构造思想在解题中的应用

用构造思想解决问题具有一定的创造性和启发性。一些数学问题用构造思想作为辅助手段来解决 ,使解题变得简单、快捷。本文第举一些实例对构造思想解题做一些探讨。一、构造函数解题构造函数法是运用函数思想 ,对问题进行观察、分析 ,构造也与问题有一定联系的函数 ,利用函数的知识来解决问题的一种方法。1、构造函数证明不等式构造二次函数模型Ｆ(ｘ) =(ａ1 ｘ -ｂ1 ) 2 +(ａ2 ｘ -ｂ2 ) 2 +… +(ａｎｘ -ｂｎ) 2 考虑到Ｆ(ｘ)≥ 0 ,有△≤ 0 ,即 (ａ1 ｂ1 +ａ2 ｂ2+… +ａｎｂｎ) 2 ≤ (ａ12 +ａ22 +… +ａｎ2 )·(ｂ12 +ｂ22 +… +ｂｎ2 )…     

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.     

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.     

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.     

A working memory model applied to mathematical word problem solving

The main objective of this study is (a) to explore the relationship among cognitive style (field dependence/independence), working memory, and mathematics anxiety and (b) to examine their effects on students’ mathematics problem solving. A sample of 161 school girls (13–14 years old) were tested on (1) the Witkin’s cognitive style (Group Embedded Figure Test) and (2) Digit Span Backwards Test, with two mathematics exams. Results obtained indicate that the effect of field dependency, working memory, and mathematics anxiety on students' mathematical word problem solving was significant. Moreover, the correlation among working memory capacity, cognitive style, and students’ mathematics anxiety was significant. Overall, these findings could help to provide some practical implications for adapting problem solving skills and effective teaching/learning.     

有关数学问题的解题思想与方法

我们从中学就开始接触各类数学问题,而要解决这些数学问题,最重要的就是找出问题的精髓也就是所运用的思想与方法,并且这些思想与方法在实际应用中也非常广泛,因此,在这里我们主要介绍几种重要的解决数学问题的思想与方法。     

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.     

Towards efficient measurement of metacognition in mathematical problem solving

Metacognitive monitoring and regulation play an essential role in mathematical problem solving. Therefore, it is important for researchers and practitioners to assess students?? metacognition. One proven valid, but time consuming, method to assess metacognition is by using think-aloud protocols. Although valuable, practical drawbacks of this method necessitate a search for more convenient measurement instruments. Less valid methods that are easy to use are self-report questionnaires on metacognitive activities. In an empirical study in grade five (n?=?39), the accuracy of students?? performance judgments and problem visualizations are combined into a new instrument for the assessment of metacognition in word problem solving. The instrument was administered to groups of students. The predictive validity of this instrument in problem solving is compared to a well-known think-aloud measure and a self-report questionnaire. The results first indicate that the questionnaire has no relationship with word problem solving performance, nor the other two instruments. Further analyses show that the new instrument does overlap with the think-aloud measure and both predict problem solving. But, both instruments also have their own unique contribution to predicting word problem solving. The results are discussed and recommendations are made to further complete the practical measurement instrument.     

Toward a design theory of problem solving

Problem solving is generally regarded as the most important cognitive activity in everyday and professional contexts. Most people are required to and rewarded for solving problems. However, learning to solve problems is too seldom required in formal educational settings, in part, because our understanding of its processes is limited. Instructional-design research and theory has devoted too little attention to the study of problem-solving processes. In this article, I describe differences among problems in terms of their structuredness, domain specificity (abstractness), and complexity. Then, I briefly describe a variety of individual differences (factors internal to the problem solver) that affect problem solving. Finally, I articulate a typology of problems, each type of which engages different cognitive, affective, and conative processes and therefore necessitates different instructional support. The purpose of this paper is to propose a metatheory of problem solving in order to initiate dialogue and research rather than offering a definitive answer regarding its processes. This paper represents an effort to introduce issues and concerns related to problem solving to the instructional design community. I do not presume that the community is ignorant of problem solving or its literature, only that too little effort has been expended by the field in articulating design models for problem solving. There are many reasons for that state of affairs. The curse of any introductory paper is the lack of depth in the treatment of these issues. To explicate each of the issues raised in this paper would require a book (which is forthcoming), which makes it unpublishable in a journal. My purpose here is to introduce these issues in order to stimulate discussion, research, and development of problem-solving instruction that will help us to articulate better design models.     

Non‐mathematical problem solving in organic chemistry

Differences in problem‐solving ability among organic chemistry graduate students and faculty were studied within the domain of problems that involved the determination of the structure of a molecule from the molecular formula of the compound and a combination of IR and ¹H NMR spectra. The participants' performance on these tasks was compared across variables that included amount of research experience, year of graduate study, and level of problem‐solving confidence. Thirteen of the 15 participants could be classified as either “more successful” or “less successful.” The participants in this study who were “more successful” adopted consistent approaches to solving the problems; were more likely to draw molecular fragments obtained during intermediate stages in the problem‐solving process; were better at mining the spectral data; and were more likely to check their final answer against the spectra upon which the answer was based. Experience from research, teaching, and course work were found to be important factors influencing the level of participants' success. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 47:643–660, 2010     

数学思想及解题技巧初探

数学是一门运用非常广泛的基础性学科,数学思想是教学精髓所在,在解题教学中应加强数学思想方法的引导渗透,从引导建立直觉认识开始,逐步开启学生的理性认知,培养灵活应用数学思想的素养。教师要根据不同阶段、不同水平层次学生的实际情况,分类分批地渗透数学思想方法。在倡导素质教育的背景下,让学生将数学思想运用于小学习题解题,是培养学生发散性思维的重要环节。据此,建议把全班学生分成若干组,让学习成绩好的学生扮演小老师角色,在课外帮助差生解答疑难问题,可以取得一举两得的效果。     

The contribution of external representations in pre-school mathematical problem solving

Could problem solving be the object of teaching in early education? Could appropriate teaching interventions develop to scaffold children's efforts to solve problems? These were the central questions of this article. The sample consisted of 18 children attending public pre-school in Cyprus. The problem they were asked to solve was to find all solutions of the pentomino. The children's problem solving was supported by graphically representing their solutions on squared paper. The findings show that children responded positively to the problem and were successful in finding all solutions for the specific problem. The graphical representation of the solutions and the forms of teacher–children and children–children interactions played an important role in the positive outcome of the activity.     

Teaching and testing mathematical problem solving by offering optional assistance

Transfer capability is usually defined as theability to apply acquired knowledge and skillsin novel situations.The experiment reported here concerned transferin mathematics education. An experimental programme was constructed, based on strengtheningthe connection of strategic and domain specificknowledge and offering hints during teaching aswell as during testing.Subjects were first graders from secondaryeducation in the Netherlands, from two schools,two classes each. Students from these fourclasses were randomly allocated either to theexperimental or the control group.The experimental computer-supported teachingprogramme was offered once a week during sixweeks; the control group received regularmathematics instruction.After controlling for the effects of thecovariates intelligence, mathematics aptitudeand anxiety, it was shown that the experimentalsubjects performed significantly better on aposttest than subjects in the control group.The results suggest that the experimentalinstruction method enhances mathematics problemsolving ability more strongly than traditionalinstruction. Upon closer examination thiseffect appears to be restricted to subjectsalready relatively high in intelligence andmathematical ability. This finding is notuncommon in intervention research, and issometimes referred to as the Matthew orthe fan-spread effect.     

Pathways to word problem solving: The mediating roles of schema construction and mathematical vocabulary

When solving word problems, many children encounter difficulties in making sense of the information and integrate it into a meaningful schema. This is the fundamental phase on which subsequent problem solution depends. To better understand the processing underlying this fundamental phase, this study examined the roles of schema construction and knowledge of mathematical vocabularies in word problem solving. The participants were 139 Chinese third graders studying in Hong Kong. Path analysis showed that there were two kinds of pathways to word problem solving: language-related and number-related. In particular, reading fluency was related to word problem solving in two mediated language-related pathways: one via schema construction, the other via knowledge of mathematical vocabularies. In the number-related pathway, arithmetic concept was related to word problem solving via knowledge of mathematical vocabularies. These findings highlight the specific roles of schema construction and mathematical vocabulary in word problem solving, thereby providing useful implications of how best to support children in understanding and integrating the information from the problem.     

Examining the unique contributions and developmental stability of individual forms of relational reasoning to mathematical problem solving

Relational reasoning, a higher-order cognitive ability that identifies meaningful patterns among information streams, has been suggested to underlie STEM development. This study attempted to explore the potentially unique contributions of four forms of relational reasoning (i.e., analogy, anomaly, antinomy, and antithesis) to mathematical problem solving. Two separate samples, fifth graders (n = 254) and ninth graders (n = 198), were assessed on their mathematical problem solving ability and the different forms of relational reasoning ability. Linear regression analysis was conducted, with participants’ age, working memory, and spatial skills as covariates. The results showed that analogical and antithetical reasoning abilities uniquely predicted mathematical problem solving. This pattern demonstrated developmental stability across a four-year time frame. The findings clarify the unique significance of individual forms of relational reasoning to mathematical problem solving and call for a shift of research direction to reasoning abilities when exploring dissimilarity-based relations (opposites in particular).     

Surface features,representations and tutorial interventions in mathematical problem solving

The concept of instability of representation, which has developed from observing pupils who experience difficulties whilst performing complex tasks, is used to measure the impact of a certain number of hints given in order to help solve mathematical problems. The purpose of these hints is to neutralize the effect of superficial elements of information and to anchor the representation which the subject forms of the problem to be solved. The hints used in the experiments fall into two categories: the simultaneous presentation of several variants of the problem, and the accomplishment of recognition tasks in the course of the solving process. The possibility of intervening during the cognitive functioning of the pupil by allowing him maximum autonomy in his choice and application of the solving process is questioned both from a didactic and a psychological point of view.     

The mathematical discourse of 13-year-old partnered problem solving and its relation to the mathematics that emerges

This paper, written within a discursive perspective, explores the co-shaping of public and private discourse, and some of the circumstances under which one occasions the other, in the evolution of mathematical thinking by pairs of 13-year-olds. The discourse of six pairs of students, engaged in interpreting and graphing problem situations involving rational functions, was analyzed by means of recently developed methodological tools. The nature of the mathematics that emerged for each pair was found to be related to several factors that included the characteristics of the interpersonal object-level utterances both before and after the solution path had been generated, the degree of activity of the personal channels of the interlocutors, and the extent to which the thoughts of participants were made explicit in the public discourse. The analysis of the discursive interactions provided evidence that adolescents within novel problem situations can experience some difficulty in making their emergent thinking available to their partners in such a way that the interaction be highly mathematically productive for both of them. This revised version was published online in July 2006 with corrections to the Cover Date.     

Problem solving strategies and mathematical resources: A longitudinal view on problem solving in a function based approach to algebra

This study is an attempt to analyze students' construction of function based problem solving methods in introductory algebra. It claims that for functions to be a main concept for learning school algebra, a complex process that has to be developed during a long period of learning must take place. The article describes a longitudinal observation of a pair of students that studied algebra for 3 years using a function approach, including intensive use of graphing technology. Such a long observation is difficult to carry out and even more difficult to report. We watched for three years classrooms using the ‘Visual-Math’ sequence, and sampled students that exhibited various levels of mathematics achievement. The analysis method presented here is a non-standard case study of a pair of lower achievers students and their work is often juxtaposed to the work of other pairs participating in the study. The students' attempts to solve a linear break-even problem is analyzed along three interviews which present the development of the use of mathematical resources and the patterns of problem solving at different learning phases. Beyond describing solving attempts, the article offers terms for describing and explaining what and how do learners appreciate and make out of solving introductory school algebra problems over a three years course. This revised version was published online in July 2006 with corrections to the Cover Date.     

Visual spatial representation in mathematical problem solving by deaf and hearing students

This research examined the use of visual-spatial representation by deaf and hearing students while solving mathematical problems. The connection between spatial skills and success in mathematics performance has long been established in the literature. This study examined the distinction between visual-spatial "schematic" representations that encode the spatial relations described in a problem versus visual-spatial "pictorial" representations that encode only the visual appearance of the objects described in a problem. A total of 305 hearing (n = 156) and deaf (n = 149) participants from middle school, high school, and college participated in this study. At all educational levels, the hearing students performed significantly better in solving the mathematical problems compared to their deaf peers. Although the deaf baccalaureate students exhibited the highest performance of all the deaf participants, they only performed as well as the hearing middle school students who were the lowest scoring hearing group. Deaf students remained flat in their performance on the mathematical problem-solving task from middle school through the college associate degree level. The analysis of the students' problem representations showed that the hearing participants utilized visual-spatial schematic representation to a greater extent than did the deaf participants. However, the use of visual-spatial schematic representations was a stronger positive predictor of mathematical problem-solving performance for the deaf students. When deaf students' problem representation focused simply on the visual-spatial pictorial or iconic aspects of the mathematical problems, there was a negative predictive relationship with their problem-solving performance. On two measures of visual-spatial abilities, the hearing students in high school and college performed significantly better than their deaf peers.