are related to sexual selection, then sex differences should be evident in the play patterns that facilitate the acquisition of these abilities (Geary 1992; McGuinness & Pribram 1979). Moreover, these play patterns should also be influenced by sex hormones; this does not preclude social influences as well (MacDonald 1988). Finally, the associated social and cognitive skills should serve some plausible function related to reproductive success. With regard to potential sex differences in mathematical abilities, sex differences in social and cognitive styles need to be considered.

3.1. Social sex differences. Sex differences for a variety of social and sexual behaviors might be understood based on the principles of sexual selection (e.g., mate preferences; Buss 1989). However, this section only provides a brief consideration of two forms of social behavior, the relative degree of competitive versus cooperative social styles and the relative degree of object versus people preferences. A consideration of these two forms of social behavior is necessary because there is some evidence of sex differences on these dimensions, and some research that suggests that any such differences might influence the relative achievement of boys and girls in competitive and cooperative classroom environments (Peterson & Fennema 1985), as well as one's relative interest in science and mathematics-related careers (Chipman et al. 1992). Of course, any such influences would likely be applicable to academic domains other than mathematics. The goal of this article is to consider these potential influences within the context of children's mathematical development, and not to argue that they are exclusively related to mathematics.

As noted in the preceding section, males tend to be more aggressive and dominance oriented than females in most mammalian species, including humans (e.g., Daly & Wilson 1983; Pratto, Sidanius & Stallworth 1993). There are a number of factors that suggest that the male propensity toward aggression, competitiveness, and dominance seeking has its roots in intramale competition (i.e., sexual selection). Human males are verbally and physically more aggressive than females across cultures (Eibl- Eibesfeldt 1989; Rohner 1976). Sex differences in the concern for status and the use of rough-and-tumble play to establish dominance emerges early in the preschool years and at about the same age in all cultures that have been studied (Maccoby 1988). A sex difference in the relative degree of competitive versus cooperative social styles is found in most nonhuman primates as well, although these differences are also related to kin relationships within social groups (Ghiglieri 1987; Manson & Wrangham 1991). Moreover, Susman et al. (1987) found that during adolescence, adrenal and gonadal hormones were related to aggression and delinquency in human males but not females. As with males in many other mammalian species, aggression in human males is most extreme (i.e., homicide) during the courtship period, that is, late adolescence and young adulthood (Daly & Wilson 1983). During this time, homicide tends to be one male killing another male, either for resources (robbery), status, or as a result of sexual jealousy (Wilson & Daly 1985).

None of this should be taken to mean that the sex difference in competitive and aggressive behavior is not influenced by child- rearing or other sociocultural practices. In fact, the expression of many, if not all, biologically based sex differences are likely to be facultatively influenced by developmental experiences and contextual factors (Hall 1992). Indeed, the magnitude of the sex difference in the level of aggression varies across cohorts, age, context, and culture (Eagly & Steffen 1986; Eibl-Eibesfeldt 1989; Hyde 1984). Sex differences in the level of aggression are rather small and unimportant in many contexts, such as laboratory settings (Eagly & Steffen 1986). However, sex differences in competitive and aggressive behavior are often very large under conditions that represent a threat to the male's status, or might directly influence reproduction, e.g., sexual jealousy (Daly & Wilson 1988; Wilson & Daly 1985). The point is, in terms of mean differences and the ratio of extremely aggressive males to females, sex differences are expected to be largest in evolutionarily significant contexts, such as a threat to status, and much smaller in other areas. Nevertheless, these differences might influence how boys and girls respond to competitive, as contrasted with cooperative, classroom environments.

The nature of the social relationships of males and females also tends to differ, across cultures, in other ways that might create a sex difference in responsiveness to competitive and cooperative classroom environments (Eibl-Eibesfeldt 1989). During the preschool years and into adulthood, girls' social styles reflect, relative to boys' styles, a greater concern for egalitarian relationships, more cooperation, and a greater concern for the feelings of other group members (usually other girls; Buhrmester & Furman 1987; Maccoby 1988; 1990). In a longitudinal study of personality development, Block (1993) found rather interesting sex differences in the relation between the self-esteem and social relationships: "young women with high self-esteem scores seemed happy, warmly extraverted, and deeply concerned about interpersonal relationships. Young men with high self esteems seemed self-focused and defensively critical, uneasy, and unready for a connection with others (Block 1993; p. 28).

Indeed, there is reason to suspect that a more self-serving personality provided males with a reproductive advantage over less self-serving males at certain points in our evolutionary history; that is, this style might be facultatively expressed in competitive contexts. Betzig (1993), for instance, described the social structure of humanity's first six civilizations as being despotic. Across all six civilizations, "powerful men mate with hundreds of women, pass their power on to a son by one legitimate wife, and take the lives of men who get in their way" (Betzig, 1993, p. 37). No matter how socially abhorrent, the reproductive advantage that these high status males achieved over other males was likely facilitated by striving for and achieving dominance, and by an insensitivity to the effects that their exploitative behavior had on the well-being of other human beings, male and female.

Perhaps related to the above are sex differences in the relative degree to which males and females are object versus people oriented; males appear to be relatively more object oriented, and females more people oriented (McGuinness 1993; Thorndike 1911). Again, this sex difference is also found in many different cultures and early in development (Eibl-Eibesfeldt 1989). For instance, in one study it was found that people were the focus of 80% of the stories of 2-year-old girls, relative to 10% of the boy's stories; boys primarily talked about objects, such as cars and trains. By 4- years-of-age, 100% of the girls and about 60% of the boys talked about people, as opposed to objects (Goodenough 1957). If the sex difference in object versus people interests was simply due to socialization, then the interests of boys and girls should have diverged not converged with development (McGuinness 1985; McGuinness & Pribram 1979). This is not to say that this pattern necessarily supports a proximate mechanism associated with sexual selection, but rather suggests that parental socialization is not likely to be the sole cause of the sex difference in object versus people preferences.

One might argue, for instance, that the pattern found in this study (Goodenough 1957) simply reflects a sex difference in the rate of the psychological development of boys and girls, favoring girls. However, the sex difference in object versus people orientations is found in throughout the lifespan (McGuinness & Pribram 1979; McGuinness 1993); the sex difference is not likely to be due to a "developmental delay" in boys. McGuinness and Pribram, for instance, noted that female infants are preferentially attentive to communicative signals (e.g., facial expressions), as compared to male infants' preference for "blinking lights, geometric patterns, colored photographs of objects and three- dimensional objects" (p. 19).

While levels of competitiveness, degree of social cooperation, and object orientation will almost certainly be influenced by cultural and child-rearing practices (MacDonald 1988; Rohner 1976), the overall pattern suggests that we should entertain the possibility that at least a portion of the sex difference on these social dimensions is biologically primary, and perhaps related to sexual selection. Indeed, there have been a number of models that have attempted to explain these differences based on the principles of sexual selection. For instance, as described above, it has been argued that males compete with one another more intensely than females (females compete as well, but differently; see Buss 1989) in order to gain social status and resources, which influence mate value and reproductive success more strongly in males than in females (Betzig 1993; Buss 1989; Symons 1979). The object oriented bias of males might reflect an inherent interest in resource acquisition or perhaps, under some circumstances, allow for a depersonalized, and therefore potentially exploitative, view of other human beings. Indeed, even for preschool boys it has been noted that "people appear to be assessed ... more on the basis of their function or utility (can they play games or build a tree house?) than on the basis of their personal characteristics" (McGuinness & Pribram 1979, p. 21). Nevertheless, it is more likely that the object preferences of boys reflect a bias toward learning about the physical, as opposed to the social, environment.

Eibl-Eibesfeldt (1989) suggested that throughout human evolution, the social style of females provided the nucleus for maintaining the long-term stability of social groups. In particular, the cooperative and sociable nature of female groups, relative to male groups, might have provided an important context for the socialization of children and for the sharing of child care responsibilities. Whatever the reason, the consistency of these sex differences across historical periods and cultures, combined with the finding that some of these differences fluctuate with sex and adrenal hormones and are most pronounced during the courtship years, suggest that they might reflect the operation of sexual selection during our evolutionary history. More to the point of the present discussion, these sex differences might contribute to the later described sex differences in an interest in pursuing math- intensive careers (e.g., engineering) and in responsiveness to competitive and cooperative classroom environments.

3.2. Cognitive sex differences. It has been proposed that biologically-primary sex differences exist for some cognitive abilities, in particular language fluency and certain forms of spatial cognition (e.g. Buffery & Gray 1972; McGuinness & Pribram 1979). The focus of this section is on potential sex differences in spatial skills, because it has been frequently argued that the male advantage in certain areas of mathematics (e.g., problem solving) is related to a male advantage in spatial abilities (e.g., Benbow 1988; McGee 1979).

Across various types of spatial ability measures, the magnitude and significance of the male advantage varies considerably (e.g., Caplan et al. 1985). In fact, the lack of consistency across spatial tasks, among other things, led Caplan et al. to conclude "sex differences in spatial abilities do not exist or at least it is by no means clear as yet" (p. 797). Despite this claim, consistent and substantive sex differences, favoring males, have been found for spatial tests that require the mental manipulation of images in 3-dimensional space, and for more dynamic measures of complex spatial abilities (Linn & Petersen 1985; Masters & Sanders 1993; Gilger & Ho 1989; Law et al. 1993; Voyer, Voyer & Bryden 1995). For instance, in a meta-analysis of sex differences on the Mental Rotation Test (MRT; Vandenberg & Kuse 1978), a measure of the ability to mentally rotate 3- dimensional geometric figures, Masters and Sanders found a substantial male advantage (3/4 to 1 1/4 standard deviations) in 14 of the 14 studies assessed. The magnitude of the sex difference was not related to the year the study was published, suggesting that the overall male advantage on the MRT does not fluctuate with short-term cultural changes; although the magnitude of the sex difference on some other spatial tests does appear to vary from year to year (Voyer et al. 1995).

The sex difference on the MRT is especially interesting because of the nature of the cognitive processes used in the rotational task from which the MRT was developed. Based on the results of a detailed series of studies of the strategic and cognitive processes underlying spatial test performance, Just and Carpenter (1985) stated that the rotation of 3-dimensional geometric figures (the figures used to develop the MRT) "is not open to alternative strategies...(and) the cognitive coordinate system within which the figures are represented is the standard environmentally defined one" (p. 165). Stated differently, there appears to be little across subject and across item variability in the processes used to rotate 3- dimensional figures and the rotation of these figures appears to involve the engagement of the same neurocognitive systems that support navigation in three dimensional space (Shepard 1994). In addition to the sex difference in skill at rotating 3-dimensional geometric figures, sex differences are also evident on more dynamic measure of spatial cognition. Law et al. (1993), for instance, found that male college students were better at judging the relative distance and the relative velocity of moving objects than were female college students. Performance on these spatial tasks improves with practice and experience for both males and females. However, the magnitude of the male advantage on these tasks does not diminish with practice (Baenninger & Newcombe 1989; Law et al. 1993).

Furthermore, "there is now substantial evidence that cognitive patterns may vary with phases of the menstrual cycle in normally cycling women and with seasonal variations in androgens in men" (Kimura & Hampson 1994; p. 57). For normally cycling females, performance on spatial tests, including the MRT, is at its lowest when estrogen and progesterone levels are at their highest (Hampson & Kimura 1988; Kimura & Hampson 1994; Silverman & Phillips 1993). For young males, performance on spatial tests varies with testosterone levels (though not linearly) and peaks in the Spring (Kimura & Hampson 1994). Prenatal exposure to sex hormones (e.g., androgens; however, the process is complex and is not always related to androgens in the straightforward manner) increases the spatial abilities of human females and females of other species (Diamond et al. 1979; Resnick et al. 1986). In the laboratory rat, for instance, hormonally treated females and normal males often outperform castrated males and normal females on spatial tasks (e.g., Williams et al. 1990).

In one study it was found that "rats exposed neonatally to gonadal steroids normal males and (hormonally) treated females| selectively attended to geometric cues when they (were) presented in compound with other types of cues; landmarks (were) overshadowed by a coordinate system obtained by the geometry of the room" (Williams et al. 1990; p. 95). Castrated males and normal females did not selectively attend to geometric cues, although they did attend to these cues at times. Gaulin (1992) found that the male advantage on laboratory-based spatial tasks was evident in polygynous but not monogamous species of voles (Microtus). In natural settings, males in polygamous species have much larger home ranges than females, but only during the mating season. For monogamous vole species, the male and female home ranges are equivalent.

In preliterate societies, and for children in Western societies, human males also tend to have larger "ranges" (or play areas) than human females (Eaton & Enns 1986; Gaulin 1992; Maccoby 1988). It has been suggested that for humans, intramale competition favored males with well developed spatial and navigational abilities (Symons 1979). The selection for strong spatial abilities in human males might have been directly related to male-male aggression, which often occurs in the context of small scale warfare between kin-based groups (Alexander 1979; Geary in press). During these periods of conflict, groups of related males often travel relatively long distances to ambush males from other groups or to capture females, which makes foraging far from the home base dangerous for females in some societies (e.g., Chagnon 1977). Moreover, skill in the use of primitive weapons, such as a bow and arrow, might be facilitated by more dynamic spatial skills, such as tracking moving trajectories (Kolakowski & Malina 1974).

Alternatively, the sex difference in spatial and navigation abilities might have been indirectly related to male-male competition through their relation to hunting success (e.g., Hill 1982). In many preliterate societies, "a man's hunting prowess is directly related to the number of wives he can obtain" (Symons 1979; p. 159), and, as a result, skilled hunters often have more offspring than their less skilled peers. Hunting skill is presumably dependent, in part, on the neurocognitive systems that support habitat navigation, as well as on more dynamic forms of spatial cognition (e.g., tracking moving trajectories). This is not to say that hunting is the only way that males can achieve high status in preliterate societies. The point is---if males with superior spatial abilities (which facilitate hunting and, for example, group migration or warfare) had even a slight reproductive advantage over their low-ability peers, then a sex difference, favoring males, in certain spatial abilities (e.g., those involving the processing of 3- dimensional information) would have emerged over the course of human evolution.

Whatever the ultimate reason, for humans, it is likely that hormones have direct and indirect effects on the development of spatial abilities (Geary 1989). Direct effects would include the influence of sex hormones on the development of the neural substrate that supports spatial cognition (Diamond et al. 1979). Indirect influences would involve engagement in behaviors that are likely to provide spatial-related experiences to the individual. Engagement in behaviors that are likely to influence spatial ability development (e.g., Serbin & Connor 1979) appear to be more strongly influenced by early exposure to sex hormones (Berenbaum & Hines 1992) than to parental reinforcement of sex-typed behavior (Lytton & Romney 1991), although the definitive study of these relationships has yet to be conducted (Baenninger & Newcombe in press). In addition, sex differences that are likely to facilitate some component skills associated with hunting/fighting are evident in children's play. In many preliterate societies, boys practice throwing sticks and rocks (often at small animals) about three times as often as girls (Eibl-Eibesfeldt 1989). Two of the largest (often 1.5 to 3.5 standard deviations) sex differences that have been documented are the velocity and distance with which objects can be thrown. A male advantage in throwing velocity and distance is even evident in the preschool years, before large sex differences in sports-related activities (e.g., baseball; Thomas & French 1985).

In summary, human males have a substantial and robust advantage over human females on dynamic spatial measures and for measures that require the representation of information in 3- dimensional space, though much smaller or no sex differences are found for some less complex spatial measures (e.g., rotating images in 2-dimensional space). For complex spatial measures, the male advantage is found across historical periods and cultures, is influenced by prenatal exposure to sex hormones, and fluctuates with circulating levels of sex hormones. At the same time, early exposure to gonadal steroids appears to predispose boys to engage in activities that will facilitate the development of spatial skills. At least for rats, sex hormones also appear to make males especially sensitive to geometric relationships in their habitat, and might explain the earlier described sex difference in human infants' preferences for social versus environment (e.g., geometric shapes) stimuli (McGuinness & Pribram 1979). Finally, consistent with the principles of sexual selection (Trivers 1972), the work of Gaulin (1992) suggests that the male advantage in spatial abilities is restricted to polygynous species, that is, those species, which include humans, where males are typically larger than females, have higher mortality rates, etc. The overall patten of results suggests that sexual selection should be seriously considered as one potential source of human sex differences in complex spatial abilities, especially those supporting navigation in the 3- dimensional physical universe.

This position should not be taken to mean that psychological factors, such as sex-roles, cannot influence performance on spatial tests (e.g., Antill & Cunningham 1982). Baenninger and Newcombe (in press) have recently argued that the just described sex differences in spatial abilities, and the later described sex differences in certain mathematical domains, reflects, at least in part, the differential experiences of boys and girls. They argue that the sex difference in early experiences is driven, in part, by sociocultural influences such as sex role stereotypes, patterns of parental expectations and encouragement, etc. In fact, the emergence of the just described sex differences likely represent an interaction between biological biases and cultural influences. The position here is that at least a portion of the sex difference in spatial abilities appears to reflect biological influences that have been shaped by sexual selection. The proximate mechanisms governing the emergence of these sex differences include sex hormones and a biological bias in the spatial-related activities of boys and girls, a bias that is also likely to be influenced by sociocultural factors.

3.3 Intrasexual variability. Although not a primary focus of this article, the position that sexual selection might be related to the tendency of males to show more variability in some cognitive domains than females is relevant to the issue of sex differences in mathematics (Benbow 1988; Feingold 1992; Thorndike 1911). Although it was noted earlier that human males tend toward polygyny, at least under some circumstances, there is also evidence to suggest that more monogamous pair-bonding, as well as polygyny, has been selected for at some points in our evolutionary history (Smuts & Gubernick 1992; Lovejoy 1981). It is very possible that human males are simply more variable than human females in terms of reproductive strategies, some males favoring short-term mating opportunities and others showing high levels of investment in the martial relationship and the resulting offspring (Buss & Schmitt 1993). Greater variability in reproductive strategies would presumably result in greater variability in any associated social or cognitive domains. In fact, it appears that for many cognitive domains there is greater variability in the scores of males than in the scores of females (Lubinski & Benbow 1992; Lubinski & Dawis 1992). In other words, there are more males than females at the high and low ends of many ability distributions, even for domains where there are no mean sex differences.

The greater male variability across many cognitive domains has often been interpreted to reflect something fundamental about males (Ounsted & Taylor 1972). Fausto-Sterling (1985), in contrast, has argued that any such differences in variability are rather small and when they are found can be attributed to social bias (e.g., differences in the numbers of boys and girls referred to remedial or gifted programs) or more males than females at the lower end of the ability distribution. A this point, the source, or sources, of the sex difference in cognitive variability is not clear.

If sexual selection were somehow related to this sex difference, then larger heritability estimates should be found for males than females on any associated attributes (Wilcockson, Crean & Day 1995). In theory, there might be two ways in which sexual selection might be related to this sex difference intrasexual variability. First, the proximate mechanisms, presumably sex hormones, that induce sex-dimorphic behaviors and cognitive styles might also contribute to the greater variability in males. For instance, not only are males more variable in many cognitive abilities, they are also at much greater risk than females for an array of neurodevelopmental and other physiological disorders (e.g., Gualtieri & Hicks 1985; Stillion 1985). The greater morbidity and mortality of males is not restricted to humans, but is seen in most polygynous species (Daly & Wilson 1983; Ferguson 1985; Mitchell 1981). Sex hormones have been implicated as one source of these physiological sex differences, although other models have been presented as well (e.g., Gualtieri & Hicks 1985). In other words, "androgens not only induce males to violent and risky behavior but probably also hasten degeneration and senescence (Ferguson 1985; p. 448). Such deleterious effects of male hormones might explain the greater number of males than females at the low end of many ability distributions, but does not explain the greater number of males at the high end of these distributions. Alternatively, it has been suggested that males are more sensitive than females to environmental effects, perhaps due to a slower developmental rate which, in turn, might be influenced by sex hormones (e.g., Ounsted & Taylor 1972; Juraska 1986). If so, then poor environments would negatively impact larger numbers of males than females, whereas larger numbers of males than females would benefit from enriched environments. The overall result would be greater male variability in the distribution of many social and cognitive attributes.

Second, as noted above, the relatively greater intrasexual competition among males in comparison to females might have resulted in greater intermale variability in reproductive strategies. In most mammalian species and in most polygynous human societies, there is much greater variability in the number of offspring produced by different males, in comparison to females (Daly & Wilson 1983; Symons 1979; Wade 1979). The greater variability in the reproductive success of males, relative to females, might have created pressures for males who were not successful in modal forms of intramale competition to develop alternative reproductive strategies (Le Boeuf 1974). For instance, although the reproductive success of male elephant seals (Mirounga angustirostris) is directly related to social status, which is determined by means of intermale aggression, some low-status males also father offspring (Le Boeuf 1974; Le Boeuf & Peterson 1968). These males "sneak into the harem and occasionally succeed in copulating with females...by apparently passing for females" (Le Boeuf 1974; p. 173). In addition to the above described differences across males in the level of paternal investment in marriage and children, there is further evidence for variability in the reproductive strategies of human males. For instance, Symons (1979) stated that in many preliterate societies, some high ranking males were good hunters, others were skilled at leading group migrations, and still others had well developed oratory skills. It seems that there are many different routes to high status for males. Presumably, different routes to high status would have led to different patterns of cognitive abilities being selected for in males who used different reproductive strategies. Any such differences in reproductive strategy would presumably have resulted in greater intra- and inter-individual variability in the cognitive profiles of males, relative to females. However, any such sex difference in variability would presumably be restricted to those domains, such as 3-dimensional spatial abilities, associated with intramale competition, rather than across all ability domains. Males are, in fact, more variable than females on spatial-related ability measures but are not more variable on many verbal tests (e.g. Feingold 1992).

Whatever the reason for the sex difference in variability, there are two important points. First, understanding sex differences in the variability of cognitive abilities is probably as important as understanding mean sex differences, because this variability has implications for understanding any difference in the numbers of males and females at the high and low ends of ability distributions (Feingold 1995; Humphreys 1988). Here, sexual selection and its consequences might be one theoretical perspective from which the issue of the sex difference in intrasexual variability can be addressed. Second, sex differences in the distributions of cognitive abilities has important implications for interpreting mean sex differences in mathematical abilities, because mathematics is a domain where a consistent sex difference in variability is found; males are more variable than females (e.g., Feingold 1992). For measures on which there is a mean male advantage, such as mathematics, there could be no sex difference or a female advantage for low ability samples, or for samples with poorly developed skills. Moreover, relative to average-ability samples, there could be larger mean differences, favoring males, for higher ability samples, and greater numbers of males than females in these samples (Benbow 1988).

4. Sex differences in mathematical abilities

This section focuses on potential sex differences, or a lack thereof, in biologically-primary and biologically-secondary mathematical abilities. In preview, research in these domains suggests no sex differences in biologically-primary mathematical abilities, but consistent differences, favoring males, in some biologically secondary areas, in particular mathematical problem solving and geometry.

4.1. Biologically-primary abilities. Although there have been many reviews of sex differences in mathematical abilities, these reviews have been largely confined to kindergarten and older children (Hyde et al. 1990; Kimball 1989). As a result, there have been no systematic assessments of sex differences in biologically- primary mathematical domains, most of which can be assessed in infancy and during the preschool years.

Of particular interest with the infancy studies is whether boys and girls differ in their sensitivity to numerosity, and in their basic understanding of the effects of addition and subtraction on quantity (Starkey et al. 1983; Wynn 1992); sex effects were not assessed in the ordinality studies. Many of the associated studies did not examine sex differences in infant's sensitivity to numerosity (Starkey & Cooper 1980). Those studies that did examine sex differences are very consistent in their findings: Boy and girl infants do not differ in their ability to discriminate small numerosities (Antell & Keating 1983; Starkey et al. 1990; Strauss & Curtis 1981). Wynn (1992), in her assessment of 5-month-old infants' understanding of the effects of addition and subtraction on quantity, did not examine sex differences. In a personal communication, however, she stated that there was no sex difference at all on this task (Wynn, personal communication, August 20, 1993). In a set of related studies, with 18- to 42- month-olds, Starkey (1992) also found no sex difference in the basic understanding of the effects of addition and subtraction on quantity.

As described in Section 2.1, an array of important and potentially biologically-primary numerical abilities emerge during the preschool years. Ginsburg and Russell (1981), in an extensive study of social class and racial differences in the basic numerical skills of 4- to 5-year-olds, found only a single sex difference on measures that appear to assess biologically-primary abilities; girls were slightly better than boys on a basic addition and subtraction task. There were no sex differences, across race or social class, in the ability to count and enumerate, or on any task that assessed basic counting and number knowledge. In a similar study, Song and Ginsburg (1987) examined the basic numerical skills of 4- to 8- year-old children from Korea and the United States. Again, no sex differences for either Korean or American 4- to 5-year-old children were found for measures that appeared to assess biologically- primary numerical abilities. Finally, Lummis and Stevenson (1990) examined the pattern of sex differences in reading and mathematical skills from three cross-national studies of kindergarten, and first- and fifth-grade children from the United States, Taiwan, and Japan. The results showed no sex differences in the counting skills, conceptual knowledge, or the simple arithmetic skills of kindergarten children in any of the three cultures.

The pattern of results suggests that there are no sex differences in biologically-primary mathematical abilities. This conclusion seems to be especially sound for preschool and kindergarten children, because the results are robust across studies and across cultures. For the infancy research, however, this conclusion must be considered tentative, because the measures used in these studies, combined with the small sample sizes, might not be sensitive enough to detect any potentially more subtle differences. Nevertheless, given the results for preschool children, there appears to be little reason to suspect that more subtle sex differences exist in these basic skills. The overall pattern of findings for the infancy and preschool studies suggest that boys are not biologically primed to outperform girls in basic mathematics. In other words, the later sex differences in mathematical problem solving and geometry do not appear to have their antecedents in fundamental numerical abilities.

4.2. Biologically-secondary abilities. This section focuses on sex differences in mathematical problem solving, geometry, and other spatially-related areas. But first, a brief overview of sex differences in complex arithmetic skills (e.g., solving 35 + 97), but not arithmetic word problems, is presented, because differences are sometimes found in this area (Hyde et al. 1990).

4.2.1. Complex arithmetic. A sex difference, favoring girls, is often found on arithmetic achievement tests, especially in the elementary-school and junior-high-school years (Hyde et al. 1990; Marshall & Smith 1987). Hyde et al., in a meta-analysis of sex differences in mathematical skills, found that this advantage is modest (about 1/5 of a standard deviation) but robust, at least for studies conducted in the United States. Cross-national studies that have examined sex differences in complex arithmetic suggest that the advantage of girls in this area is largely an American (i.e., U.S.) phenomenon (Husn 1967). It appears that the sex difference for complex arithmetic skills is found primarily in countries with low-achieving children, which includes the United States, and some other nations, such as Sweden (Husn 1967). Carefully conducted cross-national studies, in nations with higher-achieving children, typically do not find a sex difference in this area, or sometimes find a male advantage (Husn 1967; Lummis & Stevenson 1990). Finally, it should be noted that Hyde et al. (1990) found no sex differences in the understanding of arithmetical concepts in elementary school through high school. Even though most of the studies used in the Hyde et al. meta-analysis were conducted in the United States, cross-national studies support the same conclusion (Fuson & Kwon 1992; Ginsburg et al. 1981a; 1981b; Lummis & Stevenson 1990; Song & Ginsburg 1987).

4.2.2. Mathematical problem solving and geometry. The issue of sex differences in mathematical problem solving, or mathematical reasoning, and geometry is complex, because there are many different types of component skills associated with these domains, some of which show a sex difference, favoring males, and some of which do not. Moreover, in keeping with patterns of intrasexual variability described in Section 3.3, the magnitude and the practical importance of any such sex difference appears to vary with the level of ability of the sample (Hyde et al. 1990). For these reasons, the issue of sex differences in mathematics is addressed in two sections. The first focuses on sex differences in general samples, and separately examines performance on word problems and geometry (and other spatially-based domains, such as measurement). The second section focuses on the issue of sex differences in gifted samples (Benbow 1988; Benbow & Stanley 1983). In both sections, the assessment of sex differences is based on performance on ability and achievement measures, because performance on these measures is predictive of later academic and job-related performance (Benbow 1992; Humphreys et al. 1993; McGee 1979; Rivera-Batiz 1992).

4.2.3. Sex differences in general samples. The results described in this section are from large-scale studies of what appear to be representative samples of boys and girls from many different nations. These studies represent differences across a broad range of abilities, including the gifted. Hyde et al. (1990), and others before them (e.g., Dye & Very 1968), argued that the sex difference in mathematical problem solving, for instance, is not evident until adolescence. As described in Section 2.2, factor- analytic studies suggest that the Mathematical Reasoning factor does not emerge until high school, and it is the emergence of this factor that is typically associated with the emergence of sex differences in mathematical skills, at least in the United States (Very 1967).

In contrast, a number of cross-national studies of elementary-school children have found that boys have a performance advantage over girls in several areas of mathematical problem solving, such as the solving of arithmetic word problems. Lummis and Stevenson (1990) found a reliable sex difference, favoring boys, for the solving of arithmetic word problems for first- and fifth-grade children from the United States, Taiwan, and Japan. Stevenson and his colleagues also found a reliable sex difference, again favoring boys, for performance on arithmetic word problems for fifth-grade children from mainland China and the United States (Stevenson et al. 1990). In all of the comparisons, the Asian children outperformed their American peers, but the magnitude of the sex difference (about 1/5 of a standard deviation) was about the same across nations. Similarly, Marshall and Smith (1987) found that sixth-grade boys, in the United States, committed fewer errors than sixth-grade girls when solving arithmetic word problems. Error patterns suggested that girls found translating relational information into appropriate equations more difficult than boys (Marshall & Smith 1987).

Harnisch et al. (1986), in a reanalysis of data from a large- scale international study of the mathematics of achievement of adolescents (Husn 1967), found a male advantage on a mathematics achievement measure in each of the ten nations included in this reassessment, although the magnitude of the difference varied greatly across nations. The smallest overall sex difference was found in the United States and Sweden, and the largest in Great Britain and Belgium. For 13-year-olds, sex differences were the largest for solving word problems and for geometry in all ten nations (Steinkamp et al. 1985); for 17-year- olds, this same pattern was found in 8 of the 10 countries. Johnson (1984) found that male college students consistently (i.e., across 9 experiments) outperformed their female peers for solving algebraic word problems. The problem set used in this study was largely the same as that used in a series of studies conducted in the 1950s. Across studies, the overall magnitude of the sex difference (about 1/2 of a standard deviation) was unchanged comparing the results from the 1950s to those in the 1980s, although the male advantage in solving word problems did not generalize to other non- mathematical problem-solving tasks (Johnson 1984; Maccoby & Jacklin 1974). In all, these studies indicate that beginning in the elementary-school years and continuing into adulthood, boys often show an advantage over girls in the solving of arithmetical and algebraic word problems.

Except for the solving of algebraic word problems, there appear to be no other consistent sex differences in algebraic skills (Hyde et al. 1990), but modest sex differences are found in geometry and calculus. As noted earlier, Harnisch et al. (1986) found a consistent male advantage (of about 1/2 of a standard deviation) in geometry for both 13- and 17-year-olds across ten nations. Similarly, Stevenson and his colleagues have consistently found sex differences on many mathematical tasks that, like geometry, can be solved by co-opting spatial skills (Lummis & Stevenson 1990; Stevenson et al. 1990). The associated tasks involved measurement, estimation, and the visualization of geometric figures. Differences on these tasks, which always favored boys, emerged as early as the first grade and were found in mainland China, Taiwan, Japan, the United States, as well as in Great Britain (Wood 1976). In all, these studies show a consistent advantage of boys over girls in geometry and other spatially-related mathematical areas, such as measurement, beginning as early as the first grade.

4.2.4. Sex differences in gifted samples. Perhaps the most striking sex difference in mathematical problem solving (or mathematical reasoning) emerges in studies of gifted adolescents (Benbow 1988; Benbow & Stanley 1980; 1983). In the early 1970s, Stanley began a project designed to identify mathematically precocious adolescents (Study of Mathematically Precocious Youth; SMPY). To achieve this end, children who scored in the top 2 to 5% on standard mathematics achievement tests in the seventh grade were invited to take the SAT. The mathematics section of the SAT, the SAT-M, assesses the individual's knowledge of some arithmetic concepts, such as fractions, as well as basic algebraic and geometric skills (Stanley et al. 1986). Twelve- and thirteen-year-olds who scored above the mean for high school girls on the SAT-M were considered to be mathematically gifted, that is, in the top 1% of mathematical ability (for an overview see Benbow 1988).

The consideration of the sex difference in SAT-M performance for SMPY adolescents has been on two levels, mean differences and the ratio of boys to girls at different levels of performance (Benbow 1988; Benbow & Stanley 1983). Across cohorts, American boys, on average, have been found to consistently outperform American girls on the SAT-M by about 30 points (about 1/2 of a standard deviation). This sex difference has also been found in the former West Germany and in mainland China (Benbow 1988; Stanley et al. 1986), although it is of interest to note that the mean performance of gifted Chinese girls (M = 619) was between 50 and nearly 200 points higher, depending on the cohort, than the mean of the American boys identified through SMPY (Stanley et al. 1986). Stanley et al. argued that the advantage of gifted Chinese children over gifted American children on the SAT-M was probably due to more homework in China and the fact that some of the material covered on the SAT-M is introduced in the seventh grade in China, but not until high school in the United States. The finding that Chinese individuals do not have better developed spatial abilities than Americans indicates that this national difference in SAT-M performance is not related to a national difference in spatial abilities (Stevenson et al. 1985).

In the United States, the sex difference in the ratio of boys to girls at different levels of mathematical reasoning ability, as measured by the SAT-M, increases dramatically as the level of mathematical performance increases. The ratio of boys to girls at the lower end of SAT-M scores is a rather modest 1.5:1, but increases to 13:1 for those scoring > 700 (Benbow & Stanley 1983). The over-representation of boys at the high end of SAT-M performance is not limited to SMPY samples. Dorans and Livingston (1987), for instance, reported that across two administrations of the SAT to high-school seniors, 96% of the perfect scores (i.e., 800) on the SAT-M were obtained by males.

A more recent study indicated that the sex difference in the mathematical problem- solving abilities of mentally gifted children is evident even during the elementary school years (Mills et al. 1993). In this study, relatively large samples (ns greater than 400) of mentally gifted boys and girls in grades 2, 3, 4, 5, and 6 were administered a quantitative test that assessed the ability to problem solve in mathematics (no computations were needed). The content of the test included fractions, proportions, and geometry. For all five grade levels, the boys outperformed girls on the overall measure by roughly 1/2 of a standard deviation. A component analysis indicated that boys and girls were equally skilled at determining whether or not enough information was available to answer the question, but boys outperformed girls in the remaining areas.

4.2.5. Summary and conclusion. Consistent sex differences in mathematical performance are found in some domains, such as geometry and word problems, but not other domains, such as algebra (Hyde et al. 1990). It has generally been argued that when a sex difference in mathematical skills is found, it is typically not found until adolescence (Benbow 1988; Hyde et al. 1990). This conclusion has been based, for the most part, on comparisons of American children. Multinational studies, in contrast, show that a male advantage in the solving of arithmetic word problems and on tasks that are solvable through the use of spatial skills, such as visualizing geometric shapes, is often evident in elementary school (Lummis & Stevenson 1990). Differences across studies primarily conducted in the United States and those conducted in other countries might be related to the overall level of achievement of the associated samples. Harnisch et al. (1986) found that sex differences tended to be smallest in nations with the lowest achieving children, which includes the United States. Thus, the relatively poor mathematical skills of American children appear to mask differences that might otherwise be found (see Section 5.2.1).

Even in areas where a sex difference in performance exists across cultures, the difference tends to be selective. For instance, the results of Marshall and Smith (1987) suggest that the advantage of elementary-school boys over girls for solving arithmetical word problems might be more pronounced for problems that require the translation of important relationships, rather than being a general male advantage in solving word problems. Similarly, Senk and Usiskin (1983), in a large-scale national (U.S.) study, found no sex difference in high-school students' ability to write geometric proofs, after taking a standard high-school geometry course, even though adolescent males typically perform better than their female peers on geometric ability tests (Hyde et al. 1990). Thus, the male advantage in geometry also appears to be selective, that is, associated with certain features of geometry rather than the entire domain. The apparent selectivity of the male advantage in secondary mathematical domains is important, because it suggests that the finding of no sex differences in primary domains and select sex differences in secondary domains is not simply due to differences in the complexity of primary and secondary mathematical tasks. Finally, where sex differences emerge, the magnitude of the difference appears to increase as the level of skill of the associated sample increases, at least for adolescents and adults.

5. Sex differences in mathematical abilities: Potential causes

In this section, proximate cognitive and psychosocial factors that appear to contribute to the just described sex differences in mathematical performance are considered. The relationship between these factors and the biologically-primary sex differences discussed in Section 3 are explored in Section 6.

5.1. Cognitive style. The relationship between spatial skills and mathematical performance is complex and inconsistent (Pattison & Grieve 1984; McGee 1979; Sherman 1967). Spatial skills might be useful for some types of mathematical tasks, such as visualizing geometric shapes, but are probably relatively unimportant for seemingly similar tasks, such as providing geometric proofs. Also, many mathematical problems that could be solved with the use of spatial representations can also be solved with the use of non- spatial strategies (Fennema & Tartre 1985; McGuinness 1993), while performance in other areas of mathematics, such as solving algebraic equations, might not be strongly facilitated by spatial skills at all (Halpern 1992). In other words, the relationship between spatial and mathematical skills is probably selective, even within mathematical areas that appear to have important spatial components. To illustrate, Ferrini-Mundy (1987) found that spatial skills were important for solving solids of revolution problems, but less important for solving other types of calculus problems. Thus, it is not surprising that correlational studies sometimes find a relationship between spatial skills and mathematical performance (see McGee 1979), and sometimes do not (Armstrong 1981). It is likely that the content of the mathematical and spatial tests, and the skill level of the sample strongly influence the degree of correlation between tests of mathematical and spatial abilities.

Nevertheless, there appears to be a moderate relationship between certain spatial skills and certain mathematical skills (e.g., Burnett et al. 1979; Fennema & Sherman 1977; Friedman 1995; Sherman 1980). Johnson (1984), for instance, found spatial skills and word problem-solving skills to be correlated .52 and .63, respectively, for male and female college students. Moreover, providing diagrams (i.e., spatial representations) of the mathematical relationships presented in the word problems improved the problem-solving performance of females but not males. Burnett et al. statistically eliminated an advantage of male college students over their female peers on the SAT-M, by partialing a sex difference on a spatial visualization test, which included 2- and 3-dimensional spatial features. Cross-national studies also show that sex differences in mathematical problem solving primarily reside on tasks that appear to be solvable through the use of spatial representations (Harnisch et al. 1986; Lummis & Stevenson 1990). Moreover, although not intuitively obvious, the male advantage in solving mathematical word problems might also be related to the sex difference in spatial abilities. Recall that elementary-school girls appear to have particular difficulty in solving word problems that involve relational comparisons (Marshall & Smith 1987). As noted in Section 2.2, the errors that are associated with solving these types of problems can often be avoided, if important relationships described within the problem are diagramed as part of the problem-solving process (Lewis 1989).

This should not be taken to mean that all cognitive psychologists agree that the male advantage is spatial abilities contributes to the sex difference in mathematical abilities. Lubinski and Humphreys (1990a), for instance, argued that the relationship between sex differences in spatial and mathematical abilities is a statistical artifact. Friedman (1995) showed that mathematical achievement scores are just as highly, and sometimes more highly correlated, with verbal ability than with spatial ability. Clearly, spatial abilities are not the only source of individual differences in performance on mathematical problem solving and geometry tests. Nevertheless, the finding that experimental manipulations, that is, teaching subjects to spatially represent mathematical relationships, reduces the frequency of errors associated with the solving of mathematical word problems (Lewis 1989), especially for females (Johnson 1984), suggests that there is a direct relation between spatial abilities and some mathematical skills.

The teaching of diagramming skills facilitates the performance of females more than males, presumably because more males than females spontaneously use spatial representations to aid in their mathematical problem solving. Indeed, McGuinness (1993) has recently reported that males, from the age of 4-years, are much more likely to resort to spatial-related strategies in problem-solving situations than are females; Johnson and Meade (1987) also found an early (i.e., elementary school) sex difference, favoring males, in spatial abilities. In particular, males spontaneously use dynamic 3- dimensional representations of problem situations much more frequently than do females. This sex difference in strategic approaches to problem solving easily accommodates the cross- national male advantage in certain areas of geometry, as well as performance on estimation and measurement tasks (Harnisch et al. 1986; Lummis & Stevenson 1990), and for solving word problems.

In fact, the final answer to this issue will likely require detailed trial-by-trial assessments of problem solving strategies across mathematical domains, as has been done with children's arithmetic (Siegler 1986). Any such studies will likely show that both males and females use a variety of problem solving strategies to solve mathematics problems. The prediction is that such studies will also show that males and females differ in the frequency with which they use spatial-related strategies for solving certain classes of problem, such as word problems that involve relational comparisons. Indeed, many of the contradictory results in this area are quite likely the result of using measures, such as complex multi-item achievement tests, that reflect an averaging of performance across items for which different strategies have been used for problem solving (Siegler 1987).

5.2. Psychosocial factors. In this section, psychosocial influences on the magnitude of the sex differences in mathematical performance are considered. Any such influence should be considered as complimentary to the influence that the sex difference in spatial skills appears to have on mathematical performance, rather than an alternative explanation. Psychosocial factors appear to influence the level of participation in mathematics and mathematics-related activities. A sex difference in the level of participation in mathematical activities might increase the male advantage in mathematical and spatial performance, but is not likely to create a sex difference in the spontaneous use of spatial strategies in problem-solving situations, especially in 4-year-olds (McGuinness 1993). In all, three general psychosocial influences on the sex differences in mathematical performance are considered; historical trends, perceived competence and perceived usefulness of mathematics, and classroom experiences.

5.2.1. Historical trends. Several recent meta-analyses have suggested that the magnitude of the sex differences in mathematical performance has declined over the last several decades (Feingold 1988; Hyde et al. 1990). These trends support the conclusion that the magnitude of the sex differences in certain mathematical skills is responsive to social changes, such as increased participation of girls in mathematics courses (Travers & Westbury 1989). Hyde et al. (1990) reported that the overall, averaged across all areas, male advantage in mathematics had been reduced by about 1/2, comparing studies published before 1973 with studies published in 1974 or later. The authors did not provide analyses for separate mathematical areas, such as arithmetic or geometry, so it is unclear whether this represents a general or selective phenomenon. Also, two data points are certainly not enough to argue for any type of trend. Feingold, on the other hand, did report separate historical trends, across four time periods, for the Preliminary SAT (PSAT), taken by high-school juniors, and the SAT. For the PSAT, there was a 65% decline in the relative advantage of boys over girls from 1960 to 1983. For the SAT-M, in contrast, the male advantage remained relatively constant from 1960 to 1983. Benbow and her colleagues have found no evidence for a declining male advantage on the SAT-M for SMPY adolescents assessed in different years (Benbow 1988). Similarly, as noted earlier, Johnson (1984) found that the overall magnitude, across nine studies, of the male advantage for solving algebraic word problems remained unchanged from the 1950s to 1980s.

The pattern of results in this area is obviously conflicting. It seems likely that, in the United States, the magnitude of the sex differences in mathematical performance is declining in some areas, especially for the general population (Hyde et al., 1990). This general trend probably reflects a variety of influences, including the greater participation of girls in mathematics-related activities and courses (Linn & Hyde 1989; Travers & Westbury 1989), changes in test items, and potential "floor effects" (Harnisch et al. 1986; Steinkamp et al. 1985). Recall that Harnisch et al. found that the advantage of 17-year-old adolescent boys over same-age girls was smaller in the United States and in Sweden than in eight other nations. The mean difference in the mathematical performance of American boys and the performance of boys in the top eight countries was 1.7 standard deviations, whereas the same comparison for girls produced a difference of 1.2 standard deviations. In other words, the lack of emphasis on mathematics education in American culture (Stevenson & Stigler 1992) appears to have a larger impact on the mathematical achievement of boys than girls.

Moore and Smith (1987) found a similar pattern of sex differences even within the United States, that is, poorly educated adult females showed better mathematical skills than their male peers, but for better educated adults, males had higher mathematics achievement scores than their female peers. Thus, the historical trend for the magnitude of the sex differences in mathematical performance, which has been generated based primarily on data collected within the United States, needs to be interpreted cautiously. In addition to the increase in the percentage of girls in higher-level mathematics courses (see Section 5.2.2), it is very possible that the "disappearance" of some mathematical sex differences might reflect a more general cultural trend in education that is affecting the achievement of boys more than girls. From this perspective, and based on patterns of intrasexual variability on cognitive tests, the sex differences in mathematical problem solving and geometry should increase as the level of mathematical performance increases. This is exactly the pattern that has been found with gifted and general samples within the United States (Benbow 1988; Dorans & Livingston 1987; Moore & Smith 1987), as well as in cross-national comparisons (Steinkamp et al. 1985).

Nevertheless, these same cross-national comparisons show that 17-year-old girls from eight nations had higher mean mathematics achievement scores than boys in the United States (Harnisch et al., 1986). The cross-national pattern of results suggests that with appropriate educational experiences, American girls, on average, can achieve a level of mathematical competence that far exceeds the current skill level of their male peers. This appears to be the case for average as well as gifted children (Stanley et al. 1986). These data also suggest that with any such improved educational experiences, the magnitude of the male advantage in certain areas of mathematics will likely "reappear" or increase.

5.2.2. Perceived competence and usefulness of mathematics. In addition to spatial skills, attitudes about mathematics appear to be related to the magnitude of the sex differences in mathematical performance. Of particular importance is the influence that attitudes have on participation in mathematics courses. In this section, the sex difference in mathematics course taking is first reviewed, and is followed by a consideration of psychosocial factors that influence this sex difference.

Participation in mathematics courses is important, because the number of mathematics courses taken in high school and beyond will influence the development of mathematical skills and will influence career options in adulthood (Sells 1980; Wise 1985). Many studies, across many different nations, have shown that girls take fewer advanced mathematics courses in high school than do boys (e.g., Armstrong 1981; Husn 1967; Nevin 1973; Sherman 1981; Travers & Westbury 1989). This trend lessened somewhat from the 1960s to the 1980s, although in the 1980s male high school students still outnumbered female high school students in advanced courses by about 2 to 1 in the United States and in many other nations (e.g., Chipman & Thomas 1985; Travers & Westbury 1989). Overall, it appears that the greater participation of males in mathematics and mathematics-related courses (e.g., physics) contributes to the sex differences in mathematical performance, at least in high school (Armstrong 1981; Fennema & Sherman 1977; Husn 1967). Thus, it is important to understand why women participate in mathematics courses less frequently than men.

Two important influences on the sex difference in mathematics course taking in high school include sex differences in perceived mathematical competence and in the perceived usefulness of mathematics. Eccles et al. (1993) recently reported that as early as the first grade, children differentiate between their perceived competence in various academic areas, including math, and the value or usefulness of the area. Eccles et al. (1993) found no sex difference in the perceived value or usefulness of mathematics for elementary-school children. However, during the high-school years female students begin to value English courses more highly than mathematics courses, whereas male students show the opposite pattern, valuing mathematics more than English (Eccles et al. 1984; Lubinski et al. 1993). Eccles et al. (1984) also found that the sex difference in the relative valuation of English and mathematics mediated a sex difference, favoring males, in the number of upper- level mathematics courses taken in twelfth grade. Husn (1967) found the same pattern for male and female adolescents across twelve nations. Thus, during the high-school years, male students consistently perceive mathematics as being a more useful skill than female students.

The perceived usefulness of mathematics appears to be largely related to long-term career goals (Chipman & Thomas 1985; Wise 1985). Not surprisingly, those students who aspire to professions that are math intensive, such as engineering or the physical sciences, take more mathematics courses in high school and college than those students who have aspirations for less math- intensive occupations. Chipman and Thomas showed that women were much less likely to enter math-intensive professions than equal ability males. For example, in the United States, by the 1970s, women received just over 3% of the undergraduate engineering degrees, and fewer than 20% of the degrees in computer science and the physical sciences (i.e., chemistry and physics). The proportion of women earning graduate degrees in these areas is even lower (Chipman & Thomas 1985). Longitudinal studies of mathematically gifted individuals show a similar trend. "Less than 1% of females in the top 1% of mathematical ability are pursuing doctorates in mathematics, engineering, or physical science. Eight times as many similarly gifted males are doing so" (Lubinski & Benbow 1992; p. 64).

The relatively small number of women entering these math- intensive areas is not simply due to the fact that they are male- dominated professions, although this likely makes some women hesitant to enter, or remain in, these areas. It also appears that many girls and women do not believe that work in these areas will be especially interesting, even women with very high SAT-M scores (Chipman & Thomas 1985; Lubinski et al. 1993). Sherman (1982), for instance, assessed the attitudes of ninth-grade girls toward mathematics and science-related careers. One interview question asked the girls to imagine working as a scientist for a day. "A clear majority of girls (53%) disliked that day somewhat or very much" (Sherman 1982; p. 435). Even those girls who found working as a scientist for a day acceptable did not consider it to be a preferred activity. This same question was not posed to ninth- grade boys, so it is not known how many boys of this age would have also imagined that working as a scientist would be unrewarding. Either way, by mid-adolescence there is a clear sex difference, favoring boys, in intentions to pursue scientific or other math-intensive careers (Wise 1985).

The relative lack of interest in math-intensive careers on the part of girls, in turn, appears to be related to the sex difference in the relative orientation toward people and human relationships (McGuinness 1993; Maccoby 1988; 1990). As noted in Section 3.1, girls show a greater preference for and interest in social relationships than boys, whereas boys show a greater tendency than girls to be object oriented (Eibl-Eibesfeldt 1989; Thorndike 1911). Object-oriented preferences are related to interests in math- intensive careers, such as engineering or the physical sciences (Chipman & Thomas 1985; Roe 1953). Chipman et al. (1992) found that among college women, an orientation toward objects, rather than people, was related to an interest in a career in the physical sciences. This is not the whole story, however, since female college students are less likely to major in physical science areas than their male peers, even after controlling for sex differences in object versus people preferences and the number of mathematics courses taken in high school (Chipman & Thomas 1985). The fact that most math-intensive areas are male dominated likely dissuades some women from pursuing careers in these areas (Halpern 1992). Nevertheless, it also appears that many girls perceive math-intensive careers (e.g., engineering) to be incompatible with their interests in human relationships, which, in turn, appears to contribute to the sex difference in the relative valuation of mathematical skills, and the resulting sex difference in the number of mathematics courses taken in high school (Lubinski et al. 1993).

The second important psychosocial factor that appears to influence participation in mathematics is perceived competence (Meece et al. 1982). At a general level, perceived competence in academic areas appears to be related to at least two factors (Marsh et al. 1985). The first is one's performance relative to peers; the better the relative performance, the higher the perceived competence. The second factor is intraindividuality. Children who are relatively better at mathematics than reading tend to have a higher perceived competence in mathematics than in reading, independent of their skills relative to other children (Marsh et al. 1985). In the United States, adolescent males are typically more confident of their mathematical abilities than are their female peers (Eccles et al. 1984). Harnisch et al. (1986) found that male 17- year-olds had generally more positive attitudes toward mathematics than female 17-year-olds in 8 of the 10 countries included in their assessment. Eccles et al. (1993) and Marsh et al. (1985) have found that elementary-school boys feel better about their mathematical competence than elementary-school girls, despite the finding that the girls sometimes had higher achievement scores in mathematics.

Lummis and Stevenson (1990) also found that elementary- school boys in the United States, Taiwan, and Japan tended to believe that they were better at mathematics than reading, whereas girls showed the opposite pattern. Nevertheless, in Taiwan and the United States, most of the children thought that boys and girls were equally skilled in mathematics. In Japan, roughly 1/3 of the boys and girls thought that there was no sex difference in mathematical ability, whereas 1/3 of the children thought that boys were better, and the remaining 1/3 thought that girls were better. Thus, the finding that elementary-school boys have better perceptions of their mathematical skills than girls might be related to intraindividual comparisons, that is, because girls tend to prefer reading over math, whereas boys tend to prefer math over reading (Lummis & Stevenson 1990). It is likely that for adolescents, performance differences in high school reinforce these perceptions, as do sex- role stereotypes that portray mathematics as a male domain (Becker 1981; Linn & Hyde 1989).

Regardless of why a sex difference in perceived mathematical competence emerges, it has been argued that perceived competence and the associated expectancies for success might influence task persistence after experiencing failure, and influence the likelihood that the individual will aspire to a math- intensive career (Chipman et al. 1992). Eccles et al. (1984), however, found no sex difference in the tendency to persist on a mathematical problem-solving task following failure, although Kloosterman (1990) found that for algebraic problem solving, high- school boys tended to increase their efforts following failure, whereas high-school girls tended to show less effort following failure. Meece et al. (1990) found that basic abilities, as indexed by earlier grades, and expectancies both contributed to mathematical performance. In all, it seems likely that the sex difference in perceived mathematical competence likely contributes to the sex difference in aspirations to math-intensive careers and the associated sex difference in mathematics course taking in high school (Chipman et al. 1992).

5.2.3. Classroom experiences. It has been suggested that the classroom experiences of boys and girls might impact the development of reading and mathematical skills (Becker 1981; Kimball 1989). In high-school mathematics courses, for instance, a number of studies have found that boys tend to receive more personal contact with teachers than girls. Becker (1981), for instance, collected quantitative and qualitative data on student- teacher interactions across 100 geometry lessons and found some important differences in the way in which teachers interacted with males and females. For example, in comparison to females, males received many more encouraging comments and were asked more process oriented questions. The asking of process oriented questions is important because these questions tend to facilitate the student's conceptual understanding of the material (Geary 1994). Thus, it is likely that the mathematics classroom experiences differ for some males and females, especially in high school, and might contribute to the sex difference in perceived mathematical competence. Though problematic, the nature of these student- teacher interactions is not likely to be the only source of the sex difference in perceived mathematical competence. This is so because females typically receive higher grades than males in these classes which presumably would facilitate their mathematical competence (Kimball 1989).

Indeed, an intriguing study conducted by Peterson and Fennema (1985) suggests that more general teaching styles might have stronger influences on the mathematical, and probably general academic, development of boys and girls than individual interactions between students and teachers. In this study, the activities of teachers and students in 36 fourth-grade classrooms were recorded during mathematics instruction for at least 15 days. Student-teacher interactions were observed, such as the amount of time the teacher spent helping individual students, as well as the amount of time each student spent in mathematical and nonmathematical (e.g., socializing) activities. There was no sex difference in the amount of time boys and girls spent learning mathematics, and no overall sex difference in the change in mathematics achievement scores from the beginning to the end of the school year. However, there were important across-class differences in the relative gains of boys and girls in mathematics achievement. These gains appeared to be related to frequency with which the children engaged in competitive or cooperative classroom activities. Engagement in competitive activities had a considerable negative impact on the mathematics achievement of girls, but slightly improved the performance of boys. Engagement in cooperative mathematical activities (e.g., problem solving in small groups), in contrast, was associated with improvements, across the academic year, in the mathematical performance of girls, but poorer performance for boys.

6. Culture, sexual selection, and sex differences in mathematical abilities: An integrative model

In this section, an integrative consideration of cultural and biological influences on mathematical development and the earlier described sex differences in mathematical performance is presented. 




Figure 1: Schematic  representation of the relations among biological, cognitive, and psychosocial influences in mathematical performance.

The general pattern of relations is show in Figure 1. Here, I have divided biologically-secondary mathematical domains into two general areas, because different factors might influence sex differences in these areas. The first includes the solving of mathematical word problems (i.e., mathematical problem solving), geometry, and other spatially-related areas, such as measurement. The second includes other areas of mathematics that should not be as strongly influenced by spatial abilities, such as algebra (except word problems), probability, etc.

Before examining the pattern of relations shown in Figure 1, I want to suggest that the general level of mathematical achievement within any given culture influences the degree to which sex differences in biologically-secondary mathematical domains (e.g., mathematical problem solving) are expressed, but is not the primary cause of these sex differences. In fact, given that sex differences only appear to emerge for secondary, and not primary, mathematical domains, these differences will only emerge in societies with prolonged formal mathematical instruction. Although the mode of instruction (i.e., competitive versus cooperative classrooms) appears to differentially influence the mathematical development of boys and girls, instructional differences are not likely to be the only source of the sex differences in mathematical abilities. Rather, math instruction in school provides an important context within which more primary sex differences (e.g., spatial abilities) can be expressed.

Moreover, the finding that males tend to be more variable in mathematical and other cognitive abilities and more sensitive to environmental influences (see Section 3.3) in comparison to females, suggests that the degree to which sex differences in mathematical performance are expressed should vary directly with the overall degree of mathematical development. The magnitude of the sex differences in mathematical performance should, and appears to be, the largest for samples (gifted and nongifted) with well developed mathematical skills (Benbow 1988; Harnisch et al. 1986). At the other end of the continuum, it appears that these sex differences disappear or are reversed in some cases (Moore & Smith 1987). Thus, the overall level of skill development influences the expression of sex differences in mathematical abilities, but is not likely to be the primary causes of these differences. Stated differently, sex differences in mathematical performance appear to be the largest, especially by adolescence, in those cultures that greatly facilitate the mathematical development of children. As the emphasis on mathematics achievement declines within a culture, such as the United States, the performance of boys appears to fall more quickly than that of girls. As a result, any sex differences in component abilities will likely be masked.

Nevertheless, there do appear to be some more specific cultural influences that contribute to the magnitude of the sex differences in mathematical performance, that is, sex-role stereotypes and cultural influences on perceived mathematical competence. These factors are shown in the bottom left-hand corner of Figure 1 and are important because they appear to influence participation in mathematics course taking and related activities (Becker 1981; Chipman et al. 1992; Linn & Hyde 1989). However, in comparison to the sex difference in social preferences (described below), stereotypes and perceived competencies probably have a relatively minor, but nonetheless important, influence on participation in mathematics course taking and related activities. Fennema et al. (1981), for instance, found that focusing on the utility of mathematics had a stronger effect on increasing girl's mathematics course taking than did changing sex-role stereotypes. Moreover, stereotypes are the smallest in those groups (i.e., high-ability groups) where the sex differences in mathematics are the largest (Lubinski & Humphreys 1990b; Raymond & Benbow 1986). Stated differently, high-ability girls tend to be less sex-typed than other girls, but still participate much less frequently in mathematics-related activities than their male peers (Lubinski & Benbow 1992). Nevertheless, these findings do not preclude the possibility that some high-ability girls (and other girls) who do not have strong sex-role stereotypes might be discouraged from entering mathematical areas by individuals with sex-role stereotypes (Becker 1981).

Sex differences in social preferences and general interests appear to be important influences on the sex difference in the perceived utility of mathematics and the resulting sex difference in mathematics course taking (Lubinski et al. 1993). Lubinski et al. argued that "the personal attributes of females compared to males will lead them to choose scientific (and mathematical) careers less frequently, as a group, and to distribute their educational development across artistic, social and investigative areas more evenly" (Lubinski et al. 1993; p. 704). Moreover, "preference and ability profiles are in place long before high school" (Lubinski et al. 1993; p. 704), that is, before the sex difference in mathematics course taking emerges. It is not unreasonable to argue that some of these sex differences in social preferences are influenced by stereotypes and perceived competencies; in fact, it is likely that preferences and stereotypes have bidirectional influences on one another, as shown in Figure 1. However, the finding that interest in math-intensive careers is related to object rather than people orientations (e.g. Chipman et al. 1992), and that the sex difference in object versus people orientations is evident in cultures where there are no scientists, mathematicians, or high-school mathematics curriculums, suggests that sex differences in social preferences and interests largely transcend specific cultural influences. Culture, will of course, influence how these preferences can be expressed, but is probably not the primary cause of these differences.

In fact, as depicted in Figure 1, I am arguing, as others have (e.g., Thorndike 1911), that the sex difference in social preferences, that is, object versus people orientations, and social styles might arise, at least in part, from biological sex differences. I argued in Section 3.1 that sexual selection might have operated to make the nature of social relationships different for males and females. For instance, developing intimate social relationships seems to be an end in itself for young females but not young males (Block 1993). Young males, of course, have companions and develop social alliances, but these might be more of a means to an end, rather than an end in itself. Regardless, the point is that males and females differ in terms of the importance of social relationships and interests in objects, which appear to influence career aspirations. Career aspirations appear to influence the perceived utility of mathematics, which, in turn, influences participation in mathematics-related activities (and other types of activities, as well).

It was also suggested that sexual selection operated to make males more competitive than females, and, as such, might influence how boys and girls perform, in mathematics and other academic areas, in competitive and cooperative classroom environments. In keeping with this view, is the finding that the mathematical achievement of girls is the highest in cooperative settings (e.g., problem solving small groups); the performance of boys, however, drops in these setting (Peterson & Fennema 1985). Similarly, the mathematical achievement of girls drops in competitive classrooms, while the achievement of boys improves slightly. Sociocultural models for these social sex differences could be developed, I'm sure. The overall pattern, however, seems to be consistent with the earlier described sex differences in social styles (Section 3.1). At the very least, we should consider the possibility that fundamental differences between males and females, perhaps shaped by sexual selection, influence social styles and preferences and, as such, are indirect sources of the sex difference in participation in mathematics courses and in related activities.

I also argued that one consequence of sexual selection was different cognitive ability patterns in males and females. Most relevant to the current discussion is the argument that sexual selection might be the primary source of the male advantage in complex spatial abilities. Specifically, it was argued that sexual selection, and any associated proximate mechanisms (e.g., sex hormones) resulted in the more elaborate development of the neurocognitive systems that evolved to support navigation in the three-dimensional physical universe in males than in females. >From this perspective, sex differences in spatial abilities are expected, and appear to be the largest, for tasks that require the mental manipulation or tracking of information in 3-dimensional space (e.g., Masters & Sanders 1993). Sex differences in the systems that support habitat navigation are expected to show a selective relationship to mathematical performance. In particular, knowledge implicit in the systems that support habitat navigation should result in a sex difference, favoring males, in an intuitive understanding of the basic principles of Euclidean geometry. This is because Euclid's basic postulates seem to reflect, to some extent, knowledge that appears to be implicit (i.e., skeletal principles) to the neurocognitive systems that support habitat navigation.

Moreover, it is expected that the systems that have evolved to support habitat navigation can be co-opted by both males and females; the use of spatial representations of mathematical relationships to solve mathematical word problems is one example of co-optation (Lewis 1989). Sex differences are anticipated, however, in the complexity of the spatial representations that can be spontaneously developed by males and females, and perhaps in the frequency with which these representations are spontaneously used by males and females during problem solving (McGuinness 1993). The relatively better developed spatial abilities of males, on average, in comparison to females might also contribute to the sex difference, favoring males, in participation in mathematics-related courses (e.g., drafting) and activities, as shown in Figure 1. A sex difference in such activities is also likely to further increase the male advantage in complex spatial abilities.

Either way, the cross-national pattern of sex differences in mathematical abilities is generally consistent with the argument that these differences emerge primarily for geometry and other areas where the use of spatial representations might facilitate performance. However, none of the studies described in Section 4 assessed the sex differences in mathematical performance with the fidelity that I am proposing; for example, the sex differences in spatial abilities should be the largest for 3-dimensional tests, and that performance on 3-dimensional tests of spatial ability should be more strongly related to performance in specific areas of geometry (e.g., solving problems represented in 3-dimensional space) than, for instance, algebra or even other areas of geometry (e.g., writing proofs). Thus, although the overall pattern of sex differences in mathematical abilities seems to be consistent with the argument that these differences are, in part, due to sex differences in the degree to which the neurocognitive systems that support habitat navigation are elaborated, the validity of the specific predictions presented here must await further research.

The model presented in Figure 1 also predicts some sex differences in mathematical areas that are not facilitated by spatial abilities. For these areas, sex differences would be primarily driven by sex differences in mathematics course taking and related experiences. As a result, sex differences in these areas, if they arise, are expected to be smaller than for geometry and mathematical word problems, and emerge only after the emergence of sex differences in mathematics course taking (i.e., the latter part of high school).

Finally, in closing, even though I have argued that biological sex differences are an important source, though not the only source, of the sex differences that emerge for some biologically-secondary mathematical domains, this should not be taken to mean that the mathematical development of girls cannot be improved. In fact, there are a number of steps that can be taken to improve the mathematical development of girls. First, stressing the utility of mathematics for future career options increases girl's mathematics course taking (Fennema et al. 1981). Second, even though girls do not appear to spontaneously use spatial representations in problem-solving situations as frequently as boys do, they can be taught to do so (Lewis 1989; Johnson 1984). Teaching girls to use diagrams during mathematical problem solving significantly improves their performance, but the male advantage does not disappear (Johnson 1984). Third, highly competitive classroom environments should be avoided; highly cooperative environments should be avoided as well, since the achievement of boys appears to drop in these environments. Classroom teaching styles should either be gender neutral, or boys and girls should be educated in separate classrooms. Finally, relative to international standards, the mathematical development of American children, boys and girls, is very poor. As noted earlier, with improved mathematical instruction, American girls should be able to develop a level of mathematical skill that far exceeds the current level of their male peers. At the same time, a greater emphasis on mathematical instruction in the United States will not likely result in disappearing sex differences. In fact, based on cross-national studies, sex differences in mathematics are likely to increase, as the level of mathematical achievement increases.

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