浅析高等数学中蕴含的思维观和方法论

The Methodologies and Thinking Strategies in Advanced Mathematics

本文着眼于高等数学教学,以空间解析几何为切入点,剖析其中所蕴含的研究方法和科学思维,包括坐标、数形结合、动点、向量、降维,以及空间想象、相似类比、哲学思维等等.这些方法和思维不仅蕴含在高等数学课程中,而且在线性代数和概率论与数理统计课程中也有很多体现.通过对空间解析几何这一模块的分析和梳理,本文展示如何在数学基础课的教与学中,有意识地以知识内容为载体,挖掘和培养科学的研究和思维方法.

Focusing on Advanced Mathematics, this paper explores the methodologies and thinking used in the course of Analytic Geometry of Space, such as coordinates, vectors, moving points, numbers and shapes, dimension reduction, spatial imagination, analogical reasoning, and philosophical thinking. These can also be found in the Advanced Mathematics, as well as in the Linear Algebra and the Probability and Statistics. We provide an example for exploring and cultivating the thinking abilities through the study of the mathematical foundation courses.