Towers of Hanoi and London: is the nonshared variance due to differences in task administration?

Although it has been assumed that the Tower of Hanoi and Tower of London are more or less interchangeable tasks dependent on executive function, a series of studies in our laboratory have indicated substantial nonshared variance between the performances on the two tasks. The purpose of the present study was to examine how much methods of administration, such as number of trials per problem, contribute to this nonshared variance. A new one-trial version of the Tower of Hanoi was developed to be identical to the Tower of London in four procedural characteristics. The one-trial version of the Tower of Hanoi was administered to 39 normal adults along with the traditional Tower of Hanoi and the Tower of London-Revised in two test sessions 5-7 weeks apart. The correlations between the two tasks were in the same range as found previously with the traditional task, indicating that administration differences do not account for the nonshared variance between the tasks. A reliability analysis of the one-trial tasks showed poor internal consistency. Also, the internal consistency of the 6-trial tower was artificially inflated by aspects of the administration and scoring procedures. Moreover, this task exhibited a ceiling effect on repeated testing. These results suggest that it would be of value to redesign the one-trial Tower of Hanoi systematically to increase its reliability and, potentially, its validity as a measure of executive functions.     

Problem-Solving Restructuration: Elimination of Implicit Constraints

A general model of problem-solving processes based on misconception elimination is presented to simulate both impasses and solving processes. The model operates on goal-related rules and a set of constraint rules in the form of “if (state or goal), do not (Action)” for the explicit constraints in the instructions and the implicit constraints that come from misconceptions of legal moves. When impasses occur, a constraint elimination mechanism is applied. Because successive eliminations of implicit constraints enlarge the problem space and have an effect on planning, the model integrates “plan-based” and “constraint-based” approaches to problem-solving behavior. Simulating individual protocols of Tower of Hanoi situations shows that the model, which has a proper set of constraints, predicts a single move with no alternative on about 61% of the movements and that protocols are quite successfully simulated movement by movement. Finally, it is shown that many features of previous models are embedded in the constraint elimination model.     

A longitudinal study of the performance of the elderly and young on the Tower of Hanoi puzzle and Rey recall

Normal performance on the Tower of Hanoi puzzle by amnesic patients has been taken as support for viewing this problem solving task as having a nondeclarative memory component. Individuals in each decade of life between the 20s and 80s were asked to solve this puzzle four times in four sessions with intersession intervals from 1 to 7 days (Davis & Keller, 1998). Participants in their 70s and 80s were significantly impaired compared to participants in their 20s and 30s. The elderly were also significantly impaired on five immediate trials of a 15 words verbal recall test. Participants were readministered these tests an average of 6.6 years later for the elderly (n = 12) and 7.7 years later for the young (n = 11). For the Tower of Hanoi, the performance of the elderly, but not the young individuals, was significantly poorer than their original performance. For the verbal recall test, no significant change over time was detected for the young or elderly participants. These findings support the view that some nondeclarative and/or problem solving tasks demonstrate as great or greater decline with age than some declarative tasks.     

Gesture offers insight into problem‐solving in adults and children

When asked to explain their solutions to a problem, both adults and children gesture as they talk. These gestures at times convey information that is not conveyed in speech and thus reveal thoughts that are distinct from those revealed in speech. In this study, we use the classic Tower of Hanoi puzzle to validate the claim that gesture and speech taken together can reflect the activation of two cognitive strategies within a single response. The Tower of Hanoi is a well‐studied puzzle, known to be most efficiently solved by activating subroutines at theoretically defined choice points. When asked to explain how they solved the Tower of Hanoi puzzle, both adults and children produced significantly more gesture‐speech mismatches—explanations in which speech conveyed one path and gesture another—at these theoretically defined choice points than they produced at non‐choice points. Even when the participants did not solve the problem efficiently, gesture could be used to indicate where the participants were deciding between alternative paths. Gesture can, thus, serve as a useful adjunct to speech when attempting to discover cognitive processes in problem‐solving.     

Why are some problems hard? Evidence from Tower of Hanoi

This paper analyzes the causes for large differences in difficulty of various isomorphic versions of the Tower of Hanoi problem. Some forms of the problem take 16 times as long to solve, on average, as other versions. Since isomorphism rules out size of task domain as a determinant of relative difficulty, these experiments seek and find causes for the differences in features of the problem representation. Analysis of verbal protocols and the temporal patterning of moves allows the problem-solving behavior to be divided into exploratory and final-path phases. Entry into the final-path phase depends on acquisition of the ability to plan pairs of moves, an achievement made difficult by the working memory load it entails. This memory load can be reduced by automating the rules governing moves, either through problem exploration or training. Once automation has occurred, the solution is obtained very rapidly. Memory load is also proposed as the locus of other differences in difficulty found between various problem representations.     

Perceived self-efficacy and everyday problem solving among young and older adults

This research tested the hypothesis that age differences in both self-efficacy perceptions and problem-solving performance would vary as a function of the ecological relevance of problems to young and older adults. The authors developed novel everyday problem-solving stimuli that were ecologically representative of problems commonly confronted by young adults (young-adult problems), older adults (older adult problems), or both (common problems). Performance on an abstract problem solving task lacking in ecological representativeness (the Tower of Hanoi problem) also was examined. Although young persons had higher self-efficacy beliefs and performance levels on the Tower of Hanoi task problem and the young-adult problems, this pattern reversed in the domain of older adult problems, where the self-efficacy beliefs and performance of older persons exceeded those of the young.     

Towers of Hanoi and London: contribution of working memory and inhibition to performance

The Tower of Hanoi and Tower of London have become well-established executive function tasks that presumably tap cognitive skills mediated by the frontal cortex. It has been assumed that the two tower tasks are more or less interchangeable and that both measure working memory and inhibition processes. These assumptions were tested in a study involving 37 normal college volunteers (M age = 20 years). Participants were administered the Tower of Hanoi (TOH), Tower of London-Revised (TOL-R), two working memory tests, and two tests of inhibition. The two tower tasks correlated significantly (r = .39), but only moderately. The working memory and inhibition variables explained over one-half of the variance in TOL-R performance; however, there was a relatively weaker contribution of inhibition to TOH performance.     

Control mechanisms in problem solving

Hypothesis-testing theories and information processing theories have both been used to explain the results of problem solving experiments. The theories have, in general, been applied to different classes of problems, resulting in little overlap or interaction between them. Information processing theories have tended to emphasize problems in which the problem solver’s primary source of information concerning the correctness of his moves is the problem goal and, as a result, the goal plays a major role in the control mechanisms determining moves. Hypothesis theories have emphasized problems in which direct feedback is the primary control mechanism available. It is suggested that altering a problem solver’s major source of information from a goal to feedback or vice versa will have important implications for the course of problem solving and what is learned during the solution process. Specifically, changes in the information feedback and/or goal-specificity characteristics of a problem should, in a predictable fashion, alter problem solving strategies, which should, in tum, through the presence or absence of information compression devices such as rule induction, strongly affect transfer. Experiments using the Tower of Hanoi and visual maze-tracing problems indicated that altering the density of subgoals resulted in alterations in subsequent transfer performance.     

Social audiences can disrupt learning by teaching

To investigate the effect of a social audience on learning-by-teaching, we examined participants' solutions of the 4-ring Tower of Hanoi problem after they demonstrated the 3-ring problem to a social agent (a person) or a non-social agent (a computer). In Introduction, Discussion participants produced less optimal solutions of the 4-ring problem after demonstrating the 3-ring problem to a social agent. An analysis of pointing behavior demonstrated that social highlighting contributed substantially to this effect. Together, these findings indicate that more social highlighting may produce a cost, rather than a benefit, on how deeply the demonstrator encodes the problem solution. Experiment 3 clarified that these results were not simply caused by the disruptions inherent to social highlighting. Taken together, the results suggest that social highlighting does not come for free — producing the highlighting may lead to more shallow encoding of demonstrated actions.     

Move evaluationasa predictor and moderator of success in solutions to well-structured problems

This paper presents three studies concerned with the evaluation of moves in solutions to Tower of Hanoi problems and the effect that such evaluation processes have on solution success. The existing literature on problem solving suggests that verbalizing whilst solving a problemcan havea positive effect upon performance. However,such verbalization has to be directed toward an explicit evaluation of particular moves. What remains unclear is whether evaluation without verbalization has the same effects or whether some characteristic of the process of verbalization gives rise to improved performance on such tasks. For example, the act of verbalizing per se may simply mean that more processing time is directed toward the problem-solving process. The studies reported in this paper suggest that the process of evaluation may be independent of verbalization processes and that non-verbal evaluation of moves (indicated by a key press) produces the same effects as a verbal evaluation of such moves. Moreover, the process of evaluating moves appears to producea form of behaviour that is prone to disruption via the administration of secondary tasks, whereas non-evaluated solutions are not. This may suggest that problem solvers who engage in evaluation processes develop an explicit representation of the strategies used to solve the problem.     

The assessment of cognitive procedural learning in amnesia: why the tower of Hanoi has fallen down

The Tower of Hanoi has been widely accepted as an evaluation of cognitive procedural learning in amnesia but inconsistent findings have raised questions about the nature of the learning process involved in this task. This article presents the performance of a hippocampal amnesic, MS, who, showing poor learning across daily sessions of a formal evaluation, subsequently solved the puzzle through spontaneous use of a declarative-level strategy (the odd-even rule), suggesting that his primary approach to the task was the deployment of declarative solution-searching strategies. The presented data suggest normal learning within daily sessions, but subnormal learning across daily sessions due to the forgetting of acquired declarative information. It is suggested that tasks that are potentially solvable by an algorithm or rule, as is the Tower of Hanoi, be regarded as inappropriate for use in cognitive procedural assessments.     

The effect of conscious controlled verbalization cognitive strategy on transfer in problem solving

The effect of controlled verbalization on learning to solve complex problems was investigated. Fifty participants individually solved the six-disk Tower of Hanoi problem as a criterion task, following one of the five treatments represented in a 2 × 2 factorial design with an appended control group. One factor was the presence or absence of a practice series which required participants to provide verbal rationales for their moves. The other was the presence or absence of verbalization on the six-disk criterion task. The control participants performed the practice tasks and the criterion task without verbalization. Although practice tended to be more effective than no practice for improving performance, its strongest effect occurred when it was coupled with controlled verbalization. Controlled verbalization during the criterion task facilitated performance, but only for subjects who received no prior practice. It was concluded that verbalization was most helpful during the initial flexible stages of learning to solve problems before the skill had become organized. The discussion indicated that performance is facilitated by the quality and timing of the use of verbalizations rather than by the mere activity of verbalizing.     

Practice effects on visuomotor and problem-solving tests by children

Practice effects on a visuomotor test (the Developmental Test of Visuo-Motor Integration), a timed visual discrimination test (the Underlining Test), and two problem-solving tests (the Porteus Mazes Test and the Tower of Hanoi Test) were analyzed. Children of two age groups (Ms: 7.7 and 11.6 yr.) were chosen to study the effect of age on practice effects. The tests were repeated nine times with test-retest intervals of 2 mo. The Developmental Test of Visuo-Motor Integration showed no practice effects, while the Porteus Mazes Test, the Underlining Test, and the Tower of Hanoi Test showed significant practice effects. Practice effects were larger for the older age group on all the tests, except the Developmental Test of Visuo-Motor Integration. The Developmental Test of Visuo-Motor Integration and the Underlining Test showed good reliability, but those of the problem-solving tasks were less satisfactory. The stability of all the tests, except the Tower of Hanoi Test, was good.     

Relationship between choice reaction time and the Tower of Hanoi test.

The aim of this study was to examine a possible relation between the speed of information processing, as measured by simple and choice visual RT, and problem-solving, as measured by the Tower of Hanoi test. For 20 normal teenagers, performing all tests, significant correlations were found between choice RT and both measures of performance on the Tower of Hanoi, number of disk moves, and time taken to complete the task. Simple RT was correlated with Completion time but not with the number of moves, while the reverse pattern was discerned for decision time. Choice movement time was also associated with both measures, but simple movement time was not. These results are consistent with the hypothesis of a common neurobiological basis to information-processing speed and executive functions.     

In which direction to move? Facilitative and interference effects of gestures on problem solver's thinking

Recent research shows that co-speech gestures can influence gesturers’ thought. This line of research suggests that the influence of gestures is so strong, that it can wash out and reverse an effect of learning. We argue that these findings need a more robust and ecologically valid test, which we provide in this article. Our results support the claim that gestures not only reflect information in our mental representations, but can also influence gesturer's thought by adding action information to one's mental representation during problem solving (Tower of Hanoi). We show, however, that the effect of gestures on subsequent performance is not as strong as previously suggested. As opposed to what previous research indicates, gestures' facilitative effect through learning was not nullified by the potentially interfering effect on subsequent problem-solving performance of incompatible gestures. To conclude, using gestures during problem solving seems to provide more benefits than costs for task performance.     

Representations in Distributed Cognitive Tasks

In this article we propose a theoretical framework of distributed representations and a methodology of representational analysis for the study of distributed cognitive tasks—tasks that require the processing of information distributed across the internal mind and the external environment. The basic principle of distributed representations Is that the representational system of a distributed cognitive task is a set of internal and external representations, which together represent the abstract structure of the task. The basic strategy of representational analysis is to decompose the representation of a hierarchical task into its component levels so that the representational properties at each level can be independently examined. The theoretical framework and the methodology are used to analyze the hierarchical structure of the Tower of Hanoi problem. Based on this analysis, four experiments are designed to examine the representational properties of the Tower of Hanoi. Finally, the nature of external representations is discussed.     

Self-observation and learning: the effect of watching oneself on problem solving performance

Two experiments examined the effects of videotape feedback and self-observation on children’s problem solving. The first experiment examined children’s performance on the Tower of Hanoi problem, and demonstrated that video self-observation promotes the acquisition and transfer of procedural knowledge necessary for problem solving. The study also found that specific information presented during the video presentation was not as important as the children seeing their actual prior performance on the problem. The second experiment examined the type of information that may be operative during self-observation. These findings narrow the range of possible explanations for learning through self-observation and, in general, suggest that the positive effect of self-observation is due to active observation of one’s own actual performance.     

Gesturing during mental problem solving reduces eye movements,especially for individuals with lower visual working memory capacity

Non-communicative hand gestures have been found to benefit problem-solving performance. These gestures seem to compensate for limited internal cognitive capacities, such as visual working memory capacity. Yet, it is not clear how gestures might perform this cognitive function. One hypothesis is that gesturing is a means to spatially index mental simulations, thereby reducing the need for visually projecting the mental simulation onto the visual presentation of the task. If that hypothesis is correct, less eye movements should be made when participants gesture during problem solving than when they do not gesture. We therefore used mobile eye tracking to investigate the effect of co-thought gesturing and visual working memory capacity on eye movements during mental solving of the Tower of Hanoi problem. Results revealed that gesturing indeed reduced the number of eye movements (lower saccade counts), especially for participants with a relatively lower visual working memory capacity. Subsequent problem-solving performance was not affected by having (not) gestured during the mental solving phase. The current findings suggest that our understanding of gestures in problem solving could be improved by taking into account eye movements during gesturing.     

Linking vitamin D status,executive functioning and self‐perceived mental health in adolescents through multivariate analysis: A randomized double‐blind placebo control trial

The aim of the present randomized double‐blind placebo control trial was to investigate if vitamin D supplementation had an effect on vitamin D status, executive functioning and self‐perceived mental health in a group of Norwegian adolescents during winter time. Fifty adolescents were randomly assigned into an intervention group (vitamin D pearls) or a control group (placebo pearls). Before (pre‐test in December/January) and after (post‐test in April/May) the intervention period the participants were exposed to a test procedure, consisting of blood draw, completion of cognitive tests (Tower of Hanoi and Tower of London), and the Youth Self‐report version of the Child Behavior Checklist. Multivariate data analysis showed that participants with low vitamin D status scored worse on the Tower of London tests and the more difficult sub‐tasks on the Tower of Hanoi tests. They also had a tendency to report higher frequency of externalizing behavior problems and attention deficit. At pre‐test, the overall mean vitamin D status measured as 25‐hydroxy vitamin D was 42 nmol/L, defining deficiency (Intervention group = 44 nmol/L, Control group = 39 nmol/L). However, vitamin D supplementation caused a significant increase in vitamin D status resulting in a sufficient level in the Intervention group at post‐test (mean 62 nmol/L). The results also revealed that the intervention group improved their performance on the most demanding sub‐tasks on the ToH. Overall, the study indicates that vitamin D status in adolescents may be important for both executive functioning and mental health.     

Memory for goals: an activation‐based model

Goal‐directed cognition is often discussed in terms of specialized memory structures like the “goal stack.” The goal‐activation model presented here analyzes goal‐directed cognition in terms of the general memory constructs of activation and associative priming. The model embodies three predictive constraints: (1) the interference level, which arises from residual memory for old goals; (1) the strengthening constraint, which makes predictions about time to encode a new goal; and (3) the priming constraint, which makes predictions about the role of cues in retrieving pending goals. These constraints are formulated algebraically and tested through simulation of latency and error data from the Tower of Hanoi, a means‐ends puzzle that depends heavily on suspension and resumption of goals. Implications of the model for understanding intention superiority, postcompletion error, and effects of task interruption are discussed.