西方儿童策略研究

策略的选择、使用直接决定了儿童对相关问题的解决。本介绍了西方儿童策略研究的相关理论，分析了儿童策略发展的几个阶段，并以河内塔实验的改进和微观发生法的使用为例阐明了研究方法、实验设计在该领域的研究价值。     

Children’s solving of ‘Tower of Hanoi’ tasks: dynamic testing with the help of a robot

Abstract

The present study investigated the usefulness of a pre-programmed, teleoperated, socially assistive peer robot in dynamic testing of complex problem solving utilising the Tower of Hanoi. The robot, in a ‘Wizard of Oz’ setting, provided instructions and prompts during dynamic testing to children when they had to solve 3?D Tower of Hanoi puzzles. Participants were 37?second grade 8-year-old children, of whom half received graduated prompts training between pre-test and post-test, delivered by the robot, and half did not. It was found that children’s progression in task accuracy varied considerably, depending on whether or not children were trained in solving Tower puzzles. Trained children showed greater progression in the number of Tower problems that they could solve accurately, made considerably fewer steps, although the Tower puzzles increased quickly in difficulty level. The mean completion time of trained children decreased at a slower rate than that of the untrained children, but both groups of children took considerably more time to think and plan ahead before they started the solving process. Only moderate relations with planning behaviour were found. In general, the study revealed that computerised dynamic testing with a robot as assistant has much potential in unveiling children’s potential for learning and their ways of tackling complex problems. The advantages and challenges of using a robot in educational assessment were discussed.     

Hanoi塔演示动画在Flash中的实现

Hanoi塔是程序设计中比较经典的递归问题．递归算法一直为大家所熟识，但真正用直观的方法将其演示出来却较少被提及．本文就Hanoi塔递归算法在Flash中的实际应用进行探讨，模拟Hanoi塔的具体移动过程．     

Poor Performance in Mathematics: Is there a basis for a self‐worth explanation for women?

The self‐worth theory of achievement motivation holds that in situations in which poor performance is likely to reveal low ability, certain students (known as self‐worth protective students) intentionally withdraw effort in order to avoid the negative implications of poor performance in terms of damage to self‐worth. In this study, evidence of self‐worth protection was sought where it is perhaps least likely to be seen: in women in mathematics, where the negative effects of stereotype threat are suggested. Links between self‐worth protection and performance on three maths tasks (a computational task, an algebraic task, and the Tower of Hanoi, a game‐like task tapping into mathematical skills yet devoid of mathematical signs and symbols) and a verbal task (a remote associates task) were examined in an intellectually evaluative situation (following failure) and in an intellectually non‐evaluative situation (following success). Relative to low maths self‐worth protective students, high maths self‐worth protective students performed poorly on both mathematical tasks, but not on either the Tower of Hanoi or the remote associates test. Collectively, findings are consistent with the conclusion that a self‐worth explanation can be applied to the poor performance of women in mathematics. Critical eliciting conditions appear to be mathematical signs and symbols and the presence of evaluative threat.     

The Effects of Verbal Mediation Training on the Problem‐Solving Skills of Children with Partial Sight and Children without Visual Impairments

Thirty children with partial sight and 30 children with unimpaired sight aged between 8 and 12 years were randomly assigned to either verbal mediation or visual mediation training regimes. Participants were asked to complete four variations of the Tower of Hanoi puzzle and success on the task was judged on the basis of the number of moves and time taken to complete the puzzles. Children with visual impairments had some difficulty in adjusting to the requirements of the problem‐solving tasks. However, on the final and most difficult of the tasks (a four‐disc problem), the children with impaired sight performed very much like participants without visual impairments. The effect of verbal mediation training was superior to that of visual mediation training in the final trials of the four‐disc problem‐solving period for both children with partial sight and children without visual impairments.     

The effects of knowledge,strategies, and the interaction between the two in verbal analogy problem solving

An experiment was conducted to assess how accessibility to relevant knowledge would affect performance on verbal analogy problems as well as the efficacy, perception, and continuation of use of a componential strategy for solving these problems. Participants were either taught to use a componential strategy for solving verbal analogy problems or just given practice opportunities while the accessibility of relevant knowledge was manipulated and the performance and perception of the strategy was assessed. We also investigated whether variations in accessibility to relevant knowledge affected the continuance of strategy use. Results showed that knowledge accessibility affected both performance and the way in which the strategy was perceived. Participants rated the strategy as more useful, easier to use, and less effortful when their initial strategy experience involved solving analogy problems based on familiar vocabulary and relationships. Also, participants who had experience with easily accessible vocabulary and relationships were significantly more likely to choose to continue to use of the strategy on a subsequent task. It is also argued that accessibility to knowledge and strategy use play independent, as opposed to interactive, roles in determining performance.     

Visuospatial anatomy comprehension: The role of spatial visualization ability and problem‐solving strategies

The present study explored the problem‐solving strategies of high‐ and low‐spatial visualization ability learners on a novel spatial anatomy task to determine whether differences in strategies contribute to differences in task performance. The results of this study provide further insights into the processing commonalities and differences among learners beyond the classification of spatial visualization ability alone, and help elucidate what, if anything, high‐ and low‐spatial visualization ability learners do differently while solving spatial anatomy task problems. Forty‐two students completed a standardized measure of spatial visualization ability, a novel spatial anatomy task, and a questionnaire involving personal self‐analysis of the processes and strategies used while performing the spatial anatomy task. Strategy reports revealed that there were different ways students approached answering the spatial anatomy task problems. However, chi‐square test analyses established that differences in problem‐solving strategies did not contribute to differences in task performance. Therefore, underlying spatial visualization ability is the main source of variation in spatial anatomy task performance, irrespective of strategy. In addition to scoring higher and spending less time on the anatomy task, participants with high spatial visualization ability were also more accurate when solving the task problems. Anat Sci Educ 7: 280–288. © 2013 American Association of Anatomists.     

Mathematics practice without feedback: A desirable difficulty in a classroom setting

Recent research highlights the potential benefits of practice without feedback on learner’s strategy knowledge. However, most prior work has been conducted in one-on-one settings with short retention intervals. We compared the effects of mathematics practice with and without correct-answer feedback on immediate and 1-week delayed performance in a classroom setting. In a randomized experiment, 243 second- and third-grade children received strategy instruction and then practiced solving relevant problems in small groups within their classroom. During practice, children received immediate feedback (after each problem), summative feedback (after all the problems in the set), or no feedback. During the practice task, immediate feedback led to the best performance. However, practice without feedback led to higher levels of mastery on the 1-week knowledge retention test. Thus, instructional practices that seem less effective at first can, for some children, ultimately result in more desirable learning.     

Empowering educationally disadvantaged mathematics students through a strategies-based problem solving approach

A major impediment to problem solving in mathematics in the great majority of South African schools is that disadvantaged students from seriously impoverished learning environments are lacking in the necessary informal mathematical knowledge to develop their own strategies for solving non-routine problems. A randomized pretest–posttest control group design was used to empirically investigate the effectiveness of a strategies-based problem solving approach on the problem solving performance of 9th grade disadvantaged students. In this approach students receive explicit instruction on a wide repertoire of problem solving strategies. The results reported in this study show a significant improvement in problem solving performance when a strategies-based approach to problem solving was being implemented. Quantitative and qualitative analyses of the responses to the items showed how the treatment group students had internalized as part of their problem solving repertoire the strategies on which they had been explicitly instructed on. The findings of this study make a case for the adoption of this approach so that the gap between the student’s existing problem solving competence, and the cognitive demands of a problem solving task can be bridged.     

A COMPARATIVE STUDY OF CHILDREN'S LEARNING IN THE ZONE OF PROXIMAL DEVELOPMENT

Both IQ and age related differences in the zones of proximal development of 89 four to eight year olds were investigated. Each child was asked to learn, transfer and generalize a strategy for solving the Tower of Hanoi puzzle. The speed, efficiency, accuracy and extent of support needed were analysed. Multivariate analyses of variance were performed on the data from each task. The benefits of high IQ were not as consistent as those of chronological age. They were most apparent in the learning and generalization tasks. Some high IQ children created challenges of their own in the unchallenging transfer task, or became bored. Two practical implications became apparent: the range of learning potentials in classes of children grouped by chronological age poses a formidable challenge for educators, and the learning of highly able children may detoriorate when they are offered the same curriculum as their peers.

     

Make a drawing. Effects of strategic knowledge,drawing accuracy,and type of drawing on students’ mathematical modelling performance

Drawing strategies are widely used as a powerful tool for promoting students’ learning and problem solving. In this article, we report the results of an inferential mediation analysis that was applied to investigate the roles that strategic knowledge about drawing and the accuracy of different types of drawings play in mathematical modelling performance. Sixty-one students were asked to create a drawing of the situation described in a task (situational drawing) and a drawing of the mathematical model described in the task (mathematical drawing) before solving modelling problems. A path analysis showed that strategic knowledge about drawing was positively related to students’ modelling performance. This relation was mediated by the type and accuracy of the drawings that were generated. The accuracy of situational drawing was related only indirectly to performance. The accuracy of mathematical drawings, however, was strongly related to students’ performance. We complemented the quantitative approach with a qualitative in-depth analysis of students’ drawings in order to explain the relations found in our study. Implications for teaching practices and future research are discussed.     

The influence of self-efficacy and metacognitive prompting on math problem-solving efficiency

A regression design was used to test the unique and interactive effects of self-efficacy beliefs and metacognitive prompting on solving mental multiplication problems while controlling for mathematical background knowledge and problem complexity. Problem-solving accuracy, response time, and efficiency (i.e. the ratio of problems solved correctly to time) were measured. Students completed a mathematical background inventory and then assessed their self-efficacy for mental multiplication accuracy. Before solving a series of multiplication problems, participants were randomly assigned to either a prompting or control group. We tested the motivational efficiency hypothesis, which predicted that motivational beliefs, such as self-efficacy and attributions to metacognitive strategy use are related to more efficient problem solving. Findings suggested that self-efficacy and metacognitive prompting increased problem-solving performance and efficiency separately through activation of reflection and strategy knowledge. Educational implications and future research are suggested.     

Students’ Ability to Apply Their Knowledge in a Gaming Exercise: An Exploratory Study

This paper reports on a study exploring master students’ ability to apply their knowledge when solving an internal pricing problem in a supply chain. Analyses of 33 negotiation progress reports and 8 recordings of discussions demonstrate that most of the students were not able to apply relevant concepts and models to guide their handling of the task. The main reason for the variability in performance was the significant differences in the students’ general problem-solving abilities. Most students did not perform a thorough task analysis or develop a solution strategy before they met for negotiations, and they did not keep track of their task performance. The implications for teaching are highlighted, and directions for further research are proposed.     

Dutch sixth graders’ use of shortcut strategies in solving multidigit arithmetic problems

Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common.     

Procedural and conceptual changes in young children’s problem solving

This study analysed the different types of arithmetic knowledge that young children utilise when solving a multiple-step addition task. The focus of the research was on the procedural and conceptual changes that occur as children develop their overall problem solving approach. Combining qualitative case study with a micro-genetic approach, clinical interviews were conducted with ten 5–6-year-old children. The aim was to document how children combine knowledge of addition facts, calculation procedures and arithmetic concepts when solving a multiple-step task and how children’s application of different types of knowledge and overall solving approach changes and develops when children engage with solving the task in a series of problem solving sessions. The study documents children’s pathways towards developing a more effective and systematic approach to multiple-step tasks through different phases of their problem solving behaviour. The analysis of changes in children’s overt behaviour reveals a dynamic interplay between children’s developing representation of the task, their improved procedures and gradually their more explicit grasp of the conceptual aspects of their strategy. The findings provide new evidence that supports aspects of the “iterative model” hypothesis of the interaction between procedural and conceptual knowledge and highlight the need for educational approaches and tasks that encourage and trigger the interplay of different types of knowledge in young children’s arithmetic problem solving.     

A Math Intervention for Third Grade Latino English Language Learners at Risk for Math Disabilities

Word problems for English language learners (ELLs) at risk for math disabilities are challenging in terms of the constant need to develop precise math language and comprehension knowledge. As a result of this, ELLs may not only need math support but also reading and linguistic support. The purpose of this study was to assess the effectiveness of a word problem–solving strategy called Estratégica Dinámica de Matemáticas (EDM). This strategy was designed to provide math support in the native language based on students' math comprehension levels. A changing criterion multiple baseline design was used to instruct six second-grade Latino ELLs at risk for math disability. As compared with the baseline phase, EDM increased word problem solving for all participants. All students' level of performance were maintained and generalized during follow-up sessions. This study has implications for a native language intervention that focuses on strategy training to facilitate word problem–solving performance.     

Hanoi塔问题递归算法的一种改进

Hanoi塔问题是一个古典的数学问题,在C程序设计中常用递归算法去分析处理.但由于其递归算法中有两次递归调用,其程序执行的具体过程很复杂,很难直接看出其调用过程.本文通过运用堆栈概念深入分析Hanoi塔问题递归过程并修改其通用的C程序递归算法,使其递归算法程序执行过程具有直观易于理解性,对深入理解和灵活运用递归算法有积极意义.     

小学数学新授课“问题解决”教学的校本策略研究

文章首先讨论在新授课中引入问题解决教学的价值,并基于校本实践的多个教学案例,分析小学数学新授课常用的问题设计策略和教学引导策略,即基于新知识设计多元开放的探究问题、引导学生灵活利用已有知识去解决新问题等,最后,对小学数学新授课中的问题解决教学进行初步定义。     

Completion problems can reduce the illusions of understanding in a computer-based learning environment on genetics

Inaccurate judgments of task difficulty and invested mental effort may negatively affect how accurate students monitor their own performance. When students are not able to accurately monitor their own performance, they cannot control their learning effectively (e.g., allocate adequate mental effort and study time). Although students' judgments of task difficulty and invested mental effort are closely related to their study behaviors, it is still an open question how the accuracy of these judgments can be improved in learning from problem solving. The present study focused on the impact of three types of instructional support on the accuracy of students' judgments of difficulty and invested mental effort in relation to their performance while learning genetics in a computer-based environment. Sixty-seven university students with different prior knowledge received either incomplete worked-out examples, completion problems, or conventional problems. Results indicated that lower prior knowledge students performed better with completion problems, while higher prior knowledge students performed better with conventional problems. Incomplete worked-out examples resulted in an overestimation of performance, that is, an illusion of understanding, whereas completion and conventional problems showed neither over- nor underestimation. The findings suggest that completion problems can be used to avoid students' misjudgments of their competencies.