基于“以学为中心”的数列极限教学设计研究

本文选取数列极限的定义这一部分内容,基于“以学为中心”教学理念介绍如何设计数列极限定义的教学过程,从九个环节进行设计旨在使学生更好的理解掌握数列极限的本质和内涵,达到以学为中心的教学目标.     

知识形成过程教学个案——数列极限的ε-N定义

个案包括三部分 :教学目标的确立 ;教学过程实录 ;对个案的分析与评价 .1　教学目标的确立数列极限的ε-Ｎ定义是学生相当难掌握的内容 ,往往需要学生在相当长的学习时间内 (甚至要到学习微积分以后 )反复体会才能加深对此概念的理解 .因此 ,一开始让学生接触数列极限的ε-Ｎ定义时 ,应注重让学生体会数列极限概念的合理性 ,并为学生创立一个比较容易独立进行准确、深入思考的语境背景和图形背景 .2　教学过程2 1　数列极限的描述性定义设计思想　在生活中学生也会使用诸如“极限”、“无限接近”等词语 ,对这些词语生活化的使用有时会给准…     

一道数列极限的多种解法

利用数学分析中关于数列极限的定义、收敛数列的性质及数列极限存在的条件,介绍一道数列极限问题的多种解法.     

有限集合与数列极限的定义

极限理论是微积分学的理论基础,而数列极限是其中最基础也是最重要的一个部分,准确深入理解数列极限的概念对微积分的学习具有重要作用.本文用集合给出数列极限的另一个定义,它与数列极限的ε-N定义等价,其应用可以使数列极限的验证过程的逻辑关系更为清晰.     

运用数列极限运算法则解题的两点注意

数列极限运算法则 ,是中学生求数列极限的基础 .为了更好地掌握求数列极限的方法 ,学生在运用此法则时应注意以下两点 .1)如果ｌｉｍｎ→∞ ａｎ=Ａ .ｌｉｍｎ→∞ ｂｎ=Ｂ ,那么ｌｉｍｎ→∞(ａｎ±ｂｎ) =Ａ±Ｂ .此法则只适用于求有穷数列的极限 ,不能用于求无穷数列的极限 .例 1　ｌｉｍｎ→∞(1ｎ2 1 2ｎ2 1 … 30ｎ2 1) .分析 :此题为有穷数列求极限 ,故可直接运用极限运算的加法法则 .解　ｌｉｍｎ→∞(1ｎ2 1 2ｎ2 1 … 30ｎ2 1)　 =ｌｉｍｎ→∞1ｎ2 1 ｌｉｍｎ→∞2ｎ2 1 … ｌｉｍｎ→∞30ｎ2 1　 =0 0 … 0　 =0…     

数列极限教学中应注意的几个问题

数列极限是高中数学教学中的重要内容,这部分内容对于学生掌握数学方法、培养数学思维、解决实际问题以及进一步学习数学都具有重要作用．高中数学的数列极限尽管教学要求不高,但学生在学习上仍存在诸多困难,为了有效实施这一部分内容的教学,必须注意以下四个问题．     

一道数列极限题的注记

用几何方法分析了高等数学中的一道数列求极限的题目,直观地显示了该数列趋向于极限的方式.并把极限的结果从实数域中拓展到复数域中,指出了该数列在复数域趋向于极限的方式是螺旋的,在实数域中趋向于极限的方式是沿直线靠近的.最后类比该数列,构造出相似数列的求极限问题.     

借助函数极限判断无限数列的敛散性

本文首先指出了什么是无限数列和无限数列的敛散性的特征,数列的敛散性和连续函数的极限的求值有怎样的关系?数列的敛散性必有其特殊的地方,同时,将连续函数的求极限的方法移植到数列敛散性的判别上,有哪些需要注意的地方.文中作者将针对两者关系进行了详细的论述.无限数列在无穷远处的项具有什么特点呢?或是渐近某一个数,或渐近某几个数,或在某几个数之间来回摇摆等等.当数列渐近某一个数时,无限数列收敛.无限数列敛散性的代数验证方法就是求其在无穷远处的极限.当极限结果为一个有限数时,无穷数列收敛,当极限结果为无穷或不存在时,称其发散.既然数列是一种特殊的函数,那么是否可以借助函数极限来求解数列的极限呢?     

一类数列极限的计算

极限计算是高等数学中的基本计算,虽然计算极限的方法有很多种,但是却不能解决所有的极限问题.在十几年的高等数学教学过程中,我们经常帮助考研的学生解决一些问题,在解决问题的过程中,我们发现有一类数列极限的计算有着共同的特点,本文中我们对这类数列极限的计算方法进行了总结,并给出定理及证明.     

数列极限教学中的若干思考

在中学阶段渗透近代数学的基础知识是课程教材改革的要求之一.高中数学教材把数列极限作为必修内容,其目的是在中学阶段渗透极限思想,使学生初步接触用有限刻划无限,由已知认识未知,由近似描述精确的数学方法.本文对数列极限的教学提出几点思考,谨供大家参考.(一)基本概念、基础知识的正确理解与掌握1.数列极限的定义数列极限概念是教学上的难点,教材采用描述法定义数列极限.对数列极限定义的正确理解,是学习本章内容的基础例1等差数列{an}中,首项a1=60,公差d=-2,记Sn=a1 a2 … an,Tn=|a1| |a2| … |an|,求li mn→∞SnTn.错解:Sn=na1 n(n2…     

Challenges of maintaining cognitive demand during the limit lessons: understanding one mathematician’s class practices

In this study, we examined five limit lessons using Mathematical Tasks Framework to understand students’ opportunities to learn cognitively challenging tasks and maintain cognitive demand during limit lessons. Our analysis of Dr A’s five lessons shows that students rarely had opportunities to maintain or increase cognitive demand. There are two main factors that shaped her instructional practices, students and time. These two factors greatly influenced how she selects and implements limit tasks in her classes. To serve her students’ needs of knowing more rules, formulas and procedures, she selected and discussed those simple tasks a lot. Although Dr A thinks challenging tasks and asking demanding questions can be potentially good instructional practices, she thinks these instructional practices would not serve her students well. With these factors, we made possible recommendations to have more student-centred teaching.     

极限概念教学难点分析及其突破策略

多年来,我国不少学者就极限概念教学难的问题做了大量研究,但该问题并未得到根本解决。通过对极限概念教学进行全面系统的研究,将会发现,在我国的教材体系下,极限概念教学的最大特点是难点多而密集。具体表现在极限的精确定义被高度形式化,且逻辑结构复杂、极限精确定义种类繁多、用精确定义验证极限的证明形式独特、证明技巧性强等方面。因此,为使极限概念教学难的问题得到根本解决,需采取充分铺垫、分散难点、淡化形式、借助直观、梯式演练和因材施教等策略。     

The notion of motion: covariational reasoning and the limit concept

This paper investigates outcomes of building students’ intuitive understanding of a limit as a function's predicted value by examining introductory calculus students’ conceptions of limit both before and after instruction. Students’ responses suggest that while this approach is successful at reducing the common limit equals function value misconception of a limit, new misconceptions emerged in students’ responses. Analysis of students’ reasoning indicates a lack of covariational reasoning that coordinates changes in both x and y may be at the root of the emerging limit reached near x = c misconception. These results suggest that although dynamic interpretations of limit may be intuitive for many students, care must be taken to foster a dynamic conception that is both useful at the introductory calculus level and is in line with the formal notion of limit learned in advanced mathematics. In light of the findings, suggestions for adapting the pedagogical approach used in this study are provided.     

Problems and solutions in students’ reinvention of a definition for sequence convergence

Little research exists on the ways in which students may develop an understanding of formal limit definitions. We conducted a study to (i) generate insights into how students might leverage their intuitive understandings of sequence convergence to construct a formal definition and (ii) assess the extent to which a previously established approximation scheme may support students in constructing their definition. Our research is rooted in the theory of Realistic Mathematics Education and employed the methodology of guided reinvention in a teaching experiment. In three 90-min sessions, two students, neither of whom had previously seen a formal definition of sequence convergence, constructed a rigorous definition using formal mathematical notation and quantification equivalent to the conventional definition. The students’ use of an approximation scheme and concrete examples were both central to their progress, and each portion of their definition emerged in response to overcoming specific cognitive challenges.     

The performance of undergraduate students in the limit concept

In this work, we investigated first-year university students’ skills in using the limit concept. They were expected to understand the relationship between the limit-value of a function at a point and the values of the function at nearby points. To this end, first-year students of a Turkish university were given two tests. The results showed that the students were able to compute the limit values by applying standard procedures but were unable to use the limit concept in solving related problems.     

Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics

The interpretive cross-case study focused on the examination of connections between teacher and student topic-specific knowledge of lower secondary mathematics. Two teachers were selected for the study using non-probability purposive sampling technique. Teachers completed the Teacher Content Knowledge Survey before teaching a topic on the division of fractions. The survey consisted of multiple-choice items measuring teachers’ knowledge of facts and procedures, knowledge of concepts and connections, and knowledge of models and generalizations. Teachers were also interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. After teaching the topic on the division of fractions, two groups of 6th-grade students of the participating teachers were tested using similar items measuring students’ topic-specific knowledge at the level of procedures, concepts, and generalizations. The cross-case examination using meaning coding and linguistic analysis revealed topic-specific connections between teacher and student knowledge of fraction division. Results of the study suggest that students’ knowledge could be associated with the teacher knowledge in the context of topic-specific teaching and learning of mathematics at the lower secondary school.     

Emergent models in a reinvention activity for learning the slope of a curve

Introducing the slope of a curve as the limit of the slope of secant lines is a well-known challenge in mathematics education. As an alternative, three other approaches can be recognized, based on linear approximation, based on multiplicities, or based on transition points. In this study we investigated which of these approaches fits students most by analyzing students’ inventions during a lesson scenario revolving around a design problem. The problem is set in a context that is meaningful to students and invites them to invent methods to construct a tangent line to a curve: an implementation of the guided reinvention principle from Realistic Mathematics Education (RME). The teaching scenario is based on the phased lesson structure of the Theory of Didactical Situations (TDS). The scenario was tested with 44 groups of three students in six grade 9 or 10 classrooms. We classified the strategies used by students and, using the emergent models-principle from RME, investigated to which of the four approaches the student strategies connect best. The results show that the groups produced a variety of strategies in each classroom and these strategies contributed to a meaningful institutionalization of the notion of slope of a curve.     

Chinese (Mainland) teachers’ views of effective mathematics teaching and learning

This study investigates Chinese teachers’ cultural beliefs concerning effective mathematics teaching through semi-structured interview with nine experienced teachers. For the Chinese teachers, an effective teacher should always be passionate and committed to the teaching profession. She should not only understand the knowledge in the textbook thoroughly but also be able to carefully craft the knowledge from the textbook for teaching by predicting possible students’ difficulties. Although Chinese teachers emphasize student participation and flexible teaching, they tend to see the teacher’s ability to design and lead coherent lessons as the key for facilitating students’ understanding. The result of this study helps researchers and educators understand the teacher-designed and content-oriented teaching model in Chinese classrooms.     

基于BOPPPS教学模式的离散数学教学改革的探索与实践

针对离散数学抽象性高、逻辑性强、理论性深、跳跃性强等内容特点,以及授课中为赶进度"满堂灌",学生完全被动学习等现象,采取BOPPPS模式进行课堂教学.将教学过程分为6个阶段,根据学生参与反馈的情况进行内容的调整与优化,最后通过学生评教说明该教学模式的教学效果.