Development of expertise in mathematical problem solving.

Investigated the distinctive strategies employed by expert and novice problem solvers (forward-chaining and means–ends, respectively) in 7 experiments using 14 mathematics graduates and 162 9–12 yr olds. Exp I studied the course of development of expertise using a subset of kinematics problems. Ss demonstrated the switch from a means–ends to a forward-chaining strategy. This was associated with the conventional concomitants of expertise such as a decrease in the number of moves required for solution. Ss appeared to categorize problems according to the order in which equations would be required. Exps II and III tested the hypothesis that the means–ends strategies used by novices retarded the acquisition of appropriate schemata. The use of nonspecific rather than specific goals was found to enhance the acquisition of expertise, the number of moves required for solution, and the number of equations written without substitutions. Exps IV and V, using geometry problems, duplicated the enhanced rate of strategy alteration found with reduced goal specificity. Results of Exps VI and VII indicated that reduced goal specificity also enhanced the rate at which problem solvers induced appropriate problem categories. It is concluded that in circumstances in which the primary reason for presenting problems is to assist problem solvers in acquiring knowledge concerning problem structure, the use of conventional problems solved by means–ends analysis may not be maximally efficient. (20 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Enhancing third-grade student' mathematical problem solving with self-regulated learning strategies.

The authors assessed the contribution of self-regulated learning strategies (SRL), when combined with problem-solving transfer instruction (L. S. Fuchs et al., 2003), on 3rd-graders' mathematical problem solving. SRL incorporated goal setting and self-evaluation. Problem-solving transfer instruction taught problem-solution methods, the meaning of transfer, and 4 superficial-problem features that change a problem without altering its type or solution; it also prompted metacognitive awareness to transfer. The authors contrasted the effectiveness of transfer plus SRL to the transfer treatment alone and to teacher-designed instruction. Twenty-four 3rd-grade teachers, with 395 students, were assigned randomly to conditions. Treatments were conducted for 16 weeks. Students were pre- and posttested on problem-solving tests and responded to a posttreatment questionnaire tapping self-regulation processes. SRL positively affected performance. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Can offline metacognition enhance mathematical problem solving?

The effectiveness of a short metacognitive intervention combined with algorithmic cognitive instruction was assessed in an elementary school setting. Two hundred thirty-seven 3rd-grade children were randomly assigned to a 5-session metacognitive strategy instruction, an algorithmic direct cognitive instruction, a motivational program, a quantitative-relational condition, or a spelling condition. Children in the metacognitive program achieved significant gains in trained metacognitive skills compared with the 4 other conditions. Moreover, the children in the metacognitive program performed better on trained cognitive skills than children in the algorithmic condition, with a follow-up effect on domain-specific mathematics problem-solving knowledge. Despite the consistency of findings, no generalization effects were found on transfer of cognitive learning. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Cross-sectional and incremental changes in working memory and mathematical problem solving.

Cross-sectional and incremental age effects on cognitive processes that underlie individual differences in components of working memory (WM; phonological loop, visual-spatial sketchpad, executive processing) and mathematical problem-solving accuracy were examined in elementary schoolchildren. A battery of tests was administered that assessed problem solving, achievement, memory, and cognitive processing (inhibition, speed, phonological coding) in children in Grades 1, 2, and 3 (Wave 1) and 1 year later (Wave 2). The results showed that (a) 31% of the explainable within-person changes across testing waves and 42% of the age-related differences in word problem-solving accuracy were related to executive processing and (b) executive processing and reading performance in Year 1 were the only variables that contributed unique variance to Year 2 problem-solving performance. The results support the notion that growth in the executive system is an important predictor of children's problem solving abilities beyond the contribution of reading and calculation skills and that growth in executive processing can operate independently of individual differences in phonological processing, inhibition, and processing speed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Distributing working memory resources during problem solving.

This study examines how problem solvers distribute working memory demands over internal and external resources. Participants recorded notes while performing an arithmetic task. They recorded a majority of intermediate results and labeled many of those results (e.g., C?=?10). When more effort was required to take notes, participants recorded fewer results. Participants with a consistent goal structure recorded fewer results and with practice labeled fewer recorded results than those with varied goal structures. When notes were displayed in a consistent spatial arrangement participants labeled fewer recorded results than when notes appeared in varied locations. These findings indicate that individuals use explicit and implicit strategies for indexing intermediate results. The data support the view that individuals flexibly distribute working memory over internal and external resources in response to situational cost-benefit considerations. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Explicitly teaching for transfer: Effects on third-grade students' mathematical problem solving.

The purpose of this study was to assess the effects of explicitly teaching for transfer by (a) broadening the categories by which students group problems requiring the same solution methods and (b) prompting students to search novel problems for these broad categories. This transfer treatment was combined with instruction on solution methods. The effectiveness of a combined treatment (transfer-plus-solution instruction) was contrasted to solution instruction alone and to teacher-designed instruction. Twenty-four 3rd-grade teachers, with 375 students, were assigned randomly to conditions. Treatments were conducted for 16 weeks. Improvement on immediate- and near-transfer measures supported the utility of solution instruction. Improvement on near- and far-transfer measures revealed the added contribution of explicitly teaching for transfer. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Dynamic assessment of algebraic learning in predicting third graders' development of mathematical problem solving.

Dynamic assessment (DA) involves helping students learn a task and indexing responsiveness to that instruction as a measure of learning potential. The purpose of this study was to explore the utility of a DA of algebraic learning in predicting third graders' development of mathematics problem solving. In the fall, 122 third-grade students were assessed on language, nonverbal reasoning, attentive behavior, calculations, word-problem skill, and DA. On the basis of random assignment, students received 16 weeks of validated instruction on word problems or received 16 weeks of conventional instruction on word problems. Then, students were assessed on word-problem measures proximal and distal to instruction. Structural equation measurement models showed that DA measured a distinct dimension of pretreatment ability and that proximal and distal word-problem measures were needed to account for outcome. Structural equation modeling showed that instruction (conventional vs. validated) and pretreatment calculation skills were sufficient to account for math word-problem outcome proximal to instruction; by contrast, language, pretreatment word-problem skill, and DA were needed to forecast learning on word-problem outcomes more distal to instruction. Findings are discussed in terms of responsiveness-to-intervention models for preventing and identifying learning disabilities. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Role of self-efficacy and self-concept beliefs in mathematical problem solving: A path analysis.

Path analysis was used to test the predictive and mediational role of self-efficacy beliefs in mathematical problem solving. Results revealed that math self-efficacy was more predictive of problem solving than was math self-concept, perceived usefulness of mathematics, prior experience with mathematics, or gender (N?=?350). Self-efficacy also mediated the effect of gender and prior experience on self-concept, perceived usefulness, and problem solving. Gender and prior experience influenced self-concept, perceived usefulness, and problem solving largely through the mediational role of self-efficacy. Men had higher performance, self-efficacy, and self-concept and lower anxiety, but these differences were due largely to the influence of self-efficacy, for gender had a direct effect only on self-efficacy and a prior experience variable. Results support the hypothesized role of self-efficacy in A. Bandura's (1986) social cognitive theory. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A comparison of single and multiple strategy instruction on third-grade students' mathematical problem solving.

The purposes of this study were to assess the differential effects of a single strategy (schema-based instruction; SBI) versus multiple strategies (general strategy instruction, GSI) in promoting mathematical problem solving and mathematics achievement as well as to examine the influence of word problem-solving instruction on the development of computational skills. Eighty-eight 3rd graders and their teachers were assigned randomly to conditions (SBI and GSI). Students were pre- and posttested on mathematical problem-solving and computation tests and were posttested on the Pennsylvania System of School Assessment Mathematics test, a criterion-referenced test that measures student attainment of academic standards. Results revealed SBI to be more effective than GSI in enhancing students' mathematical word problem-solving skills at posttest and maintenance. Further, results indicate that the SBI groups' performance exceeded that of the GSI group on the Pennsylvania System of School Assessment measure. On the computation test, both groups made gains over time. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Metacognition and insight during problem solving: Comment on Metcalfe.

J. Metcalfe (see PA, Vols 73:19144, 74:3081, and 75:22269) used 2 types of metacognitive judgments, prediction of solution and rating of warmth, to support the claim that there is a nonincremental process corresponding to the Gestalt notion of insight that occurs in the solution of certain sorts of problems. However, there are logical and empirical issues that can be raised about Metcalfe's analysis and conclusions: (1) Metcalfe's finding that Ss cannot predict their performance on "insight" problems does not logically necessitate that solutions to such problems occur in sudden bursts of insight, (2) Metcalfe's feeling-of-warmth data for insight problems are problematic as support for her hypothesis, and (3) Metcalfe's definition of insight is circular and does not further an understanding of problem solving. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Conformity and problem solving.

Tests of problem solving, conformity, and intelligence were given to 77 women and 64 men introductory psychology students to investigate the possibility that conformity, a nonintellectual variable, contributes to the variability in achievement in problem solving. The results show a negative correlation between tendency to conform and achievement in problem solving when the influence of intelligence is statistically removed. 17 references. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Physiological patterns accompanying complex problem solving during warning and nonwarning conditions.

Studied physiological responses of 48 college students during both a warning and a nonwarning condition and during anagram problem solving. The warning condition was associated with heart rate deceleration and increased occurrence of galvanic skin responses (GSRs), of skin conductance, and of tonic galvanic skin potentials (GSPs). The warning condition also resulted in significantly faster solution rates. Problem solving was associated with increases in heart rate, occurrence of GSR, and skin conductance. The occurrence of phasic GSP habituated, but the occurrence of tonic GSP maintained a significant response frequency. Large increases in tonic GSP, with minimal phasic increases during problem solving, reflected the bidimensionality of electrodermal activity as a function of the task. Findings support J. I. Lacey and B. C. Lacey's (1970) hypothesis of response fractionation and suggest the utility of physiological measures in the study of putative attentional functioning. (17 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Organization and creative problem solving.

"Groups from four populations differing in their amount of experience and identification with industrial vocation, were compared in their performances on the Change of Work Procedure problem." Arranged from most to least identified, there were 179 groups. "The results are interpreted as providing support for the proposition that the formal authority relations in organizations inhibit creative problem solving. They also suggest that business may be attracting people who can work comfortably, but not creatively, in such formal authority systems." (12 ref.) From Psyc Abstracts 36:04:4LI77M. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Sex differences in problem solving.

Used a 20-item set of problems (similar to those used by E. Sweeney [1953]) in 9 experiments with 558 male and 578 female undergraduates to determine which sex was superior in problem solving, the role of previous experience, whether sex differences extended throughout the domain of problem-solving tasks, whether they extended to other word problems, the role of spatial ability and verbal ability, the role of mathematics aptitude, and the relative importance of aptitude and social learning variables. Results show that the male advantage, averaging 35% across experiments, persisted at the same level as in experiments conducted in the 1950's. Sex differences extended to other word problems. The male advantage was related to similar advantages in spatial and mathematical ability. Aptitude variables dominated attitude and mathematics experience variables in accounting for the sex difference. (48 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Models of group problem solving.

The nature of the group problem solving process was investigated with college Ss working on Maier's horse trading problem (1952). 3 theoretical models were posited to be utilized to interpret the results: an independent model (group response will not differ from those of Ss working alone), a rational model (when one group member solves the problem, the group will adopt the answer), and a consensus model (assuming pressure towards conformity). The results appeared to validate the consensus model. Factors influencing the quality of group responses were evaluated and presented. Suggestions are made regarding the design for future research. From Psyc Abstracts 36:04:4GE53T. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Depression and social problem solving.

20 depressed patients with major depressive disorder, 20 nondepressed matched control Ss, and 17 patients with anxiety disorders were compared in different measures of social problem solving. Problem solving was assessed with the Means-Ends Problem-Solving Test (Study 1), the solution of personal problems, and a problem-solving questionnaire (Study 2). Results showed that, as predicted, depressed Ss suffered from a deficit in problem solving in all 3 measures. The majority of these deficits were also displayed by the clinical control group rather than being specific to a diagnosis of depression. However, depressed Ss produced less effective solutions than did normal and clinical control Ss. The results suggest that depressed and anxious patients may have difficulties at different stages of the problem-solving process. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Content effects in problem categorization and problem solving.

In many domains, the content of a problem (i.e., its surface cover story) provides useful clues as to the type of problem it is and to its solution. Five experiments examined this role of problem content on the problem solution and categorization of algebra word problems with experienced participants. In the first experiment, when problem content was atypical for the the problem's deep structure, people were worse at solving the problem. Differences were also detected in the problem solution where the problem's content was highly correlated with its deep structure versus problems where content was neutral to their deep structure. In the other experiments, problem categorization and determination of information relevance depended on how closely the problem's content matched its deep structure. These results suggest that content may be influential even for experienced problem solvers. The discussion examines the implications for problem schema access and application. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The problem with problem solving

PURPOSE: We conducted a prospective study of patients with primary pterygia to analyze surgically-induced astigmatic changes following pterygium surgery and their relation to pterygium morphology and size. METHODS: One hundred twenty-three eyes of 123 adult patients with primary pterygia were evaluated. Pterygium was graded according to morphology and the extent of corneal encroachment. Manifest refraction was performed preoperatively and at 1, 3, and 6 months postoperatively. Patients underwent either the bare sclera technique of pterygium excision or conjunctival grafting. Surgery was performed by one surgeon. Olsen's method of vector decomposition was used to analyze surgically-induced astigmatism. RESULTS: The mean magnitude of preoperative astigmatism was 0.99 D, with vector decomposition revealing a relative flattening of the cornea in the horizontal meridian. Astigmatism of 1.00 D or more was seen when the pterygium exceeded 3.5 mm beyond the limbus. Postoperatively, a steepening of the cornea in the horizontal meridian was demonstrated, the magnitude of which was related to pterygium size (P = 0.0001). ANOVA testing showed no significant difference (P > 0.05) when pterygia were divided according to morphology. CONCLUSIONS: This study confirms that pterygium excision induces a reversal of pterygium-related corneal flattening. A strong correlation was also found between the horizontal extent of pterygium encroachment and astigmatic change following surgery.     

Instruction by mothers and fathers during problem solving with their young children.

Mothers and fathers of 163 5-year-olds were observed interacting with their children in dyads on 2 separate occasions on a familiar and unfamiliar cognitive activity. Within- and between-family comparisons were conducted. Few differences in the instruction provided by mothers and fathers appeared, and those that did were on the unfamiliar task. On this task, instruction by mothers, within and across families, was more responsive to children's changing skill than was instruction by fathers. Directive and disapproving comments by parents were related to poorer posttest performance by the child. High level of instruction by both parents was related to better posttest performance than was low level of instruction by parents. Contributions of parents to their children's cognitive development are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Variable attention facilitates creative problem solving.

Traditionally, researchers have associated creativity with defocused attention. Recent experimental evidence contradicts this notion by demonstrating that in creative people, defocused attention is a variable state rather than a stable trait. Specifically, creative people are better at adjusting their focus of attention as a function of task demands. When the task is ill defined and ambiguity is high, attention is defocused, resulting in slower processing on the task. In contrast, when the task is well defined and ambiguity is low, attention is focused, resulting in faster processing on the task. This flexibility can confer distinct advantages to creative people in the course of problem solving as changes in the structure of the problem necessitate corresponding adjustments in solution strategy. (PsycINFO Database Record (c) 2010 APA, all rights reserved)