Effective family problem solving

Effective family problem solving was studied in 97 families of elementary-school-aged children, with 2 definite-solution tasks--tower building (TWB) and 20 questions (TQ), and 1 indefinite-solution task--plan-something-together (PST). Incentive (for cooperation or competition) and task independence (members worked solo or jointly) were manipulated during TWB and TQ, yielding 4 counterbalanced conditions per task per family. On TQ, solo performance exceeded joint performance; on TWB, competition impaired joint performance. Families effective at problem solving in all conditions of both definite-solution tasks tried more problem-solving strategies during TWB and deliberated longer and reached more satisfactory agreements during PST. Family problem-solving effectiveness was moderately predicted by 2 parents' participation in the study. Parental education, parental occupational prestige, and membership in the family of an academically and socially competent child were weaker predictors. The results indicate that definitions of effective family problem solving that are based on directly observed measures of group interaction are more valid than definitions that rely primarily on family characteristics.     

Case studies of teaching problem solving

Summary Some important results that relate to classroom learning and teaching of problem solving emerge from these case studies. These are now summarized as follows. In terms of the students' potential learning experiences of problem solving, it was found that the students were mainly witnessing their teachers' demonstrations of using rules or algorithms for solution to problems. Repeated practice of solving the sorts of problems that occur in examinations was also emphatically included as part of the learning experience. The students were not exposed to a range of strategies that could possibly be used to solve the same problems. There was no explicit teaching of important problem solving skills such as translation skills (comprehending, analyzing, interpreting, and defining a given problem) and linkage skills (concept relatedness between two concepts or using cues from the problem statements to associate ideas, concepts, diagrams, etc. from memory). When teachers solve problems they use, in general, several strategies to solve the same class of problems and they are very careful and explicit about translating problem statements, making relevant linkages and checking. These absences in the teachers' teaching of problem solving (and hence in the students' range of learning experiences) are particularly interesting because they are part of the teachers' own repertoire of skills. Accordingly, it may not be too difficult to get teachers to include them in their teaching. This would mean that the students' range of learning experiences for problem solving would be very much strengthened.     

Productive failure in mathematical problem solving

This paper reports on a quasi-experimental study comparing a “productive failure” instructional design (Kapur in Cognition and Instruction 26(3):379–424, 2008) with a traditional “lecture and practice” instructional design for a 2-week curricular unit on rate and speed. Seventy-five, 7th-grade mathematics students from a mainstream secondary school in Singapore participated in the study. Students experienced either a traditional lecture and practice teaching cycle or a productive failure cycle, where they solved complex problems in small groups without the provision of any support or scaffolds up until a consolidation lecture by their teacher during the last lesson for the unit. Findings suggest that students from the productive failure condition produced a diversity of linked problem representations and methods for solving the problems but were ultimately unsuccessful in their efforts, be it in groups or individually. Expectedly, they reported low confidence in their solutions. Despite seemingly failing in their collective and individual problem-solving efforts, students from the productive failure condition significantly outperformed their counterparts from the lecture and practice condition on both well-structured and higher-order application problems on the post-tests. After the post-test, they also demonstrated significantly better performance in using structured-response scaffolds to solve problems on relative speed—a higher-level concept not even covered during instruction. Findings and implications of productive failure for instructional design and future research are discussed.     

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.

     

求解布局分配问题的群智能聚类算法

提出1种适用于求解布局分配问题的的群智能聚类算法,布局分配问题属于选址问题的1种,其数学表达为包含混合变量的非线性规划模型。根据问题的特点将聚类思想引入群体智能算法,种群中的个体再分出1级,称为子个体,同一个体具有相同的适应度,而子个体根据个体适应度与自身所获得的信息进行移动,其移动有3种方式,方式1与方式2是子个体根据周围需求点的分布而移动,方式3为结合当前种群最优位置信息移动。对7组文献数据使用算法进行50次优化运算,并给出运算的最优值、均值与标准差。运算结果表明:群智能聚类算法能够达到或接近问题的最优解,与一些智能算法相比,算法在问题规模较大时运算结果更优。     

Cognitive variables in problem solving in chemistry

Conclusion A problem solver who is successful in securing a solution will need to achieve in relation to the three tasks to which these variables relate: first, the adequate translation of the problem's statements; second, the correct recalling of prior knowledge such as rules and facts and, third, making the relevant linkage between the problem's statements and rules and facts so that a solution sequence emerges. If he or she is familiar with the problem then the tasks of linkage and translation with play the important role in predicting the problem solving performance. For a problem with which he or she is only partially familiar, the three tasks stated will all contribute significantly to the problem solving performance. For an unfamiliar problem, the task of translation will be the best predictor of the problem solver's performance.     

Facilitating problem solving in high school chemistry

The major purpose for conducting this study was to determine whether certain instructional strategies were superior to others in teaching high school chemistry students problem solving. The effectiveness of four instructional strategies for teaching problem solving to students of various proportional reasoning ability, verbal and visual preference, and mathematics anxiety were compared in this aptitude by treatment interaction study. The strategies used were the factor-label method, analogies, diagrams, and proportionality. Six hundred and nine high school students in eight schools were randomly assigned to one of four teaching strategies within each classroom. Students used programmed booklets to study the mole concept, the gas laws, stoichiometry, and molarity. Problem-solving ability was measured by a series of immediate posttests, delayed posttests and the ACS-NSTA Examination in High School Chemistry. Results showed that mathematics anxiety is negatively correlated with science achievement and that problem solving is dependent on students' proportional reasoning ability. The factor-label method was found to be the most desirable method and proportionality the least desirable method for teaching the mole concept. However, the proportionality method was best for teaching the gas laws. Several second-order interactions were found to be significant when mathematics anxiety was one of the aptitudes involved.     

Mathematics knowledge for understanding and problem solving

Two important aspects of transfer in mathematics learning are the application of mathematical knowledge to problem solving and the acquisition of more advanced concepts, both in mathematics and in other domains. This paper discusses general assumptions and themes of current cognitive research on mathematics learning, focusing on issues of the understanding thought to facilitate transfer of mathematical knowledge. Two studies illustrating these themes are presented, one concerning students' understanding of numerical relationships involved in basic addition and subtraction combinations, the other dealing with students' understanding of algebraic expressions and transformations. Implications of these cognitive perspectives for instruction are discussed.     

Schemas and mental models in geometry problem solving

Recent investigations of mathematical problem solving have focused on an issue that concerns students' ability at accessing and making flexible use of previously learnt knowledge. I report here a study that takes up this issue by examining potential links between mental models constructed by students, the organisational quality of students' prior geometric knowledge, and the use of that knowledge during problem solving. Structural analysis of the results suggest that the quality of geometric knowledge that students develop could have a powerful effect on their mental models and subsequent use of that knowledge.     

Methodologies for problem solving: An engineering approach

Learning how to approach and solve problems which relate to real world situations is an integral part of the education of many higher and further education students and is particularly relevant to students studying for a professional or vocational degree. This paper outlines the nature of problems experienced by engineers and indicates how engineering students are taught to approach the identification and solution of the types of problems which they will experience as practicing professionals. Methodologies used are of general interest and may be applicable in other, unrelated, disciplines.