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.     

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.     

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.     

Effects of a problem-solving strategy on the independent completion of vocational tasks by adolescents with autism spectrum disorder

Individuals with autism spectrum disorder (ASD) often have few employment opportunities and a lower job quality than individuals of typical development. Social deficits and lack of independence may contribute to underemployment and unemployment of individuals with ASD. The ability to solve problems might ameliorate some of these barriers. We taught four adolescents with ASD a problem-solving strategy (i.e., use of a textual activity schedule) to assist with independent completion of vocational tasks in the face of three types of problems (e.g., missing or broken items) and nonproblem situations. Following introduction of the problem-solving strategy, all four participants independently completed the tasks when a problem was presented and responding generalized to untaught vocational tasks.     

Selecting analogous problems: Similarity versus inclusiveness

Students were asked to select one of two analogous problems in order to solve algebra word problems. In Experiment 1, one problem was less inclusive and the other was more inclusive than a test problem. The students judged the complexity and similarity of problems, selected analogous problems, and used the solutions to solve test problems. They performed significantly better on the test problems when given the more inclusive solutions, but used perceived similarity rather than inclusiveness to select analogous problems. The same pattern of results occurred in Experiment 2, in which isomorphic problems replaced the more inclusive problems. The results show that students are deficient in selecting good analogies, both from the same category (Experiment 1) and from a different category (Experiment 2). Students who saw the analogous solutions (Experiment 3) or were majoring in mathematics (Experiment 4) were more likely to select an isomorphic problem over a less inclusive problem, but were not more likely to select a more inclusive over a less inclusive problem.     

From poor performance to success under stress: working memory, strategy selection, and mathematical problem solving under pressure

Two experiments demonstrate how individual differences in working memory (WM) impact the strategies used to solve complex math problems and how consequential testing situations alter strategy use. In Experiment 1, individuals performed multistep math problems under low- or high-pressure conditions and reported their problem-solving strategies. Under low-pressure conditions, the higher individuals' WM, the more likely they were to use computationally demanding algorithms (vs. simpler shortcuts) to solve the problems, and the more accurate their math performance. Under high-pressure conditions, higher WM individuals used simpler (and less efficacious) problem-solving strategies, and their performance accuracy suffered. Experiment 2 turned the tables by using a math task for which a simpler strategy was optimal (produced accurate performance in few problem steps). Now, under low-pressure conditions, the lower individuals' WM, the better their performance (the more likely they relied on a simple, but accurate, problem strategy). And, under pressure, higher WM individuals performed optimally by using the simpler strategies lower WM individuals employed. WM availability influences how individuals approach math problems, with the nature of the task performed and the performance environment dictating skill success or failure.     

Beyond Compliance: Mother–Child Joint Action During a “Do” Task

Much of the literature on parent-toddler interactions does not account for the lack of independence in data. This investigation used a theory and method appropriate for examining the moment-to-moment patterns of joint actions between mothers and toddlers during a “do” task. Using contextual action theory to guide a mixture of methods, we observed and described the joint goal-directed series of actions of mothers and their toddlers as well as mothers’ intentions, meanings, and emotions they ascribed to their actions. Additionally, the associations between the patterns of joint mother-toddler actions and children’s negative emotionality were examined. Thirty mothers and their children participated in the study. Data collection included video-recorded activity, video recall interviews, and self-report questionnaires. Quantitative and qualitative analysis of the data revealed that (a) children were mostly engaged in the task alongside with their mothers’ on-going involvement; (b) mothers attributed a range of meanings to their interactions with their children; and (c) mothers’ perceptions of their toddlers’ dimensions of emotional negativity were associated with the organization of dyads’ joint goal directed actions—both when children were engaged in the task or self-focused.     

Social-Psychological Origins of Feelings of Presence: Creating Social Presence With Machine-Generated Voices

Two experiments were conducted to test several questions regarding very young children's (6–24 months) learning (i.e., simple action imitation and word learning) from video. Specifically, this study tested the video deficit, which is the tendency for infants and toddlers to learn significantly more effectively from live information than they do when identical information is presented on a screen. First, the video deficit was explored using two different tasks. Overall, the pattern of results was similar for action imitation and word learning. Specifically, the video deficit was present for both simple action imitation and for word learning in the middle cohort, but not present for younger and older children. Second, there was some mitigation of the video deficit from seeing socially meaningful actors for action imitation; however for word learning the effect only approached significance. Third, repetition helped children learn words more effectively, especially for the youngest and oldest cohort; however, repetition did not help for simple task imitation.     

On Distinguishing Epistemic from Pragmatic Action

We present data and argument to show that in Tetris—a real-time, interactive video game—certain cognitive and perceptual problems are more quickly, easily, and reliably solved by performing actions in the world than by performing computational actions in the head alone. We have found that some of the translations and rotations made by players of this video game are best understood as actions that use the world to improve cognition. These actions are not used to implement a plan, or to implement a reaction; they are used to change the world in order to simplify the problem-solving task. Thus, we distinguish pragmatic actions—actions performed to bring one physically closer to a goal—from epistemic actions—actions performed to uncover informatioan that is hidden or hard to compute mentally. To illustrate the need for epistemic actions, we first develop a standard information-processing model of Tetris cognition and show that it cannot explain performance data from human players of the game—even when we relax the assumption of fully sequential processing. Standard models disregard many actions taken by players because they appear unmotivated or superfluous. However, we show that such actions are actually far from superfluous; they play a valuable role in improving human performance. We argue that traditional accounts are limited because they regard action as having a single function: to change the world. By recognizing a second function of action—an epistemic function—we can explain many of the actions that a traditional model cannot. Although our argument is supported by numerous examples specifically from Tetris, we outline how the new category of epistemic action can be incorporated into theories of action more generally.     

The Effects of Presenting Multidigit Mathematics Problems in a Realistic Context on Sixth Graders' Problem Solving

Mathematics education and assessments increasingly involve arithmetic problems presented in context: a realistic situation that requires mathematical modeling. This study assessed the effects of such typical school mathematics contexts on two aspects of problem solving: performance and strategy use. Multidigit arithmetic problems presented in two conditions—with and without a realistic context—were solved by 685 sixth graders from The Netherlands. Regarding performance, the same (latent) ability dimension was involved in solving both types of problems, and the presence of a context increased the difficulty level of the division problems but not of other operations. Regarding strategy use, strategy choice and strategy accuracy were not affected by the presence of a problem context. In sum, the presence of a typical context in multidigit arithmetic problems had no marked effects on students' problem-solving behavior, which held for different subgroups of students with respect to language ability and gender.     

Acquiring an understanding of design: evidence from children's insight problem solving

The human ability to make tools and use them to solve problems may not be zoologically unique, but it is certainly extraordinary. Yet little is known about the conceptual machinery that makes humans so competent at making and using tools. Do adults and children have concepts specialized for understanding human-made artifacts? If so, are these concepts deployed in attempts to solve novel problems? Here we present new data, derived from problem-solving experiments, which support the following. (i) The structure of the child's concept of artifact function changes profoundly between ages 5 and 7. At age 5, the child's conceptual machinery defines the function of an artifact as any goal a user might have; by age 7, its function is defined by the artifact's typical or intended use. (ii) This conceptual shift has a striking effect on problem-solving performance, i.e. the child's concept of artifact function appears to be deployed in problem solving. (iii) This effect on problem solving is not caused by differences in the amount of knowledge that children have about the typical use of a particular tool; it is mediated by the structure of the child's artifact concept (which organizes and deploys the child's knowledge). In two studies, children between 5 and 7 years of age were matched for their knowledge of what a particular artifact "is for", and then given a problem that can only be solved if that tool is used for an atypical purpose. All children performed well in a baseline condition. But when they were primed by a demonstration of the artifact's typical function, 5-year-old children solved the problem much faster than 6-7-year-old children. Because all children knew what the tools were for, differences in knowledge alone cannot explain the results. We argue that the older children were slower to solve the problem when the typical function was primed because (i) their artifact concept plays a role in problem solving, and (ii) intended purpose is central to their concept of artifact function, but not to that of the younger children.     

Schema induction in problem solving: a multidimensional analysis

The present research examined the processes of schema formation in problem solving. In 4 experiments, participants experienced a series of tasks analogous to A. S. Luchins' (1942) water jar problems before attempting to solve isomorphic target problems. Juxtaposing illustrative source instances varying in procedural features along multiple dimensions promoted the construction of a general schema that facilitated solving an isomorphic problem requiring a novel procedure. Exposure to less variant problems led to faster initial learning, but narrower and fixed schemas (mental set), whereas exposure to variant procedures led to slower initial learning, but broader and more flexible schemas. The findings support the dimensional specificity hypothesis: Generalization along 1 dimension facilitates transfer to a target problem differing from the source problems in that dimension.     

Goals and Everyday Problem Solving: Examining the Link between Age-Related Goals and Problem-Solving Strategy Use

ABSTRACT

Qualitative interviews on family and financial problems from 332 adolescents, young, middle-aged, and older adults, demonstrated that developmentally relevant goals predicted problem-solving strategy use over and above problem domain. Four focal goals concerned autonomy, generativity, maintaining good relationships with others, and changing another person. We examined both self- and other-focused problem-solving strategies. Autonomy goals were associated with self-focused instrumental problem solving and generative goals were related to other-focused instrumental problem solving in family and financial problems. Goals of changing another person were related to other-focused instrumental problem solving in the family domain only. The match between goals and strategies, an indicator of problem-solving adaptiveness, showed that young individuals displayed the greatest match between autonomy goals and self-focused problem solving, whereas older adults showed a greater match between generative goals and other-focused problem solving. Findings speak to the importance of considering goals in investigations of age-related differences in everyday problem solving.     

Language Format and Analogical Transfer among Bilingual Problem Solvers in the Philippines

Is analogical problem solving of bilinguals affected by the language in which the problem is given? Three experiments study whether the language used for the problem affects analogical transfer between word problems among Filipino-English bilinguals. Subjects were given a set of study problems all of which were either in Filipino or English. They were then given a set of analogous transfer-test problems either in Filipino or in English. Experiment 1 showed a language-compatibility effect; transfer was better when the language of the study and transfer-test problems were the same. Experiments 2 to 3 studied the locus of the effect. The data suggest that language compatibility facilitates the retrieval of relevant, superficially similar problem information but does not affect the process of applying the same problem information to the new problem. The results are discussed in terms of their implications to problem-solving behaviour of bilinguals and to the education of bilingual students.     

Creative Failure: Why Can't People Solve Creative Problems

Although people attempt many creative problem solutions, in general, most creative problem-solving efforts, at least real-world efforts, fail. In the present effort, we examine the reasons creative problem-solving efforts typically fail. We argue that creative problem-solving efforts fail, in part, due to the fundamental nature of the kinds of problems that call for creative thought. However, the nature of people's creative problem-solving skills and the context in which they attempt to develop and implement creative problem solutions also results in failed attempts to solve creative problems. Based on these observations, we discuss how one might seek to develop people to encourage more success in creative problem-solving efforts.     

Discovery Learning and Transfer of Problem-Solving Skills

A framework for understanding the effects of discovery learning on the transfer of problem-solving skill is presented. A distinction is drawn between applying a learned strategy on a transfer problem versus having to generate a novel strategy to solve a transfer problem. The main premise of the framework is that requiring discovery of a strategy while in training encourages the activation or refinement of procedures that are useful for generating a novel strategy. In general, then, the primary benefit of discovery is that it should facilitate transfer to tasks requiring a novel strategy. Learning by discovery, however, may provide little benefit for tasks that can be completed only by applying the learned strategy. Two experiments provided support and further refinement of this hypothesis. Experiment 1 used a transfer problem that could be solved with the general strategy learned in training but required new move sequences to instantiate the strategy. The results indicated that, when transfer required new move sequences to implement a general strategy learned previously, discovery did not enhance transfer of that strategy. In experiment 2, some transfer problems required using a strategy other than that learned in training. As predicted, in this transfer situation, having to discover a strategy while in training produced better transfer than being provided with a strategy in training. Thus, discovering a strategy provided benefits when a new strategy had to be generated to solve a transfer problem but not when the learned strategy could be applied, albeit with new moves, to the transfer problem. Educational implications are discussed.     

目标启动对类比问题解决策略迁移的影响

运用实验法和问卷法对小学2年级和5年级学生的类比问题解决策略的迁移进行了研究。结果表明:无论儿童年龄大小,启动掌握目标都可以有效地促进类比问题解决策略的迁移。能力知觉与回避型表现目标存在交互作用,在回避型表现目标启动条件下,能力知觉低的儿童迁移成绩较差。儿童自身的成就目标取向未对迁移产生影响。     

Decision theory and artificial intelligence II: The hungry monkey

This paper describes a problem-solving framework In which aspects of mathematical decision theory are incorporated into symbolic problem-solving techniques currently predominant in artificial intelligence. The utility function of decision theory IS used to reveal tradeoffs among competing strategies for achieving various goals, taking into account such factors as reliability, the complexity of steps in the strategy, and the value of the goal. The utility function on strategies can therefore be used as a guide when searching for good strategies. It is also used to formulate solutions to the problems of how to acquire a world model, how much planning effort is worthwhile, and whether verification tests should be performed. These techniques are illustrated by application to the classic monkey and bananas problem.     

Seeing the Problem: An Explanation From Pólya

In this article, we focus on the instructional explanation of guessing as a heuristic for solving the Five Planes Problem (FPP), given in a lesson by George Pólya. We use the analysis of the lesson to test the applicability of what is known about instructional explanations in elementary mathematics to higher mathematics. The most salient characteristic of Pólya's lesson is the profusion of models and representations used to develop a sense of the problem and to support the instructional explanation. Pólya used analogical models to transform the complex FFT to a simpler one, and he used representations to extend the perspective on the problem. Introducing such models and representations requires keeping track of the links between them and the original problem. Pólya excelled in this endeavor, and his passage to each new representation or model was generally justified in terms of the goals of the explanation. Another very important feature of an instructional explanation is problem identification, a fragile goal state that needs to be constantly maintained when the problem being explained is complex. For this lesson, we examine how Pólya established and maintained that goal. Finally, we offer some insight on instructional explanations in a situation in which the teacher needs to fulfill two goals at the same time. Here, the first goal was to teach students both how to solve FPP and how to use guessing as a problem-solving heuristic or strategy; the second goal was to show us how to teach guessing. Keeping these goals in mind, we offer some suggestions on how to teach metaskills through mathematical problems in accordance with the current understanding of constructivist approaches to learning.     

Randomized controlled trial of a brief problem-orientation intervention for suicidal ideation

Empirical evaluations suggest that problem orientation, the initial reaction to problems, differentiates suicidal youth from nonchnical controls and nonideating psychiatric controls. One promising area for intervention with suicidal youth relates to enhancing this specific coping skill. Nonclinical participants (N = 110) with active suicidal ideation were randomly assigned to receive a brief problem-orientation intervention or a control procedure. The current study evaluated the benefits of a brief prevention intervention based on the Problem-Solving Therapy model. Exposure to a brief video intervention regarding problem solving and coping skills was sufficient to elicit significant decreases in suicidal ideation and depression, but the intervention did not elicit improvements in problem orientation or other problem-solving abilities. These findings provide preliminary support for a brief, video-based problem-solving and coping skills module. However, considerable work in developing secondary prevention interventions remains.