例谈“结构分析法,形式统一法”在微积分问题解决中的应用

Illustrating the“Structure Analysis”and“Formal Unification”in Calculus

波利亚在其名著《怎样解题》中,对如何解题给出了一般性的解题思维程序:(1)理解题目;(2)拟定方案;(3)执行方案;(4)回顾,其中建立已有知识与未知量之间的联系,拟定解题方案是解题过程的关键,但是如何建立题目中未知量与已有知识联系,形成有效的解题思路和解题路径呢?本文提出并详细介绍了能有效沟通已有知识与当前题目,进而能有效制定具体解题路径的数学问题解决方法——“结构分析法,形式统一法”.

In his masterpiece“How to solve problems”,George Polya provided a general program of thinking for problem solving:(1)understanding the subject;(2)formulating the scheme;(3)implementing the scheme;(4)reviewing,among which establishing a bridge between the existing knowledge and the unknown quantities,formulating the scheme are the core segments in the whole process of problem solving.But how to effectively search for the useful information related to the current subject with the existing knowledge and how to use them to solve the current subject?This paper presents,in details and with examples in Calculus,the methods of“structural analysis”and“formal unification”for effectively communicating between existing knowledge and current problem,and formulating problem-solving paths.