谈初中数学反例的教学功能

一、数学反例的功能数学反例贯穿于整个数学学习阶段 ,通过学习数学反例可加深学生对数学概念的理解 :培养学生对数学知识归纳、提炼 ;还养成严密的逻辑思维能力和正确运用数学语言 ,通过学习数学反例可以提高学生作图技能 .教学中恰当地利用反例 ,可以促进学生数学概念的形成、数学内涵的理解 ,使学生全面掌握数学知识 ,解决数学问题 .除此之外 ,学会举反例 ,有助于学生形成批判意识 ,这也是二期课改提出的要求 .显而易见 ,数学反例具有独特的教学功能 ,所以 ,在教学中既要重视解答数学命题的能力 ,又要加强数学反例的教学 .二、数学反例与…     

试谈反例在解题中的作用

反例——即问题反面的例子.那么它在数学中有什么作用呢?1利用反例纠正错误,提高认识在教学中,每当学生对一些概念、性质、定理等认识不足、理解不透时,教师经常会举出反例,对知识内容进行阐述、澄清、剖析,这样一来,反例就能起到正面解释所达不到的领悟效果.除此以外,对于学生学习时出现的典型错误,还可以用反例来纠正,这     

例谈反例的教学功能

在数学的发展史中 ,反例和证明占有同等重要的地位 .一个正确的数学命题需要严密的证明 ,谬误则靠反例即可否定 .因此 ,在中学数学的教学中 ,反例也有着极为重要的意义 ,它在发现和认识数学真理 ,强化数学基础知识的理解和掌握 ,培养学生思维能力和创造能力 ,以及提高学生解题速度等方面的意义和作用是不可低估的 .本文就此谈谈反例教学的几点认识 ,以供参考 .1　利用反例 ,深化学生对知识的理解在中学数学教学中 ,我们不仅要运用正确的例子深刻阐明知识点 ,而且要运用恰当的反例从另一个侧面抓住概念或规则的本质 ,弥补正面教学的不足 ,从…     

反例在中学数学解题中的应用

反例教学是指教师根据教学内容和目标,采用概念和例题的典型错误认识或错误解法组织学生探讨错误的原因,从而达到真正掌握数学概念和性质的一种教学方法.本文中通过论述反例在数学解题教学中的作用,探索如何恰当运用反例,引导学生从反面视角看待问题,提高数学课堂效率和教学质量,从而提升学生的逻辑思维能力与数学核心素养.     

谈数学教学中的"反例"效用

在数学教学中适当运用反例,可以收到事半功倍的效果.1反例是理解概念的工具数学概念是整个数学大厦的基石.教师要善于利用反例把“死”知识教活.例如,函数的概念对于初学者来说是比较难理解的,利用反例可加深学生对反函数的理解.现举例如下:例1下列图形中,不可能是函数y=f(x)的     

反例反例 一把利器

《普通高中数学课程标准》(实验)指出:“教学中应通过实例,引导学生运用合情推理去探索、猜测一些数学结论,并用演绎推理确认所得结论的正确性,或者用反例推翻错误的猜想.”教育理论认为“概念或规则的正例传递了最有利于概括的信息,反例则传递了最有利于辨别的信息”,心理学同样对反例有所推崇:“在教学中利用反例是一种比较.比较能确定其与被比较对象的共同点和不同点.有比较才有鉴别,才能容易把握住所研究对象的本质特征.”     

类比法在多元函数微分学构造反例教学中的应用

通过构造反例的类比法,总结了多元函数与一元函数的同名(相近)概念的区别与联系,让学生能够在学习新的知识体系时,学会通过具体反例的构造,运用类比法对照思考新旧知识的异同.     

重视反例教学 提升教学品质

反例教学在促进知识深化,提升学生纠错、防错能力,培养思维严谨性、深刻性等方面发挥着不可替代的作用.在教学中,教师要从教学实际出发,重视整理归纳反例教学资源,引导学生通过对比、辨析、纠错等活动更好地理解知识,应用知识,使学生的发散性、逆向性、辩证性思维得到训练和提升,有效提高数学教学品质.     

关于条件极值的注记

文中结合反例,分析学生在求条件极值时的常见错误,给出求条件极值的正确方法.     

反例教学——中学数学探究性学习的重要环节

众所周知,要判断一个命题是真命题,必须经过严格的证明,而判断一个命题是假命题,只要举出一个反例.所谓举反例就是举出符合命题的题设,而不满足命题结论的例子.因其具有构造性,所以举反例实际上是一种创造性思维的体现.但在中学数学的教学中,强调证明有余,而对反例教学却明显重视不够.其实,反例和证明在知识发现的过程中具有同等地位,是"观察——归纳——猜想——证明(反例)"这一数学知识探究过程中的重要环节.可以说,反例     

Computation of eigenpair partial derivatives by Rayleigh–Ritz procedure

Based on Rayleigh–Ritz procedure, a new method is proposed for a few eigenpair partial derivatives of large matrices. This method simultaneously computes the approximate eigenpairs and their partial derivatives. The linear systems of equations that are solved for eigenvector partial derivatives are greatly reduced from the original matrix size. And the left eigenvectors are not required. Moreover, errors of the computed eigenpairs and their partial derivatives are investigated. Hausdorff distance and containment gap are used to measure the accuracy of approximate eigenpair partial derivatives. Error bounds on the computed eigenpairs and their partial derivatives are derived. Finally numerical experiments are reported to show the efficiency of the proposed method.     

Flux-splitting schemes for parabolic equations with mixed derivatives

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.     

Extremum of second-order directional derivatives

In this paper, the extremum of second-order directional derivatives, i.e. the gradient of first-order derivatives is discussed. Given second-order directional derivatives in three nonparallel directions, or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions, the formulae for the extremum of second-order directional derivatives are derived, and the directions corresponding to maximum and minimum are perpendicular to each other.     

The interpolation method of Sprague-Karup

The usual interpolation method is that of Lagrange. The disadvantage of the method is that in the given points the derivatives of the interpolating polynomials are not equal one to the other. In the method of Hermite, polynomials of a higher degree are used, whose derivatives in the given points are supposed to be equal to the derivatives of the function at the given points. This means that those derivatives must be known.If those derivatives are not known, then in the given points the derivatives may be replaced by approximative values, e.g. based on the interpolating polynomials of Lagrange. Such a method has been described by T. B. Sprague (1880) and in a simplified form by J. Karup (1898). In this paper the formulae are derived. Both methods are illustrated with an example. Some properties and theorems are stated. Tables to simplify the computational work are given. Subroutines for these interpolation methods will be published in a next article.     

A new iterative method for many eigenpair partial derivatives

A new iterative method is proposed for computing partial derivatives of many eigenpairs. This method simultaneously computes a few eigenpair partial derivatives. For each origin shift, partial derivatives of eigenpairs whose eigenvalues are closest to the origin shift can be computed. Hence, this method may be used for partial derivatives of all eigenpairs. Convergence of the proposed method is established. Finally numerical experiments are given to show the effectiveness of the proposed method.     

一类新的二阶组合切导数及其应用

引进了一种新的切锥,讨论它与相依切锥的关系.借助这种新的切锥引进了一类新的二阶组合切导数,并讨论了它与其他二阶切导数的关系.利用这类新的二阶组合切导数,建立了集值优化分别取得Henig有效元和全局有效元的最优性必要条件.     

Stability of optimal filter higher-order derivatives

In many scenarios, a state-space model depends on a parameter which needs to be inferred from data. Using stochastic gradient search and the optimal filter first-order derivatives, the parameter can be estimated online. To analyze the asymptotic behavior of such methods, it is necessary to establish results on the existence and stability of the optimal filter higher-order derivatives. These properties are studied here. Under regularity conditions, we show that the optimal filter higher-order derivatives exist and forget initial conditions exponentially fast. We also show that the same derivatives are geometrically ergodic.     

Stationary waiting time derivatives

We investigate the stability of waiting-time derivatives when inputs to a queueing system-service times and interarrival times-depend on a parameter. We give conditions under which the sequence of waiting-time derivatives admits a stationary distribution, and under which the derivatives converge to the stationary regime from all initial conditions. Further hypotheses ensure that the expectation of a stationary waiting-time derivative is, in fact, the derivative of the expected stationary waiting time. This validates the use of simulation-based infinitesimal perturbation analysis estimates with a variety of queueing processes.We examine waiting-time sequences satisfying recursive equations. Our basic assumption is that the input and its derivatives are stationary and ergodic. Under monotonicity conditions, the method of Loynes establishes the convergence of the derivatives. Even without such conditions, the derivatives obey a linear difference equation with random coefficients, and we exploit this fact to find stability conditions.     

IRAM‐based method for eigenpairs and their derivatives of large matrix‐valued functions

Based on the implicitly restarted Arnoldi method for eigenpairs of large matrix, a new method is presented for the computation of a few eigenpairs and their derivatives of large matrix‐valued functions. Eigenpairs and their derivatives are calculated simultaneously. Equation systems that are solved for eigenvector derivatives are greatly reduced from the original matrix size. The left eigenvectors are not required. Hence, the computational cost is saved. The convergence theory of the proposed method is established. Finally, numerical experiments are given to illustrate the efficiency of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical differentiation with noisy signal

The aim of this paper is to present a new numerical method, which ables one to filter and compute numerical derivatives of a function whose values are known in some points from experimental measurements, inducing noisy data. We use a piecewise cubic spline interpolation to generate a function whose Fourier coefficients give an approximation of the numerical derivatives we are looking for. Error and stability analysis of this numerical algorithm are provided. Numerical results are presented for data smoothing and for the first and second derivatives computed from noisy data. They show that this method gives good numerical results. Comparison with other methods is done.