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Children’s Block Balancing 1
Running head: CHILDREN’S BLOCK BALANCING
Children's Block Balancing Revisited:
No Evidence for Representational Redescription
Horst Krist, Holger Horz, and Tina Schönfeld
Department of Psychology, University of Greifswald, Germany
April 11, 2005
Children’s Block Balancing 2
Abstract
According to the theory of representational redescription (RR theory, Karmiloff-Smith,
1992), children's reasoning in a microdomain is grounded on implicit knowledge. This initial
knowledge is first consolidated and then subjected to reiterative cycles of representational
redescription, leading to knowledge of increasing accessibility. One of the microdomains for
which evidence supporting RR theory has been reported is block balancing. This evidence
consisted in a U-shaped developmental trend regarding children's ability to balance asymmetrical
blocks. To assess whether this trend is a robust phenomenon reflecting spontaneous
developmental change, we investigated how 4-, 5-, 6-, and 8-year-olds (N=65) attempted to
balance symmetrical and asymmetrical blocks on a narrow support. Independent from block
type, we found quasi-linear improvements with age in all performance measures. These results
question the robustness of the U-shaped developmental trend regarding children's block
balancing and fail to provide any evidence for representational redescription.
Key words: conceptual development, intuitive physics, representational redescription, U-shaped
growth
Children’s Block Balancing 3
Children’s Block Balancing Revisited:
No Evidence for Representational Redescription
Representational Redescription Theory
One of the most influential modern theories of cognitive development is Karmiloff-
Smith’s (1992) theory of representational redescription (RR theory). In her theory, she
synthesises the constructivist tradition of Piagetian provenience and recent nativist accounts
originating from infancy research, mainly on infants’ physical reasoning abilities (e.g., Spelke,
1994). From the constructivist perspective, she adapts the central assumption that cognitive
development is an active process, and from the nativist perspective, she adapts the claim that the
infant is equipped with pre-adapted structures and reasoning capabilities constituting certain core
domains, such as an intuitive physics or an intuitive psychology. Karmiloff-Smith resolves the
tension that the combination of these contrasting perspectives creates essentially by introducing
two main ideas: The first idea is that children are born with predispositions to attend to and
process certain inputs. These predispositions serve as constraints that lead to an increasing
modularity of the cognitive system, a gradual process that Karmiloff-Smith calls modularization.
The second, somehow opposite idea is that early knowledge is implicit but becomes more and
more generally accessible with development. Increasing accessibility of initially implicit
knowledge is achieved by the active and reiterative process that gives RR theory its name:
representational redescription.
Representational redescription is active because it is (mainly) internally driven and it is
reiterative because it repeats itself on increasingly higher levels of representation. According to
RR theory, repeated success, rather than failure, is most important for cognitive growth. More
specifically, behavioral mastery marks the beginning of representational development rather than
Children’s Block Balancing 4
its endpoint. Success on the behavioral plane is possible without any explicit representation of
the how and why of the successful behavior. For humans, this is evidenced by perceptual-motor
skills like riding a bicycle or throwing a ball at a target (e.g., Krist, Fieberg, & Wilkening, 1993).
Karmiloff-Smith conceptualizes the know-how underlying such skills, as well as infants’ early
cognitive competencies, as procedural rather than declarative knowledge. Procedural knowledge,
or implicit knowledge, in Karmiloff-Smith’s sense can only be used by running the respective
procedure, e.g., by riding a bicycle or throwing a ball. It is not available to consciousness or
verbal report, and it cannot be combined with or be influenced by other pieces of knowledge—it
is thus encapsulated or „bracketed”. Once the implicit knowledge has been consolidated,
representational redescription may occur, which leads to a re-representation of the implicit
knowledge on an explicit level. Besides the level of implicit representations, called level I,
Karmiloff-Smith postulates three levels of explicit representations: E1, E2, and E3.
On level E1, representations are available to cognitive operations, such as comparison,
negation, or abstraction, but not yet to conscious access or verbal report. For example, pretend
play requires this level of representation. On level E2, representations are encoded in a format
that is available to conscious access but not yet for verbal report. Physical imagery, for example,
can be considered as operating on level-E2 representations (although Karmiloff-Smith states that
empirical evidence for level-E2 representations is hard to find). It is only on level E3 that
representations are encoded in a format similar to linguistic expressions, so that they become
available to verbal report and hence to operations across different domains, such as physics and
mathematics. According to Karmiloff-Smith, level-I representations are first redescribed into
level-E1 representations, which are later redesribed into level-E2 representations, and eventually
into level-E3 representations. As with the transition from level-I format to level E1, the
Children’s Block Balancing 5
transitions from E1 to E2 and from E2 to E3 become possible only after consolidation of the
lower-level representation.
Representational redescription as conceived by Karmiloff-Smith is a domain-general
process, i.e., it is not restricted to certain domains such as language or physics. However, the
onset and timing of this process is not only assumed to be specific to domains but even to
microdomains, e.g., acquiring past tense verb forms or balancing blocks. This means that
representational redescription may occur at different ages in different microdomains. In one
microdomain, a child may only have level-I representations, while, in another microdomain, she
may have acquired representations up to level E3, for example.
RR theory offers a unifying framework for understanding both the amazing, yet
piecemeal competencies of young infants and the protracted, domain-specific course of
development culminating in theory-like knowledge structures and the potential for creative and
scientific thinking. But how can the theory be validated, and what empirical evidence supports
it? Two types of evidence may be distinguished, one is relatively weak and the other is relatively
strong. The weaker evidence concerns results suggesting a dissociation between different kinds
of knowledge within the same individual or a developmental progression from context-specific
(nonverbal) knowledge to more generally accessible (verbal) knowledge. We will not deal with
this kind of evidence here (see, e.g., Alibali & Goldin-Meadow, 1993; Clements & Perner, 1994;
Krist et al., 1993). Stronger evidence comes from studies showing that children go beyond
behavioral mastery, especially if this results in U-shaped behavioral growth. RR theory explains
certain U-shaped growth functions, i.e., the temporary decline of performance, by positing that,
after having achieved behavioral mastery, children no longer focus on the external data but rather
on their internal representations (for other accounts of U-shaped behavioral change, see Strauss,
Children’s Block Balancing 6
1982; Siegler, 2004). On the one hand, this shift of attention enables representational change, and
on the other hand it can lead to new errors and inflexibilities.
Among the stronger pieces of evidence Karmiloff-Smith (1992) refers to in her book, one
stands out as being particularly compelling and illustrative. It concerns the development of
children’s block balancing skills and concepts. In this microdomain a U-shaped behavioral
growth pattern was observed by Karmiloff-Smith and Inhelder (1974) suggesting
representational redescription. Children first balanced both symmetrical and asymmetrical
objects successfully (on a narrow support), then exhibited systematic errors with asymmetrical
objects by trying to balance them at their geometric center, before they mastered both types of
balancing problems again. Karmiloff-Smith (1992) interprets this U-shaped behavioral sequence
as follows: In the first phase, children deal with each object as a separate item by applying
implicit knowledge. The level-I knowledge consists of sensorimotor routines for balancing
individual objects based on proprioceptive feedback. In the second phase, a level-E1
representation is abstracted from the level-I knowledge. This representation comprises a simple
rule that correctly applies to most objects but leads to systematic errors in the case of
asymmetrical objects. It is only in the third phase that these objects are treated correctly again. In
the process of redescribing the level-E1 representation into an E2 or E3 format, this is achieved
by considering the anomalous items as such, i.e., as exceptions to the rule or as demanding a
more complex rule.
The present study focuses on the balancing task introduced by Karmiloff-Smith and
Inhelder (1974). Therefore, before presenting the aims of our own study, a closer look will be
taken at the original work, followed by a short review of follow-up studies reported in the
literature.
Children’s Block Balancing 7
The Development of Children’s Knowledge About Block Balancing
The research conducted by Karmiloff-Smith and Inhelder (1974) on children’s block
balancing is a pioneering work for the Genevan school, if not for cognitive developmentalists as
a whole. It has still great appeal for leading theoreticians in the field, as exemplified by a recent
reference to it by Siegler (2004, p. 7). Given the great theoretical influence of Karmiloff-Smith
and Inhelder’s original work, it is striking both how preliminary this work has been and how few
follow-up studies have been conducted during the past three decades.
The preliminary and exploratory character of the original study is acknowledged by the
authors themselves. With respect to the lack of a standardized procedure, they write:
"Indeed, just as the child was constructing a theory-in-action in his endeavor to balance
blocks, so we, too, were making on-the-spot hypotheses about the child’s theories and providing
opportunities for negative and positive responses in order to verify our theories!” (Karmiloff-
Smith & Inhelder, 1974, p. 197).
The research strategy chosen was thus similar to Piaget's method of the clinical interview
(Ginsburg, 1997) with the important difference that children's actions were the primary focus of
inquiry, rather than their verbal responses. Probably due to the "clinical" nature of Karmiloff-
Smith and Inhelder's research strategy, neither procedural details nor quantitative results are
reported in their article. This caveat has to be kept in mind when trying to interpret their
observations in terms of the RR theory or any other theoretical account. Nevertheless, these
observations have told a rich and interesting story about children's learning and development, the
main themes of which will be summarized next.
In the main part of their study, Karmiloff-Smith asked 23 children between 4 and 9 years
of age to balance various blocks on a narrow bar "so that they do not fall". There were various
Children’s Block Balancing 8
types of blocks belonging to four different categories: length blocks, conspicuous weight blocks,
inconspicuous weight blocks, and blocks that were only balanceable by using counterweights. In
length blocks, the center of gravity coincided with the block's geometric center, i.e., the weight
was evenly distributed in these blocks. Conspicuous weight blocks were asymmetrical along
their length axis, i.e., their center of gravity was shifted towards the heavier side. The latter was
also true for inconspicuous weight blocks, due to invisible weights inserted into one end of the
otherwise symmetrical blocks.
Although Karmiloff-Smith and Inhelder do not give exact figures regarding the mean age
or time of occurrence within a session at which particular behaviors were observed, three levels
can be discerned that were later taken as indicative of level-I, level-E1, and level-E2/3
representations by Karmiloff-Smith (1992). On the first level, characteristic of 4- and 5-year-
olds, children were highly successful in balancing both length and weight blocks irrespective of
their type. They appeared to treat each balancing trial as a separate task by using proprioceptive
information to adjust the point of support until equilibrium was achieved. Corrections were
rarely made in the wrong direction, i.e., away from the center of gravity. On the second level,
characteristic of 6- and 7-year olds, children appeared to have generalized the naive rule that all
blocks balance at their geometric center, the so-called geometric-center theory. They now
systematically tried all blocks first at their geometric center and, with conspicuous and
inconspicuous weight blocks, sometimes made adjustments in the direction of the geometric
center even if this lead to greater imbalance. Often children abandoned their attempts to balance
a weight block altogether, after having tried only a small region around the geometric center, and
declared the block as impossible to balance. Interestingly, the same children were reported to be
able to balance inconspicuous weight blocks easily with their eyes closed relying on
Children’s Block Balancing 9
proprioceptive feedback only. The rigid application of the geometric-center theory (with eyes
open) actually led to a decline in the number of successful trials at the age of approximately 5.5
years. Older children, 8 and 9 years of age, typically performed on the highest level. They were
again successful with both length and weight blocks, assessing the weight distribution of each
block in advance by lifting it, and immediately placing it close to the correct point of support.
Surprisingly few studies have tried to replicate or extend the original observations by
Karmiloff-Smith and Inhelder. To the best of our knowledge, there is only one line of such
research: a series of studies conducted by Pine and Messer (Pine & Messer, 1998, 1999, 2000,
2003; Pine, Lufkin, & Messer, 2004). While the focus of most of their research has been on
instructional issues, they have also studied children's block balancing from the developmental
perspective of RR theory. Only the latter part of their research is considered here. Pine and
Messer adapted the balancing paradigm and the stimuli from Karmiloff-Smith and Inhelder
(1974) using a semi-standardized procedure and objective criteria for identifying different levels
of representation. Children were asked to balance different symmetrical and asymmetrical
blocks. Additionally, they were encouraged to give explanations about how each block balanced
or not. In some of the studies (Pine & Messer, 1999, 2003) children were also asked for
predictions of whether a block would balance or not. Based on children's number of successful
trials with symmetrical and asymmetrical blocks, their tendency to try to balance all blocks at
their geometric center, and their verbal responses, Pine and Messer defined six levels of
representation that could be reliably coded by independent raters. Besides two transition levels,
they distinguished the following levels: Implicit (I), Abstraction Nonverbal, Abstraction Verbal,
and Explicit (E3). While the I and E3 levels correspond with those postulated by Karmiloff-
Smith, no evidence for level E2 was found, and level E1 was replaced by the two Abstraction
Children’s Block Balancing 10
levels. As already noted by Karmiloff-Smith and Inhelder (1974), but not predicted by RR
theory, some children are able to verbalize their geometric-center theory (e.g., "All things
balance in the middle"). These children are classified, according to Pine and Messer's scheme, as
having reached the Abstraction Verbal level, whereas children who only exhibit this theory in
their actions are classified as still operating with representations on the Abstraction Nonverbal
level.
In an earlier cross-sectional study conducted with more than 168 children aged between 5
and 9 years, Pine and Messer (1999) categorized children's verbal and nonverbal behavior into
the levels of representation defined by Karmiloff-Smith (1992). They found that about every
second child exhibited the naive center strategy, being successful with the symmetrical but not
with the asymmetrical blocks, placing all blocks onto the support at their midpoint, and often
declaring asymmetrical blocks as impossible to balance. However, there was only mixed
evidence for a developmental trend along the lines suggested by RR theory. Most 4-year-olds'
behavior was unclassifiable, evidence for implicit knowledge was most frequent among the 7-
year-olds, and the geometric-center theory was found with about the same frequency in 5-
through 9-year-olds. Only the frequency of level-E3 classifications exhibited the predicted age
trend, beginning in the group of 6-year-olds and increasing through the group of 9-year-olds.
Using their own, modified classification system, Pine and Messer (2003) obtained at least
partial support for the sequential ordering of the postulated levels of representation in a
longitudinal study covering a 5-day period. Twenty-five 5- to 6-year-olds were given the
balancing task on each of five consecutive days. On the second, third, and fourth day, the
balancing task was part of a free play session in which the children were encouraged to talk
about what they were learning. Apart from one child, all children changed levels at least once. Of
Children’s Block Balancing 11
all transitions 88% were to a higher and 12% to a lower level. By the fifth day, 80% of the
children had moved to a higher level, 8% had stayed on the same level, and 12% had moved to a
lower level. Skipping of at least one level occurred frequently, in 44% of the children, but no
child progressed directly from level I to the Explicit level (E3).
The Present Research
Pine and Messer have presented some evidence in support of RR theory and have shed
new light on the development of children's block balancing. However, their work has left open at
least the following important issue: Do children spontaneously develop the geometric-center
theory and regress to a lower rate of success with asymmetrical or invisibly weighted objects? In
other words, is there evidence for representational redescription occurring (within the
microdomain of block balancing but) outside the laboratory? Karmiloff-Smith and Inhelder's
original study does not answer this question because of its exploratory character and the fact that
children had extensive opportunities to explore the blocks and practice the task. Pine and
Messer's research does not answer this question either because their results are contaminated by
possible effects of repeatedly asking the child for a verbal explanation or prediction.
In the present study, we therefore combined a detailed observation of children's block
balancing with a standardized procedure of data collection, uncontaminated by repeated verbal
questioning. To cover the age range, during which the presumed developmental changes should
occur, we assessed the block balancing behavior of children between 4 and 9 years of age in a
cross-sectional study. Using a subset of the blocks originally employed by Karmiloff-Smith and
Inhelder (1974), including an invisibly weighted block, we carefully analyzed children's
balancing behavior in two series of five trials. The following empirical hypotheses were derived
from Karmiloff-Smith and Inhelder’s findings and tested in the present study:
Children’s Block Balancing 12
(1) The rate of success should follow a U-shaped age trend with asymmetrical (or
invisibly weighted) blocks, whereas it should remain constant (at a high level) or increase with
age with symmetrical blocks. (2) The time needed for successful performance should exhibit an
inverted U-shaped age trend with the asymmetrical but not the symmetrical blocks. (3) An
inverted U-shaped age trend should also be obtained for the frequency at which asymmetrical
blocks are first tried at their midpoint. (4) The relative frequency of correct adjustments should
again follow a U-shaped age trend, as far as the asymmetrical blocks are concerned.
In terms of RR theory, each of these four non-monotonic age trends occurs because the
externally driven level-I representations become redescribed into the level-E1 format—the
geometric-center theory—thereby disregarding negative action outcomes as well as
proprioceptive feedback. This leads to a temporary decline in performance, which is later
reversed when children begin to redescribe their level-E1 knowledge into the more flexible,
better integrated, and more adaptive E2 or E3 format.
Method
Participants. Sixty-five children participated in this experiment. There were four age
groups: 4-year-olds (12 boys, 3 girls, mean age: 4 years, 5 months), 5-year-olds (7 boys, 11 girls,
mean age: 5 years, 5 months), 6-year-olds (6 boys, 10 girls, mean age: 6 years, 8 months), and 8-
year-olds (6 boys, 10 girls, mean age: 8 years, 5 months). The 4- and 5-year-olds were recruited
from two kindergartens and the 6- and 8-year-olds from two elementary schools in Greifswald,
Germany. All children participated on a voluntary basis and with the consent of their parents.
One 4-year-old child had to be replaced because she obviously did not understand the task.
Materials. The balancing task was adopted from Karmiloff-Smith and Inhelder (1974).
Five different wooden blocks were to be balanced on a narrow bar (25 cm long, 1 cm wide, 3.5
Children’s Block Balancing 13
cm high) mounted onto a board (Figure 1). There were two symmetrical blocks (A and B), two
asymmetrical blocks (C and D), and one inconspicuous weight block (E). The symmetrical
blocks balanced at their geometric center, whereas the other blocks balanced off-center (block C:
2.8 cm; block D: 5.2 cm; block E: 1.7 cm). Block A was a single beam (25 cm long, 5.5 cm
wide, 2.5 cm high). Block B was made by overlapping two flat beams of the same size as block
A (total length: 37.5 cm). The asymmetrical blocks C and D consisted of a beam (block C: 25 cm
long, 5.5 cm wide, 2.8 cm high; block D: 25 cm long, 6 cm wide, 1 cm high) onto which a block
(6 cm long, 5.5 cm wide, 6 cm high) was glued on one end. The inconspicuous weight block E
was a single beam (23.5 cm long, 9 cm wide, 2.8 cm high) into which a metal plate was inserted
at one end. Blocks A and B were colored red; block C was yellow; block D was blue; and block
E was green. To allow for accurate readings of the blocks' placement on the bar from the video
recordings, each block had a measuring scale attached to it consisting of alternating black and
white stripes (0.5 cm wide; see Figure 1).
Video analysis. Children's balancing behavior was videotaped using a digital camcorder
(Sony DSR-PD150P) mounted on a tripod opposite to the child. A first observer coded the video
data according to a predefined coding system and by means of the computer program "Interact"
(Mangold Software & Consulting GmbH). To evaluate the inter-observer reliability, a second
observer coded a subset of the data (all trials from 4 children). The coding system included the
following categories: (1) Success was defined as the successful balancing of a block. (2) A
midpoint placement was coded if a block was first placed onto the support close to its geometric
center (+/- 1.25 cm). (3) Adjustments were defined as any corrections of the block's position
along its length axis. They were coded as correct if children moved the block towards the point
of balance, otherwise they were coded as incorrect. Additionally, response times were calculated
Children’s Block Balancing 14
for successful trials from the video data. The response time was defined as the time interval
between the child's first touch and her final release of the block.
Inter-observer reliability was greater than κ = .93 for the success and midpoint categories
and greater than r = .98 for the number of correct and incorrect adjustments and the response
times.
Procedure. Each child was tested individually in a suitable room of her kindergarten or
school by one experimenter. An experimental session took about 5 to 15 minutes. The child was
seated at a table at a right angle to the experimenter. The balancing board was placed on the table
in front of the child so that the length axis of the fulcrum was perpendicular to the child's frontal
plane. At the beginning of an experimental session, only block A was visible, and the child was
instructed as follows: "Today we are going to play with blocks. Did you ever play with blocks?
It's a special game and a little bit tricky. I am going to give you several blocks one at a time.
Please, try to put each block on this rod so that it will not fall down. I am not going to help you.
If you can't do it the first time, that's not a problem. You can try it again if you want."
Demonstrating and commenting on the orientation at which each block had to be put onto the bar
(i.e., with the block's length axis perpendicular to the fulcrum), the experimenter then presented
block A without giving any feedback about its equilibrium position. The child was asked to try to
balance the five blocks in the order A, B, C, D, E in two series of trials. The blocks were
presented separately with the remaining blocks remaining out of view. If a child paused after
having failed to balance a block, she was encouraged to try again. The next block was presented
when (a) success had been achieved, (b) the child gave up, or (c) she did not change the block's
position for at least two minutes.
Children’s Block Balancing 15
Results
The values of the performance scores obtained by analyzing the video recordings were
submitted to 2 (Symmetry) x 2 (Series) x 4 (Age) ANOVAs with repeated measures on the first
two factors. As a part of the ANOVAs, polynomial contrasts were computed for the age factor,
to assess the hypotheses of (inverted) U-shaped age trends for asymmetrical and invisibly
weighted blocks. A U-shaped trend should be revealed by a large and significant quadratic trend.
Success scores. For each series of trials, success scores were computed by counting the
successful balancing trials for each type of block (symmetrical: A1+B1, A2+B2; asymmetrical:
C1+D1, C2+D2). Therefore, success scores ranged between 0 and 2.
An ANOVA performed on the success scores revealed a large and significant age effect,
F(3, 61) = 8.334, p < .001, η 2
= .291. Four-year-old children had the lowest success scores
(averaged over both series: Msym = 1.67, Masym = 1.00), followed by the group of 5-year-olds
(Msym = 1.83, Masym = 1.69), whereas the 6-year-olds (Msym = 2.00, Masym = 1.81) and 8-year-olds
(Msym = 1.97, Masym = 2.00) performed close to ceiling.
As illustrated in Figure 2, a large effect of symmetry, F(1, 61) = 14.364, p < .001, η 2
= .
191, and a significant interaction between symmetry and age, F(3, 61) = 5.091, p = .003, η 2
= .
200, were also obtained. Overall, symmetrical blocks were more often balanced successfully
than asymmetrical blocks. Older children, especially the 8-year-olds, performed close to ceiling
with both types of blocks, however. The interaction of age and symmetry was largely attributable
to different slopes of the linear age trend, with the slope being steeper for asymmetrical than for
symmetrical blocks, p = 0.01. Neither the quadratic component of the age by symmetry
interaction, p = .265, nor the quadratic trend calculated for the asymmetrical blocks alone were
reliably different from chance, p = .099.
Children’s Block Balancing 16
Response times. The times children needed to complete a balancing trial successfully
were analyzed analogously to the success scores. Where only one of the two trials per series and
block type was performed successfully, the mean was calculated as the value from the other trial.
Children who failed on both symmetrical or both asymmetrical trials within a series where
excluded from the analysis. Among the remaining children, there were 7 4-year-olds, 17 5-year-
olds, 15 6-year-olds, and 16 8-year-olds.
Whereas the age effect just missed the significance level, F(3, 51) = 2.747, p = .052, there
was a significant series effect, F(1, 51) = 5.877, p = .019, η 2
= .103, as well as a significant
interaction between series and symmetry, F(1, 51) = 4.959, p = .030, η 2
= .089 (see Figure 3).
Mean response times (adjusted for age) were almost identical for symmetrical and asymmetrical
blocks in the first series (M1st,sym = 20.93 s, M1st,asym = 18.80 s). From the first to the second series,
the mean response time decreased substantially for symmetrical blocks but not for asymmetrical
blocks (M2nd,sym = 10.06 s, M2nd,asym = 18.36 s).
Midpoint scores. As described in the Method section, the initial placement of each block
was assessed by analyzing the video recordings. Midpoint scores were calculated, analogously to
the success scores, by counting the frequency of trials in which a block was first placed with its
geometric center close to the support.
Figure 4 depicts the mean midpoint scores as a function of age, symmetry, and series. An
ANOVA performed on the midpoint scores revealed no significant age trend, F(3, 61) = 1.721, p
= .172. Symmetry exhibited a very large effect, F(1, 61) = 80.708, p < .001, η 2
= .570, and the
interaction between age and symmetry also proved significant, F(3, 61) = 5.097, p = .003, η 2
= .
200. Midpoint scores were positively correlated with age for symmetrical blocks (M = 1.63,
1.75, 1.78, and 1.91 for the 4-, 5-, 6-, and 8-year-olds, respectively) and negatively correlated
Children’s Block Balancing 17
with age for asymmetrical blocks (M = 1.30, .972, .844, and .50 for the 4-, 5-, 6-, and 8-year-
olds, respectively). There was no significant quadratic component of the age by symmetry
interaction, p = .950, and there was no significant quadratic trend for the asymmetrical blocks
alone either, p = .959. The significance criterion was just missed by the series factor, F(1, 61) =
3.767, p = .057, and the interaction between series and symmetry, F(1, 61) = 3.535, p = .065.
Number of adjustments and percentage of correct adjustments. In most trials, children
adjusted their initial placement of the block in their balancing attempts. The number of
adjustments as well as the percentage of correct adjustments was calculated for symmetrical and
asymmetrical blocks of each series. Children who did not adjust in two trials of the same type
within a series had to be excluded from the analysis of the percentage of correct adjustments.
This pertained to two 4-year-olds, one 5-year-old, two 6-year-olds, and three 8-year-olds.
Figure 5 summarizes the results for the number of adjustments and the percentage of
correct adjustments. ANOVAs performed on both dependent measures revealed a significant age
effect in both cases, F(3, 61) = 5.311, p = .003, η 2
= .207, and F(3, 53) = 4.819, p = .001, η 2
= .
214, respectively. Four-year-olds (M = 14.63) and 5-year-olds (M = 14.58) made nearly twice as
much adjustments as 6-year-olds (M = 7.53) and 8-year-olds (M = 7.31). The percentage of
correct adjustments also increased with age (M = 73.4%, 76.3%, 85.4%, and 86.1% for the 4-, 5-,
6-, and 8-year-olds, respectively). Symmetry also had a significant effect both on the number of
corrections, F(1, 61) = 29.893, p < .001, η 2
= .329, and the percentage of correct adjustments,
F(3, 53) = 14.193, p < .001, η 2
= .211. Averaged over age and series, children adjusted less often
with symmetrical than with asymmetrical blocks (Msym = 7.45, Masym = 15.59) and had a higher
rate of correct adjustments (Msym = 83.7%, Masym = 76.9%).
Children’s Block Balancing 18
Regarding the number of adjustments, there was a significant interaction between
symmetry and age, F(3, 61) = 3.718, p = .016, η 2
= .115. The symmetry effect was noticeably
larger in the groups of 4-year-olds (Msym = 11.17, Masym = 18.10) and 5-year-olds (Msym = 7.58,
Masym = 21.58) than in the groups of 6-year-olds (Msym = 5.13, Masym = 9.94) and 8-year-olds
(Msym = 5.91, Masym = 8.72). There was neither a significant quadratic component of this
interaction, p = .088, nor a significant quadratic trend for the asymmetrical blocks considered
alone, p = .386. No significant age by symmetry interaction was found with respect to the
percentage of correct adjustments, F(3, 53) = 1.406, p = .251.
There was a significant main effect of series on the number of adjustments, F(1, 61) =
9.107, p = .004, η 2
= .130. Adjustments were more frequent in the first series (M = 12.71) than in
the second series (M = 9.32).
Inconspicuous weight block. Trials with the invisibly weighted block were analyzed like
the other trials, except that there was no symmetry factor and only one trial per series. For the
latter reason, success and midpoint scores were analyzed by applying χ 2
-tests for each series
separately. From the ANOVAs and χ 2
-tests performed on the success scores, response times,
midpoint scores, number of adjustments and percentage of correct adjustments, only a single
effect emerged which was statistically reliable: The percentage of correct adjustments tended to
increase with age (M = 73.6%, M = 78.0%, M = 89.9%, M = 85.3%, for the 4-, 5-, 6-, and 8-
year-olds, respectively), F(3, 50) = 3.497, p = .022, η 2
= .173. The small quadratic component of
this trend did not point in the predicted direction and remained statistically insignificant, p = .
241.
Discussion
Children’s Block Balancing 19
The results of this study call into question the robustness of the U-shaped developmental
trend regarding children's block balancing as originally reported by Karmiloff-Smith and
Inhelder (1974). Where performance was not already close to ceiling, we observed significant
quasi-linear improvements from age group to age group in almost all analyses: The rate of
success and the percentage of correct adjustments tended to increase, while the time to success
and the number of adjustments tended to decrease with age. As expected, symmetrical blocks
were easier to balance than asymmetrical blocks. However, there was no indication of any U-
shaped age trend as predicted for the asymmetrical blocks and the invisibly weighted block.
Neither the rate of success nor the percentage of correct adjustments followed a U-shaped age
trend; and neither the time needed for successful performance nor the frequency at which these
blocks were first tried at their geometric center exhibited an inverted U-shaped age trend.
Furthermore, there was no age by symmetry interaction effect on these four dependent measures,
as far as the quadratic age component (indicative of a U-shaped trend) was concerned.
Taken together, our results are at odds with each of the predictions derived from
Karmiloff-Smith and Inhelder's (1974) findings and the interpretation of these findings in terms
of RR theory. In the present study, the geometric-center theory, i.e., the naive rule that all blocks
balance in the middle, was apparently not used more frequently in 5- or 6-year-olds than in the
other age groups. How can this conclusion be reconciled with Karmiloff-Smith's original
demonstration of a U-shaped developmental trend and the replication of this result in Pine and
Messer's research (Pine & Messer, 1999, 2003)? In principle, two possibilities have to be
considered: First, the previous findings and/or the present results may lack internal validity; and
second, the discrepancies may be due to procedural differences. Each possibility will be
considered in turn.
Children’s Block Balancing 20
Concerning the issue of internal validity, our own results appear to be the least
questionable. We have employed a highly standardized procedure combined with detailed video
analyses of different aspects of children's actions, and the sample size was large enough (N=65)
to detect moderate age trends in the predicted direction. By contrast, Karmiloff-Smith and
Inhelder (1974) used a "clinical" procedure with a small sample (N=23) and analyzed children's
actions on the spot, apparently without any specific coding system. However, it would be
premature to dismiss the original findings by simply criticizing them on methodological grounds.
After all, the comprehensive research published by Pine and Messer has lent substantial support
to the developmental trend claimed by Karmiloff-Smith and Inhelder. Although Pine and Messer
did not analyze U-shaped age trends directly, such trends are implied by their definition of
different levels of representation in conjunction with evidence suggesting that these levels
constitute a developmental sequence (see Introduction). We see no reason to seriously question
the internal validity of the studies conducted by Pine and Messer. Therefore, we tentatively
conclude that the discrepancy between our results and previous findings is real and thus must
originate from procedural differences between the studies.
Although only few procedural details are reported by Karmiloff-Smith and Inhelder
(1974), it is obvious that their study differed in many respects from the present one. The most
important differences appear to concern the amount of practice children were given and the order
of presentation of the blocks. In Karmiloff-Smith and Inhelder's study, children practiced the
balancing task extensively using a variety of blocks, and they were more or less free to choose
which block to use on each trial. In Pine and Messer's research, the amount of practice given
varied from study to study, but evidence for inflexible use of the geometric-center theory was
also obtained when the number of trials was comparable to the present study (e.g., Pine &
Children’s Block Balancing 21
Messer, 2000). Similarly to the original study, children were free to select which block to try
next, but were asked to deal with each block. In another respect, Pine and Messer's procedure
differed, however, from both the original and the present study. To be able to infer the level of
representation a child used, they repeatedly asked the child for explanations and, in some studies
(Pine & Messer, 1999, 2000, 2003) also for predictions, e.g., "Why do you think that one
wouldn't balance?" (Pine & Messer, 1999, p. 21), or "Do you think it can be balanced?" (Pine &
Messer, 2000, p. 41).
Whether the less constrained (and perhaps the simultaneous) presentation of the blocks,
extensive practice, or repeated verbal questioning, either alone or in combination, are sufficient
to induce a U-shaped age trend in one or the other dependent measure, can and should be
clarified in future research. At present, one can only speculate about the reason why many
children of about 5 to 6 years of age appear to apply the geometric-center theory in one
experimental setting but not the other. One possibility, particularly suggested by the difference
between the present study and Pine and Messer's research, is that children of this age tend to
elaborate the geometric-center theory or a similar rule if they are induced to think about the
question of why some blocks balance and some do not. Note that such a question is actually
misleading because it erroneously presupposes that there are blocks which cannot be balanced,
and it probably rarely occurred to the children investigated in our study. The question that was
more likely induced by our procedure was about how blocks balance. Thus, even if children used
or elaborated the rule that some blocks balance in the middle, in our case, they apparently did not
generalize this rule to asymmetrical blocks and not even to the invisibly weighted block. Had
they been encouraged to categorize blocks into balanceable and not balanceable ones, they might
have framed a rule like "blocks balance in the middle, or not at all." The application of such a
Children’s Block Balancing 22
rule (combined with a notion of symmetry) would lead to just the kind of inflexibilities and
errors reported in the previous studies, although it is different from the presumed geometric-
center theory which says that all blocks balance in the middle.
Independently from these still speculative considerations, the present results strongly
suggest that the geometric-center theory does not develop spontaneously outside the laboratory.
Theoretically, it is possible, that children do develop such a representation but do (a) not readily
or (b) not exclusively apply it in our task context. However, at least, the former of these two
possibilities does not seem very likely, because the present task was well-suited to activate a
representation of how blocks balance, if it existed prior to the experiment. In sum, we can
conclude that there appears to be no stage in the development of knowledge about block
balancing in which children adhere to the geometric-center theory in an inflexible and exclusive
way. Hence, we were not able to find evidence for representational redescription in the
microdomain studied.
What are the implications for RR theory? To be sure, a single experiment cannot refute a
complex theory. Yet, strong evidence for representational redescription is sparse, and the present
results have severely undermined one important piece of evidence in support of RR theory. To
the extent that evidence for representational redescription can only be found in a few
microdomains and, within these microdomains, only under limited circumstances, RR theory
looses its explanatory power and heuristic value. It appears, however, both premature and unwise
to give up RR theory right away. Giving up RR theory, would be premature because RR theory
does not comprise a fixed set of axioms but a few core assumptions which can be specified in
different ways and, of course, because more pertinent research is needed. Giving up RR theory
would also be unwise because alternatives are almost completely lacking. Karmiloff-Smith
Children’s Block Balancing 23
herself seems to view modern connectionism combined with developmental neuropsychology as
an advancement of RR theory (Elman, Bates, Johnson, Karmiloff-Smith, Parisi, & Plunkett,
1996; Karmiloff-Smith & Inhelder, 1992). We are not so sure about that. The brilliance and
power of connectionist simulations of developmental change notwithstanding, it is not yet clear
how they could give a principled account of different levels of knowledge and the process of
representational redescription. Particularly the idea that development is mainly driven by success
rather than failure runs counter to the fact that standard simulation networks stop learning once
success has been achieved.
Better alternatives to RR theory appear to be offered by approaches which acknowledge
different kinds of knowledge or processes but do not claim strong ontogenetic relations between
these kinds of knowledge or processes (e.g., Gershkoff-Stowe & Thelen, 2004). In particular, the
constructivist assumption that conceptual knowledge is derived from action knowledge is
questionable. If at all, conceptual generalization of action knowledge appears to be achieved only
indirectly, via perceptual and imagery processes concerning actions (Krist, 2003). Applied to the
balancing paradigm, this would mean that the geometric-center theory is not directly abstracted
from the procedural skill of balancing blocks based on proprioceptive and visual feedback. In
fact, it is not clear how a generalization like "all objects balance in the middle" could ever be
derived from a negative-feedback loop alone. It is more plausible that the geometric-center
theory or a similar rule is derived from an analysis of the outcomes of successful balancing
attempts (cf., Mandler, 1988, 1992). In this sense, children may indeed go beyond behavioral
mastery. Exploring under which conditions they actually do and in which cases this results in U-
shaped behavioral change, is worth further scrutiny.
Children’s Block Balancing 24
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Children’s Block Balancing 26
Author Note
We wish to thank Alexander Koch for assistance in data collection and coding.
Correspondence concerning this article should be addressed to Horst Krist, Institut für
Psychologie, Universität Greifswald, Franz-Mehring-Str. 47, D-17487 Greifswald, Germany. E-
mail: krist@uni-greifswald.de
Children’s Block Balancing 27
Figure Captions
Figure 1. Materials used in this study: Blocks A-E (left panel: front view; right panel: top view)
and balancing board (top panel, with block A).
Figure 2. Means and standard errors of the success scores achieved by 4-, 5-, 6-, and 8-year-old
children trying to balance symmetrical and asymmetrical blocks in two series of trials. The
maximum score is two.
Figure 3. Means and standard errors of the response times of 4-, 5-, 6-, and 8-year-old children
for successful attempts to balance symmetrical and asymmetrical blocks in two series of trials.
Figure 4. Means and standard errors of the midpoint scores of 4-, 5-, 6-, and 8-year-old children
trying to balance symmetrical and asymmetrical blocks in two series of trials. A score of one was
given for each trial in which the child first placed the block onto the support at its geometric
center. The maximum score is two.
Figure 5. Means and standard errors for number of adjustments (top panel) and percentage of
correct adjustments (bottom panel) observed with 4-, 5-, 6-, and 8-year-old children trying to
balance symmetrical and asymmetrical blocks in two series of trials.
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