数学思维,数学教学与问题解决

问题是数学的心脏,问题是引导研究的,提出和发现数学问题是数学思维的起步.数学问题解决体现了数学思维的目的、过程和基本方法,是创造性的思维活动.问题解决作为教学方法,能体现知识的涵义和应用价值.     

通过综合实践活动提高学生解决数学实际问题能力的案例研究

生活是数学的源泉,数学实际问题从生活中来;数学最后也要回归生活,解决数学实际问题是数学学习的最终目的.提高学生解决数学实际问题的能力也是数学教学的主要目标之一.通过2个案例介绍了通过综合实践活动提高学生解决数学实际问题能力的具体做法,以供同仁参考.     

谈垂直化与水平化在数学教学中的应用

水平数学化是指由现实问题到数学问题的转化,是把情景问题表述为数学问题的过程,亦指数学问题的水平发展.垂直数学化是从符号到概念的转化或用符号解决数学问题,亦指数学问题的梯度发展,类似演绎推理.本文拟举例来谈谈垂直化与水平化在数学教学中的应用.     

巧用转化思想 妙解数学问题

数学思想是数学知识的升华,是解决数学问题的灵魂,它渗透于整个数学的学习过程.数学思想方法理解掌握的好,对于提高我们的教学效果,促进学生解题能力的提升都有着不可小觑的作用.转化思想是解决数学问题的一种最基本的数学思想,在研究问题时,我们通常是将未知问题转化为已知问题,将复杂的问题转化为简单问题,将抽象的问题转化为具体问题,将实际问题转化为数学问题.下面就转化思想在教学中的应用作具体阐述.     

深入问题本质 激活学生思维

“为思维而教”.数学教学要深刻理解数学问题的本质,激活学生思维,帮助学生插上思维飞翔的翅膀,沟通数学问题内部多层次的联系,让学生对问题“不仅知其然,更知所以然”,努力提高数学课堂教学的有效性,使学生的数学核心素养得到充分的发展.如何深入数学问题的本质,激活学生思维?结合例子谈“三点”做法.     

巧用数学方法 妙解数学问题

数学思想方法就是指从某具体数学内容和对数学的认识过程中抽象概括出的观点,是对数学知识内容的本质认识.教学实践也证明,数学思想方法(转化思想、函数思想、构造思想、分类思想、数形结合思想等方法)是解决实际问题的重要途径,而数学习题浩瀚无边,问题又可变式发散,问题千千万万,但是蕴涵数学思想方法总是不变的.为此,在数学学习中,我们要巧用数学思想方法,妙解数学问题,不断提高学习效果.下面,现举一些案例,以供读者参考.     

数学建模素养发展:从“咬文嚼字”开始——一道例题的教学感悟

数学建模是对现实问题进行数学抽象,用数学语言表述问题,用数学方法构建模型解决问题的素养.对于初中数学教学而言,让学生从数学问题中抽象出数量关系,建构出方程、不等式、函数等模型来解决问题,是本学段中发展数学建模素养的基本途径.     

结构:数学解题的切入点

数学问题结构是指影响和决定问题本质、条件与结论相互作用方式、解决问题的策略和方法的深层特征.明晰数学问题结构的常用策略与方法有:寻找、发现问题表现形式的共性;透过问题的表现形式,发现其本质;追问条件和目标的数学意义;寻找与问题的条件或目标相关联的数学概念;分析相关数学知识的结构特征.应以问题结构为抓手,改进和优化数学解题教学.     

深度学习背景下初中数学问题引领教学策略——以“因动点产生面积问题”的教学为例

学生数学学习的过程,是学生经历数学问题,并在探究、解决问题过程中不断发现新的数学问题的过程.“问题”是数学教学的核心,是推动和引领学生深度思考、主动探究的有效载体.数学教学中实施问题引领教学,将数学学习置于有意义的、真实的问题情境中,可以使得学生的数学思维更加广泛又活跃,满足学生数学学习关联性、批判性和深度性的要求,将浅性思考转化为深度设疑,从而把学生思维不断向深处推进,促进学生深度探究学习,提升学生的数学创造力和数学学科核心素养.     

例析补偿原理在解决不良结构数学问题中的运用

著名数学教育家波利亚曾说过:问题是数学的心脏.问题可分为结构良好问题和结构不良问题,在中学数学解题中大量出现的是结构良好的数学问题.所谓结构良好,是指提供的信息完整,数学结构(研究对象、输入过程)理想,问题目标明确,解决过     

Discrete pseudo-integrals

Integration of simple functions is a corner stone of general integration theory and it covers integration over finite spaces discussed in this paper. Different kinds of decomposition and subdecomposition of simple functions into basic functions sums, as well as different kinds of pseudo-operations exploited for integration and sumation result into several types of integrals, including among others, Lebesgue, Choquet, Sugeno, pseudo-additive, Shilkret, PAN, Benvenuti and concave integrals. Some basic properties of introduced discrete pseudo-concave integrals are discussed, and several examples of new integrals are given.     

Absolute continuity theorems for abstract Riemann integration

Absolute continuity for functionals is studied in the context of proper and abstract Riemann integration examining the relation to absolute continuity for finitely additive measures and giving results in both directions: integrals coming from measures and measures induced by integrals. To this end, we look for relations between the corresponding integrable functions of absolutely continuous integrals and we deal with the possibility of preserving absolute continuity when extending the elemental integrals.     

Computation of Feynman loop integrals

We address multivariate integration and extrapolation techniques for the computation of Feynman loop integrals. Loop integrals are required for perturbation calculations in high energy physics, as they contribute corrections to the scattering amplitude and the cross section for the collision of elementary particles. We use iterated integration to calculate the multivariate integrals. The combined integration and extrapolation methods aim for an automatic calculation, where little or no analytic manipulation is required before the numeric approximation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.     

Evaluation of regular and singular domain integrals with boundary-only discretization—theory and Fortran code

In this paper, a set of boundary integrals are derived based on a radial integration technique to accurately evaluate two dimensional (2D) and three dimensional (3D), regular and singular domain integrals. A self-contained Fortran code is listed and described for numerical implementation of these boundary integrals. The main feature of the theory is that only the boundary of the integration domain needs to be discretized into elements. This feature cannot only save considerable efforts in discretizing the integration domain into internal cells (as in the conventional method), but also make computational results for singular domain integrals more accurate since the integrals have been regularized. Some examples are provided to verify the correctness of the presented formulations and the included code.     

The double-exponential transformation in numerical analysis

The double-exponential transformation was first proposed by Takahasi and Mori in 1974 for the efficient evaluation of integrals of an analytic function with end-point singularity. Afterwards, this transformation was improved for the evaluation of oscillatory functions like Fourier integrals. Recently, it turned out that the double-exponential transformation is useful not only for numerical integration but also for various kinds of Sinc numerical methods. The purpose of the present paper is to review the double-exponential transformation in numerical integration and in a variety of Sinc numerical methods.     

Integration by parts

Many theorems of integration theory are true for a wide range of definitions of integrals. One such theorem is that giving integration by parts, and we discuss it in this paper.     

A New Approach to Numerical Integration

A novel method of estimating integrals is introduced using thetheory of measure and Lebesgue integration. It is shown thatmultiple integrals reduce to the evaluation of a one-dimensionalintegral of a measure function. Comparison of the method andvarious conventional techniques is carried out for several integrals.     

Pedagogical alternatives for triple integrals: moving towards more inclusive and personalized learning

This paper is based on the presumption that teaching multiple ways to solve the same problem has academic and social value. In particular, we argue that such a multifaceted approach to pedagogy moves towards an environment of more inclusive and personalized learning. From a mathematics education perspective, our discussion is framed around pedagogical approaches to triple integrals seen in a standard multivariable calculus curriculum. We present some critical perspectives regarding the dominant and long-standing approach to the teaching of triple integrals currently seen in hegemonic calculus textbooks; and we illustrate the need for more diverse pedagogical methods. Finally, we take a constructive position by introducing a new and alternate pedagogical approach to solve some of the classical problems involving triple integrals from the literature through a simple application of integration by parts. This pedagogical alternative for triple integrals is designed to question the dominant one-size-fits-all approach of rearranging the order of integration and the privileging of graphical methods; and to enable a shift towards a more inclusive, enhanced and personalized learning experience.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.