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。     

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。     

3~5岁儿童理解和使用空间表征的特点

以96名3~5儿童为被试,采用自编的儿童空间表征实验任务,在语言表征、模型表征和图画表征三种空间表征形式上,考察了儿童理解和使用空间表征的发展特点。结果表明:(1)总体上,3~5岁儿童理解空间表征的发展水平均显著高于使用空间表征的发展水平。(2)3~5岁儿童理解和使用空间表征的发展表现出显著的年龄效应。(3)从不同的空间表证形式来看,3岁儿童理解语言表征的能力与理解模型表征的能力之间差异显著,4岁儿童使用语言表征的能力与使用模型表征的能力之间的差异、使用语言表征的能力与使用图画表征的能力之间的差异、以及使用模型表征的能力与使用图画表征的能力之间的差异都显著,其余形式的空间表征理解之间的差异或空间表征使用之间的差异在各年龄段中均不显著。     

问题表征、工作记忆对小学生数学问题解决的影响

以104名小学六年级学生为被试, 采用4个相对独立的研究探讨了工作记忆广度、问题表征方式与小学生数学应用题解决的关系。结果表明：(1)言语工作记忆广度只影响高难度应用题的解决, 在低难度、中等难度的应用题解决上, 高、低言语工作记忆广度者不存在显著差异; (2)视觉-空间工作记忆广度对低难度、中等难度、高难度应用题的解决都存在影响; (3)问题表征方式影响数学应用题的解决, 应用题的解题成绩与问题表征方式的使用有关; (4)言语工作记忆广度对应用题的表征方式没有影响, 高、低言语工作记忆广度者在三种难度水平应用题的表征方式上不存在显著差异; (5)视觉-空间工作记忆广度对应用题的表征方式存在影响, 高、低视觉-空间工作记忆广度者在三种难度水平应用题的表征方式上均存在组间差异。     

学习不良儿童元记忆监测与控制的发展

采用2×3×2的混合设计,在自定步调和对项目逐项评定的学习条件下,对小学四~六年级学习不良儿童的元记忆监测与时间分配策略进行了实验研究。结果表明：在对学习材料难度的有效区分上,学习不良儿童与对照组儿童间无差异,不论学习不良组还是对照组,四年级儿童均不能对实验中配对学习材料难度性质做出明确区分,五、六年级儿童能够很好的区分学习材料难度;在学习判断水平上,五、六年级学习不良儿童均低于对照组儿童,四年级两组之间无差异;从对不同难度学习材料的时间分配来看,四年级学习不良儿童与对照组儿童分配在不同难度材料上的学习时间均无显著差别。五、六年级对照组儿童能够根据学习材料的不同难度分配不同学习时间,而且学习时间分配与难度判断之间存在显著相关。五、六年级学习不良儿童能在一定程度上根据学习材料的不同难度分配不同的学习时间,但这种时间分配与难度判断之间相关未达到显著水平,提示他们尚不能在有效元记忆监测基础上对不同难度学习材料进行合理的时间分配     

4-5年级学生的空间表征与几何能力的相关性研究

本研究是“学生空间能力和几何能力关系”研究的一部分,以小学4、5年级学生为被试(共117人),以空问表征能力测验和几何能力测验为测验工具,初步探讨了学生空间表征与几何能力的关系。对数据结果的分析表明:①就总成绩而言,五年级学生的空间表在成绩明显高于四年级学生的成绩,但并不是空间表征的所有方面都存在着显著的年级差异;②就空间测验的总成绩而言,无论是四、五年级分别考察还是总起来考察,空间成绩与几何成绩之间的相关显著,但就各个分项而言,并不是空间测验的各项都与几何测验成绩有显著相关;学生的几何成绩在一定程度上可通过回归方程:Y_i=0.5736X_i+0.7635加以预测。     

汉语发展性阅读障碍儿童视觉同时性加工技能子成分的发展及其与阅读的关系

通过分别以高频汉字(实验1)和图形非言语材料(实验2)为刺激的两个联合视觉注意任务, 并采用基于Budensen视觉注意理论的参数估计方法, 系统地探查小学三~六年级汉语发展性阅读障碍儿童的视觉同时性加工技能缺陷的内在机制。以43名汉语发展性阅读障碍儿童和46名生理年龄匹配典型发展儿童为被试, 每类被试均被分为小学中年级组(三、四年级)和高年级组(五、六年级)。两个实验均发现不同年级组的阅读障碍儿童在知觉加工速度参数上显著小于控制组儿童。在空间注意分布权重参数上, 实验1的结果显示, 不同于控制组儿童向左侧化发展的注意分布模式, 两个年级组的阅读障碍儿童均表现为无偏的注意分布; 而实验2未发现显著组别差异。且这两种同时性加工子技能分别与不同水平的汉语阅读技能密切相关。结果表明, 汉语阅读障碍儿童在同时加工多个视觉刺激时存在持续的知觉加工速度缓慢的问题, 在同时加工言语类刺激时还表现出异常的空间注意分布模式。本研究有助于从基础认知层面揭示汉语发展性阅读障碍儿童的缺陷机理, 为进一步设计相关的提高阅读效率的干预方案提供理论依据。     

小学儿童在线和离线元认知监控的发展特点及其对问题解决的影响

本研究以小学二、四、六年级儿童为被试,在不同难度的拼图任务中考查其在线和离线元认知监控的发展特点及其对问题解决的影响。结果表明:(1)小学儿童的在线元认知监控能力随年级升高而提高,六年级儿童的计划时间显著长于二、四年级儿童。(2)准确预测和评价任务执行时间的人数百分比随年级升高而依次增加;而难度评价准确性在四、六年级间发展较平稳,并高于二年级。(3)在线和离线元认知监控对问题解决的影响与任务难度有关,在简单和中等难度任务中,离线元认知判断有助于问题解决;而在复杂任务中,在线元认知监控促进问题解决。     

非言语型学习障碍儿童视觉空间工作记忆的研究

目的探讨非言语型学习障碍儿童视觉空间工作记忆的特点.方法采用分别考察视觉空间工作记忆的存储和加工能力的被动记忆、主动记忆、图片记忆和词义联想四个实验任务,比较了非言语型学习障碍儿童(NLD)与言语型学习障碍儿童(VLD)及正常儿童视觉空间工作记忆的差异.结果NLD儿童在四个实验任务上的成绩均显著低于VLD儿童和正常儿童,而VLD与正常儿童之间差异不显著.结论NLD儿童的视觉空间工作记忆存在缺陷.     

不同亚型数学学习困难儿童应用题问题表征过程的研究

采用自编数学应用题解决能力测验题对小学四年级单纯型数困儿童、混合型数困儿童和普通儿童问题理解阶段、问题整合阶段的差异,以及问题表征能力与数学问题解决之间的关系进行探究。结果表明：（1）单纯型与混合型数困儿童有效识别信息的能力弱,难于利用相关信息和排除干扰信息。（2）单纯型数困儿童比混合型数困儿童更擅于运用图式表征策略。（3）图式表征策略能促进数困儿童应用题的解决。     

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.     

Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving

Background. High school and college students demonstrate a verbal, or textual, advantage whereby beginning algebra problems in story format are easier to solve than matched equations ( Koedinger & Nathan, 2004 ). Adding diagrams to the stories may further facilitate solution ( Hembree, 1992 ; Koedinger & Terao, 2002 ). However, diagrams may not be universally beneficial ( Ainsworth, 2006 ; Larkin & Simon, 1987 ). Aims. To identify developmental and individual differences in the use of diagrams, story, and equation representations in problem solving. When do diagrams begin to aid problem‐solving performance? Does the verbal advantage replicate for younger students? Sample. Three hundred and seventy‐three students (121 sixth, 117 seventh, 135 eighth grade) from an ethnically diverse middle school in the American Midwest participated in Experiment 1. In Experiment 2, 84 sixth graders who had participated in Experiment 1 were followed up in seventh and eighth grades. Method. In both experiments, students solved algebra problems in three matched presentation formats (equation, story, story + diagram). Results. The textual advantage was replicated for all groups. While diagrams enhance performance of older and higher ability students, younger and lower‐ability students do not benefit, and may even be hindered by a diagram's presence. Conclusions. The textual advantage is in place by sixth grade. Diagrams are not inherently helpful aids to student understanding and should be used cautiously in the middle school years, as students are developing competency for diagram comprehension during this time.     

Spatial visualization in physics problem solving

Three studies were conducted to examine the relation of spatial visualization to solving kinematics problems that involved either predicting the two-dimensional motion of an object, translating from one frame of reference to another, or interpreting kinematics graphs. In Study 1, 60 physics-naíve students were administered kinematics problems and spatial visualization ability tests. In Study 2, 17 (8 high- and 9 low-spatial ability) additional students completed think-aloud protocols while they solved the kinematics problems. In Study 3, the eye movements of fifteen (9 high- and 6 low-spatial ability) students were recorded while the students solved kinematics problems. In contrast to high-spatial students, most low-spatial students did not combine two motion vectors, were unable to switch frames of reference, and tended to interpret graphs literally. The results of the study suggest an important relationship between spatial visualization ability and solving kinematics problems with multiple spatial parameters.     

The Effects of Presenting Multidigit Mathematics Problems in a Realistic Context on Sixth Graders' Problem Solving

Mathematics education and assessments increasingly involve arithmetic problems presented in context: a realistic situation that requires mathematical modeling. This study assessed the effects of such typical school mathematics contexts on two aspects of problem solving: performance and strategy use. Multidigit arithmetic problems presented in two conditions—with and without a realistic context—were solved by 685 sixth graders from The Netherlands. Regarding performance, the same (latent) ability dimension was involved in solving both types of problems, and the presence of a context increased the difficulty level of the division problems but not of other operations. Regarding strategy use, strategy choice and strategy accuracy were not affected by the presence of a problem context. In sum, the presence of a typical context in multidigit arithmetic problems had no marked effects on students' problem-solving behavior, which held for different subgroups of students with respect to language ability and gender.     

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.     

A visualization technique for Bayesian reasoning

We tested a method for solving Bayesian reasoning problems in terms of spatial relations as opposed to mathematical equations. Participants completed Bayesian problems in which they were given a prior probability and two conditional probabilities and were asked to report the posterior odds. After a pretraining phase in which participants completed problems with no instruction or external support, participants watched a video describing a visualization technique that used the length of bars to represent the probabilities provided in the problem. Participants then completed more problems with a chance to implement the technique by clicking interactive bars on the computer screen. Performance improved dramatically from the pretraining phase to the interactive‐bar phase. Participants maintained improved performance in transfer phases in which they were required to implement the visualization technique with either pencil‐and‐paper or no external medium. Accuracy levels for participants using the visualization technique were very similar to participants trained to solve the Bayes theorem equation. The results showed no evidence of learning across problems in the pretraining phase or for control participants who did not receive training, so the improved performance of participants using the visualization method could be uniquely attributed to the method itself. A classroom sample demonstrated that these benefits extend to instructional settings. The results show that people can quickly learn to perform Bayesian reasoning without using mathematical equations. We discuss ways that a spatial solution method can enhance classroom instruction on Bayesian inference and help students apply Bayesian reasoning in everyday settings.