数学学习不良儿童视觉-空间表征与数学问题解决

采用临床访谈的方法,考察了30名数学学习不良(MD)儿童和31名一般儿童的数学问题解决、视觉-空间表征策略和空间视觉化能力。结果发现：图式表征能促进数学问题的解决,图像表征则起妨碍作用;空间视觉化能力与解题正确率及图式表征策略有显著正相关,与图像表征策略有显著负相关。MD儿童的解题正确率以及使用图式表征策略的程度显著低于一般儿童,使用图像表征策略的程度则显著高于一般儿童。在解题正确率和图式表征策略这两个变量上,MD儿童和一般儿童的年级发展趋势相同,都随年级的升高而提高。但在图像表征策略的使用上,一般儿童有随年级的升高而下降的趋势,MD儿童却没有下降的趋势。两类儿童的空间视觉化能力都随年级的升高而提高。     

Hypothetico-nomological aspects of medical diagnosis Part I: General structure of the diagnostic process and its hypothesis-directed stage

In medical diagnostic examination three main stages may be distinguished: (a) initial exploration, (b) hypothesis-directed investigation, and (c) final diagnosis making. The purpose of this work is to study some methodological problems concerning the second of the above stages of the diagnosis and to prepare a background for a mathematical model [30] of this process. In diagnostic problem solving, the reasoning proceeds along the main lines traced by some initial suggestions and passes through various intermediate elements which are connected with one another forming ramifying chains and nets of inferences and hypotheses. Such a complex mental construction is based on laws which form medical knowledge and reflect various regularities and relations, causal, structural, functional, and others. The main components of diagnostic reasoning may be divided into several classes according to their function and content: leading hypotheses, working hypotheses, main diagnostic hypotheses, statements accepted as certain, intermediary and reserve hypotheses, therapeutic suggestions of immediate consequence. In an example of diagnostic problem solving these types of propositions are defined and analysed. In diagnostic reasoning, as in every other process of rational problem solving, explanation of the observed symptoms and signs and testing of the explaining hypotheses play a predominant role. These procedures form successive, frequently numerous and diversified steps and stages of the reasoning, leading to the construction of a mental model of the patient's state. Some problems relative to the scheme of explanation, especially to that which is based on causal laws, are discussed.     

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。     

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.     

Responses to Depression in Children: Reconceptualizing the Relation Among Response Styles

We propose that researchers should utilize ratio scores when examining the response styles theory rather than examine each of the response styles separately. Higher ratio scores indicate a higher probability of engaging in ruminative, as opposed to distracting and problem solving, behaviors. In Phase One, we examined the factor structure and reliability of the Children's Response Styles Questionnaire (CRSQ) in a sample of 287 third through sixth grade schoolchildren. A two factor solution was obtained: (1) Rumination and (2) Distraction and Problem Solving. Both factors exhibited strong psychometric properties. In Phase Two, 140 children completed the CRSQ and the Children's Depression Inventory (CDI). Six weeks later, children completed the CDI. Confirmatory factor analysis indicated that the two-factor solution fit the data well. In line with hypotheses, CRSQ ratio scores were more strongly associated with CDI residual change scores than were either CRSQ rumination or distraction and problem solving scores.     

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.     

客体投影方式对空间问题解决和再认的影响

通过设置四种实验条件,旨在阐明空间问题解决和再认的水平是如何受到客体不同投影方式的影响。除进一步支持了正投影问题的解决是以一个具有三维结构特性的心理表征为基础而并非是对二维正投影信息进行了充分识别的观点外,研究结果还表明：（１）这个具有客体三维结构特性的心理表征是建构于对正投影问题解决而非轴测投影图的再认过程中;（２）正投影问题解决的复杂性显著地高于轴测投影图问题解决而且一个建构于正投影问题解决中     

数学解题过程的眼动研究

数学解题过程的眼动研究可划分为3个阶段：第一阶段是对数字运算过程的眼动研究,第二阶段是对数学应用题解题过程的眼动研究,第三阶段是对几何题解题过程的眼动研究。在我国,对数学解题过程的眼动研究尚为空白。加强这一领域的研究,可以深化对数学问题表征的认识,对数学学科的教与学具有重要意义。     

Knowledge structures of expert-novice gymnasts

Differences in eye movements, problem solving, and verbalization were found for three groups of female gymnasts (N=30) differentiated by skill level (elite, intermediate, novice). In the first study, eye fixations to body parts were recorded during slide presentations of six gymnastic sequences. In a second study, time and error scores were recorded in a problem solving task requiring the resequencing of photographs derived from the sequences used in the eye movement study. In the third study, verbal reports identified the photograph in each sequence and body part thought most important to a successful performance. Consistent differences between groups were recorded in fixations to body parts, time and error scores, and verbal reports; relationships between eye movement and time and error scores, and eye movements and verbal reports were also determined. All three studies supported the general hypotheses that gymnasts with different skill levels are attuned to different type of gymnastic information. These differences are discussed within the context of a knowledge structure of gymnastics, which is defined as an internal representation for a physical skill that is developed in concert with the gymnast's motor system. It is suggested that gymnasts, depending upon skill level, are drawn to perceive certain types of gymnastic information while being oblivious to other equally available types of information. Attunement to gymnastic information appears to be linked up to physical skill level in ways not yet understood.     

Is conditional reasoning related to mathematical problem solving?

The current study aimed to investigate the relation between conditional reasoning, which is a common type of logical reasoning, and children's mathematical problem solving. A sample of 124 fourth graders was tested for their conditional reasoning skills and their mathematical problem solving skills, as well as a list of control variables (e.g., IQ, working memory, reading) and potential mediators (number sentence construction and computation). The children's ability to make modus ponens (MP) inferences significantly predicted their mathematical problem solving skills, even after controlling for the potential confounding variables. The relation was mediated by the number sentence construction skills. The findings, in addition to supporting the link between conditional reasoning and mathematics, further indicate that the ability to process relations may be the mechanism underlying the relation. Theoretical and practical implications 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.     

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 Microevolution of Mathematical Representations in Children's Activity

In this article, I discuss children's design of mathematical representations on paper, asking how material displays are constructed and transformed in activity. I show that (a) the design of displays during problem solving shapes one's mathematical activity and sense making in crucial ways, and (b) knowledge of mathematical representations is not simply recalled and applied to problem solving but also emerges (whether constructed anew or not) out of one's interactions with the social and material settings of activity. A detailed characterization of student-designed tables of values to solve problems about linear functions is also presented.     

Working memory components as predictors of children's mathematical word problem solving

This study determined the working memory (WM) components (executive, phonological loop, and visual–spatial sketchpad) that best predicted mathematical word problem-solving accuracy of elementary school children in Grades 2, 3, and 4 (N = 310). A battery of tests was administered to assess problem-solving accuracy, problem-solving processes, WM, reading, and math calculation. Structural equation modeling analyses indicated that (a) all three WM components significantly predicted problem-solving accuracy, (b) reading skills and calculation proficiency mediated the predictive effects of the central executive system and the phonological loop on solution accuracy, and (c) academic mediators failed to moderate the relationship between the visual–spatial sketchpad and solution accuracy. The results support the notion that all components of WM play a major role in predicting problem-solving accuracy, but basic skills acquired in specific academic domains (reading and math) can compensate for some of the influence of WM on children’s mathematical word problem solving.     

The contribution of working memory to children's mathematical word problem solving

The study explored the contribution of working memory to mathematical word problem solving in children. A total of 69 children in grades 2, 3 and 4 were given measures of mathematical problem solving, reading, arithmetical calculation, fluid IQ and working memory. Multiple regression analyses showed that three measures associated with the central executive and one measure associated with the phonological loop contributed unique variance to mathematical problem solving when the influence of reading, age and IQ were controlled for in the analysis. In addition, the animal dual‐task, verbal fluency and digit span task continued to contribute unique variance when the effects of arithmetical calculation in addition to reading, fluid IQ, and age were controlled for. These findings demonstrate that the phonological loop and a number of central executive functions (shifting, co‐ordination of concurrent processing and storage of information, accessing information from long‐term memory) contribute to mathematical problem solving in children. Copyright © 2006 John Wiley & Sons, Ltd.     

Mental representation and hypothetico-deductive reasoning: The case of the THOG problem

Summary This article reports three experiments that deal with the source of the difficulty of Wason's (1977) THOG problem. The solution of this problem demands both the postulation of hypotheses and a combinatorial analysis of their consequences. Experiment 1 showed that the generation of the hypotheses is not in itself sufficient to solve the problem. Experiment 2 showed that a version presenting a plausible context for separating the level of data from that of hypotheses produced a better performance than both the original abstract version and a thematic version lacking the plausible context separating the levels. Experiment 3 gave evidence that this context can produce facilitation even with the geometric material of the classic version. This experiment also showed that a pictorial presentation of data and a verbal presentation of hypotheses affect performance negatively. The results demonstrate the role of problem representation in problem solving, and, in particular, the role of homogeneity in representing data and hypotheses in hypothetico-deductive reasoning.     

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.     

Odour and Proustian memory: reduction of context‐dependent forgetting and multiple forms of memory

In Experiment 1, the reintroduction of the same ambient odour (lemon or lavender) improved performance four weeks later in both free recall and recognition of a word list. This was a cross‐over design that allowed direct comparison between congruent and incongruent odour conditions. A further comparison with an additional group showed that memory was not improved by the presence of a different odour. Experiment 2 investigated the effect of two odour cues (lemon and lavender) in the same cross‐over design using three learning and memory tests: (1) free recall of a word list; (2) problem solving; and (3) spatial learning. While recall of the word list and spatial learning were best when the same odour was present at both learning and test, there was no such context‐dependent effect for the problem‐solving task. However, the presence of the lavender odour at test improved performance in the problem‐solving task, irrespective of the odour present at the first exposure. Thus although lavender had some effect on problem solving, we saw context‐dependent retrieval only in free recall and spatial learning. We discuss the implications of this dissociation. Copyright © 2001 John Wiley & Sons, Ltd.     

Can Students Predict When Imagery Will Allow Them to Discover the Problem Solution?

Three studies have been carried out to identify to what extent and on what grounds undergraduates realise that visual imagery can be useful to solve a problem. A series of problems had been selected according to the kind of task (logical, mathematical, geometrical, and practical) and to the objective level of imagery efficacy (as reported by previous experiments aimed at assessing the effects of mental visualisation on problem solving). Then, by means of different procedures, students were asked to rate how useful, in their opinion, mental visualisation is in solving each of such problems and to explain why they thought it is useful. Results showed that usefulness scores were highly consistent and were not affected by gender, course of study attended, and individual differences in imagery (vividness, control, and cognitive style). Undergraduates rated imagery as useful above all when problems deal with concrete situations, whereas mental images were not perceived as possible aids for abstract or conceptual problems. The subjective evaluation did not correspond to the objective effect. However, when a specific imagery strategy was described, the discrepancy between the subjective and objective measures decreased.     

顿悟问题解决中抑制功能的作用

探讨了抑制功能在顿悟问题解决过程中的作用。在实验1中,以148名大学生为被试,使用Stroop任务、潜伏抑制任务、提取抑制任务等多种抑制功能任务和若干初始搜索空间不同的言语类顿悟问题,探讨了抑制功能在言语顿悟问题解决中的作用;在实验2中,以159名大学生为被试,使用Stroop任务等抑制功能任务和初始搜索空间不同的空间类顿悟问题,探讨了抑制功能在空间顿悟问题解决中的作用。研究结果表明：（1）自发性侧抑制功能和主动性抑制功能对于顿悟问题解决均有不利影响;（2）抑制功能主要作用于顿悟问题解决的重构阶段。这些结果表明抑制功能对于顿悟问题解决的重构过程具有重要影响。