数学学习是知识建构的过程

掌握数学的知识结构有助于对数学知识的理解、记忆和迁移.本文主要阐述数学学习的过程就是数学知识结构经由学生的感觉、知觉、记忆、思维等认知操作系统和自我认知、自我监控、反思等元认知系统以顺应、同化等方式内化为自己的认知结构的过程,以及这一理论对数学教学的启示.     

2007年北京理科试卷压轴题的数学背景

<正>2007年的高考结束了,北京理科试卷的最后三道题目分别涉及了概率论,条件极值和代数结构问题.三题共39分,占总分数的26%.而这三道题目都是具有一定的难度和数学背景的,尤其是最后一题涉及代数结构问题,它有一定的高等代数背景,中学生很少接触这方面的问题,现对此题进行一些分析和讨论.     

数学三个世界的理论简介

建构主义认为学习即发展,教学的作用在于根据学生实际需要帮助其发展.2004年英国Warwick大学教授David Tall以建构主义理论为基础,结合当代认知科学、信息科学、新皮亚杰主义等研究成果创立了数学的三个世界理论.该理论主要以中学生和大学生为研究对象,重新划分了认知层次,更加合理地解释了人的认知发展过程.本文旨在阐述该理论在促进学生发展过程中的作用.1 数学的三个世界理论结构概念—具体化世界:以对世界的感知为基础,通过反思利用语言形成精致的意义.不仅包括对外部世界的认识,也包括对主体内部世界的感知.在这个世界中数学学习的对象是具体的、形象的、可见的,简称具体化世界.     

初中数学分层递进教学的基本策略

1 动态分层隐形化动态分层隐形化是指在分层递进教学中,学生的所处的层次是动态变化的,隐形化是分层结果不进行公示.把班内学生分成A,B,C三个层次,用A,B,C来代表低、中、高三层,其中,A层和C层学生的人数各控制在全班学生的10％左右.学生差异既包括现有水平差异,也包括潜在水平差异,因此要动态分层.当学生有了进步,就可以从一个层次上升为另一个层次,使学生体验成功,以调动学生的学习积极性.哪些学生处于哪个层次水平,教师要心中有数,以便对他提出适当要求,因材施教,但这个层次不对外公开,以免形成标签效应.     

On the Structure of Order Domains

The notion of an order domain is generalized. The behaviour of an order domain by taking a subalgebra, the extension of scalars, and the tensor product is studied. The relation of an order domain with valuation theory, Gröbner algebras, and graded structures is given. The theory of Gröbner bases for order domains is developed and used to show that the factor ring theorem and its converse, the presentation theorem, hold. The dimension of an order domain is related to the rank of its value semigroup.     

On Bers Generating Functions for First Order Systems of Mathematical Physics

Considering one of the fundamental notions of Bers’ theory of pseudoanalytic functions the generating pair via an intertwining relation we introduce its generalization for biquaternionic equations corresponding to different first-order systems of mathematical physics with variable coefficients. We show that the knowledge of a generating set of solutions of a system allows one to obtain its different form analogous to the complex equation describing pseudoanalytic functions of the second kind and opens the way for new results and applications of pseudoanalytic function theory. As one of the examples the Maxwell system for an inhomogeneous medium is considered, and as one of the consequences of the introduced approach we find a relation between the time-dependent one-dimensional Maxwell system and hyperbolic pseudoanalytic functions and obtain an infinite system of solutions of the Maxwell system. Other considered examples are the system describing force-free magnetic fields and the Dirac system from relativistic quantum mechanics.     

On the Structure of the Set of E-Functions Satisfying Linear Differential Equations of Second Order

We prove a special case of Siegel's conjecture concerning the representability of E-functions in the form of polynomials in hypergeometric functions. We prove several assertions (formulated earlier by A. B. Shidlovskii) about the transcendence and linear independence of values of E-functions.     

关于工科数学教改中的数学实验

本文就开设数学实验课的必要性、作用和意义以及数学实验课与其它数学课之间的关系、如何开设数学实验课等问题进行讨论 .     

On the Restricted Order of Asymptotic Bases of Order Two

The restricted order of an asymptotic basis A is the least integer h, if it exists, such that every sufficiently large integer is the sum of h or fewer distinct elements of A. We show that any asymptotic basis of order 2 has a restricted order at most equal to 4. We also provide an example of an additive basis of order 2 whose restricted order is 4.To Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B13     

On the quartic curve of Han

The quartic curve of Han [X. Han, Piecewise quartic polynomial curves with shape parameter, Journal of Computational and Applied Mathematics 195 (2006) 34–45] can be considered as the generalization of the cubic B-spline curve incorporating shape parameters into the polynomial basis functions. We show that this curve can be considered as the linear blending of the original cubic B-spline curve and a fixed quartic curve. Moreover, we present the Bézier form of the curve, which is useful in terms of incorporating the curve into existing CAD systems. Geometric effects of the alteration of shape parameters is also discussed, including design oriented computational methods for constrained shape control of the curve.     

数学建模思维方法论

数学建模过程是一个复杂的思维过程,全国大学生数学建模竞赛又加以竞赛的形式,有其特殊性,其中如何研究问题、如何做出成果、如何争取获奖,本文探讨其思维规律、方法技巧.     

结构的延拓及三斜结构系统的代数弹性运动的数学性质

本文的目的并非简单地评述弹性静力学。著名的Cauchy六方程,其命题是由位移函数(u_i, u_j, u_k)=u(xi, xj, xk)的九个偏导数线性表达的,但其逆命题是该六个方程不可能表达阵(?(u_i, u_j, u_k)/?(xi, xj, xk))的九个元素,这是由于在给定点上的变形的几何表示至今尚不完全[1]。用几何语言来说,其逆命题的含意就是:在空间中任意三角形(正交除外)边的“平方长”运算用Pythogora's定理的结论是不真的[2]。本文将叙述代数弹性运动的某些数学规律及其与上述问题的关系。