Decomposing intuitive components in a conceptual problem solving task

Research into intuitive problem solving has shown that objective closeness of participants' hypotheses were closer to the accurate solution than their subjective ratings of closeness. After separating conceptually intuitive problem solving from the solutions of rational incremental tasks and of sudden insight tasks, we replicated this finding by using more precise measures in a conceptual problem-solving task. In a second study, we distinguished performance level, processing style, implicit knowledge and subjective feeling of closeness to the solution within the problem-solving task and examined the relationships of these different components with measures of intelligence and personality. Verbal intelligence correlated with performance level in problem solving, but not with processing style and implicit knowledge. Faith in intuition, openness to experience, and conscientiousness correlated with processing style, but not with implicit knowledge. These findings suggest that one needs to decompose processing style and intuitive components in problem solving to make predictions on effects of intelligence and personality measures.     

An investigation of the relationships between problem solving strategies, representation and memory

There has been considerable debate in recent years about the status of “imagery” in problem solving. The present experiment attempts to show that while subjects may employ representational strategies when they first encounter a class of problems, they abandon such strategies as they gain experience with the problems. It does this by asking subjects to answer unexpected questions which are based upon the information which they have just used to solve a problem. The hypothesis, which is supported by the results, is that increasing experience with problems will be paralleled by a decreasing ability to answer unexpected questions. The experiment also shows that such effects are not attributable to a build-up in proactive interference.     

Visual representation in analogical problem solving

Analogical reasoning has been shown to be effective in the process of solving Dunker’s radiation problem. The spatial nature of the solution to this problem suggests that a visually represented analogue should be particularly effective. However, recent work seems to indicate that a visual analogue does not assist in solving the radiation problem. This paper reports a detailed experimental analysis of the effectiveness of visually represented analogues to the radiation problem. The results show that visual analogues can be effective ff they represent the appropriate features of the problem-solution relationship. The paper also reports on the use of an appropriate visual representation within the problem as a facilitator of analogical reasoning. The results indicate that a visual representation within the problem can act as a facilitator of analogical reasoning, possibly by acting as a retrieval cue.     

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.     

Visual problem solving in the absence of lexical semantics: Evidence from dementia

A case of dissociation between visual and praxic skills and linguistic, referential, and semantic skills is described in a case of dementia. Testing revealed that her visual processing was intact and that she was capable of a wide range of visually based tasks. These included copying geometric figures and block designs; matching items on the basis of number, color, or relative size, matching upper and lower case letters, sequencing numerals, orienting pictures, coins, and letters appropriately, and solving problems based on the identification of geometric patterns or on the identification of logical sequences based on pattern alternation, number, and size increases. In contrast, she was completely incapable of matching pictures on the basis of their semantic or referential meaning; similarly she had lost all comprehension of language. It was argued that this case demonstrates the capacities of a visual system cut off from all symbolic and semantic processes. When examined with other case studies this case provides information about the nature of and interaction between modality-specific perceptual processing and semantic processing, and information about the various cognitive factors involved in visual gnosis. The CT scan indicated frontal atrophy and marked anterior temporal atrophy, a pattern that is consistent with Pick's disease. The case was discussed in terms of the correlation between the pattern of observed strengths and deficits and the neurological pattern of atrophy.     

Language and mathematical problem solving among bilinguals

Does using a bilingual's 1st or 2nd language have an effect on problem solving in semantically rich domains like school mathematics? The author conducted a study to determine whether Filipino-English bilingual students' understanding and solving of word problems in arithmetic differed when the problems were in the students' 1st and 2nd languages. Two groups participated-students whose 1st language was Filipino and students whose 1st language was English-and easy and difficult arithmetic problems were used. The author used a recall paradigm to assess how students understood the word problems and coded the solution accuracy to assess problem solving. The results indicated a 1st-language advantage; that is, the students were better able to understand and solve problems in their 1st language, whether the 1st language was English or Filipino. Moreover, the advantage was more marked with the easy problems. The theoretical and practical implications of the results are discussed.     

The effect of problem size on representation in deductive problem solving

Adult Ss attempted to solve logical deductive reasoning problems that varied systematically in amount of information presented. Methods Ss employed in representing the problem were classified into five main types or modes. The proportion of Ss using a matrix mode of problem representation increased moderately with problems containing large amounts of information. The performance of Ss using a matrix mode of representation suggested that this advantage is related to ease of applying and/or storing the results of logical operations in such a problem space rather than to any facilitative effects of encoding processes. Specific limits to normal processing are hypothesized.     

Cognitive skills in mathematical problem solving in Grade 3

Background. Research on the relationship between cognitive skills and mathematical problem solving is usually conducted on adults or on participants with acquired deficits associated with brain injury (e.g. Cipolotti, 1995 ; Cohen, Dehaene, & Verstichel, 1994 ; McCloskey, 1992 ). Aims. In these studies we wanted to make a contribution to the field of children's mathematical problem solving. The first aim of this study was to investigate whether mathematical problem solving in children is merely determined by semantic elaboration, as hypothesized in some of the models of adult processing (semantic hypothesis). In addition, we aimed to investigate whether there is a continuum from very good to very poor mathematical problem solving among children with mathematical learning disabilities showing immature cognitive skills (maturational lag hypothesis). Sample. The participants were 376 third graders and 107 second graders. Method. The internal structure of the data was analysed with a principal components analysis. In addition, two MANOVA were conducted to compare children with learning disabilities or problems with age‐matched and performance‐matched subjects. Results. Two components, a semantic and a non‐semantic one, were needed to account for an adequate fit of the dataset. In addition, children with mathematical learning disabilities had less‐developed cognitive skills compared with peers without learning disabilities, but they did not differ from younger children on seven of the nine cognitive skills. Conclusions. This study highlighted that children's mathematical problem solving is not determined by one general component. The picture is more complex, since two mathematics components were found. In addition, although our findings point in the direction of the maturational lag hypothesis it may be important to assess the different cognitive skills and especially assess the number system knowledge, since it seems below average in children with mathematical learning disabilities, compared with the knowledge of younger children with comparable skills in mathematics.     

Hindsight bias in insight and mathematical problem solving: Evidence of different reconstruction mechanisms for metacognitive versus situational judgments

This article presents two experiments that used insight and mathematical problems to investigate whether different factors would affect hindsight bias on metacognitive and situational judgments. In both studies, participants initially rated their likelihood of solving each problem within a certain amount of time (metacognitive judgments) and rated the importance of each component of the problem for finding the solution (situational judgments). Next, participants attempted to solve each problem. In Experiment 1, all participants were given solution feedback information, but in Experiment 2, participants were not given any solution feedback. After 1 week, participants were asked to recall their original judgments. Hindsight bias was assessed by comparing the initial with the final ratings. Insight problems and math problems showed different patterns of hindsight bias effects on the metacognitive and situational judgments. The results suggest that two competing models of hindsight effects are actually complementary explanations for judgment reconstruction on different types of judgment tasks.     

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.     

Arithmetic word problem solving: evidence for a magnitude-based mental representation

Previous findings have suggested that number processing involves a mental representation of numerical magnitude. Other research has shown that sensory experiences are part and parcel of the mental representation (or “simulation”) that individuals construct during reading. We aimed at exploring whether arithmetic word-problem solving entails the construction of a mental simulation based on a representation of numerical magnitude. Participants were required to solve word problems and to perform an intermediate figure discrimination task that matched or mismatched, in terms of magnitude comparison, the mental representations that individuals constructed during problem solving. Our results showed that participants were faster in the discrimination task and performed better in the solving task when the figures matched the mental representations. These findings provide evidence that an analog magnitude-based mental representation is routinely activated during word-problem solving, and they add to a growing body of literature that emphasizes the experiential view of language comprehension.     

More on the fragility of performance: choking under pressure in mathematical problem solving

In 3 experiments, the authors examined mathematical problem solving performance under pressure. In Experiment 1, pressure harmed performance on only unpracticed problems with heavy working memory demands. In Experiment 2, such high-demand problems were practiced until their answers were directly retrieved from memory. This eliminated choking under pressure. Experiment 3 dissociated practice on particular problems from practice on the solution algorithm by imposing a high-pressure test on problems practiced 1, 2, or 50 times each. Infrequently practiced high-demand problems were still performed poorly under pressure, whereas problems practiced 50 times each were not. These findings support distraction theories of choking in math, which contrasts with considerable evidence for explicit monitoring theories of choking in sensorimotor skills. This contrast suggests a skill taxonomy based on real-time control structures.     

Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns

Students were taught an algorithm for solving a new class of mathematical problems. Occasionally in the sequence of problems, they encountered exception problems that required that they extend the algorithm. Regular and exception problems were associated with different patterns of brain activation. Some regions showed a Cognitive pattern of being active only until the problem was solved and no difference between regular or exception problems. Other regions showed a Metacognitive pattern of greater activity for exception problems and activity that extended into the post-solution period, particularly when an error was made. The Cognitive regions included some of parietal and prefrontal regions associated with the triple-code theory of (Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506) and associated with algebra equation solving in the ACT-R theory (Anderson, J. R. (2005). Human symbol manipulation within an 911 integrated cognitive architecture. Cognitive science, 29, 313–342. Metacognitive regions included the superior prefrontal gyrus, the angular gyrus of the triple-code theory, and frontopolar regions.     

Insight versus analysis: Evidence for diverse methods in problem solving

Since the Gestalt psychologists made the distinction approximately 100 years ago, psychologists have differentiated between solving problems through analysis versus insight. The present paper presents evidence to support the idea that, rather than conceptualising insight versus analysis as distinct modes of solving problems, it is more useful to conceive of insight and analysis as two approaches within a set of possible solving methods. In the present research, 60 participants solved insight problems while thinking aloud, which provided evidence concerning the processes underlying problem solution. Comparison with performance of a nonverbalisation control group (n = 35) indicated no negative effects of thinking aloud on insight in problem solving. The results supported the idea that various methods are utilised in solving insight problems. The “classic” impasse–restructuring–insight sequence occurred in only a small minority of solutions. A number of other solution methods were found, ranging from relatively direct applications of knowledge, to various heuristic methods, to restructuring arising from new information gleaned from a failed solution. It is concluded that there is not a sharp distinction between solving a problem through analysis versus insight, and implications of that conclusion are discussed.     

Effects of semantic cues on mathematical modeling: Evidence from word-problem solving and equation construction tasks

Mathematical solutions to textbook word problems are correlated with semantic relations between the objects described in the problem texts. In particular, division problems usually involve functionally related objects (e.g., tulips-vases) and rarely involve categorically related objects (e.g., tulips-daisies). We examined whether middle school, high school, and college students use object relations when they solve division word problems (WP) or perform the less familiar task of representing verbal statements with algebraic equations (EQ). Both tasks involved multiplicative comparison statements with either categorically or functionally related objects (e.g., "four times as many cupcakes [commuters] as brownies [automobiles]"). Object relations affected the frequency of correct solutions in the WP task but not in the EQ task. In the latter task, object relations did affect the structure of nonalgebraic equation errors. We argue that students use object relations as "semantic cues" when they engage in the sense-making activity of mathematical modeling.     

Mental representation of three-dimensional objects in visual problem solving and recognition

Subjects inspected sets of flat, separated orthographic projections of surfaces of potential three-dimensional objects. After solving problems based on these orthographic views, subjects discriminated between isometric views of the same objects and drawings of distractor structures. Recognition of the isometrics, which had never been shown during the problem solving phase of the experiment, was excellent. In addition, recognition of isometrics corresponding to problems that had been solved correctly when presented in orthographic form was significantly superior to recognition of isometrics based on problems solved incorrectly. In Experiment 2, conditions were included in which either orthographic or isometric views functioned as problem solving or recognition displays. Only in the case of orthographic problem solving followed by isometric recognition (Experiment 1) was the superiority of recognition for correctly-solved problems over incorrectly-solved problems obtained. The pattern of results suggests that viewers construct mental representations embodying structural information about integrated, three-dimensional objects when asked to reason about flat, disconnected projections.