尝试错误--学习的"催化剂"

1　问题的提出错误是学生在学习过程中自然存在的现象 ,也是不可避免的 ,在数学教学中企图让学生完全避免错误是不可能的 ,也是没有必要的 .相反 ,在某些情况下却需要有意识地让学生专门进行“尝试错误”的活动 ,这样 ,一方面可充分暴露学生思维的薄弱环节 ,有利于对症下药 ;另一方面 ,错误是正确的先导 ,有时错误比正确更具有教育价值 .正如当代科学家、哲学家波普尔所说 :“错误中往往孕育着比正确更丰富的发现和创造因素 ,发现的方法就是试错方法” .因此 ,在教学中通过暴露学生学习过程中的错误 ,为学生提供以错误为源泉的学习反应后刺…     

195 出错本无心就错却有意--一道课本例题的研究性学习

[主持人按错误是一种可以利用的教学资源,它有着正面教育所不可替代的功用.有些错误的经历则会在学生心中产生巨大的冲击波:既使学生知识中的缺陷一下子暴露无遗,又让学生经过了这样的错误的洗礼,他的思维也会在瞬间得到升华.所以,教师对课堂上学生发生的某些错误,常常要考虑如何利用它来教育好学生自己的问题.     

模糊错误逻辑的组合转化词分析

模糊错误逻辑研究的是错误传递、转化的规律,它为决策中避免错误,消除错误,减小错误的危害提供了新的方法.组合转化词是模糊错误逻辑中的一个重要转化词,定义了组合转化词的概念.分析了这个转化词的性质和运算规律.     

一道考研数学题的深度思考

本文通过对一道考研数学题错误的解法加以分析,引出《高等数学》中一系列问题也会犯同样的错误,进而找出出错的根源.另一方面作者发现,同济版《高等数学》内容上存在一些不足,为以后《高等数学》的教学、教材的编撰和修改提供了一定的理论参考.     

纠错赏析两则

<正>人们常说:"发现问题比解决问题更重要","发现问题是能力问题,而解决问题只是水平问题"由此可见,发现别人解的错误,对我们学习能力与学习品质的提升有多么的重要!下面就两例典型的的纠错进行肤浅的赏析,以期达到抛砖引玉的效果.     

经典错误逻辑的相似转化词

经典错误逻辑研究的是错误传递、转化的规律,它为决策中避免错误,消除错误,减小错误的危害提供了新的方法.相似转化词是经典错误逻辑中的一个重要转化词,定义了相似转化词的概念,研究了这个转化词所具有的性质和规律.     

对现行初中教材中几个问题的异议

近几年在进行初中数学教学中,我发现统编教材中个别地方存在错误,而教材由试用本改为正式本后有些错误依然如故,并又出现一些新错误,又未见报刊对这些问题提出非议.最近又获悉现行教材将要使用到一九九二年,笔者感到很有必要将这些问题提出来,与同行们商讨,供专家们评判.其一,初中现行教材中的概念错误.     

解最值问题忽视等号成立致错举例

最值问题往往涉及的知识点多、覆盖面广、综合性强,它是高考考查的一项重要内容.利用不等式中的等号成立求最值是解决最值问题的主要方法.运用这种方法,往往需要对相关对象进行适当的放大、缩小,或不等式之间进行传递、相加、相乘等变形.在此过程中,学生常常因忽视等号成立而导致错误,而且错误不易察觉.下面介绍几例,以引起复习中足够的重视.     

重视错解的再思考

<正>众所周知,学习上的进步总是伴随着错误的产生和对错误的不断改正,而困扰我们的并不是如何改正错误,而是错误产生的原因.没有对错因深究与分析,就无法发现我们自身在知识上和认识上的漏洞,从而阻碍学习上的进步.下面,我们以一道数列问题尝试一下错因分析.     

有错检查模型

检查是最广泛应用的统计方法之一.它在质量控制,医学诊断以及其它科学,生产和商业贸易活动中起着基本的作用.然而,几乎所有的有关的规则或标准都忽略了检查本身的错误.这些错误可能是灾难性的,使检查结果毫无意义.本文介绍了最近的关于检查错误影响的工作,重点是接受抽样和群筛选,也简介了有关的估计和分等问题.     

New results on the algebraic theory of linear systems: the solution of the cover problems

We give the complete solution of a large class of problems in linear system theory, the so-called cover problems. These problems are formulated and solved both in the state-space and in the input-output frameworks. The key concept, which allows the effective parametrization of all solutions of the cover problems, is that of the partial realizations of a sequence of matrices. It is shown that the solutions of the state-space cover problems can be expressed as state spaces of the partial realizations of appropriately defined sequences of matrices, and the solution of the input-output cover problems can be expressed as a simple function of the functions of the partial realizations of the sequences of mentioned above.     

On the performance of the algebraic optimized Schwarz methods with applications

We investigate the performance of algebraic optimized Schwarz methods used as preconditioners for the solution of discretized differential equations. These methods consist on modifying the so-called transmission blocks. The transmission blocks are replaced by new blocks in order to improve the convergence of the corresponding iterative algorithms. In the optimal case, convergence in two iterations can be achieved. We are also interested in the behavior of the algebraic optimized Schwarz methods with respect to changes in the problems parameters. We focus on constructing preconditioners for different numerically challenging differential problems such as: Periodic and Torus problems; Meshfree problems; Three-dimensional problems. We present different numerical simulations corresponding to different type of problems in two- and three-dimensions.     

Identification of distributed systems and the theory of the regularization

The identification problems, i.e., the problems of finding unknown parameters in distributed systems from the observations are very important in modern control theory. The solutions of these identification problems can be obtained by solving the equations of the first kind. However, the solutions are often unstable. In other words, they are not continuously dependent on the data. The regularization or Tihonov's regularization is known as one of the stabilizing algorithms to solve these non well-posed problems. In this paper is studied the regularization method for identification of distributed systems. Several approximation theorems are proved to solve the equations of the first kind. Then, identification problems are reduced to the minimization of quadratic cost functionals by virtue of these theorems. On the other hand, it is known that the statistical methods for identification such as the maximum likelihood lead to the minimization problems of certain quadratic functionals. Comparing these quadratic cost functionals, the relations between the regularization and the statistical methods are discussed. Further, numerical examples are given to show the effectiveness of this method.     

On the Necessary Conditions for the Solubility of Algebraic Inverse Eigenvalue Problems

In this paper we give some necessary conditions for the solubility of additive inverse eigenvalue problems, multiplicative inverse eigenvalue problems and general inverse eigenvalue problems.     

On the solvability of some boundary value problems for the inhomogeneous polyharmonic equation with boundary operators of the Hadamard type

We study the properties of fractional integro-differential operators. As an application, we analyze the solvability of some boundary value problems for the inhomogeneous polyharmonic equation in the unit ball. These problems generalize the Dirichlet and Neumann problems to the case of fractional boundary operators.     

On the Stability of the Feasible Set in Linear Optimization

This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower semicontinuity property of the feasible set mapping and the boundedness of the optimal set of the corresponding coupled problem. Other continuity properties of the feasible set mapping are also derived. This stability theory extends some well-known theorems of Williams and Robinson on the stability of ordinary linear programming problems to linear optimization problems with infinitely many variables or constraints.     

On the approximation and the optimization of plates

This work is devoted to the study of simply supported and of clamped plates together with related variational inequalitiesand optimization problems. We introduce a new unitary approach based on distributed optimal control problems governed by second order elliptic boundary value problems and their penalization. This approach gives the possibility to approximate the solution via piecewise linear continuous finite elements and is simpler than other methods considered in the literature.The convergence with respect to the penalization parameter (?) is proved under very general assumptions.

In order to solve the obtained control problems, optimization procedures of steepest descent type are considered. Relevantnumerical examples illustrate the applicability of the proposed methods.     

Convexity of the Implicit Lagrangian

The implicit Lagrangian has attracted much attention recently because of its utility in reformulating complementarity and variational inequality problems as unconstrained minimization problems. It was first proposed by Mangasarian and Solodov as a merit function for the nonlinear complementarity problem (Ref. 1). Three open problems were also raised in the same paper. This paper addresses, among other issues, one of these problems by giving the properties of the implicit Lagrangian and establishing its convexity under appropriate assumptions.     

Variational problems in the theory of optimum processes

This paper presents a review of the optimization problems for control processes described by ordinary differential equations and of the variational methods for solving these problems. The following cases are studied: problems with constraints on the controls or the coordinates, problems described by equations with discontinuous right-hand sides, problems with functionals depending on intermediate coordinates, and problems with given discontinuities in the coordinates. Variational problems of synthesis of optimal systems are also discussed. The method of solution is based on the multiplier rule and the Weierstrass necessary condition for the strong minimum of a functional. In some cases, the Legendre-Clebsch necessary condition for the weak minimum of a functional is used.     

On the Use of Augmented Lagrangians in the Solution of Generalized Semi-Infinite Min-Max Problems

We present an approach for the solution of a class of generalized semi-infinite optimization problems. Our approach uses augmented Lagrangians to transform generalized semi-infinite min-max problems into ordinary semi-infinite min-max problems, with the same set of local and global solutions as well as the same stationary points. Once the transformation is effected, the generalized semi-infinite min-max problems can be solved using any available semi-infinite optimization algorithm. We illustrate our approach with two numerical examples, one of which deals with structural design subject to reliability constraints.