函数单调性反映函数图象与变化中的规律性，是函数的基本性质之一。函数单调性概念较为抽象，是学生学习的难点。目前国内以某一概念为载体，对学生抽象能力以及具体抽象过程的深入研究还比较少。本文选取山东省某普通高中的高一学生作为研究对象，对函数单调性概念的抽象能力进行研究。

通过对函数单调性概念和数学抽象相关文献的研究，本研究基于 AiC 框架建立对于函数单调性概念的学生抽象能力分析框架，将函数单调性概念抽象过程划分为识别阶段、建构阶段、巩固阶段。在识别阶段，从具体函数图象中识别与函数单调性相关的已有的知识结构。在建构阶段，重组已识别的知识结构得到完成对函数单调性概念的形式化抽象。在巩固阶段，巩固函数单调性的概念，以便学生在不同情境下自由、灵活地应用。依据抽象能力分析框架和函数单调性概念的抽象，选取能涵盖分析框架所有抽象阶段的测试题，使得测试能准确评定出样本学生函数单调性概念的抽象能力。选取山东省某中学高一6个班级共 301名学生进行测试，并对抽象能力不同的三组学生进行访谈。所得研究结论如下：

第一，样本学生函数单调性概念的抽象能力具体表现是：根据测试结果表明平均得分率为59%；低于均分人数较多，占测试学生总人数的54％；测试总分标准差为8.88，学生之间抽象能力差异较大。

第二，样本学生在函数单调性概念的抽象过程中，18％的学生未达识别阶段；37.7％的学生处于识别阶段，能识别函数图象的变化规律，用自然语言描述自变量x和对应函数值y之间的关系；29.3％的学生处于建构阶段，能使用数学符号准确建构函数单调性的形式化定义；仅有15.1％的学生达到了巩固阶段，能在具体情境中运用函数单调性的概念解决实际问题。

第三，样本学生在函数单调性概念的抽象过程中，识别阶段的难点是不能准确感知函数图象的变化规律，抽象出特殊函数的单调性；建构阶段的难点是学生对“y随x增大而增大”的符号化抽象以及“任意”一词的使用；巩固阶段的难点是学生没有在实际问题情境中运用函数单调性的意识。

最后，基于在本研究中样本学生函数单调性概念抽象过程中存在的困难和问题，给出相关的建议以供参考。

The monotonicity of a function reflects the regularity in the graph and change of the function, and is one of the basic properties of the function. The concept of function monotonicity is abstract and is a difficult task for students to learn. At present, there are few in-depth studies on the abstraction ability of students and the concrete abstraction process with a certain concept as a carrier in China. In this paper, the abstraction ability of the concept of monotonicity of a function is studied by taking senior students of a general high school in Shandong Province as the object of study.

Based on the AiC framework, this study establishes a framework for the analysis of students' abstraction ability of the concept of function monotonicity, and divides the abstraction process of the concept of function monotonicity into the identification stage, the construction stage and the consolidation stage. In the identification stage, the existing knowledge structure related to function monotonicity is identified from the graph of a specific function. In the construction stage, the identified knowledge structure is reorganised to complete the formal abstraction of the concept of function monotonicity. In the consolidation stage, the concept of monotonicity of functions is consolidated so that students can apply it freely and flexibly in different contexts. Based on the abstraction framework and the abstraction of the concept of function monotonicity, test questions were selected to cover all the stages of abstraction in the framework, so that the test could accurately assess the abstraction ability of the sample students in the concept of function monotonicity. A total of 301 students from six classes of senior secondary schools in Shandong Province were selected for the test, and three groups of students with different abstraction abilities were interviewed. The conclusions of the study obtained are as follows:

Firstly, the abstraction ability of the sample students in the concept of function monotonicity is shown as follows: according to the test results, the average score rate is 59%; the number of students below the mean score is higher, accounting for 54% of the total number of students tested; the standard deviation of the total test score is 8.88, and the abstraction ability varies greatly among students.

Secondly, in the abstraction process of the concept of function monotonicity, 18% of the students in the sample did not reach the identification stage; 37.7% of the students were at the identification stage, being able to identify the pattern of change in the graph of a function and describe the relationship between the independent variable x and the corresponding function value y in natural language; 29.3% of the students were at the construction stage, being able to use mathematical notation to accurately construct a formal definition of function monotonicity; only 15.1% of the students reached the consolidation stage, where they were able to apply the concept of function monotonicity in concrete situations to solve practical problems.

Thirdly, in the abstraction process of the concept of monotonicity of functions, the difficulty in the identification stage is that students cannot accurately perceive the law of change of the graph of a function and abstract the monotonicity of a special function; the difficulty in the construction stage is the students' symbolic abstraction of "y increases with x" and the use of the term "arbitrary "The difficulty in the consolidation stage is that students do not have the awareness to use the monotonicity of functions in practical problem situations.

Finally, based on the difficulties and problems that existed in the abstraction of the concept of monotonicity of functions for the sample students in this study, relevant suggestions are given for reference.