Appendix C
Individual Differences in Spatial Thinking: The Effects of Age, Development, and Sex

Differences in domain-specific expertise are not the only way of characterizing distinctions among learners. This appendix addresses other ways in which differences among learners are relevant for spatial thinking. It begins by discussing the notion of learner differences in general, and then considers the links between learner characteristics and spatial thinking. In turn the roles of three factors are discussed: chronological age, developmental level, and biological sex and cultural gender.

CONCEPTUALIZING DIFFERENCES AMONG LEARNERS

At any given moment, a learner approaches a learning situation with particular skills, knowledge, experience, talents, motivation, and so on. These factors may affect the degree to which—and perhaps even the way in which—the learner acquires new knowledge and skills, and thus develops expertise. How can we understand the basis for differences among learners in general?

First, and contrary to common interpretations of statements about such differences, to argue that individuals differ in what they bring to a learning situation at a particular moment in no way prejudges the origins or causal explanations of those differences. For example, consider the observation that Learner A has better mental rotation skills than Learner B, and the suggestion that, as a result, Learner A can more easily imagine what a rock formation seen from one vantage point would look like from another. This statement says nothing about why Learner A has better mental rotation skills as he or she encounters a new learning opportunity. Explanations might include (1) biological mechanisms (e.g., genetic inheritance of talent); (2) experiential mechanisms (e.g., a history of differential experience with building toys such as Tinker Toys or model airplanes, both of which would be expected to call upon and hence develop mental rotation skills); and (3) biological-environmental interactions (e.g., differential inherited talents that are channeled differently depending upon environmental socialization practices).

Second, and again contrary to common interpretations, statements about differences among individuals (e.g., in their performance on a spatial abilities test such as the Block Design test of the Wechsler Intelligence Scale for Children; see Wechsler, 1992) or between groups of individuals



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Learning To Think Spatially Appendix C Individual Differences in Spatial Thinking: The Effects of Age, Development, and Sex Differences in domain-specific expertise are not the only way of characterizing distinctions among learners. This appendix addresses other ways in which differences among learners are relevant for spatial thinking. It begins by discussing the notion of learner differences in general, and then considers the links between learner characteristics and spatial thinking. In turn the roles of three factors are discussed: chronological age, developmental level, and biological sex and cultural gender. CONCEPTUALIZING DIFFERENCES AMONG LEARNERS At any given moment, a learner approaches a learning situation with particular skills, knowledge, experience, talents, motivation, and so on. These factors may affect the degree to which—and perhaps even the way in which—the learner acquires new knowledge and skills, and thus develops expertise. How can we understand the basis for differences among learners in general? First, and contrary to common interpretations of statements about such differences, to argue that individuals differ in what they bring to a learning situation at a particular moment in no way prejudges the origins or causal explanations of those differences. For example, consider the observation that Learner A has better mental rotation skills than Learner B, and the suggestion that, as a result, Learner A can more easily imagine what a rock formation seen from one vantage point would look like from another. This statement says nothing about why Learner A has better mental rotation skills as he or she encounters a new learning opportunity. Explanations might include (1) biological mechanisms (e.g., genetic inheritance of talent); (2) experiential mechanisms (e.g., a history of differential experience with building toys such as Tinker Toys or model airplanes, both of which would be expected to call upon and hence develop mental rotation skills); and (3) biological-environmental interactions (e.g., differential inherited talents that are channeled differently depending upon environmental socialization practices). Second, and again contrary to common interpretations, statements about differences among individuals (e.g., in their performance on a spatial abilities test such as the Block Design test of the Wechsler Intelligence Scale for Children; see Wechsler, 1992) or between groups of individuals

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Learning To Think Spatially (e.g., relative performance of males versus females or of older versus younger children) in no way prejudge the immutability of those differences. To continue with the example of mental rotation, the statement that Learner A has better mental rotation skills than Learner B says nothing about whether or not Learner B’s mental rotation skills might be improved (e.g., through repetition) to be just as good as Learner A’s. Third, a statement about a difference between two learners on one specific skill need not imply something about the availability of another skill. Thus, although Learner B may be less adept than Learner A with respect to rotating a visual image mentally, Learner B may be more skilled in using a different strategy that may be just as effective (e.g., reasoning verbally about the relative locations of different sections of the geological formation). In short, a statement about differential abilities or strategies carries no implications that differences are inherent (the first point ), fixed (the second point ), or pervasive (the third point). Fourth, differences among learners may be considered either at the level of “group differences” or at the level of “individual differences.” The observation that there are group differences with respect to spatial learning means that—on average—groups differ in their level of or strategies for spatial thinking. Groups may be defined along a virtual infinity of dimensions: for example, biological sex (boys versus girls; men versus women), educational focus (e.g., engineering versus education majors), chronological age (e.g., elementary versus middle school students), or socioeconomic background (e.g., children from professional versus working class backgrounds). A statement that there are “significant group differences” merely means that, overall, one group performs differently than the other group, and that statistical analyses show the observed differences are unlikely to be attributed to chance, but are instead reliable differences. Even when groups do differ significantly, it is almost never the case that every single member of one group differs from every single member of the other group on the relevant characteristic. In real life, distributions overlap. For example, consider group membership defined by biological sex with respect to two variables, first, a familiar and uncontroversial one, physical height. On average, men are taller than women. However, among the group of men, some are short; among the group of women, some are tall so that a particular man may well be shorter than a particular woman. Thus, if we want to know the relative heights of a pair comprised of one man and one woman, we would be better off actually measuring them than we would simply assuming that the man was the taller of the two. The identical reasoning holds for a second, potentially more controversial, variable—the ability to rotate mental images of two- or three-dimensional shapes. Again, at the group level of analysis, on average, boys perform better on tests of mental rotation than do girls, on average (e.g., Linn and Petersen, 1985). Nevertheless, many individual girls perform better than many individual boys. Thus, if one were planning to teach some spatial concept that would be taught differently depending on the level of the learner’s mental rotation skills, it would be misguided to assume that all boys should get one form of instruction and all girls should get another. Rather, different instructional methods should be assigned to children based on a direct measure of mental rotation ability, not on the basis of their biological sex. Similar reasoning holds for other differences between or among groups, including those related to chronological age. For example, on average, first-grade children may be expected to find it more difficult than fifth-grade children to employ projective spatial concepts (see Piaget and Inhelder, 1956). Yet a given first-grader may be particularly advanced and a given fifth-grader may be particularly delayed with respect to that mastery. As a result, decisions about the best ways to teach individual learners are better informed by obtaining information about specific prerequisite skills or concepts than by knowing chronological age alone. Alternatively, one may find means of presenting material at multiple levels of complexity and with multiple strategies so that the material will be appropriate for learners who have a range of strengths.

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Learning To Think Spatially Research on individual differences is focused on the variability in performance that exists across the members of any group. Within a group defined by one or more dimensions (e.g., all first-grade boys), not everyone performs identically. Furthermore, if more than one skill is relevant for a learning task, the profiles of skills may vary from child to child. One boy might have relatively poor mental rotation skills but an excellent sense of his body in physical space, whereas another boy might have the reverse profile. THE CONCEPT OF GROUP DIFFERENCES Given the variability among individuals within any kind of group (such as those defined by the learner’s biological sex or age), why should one bother with identifying differences at the level of groups? There are several reasons why it is useful to identify patterns of skills and behaviors across groups, even though knowledge of these group differences can provide only probabilistic information about the characteristics of any given learner. The first reason is practical and stems from the way educational systems are structured. In particular, the two dimensions along which spatial skills and abilities are most commonly examined in psychological research—chronological age and gender—are also the dimensions along which many educational opportunities are differentiated. Of the two, chronological age is more pervasive in the American education system. With few exceptions (e.g., in very small school districts or home schooling), most education is delivered in age-segregated schools and classrooms. Educational distinctions also occur with respect to gender despite the fact that there are no longer divisions between required curricula for boys and girls as was once the case (girls were required to take home economics, boys were required to take shop, and neither group was allowed into the other’s classes). Nevertheless, proportions of males and females enrolled in elective subjects (e.g., computer science) continue to differ. Furthermore, private and public schools may offer intentional opportunities for single-sex classes, and even in classrooms containing students of both sexes, educational experiences may differ if not by design then by practice (as when disproportionate numbers of boys are found at classroom computers). Although group membership will not ensure that every member of the group has a particular set of characteristics, knowledge of group membership is useful, given that it is—for practical reasons—impossible to measure individuals on every relevant dimension. In the absence of individually administered assessments of relevant variables, knowledge of group characteristics can provide hints about what kinds of experiences and skills learners bring to a new educational situation. However, the predictive power of gender, for example, is insufficient to use gender alone as a basis for recommending differential educational opportunities (see Box C.1). A second reason to identify group differences is that they may suggest factors that account for individuals’ development of different or better spatial skills and strategies. This knowledge could, in turn, provide educational guidance as to learning strategies that might be easier and/or more effective. Thus, the finding that spatial skills or strategies are not randomly distributed across groups may be useful in identifying what kinds of factors account for differential outcomes. For example, if boys as a group do better than girls as a group in learning to assemble car motors, we can look for differences between boys and girls that are potentially relevant. We might find that boys, on average, have been given 10 times more model cars to put together during childhood than have girls. We might hypothesize that something that helps people figure out how pieces of motors fit together is “model experience.” To test this hypothesis, we could examine whether those girls who were given model cars to assemble during childhood do better at assembling car motors than do girls who were not given model cars. Of course, an affirmative answer would not prove the causal connection because it could be that children with better spatial skills begged for, and

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Learning To Think Spatially BOX C.1 Use of the Terms Sex and Gender The words “sex” and “gender” are differentiated by some scholars. Those who make a distinction usually argue that the term sex should be used to refer to biological sex (i.e., whether an individual is biologically male or female), whereas gender should be used to refer to societal variables and influences (i.e., to culturally defined characteristics, such as masculine or feminine identity or the degree to which an individual endorses gender-related attitudes of the culture). At first glance, one might assume that the former is a clean, dichotomous distinction, whereas the latter is a fuzzy, continuous one. However, in actuality, both sex-related and gender-related characteristics fall along a continuum. At the biological level, within groups of genetically male or genetically female individuals (i.e., those who have XY versus XX sex chromosomes), there is, for example, variation with respect to the amount of exposure to androgen and estrogen experienced during prenatal development or with respect to the quantity of sex steroids circulating during adolescence (e.g., see Liben and Bigler, 2002). At the societal level, individuals within such groups vary in the degree to which they self-identify or are identified by others as being representative of societal gender roles, as illustrated by the concept of “tomboys.” Typically, respondents self-report, or researchers judge, physical features of participants to assign them to categories variously labeled as “boy or girl,” “man or woman,” or “male or female.” It is rare that either biological sex or cultural gender is actually measured (e.g., by assessing relative amounts of circulating sex hormones or asking respondents to complete some measure of self-endorsement of culturally masculine and feminine traits). Because it is rare that investigators actually use these measures and because there are complex interconnections between biological and social constructs, the distinction between the terms sex and gender is not always a clean one. In general, the word sex is used to distinguish between males and females, with gender reserved for distinctions linked to culturally defined roles, but the boundaries are admittedly not sharp ones. received, more model cars than did children with worse spatial skills. Additional support for the hypothesized link could come from research demonstrating that individuals randomly assigned to receive model-building experience later perform better on motor-assembly tasks than do individuals not given that experience. THE VALUE OF STUDYING GROUP AND INDIVIDUAL DIFFERENCES In summary there are several reasons for studying group and individual differences. First, they remind us that different learners approach tasks with different skills and experience and, hence, it is important to provide a range of educational opportunities. Second, they are valuable for suggesting hypotheses about the kinds of experiences that might promote spatial skills, and results from testing those hypotheses would be useful for the basic scientific study of learning and for the application of that science to the design of educational interventions. Throughout these efforts, it is, of course, important to remember that acknowledging group or individual differences in no way implies anything about either their origins or their immutability. Thus, rather than shying away from the identification of differences in the ease with which learners develop and apply spatial thinking to new learning challenges or in the types of strategic approaches that are employed, we should catalogue and understand these differences. In keeping with this position, the committee turns first to discussions of group differences concerning age and sex. The former was selected because much of our educational system is age based; the latter, because gender differences in spatial skills have often been observed (Linn and Petersen, 1985).

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Learning To Think Spatially THE ROLES OF CHRONOLOGICAL AGE AND DEVELOPMENTAL LEVEL, AND THE CONCEPT OF EDUCATIONAL APPROPRIATENESS The Role of Chronological Age. As is apparent from our age-graded educational system, chronological age is an important characteristic for many aspects of thinking and learning. As a group, older children are generally better prepared for learning more complex ideas and skills than younger children. Indeed, one way of thinking about very young children is that they are “universal novices” (Brown and DeLoache, 1978, p. 14). Growing older translates into becoming increasingly better prepared to acquire knowledge and understanding. As a result, when exposed to the same lessons or experiences, older learners are generally prepared to learn more from the experience than are younger learners. Conceptualizing Developmentally Appropriate Education. Many educators and scholars have used the term “developmentally appropriate” to capture the notion that educational efforts should take age-linked changes into account. At this level of generality, the committee endorses the importance of the notion (also captured by the phrase “educationally appropriate”). It is important to caution, however, that the term developmentally appropriate has often been assigned extremely narrow meanings in developmental psychology and education. Thus, we must clarify what we do and do not mean when we suggest that learners of different ages have typically moved to different points along the novice-to-expert path. From the perspective of developmental psychology, the committee means that age is an excellent predictor of how advanced an individual is in a variety of arenas—physical, cognitive, and social. It does not, however, mean that simple chronological age itself determines that level of preparation. Thus, for example, the committee is not suggesting that every child of age X is unprepared to learn concept Y. It would argue that there are some developmental constraints to learning: some aspects of understanding evolve gradually and sequentially. However, there is usually a broad range of possibility within any given age. From the perspective of education, the committee does mean that a developmentally appropriate curriculum takes the learner’s preparedness into account in what and how it teaches. It does not mean that structured, teacher-directed programs that follow a relatively fixed sequence (as opposed to child-directed, “discovery learning” approaches) are necessarily doomed to failure, a way in which developmentally appropriate has often been interpreted in education (see Golbeck, 2001, for a discussion of this controversy in early childhood education). Developmental Theories and Learning. Of the theoretical perspectives that are useful for conceptualizing developmentally appropriate learning, the committee focuses on three that have the broadest application and impact: those of Vygotsky, Piaget, and Bruner. Vygotsky (1935, 1978) focused on children’s cognitive development within the broader social context. More specifically, he suggested that at any given point in development, a child might evidence one level of skill independently, but with appropriate supports by the surrounding social context, the same child could show a higher level of functioning than had first been evident. For example, parents, teachers, or even peers might act as supports to extend the child’s functioning. Supports would help to push toward the higher end of what that child is positioned to accomplish—what Vygotsky called the zone of proximal development. Piaget (1970) offered similar ideas in his concepts of assimilation and accommodation. He argued that as the child encounters new experiences that go beyond existing levels of understanding, the existing concepts are stretched. Later investigators demonstrated the viability of using these ideas to facilitate learning (e.g., Inhelder et al., 1974). Bruner (1960) proposed a “spiral curriculum” that “respects the ways of thought of the growing child.” Although he argued that instruction needed to be appropriate to the child’s logical abilities available at any given time, he also argued that “any subject can be taught effectively in some

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Learning To Think Spatially intellectually honest form to any child at any stage of development.” Thus, the same subject should be introduced early but reintroduced repeatedly in later grades. Again, consistent with the point about the imperfect relation between intellectual progress and chronological age was Bruner’s point that [t]he intellectual development of the child is no clockwork sequence of events; it also responds to influences from the environment, notably the school environment. Thus instruction in scientific ideas, even at the elementary level, need not follow slavishly the natural course of cognitive development in the child. It can also lead to intellectual development by providing challenging but usable opportunities for the child to forge ahead in his development. (Bruner, 1960 p. 147) These developmental theories converge on three points: (1) children engage their current levels of understanding as they encounter new material; (2) the process of encountering new material can result not only in the child’s acquiring knowledge about the topic at hand (e.g., information about plate tectonics and earthquakes) but also in the child’s coming to understand broader conceptual principles and skills (e.g., understanding how to interpret spatial representations to show sequential states of some phenomenon); and (3) although cognitive functioning increases with age, this increase is not some precise, fixed, overdetermined function of the child’s chronological age, but rather the result of complex interactions among organismic and experiential factors. Age-Linked Spatial Development. The argument concerning age-linked (but not chronologically locked) changes in cognitive growth also applies to changes in spatial thinking. Certain kinds of spatial thinking are exhibited more uniformly by children at older ages than by children at younger ages. Again, however, within any age group there can be vast differences among children with respect to how far they have progressed in their spatial development. For example, children were asked to show where someone was standing in their classroom by placing stickers on a map of the classroom (Liben and Downs, 1993). Much as often occurs with “you-are-here” maps (Levine et al., 1984), in one condition the map was out of alignment with the room. In unaligned conditions, people must reconcile different frames of reference (one being their own bodily experience in the space; the other being the frame of reference as defined on the map). Based on six items, the average correct numbers for children in grades K, 1, 2, and 5, respectively, were 1.3, 3.2, 4.1, and 5.1. Clearly, performance improved with age. However, these summaries of mean performance at the group level obscure the marked range in performance within each age group. For example, 18 percent of kindergarten children performed perfectly or nearly perfectly (six or five correct items), despite the low average performance of the kindergarten group as a whole. There remain controversies about whether children of a given age who do not demonstrate some kind of spatial thinking (reconciling frames of reference in the map example) are actually incapable of that kind of thinking. An alternative is that they have the skill, but are unable to demonstrate it on some particular task. Examples of this issue abound in work on spatial thinking in infancy. To illustrate, some developmental psychologists (e.g., Piaget, 1954) suggested that it is only gradually and through self-directed interactions with the physical world that infants come to link self body space with external space. This view is compatible with observations that in the first months of life, infants have difficulty in reaching out for and grabbing objects accurately (e.g., Pick and Lockman, 1981). Others (e.g., Spelke and Newport, 1998), however, have argued that fundamental spatial concepts are part of human infants’ biological endowment. Recent work has shown that infants’ skills may indeed be demonstrated early once the motor (or physical) demands of the task are reduced. Rönnqvist and von Hofsen (1994) showed that 2-day-old infants are able to make sweeping motions toward moving objects that are in synchrony with the objects’ trajectories, even though infants at this age are completely unable to grasp moving objects. Despite the debate over the levels of spatial thinking available during infancy, well before they enter preschool, children have mastered basic spatial relations in physical space, understanding

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Learning To Think Spatially notions such as on, in, and into, and understanding how to effect skilled movements in space. What remains less well developed by school entry are skills in manipulating mental or graphic spatial representations. Again, ongoing research is aimed at learning whether young children are completely unable to use certain conceptual systems in their spatial thinking (such as flexibility in frame of reference in the mapping example) or instead they simply have difficulty implementing them in particular tasks. Irrespective of which position one endorses, or which ultimately turns out to be more widely accepted, many more younger children than older children find certain kinds of spatial thinking challenging. What are some of the ways in which spatial thinking develops during the K–12 period? Theoretical Approaches to Spatial Development. There are many ways in which spatial thinking becomes increasingly advanced in most children as they move from early, to middle, to late childhood. There is no one theory of spatial development that captures all relevant progressions, and therefore, this appendix merely samples ways in which age-linked changes in spatial thinking have been conceptualized. When taken together, theory and empirical work have identified a number of general spatial thinking skills and concepts that are relevant for education in specific domains (including but not limited to GIS). One of the earliest and most far-reaching ways of describing the development of spatial thinking is Piaget and Inhelder’s (1956) book, The Child’s Conceptualization of Space. Piaget and Inhelder focus on the spatial concepts that children are able to understand and think about. They suggest that children initially understand topological concepts such as in, on, next to, between, open, and closed. Children at this developmental level (generally preschoolers) are, for example, able to distinguish open from closed figures (e.g., a U from a circle) but cannot distinguish figures that differ only metrically (e.g., a square from a trapezoid). Children at this stage are unaware of point of view and, as a consequence, are unable to distinguish how a scene would appear if one approached it from one side rather than from the other side. Beginning in early elementary school years, and developing gradually through adolescence, children are said to master projective and Euclidean concepts. With an understanding of point of view, children are able to understand projections (thus, for example, to understand and represent ways that cast shadows change as an object is rotated, or to predict the shapes of cross sections made by slicing three-dimensional objects) (see Box C.2). Likewise, children can understand Euclidean representations—for example, using a Cartesian coordinate system to represent physical phenomena (e.g., the surface of the liquid is horizontal). Although subsequent research has placed serious doubt on characterizing spatial cognition as following a fixed, age-linked trajectory from topological to projective and Euclidean thinking, it has confirmed that the kinds of concepts studied by Piaget and Inhelder do challenge individuals throughout childhood and adolescence. Newcombe and Huttenlocher (2000) present a conceptualization of the development of spatial thinking. Their work focuses on spatial location, and they present data showing that even infants are sensitive to metric information about location. Furthermore, they suggest that the kinds of processes used by young children to represent locations are fundamentally similar to those used by adults. These processes include identifying locations in relation to some space or landmark (similar to the concepts of near or next to in the topological formulation by Piaget and Inhelder) and then fine-tuning precise location (similar to the metric concepts discussed under Euclidean spatial understanding). Both Piaget and Inhelder and Newcombe and Huttenlocher, however, characterize the child’s developing understanding of space as resting on a general, categorical system (e.g., in regions, or near landmarks) as well as on a fine-tuned metric system. Furthermore, embedded within the approaches is another important component of spatial development—the use of alternative frames of reference. In Piaget and Inhelder’s formulation, frame of reference is most explicit in the discussion of projective spatial concepts. It is, for example, tapped by the “three-mountains task” in

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Learning To Think Spatially which children are asked to identify what is seen by someone observing the model mountains from a different vantage point. In Newcombe and Huttenlocher’s work, frame of reference is tapped when a child must figure out where to search for an object after moving to the opposite side of the room from where the child was when the object was being hidden. Again, we have to resolve the question of how early children are able to put aside their egocentric frame of reference to answer questions such as these, but it is clear that tasks involving conflicts across frames of reference are more demanding and difficult, and lead to multiple errors. Thus, in the case of misaligned you-are-here maps (Levine et al., 1984), even adults commonly walk in the wrong direction when the maps are not aligned with the environment, a situation that requires an understanding of alternative frames of reference. These examples of how spatial development has been conceptualized are based on conceptual analyses of what spatial tasks demand, and on theories that explore how abilities to meet such demands change with age in concert with general cognitive progressions. An alternative approach to characterizing spatial development is one in which individuals are given a wide-ranging, diverse assortment of spatial tasks and their responses are analyzed statistically to reveal the implicit, underlying structure of component spatial skills. Insofar as tasks are given to individuals of different ages, the data allow for cataloging age-linked differences in levels of performance as well as the underlying structure of component spatial skills. An illustration of this approach is the “psychometric testing” movement (Eliot, 1987) that began in the early part of the twentieth century, inspired less by theoretical goals than by practical ones. The practical concerns focused on finding ways to make decisions such as which people should be allowed to enter the United States as immigrants, which children should be steered toward academic education and which toward vocational education, who should be institutionalized, and who should be inducted into the armed forces. Tests of intellectual capacity and functioning were developed, and by amassing large data sets from people who took various tests, psychometricians catalogued age norms against which an individual’s performance could be judged. People’s performance on these tasks was analyzed (e.g., via factor analysis) to identify subcomponents of intellectual functioning. Significant from the perspective of spatial thinking is that some kind of spatial component emerged from analyses of virtually every intelligence test (Sternberg, 1994). More recently, “spatial intelligence” is included in the “Theory of Multiple Intelligences” proposed by Gardner (1983) that serves as a foundation for many contemporary educational programs. Much of the psychometric study of spatial intelligence or spatial abilities has been directed to identifying specific components of the broader category (e.g., French et al., 1963; Lohman, 1979) and then linking the more specific abilities to specific educational and occupational decisions. As a broad generalization, and as may be seen from age-reported normative data for performance on these tasks, overall, performance improves over the K–12 period. Patterns of performance on separate components of spatial skills may also be used to examine the second kind of group difference, that linked to sex of learners. GROUP DIFFERENCES: BIOLOGICAL SEX AND CULTURAL GENDER Sex Differences in Spatial Performance: Descriptions As noted above, factor analytic studies typically reveal subcomponents of spatial skills. In a study addressed to both developmental and sex-related issues, Linn and Petersen (1985) combined an analytical approach of this kind (i.e., a data-driven approach) with a conceptual analysis of what particular spatial tasks appear to demand of the respondent. Their work has thus been seminal both in the identification of component spatial skills and in the identification of sex differences within

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Learning To Think Spatially BOX C.2 Shadow Projection Task A dramatic illustration of the incomplete association between chronological age and spatial performance comes from research using a “shadow projection task,” originally developed by Piaget and Inhelder (1956) as one method to study children’s developing projective spatial concepts. In this task, children were asked to predict (through drawings or via selections among response alternatives) the shadows that would be cast on a screen if a light source were directed at an object. Of particular interest were children’s predictions about shadows as the object was systematically rotated. Illustrative is a pencil that was initially positioned so that the side of the pencil was in the path of the light (thus casting a rectangular, horizontal line-like shadow) and then gradually rotated toward the light source so that the shadow line became shorter, and eventually became a small circle (once the light source directly faced the pencil point or pencil eraser end). Protocols from children under 9 or 10 years of age were used to demonstrate that young children have difficulty in using projective geometry to represent spatial transformations. For example, “knowing” that pencils are long and thin made it difficult for young children ever to give up the idea that the shadow would be line like. Although the original reports suggested that children gradually and universally mastered projective spatial concepts and hence performed well on the shadow projection task, later research with an adult college population (Merriwether and Liben, 1997) demonstrates that many adults (more women than men) have the same kinds of difficulties on the shadow projection task that Piaget and Inhelder (1956) had reported for children some 40 years earlier. For example, adults were asked to draw the cast shadow of 10-mm-thick plastic shapes—circular, triangular, and hexagonal—when the shape was rotated 30, 60, and 90 degrees towards and away from the screen. Adults of both sexes averaged only about one correct answer for the 30 and 60 degree rotations, and although most males were generally correct ( in about five of the six trials) on the 90 degree rotation (which casts a simple thick line [rectangular] shadow), females fared far worse (averaging only about three of six correct). Furthermore, the errors were not trivial. Figure C.1, from Downs and Liben (1991), shows three participants’ responses. These data provide a striking illustration of the point that just as some young children succeed especially well on spatial tasks (as in the classroom location and direction task), some adults appear to have difficulty in visualizing or representing events that draw on spatial concepts.

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Learning To Think Spatially FIGURE C.1 Shadow projection drawing tasks. Source: Downs and Liben, 1991.

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Learning To Think Spatially those skills. As Linn and Petersen (1985, p. 1480) argued, “explanations of sex differences in spatial ability depend, to some extent, on when these differences first occur.” They addressed questions concerning first, the size of sex differences in spatial ability; second, whether the existence or size of sex differences varies among component spatial skills; and third, for any skill revealing sex differences, when in the life span differences emerge. Based on psychometric studies and on their conceptual analysis of the tasks within empirically derived groups of abilities, Linn and Petersen (1985) identified three categories of spatial ability: spatial perception, mental rotation, and spatial visualization. The first, spatial perception, refers to an individual’s ability to identify spatial relations with respect to one’s own bodily location or position in relation to something in the external space. One task falling into this category is the “rod and frame task, designed to study field-dependence/independence” (Witkin and Goodenough, 1981). In this task, people are asked to identify “vertical” in the face of conflicting spatial cues. A luminous square angled at 22 degrees appears in an otherwise darkened room, and the respondent is asked to adjust a rod (also positioned at 22 degrees and thus parallel to a side of the square) to vertical. Because the horizontal and vertical cues normally available are hidden from sight, the respondent must rely upon his or her own body position (gravitational upright) to solve the problem. Another illustrative spatial perception task is the “water-level task,” which was originally designed by Piaget and Inhelder (1956) to assess children’s developing ability to use a systematic coordinate system of horizontal and vertical axes with which to represent abstract or concrete phenomena. In a typical version, respondents are shown a picture of an empty, upright bottle and asked to draw a line to show how it would look if it were about half full. Even by kindergarten, children perform well, correctly drawing a horizontal line. Participants are then given a picture of the bottle in a tipped position (e.g., held at a 45 degree angle), told the bottle is being held steady in the position shown, and again asked to show the way the liquid would look in the bottle if it were about half full. Typically, children in early elementary grades err, at first placing the liquid line so it remains parallel to the base of the bottle. At slightly older ages they recognize that there is some change in position relative to the sides of the container, but do not yet recognize the invariance in relation to the context. Thus, for example, they may draw the line at a diagonal. Only later do they recognize the invariant horizontal position of the liquid, irrespective of the degree of bottle tilt. The second component is mental rotation or the mental manipulation of an imagined figure or object. This skill permits one to imagine how something (a two-dimensional drawing such as a letter or a three-dimensional object such as a block construction) would look if rotated in the plane or in three-space (see Chapter 4). A classic mental rotation test is a block design task by Vandenberg and Kuse (1979), modeled after Shepard and Metzler (1971). A drawing of a construction made of blocks is shown as the model or standard (see Figure 4.1). Respondents are asked to look at additional drawings to decide whether or not they depict the same block construction except in a different position. Because respondents are given only a finite (and relatively brief) amount of time to work on the problems, the ease with which one can rotate images mentally is reflected in the total number answered correctly. Given limitless time, most people get the right answers, however. Finally, the third component skill identified by Linn and Petersen, spatial visualization, involves the multistep manipulation of spatial information, which is thus conducive to the use of multiple solution strategies. Spatial visualization is tapped by “tasks that involve complicated, multistep manipulations of spatially presented information” (Linn and Petersen, 1985, p. 1484). This final component is less clear-cut and may be thought of as a conglomeration of strategies used in solving spatial tasks. An illustrative task is the Paper Folding Task developed by the Educational Testing Service (Ekstrom et al., 1976) in which respondents are shown a piece of paper that has been folded, and through which holes have been punched. Respondents must select from among five choices that show the way the paper would look once it is again unfolded. To solve the

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Learning To Think Spatially problem, one might use mental rotation strategies, verbal, logical strategies, and so on. Sample items from each of the three types of tasks are shown in Figure 2.1. Sex Differences: Developmental Emergence Having identified these three component skills, Linn and Petersen (1985) used meta-analyses to examine sex differences and their age of emergence. Specifically, they gathered data from prior investigations, calculated effect sizes, and then tested for homogeneity of effect sizes for groups of studies. The groupings began by including data from males and females and across broad ages. In cases in which homogeneity of effect sizes was not found, studies were partitioned into subgroups based on age and sex, and further partitioned into specialized groups until homogeneous or nearly homogeneous groups of studies were identified. They identified sex differences—favoring males—in the first two skills—spatial perception and mental rotation—but not the third—spatial visualization. The sex difference on mental rotation was large, whereas that on spatial perception was small. For those skills that show sex differences, the differences appeared as early as the skill has been measured (e.g., at least as early as seven years of age). This suggests that at least some of the explanation for sex differences in spatial skills rests in factors that operate very early in life, potentially including early biological experiences such as differential exposure to hormones during the prenatal period. Sex Differences in Memory for Spatial Location In addition to the spatial abilities tested in the psychometric literature, there has been growing attention to another kind of spatial ability, namely the ability to remember spatial location. In a typical paradigm, participants are brought into a room that contains a desk on which are scattered miscellaneous objects (e.g., a stapler, scissors). Participants are asked to memorize the objects and are later asked to recall where the objects had been located. On such tasks, performance by females is typically better than performance by males (McBurney et al., 1997; Montello et al., 1999; Silverman and Eals, 1992). In many of the tasks, performance may be arguably said to be “better” or “worse” in either males or females. There are, however, situations in which performance is probably best conceptualized as “different.” For example, both women and men may be successful in getting from one location to another in a real environment, but they may use different strategies. Illustrative is the finding that as a group, men are more likely to invoke cardinal direction, whereas as a group, women are more likely to invoke relations to landmarks (Ward et al., 1996). In summary, there is considerable evidence for spatial tasks on which males and females perform differently. In some cases, especially those involving mental rotation and spatial perception, males as a group perform better than females as a group. There are tasks on which the group difference is reversed and some in which the strategy or speed, but not the accuracy, of performance differs. Given that these findings, taken together, reveal at least some sex-related group differences, the next section reviews some of the explanations that have been offered for them. Origins of Sex Differences As in every other domain of sex-related differences, explanations for sex differences in spatial performance include experiential factors (nurture), biological factors (nature), and the interactive effects of both. The accumulated evidence suggests that all three factors play some role in the observed group differences.

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Learning To Think Spatially The argument for nurture effects is a strong one, based on evidence showing different socialization patterns and different experiences encountered by boys and girls. Importantly, small differences in early years may have snowballing effects on choices and opportunities later in life, which in turn may have powerful effects on spatial outcomes. Only a sample of the research literature can be discussed here; detailed accounts are available (e.g., see Halpern, 2000). At early ages, there is evidence that boys in Western cultures are given and encouraged to play with toys such as blocks and construction kits, whereas girls are given and encouraged to play with toys such as dolls and role play. The former are associated with physically more active play and with involvement in spatial actions (such as the need to rotate construction toy pieces to connect them). There are data showing that boys who engage in more culturally “masculine” play perform relatively better on spatial tasks (Connor and Serbin, 1977; O’Brien and Huston, 1985; Serbin and Connor, 1979). Similarly, girls and adolescents who show higher levels of participation in activities viewed as more spatial also score relatively better on spatial tasks (e.g., Newcombe et al., 1983). These data are consistent with the idea that participation in certain kinds of activities—those that are traditionally associated with boys and men in our culture—promotes better spatial skills. Of course, these correlational data are also consistent with the inverse causal possibility, that is, children who have higher spatial skills to begin with are more likely to gravitate toward spatially challenging activities. One means of exploring the hypothesis that experience causes better spatial skills has been to conduct experiments in which experiences hypothesized to promote spatial skills are provided to some, but not other, children and to observe later differences in performance on spatial tasks. In a classic example of this approach, Connor et al. (1978) tested first-grade children’s performance on the Children’s Embedded Figures Test (Witkin et al., 1971) and found the traditional sex-related difference. They then provided related experience. Both boys’ and girls’ performance improved significantly, with improvement among girls being large enough that the sex difference was no longer significant after training. Although other investigations using interventions have also shown significant improvements from training, many have failed to overcome the initial sex difference. A meta-analysis by Baenninger and Newcombe (1989) concluded that training has positive effects on both males’ and females’ performance, but does not overcome initial differences. We do not know, however, whether group differences remain after interventions because of some inherent sex-related difference or whether the experiences provided by some constrained educational intervention are simply not powerful enough to overcome the accumulated effects of prior years of differential experiences. The argument for nature effects has also generated a vast amount of research. A number of potential biological mechanisms have been considered as possible factors in the observed sex-related differences in spatial performance. One possibility is that some spatial abilities are controlled by an X-linked recessive gene. This account holds that—as is true for any X-linked recessive inheritance pattern such as hemophilia—a male’s recessive gene cannot be obscured by a dominant gene on the other chromosome because there is no counterpart genetic material on the Y chromosome. This means that if a male and a female each had inherited a recessive gene that is favorable to the development of spatial skills on an X chromosome, for males, that gene would be seen in the phenotype, whereas for females the gene would be seen if and only if that female had also inherited the recessive on her other X chromosome. Had she inherited the gene that is not favorable to the development of spatial skills at that locus, the gene that is favorable (as a recessive) would be obscured in the phenotype. Investigators have reported data on family patterns of spatial abilities that are consistent with this notion (e.g., Bock and Kolakowski, 1973; Thomas, 1983; Thomas and Kail, 1991; Vandenberg and Kuse, 1979), although these data have not gone unchallenged (see Wittig et al., 1981). A second biological mechanism proposed as playing a causal role in sex-related differences in

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Learning To Think Spatially spatial performance is exposure to different sex hormones in males and females during early and late portions of the life span. The effects of sex hormones are typically discussed under two major headings. One, organizational effects, concern long-lasting, structural influences of hormones on the individual. These occur primarily during the prenatal and perinatal periods. The second, activation effects, concern more transient effects of sex steroids, for example, effects that occur as hormonal levels change with the menstrual cycle or in relation to environmental stress. The term “activation” is used because it is presumed that circulating hormones have their effects on an already prepared organism. Although there is a tendency to equate early effects with immediate structural changes in the brain, there is increasing evidence that effects of sex steroids early in life may affect neural changes at later portions of the life span as well (see Arnold and Breedlove, 1985; Collaer and Hines, 1995; Liben et al., 2002). Irrespective of the degree to which the two kinds of effects are cleanly separable, hormones may have both long-lasting effects that are traceable to much earlier portions of development and as short-lived effects susceptible to variations that may occur with the seasons, within a month, within a day (see reviews in Alexander and Peterson, 2001; Kimura, 1999; Liben et al., 2002), or even within minutes, with changing environmental contexts (e.g., competitive experiences such as sports events; see Mazur and Booth, 1998). There is considerable evidence that prenatal exposure to testosterone affects later spatial behavior, although there is controversy about what levels of prenatal exposure lead to the highest levels of spatial performance and whether these follow different patterns in boys and girls. Because ethical concerns prohibit manipulation of prenatal hormones experimentally, most human research on early hormonal effects on spatial skills has been conducted with samples that have some atypical hormonal history. Illustrative is research on girls with congenital adrenal hyperplasia (CAH), a condition in which girls are exposed prenatally to excessive levels of androgen. These girls have greater spatial skills than do non-CAH girls, including blood relatives (Hampson et al., 1998; Resnick et al., 1986). Investigators have also examined the effects of levels of circulating sex steroids on behavior, with some evidence that changes in spatial performance may depend on cyclical hormonal changes. To illustrate, Hampson (1990) reported that women who were in the low-estrogen phase of the menstrual cycle performed better on spatial tasks than when they were at their estrogen peak. However, no evidence of an effect of hormones on spatial behavior was found in a clinical investigation in which estrogen and placebo were administered in an alternating schedule to a group of adolescents being treated for pubertal delay (Liben et al., 2002). Taken together with other studies (see reviews in Collaer and Hines, 1995; Fitch and Bimonte, 2002; Hines, 2000; Kimura, 1999; Liben et al., 2002), it is probably most conservative to conclude that sex hormones do have an influence on spatial behaviors, but that the details of this influence are far from certain. In addition to saying that experiential and biological factors have effects on spatial outcomes individually, there is good evidence that they also have effects interactively. It appears likely that initial differences—even small ones—among individuals may affect the experiences that they seek out or are encouraged to pursue, which in turn have further effects on ultimate outcomes. It is not only that nature and nurture interact in the statistical sense that both components contribute in some additive sense to the ultimate outcome (what has been referred to as “statistical interaction;” see Sameroff, 1975). It is also that each can affect the way in which the other operates (interaction referred to as “transaction;” Sameroff, 1975). For example, consider a very young child who begins with some inherent predisposition to explore, observe, or represent spatial qualities of the environment. This child may be expected to behave in ways that elicit different kinds of behaviors from adults than those that would be elicited by a child without this predisposition. Furthermore, even when an identical physical and interpersonal environment is encountered by these hypothetical

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Learning To Think Spatially children, they may be expected to take something quite different from their experiences with that environment. Given the likelihood of statistical and transaction interactions, it is difficult to tease apart the effects of nature and nurture. Two illustrations of an interactive approach follow. In the first, Casey (1996) hypothesized that girls with different brain organization patterns might be expected to respond differently to similar kinds of spatial experiences. She operationalized brain organization on the basis of handedness, which has been linked to different patterns of hemispheric dominance (Annett, 1985). She operationalized spatial experience as exposure to a math-science college curriculum, a curriculum that provides significant exposure to spatial thinking. Although Casey acknowledged that college major is not purely an environmental variable (students self-select into majors; hence those who self-select into math and science may be expected to have relatively strong spatial skills), the self-selection component was held constant because all participants had self-selected into the major. Hence, the critical question was whether there would be differential effects of exposure to that college curriculum as a function of brain organization (measured by handedness). Consistent with the interactive hypothesis, her data show that the same curriculum enhanced spatial performance (measured by a mental rotation task) differentially in relation to students’ handedness. A second illustration comes from work by Berenbaum and colleagues (e.g., Berenbaum et al., 1995; Resnick et al., 1986). Although this work is often taken as a demonstration of direct effects of biology (hormones) on spatial outcomes, it simultaneously speaks to the manner in which biology and experience may interact. For example, CAH girls, subjected to unusually high androgen levels prenatally, show greater spatial skills than do their non-CAH relatives (Resnick et al., 1986). One possible interpretation is a direct effect of prenatal androgens (and hence brain organization) on spatial skills. However, another possibility is that there is only an indirect effect. Berenbaum has shown that CAH girls prefer toys and activities more typical of boys than of girls (Berenbaum and Hines, 1992; Berenbaum and Snyder, 1995). Perhaps these preferences are driven by atypically high activity levels that lead CAH girls to prefer objects and activities that allow active, rather than passive, play. Under this scenario, CAH girls’ gravitation toward “boys’” toys stems from the fact that these toys support more action or manipulation, rather than from the fact that they support spatial play in particular; however, playing with these toys provides experience conducive to developing higher-level spatial skills.