在新一轮的课程改革中，重视过程性评价成为学界普遍的共识，SOLO 分类理论因其理论成熟、应用广泛等优势而受到研究者的青睐。在高一年级，基本不等式是不等式内容中非常重要的一部分，在生活中有着非常广泛的应用，其内容形式多样、灵活性强、使用条件严苛因而成为许多学生学习的难点，以此为切入点，研究学生对此内容的掌握情况就显得格外必要。

基于 SOLO 分类理论，依据《普通高中教科书数学必修（第一册）》以及教学参考书等文献资料，确定了本研究内容分为基本不等式的概念、几何解释、证明方法、基本不等式的应用这四个维度，并且根据《普通高中数学课程标准（2017 年版）》（简称课标）确定了以上内容的最高水平应分别不低于多点结构水平、关联结构水平、抽象拓展结构水平、抽象拓展结构水平。依照上述四个维度和对应的水平层次编制了测试卷，经过预测试及调整，最终形成信度为 0.912、KMO 值为 0.86 的正式测试卷。以山西省某中学四个不同类型的班级共计 217 名学生为研究对象，并选取了具有典型性的教师及学生进行访谈，对高一年级学生的学情进行了分析与研究，其结论为以下内容。

1.高一年级学生的学情统计水平
（1） 基本不等式的概念中，不论是基本不等式的基础概念还是基本不等式的深入内涵均有超过 75%的比例的学生达到了《课标》中的要求，但仍有不少学生对此内容的掌握存在着漏洞。
（2） 基本不等式的几何解释中，有 45.62%的学生位于抽象拓展结构水平，可以根据所学知识进行灵活运用。
（3） 在基本不等式的证明方法中，有 29.95%的人位于多点结构水平，对证明题有思路但是无法书写出详细完整的证明过程；有 38.25%的人位于最高水平（抽象拓展结构水平）。
（4） 在基本不等式的应用这一维度下，涉及到在数学背景的应用中，简单的求最值题目中有 72.81%的人位于关联结构水平，在综合性求最值的题目中有 48.85%的人位于最高水平（抽象拓展结构水平）。而在生活背景的应用中，简单的应用题目里有 61.75%的学生位于关联结构水平，稍复杂一些的题目中有 34.56%的学生位于抽象拓展结构水平。

2.高一年级学生在测试卷中存在的问题及原因

（1） 在基本不等式的概念中，学生存在的问题主要是对这一概念未充分地理解，多依赖于机械记忆。出现这一现象的主要原因是教师上课时较少关注到知识的生成过程。
（2） 在基本不等式的几何解释中，学生存在的问题主要是无法有效地做到举一反三，对拓展及延伸的内容欠缺深刻的思考，并且对细节的关注较少。出现这一问题的主要原因是学生对知识的迁移能力较弱。
（3） 基本不等式的证明方法中存在的主要问题是证明思路不严谨，证明方法存在漏洞。这一问题主要是教师对解题思路的引导不到位以及批阅作业不严格导致的。
（4） 基本不等式的应用中存在的主要问题是解题过程缺少关键步骤，理解题意以及计算的能力较弱。这一问题主要是由于学生对带有数学背景的问题情境相对比较陌生且缺乏耐心，疏于计算。

针对上述研究，本人对教师教学提出如下建议：（1）注重概念的生成过程，引导学生准确理解概念中的细节，辅助学生对易犯错的地方有更深入的认识；（2）注重逻辑推理素养的提升，关注学生的解题思路，加强对学生迁移能力的培养，引导学生学会分析题目并找到解题思路；（3）关注准确书写解题过程的能力，培养文字语言到数学语言的转化能力；（4）引导学生学会阅读，抓住关键信息并准确理解题意。

In the new round of curriculum reform, emphasis on process evaluation has become a general consensus in the academic. SOLO classification theory is favored by researchers due to its advantages of mature theory and wide application. In the first year of high school, basic inequalities are very important parts of inequality content, and have a very wide range of applications. It is difficult for many students to learn because of its diversified forms, strong flexibility and strict use conditions. So, it is particularly necessary to study the learning situation of students.

Based on the SOLO theory, according to the general High School Textbook Mathematics Required (Volume 1) and teaching reference books and other materials, this research is divided into four dimensions: the concept of basic inequality, geometric interpretation, proof method, and the use of basic inequality. And according to the "general high school Mathematics curriculum standards" determined that the highest level of the above content should be not lower than the level of multi-point structure, related structure, abstract extended structure, abstract extended structure. According to the above four dimensions and the corresponding level, the test was compiled. After the pre-test and adjustment, the formal test with 0.912 reliability and 0.86 KMO value was finally formed. A total of 217 students from four different types of classes in a middle school in Shanxi Province were selected as the research object, and typical teachers and students were selected to interview. This paper analyzes and studies the learning situation of freshmen in senior high school. The conclusions are as follows.

1. The statistical level of students' academic situation

 (1) In the concept of basic inequality, both the basic concept of basic inequality and the in-depth connotation, more than 75% of the students have reached the requirements in the curriculum standard, but there are still some loopholes in many students' grasp of this content.(2) In the geometric interpretation of basic inequalities, 45.62% of the students are at the level of abstract IV extended structure and can flexibly use the knowledge they have learned.(3) In the proof method of basic inequality, 29.95% of the people are at the multi-point structure level, and have ideas about the proof problem but cannot write a detailed and complete proof process; 38.25% were at the highest level (level of abstract extended structure).(4) In the dimension of the application of basic inequality, 72.81% of the people are at the level of related structure in simple mathematical applications, and 48.85% of the people in the comprehensive problem are at the highest level (the level of abstract extended structure). In the application of life background, 61.75% of the students in the simple application questions were at the level of relevance structure, while 34.56% of the students in the more complex questions were at the level of abstract extension structure.

2. Problems and reasons in the test papers of senior one students

(1) In the concept of basic inequality, the problem of students is that they do not fully understand this concept and rely on mechanical memory. The main reason for this phenomenon is that teachers pay less attention to the process of knowledge generation.(2) In the geometric interpretation of basic inequalities, the main problems of students are that they can not effectively draw inferences from one example to another, lack of deep thinking on the extension and extension content, and pay little attention to details. The main reason for this problem is that students' ability to transfer knowledge is weak.(3) The main problem of the proof method of basic inequality is that the proof idea is not rigorous and there are loopholes in the proof method. This problem is mainly caused by teachers' inadequate guidance to solve problems and lax marking.(4) The main problem sin the application are the lack of key steps in the process of solving problems, and the weak ability to understand the meaning of problems and calculate problems. This problem is mainly due to the fact that students are relatively unfamiliar with the problem situation with mathematical background and lack of patience, and are negligent in calculation. 

In view of the above research, I put forward the following suggestions for teachers' teaching:(1) pay more attention to the generation process of concepts, and pay more attention to mistakes;(2) Pay attention to the improvement of logical reasoning literacy, pay attention to students' problem-solving ideas, strengthen the cultivation of students' migration ability, and guide students to learn to analyze problems and find problem-solving ideas; (3) Pay attention to the ability to accurately write the problem solving process, and cultivate the transformation ability from written language to mathematical language; (4) Guide the students to learn to read, grasp the key information and accurately understand the meaning of the questions.