代数基本定理的证明方法

对代数基本定理的证明，进行了多种方法的分桥，运用初等方法、Cauchy积分定理和Brouwer不动点定理，给出另外3种方法进行论证。     

浅谈微积分基本定理

在一元微积分中,牛顿-莱布尼兹公式是最重要的公式,它建立了微分学与积分学之间的联系.在多元微积分中,也有类似的公式.通过研究场论中三个基本公式的关系,可统一处理多元函数中的相关内容.     

Images of rate and operational understanding of the fundamental theorem of calculus

Conceptual analyses of Newton's use of the Fundamental Theorem of Calculus and of one 7th-grader's understanding of distance traveled while accelerating suggest that concepts of rate of change and infinitesimal change are central to understanding the Fundamental Theorem. Analyses of a teaching experiment with 19 senior and graduate mathematics students suggest that students' difficulties with the Theorem stem from impoverished concepts of rate of change and from poorly-developed and poorly coordinated images of functional covariation and multiplicatively-constructed quantities.Research reported in this paper was supported by National Science Foundation Grants No. MDR 89-50311 and 90-96275, and by a grant of equipment from Apple Computer, Inc., Office of External Research. Any conclusions or recommendations stated here are those of the author and do not necessarity reflect official positions of NSF or Apple Computer. Also, I wish to thank Paul Cobb and Guershon Harel for their helpful reactions to an earlier draft of this article.     

质量互变定理用于识别可拓学与集对分析的基础性错误

根据2007年笔者创立的可变集及其辩证法基本规律数学定理,识别可拓学(物元分析)与集对分析的基础性错误。指出可拓学的基础可拓集合关联函数等于零等概念与定义的错误。指出集对分析的中介不确定性等概念与定义,同异反联系度公式计算概率可能性的错误。     

以心观心:中国传统道德教育思想的内在理路

中国“修身”的道德教育思想有注重外在规范与行为养成的传统,同时内在蕴含“修心”的思路。中国文化中的“心”概念与西方哲学对认知与理性的强调相对,表现出明显的情感取向,是人之为人的本质所在。心作为内在意义世界,标画出人的道德意义系统,该意义世界和意义系统具有在世品格与开放自主性。从这一心概念出发,中国传统道德教育思想的内在理路可以被概括为“以心观心”。“以心观心”的内在理路是:首先引导人从心的角度观照道心和他心,从而进入道德意义系统;以前者为前提,指引人从我心走向道心和他心,并在这一行动过程中进一步体认道德意义,如此循环往复,不断充实意义世界,共创和谐美善的人间。     

Pathways to arithmetic: The role of visual-spatial and language skills in written arithmetic,arithmetic word problems,and nonsymbolic arithmetic

This study set out to develop and test a pathway model of the relations between general cognitive skills, specifically visual-spatial and spoken and written language skills, and competence in three forms of arithmetic that vary in modes of number representation. A total of 88 Chinese 4-year-olds participated and were tested first in kindergarten second grade (K2) and then in kindergarten third grade (K3). Language skills, including phonological, morphological, and visual-orthographic skills, and visual-spatial skills were measured at K2, and arithmetic outcomes, including written arithmetic, word problems, and nonsymbolic arithmetic, at K3. The results generally supported our model. Specifically, visual-spatial skills contributed to the prediction of all three types of arithmetic outcomes. Morphological skills predicted word problems, whereas phonological skills predicted written arithmetic. Finally, visual-orthographic skills contributed to both written and nonsymbolic arithmetic. These findings underscore the importance of delineating the specificity of cognitive processes in learning diverse forms of arithmetic.     

Persistent arithmetic, reading, or arithmetic and reading disability

The achievement of 1,075 school children was followed to the end of grade 7 or 8. Retrospective details of birth, medical history, and infant adaptive behavior, as well as current information about preschool behavior, were provided by parents before kindergarten entry when their children were ages four to five. At that time, each child was given language, preacademic, and visual-motor tests, and a short-form verbal IQ was obtained. For grades 1 to 8, a mean 6.9 percent qualified as low in arithmetic (including 3.9 percent low only in arithmetic) and 9 percent as low in reading (including 6 percent low only in reading). Groups of children with persistent disability in arithmetic only (2.3 percent), reading only (6.6 percent), and arithmetic plus reading (3.4 percent) were compared on all variables, and gender differences were examined. The specific arithmetic group was superior to the other two groups in preschool verbal IQ and language, and had suffered more birth problems and illness. As found by Rourke and his colleagues (e.g., Rourke and Finlayson 1978) for older children, boys with a specific arithmetic disability showed a preschool profile with high verbal scores and low nonverbal scores. Many gender differences were found for the total sample, but few for the learning disability subgroups.     

Foundation of basic arithmetic

This is the second of a series of articles, in which we demonstrate the utility of the place value system of representing numbers.     

The arithmetic connection

From test and interview data obtained during an investigation into Grade 10 students' conceptions of algebraic generalisation and justification, we have extracted evidence of the extent to which these students have coordinated the worlds of arithmetic and algebra, and can move freely between them. The data show more dissociation than we expected, even among students who were successful at standard algebraic tasks. Conceiving algebra as generalised arithmetic may obscure the many genuine obstacles that the learner has to overcome in moving from fluent performance in arithmetic to fluent performance in algebra while achieving and maintaining a smooth coordination of both modes of action.     

Foundation of basic arithmetic

This is the first of a series of articles, in which we shall briefly outline the early evolution of numerals as we currently know it.     

Piaget's theory and the teaching of arithmetic

Conclusion For centuries, education has been a folk art based on tradition and opinions called philosophies. With the advent of associationism and behaviourism, however, education began to enter a scientific era. Now that we have Piaget's constructivism—a more adequate scientific theory—it is time to change the way arithmetic is taught in elementary schools.I hope educators and researchers in other countries will also experiment in classrooms to test the hypothesis that I tested. Because logico-mathematical knowledge is universal and the same in all cultures, I expect the same kinds of results in other countries.Constructivist teaching is much harder than the teaching of algorithms and correcting of worksheets. However, in spite of this difficulty, an increasing number of teachers in the United States are becoming convinced of the truth of Piaget's constructivism. When these teachers encounter children's originality and intelligence that they have not observed before, they are truly overjoyed. Once they have seen the possibility that exists in each child, these teachers do not think of going back to their old ways of teaching.Constance Kamii studied under Jean Piaget, Barbel Inhelder and Hermina Sinclair in the late 1960s and during the 1970s and subsequently developed a pre-school curriculum based on Piaget's theory. She later extended this work to mathematics education in the first three grades and is now working at the fourth and fifth grade level. She has taught at the University of Illinois, Chicago, and the University of Geneva, and is now a professor of education at the University of Alabama, Birmingham, in the United States.I am grateful to Janice K. Ewing for critically reading a draft of this paper and making helpful suggestions.     

Stability of arithmetic disability subtypes

Cross-sectional research has identified subtypes of children with learning disabilities who may have distinctive cognitive ability patterns. This study examined the stability over 19 months of academic subtyping classifications for 80 children ages 9 to 13 representing four subtypes of arithmetic disabilities (AD), using three criteria for learning disability identification. Approximately half of the sample retained AD regardless of identification method. Children with pervasive deficits in arithmetic, reading, and spelling displayed the greatest subtype stability. Only one third of the children with the other subtypes, including those with isolated arithmetic deficits, retained their original subtypes. Thus, drawing conclusions and making recommendations based on academic subtyping at a single point in time may be unwise.     

泰勒定理的推广

从泰勒定理余项的积分表示法推证泰勒定理的推广.     

一个定理的改进及拓展

对被3(或9)整除的整数特征(即文中预备定理)进行改进,对于解决此类问题更便捷.