Encouraging problem solving

Children seem to be natural problem solvers and delight in the challenges that are provided for them. Teachers who are careful observers of what children do can begin to provide many opportunities for helping them build their skills in problem solving. At the same time, it is important to let children create and solve some of their own newly discovered problems. A balance of both seems to be important to solving problems.Janis Bullock is Instructor of Early Childhood Education at Montana State University in Bozeman.     

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.     

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.     

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.

     

Technological problem solving in two science classrooms

This paper reports on the analysis of student (aged 13–15) technological capability as they undertook technological tasks in science classrooms. The activities covered a number of different contexts, had differing degrees of openness, and methods of presentation. An holistic approach to analysing student performance was developed and this provided insights into the approaches adopted by the students. The focus of students on an end-product meant that students did not fully consider the process that might be required to solve the problem. The strategies, skills and knowledge they brought to bear were often not appropriate. Present classroom cultures and contexts need to be understood as greatly affecting performance in technological problem solving. Specializations: science and technology education.     

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.     

The effect of a problem solving in-service program on the classroom behaviors and attitudes of middle school science teachers

This study examines the effect of an in-service education program emphasizing problem solving on teacher attitudes toward teaching science and on teaching behaviors. Twenty-two middle school science teachers participated in the program and another 22 served as the control group. The two groups were similar in terms of gender, teaching status, educational background, and professional activity during the treatment period. Before and after the eight-month project, subjects completed attitude surveys and recorded videotapes of themselves teaching science lessons. No difference was noted between the groups on the attitude measure. The videotapes were analyzed using a coding scheme developed for use in this study. A multivariate analysis of variance performed on the observational data showed a significant difference between the groups on the postworkshop measure. The experimental-group teachers shifted to more student-centered classrooms, with less lecture and procedural talk. This study provides evidence that an extended in-service education program can affect the teaching behaviors of science teachers in the middle grades.     

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.     

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.     

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.