一道收敛数列均值极限问题的探究式教学

An Inquiry-based Teaching of the Mean Limit Problem of Convergent Sequence

基于波利亚的阶段序进原理,本文给出全国大学生数学竞赛中的一道收敛数列均值极限问题的教学设计方案.该方案分为解题思路讲授与问题拓展教学两个部分.在第一部分中,举出一些具体实例使学生对问题形成直观感受,并引出初步的解题思路,进而通过归纳法引导学生写出一般的解题步骤,以培养学生良好的解题习惯;在第二部分,从原问题出发,不断延伸,最终拓展至依概率收敛的随机变量序列的期望极限问题,以强化学生对证明思想及问题本质的理解与掌握能力,并提升学生综合运用知识能力.

Based on Polya s principle of stage sequence progression,this paper presents a teaching design scheme for the problem of the mean limit of a convergent sequence in the National Mathematics Competition for college students.The scheme is divided into two parts:the teaching of problem-solving ideas and the teaching of problem expansion.In the first part,some concrete examples are cited to make students form intuitive feelings to the problem,and lead to the preliminary solution ideas,and then through induction to guide students to write the general solution steps,in order to cultivate students good problem-solving habits.In the second part,starting from the original problem,it is extended continuously and finally extended to the expectation limit problem of a sequence of random variables converging in probability.