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The Construction of Symmetry in Children and Adults a
William A. Zingrone a
Murray State University Published online: 31 Jan 2014.
Click for updates To cite this article: William A. Zingrone (2014) The Construction of Symmetry in Children and Adults, The Journal of Genetic Psychology: Research and Theory on Human Development, 175:2, 91-104, DOI: 10.1080/00221325.2013.799058 To link to this article: http://dx.doi.org/10.1080/00221325.2013.799058
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THE JOURNAL OF GENETIC PSYCHOLOGY, 175(2), 91–104, 2014 C Taylor & Francis Group, LLC Copyright ISSN: 0022-1325 print / 1940-0896 online DOI: 10.1080/00221325.2013.799058
The Construction of Symmetry in Children and Adults William A. Zingrone
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Murray State University
ABSTRACT. The development of the concept of symmetry is important to an overall understanding of cognitive development in children and to spatial cognition in particular. Age differences in the construction of the 3 types of symmetry (bilateral, translational, and radial) were investigated in children and adults engaged in block construction. Children 2–4.5 years old produced bilateral symmetry in low frequencies independent of their precise vertical alignment of blocks. Children 4–12 years old and adults produced all 3 types of symmetry. The hypothesis predicting the sequence and frequency of the 3 types of symmetry based on an analysis of spatial complexity was partially supported. Bilateral symmetry was produced at significantly higher frequencies than the other 2 types across all age groups. Children 5–12 years old produced adult levels of bilateral symmetry while children 9–12 years old reached adult levels of construction of translational and radial symmetry. Keywords block construction, spatial cognition, symmetry
Symmetry is a pervasive feature of natural and artificial objects. Symmetric construction is prevalent in all architectural structures and is a ubiquitous component of human artifacts. Despite this fact and the presence of a large literature on symmetry in art, architecture, mathematics, and design, the development of symmetric construction abilities in children has remained largely uninvestigated. There has been no published data or analysis of this aspect of children’s spatial development, despite the importance of symmetry in our manufactured, constructed world and its importance to an understanding of children’s spatial and conceptual development. The present studies were designed to be the first in the developmental literature to systematically define and quantitatively examine children’s symmetric construction abilities. There are three types of symmetry: bilateral, translational, and radial (Weyl, 1952). A spatial configuration is bilaterally symmetric if when it is bisected points on one side of the midline can be mapped to spatially equivalent points on the other side. Bilateral or reflection symmetry is the most common of the three types of symmetry. Translational symmetry is the repetition of a pattern of equal elements interspersed by different equal elements or equal spaces along a given dimension, often along the horizontal. Radial or rotational symmetry repeats a pattern of equal elements or equal spaces around a central point. Examples of all three types of symmetry are depicted using standard wooden play blocks in Figure 1.
Received December 14, 2012; accepted March 4, 2013. Address correspondence to William A. Zingrone, Department of Psychology, 203 Wells Halls Murray State University, Murray, KY 42071-3318, USA; [email protected] (e-mail).
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FIGURE 1 The three types of symmetry. Front row: bilateral symmetries (bisection and reiterative [right to left]). Middle rows: translational symmetries. Back row: radial symmetries (color figure available online).
The perception of symmetry has been studied since the Gestalt psychology experiments of the late 1800s and early 1900s (Koffka, 1935; Mach, 1886/1959). Adults exhibit a preference for bilateral symmetry in visual perception (Julesz, 1971). Patterns and forms that exhibit bilateral, or mirror, symmetry of equivalent left and right sides are more easily detected, copied, and remembered than those displaying either asymmetry or other kinds of symmetry (Attneave, 1955; Julesz, 1971; Palmer & Hemenway, 1978; Wagemans, 1997, 2002). A preference for symmetry also has been documented in adult constructions. Adults unconsciously center and balance their designs when allowed to freely compose a visual display using geometric and natural shapes (Locher, Stappers, & Overbeeke, 1998). Similarly, when adults are asked to place objects in one-, two-, or three-dimensional arrays in an aesthetically pleasing manner perfect symmetry is the most common pattern produced (Szilagyi & Baird, 1977). Children’s ability to make symmetric compositions of their own has never been investigated, although children do prefer to look at symmetric designs as do adults. Daniels (1933) tested 2–5-year-olds on a symmetric arrangement preference task and found a significant preference for symmetry over asymmetry. Attention to symmetry begins early in life and continues throughout childhood and adulthood (Bornstein, Ferdinandsen, & Gross, 1981; Bornstein & Krinsky, 1985; Bornstein & Stiles-Davis, 1984; Fisher, Ferdinandsen, & Bornstein, 1981). By 5 years of age the structures children build out of blocks are multilayered, expansive arrangements that exhibit careful alignment, balance, and multiple examples of symmetric placement (Bayley, 1933; Gesell, 1940). Bayley observed an overall increase in the symmetry of
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children’s block building over the ages of 2–5 years, without reporting specific types of symmetry or their frequency. Langer (1986) described one 21-month-old child as composing bilateral symmetry by centering a cylindrical block on the middle of a horizontal rectangular slab, and Forman (1982) incidentally described instances of construction of bilateral symmetry by children 18–24 months of age. These few reports are representative of the sporadic and anecdotal literature on children’s block building, with Forman’s brief description of symmetry construction being the most current. Symmetric construction is an important component of the child’s spatial cognitive development. From the early exploratory behavior of infants who universally begin to manipulate and mouth objects at 4–5 months of age to understand their properties and affordances (Gibson, 1988), the 2-year-old advances beyond composition of objects in unstable arrangements (Langer, 1980) to deliberately building stabile constructions. Standard children’s blocks provide multiple affordances allowing the child to construct a variety of arrangements by matching surfaces and edges to each other. Using just two different blocks the child can arrange them in dozens of ways, with the vast majority being nonsymmetric. Affordance theory does not explain the onset or selection of symmetry in children’s building activities. Do children extend their perceptual preference for symmetry by choosing from among a nearly unlimited set of affordances to build symmetries of their own? A key research question is when and how often and in what forms does the ubiquitous symmetry of our constructed world begin to appear in the child’s developing ability to construct, and when does that ability equal that of adults?
The Spatial Complexity Hypothesis To develop a theoretical standpoint with which to investigate the developmental sequence of the different types of symmetry in children’s constructions, the three types, bilateral, translational, and radial, were analyzed in terms of increasing spatial complexity (Zingrone, 2010). The spatial complexity hypothesis proposes that children begin constructing less spatially complex symmetries before more complex ones and continuing building simpler symmetries with greater frequency throughout development. Complexity was defined as the number of elements (blocks in these studies) and spaces that would need to be attended to and manipulated to construct one of the three types of symmetry. More complex symmetries should present a greater task demand on the child. Symmetric arrangements consist of equal intervals that can be filled by equal elements such as blocks or by equal spaces. Block constructions using cubes and slabs were used to examine the complexity of each form of symmetry. Following Forman (1982), bilateral symmetry can be broken down into two subtypes: bisection and reiterative. Bisection symmetry is produced by placing a cube onto the middle of a horizontal slab: when two equal blocks such as two cubes are placed at the ends of a horizontal slab it is termed reiterative symmetry (see Figure 1). Bilateral symmetries seem to be the simplest of symmetries for young children to construct requiring the placement of just one or two blocks in a simple and perceptually salient arrangement. The production of translational symmetry is more complex than bilateral in two ways. Translational symmetry requires (a) placing more than one element and (b) creating equal intervals filled by equal blocks or equal spaces. To construct a repeating pattern such as the examples of translational symmetry depicted in Figure 1, all five blocks must be precisely positioned and adjusted and
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the spaces between them compared and made equal. Radial symmetry production demands even more of the child. In addition to the placement and comparison of multiple blocks, two sets of equal intervals must be created: (a) the equivalent intervals between elements and (b) the equal arrangement of each element around a central point. The perceptual properties of the three types of symmetry have been compared in previous research (Wagemans, 1997). Bilateral symmetry appears to be easiest to perceive as the axis of symmetry is aligned with the focal point of visual perception. The two halves of the bilaterally symmetric figure can be readily compared as a result. Translational and radial symmetries in contrast must be scanned visually along or around the figure to detect symmetry. Translational and radial symmetry then require additional effort in the form of deliberate attention and search to be perceived (Julesz, 1971). Bilateral symmetry is not only perceptually salient but according to the spatial complexity hypothesis it also is the simplest to construct. Accordingly, bilateral symmetry should be the first to appear in children’s constructions. The development of the construction of bilateral, translational and radial symmetry was investigated in two studies. The first study examined only bilateral symmetry in young children and the second investigated the construction of all three types of symmetry in children and adults. Study 1 was designed to quantify observations of bilaterally symmetric constructions reported by Forman (1982) and to examine his hypothesis explaining the onset of symmetric construction. Forman claimed that bisection symmetry resulted from an overgeneralization by children of their emerging knowledge of support, that is, children placed a block directly on the middle of another one for purposes of better support. Centering of one block on another was thought to be similar in intention to children’s precisely aligning the edges in a vertical stack of blocks, ostensibly to make the stack more stable against falling down. Children in Forman’s study were observed deliberately adjusting the edges of blocks in vertical stacks just prior to their first creations of bisection symmetry. Forman (1982) suggested, “Bisection symmetry most likely evolves from the need to balance vertical structures against the tangible pull of gravity. It is not implausible that the child centers the small block . . . because he anticipates its fall” (p. 125). Vertical precise alignment might not be performed to balance against the pull of gravity. Precise aligning of block edges may merely demonstrate a preference for neatly arranged blocks both in vertical and horizontal constructions. If children carefully align a horizontal row of blocks, it cannot be for support against gravity. Forman recorded only vertical precise alignment. Study one was designed to investigate his proposed relationship of vertical precise alignment and bisection symmetry in 2–4.5-year-old children by testing the following hypotheses: Hypothesis 1: Children would produce bisection symmetry independently of precise vertical alignment. That is, bisection symmetry would (a) occur before vertical precise alignment and (b) be made by children who do not also produce vertical precise alignment. Hypothesis 2: Children would produce bisection symmetry along with horizontal precise alignment, with or without also making vertical precise alignments. Hypothesis 3: Children would produce alignment similarly along both the vertical and horizontal dimensions that would undermine the idea that alignment is produced for better vertical support.
Study 2 then examined the sequence and frequency of all three types of symmetry in 3–12-yearold children and adults to determine if the specific predictions of the spatial complexity hypothesis
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were supported: that bilateral symmetry should appear first in development, translational second, and radial last, with bilateral remaining at the highest frequency through all age groups with adult levels of production on all three types being reached late in childhood.
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General Method Both studies employed a free block play procedure as it has been shown to elicit construction activity and demonstrate spatial competence in children (Case, 1985; Gesell, 1940; Stiles-Davis, 1988). Free block play has some distinct advantages over free drawing or copying. Children are able to construct spatial relations and arrange objects in ways that would be impossible to draw at younger ages. Children’s drawings are generally lacking in symmetry until the age of eight (Harris, 1963), and the copying of geometric shapes using exemplars by drawing is difficult for the child: symmetric shapes are not easily reproduced (Piaget & Inhelder, 1956). STUDY 1 METHOD Participants A total of 81 children were recruited for this study. Three were dropped due to lack of proper consent or video malfunction. Of the remaining 78, 60 participants were chosen at random to include 10 children each of ages 2 years, 2 years 6 months, 3 years, 3 years 6 months, 4 years, and 4 years 6 months (five boys and five girls were in each age group, excepting the 2-year-olds, which comprised eight boys and two girls). The children’s ages were within three months of the age group they were assigned to. Through pilot studies and a review of previous research it was determined not to include children under the age of two as many at that age do not construct with blocks. Children were recruited from private day care centers in Northern Illinois. Parental consent forms for participation and videotaping of the free block play study were obtained for each child. Stimulus Materials Twelve standard wooden play blocks including six cubes and six slabs finished in a natural color were presented in six different sets in two presentation orders. Presentation order 1 comprised Set 1 (one cube, one slab), Set 2 (six cubes), Set 3 (six slabs), Set 4 (one cube, two slabs), Set 5 (two cubes, one slab), and Set 6 (six cubes, six slabs). Presentation order 2 was Set 1, Set 4, Set 5, Set 2, Set 3, Set 6. The dimensions of the cubes were 3 × 3 × 3 cm, and the slabs were 3 × 9 × 1.5 cm. Procedure Data collection for this study was done at private day care facilities in a separate area away from the classroom with the child seated at a child-sized table. Participants were video- and audiorecorded. A warm-up activity was not included. Children were presented with each set of blocks dispersed randomly on the table, with the instructions “See what you can make with these.”
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Each presentation set lasted for 1 min or when four different constructions were produced, which was easily noted when the child moved their hands away or stopped arranging. The videotaped constructions were coded as one of the following three construction product types: vertical, horizontal, or combinations of vertical stacks and horizontal rows. Each of these products was coded for displaying bisection or reiterative symmetry and precise alignment, which was operationally defined as the display of flush edges in a construction. Nearly flush edges were also counted, allowing for variability in the child’s motor skill. The nine measures of construction products were (a) vertical without precise alignment, (b) horizontal without precise alignment, (c) combinations without precise alignment, (d) vertical with precise alignment, (e) horizontal with precise alignment, (f) combinations with precise alignment, (g) bisection symmetry, (h) reiterative symmetry, and (i) total constructions. A second coder (also blind to the hypotheses) recoded a random selection of 20% of the tapes. Reliability was excellent: agreement averaged 96.5% with a range of 85–100%. All discrepancies were discussed and corrected to 100% agreement (κ = 1).
RESULTS All nine measures of the construction products were summed across the six presentation sets for each child. Preliminary one-way analyses of variance (ANOVAs) on each of the nine measures with gender as a between-subjects factor revealed no sex differences in construction products, precise alignment or in the production of bisection or reiterative symmetry. These findings were in agreement with the previous free block play literature where no sex differences had been reported. To examine the relation between bisection symmetry and vertical precise alignment children across all age groups were grouped into four categories: (a) children who produced bisection symmetry with vertical precise alignment, (b) those who produced bisection symmetry without vertical precise alignment, (c) those who produced vertical precise alignment without bisection symmetry, and (d) those children who produced neither (see Table 1). The data were combined from all age groups as the youngest children produced bisection symmetry and vertical precise alignment either at very low frequencies or not at all. A 2 × 2 chi-square analysis yielded a significant association between bisection symmetry and vertical precise alignment with most children producing neither, χ 2(3, N = 60) = 31.33, p < .001. When children did produce bisection symmetry they were equally likely to produce it without vertical precise alignment as they were with it. These findings appear to undermine Forman’s (1982) proposed hypothesis wherein vertical precise alignment was said to mediate the production of bisection symmetry, TABLE 1 Number of Children Producing Bisection Symmetry and Vertical Precise Alignment Category
Vpa
No Vpa
Bisym No Bisym
12 4
11 33
Note. Bisym = bisection symmetry; Vpa = vertical precise alignment.
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TABLE 2 Number of Children Producing Bisection Symmetry and Vertical and Horizontal Precise Alignment, by Age Group Bisym Age (months)
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24 30 36 42 48 54
No Pa
w/Vpa
w/Hpa
w/Vpa/Hpa
2 0 0 0 1 2
0 0 0 0 2 1
0 0 3 3 0 0
0 0 1 1 4 3
Note. Bisym = bisection symmetry; Pa = precise alignment; Vpa = vertical precise alignment; Hpa = horizontal precise alignment.
as nearly 50% of the children produced bilateral symmetry without producing vertical precise alignment. Table 2 shows the number of children per age group who produced bisection symmetry and precise alignment. The two 24-month-olds who produced bisection symmetry did so without producing any vertical precise alignment in their constructions. In fact, no 24-month-old produced any vertical precise alignment. No 30-month-old children in the present study produced bisection symmetry while one produced vertical precise alignment and one produced horizontal precise alignment. At 36 months the dissociation of vertical precise alignment and bisection symmetry was even clearer. Of the four children who produced bisection symmetry at this age, three did so with only horizontal precise alignment evident in their constructions. The remaining child produced bisection symmetry with both horizontal and vertical precise alignment. At 42 months, four children produced bisection symmetry with the same pattern of results, three with horizontal precise alignment alone and one with vertical and horizontal. At 48 and 54 months, three children produced bisection symmetry with only vertical precise alignment, while 10 others in the same age groups produced bisection symmetry alone or with both horizontal and vertical precise alignment. To investigate the second hypothesis that there may be a relationship of bisection symmetry and horizontal precise alignment, children across all ages were grouped into four categories: (a) those who produced bisection symmetry with horizontal precise alignment, (b) those who produced bisection symmetry without horizontal precise alignment, (c) those who produced horizontal precise alignment but no bisection symmetry, and (d) those children who produced neither horizontal precise alignment nor bisection symmetry. The frequencies are shown in Table 3. A 2 × 2 chi-square analysis revealed a significant association of bisection symmetry and horizontal precise alignment, χ 2(3, N = 60) = 22.53, p < .001. When children produced bisection symmetry they were almost twice as likely to produce it with horizontal precise alignment as without. This association appears to undermine the idea that bisection symmetry is mediated by the knowledge of support evidenced in vertical precise alignment. Horizontal alignment is not done for support yet is associated with bisection symmetry. The third hypothesis proposed in this study predicted that children would produce horizontal precise alignment as frequently as vertical precise alignment. A one-way ANOVA of horizontal
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TABLE 3 Number of Children Producing Bisection Symmetry and Horizontal Precise Alignment Category
Hpa
No Hpa
Bisym No Bisym
15 7
8 30
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Note. Bisym = bisection symmetry; Hpa = horizontal precise alignment.
precise alignment with age as the between subjects factor was performed, yielding a significant effect of age, F(5, 54) = 3.49, MSE = 2.05, p < .05, η2 = .244. Children in the 42-month age group produced significantly more horizontal precise alignment than those of 24, 30, 48, and 54 months, Tukey’s HSD, p < .05. In contrast, the ANOVA of vertical precise alignment with age as the between subjects variable did not find significant age differences, F(5, 54) = 2.31, MSE = 0.93, p = .057, η2 = .176. To specifically test the hypothesis that horizontal precise alignment may be produced as frequently as vertical precise alignment a series of paired sample t-tests were performed to compare the means for vertical precise and horizontal precise alignment for each age group. Mean differences (Table 4) were not significant for any of the age groups (p > .05). Examination of the means suggests that children produce precise alignment both vertically and horizontally at the same time and with comparable frequency. Children show a similar tendency for aligning blocks in both dimensions across all age groups. These findings question whether precise alignment was performed for the purpose of better support in two ways: (a) the frequency was quite low and most constructions are made without precise alignment, and (b) horizontal precise alignment occurred at similar frequency as vertical precise alignment.
STUDY 2 To further investigate the development of the construction of symmetry, a second study examined the constructions of children and adults for all three types of symmetry; bilateral, translational TABLE 4 Mean Vertical and Horizontal Precise Alignments, by Age Vpa Age (months) 24 30 36 42 48 54
Hpa
M
SD
M
SD
0.00 0.10 0.90 0.40 1.20 0.60
0.00 0.32 1.52 0.97 1.23 0.84
0.00 0.20 1.00 2.30 0.40 0.40
0.00 0.63 1.25 3.02 0.52 0.97
Note. Vpa = vertical precise alignment; Hpa = horizontal precise alignment.
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and radial. Based on the spatial complexity hypothesis, (a) children should demonstrate a three step developmental sequence in the production of symmetry, with bilateral symmetry appearing first, followed by translational, with radial symmetry last; (b) frequencies of all three types of symmetry should increase to adult levels by late childhood; and (c) the frequency of the three types of symmetry should differ within age groups with bilateral being produced at highest frequencies by all age groups. Translational should be produced significantly less than bilateral and radial symmetry should be produced at significantly lower levels by all age groups.
METHOD
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Participants A total of 131 children and adults in six age groups (children 3–4, 5–6, 7–8, 9–10, and 11–12 years old and adults) were recruited. The sample included 19 children 3–4 years old (M age = 4 years 6 months, SD = 0.42 years; age range = 3 years 8 months to 4 years 10 months; nine girls and 10 boys), 26 children 5–6 years old (M age = 6 years 3 months, SD = 0 years; age range = 5 years 6 months to 6 years 10 months; 10 girls and 16 boys), 26 children 7–8 years old (M age = 7 years 11 months, SD = 0.60 years; age range = 7 years 8 months to 8 years 11 months; 13 girls and 13 boys), 26 children 9–10 years old (M age = 10 years, SD = 0.51 years; age range = 9 years 3 months to 10 years 11 months; 11 girls and 15 boys) 14 children 11–12 years old (M age = 11 years 5 months, SD = 0.16 years; age range = 11 years 2 months to 11 years 7 months; five girls and nine boys) and 20 adults (M age = 25 years 9 months, SD = 9.10 years; age range = 20–52 years; 11 women and nine men). Participants were recruited from preschools, elementary schools, and a large regional university. Both adults and children were given the same nondirective explanation that they were going to build whatever they liked with a series of blocks, with the adults told that their data was to be compared with what young children were able to produce. The adults were debriefed afterwards as to the investigation into symmetry.
Stimulus Materials The same 12 wooden play blocks used in Study 1 were presented (six cubes and six rectangular slabs). However, eight different sets drawn from the 12 blocks were presented to each child in two presentation orders. Presentation order 1 comprised Set 1 (one cube, one slab), Set 2 (six cubes), Set 3 (six slabs), Set 4 (one cube, two slabs), Set 5 (two cubes, one slab), Set 6 (four slabs, one cube), Set 7 (three slabs, two cubes), and Set 8 (six cubes, six slabs). Two new sets were added to the original six from Study 1. They were designed to promote construction of translational and radial symmetries. Set 6 was designed to elicit radial symmetry and Set 7 was designed to elicit translational symmetry. Presentation order 2 was Set 1, Set 7, Set 5, Set 3, Set 6, Set 2, Set 4, Set 8. The presentation orders began with the simplest set, just two blocks of one cube and one slab each, and ended with all 12 blocks. The purpose of this set order was to allow the younger children to get used to the task of putting blocks together and to leave the largest set as a reward at the end of 8–10 min of construction. The intervening sets were varied in the two presentations to control for effects of order. Presentation order 1 provided six
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cubes, then six slabs in Sets 2 and 3, respectively. These sets had enough blocks to allow for all three types of symmetry, whereas Sets 1, 4, and 5 only allowed for bilateral symmetries.
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Procedure Participants were tested in a quiet corner of their classroom or separate area of the school. Preschoolers were seated at low tables appropriate for their height. The block sets were presented one set at a time, for approximately 1 min or a total of four constructions. The last set, Set 8, using all 12 blocks, was limited to two constructions. Videotapes of each participant’s constructions were coded for construction product and symmetry type. Each construction was coded exclusively as displaying one the three types of symmetry, bilateral, translational, or radial. Two separate coders blind to the hypotheses of the study scored the videotapes. The primary coder scored all the videotapes and a random selection of 20% of the videotapes were rescored by a second coder. Agreement was excellent (κ = .86).
RESULTS Preliminary analyses examined the possible effects of gender and order on the free block play construction task. For each of these analyses six measures were computed: (a) the three construction product measures and (b) three symmetry measures. Totals for each of the measures were summed across all eight presentation sets for each participant. There was a significant effect of gender on combination constructions, t(129) = –2.57, p < .05, d = 0.45 (two-tailed), with girls building more combination constructions (M = 10.25, SD = 3.43) than boys (M = 8.60, SD = 3.87). There were no effects of gender on the other construction products and symmetry types and no effect of order on any of the measures. The analyses of the three hypotheses generated by the spatial complexity hypothesis were performed only on the sets of blocks that allowed for all three types to be produced. Set 1 for example, consisting of one cube and one slab was excluded as it allowed only for the construction of bilateral symmetry. Data from Sets 1, 4, and 5 were not included in these analyses. Only data from the five remaining presentation sets (2, 3, 6, 7, and 8) were included. Means of symmetry type by age are shown in Table 5. The 6 × 3 (Age × Symmetry) ANOVA yielded a main effect of age, F(5, 125) = 13.46, MSE = 25.80, p < .001, η2 = .04, and a main effect of symmetry, F(2, 125) = 797.47, MSE = 45.92, p < .001, η2 = .87. These main effects were qualified by a significant age by symmetry interaction, F(5, 125) = 9.80, MSE = 45.92, p < .001, η2 = .28 (see Figure 2). To examine differences in the construction of the three types of symmetry, for each age group the relative frequency of bilateral, translational, and radial symmetry were compared using Tukey’s HSD post hoc comparisons. Participants in all six age groups produced bilateral symmetry significantly more frequently than translational or radial symmetry; however, none of the age groups produced translational symmetry significantly more frequently than radial symmetry (3–4-year-olds: Tukey’s HSD = 3.61, p < .05; 5–6-year olds: Tukey’s HSD = 3.97, p < .05; 7–8-year olds: Tukey’s HSD = 4.23, p < .05; 9–10-year olds: Tukey’s HSD = 2.88, p < .05; 11–12-year olds: Tukey’s HSD = 5.02, p < .05; adults: Tukey’s HSD = 4.24, p < .05).
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TABLE 5 Mean Symmetry Production, by Age Group Bilateral Age (years)
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3–4 5–6 7–8 9–10 11–12 Adults
Translational
Radial
M
SD
M
SD
M
SD
10.21 20.46 23.73 23.31 28.57 26.85
5.14 8.89 9.93 6.65 8.55 8.99
0.58 1.46 1.31 2.15 3.07 2.50
1.02 2.23 1.44 1.59 2.16 1.70
0.21 0.58 0.23 0.42 0.64 1.20
0.42 0.86 0.43 1.14 1.45 1.24
The main effect of age and the age by symmetry interaction are consistent with Hypothesis 2, which predicted the frequency of all three types of symmetry would reach adult levels by late childhood. These effects were examined further with separate one-way ANOVAs with age as a between-subjects factor for each of the three types of symmetry. For bilateral symmetry, the one-way ANOVA yielded a significant effect of age, F(5, 125) = 11.50, MSE = 67.81, η2 = .32. Tukey’s HSD (7.41) revealed that the age groups 5–6, 7–8, 9–10, and 11–12 years old and adults all made significantly more bilateral symmetry than 3–4-year-olds (p < .05, d = 2.31), but did not differ significantly from each other. Thus, bilateral symmetry production reached adult levels as early as 5–6 years old. For translational symmetry, the ANOVA yielded a significant effect of age, F(5, 125) = 4.89, MSE = 3.00, η2 = .16. The age groups 9–10 and 11–12 years old and adults all made significantly more translational symmetry than the 3–4-year-olds (p < .05, Tukey’s HSD = 1.56, d = 1.57) and did not significantly differ from each other. Translational symmetry production then reached adult levels by 9–10 years old. For radial symmetry a significant effect
FIGURE 2
Mean symmetry type, by age group (color figure available online).
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of age was found, F(5, 125) = 3.00, MSE = 0.92, η2 = .11. Only the adults made significantly more radial symmetry than the 3–4-year-olds and 7–8-year-olds (Tukey’s HSD = 0.86, p < .05, d = 1.09). Adults, 9–10-year-olds, and 11–12-year-olds did not produce radial symmetry in significantly different frequencies from each other. Radial symmetry production then reached adult levels by age 9–10 years. Even though adults did produce nearly three times as much radial symmetry as 9–10-year-olds the frequencies were too small in the adults and older children to reach significance. These findings support the prediction that production of all three types of symmetry would reach adult levels by late childhood.
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GENERAL DISCUSSION Study 1 critically examined Forman’s (1982) hypothesis that bilateral symmetry was related to an emerging knowledge of support. The results did not support his hypothesis and may be more readily explained by the perceptual salience and spatial simplicity of bilateral constructions. In Study 2, as predicted, participants of all ages produced more bilateral symmetry than translational or radial symmetry. However, contrary to the prediction that older children and adults would produce translational symmetry more than radial symmetry, the frequency of translational and radial symmetry did not differ for any of the age groups. Based on the spatial complexity hypothesis, predictions were made about the relative frequencies of the three types of symmetry among children from preschool through adolescents and adults. In experiment two, the 3–4-year-olds made significantly more bilateral symmetry than translational or radial symmetry. All 3–4-year-olds made bilateral symmetry while only a third of these children made translational or radial symmetry. This supported Hypothesis 1, which predicted that bilateral symmetry would be produced first by the youngest children. The second hypothesis predicted that the frequencies of all three types of symmetry would reach adult levels by late childhood. By age 9–10 years, children were producing all three symmetries at frequencies not significantly different from adults. Finally, as predicted by Hypothesis 3, bilateral symmetry remained at significantly higher frequencies than translational and radial throughout all age groups. Translational and radial symmetries were produced at significantly lower levels than bilateral in all age groups but were not significantly different in their frequencies. Although the spatial complexity hypothesis predicted a three-step sequence of symmetry production in development, the results appear to support a two-step sequence. Both children and adults produced bilateral symmetry more frequently than translational or radial and the frequency of bilateral construction increased between age groups 3–4 years and 5–6 years. Translational and radial remained at comparable levels of production throughout development. Translational symmetry did not appear earlier and was not produced at higher frequencies than radial symmetry, contrary to the initial prediction. From the analysis of spatial complexity it was expected that the youngest children would make bilateral symmetry exclusively. A limitation of Study 1 was that only bilateral symmetry construction was examined in 2–4.5-year-olds. The other two types of symmetry were not investigated. A limitation of Study 2 was that the 3–4-year-old group averaged 4.5 years of age, ranging from 3.66 to 4.83 years old. To properly analyze the three-step sequence as proposed by the spatial complexity hypothesis, 2–4-year-olds should be investigated for all three types of symmetric production. An additional limitation of both studies was their being cross-sectional in
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design, and the developmental sequences observed would preferably be verified through followup longitudinal designs. Ideally, these results should be replicated not only in similar age groups from industrialized developed cultures, but in less developed countries from populations who do not live in a carpentered world full of highly symmetric manufactured buildings and objects in order to generalize as to the universality of these developmental sequences. These findings are an important addition to our knowledge of the spatial development of children as they are consistent with other important aspects of children’s developing spatialrepresentational skills. Willats’s (2005) theory explaining the development of children’s drawing behavior can also explain the sequence of symmetry construction observed in study two based on increasing spatial complexity. The development of drawing skills is proposed to be the result of the child acquiring increasingly complex internal descriptions of objects and their spatial relations, and not in getting better at reading off the features depicted in some image-like mental representation. Willats’s proposed that children at any given age have a set of only partial descriptions of the features and spatial relations of objects in their visual field. A description contains information about the features of objects and their spatial relations. This set of descriptions improves with age as the child assimilates more conceptual knowledge of objects and their relations into these descriptions. Three- and 4-year-olds draw people and objects with incomplete features and incorrect relations of those features as in the familiar tadpole drawings of people made by children of this age (Freeman, 1975). From the perspective of Willats’s theory the lack of spontaneous production of more complex symmetries does not reflect a deficiency of memory or lack of copying skill from a veridical image–like representation in the child’s head. Instead the child’s lack of production of more complex symmetries would belie an incomplete representational set of descriptions of those symmetries. The partial description of the elements and their spatial relations that define translational and radial symmetries might not be as complete as those for less complex bilateral symmetries in the 3–4- or 5–6–year-olds’ mind any more than their spatial descriptions would be complete for other objects or people. As children begin to make both subtypes of bilateral symmetry; bisection and reiterative from age 2 years on and although in low frequencies averaging 11% or less of total constructions, a question to be asked is, why do they begin to produce symmetry at all? The literature is clear on the perceptual salience of symmetric arrangements from infancy, with bilateral symmetry being easy to detect and remember. However, children begin making symmetry regularly and eventually the other two types of symmetry are produced often enough along with the bilateral that by late childhood and beyond, 60–80% of the constructions children and adults make are symmetric. The literature suggests a strong preference for symmetry in faces, architecture, art, and all manner of manufactured items, and it appears we prefer to construct it ourselves when given the chance. How did symmetry get to be such a significant part of the child’s spatial development? Answers to this question are beyond the scope of this initial investigation into symmetry construction, but warrant further investigation.
AUTHOR NOTE William A. Zingrone is a developmental psychologist and an assistant professor of psychology at Murray State University. He is interested in both cognitive development and cognitive evolution. His work has focused on the conceptual development of children, specifically with regards spatial
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concepts such as symmetry and equivalence. The cognitive evolution side of my research also looks at the first signs of symmetry and equivalence concepts in the archaeological record of stone tools, artifacts, and art.
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REFERENCES Attneave, F. (1955). Symmetry information and memory for patterns. American Journal of Psychology, 68, 209–222. Bayley, N. W. (1933). A scale of block constructions for young children. Child Development, 4, 121–140. Bornstein, M. H., Ferdinandsen, K., & Gross, C. G. (1981). Perception of symmetry in infancy. Developmental Psychology, 17, 82–86. Bornstein, M. H., & Krinsky, S. J. (1985). Perception of symmetry in infancy: The salience of vertical symmetry and the perception of pattern wholes. Journal of Experimental Child Psychology, 39, 1–9. Bornstein, M. H., & Stiles-Davis, J. (1984). Discrimination and memory for symmetry in young children. Child Development, 20, 637–649. Case, R. (1985). Intellectual development from birth to adulthood. Orlando, FL: Academic Press. Daniels, P. C. (1933). Discrimination of compositional balance at the pre-school level. In N. C. Meier & H. S. Langfield (Eds.), Studies in the psychology of art, psychological monographs (Vol. 45, pp. 1–11). Washington, DC: American Psychological Association. Fisher, C. B., Ferdinandsen, K., & Bornstein, M. H. (1981). The role of symmetry in infant form perception. Child Development, 52, 457–462. Forman, G. (1982). A search for the origins of equivalence concepts through a microanalysis of block play. In G. Forman (Ed.), Action and thought, from sensorimotor schemes to symbolic operations (pp. 97–135). New York, NY: Academic Press. Freeman, N. H. (1975). Do children draw men with arms coming out of their heads? Nature, 254, 416–417. Gesell, A. (1940). The first five years of life. New York, NY: Harper & Brothers. Gibson, E. J. (1988). Exploratory behavior in the development of perceiving, acting, and the acquiring of knowledge. Annual Review of Psychology, 39, 1–42. Harris, P. B. (1963). Children’s drawings as measures of intellectual activity. New York, NY: Harcourt and Brace. Julesz, B. (1971). Foundations of cyclopean perception. Chicago, IL: University of Chicago Press. Koffka, K. (1935). Principles of Gestalt psychology. New York, NY: Harcourt Brace. Langer, J. (1980). The origins of logic, six to twelve months. New York, NY: Academic Press. Langer, J. (1986). The origins of logic, one to two years. New York, NY: Academic Press. Locher, P. J., Stappers, P. J., & Overbeeke, K. (1998). The role of balance as an organizing design principle underlying adults’ compositional strategies for creating visual displays. Acta Psychologica, 99, 141–161. Mach, E. (1959). The analysis of sensations. Dover, England: Open Court. (Original work published 1886) Palmer, S. E., & Hemenway, K. (1978). Orientation and symmetry: Effects of multiple, rotational, and near symmetries. Journal of Experimental Psychology: Human Perception and Performance, 4, 691–702. Piaget, J., & Inhelder, B. (1956). The child’s conception of space. London, England: Routledge, Kegan Paul. Stiles-Davis, J. (1988). Developmental change in young children’s spatial grouping activity. Developmental Psychology, 24, 522–531. Szilagyi, P. G., & Baird, J. C. (1977). A quantitative approach to the study of visual symmetry. Perception and Psychophysics, 22, 287–292. Wagemans, J. (1997). Characteristics and models of human symmetry detection. Trends in Cognitive Sciences, 1, 346–352. Wagemans, J. (2002). Detection of visual symmetries. In C. W. Tyler (Ed.), Human symmetry perception and its computational analysis (pp. 25–48). Mahwah, NJ: Erlbaum. Weyl, H. (1952). Symmetry. Princeton, NJ: Princeton University Press. Willats, J. (2005). Making sense of children’s drawings. Mahwah, NJ: Erlbaum. Zingrone, W. A. (2010). Symmetry and spatial complexity in children’s constructions. Unpublished doctoral dissertation.
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The Construction of Symmetry in Children and Adults a
William A. Zingrone a
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Click for updates To cite this article: William A. Zingrone (2014) The Construction of Symmetry in Children and Adults, The Journal of Genetic Psychology: Research and Theory on Human Development, 175:2, 91-104, DOI: 10.1080/00221325.2013.799058 To link to this article: http://dx.doi.org/10.1080/00221325.2013.799058
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THE JOURNAL OF GENETIC PSYCHOLOGY, 175(2), 91–104, 2014 C Taylor & Francis Group, LLC Copyright ISSN: 0022-1325 print / 1940-0896 online DOI: 10.1080/00221325.2013.799058
The Construction of Symmetry in Children and Adults William A. Zingrone
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Murray State University
ABSTRACT. The development of the concept of symmetry is important to an overall understanding of cognitive development in children and to spatial cognition in particular. Age differences in the construction of the 3 types of symmetry (bilateral, translational, and radial) were investigated in children and adults engaged in block construction. Children 2–4.5 years old produced bilateral symmetry in low frequencies independent of their precise vertical alignment of blocks. Children 4–12 years old and adults produced all 3 types of symmetry. The hypothesis predicting the sequence and frequency of the 3 types of symmetry based on an analysis of spatial complexity was partially supported. Bilateral symmetry was produced at significantly higher frequencies than the other 2 types across all age groups. Children 5–12 years old produced adult levels of bilateral symmetry while children 9–12 years old reached adult levels of construction of translational and radial symmetry. Keywords block construction, spatial cognition, symmetry
Symmetry is a pervasive feature of natural and artificial objects. Symmetric construction is prevalent in all architectural structures and is a ubiquitous component of human artifacts. Despite this fact and the presence of a large literature on symmetry in art, architecture, mathematics, and design, the development of symmetric construction abilities in children has remained largely uninvestigated. There has been no published data or analysis of this aspect of children’s spatial development, despite the importance of symmetry in our manufactured, constructed world and its importance to an understanding of children’s spatial and conceptual development. The present studies were designed to be the first in the developmental literature to systematically define and quantitatively examine children’s symmetric construction abilities. There are three types of symmetry: bilateral, translational, and radial (Weyl, 1952). A spatial configuration is bilaterally symmetric if when it is bisected points on one side of the midline can be mapped to spatially equivalent points on the other side. Bilateral or reflection symmetry is the most common of the three types of symmetry. Translational symmetry is the repetition of a pattern of equal elements interspersed by different equal elements or equal spaces along a given dimension, often along the horizontal. Radial or rotational symmetry repeats a pattern of equal elements or equal spaces around a central point. Examples of all three types of symmetry are depicted using standard wooden play blocks in Figure 1.
Received December 14, 2012; accepted March 4, 2013. Address correspondence to William A. Zingrone, Department of Psychology, 203 Wells Halls Murray State University, Murray, KY 42071-3318, USA; [email protected] (e-mail).
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FIGURE 1 The three types of symmetry. Front row: bilateral symmetries (bisection and reiterative [right to left]). Middle rows: translational symmetries. Back row: radial symmetries (color figure available online).
The perception of symmetry has been studied since the Gestalt psychology experiments of the late 1800s and early 1900s (Koffka, 1935; Mach, 1886/1959). Adults exhibit a preference for bilateral symmetry in visual perception (Julesz, 1971). Patterns and forms that exhibit bilateral, or mirror, symmetry of equivalent left and right sides are more easily detected, copied, and remembered than those displaying either asymmetry or other kinds of symmetry (Attneave, 1955; Julesz, 1971; Palmer & Hemenway, 1978; Wagemans, 1997, 2002). A preference for symmetry also has been documented in adult constructions. Adults unconsciously center and balance their designs when allowed to freely compose a visual display using geometric and natural shapes (Locher, Stappers, & Overbeeke, 1998). Similarly, when adults are asked to place objects in one-, two-, or three-dimensional arrays in an aesthetically pleasing manner perfect symmetry is the most common pattern produced (Szilagyi & Baird, 1977). Children’s ability to make symmetric compositions of their own has never been investigated, although children do prefer to look at symmetric designs as do adults. Daniels (1933) tested 2–5-year-olds on a symmetric arrangement preference task and found a significant preference for symmetry over asymmetry. Attention to symmetry begins early in life and continues throughout childhood and adulthood (Bornstein, Ferdinandsen, & Gross, 1981; Bornstein & Krinsky, 1985; Bornstein & Stiles-Davis, 1984; Fisher, Ferdinandsen, & Bornstein, 1981). By 5 years of age the structures children build out of blocks are multilayered, expansive arrangements that exhibit careful alignment, balance, and multiple examples of symmetric placement (Bayley, 1933; Gesell, 1940). Bayley observed an overall increase in the symmetry of
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children’s block building over the ages of 2–5 years, without reporting specific types of symmetry or their frequency. Langer (1986) described one 21-month-old child as composing bilateral symmetry by centering a cylindrical block on the middle of a horizontal rectangular slab, and Forman (1982) incidentally described instances of construction of bilateral symmetry by children 18–24 months of age. These few reports are representative of the sporadic and anecdotal literature on children’s block building, with Forman’s brief description of symmetry construction being the most current. Symmetric construction is an important component of the child’s spatial cognitive development. From the early exploratory behavior of infants who universally begin to manipulate and mouth objects at 4–5 months of age to understand their properties and affordances (Gibson, 1988), the 2-year-old advances beyond composition of objects in unstable arrangements (Langer, 1980) to deliberately building stabile constructions. Standard children’s blocks provide multiple affordances allowing the child to construct a variety of arrangements by matching surfaces and edges to each other. Using just two different blocks the child can arrange them in dozens of ways, with the vast majority being nonsymmetric. Affordance theory does not explain the onset or selection of symmetry in children’s building activities. Do children extend their perceptual preference for symmetry by choosing from among a nearly unlimited set of affordances to build symmetries of their own? A key research question is when and how often and in what forms does the ubiquitous symmetry of our constructed world begin to appear in the child’s developing ability to construct, and when does that ability equal that of adults?
The Spatial Complexity Hypothesis To develop a theoretical standpoint with which to investigate the developmental sequence of the different types of symmetry in children’s constructions, the three types, bilateral, translational, and radial, were analyzed in terms of increasing spatial complexity (Zingrone, 2010). The spatial complexity hypothesis proposes that children begin constructing less spatially complex symmetries before more complex ones and continuing building simpler symmetries with greater frequency throughout development. Complexity was defined as the number of elements (blocks in these studies) and spaces that would need to be attended to and manipulated to construct one of the three types of symmetry. More complex symmetries should present a greater task demand on the child. Symmetric arrangements consist of equal intervals that can be filled by equal elements such as blocks or by equal spaces. Block constructions using cubes and slabs were used to examine the complexity of each form of symmetry. Following Forman (1982), bilateral symmetry can be broken down into two subtypes: bisection and reiterative. Bisection symmetry is produced by placing a cube onto the middle of a horizontal slab: when two equal blocks such as two cubes are placed at the ends of a horizontal slab it is termed reiterative symmetry (see Figure 1). Bilateral symmetries seem to be the simplest of symmetries for young children to construct requiring the placement of just one or two blocks in a simple and perceptually salient arrangement. The production of translational symmetry is more complex than bilateral in two ways. Translational symmetry requires (a) placing more than one element and (b) creating equal intervals filled by equal blocks or equal spaces. To construct a repeating pattern such as the examples of translational symmetry depicted in Figure 1, all five blocks must be precisely positioned and adjusted and
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the spaces between them compared and made equal. Radial symmetry production demands even more of the child. In addition to the placement and comparison of multiple blocks, two sets of equal intervals must be created: (a) the equivalent intervals between elements and (b) the equal arrangement of each element around a central point. The perceptual properties of the three types of symmetry have been compared in previous research (Wagemans, 1997). Bilateral symmetry appears to be easiest to perceive as the axis of symmetry is aligned with the focal point of visual perception. The two halves of the bilaterally symmetric figure can be readily compared as a result. Translational and radial symmetries in contrast must be scanned visually along or around the figure to detect symmetry. Translational and radial symmetry then require additional effort in the form of deliberate attention and search to be perceived (Julesz, 1971). Bilateral symmetry is not only perceptually salient but according to the spatial complexity hypothesis it also is the simplest to construct. Accordingly, bilateral symmetry should be the first to appear in children’s constructions. The development of the construction of bilateral, translational and radial symmetry was investigated in two studies. The first study examined only bilateral symmetry in young children and the second investigated the construction of all three types of symmetry in children and adults. Study 1 was designed to quantify observations of bilaterally symmetric constructions reported by Forman (1982) and to examine his hypothesis explaining the onset of symmetric construction. Forman claimed that bisection symmetry resulted from an overgeneralization by children of their emerging knowledge of support, that is, children placed a block directly on the middle of another one for purposes of better support. Centering of one block on another was thought to be similar in intention to children’s precisely aligning the edges in a vertical stack of blocks, ostensibly to make the stack more stable against falling down. Children in Forman’s study were observed deliberately adjusting the edges of blocks in vertical stacks just prior to their first creations of bisection symmetry. Forman (1982) suggested, “Bisection symmetry most likely evolves from the need to balance vertical structures against the tangible pull of gravity. It is not implausible that the child centers the small block . . . because he anticipates its fall” (p. 125). Vertical precise alignment might not be performed to balance against the pull of gravity. Precise aligning of block edges may merely demonstrate a preference for neatly arranged blocks both in vertical and horizontal constructions. If children carefully align a horizontal row of blocks, it cannot be for support against gravity. Forman recorded only vertical precise alignment. Study one was designed to investigate his proposed relationship of vertical precise alignment and bisection symmetry in 2–4.5-year-old children by testing the following hypotheses: Hypothesis 1: Children would produce bisection symmetry independently of precise vertical alignment. That is, bisection symmetry would (a) occur before vertical precise alignment and (b) be made by children who do not also produce vertical precise alignment. Hypothesis 2: Children would produce bisection symmetry along with horizontal precise alignment, with or without also making vertical precise alignments. Hypothesis 3: Children would produce alignment similarly along both the vertical and horizontal dimensions that would undermine the idea that alignment is produced for better vertical support.
Study 2 then examined the sequence and frequency of all three types of symmetry in 3–12-yearold children and adults to determine if the specific predictions of the spatial complexity hypothesis
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were supported: that bilateral symmetry should appear first in development, translational second, and radial last, with bilateral remaining at the highest frequency through all age groups with adult levels of production on all three types being reached late in childhood.
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General Method Both studies employed a free block play procedure as it has been shown to elicit construction activity and demonstrate spatial competence in children (Case, 1985; Gesell, 1940; Stiles-Davis, 1988). Free block play has some distinct advantages over free drawing or copying. Children are able to construct spatial relations and arrange objects in ways that would be impossible to draw at younger ages. Children’s drawings are generally lacking in symmetry until the age of eight (Harris, 1963), and the copying of geometric shapes using exemplars by drawing is difficult for the child: symmetric shapes are not easily reproduced (Piaget & Inhelder, 1956). STUDY 1 METHOD Participants A total of 81 children were recruited for this study. Three were dropped due to lack of proper consent or video malfunction. Of the remaining 78, 60 participants were chosen at random to include 10 children each of ages 2 years, 2 years 6 months, 3 years, 3 years 6 months, 4 years, and 4 years 6 months (five boys and five girls were in each age group, excepting the 2-year-olds, which comprised eight boys and two girls). The children’s ages were within three months of the age group they were assigned to. Through pilot studies and a review of previous research it was determined not to include children under the age of two as many at that age do not construct with blocks. Children were recruited from private day care centers in Northern Illinois. Parental consent forms for participation and videotaping of the free block play study were obtained for each child. Stimulus Materials Twelve standard wooden play blocks including six cubes and six slabs finished in a natural color were presented in six different sets in two presentation orders. Presentation order 1 comprised Set 1 (one cube, one slab), Set 2 (six cubes), Set 3 (six slabs), Set 4 (one cube, two slabs), Set 5 (two cubes, one slab), and Set 6 (six cubes, six slabs). Presentation order 2 was Set 1, Set 4, Set 5, Set 2, Set 3, Set 6. The dimensions of the cubes were 3 × 3 × 3 cm, and the slabs were 3 × 9 × 1.5 cm. Procedure Data collection for this study was done at private day care facilities in a separate area away from the classroom with the child seated at a child-sized table. Participants were video- and audiorecorded. A warm-up activity was not included. Children were presented with each set of blocks dispersed randomly on the table, with the instructions “See what you can make with these.”
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Each presentation set lasted for 1 min or when four different constructions were produced, which was easily noted when the child moved their hands away or stopped arranging. The videotaped constructions were coded as one of the following three construction product types: vertical, horizontal, or combinations of vertical stacks and horizontal rows. Each of these products was coded for displaying bisection or reiterative symmetry and precise alignment, which was operationally defined as the display of flush edges in a construction. Nearly flush edges were also counted, allowing for variability in the child’s motor skill. The nine measures of construction products were (a) vertical without precise alignment, (b) horizontal without precise alignment, (c) combinations without precise alignment, (d) vertical with precise alignment, (e) horizontal with precise alignment, (f) combinations with precise alignment, (g) bisection symmetry, (h) reiterative symmetry, and (i) total constructions. A second coder (also blind to the hypotheses) recoded a random selection of 20% of the tapes. Reliability was excellent: agreement averaged 96.5% with a range of 85–100%. All discrepancies were discussed and corrected to 100% agreement (κ = 1).
RESULTS All nine measures of the construction products were summed across the six presentation sets for each child. Preliminary one-way analyses of variance (ANOVAs) on each of the nine measures with gender as a between-subjects factor revealed no sex differences in construction products, precise alignment or in the production of bisection or reiterative symmetry. These findings were in agreement with the previous free block play literature where no sex differences had been reported. To examine the relation between bisection symmetry and vertical precise alignment children across all age groups were grouped into four categories: (a) children who produced bisection symmetry with vertical precise alignment, (b) those who produced bisection symmetry without vertical precise alignment, (c) those who produced vertical precise alignment without bisection symmetry, and (d) those children who produced neither (see Table 1). The data were combined from all age groups as the youngest children produced bisection symmetry and vertical precise alignment either at very low frequencies or not at all. A 2 × 2 chi-square analysis yielded a significant association between bisection symmetry and vertical precise alignment with most children producing neither, χ 2(3, N = 60) = 31.33, p < .001. When children did produce bisection symmetry they were equally likely to produce it without vertical precise alignment as they were with it. These findings appear to undermine Forman’s (1982) proposed hypothesis wherein vertical precise alignment was said to mediate the production of bisection symmetry, TABLE 1 Number of Children Producing Bisection Symmetry and Vertical Precise Alignment Category
Vpa
No Vpa
Bisym No Bisym
12 4
11 33
Note. Bisym = bisection symmetry; Vpa = vertical precise alignment.
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TABLE 2 Number of Children Producing Bisection Symmetry and Vertical and Horizontal Precise Alignment, by Age Group Bisym Age (months)
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24 30 36 42 48 54
No Pa
w/Vpa
w/Hpa
w/Vpa/Hpa
2 0 0 0 1 2
0 0 0 0 2 1
0 0 3 3 0 0
0 0 1 1 4 3
Note. Bisym = bisection symmetry; Pa = precise alignment; Vpa = vertical precise alignment; Hpa = horizontal precise alignment.
as nearly 50% of the children produced bilateral symmetry without producing vertical precise alignment. Table 2 shows the number of children per age group who produced bisection symmetry and precise alignment. The two 24-month-olds who produced bisection symmetry did so without producing any vertical precise alignment in their constructions. In fact, no 24-month-old produced any vertical precise alignment. No 30-month-old children in the present study produced bisection symmetry while one produced vertical precise alignment and one produced horizontal precise alignment. At 36 months the dissociation of vertical precise alignment and bisection symmetry was even clearer. Of the four children who produced bisection symmetry at this age, three did so with only horizontal precise alignment evident in their constructions. The remaining child produced bisection symmetry with both horizontal and vertical precise alignment. At 42 months, four children produced bisection symmetry with the same pattern of results, three with horizontal precise alignment alone and one with vertical and horizontal. At 48 and 54 months, three children produced bisection symmetry with only vertical precise alignment, while 10 others in the same age groups produced bisection symmetry alone or with both horizontal and vertical precise alignment. To investigate the second hypothesis that there may be a relationship of bisection symmetry and horizontal precise alignment, children across all ages were grouped into four categories: (a) those who produced bisection symmetry with horizontal precise alignment, (b) those who produced bisection symmetry without horizontal precise alignment, (c) those who produced horizontal precise alignment but no bisection symmetry, and (d) those children who produced neither horizontal precise alignment nor bisection symmetry. The frequencies are shown in Table 3. A 2 × 2 chi-square analysis revealed a significant association of bisection symmetry and horizontal precise alignment, χ 2(3, N = 60) = 22.53, p < .001. When children produced bisection symmetry they were almost twice as likely to produce it with horizontal precise alignment as without. This association appears to undermine the idea that bisection symmetry is mediated by the knowledge of support evidenced in vertical precise alignment. Horizontal alignment is not done for support yet is associated with bisection symmetry. The third hypothesis proposed in this study predicted that children would produce horizontal precise alignment as frequently as vertical precise alignment. A one-way ANOVA of horizontal
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TABLE 3 Number of Children Producing Bisection Symmetry and Horizontal Precise Alignment Category
Hpa
No Hpa
Bisym No Bisym
15 7
8 30
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Note. Bisym = bisection symmetry; Hpa = horizontal precise alignment.
precise alignment with age as the between subjects factor was performed, yielding a significant effect of age, F(5, 54) = 3.49, MSE = 2.05, p < .05, η2 = .244. Children in the 42-month age group produced significantly more horizontal precise alignment than those of 24, 30, 48, and 54 months, Tukey’s HSD, p < .05. In contrast, the ANOVA of vertical precise alignment with age as the between subjects variable did not find significant age differences, F(5, 54) = 2.31, MSE = 0.93, p = .057, η2 = .176. To specifically test the hypothesis that horizontal precise alignment may be produced as frequently as vertical precise alignment a series of paired sample t-tests were performed to compare the means for vertical precise and horizontal precise alignment for each age group. Mean differences (Table 4) were not significant for any of the age groups (p > .05). Examination of the means suggests that children produce precise alignment both vertically and horizontally at the same time and with comparable frequency. Children show a similar tendency for aligning blocks in both dimensions across all age groups. These findings question whether precise alignment was performed for the purpose of better support in two ways: (a) the frequency was quite low and most constructions are made without precise alignment, and (b) horizontal precise alignment occurred at similar frequency as vertical precise alignment.
STUDY 2 To further investigate the development of the construction of symmetry, a second study examined the constructions of children and adults for all three types of symmetry; bilateral, translational TABLE 4 Mean Vertical and Horizontal Precise Alignments, by Age Vpa Age (months) 24 30 36 42 48 54
Hpa
M
SD
M
SD
0.00 0.10 0.90 0.40 1.20 0.60
0.00 0.32 1.52 0.97 1.23 0.84
0.00 0.20 1.00 2.30 0.40 0.40
0.00 0.63 1.25 3.02 0.52 0.97
Note. Vpa = vertical precise alignment; Hpa = horizontal precise alignment.
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and radial. Based on the spatial complexity hypothesis, (a) children should demonstrate a three step developmental sequence in the production of symmetry, with bilateral symmetry appearing first, followed by translational, with radial symmetry last; (b) frequencies of all three types of symmetry should increase to adult levels by late childhood; and (c) the frequency of the three types of symmetry should differ within age groups with bilateral being produced at highest frequencies by all age groups. Translational should be produced significantly less than bilateral and radial symmetry should be produced at significantly lower levels by all age groups.
METHOD
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Participants A total of 131 children and adults in six age groups (children 3–4, 5–6, 7–8, 9–10, and 11–12 years old and adults) were recruited. The sample included 19 children 3–4 years old (M age = 4 years 6 months, SD = 0.42 years; age range = 3 years 8 months to 4 years 10 months; nine girls and 10 boys), 26 children 5–6 years old (M age = 6 years 3 months, SD = 0 years; age range = 5 years 6 months to 6 years 10 months; 10 girls and 16 boys), 26 children 7–8 years old (M age = 7 years 11 months, SD = 0.60 years; age range = 7 years 8 months to 8 years 11 months; 13 girls and 13 boys), 26 children 9–10 years old (M age = 10 years, SD = 0.51 years; age range = 9 years 3 months to 10 years 11 months; 11 girls and 15 boys) 14 children 11–12 years old (M age = 11 years 5 months, SD = 0.16 years; age range = 11 years 2 months to 11 years 7 months; five girls and nine boys) and 20 adults (M age = 25 years 9 months, SD = 9.10 years; age range = 20–52 years; 11 women and nine men). Participants were recruited from preschools, elementary schools, and a large regional university. Both adults and children were given the same nondirective explanation that they were going to build whatever they liked with a series of blocks, with the adults told that their data was to be compared with what young children were able to produce. The adults were debriefed afterwards as to the investigation into symmetry.
Stimulus Materials The same 12 wooden play blocks used in Study 1 were presented (six cubes and six rectangular slabs). However, eight different sets drawn from the 12 blocks were presented to each child in two presentation orders. Presentation order 1 comprised Set 1 (one cube, one slab), Set 2 (six cubes), Set 3 (six slabs), Set 4 (one cube, two slabs), Set 5 (two cubes, one slab), Set 6 (four slabs, one cube), Set 7 (three slabs, two cubes), and Set 8 (six cubes, six slabs). Two new sets were added to the original six from Study 1. They were designed to promote construction of translational and radial symmetries. Set 6 was designed to elicit radial symmetry and Set 7 was designed to elicit translational symmetry. Presentation order 2 was Set 1, Set 7, Set 5, Set 3, Set 6, Set 2, Set 4, Set 8. The presentation orders began with the simplest set, just two blocks of one cube and one slab each, and ended with all 12 blocks. The purpose of this set order was to allow the younger children to get used to the task of putting blocks together and to leave the largest set as a reward at the end of 8–10 min of construction. The intervening sets were varied in the two presentations to control for effects of order. Presentation order 1 provided six
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cubes, then six slabs in Sets 2 and 3, respectively. These sets had enough blocks to allow for all three types of symmetry, whereas Sets 1, 4, and 5 only allowed for bilateral symmetries.
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Procedure Participants were tested in a quiet corner of their classroom or separate area of the school. Preschoolers were seated at low tables appropriate for their height. The block sets were presented one set at a time, for approximately 1 min or a total of four constructions. The last set, Set 8, using all 12 blocks, was limited to two constructions. Videotapes of each participant’s constructions were coded for construction product and symmetry type. Each construction was coded exclusively as displaying one the three types of symmetry, bilateral, translational, or radial. Two separate coders blind to the hypotheses of the study scored the videotapes. The primary coder scored all the videotapes and a random selection of 20% of the videotapes were rescored by a second coder. Agreement was excellent (κ = .86).
RESULTS Preliminary analyses examined the possible effects of gender and order on the free block play construction task. For each of these analyses six measures were computed: (a) the three construction product measures and (b) three symmetry measures. Totals for each of the measures were summed across all eight presentation sets for each participant. There was a significant effect of gender on combination constructions, t(129) = –2.57, p < .05, d = 0.45 (two-tailed), with girls building more combination constructions (M = 10.25, SD = 3.43) than boys (M = 8.60, SD = 3.87). There were no effects of gender on the other construction products and symmetry types and no effect of order on any of the measures. The analyses of the three hypotheses generated by the spatial complexity hypothesis were performed only on the sets of blocks that allowed for all three types to be produced. Set 1 for example, consisting of one cube and one slab was excluded as it allowed only for the construction of bilateral symmetry. Data from Sets 1, 4, and 5 were not included in these analyses. Only data from the five remaining presentation sets (2, 3, 6, 7, and 8) were included. Means of symmetry type by age are shown in Table 5. The 6 × 3 (Age × Symmetry) ANOVA yielded a main effect of age, F(5, 125) = 13.46, MSE = 25.80, p < .001, η2 = .04, and a main effect of symmetry, F(2, 125) = 797.47, MSE = 45.92, p < .001, η2 = .87. These main effects were qualified by a significant age by symmetry interaction, F(5, 125) = 9.80, MSE = 45.92, p < .001, η2 = .28 (see Figure 2). To examine differences in the construction of the three types of symmetry, for each age group the relative frequency of bilateral, translational, and radial symmetry were compared using Tukey’s HSD post hoc comparisons. Participants in all six age groups produced bilateral symmetry significantly more frequently than translational or radial symmetry; however, none of the age groups produced translational symmetry significantly more frequently than radial symmetry (3–4-year-olds: Tukey’s HSD = 3.61, p < .05; 5–6-year olds: Tukey’s HSD = 3.97, p < .05; 7–8-year olds: Tukey’s HSD = 4.23, p < .05; 9–10-year olds: Tukey’s HSD = 2.88, p < .05; 11–12-year olds: Tukey’s HSD = 5.02, p < .05; adults: Tukey’s HSD = 4.24, p < .05).
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TABLE 5 Mean Symmetry Production, by Age Group Bilateral Age (years)
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3–4 5–6 7–8 9–10 11–12 Adults
Translational
Radial
M
SD
M
SD
M
SD
10.21 20.46 23.73 23.31 28.57 26.85
5.14 8.89 9.93 6.65 8.55 8.99
0.58 1.46 1.31 2.15 3.07 2.50
1.02 2.23 1.44 1.59 2.16 1.70
0.21 0.58 0.23 0.42 0.64 1.20
0.42 0.86 0.43 1.14 1.45 1.24
The main effect of age and the age by symmetry interaction are consistent with Hypothesis 2, which predicted the frequency of all three types of symmetry would reach adult levels by late childhood. These effects were examined further with separate one-way ANOVAs with age as a between-subjects factor for each of the three types of symmetry. For bilateral symmetry, the one-way ANOVA yielded a significant effect of age, F(5, 125) = 11.50, MSE = 67.81, η2 = .32. Tukey’s HSD (7.41) revealed that the age groups 5–6, 7–8, 9–10, and 11–12 years old and adults all made significantly more bilateral symmetry than 3–4-year-olds (p < .05, d = 2.31), but did not differ significantly from each other. Thus, bilateral symmetry production reached adult levels as early as 5–6 years old. For translational symmetry, the ANOVA yielded a significant effect of age, F(5, 125) = 4.89, MSE = 3.00, η2 = .16. The age groups 9–10 and 11–12 years old and adults all made significantly more translational symmetry than the 3–4-year-olds (p < .05, Tukey’s HSD = 1.56, d = 1.57) and did not significantly differ from each other. Translational symmetry production then reached adult levels by 9–10 years old. For radial symmetry a significant effect
FIGURE 2
Mean symmetry type, by age group (color figure available online).
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of age was found, F(5, 125) = 3.00, MSE = 0.92, η2 = .11. Only the adults made significantly more radial symmetry than the 3–4-year-olds and 7–8-year-olds (Tukey’s HSD = 0.86, p < .05, d = 1.09). Adults, 9–10-year-olds, and 11–12-year-olds did not produce radial symmetry in significantly different frequencies from each other. Radial symmetry production then reached adult levels by age 9–10 years. Even though adults did produce nearly three times as much radial symmetry as 9–10-year-olds the frequencies were too small in the adults and older children to reach significance. These findings support the prediction that production of all three types of symmetry would reach adult levels by late childhood.
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GENERAL DISCUSSION Study 1 critically examined Forman’s (1982) hypothesis that bilateral symmetry was related to an emerging knowledge of support. The results did not support his hypothesis and may be more readily explained by the perceptual salience and spatial simplicity of bilateral constructions. In Study 2, as predicted, participants of all ages produced more bilateral symmetry than translational or radial symmetry. However, contrary to the prediction that older children and adults would produce translational symmetry more than radial symmetry, the frequency of translational and radial symmetry did not differ for any of the age groups. Based on the spatial complexity hypothesis, predictions were made about the relative frequencies of the three types of symmetry among children from preschool through adolescents and adults. In experiment two, the 3–4-year-olds made significantly more bilateral symmetry than translational or radial symmetry. All 3–4-year-olds made bilateral symmetry while only a third of these children made translational or radial symmetry. This supported Hypothesis 1, which predicted that bilateral symmetry would be produced first by the youngest children. The second hypothesis predicted that the frequencies of all three types of symmetry would reach adult levels by late childhood. By age 9–10 years, children were producing all three symmetries at frequencies not significantly different from adults. Finally, as predicted by Hypothesis 3, bilateral symmetry remained at significantly higher frequencies than translational and radial throughout all age groups. Translational and radial symmetries were produced at significantly lower levels than bilateral in all age groups but were not significantly different in their frequencies. Although the spatial complexity hypothesis predicted a three-step sequence of symmetry production in development, the results appear to support a two-step sequence. Both children and adults produced bilateral symmetry more frequently than translational or radial and the frequency of bilateral construction increased between age groups 3–4 years and 5–6 years. Translational and radial remained at comparable levels of production throughout development. Translational symmetry did not appear earlier and was not produced at higher frequencies than radial symmetry, contrary to the initial prediction. From the analysis of spatial complexity it was expected that the youngest children would make bilateral symmetry exclusively. A limitation of Study 1 was that only bilateral symmetry construction was examined in 2–4.5-year-olds. The other two types of symmetry were not investigated. A limitation of Study 2 was that the 3–4-year-old group averaged 4.5 years of age, ranging from 3.66 to 4.83 years old. To properly analyze the three-step sequence as proposed by the spatial complexity hypothesis, 2–4-year-olds should be investigated for all three types of symmetric production. An additional limitation of both studies was their being cross-sectional in
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design, and the developmental sequences observed would preferably be verified through followup longitudinal designs. Ideally, these results should be replicated not only in similar age groups from industrialized developed cultures, but in less developed countries from populations who do not live in a carpentered world full of highly symmetric manufactured buildings and objects in order to generalize as to the universality of these developmental sequences. These findings are an important addition to our knowledge of the spatial development of children as they are consistent with other important aspects of children’s developing spatialrepresentational skills. Willats’s (2005) theory explaining the development of children’s drawing behavior can also explain the sequence of symmetry construction observed in study two based on increasing spatial complexity. The development of drawing skills is proposed to be the result of the child acquiring increasingly complex internal descriptions of objects and their spatial relations, and not in getting better at reading off the features depicted in some image-like mental representation. Willats’s proposed that children at any given age have a set of only partial descriptions of the features and spatial relations of objects in their visual field. A description contains information about the features of objects and their spatial relations. This set of descriptions improves with age as the child assimilates more conceptual knowledge of objects and their relations into these descriptions. Three- and 4-year-olds draw people and objects with incomplete features and incorrect relations of those features as in the familiar tadpole drawings of people made by children of this age (Freeman, 1975). From the perspective of Willats’s theory the lack of spontaneous production of more complex symmetries does not reflect a deficiency of memory or lack of copying skill from a veridical image–like representation in the child’s head. Instead the child’s lack of production of more complex symmetries would belie an incomplete representational set of descriptions of those symmetries. The partial description of the elements and their spatial relations that define translational and radial symmetries might not be as complete as those for less complex bilateral symmetries in the 3–4- or 5–6–year-olds’ mind any more than their spatial descriptions would be complete for other objects or people. As children begin to make both subtypes of bilateral symmetry; bisection and reiterative from age 2 years on and although in low frequencies averaging 11% or less of total constructions, a question to be asked is, why do they begin to produce symmetry at all? The literature is clear on the perceptual salience of symmetric arrangements from infancy, with bilateral symmetry being easy to detect and remember. However, children begin making symmetry regularly and eventually the other two types of symmetry are produced often enough along with the bilateral that by late childhood and beyond, 60–80% of the constructions children and adults make are symmetric. The literature suggests a strong preference for symmetry in faces, architecture, art, and all manner of manufactured items, and it appears we prefer to construct it ourselves when given the chance. How did symmetry get to be such a significant part of the child’s spatial development? Answers to this question are beyond the scope of this initial investigation into symmetry construction, but warrant further investigation.
AUTHOR NOTE William A. Zingrone is a developmental psychologist and an assistant professor of psychology at Murray State University. He is interested in both cognitive development and cognitive evolution. His work has focused on the conceptual development of children, specifically with regards spatial
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concepts such as symmetry and equivalence. The cognitive evolution side of my research also looks at the first signs of symmetry and equivalence concepts in the archaeological record of stone tools, artifacts, and art.
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