Influence of Item Content and Stereotype Situation on Gender Differences in Mathematical Problem Solving

On standardized tests of mathematical problemsolving, the typical finding has been that women scorelower than men. Experiment 1 manipulated gender labeling(female character, male character, or gender neutral) within the problem question to seewhether this accounted for gender differences inmathematical problem solving. Sixty-four seventh andeighth graders were tested on modified versions of theCanadian Test of Basic Skills (CTBS) with the resultsshowing that although gender labeling affected studentsperformance, it did not account for gender differences.Experiment 2 manipulated both gender labeling and gender stereotype threat for 174 universitystudents writing modified versions of a modelStandardized Achievement Test (SAT). Again, genderlabeling within problem questions did not account forgender differences. However, women scored lower thanmen when they believed that the test had previouslyshown gender differences. There was no gender differencein the performance of the same women and men when they believed that the test was merelycomparing Canadian students with American students. Thissuggests that gender stereotype threat could be a keyfactor in explaining gender differences in mathematical problem solving.     

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.     

Gender Differences in Problem Solving: Expectancy and Problem Context

In this study, we investigated gender differences in problem solving as a function of problem context and expectations for success. Subjects were 90 women and 56 men from introductory psychology classes who were administered a set of mathematical problems varying in gender context under male-expectancy, female-expectancy, and neutral-expectancy conditions. No significant gender differences or interaction of gender with problem context were found. However, significant effects of problem context and expectancy were found. Both male and female subjects made higher scores on female-context problems and lower scores in the female-expectancy condition. When mathematics aptitude was used as a covariate, an interaction of gender with expectancy was found. Men in the female-expectancy condition made significantly lower scores than either men or women in the male-expectancy condition.     

Gender and Mathematical Problem Solving

The relationship between gender and mathematical problem-solving among high ability students depends on the attributes of the problem solving questions. This was evident in the present study of 12-year-olds. The children were from predominately White families. Eighty-three males and 76 females were tested in both the fall and the spring on the Fennema-Sherman Mathematics Attitudes Scales and on the Canadian Test of Basic Skills (CTBS). In the Spring, students were also tested on the GAUSS. Both the CTBS and the GAUSS measure mathematical problem solving. Among high ability students, there were gender differences on the problem-solving scale of the CTBS but not on the GAUSS, even though the GAUSS was independently rated as the more abstract and difficult of the two tests. The present study describes the implications of this for the question of the origin of gender differences in mathematics, and also looked at the relationship between attitudes toward mathematics and mathematical problem-solving performance.     

小学三年级数学学优生与学困生解决比较问题的差异

运用实验法和临床访谈法研究了数学学优生与学困生在解决比较问题时的差异及元认知对解题成绩的影响。被试为40名小学三年级学生。比较问题分为一致问题和不一致问题。元认知包括元认知知识和元认知监控技能。结果表明：(1)学优生与学困生解决比较问题的成绩差异显著,学优生在一致问题和不一致问题上的解题成绩均优于学困生。这种差异与其解题时所运用的表征策略有关。(2)学生在解决比较问题中出现的主要错误为转换错误,在不一致问题中出现的错误多于一致问题中出现的错误。(3)学优生与学困生在元认知知识和监控技能上均有显著差异,元认知监控技能对解决比较问题的成绩有显著预测作用。     

Flexible strategy use: A characteristic of gifted problem solving

The aim of the study was to investigate the relationship between flexible strategic thinking and problem solving performance. In the first experiment, gifted, average, and poor problem solvers performed a number of tasks which were designed to provoke rigidity. The results indicated that the average and poor performance groups did not differ in the number of answers indicating response and perceptual set. On the other hand, creative and intelligent students showed fewer rigid answers than the other two groups. In the second experiment, gifted and average students were asked to think aloud while solving different problems with open and closed solution situations. The thinking aloud protocols were analyzed by classifying the statements into different strategy types. Gifted students used a variety of strategies when solving problems, as well as different strategies for different problem types. No such differences were observed with the average problem solvers. The results of both experiments demonstrated consistent evidence for the role of flexible strategic thinking in gifted problem solving, as well as less direct evidence of the importance of metacognitive knowledge.     

Mental models in word problem-solving: A comparison between American and Korean sixth-grade students

Mental model analysis was conducted in word problem solving using American (n = 42) and Korean (n = 44) sixth-graders. Two levels of mental models–the problem model and the mathematical model–constructed in the process of word problem solving were investigated. Categories for correct and incorrect models were developed to be used in think-aloud protocol analysis. The majority of students who constructed correct problem models classified the problems rapidly according to solution procedures as they read the problems, or restated the problems focusing on the specific words relating them to other statements in the problems. However, significant differences were found between American and Korean sixth-graders' in some categories of problem and mathematical models. While many Korean sixth-graders' problem-solving process steps were proceduralized, this was not the case for American sixth-graders, even for those who had a readily accessible knowledge base of basic mathematical facts.     

小学生视觉-空间表征类型和数学问题解决的研究

本研究考察并比较了四至六年级儿童的视觉－空间表征策略、数学问题解决和空间视觉化能力。结果表明：五、六年级儿童的解题正确率、使用图式表征策略的程度显著高于四年级儿童;使用图像表征策略的程度各年级无显著差异。将数学问题分成三个难度等级,发现年级差异主要表现在难度等级1的题目上。另外,六年级儿童的空间视觉化能力显著高于四年级儿童。     

Type of stimulus mediates the relationship between working-memory performance and type of precocity

The relationship between type of stimulus (numeric and verbal) and type of precocity (mathematical and verbal) was examined in tasks designed to tap three aspects of working memory: encoding, capacity, and manipulation of information. The tasks included semantic categorization, odd-even categorization, recall of five-item lists after semantic categorization, and recall of items in a continuous paired-associates task. Correlations between task performance and the mathematical and verbal portions of the Scholastic Aptitude Test (SAT-M and SAT-V) were computed for gifted youth. There were no sex differences in the performance measures or in the pattern of correlations between performance and SAT scores. The analysis revealed positive relationships between SAT-M scores and numeric categorization latency, recall in the continuous paired-associate task with words and digits, and recall of digit lists. SAT-V scores were related only to word recall in the continuous paired-associate task and recall of word lists. In the working-memory tasks used, mathematical precocity is more strongly related to performance than is verbal precocity. The relationship is especially strong with numeric stimuli, even when the numeric stimuli are simply items to be remembered. The relationship between type of stimulus and type of precocity suggests underlying differences between verbally and mathematically precocious youth in how different types of stimuli are represented in memory.     

小学生数学问题解决中视觉空间表征的研究

本研究区分了两类数学应用题:非视觉化题目与视觉化题目,采用数学测验与个别访谈相结合的方法,考察了54名小学四、五、六年级不同学业水平学生的视觉空间表征。结果表明:图式表征在非视觉化题目与视觉化题目上都极大地促进了问题解决,图像表征妨碍非视觉化题目的解决但与视觉化题目的解决无关,并提出图式表征和图像表征在两类题目上有不同的含义。六年级学生的解题成绩及图式表征有显著的提高,但图像表征与年级因素无关。差生的图式表征能力很差,而在视觉化题目上使用图像表征显著地多于优生及中等生。在非视觉化题目的非视觉空间表征与图式表征之间的转换灵活性上,优生表现了明显的优势。     

Teaching students to solve insight problems: Evidence for domain specificity in creativity training

The purpose of this research was to investigate whether insight problem solving depends on domain‐specific or domain‐general problem‐solving skills, that is, whether people think in terms of conceptually different types of insight problems. In Study 1, participants sorted insight problems into categories. A cluster analysis revealed 4 main categories of insight problems: verbal, mathematical, spatial, and a combination of verbal or spatial. In Studies 2 and 3, participants received training in how to solve verbal, spatial, or mathematical problems, or all 3 types. They were taught that solutions to verbal insight problems lie in defining and analyzing the terms in the problem, solutions to mathematical insight problems lie in a novel approach to numbers, or solutions to spatial insight problems lie in removing a self‐imposed constraint. In both studies, the spatial trained group performed better than the verbal trained group on spatial problems but not on other types of problems. These findings are consistent with the idea that people mentally separate insight problems into distinct types. Implications for instruction in insight problem solving are discussed.     

Evidence of a detrimental effect of extrinsic incentives on breaking a mental set

College students were either rewarded ($1.50) or not for solving a series of 10 water-jar problems. The first 9 of these were designed to establish a set for a relatively complicated, three-jar solution (B ? A ? 2C). The 10th problem was a set breaker which required that subjects discover a simple (A-C) solution. The mental activities needed to produce a solution to the two types of problems (Problems 1 to 9 and Problem 10) were, therefore, different and one interest was in seeing whether reward would be detrimental for performance on both types of problems. It was not. Poorer reward group performance was obtained only on Problem 10. This result was not due to between-group differences in mathematical ability, and explanations in terms of differences in time taken to check answers or concern over quality of performance were considered and discounted. Measures of intrinsic interest (subject evaluations of the task and a measure of willingness to volunteer for future research) failed to support the belief that interest differences account for performance differences.     

Distribution of Spatial Visualization and Mathematical Problem Solving Scores: A Test of Stafford's X-linked Hypotheses

This study investigated distribution of spatial visualization scores (Space Relations test of the Differential Aptitude Test) and mathematical problem solving scores (Mental Arithmetic Problems) obtained by 161 male and 152 female, 9th grade, white students for fit to the distributions predicted by the X-linked hypotheses of recessive inheritance of these skills. Data did not support the X-linked hypotheses. No significant sex-related differences were found between mean scores of tests of spatial visualization or mathematical problem solving.     

Gender,status and the use of power strategies

The present study examined the effects of gender and status on the use of power strategies. The experiment consisted of a computer‐based problem‐solving task performed in pairs, where participants interacted with simulated long‐distance partners. Participants were 36 female and 38 male undergraduate students, who were assigned to be influencing agents and were required to convince their partners to accept their help in the problem‐solving process. Status was manipulated by the extent to which partners were dependent upon the participants' resources. Partners were either same sex or other sex. Results indicated an interactive effect of agent gender by status. Men used more frequently ‘masculine’‐typed and less frequently ‘feminine’‐typed strategies than did women in low status positions, whereas in high status positions no significant gender differences in power strategy choices were found. These findings suggest that gender differences and similarities vary according to social contexts. Implications of the findings for both theory and practice are discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

中学生自我觉知的日常问题解决能力研究

采用问卷法对907名中学生自我觉知的日常问题解决能力进行研究。结果表明,自我觉知的日常问题解决能力存在显著的领域差异;对不同领域的日常问题,初中生与高中生自我觉知的能力有不同的差异模式,我向思维会导致初中生在某些领域高估其能力水平;性别差异以及独生与非独生子女差异在多项能力评估中显著存在;大多数自我觉知能力与自评健康状况、家庭经济状况、自评外在形象、做家务事的次数及父母管教严格程度有显著的正相关。     

Working memory and phonological processing as predictors of children’s mathematical problem solving at different ages

The study explored the contribution of working memory (WM) to mathematical problem solving in younger (8-year-old) and older (11-year-old) children. The results showed that (1) significant age-related differences in WM performance were maintained when measures of phonological processing (i.e., digit naming speed, short-term memory, phonological deletion) were partialed from the analysis; (2) WM predicted solution accuracy of word problems independently of measures of problem representation, knowledge of operations and algorithms, phonological processing, fluid intelligence, reading, and math skill; and (3) a second-order WM factor was correlated with problem solving, suggesting that a general or executive system underlies age-related performance. The results were interpreted as support for the notion that the executive system was an important predictor of age-related changes in problem solving beyond the contribution of math and reading skills, and this system operates independently of the phonological system and domain-specific knowledge in predicting solution accuracy.     

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.     

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.     

中等生与学困生解决故事问题的认知差异

通过对40名小学生解决故事问题的过程分析,研究表明:(1)中等生与学困生在解题结果和解题过程2个方面都存在显著差异;(2)中等生和学困生的认知差异体现在,前者的特点主要是"比较"、"编码"和"目标定向",而后者的特点是"联结"和"比较"。     

Working in Dyads and Alone: Examining Process Variables in Solving Insight Problems

This study investigated the effects of working in dyads and their associated gender composition on performance (solution rate and time) and process variables (number of impasses, number of passed solutions, and number of problem solving suggestions and interactions) in a set of classic insight problem solving tasks. Two types of insight problems were used: multiple moves available (MMA) and few moves available (FMA) problems. The results showed no effects on number of problems solved or solving time, but group and gender composition effects emerged when looking at process variables such as number of impasses, time to impasse, number of passed solutions, number of problem solving suggestions, and number of interactions between dyad members. Results support prior evidence that process loss in creative problem solving may be due to differences in interpersonal interactions during group problem solving that do not necessarily affect overall problem solving success.