探索HPM与PME的结合之桥：等比数列前n项和的教学设计
Exploring the Bridge of Combining HPM and PME: Teaching Design for the First n Sum of Proportional Sequence

在对HPM与PME相结合的可行性与必要性分析的基础上，以“等比数列前n项和”为例，对该理论在等比数列求和教学过程中的应用进行了分析，并开发了相应的教学片段设计。该教学片段通过数学史导入新课，利用五种推导方法得到等比数列前n项和公式，并从PME视角分析HPM对学习该内容的作用及机理，使学生体会数学家们当时的思维方式并收获数学知识。该教学方法在理解和掌握知识的同时有效培养学生的核心素养，为教师提供了一种新的教学方法和理念，以期改进等比数列的教学设计从而提高课堂教学质量与效果。
Based on the feasibility and necessity analysis of the combination of HPM and PME, taking “the first n terms and formulas of proportional sequence” as an example, the application of this theory in the teaching process of proportional sequence summation was analyzed, and a corresponding teaching segment design was developed. This teaching segment was introduced into a new lesson through mathematical history, and five deduction methods were used to obtain the first n terms and formulas of proportional sequence. The role and mechanism of HPM in learning this content were analyzed from the perspective of PME, enabling students to experience the thinking patterns of mathematicians at that time and gain mathematical knowledge. This teaching method effectively cultivates students’ core competencies while understanding and mastering knowledge, providing teachers with a new teaching method and concept to improve the teaching design of proportional sequences and enhance the quality and effectiveness of classroom teaching.