“变量间相关关系”作为高中统计教学中的一个重要内容，是学生进入统计学领域的重要起点。然而学生在学习这一知识时常常遇到困难。认知诊断研究打破了传统测试仅关注总成绩的局限性，实现了对每个学生的具体认知模式进行诊断，同时对学生当前的认知现状进行反馈并提供补救建议。因此本研究基于认知诊断理论来研究高中生“变量间相关关系”的学习现状，研究问题如下：“变量间相关关系”的认知属性及其层级关系是怎样的？如何利用认知诊断工具得到学生对于“变量间相关关系”的知识掌握情况？不同班级、不同性别学生的认知现状是怎样的？如何根据学生的认知诊断结果制定补救教学方案？

研究共分为两部分：第一部分是对高三学生“变量间相关关系”学习的认知诊断研究，该部分首先确定了“变量间相关关系”的认知属性并构建属性层级关系。其次根据Q矩阵理论编制单元测试卷作为测验工具，以湖南省长沙市某学校高三学生为研究对象，进行实地测验调查；第二部分是运用G-DINA模型对调查回收的测试卷进行认知诊断测量，并对被试的认知属性掌握模式进行识别分类，最后针对某一类被试设计个性化补救教学方案。本研究主要结论如下：

（1）基于文献回顾与专家研究，确定了高中生“变量间相关关系”学习的7个认知属性：其中内容属性为A1相关关系的定义；过程属性为A2散点图的理解、A3相关方向的判断；技能属性为A4线性相关的理解、A5相关系数的含义、A6相关系数的计算、A7相关程度的判断。在此基础上确定认知属性层级关系。

（2）以高中数学教材为载体，根据建立好的“变量间相关关系”属性间层级关系以及Q矩阵理论，编制了符要求的单元测试卷用于预测试。紧接着对测试工具的合理性进行定性与定量的检验，包括信效度、难度以及区分度等指标，并对模型参数进行多次修正，最终形成正式的认知诊断测验卷，共13个项目，采用二级计分的模式，利用G-DNIA模型对正式测试的390名被试进行认知诊断研究，得到其认知属性掌握概率及模式等信息。

（3）运用flexCDMs、SPSS、R语言、Excel等软件，对被试知识掌握情况进行分类；分析不同类型班级和不同性别在认知上的差异，得到高三学生对于“变量间相关关系”的认知现状如下：

首先，根据被试掌握认知属性的概率发现：有86%同学掌握了A1相关关系的定义这一属性，而超过一半的学生没有掌握A6相关系数的计算这一属性，还有50%同学对于A4线性相关和A7相关程度的判断较为模糊。根据被试掌握认知属性的模式发现：掌握模式为AMP10（1111101）的被试人数最多，AMP3（110000）和AMP4（1010000）其次；物化生班级大部分学生的认知属性掌握模式为AMP10；物化地班级中，大部分被试的掌握模式为AMP8（1111001），其次是AMP3；历史类班级中，较为常见的认知属性掌握模式为AMP4和AMP3。

根据对被试平均得分与认知属性掌握模式的分析发现：男生和女生分数和认知属性掌握模式差异大，究其原因是女生大多数所在的历史类班级与物化生、物化地班级的组间差异较大。而在同类型班级中，虽然得分可能有较大的差异，但是学生的认知属性掌握模式较为类似，因此可以针对不同类型的班级进行学习路径的分析。并且从箱线图中可知，学生总得分并不等同于其认知属性掌握模式。

（4）基于认知诊断结果进行个性化补救教学设计。首先选出物化生班级被试的属性掌握概率，利用探索性聚类分析，将全部的被试恰好分为8种潜在知识状态。再依照属性层级关系并根据包含原则设计出4条学习路径。其次选择掌握了属性A1与A2的这类学生群体，按照学习路径中最长的一条，也就是“PKS5-PKS4-PKS1-PKS6-PKS8-PKS3-PKS7”的顺序进行补救教学设计。补救的步骤分为3步，先补救属性A3和A4，接着补救A7属性，最后补救属性A5与A6。补救内容为设计分层作业：一是要涵盖补救属性的基本概念、基本原理以及基本思想方法；二是要使学生在真实题目情境中按照学习路径逐步迁移和应用认知属性，从而帮助老师和学生更快速精准地进行补救。

As an important content in senior high school statistics teaching, "correlation between variables" is an important starting point for students to enter the field of statistics. However, students often have difficulties in learning this knowledge. Cognitive diagnosis research breaks the limitation of the traditional test that only focuses on the total score, realizes the diagnosis of each student's specific cognitive mode, and gives feedback on the current cognitive status of students and provides remedial suggestions. Therefore, this study is based on the cognitive diagnosis theory to study the learning status of "inter-variable correlation" among high school students. The research questions are as follows: What are the cognitive attributes and hierarchical relationships of "inter-variable correlation"? How to use cognitive diagnostic tools to get students' knowledge of "correlation between variables"? What is the cognitive status of students in different classes and different genders? How to develop remedial teaching programs according to the cognitive diagnosis results of students?

The research is divided into 2 parts: the former part is the cognitive diagnostic research on the learning of "correlation between variables" of senior high school students, which first determines the cognitive attribute of "correlation between variables" and constructs the attribute hierarchy relationship. Secondly, based on the Q matrix theory, the unit test paper is prepared as a test tool, and the senior three students in a school in Changsha City of Hunan Province are taken as the research objects. The second part is to use the G-DINA model to measure the cognitive diagnosis of the recovered test paper, identify and classify the cognitive attribute mastery pattern of the subjects, and finally design a personalized teaching remedy plan for a certain class of subjects. This paper has the following four conclusions:

(1) Based on literature review and expert research, seven cognitive attributes of high school students' "correlation between variables" learning are identified: content attribute is the definition of A1 correlation; The process attributes are the understanding of A2 scatter plot and the judgment of A3 correlation direction. The skill attribute is the understanding of A4 linear correlation, the meaning of A5 correlation coefficient, the calculation of A6 correlation coefficient, and the judgment of A7 correlation degree. Based on the above analysis, the hierarchical relationship of cognitive attributes is determined.

(2) Based on the high school mathematics textbook as the carrier and according to the established hierarchical relationship between the attributes of "correlation between variables" and the Q matrix theory, the unit test paper with character requirements was prepared for pre-test. Then, the rationality of the test tool was tested qualitatively and quantitatively, including reliability and validity, difficulty, differentiation and other indicators, and the model parameters were revised several times. Finally, a formal cognitive diagnostic test paper with 13 items was formed. The G-DNIA model was used to conduct cognitive diagnostic research on 390 subjects who were formally tested by adopting the two-level scoring mode. Get its cognitive attributes and grasp the probability and pattern information.

(3) flexCDMs, SPSS, R language, Excel and other software were used to classify the knowledge of the subjects; By analyzing the cognitive differences between different types of classes and different genders, the cognitive status of senior high school students on the "correlation between variables" is as follows:

First of all, it can be found by analyzing the probability of subjects mastering cognitive attribute that 86% of the students have mastered the attribute of the definition of A1 correlation, while more than half of the students have not mastered the attribute of the calculation of A6 correlation coefficient, and 50% of the students have vague judgments about the degree of A4 linear correlation and A7 correlation. According to the pattern of subjects' mastery of cognitive attributes, it was found that AMP10 (1111101) had the largest number of subjects, AMP3 (110000) and AMP4 (1010000). Secondly, the cognitive attribute mastery mode of most students in the materialized class is AMP10; In the physical class, the mastery mode of most subjects was AMP8 (1111001), followed by AMP3; In the history class, the more common cognitive attribute mastery modes are AMP4 and AMP3.

According to the analysis of the average scores and cognitive attribute mastery patterns of the subjects, it is found that there are great differences between the scores and cognitive attribute mastery patterns of the boys and the girls, and the reason is that there are great differences between the groups of the history class and the materialized students and the materialized land class. In the same type of classes, although there may be a big difference in scores, students' cognitive attribute mastery mode is similar, so the learning path analysis can be carried out for different types of classes. Moreover, it can be seen from the boxplot that the total score of students is not equal to their cognitive attribute mastery mode.

(4) Personalized teaching remedy design based on cognitive diagnosis results. Firstly, the attribute mastery probability of the subjects in the class of materialization students was selected, and all the subjects were classified into 8 potential knowledge states by exploratory cluster analysis. Then four learning paths are designed according to the attribute hierarchy and the inclusion principle. Secondly, select the students who have mastered the attributes A1 and A2, and carry out remedial teaching design according to the longest learning path, that is, "PKS5- PKS4- PKS1- PKS1- PKS6- PKS8-PKS3-PKS7". The remedial steps are divided into three steps, first remediating attributes A3 and A4, then remediating attributes A7, and finally remediating attributes A5 and A6. The remedial content is to design hierarchical work: first, it should cover the basic concept, basic principle and basic thought method of remedial property; The second is to enable students to gradually transfer and apply cognitive attributes according to the learning path in the real problem situation, so as to help teachers and students to remedy more quickly and accurately.