Spatial Understanding Vantage Point

As discussed earlier and as illustrated in Figure 12.2, any given representation has a particular vantage point that is determined by the combination of viewing distance, viewing angle, and viewing azimuth. Our empirical work has addressed children's developing appreciation of vantage point by examining their ability to distinguish among images that depict the same referent from different vantage points, and to produce representations that fit certain vantage-point specific qualities.

As one means of testing children's appreciation of the elements of vantage point shown in Figure 12.2, we (Liben & Szechter, 2001) prepared pairs of photographs with the same referent. Children were shown each pair of photographs and asked whether they were identical. Whenever children judged the photographs to be different, they were asked to say whether the photographs differed because something had changed in the scene itself, or because of something that had been done by the person who took the photograph. Irrespective of which attribution was given, the child was asked to explain what had happened, that is, what had changed, or what the photographer had done. Critical items were 15 pairs in which the photographer's vantage point had changed by altering viewing distance, viewing angle, or viewing azimuth (illustrated in Figure 12.4). To ensure that correct answers varied, there were also filler items in which either something in the scene had changed or in which the photographs were identical.

This spatial photo-pair task was given to children aged 3-, 5-, and 7-years, as well as to a comparison group of college students. For pairs that differed by viewing distance (so that a correct response would include explaining that the photographer had moved closer to, or further away from the depicted subject), only about 25% of the 3-year-old children were correct even on a single one of the five items; the rest were completely unable to explain anything about viewing distance. About 50% of the 5-year-old children were correct on four or all five items, with the rest distributed fairly evenly across the remaining lower scores; just over 75% of the 7-year-old children were correct on all or all but one items, and virtually all adults had errorless performance.

For pairs that differed by viewing angle or viewing azimuth, the pathway to mastery was more protracted. More specifically, for viewing-angle pairs, none of the 3-year-old children explained even a single item. Indeed, children of this age commonly failed to even notice that anything differed in the two photographs, focusing instead on the shared referential content. For example, when commenting on the viewing-angle pair (tulips) shown in Figure 12.4, one 3-year-old child said, "They're both the same." When the interviewer continued by saying, "Well, they're both of tulips, but is there anything different about the pictures?" the child answered in the negative, saying, "Nope. This one [pointing to one on the right] has the same stuff."

The 5-year-old children were more likely to notice a difference between the two images, but their explanations suggested they inferred that there had been a change in what had been in the scene rather than in the way that was photographed. Thus, for example, 5-year-olds accounted for the difference

Figure 12.4 Sample pairs of photographs used in the photograph-pair task illustrating (top to bottom) changes in viewing distance, viewing angle, viewing azimuth, and referent. Reproduced from Liben (2003a) with permission.

between the tulip photographs by explaining that the photographer took "this one when [the tulips] were all curled up and those ones when they were all blooming," that the photographs were taken "in the spring [when] they were closed [and] then in the summer [when] they came out again," and that the photographer "took one sometime when they were closed and one sometime when they were open." About half of the 5-year-old children were able to provide correct explanations of at least one angle pair, sometimes extremely clearly as one child explained, "Um, that one you're looking that way [points straight ahead] and that one you're looking down, this way [bends over the picture]" or another who said, "Oh I like these . . . He [the photographer] went sort of on the side of them and then like up above them to get the middle [points hand down on top of the tulips and rises in chair] . . . because um, like this one is like straight across and this one's like you're looking down [flexes hand to point all fingers down onto the tulips]." By the age of 7, responses like these were common, and by adulthood, they were virtually universal.

The azimuth pairs showed an even more protracted period for mastery. Again, many 3-year-old children failed to note the difference between the two photographs, but even when they did, they commonly believed that there had been a change in the referent objects rather than a change in the photographer's position. To illustrate by reference to the rooster-tile azimuth example shown in Figure 12.4, many children (and even some adults) thought that what had changed was the orientation of the tile rather than the vantage point of the photographer. That it was, instead, the photographer's position that had changed is apparent from looking at the grain pattern of the wood surface on which the tile rests.

In a second task used to study the developing ability to understand vantage point, we gave digital cameras to 8- to 10-year-old children and adults, and asked them to reproduce model photographs. Specifically, respondents were asked to "try to take a picture so that yours will look as much like this photograph as you can make it" and then, after viewing the result on the display screen of the digital camera, to take a second photograph to try to improve the match. Figure 12.5 shows a sample of children's initial photographs for one model. Photographs were scored by assigning points to qualities of the photograph that reflected correct viewing distance, viewing angle, and viewing azimuth. An analysis of variance on scores totaled over four photographs revealed significant effects of age and trial, and a significant interaction showing that the improvement over trials was significant in children only.

We also studied understanding of vantage point by asking participants to create photographs that were consistent with some view-specific verbal description. For example, as participants approached an art museum at which two lion paw sculptures flanked the entrance they were asked to take a photograph with "just one paw in the picture." Adults were almost universally successful in implementing this request, usually by adjusting the direction the camera pointed (azimuth). Only 10% of the adult participants produced an image in which a second paw showed at the edge of the image, an error presumably accounted for by the fact that the area recorded on the image is slightly larger than that

Figure 12.5 Four sample responses to a request to reproduce a photographic model that was almost identical to the bottom right photograph. Reproduced from Liben (2003a) with permission.

Figure 12.6 Illustrative responses by children asked to create a photograph showing only one paw. Solutions involved (1) canonical azimuth (front view) but noncanonical distance (close up), (2) nonca-nonical azimuth (building entrance viewed from side), (3) noncanonical azimuth (back view), (4) non-canonical distance and azimuth (close and to the side), (5) noncanonical angle (overhead rather than eye-level), and (6) change in referent (child said he "waited for people walking by to block one of the paws from view.") Reproduced from Liben (2003a) with permission.

seen through the viewfinder. Errors were more common (32%) among children. Unlike adults, when children erred, the second paw typically appeared well within the frame of the image. The children who were successful used a variety of strategies involving distance, angle, and azimuth as illustrated in Figure 12.6.

Linking Location and Direction Information Across Spatial-Graphic Representations

Another way that we have studied the developing ability to understand the spatial meaning contained in graphic representations is to ask participants to transfer location or direction information gleaned from one spatial-graphic representation to another. Table 12.4 lists four representation-to-representation tasks discussed next. In each, participants were first given some kind of representation (such as an eye-level photograph, aerial photograph, or line drawing) of a space (such as a park, city, or local region) that was not currently in view. They were then asked to demonstrate their understanding of some information that could be gleaned from that representation (e.g., the location of a particular landmark in the represented park) by performing some task on another representation of that same referent space (e.g., placing a location dot on a plan map of the park). Success on tasks like these draws simultaneously and interactively on understanding the initial depiction (e.g., the eye-level photograph), on understanding of the representation on which responses must be made (e.g., the plan map), and on the skill (and care) needed to implement the required response (e.g., precision in placing the sticker on the plan map to show the landmark's location).

In the city aerial task (Liben & Downs, 1991), first and second graders were given individual copies of an aerial photograph of Chicago. After class discussion of the photograph and its general referent, children were asked to use it as a basis to draw (but not trace) a map of Chicago. Next, children were given a plan map, and were asked to identify what area of the photograph was depicted. An acetate sheet was then placed over each child's copy of the aerial photograph to outline the correct area and to

TABLE 12.4 Representation-To-Representation Tasks

Task name

Spatial information obtained from . . .

Response indicated on . . .

Query: Location?

Query: Orientation?

City aerial

Vertical aerial photo of large city

Plan map

Y

School aerial

Oblique aerial photo of school neighborhood

Plan map

Y

Terrain perspective

Line drawings of local topography in oblique view

Contour map

Y

Photo-map

Eye-level photos of room, playground, & campus scenes

Plan map; oblique map

Y

Y

indicate eight locations. Children were asked to place stickers on the plan map to show the locations indicated on the aerial photograph.

With respect to location variables, and as hypothesized in the selection of the locations that were queried, items that allowed solution via topological concepts such as "on" or "next to" generally elicited better performance than items that required metric or quantitative reasoning. For example, the item at a clear and unique bend of the breakwater was answered correctly by 29% and 70% of the first and second graders, respectively, whereas an item located on one of many similar buildings whose identification required metric concepts (e.g., judging distances from some other distinct location) was answered correctly by only 0% and 19% of the same age groups.

With respect to user variables, one generalization that emerges from the data is clear improvement in performance with age. For example, the modal response (42%) in Grade 1 was failing to be correct on even a single item, a score that was almost unheard of (4%) by Grade 2. These data suggest that handling the simultaneous changes in graphic medium, scale, and different representational content of the two representations (i.e., the fact that the map represented only a portion of the area shown in the photograph) was completely overwhelming for many of the first but not the second graders. In addition, the maximum score was four correct in Grade 1 but seven correct in Grade 2, and an analysis of variance revealed significantly higher mean scores in the older children. The age difference is unlikely to be attributed to formal instruction given that neither aerial photographs nor mapping tasks like these were included in the children's elementary school curriculum. Instead, it is more likely to reflect general development in both representational and spatial skills during middle childhood. Equally or perhaps even more striking than the age-linked differences, is the wide range of performance within a given grade. For example, within Grade 2, performance ranged from erring on every single item out of eight, to erring on only one item.

The city aerial task could be expected to be challenging because the referent space (Chicago) was entirely unknown, the vertical perspective of both photograph and map which—although shared—was unfamiliar, and the viewing distance of the photograph was great. A different set of challenges was presented by the school aerial task given to first and second grade children (Liben & Downs, 1986). Introductory tasks familiarized children with directional arrow stickers, a map of the school neighborhood, and an aerial photograph of their school (see Figure 12.7). Children were then shown slides of eight aerial photographs of the school, and asked to place arrows on the map to show the direction from which the building had been photographed. This task could be expected to be somewhat easier than the city aerial task because the photographs depicted a familiar rather than an unfamiliar referent and used a smaller viewing distance with a somewhat more familiar viewing angle (oblique rather than nadir). However, the task could be expected to prove even more challenging because it required linking spatial information across representations with two different viewing angles (the oblique angle photograph and the vertical angle map) and queried children's understanding of viewing direction (azimuth) rather than simply location.

Figure 12.7 Aerial photograph and neighborhood map of school used in school aerial task.

Performance was scored by measuring the angular displacement from the correct arrow placement. Even using a lenient margin of error (± 45°), most children in both grades (93% and 87%, respectively) were correct on fewer than half the items, with no age-linked increase, and with no child performing perfectly or almost perfectly. These findings are consistent with the observation that projective (point-of-view) tasks continue to be challenging throughout childhood and even during adulthood.

The pattern of differential performance across items shows that although performance was low overall, it was not random. For example, there appeared to be some systematic tendency to mistake another centrally-located building for the school, or to mistake the rear of the school for its side. Unfortunately, we were unable to vary desired image variables (e.g., azimuth, distance) systematically because of constraints imposed by terrain, cloud cover, and possible flight paths at the time the aerial photographs were taken. Systematic control of viewing angle and azimuth was, however, possible in the terrain perspective task discussed next.

In this task, nine computer-generated line drawings of the local topography were created from different viewing azimuths and angles (see Liben & Downs, 1992). First and second grade children were asked to indicate the direction from which the topographical region had been drawn by placing an arrow sticker on a contour map, and, responses were scored for the degree of divergence from the correct angle. Although there was some slight age-linked advance in performance, again what was most striking were the similarly low scores in both grades (averaging, respectively, only 1.9 and 2.0 correct out of 9 possible). When the quality of the errors is taken into account, however, an age-linked difference is evident: in Grade 1, errors tended to be dispersed evenly over 360° whereas in Grade 2, errors tended to cluster near the correct response. Apparently older children are better able to understand the directional information, but still have difficulty in understanding or in demonstrating that understanding with precision. Interestingly, and consistent with earlier observations of the wide range of performance within any given age is the fact that even as roughly one-fifth of the children at each grade were correct on none of the items, a single child at each grade performed perfectly or nearly perfectly on all of them.

As on other tasks, different items elicited different performance systematically rather than randomly. In general, performance was better on the items for which the correct response faced toward one of the corners of the contour map than for items that faced toward one of the sides. Further research is needed to disentangle the extent to which the corner advantage is due to the match between the corners of both representations, and whether it would still be evident if only one or neither of the two representations had graphically-defined corners (implemented, for example, by using a circular rather than a square contour map).

The final task discussed in this section is the photo-map task (Liben, Kastens, & Stevenson, 2002). In this task, participants were shown eye-level photographs of everyday environments like parks, rooms, and campus vistas. With the photograph in view, participants were asked to place an arrow on a plan or oblique map to indicate where the photographer was standing, and the direction in which his camera was pointing when the picture was taken.

To date, the photo-map task has been given to several samples of fourth-grade children who have been participants in research designed to evaluate the efficacy of a map-use curriculum (see Liben et al., 2002), and to several samples of college students who have been participants in studies examining the role of individual differences and task variables on map use. To provide illustrative data, Figure 12.8 shows one photograph and composite maps of responses from one illustrative sample of each age.

Among the adults, a large cluster of responses (marked as cluster A in Figure 12.8 is correct within a generous margin of error. With two exceptions (responses E and F), even the erroneous arrow placements appear to reflect at least an understanding that the camera was aimed down a path between a building and a tower. Errors suggest inattention to or confusion about viewing direction, about which building appears in the photograph, or both. For example, responses B, C, and cluster G suggest a failure to appreciate that the camera must be positioned so that in the photograph, a building would be to the right rather than to the left of the tower. If one were inattentive to that right-left relation, it would be possible that building #2 or #3 could have been the building appearing in the photograph

Figure 12.8 Composite of arrows placed by adults (left) and children (right) to show position and direction of camera for photograph shown above. Letters (for responses) and numbers (for key buildings) have been added to adult composite to facilitate discussion. Images reproduced from Liben et al. (2002) with permission.

(responses B and C), or that the camera was positioned near a different face of building #4 (response cluster G).

Among the children, not only are there very few correct responses, but performance is far more varied, and strategies are harder to discern. One clear contrast between the children's and adults' responses is that the children are not even uniformly able to identify the critical referential links between the photograph and map. For example, it appears that many of the children had difficulty identifying the representation of the free-standing, tower-like building on the map (labeled #6 on the adult composite). Several children placed their arrows to point at the tower-like part of the large building (#4). It should have been apparent from the photograph that it was the former (#6) rather than the latter (#4) given that the tower in the photograph goes down to ground level and is not embedded within a building. At least one child placed the arrow facing directly toward a rectilinear building with no tower-like structure in sight, and many showed right-left confusions like those discussed for adult respondents.

Taken together, these data suggest that as late as fourth grade, many children have difficulty interpreting the meaning of spatial-graphic representations, and that not only children, but also adults find it difficult to understand the links between representations of given referents when the task challenges point-of-view (or projective) concepts.

Insofar as the photograph is meant to be a proxy for the kind of visual experience one would have if one had actually been standing in the real environment, the photo-map task is transitional to the next category of tasks, that is, to tasks that assess respondents' ability to extract information from a real, currently experienced space, and communicate that information by performing some response on a spatial-graphic representation.

Linking Experienced Environments to Spatial-Graphic Representations

The research discussed in the prior section involves tasks in which there are representational challenges at both input and output. That is, to perform successfully on the described tasks, participants must be able to extract spatial meaning from one graphic representation and then demonstrate that understanding on another graphic representation. In contrast, the research discussed in the current section requires participants to use graphic representations for the output stage only; spatial information is gathered by looking at or moving around in the real, physical environment.

In a taxonomy of methods used to study spatial representation (Liben, 1997a), the former tasks would fall under the rubric of "Representational Correspondence Methods" because they require participants to relate two representations (e.g., a photograph and a plan map). The latter would fall under the rubric of "Production Methods" because they require participants to produce [or modify] a representation by using knowledge collected from a real, physical space. Before turning to illustrative "production" research, it is important to acknowledge that it is also possible to use tasks in which participants gather spatial information from a spatial-graphic representation and then demonstrate their understanding by an action in a real, physical space (called "Comprehension Methods"). To date there has been relatively little research of this kind, probably because of the greater practical difficulties it presents. For example, consider a task in which children are asked to record the location of a novel object placed in their classroom on a copy of a map (a production task) versus the reciprocal task in which children are asked to place the object at a location indicated on a map (a comprehension task). Only the first task permits many children to perform the task simultaneously and independently, and only the first task provides an automatic record of the child's response. Recording where the child places the real object in the real room creates what is in essence a comprehension task for the researcher, who may or may not have advanced enough spatial skills to record responses accurately him or herself. Ongoing research employs comprehension methods (e.g., Kastens & Liben, 2004) but until more data are available, it is not yet possible to know whether or not the two kinds of tasks—comprehension and production—are truly reciprocal. Table 12.5 lists three tasks discussed next to illustrate research in

TABLE 12.5 Environment-To-Representation Tasks

Task name

Stimulus (information obtained from . . . )

Response (indicated on .

Query: . . ) Location?

Query: Orientation?

Object location

Surrounding familiar classroom; familiar objects

Plan map

Y

Person location, direction

Surrounding familiar classroom; person standing and pointing

Plan map

Y

Y

Flag field task

Surrounding unfamiliar urban or rural campus; flags on campus

Plan map

Y

which participants gather spatial information available in the surrounding environment and record it on a graphic representation.

The object location task (see Liben & Downs, 1986) was given to kindergarten, first- and second-grade children in their own classrooms. Children were first introduced to maps in general and then shown overhead transparencies of a map of their classroom. The first contained a scale map of the walls, door, and windows of the classroom, and the second added scale symbols for the classroom furniture. Group discussions involved collaborative identification of key features on the map (such as doors and windows) and of their link to the analogous features in the actual room (e.g., linking map lines to corresponding room walls). After general correspondences were established, a map was placed on each child's desk so that it was aligned with the room. Children were then asked to show on their map the location of six objects in their classroom. Because this task was designed to test children's ability to use a spatial-graphic representation to convey information about locations that were perceptually available (rather than, say, stored environmental knowledge about the room), children were first asked to point to a given object (e.g., the pencil sharpener), and only after all children were pointing correctly, to indicate its location on the map.

As in most earlier tasks, what is striking from the resulting data is less that there was an increment in performance with age (correct object locations were, by grade, 50%, 78%, and 87%), but rather the range of performance within each grade such that even some kindergarten children were correct on every item, and even some second-grade children were incorrect on every item.

Also as in earlier tasks, there was considerable variability in success across items. The most likely interpretation of the differences is that they reflect the degree to which similarly shaped, nearby landmarks are available for possible confusion. For example, in one classroom, children were very accurate (73% correct) in locating the blue box which was on the piano. The only similarly shaped and isolated piece of furniture was the teacher's desk which was at the opposite end of the room. In contrast, children in that class were very inaccurate (4% correct) in locating the red phone which was on a small shelf unit. In this case, there were several other nearby pieces of furniture of a similar size and shape that offered confusing alternatives.

The second illustrative task, the person location and direction task, was given to children in Kindergarten, Grade 1, Grade 2, and in a combined Grade 5/6 (Liben & Downs, 1993). This task was conceptually similar to the object location task, but extended it in several ways. First, rather than identifying locations of familiar objects already present in the classroom, target locations were defined by a person who moved to various places in the room. Because the person stood on the floor, this meant that locations were on undifferentiated areas of the map, thus involving metric definitions of location (e.g., "about a third of the way across the room") rather than only topological ones (e.g., "on the piano"). Second, children were asked to indicate the person's orientation as well as his location. Specifically, at each location, the person pointed straight ahead, and said, "Now I am the [color, e.g., blue] arrow. Put your [blue] arrow on the map to show where I am standing and which direction I am pointing." Finally, children were asked to complete the task twice, first when the map was aligned with the room, and second, after the map had been rotated by 180°. Under the latter condition, when a person was, say, to a child's right in the actual space, the correct location for the sticker on the map would be to the child's left. Whenever a map is out of alignment with the space (as often occurs in the real world, see Levine, Marchon, & Hanley, 1984), users must draw on projective spatial concepts to use the map successfully.

Performance was generally worse among younger children, but particularly so in the unaligned condition (Liben & Downs, 1993). This finding is consistent with the notion that reading or communicating point-of-view information (projective spatial concepts) is mastered later in childhood than is landmark or topological information. Again, data revealed a wide range of performance within grades. For example, even on the most difficult unaligned condition, a small percentage of the kindergarten children performed very well.

Patterns of responses to individual items are again useful in suggesting strategies, and in identifying what aspects of these tasks are difficult. Figure 12.9 shows the placement of arrows for one particular item in one of the first-grade classrooms. What is apparent is that under the aligned condition, most children place their stickers in the correct quadrant of the map, and almost all children orient the arrows in roughly the correct direction. They apparently have little difficulty differentiating what part of the map refers to the floor, and what part to furniture (only a single arrow is placed on a piece of furniture). In contrast, in the unaligned condition, few responses are in the correct quadrant and facing in the correct direction. There are some responses that appear to be the result of children placing their arrows on the map in relation to their own bodies, rather than in relation to the represented space. These arrows are in the opposite corner of the map (lower right, rather than upper left) and point in the opposite direction (facing the upper left rather than the lower right). Other children appear to realize that they need to take the unaligned position of the map into account as they answer and thus respond differently, but they do not understand how to adjust. Many responses suggest deep confusions: several arrows were placed on furniture (impossible, given that the person had not climbed on top of a desk) and some directions were correct neither in relation to the piece of paper, nor in relation to the child's own body (e.g., see arrows pointing to the upper right corner of the map).

Figure 12.9 Composite of arrow placements (black arrows) by first-grade children asked to show the location and orientation of an adult (open arrow) when map was aligned (left) and unaligned (right) with the room. Reproduced from Liben and Downs (1993) with permission.

Response strategies are revealed not only by looking at arrow placements on individual items, but also by examining the distributions of responses across items. First, those items in which the pointing direction was parallel to one of the classroom walls tended to elicit better performance than those items that were at some oblique angle, perhaps because in the former case, the wall lines on the map provided guides for children's responses. Second, within the oblique items, the types of errors change dramatically by age. The oldest children's errors were virtually exclusively ones of precision. That is, when children in Grade 5/6 made a mistake, it was generally because they placed their arrow only a little bit off from the correct direction. In contrast, when the kindergarten children erred, their placements were generally distributed throughout 360° suggesting random responding. Children between these two ages (Grades 1 and 2) tended to err by placing their arrows directly opposite the correct orientation, suggesting that they were failing to compensate for the map rotation.

Because of practical constraints, most developmental map research (including our work) has employed maps of very small environments such as rooms or hallways rather than of large environments such as parks, campuses, or towns. The former are often not even considered to be maps by many children and adults (see Downs, Liben, & Daggs, 1988). Even more important, maps of such small spaces do not present many of the cognitive challenges presented by the kinds of maps that are typically used in daily life (e.g., for wayfinding or route planning) or in professional tasks (e.g., for recording distributions of regions affected by acid rain). For example, it is often difficult to orient maps of large environments because they may have less-differentiated landmarks (e.g., many similar paths, roads, buildings, vegetation, hills) than might be found in a small contained space such as a classroom, it is challenging to integrate information acquired sequentially (e.g., as in walking or driving through an environment) rather than simultaneously (as looking around a classroom while seated at a particular desk), and it is challenging to understand the relative scales of a very large, navigable environment and its scaled representation.

Many of these ecological challenges were incorporated into the flag location task designed as part of research (Liben et al., 2002) on the Where Are We? map-skills curriculum designed by Kastens (2000). First, eight colored flags were placed at various locations on campus field sites (either urban or rural settings). Fourth-grade children, unfamiliar with the sites, were then brought to the campus as part of a field trip, and as a group, were introduced to the map and the task in general. Each child was then launched on the task individually by an adult who gave the child a copy of the map, aligned the map with the space, and pointed out the child's current location and facing direction on the map. Children were then asked to explore the area, and when they found flags, to place colored stickers on their map to show the flags' locations.

The patterns of data showed that understanding the links between a real space and a graphic representation remains difficult during middle childhood. Scores ranged from zero correct to perfect performance, with the average in most samples at roughly 50% correct. As had been true for tasks described earlier, performance on individual items suggested differential challenges depending on relevant spatial concepts. For example, sticker placements for the black flag were closely clustered around the correct location whereas those for the red flag were scattered, not only near the correct location, but even in other sections of the map entirely. The former was on a statue that was the only statue symbolized on the map and thus topological concepts could be used to identify its location correctly ("on the statue"). In contrast, the latter was on a road which required projective concepts (to identify the correct end of the map) and Euclidean concepts (to identify the precise location along the extended linear symbol).

Conclusions

I have reviewed an assortment of tasks addressing children's developing ability to decode or communicate spatial information with spatial-graphic representations. It is important to point out explicitly that all the tasks discussed intentionally involve these representations. That is, they are not addressed to investigating what information participants may have about a space independently of these repre sentations. For example, even without an external representation, individuals may be able to figure out where they are (e.g., in front of the library) and how to get from one place to another (e.g., from the library to the movie theatre) by using mental imagery, an intuitive sense of direction, a learned motor sequence, or other strategies. At the same time, when participants perform imperfectly on some task that uses spatial-graphic representations, their difficulty might be traced to incomplete or inaccurate "real-space" knowledge or skills, rather than to difficulty in meeting the representational challenges of the task. For example, participants who place stickers incorrectly on the map for the flag location task might err not because they have difficulty relating a location in the environment to a spot on the map, but rather because they have difficulty understanding where they are in the environment in the first place. Much additional research is needed to tease apart where the challenges lie for participants, and to understand the ways in which the use of spatial-graphic representations may aid individuals' knowledge of environments just as familiarity with environments may enhance individuals' facility in using spatial-graphic representations (Liben, 2000, 2002; Liben & Downs, 2001; Uttal, 2000).

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