基于牟合方盖体积计算的教学思考

Teaching Reflections on the Calculation of the Volume of Mouhefanggai

摘要: 数学概念的高度抽象性与逻辑结构的严密性，常导致教学过程中出现“认知断层”现象，而将数学史中相关数学思想的发展历程融入数学教学，有助于学生攻克数学难题，促进数学思想的深度理解与精准掌握。本文通过对牟合方盖体积的计算古今方法的研究，系统阐述了牟合方盖的由来，详细介绍了祖暅方法、定积分截面法及二重积分三种方法计算牟合方盖体积的过程。通过在课堂上实现数学思想的古今碰撞，直观呈现应用截面法计算不规则立体体积思想的演进，让学生了解数学思想的形成、发展，深入洞悉数学知识背后蕴含的深邃思想，从而提升学生的数学思维，培养学生的创造性思想。

Abstract: The high degree of abstraction of mathematical concepts and the rigor of their logical structure often lead to the phenomenon of “cognitive disconnection” in the teaching process. Integrating the development of relevant mathematical ideas from the history of mathematics into teaching can help students overcome mathematical difficulties and promote a deeper understanding and accurate mastery of mathematical ideas. This paper systematically elaborates on the origin of the Mouhefanggai by studying the ancient and modern methods for calculating its volume. It also provides a detailed introduction to the processes of calculating the volume of Mouhefanggai using Zu Geng’s method, the cross-sectional method of definite integrals, and double integrals. By facilitating a collision of ancient and modern mathematical ideas in the classroom and intuitively presenting the evolution of the idea of using the cross-sectional method to calculate the volume of irregular solids, students can gain an understanding of the formation and development of mathematical ideas. They can also gain a deeper insight into the profound ideas behind mathematical knowledge, thereby enhancing their mathematical thinking and fostering their creative thinking.