基于过程性变式教学的一类函数恒为零问题

A Study on the Problem of Functions Identically Zero Based on Process-Oriented Variant Teaching Methodology

摘要: 变式教学是一种常见的教学策略，尤其在数学课堂的一类专题中被广泛使用。基于过程性变式教学，对一类函数恒为零的问题进行了研究。针对华东师范大学数学系编写的《数学分析》第五版中的一道习题和数学竞赛中的两道题目，从分析习题与过程性变式两方面进行阐述，层层递进，多角度变式，遵循学生的认知发展规律，总结了此类问题的解决方法。通过对证明一类函数恒为零问题的过程性变式设计，旨在为教师提供过程性变式设计的启示。

Abstract: Variant teaching is a common teaching strategy, especially widely used in a certain type of topic in mathematics classes. Based on process-oriented variant teaching, a study is conducted on the problem of a certain type of function always being zero. Focusing on an exercise problem in the fifth edition of “Mathematical Analysis” compiled by the Department of Mathematics of East China Normal University and two problems in mathematics competitions, this paper elaborates from the aspects of analyzing exercises and process-oriented variants, progressing step by step and varying from multiple perspectives, following the cognitive development laws of students, and summarizes the solution methods for such problems. Through the design of process-oriented variants for proving that a certain type of function is always zero, the aim is to provide teachers with inspiration for the design of process-oriented variants.