We investigate problem solving by analogy in order to see whether solving a similar, easier problem beforehand aids in deciphering a more complicated one, given that the two problems have a similar (homomorphic) problem state. The two problems that have been used in this experiment are the historic Missionary & Cannibals problem (M&C) as well as its visual representation - referred to as "Counter-Moving" (CM) in the following.
The use of analogy in problem solving requires restructuring the problem in order to create an insight - as can be measured by "warmth" ratings - in contrast to problem solving based on trial-and-error learning (mostly common in elementary algebra) as proposed by behaviourists such as Thorndike. The restructuring of the problem relies on mapping from a source domain to a target domain. Since there is an infinite number of possible source domains, many humans have difficulies at spontaneous use of analogy and need its relevance to be highlighted. We have shown that the success rate on the counter-moving problem does differ significantly from the success rate on the M&C problem, which implies that the CM problem is conceptually easier to understand. Hence, we could show that solving a similar, easier problem with a homomorphic problem state does indeed increase the success rate of solving a more complicated one later by comparing the success rates between Groups A and B that solved the problems in opposite orders.
Table of Contents
1. Introduction
2. Methods
2.1. Subjects
2.2. Materials
2.3. Procedure
3. Results
4. Discussion
5. Conclusion
Research Objectives and Themes
This report investigates whether solving a simpler, homomorphic problem prior to a more complex one facilitates insight and improves problem-solving success rates, specifically focusing on the Missionary & Cannibals (M&C) problem and its visual counterpart, the Counter-Moving (CM) problem.
- Cognitive mechanisms of insight and analogical problem solving.
- Restructuring problem representations to overcome impasses.
- Comparison of success rates between M&C and CM problem variants.
- Impact of task ordering on analogical transfer and learning.
- Statistical validation of problem-solving strategies using Chi-squared analysis.
Excerpt from the Book
1. Introduction
Insight was first studied by Gestalt Psychology, in the first half of the 20th century, partly as reaction to the associationistic view of learning proposed by behaviourists such as Thorndike. [10] The most famous experiments are Maier’s pendulum problem [5] and Duncker’s candle problem [1]. Some proposed potential mechanisms for insight include: suddenly seeing the problem in a new way, connecting the problem to another relevant solved problem (solving by analogy – as investigated here), releasing past experiences that are blocking the solution (overcoming functional fixedness), or seeing problem in a larger, coherent context. [9] One more recent example is Knoblich’s matchstick experiment [4], whereby the constraints of algebra need to be relaxed. This implies that impasses need to be broken by changing the problem representation. This breakage of impasses is crucial for solving insight problems as pointed out by Ohlsson [8].
In the following, we are going to investigate the Missionary & Cannibals problem (M&C) problem state in more detail. This problem has various homomorphic appearances, the first one of which is the jealous husbands problem in the medieval text “Propositiones ad Acuendos Juvenes” (Alcuin), whereby couples of husbands and wifes are considered and the constraint is that no woman can be in the company of another man unless her husband is present. The problem can be translated into a visual representation in a 2D-space, whereby certain vectors lead from one problem state to the next one. One of the main impasses in the two appearances of the problem is a goal-subgoal conflict, in which an intermediate step seems to increase the distance to finding the solution – as in the Tower of Hanoi problem. [7] Hence, problem solving by analogy [2, 3] might be necessary in order to overcome this obstacle.
Summary of Chapters
1. Introduction: Discusses the theoretical background of insight problems, the role of Gestalt Psychology, and the concept of problem solving by analogy as a mechanism to overcome goal-subgoal conflicts.
2. Methods: Details the experimental setup, involving 204 students over three years, where subjects attempted M&C and CM problems in different orders using pen and paper.
3. Results: Presents the statistical analysis of the experimental data using Chi-squared tests, showing significant differences in success rates between problem types and solving orders.
4. Discussion: Explores potential biases in the experiment, specifically regarding prior knowledge of the radiation problem, and suggests improvements for future research designs.
5. Conclusion: Summarizes the finding that the CM problem is conceptually easier and that solving a similar, simpler problem aids in tackling more complex tasks.
Keywords
Problem solving, Analogy, Insight, Missionary & Cannibals problem, Counter-Moving problem, Homomorphic problem state, Gestalt Psychology, Cognitive science, Goal-subgoal conflict, Statistical analysis, Chi-squared test, Analogical transfer, Problem representation, Impasse, Thorndike.
Frequently Asked Questions
What is the core focus of this research paper?
The paper examines whether individuals can improve their problem-solving performance on complex tasks by first solving a simpler, structurally similar problem through the mechanism of analogy.
Which specific problems are analyzed in this study?
The study centers on the historic Missionary & Cannibals (M&C) problem and its visual representation, referred to as the Counter-Moving (CM) problem.
What is the primary research goal?
The primary goal is to determine if analogical transfer occurs when subjects encounter two problems with a homomorphic (similar) problem state, thereby facilitating a breakthrough or "insight" in the harder task.
What scientific methodology was employed?
The researcher conducted a practical experiment with 204 psychology students, using a counter-balanced design where participants attempted the problems in different orders, followed by Chi-squared statistical analysis.
What does the main body cover?
The body covers the historical context of insight research, the specific experimental methodology (subjects, materials, procedure), the statistical results obtained from the experiments, and a discussion on potential experimental biases.
Which keywords define this work?
Key terms include problem solving, analogy, insight, homomorphic problem state, and analogical transfer.
Why is the "Counter-Moving" problem considered easier than the M&C problem?
The experimental results indicated a significantly higher success rate for the CM problem, suggesting that its visual representation makes it conceptually more accessible to participants than the traditional M&C formulation.
What role does the "goal-subgoal conflict" play in these experiments?
It is identified as a major impasse where intermediate steps appear to move the subject further from the goal; the study suggests analogy helps bypass this mental block.
Did the order of the problems affect the success rate?
Yes, the results showed that the order in which the M&C and CM problems were presented had a statistically significant impact on performance, as analyzed in Table 2.
What is the author's suggestion for future experimental iterations?
The author suggests varying the timing of the theoretical instruction regarding analogical problem solving to control for potential biases caused by students' prior awareness of similar problems like the "radiation problem."
- Quote paper
- Laura Imperatori (Author), 2014, Problem Solving by Analogy, Munich, GRIN Verlag, https://www.hausarbeiten.de/document/276480
