Applied mathematical problem solving

A case is presented for the importance of focusing on (1) average ability students, (2) substantive mathematical content, (3) real problems, and (4) realistic settings and solution procedures for research in problem solving. It is suggested that effective instructional techniques for teaching applied mathematical problem solving resembles mathematical laboratory activities, done in small group problem solving settings.The best of these laboratory activities make it possible to concretize and externalize the processes that are linked to important conceptual models, by promoting interaction with concrete materials (or lower-order ideas) and interaction with other people.Suggestions are given about ways to modify existing applied problem solving materials so they will better suit the needs of researchers and teachers.     

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.     

Metacognitive macroevaluations in mathematical problem solving

This paper focuses on the role of evaluation in mathematics in 749 elementary school children. The macroevaluative skills and calibration scores of high versus low mathematical problem solvers were contrasted as measures of metacognition. No relevant calibration differences were found for gender. In addition, the performances of children with mathematics learning disabilities could not be explained according to the maturational lag hypothesis. Finally, although macrometacognitive evaluation and calibration seem attractive alternatives for time-consuming on-line metacognitive assessment techniques, our data show that a global and retrospective assessment of the macroevaluation is not always enough to get the picture of mathematical problem solving in young children.     

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.     

Mathematical problem solving by pre-school children

In this article we provide new evidence for mathematical problem-solving abilities of pre-school children. These problem-solving behaviours occurred in a study of sharing of discrete items by dealing, in which we examined the abilities of three categories of counters to solve a discrete re-distribution problem. We detail the problem solving strategies used in the context of sharing by dealing as a common action scheme of pre-school children in clinical interviews.     

Mathematical problem solving and young children

Educators of young children can enhance the development of a problem-solving thought process through daily activities in their classrooms. An emphasis should be placed on the actual thought process needed to solve problems that occur in everyday living. Educators can follow simple suggestions to create problem-solving situations for all ages of children. The process of thinking through a problem and finding a solution is more important than traditional mathematics counting and memorizing useless facts. Even very young children are capable of a problem-solving process that is on the appropriate developmental level. The problem-solving process is constructivist in nature, as each individual perceives problems according to her or his background and developmental levels. Educators need to make a conscious effort to capitalize on all stages of problem-solving thinking to enhance future mathematical development.     

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.     

Visual processing during mathematical problem solving

This study investigated, in the context of mathematical problem solving by secondary school students, the nature of the visual schemata which Johnson (1987) hypothesises mediate between logical propositional structures and rich specific visual images. Four groups of grade 10 students were studied, representing all combinations of high and low operational ability in mathematics (equivalent to Johnson's logical propositional structures) and high and low vividness of visual imagery (corresponding to Johnson's rich images). The results suggested first, that success at problem solving was related to logical operational ability, but not to vividness of visual imagery. Second, a variety of visually based strategies were used during problem solving which differed in their level of generality and abstraction, and use of these strategies appeared related to either logical operational ability or vividness of visual imagery, depending on their level of abstraction. The results supported Presmeg's (1992b) continuum of abstraction of image schemata.Throughout the paper, the first High or Low denotes logical operational ability, and the second, vividness of visual imagery.     

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.

     

Enhancing spatial skills through mechanical problem solving

Higher spatial skills are associated with increased interest, performance, and creativity in STEM fields (Science, Technology, Engineering, Mathematics). However, evidence for causal relations between spatial skills and STEM performance remains scarce. In this study, we test the extent to which mechanical problem solving, a spatially demanding STEM activity, facilitates spatial performance. Participants (N = 180) were randomly assigned to one of four training conditions: mechanical reasoning with a hands-on component; mechanical reasoning without a hands-on component; an active control condition involving spatial training with cross-sectioning; and an active control group involving a reading exercise. All participants were tested immediately before, after, and one-week following training. Both mechanical conditions were associated with enhanced spatial visualization performance, an effect that was similar for both conditions and remained stable across immediate and delayed post-tests. These findings suggest that mechanical problem solving is a potentially viable approach to enhancing spatial thinking.     

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。     

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.