代数应用题视觉化表征的理论模型及影响因素

视觉化表征是代数应用题表征的一大类型, 在问题解决过程中起着重要作用。视觉化表征的理论主要有视觉-逻辑二维模型、理解-转换模型、表象表征理论和图式-图像表征论, 四种理论模型各有其特殊的方法学意义。代数应用题的视觉化表征主要受到题目视觉化程度和认知负载等刺激因素以及自我效能感等认知因素的影响, 最后, 本文指出了在代数应用题视觉化表征理论、研究方法和影响因素探索中的进步以及发展空间。     

代数应用题项目生成中的认知过程与任务特征分析

认知项目设计起源于理论驱动的测量模式,要求在详尽阐述测量建构理论的基础上,鉴别测验项目类型的任务特征,建立和标定与测量建构认知变量之间的对应关系,从而通过系统变换任务特征生成项目.本文以Greeno (1994)提出的代理者-情境交互作用(agent-situation interaction)的问题解决观为基础,对代数应用题解决的认知过程和任务蕴含性(affordance)进行了系统分析.具体而言,代数应用题被视为以文本形式呈现的,嵌套在现实情境中的某种量化结构.在代数应用题解决的转译、整合、计划、执行的认知过程中需要形成三种不同的表征,即文本基、情境模型和问题模型.其形成及解决过程受到个体问题图式知识的深刻影响.对应于不同表征方式,对代数应用题语法/语义结构、量化元素及结构、问题情境特征进行了系统分析和综合.该分析表明,为了保证代数应用题项目生成的科学性,当前研究的关键在于建立任务特征与问题解决认知变量之间的经验关系,并在问题领域整体水平上检验这种关系的可推广性.     

共代数模态逻辑研究述评

共代数(coalgebra,数学中常译为余代数)是代数的对偶概念。数学、逻辑学和理论计算机科学中的许多结构都能够很自然地看做共代数。奥采尔(P.Aczel)在1988年给出的第一个例子将转换系统和     

小学3-6年级学生解决比较问题的研究

本研究对465名小学3—6年级学生解决不同类型的比较问题进行了研究。结果表明：(1)小学生解决比较问题的成绩受问题类型及年级的交互影响。儿童在一致算术问题上的成绩都很好,且显著优于不一致算术问题。在不一致算术问题、一致代数问题及不一致代数问题上,存在年级差异。5、6年级学生优于4年级学生。(2)小学生在算术问题上的通过率高于代数问题,5、6年级学生的通过率高于3、4年级。除一致算术问题外,其它类型比较问题的通过率较低。(3)小学生解决比较问题的成绩无性别差异。     

不同概化的问题原型对问题归类和解决的影响

探讨原理学习过程中形成不同概化程度／问题原型对解决不同适宜程度问题的影响。实验1探讨个体形成不同概化程度问题原型对解决表面内容不同适宜程度的问题的效果;实验2探讨形成不同概化程度问题原型的被试解决问题的信息加工方式与机制;实验3进一步探讨形成不同概化程度问题原型的个体解决复杂问题的效果与机制。结果表明：(1)形成概化程度较高的问题原型有利于个体对表面内容不适宜题目的解答;(2)形成概化程度较高问题原型的被试在解答适宜或不适宜问题时均能迅速激活问题原型而通达原理图式,因此,更多地采用图式的方法解决问题;(3)形成概化程度较高的问题原型会有助于被试排除题目的无关信息,正确进行解答。     

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。     

代数图式相似性对样例迁移中原理通达的影响

应用题深层抽象结构有几个不同层次,被试对概念图式的相似性更敏感,而不是方程结构,概念图式的相似性更容易引起原理的通达。学生掌握了代数图式对解题有很大的促进作用,较少受表面内容等因素的影响。该研究对代数应用题教学模式的改革提供了理论支持。     

语境同义性逻辑(英文)

众所周知,基于可能世界语义的内涵逻辑由于对意义的刻画过于粗粝而导致了所谓的"超内涵问题"。为了解决超内涵问题,出现了各种超内涵逻辑,其中由Suszko提出的带等词的命题逻辑(SCI)是超内涵逻辑中最基本的一种。本文是对SCI的精炼,其动机是语境同义性论题(CST)。该论题认为,同义性标准具有语境依赖性。基于认知语境主义,我们给出了CST的一个论证。通过将SCI中的二元等词修改为一个三元结构,用来表示两个陈述相对某个语境表达同一命题,我们给出了CST的希尔伯特式公理系统。我们证明了该系统相对一个代数模型类是可靠的和完全的。该代数模型的论域由命题构成,同时附带一组命题上的全等关系,用以刻画相对于语境的命题同一性。我们运用该逻辑部分解决了分析悖论这一困扰逻辑学家多年的问题。与我们之前的基于相同动机的论文[17]相比,本文给出的形式语言更加丰富,从而能够表达不同语境之间以及不同语境的同义性之间的关系。     

集体问题解决中的知识、惯例和绩效

在Cohen和Bacdayan发明的两人TTT纸牌游戏的基础上开发了三人TTT游戏,并以此为实验任务,考察了三人集体问题解决活动能否形成惯例,比较了有问题图式组和无问题图式组之间在惯例和绩效上的差异。实验结果表明：（1）三人问题解决能够形成惯例;（2）有图式组的惯例化程度总体上低于无图式组,而其惯例化程度提高的速度则快于后者;（3）无图式组的绩效高于有图式组,但前期差异显著,后期不显著。这些结果将前人基于两人TTT游戏所做的惯例研究进一步拓展到三人情境,证实了三人互动中惯例的生成,为研究涉及更多人互动的真正意义上的组织惯例奠定了基础;阐明了知识,特别是前人在组织惯例研究中所忽视的陈述性知识,对于惯例形成和绩效提升的影响,并揭示了其在组织管理和决策活动中的实际意义。     

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。     

Effects of Variation and Prior Knowledge on Abstract Concept Learning

Learning abstract concepts through concrete examples may promote learning at the cost of inhibiting transfer. The present study investigated one approach to solving this problem: systematically varying superficial features of the examples. Participants learned to solve problems involving a mathematical concept by studying either superficially similar or varied examples. In Experiment 1, less knowledgeable participants learned better from similar examples, while more knowledgeable participants learned better from varied examples. In Experiment 2, prior to learning how to solve the problems, some participants received a pretraining aimed at increasing attention to the structural relations underlying the target concept. These participants, like the more knowledgeable participants in Experiment 1, learned better from varied examples. Thus, the utility of varied examples depends on prior knowledge and, in particular, ability to attend to relevant structure. Increasing this ability can prepare learners to learn more effectively from varied examples.     

样例表面内容对问题解决类比迁移过程的影响

探讨样例与新问题表面内容不同的相似关系对类经迁移过程的影响。实验１要求被试学习概率问题的样例后,解决与样例表面内容相似关系不同的新问题。结果表明,在新问题的内在原理与样例相同的情况下,两者表面根貌相似有利于被试对新问题的类化,表面对应相似则促进被试对公式的正确运用;     

算术应用题二重变异样例学习的迁移效果

设计并运用了三种表面特征变异的二重样例和三种结构特征变异的二重样例,对二年级小学生解决算术应用题的迁移效果进行了实验研究。结果表明：一个原样例的学习就能明显促进了二年级小学生解决与原样例表面特征不同而结构特征相同的近迁移问题;学习三种表面特征变异的二重样例后没有产生明显的远迁移效果;而学习了三种结构特征变异的二重样例后产生了不同程度的远迁移效果,其中,规则平行组合二重样例学习的远迁移效果最好,其次是规则变异二重样例学习的远迁移效果,规则镶嵌组合变异二重样例学习的远迁移效果最差     

Judging a book by its cover: Interpretative effects of content on problem-solving transfer

We examine how cover stories of isomorphic problems affect transfer. Existing models posit that people retain content in problem representations and that similarities and differences between the “undeleted” cover stories might interfere with recognition of structural similarities.We propose that cover stories can affect transfer in another way—by inducing semantic knowledge that modifies problem structures. Two experiments examined how people represent and solve permutation problems dealing with random assignment of elements from one set to elements from another set. Although the problems were structurally isomorphic, cover stories involving different pairs of element sets led subjects to abstract different “interpreted structures.” Problems involving objects and people (e.g., prizes and students) led subjects to abstract an asymmetric structure (“get”) and problems involving similar sets of people (e.g., doctors and doctors) led subjects to abstract a symmetric structure (“pair”). Transfer was mediated by similarities and differences between the interpreted structures of the learned and the novel problems.     

Selecting analogous problems: Similarity versus inclusiveness

Students were asked to select one of two analogous problems in order to solve algebra word problems. In Experiment 1, one problem was less inclusive and the other was more inclusive than a test problem. The students judged the complexity and similarity of problems, selected analogous problems, and used the solutions to solve test problems. They performed significantly better on the test problems when given the more inclusive solutions, but used perceived similarity rather than inclusiveness to select analogous problems. The same pattern of results occurred in Experiment 2, in which isomorphic problems replaced the more inclusive problems. The results show that students are deficient in selecting good analogies, both from the same category (Experiment 1) and from a different category (Experiment 2). Students who saw the analogous solutions (Experiment 3) or were majoring in mathematics (Experiment 4) were more likely to select an isomorphic problem over a less inclusive problem, but were not more likely to select a more inclusive over a less inclusive problem.     

The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra

The knowledge required to solve algebra manipulation problems and procedures designed to hasten knowledge acquisition were studied in a series of five experiments. It was hypothesized that, as occurs in other domains, algebra problem-solving skill requires a large number of schemas and that schema acquisition is retarded by conventional problem-solving search techniques. Experiment 1, using Year 9, Year 11, and university mathematics students, found that the more experienced students had a better cognitive representation of algebraic equations than less experienced students as measured by their ability to (a) recall equations, and (b) distinguish between perceptually similar equations on the basis of solution mode. Experiments 2 through 5 studied the use of worked examples as a means of facilitating the acquisition of knowledge needed for effective problem solving. It was found that not only did worked examples, as expected, require considerably less time to process than conventional problems, but that subsequent problems similar to the initial ones also were solved more rapidly. Furthermore, decreased solution time was accompanied by a decrease in the number of mathematical errors. Both of these findings were specific to problems identical in structure to the initial ones. It was concluded that for novice problem solvers, general algebra rules are reflected in only a limited number of schemas. Abstraction of general rules from schemas may occur only with considerable practice and exposure to a wider range of schemas.     

Analogical problem solving: a hierarchical analysis of procedural similarity

Similarity between source analogues and target problems is a central theme in the research on analogical transfer. Much of the theorizing and research has focused on the effects of superficial and structural similarity on transfer. The present research is an attempt to analyze systematically another critical type of similarity, namely, procedural similarity, and to examine its effects on the executing process. Participants viewed a schematic picture as a source model, interpreted its conceptual meaning, and then attempted to solve a problem to which the conceptual information from the source model could be applied. The results indicate that the ease with which a source solution was implemented was largely determined by the abstraction level at which a solution was shared by a source analogue and the target problem. The degree of procedural similarity was also found to influence the executing process in analogical transfer. A conceptual model concerning the function of procedural similarity as a utilizational constraint in analogical problem solving is proposed.     

Learning subgoals and methods for solving probability problems

We hypothesize that typical example problems used in quantitative domains such as algebra and probability can be represented in terms of subgoals and methods that these problems teach learners. The "quality" of these subgoals and methods can vary, depending on the features of the examples. In addition, the likelihood of these subgoal's being recognized in novel problems and the likelihood of learners' being able to modify an old method for a new problem may be functions of the training examples learners study. In Experiment 1, subjects who studied examples predicted to teach certain subgoals were often able to recognize those subgoals in nonisomorphic transfer problems. Subjects who studied examples demonstrating two methods rather than one exhibited no advantages in transfer. Experiment 2 demonstrated that if the conditions for applying a method are highlighted in examples, learners are more likely to appropriately adapt that method in a novel problem, perhaps because they recognize that the conditions do not fully match those required for any of the old methods. Overall, the results indicate that the subgoal/method representational scheme may be useful in predicting transfer performance.     

Analogical transfer: From schematic pictures to problem solving

A series of experiments was conducted to explore whether individuals can solve problems by transferring conceptual information gained from schematic pictures and to examine the mechanisms involved in this transfer process. Subjects viewed a schematic picture and then attempted to solve an insight problem to which the conceptual information from the picture could be applied. The results indicate that the degree of similarity—specifically, superficial and procedural similarity—between a source schematic picture and the target problem determined transfer performance. Discussion focuses on the relationship between these two types of similarity and the two key cognitive components involved in transferring pictorial analogies to solve problems: accessing the pictorial analogues and executing the solutions. Portions of this research were presented at the 33rd meeting of the Psychonomic Society in St. Louis, November 1992.