American Psychologist

© 1995 by the American Psychological Association

January 1995 Vol. 50, No. 1, 24-37

For personal use only--not for distribution.

Reflections of Evolution and Culture in Children's Cognition
Implications for Mathematical Development and Instruction

David C. Geary
University of Missouri at Columbia

ABSTRACT


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


Models derived from the principles of evolutionary selection have provided important insights into the architecture of human perception and cognition ( Anderson & Schooler, 1991 ; Cosmides, 1989 ; Shepard, 1994 ), and, given this, evolution might also be considered a viable theoretical perspective for understanding children's cognitive and academic development. Shepard (1994) has argued quite eloquently for the potency of such an approach:

The ways in which genes shape an individual's perceptual and cognitive capabilities influence the propagation of those genes in the species' ecological niche just as much as the way in which those genes shape the individual's physical size, shape, and coloration. (p. 2)

Even so, cultural factors influence perception and cognition in ways that are independent of the pressures that molded the evolution of the associated systems. For instance, the visual system appears to have evolved so that the perception of object color remains constant across variations in natural lighting (e.g., sunny vs. cloudy days; Shepard, 1994 ). Color constancy breaks down, however, with many forms of artificial lighting (e.g., mercury vapor street lamps; Shepard, 1992 , 1994 ). In an analogous fashion, cultural expectations likely influence children's cognitive and academic development in ways that are independent of the selection pressures that were associated with the evolution of sensory and cognitive systems. A full understanding of children's cognitive and academic growth should then include both biological as well as cultural influences on children's cognition.

In this article, I present a framework for considering biological and cultural influences on children's cognitive and academic growth and then use this framework for organizing research on children's mathematics and evaluating current approaches to educational reform in the United States. Even though these issues are primarily illustrated within the context of children's mathematics, the general conclusions are very likely to be applicable to other academic areas as well. In all, the article includes three general sections. In the first section, a general evolution-based approach to cognitive development is presented. In the second section, this framework is applied to research on children's mathematics, and in the final section, to mathematics instruction.

Evolution-Based Model

The argument that cognitive and academic development reflects both biological and cultural influences is probably stating the obvious and, as such, adds little to our understanding of children's cognition. However, one can begin to disentangle the relative contribution of biological and cultural influences on children's cognitive and academic growth if the mechanisms underlying the acquisition of biologically based and culturally taught skills can be identified. In the first of two sections that follow, a distinction between biologically based and culturally taught skills is made, classes of ability that are termed biologically primary and biologically secondary , respectively, in this article. In the second section, I argue that at least some of the mechanisms underlying the acquisition of these two classes of ability differ.

Biologically Primary and Biologically Secondary Abilities 1

In one respect, most forms of human and animal cognition rely on the functioning of neurobiological systems that have evolved in a particular ecological or social niche and that serve some function or functions related to reproduction or survival. However, one important difference between human beings and, arguably, most other species of animal appears to involve the ease with which these highly specialized neurocognitive systems can be co-opted for tasks unrelated to their original evolution-based function ( Rozin, 1976 ; Rozin & Schull, 1988 ; cf. S. J. Gould & Vrba, 1982 ; Hall, 1992 ). In fact, for human beings it is likely that culture and biology interface in the process of co-opting evolved, biologically primary neurocognitive systems. Stated differently, cultural practices can instill in children a mix of cognitive abilities that are in some respects unrelated to evolutionary pressures because the highly specialized neurocognitive systems that support biologically primary abilities can be used for purposes other than the original evolution-based function. 2

The implications of this perspective are important: First, biologically primary cognitive abilities should be found pan-culturally and should serve a plausible evolutionary function, and analogous abilities and functions should be found across related species ( Pinker & Bloom, 1990 ). Secondary abilities, in contrast, are likely to be found in some cultures and not in others. More important, the extent to which children in various cultures, or individuals across generations, acquire secondary abilities should vary directly with the extent to which formal cultural institutions, such as schools, emphasize the development of such abilities. 3 The distinction between primary and secondary abilities will be illustrated first for the domain of language and then for mathematics.

Human language in one form or another is found throughout the world, but the ability to read is not ( Pinker & Bloom, 1990 ). Reading should therefore be considered a biologically secondary cognitive domain. Moreover, the acquisition of reading-related abilities (e.g., word decoding) appears to involve the co-optation of primary language and language-related systems, among others (e.g., visual scanning; Luria, 1980 ; Rozin, 1976 ). Wagner, Torgesen, and Rashotte (1994) , for instance, reported that individual differences in the fidelity of kindergarten children's phonological processing systems, which are basic features of the language domain, are strongly predictive of the ease with which basic reading abilities (e.g., word decoding) are acquired in first grade. In other words, the evolutionary pressures that selected for phonological processing systems, such as the ability to segment language sounds, were unrelated to reading ( Pinker & Bloom, 1990 ), but these systems are used, or co-opted, when children learn how to read.

For a mathematics example, consider the possibility that explicit knowledge of Euclidean geometry involves, in part, the co-optation of the knowledge that is implicit in the design of the neurocognitive systems that have evolved to support habitat navigation. Map-like representations of the environment are used by many animal species for habitat navigation ( Cheng & Gallistel, 1984 ; Gallistel, 1990 ), and the underlying neurocognitive systems are highly responsive to Euclidean features of the three-dimensional physical universe ( Shepard, 1994 ). For instance, the navigational system of the honey bee ( Apis mellifera ) appears to involve the use of a map-like analog representation of their proximate ecology ( J. L. Gould, 1986 ). This representation allows for efficient movement to and from the hive and food sources and is used in the communication of the location of the food source to other members of the colony ( von Frisch, 1967 ). Similarly, digger wasps can use geometric configurations, formed by rocks or other objects in the environment, to locate their nest ( Gallistel, 1990 ). Cheng and Gallistel showed that 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. Landau, Gleitman, and Spelke (1981) showed that a congenitally blind 2 1/2-year-old child was able to develop a map-like representation of the relative location of four objects in a room and use this representation to move from one object to the next.

These and other examples suggest that many species of animal, including humans, have an implicit, although imprecise, understanding of some fundamental features of Euclidian geometry (e.g., that a line from one point to another is straight; Gallistel, 1990 ). However, this implicit knowledge appears to reflect the evolution of sensory and cognitive systems that are sensitive to basic geometric relationships among objects in the physical universe and does not mean that individuals have an explicit understanding of the formal principles of Euclidian geometry.

The development of geometry as a formal discipline might have been initially based on early geometers' access to, or co-optation of, the knowledge that is implicit in the basic sensory and cognitive systems that have evolved for navigation in the three-dimensional physical universe. In fact, in the development of the basic principles of classical geometry, Euclid apparently "started with what he thought were self-evident truths and then proceeded to prove all the rest by logic" ( West, Griesbach, Taylor, & Taylor, 1982, p. 220 ). Thus, the implicit understanding 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., p. 221 ). From an evolutionary perspective, the former represents implicit knowledge, or a skeletal principle, that is likely built into the neurocognitive systems that support habitat navigation, whereas the latter represents the co-optation and formalization of this knowledge as part of a formal academic discipline or disciplines (e.g., geometry, architecture).

Individuals in preliterate societies can certainly navigate in their ecological niches and might even explicitly use knowledge of spatial relationships for navigation, but this knowledge is typically found without a formalized explicit system of Euclidean geometry or formal institutions, such as schools, to teach this knowledge to their children ( Gallistel, 1990 ). Moreover, it is not likely that navigational skills vary systematically in children from different industrialized nations. Yet, international comparisons show substantial variability in the geometric abilities of children in industrialized nations ( Huseń, 1967a , 1967b ). More important, "national differences [in mathematics achievement] can in part be explained by differences in emphasis in curriculum" ( Huseń, 1967b, p. 300 ). Stated differently, cross-national differences in achievement in complex mathematics vary directly with the relative degree of emphasis in each nation's mathematics curriculum.

Thus, when one considers the pattern of ability development in children across cultures, it becomes clear that many cognitive abilities (e.g., language comprehension, habitat representation) are universal, whereas other abilities (e.g., word decoding in reading, geometry) only emerge in specific cultural contexts. In the next section, a general framework for considering the conditions under which children acquire such biologically primary and biologically secondary cognitive abilities is presented.

Developmental Contexts and Mechanisms

The argument that certain cognitive abilities are biologically primary should not be taken to mean that experience is not necessary for the normal development of the associated competencies. In fact, the opposite is probably the case, that is, the development of both primary and secondary abilities requires experience with the content in question ( Greenough, Black, & Wallace, 1987 ). The basic issue is whether the pattern of experiences necessary for the development of biologically primary and biologically secondary cognitive abilities differs. A parallel question is, What are the factors that motivate engagement in the activities that will lead to the acquisition of primary and secondary cognitive abilities? In this section, I provide some speculation, a starting point, for disentangling the principal influences, experiential and motivational, on the acquisition of biologically primary and biologically secondary cognitive abilities. The section closes with a brief consideration of some similarities between the acquisition of primary and secondary abilities.

Acquisition of biologically primary cognitive abilities.

In this section, a brief overview of the likely features of the systems that support biologically primary cognitive abilities is followed by a discussion of potential experiential and motivational factors that might contribute to the acquisition of primary abilities. First, it is likely that many biologically primary cognitive abilities are supported by neurobiological systems that are specialized for the processing of domain-specific information ( Witelson, 1987 ). Second, it appears that the associated information-processing systems include basic implicit knowledge or skeletal principles of the domain ( Gelman, 1990 , 1993 ). Among other things, these principles appear to orient the child to relevant features of the environment and guide the processing of these features.

For example, Trick (1992) has recently argued that the determination of the numerosity, or the quantity, of small sets of visually presented objects involves preattentive processing in the visual system. Initially, viewed objects are processed in terms of edges, orientations, contours, and so on. In a later step but before the items are actually perceived, the edges, contours, and so on are grouped as individual items. The process of constructing individual items is necessary for object perception, but the associated mechanisms also appear to simultaneously provide information on the number of objects being viewed, although this information appears to be limited to sets of about four objects. In this view, a limited amount of quantitative information (i.e., number of objects) is automatically extracted from the environment by the visual system.

Gelman (1990 ; Gelman & Meck, 1992) referred to such features of neurocognitive systems as skeletal principles because they provide only the initial structure for the acquisition of biologically primary cognitive abilities. These skeletal principles appear to be fleshed out by the processing of domain-relevant information as well as by more top-down inductions that children might make about the domain in question. Thus, the development of biologically primary cognitive abilities must involve a combination of skeletal principles and a motivation to engage in the activities that are necessary to flesh out these principles. The motivation to engage in these arguably evolutionarily expectant activities probably involves an affective component ( Campos, Campos, & Barrett, 1989 ; Greenough et al., 1987 ); in fact, many of these activities are likely to be inherently enjoyable.

One activity, although probably not the only activity, that should be considered a primary candidate for the fleshing out of skeletal principles is children's play. The function of play across mammalian species and across human cultures appears to be the acquisition of adult-like abilities ( Eibl-Eibesfeldt, 1989 ; Panksepp, Siviy, & Normansell, 1984 ; Rubin, Fein, & Vandenberg, 1983 ; Stevens, 1977 ). Eibl-Eibesfeldt described play as "self-activated practice" (p. 585) that enables children to rehearse and experiment with social roles and to begin to acquire functional abilities. This is not to say that all forms of play will be the same across cultures; Americans, for instance, do not typically encourage their children to engage in play hunting. Nevertheless, certain types of play activities are likely to be universal and therefore probably serve similar functions across cultures ( Piaget, 1962 ; Stevens, 1977 ).

If play is an important feature in the development of biologically primary mathematical abilities, then certain forms of mathematics-related play should be found across cultures. Indeed, children appear to engage in number-related activities and games throughout the world (e.g., Saxe, 1982 ; Saxe, Guberman, & Gearhart, 1987 ; Zaslavsky, 1973 ). Saxe et al., for instance, showed that children as young as two years of age often engage in solitary or social play that involves numerical activities, such as counting toys. Similarly, Zaslavsky described many common numerical games engaged in by children in Africa. This is not to say that there will not be cultural differences in the representation of basic numerical knowledge (see Saxe, 1991 ). The point is that the goal of learning about numerical features of the environment is evident in the activities of children across many different types of cultures, and engagement in these activities is likely to flesh out the skeletal principles associated with the biologically primary mathematical abilities described later.

At the same time, the fact that there are cognitive abilities that emerge in some cultures but not in others suggests that many abilities do not have the advantage of skeletal principles or an associated inherent bias for engaging in activities that will assist in their acquisition. Given this, the mechanisms that contribute to the acquisition of these biologically secondary abilities must, perforce, differ in some respects from those associated with the acquisition of biologically primary abilities, an issue that will be briefly considered.

Acquisition of biologically secondary cognitive abilities.

In the section above, it was argued that the development of biologically primary cognitive abilities appears to require a tightly choreographed interplay between specialized neurobiological and inchoate neurocognitive systems and an inherent enjoyment of the activities that are necessary for the development of these systems. The development of biologically secondary abilities, in contrast, does not appear to have these biological advantages, and, as a result, their acquisition is generally slow, effortful, and occurs only with sustained formal or informal instruction ( Gelman, 1993 ; Siegler & Crowley, in press ). For instance, Ericsson, Krampe, and Tesch-Römer (1993) have shown that the development of expertise in musical, athletic, and academic domains requires high levels of sustained, deliberate practice. "Deliberate practice includes activities that have been specifically designed to improve the current level of performance" (p. 368).

The primary context within which children receive sustained exposure to biologically secondary cognitive domains, such as reading, writing, and complex arithmetic, is school ( Ceci, 1991 ). It is not a coincidence that universal schooling is found only in technologically and socially complex societies and that as the level of the technological and social complexity of the society increases, the level of formal schooling required for children increases ( Flynn, 1987 ; Whiting & Whiting, 1975 ). It is also not a coincidence that biologically secondary cognitive abilities emerge primarily in these societies. Stated differently, complex societies have developed formal institutions, that is, schools, that organize the activities of children so that children acquire social and cognitive skills that would not otherwise emerge.

One implication of this view is that cross-cultural differences in the degree to which biologically secondary cognitive abilities are developed directly reflect differences in the degree to which the domain in question is emphasized in school, rather than, for example, differences in the level of intelligence. A second implication is that any cross-cultural differences in the degree to which biologically secondary cognitive abilities are acquired should be independent of the development of biologically primary cognitive abilities. In support of this argument is the finding of apparently no differences in the biologically primary mathematical abilities of East Asian and American children, but a substantial Asian advantage in biologically secondary mathematical domains ( Geary, 1994 ). More important, except for the influence of the structure of Asian- and European-derived number words on early mathematical development, the advantage of East Asian children over American children in mathematics is coincident with the advent of formal schooling (e.g., Song & Ginsburg, 1987 ). 4

A third implication of this view is that the motivation to acquire complex biologically secondary cognitive abilities is based on the requirements of the larger society and not on the inherent interests of children. Given the relatively recent advent of near universal schooling in complex societies, there is no reason to suspect that the skills that are taught in school are inherently interesting or enjoyable for children to learn. Indeed, Ericsson et al. (1993) showed that deliberate practice–practice that improves performance–is not inherently enjoyable, even for experts in the domain. Thus, in addition to the practices that occur in formal school settings, one important difference between the acquisition of biologically primary and biologically secondary cognitive abilities is the level and source of motivation to engage in the activities that are necessary for their acquisition. Nevertheless, this does not preclude the self-motivated engagement in some biologically secondary activities.

Even though I have suggested that reading represents a biologically secondary cognitive domain that involves the co-optation of primary language systems, many children and adults will of course choose reading over other activities. The motivation to read, however, is probably driven by the content of what is being read rather than by the process itself. In fact, the content of many stories and other secondary activities (e.g., video games, television) might reflect evolutionarily relevant themes that motivate engagement in these activities (e.g., social relationships, competition; MacDonald, 1988 ). Furthermore, the finding that intellectual curiosity is a basic dimension of human personality ( Goldberg, 1993 ) suggests that, in any given society, there will be a number of intellectually curious individuals who will pursue biologically secondary activities. Euclid's investment in formalizing and proving the principles of geometry is one example. However, one should seriously consider the possibility that this type of discovery reflects the activities and insights of only a few individuals and that any such advances spread through the larger society only by means of formal instruction. Either way, the point is that the motivation to engage in the activities that will promote the acquisition of biologically secondary abilities is not likely to be universal.

General influences on cognitive development.

The argument that biologically primary and biologically secondary cognitive abilities emerge in different contexts should not be taken to mean that there are no cognitive factors that similarly influence the acquisition of both primary and secondary abilities. One such general influence appears to be the goal structure of the activity ( Siegler & Crowley, in press ); for example, the goal of counting is to quantify how many items are in a set. Knowing the goal of the activity seems to be an important precondition for shifts from the use of relatively unsophisticated problem-solving strategies to the use of more adult-like problem-solving strategies in both primary and secondary domains ( Siegler & Crowley, in press ). Even so, it might be the case that any such goals are implicit features of the neurocognitive systems that support biologically primary cognitive abilities, whereas the goal structure for secondary skills must either be induced or learned from other people.

Evolution and Mathematics

I have just argued that the ease and conditions under which biologically primary and biologically secondary cognitive abilities are acquired are likely to differ. Thus, in order to fully understand mathematical development and to relate this knowledge to mathematics instruction, a distinction must be made between biologically primary and biologically secondary mathematical abilities, a task that is addressed in the following two sections.

Biologically Primary Mathematical Abilities

Potential biologically primary mathematical abilities are shown in Appendix A . Each of these abilities emerge early in development, appear to be found pan-culturally, and are evident in nonhuman primates and some other animal species.

Numerosity is the first of these abilities and represents the ability to quickly determine the quantity of a set of items without the use of counting or estimating, as shown in Appendix A . Studies from different laboratories indicate that human infants are sensitive to the numerosity of an array of up to three, and sometimes four, items as early as the first week of life, with homogeneous versus heterogeneous collects of objects, with displays in motion, and intermodally ( Antell & Keating, 1983 ; Starkey, 1992 ; Starkey, Spelke, & Gelman, 1983 , 1990 ; Trick, 1992 ; van Loosbroek & Smitsman, 1990 ). The intermodal studies are especially important because they suggest that the infant's sensitivity to numerosity is based on an abstract representation ( Starkey et al., 1983 ); the infant's knowledge that a set of two items differs from a set of three items is not dependent on whether the items are seen or heard (e.g., as in a series of two or three drumbeats). Consistent with this research 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 (e.g., a number of flashes), auditory, or tactile modalities ( Thompson, Mayers, Robertson, & Patterson, 1970 ). In addition to the cat, it appears that a sensitivity to numerosity is found in many other species ( Davis & Peŕrusse, 1988 ), including the laboratory rat, an African grey parrot ( Psittacus erithacus ), and the common chimpanzee ( Pan troglodytes ; Boysen & Berntson, 1989 ; Davis & Memmott, 1982 ; Pepperberg, 1987 ).

Human infants also show a sensitivity to ordinal relationships, for example, that three is more than two and two is more than one, by 18 months of age ( R. G. Cooper, 1984 ; Strauss & Curtis, 1984 ). A general sensitivity to more than and less than is also evident in many animal species ( Davis & Peŕrusse, 1988 ). Moreover, well-controlled experimental 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 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.

There is considerable evidence to support the view that counting is a pan-cultural human activity and that human infants have a set of skeletal principles that guide counting behavior before they learn to use number words ( Crump, 1990 ; Geary, 1994 ; Gelman & Gallistel, 1978 ; Ginsburg, Posner, & Russell, 1981 ; Saxe, 1982 ; Starkey, 1992 ; Zaslavsky, 1973 ). For example, one basic principle that appears to constrain counting is one—one correspondence ( Gelman & Gallistel, 1978 ). Implicit knowledge of this skeletal principle is reflected in counting when each item is tagged (e.g., with a number word) or pointed to once and only once ( Gelman & Gallistel, 1978 ). Although the mechanisms underlying tagging during preverbal counting are not clear, it is clear that some human infants as young as 18 months of age are able to use some form of tag to determine the numerosity of sets of up to three items ( Starkey, 1992 ). There is also some evidence that chimpanzees can count in a similar manner. In one study, a chimpanzee named Sheba was required to point to the Arabic number that corresponded to the number of food pellets on a food tray ( Boysen, 1993 ). During this task, Sheba often pointed to the food pellets in succession and then pointed to the corresponding Arabic numeral; Rumbaugh and Washburn (1993) reported similar results using a different experimental procedure.

Recent research also suggests that infants as young as 5 months of age are aware of the effects that the addition and subtraction of one item has on the quantity of a small set of items ( Wynn, 1992 ). Similar results have been reported for 18-month-olds ( Starkey, 1992 ) and for the common chimpanzee ( Boysen & Berntson, 1989 ). As with basic counting skills, the pattern of simple arithmetic skills appears to be qualitatively similar in the chimpanzee and human infants and young children ( Gallistel & Gelman, 1992 ). For instance, preschool children appear to be able to add quantities up to and including three items using some form of preverbal counting, whereas Sheba appears to be able to add items up to and including four items also by means of preverbal counting ( Boysen & Berntson, 1989 ; Starkey, 1992 ).

Finally, it appears that these potential biologically primary mathematical abilities cluster together in a coherent numerical domain. An implicit understanding of the relations between counting, number, and simple arithmetic is found pan-culturally (e.g., Ginsburg et al., 1981 ; Saxe, 1982 ) and apparently in the chimpanzee ( Boysen, 1993 ). For example, preverbal and later verbal counting (in humans) appears to be used for enumeration and later for simple arithmetic throughout the world, even when the surface structure of counting systems (e.g., whether or not it is a base-10 system) varies ( Geary, 1994 ). Psychometric or factor analytic studies also support the argument that some numerical and arithmetical skills are biologically primary and cluster together. For 5-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, a Numerical Facility, or Number, factor has been consistently identified through decades of psychometric research (e.g., Coombs, 1941 ; Thurstone, 1938 ; Thurstone & Thurstone, 1941 ) and has been found with studies of American, Chinese, and Filipino students ( Guthrie, 1963 ; Vandenberg, 1959 ). Even Spearman (1927) , a staunch supporter of the position that individual differences in human abilities are best explained by a single factor called general intelligence (g), stated that basic arithmetic skills "have much in common over and above...g" (p. 251). Finally, behavioral genetic studies of individual differences on numerical facility tests indicate that roughly one half of the variability in some components of arithmetical abilities is due to genetic differences across people ( Vandenberg, 1962 , 1966 ).

Biologically Secondary Mathematical Abilities

Some potential biologically secondary mathematical abilities are shown in Appendix B . The abilities listed are restricted to counting, number, and arithmetic to provide a contrast with the biologically primary abilities presented in Appendix A . The features of counting and arithmetic listed in Appendix B represent abilities that are taught by parents (e.g., number names), induced by children during the act of counting (e.g., that counted objects are usually tagged from left to right), or are formally taught in school (e.g., the base-10 system; Briars & Siegler, 1984 ; Fuson, 1988 ; Geary, 1994 ; Ginsburg et al., 1981 ). In addition to the competencies listed in Appendix B , most features of complex mathematical domains, such as algebra, geometry (except perhaps for an implicit understanding of basic Euclidean postulates), and calculus, are probably biologically secondary.

The focus of this section is on the solving of arithmetical and algebraic word problems, which is called mathematical problem solving by educational researchers and mathematical reasoning by psychometric researchers. The focus is on mathematical problem solving or mathematical reasoning because one important goal of current reforms in mathematics education is to improve the problem-solving competencies of American children. Strong mathematical problem solving appears to be associated with, among others, the ability to spatially represent mathematical relations, the ability to translate word problems into appropriate equations, and an understanding of how and when to use mathematical equations ( Geary, 1994 ; Mayer, 1985 ; Schoenfeld, 1985 ).

The results of psychometric research are consistent with the view that mathematical reasoning, as defined by the solving of word problems, is a biologically secondary cognitive domain. Although psychometric researchers have identified a Mathematical Reasoning factor in many studies ( Dye & Very, 1968 ; Thurstone, 1938 ), a distinct Mathematical Reasoning factor does not emerge in all samples, not even in some samples of college students (e.g., Guthrie, 1963 ). When a distinct Mathematical Reasoning factor is found, it is 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 ). The finding that a Mathematical Reasoning factor emerges only in samples with prolonged mathematical instruction suggests that, unlike biologically primary mathematical abilities, many individuals do not easily acquire the competencies that are associated with complex mathematical problem solving.

The one exception to this pattern appears to be the performance of the mathematically precocious youths studied by Benbow (1988) and her colleagues. However, the precocious mathematical development of these adolescents appears to involve the co-optation of biologically primary systems and abilities during the solving of mathematical word problems and not an inherent understanding of mathematical problem solving. For instance, Dark and Benbow (1991) found that mathematical precocity was associated with exceptional spatial and working memory skills, which are co-opted during mathematical problem solving. Spatial abilities appear to aid in the solving of algebraic word problems through, for example, the diagramming of important relationships in the problem ( Geary, 1994 ; Johnson, 1984 ), but the solving of word problems is biologically secondary with respect to spatial abilities.

In other words, the evolution of spatial abilities was likely related to habitat navigation and unrelated to algebraic problem solving, but, nevertheless, spatial abilities can be co-opted to aid in the solving of algebraic word problems. Regardless, except for mathematically gifted youths, the utility of using spatial representations of mathematical relationships is not likely to be intuitively obvious to many children and adolescents, given the secondary relationship between spatial abilities and the solving of mathematical word problems. As a result, the use of spatial representations in mathematical problem solving needs to be explicitly taught to most individuals ( Lewis, 1989 ).

In sum, most of children's knowledge of complex arithmetic and complex mathematics only emerges in formal school settings ( Ginsburg et al., 1981 ) and only as a result of deliberate and sustained practices that are explicitly designed to teach this knowledge. The conditions under which such biologically secondary mathematical abilities emerge are thus very different from the conditions that lead to the arguably ubiquitous emergence of biologically primary mathematical abilities. The educational implications of these differences are explored in the next section.

Mathematics Instruction

American culture produces some of the most poorly educated children in mathematics in the industrialized world ( Huseń, 1967b ; Stevenson, Chen, & Lee, 1993 ). The basic quantitative abilities of America's children will influence their later productivity in the workplace as well as their later wages and employability ( Boissiere, Knight, & Sabot, 1985 ; Rivera-Batiz, 1992 ). The academic capabilities of the work force, in turn, not only influence the well-being of individual workers but also have wider social consequences. Bishop (1989) , for instance, estimated that the poor reading and mathematical competencies of much of the work force will cost the U.S. economy nearly 170 billion dollars each year by the year 2000. Improving the basic academic capabilities of America's children and subsequent labor force might therefore be considered one of this nation's most pressing needs. In fact, the importance of improving the mathematical competencies of American children was highlighted in America 2000: An Education Strategy . One of the six primary goals stated that "U.S. students will be first in the world in science and mathematics achievement" by the year 2000 ( U.S. Department of Education, 1991, p. 3 ). Given the clear importance of children's mathematical and general academic development for the well-being of this society, a principled consideration of instructional issues seems to be in order.

The instructional section begins with an overview and critique of basic philosophical themes that currently guide educational practice in the United States, in particular the constructivist view of mathematics education. The constructivist view of mathematics education is targeted because this view is growing in influence and might very well dominate educational practice in the United States in the near future. In fact, these educators have adopted the worldviews of several eminent psychologists (e.g., Piaget and Vygotsky) and have used these views as a foundation for educational reform in the United States. At the same time, many of these educational researchers have ignored or dismissed a large body of relevant psychological research and theory.

The selective use of psychological theory by some educational researchers is not in the best interest of our discipline or in the best interest of education in general in the United States. The broad expertise represented by psychology has much to offer American education, but in order to ensure that it actually has an impact, we must begin to spell out the educational implications of our research more assertively and directly and more forcefully challenge the naive use of psychological principles ( Penner, Batsche, Knoff, & Nelson, 1993 ). This section provides a theoretical critique of educational reform, based on the just described evolution-based framework, and highlights some of the ways in which contemporary biological and psychological models can be used in educational practice.

Educational Philosophy and Constructivism

von Glaserfeld and Steffe (1991) , and other constructivists, argued that there are two general approaches to educational research and practice, mechanistic and organismic (e.g., Cobb, Yackel, & Wood, 1992 ). The mechanistic approach is exemplified by traditional learning theory and, in their view, by many contemporary information-processing approaches to cognitive development. Their view of the mechanistic approach is that the learner passively receives information from his or her environment, most notably from the teacher. The information results in reflexive changes in the child's overt behavior, such as the number of problems solved correctly or in the child's mental representations of mathematical information; the latter might be illustrated by the development of memory representations for specific arithmetic facts ( Siegler, 1986 ). It is further assumed that any such changes in overt behavior or mental representations generally occur without the child conceptually understanding the material ( Cobb et al., 1992 ). It follows from this assumption that many features of the information-processing system that are emphasized by cognitive psychologists, such as automaticity (that overlearned processes occur automatically without conscious effort), are viewed as detrimental to the child's academic development, as is the route to automaticity, that is, drill and practice.

The constructivist perspective offers an alternative, organismic approach to mathematical development. As noted above, this organismic approach is exemplified by the worldviews of Piaget and Vygotsky. The basic assumption is that children are active learners and must construct mathematical knowledge for themselves. In order to completely understand mathematical material, the child must rediscover basic mathematical principles. The teacher provides appropriate materials and a social context within which the material is discussed but does not lecture or guide discussion in the traditional sense ( Cobb et al., 1992 ; Lampert, 1990 ).

In constructivism, a zone of potential construction of a specific mathematical concept is determined by the modifications of the concept children might make in, or as a result of, interactive communication in the mathematical learning environment. ( Steffe, 1992, p. 261 )

In other words, mathematical learning is a social enterprise. Social disagreements about the meaning of mathematical materials or concepts provide the grist for mathematical development, as these disagreements provide the impetus to change or accommodate one's understanding of such concepts. Any such change serves to make the child's understanding of mathematics more consistent with the understanding of the larger social community ( Steffe, 1990 ). With the development of appropriate social—mathematical environments, "it is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve" ( Cobb et al., 1992, p. 28 ).

The basic assumptions that guide constructivist-based instruction appear to be well suited for the acquisition of biologically primary mathematical abilities, such as number and counting. However, constructivist philosophers and researchers fail to distinguish between biologically primary and biologically secondary mathematical abilities and, as a result, treat all of mathematics as if it were a biologically primary domain. That is, given an appropriate social context and materials, children will be motivated and able to construct mathematical knowledge for themselves in all areas ( Cobb et al., 1992 ). The adoption of these assumptions and the associated instructional techniques appear to reflect wider cultural values and only weakly follow empirical and theoretical work in contemporary developmental and cognitive psychology, much less a consideration of evolutionary issues.

Relative to cultures that foster the mathematical development of their children, such as Asian culture, American culture allows for much greater freedom in the extent to which individuals are allowed to pursue their own self-interests or engage in activities that are inherently interesting. Unfortunately, except for basic number and counting activities, engagement in most other mathematics-related activities is not likely to be inherently interesting for most individuals, Asian or American. It is not likely that the acquisition of complex biologically secondary mathematical abilities will occur for a large segment of any given society without strong cultural values that reward mathematical development and a strong emphasis on mathematics education in school ( Stevenson & Stigler, 1992 ). Given the current American cultural milieu that allows for the pursuit of inherently more enjoyable activities than complex mathematics, it is not surprising that very few American children develop a level of mathematical competence that is achieved by children in many other cultures ( Stevenson et al., 1993 ). In sum, constructivism is largely a reflection of current American cultural beliefs and, as such, involves the development of instructional techniques that attempt to make the acquisition of complex mathematical skills an enjoyable social enterprise that will be pursued on the basis of individual interest and choice.

As noted above, the associated social—constructivist techniques probably work rather well for fostering the acquisition of biologically primary abilities because these constructivist activities appear to be very similar to the social contexts within which many biologically primary abilities naturally emerge ( Eibl-Eibesfeldt, 1989 ). However, these same techniques are probably not sufficient for the development of biologically secondary abilities. This is not to say that mathematics instruction should not attempt to engage student's interest and curiosity with the material; they should and probably must. The point is, creating a learning environment that mimics the learning environment that has evolved to support the acquisition of biologically primary abilities might be necessary but is not sufficient to support the acquisition of complex biologically secondary abilities.

To illustrate, it was noted earlier that many constructivist researchers reject outright the use of drill-and-practice for acquiring mathematical skills. Indeed, formal drill-and-practice does not appear to be necessary for the acquisition and maintenance of many biologically primary cognitive abilities. The practice of language skills, for example, is built into natural social discourse; there is no need for young children to formally practice speaking skills. The evolved natural activities of humans, however, do not include embedded practice of the abilities that are associated with biologically secondary domains (e.g., complex mathematics or reading; Eibl-Eibesfeldt, 1989 ). The acquisition and maintenance of biologically secondary abilities over the long-term almost certainly require some amount of sustained practice ( Bahrick & Hall, 1991 ). Cultural values that support student involvement in this practice are essential ( Stevenson & Stigler, 1992 ). This is so because evolution has not provided children with a natural enjoyment of the activities, such as drill-and-practice, that appear to be needed in order to master the abilities that are associated with complex secondary domains ( Eibl-Eibesfeldt, 1989 ).

Instructional Implications

As noted earlier (see Footnote 1 ), in addition to a set of goals, the abilities that are associated with both biologically primary and biologically secondary domains appear to encompass procedural skills and conceptual knowledge ( Gelman, 1993 ). The primary goal of this section is to consider potential differences between the acquisition of procedural and conceptual competencies in primary and secondary domains and the associated instructional implications. But first, consider a few examples that illustrate the difference between procedural and conceptual competencies.

Biologically primary features of counting include a conceptual understanding that small sets of objects can be enumerated, or counted, as well as rudimentary procedures for the act of counting ( Gelman & Gallistel, 1978 ). The conceptual knowledge reflects a basic understanding of what can be achieved and constraints on how the goal can be achieved. For instance, one principled constraint (i.e., skeletal principle) on counting involves an implicit understanding that each item in the set can be counted once and only once. The procedural feature of this system often reflects behavioral skills, such as pointing to each object as it is counted. For an example in a biologically secondary mathematical area, consider children's ability to trade, or carry, as in the problem 36 + 78. The accurate solution of such a problem requires a conceptual understanding of the base-10 system, that the 1 traded from the units to the 10s column (from 6 + 8 = 14) is in fact 10 units, not 1 unit ( Fuson & Kwon, 1992 ), as well as a procedure to implement the trade. For this example, if the problem is presented vertically (i.e., the 36 is placed over the 78), the procedure might involve writing a 4 under the units-column values, placing a 1 above the 10s-column values (i.e., above 3 and 7), and then adding 10 to the sum of the 10s-column values. In short, the trading procedure involves "knowing how" to trade, and the associated conceptual knowledge reflects "knowing that" sets of 10 are being traded.

There are likely to be several important differences between the development of procedural and conceptual competencies in primary and secondary domains. For primary domains, the initial conceptual competencies of children appear to be implicit; that is, conceptual competencies are reflected in children's performance, but children cannot articulate the associated principles. For secondary domains, conceptual knowledge is more accessible. In fact, the way in which we assess the relative degree of mastery of secondary domains (i.e., through tests) is based on an assumption that conceptual knowledge can be articulated. Moreover, for biologically primary domains, procedural and conceptual competencies will likely be fleshed out with the child's natural activities, although formal education will almost certainly contribute to this process. As noted earlier, the activities that are associated with the acquisition of procedural and conceptual competencies in biologically primary domains are not likely to be sufficient for the acquisition of the procedural and conceptual competencies that are associated with biologically secondary domains. In fact, it is with the understanding of the mechanisms that are associated with the development of procedural and conceptual competencies in biologically secondary domains, those primarily learned in school, that psychological research has the most to offer American education.

Even though procedural skills and conceptual knowledge likely influence one another, from an instructional perspective they should probably be considered distinct ( Silver, 1987 ). This is so because psychological research suggests that the acquisition of procedural skills and conceptual knowledge probably require different forms of instruction ( G. Cooper & Sweller, 1987 ; Novick, 1992 ). Regardless, as noted before, current educational philosophy focuses on the acquisition of conceptual knowledge and often deems the acquisition of procedural skills as unnecessary and even detrimental to children's mathematical development (e.g., Cobb et al., 1992 ). Nevertheless, children need to understand mathematical concepts, and they need to know how and when to use mathematical procedures ( Gelman, 1993 ).

Briefly, procedural learning requires extensive practice on the whole range of problems on which the procedure might eventually be used. Practice should not, however, involve the use of the same procedure on the same type of problem for an extended period of time. Wenger (1987) argued that this form of practice results in the development of procedural bugs, that is, procedures that are correct for some problems but are incorrectly extended to other problems. Rather, practice should involve the use of a mixture of procedures that are practiced on a variety of different types of problems. Practice should also occur in small doses (e.g., 20 minutes/day) and over an extended period of time ( H. Cooper, 1989 ). For example, Bahrick and Hall (1991) showed that the retention of basic algebraic skills over a 50-year period was related to repeated exposure to algebra in high school and college. A procedure that is practiced on one or two work sheets for a day or two will likely be forgotten rather quickly ( Bahrick, 1993 ). Basically, the procedure should be practiced until the child can automatically execute the procedure with the different types of problems that the procedure is normally used to solve. Once procedures are automatized, they require little conscious effort to use, which, in turn, frees attentional and working memory resources for use on other, more important features of the problem ( Geary & Widaman, 1992 ; Silver, 1987 ).

I am recommending, of course, a modified form of drill-and-practice. Although drill-and-practice is the bane of many contemporary educational researchers, it is probably the only way to ensure the long-term retention of basic, biologically secondary procedures. The bottom line is, "if you want somebody to know something, you teach it to them" ( Detterman, 1993, p. 15 ). If you want somebody to know something and retain it for a long period of time, then you have them practice it ( Symonds & Chase, 1929 ). The practice of basic procedures, especially when the practice is mixed with other types of procedures, should also provide the child with an opportunity to come to understand how the procedure works. In other words, practice provides an environment within which children can flesh out their understanding of the procedure and any associated conceptual knowledge.

Psychological research also suggests that a deep conceptual understanding of a mathematical domain requires a lot of experience but does not appear to require drill-and-practice per se ( G. Cooper & Sweller, 1987 ). Conceptual knowledge reflects the child's understanding of the basic principles of the domain and allows the child to see similarities across problems that have different superficial features (e.g., Morales, Shute, & Pellegrino, 1985 ; Perry, 1991 ). A child might demonstrate a good conceptual understanding of counting, for instance, when she knows that counting can occur from left to right, or right to left, or haphazardly, and, as long as all of the items are counted, still yield the same answer. One way that appears to be useful for promoting conceptual knowledge is to ask students to come up with as many ways as possible to solve a particular problem or class of problems ( Sweller, Mawer, & Ward, 1983 ). For this example, instruction might involve having the children count in as many different ways as they can (e.g., left to right and right to left). Asking children to count from left to right and then from right to left will not likely improve their procedural skills (e.g., pointing at counted objects) but should allow them to induce that objects can be counted in any order, as long as each object is counted once and only once ( Briars & Siegler, 1984 ). Another important instructional feature that might facilitate children's development of conceptual knowledge involves presenting problems in familiar contexts, those that the child can relate to personal experiences ( Perry, VanderStoep, & Yu, 1993 ).

Moreover, it is with the teaching of conceptual principles that the evolutionary perspective might be particularly useful. To illustrate this point, consider again the potential relation between the neurocognitive systems that support habitat navigation and implicit knowledge of the basic principles of Euclidean geometry. If the development of geometry as a formal discipline was in fact aided by implicit knowledge of the principles of spatial navigation, then the use of cognitive research on animal and human navigation might be a useful method to illustrate basic geometric principles. The use of navigational examples to illustrate geometric principles might have two benefits. First, these examples might provide a link between what is often considered an abstract discipline and real-world contexts. Second, these illustrations might help to draw on students' implicit knowledge of geometric relationships. These examples should make geometry more interesting and engaging for students but, in and of themselves, are probably not enough to master geometry. This is so because implicit knowledge of habitat navigation is imprecise and probably limited ( Gallistel, 1990 ). For instance, a basic understanding of angles, which appears to be implicit to the navigational system, is not enough to master analytic geometry. The mastery of analytic geometry will require an extensive fleshing out of any implicit knowledge about habitat navigation (e.g., most individuals need to be explicitly taught that the sum of the angles in a triangle is always 180°) and the practice of basic procedures (e.g., graphing equations).

Conclusion

The principles of evolutionary selection can provide an important theoretical framework for models of human cognition and developmental (e.g., Shepard, 1994 ). These principles are not only useful for making inferences about the evolution of perceptual and cognitive systems but might also enable a clearer demarcation between biological and cultural influences on human cognition and cognitive development. For instance, in this article it was argued that some cognitive abilities, such as habitat representation, are acquired more readily than other abilities, such as geometry, and that differences between these classes of cognitive ability reflect biological biases that can be understood within the context of human evolution. A distinction between these different classes of ability, which were termed biologically primary and biologically secondary, also has implications for how we educate our children as well as for understanding individual and group differences in academic achievement.

Approaching these issues from the perspective of human evolution highlights the importance of formal schooling on children's cognitive and academic growth ( Ceci, 1991 ), especially for biologically secondary domains. This is so because, unlike biologically primary abilities, secondary cognitive abilities probably do not have the advantage of skeletal principles or an associated bias to engage in activities that will facilitate their development. The acquisition of most secondary abilities will occur in school and only with sustained and deliberate instructional practices that are explicitly designed to foster their acquisition. Moreover, we cannot expect that the acquisition of all secondary abilities will be particularly enjoyable for children. The motivation to acquire these abilities comes from the requirements of the wider and increasingly complex society, and not from the inherent interests of children. This distinction between biologically primary and biologically secondary cognitive abilities also points to formal schooling and cultural attitudes about academic achievement as a primary source of, for instance, cross-national differences in mathematical abilities, rather than differences in the level of intelligence ( Geary, 1994 ).

In closing, children's cognitive growth perforce reflects evolutionary and cultural influences. However, it is not enough to simply acknowledge these multiple influences; we must begin to more closely examine how biological and cultural mechanisms might operate in the acquisition of cognitive abilities. The general framework presented in this article is a small, and in places a speculative, step toward trying to disentangle biological and cultural influences on children's cognition. Even though this framework is not likely to provide the complete picture, a consideration of the associated issues gives us reason to evaluate how we teach our children and how we currently understand cognitive and academic development.

APPENDIX A

A Potential Biologically Primary Mathematical Abilities



Table 1

Numerosity

The ability to accurately determine the quantity of small sets of items, or events, without counting. In humans, accurate numerosity judgments are typically limited to sets of four or fewer items.

Ordinality

A basic understanding of more than and less than and, later, an understanding of specific ordinal relationships. For example, understanding that 4 > 3, 3 > 2, and 2 > 1. For humans, the limits of this system are not clear, but is probably limited to quantities < 5.

Counting

Early in development there appears to be a preverbal counting system that can be used for the enumeration of sets up to three, perhaps four, items. With the advent of language and the learning of number words, there appears to be a pan-cultural understanding that serial-ordered number words can be used for counting, measurement, and simple arithmetic.

Simple Arithmetic

Early in development there appears to be a sensitivity to increases (addition) and decreases (subtraction) in the quantity of small sets. This system appears to be limited to the addition or subtraction of items within sets of three, perhaps four, items.

APPENDIX B

B Potential Biologically Secondary Mathematical Abilities



Table 2

Counting and Number

Number names are arbitrary, vary from culture to culture, and must be memorized. Even though a basic understanding of counting appears to be biologically primary, the extension of this knowledge to larger valued numbers is probably biologically secondary, as is learning the relationship between number names and the associated quantity. Another important feature of number and counting systems that is culturally specific is the use of the base-10 system, which is not used in preliterate cultures. Finally, even though a basic understanding of counting and ordinality appears to be biologically primary, much of children's understanding of number and counting concepts, especially for larger valued numbers, appears to be induced during the act of counting.

Arithmetic

Even though a basic understanding of the effects of additions and subtractions on quantity appears to be biologically primary, many other forms of arithmetical knowledge emerge only with formal schooling and should therefore be considered biologically secondary. These features include fractions, multicolumn addition and subtraction, trading (i.e., carrying or borrowing), multiplication, and division, as well as the use of radicals and exponents.

References


Anderson, J. R. & Schooler, L. J. (1991). Reflections of the environment in memory. Psychological Science, 2, 396-408.
Antell, S. E. & Keating, D. P. (1983). Perception of numerical invariance in neonates. Child Development, 54, 695-701.
Bahrick, H. P. (1993). Extending the life-span of knowledge.(In L. A. Penner, G. M. Batsche, H. M. Knoff, & D. L. Nelson (Eds.), The challenge in mathematics and science education: Psychology's response (pp. 61—82). Washington, DC: American Psychological Association.)
Bahrick, H. P. & Hall, L. K. (1991). Lifetime maintenance of high school mathematics content. Journal of Experimental Psychology: General, 120, 22-33.
Benbow, C. P. (1988). Sex differences in mathematical reasoning ability in intellectually talented preadolescents: Their nature, effects, and possible causes. Behavioral and Brain Sciences, 11, 169-232.
Bishop, J. H. (1989). Is the test score decline responsible for the productivity growth decline? American Economic Review, 79, 178-197.
Boissiere, M., Knight, J. B. & Sabot, R. H. (1985). Earnings, schooling, ability, and cognitive skills. American Economic Review, 75, 1016-1030.
Boysen, S. T. (1993). Counting in chimpanzees: Nonhuman principles and emergent properties of number.(In S. T. Boysen & E. J. Capaldi (Eds.), The development of numerical competence: Animal and human models (pp. 39—59). Hillsdale, NJ: Erlbaum.)
Boysen, S. T. & Berntson, G. G. (1989). Numerical competence in a chimpanzee ( Pan troglodytes ). Journal of Comparative Psychology, 103, 23-31.
Briars, D. & Siegler, R. S. (1984). A featural analysis of preschoolers' counting knowledge. Developmental Psychology, 20, 607-618.
Campos, J. J., Campos, R. G. & Barrett, K. C. (1989). Emergent themes in the study of emotional development and emotion regulation. Developmental Psychology, 25, 394-402.
Ceci, S. J. (1991). How much does schooling influence general intelligence and its cognitive components? A reassessment of the evidence. Developmental Psychology, 27, 703-722.
Cheng, K. & Gallistel, C. R. (1984). Testing the geometric power of an animal's spatial representation.(In H. L. Roitblat, T. G. Bever, & H. S. Terrace (Eds.), Animal cognition (pp. 409—423). Hillsdale, NJ: Erlbaum.)
Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2-33.
Coombs, C. H. (1941). A factorial study of number ability. Psychometrika, 6, 161-189.
Cooper, G. & Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79, 347-362.
Cooper, H. (1989). Synthesis of research on homework. Educational Leadership, 47, 85-91.
Cooper, R. G. (1984). Early number development: Discovering number space with addition and subtraction.(In C. Sophian (Ed.), Origins of cognitive skills: The Eighteenth Carnegie Symposium on Cognition (pp. 157—192). Hillsdale, NJ: Erlbaum.)
Cosmides, L. (1989). The logic of social exchange: Has natural selection shaped how humans reason? Studies with the Wason selection task. Cognition, 31, 187-276.
Crump, T. (1990). The anthropology of numbers. (New York: Cambridge University Press)
Dark, V. J. & Benbow, C. P. (1991). Differential enhancement of working memory with mathematical versus verbal precocity. Journal of Educational Psychology, 83, 48-60.
Davis, H. & Memmott, J. (1982). Counting behavior in animals: A critical examination. Psychological Bulletin, 92, 547-571.
Davis, H. & Pe´russe, R. (1988). Numerical competence in animals: Definitional issues, current evidence, and a new research agenda. Behavioral and Brain Sciences, 11, 561-615.
Detterman, D. K. (1993). The case for the prosecution: Transfer as an epiphenomenon.(In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction (pp. 1—24). Norwood, NJ: Ablex.)
Dye, N. W. & Very, P. S. (1968). Growth changes in factorial structure by age and sex. Genetic Psychology Monographs, 78, 55-88.
Eibl-Eibesfeldt, I. (1989). Human ethology. (New York: Aldine de Gruyter)
Ericsson, K. A., Krampe, R. T. & Tesch-Römer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100, 363-406.
Flynn, J. R. (1987). Massive IQ gains in 14 nations: What IQ tests really measure. Psychological Bulletin, 101, 171-191.
Fuson, K. C. (1988). Children's counting and concepts of number. (New York: Springer-Verlag)
Fuson, K. C. & Kwon, Y. (1992). Korean children's understanding of multidigit addition and subtraction. Child Development, 63, 491-506.
Gallistel, C. R. (1990). The organization of learning. (Cambridge, MA: MIT Press/Bradford Books)
Gallistel, C. R. & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43-74.
Geary, D. C. (1994). Children's mathematical development: Research and practical applications. (Washington, DC: American Psychological Association)
Geary, D. C. & Widaman, K. F. (1992). Numerical cognition: On the convergence of componential and psychometric models. Intelligence, 16, 47-80.
Gelman, R. (1990). First principles organize attention to and learning about relevant data: Number and the animate—inanimate distinction as examples. Cognitive Science, 14, 79-106.
Gelman, R. (1993). A rational—constructivist account of early learning about numbers and objects.(In D. L. Medin (Ed.), The psychology of learning and motivation: Advances in research and theory (Vol. 30, pp. 61—96). San Diego, CA: Academic Press.)
Gelman, R. & Gallistel, C. R. (1978). The child's understanding of number. (Cambridge, MA: Harvard University Press)
Gelman, R. & Meck, B. (1992). Early principles aid initial but not later conceptions of number.(In J. Bideaud, C. Meljac, & J. P. Fischer (Eds.), Pathways to number: Children's developing numerical abilities (pp. 171—189). Hillsdale, NJ: Erlbaum.)
Ginsburg, H. P., Posner, J. K. & Russell, R. L. (1981). The development of mental addition as a function of schooling and culture. Journal of Cross-Cultural Psychology, 12, 163-178.
Goldberg, L. R. (1993). The structure of phenotypic personality traits. American Psychologist, 48, 26-34.
Gould, J. L. (1986). The locale map of honey bees: Do insects have cognitive maps? Science, 232, 861-863.
Gould, S. J. & Vrba, E. S. (1982). Exaptation–A missing term in the science of form. Paleobiology, 8, 4-15.
Greenough, W. T., Black, J. E. & Wallace, C. S. (1987). Experience and brain development. Child Development, 58, 539-559.
Guthrie, G. M. (1963). Structure of abilities in a non-Western culture. Journal of Educational Psychology, 54, 94-103.
Hall, B. K. (1992). Evolutionary developmental biology. (London: Chapman & Hall)
Huse´n, T. (1967a). International study of achievement in mathematics: A comparison of twelve countries (Vol. I).(New York: Wiley)
Huse´n, T. (1967b). International study of achievement in mathematics: A comparison of twelve countries (Vol. II).(New York: Wiley)
Johnson, E. S. (1984). Sex differences in problem solving. Journal of Educational Psychology, 76, 1359-1371.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63.
Landau, B., Gleitman, H. & Spelke, E. (1981). Spatial knowledge and geometric representation in a child blind from birth. Science, 213, 1275-1278.
Lewis, A. B. (1989). Training students to represent arithmetic word problems. Journal of Educational Psychology, 81, 521-531.
Luria, A. R. (1980). Higher cortical functions in man (2nd ed.).(New York: Basic Books)
MacDonald, K. B. (1988). Social and personality development: An evolutionary synthesis. (New York: Plenum Press)
Mayer, R. E. (1985). Mathematical ability.(In R. J. Sternberg (Ed.), Human abilities: An information processing approach (pp. 127—150). San Francisco: Freeman.)
Miller, K. F. & Stigler, J. W. (1987). Counting in Chinese: Cultural variation in a basic cognitive skill. Cognitive Development, 2, 279-305.
Morales, R. V., Shute, V. J. & Pellegrino, J. W. (1985). Developmental differences in understanding and solving simple mathematics word problems. Cognition and Instruction, 2, 41-57.
Novick, L. R. (1992). The role of expertise in solving arithmetic and algebra word problems by analogy.(In J. I. D. Campbell (Ed.), The nature and origins of mathematical skills (pp. 155—188). Amsterdam: North-Holland.)
Osborne, R. T. & Lindsey, J. M. (1967). A longitudinal investigation of change in the factorial composition of intelligence with age in young school children. Journal of Genetic Psychology, 110, 49-58.
Panksepp, J., Siviy, S. & Normansell, L. (1984). The psychobiology of play: Theoretical and methodological perspectives. Neuroscience and Biobehavioral Reviews, 8, 465-492.
Penner, L. A., Batsche, G. M., Knoff, H. W. & Nelson, D. L. (Eds.) (1993). The challenge of mathematics and science education: Psychology's response. (Washington, DC: American Psychological Association)
Pepperberg, I. M. (1987). Evidence for conceptual quantitative abilities in the African grey parrot: Labeling of cardinal sets. Ethology, 75, 37-61.
Perry, M. (1991). Learning and transfer: Instructional conditions and conceptual change. Cognitive Development, 6, 449-468.
Perry, M., VanderStoep, S. W. & Yu, S. L. (1993). Asking questions in first-grade mathematics classes: Potential influences on mathematical thought. Journal of Educational Psychology, 85, 31-40.
Piaget, J. (1962). Play, dreams and imitation in childhood. (New York: Norton)
Pinker, S. & Bloom, P. (1990). Natural language and natural selection. Behavioral and Brain Sciences, 13, 707-784.
Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of employment among young adults in the United States. Journal of Human Resources, 27, 313-328.
Rozin, P. (1976). The evolution of intelligence and access to the cognitive unconscious.(In J. M. Sprague & A. N. Epstein (Eds.), Progress in psychobiology and physiological psychology (Vol. 6, pp. 245—280). New York: Academic Press.)
Rozin, P. & Schull, J. (1988). The adaptive—evolutionary point of view in experimental psychology.(In R. C. Atkinson, R. J. Herrnstein, G. Lindzey, & R. D. Luce (Eds.), Steven's handbook of experimental psychology (2nd ed., Vol. 1, pp. 503—546). New York: Wiley.)
Rubin, K. H., Fein, G. G. & Vandenberg, B. (1983). Play.(In P. Mussen & E. M. Hetherington (Eds.), Handbook of child psychology: Socialization, personality, and social development (Vol. 4, pp. 693—774). New York: Wiley.)
Rumbaugh, D. M. & Washburn, D. A. (1993). Counting by chimpanzees and ordinality judgments by macaques in video-formatted tasks.(In S. T. Boysen & E. J. Capaldi (Eds.), The development of numerical competence: Animal and human models (pp. 87—106). Hillsdale, NJ: Erlbaum.)
Saxe, G. B. (1982). Culture and the development of numerical cognition: Studies among the Oksapmin of Papua New Guinea.(In C. J. Brainerd (Ed.), Children's logical and mathematical cognition: Progress in cognitive development research (pp. 157—176). New York: Springer-Verlag.)
Saxe, G. B. (1991). Culture and cognitive development: Studies in mathematical understanding. (Hillsdale, NJ: Erlbaum)
Saxe, G. B., Guberman, S. R. & Gearhart, M. (1987). Social processes in early number development. Monographs of the Society for Research in Child Development, 52, (2, Serial No. 216).
Schoenfeld, A. H. (1985). Mathematical problem solving. (San Diego, CA: Academic Press)
Shepard, R. N. (1992). The perceptual organization of colors: An adaptation to regularities of the terrestrial world?(In J. H. Barkow, L. Cosmides, & J. Tooby (Eds.), The adapted mind: Evolutionary psychology and the generation of culture (pp. 495—532). New York: Oxford University Press.)
Shepard, R. N. (1994). Perceptual—cognitive universals as reflections of the world. Psychonomic Bulletin & Review, 1, 2-28.
Siegler, R. S. (1986). Unities across domains in children's strategy choices.(In M. Perlmutter (Ed.), Perspectives for intellectual development: Minnesota Symposia on Child Psychology (Vol. 19, pp. 1—48). Hillsdale, NJ: Erlbaum.)
Siegler, R. S. & Crowley, K. (in press). Constraints on learning in non-privileged domains.(Cognitive Psychology. )
Silver, E. A. (1987). Foundations of cognitive theory and research for mathematics problem-solving instruction.(In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 33—60). Hillsdale, NJ: Erlbaum.)
Song, M. J. & Ginsburg, H. P. (1987). The development of informal and formal mathematical thinking in Korean and U.S. children. Child Development, 58, 1286-1296.
Spearman, C. (1927). The abilities of man. (London: MacMillan)
Starkey, P. (1992). The early development of numerical reasoning. Cognition, 43, 93-126.
Starkey, P., Spelke, E. S. & Gelman, R. (1983). Detection of intermodal numerical correspondences by human infants. Science, 222, 179-181.
Starkey, P., Spelke, E. S. & Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36, 97-127.
Steffe, L. P. (1990). Inconsistencies and cognitive conflict: A constructivist's view. Focus on Learning Problems in Mathematics, 12, 99-109.
Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4, 259-309.
Stevens, P. (1977). Studies in the anthropology of play: Papers in the memory of B. Allan Tindall. (West Point, NY: Leisure Press)
Stevenson, H. W., Chen, C. & Lee, S. Y. (1993). Mathematics achievement of Chinese, Japanese, and American children: Ten years later. Science, 259, 53-58.
Stevenson, H. W. & Stigler, J. W. (1992). The learning gap: Why our schools are failing and what we can learn from Japanese and Chinese education. (New York: Summit Books)
Strauss, M. S. & Curtis, L. E. (1984). Development of numerical concepts in infancy.(In C. Sophian (Ed.), Origins of cognitive skills: The Eighteenth Carnegie Symposium on Cognition (pp. 131—155). Hillsdale, NJ: Erlbaum.)
Sweller, J., Mawer, R. F. & Ward, M. R. (1983). Development of expertise in mathematical problem solving. Journal of Experimental Psychology: General, 112, 639-661.
Symonds, P. M. & Chase, D. H. (1929). Practice vs. motivation. Journal of Educational Psychology, 20, 19-35.
Thompson, R. F., Mayers, K. S., Robertson, R. T. & Patterson, C. J. (1970). Number coding in association cortex of the cat. Science, 168, 271-273.
Thurstone, L. L. (1938). Primary mental abilities. Psychometric Monographs, , (No. 1).
Thurstone, L. L. & Thurstone, T. G. (1941). Factorial studies of intelligence. Psychometric Monographs, , (No. 2).
Trick, L. M. (1992). A theory of enumeration that grows out of a general theory of vision: Subitizing, counting, and FINSTs.(In J. I. D. Campbell (Ed.) The nature and origins of mathematical skills (pp. 257—299). Amsterdam: North-Holland.)
U.S. Department of Education. (1991). America 2000: An education strategy. (Washington, DC: Author)
Vandenberg, S. G. (1959). The primary mental abilities of Chinese students: A comparative study of the stability of a factor structure. Annals of the New York Academy of Sciences, 79, 257-304.
Vandenberg, S. G. (1962). The hereditary abilities study: Hereditary components in a psychological test battery. American Journal of Human Genetics, 14, 220-237.
Vandenberg, S. G. (1966). Contributions of twin research to psychology. Psychological Bulletin, 66, 327-352.
van Loosbroek, E. & Smitsman, A. W. (1990). Visual perception of numerosity in infancy. Developmental Psychology, 26, 916-922.
Very, P. S. (1967). Differential factor structures in mathematical ability. Genetic Psychology Monographs, 75, 169-207.
von Frisch, K. (1967). The dance language and orientation of bees. (Cambridge, MA: Harvard University Press)
von Glaserfeld, E. & Steffe, L. P. (1991). Conceptual models in educational research and practice. The Journal of Educational Thought, 25, 91-103.
Wagner, R. K., Torgesen, J. K. & Rashotte, C. A. (1994). Development of reading-related phonological processing abilities: New evidence of bidirectional causality from a latent variable longitudinal study. Developmental Psychology, 30, 73-87.
Washburn, D. A. & Rumbaugh, D. M. (1991). Ordinal judgments of numerical symbols by macaques ( Macaca mulatta ). Psychological Science, 2, 190-193.
Wenger, R. H. (1987). Cognitive science and algebra learning.(In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 217—251). Hillsdale, NJ: Erlbaum.)
West, B. H., Griesbach, E. N., Taylor, J. D. & Taylor, L. T. (1982). The Prentice-Hall encyclopedia of mathematics. (Englewood Cliffs, NJ: Prentice-Hall)
Whiting, B. B. & Whiting, J. W. M. (1975). Children of six cultures: A psycho-cultural analysis. (Cambridge, MA: Harvard University Press)
Witelson, S. E. (1987). Neurobiological aspects of language in children. Child Development, 58, 653-688.
Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358, 749-750.
Zaslavsky, C. (1973). Africa counts: Number and pattern in African culture. (Boston, MA: Prindle, Weber, & Schmidt)


1

Both biologically primary and biologically secondary forms of cognition can probably be hierarchically organized 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 consist of three types of competencies, goal structures, procedural skills, and conceptual knowledge ( Gelman, 1993 ; Siegler & Crowley, in press ). These competencies are illustrated later in the article.


2

This argument should not be taken to mean that cultural influences are unrestrained in the types of abilities that can be instilled in children. For instance, we have evolved neurocognitive systems that are very sensitive to basic features of the three-dimensional physical universe ( Shepard, 1994 ). As a result, we can, for instance, create art in three dimensions. However, art works in four physical dimensions cannot be created nor imagined, because the associated neurocognitive architecture is designed to process information in three (or two) dimensions, not four.


3

Note that cognitive tasks (e.g., paper-and-pencil tests) likely vary in the extent to which they draw on biologically primary and biologically secondary abilities and thus might form more of a continuum, rather than discrete classes. However, the neurocognitive systems that support biologically primary abilities appear to be highly specialized and probably evolved to serve a limited number of functions. For this reason, at this point, I prefer to consider biologically primary and biologically secondary abilities distinct. For instance, even though the ability to solve complex arithmetic problems, such as 435 + 537, probably involves a combination of biologically primary (e.g., understanding that adding increases quantity) and biologically secondary (e.g., trading) competencies, the underlying categories of ability are still likely to be discrete.


4

The structure of number words in Asian languages is regular and reflects the underlying base-10 structure of the number system, whereas number words in most European-derived languages, including English, are irregular and do not reflect the base-10 structure (e.g., Miller & Stigler, 1987 ). For example, the word for 11 is "ten one" in Chinese, whereas 21 is "two ten one." The regular structure of Asian-language number words facilitates Asian children's learning to count past 10 and many other basic counting and arithmetic skills (see Geary, 1994 ).



Lyle E. Bourne served as action editor for this article.
Preparation of this article was supported by Grant 1R01-HD27931 from the National Institute of Child Health and Human Development.
I would like to thank Craig Anderson, Jamie Campbell, Harris Cooper, Peter Frensch, Jonathan Schooler, and Robert Siegler for comments on earlier versions of this article.
Correspondence may be addressed to David C. Geary, Department of Psychology, University of Missouri, 210 McAlester Hall, Columbia, MO, 65211.

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