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。     

The Nature of External Representations in Problem Solving

This article proposes a theoretical framework for external representation based problem solving. The Tic-Tac-Toe and its isomorphs are used to illustrate the procedures of the framework as a methodology and test the predictions of the framework as a functional model. Experimental results show that the behavior in the Tic-Tac-Toe is determined by the directly available information in external and internal representations in terms of perceptual and cognitive biases, regardless of whether the biases are consistent with, inconsistent with, or irrelevant to the task. It is shown that external representations are not merely inputs and stimuli to the internal mind and that they have much more important functions than mere memory aids. A representational determinism is suggested—the form of a representation determines what information can be perceived, what processes can be activated, and what structures can be discovered from the specific representation.     

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。     

Designing Knowledge Scaffolds to Support Mathematical Problem Solving

We present a methodology for designing better learning environments. In Phase 1, 6th-grade students' (n = 223) prior knowledge was assessed using a difficulty factors assessment (DFA). The assessment revealed that scaffolds designed to elicit contextual, conceptual, or procedural knowledge each improved students' ability to add and subtract fractions. Analyses of errors and strategies along with cognitive modeling suggested potential mechanisms underlying these effects. In Phase 2, we designed an intervention based on scaffolding this prior knowledge and implemented the computer-based lessons in mathematics classes. In Phase 3, we used the DFA and supporting analyses to assess student learning from the intervention. The posttest results suggest that scaffolding conceptual, contextual, and procedural knowledge are promising tools for improving student learning.     

Intuition and Visualization in Mathematical Problem Solving

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization.     

从困境到超越：顿悟的脑机制研究

在1913~1920年间,德国心理学家苛勒进行了有关顿悟的经典实验。苛勒发现挑战了当时占主导地位的“尝试-错误”学习理论,表明问题解决行为可以以一种突发的形式产生,并在一瞬间获得对于问题情境的全新的思考和把握。但是直到最近心理学家才开始逐渐认识顿悟的大脑机制。该文介绍了近期围绕顿悟过程开展的脑成像研究,介绍了有关的研究方法,并讨论了参与顿悟过程的各个关键脑区的功能     

集体问题解决中的认知表征、行为惯例和动态能力

以王建安和张钢(2008)开发的三人TTT纸牌游戏(原游戏)及其变式(变式游戏)为实验任务, 研究了集体问解决中认知表征和行为惯例对提高动态能力的作用。共有240名被试, 分成80个三人小组, 参加了前后两场各20盘游戏。80个小组按照是否被告知游戏的问题表征(即有无认知表征), 以及是玩原游戏还是玩变式游戏, 均分成4个区组来玩第一场游戏。在第二场游戏中, 4个区组都玩变式游戏。以玩每盘游戏的平均步时和在问题表征中走过的路径为指标测量惯例, 以玩第一场原游戏习得的能力向玩第二场变式游戏的迁移率来衡量动态能力。结果发现：一方面, 适当的认知表征有助于提高集体问题解决的动态能力; 另一方面, 没有适当认知表征支持的行为惯例反而会阻碍动态能力的发展。概括地说, 动态能力的决定因素是认知表征而不是行为惯例。     

Training in Self-Explanation and Self-Regulation Strategies: Investigating the Effects of Knowledge Acquisition Activities on Problem Solving

Previous research has found positive correlations between particular strategies students use while studying to explain instructional materials to themselves and student performance on associated problem-solving tasks (Chi, Bassok, Lewis, Reimann, & Glaser, 1989; Pirolli & Bielaczyc, 1989; Pirolli & Recker, 1994). In the study reported here, we investigate the causal nature of this relation. This was accomplished by identifying a set of self-explanation and self-regulation strategies used by high-performance students in our earlier studies. We used strategy training to manipulate students' application of these strategies and examined the impact of their use on student explanations and performance. Twenty-four university students with no prior programming experience worked through a sequence of programming lessons. Following introductory lessons, participants received interventions involving explicit training in the strategies (instructional group) or received a similar set of interventions but no explicit training (control group). The instructional group showed significantly greater gains than the control group in the use of self-explanation and self-regulation strategies from the pre- to postinterventions lessons. Increased strategy application was accompanied by significantly greater performance gains. The results indicate that the particular self-explanation and self-regulation strategies used in training contribute to learning and problem-solving performance.     

Re-examining the Relationship Between Need for Cognition and Creativity: Predicting Creative Problem Solving Across Multiple Domains

Prior studies have demonstrated inconsistent findings with regard to the relationship between need for cognition and creativity. In our study, measurement issues were explored as a potential source of these inconsistencies. Structural equation modeling techniques were used to examine the factor structure underlying the 18-item need for cognition scale in three independent samples drawn from three prior studies. In addition, the relationships between need for cognition and the quality, originality, and elegance of creative problem solutions across multiple domains of performance were analyzed. Across all 3 samples, the bifactor model fit the data better than the 2-factor and 1-factor need for cognition models. After controlling for method-specific variance, the trait factor of the bifactor model showed consistently positive relationships with all 3 facets of creative performance and showed stronger relationships with these facets than the composite score results reported in prior studies. Implications for research and practice are discussed.     

Text Integration and Mathematical Connections: A Computer Model of Arithmetic Word Problem Solving

Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence-level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence-level and text integration processes. Working memory variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K-3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process-oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships.     

问题解决中图式与策略的关系:来自表征复杂性模型的说明

采用口语报告法收集了26名小学三年级儿童解决两道复杂的算术应用题时的口语报告和作业资料,探讨了问题解决中图式与策略的关系。结果发现,对二者关系的解释要以问题表征复杂性为中介：如果被试的关系图式水平较高,表征复杂性(特别是表征深度)也会较大,相应会选择领域专门性更强的快捷策略;如果缺乏必需的关系图式。表征深度就降低,甚至不能正确表征问题,这时只好使用常规策略甚至错误策略。这个结论适用于有多种表征和解题方式的问题中。     

元认知训练对提高中学生解答数学应用题能力的实验研究

采用对比实验的方法,运用自编思维训练教程及元认知监控提问单对初二年级学生进行有关解答数学应用题的思维策略训练及其元认知训练。结果表明:(1)不同层次学生(优、中、差生)的思维策略训练效果显著,中、差生的效果尤为显著。(2)在思维策略训练的基础上再加上元认知训练,能够更有效地提高解答数学应用题思维训练的效果。     

Investigating the Sect-Church-Sect Cycle: Cohort-Specific Attendance Differences Across African-American Denominations

Varying religious commitment across generations plays a key role in explaining transitions from sect to church, and the formation of sectarian movements. Within a sect, elite members of younger generations are seen to spur internal secularization that transforms otherworldly sects into world-affirming churches. In this paper I examine how cohort differences in religious commitment across denominations evidence the sect-church transformation process, and point to sources of sect formation among African-Americans. Using data from the 1972–1998 General Social Surveys, I analyze denomination-specific cohort differences in religious participation among African-Americans. Cohort-specific shifts in religious participation across denominations demonstrate the secularization of African-American mainline Methodist and Baptist groups, the continued vitality of sectarian denominations, and the nascent ascendance of "nondenominational" churches on the fringes of the religious marketplace.     

Applied Problem Solving in Children with ADHD: The Mediating Roles of Working Memory and Mathematical Calculation

The difficulties children with ADHD experience solving applied math problems are well documented; however, the independent and/or interactive contributions of cognitive processes underlying these difficulties are not fully understood and warrant scrutiny. The current study examines two primary cognitive processes integral to children’s ability to solve applied math problems: working memory (WM) and math calculation skills (i.e., the ability to utilize specific facts, skills, or processes related to basic math operations stored in long-term memory). Thirty-six boys with ADHD-combined presentation and 33 typically developing (TD) boys aged 8–12 years old were administered multiple counterbalanced tasks to assess upper (central executive [CE]) and lower level (phonological [PH STM] and visuospatial [VS STM] short-term memory) WM processes, and standardized measures of mathematical abilities. Bias-corrected, bootstrapped mediation analyses revealed that CE ability fully mediated between-group differences in applied problem solving whereas math calculation ability partially mediated the relation. Neither PH STM nor VS STM was a significant mediator. When modeled together via serial mediation analysis, CE in tandem with math calculation ability fully mediated the relation, explained 79% of the variance, and provided a more parsimonious explication of applied mathematical problem solving differences among children with ADHD. Results suggest that interventions designed to address applied math difficulties in children with ADHD will likely benefit from targeting basic knowledge of math facts and skills while simultaneously promoting the active interplay of these skills with CE processes.     

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.     

Teaching a Problem Solving Strategy to Inpatient Adolescents:

This study evaluated a recently developed program for teaching problem solving skills. Six inpatient adolescents received training in four general areas: Authority Figures; Peer Issues; Community Resources; and Alcohol and Drugs. The program featured response-specific feedback, modeling, self-monitoring, visual cues, positive reinforcement, response practice, self-correction, and individualized performance criterion levels. A multiple baseline across groups design was used in which each group (N + 3) received baseline, training, probes, and pre/post training generalization assessments. Generalization was assessed to two different sets of untrained problem situations that contained similar and dissimilar situations to those used in training. One set was assessed during probes and the other in pre/post evaluations. Both the probes and post assessment results revealed that the subjects' verbal problem solving skills had generalized to both types of situations. Furthermore, the subjects' three-month post assessment scores demonstrated partial maintenance and were comparable to those of a group of normal high school students. Issues related to these results and suggestions for future research are discussed.     

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.