函数是高中数学课程的主题之一，也是高中数学课程内容四条主线之一，贯穿必修、选择性必修和选修课程。但是在教学过程中发现大多数学生在函数中会出现各种错误，函数的学习仍存在较多困难。因此，发现和分析学生在函数学习中出现的错误，了解错误的分布特征对于提高函数的教学质量有着重要意义。基于此，本文主要研究了两个基本问题：（1）高一学生的函数学习错误在错误分析框架中各维度的分布如何？其分布情况与学生的数学水平、所在学校的教学水平有何关系？（2）高一学生在函数问题作答上容易出现哪些典型错误？这些典型错误构成了哪些错误路径？

本研究基于扎根理论，采用测试、内容分析等数据收集处理方法，以1034 名高一学生的函数测试作答内容为研究资料。在已有错误分析理论的基础上，从研究工具中选出5道解答题中的6个典型问题，进行标定编码，利用扎根理论从错误原因角度出发构建了3个一级指标，9个二级指标和23个三级指标的函数错误分类维度，又根据信息加工理论和认知水平视角，构建了信息加工维度，经反复修改后形成错误分析框架，并据此框架得出了学生的典型错误分布特征及错误类型，根据信息加工维度探讨了学生错误出现的形成过程，最后结合错误分析框架从学生和教师两个角度提出教学建议。研究结论如下：

1.在信息加工维度下，学生错误最多的出现在解决问题阶段，其次是阅读与理解阶段和分析与策略选择阶段，最少是解答与检验答案阶段，其比例曲线存在“倒U 字形”、“Z 字形”和“倒Z 字形”。学校教学水平对各阶段错误分布曲线的影响较小，而学生数学水平对其信息加工阶段错误分布曲线的具有一定的影响。数学水平较低的学生在阅读与理解阶段出现错误频率较高，数学水平较高的学生在解决问题阶段出现错误频率较高。

2.高一学生在函数中出现的错误类型一级指标包括知识性、程序性以及心理性错误，分布相对均衡。在三级指标中，高一学生函数错误的主要类型为：函数的表示方法、新定义类型的函数、函数的周期性及其运算、解不等式（组）、看错/遗漏题目条件、思维定势和猜测作答；在二级指标中，高一学生函数错误的主要类型为：函数的概念及其表示理解错误、函数的基本性质理解错误、运算错误、思维定式和畏难情绪。学校教学水平对各维度的错误分布的影响较小，其各维度错误分布情况与总体情况几乎一致。学生数学水平对错误分类维度的二级指标分布有一定影响。高数学水平的学生更容易产生思维定势；低数学水平的学生错误则更集中在函数的概念及其表示、函数的基本性质这些知识性错误上，且低数学水平学生在看到难题时更容易产生畏难情绪，从而只进行猜测作答。

3.学生出现次数较多的典型错误为运算错误、思维定势、函数的基本性质理解错误。大多数典型错误在不同教学水平学校上的分布不存在显著差异，但在三角函数“五点法”作图表格、求周期参数的运算、使用端点带入求最值、函数的单调性证明、分类讨论以及解不等式中的思维定势等典型错误在不同教学水平学校上的分布呈现显著差异。除函数的概念与性质问题，其他绝大多数问题中空白作答、正确作答在不同数学水平学生中分布存在显著差异。

4.学生的错误路径可以被划分为概念错误、方法错误、检验意识缺乏、思维定势主导下的四类错误路径。在概念错误主导下错误路径中，其错误多集中在阅读理解问题阶段；在方法错误、思维定势主导下错误路径中，其错误多集中在分析与策略选择问题阶段；在思维定势主导下的错误路径中，其错误多出现在解决问题阶段。在信息加工维度不同维度中，都存在检验意识缺乏主导下的错误路径。从对学生的访谈中，也能发现学生的检验问题的意识不足，没有形成良好的检查习惯；学生检查的效果不佳；学生对错因的反思能力不足；且学生在数学抽象、数学运算素养和推理能力上有较大提升空间。同时学生在函数学习中需要熟练掌握数形结合思想、分类讨论思想、函数与方程思想。

基于以上分析，从理论层面，我们可以得出：采用扎根理论自下而上构建错误分析框架是高中数学领域的错误分析研究切实可行的途径之一；从信息加工理论和认知水平视角构建信息加工维度能有效分析解题过程分布特征。通过错误分析框架能揭示高一学生的在函数学习中的典型错误和错误路径，并结合学生访谈能诊断出学生在学科素养、能力和思想方法上的不足之处。

从实践层面，可以从学生和教师两方面提出建议。学生应养成良好的审题、检查习惯；加强函数基础知识的习得；提高做题的心理素质。教师应加强函数概念的教学，重视高中函数概念的形成过程；加强函数基本性质的认识，重视函数解题指导；注重不同水平学生的分层教学；重视数学思想方法的渗透，提升学生数学运算素养；及时对学生的错误进行归因和矫正，提出相应的矫正策略。

Function is one of the themes of high school mathematics curriculum and one of the four main lines of high school mathematics curriculum content, running through compulsory, selective compulsory, and elective courses. However, during the teaching process, it was found that most students make various errors in functions, and there are still many difficulties in learning functions. Therefore, discovering and analyzing students' errors in function learning, as well as understanding the distribution characteristics of errors, is of great significance for improving the quality of function teaching. Based on this, this article mainly studies two basic questions: (1) What is the distribution of function learning errors among senior high school students in various dimensions of the error analysis framework? What is the relationship between its distribution and students' mathematical level and the teaching level of their school? (2) What typical mistakes are senior one students prone to when answering function questions? What are the error paths generated by these typical errors?

Based on Grounded theory, this research adopts data collection and processing methods such as testing and content analysis, and takes 1034 senior students' answering content in function test as research materials. On the basis of the existing error analysis theory, six typical answers from five questions were selected from the research tools and coded. The Grounded theory was used to construct the functional error classification dimension of three first level indicators, nine second level indicators and 23 third level indicators from the perspective of error causes. According to the information processing theory and the perspective of cognitive level, the information processing dimension was constructed, and the error analysis framework was formed after repeated modifications, Based on this framework, the typical error distribution characteristics and types of students were identified, and the formation process of student errors was explored based on information processing dimensions. Finally, teaching suggestions were proposed from the perspectives of both students and teachers using the error analysis framework. The research conclusion is as follows:

1. In the dimension of information processing, the most common errors made by students occur in the problem-solving stage, followed by the reading and comprehension stage, analysis and strategy selection stage, and the least is the answer and test answer stage. The proportion curve has "inverted U", "Z", and "inverted Z" shapes. The influence of school teaching level on the error distribution curve of each stage is relatively small, while students' mathematical level has a certain impact on the error distribution curve of their information processing stage. Students with lower levels of mathematics have a higher frequency of errors in the reading and comprehension stage, while students with higher levels of mathematics have a higher frequency of errors in the problem-solving stage.

2. The first level indicators of the types of errors that high school students encounter in functions include knowledge-based, procedural, and psychological errors, with a relatively balanced distribution. In the third level indicators, the main types of function errors among senior high school students are: representation methods of functions, newly defined types of functions, periodicity of functions and their operations, solving inequalities (groups), conditions for misreading/missing questions, fixed thinking patterns, and guessing answers; In the second level indicators, the main types of functional errors among senior high school students are: misconceptions in the concept and representation of functions, misconceptions in the basic properties of functions, arithmetic errors, fixed thinking patterns, and fear of difficulty. The impact of school teaching level on the distribution of errors in each dimension is relatively small, and the distribution of errors in each dimension is almost consistent with the overall situation. The mathematical level of students has a certain impact on the distribution of secondary indicators for misclassification dimensions. Students with high mathematical proficiency are more likely to develop fixed thinking patterns; Students with low mathematical proficiency tend to make more errors in the concept and representation of functions, as well as the basic properties of functions. Moreover, students with low mathematical proficiency are more likely to develop a fear of difficulty when faced with difficult problems, thus only guessing answers.

3. Typical errors that students make frequently are arithmetic errors, fixed thinking patterns, and misunderstandings of basic properties of functions. There is no significant difference in the distribution of most typical errors in schools of different teaching levels, but there is a significant difference in the distribution of typical errors in schools of different teaching levels, such as the "five point method" of trigonometric functions to draw tables, the operation of solving periodic parameters, the use of endpoints to calculate the maximum value, the proof of monotonicity of functions, classification discussions, and the thought set in solving inequalities. Except for the concept and properties of functions, there are significant differences in the distribution of blank answers and correct answers among students of different mathematical levels in the vast majority of other questions.

4. Students' error paths can be divided into four types: conceptual errors, methodological errors, lack of awareness of inspection, and error paths dominated by fixed thinking patterns. In the error path dominated by conceptual errors, the errors are mostly concentrated in the stage of reading comprehension problems; In the wrong path dominated by methodological errors and fixed thinking patterns, the errors are mostly concentrated in the analysis and strategy selection stage; In the wrong path dominated by fixed thinking, errors often occur in the problem-solving stage. In different dimensions of information processing, there are error paths led by a lack of awareness in testing. From interviews with students, it can also be found that students lack awareness of inspection issues and have not formed good inspection habits; The effectiveness of student inspections is poor; Students lack the ability to reflect on the causes of mistakes; Moreover, students have significant room for improvement in mathematical abstraction, mathematical operation literacy, and reasoning ability. At the same time, students need to proficiently master the idea of combining numbers and shapes, the idea of classification discussion, and the idea of functions and equations in the learning of functions.

Based on the above analysis, from the theoretical level, we can draw the following conclusions: using Grounded theory to build a bottom-up error analysis framework is one of the practical ways of error analysis research in the field of high school mathematics; Constructing an information processing dimension from the perspective of information processing theory and cognitive level can effectively analyze the distribution characteristics of problem-solving processes. The error analysis framework can reveal the typical errors and error paths of senior high school students in function learning, and diagnose the shortcomings of students in subject literacy, abilities, and thinking methods through student interviews.

From a practical perspective, suggestions can be provided to students and teachers. Students should develop good habits of reviewing and checking questions; Strengthen the acquisition of basic knowledge of functions; Improve the psychological quality of problem-solving. Teachers should strengthen the teaching of function concepts and attach importance to the formation process of high school function concepts; Strengthen the understanding of the basic properties of functions and attach importance to the guidance of function problem-solving; Emphasize hierarchical teaching for students of different levels; Emphasize the infiltration of mathematical thinking methods and enhance students' mathematical literacy; Timely attribution and correction of students' mistakes, and propose corresponding correction strategies.