等比数列作为高中数学的重要知识点，不仅具有丰富的数学内涵，而且在
解决实际问题中具有广泛的应用价值。高考题对于等比数列的考察具有很强的综合性，对学生的逻辑推理和数学运算能力有较高的要求。然而，由于部分学生对于等比数列的理解和应用不够深入，可能导致学习效果不够理想。《普通高中数学课程标准（2017 年版 2020 年修订)》对数学文化融入教学较为重视，而现有教材中数学史料所能发挥的教育价值相对有限。基于以上背景，开展 HPM 视角下等比数列的教学研究能够很好地弥补传统教学的不足，具有重要的理论意义和实践价值。
本研究首先通过文献分析和史料研究，探究与等比数列相关的史料内容，
并进一步明确哪些史料适合融合到等比数列的教学中。在此基础上，根据研究者对于史料的掌握情况，基于 HPM 视角设计等比数列的教学案例。
研究发现：
（1）“国王棋盘问题”“欧拉‘错位相减法’”等七个中外史料适合融入等比数列前 n 项和公式的教学中。选取时，主要考虑的教学情境因素有：1、知识本身的生成规律；2、学生的认知规律；3、教学目标；4、文化因素。
（2）与传统教学设计相较，HPM 视角下的等比数列前 n 项和公式的教学设计具有以下优势：首先，数学史的融入让等比数列公式教学更为丰富、生动、有趣。其次，数学史的融入深刻揭示了等比数列求和公式的生成过程。然后，数学史的融入有助于培养学生的数学核心素养。最后HPM 视角下的教学设计有助于培养学生的数学素养和人文素养。
（3）现阶段数学史融入数学教学仍在一些问题：一是数学史的素材比较
匮乏；二是在现行的高考评价制度下，教学目标往往呈现出一种畸形倾向；三是数学史融入数学教学方式单一。针对以上问题，给出以下建议：1、提高数学史素养并尝试在教学中应用；2、调整教学评价制度并明确数学史的教育价值；3、探索多样化的数学史融入方式。

As an important knowledge point in high school mathematics, geometric progression not only has rich mathematical implications but also has extensive application value in solving practical problems. The college entrance examination questions on geometric progression are highly comprehensive and have high demands on students' logical reasoning and mathematical calculation abilities. However, due to the lack of in-depth understanding and application of geometric progression by some students, their learning effect may not be ideal. The "General High School Mathematics Curriculum Standards (2017 Edition, Revised in 2020)" attaches great importance to the integration of mathematical culture into teaching, while the educational value of mathematical historical materials in existing textbooks is relatively limited. Against this background, conducting teaching research on geometric progression from the perspective of HPM can effectively compensate for the shortcomings of traditional teaching and has important theoretical and practical value.
This study first explores the historical materials related to geometric progression through literature analysis and historical research, and further clarifies which historical materials are suitable for integrating into the teaching of geometric progression. Based on this, teaching cases of geometric progression are designed from the perspective of HPM according to the researcher's mastery of historical materials.
The study found that:
(1) Seven Chinese and foreign historical materials, such as "the chessboard problem of the king" and "Euler's 'method of subtraction by shifting positions'", are suitable for integrating into the teaching of the sum formula for the first n terms of geometric progression. The main teaching context factors considered in the selection are: 1) the generation law of the knowledge itself; 2) students' cognitive laws; 3) teaching objectives; 4) cultural factors.
(2) Compared with traditional teaching design, the teaching design of the sum formula for the first n terms of geometric progression from the perspective of HPM has the following advantages: Firstly, the integration of mathematical history makes the teaching of geometric progression formulas more enriched, vivid, and interesting. Secondly, the integration of mathematical history profoundly reveals the generation process of the summation formula for geometric progression. Then, the integration of mathematical history helps cultivate students' core mathematical literacy. Finally, the teaching design from the perspective of HPM contributes to cultivating students' mathematical and humanistic literacy.
(3) There are still some issues with integrating mathematical history into mathematics teaching at the current stage: First, there is a lack of materials on mathematical history; second, under the current college entrance examination evaluation system, teaching objectives often exhibit a distorted tendency; third, the integration of mathematical history into mathematics teaching methods is single. In response to these issues, the following suggestions are given: 1) Improve mathematical history literacy and attempt to apply it in teaching; 2) Adjust the teaching evaluation system and clarify the educational value of mathematical history; 3) Explore diverse ways of integrating mathematical history.