高职教学中关于数学概念的教学浅议

掌握数学概念是学好数学的重要一环。在教学中,教师应从实际事例和学生已有知识出发引入新概念。利用揭示概念的形成过程及巩固和应用概念,来完善学生的认知结构,培养学生的逻辑思维能力,提高教学质量。     

Two Mechanisms for Understanding Mathematical Concepts in Terms of Fictive Motions

This article discusses two mechanisms through which understanding static mathematical concepts (basic and more advanced mathematical concepts) in terms of fictive motions or motion events enhance our understanding of these concepts. It is suggested that at least two mechanisms are involved in this enhancing process. The first mechanism enables us to employ both the motor system and the visual system as two contributing cognitive resources to process the static concept. When one representation of a mathematical concept is transformed into another representation, there is a shift in the mode of processing. This shift facilitates the process of employing new cognitive resources such as the motor and visual systems. The second mechanism, which is a special form of mental simulation, enables us to simulate the process of formation of the static concept, which, in turn, makes it easier for us to understand the structure and properties of the static concept.     

新角度看修辞格Zeugma和Syllepsis

修辞格Zeugma和Syllepsis的研究尽管已进入认知阶段,但目前国内很多研究对两者的认识还并不是很全面。不同于国内的很多研究,这两个修辞格的概念、分类、修辞功能以及两者之间的异同还可以从狭义的角度来探讨,并在此基础上厘清了与汉语拈连辞格的异同。     

Mathematics and learning disabilities

Between 5% and 8% of school-age children have some form of memory or cognitive deficit that interferes with their ability to learn concepts or procedures in one or more mathematical domains. A review of the arithmetical competencies of these children is provided, along with discussion of underlying memory and cognitive deficits and potential neural correlates. The deficits are discussed in terms of three subtypes of mathematics learning disability and in terms of a more general framework for linking research in mathematical cognition to research in learning disabilities.     

Mathematical language and mathematical abilities in preschool: A systematic literature review

It is well established that early general language during preschool is critical for children's mathematical abilities. In an attempt to further characterize this association between language and mathematics, an increasing number of studies show that one specific type of language, namely mathematical language or the key linguistic concepts that are required for performing mathematical activities, is even more critical to children's mathematical abilities. The purpose of this systematic review is to summarize the evidence on mathematical language and mathematical abilities. We focus on preschool children as nearly all of the existing work has been done at this age. We first explain how mathematical language has been defined across studies, and report how it has been evaluated in studies in preschool. Next, we present the results of our systematic review. Following the PRISMA guidelines and after a critical appraisal, we ended with a set of 18 papers that were all of sufficient methodological quality. In these studies, mathematical language was defined as terms that are about numbers and operations on numbers (e.g., nine), but also included linguistic terms that do not directly refer to numbers, yet are important to understand mathematical concepts (i.e., quantitative and spatial terms such as fewest and middle, respectively). Some of these studies evaluated children's performance on mathematical language tasks, while others evaluated the mathematical language input provided to the child by their (educational) environment (teachers/parents/interventionists). Mathematical language correlated positively with children's mathematical abilities, concurrently and longitudinally. It also directly affected children's mathematical abilities, as was shown by intervention studies. We discuss potential directions for future research and highlight implications for education, arguing for more support for teachers and parents to improve the use of mathematical language in the classroom and in home settings.     

数学概念的理解问题

对数学概念的理解问题是数学学习理论中的重要理论问题．就刻划数学概念理解的重要概念——数学概念的心理表征和数学概念理解的认知基础进行分析，从而对数学概念的理解问题有更深入的认识。     

几个计数问题的映射解法

剖析了映射与函数概念的内涵，通过建立集合间映射的技巧，给出了交替和，树，好子集，对称数等计数问题的求解方法，强调了数学概念在解决理论和实际问题中的作用。     

形成数学概念体系对掌握数学概念的作用

提出数学概念体系的特点。运用现代认知同化理论,分析了数学概念体系对数学概念的理解、保持及运用的作用,并得出了这一过程的结构关系。     

The Impact of an Ongoing Professional Development Program on Prekindergarten Teachers' Mathematics Practices

Mathematics is a natural part of daily life for young children as they explore and investigate the world around them. To build on these experiences, and to begin establishing a mathematical foundation, early childhood educators must not only be knowledgeable about mathematical concepts, they must also be aware of the most developmentally appropriate ways in which to teach these concepts to young children. After participation in an ongoing professional development program, specifically targeting teachers of prekindergarten children in public school, Preschool Programs for Children with Disabilities (PPCD), Head Start, and child care settings, teachers reported positive changes in math practices. Specifically, teachers reported a stronger alignment to national mathematics standards and increased awareness pertaining to developmentally appropriate mathematics practices as they apply to early childhood classrooms. Teachers reported a shift towards more hands-on activities and a shift away from the use of worksheets in their prekindergarten classrooms. Implications from this study suggest that ongoing professional development that is designed to meet the specific needs of early childhood educators can have a positive impact on reported mathematics content knowledge and instructional practices.     

词义扩展的内在机制及认知思维培养

词义扩展是人类认知思维的体现。隐喻和转喻是人类基本的认知思维,分别基于事物和概念间的相似性和邻近性而构建,是实现词义扩展的重要手段。传统的多义词教学往往将其各个义项独立,忽略彼此之间的认知关联,教学效果差强人意。从认知角度来分析词义扩展的内在机制,并对英语词汇教学中如何培养多义词的认知思维进行研究。     

APOS理论下高职化工专业高等数学概念教学方法探究

美国数学教育家杜宾斯基提出的APOS理论是一种建构主义的数学学习理论,他将数学概念的建构分为Action、Process、Object、Scheme四个阶段.在对该理论的认识基础上,结合高职学生数学学习认知的心理特点,对化工专业高等数学概念的教学进行探讨,并就如何进行数学概念教学设计作了探索,使学生主动建构其概念体系.     

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.     

Knowledge construction and diverging thinking in elementary & advanced mathematics

This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a number of studies on children's arithmetic shows a divergence in performance. The less successful seem to focus more on perceptions of their physical activities than on the flexible use of symbol as process and concept appropriate for a conceptual development in arithmetic and algebra. Advanced mathematical thinking introduces a new feature in which concept definitions are formulated and formal concepts are constructed by deduction. We show how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success.This revised version was published online in September 2005 with corrections to the Cover Date.     

Letters

Conclusions One of the things particularly disturbing about managing our doctoral programs is the growing realization that our professionals must be more highly trained than those working in the traditional social or physical sciences. It takes longer (and I suspect it is more difficult) to produce the hybrid professional capable of and committed to contributing to educational practice. Our students must be as knowledgeable as the doctoral candidate in (for example) psychology and in addition must acquire a set of skills that deal with enhancing practice in a great variety of settings. It also is disturbing to realize that our doctoral students probably won’t acquire this capability unless faculty members begin to model the skills and approaches they expect their students to acquire. I have consciously ignored a number of areas (curriculum, focus for courses and assessment, and delineation of desirable competencies and performances). At the moment my concern is to communicate my strong belief that we must grow away from the training of people primarily concerned with technical skills in developing instruction, evaluating programs, managing resource centers, and producing films and television programs and focus on the training of people skilled in inquiring about problems and their solutions. Presumably these people would be able to arrive at some solutions that would be useful regardless of their career choices—research, development, production, and/or administration. All student activities involving the actual development of instruction should be conducted in an atmosphere of constant inquiry and critical discussion of the usefulness of the concepts being acquired and the process being employed.     

数学概念表征缺失引起的理解障碍探析

数学概念学习中,概念理解是首要的;认知心理学研究表明,学生数学概念的获得是一个对概念心理表征的构建过程;相关的数学概念表征的调查研究也证明了数学概念表征与概念理解是相互促进、相互制约的;根据学生在数学概念学习中,对因概念表征缺失引起的概念理解障碍进行认知分析。     

数学知识观与数学课堂

传统的数学知识观把数学知识看成是概念、原理、定理，符号等和静态集合体，将数学教学看成是“传授-接受”，有必要对其进行教育学的批判，而现代数学知识观测把数学和知识看成是一种动态的生成过程，将数学教学看成是“活动-建构”强调学生自身的经验与体验，数学学习是一个主动建构的过程。     

Using what matters to students in bilingual mathematics problems

In this study, the author represented what matters to bilingual students in their everyday lives—namely bilingualism and everyday experiences—in school-based mathematical problems. Solving problems in pairs, students demonstrated different patterns of organizing and coordinating talk across problem contexts and across languages. Because these patterns bear consequences for how bilinguals experience mathematics learning, the author takes these patterns as the basis to argue that mathematics education for bilingual students should capitalize on bilingualism and experiences as cognitive resources.     

Some Cognitive Difficulties Related to the Representations of two Major Concepts of Set Theory

The main focus of this paper is on the study of students' conceptual understanding of two major concepts of Set Theory – the concepts of inclusion and belonging. To do so, we analyze two experimental classroom episodes. Our analysis rests on the theoretical idea that, from an ontogenetic viewpoint, the cognitive activity of representation of mathematical objects draws its meaning from different semiotic systems framed by their own cultural context. Our results suggest that the successful accomplishment of knowledge attainment seems to be linked to the students' ability to suitably distinguish and coordinate the meanings and symbols of the various semiotic systems (e.g. verbal, diagrammatic and symbolic) that encompass their mathematical experience.     

Complementarity, sets and numbers

Niels Bohr's term‘complementarity' has been used by several authors to capture the essential aspects of the cognitive and epistemological development of scientific and mathematical concepts. In this paper we will conceive of complementarity in terms of the dual notions of extension and intension of mathematical terms. A complementarist approach is induced by the impossibility to define mathematical reality independently from cognitive activity itself. R. Thom, in his lecture to the Exeter International Congress on Mathematics Education in 1972,stated ‘‘the real problem which confronts mathematics teaching is not that of rigor,but the problem of the development of‘meaning’, of the ‘existence' of mathematical objects'. Student's insistence on absolute ‘meaning questions’, however,becomes highly counter-productive in some cases and leads to the drying up of all creativity. Mathematics is, first of all,an activity, which, since Cantor and Hilbert, has increasingly liberated itself from metaphysical and ontological agendas. Perhaps more than any other practice,mathematical practice requires acomplementarist approach, if its dynamics and meaning are to be properly understood. The paper has four parts. In the first two parts we present some illustrations of the cognitive implications of complementarity. In the third part, drawing on Boutroux' profound analysis, we try to provide an historical explanation of complementarity in mathematics. In the final part we show how this phenomenon interferes with the endeavor to explain the notion of number. This revised version was published online in July 2006 with corrections to the Cover Date.