数学焦虑是指个体在面对与数学相关的场景时所产生的紧张、焦虑或者恐惧情绪。研究发现数学焦虑广泛存在于不同年龄段的学生乃至成年人之中，目前已经成为了一种世界范围内的数学学习中的情绪问题。多项元分析研究一致发现，学生的数学焦虑与数学成绩之间存在稳定的负相关关系，数学焦虑对学生的数学学习和学业发展有着深远的影响。因此，理解数学焦虑发生发展的内在机制，制定相应的干预方案以降低数学焦虑，是当今数学教育领域的重要课题。这对于推动数学教育改革，促进儿童青少年数学能力的良好发展以及提升国民数学素养都具有重要意义。过往的研究从个体情绪态度发展以及从个体数学能力发展的角度，对影响数学焦虑发生发展的因素进行了一系列的探索，并基于这些研究成果开发了相应的干预方案和方法。然而，这些方案和方法在干预场景以及在干预效果上还存在着一些局限。针对这一问题，当前研究有必要从新的视角对数学焦虑的来源进行思考，更加深入地理解数学焦虑发生发展的内在机制，进而开发出更加切实可行、具有长期效果的干预方案和方法，以解决数学焦虑这一难题。

从数学焦虑的定义来看，它是由“个体在面对与数学相关的场景”时所产生的情绪反应。以往的研究主要从“个体”（包括情绪态度发展和数学能力发展）的角度探索了数学产生的原因，但是忽视了“数学”本身这一根源。相比于从个体发展的角度探索影响数学焦虑发生发展的因素，从数学知识本身的角度对数学焦虑的来源进行探索，可能为解决数学焦虑这一难题提供新的视角。数学是研究数量关系和空间形式的科学。与其他学科相比，数学具有相对独特的知识结构和表示方式，具有高度抽象、逻辑严密的特点。但是在当前数学教育领域中，数学知识抽象性的表达还存在“符号化”和“情境化”之争。“符号化”强调数学对一般规律的概括和归纳，使用纯粹的数量关系和空间形式对知识进行表达。而“情境化”则强调的是将数学知识融入到具体的生活、学习场景之中，注重数学知识和现实生活的结合。基于这些争论，有研究者提出了“三元数学”理论，从数学知识特征的角度，区分出“符号数学”、“言语数学”和“情境数学”这三种不同的知识表示，认为学生对不同知识表示之间转化关系的掌握是数学学习的核心。而这些不同的数学知识表示很可能存在接受、理解难易程度上的差异，相比于言语数学和情境数学，符号数学这种表示方式强调对抽象思维和逻辑思维的运用，具有高度抽象的特点，因此在数学学习的过程中更可能造成学生的理解困难，进而更有可能引起学生的畏难、害怕、焦虑等情绪。基于以上原因，本论文结合“三元数学理论”提出了“数学知识表示关联于数学焦虑”的假设，认为“相比于言语数学和情境数学，符号数学与数学焦虑的关联更密切”。本论文设计了基于“符号”、“言语”、“情境”不同类型的数学问题，并采用了问卷调查、行为实验、事件相关电位 (ERP, event-related potentials) 和功能磁共振 (fMRI, functional magnetic resonance imaging) 不同的方法技术，对研究假设进行了多个角度的验证，探索了数学知识表示和数学焦虑的关联机制。

研究一首先通过主观自评报告的方法，对被试面对不同类型的数学知识或数学问题场景下的数学焦虑水平进行了调查。基于“三元数学”理论中对数学知识表示的三种不同的分类，研究一设计了具有“符号数学焦虑”、“言语数学焦虑”、“情境数学焦虑”三个子维度的数学焦虑问卷。结果发现，被试对于不同知识表示的数学问题的主观感受存在明显的差异。相比于基于数学情境所表示的数学知识和问题，基于数学符号的以及基于数学语言的数学知识和问题更容易引起学生的焦虑情绪。更重要的是，研究结果还表明，相比于言语数学和情境数学，基于符号数学的数学知识和问题所引起的焦虑情绪对他们数学表现的影响更大。

研究二则从教育实践的视角，对数学知识表示与数学焦虑的这种关系进行了进一步的验证。在学生日常的学习生活中，并没有接触“三元数学”理论以及相关的概念。因此研究二试图通过对学生试卷中题目的“符号化程度”进行衡量，以“符号化程度”来反映数学题目在表示方式上的差异。数学题目的“符号化程度”越高，则说明数学题目在信息表达上的符号特征越强。具体而言，研究二针对2019年国家基础教育质量监测中数学监测的大样本数据进行了分析，采用了文本分析的方法，对数学试卷中综合知识点下的数学题目的符号数量进行了文本统计。结果发现，在八年级学生群体中，数学题目中符号的数量越多、数学符号特征越明显，其与数学焦虑的相关越高。这些结果说明了在八年级学生群体中，符号类型的数学知识表示与数学焦虑存在关联。

在研究一和研究二的基础上，研究三从实验设计的角度，针对五、六年级小学生数学知识学习的重点和难点，设计了同一知识点（“分数”）下的不同数学知识表示的数学问题（符号、言语、情境），并将学生在数学问题上的表现与他们的数学焦虑的关系进行了分析。结果发现，学生的数学焦虑与符号类型的数学问题的相关最高（高于与言语、情境数学问题的相关）。并且进一步的分析发现，由符号类型数学问题引起的焦虑要高于由言语或是情境类型数学问题所引起的焦虑。

研究四从数学问题加工的不同阶段（“问题识别阶段”和“问题解决阶段”），进一步检验了数学知识表示与数学焦虑之间的关联。在探究数学知识表示与数学焦虑的关系的过程当中，数学任务内容以及学生在相应的数学任务中的表现会影响学生数学焦虑的高低，是潜在的干扰变量。因此研究四通过研究设计将“问题识别阶段”和“问题解决阶段”进行分离，直接在“问题识别阶段”中对数学知识表示与数学焦虑的关联予以验证。研究四通过实验范式的改编，在控制了个体实际任务中的表现之后，测量了被试对不同类型数学题目的选择偏好。结果发现，高、低数学焦虑两组被试在选择符号类型题目个数上不存在显著的差异（可能受到了文本信息长度的影响），但是发现高数学焦虑被试选择符号类型题目的个数与他们数学焦虑存在显著的负相关，而言语、情境类型题目的选择个数与数学焦虑则没有相关。这些结果同样证明了数学知识表示与数学焦虑的关联。但由于研究四中的结果可能受到了额外变量的影响（文本信息长度），后续还需要设计更完善的实验研究对该研究四中的结果进行进一步的验证。

研究五结合启动范式和ERP技术，从时间进程的角度对数学知识表示与数学焦虑的关联机制进行了探索。研究五采集了被试在进行数学问题解决任务时的脑电信号，发现相比于基于言语、情境类型的数学问题，被试在面对符号类型的数学问题时，与注意相关的脑电成分的波幅 (N200) 更大，与心理压力相关的频谱信号强度 (beta频段27-31Hz) 也更高。这些结果说明，被试在面对符号类型的数学问题时，内心所产生的压力可能会更高，会将更多的注意资源分配到当前任务中，从而导致加工效率的降低。

研究六从大脑功能活动空间定位的角度，采用fMRI技术对知识表示关联数学焦虑的神经机制进行了进一步的探索。研究六采集了被试在进行不同类型数学问题时的大脑血氧水平的信号变化，发现被试在面对符号类型数学问题时，与情绪相关脑区（脑岛、扣带回前部、杏仁核、内侧前额叶等）的激活强度比面对言语、情境类型数学问题时更高，并且也发现了情绪相关脑区与数学加工相关脑区的功能连接更低。进一步，研究六基于大脑功能活动的特征，构建了数学知识表示与数学焦虑关联的大脑网络模型，划分了基于数学加工和情绪加工的5个子网络。对大脑功能网络内部、以及网络之间的功能连接进行差异分析发现，被试在解答符号类型数学问题时的功能连接要低于解答言语和情境类型数学问题时的功能连接。这些结果发明了，符号类型数学问题引起的情绪加工网络与数学加工网络的失连接，可能是数学知识表示与数学焦虑关联差异的神经机制。

综上所述，本论文的一系列研究从主观判断、认知行为、神经机制不同的层面证明了数学知识表示与数学焦虑的关联性，发现了以“符号数学”为主要特征的数学知识表示比以“言语数学”、“情境数学”为主要特征的数学知识表示与数学焦虑的关联更密切。这一发现从理论上说明了数学知识表示是数学焦虑发生发展的重要来源，也为未来的数学教育教学改革、数学焦虑的干预提供了新的视角。

Math anxiety refers to the tension, anxiety, or fear that individuals experience when facing situations related to mathematics. Research has found that math anxiety is widely present among students of different age groups and even adults, and it has become an emotional problem in mathematics learning worldwide. Multiple meta-analysis studies have consistently found a stable negative correlation between math anxiety and mathematical performance, which has a profound impact on students' mathematical learning and academic development. Therefore, understanding the internal mechanisms of the occurrence and development of math anxiety, and formulating corresponding intervention projects to reduce math anxiety, is an important issue in the field of mathematics education today. This is of great significance for promoting the reform of mathematics education, promoting the good development of children and adolescents' mathematical abilities, and enhancing the national mathematical literacy. Past research, from the perspectives of individual emotional attitudes and mathematical ability development, has explored factors contributing to the occurrence and growth of math anxiety, leading to the development of intervention programs and methods based on these findings. However, these interventions exhibit limitations in terms of application scenarios and effectiveness. In light of this, current research necessitates a novel perspective on the origins of math anxiety, aiming for a deeper comprehension of its intrinsic mechanisms and the formulation of more practical and enduring intervention strategies to tackle this challenge.

From the definition of math anxiety, which arises when individuals confront math-related scenarios, previous studies have primarily focused on the "individual" (emotional attitudes and mathematical ability development) as the source, overlooking the fundamental role of "mathematics" itself. Exploring the sources of math anxiety from the aspect of mathematical knowledge, as opposed to individual development, may offer fresh insights into addressing this issue. Mathematics, as the science of quantity relationships and spatial forms, possesses a unique knowledge structure and representation mode characterized by high abstraction and rigorous logic. However, in the current field of mathematics education, the expression of mathematical knowledge abstractness still has the contention of "symbolization" and "contextualization". "Symbolization" emphasizes the generalization and induction of general laws in mathematics, and the expression of knowledge in pure quantitative relations and spatial forms. However, "contextualization" emphasizes the integration of mathematical knowledge into specific life and learning scenes, focusing on the combination of mathematical knowledge and real life. Based on these arguments, some researchers put forward the theory of "Three-component mathematics ", which distinguishes three kinds of knowledge representations: "symbolic mathematics", "verbalized mathematics" and "situational mathematics" from the perspective of mathematical knowledge characteristics, and holds that students' mastery of the transformation relationship between different knowledge representations is the core of mathematics learning. Compared with verbalized mathematics and situational mathematics, symbolic mathematics is more abstract and emphasizes the application of abstract thinking and logical thinking. Therefore, it is more likely to cause students' understanding difficulties in the process of mathematics learning, thus causing students' anxiety. Based on the above reasons, this paper puts forward the hypothesis that "mathematical knowledge representation is related to math anxiety" in combination with "Three-component theory", and holds that "compared with verbal mathematics and situational mathematics, symbolic mathematics is more closely related to mathematical anxiety". This paper designs different types of mathematical problems and math anxiety questionnaires based on "symbolic", "verbalized" and "situational", and used multiple methods and technologies, including questionnaire survey, behavior experiment, event-related potentials (ERP) and functional magnetic resonance imaging (fMRI) to verify the hypothesis, and explore the correlation mechanism between mathematical knowledge representation and math anxiety.

       In the first study, the level of math anxiety of subjects faced with different types of math knowledge or math problems was investigated through the method of subjective self-assessment report. Based on three different classifications of mathematical knowledge representation in Three-component mathematics, a mathematical anxiety questionnaire with three sub-dimensions of symbolic mathematical anxiety, verbal mathematical anxiety and situational mathematical anxiety was designed in Study 1. The results show that there are obvious differences in subjects' subjective feelings about mathematical problems with different knowledge representation. Compared with mathematical knowledge expressed in mathematical situations, mathematical knowledge based on mathematical symbols and mathematical language is more likely to cause students' anxiety. More importantly, the results also showed that anxiety caused by symbolic math knowledge had a greater impact on their math performance than verbalized math and situational math.

The second study further verifies the relationship between mathematical knowledge representation and math anxiety from the perspective of educational practice. In the daily learning life of students, there is no contact with the "Three-component mathematics " theory and related concepts. Therefore, the second study attempts to measure the "degree of symbolization" of the questions in the students' examination papers, and use the degree of symbolization to reflect the differences in the representation of mathematical questions. The higher the degree of symbolization of mathematical problems, the stronger the symbolic characteristics of mathematical problems in information expression. Specifically, the second study adopts the method of text analysis based on large sample data to carry out text statistics on the number of symbols in math questions under comprehensive knowledge points in math papers. The results show that in the eighth grade students, the higher the number of symbols in math problems and the more obvious the characteristics of mathematical symbols, the higher the correlation with math anxiety. These results suggest that representation of math knowledge is associated with math anxiety in eighth grade students.

       On the basis of the first and the second study, the third study designed mathematical problems (symbolic, verbalized, situational) representing different mathematical knowledge under the same knowledge point (" fraction ") from the perspective of experimental design, aiming at the key and difficult points of mathematical knowledge learning for students in grade 5 and 6, and analyzed the relationship between students' performance in mathematical problems and their mathematical anxiety. The results show that students' math anxiety is most correlated with symbolic math problems (higher than that with verbalized and situational math problems). Further analysis shows that the anxiety caused by symbolic math problems is higher than that caused by verbalized or situational math problems.

Study 4 further examines the relationship between mathematical knowledge representation and mathematical anxiety from the perspective of different stages of mathematical problem processing ("problem identification stage"and "problem solving stage"). In the process of exploring the relationship between mathematical knowledge representation and mathematical anxiety, the content of mathematical tasks and students' performance in corresponding mathematical tasks will affect the level of students' mathematical anxiety, which is a potential interfering variable. Therefore, in the fourth study, the "problem identification stage" and "problem solving stage" are separated by research design, and the correlation between mathematical knowledge representation and mathematical anxiety is directly verified in the "problem identification stage". By adapting the psychological experiment paradigm, we measured the subjects' preference for different types of mathematical problems after controlling their performance in actual tasks. The results showed that there was no significant difference between the two groups of high and low math anxiety in the number of questions chosen by symbolic type (which may be affected by the length of text information), but there was a negative correlation between the number of questions chosen by the subjects with high math anxiety and their math anxiety. However, there was no significant correlation between the number of verbalized and situational questions and math anxiety. These results also demonstrate the association between mathematical knowledge representation and mathematical anxiety. However, since the results in study 4 may be affected by additional variables (text information length), more complete experimental studies are needed to further verify the results in Study 4.

Study 5 explores the correlation mechanism between mathematical knowledge representation and mathematical anxiety from the perspective of time process by combining priming paradigm and ERP technology. In study 5, the EEG electrical signals of the subjects were collected during the mathematical problem-solving task, and it was found that compared with the mathematical problems based on verbal mathematics and situational mathematics, the amplitude (N200) of the brain electrical components related to attention was larger when the subjects were faced with symbolic mathematical problems. The signal strength of the spectrum associated with psychological stress (beta band 27-31Hz) was also higher. These results indicate that when subjects are faced with symbolic mathematical problems, the inner pressure may be higher, and more attention resources will be allocated to the current task, resulting in the reduction of processing efficiency.

       From the perspective of spatial localization of brain functional activities, fMRI technology was used in study 6 to further explore the neural mechanism of knowledge representation and math anxiety. Specifically, study 6 collected the signal changes of brain blood oxygen level when subjects performed different types of math problems, and found that the activation intensity of emotion-related brain areas (insula, anterior cingulate gyrus, amygdala, medial prefrontal cortex, etc.) was different from that of verbal and contextual math problems. Moreover, differences in functional connectivity between emotion-related brain regions and mathematical processing brain regions were also found. Further, based on six features of brain functional activities, a brain network model of mathematical knowledge representation and mathematical anxiety is constructed, and five sub-networks based on mathematical processing and emotional processing are divided. By analyzing the differences in functional connections within and between brain functional networks, it is found that the disconnection between emotional processing network and mathematical processing network caused by symbolic type math problems may be the neural mechanism of the difference in association between mathematical knowledge representation and mathematical anxiety.

To sum up, a series of studies in this paper prove the correlation between mathematical knowledge representation and mathematical anxiety from different levels of subjective judgment, cognitive behavior and neural representation. Mathematical knowledge representation with symbol as the main feature is more closely related to mathematical anxiety than mathematical knowledge representation with verbal and situational mathematics as the main feature. This finding theoretically shows that mathematics knowledge representation is an important source of mathematics anxiety, and provides a new perspective for the future reform of mathematics education and intervention of mathematics anxiety.