Problem representation and solution strategies in solitaire chess

Abstract

Experimental and computer simulation were used to study problem representation, solution strategies, and the relation between the two when solving problems in solitaire chess. Experimental results showed that subjects produced three kinds of problem representations, corresponding to three kinds of solution strategies; namely, trial-and-error, concentration and decomposition. The three different problem representations reflected different cognitive levels achieved by the subjects in structuring the problem and determined which solution strategy the subjects employed. The three kinds of solution strategies exhibited different kinds of selectivity and different degrees of effectiveness.     

Tower of Hanoi performance of retarded young adults and nonretarded children as a function of solution length and goal state

Three experiments were conducted with the Tower of Hanoi task to assess problem solving ability in 6-, 7-, 8-, and 10-year-old nonretarded children and mentally retarded young adults of varying maturational ages. In Experiment 1 we gradually reduced the number of moves required for solution until subjects could solve the 3-disk tower-ending problem. Although all groups experienced difficulty with the standard 7-move problem, all but the trainable retarded group readily solved the 6-move problem. The trainable group did not reach a comparable level of success until the 4-move problem. On the 7-move problem the retarded groups performed at the level of nonretarded groups that were maturationally $\text{1}\text{1}\text{2}$ to 3 years younger. An analysis of first moves indicated that subject groups differed in the strategies they used to solve the problems. In Experiment 2, practice effects were ruled out as a source of the superior performance on the 6- than on the 7-move problem. In Experiment 3, 7- and 10-year-old nonretarded children and mentally retarded young adults did not differ on 5-move problems in which configuration of the goal states was varied. A comparison of all 5-move problems judged to have the same depth of search requirements indicated that the tower-ending problems were significantly easier to solve than the partial-tower-ending problems, which in turn were easier than the flat-ending problems. A limited depth of search capacity sets boundaries on the use of sophisticated strategies and, to a large extent, accounts for the retarded groups' maturational lag.     

The Impact of Goal Specificity on Strategy Use and the Acquisition of Problem Structure

Theories of skill acquisition have made radically different predictions about the role of general problem-solving methods in acquiring rules that promote effective transfer to new problems. Under one view, methods that focus on reaching specific goals, such as means-ends analysis, are assumed to provide the basis for efficient knowledge compilation (Anderson, 1987), whereas under an alternative view such methods are believed to disrupt rule induction (Sweller, 1988). We suggest that the role of general methods in learning varies with both the specificity of the problem solver's goal and the systematicity of the strategies used for testing hypotheses about rules. In the absence of a specific goal people are more likely to use a rule-induction learning strategy, whereas provision of a specific goal fosters use of difference reduction, which tends to be a non-rule-induction strategy. We performed two experiments to investigate the impact of goal specificity and systematicity of rule-induction strategies in learning and transfer within a complex dynamic system. The results of Experiment 1 indicated that during free exploration of a problem space, greater learning occurred if participants adopted more systematic strategies for rule induction, and that participants come to favor such strategies. Experiment 2 revealed that participants who were provided with a specific goal performed well on the initial problem but were impaired on a transfer test using a similar problem with a different goal. Instruction on a systematic rule-induction strategy facilitated solution for both the initial and transfer problems, but participants' use of this strategy declined if they had a specific goal. Our results support Sweller's (1988) proposal that general problemsolving methods applied to a specific goal foster acquisition of knowledge about an isolated solution path but do not provide an effective way of learning the overall structure of a problem space. We interpret these results in terms of dualspace theories of search through problem space.     

Creativity,consciousness, and “becoming wise fast enough”

Thirteen 11th-grade students in enriched mathematics and 9 in a regular course solved nine word problems that varied in the ease with which they could be solved by verbal-logical or visual strategies. A pervasive use of verbal-logical solution methods was noted for both groups. Significant differences were found in the final solution strategies. In the course of solving each problem, students tried different strategies. Enriched students alternated almost exclusively between verbal-logical and visual solution methods, whereas regular students alternated equally between verbal-logical and visual solution methods or between verbal-logical strategies and trial and error.     

Aggregating preference rankings: An AHP-based approach

The problem considered here involves the selection of an alternative by a group of decision makers in a context that involves the aggregating of votes from a preferential ballot. A popular approach for solving such problems is the well-known weighting problem formulation. However, with such a formulation two issues become of critical importance: (a) the weights should reflect the decision-making group's belief on the importance of each rank position and (b) the weights should accord each alternative a fair (if not its best) assessment. Many researchers have studied this problem and proposed various solution strategies. None of these strategies, however, appears to have adequately addressed both issues. The objective of this research is to develop a group decision-making model that solves the problem in a manner that adequately addresses the two critical issues identified, as well as allows for sensitivity analysis and is fairly robust with respect to the various paradoxes identified in voting systems.     

Navigation in multiobjective optimization methods

Building on previous work of the authors, this paper formally defines and reviews the first approach, referred to as navigation , towards a common understanding of search and decision‐making strategies to identify the most preferred solution among the Pareto set for a multiobjective optimization problem. In navigation methods, the decision maker interactively learns about the problem, whereas the decision support system learns about the preferences of the decision maker. This work introduces a detailed view on navigation leading to the identification of integral components and features. A number of different existing navigation methods are reviewed and characterized. Finally, an overview of applications involving navigation is given, and promising future research direction are discussed.     

Transformational and derivational strategies in analogical problem solving

Analogical problem solving is mostly described as transfer of a source solution to a target problem based on the structural correspondences (mapping) between source and target. Derivational analogy (Carbonell, Machine learning: an artificial intelligence approach Los Altos. Morgan Kaufmann, 1986) proposes an alternative view: a target problem is solved by replaying a remembered problem-solving episode. Thus, the experience with the source problem is used to guide the search for the target solution by applying the same solution technique rather than by transferring the complete solution. We report an empirical study using the path finding problems presented in Novick and Hmelo (J Exp Psychol Learn Mem Cogn 20:1296–1321, 1994) as material. We show that both transformational and derivational analogy are problem-solving strategies realized by human problem solvers. Which strategy is evoked in a given problem-solving context depends on the constraints guiding object-to-object mapping between source and target problem. Specifically, if constraints facilitating mapping are available, subjects are more likely to employ a transformational strategy, otherwise they are more likely to use a derivational strategy.     

Converging evidence for the functional significance of imagery in problem solving

Two kinds of explanations have been offered for the process by which three-term series problems are solved, one in terms of linguistic principles and the other in terms of visual-spatial imagery. Two experiments are reported in which three different classes of operations are brought to bear on the problem: (1) Manipulation of stimulus attributes (characteristics of problems), (2) manipulation of variables that selectively encourage or inhibit the use of imagery (facilitating instructions; the suppression of visualization by reading), and (3) measurement of relevant individual differences (spatial-reasoning ability). All of the results indicate that imagery plays a functional but not a necessary role in the solution of three-term series problems; it is suggested that imaginal representation is functional because it reduces the load on memory. An adequate explanation of problem-solving will have to address certain general issues, such as the diversity of forms of cognitive representation and differences within and between individuals in the choice of problem-solving strategies.     

Decomposing the reaction time distribution: Pure insertion and selective influence revisited

This paper investigates the consequences of extending the assumptions of pure insertion and selective influence (popular in RT theorizing) to the level of the distribution. In the case of pure insertion and under the additional assumption that the additive random variable is exponentially distributed, a solution is obtained which not only allows estimation of the exponential-rate parameter but also provides a test of the assumptions. The result is shown to be applicable not only when processing is serial but also for certain parallel models. In addition, discrimination between self-terminating and exhaustive search strategies is provided, and in the case of either, both parameter estimation and tests of the model are possible. Extensions to nonexponential models are investigated and a general method of moments solution is outlined. In the case of selective influence a general nonparametric alternative to Sternberg's additive factor method is developed. The problem of empirical estimation and application is then considered. Simulations which place bounds on the type I and II error are reported. Finally the first theorem is provided an illustrative application with data from a memory scanning experiment. The results provide some support for the double assumption of pure insertion and that the additive random variable is distributed exponentially.     

THE STRUCTURE OF JUSTIFICATION IN POLITICAL CONSTRUCTIVISM

Abstract: In this article the author develops the view, held by some, that political constructivism is best interpreted as a pragmatic enterprise aiming to solve political problems. He argues that this interpretation's structure of justification is best conceived in terms of two separate investigations—one develops a normative solution to a particular political problem by working up into a coherent whole certain moral conceptions of persons and society; and the other is an empirically based analysis of the political problem. The author argues that the empirically based analysis can generate criteria for assessing whether the normative theory successfully works out a solution, thereby developing a functionalist structure of justification. He further argues that this interpretation overcomes a longstanding criticism of constructivism, namely, that the use of substantive moral concepts in the hypothetical choice procedure biases the defense of principles in a particular direction and therefore begs important philosophical questions.     

A new strategy for studying spatial aptitude

A new approach to testing for and training in spatial aptitude is described. Figurai analogy problem terms from the cognitive abilities test are photodigitized and arrayed in zones on the computer screen; they can be seen only when selected with the mouse (i.e., one at a time). Viewing patterns and durations can then be analyzed to assess problem encoding and solution strategies. Implications are discussed.     

Patterns of coping in rheumatoid arthritis

Abstract

One hundred and fifty eight outpatients were assessed on a 36 item coping questionnaire in order to examine the impact of coping on the disability and psychological well being of individuals with rheumatoid arthritis. To consider the total pattern of coping strategies accepted and rejected by these individuals, the data was subjected to a cluster analysis. Hierarchical clustering techniques suggested the four group solution as optimum. Following a k means iterative solution external validity for the four groups was considered. The main findings indicate that the pattern of coping exerts a significant influence on symptom perception, disability and psychological well being. The findings are discussed in relation to two approaches to the impact of coping on chronic illness.     

Gender differences in advanced mathematical problem solving

Strategy flexibility in mathematical problem solving was investigated. In Studies 1 and 2, high school juniors and seniors solved Scholastic Assessment Test-Mathematics (SAT-M) problems classified as conventional or unconventional. Algorithmic solution strategies were students' default choice for both types of problems across conditions that manipulated item format and solution time. Use of intuitive strategies on unconventional problems was evident only for high-ability students. Male students were more likely than female students to successfully match strategies to problem characteristics. In Study 3, a revised taxonomy of problems based on cognitive solution demands was predictive of gender differences on Graduate Record Examination-Quantitative (GRE-Q) items. Men outperformed women overall, but the difference was greater on items requiring spatial skills, shortcuts, or multiple solution paths than on problems requiring verbal skills or mastery of classroom-based content. Results suggest that strategy flexibility is a source of gender differences in mathematical ability assessed by SAT-M and GRE-Q problem solving.     

中学生连环数运算中的工作记忆资源分配策略

以初一到高三274名中学生为被试,以连环数认知运算为实验材料,对中学生工作记忆资源的分配策略进行了考察,结果表明:中学生工作记忆资源的分配存在多种策略,且具有多样性;外部策略的正确性高于内部策略和内外部结合策略;中学生使用不同策略解决问题的反应时差异显著;以学生的策略选择的百分比和策略的正确率与解题速度为指标综合考察学生的策略选择来看,中学生较倾向于选择正确率较高的策略。     

Using dual-process theory and analogical transfer to explain facilitation on a hypothetico-deductive reasoning task

Using the analogical transfer paradigm, the present study investigated the competing explanations of Girotto and Legrenzi (Psychological Research 51: 129–135, 1993) and Griggs, Platt, Newstead, and Jackson (Thinking and Reasoning 4: 1–14, 1998) for facilitation on the SARS version of the THOG problem, a hypothetico-deductive reasoning task. Girotto and Legrenzi argue that facilitation is based on logical analysis of the task [System 2 reasoning in Evans’s (Trends in Cognitive Sciences 7: 454–459, 2003) dual-process account of reasoning] while Griggs et al. maintain that facilitation is due to an attentional heuristic produced by the wording of the problem (System 1 reasoning). If Girotto and Legrenzi are correct, then System 2 reasoning, which is volitional and responsible for deductive reasoning, should be elicited, and participants should comprehend the solution principle of the THOG task and exhibit analogical transfer. However, if Griggs et al. are correct, then System 1 reasoning, which is responsible for heuristic problem solving strategies such as an attentional heuristic, should occur, and participants should not abstract the solution principle and transfer should not occur. Significant facilitation (68 and 82% correct) was only observed for the two SARS source problems, but significant analogical transfer did not occur. This lack of transfer suggests that System 1 reasoning was responsible for the facilitation observed in the SARS problem, supporting Griggs et al.’s attentional heuristic explanation. The present results also underscore the explanatory value of using analogical transfer rather than facilitation as the criterion for problem understanding.     

Learning new problem-solving strategies leads to changes in problem representation

Children sometimes solve problems incorrectly because they fail to represent key features of the problems. One potential source of improvements in children's problem representations is learning new problem-solving strategies. Ninety-one 3rd- and 4th-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 6 = 3 + __) and completed a representation assessment in which they briefly viewed similar problems and either reconstructed each problem or identified it in a set of alternatives. Experimental groups then received a lesson about one or both of two solution strategies, the equalize strategy and the add–subtract strategy. A control group received no instruction. All children completed posttest assessments of representation and problem solving. Children taught the equalize strategy improved their problem representations more than those not taught it. This pattern did not hold for the add–subtract strategy. These results indicate that learning new strategies is one source of changes in problem representation. However, some strategies are more effective than others at promoting accurate problem representation.     

Correlational Data,Causal Hypotheses,and Validity

A shared problem across the sciences is to make sense of correlational data coming from observations and/or from experiments. Arguably, this means establishing when correlations are causal and when they are not. This is an old problem in philosophy. This paper, narrowing down the scope to quantitative causal analysis in social science, reformulates the problem in terms of the validity of statistical models. Two strategies to make sense of correlational data are presented: first, a ‘structural strategy’, the goal of which is to model and test causal structures that explain correlational data; second, a ‘manipulationist or interventionist strategy’, that hinges upon the notion of invariance under intervention. It is argued that while the former can offer a solution the latter cannot.     

Individual response profiles in the behavioral assessment of personality

A long-standing problem in the behavioural assessment of personality is the individual specificity of responses. Often, different persons externalize the same trait in different responses. One solution to this problem is to aggregate many different responses. The paper compares the power of response aggregation for predicting self-and other-ratings of personality with two alternative strategies of response selection: the nomothetic strategy of selecting the response with the highest overall predictive power, and the more idiographic strategy of selecting, for each individual, the most extreme response. Seventy subjects were videotaped in a sequence of social situations inducing shyness to various degrees. Five different nonverbal measures of shyness all correlated significantly with the subjects' self- or other-rated shyness, and showed low correlations across subjects and a substantial cross-situational consistency of response profiles. Response aggregation and both strategies of response selection were found to be equally powerful in predicting the subjects' self- and other-ratings of shyness from the five behavioural responses. Therefore, these findings somewhat dampen the hope often expressed in recent theoretical discussions of personality assessment that more respect for the individual case may improve nomothetic assessment procedures.     

Multi‐criteria minisum facility location problems

This paper deals with the multi‐criteria Weber (minisum) problem. Several new solution concepts are introduced, related to ideas of equilibrium between the different aspects covered by the objectives. Structural results are presented characterizing the proposed solution sets and showing the relationships existing with the efficient sets in the point objective location problem. Also, algorithms to solve this problem for the polyhedral norm case are developed. The proposed approach is more general than previously studied on multi‐criteria location problems and is particularly applicable in the real world when different scenarios must be compared. Copyright © 1999 John Wiley & Sons, Ltd.