对学生解题错误探究一文的再探究

<化学教学>2003年第3期<学生解题错误探究>一文读后受益匪浅,但其中二道例题解法缺乏科学性和严密性,特提出与原文作者商榷并供同行参考.     

Assessing problem solving in mathematics: Some variables related to student performance

A mathematical problem is defined here as a question not dependent on specific syllabus content, and one sufficiently new to the student such that it cannot be solved by a previously known method. With increased attention being paid to this type of mathematical problem solving at the primary school level, the need for reliable and valid methods of assessment has become more apparent. This paper reports the results of using a new problem solving test, developed for use in the upper primary school, with 371 students in Years 4,5 and 6 at government schools in Melbourne. Particular attention is given to the effects of year level, sex and the method of test administration on student performance for different types of items and different problem solving processes. The performance of Year 4 students was generally lower than that of other students, but differences were small for most items and processes between Years 5 and 6. Although most of the differences in performance between the sexes were not significant, the girls had higher scores than the boys for the total score, for all processes and for all items except the spatial item. The method of administration was important for performance, especially for the girls. The marking schedule developed enabled high intra- and inter-marker reliabilities to be obtained.     

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.     

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.     

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.     

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.     

Metacognition and mathematical problem solving in grade 3

This article presents an overview of two studies that examined the relationship between metacognition and mathematical problem solving in 165 children with average intelligence in Grade 3 in order to help teachers and therapists gain a better understanding of contributors to successful mathematical performance. Principal components analysis on metacognition revealed that three metacognitive components (global metacognition, off-line metacognition, and attribution to effort) explained 66% to 67% of the common variance. The findings from these studies support the use of the assessment of off-line metacognition (essentially prediction and evaluation) to differentiate between average and above-average mathematical problem solvers and between students with a severe or moderate specific mathematics learning disability.     

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.     

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.     

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.     

Inquiry training and problem solving in elementary school children

An experiment designed to determine whether children can benefit from directed instruction in the strategies of problem solving.     

Processing errors in conditional and biconditional problem solving behavior

Students (N = 237) in each of grades 8, 10, 12, and 14 were randomly divided into three groups and administered either a 16-item multiple-choice test of conditional syllogisms, or a 16-item test of biconditional syllogisms, or a 32-item test with both conditional and biconditional syllogisms. Results provided within experiment and within-subjects comparison of responses on conditional and biconditional syllogisms. A comparison of response patterns on conditional items with responses on actual biconditional items provided a direct test of the previously hypothesized biconditional misinterpretation of conditional problems. These analyses confirmed the strong tendency of subjects (across grades) to interpret conditional syllogisms biconditionally. Surprisingly, performance on biconditional problems does not improve systematically with age; in fact, college sophomores perform only slightly better than eighth graders. With regard to conditional syllogisms, results confirmed previously described performance variations across forms of both major and second premises and also replicated an unusual reversed developmental trend on the problem which involves denying the consequent.     

Successful and unsuccessful problem solving in classical genetic pedigrees

Using the think-aloud interview technique, 16 undergraduates and 11 genetics graduate students and biology faculty members were asked to solve from 1 to 3 classical genetics problems which require pedigree analysis. Subjects were classified as either successful or unsuccessful and the performances of these groups were analyzed from videotaped recordings of the interviews. A number of previously reported findings were corroborated. Additional observations are discussed in terms of genetic knowledge, use of production rules, strategy selection, use of critical cues, hypothesis testing, use of logic, understanding of issues of probability, and the thinking process itself. Taken collectively, these findings evidence a remarkable similarity between the successful solution of pedigree problems and the processes of medical diagnosis and scientific investigation. This convergence of research findings suggests a qualitative advance in the understanding of problem solving. Based on this understanding, recommendations for classroom instruction are presented.     

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.