How handwriting behaviors during problem solving are related to problem-solving success in an engineering course

If we carefully observe the spatial and temporal organization of students' pen strokes as they solve an engineering problem, can we predict their ability to achieve the correct answer? To address this question, 122 college students were asked to solve exam problems in an engineering course using a smartpen that recorded their writing as digitized timestamped pen strokes. The pen stroke data was used to compute a collection of 10 metrics characterizing various elements of problem-solving fluency including the tendency to progress down the page without revisions, the amount of time with no activity, and the frequency of constructing and using equations. The primary finding is that, on average across 13 different exam problems, these elements of problem-solving process explained 40% of the variance in scores of the correctness of the problem solution. In short, success on generating correct solutions was related to the fluency of the student's problem-solving process (i.e., working sequentially from the top to the bottom of the page, working without detours or long pauses, and working by constructing equations). This work is consistent with the idea that expertise in solving common engineering problems involves being able to treat them like routine rather than non-routine problems.     

Strategic development of multiplication problem solving: Patterns of students' strategy choices

Low mathematics achievement is a persistent problem in the United States, and multiplication is a fundamental area in which many students manifest learning difficulties. This study examined the strategic developmental levels of multiplication problem solving among 121 elementary school students in Grades 3 through 5. A latent class analysis modeling was used to identify three valid groups representing different patterns of strategy choices for each of three types of multiplication problems. Findings indicated intra-group variability for problem-solving accuracy, for frequency of using different strategies, and for accuracy of executing direct retrieval/algorithm (DR/AG) strategies. Students demonstrated relative consistency in their strategy choices for solving the three problem types. Students who used DR/AG strategies most frequently showed the highest problem-solving accuracy and the highest accuracy of executing the DR/AG strategies. Students who most frequently relied on incorrect operations or who indicated they did not know how to solve problems demonstrated the lowest problem-solving accuracy among the three groups; the number of students in this group increased with problem difficulty levels. Implications are discussed in terms of identifying students' strategic developmental levels and providing differentiated instruction based on the identified levels.     

网络环境下PBL的模式研究

PBL(Problem-based Learning),即基于问题的学习。该学习模式的主要特点是学习者围绕着复杂问题的解决而结合成小组,小组成员在自主学习的基础上,相互间进行合作,共同为解决问题而努力。在现代远程教育领域中,PBL有着广泛的应用前景。我们研究认为,Internet环境下PBL的模式大体具有以下六个环节,依次是:设计并开发问题,把学习者组织成小组,传输问题并引导学习者确定学习目标,小组成员独立自主的学习,小组成员相互汇报和讨论各自学习的结果,师生对于解决问题的过程进行总结和评价。     

The Effect of Hints and Model Answers in a Student-Controlled Problem-Solving Program for Secondary Physics Education

Many students experience difficulties in solving applied physics problems. Most programs that want students to improve problem-solving skills are concerned with the development of content knowledge. Physhint is an example of a student-controlled computer program that supports students in developing their strategic knowledge in combination with support at the level of content knowledge. The program allows students to ask for hints related to the episodes involved in solving a problem. The main question to be answered in this article is whether the program succeeds in improving strategic knowledge by allowing for more effective practice time for the student (practice effect) and/or by focusing on the systematic use of the available help (systematic hint-use effect). Analysis of qualitative data from an experimental study conducted previously show that both the expected effectiveness of practice and the systematic use of episode-related hints account for the enhanced problem-solving skills of students.     

问题解决在线协作学习中的问题设计研究

问题设计是问题解决学习中知识建构的基础,是问题解决学习得以开展的前提条件。首先,研究分析了问题解决在线协作学习所针对的问题,在性质、组成信息等方面的基本特征,并针对上述特征,提出由意义信息、支持信息和知识线索组成的问题设计结构。其次,实证研究部分利用准实验研究的方法,验证了问题设计结构的有效性。实证研究的结论认为,问题设计结构能够增强学习者在线问题解决学习的效果,并提升学习者对学习的主观感知。最后,结合具体的设计经验,研究给出了问题设计的具体策略。     

问题解决教学的四种模式

文章依据问题提出和问题解决主体的不同组合,将问题解决教学法在课堂中的应用分为四种模式——教师提问教师解决模式、教师提问学生解决模式、学生提问教师解决模式、学生提问学生解决模式,并详细分析每种模式的特征,提出相应的教学策略,以供教育工作者教学和研究参考。     

大学生创造性社会问题解决的结构及其与心理健康的关系

以分层取样的方式选取541名大学生,采用社会性问题（故事）情境和SCL-90,考察了创造性的社会问题解决的结构及其与心理健康的关系。结果表明,（1）大学生创造性的社会问题解决策略的独创性、流畅性、变通性、适当性、有效性具有良好的内部一致性,通过探索性因素分析可抽取发散思维能力和适宜性两个主因素,累积解释率为89.68%;（2）创造性的社会问题解决与心理健康不存在显著相关;（3）计算机专业在SCL-90上属低分组的学生的发散思维能力显著高于高分组,管理专业学生问题解决总体上显著高于艺术和计算机专业。专业类型在创造性问题解决与心理健康之间具有调节作用,特定专业（如管理专业）的训练更可能改善创造性社会问题解决的策略（包括适宜性和发散思维能力）。     

Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving

In Singapore, 6–12 year-old students are taught to solve algebra word problems with a mix of arithmetic and pre-algebraic strategies; 13–17 year-olds are typically encouraged to replace these strategies with letter-symbolic algebra. We examined whether algebra problem-solving proficiency amongst beginning learners of letter-symbolic algebra is correlated with the ability to inhibit intrusions from the earlier arithmetic strategies. Similar to typical school practice in Singapore, we asked 14 year-old students (N = 157) to use only letter-symbolic algebra to solve 9 algebra word problems. After having controlled for algebraic knowledge, working memory, and intelligence, better inhibitory ability still predicted fewer arithmetic intrusions and higher problem solving accuracy. Path analysis revealed 2 types of inhibition. Inhibition-of-reified-processes predicted accuracy through arithmetic intrusions. Inhibition-of-recently-learned-associations predicted accuracy through intelligence. Findings suggest establishing pedagogical links between arithmetic and algebraic methods may facilitate students' transition to letter-symbolic algebra.     

Problem Solving and Teaching How to Solve Problems in Technology-Rich Contexts

ABSTRACT

Problem solving is perhaps the key characteristic that makes us human. Given the kinds of problems that we face in a competitive economy and society, the new generation of learners is ever more required to have problem-solving abilities. By drawing from the literature on technological pedagogical content knowledge, design thinking, general and specific methods of problem solving, and role of technologies for solving problems, this article highlights the importance of problem solving for future teachers and discusses strategies that can help them become good problem solvers and understand the requirements of teaching their students problem solving in technology-rich contexts. This article consists of two main parts. Part 1 focuses on strategies required to help preservice teachers to be better problem solvers, and Part 2 summarizes approaches to introduce preservice teachers to the methods of teaching problem solving. The strategies reviewed provide a tangible guidance for teacher education programs regarding how to promote future teachers’ problem-solving skills.     

Using a computer simulation to compare expert/novice problem-solving subroutines

Hierarchical problem-solving strategies employed in solving exercise science problems were examined in this study, which also tested the validity of an educational computer simulation. Hypothesis testing was used as the theoretical base for the study of differences in problem-solving within the computer simulation. In a previous study two groups of undergraduate (novices) and graduate students were compared in their ability to solve exercise science problems. The present study added a group of faculty (experts) who were presented with the same simulation protocol as the other subjects. Protocol analysis and the Pitt coding system were used to analyse verbal data. Group differences were examined statistically. The faculty were superior in interpreting data and used the Basic Heuristic and Pattern Extraction strategies for the generation and use of algorithms. The problem-solving strategies varied for each group based on the perceived difficulty of the problem, the knowledge base available, and the similarity of the given problem to previous problems.     

Increasing explanatory behaviour,problem-solving,and reasoning within classes using cooperative group work

The present study builds on research that indicates that teachers play a key role in promoting those interactional behaviours that challenge children’s thinking and scaffold their learning. It does this by seeking to determine whether teachers who implement cooperative learning and receive training in explicit strategic questioning strategies demonstrate more verbal behaviours that mediate children’s learning than teachers who implement cooperative learning only. The study also sought to determine whether students who receive training in explicit questioning strategies demonstrate more explanatory behaviour than their untrained peers, and, as a consequence, do these same students demonstrate more advanced reasoning and problem-solving skills on follow-up reasoning and problem-solving tasks. The study involved 31 teachers in two conditions, the cooperative + strategic questioning condition and the cooperative condition, and two groups of students from each teacher’s classroom. The results show that the teachers in the cooperative + strategic questioning condition used significantly more mediating behaviours than their peers in the cooperative condition. The study also showed that the children in these teachers’ classes engaged in more elaboration and obtained significantly higher scores on the follow-up reasoning and problem-solving tasks. The study demonstrates the importance of explicitly teaching strategic questioning strategies to children during cooperative learning.     

Learning from Their Mistakes: Glimpses of symbolic functioning in two-and-a-half to three-year-old children

In this cross-sectional study done in the United States, 19 female and 17 male children ( N =36) ranging in age from 30- to 36-months ( M =33 months, 8 days) were presented with two different search and retrieval type problems to solve. Examination of the data revealed clear age-related differences between children who fluidly and efficiently problem solve using symbolically mediated knowledge and those who do not. Younger children tended to rely on a trial and error, motor-based strategy for solving the problems presented, whereas older children consistently used symbolic strategies. This conclusion supports findings from other similar studies. Further analyses of the data, however, revealed an interesting phenomenon in the form of a patterned, non-random error. This error pattern, a perseverative-type error, and the other problem-solving strategies used by the two-year-olds and young threes in this study offer insight into how young children become efficient users of symbols. Facilitating the development of symbolic problem solving in young children is discussed.     

Approach to mathematical problem solving and students’ belief systems: two case studies

The goal of the study reported here is to gain a better understanding of the role of belief systems in the approach phase to mathematical problem solving. Two students of high academic performance were selected based on a previous exploratory study of 61 students 12–13 years old. In this study we identified different types of approaches to problems that determine the behavior of students in the problem-solving process. The research found two aspects that explain the students’ approaches to problem solving: (1) the presence of a dualistic belief system originating in the student’s school experience; and (2) motivation linked to beliefs regarding the difficulty of the task. Our results indicate that there is a complex relationship between students’ belief systems and approaches to problem solving, if we consider a wide variety of beliefs about the nature of mathematics and problem solving and motivational beliefs, but that it is not possible to establish relationships of causality between specific beliefs and problem-solving activity (or vice versa).     

Impacts of learning inventive problem-solving principles: students’ transition from systematic searching to heuristic problem solving

This paper presents the outcomes of teaching an inventive problem-solving course in junior high schools in an attempt to deal with the current relative neglect of fostering students’ creativity and problem-solving capabilities in traditional schooling. The method involves carrying out systematic manipulation with attributes, functions and relationships between existing components and variables in a system. The 2-year research study comprised 112 students in the experimental group and 100 students in the control group. The findings indicated that in the post-course exam, the participants suggested a significantly greater number of original and useful solutions to problems presented to them compared to the pre-course exam and to the control group. The course also increased students’ self-beliefs about creativity. Although at the beginning of the course, the students adhered to ‘systematic searching’ using the inventive problem-solving principles they had learned, later on they moved to ‘semi-structured’ and heuristic problem solving, which deals with using strategies, techniques, rules-of-thumb or educated guessing in the problem-solving process. It is important to note, however, that teaching the proposed method in school should take place in the context of engaging students in challenging tasks and open-ended projects that encourage students to develop their ideas. There is only little benefit in merely teaching students inventive problem-solving principles and letting them solve discrete pre-designed exercises.     

Deaf and hard of hearing students' performance on arithmetic word problems

There has been limited research into the intersection of language and arithmetic performance of students who are deaf or hard of hearing, although previous research has shown that many of these students are delayed in both language acquisition and arithmetic performance. The researchers examined the performance on arithmetic word problems of deaf and hard of hearing students in the South-East Queensland region of Australia; they also examined these students' problem-solving strategies. It was found that performance on word problems was similar for deaf and hearing students, but that deaf students experienced delays in achieving successful performance on word problems relative to their hearing peers. The results confirm the findings of other studies showing that students who are deaf or hard of hearing experience delayed language acquisition, which affects their capacity to solve arithmetic word problems. The study conclusions stress the need for greater use of direct teaching of analytic and strategic approaches to arithmetic word problems.     

Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography

This paper introduces an applied problem-solving task, set in the context of cryptography and embedded in a network of computer-based tools. This designed learning environment engaged students in a series of collaborative problem-solving activities intended to introduce the topic of functions through a set of linked representations. In a classroom-based study, students were asked to imagine themselves as cryptanalysts, and to collaborate with the other members of their small group on a series of increasingly difficult problem-solving tasks over several sessions. These tasks involved decrypting text messages that had been encrypted using polynomial functions as substitution ciphers. Drawing on the distinction between viewing functions as processes and as objects, the paper presents a detailed analysis of two groups’ developing fluency with regard to these tasks, and of the aspects of the function concept underlying their problem-solving approaches. Results of this study indicated that different levels of expertise with regard to the task environment reflected and required different aspects of functions, and thus represented distinct opportunities to engage those different aspects of the function concept.     

Cultivating divergent thinking in mathematics through an open-ended approach

The purpose of this study was to develop a program to help cultivate divergent thinking in mathematics based on open-ended problems and to investigate its effect. The participants were 398 seventh grade students attending middle schools in Seoul. A method of pre- and post-testing was used to measure mainly divergent thinking skills through open-ended problems. The results indicated that the treatment group students performed better than the comparison students overall on each component of divergent thinking skills, which includes fluency, flexibility, and originality. The developed program can be a useful resource for teachers to use in enhancing their students’ creative thinking skills. An open-ended approach in teaching mathematics suggested in this paper may provide a possible arena for exploring the prospects and possibilities of improving mathematical creativity.     

Exploring problem conceptualization and performance in STEM problem solving contexts

Problem solving abilities are critical components of contemporary Science, Technology, Engineering and Mathematics (STEM) education. Research in the area of problem solving has uncovered much about the representation, processes and heuristic approaches to problem solving. However, critics claim this overemphasis on the process of solving problems has led to a dearth in understanding of the earlier stages such as problem conceptualization. This paper aims to address some of these concerns by exploring the area of problem conceptualization and the underlying cognitive mechanisms that may play a supporting role in reasoning success. Participants (N?=?12) were prescribed a series of convergent problem-solving tasks representative of those used for developmental purposes in STEM education. During the problem-solving episodes, cognitive data were gathered by means of an electroencephalographic headset and used to investigate students’ cognitive approaches to conceptualizing the tasks. In addition, interpretive qualitative data in the form of post-task interviews and problem solutions were collected and analyzed. Overall findings indicated a significant reliance on memory during the conceptualization of the convergent problem-solving tasks. In addition, visuospatial cognitive processes were found to support the conceptualization of convergent problem-solving tasks. Visuospatial cognitive processes facilitated students during the conceptualization of convergent problems by allowing access to differential semantic content in long-term memory.