1. Actual tower vs. computer simulation
The Effects of Practice, Verbalization and Mode of Presentation on Problem Solving Strategies for the Tower of Hanoi Problem Lucas L. Wong, Noah P. Cohen, & Gary Kose Department of Psychology, Long Island University-Brooklyn
Introduction
The Tower of Hanoi problem is known as the “perfect problem,” having a clear starting point, a clear end point and discrete steps to achieve a solution. Difficulty solving the problem involves mastering a recursive move sequence. Performance on the problem has been linked to higher cognitive processes, planning and metacognitive awareness (Simon and Anzai, 1979).
Solving the problem involves moving a set of graduated disks across a group of three pegs to reach a predefined end state. Progress towards the goal configuration is limited by two rules: move only one disk at a time and larger disks cannot be placed on top of smaller disks. The problem becomes more difficult by increasing the number of discs to be moved: moving two discs takes a minimum of 3 moves, moving 3 discs takes a minimum of 7, moving 4 discs requires at least 15 moves.
How do these factors affect problem solving skills?
1. Practice: participants practiced on a 3-disc problem and then preformed a 4-disc problem (Ahlum-Heath and Di Vesta, 1986).
2. Mode of presentation: participants either solved the problem using a wooden model or solved the task on a computer-simulated version (Noyes and Garland, 2003) .
3. Verbalization: participants either verbalized their move sequence or solved the problem in silence (Zhang and Norman, 1994).
Note: All participants were tested on a 4-disc problem and time of completion, number of moves and the proportion of optimal moves were the dependent variables.
Hypotheses: It was expected that practice would improve performance, performance on the actual wooden puzzle would be better than on the computer-simulated version and that describing the moves while solving the problem would facilitate performance.
Method
Participants. Ninety-five undergraduate students participated in this study.
Materials. The physical model of the Tower of Hanoi puzzle is composed of three vertical pegs that hold a predetermined number of disks of graduated diameters attached to a board. The computer model of the Tower of Hanoi puzzle was presented on a laptop, using a program called PEBL (Mueller, 2012).
Procedure. A 2 x 2 x 2 between subjects design was used. Participants were randomly assigned to eight conditions: mode of presentation (computer vs. physical), experience (practice vs. no practice) and verbalization (concurrent verbalization vs. no verbalization). In the training phase participants were given their respective instructions and completed between 1 (no practice) and 5 (practice) multi-disk problems. Then, in the test phase, the final 4-disk criterion puzzle was completed.
Results
MANOVA showed a significant main effect for modality (Wilks’ Λ = .846, F(2,85) = 7.764, p = .001, η² = .154); a marginally significant effect for verbalization (Wilks’ Λ = .933, F(2,85) = 3.071, p = .052, η² = .067); and there was no significant main effect for practice; however, there was a significant interaction effect for experience and verbalization (Wilks’ Λ = .94, F(2, 85) = 4.017, p = .022, η² = .086).
Follow up ANOVAs found significant main effect of modality on the number of moves used to solve the problem (F(1,93) = , p = .002, η² = .107) and the proportion of optimal moves (F(1,93) = , p = .002, η² = .110). Interestingly, it required move moves with the actual problem than with the computer simulation, but there were more optimal moves with the actual problem.
There was significant main effect of verbalization on the number of moves (F(1,93) = 5.748, p = .019, η² = .063). Also, significant interaction between experience and verbalization for the number of moves, F(1,93) = 7.503, p = .007, η² = .080).
Finally, there was a marginally significant interaction between modality and experience for the number of moves (F(1,93) = 3.513, p = .064, η² = .039).
Actual tower vs. computer simulation
Conclusions
Practice on a 3-disc problem did not improve performance on a more difficult 4-disc problem. Although, practice may be effective when verbalizing while solving the problems.
The modality of presenting the problem did have an impact on performance. While the actual problem required more moves than the computer simulated version, more optimal moves were made with the actual problem. Performance on the physical model of the Tower of Hanoi puzzle was improved for the total number of moves and the proportion of optimal to total moves. With practice, no effects on performance were found for the physical task, but a marginally significant effect was found for the computer task on number of moves.
References
Ahlum-Heath, M. E., & Di Vesta, F. J. (1986). The effect of conscious controlled verbalization of a cognitive strategy on transfer in problem solving. Memory & Cognition, 14 (3),
Anzai, Y., & Simon, H. A. (1979). The Theory of Learning by Doing. Psychological Review, 86 (2),
Mueller, S. T. (2012). PEBL: The psychology experiment building language (Version 0.12) [Computer experiment programming language]. Retrieved from
Noyes, J. M., & Garland, K. J. (2003). Solving the Tower of Hanoi: does mode of presentation matter? Computers in Human Behavior, 19,
Zhang, J., & Norman, D. A. (1994). Representations in Distributed Cognitive Tasks. Cognitive Science, 18,