The Use of Geometry and Reading a Map

Child Dev. Author manuscript; available in PMC 2015 Jan 1.

Published in final edited form as:

PMCID: PMC3751975

NIHMSID: NIHMS452183

Reading Angles in Maps

Véronique Izard

^oneLaboratoire Psychologie de la Perception, Université Paris Descartes, Sorbonne Paris Cité, 75006 Paris, France

^twoCNRS UMR 8158, 75006 Paris, France

³Department of Psychology, Harvard Academy, Cambridge MA 02138, USA

Evan O'Donnell

^threeDepartment of Psychology, Harvard University, Cambridge MA 02138, USA

Elizabeth S. Spelke

³Department of Psychology, Harvard University, Cambridge MA 02138, Usa

Abstruse

Preschool children can navigate by simple geometric maps of the environment, only the nature of the geometric relations they use in map reading remains unclear. Here, children were tested specifically on their sensitivity to bending. Forty-eight children (age 47:15–53:thirty months) were presented with fragments of geometric maps, in which angle sections appeared without whatever relevant length or distance information. Children were able to read these map fragments and compare 2D to 3D angles. However, this ability appeared both variable and frail among the youngest children of the sample. These findings suggest that 4-year-old children begin to class an abstruse concept of angle that applies both to 2D and 3D displays and that serves to interpret novel spatial symbols.

Keywords: geometry, map reading, spatial cognition, cerebral evolution, preschool children

Geometry defines abstruse concepts that apply to various types of spatial entities. For example, we may find angles in 2-dimensional line drawings, in the contours of 3-dimensional objects, in the arrangement of the walls surrounding usa, or even in the projections of star patterns in the sky. Similarly, in geometry the length of an object and the distance between two landmarks or walls are captured by the aforementioned abstract concept of metric altitude. Comparing shapes of unlike entities is challenging, because ane must abstract away from the item media in which these shapes announced. In immature children, this ability might take time to develop: for example, 1 ½-year-old toddlers still fail to recognize the matching shapes of blocks and holes or apartment patterns (Shutts, Örnkloo, von Hofsten, Keen, & Spelke, 2009), and even in adults, the presence of convex or concave segments induce unlike descriptions of interlocking shapes (Cohen & Singh, 2007). In an effort to understand children'south developing sensitivity to abstruse geometry, here we focus on angle, a central concept in Euclidean geometry. We inquire whether preschool children possess an abstruse representation of angle that applies both to large 3-dimensional surface arrays and to minor 2-dimensional figures.

Angle is a specially interesting case in the study of abstruse geometric concepts, because children'due south performance with angle appears greatly affected by stimulus format. On one mitt, young preschool children can find angle variations in small, ii-dimensional figures, even when these figures too vary in size (Izard & Spelke, 2009). Preschoolers' sensitivity to angles and metric properties of 2-dimensional figures is then pervasive that, when they start learning the names of geometric shapes (east.g. "triangle", "square"), they do not apply these names to non-prototypical figures (ex: an irregular triangle) (Clements, Swaminathan, Hannibal, & Sarama, 1999), yet are willing to generalize the categories to disrupted figures (ex: a triangle with a corner cutting off, or with an interrupted side), provided that they retain the metric backdrop of the paradigm (Satlow & Newcombe, 1998). Sensitivity to angles in two-dimensional figures may already be nowadays in infants (Lourenco & Huttenlocher, 2008; Schwartz & Solar day, 1979; Younger & Gotlieb, 1988), and fifty-fifty newborns (Slater, Mattock, Brown, & Bremner, 1991); although in all these studies, it is unclear whether infants represented angles per se, or reacted to length and distance variations (see below). Also 2-dimensional figures, children in their second year of life can compute heading angles and detect shortcuts between locations, even without visual input (Landau, Gleitman, & Spelke, 1981; Landau, Spelke, & Gleitman, 1984); an ability that subsequently improves with age (Morrongiello, Timney, Humphrey, Anderson, & Skory, 1995). When tested with configurations of iii-dimensional surfaces, nevertheless, navigating children appear to ignore the angles formed between surfaces until the age of four (Hupbach & Nadel, 2005; Lee, Sovrano, & Spelke, 2012), even though they can process other geometric properties of a configuration of walls, such as the distances that separate them (Hermer & Spelke, 1994; Lee et al., 2012).

Comparing two-dimensional figures to the three-dimensional environment is a central component of map reading (Shusterman, Lee, & Spelke, 2008; Uttal, 1996, 2000; Vasilyeva & Bowers, 2006). In everyday life, nearly of the maps we use contain a mixture of geometric and non-geometric information. Moreover, unlike maps may convey spatial data at different levels: road maps ordinarily conserve metric information, whereas subway maps sometimes carry merely topological information nearly the connections between stations, distorting distances and angular relations. Prototypical maps (Uttal, 2000) are locally Euclidean: they conserve metric information from an overhead projection of the navigable layout. In terms of geometric data, such maps provide three types of geometric cues to the reader: angles (between axes and within shapes), relative distances (information pertaining both to the lengths of objects and other structures, and to the distances separating objects and other structures), and sense (left-right directions betwixt and inside objects and other structures).

Because map reading requires matching the geometric properties of two- and three-dimensional spatial arrays, researchers have used maps to probe the development of children'south competence in Euclidean geometry. Preschoolers are able to read simple geometric maps (Huttenlocher, Newcombe, & Vasilyeva, 1999; Huttenlocher, Vasilyeva, Newcombe, & Duffy, 2008; Landau et al., 1984; Shusterman et al., 2008; Vasilyeva & Bowers, 2006; Vasilyeva & Huttenlocher, 2004), with no credible price for reading a flat 2-dimensional map compared to a 3-dimensional scale model (Huttenlocher et al., 2008). By varying the configurations used, these studies have shed some light on the type of geometric relations that children are able to process. In one blazon of study, the configuration was a rectangular form varying only in one dimension or a trio of objects bundled in a line, and the cued position was specified solely in terms of length (along the rectangle) or distance (between the aligned objects) (Huttenlocher et al., 1999; Shusterman et al., 2008). Children equally young as 3 or 4 years of historic period were constitute to find and utilise length and distance relations from these linear maps (Huttenlocher et al., 1999) and they did so spontaneously, with no instruction to employ geometry or feedback concerning their operation (Shusterman et al., 2008).

Other tests used an isosceles triangular configuration: either three objects placed to marker the corners of an isosceles triangle, or boards attached to class a continuous triangle. With these configurations, a combination of length, altitude or angle with sense is needed to distinguish amongst all three corners of the triangle. More specifically, length (in the instance of the continuous triangle), distance (in both cases) and angle are each sufficient to ascertain the unique corner with respect to the two non-unique corners, while sense relations are necessary to distinguish amongst the ii not-unique corners. Tested with isosceles displays made of iii divide objects, members of a remote Amazonian customs responded reliably to all iii positions on the map (Dehaene, Izard, Pica, & Spelke, 2006), thus manifesting an power to employ either altitude or angle, and too sense. In the same test, four-year-sometime children from the U.S. successfully located the unique corner of the triangle from the map as well, but, in contrast to the older participants from the Amazon or from the U.S., they failed to distinguish between the other 2 similar corners (Shusterman et al., 2008). In fact, even half-dozen-year-old U.Southward. children do not reliably discriminate betwixt two similar corners of an isosceles triangle (Vasilyeva & Bowers, 2006). This failure indicates that children do non use sense relations on a map to guide their navigation in a 3D assortment. On the other mitt, because the apex of an isosceles triangle can be defined in terms of either angle, length, or distance (or all three), these experiments do not reveal whether children used angle at all to brand this distinction.

The failure to distinguish between children's utilise of angle and their utilize of length/altitude reflects a cardinal limitation of any task presenting purely geometric, complete planar figures. In Euclidean geometry, variations in bending are inevitably accompanied by variations in length or distance. Every bit the angular size of 1 corner of a triangle increases, for example, and so does the length of the opposite side (Euclid'south Proposition 25: Heath, 1956). Similarly, if a rectangle is changed into a parallelogram past modifying the angles between the sides, the distances between the sides change as well. Because all 2-dimensional geometric forms tin be constructed purely from an array of triangles, the covariation of length/altitude and angle found in triangles applies to all forms. Thus, it is in principle not possible to dissociate angle from length/distance in whatsoever complete 2-dimensional map of an assortment.

Spelke, Gilmore & McCarthy (2011) attempted to probe children'south utilize of each type of geometric cue separately, by introducing a new blazon of display and map task to separate bending, length and sense from each other. Hither, instead of a triangle, at each trial they installed two wall configurations shaped as 50's, which differed from each other in terms of the angle betwixt the two branches, the length of the branches, or in the sense relation across branches. For example, in angle trials, the ii Fifty's had branches of similar length, pointing in the same direction, and forming different angles. In length trials, the branches of the ii structures were oriented in the same mode (aforementioned bending, same sense) only their lengths differed. In sense trials, the 2 structures were identical except that 1 pointed to the left and the other to the right. Instead of presenting a complete map of this array, moreover, the experimenter presented children with a depiction of just ane of the Fifty-shapes, and recorded whether the children were able to recognize which of the two arrays was represented on the map. The authors plant that v- and 6-twelvemonth-old children successfully recognized an Fifty-shape construction from 2-dimensional drawings in both `bending' and `length' trials. However, success at the angle trials does not prove that children represent angle relations, considering the arrays with different angles differed also with respect to the distances between branches: with the length of the branches held constant, any increment in bending is accompanied past an increase in the distance between the branches' endpoints. Ascribing the children's responses in Spelke et al.'s written report to distance seems especially plausible in low-cal of two findings. First, 4-year-old children can read distances from a map of objects arranged in a line, where altitude is the simply available cue (Shusterman et al., 2008). Second, subsequently being disoriented by repeated turning without vision, 2-year-old toddlers reorient in a sleeping accommodation using exclusively the distances between the walls, fully ignoring wall angles and lengths (Lee et al., 2012).

Here, we introduce a new experimental prototype that isolates bending from both length and distance relations in a map reading task. The children were tested on a placement task where they needed to choose betwixt ii locations, marked past buckets situated in two of the corners of a large wooden triangular structure. The correct bucket was indicated to them on a 2-dimensional map, representing the blueprint of the wooden triangle. In some critical trials, the map was taken apart before the experimenter indicated the placement location, such that the angle information was presented in the absence of whatever informative length or distance relations. If children are sensitive to angle in maps, then they should succeed on these fragmented map trials besides as on the trials with complete maps.

Experiment 1

Children were tested on a map chore using isosceles triangle configurations. Experiment 1 included two types of configuration, allowing the extraction of precut fragments from the maps which contained either only angle information (in experimental trials) or only length information (in control trials). We chose a placement task rather than a search task, because the former has been found to be easier for young children (Huttenlocher et al., 2008) and because placement tasks provide no corrective feedback over the course of the experiment.

2 target buckets were placed either in two corners or along ii of the sides of an isosceles triangle. Depending on the configuration, fragments were cutting out of a map of the array either effectually the corners (angle fragments trials) or along the sides of the triangle (length fragments trials) (effigy 1). The angle fragments were round and centered on the corners of the triangle, thus presenting angle information in the absence of any informative length or distance information. These trials tested whether preschool children are able to generalize angle across 2-dimensional and 3-dimensional displays. The length fragment trials were rectangular, of similar width only different elongation, and presented a portion of two of the triangle sides without showing its corners, thus presenting length information in the absence of any informative bending data. Given previous reports that four-twelvemonth-old children can successfully read length from maps, we expected success at the length trials and included them every bit a control to bank check that the children were able to understand the fragmentation manipulation.

Displays for Experiment 1. The photograph shows the blazon of arrays used in a familiarization configuration: wooden triangular structure, inverted bucket, circular lath serving equally map, and toy frog. The drawings presented on the right show the configurations used in the test, as they appeared on the map: two configurations for bending (right triangle, acute isosceles triangle) and i for length. The dashed lines were not visible on the maps only are added here to evidence the fragments precut to be extracted from the map in each condition. The bottom line shows the fragments aligned as they were presented to the children.

Method

Participants

Thirty-two children (mean age l.7 months, 47:fifteen–53:xxx, 14 females) participated in the study. Children came generally from Caucasian center-form families. Birth records were obtained from local city halls in the greater Boston area, and families were invited to come to the lab and participate by phone or by post. The parents were reimbursed $5 for their travel expenses, and the children were thanked of their participation with a pocket-sized toy. An additional three children were tested just excluded from the final sample for experimenter error (1) or excessive distraction (ii).

Displays

Children were tested in a perfectly round room in the laboratory (diameter 12.v feet) containing a large orange wooden triangular structure placed at its center (longest side 60 inches, superlative 12 inches). The structure was fabricated of boards rather than isolated objects to create configurations allowing map fragmentation. Two light-green inverted buckets were placed either at two interior corners of the triangle, or forth ii of its sides, and served as targets in the placement task. A circular board made of foam cadre, with an orange triangle identical in shape to the wooden triangle and two greenish dots indicating the buckets, served equally the map of the room (figure i). Two fragments could be extracted from the pictures and then every bit to show only parts of the setting in isolation (walls or corners), exhibiting just i geometric feature (length or angle). A toy frog served as the object used in the placement trials.

Two isosceles triangles were presented. The starting time triangle had an acute angle of 36° at its unique corner, and two angles of 72° at not-unique corners (length of sides: 60×60×37 inches). The second triangle had a right angle at its unique corner (ninety°) and two 45° angles at non-unique corners (length of sides: sixty×42.5×42.5 inches). In the angle configurations, i bucket was placed in the unique corner and i was placed in a non-unique corner (effigy 1). The acute isosceles triangle served also in a length configuration, where the ii buckets were placed respectively along the brusk side and along one of the long sides of the triangle. In the angle configurations, the two precut map fragments were round and centered on the two corners with buckets. In the length configuration, the fragments were rectangular and ran alongside the sides with buckets, with the endpoints of the sides excluded and then every bit to remove all angle information.

Design

Children were outset given 3 familiarization trials to introduce the task without focusing on any metric relations of length or bending (see Procedure below) and so tested with 3 blocks of trials corresponding to the three test configurations described above and in figure 1. Each block contained ii trials with the full maps (one for each bucket), followed by 2 trials with the map fragments (i for each bucket), for a total of 12 experimental trials. Across children, the trials were presented in four fixed orders so as to counterbalance the club of presentation of the configurations, with the constraint that the two configurations with the acute triangles followed each other, thus minimizing changes in setting. The four orders counterbalanced likewise the position of the correct bucket in the get-go trial for each configuration, both in the full map condition and in the map fragments condition.

Process

The familiarization procedure used either the acute or right triangle depending on the condition that would appear first in testing, and it focused on the topological relation of containment. In the beginning familiarization trial, there was only one bucket at the center of the triangle. The corresponding map showed one dot at the eye of a triangle, which was geometrically identical to the wooden triangle. When kickoff introducing the map, the experimenter stressed the correspondence between the movie and the room display by pointing successively to the triangle (the "house") and the dot (the "chair") in the picture and and then in the room. The experimenter pointed to the dot on the map, and asked to child to help the frog get sit on her favorite chair. In the second familiarization trial, a second bucket was added outside the triangle and a new corresponding map was produced. The experimenter pointed to the map to asking that the kid place the frog on the outside bucket. Lastly, the third familiarization trial served to introduce the map fragmentation manipulation. The map was precut so that two circular sections, centered on each bucket, could exist extracted. One section showed a dot in the heart of a triangle, the other showed an isolated dot. The fragments were shuffled in front of the child, and and so the fragment with the dot inside the triangle was indicated as the target for the placement task. Throughout familiarization, children were provided with feedback if they placed the toy in the wrong location (the need for feedback was rare).

Following familiarization, the children were tested on the iii unlike configurations. At the offset of each cake, the experimenter start introduced the configuration by pointing to the triangle ("house") and the dots ("chairs") on the map, then in the room, stressing the correspondence between the map and the room. For each trial, the experimenter pointed to one of the two possible locations on the map. The child had to place a sticker on the map (to check that they had encoded the location of the right saucepan on the map), and then go place the toy on the respective saucepan. The experimenter looked either to the floor or straight towards the child, to avoid cuing to one of the response locations. The children were allowed to come back and look at the map if they wanted. In the fragments trials, before the experimenter indicated the position of the target bucket, the two precut fragments were taken apart from the map, shuffled, and so placed in front of the children, with the residual of the map removed from view. The two fragments were presented aligned with each other such that their position did non reproduce the position of these elements in the whole configuration (figure one). The relative orientation of the map and the room was randomized by walking the kid around the display and stopping at various locations across trials, while the map was presented in a constant orientation with respect to the child. Children were given neutral feedback throughout the chore.

Data coding and analyses

The sessions were videotaped from a photographic camera built in the ceiling for later on coding. The videos showed the child transporting the toy in the large triangle, but the resolution was not sufficient to perceive the details on the map. Each session was replayed offline without the sound and the experimenter, who was blind to the location of the right response, coded the location where the kid left the toy in each trial.

For the purpose of the analyses, the trials were grouped co-ordinate to the type of relations that were available (full maps: bending, distance and length relations; angle fragments: angle relations only; length fragments: length relations just). A preliminary analysis indicated that gender had no effect on operation, and then this factor was dropped from further analyses. Accuracy data were start analyzed in an ANCOVA with 1 within-subject gene for condition and one covariate for historic period. Next, performance in each condition was compared to 50% chance level in a t-examination analysis. Finally, to compare performance across weather condition, planned comparisons were conducted beyond all 3 conditions using t tests (Bonferroni corrected).

Results

The ANCOVA revealed a meaning main effect of condition (F(2,lx)=4.2, p=0.020, η_p ²=0.12) (Figure 2A). Children performed to a higher place take a chance on the full map trials (72.9%, t(31)=six.iii, p<0.001), likewise as on the fragmented trials testing angle (63.3%, t(31)=2.5, p=0.019). However, they were at chance on the fragmented trials testing length (50.0%, t(31)=0.0, p=1.0), thus yielding a pregnant deviation between the total map and the length fragments trials (t(31)=2.eight, p=0.026 Bonferroni corrected). The operation on the angle fragments trials did not differ significantly either from the full map functioning or the length fragments functioning (p values > 0.xxx, Bonferroni corrected). Performance besides improved with historic period (F(one,30)=4.ii, p=0.049, η_p ⁱⁱ=0.12), only the factors of age and status did not collaborate (F(2,lx)<i).

Results of Experiment 1. For each condition (full map, angle fragments, length fragments) the functioning was tested against chance (50%) by ways of a t exam. Error confined represent the standard error of the mean.

Because the analyses of Experiment 1 were based on chiselled data whose conformity to parametric statistical assumptions are difficult to verify, nosotros performed the same analysis using a mixed model logistic regression. This technique is specifically designed for the analysis of categorical data, and is more than robust over variations in the distribution of data beyond conditions and subjects (Jaeger, 2008). The findings of this analysis, reported in the supplementary online material (see Appendix S1, available at XX_WILEY URL_XX), fully accorded with the findings reported hither.

To explore the effect of age in further detail, nosotros separated the children into ii subgroups of 16 participants, based on historic period (group ane: mean age 49.two months, range 47:15–50:30; group 2: hateful age 52.2 months, range 51:04–53:30) and tested functioning confronting risk in each condition for each subgroup. The older children succeeded on the full map trials (79.2%, t(15)=v.seven, p<0.001) and on the angle fragments trials (71.nine%, t(xv)=3.2, p=0.0058), but their performance did not reach significance on the length fragments trials (62.5%, t(15)=1.i, p=0.thirty). The younger children performed higher up chance when presented with the total map (68.8%, t(15)=3.5, p=0.0035), simply failed to solve either the length fragments trials (37.5%, t(15)=−1.2, p=0.26) or the angle fragments trials (54.vii%, t(15)=0.59, p=0.57). The departure in performance betwixt the 2 subgroups did not achieve significance in any of the weather condition, however (tdue south(thirty)<1.nine, ps>0.08). Every bit an analogy, scatterplots of the functioning past historic period are presented in the supplementary material (figure S1).

Discussion

Experiment one confirmed that iv-year-former children are able to read a consummate geometric map. Children performed above gamble when they were presented with the total maps, which redundantly combined cues of angle, length and distance. Withal, operation was more than mixed when children were tested with a partial map. As a group, participants were able to match 2-dimensional angle map fragments with the corresponding corners of a 3-dimensional triangle, thus providing evidence that by 4 ½ years of age, children tin compute abstract angles over 2-dimensional and 3-dimensional arrays. However, this power appeared fragile and variable beyond individuals, especially amidst the youngest children tested (effigy S1C). Lastly, the children failed to solve the task when the map fragments presented two different lengths.

The results of the length fragments condition were unexpected, given a previous report that even 3-twelvemonth-one-time children use length relations in consummate maps (Huttenlocher et al., 1999). Iii different explanations can be considered to account for this finding. First, peradventure some children failed to empathize our fragmentation manipulation altogether. Although almost all children understood the perforation in the third familiarization trial (27/32 children succeeded at the first attempt), this configuration may accept been intrinsically easier, because no portion of the figure was cut in the fragmentation. In contrast, to understand how the angle or length fragments related to the triangle construction, children needed to compare a whole blueprint with its isolated parts, a task that may exist difficult for iv-year-olds. This explanation may business relationship for the performance of those children who failed at both our fragments conditions, just a different caption is required for the children who succeeded with bending fragments.

Second, perhaps the children were misled by the stimuli used as length fragments, and failed to understand that the length of the pieces was related to the length of the sides of triangle. Indeed, in the fragmentation manipulation, the endpoints of the segments were cut off on purpose to avoid including angle cues, merely their absence could take unfortunately led the children to remember that the length of the pieces was not necessarily related to the length of the sides of the triangle. Afterward all, it is entirely possible to cut off a longer segment from a shorter side, and a shorter segment from a longer side.

Third, the children's failure at the length fragments trials may reveal essential limitations to their representation of length. Indeed, although children's ability to read length from maps has been clearly established for one-dimensional displays, reports accept been mixed for configurations extending in ii dimensions (Uttal, 1996; Vasilyeva & Huttenlocher, 2004). With two-dimensional displays, it is possible that children do non represent and compare length or distance ratios but simply use distance to parse the configuration into subgroups of close items, a strategy that was not bachelor in our task.

In the context of our experiment, the children's failure in the length fragments condition raises a question: Did children fail to compare 2-dimensional to 3-dimensional angles because they lacked an abstract representation of angles, or because they failed to sympathize the fragmentation manipulation birthday? To address this question, we first analyzed whether children's performance at the angle fragment trials correlated with their performance with the unlike full-map configurations. In the total map trials, length/distance cues conveyed less distinctive information in the right triangle configuration, compared to the acute triangle configuration, considering the length and altitude ratios were larger in the latter condition (length ratio between sides: right triangle 1.4, acute triangle 1.6; ratio of the distances between sides, measured at the middle point of each sides: right triangle one.ane, acute triangle one.8). Therefore, children's ability to read the full right triangle map should correlate with their ability to read angles, whereas they may succeed at the full acute triangle even if they cannot read bending cues. Consistently with this prediction, a multiple regression assay revealed a correlation between performance on the bending fragment trials and on the right-bending full map status (t=2.iii, p=0.030), whereas performance on the other total map weather did not contribute to the fit (ts<0.29, ps>0.78). Therefore, the variations in children'due south ability to solve the angle fragment status announced to reflect their individual competencies at reading angle from maps in general, rather than their agreement of map fragmentation.

To explore further the source of the variability in young 4-year-onetime children's generalization of angles across two- and 3-dimensional stimuli, we designed a new experiment, this time using color as a control condition rather than length.

Experiment 2

In this experiment, the length configuration was replaced by two configurations where the location of the right bucket was specified by colour. In both the bending and the colour fragment weather condition, nosotros cut round fragments of the map centered on each bucket: these 2 fragments differed from each other either in terms of angle (in the bending condition) or in terms of colour (in the color condition). If the failure of some children to read angle map fragments stems from a full general difficulty with fragmentation, these children should fail with colour map fragments as well. Accordingly, in this experiment we narrowed the age range to span but the younger end of the range used in Experiment ane, in an try to test more children who failed the fragmented angle map task.

Method

Participants

Sixteen children (mean age 49.15 months, 47:24–50:18, 8 females) participated in Experiment 2. They were recruited from the aforementioned population as in Experiment i. Three additional children were tested but excluded from the sample for video equipment failure (ii), or refusal to participate (1).

Stimuli, pattern and procedure

The procedure was identical to Experiment 1, except that the children were tested on four unlike configurations (iv trials each), for a total of 16 experimental trials. The order of presentation of the four blocks was controlled across children in a Latin square design. There were ii configurations testing the utilise of colour, and two configurations testing the use of bending (figure 3). The two angle configurations were identical to those used in Experiment 1: an acute isosceles triangle, and a right isosceles triangle. The commencement color configuration corresponded to an equilateral triangle (length of sides: 48×48×48 inches), in which one of the corners was painted in blue. Correspondingly, on the map ane of the corners of the equilateral triangle appeared in blueish. Two buckets were placed respectively in the blue corner and in 1 of the orange corners. In the second color configuration, the same three boards were placed as three sides of a square, making ii corners, one of which was painted in blueish. To isolate the color information in the map fragments, round sections were precut in the map around each corner with saucepan: the ii corners had a similar bending but i appeared entirely orange and the other entirely blueish. In the equilateral condition, the corners measured both threescore°, and in the square status, the corners measured xc°.

Configurations used in Experiment 2. The ii upper configurations test the use of angle relations, and are similar to the angle configurations of Experiment ane. The two lower configurations test the use of color. Equally in effigy ane, the dashed lines were non visible on the maps merely are added here to point the fragments presented in the fragmented status. Fragments are shown with the same orientation as when they were presented to the children.

Data analysis

In this experiment, the total map conditions were non comparable as they were in Experiment 1: whereas the children needed to nourish to geometric cues to interpret the total map in the bending configurations, they could only interpret the colored full maps by attending to color. Accordingly, in Experiment ii, we divers two within-subject area factors for configuration (color / angle) and fragmentation (full map / map fragments). Preliminary analyses showed that children's performance was not affected by either gender or historic period. Therefore, the data were analyzed using an ANOVA with two inside-subject factors for configuration and fragmentation, equally described above. Operation in each condition was tested confronting adventure (50%) past means of a t-test.

In order to exam whether failures with angle fragments could be explained in terms of a general difficulty of some children to understand map fragments, we used a linear regression assay to compare the performance beyond the two fragments conditions (angle and color) across subjects. Lastly, we also extracted a subgroup of children who clearly failed at the bending fragments condition (performance 50% or less), and tested whether this subgroup succeeded at the colour fragment status by ways of a t-test.

Results

We observed a significant effect of status (F(1,15)=8.6, p=0.0083, η_p ⁱⁱ=0.38), which was qualified by a interaction between condition and fragmentation (F(1,fifteen)=4.viii, p=0.045, η_p ⁱⁱ=0.24) (effigy 4). The functioning was identical in the full map conditions across the angle and the color configurations (81.3% vs 81.3%; t(15)=0.0, p=i), only children had more difficulties solving the angle fragments than the color fragments (64.1% vs 89.1%; t(15)=3.2, p=0.0064). All the same, children performed higher up hazard in all conditions (all p values <0.05). As for Experiment 1, analyses using a mixed model logistic regression yielded the same results (come across supplementary material).

Results of Experiment ii. The performance was tested against chance (l%) in each condition. Error bars represent the standard error of the mean.

Did the children who failed to solve the angle fragment trials nonetheless succeed on the colour fragment weather condition? Across subjects, at that place was no correlation between the functioning at the angle and colour fragments trials (F(1,xiv)=0.28, R²=0.02, p=0.60). At that place was nonetheless some variability in the performance for the angle fragments trials, since 9 of the 16 children tested performed at or beneath 50% in this status. When the analyses were restricted to this subgroup of children, performance at the colour fragments condition was still in a higher place adventure (83.3% correct, t(8)=3.0, p=0.016 – merely 1 child scored below 75%) and did not differ from the colour full map functioning (72.2% vs. 83.3%, t(8)=0.84, p=0.43).

Discussion

In Experiment 2, children succeeded at reading both full maps and fragmented maps, both when the maps were purely geometric maps and when they contained a non-geometric color cue. In the instance of the fragmented maps in particular, young children succeeded at mapping angle information from the partial ii-dimensional array to the full 3-dimensional layout, and at using this mapping to navigate past purely geometric bending information. These findings confirm that young four-twelvemonth-old children tin can sympathize our fragmentation manipulation, and that they can apply abstract angle in maps.

Fifty-fifty though iv-year-old children were able to read total geometric maps, their ability to read angles from fragments again was variable. The findings of Experiment two shed calorie-free on the source of children'south difficulty, revealing that it did not stem from a general misunderstanding of fragmentation in maps. First, there was no correlation between the children's performance at reading map fragments showing dissimilar angles, and their performance with another type of fragment, which provided not-geometric information (colour). Moreover, in Experiment 2 the children who failed to solve the angle fragment trials were nonetheless able to discover the right saucepan based on color map fragments, thus providing evidence that they understood the fragmentation procedure, and understood that the fragments still referred to the 3-dimensional layout. Third, and more than generally, for the whole sample included in Experiment 2, children were more than affected by the fragmentation in the bending condition than in the color condition: generalizing angles across the map fragments and the 3-dimensional wooden structure poses a bigger challenge to the children than generalizing color.

Opposite to the younger subgroup of children in Experiment 1, in Experiment ii the participants performed above chance on the angle fragment trials. More more often than not, when considering all the trials at the angle configurations together (the angle configurations were identical in both experiments), young children tended to perform better in Experiment two than in Experiment ane (72.7% vs 58.half-dozen%; effect of Experiment in an ANOVA with two factors for experiment and fragmentation: F(1,30)=3.9, p=0.058, η_p ²=0.12) – a trend that was present both with the map fragments (62.five% vs 54.7%) and the full map (81.three% vs 64.one%). Could the blueprint of this 2d experiment have contributed to the amend success of the children? Given that the improvement on angle configurations generalized to the full map trials, it seems unlikely that the improvement resulted but from a better understanding of fragmentation. Peradventure the easy color trials contributed to maintain a high motivation throughout the task. Peradventure also, the inclusion of configurations where the corners of the triangle were painted in different colors helped the children past directing their attention to the corners of the triangle, therefore leading them to notice how they sometimes differed in angle. Still, given that the performance divergence across experiments was but marginally meaning, these hypotheses should be regarded with caution. Farther enquiry is needed to sympathise the type of experience that can heave children's abilities to read angles in maps.

General analysis of the full-map and fragments angle conditions

The results of Experiments 1 and ii indicate that children'due south ability with angles is quite variable at the onset of their v^th twelvemonth of age. Beginning, in Experiment 1 the subgroup of the 16 oldest children succeeded reliably at reading map fragments, whereas the 16 youngest children did not. Second, the younger children'due south operation fluctuated beyond samples, every bit functioning was above chance in Experiment 2 but not Experiment ane.

To explore further the hypothesis that children's ability to read maps with angles improves during the offset one-half of their five^th twelvemonth, we pooled the data from the full map and fragmented angle conditions of the two experiments, as these conditions used the same geometric configurations (Figures i and ​ iii). Data from the 48 participants were sorted according to the participants' ages and and then smoothed by taking a running average over subgroups of sixteen children. Using these processed data, we tested (1) whether performance with angles improved with age, for each of the full-map and fragments status; (2) at which ages performance exceeded take a chance reliably – whether children were above chance beyond the whole historic period range, or only in the older end of the age range.

The data are presented in figure 5. Beginning, we tested for the presence of an age effect on these smoothed information using a linear regression betwixt the mean historic period and mean performance of the subgroups. For both the full map and the fragmented map angle weather, boilerplate functioning increased with the average age of the subgroups (R²s= 0.64 respectively 0.63, Fs(1,31)=55.3 respectively 51.7; psouth<0.001). Next, we tested whether the performance of each running subgroup exceeded chance by means of a t test (p<0.05). All subgroups performed reliably higher up take chances in the full map condition. In the fragmented condition, in contrast, performance exceeded chance only in the subgroups spanning the college end of the age range (mean age above 50 months).

Running average of functioning; information pooled across the angle weather condition of Experiments 1 and 2. The shaded expanse indicates the standard error of the mean. The bar on the lesser indicates the ages at which the performance of the subgroups exceeds chance level (t test, p=0.05; North=16 in each subgroup).

This analysis confirms that children's abilities with angles are frail effectually their 4^th altogether, despite their success at reading total maps. In addition, children'south performance increased with age on both the total-map and the fragments conditions. Both furnishings may exist related to the children's increasing power to read angles (correlation between the mean operation at the full-map and fragment conditions: R²=0.66, F(1,31)=61.v, p<0.001). By using angle data along with altitude and length information, older children may raise their map reading performance.

General discussion

Our results add together to electric current cognition of immature children'southward competence at reading maps in two ways. Beginning, preschoolers are able to read maps non only when they describe a whole figure, but as well when they represent but parts of an array. Second, children in their 5^thursday year of life tin can compare angles across 2- and 3-dimensional displays so every bit to use this information in their map reading. These two points shed light on preschooler's representations of bending, which must be abstract enough to encompass angle in spaces of different dimensions, presenting different geometric configurations (figures or isolated sectors), and made of different materials (2D drawings and 3D boards).

Could the children have succeeded in our task without representing angles – by using exclusively representations of distance and length? Our experimental procedure was designed so as to exclude this possibility. In the angle fragments condition, the children were presented with two circular fragments of the map, cut around two of the corners of the triangle. First, the length backdrop of these 2 fragments were equivalent, since the two circles cut from the map were on the aforementioned size (note that, since the size of the two fragments was identical and the fragments were shuffled before the experimenter indicated the target, children could also non solve the chore by remembering where in the full map each fragment came from). Moreover, the length of the two branches of each sector, which were identical, did non correspond to the length relations between the sides of the full triangle. On the other hand, since the two fragments cut were of identical size, for the larger angle, the endpoints of the two branches were further apart. This difference could not take served as a cue in our chore, however, equally the endpoints of the sectors were not marked on the 3D triangle, where the sides of the triangles were continuous. To use these cues, the children needed to normalize the altitude betwixt sides past the altitude from the corner – in brusk, they needed to compute angles.

Our experiments likewise reveal that individual 4-yr-old children vary in their ability to read angles from maps. In both Experiments ane and 2, fifty-fifty though the group performance was above gamble in the angle fragments condition, a considerable proportion of children performed at take chances level (50% accuracy or less, fourteen/32 children in Experiment 1, 9/16 children in Experiment 2). Experiment 2 was undertaken to better understand the reason for these failures. It revealed that all the children who failed with angle fragments were able to read map fragments with color information: thus, children's difficulty did not stem from the fragmentation procedure in itself. Moreover, Experiment 2 revealed a pregnant difference betwixt young 4-year-former children's competence with colour and with angle: even though our participants succeeded at using angle on the maps, they performed better with color. Both findings suggest that angle poses a particular challenge for young children, and that children's representation of abstruse angle undergoes a developmental change around their 4^thursday altogether.

This suggestion accords with numerous findings concerning young children's operation in other tasks. Although children and infants are sensitive to angles in 2-dimensional figures at ages younger than those tested here (Izard & Spelke, 2009; Slater et al., 1991; Younger & Gotlieb, 1988), young children take far greater difficulty encoding the angles formed by 3-dimensional arrays of surfaces (Hupbach & Nadel, 2005; Lee et al., 2012), be they large surfaces such as walls or smaller surfaces bundled on tabular array-top arrays. Two unlike tasks provide show that the ability to navigate past the angular relations betwixt surfaces emerges during the five^thursday yr of age: a reorientation job in a large enclosed arena whose walls were arranged in the shape of a rhomb, and a search job with a rhombic tabular array top brandish that rotated between trials (Hupbach & Nadel, 2005). These studies advise that before their 4^th birthday, children have a partial concept of bending, which encompasses only ii-dimensional figures and not angles formed by surfaces (Spelke, Lee, & Izard, 2010). As indicated by our findings, near children overcome this limitation and develop a more abstract concept of bending by the age of four ½.

Our suggestion of a developmental change raises the question of how children gain wider, more abstract representations of angles. On one hand, the children could learn to encode 3D surfaces in the same format as they encode 2d drawings, past encoding a project of these surfaces (akin to taking a footprint of a structure). Past doing and so, they would then be able to generalize the kind of analysis they apply to planar figures to a greater multifariousness of arrays. Alternatively, children might develop an power to encode angles from 3-dimensional displays directly, using the original format of representation they employ to encode other geometric properties of surfaces, and restructuring this representation to include angles.

A related question concerns whether the ability to perceive angles in two-dimensional figures fosters the evolution of the perception of angles in 3-dimensional displays. In particular, exposure to maps may heave the development of an integrated concept of space and of angles (Landau & Lakusta, 2009; Uttal, 2000). Past analyzing 2-dimensional angular relations in maps, children may either develop an abstract concept of angle on the way to understanding angles in 3D stimuli, or they may acquire this concept equally a 2nd step after learning to perceive three-dimensional angles.

Children's competence with angles eventually extends beyond that of comparing and matching forms and objects, when children start being able to reason about the abstract geometric backdrop of shapes. For example, Izard et al. (2011) recently presented children with the task of amalgam the missing angle from an incomplete triangle, and observed that five- and half-dozen-yr-former children from the US are deeply misled past a faulty heuristic. Instead of taking into account the base angles of the triangle, which fully determine the size of the third angle on a aeroplane, young children based their estimations exclusively on the length of the triangle base of operations, which is irrelevant to the third angle's size. Interestingly, this limitation was overcome in older children (aged 7–13 years) living either in an industrialized society (France) or in an indigene grouping from the Amazon (the Mundurucu).

In conclusion, at the historic period of four years, children are already able to generalize angles both from small planar maps to larger surface layouts, and from fragments to complete figures. The fragility of the youngest children'southward performance, and the limitations on older children's grasp of abstruse geometry, suggest that the preschool years are a pivotal time for the evolution of abstract geometrical concepts. Further studies of children should be undertaken to probe the nature of the changes in sensitivity to geometry in tasks of map-guided navigation and abstruse form analysis, in order to address primal questions apropos the nature and development of this organization of cognition.

Supplementary Material

Supp Appendix S1

Acknowledgments

The authors give thanks Danielle Hinchey and Christine Size for their help with recruiting participants and constructing stimuli; Susan Carey, Sang Ah Lee, Nathan Winkler-Rhoades and T.R. Virgil for helpful comments and discussions; as well as all the families who participated to this study. Supported by grants to ESS from NIH (HD23103) and NSF (0633955), and by a Starting Grant from the European Research Council to Half dozen (FP7 Project 263179 MathConstruction).

This research was supported grants to E.S.S. from NIH (HD23103) and from NSF (0633955) and by an ERC Starting Grant to V.I. (FP7 Project 263179 MathConstruction).

References

• Clements DH, Swaminathan S, Hannibal MAZ, Sarama J. Immature children's concepts of shape. Journal for Research in Mathematics Education. 1999;thirty:192–212. [Google Scholar]
• Cohen EH, Singh M. Geometric determinants of shape segmentation: Tests using segment identification. Vision Research. 2007;47:2825–2840. doi: http://dx.doi.org/ten.1016/j.visres.2007.06.021. [PubMed] [Google Scholar]
• Dehaene S, Izard Five, Pica P, Spelke ES. Core knowledge of geometry in an amazonian indigene group. Science. 2006;311:381–384. doi: 10.1126/scientific discipline.1121739. [PubMed] [Google Scholar]
• Heath TL. The thirteen books of euclid's elements. 2 ed. Dover Publications; New York: 1956. [Google Scholar]
• Hermer L, Spelke ES. A geometric process for spatial reorientation in immature children. Nature. 1994;370:57–59. doi: ten.1038/370057a0. [PubMed] [Google Scholar]
• Hupbach A, Nadel L. Reorientation in a rhombic environment: No evidence for an encapsulated geometric module. Cerebral Development. 2005;twenty:279–302. doi: 10.1016/j.cogdev.2005.04.003. [Google Scholar]
• Huttenlocher J, Newcombe N, Vasilyeva K. Spatial scaling in young children. Psychological Science. 1999;10:393–398. [Google Scholar]
• Huttenlocher J, Vasilyeva Yard, Newcombe Northward, Duffy Southward. Developing symbolic capacity one footstep at a time. Cognition. 2008;106:one–12. doi: 10.1016/j.cognition.2006.12.006. [PubMed] [Google Scholar]
• Izard 5, Pica P, Spelke ES, Dehaene S. Flexible intuitions of euclidean geometry in an amazonian indigene group. Proceedings of the National Academy of Sciences. 2011;108:9782–9787. doi: 10.1073/pnas.1016686108. [PMC complimentary article] [PubMed] [Google Scholar]
• Izard 5, Spelke ES. Development of sensitivity to geometry in visual forms. Human being Evolution. 2009;23:213–248. [PMC gratuitous article] [PubMed] [Google Scholar]
• Jaeger TF. Categorical data analysis: Away from anovas (transformation or not) and towards logit mixed models. Journal of Retentiveness and Linguistic communication. 2008;59:434–446. doi: 10.1016/j.jml.2007.eleven.007. [PMC free article] [PubMed] [Google Scholar]
• Landau B, Gleitman H, Spelke E. Spatial knowledge and geometric representation in a child blind from birth. Science. 1981;213:1275–1278. doi: x.1126/science.7268438. [PubMed] [Google Scholar]
• Landau B, Lakusta L. Spatial representation across species: Geometry, language, and maps. Electric current Stance in Neurobiology. 2009;19:12–19. doi: x.1016/j.conb.2009.02.001. [PMC gratis article] [PubMed] [Google Scholar]
• Landau B, Spelke East, Gleitman H. Spatial knowledge in a young blind child. Knowledge. 1984;16:225–260. doi: http://dx.doi.org/10.1016/0010-0277(84)90029-v. [PubMed] [Google Scholar]
• Lee SA, Sovrano VA, Spelke ES. Navigation every bit a source of geometric cognition: Young children's use of length, angle, distance, and direction in a reorientation task. Knowledge. 2012;123:144–161. doi: doi:x.1016/j.cognition.2011.12.015. [PMC gratis commodity] [PubMed] [Google Scholar]
• Lourenco SF, Huttenlocher J. The representation of geometric cues in infancy. Infancy. 2008;thirteen:103–127. doi: x.1080/15250000701795572. [Google Scholar]
• Morrongiello BA, Timney B, Humphrey GK, Anderson S, Skory C. Spatial knowledge in blind and sighted children. Periodical of Experimental Kid Psychology. 1995;59:211–233. doi: 10.1006/jecp.1995.1010. [PubMed] [Google Scholar]
• Satlow E, Newcombe N. When is a triangle not a triangle? Young children's developing concepts of geometric shape. Cognitive Evolution. 1998;13:547–559. [Google Scholar]
• Schwartz Chiliad, Day RH. Visual shape perception in early infancy. Monographs of the Lodge for Inquiry in Child Development. 1979;44(series No. 182) [PubMed] [Google Scholar]
• Shusterman A, Lee SA, Spelke ES. Young children'south spontaneous utilise of geometry in maps. Developmental Science. 2008;11:F1–vii. doi: ten.1111/j.1467-7687.2007.00670.x. [PMC free article] [PubMed] [Google Scholar]
• Shutts K, Örnkloo H, von Hofsten C, Keen R, Spelke ES. Young children'due south representations of spatial and functional relations betwixt objects. Child Development. 2009;lxxx:1612–1627. doi: x.1111/j.1467-8624.2009.01357.x. [PMC free commodity] [PubMed] [Google Scholar]
• Slater A, Mattock A, Dark-brown Due east, Bremner JG. Grade perception at birth: Cohen and younger (1984) revisited. Journal of Experimental Child Psychology. 1991;51:395–406. doi: 10.1016/0022-0965(91)90084-6. [PubMed] [Google Scholar]
• Spelke ES, Gilmore CK, McCarthy S. Kindergarten children'due south sensitivity to geometry in maps. Developmental Science. 2011;xiv:809–821. doi: 10.1111/j.1467-7687.2010.01029.ten. [PMC gratuitous commodity] [PubMed] [Google Scholar]
• Spelke ES, Lee SA, Izard V. Beyond core knowledge: Natural geometry. Cerebral Science. 2010;34:863–884. doi: 10.1111/j.1551-6709.2010.01110.x. [PMC free article] [PubMed] [Google Scholar]
• Uttal DH. Angles and distances: Children'due south and developed'due south reconstruction and scaling of spatial configurations. Child Development. 1996;67:2763–2779. doi: 10.1111/1467-8624.ep9706244832. [PubMed] [Google Scholar]
• Uttal DH. Seeing the big motion picture: Map use and the development of spatial cognition. Developmental Science. 2000;3:247–286. [Google Scholar]
• Vasilyeva Chiliad, Bowers E. Children'south employ of geometric data in mapping tasks. Journal of Experimental Child Psychology. 2006;95:255–277. doi: x.1016/j.jecp.2006.05.001. [PubMed] [Google Scholar]
• Vasilyeva M, Huttenlocher J. Early development of scaling ability. Developmental Psychology. 2004;twoscore:682–690. doi: 10.1037/0012-1649.40.v.682. [PubMed] [Google Scholar]
• Younger BA, Gotlieb Due south. Evolution of categorization skills: Changes in the nature or structure of baby course categories? Developmental Psychology. 1988;24:611–619. doi: 10.1037/0012-1649.24.5.611. [Google Scholar]

The Use of Geometry and Reading a Map

Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3751975/

Share this post