对教材中0~0型极限内容的建议

"00型是一种结果不确定的类型,并且可以使用洛必达法则来转化解决"是高等数学大纲要求中的一个重点.现行"高等数学"教材和参考书中都指出了这一点,但给出的有关00型极限的例子和习题的结果都为1.这严重影响了教学效果和教学目的.通过实例可对该内容的教学和教材组织提供一种富有建设性的建议.     

错漏的解法 侥幸的结果

众所周知,数学解题的真正目的不在一个正确答案,主要在解题的思维过程中,培养学生分析和解决问题的能力。因此,如果利用一种错漏的方法,侥幸地得到了问题的正确答案,这对教学工作来讲,不仅是无意义,而是有害的事。以下是出现在本刊有关专栏上的两个例子,提出来供大家讨论。     

主观几何中一组余弦定律方程的解*

本文讨论在主观几何应用例子[1]中出现的由余弦定理建立的一组六元二次带根式的代数方程的解.应用隐函数存在定理,本文证明这组方程存在有唯一的实解.把求解问题转化为无约束非线性优化问题,可以用已知的诸法来求解.文中给出了用下降法求解的数值例子.     

非负矩阵的Perron补与Perron根的估计

对于非负矩阵A,主要讨论其谱半径即Perron根的估计.这里提出了一种利用非负矩阵的Perron补矩阵与Perron根关系来估计其Perron根上下界的新方法,并且给出例子来说明这种方法的有效性.     

最优化—"导数的应用"教学单元

数学建模的思想和方法可以有机地融入微积分教学中去,在讲述导数的应用时切入最优化的例子来学习是相当适合的.     

对一道错误习题的纠正

一、问题的提出我们认为对于工科院校的大学生来说,高等数学课的教学只注重计算而不甚注意严密的思维训练是不行的。举一个例子,同济大学樊映川等编《高等数学讲义》在全国大多数院校已延用了三十来     

基于谱密度评价随机性

为了评价一个平稳过程的随机性, 我们基于谱密度提出了一个图方法. 当图中的散点呈现线性关系的时候, 我们可以判定这个序列是随机的. 为了说明这个思想, 我们用模拟的办法来检验伪随机数的随机性. 另外, 我们也用了一个实际数据来考察数据的相关性. 这两个例子都说明了我们的图方法是非常有效的.     

推行高等数学目标教学法 培养创造性数学思维能力

阐述了高等数学教学改革的必要性和迫切性 .提出了一种新的教学方法 :目标教学法 ,给出了它的内容和具体实施方案 .论述了培养学生创造性思维的必要性和重要性 .结合例子 ,叙述了在教学过程中培养学生发散思维和逆向思维的方法 .通过教学实践 ,论证了此教学法取得了很好效果 ,并具有很好的推广价值     

概率统计教学的几点建议

概率统计是大学数学的基础课程之一,我们通过具体的例子分析了概率统计课程教学应该重视的几个问题,并提出了解决问题的建议．这对概率统计课程的教学有一定的借鉴意义．     

例题、习题与强化概念——指数与对数教学的一点体会

著名的力学家和数学家牛顿说过:例子有时比定律还要重要.毫无疑问,选用一些好的例题、习题来阐述、理解和应用定义、定理、定律、公式都是很有益处的.这是在于一些好的例题、习题能卓著地帮助概念的强化,而概念的强化则能更为有效地敦促解题思路的多线化、明朗化.下面仅就我教学《指数与对数》这一章,对强化、巩固对数的定义和一些性质所采取的措施,作一点汇报.     

探索错误资源在高等数学教学中的应用

学生在学习高等数学的过程中,难免会犯一些错误.这些错误,实际上是一种宝贵的教学资源.本文探讨了高等数学教学中如何利用这些错误资源,提高学生分析问题和解决问题的能力.     

关于《数值分析》教材中一道习题的商榷

指出研究生教材<数值分析>中习题的一处疏忽性错误,并给出几种可供参考的解法和必要的讨论.     

A model for understanding numerical stability

^** Email: bornemann{at}ma.tum.de We present a model of roundoff error analysis that combinessimplicity with predictive power. Though not considering allsources of roundoff within an algorithm, the model is relatedto a recursive roundoff error analysis and therefore is capableof correctly predicting stability or instability of an algorithm.By means of nontrivial examples, such as the componentwise backwardstability analysis of Gaussian elimination with a single iterativerefinement step, we demonstrate that the model even yields quantitativebackward error bounds that show all the known problem-dependentterms with the exception of dimension-dependent constants. Themodel can serve as a convenient tool for teaching or as a heuristicdevice to discover stability results before entering a furtherdetailed analysis.     

Mathematics-related teaching competence of Taiwanese primary future teachers: evidence from TEDS-M

This paper draws on data from the international TEDS-M study, organized by the IEA, and utilizes a conceptual framework describing the Taiwanese perspective of mathematics and mathematics teaching competences (MTCs) with regard to investigating the uniqueness and patterns of Taiwanese future primary teacher performance in the international context. The framework includes content-oriented and thought-oriented categories of mathematics competence. The latter category contains subcategories adopted and revised from (3rd Mediterranean conference on mathematical education. Hellenic Mathematical Society, Athens, 2003) the competence approach by Niss. Hsieh’s (Research on the development of the professional ability for teaching mathematics in the secondary school level (3/3). Taiwan: National Science Council, 2009) model is also adopted and revised to serve as an analytical framework, including four categories relating to MTCs, representations, language, and misconceptions or error procedures. This paper shows that in thought-oriented mathematics competences Taiwan and Singapore share a unique pattern of higher percent correct in competences related to formalization, abstraction, and operations in mathematics than in those related to the way of thinking, modelling and reasoning in and with mathematics. The paper also addresses weak teaching competences claimed in domestic studies, which conflict with the TEDS-M results. Namely, in contrary to the international trend, Taiwanese future primary teachers are weak at judging mathematics competences required by students to learn mathematical concepts or solve problems, and superior at diagnosing and dealing with student misconceptions and error procedures.     

Lost in Translation: Examining Translation Errors Associated With Mathematical Representations

Translation errors and conceptual misunderstandings made by students translating among graphical, tabular, and symbolic representations of linear functions were examined. The study situated student errors in the context of the “Translation‐Verification Model” developed specifically for the purpose of explaining student behavior during the process of translating relationships from one mathematical representation to another. Three distinct error types were identified to explain student performance. An examination of the error types revealed that specific translation errors tend to occur at different stages of the translation process. Translation errors are also related to “attribute density,” the amount of information inherently encoded in a given representation. The findings of the study have implications for teaching linear relationships—student weaknesses and strengths are identified.     

The Effects of a Teacher In‐Service on Low‐Achieving Grade 7 and 8 Mathematics Students

Previous research (e.g., Woodward & Baxter, 1997 ) found that Standards‐based mathematics teaching provides marginal or no benefits for low achievers, in contrast with positive effects for middle and high ability students. A randomized quasi‐experiment in 52 Canadian schools found that low achieving grade 7 and 8 students who received support consisting of placement on a learning continuum, instruction focused on their specific learning needs, and concrete materials to represent mathematical constructs, benefited from teaching that emphasized construction over transmission of knowledge. Treatment students showed small but statistically significant improvements over controls in student achievement, and controversially, in mathematical beliefs, and attitudes. The latter finding raised issues of the appropriate balance between Type I and Type II error in educational research.     

Preservice Teachers' Analysis of Children's Work to Make Instructional Decisions

Teaching is an interactive process in which teachers gather information, analyze the results, and construct a response based on this diagnosis ( Cooney, 1988 ). Considering alternatives in constructing a response, that is, making an instructional decision, is of great importance in teaching. How might mathematics teacher educators provide experiences for preservice teachers to begin the development of this skill? In an attempt to determine how these experiences might reveal the level of understanding preservice teachers have in regards to children's mathematical thinking, a study was conducted over three semesters. During the mathematics methods course, preservice teachers were involved in analyzing children's work through the review and discussion of several samples. They were required to determine the error pattern, discuss what might have lead to this misconception, and suggest appropriate instructional strategies that might help this student. Although most preservice teachers could correctly identify the computational error patterns, they had difficulty in determining what might have led to the misconceptions and proposing effective instructional strategies.     

The Impact of Teaching on Hospital Costs: A Budgetary Approach to Non-market Institutions

NHS hospitals contribute to medical education, training nurses and research, as well as to the care of patients. In the past they have been funded largely on the basis of resources employed, with additional funding for medical education and training nurses. The intellectual basis for the funding of medical education is a single econometric study of English hospitals in the financial year 1969-70. The methodology used has since been criticized, and it has been suggested that actual expenditure has been very much less than that earmarked by the health departments. New estimates are obtained using Scottish data for the financial year 1985-86. The method used is to proceed in a two-stage fashion, identifying via regression techniques variables measuring hospital activity and resources which contribute significantly to hospital costs. We then assess the significance of medical education, nurse training and hospitals' teaching status against this background. Our conclusions include: (1) actual expenditure on medical education was probably less than the funding formula allowed, but the error of margin is too large to suggest overfunding; (2) training nurses incurs significant financial costs, even after the explicit allowances made; and (3), major teaching hospitals tended to cost more, but not significantly more than their non-teaching counterparts. These financial implications for NHS hospitals should be borne in mind given the current NHS review.     

Teaching computational science with computer algebra

Computer algebra systems can be used to generate algorithms for numerical computations. We have computed and shown examples in [1] and more recently in [2]. In this paper we emphasize the teaching aspect and use essentially the capability of computer algebra system to construct and manipulate the interpolating polynomial and to compute a series expansion of a function. We automatically generate formulas for integration with error terms and also generate multi-step methods for integrating differential equations. Furthermore we discuss the use of multiple precision. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generating numerical algorithms using a computer algebra system

An useful application of computer algebra systems is the generation of algorithms for numerical computations. We have shown in Gander and Gruntz (SIAM Rev., 1999) how computer algebra can be used in teaching to derive numerical methods. In this paper we extend this work, using essentially the capability of computer algebra system to construct and manipulate the interpolating polynomial and to compute a series expansion of a function. We will automatically generate formulas for integration and differentiation with error terms and also generate multistep methods for integrating differential equations. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D25, 65D30, 65D32, 65L06