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Keywords:

  • Cognitive Development;
  • Planning;
  • Tower of London;
  • Problem Structure

Abstract

  1. Top of page
  2. Abstract
  3. Background
  4. Materials and methods
  5. Results
  6. Discussion
  7. References

Planning ability gradually increases throughout childhood. However, it remains unknown whether this is attributable to global factors such as an increased ability and willingness to inhibit premature, impulsive responding, or due to the availability of specific planning operations, such as being able to mentally plan ahead more steps (‘search depth’) or to derive a clear temporal order of goals by the task layout (‘goal hierarchy’). Here, we studied the development of planning ability with respect to these global and problem-specific aspects (search depth and goal hierarchy) of performance in 178 children from 6 to 13 years using the Tower of London task. As expected, global performance gradually developed with age. In accordance, planning durations increasingly reflected global problem demands with longer pre-planning in harder problems. Furthermore, specific planning parameters revealed that children were increasingly capable of mentally searching ahead more steps. In contrast, the ability to derive a goal hierarchy did not show age-related changes. While the global development of planning performance and adaptive planning durations were proposed to primarily reflect enhanced self-monitoring, the specific increase in search depth across childhood that most likely proceeds until young adult age represents more directly planning-related processes. Thus, development of planning ability is supported by multiple contributions.


Background

  1. Top of page
  2. Abstract
  3. Background
  4. Materials and methods
  5. Results
  6. Discussion
  7. References

In many situations beyond everyday routine, the ability to plan ahead is essential for effective performance. As planning entails the mental modelling and anticipation of the consequences of actions prior to their execution in the real world (Goel & Grafman, 1995; Ward & Morris, 2005), it is among the most complex cognitive functions and is known to be tightly coupled to a functioning prefrontal cortex (Shallice, 1982; Unterrainer & Owen, 2006). Correspondingly, the emergence of mental planning is essential for school performance (Meltzer, 2007), and planning insufficiencies are also observed in developmental disabilities such as autistic spectrum disorders (Hill, 2004).

Only recently, the development of planning abilities across childhood has been studied with large sample sizes (N > 800; Albert & Steinberg, 2011; see also Korkman, Kemp, & Kirk, 2001). The main questions in these studies concerned (1) the age at which children or adolescents finally reach adult performance (see also Asato, Sweeney, & Luna, 2006; Huizinga, Dolan, & van der Molen, 2006) and (2) whether development proceeds gradually or, alternatively, whether specific age ranges can be identified where the most prominent changes in development take place (e.g., Klenberg, Korkman, & Lahti-Nuuttila, 2001; Korkman et al., 2001). In all these studies, planning ability was assessed with computerized versions of the Tower of London task (Shallice, 1982). The Tower of London disc-transfer paradigm requires to rearrange three coloured balls on three pegs from an initial state into a given goal state (Berg & Byrd, 2002) in an optimal manner, that is, using as few moves as possible.

In the developmental studies cited above, planning performance on the Tower of London was measured as the number of perfect solutions (i.e., trials solved in the minimum possible number of moves) across different difficulty levels defined by their minimal number of moves and ranging from three to seven moves (e.g., Albert & Steinberg, 2011). Results revealed significant gains in performance with age across childhood and the attainment of adult performance level for the hardest problems in the early 20s. Thus, summation of correctly solved trials across different levels of minimum number of moves captured developmental progress in planning performance, even though it remains a rather global performance indicator.

In the last decade behaviourial studies have, however, identified more specific indicators of planning ability that are inherent to the structure of Tower of London problems and have demonstrated their vast impact on problem difficulty (e.g., Kaller, Unterrainer, Rahm, & Halsband, 2004). These studies have mainly examined two parameters, goal hierarchy and search depth (Kaller, Rahm, Köstering, & Unterrainer, 2011; Figure 1) that draw on separable processes in the course of planning. Goal hierarchy refers to the ambiguity of subgoal orderings, that is, the degree to which the sequence of goal moves can be derived from the configuration of the goal state (e.g., Berg, Byrd, McNamara, & Case, 2010; Kaller et al., 2004; Klahr & Robinson, 1981; McKinlay et al., 2008; Newman & Pittman, 2007; Ward & Allport, 1997). Problems with ‘tower’ goal states, where all three balls are stacked on the tallest peg, provide an unambiguous goal hierarchy as the ball at the bottom clearly has to be in its goal position before the second from the bottom and so on. In contrast, in problems having ‘flat’ goal states, where there is one ball on each peg, the sequential prioritization of goal moves is completely ambiguous. Ambiguous goal states hence demand active identification of the correct sequence of goal moves, which may entail a broad search among alternative sequences.

image

Figure 1. Experimental design exemplified in five-move problems (cf. Köstering et al., 2012). In five-move Tower of London problems, two predominant structural patterns are evident leading to either medium or high demands on search depth with either one or two initial intermediate moves, respectively (cf. Kaller et al., 2011). Despite three different levels of goal hierarchy (unambiguous, partially ambiguous, and completely ambiguous), not all possible combinations with search depth exist in the Tower of London problem space. Circles around states denote goal moves.

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The second parameter, search depth, represents a core feature of planning, the anticipatory mental generation of move sequences taking into account interdependencies between moves (e.g., Kaller, Rahm, Spreer, Mader, & Unterrainer, 2008; Kaller et al., 2004, 2011). In tower transformation tasks, search depth is operationalized by the number of intermediate moves1 to be considered before execution of the first goal move (Borys, Spitz, & Dorans, 1982; Spitz, Webster, & Borys, 1982). Higher search depths hence require that a longer series of intermediate moves leading to the first goal move has to be identified under consideration of move interdependencies, thereby increasing the required number of mentally anticipated steps along the solution path.

Previous research in children has indicated that performance on problems with a high search depth undergoes substantial developmental change during late kindergarten age (Kaller et al., 2008; see also Klahr, 1985; McCormack & Atance, 2011). For instance, Kaller et al. (2008) showed that 4- and 5-year-old children mastered three-move problems equally well when each ball could be placed into its goal position directly. In sharp contrast, 4-year-olds’ accuracy declined specifically in three-move problems that imposed demands on search depth (Kaller et al., 2008).

However, previous developmental research on structural properties of the Tower of London task has focused mainly on early childhood and easy problems, whereas the development of these specific planning abilities during later childhood have remained a blind spot. This study is the first to examine the development of planning ability in the Tower of London task from childhood until the transition to adolescence (age 6–13) while taking into account both global performance and specific planning ability. Systematic manipulations of structural problem properties might disentangle the trajectories underlying developmental changes in specific planning abilities as these may progress gradually or exhibit phases of accelerated growth during specific time windows. Moreover, it may reveal whether gains in global performance are attributable to increased planning-specific abilities, as captured by search depth and goal hierarchy.

Materials and methods

  1. Top of page
  2. Abstract
  3. Background
  4. Materials and methods
  5. Results
  6. Discussion
  7. References

Participants

An overall sample of 178 children (6.2–13.9 years) with normal or corrected-to-normal vision participated after their parents had given written informed consent. The study protocol was approved by the local ethics committee. Present data were derived from two different experiments. In the first experiment (N = 106; 53 males, 53 females) participants were tested behaviourally only, whereas in the second experiment (N = 79; 48 males, 31 females) behaviourial testing was followed by neurophysiological and neuroanatomical measurements, which will not be reported in this study. None of the participants of the first experiment was enroled in the second experiment and vice versa.

Both samples completed the same Tower of London problems under comparable conditions (see also below) so that data from the two experiments were pooled for the present analyses. In Study 1, the children were recruited by word of mouth from various primary schools and thus contained a large variety of performance levels. Children of Study 2 were found via newspaper advertisement seeking participants for a larger project, including anatomical and functional neuroimaging, and multiple testing sessions. Thus, Study 2 may have posed a higher threshold that possibly was passed to a higher proportion by more motivated or more skilled children, potentially resulting in a selection bias. To detect potential differences between both experimental groups and sex differences, all the presented results were also analysed with the additional factors ‘study’ and ‘sex’. As these variables did not contribute significantly to any of the central findings, for reasons of clarity and readability, we report analyses without these factors.

The combined final sample was separated into four age groups: The first group comprised children from 6 to 7 years (group 6/7: n = 48, 24 girls; M = 7.0 years, SD = .42; range = 6.2–7.8 years), the second group from 8 to 9 years (group 8/9: n = 47, 20 girls; M = 8.9 years, SD = .55; range = 8.0–9.9 years), the third group from 10 to 11 years (group 10/11: n = 50, 23 girls; M = 10.9 years, SD = .54; range = 10.0–11.9 years), and the fourth group from 12 to 13 years (group 12/13: n = 33, 15 girls; M = 12.6 years, SD = .46; range = 12.0–13.9 years). This binning of age groups enables detection of non-linear developmental processes and was thus preferred to a continuous age variable using a linear regression approach.

Planning task, set-up, and instructions

Children were tested individually in a quiet room with a computerized three-ball version of the Tower of London task, developed in-house. The experimenter stayed in the room for the complete duration of the experiment. Start state and goal state were presented in the lower and upper half of the screen, respectively. Children were told to transform the start state into the goal state using a computer mouse while following three rules: (1) Only one ball may be moved at a time; (2) a ball cannot be moved while another is on top of it; and (3) three balls may be placed at maximum on the highest peg to the left, two balls on the middle peg, and one ball on the peg to the right. To match the goal state, children had to operate on the start state and the number of minimum moves was provided. Children moved the balls with a computer mouse using clicks to pick up and release a ball. After the child had finished a problem, the next problem was initiated by the experimenter by keyboard button press. After completing the last move required to achieve the goal state the subject received a verbal acoustic feedback indicating that the problem had been solved (cf. Kaller et al., 2008). The feedback's valence was always positive, irrespective of solution accuracy, to maintain the children's motivation throughout the experiment. Problems that had not been solved within the timeout limit of 1 min were automatically aborted (8.2% of the problems in the first and 7.1% in the second sample). If the time limit was exceeded on three consecutive problems, the task was automatically aborted. These problems were analysed as not perfectly solved. Before displaying the next problem, the instruction to plan ahead first was acoustically prompted by the programme.

To become familiar with the task and ensure comprehension of instructions and task rules, children started with four one-move (first sample only) and four two-move practice problems. Then – in ascending order – 8 three-, four-, and five-move problems followed. Children in the first sample were also presented with six-move problems not included here. The number, types, and order of the three- to five-move problems were absolutely identical for both samples.

Design and problem set

Overall, 24 three- to five-move problems (8 problems per difficulty level, respectively) were analysed with respect to global planning ability and manipulations of search depth and goal hierarchy. Problems comprised the subset of three- to five-move problems from the standard Tower of London problem set suggested by Kaller et al. (2011) presented in ascending order with respect to the minimum number of moves.

As performance and/or latency effects following experimental manipulations of goal hierarchy in simple three-move problems and more difficult four- and five-move problems may not be attributable to the same underlying cognitive operations (cf. Kaller et al., 2011), analyses of variance (assessing unambiguous to completely ambiguous states of goal hierarchy and low to high levels of search depth; see below) were separately computed for three-move problems on the one hand and four- and five-move problems on the other hand.

Search depth for three-move problems – as depicted in Table 1 – concerns whether an intermediate move is needed (medium search depth) or not (low search depth) to optimally solve a problem. In contrast to three-move problems, all four- and five-move problems inherently need at least one intermediate move. Thus, in four- and five-move problems, (1) high search depths are defined by sequences of one and two initial intermediate move(s), respectively, followed by three goal moves; and (2) medium search depths are defined by sequences of one goal move, one intermediate move, and two final goal moves in four-move problems, preceded by another intermediate move in five-move problems (Table 2). As a result, four-move problems feature search depths of either one or zero initial intermediate moves, and five-move problems of either one or two initial intermediate moves, respectively.

Table 1. Experimental design for three-move problems concerning the four resulting problem indices (P1–P4), search depth and its move patterns (0 = intermediate move, 1 = goal move), and goal hierarchy
Problem indexSearch depthMove patternGoal hierarchy
P1Low (no intermediate move)111Unambiguous
P2111Partially ambiguous
P3Medium (one intermediate move)011Partially ambiguous
P4011Completely ambiguous
Table 2. Experimental design for four- and five-move problems concerning the four resulting problem indices (P1–P4), search depth and its move patterns (0 = intermediate move, 1 = goal move), and goal hierarchy. Numbers in brackets provide the extension of four- to five-move problems
Problem indexSearch depthMove patternGoal hierarchy
P1High (one (two) initial intermediate move(s))(0)0111Unambiguous
P2(0)0111Partially ambiguous
P3Medium (zero (one) initial intermediate move(s))(0)1011Partially ambiguous
P4(0)1011Completely ambiguous

For goal hierarchy, a problem's goal state can be either unambiguous (tower state – all three balls are on the tallest peg), partially ambiguous (two balls are on one peg and one ball on another), or completely ambiguous (all balls are on a different peg). As the Tower of London's problem space does not allow for a full factorial combination of search depth and goal hierarchy, here we decided to disentangle the impact of problem structure in three separate analyses (cf. Kaller, Unterrainer, & Stahl, 2012; Kaller et al., 2004). Thus, independent of the minimum number of moves, analyses of low, medium, and high search depth were limited to problems with a partially ambiguous goal hierarchy so as to analyse variations in search depth independent of goal hierarchy (see cells P2/P3 in Tables 1 and 2; Figure 1). Likewise, to avoid confounds, analysis of unambiguous, partially ambiguous, and completely ambiguous goal hierarchies were performed at constant levels of search depth, resulting in two separate comparisons. For three-move problems, unambiguous versus partially ambiguous problems were compared at low search depth (P1/P2 in Table 1) and partially ambiguous versus completely ambiguous problems were contrasted at medium search depth (P3/P4 in Table 1). For four- and five-move problems, differences between unambiguous versus partially ambiguous problems were assessed at high search depth (P1/P2 in Table 2) and partially ambiguous versus completely ambiguous problems were compared at medium search depth (P3/P4 in Table 2).

Results

  1. Top of page
  2. Abstract
  3. Background
  4. Materials and methods
  5. Results
  6. Discussion
  7. References

Percentage of perfect solutions and initial thinking time were the measures of interest and were consequently entered as dependent variables in the analyses reported below. Perfect solutions concerned whether a problem was correctly solved in the minimum number of moves. Initial thinking time was defined as the time elapsed between the presentation of the problem and the onset of the first move. For initial thinking time analyses, both correctly and incorrectly solved problems were included.

Analyses were conducted in three steps: First, to replicate previous notions of a gradual performance increase across childhood, age effects were examined across the three difficulty levels defined by three- to five-move problems irrespective of specific problem parameters (global analyses). Second, for analysis of specific planning parameters, planning performance in three-move problems and four/five-move problems was analysed with respect to search depth and age. Finally, specific age-related changes in planning performance were analysed with respect to goal hierarchy in three- and four/five-move problems.

Global analyses of planning ability

Individual data, mean-aggregated across trials, of perfect solutions were entered into a 4 × 3 repeated measurements ANOVA with age group (6/7, 8/9, 10/11, and 12/13 years) as between-subjects factor, and minimum number of moves (three- to five-move problems) as within-subjects factor. Results revealed significant main effects for minimum number of moves, F(2, 348) = 568.65, p < .001, ηp2 = .766 (linear contrast p < .001), and age group, F(3, 174) = 14.72, p < .001, ηp2 = .202 (linear contrast p < .001). As depicted in Figure 2a, performance increased with age, whereas perfect solutions decreased from three- to five-move problems.

image

Figure 2. (a) Planning performance and (b) initial thinking times for three- to five-move problems and age groups, coloured from light to dark grey starting from the youngest to the oldest group, respectively.

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For mean-aggregated initial thinking times, a 4 × 3 repeated measurements ANOVA with age group as between-subjects and minimum number of moves as within-subjects factor revealed a significant main effect for minimum number of moves, F(2, 310) = 13.57, p < .001, ηp2 = .080, and an interaction of minimum number of moves and age group, F(6, 310) = 10.46, p < .001, ηp2 = .168. In contrast to the general increase in processing time from three- to five-move problems (linear contrast p < .001), the interaction results showed a reversed effect of difficulty level across age groups: The youngest children showed the longest initial thinking times in three-move problems compared with the older ones which was in contrast to five-move problems, where the older children spent more time planning than the younger ones (Figure 2b). Accordingly, evaluating effects of number of moves for each age group separately revealed significant effects in 6/7-year-olds, F(2, 76) = 4.88, p = .010, as well as in 10/11-, F(2, 92) = 8.88, p < .001, and 12/13-year-olds, F(2, 62) = 19.27, p < .001, but not in 8/9-year-olds, F(2, 80) = 0.61, p = .545. No main effect was found for age group, F(3, 155) = 1.54, p = .206, ηp2 = .029.

After mean-aggregating initial thinking times for three- to five-move problems, correlations with overall perfect solutions were moderate but significant (r = .29, p < .001). When repeating this analysis for the individual age groups, no significant associations could be found for the two younger groups (6/7-year-olds: r = −.055; p = .737; 8/9-year-olds: r = .252; p = .113), but for the older groups (10/11-year-olds: r = .394; p = .006; 12/13-year-olds: r = .429; p = .014). Hence, better performance was associated with longer initial thinking times, a phenomenon that increased with age as moderating factor.

Specific analyses of planning ability

For an overview, descriptives of all specific analyses are provided in Table 3.

Table 3. Descriptives of specific planning analyses. Mean and standard errors (SEM) for perfect solutions (Per. Sol) and initial thinking times (ITT; in seconds) separated for minimum number of three-, four-, and five-move problems (Min. move), age groups, as well as for low, medium, and high search depth (SD) and unambiguous (Unambig), partially ambiguous (Par. Ambig), and completely ambiguous (Com. Ambig) goal hierarchy (GH)
Min. moveAge 6–7Age 8–9Age 10–11Age 12–13
Per. SolITTPer. SolITTPer. SolITTPer. SolITT
MeanMeanMeanMeanMeanMeanMeanMean
SEM SEM SEM SEM SEM SEM SEM SEM
Search depths
Three
SD86.55.593.64.394.03.695.53.4
Low 2.7 0.2 2.8 0.2 2.7 0.2 3.3 0.3
SD71.97.278.75.489.05.598. 54.4
Medium 3.9 0.4 3.9 0.4 3.8 0.4 4.7 0.5
Four
SD76.04.680.93.996.03.796.93.9
Medium 3.6 0.3 3.6 0.3 3.5 0.2 4.3 0.3
SD42.75.161.75.161.05.556.15.1
High 4.6 0.5 4.6 0.5 4.5 0.5 5.5 0.6
Five
SD37.55.443.64.951.07.065.29.1
Medium 4.8 0.9 4.8 0.9 4.7 0.8 5.8 1.0
SD3.14.717.06.213.07.227.39.8
High 3.3 1.0 3.4 1.0 3.3 0.9 4.0 1.1
Goal hierarchy
Three
GH85.45.591.54.294.03.693.93.5
Unambig 3.2 0.2 3.2 0.2 3.1 0.2 3.8 0.3
GH86.55.693.64.394.03.695.53.4
Par. ambig 2.7 0.2 2.8 0.2 2.7 0.2 3.3 0.3
Three
GH71.97.378.75.489.05.598.54.4
Par. ambig 3.9 0.4 3.9 .4 3.8 .4 4.8 0.5
GH84.46.384.05.386.04.992.44.4
Com. ambig 3.9 0.3 3.9 0.3 3.8 0.3 4.7 0.4
Four
GH83.35.282.95.284.05.083.34.7
Unambig 4.0 0.3 4.0 0.3 3.9 0.3 4.8 0.4
GH42.75.361.75.361.05.556.15.1
Par. ambig 4.6 0.5 4.6 0.5 4.5 0.5 5.5 0.6
Five
GH51.05.865.95.661.06.265.28.1
Unambig 4.1 0.6 4.2 0.6 4.0 0.6 5.0 0.7
GH3.14.717.06.213.07.227.39.8
Par. ambig 3.3 1.0 3.4 1.0 3.3 0.9 4.0 1.1
Four
GH76.04.480.94.096.03.897.04.1
Par. ambig 3.6 0.3 3.6 0.3 3.5 0.3 4.3 0.3
GH54.25.458.54.871.05.183.35.4
Com. ambig 5.1 0.6 5.2 0.6 5.0 0.5 6.2 0.7
Five
GH37.55.243.64.551.07.165.28.7
Par. ambig 4.8 0.8 4.8 0.8 4.7 0.7 5.8 0.9
GH31.36.223.46.236.07.234.98.7
Com. ambig 4.6 0.8 4.6 0.8 4.5 0.7 5.5 0.9

Search depth

Three-move problems

To keep the influence of goal hierarchy constant while testing for effects of search depth in three-move problems, we analysed problems with a partial-tower goal state that either needed an intermediate move for the optimal solution or not (medium vs. low search depth). A 4 × 2 repeated measurements ANOVA with age group (6/7, 8/9, 10/11, and 12/13 years) as between-subjects and search depth (low vs. medium) as within-subjects factors and perfect solution as dependent variable revealed significant main effects for age group and search depth (Table 4). Number of correctly solved problems increased with age (linear contrast p < .001) and problems that required an intermediate move were more difficult to solve than those without. An interaction between age group and search depth additionally showed that the ability to resolve intermediate moves significantly increased with age (Figure 3; Table 4): An effect of search depth in three-move problems was evident in younger children – 6/7-year-olds: F(1, 47) = 8.76, p = .005; 8/9: F(1, 46) = 8.79, p = .005 – but not in the older two groups – 10/11-year-olds: F(1, 49) = 1.96, p = .168; 12/13: F(1, 32) = 1.00, p = .325.

Table 4. Specific analyses of planning: ‘Search depth’. Statistical values of the repeated measurements ANOVAs for perfect solutions and initial thinking times separated for three-move and four- to five-move problems. Letters A, B, and C refer to the factors age group, search depth, and minimum number of moves (for four- to five-move problems only). Significant effects are highlighted in bold type. Effect sizes are reported as partial eta square (η2)
EffectsThree-move problemsFour- and Five-move problems
Perfect solutionInitial thinking timePerfect solutionInitial thinking time
F (df) p η2F (df) p η2F (df) p η2F (df) p η2
Age group (A)7.30 (3,174) <.001 .11214.72 (3,174) <.001 .20214.07 (3,174) < .001 .1952.79 (3,157) .043 .051
Search depth (B)12.39 (1,174) .001 .06643.88 (1,174) <.001 .201241.81 (1,170) <.001 .5828.29 (1,157) .005 .050
Min. move (C)      352.47 (1,174) <.001 .66935.87 (1,157) <.001 .186
A × B3.35 (3,174) .020 .0551.16 (3,174).326.0202.88 (3,174) .037 .0471.11 (3,157).346.021
A × C      2.34 (3,174).075.0397.52 (3,157) <.001 .126
B × C      .27 (1,174).606.0023.96 (1,157) .048 .025
A × B × C      .26 (3,174).853.004.88 (3,157).454.016
image

Figure 3. Planning performance in three-move problems for the different age groups with respect to search depth, which is either low (no intermediate move) or medium (with one intermediate move).

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For initial thinking times we observed a main effect for search depth and age group (Table 4). As expected, problems requiring an intermediate move entailed longer processing times. In addition, older children were faster in pre-planning simple three-move problems than younger children (linear contrast p < .001).

Four-move and five-move problems

For analysis of search depth effects, mean-aggregated data of perfect solution were entered into a 4 × 2 × 2 repeated measurements ANOVA with age group (6/7, 8/9, 10/11, and 12/13 years) as between-subjects factor and minimum number of moves (four- and five-move problems) and search depth (high vs. medium) as within-subjects factors. As obvious from Table 4, results revealed significant main effects for minimum number of moves, search depth, and age group (linear contrast p < .001). Consequently, five-move problems were more difficult to solve than four-move problems, problems with a high search depth were also more difficult than problems with a medium search depth, and older children solved more problems than younger one.

A significant interaction was observed for search depth and age group (Table 4). As illustrated in Figure 4, 6/7-year-old children had the worst performance of all groups both in problems with medium and high search depth. In contrast, 8- to 13-year-olds had a similar performance level for high search depth, but differed with increasing performance across age for medium search depth problems, as expressed in the interaction effect. Bonferroni-corrected post-hoc tests for problems with medium search depth between age groups revealed significant performance differences in 6/7- compared with 10/11- and 12/13-year-olds (ps = .002 and <.001) as well as a trend for 8/9- compared with 10/11-year-olds (p = .082), and again a significant effect for 8/9- compared with 12/13-year-olds (p = .002). For problems with high search depth, only the performance of 6/7-year-olds differed compared with the other groups (ps < .001, =.003, and <.001 for 8/9-, 10/11-, and 12/13-year-olds).

image

Figure 4. Planning performance in four- and five-move problems for the different age groups with respect to search depth, which is either medium (zero/one initial intermediate move(s) for four/five-move problems) or high (one/two initial intermediate move(s) for four/five-move problems).

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Performing the same analyses with initial thinking time (Table 4) revealed significant main effects for minimum number of moves, search depth, and age group (linear contrast p = .007), indicating that five-move problems required longer initial thinking times than four-move problems, problems with high search depth resulted in longer processing time than those with medium search depth, and older children worked longer on problem solution than younger ones. An interaction effect of age and number of moves showed that there was a stronger increase in planning duration from four- to five-move problems in older children (effect of number of moves in 6/7-year-olds: F(1, 39) = .19, p = .669; 8/9: F(1, 41) = 2.71, p = 0.107; 10/11: F(1, 46) = 9.78, p = .003; 12/13: F(1, 31) = 25.59, p < .001). An additional interaction of search depth and number of moves revealed that the differences in initial thinking time between high and medium search depth were more pronounced in four-move than in five-move problems.

Goal hierarchy

As already outlined in the Methods section, to avoid confounds, analysis of unambiguous, partially ambiguous, and completely ambiguous goal hierarchies were performed at constant levels of search depth, resulting in two separate models. For three-move problems, unambiguous versus partially ambiguous problems were compared at low search depth (P1/P2 in Table 1) and partially ambiguous versus completely ambiguous problems were contrasted at medium search depth (P3/P4 in Table 1). For four- and five-move problems, differences between unambiguous versus partially ambiguous problems were assessed at high search depth (P1/P2 in Table 2) and partially ambiguous versus completely ambiguous problems were compared at medium search depth (P3/P4 in Table 2).

Three-move problems

In the first model, performance in full-tower versus partial-tower goal states was compared in three-move problems without an intermediate move (low search depth). A 4 × 2 repeated measurements ANOVA with age group as between-subjects factor, goal hierarchy (unambiguous vs. partially ambiguous goal hierarchy) as within-subjects factor, and perfect solution as dependent variable showed only one significant effect, namely for age group (linear contrast p = .016). A similar picture emerged in initial thinking times: only age groups differed significantly (linear contrast p < .001). Statistical values are presented in Table 5.

Table 5. Specific analyses of planning: ‘Goal hierarchy’ in three-move problems. Statistical values of the repeated measurements ANOVAs for perfect solutions and initial thinking times for three-move problems, separated for the two models (Model 1 = full-tower vs. partial-tower goal states, Model 2 = partial-tower vs. flat goal states). Letters A and B refer to the factors age group and goal hierarchy. Significant effects are highlighted in bold type. Effect sizes are reported as partial eta square (η2)
EffectsModel 1Model 2
Perfect solutionInitial thinking timePerfect solutionInitial thinking time
F (df) p η2F (df) p η2F (df) p η2F (df) p η2
Age group (A)2.81 (3,174) .041 .04624.09 (3,174) <.001 .2936.33 (3,174) <.001 .09810.44 (3,174) <.001 .153
Goal hierarchy (B).37 (1,174).540.002.01 (1,174).953.000.56 (1,174).540.0032.65 (1,174).106.015
A × B.06 (3,174).980.001.18 (3,174).911.0032.03 (3,174).111.034.79 (3,174).503.013

In the second model, partial-tower versus flat goal states (partially ambiguous vs. completely ambiguous goal hierarchy) were compared in three-move problems with an intermediate move (medium search depth). Similar to the first model, for perfect solution there was a significant main effect for age, showing that performance increased with age (linear contrast p < .001). When analysing this model for initial thinking times, a main effect could be found only for age (linear contrast p < .001), illustrating that the youngest group had the slowest processing times and that the 12- to 13-year-olds were the fastest problem solvers (Tables 3 and 5).

Four-move and five-move problems

When looking for effects of goal hierarchy on children's planning performance at a constant search depth, the first model in analogy to the three-move problems for perfect solution – full-tower versus partial-tower goal states (unambiguous vs. partially ambiguous goal hierarchy) – revealed significant main effects of minimum number of moves, goal hierarchy, and age (Table 6). Again, four-move problems were easier to solve than five-move problems, as were problems with unambiguous versus partially ambiguous goal hierarchy, and older children gained higher perfect solution values than younger ones (linear contrast p = .001). An interaction of minimum number of moves and goal hierarchy demonstrated that the difference between unambiguous and partially ambiguous goal hierarchy was stronger in five- than in four-move problems (Table 3).

Table 6. Specific analyses of planning: ‘Goal hierarchy’ in four- and five-move problems. Statistical values of the repeated measurements ANOVAs for perfect solutions and initial thinking times for four- to five-move problems, separated for the two models (Model 1 = full-tower vs. partial-tower goal states, Model 2 = partial-tower vs. flat goal states). Letters A, B, and C refer to the factors age group, goal hierarchy, and minimum number of moves (for four- vs. five-move problems), respectively. Significant effects are highlighted in bold type. Effect sizes are reported as partial eta square (η2)
EffectsModel 1Model 2
Perfect solutionInitial thinking timePerfect solutionInitial thinking time
F (df) p η2F (df) p η2F (df) p η2F (df) p η2
Age group (A)6.75 (3,174) <.001 .1041.99 (3,160).118.03610.60 (3,174) <.001 .1542.21 (3,156).089.041
Goal hierarchy (B)307.59 (1,174) <.001 .6392.92 (1,160).089.01874.62 (1,174) <.001 .30024.29 (1,156) <.001 .135
Min move (C)201.68 (1,174) <.001 .53728.70 (1,160)<.001.152262.70 (1,174) <.001 .60244.47 (1,156) <.001 .222
A × B1.51 (3,174).215.0251.85 (1,160).140.034.67 (3,174).569.011.64 (3,156).590.012
A × C1.50 (3,174).218.0258.61 (3,160) <.001 .139.99 (3,174).397.0175.29 (3,156) .002 .092
B × C22.92 (1,174) <.001 .116.42 (1,160).516.003.43 (1,174).511.002.79 (1,156).374.005
A × B × C2.00 (3,174).115.0331.19 (3,160).317.0222.45 (3,174).066.0401.30 (3,156).276.024

Repeating these analyses for initial thinking times revealed a significant main effect only for minimum number of moves and an interaction of minimum number of moves and age groups. As obvious from the overall analysis (Figure 2b), higher number of moves entailed longer initial thinking times particularly in older children – 6/7-year-olds: F(1, 41) < .01, p = .948; 8/9: F(1, 42) = 1.37, p = .248; 10/11: F(1, 46) = 6, 39, p = .015; 12/13: F(1, 31) = 23.77, p < .001.

Analyses on perfect solution for the second model – partial-tower versus flat (partially ambiguous vs. completely ambiguous goal hierarchy with medium search depth) – revealed significant main effects of minimum number of moves, goal hierarchy, and age (Table 6). Similar to the first model, four-move problems were easier to solve than five-move problems, as were problems with partially ambiguous versus completely ambiguous goal hierarchy, and older children gained higher perfect solution values than younger ones (linear contrast p < .001).

For initial thinking times, significant effects could be observed for minimum number of moves, goal hierarchy, and an interaction for minimum number of moves and age groups (Table 6). Again similar to the first model, five-move problems took more time to solve than four-move problems, more ambiguous problems also entailed longer solution times, and with increasing age, the initial thinking time was more strongly prolonged in five-move problems – 6/7-year-olds: F(1, 38) = 6.20, p = 0.017; 8/9: F(1, 39) = 6.71, p = .013; 10/11: F(1, 48) = 11.99, p = .001; 12/13: F(1, 31) = 24.62, p < .001.

Discussion

  1. Top of page
  2. Abstract
  3. Background
  4. Materials and methods
  5. Results
  6. Discussion
  7. References

Previous studies have established that planning development proceeds in a gradual increase in global planning performance. The present analyses suggest that this overall improvement results from two separate contributions. First, a global increase in performance along with age-dependent adaptivity of planning durations replicated previous results that were interpreted as reflecting improvement of self-monitoring and control of impulsivity (Albert & Steinberg, 2011; De Luca et al., 2003; Klenberg et al., 2001; Korkman et al., 2001). Second, the novel finding of a gradually increasing capability to search along greater depths characterized the development of specific planning ability. In contrast, as previously observed in younger children working on three-move problems (Kaller et al., 2008), goal hierarchy was unaffected by age.

Age-related global changes in planning ability

Results on global planning performance, that is the percentage of perfectly solved trials at each level of minimum number of moves, fully complied with previous suggestions of a continuous improvement of global planning that stretches out over childhood and adolescence (Klenberg et al., 2001; Korkman et al., 2001) and reaches its maximum at young adult age (Albert & Steinberg, 2011; De Luca et al., 2003). For perfect solutions, no interaction effect was found for age and minimum number of moves. As obvious from Figure 2a, for each move level we observed a linear performance increase across the four age bins. This is in sharp contrast to initial thinking times. Specifically, examining the associated planning durations revealed that this development most likely is not merely a process of planning more efficiently. Rather, in simple three-move problems, the youngest children were the slowest, whereas in five-move problems, older children spent more time planning than younger ones. Thus, with increasing age, children more strongly adapt their planning behaviour to the actual task demands. In adults, better planners were previously found to spend longer planning times (Unterrainer, Rahm, Leonhart, Ruff, & Halsband, 2003; Unterrainer et al., 2004), an effect that increased with the complexity of the involved problems. Likewise, Albert and Steinberg (2011) who examined children and adults (age span 10–30 years), reported a highly significant association between the number of perfect solutions and initial thinking times. In addition, Steinberg et al. (2008) already showed that time to start for the first move considerably increased with age for more complex problems. Similar results were reported by Luciana, Collins, Olson, and Schissel (2009) who examined 9- to 20-year-olds: For three-move problems, 9- to 11-year-olds were the slowest, but for five-move problems, 9- to 11-year-olds were faster than 15- to 17- and 18- to 20-year-olds. Integrating these results with the present ones, the trend towards adjusting planning duration to problem difficulty (shorter on easy, longer on difficult) seems to develop continuously over childhood and adolescence to finally reflect the adult pattern. Comparing the present results with the data of Kaller et al. (2012) obtained in adults using exactly the same problems as employed here allows for a descriptive comparison of planning durations. Whereas the oldest group in this study took about 9–10 s to pre-plan five-move problems, adults worked about 16 s on the same problems. So adults still take considerably longer to elaborate the solution of complex problems than do children up to 13 years. Hence, again consistent with the data by Steinberg et al. (2008), the development towards adaptive adjustment of planning time to global task demands most likely continues through adolescence until young adult age.

Interestingly, Steinberg et al. (2008) found that self-reported impulsivity was associated with solution accuracy. Self-reported inattention and hyperactivity were associated with low performance also in the study of Luciana et al. (2009). Consequently, both Steinberg et al. (2008) as well as Luciana et al. (2009) regard sustained cognitive control of behaviour towards a goal (strategic self-monitoring) as a determinant of planning performance. Remarkably, this interpretation of developmental gains in planning performance does not refer to planning operations per se, but rather to important prerequisites of planning that ensure focusing on the task until goal attainment.

Age-related changes in specific planning abilities

Genuine contributions of specific planning abilities to developmental gains may be revealed by evaluating the performance across structural task parameters that impose specific cognitive demands for successful solution, as done here with search depth and goal hierarchy. Search depth was found to evolve across childhood in simple three-move as well as in four- to five-move problems. As shown by Kaller et al. (2008) the basic ability to solve problems with an intermediate move emerges between the ages 4–6 years in three-move problems. Here, we extend these findings by showing that over the ensuing years performance improves continuously, but reaches adult level surprisingly late in childhood: Only the oldest group yielded results comparable with adults. In contrast, already 8- to 9-year-olds highly efficiently coped with problems that did not demand search ahead, reaching about 95% of correct solutions. Consequently, performance did not change noticeably between 10 and 13 years of age.

Turning to more complex four- and five-move problems, again an interaction effect for search depth and age was observed. From Figure 4 it becomes obvious that performance increased gradually with age, but mostly in problems with medium search depth. Here, the oldest group attained a performance level of 80% correct solutions, rising from 57% at age 6. In contrast, when search depth was high, there was a 20% step in performance from the youngest (around 20% correct) to the 8- to 9-year-old group, but performance remained at around that level of 40% correct solutions even for the oldest group, without significant improvement.

Thus, continuous performance increases from 8 to 12 years were found for four- and five-move problems that contained the ‘easier’ pattern of search depth, but not for those that required a high search depth. To be specific, these easier four-move problems can be conceived as three-move problems that require an intermediate move (sequence intermediate-goal-goal), but with a trailing goal move (sequence goal-intermediate-goal-goal). Thus, in these problems it is sufficient for problem solution to assess whether the current candidate move – if it is not a goal move itself – leads to a state that allows for a goal move. In five-move problems following the easier search depth pattern this was extended by one preceding intermediate move that enabled an immediately following goal move, again followed structurally by a three-move problem that began with another intermediate move (sequence Intermediate-goal-intermediate-goal-goal). Thus, similar to three-move problems with an intermediate move (sequence intermediate-goal-goal), these problems were solvable by considering only the immediate consequences of the current move. In contrast, for example, the topmost problem depicted in Figure 1 puts higher demands on search depth: The only path towards solving this problem within five moves requires the problem solver to start with an intermediate move (moving the white ball from the left to the middle peg) that does not enable a goal move, but another intermediate move (grey ball from left to right peg). Only hereafter the first goal move can be taken (white ball from middle to left peg). Thus, in these problems consideration of at least the subsequent two moves and their consequences is required for an optimal solution (sequence intermediate-intermediate-goal-goal-goal).

In summary, results revealed that problems that do not require search ahead (three-move problems without a subgoal, low search depth) are almost perfectly solved even by the youngest group. In contrast, problems requiring consideration of one initial intermediate move (three-move and four/five-move problems with medium search depth) continuously developed over the age range examined here. Finally, mastery in highest search depths (four/five-move problems starting with one/two intermediate moves) is reached in early adulthood. The implications are twofold: (1) Increasing capability of searching ahead seems to develop gradually across childhood; and (2) the observed substantial increase in difficulty due to rising demands on search depth validate the proposed classification of problems according to the number of trailing intermediate moves that have to be taken before the first goal move is taken (Borys et al., 1982; Spitz et al., 1982). Likewise the results are consistent with results reported by Borys et al. (1982) on the Tower of Hanoi task. Whereas even 6-year-old children correctly solved more than 70% of the presented six-move problems when low search ahead was sufficient for solution, only about 50% of 10-year-olds could cope with searching ahead three moves (seven-move problems).2

Together, gradually developing planning performance across childhood is partly attributable to the increasing ability to mentally search ahead. Or in other words, the findings may be attributable to a lack of maturation in the capacity to search ahead. Reanalysing the data of Kaller et al. (2012), adults reached a performance of 80% and 70% for medium and high search depth demands, respectively, in the same four- and five-move problems that were used here. Hence, the oldest children in this study already attained adult performance when medium search depth was adequate for successful problem completion, but performed much worse than adults (42% vs. 70% correct) in problems with high search depth. This strongly suggests that the ability to master highest search depths evolves during adolescence.

In contrast to perfect solution, we did not observe an interaction between search depth and age for initial thinking time duration neither in three-move nor in four- and five-move problems. In three-move problems, a significant main effect for age groups indicated that the youngest children were slowest and that with increasing age the processing time decreased, as already observed in the general analyses for three-move problems. Moreover, the oldest children group with ages 12–13 revealed similar initial thinking times for problems with and without an intermediate move (3.5 vs. 4.5 s) as adults in the Kaller et al. (2012) sample. In four- and five-move problems, there was only an interaction for search depth and minimum number of moves showing that in five-move problems the planning durations in medium versus high search depth differed less than in four-move problems. The reason for this might be the rather low performance level in five-move problems, especially in younger children, which could have prevented a clear differentiation as we have usually observed in studies with adults. In these no interaction effect of minimum number of moves and search depth was found (Kaller et al., 2012).

Thus, effects of search depth on planning durations were similar across age groups, although their performance actually differed. This strikingly contrasts with the development of global planning durations that clearly showed age-related changes in planning times. These findings may be interpreted as further support for the assumption that the development of strategic self-monitoring and of planning as such forms separable contributors to age-related performance gains.

When analysing for goal hierarchy in three-move problems, we only observed overall age effects, both for perfect solution as well as for initial thinking time. The youngest children showed the worst performance and needed longest to solve the problems, but initial thinking times were not significantly longer when the ambiguity of the goal configuration increased. This might be explained by the generally prolonged processing times in three-move problems, especially in younger children, which may have blurred significant effects of goal hierarchy as usually observed in adults.

Analyses of goal hierarchy in four- and five-move problems showed main effects of the ambiguity of the goal tower configuration. Problems with an unambiguous goal hierarchy were easier and faster to solve than when the degree of ambiguity increased. Most strikingly – and in strong contrast to search depth – goal hierarchy did not interact with age. Thus, while search depth strongly and continuously evolved over childhood, differential effects of goal hierarchy did not develop noticeably. Rather, effects of goal hierarchy were similarly present at all ages.

These results fully conform to a previously reported dissociation in the development of search depth and goal hierarchy (Kaller et al., 2008). In that study, even the youngest, 4-year-old children were able to gather some information from the goal hierarchy of problems, whereas they were severely affected in coping with three-move problems that required mental search ahead. Similarly, perfect solution of old adults is also differentially sensitive to high demands on search depth over the course of healthy ageing, whereas perfect solution for high demands on goal hierarchy is age invariant (Köstering, Stahl, Leonhart, Weiller, & Kaller, 2013). The disparate development of search depth and goal hierarchy corroborated by these studies lends further support to the suggestion that goal hierarchy and search depth pose dissociable cognitive demands. Whereas search depth denotes the required number of mentally anticipated steps (including their interdependencies) along the solution path, goal hierarchy refers to the appearance of the goal configuration and its cognitive implications. In contrast to ‘tower’-like goal configurations where the correct sequence of goal moves is readily available, especially flat endings require the active generation of the correct sequence. In these, the ambiguity of which ball should be selected as a goal first, second, and last cannot be resolved by examining the goal configuration alone, but only by relating it to the start configuration to identify moveable balls, selecting one of the alternatives as a goal, and starting to plan ahead. Given the higher number of balls to select from in flat endings, these solution attempts are more likely to get stuck in dead ends or result in suboptimal solutions and may thus require multiple attempts. In consequence, goal hierarchy can be conceived as reflecting demands for a broad search among multiple alternatives and over multiple solution attempts, thus constituting a dimension of planning that is conceptually and empirically dissociable to search depth (see also Köstering, McKinlay, Stahl, & Kaller, 2012; Köstering et al., 2013).

With respect to ageing, Köstering et al. (2013) interpreted the dissociation in development between search depth and goal hierarchy in the light of the flexibility versus stability of mental representations in working memory (Cools, 2006). According to this conception, and previous results on healthy elderly subjects as compared with Parkinsonian patients (Köstering et al., 2012; McKinlay et al., 2008), greater search depths may impose higher demands on stable mental representation because the current state and the goal have to be maintained in working memory and protected against decay and interference while devising and evaluating the intermediate moves necessary to accomplish subsequent goal moves. In contrast, according to Köstering et al. (2012, 2013) the critical cognitive demand in representing and resolving ambiguous goal hierarchies may lie in the flexibility of generating alternative goal move sequences.

When translating these interpretations to this children study, one can draw parallels between the heterochronous development of cognitive stability versus flexibility and the present differences observed for the development of search depth versus goal hierarchy: Huizinga and colleagues (Huizinga et al., 2006) reported that in their developmental study of executive functions (age range 7–21 years) cognitive shifting matured earlier than working memory. If working memory is an important prerequisite of search depth, then the ongoing development of working memory until early adulthood may strongly contribute to ongoing increases in the ability to deal with higher search depth demands. In contrast, if cognitive shifting should be the determining factor for processes tapping goal hierarchy, then the already highly developed function might gradually accompany the development of high versus low demands on goal hierarchy. Even though these interpretations for the dissociation of age-related changes in search depth and goal hierarchy are rather speculative, future studies may address these questions by additionally assessing executive functions like working memory and set shifting.

Our present findings support the notion of gradual performance gains across childhood. Importantly, they extend previous evidence by delineating two contributors to this development: First, children's ability to monitor their performance and act in a controlled instead of an impulsive way may lead to a global increase in performance and more and more task-adaptive planning durations. Second, children learn to cope with increasing search depths throughout childhood, adolescence, and most likely early adulthood. As the ability to mentally search ahead the problem space to detect, integrate, and resolve interdependencies between moves is considered the core process of planning, search depth may be the most specific indicator of planning ability available.

In future studies, it would be of interest to complete the examination of planning ability with the same problem set in older children and adolescents aged 14–18, thus filling the gap between this study and the data from Kaller et al. (2012). In this study, children up to the age of 13 were severely limited in four- and five-move problems imposing high search depth, showing barely noticeable improvements from just below to just above 40% between the ages of 8–13 years. Clearly, a phase of relatively accelerated improvement must follow hereafter, as only 6 years later young adults reach about 70% correct solutions in the same problems.

Footnotes
  1. 1

    Contrary to a goal move which places a ball into its goal position, an intermediate move does not place a ball into its goal position, but is essential to problem solution nonetheless (Kaller et al., 2011).

  2. 2

    Note that due to differences in the task layout, instructions, and structural problem parameters, percentage of correct solutions on the Tower of Hanoi cannot be related directly to performance on the Tower of London task.

References

  1. Top of page
  2. Abstract
  3. Background
  4. Materials and methods
  5. Results
  6. Discussion
  7. References