# Teaching Evaluation Method Based on Fuzzy Support Vector Machine Algorithm

Authors: Zhihong Wang; Tao Xu; Lixia Wang
Journal: Mobile Information Systems
 (2022)

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## Abstract

For schools, the quality of teaching and learning is an important guarantee for achieving educational goals. There are very many factors affecting teaching quality, including hard and soft factors. Among them, the soft factors mainly refer to the teaching quality of teachers and the teaching management mechanism of schools. The teaching quality of teachers is the most critical among all factors. Traditional teaching quality evaluation (TQE) mainly adopts the way of manual scoring. This way of TQE lacks data support, and there is no evaluation mechanism based on multiple data sources. Therefore, the TQE results obtained in this way are subjective and inaccurate. Considering that the main subjects of the whole teaching session are teachers, students, and schools, each subject generates massive information in the whole teaching process. In our study, a novel evaluation method based on fuzzy classification algorithm is proposed to be applied in teaching evaluation. Firstly, this paper collects the relevant data generated by the three subjects in the teaching process, respectively. Secondly, after removing the unsuitable data, principal component analysis is used to extract the main features of the applicable data. Finally, a fuzzy support vector machine (FSVM) is used to classify and analyze the feature data in order to derive each teacher’s teaching evaluation results. The final evaluation results in this paper are divided into five grades, i.e., five categories: excellent, good, moderate, passing, and failing. The comparison with other algorithms demonstrates that the teaching evaluation model based on fuzzy comprehensive evaluation algorithm used in this paper has the advantages of high evaluation accuracy and objectivity, and we hope that this study will be useful in the field of teaching evaluation.


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## Body

## 1. Introduction

The quality of a school’s teaching is extremely important to students, the school, and the country. As the main output place of high-end talents, universities naturally assume the pivotal responsibility of cultivating talents for the country and society. How well a college does in cultivating talents depends on how well its teachers do in teaching; that is to say, whether a college can cultivate qualified talents is determined by its teaching quality. Only when colleges and universities pay attention to teaching quality, know their own teaching quality clearly, and improve and perfect their teaching work accordingly, can they continuously cultivate and deliver high-end talents to the society. As an indispensable part of education work, students’ evaluation of teaching quality is the feedback of talent cultivation work of colleges and universities and also an important reference basis for college reform. For college administrators, analyzing and summarizing students’ evaluation of teaching quality can help them understand the reasonable and insufficient aspects of the current education work arrangement and serve as a reference for future work; for teachers, students’ evaluation can spur them to improve their teaching and academic level, which is beneficial to their development. For students, evaluation of teaching quality is a way to communicate with the school, which can promote the school to arrange the teaching work reasonably, so as to improve their learning efficiency and knowledge level. In general, TQE is a kind of resource that can promote many parties. Different approaches to designing the membership function have a significant impact on the algorithm’s ease of implementation and the final classification results. The membership function can be built in a variety of ways. To measure membership, the dominant one uses the distance from the sample point to the class center. It also can regulate teaching behavior effectively, optimize the structure of teachers, promote the improvement of teachers’ teaching level, and make the management of teachers systematic and scientific [1, 2]. The traditional way of teaching quality is mostly carried out by means of questionnaires, voting, and scoring, which has several problems: one is the existence of subjectivity. People tend to score according to their impressions rather than according to the teacher’s real teaching quality. Second, the data is inaccurate. The person scoring may be unprofessional and there are cases of random scoring. This approach has been criticized by the public in real production.With the research on TQE system, there have been many studies to introduce some intelligent analysis methods into the TQE system. Reference [3] proposed an online midterm evaluation method for teacher learning and teaching. The method uses university undergraduate students as the research subjects and proves that the study has a positive effect on teachers’ online teaching. Reference [4] applied a neural network model to the assessment of teachers’ informational design skills. Reference [5] proposed an electronic assessment system for assessing the learning of students in professional and technical schools. The feasibility of the assessment method was demonstrated experimentally. Reference [6] applied a two-way long- and short-term memory network to the process of teaching physical education. And the optimal solution of the parameters of the model was obtained using genetic algorithm. The experiments proved that the proposed method is accurate and time-consuming. Reference [7] compared three teaching methods and finally concluded that the flipped classroom has the best teaching evaluation. Reference [8] proposed a method for assessing humor teaching ability, which was very accurate in evaluating teachers’ humor ability in the class. Reference [9] pointed out that the assessment of teaching quality lies mainly in how to assess the quality of the work and suggested the difficulties that can exist in too much time. Reference [10] studied the influence of students on the evaluation of teaching. The research summarizes and analyzes the problems that teachers will encounter in the teaching process. It reveals the potential conflicts that exist between teachers and students.The preceding studies on TQE can demonstrate that the importance of TQE is obvious in all schools at the moment, but there are numerous issues with the evaluation methods. These problems are mainly focused on two aspects. The first is the evaluation index system. The traditional teaching evaluation index system is basically based on the results, and there are fewer process indicators. On the other hand, the evaluation results are affected by each evaluation factor in the index system to different degrees, so different evaluation factors should be assigned different weights. In the actual teaching process, schools generally default to the same weight for each evaluation factor, which can be convenient and can reduce some unnecessary troubles. Or some leaders artificially add or subtract weights to some evaluation factors. These methods are not supported by data or theoretical bases. In the process of time use, this way will cause inaccurate and unjust evaluation results. Therefore, how to reasonably assign appropriate weights to each evaluation factor in the evaluation system is a key issue. The second is the evaluation method and evaluation medium. The traditional evaluation methods include “summation method” and “weighted summation method,” which are simple but make easy to reduce the reliability and credibility of the evaluation results. In some remote and backward areas, TQE even adopts the questionnaire form of paper form, which not only wastes a lot of paper, but also makes the efficiency of teaching evaluation activities greatly reduced. However, with the rapid development of computer-related technologies and the application of information technology in education, data for evaluating teaching quality can now cover the entire teaching process. And with the development of cell phone clients, the teaching evaluation systems of many universities develop not only Web clients but also cell phone clients. This not only solves the problem of networked evaluation, but also makes it convenient for the participants in the evaluation to evaluate teaching activities anytime and anywhere and for the evaluated teachers to view the evaluation results anytime and anywhere. As a result, this paper proposes a comprehensive evaluation method based on fuzzy support vector machine to improve the objectivity and accuracy of teachers’ teaching quality evaluation. Experiments show that evaluating teaching quality using the method described in this paper is more objective and fair, with solid data support.


## 2. Basic Concepts

### 2.1. Introduction to Teaching Quality Evaluation

TQE refers to the process of making value judgments on teachers’ own quality and their teaching ability by using advanced modern educational evaluation theories according to the regulations of the Ministry of Education. It is the process of guiding teachers to improve their own quality and teaching ability. The evaluation outcomes are numerical judgments. The evaluation results will be used to inform future training, evaluation, reward and punishment, and personnel decisions. Thus, teachers will be urged to improve themselves continuously and complete their job duties beyond the level. Based on the evaluation results, we can learn from experienced and excellent teachers or educational evaluation researchers to establish a scientific evaluation system that can effectively reflect the strengths and weaknesses of teachers’ teaching activities. Figure1 gives the specific contents included in the teaching quality evaluation.Figure 1
TQE index.Students rate the indicators in the evaluation system against the teacher’s performance in class. It is hoped that this method will enable teachers to understand their strengths and weaknesses in their teaching activities and to improve the areas where they are lacking in order to improve the quality of teaching. An objective and fair evaluation of teaching quality must be based on data support. The teaching process generates various kinds of data, and there are many potential correlations within these data. Therefore, these data can be used to explore the laws embedded in the teaching process data at the beginning, so as to improve the accuracy and objectivity of teaching quality evaluation. The principle of mining TQE data is shown in Figure2.Figure 2
TQE data mining.


### 2.2. The Role of Teaching Quality Evaluation

TQE is very important and has an important role for students, teachers, schools, and educational parts. Specifically, it has the following roles.(1)
Guidance Role. The teaching process is a mutual process, so whether it is based on classroom teaching or in the quality evaluation of after-school teaching, all elements of teaching and learning in teaching need to be considered comprehensively. These elements in teachers’ teaching may affect the accomplishment of teaching objectives to a greater or lesser extent. Based on the analysis of these elements, the education management part needs to develop a reasonable evaluation plan. According to the evaluation contents and evaluation subjects, scientific evaluation indexes are constructed. Thus, teachers can be guided to clarify the teaching direction and select appropriate teaching methods. In this process, it can also help teachers adjust their teaching processes in order to achieve better teaching effects and complete teaching tasks on time.(2)
Motivational Role. The motivating effect is mainly reflected in both teachers and students. For teachers, firstly, through the evaluation results, teachers can have a comprehensive understanding of their achievements in teaching and confirm their strengths. This will allow the teachers to use their strengths in future teaching. Through continuous optimization and improvement, teachers will be motivated to participate in teaching activities. Secondly, through the teaching quality evaluation, teachers can also find their own shortcomings in their teaching work. In response to the shortcomings, they analyze the causes and summarize and learn from other experienced teachers. Thus, they can improve their own teaching methods and enhance their teaching level. For students, TQE scoring is also a process of communication and interaction with teachers, which can inform teachers and relevant departments about the teaching methods that each student likes and the methods of teachers that they do not like. This feedback not only allows students to think about their own problems in the teaching process, but also stimulates students’ initiative and enthusiasm for learning.(3)
The Role of Feedback. The feedback generated by the evaluation not only enables teachers to realize the effectiveness of their own teaching, but also enables students to recognize their own learning. On the other hand, teachers can find out the problems they have in teaching in real time and, based on the question headings, analyze the causes to find countermeasures and dynamically update the teaching plan based on the feedback received in real time and in stages to help students change their own learning behavior. Thus, teaching can be continuously improved and enhanced to make it easier to complete teaching tasks.(4)
Accreditation Role. By conducting a TQE process, the education administration can learn all about the teachers’ performance in their teaching work. The effectiveness achieved in this process is quantified in terms of the improvement of the teacher’s own teaching ability as well as the improvement of the students’ performance. The vast amount of teaching data is used to identify key factors and indicators that influence the quality of teaching and learning. Teaching quality assessment is a scientific diagnosis of how teaching is unfolding, as well as a reasonable and comprehensive assessment of teachers’ teaching levels and achievement of teaching goals.


## 2.1. Introduction to Teaching Quality Evaluation

TQE refers to the process of making value judgments on teachers’ own quality and their teaching ability by using advanced modern educational evaluation theories according to the regulations of the Ministry of Education. It is the process of guiding teachers to improve their own quality and teaching ability. The evaluation outcomes are numerical judgments. The evaluation results will be used to inform future training, evaluation, reward and punishment, and personnel decisions. Thus, teachers will be urged to improve themselves continuously and complete their job duties beyond the level. Based on the evaluation results, we can learn from experienced and excellent teachers or educational evaluation researchers to establish a scientific evaluation system that can effectively reflect the strengths and weaknesses of teachers’ teaching activities. Figure1 gives the specific contents included in the teaching quality evaluation.Figure 1
TQE index.Students rate the indicators in the evaluation system against the teacher’s performance in class. It is hoped that this method will enable teachers to understand their strengths and weaknesses in their teaching activities and to improve the areas where they are lacking in order to improve the quality of teaching. An objective and fair evaluation of teaching quality must be based on data support. The teaching process generates various kinds of data, and there are many potential correlations within these data. Therefore, these data can be used to explore the laws embedded in the teaching process data at the beginning, so as to improve the accuracy and objectivity of teaching quality evaluation. The principle of mining TQE data is shown in Figure2.Figure 2
TQE data mining.


## 2.2. The Role of Teaching Quality Evaluation

TQE is very important and has an important role for students, teachers, schools, and educational parts. Specifically, it has the following roles.(1)
Guidance Role. The teaching process is a mutual process, so whether it is based on classroom teaching or in the quality evaluation of after-school teaching, all elements of teaching and learning in teaching need to be considered comprehensively. These elements in teachers’ teaching may affect the accomplishment of teaching objectives to a greater or lesser extent. Based on the analysis of these elements, the education management part needs to develop a reasonable evaluation plan. According to the evaluation contents and evaluation subjects, scientific evaluation indexes are constructed. Thus, teachers can be guided to clarify the teaching direction and select appropriate teaching methods. In this process, it can also help teachers adjust their teaching processes in order to achieve better teaching effects and complete teaching tasks on time.(2)
Motivational Role. The motivating effect is mainly reflected in both teachers and students. For teachers, firstly, through the evaluation results, teachers can have a comprehensive understanding of their achievements in teaching and confirm their strengths. This will allow the teachers to use their strengths in future teaching. Through continuous optimization and improvement, teachers will be motivated to participate in teaching activities. Secondly, through the teaching quality evaluation, teachers can also find their own shortcomings in their teaching work. In response to the shortcomings, they analyze the causes and summarize and learn from other experienced teachers. Thus, they can improve their own teaching methods and enhance their teaching level. For students, TQE scoring is also a process of communication and interaction with teachers, which can inform teachers and relevant departments about the teaching methods that each student likes and the methods of teachers that they do not like. This feedback not only allows students to think about their own problems in the teaching process, but also stimulates students’ initiative and enthusiasm for learning.(3)
The Role of Feedback. The feedback generated by the evaluation not only enables teachers to realize the effectiveness of their own teaching, but also enables students to recognize their own learning. On the other hand, teachers can find out the problems they have in teaching in real time and, based on the question headings, analyze the causes to find countermeasures and dynamically update the teaching plan based on the feedback received in real time and in stages to help students change their own learning behavior. Thus, teaching can be continuously improved and enhanced to make it easier to complete teaching tasks.(4)
Accreditation Role. By conducting a TQE process, the education administration can learn all about the teachers’ performance in their teaching work. The effectiveness achieved in this process is quantified in terms of the improvement of the teacher’s own teaching ability as well as the improvement of the students’ performance. The vast amount of teaching data is used to identify key factors and indicators that influence the quality of teaching and learning. Teaching quality assessment is a scientific diagnosis of how teaching is unfolding, as well as a reasonable and comprehensive assessment of teachers’ teaching levels and achievement of teaching goals.


## 3. TQE Model

### 3.1. Ideas of Teaching Quality Evaluation

The idea of the proposed TQE method is shown in Figure3. First, the teaching related data were obtained through survey questionnaires, system scoring, and various platform data. After the raw data are obtained, the data need to be preprocessed, including the elimination of invalid data or noisy data. The preprocessed TQE data are then separated into training and testing samples. Thirdly, the classification model is trained using the training samples. Then input the test sample to get the classification results of the test sample.Figure 3
Ideas of teaching quality evaluation.The TQE data are mainly collected from the following aspects, as shown in Table1. There are 5 primary indicators and 18 secondary indicators in total. The data of each index is quantified into specific values in order to start the calculation; therefore, the value interval of each index is [0, 100]. The scores of the secondary indicators are divided into 5 levels, namely, excellent, good, moderate, pass, and fail. The specific values corresponding to yes are [90, 100], good are [80, 89], medium are [70, 70], pass are [60, 69], and fail are [0, 59].Table 1
Data collection indicators.
First-level indicatorSecondary indicatorsTeaching ability1. Putonghua level2. Reasonable course schedule3. Board book combined with information technology toolsTeaching method4. Cultivate students’ creative thinking5 Classroom content is tightly arranged6. Rich teaching toolsTeaching attitude7. Prepare lessons carefully8. Patiently counseling and answering questions, timely correction of homework9. Require strict classroom discipline10. Interaction with studentsTeaching content11. The teaching content is vivid and interesting12. Peal teaching content with depth and breadth13. The teaching content is cutting-edgeTeaching effect14. Active classroom atmosphere15. Students are basically able to grasp what the teacher teaches in class16. Enlighten students’ thinking and improve their problem-solving skills17. Improve students’ ability to work together as a team18. Improve students  enthusiasm to participate in classroom interaction


### 3.2. Fuzzy Support Vector Machine

In supervised learning, the performance of many learning algorithms is very similar, so it is not the choice of learning algorithm A or learning algorithm B that is important, but rather the level of algorithmic dependence on which the performance of a large amount of data is usually applied to these algorithms. This is reflected in the choice of algorithm features and the choice of regularization parameters. One of the most widely used in industry and academia is Support Vector Machine (SVM) [11–13]. Compared to logistic regression and neural networks, SVMs offer a more powerful way in learning complex nonlinear equations. In machine learning [14–16], SVM is supervised learning model. A supervised learning model is one in which the data is labeled with a classification beforehand so that the machine knows which classification the data belongs to. Unsupervised learning means that the data is not labeled with a classification, because either prior knowledge is not available or the cost of labeling is high. In summary, the machine can be used for preprocessing related tasks. For example, data clustering analysis makes subsequent manual analysis of each class easier. As a supervised learning model, SVM is capable of pattern recognition, classification, and regression analysis. FSVM introduces a membership strategy based on the classic SVM [17, 18]. Membership is the soul of the FSVM algorithm. Membership can give different weights according to the size of each sample in the classification task. Good samples with large positive effects are given large weights; otherwise small weights are given.


#### 3.2.1. Fuzzy Set Concept

The analysis of many problems cannot be defined by absolute yes or no, belonging or not belonging, etc. conclusions. Therefore, fuzzy concepts are proposed by scholars to describe some specific problems. Fuzzy mathematics is mainly used to solve fuzzy class problems. Combining fuzzy mathematics with other disciplines, it has been successfully used in pattern recognition, classification, and other fields [19, 20]. For example, fuzzy algorithm is one of the typical applications. Before there is no fuzzy concept, the relationship between a sample and a set can only belong or not belong. After the fuzzy concept appears, the relationship between the sample x and the set W can be expressed as follows:(1)RWx=1,x∈W,0,x∉W.The individual data in the set can be represented by fuzzy numbers, so that the fuzzy sets evolve from the traditional 0 or 1 to all numbers between 0 and 1.SupposeW is a subset of the set ψ and ψ takes a range of values 0,1. The higher the degree to which x belongs to W, the larger the value of pWx. If x∉W, then pWx=0. Call pW,W a fuzzy set and PW a fuzzy membership of the set W.


#### 3.2.2. Fuzzy Support Vector Machine

The training set isM=A1,B1,M1,…,An,Bn,Mn, where Aj∈Rm, Bj∈−1,1, 0≤Mj≤1,j=1,2,…,n. Mj is the fuzzy membership of the training point Aj,Bj after fuzzification. The significance of fuzzy membership Mj represents the membership of the training points Aj,Bj in a specific class. Parameter ηj represents the proportion of wrong samples in the samples divided by SVM. Therefore, Mjηj can be used to measure the degree of correctness of the SVM to classify the data with different membership degrees.When the input test data is linearly divisible, the mathematical expression of the classification model to find the optimal hyperplane is given by(2)mink,b,η12k2+C∑j=1nMjηj,st⋅Bjk⋅A+b+ηj≥1,j=1,2,…,n,ηj≥0,j=1,2,…,n,where C represents the penalty parameter, η=η1,…,ηn′. To solve the quadratic programming problem shown in equation (2), construct the Lagrangian equation shown in the following equation:(3)Lk,b,η,x,y=12k2+C∑j=1nMjηj−∑j=1nxjBjk⋅A+b+ηj−1−∑j=1nyjηj.By the definition of Wolfe dual, the Lagrangian function onk,b,η is found to be minimal, i.e.:(4)∇kLk,b,η,x,y=k−∑j=1nxjBjAj=0,(5)∇bLk,b,η,x,y=−∑j=1nxjBj=0,(6)∇ηjLk,b,η,x,y=MjC−xj−yj=0,j=1,2,…,n.Substituting equations (4) and (5) into equation (6) and maximizing for x, we obtain the following equation:(7)minx∑j=1nxj−12∑i=1n∑j=1nBjxjxiAi,Aj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.To find the minimal value ofx, equation (7) is transformed into the following equation:(8)minx12∑i=1n∑j=1nBjxjxiAi,Aj−∑j=1nxj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.Solving equation (8) yields the optimal solution x∗=x1∗,...,xn∗′, and the expression for the fuzzy optimal hyperplane is obtained as follows:(9)fuzA=sgnk∗⋅A+b∗,A∈Rm,where k∗=∑j=1nxj∗BjAj , b∗=Bi−∑j=1nBixjAj⋅Ai,i∈i|0<xi∗<MiC.The training dataAi corresponding to x∗>0 in the set x∗=x1∗,...,xn∗′ is called the support vector. The support vectors are divided into two types. One is the correct test samples with the value of 0<xi∗<MiC. The other class is the incorrect test samples with the value of xi∗>MiC. The major difference between FSVM and SVM is that the former has more fuzzy membership than the latter.When dealing with nonlinear data, the kernel functionΠxi,xj can be introduced in FSVM, and then the objective function formula for solving the problem is as follows:(10)minx12∑i=1n∑j=1nBjxjxiΠAi,Aj−∑j=1nxj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.Solving equation (10) yields the fuzzy optimal classification function as follows:(11)fuzA=sgnxj∗BjΠA,Aj+b∗,A∈Rm,where b∗=Bi−∑j=1nBixjAj⋅Bi,i∈i|0<xi∗<MiC.


#### 3.2.3. Fuzzy Membership

Fuzzy membership is the probability that a sample will be classified into some class in the range of 0 to 1. 0 is assigned to determine the position that does not belong to the specified set and 1 is assigned to determine those values that belong to the specified set. Probabilities in the entire range between 0 and 1 are assigned to possible members of some class. The core of FSVM is the fuzzy membership function. The fuzzy membership is mainly used to assign weights to the samples. Different approaches to designing the membership function have a significant impact on the algorithm’s ease of implementation and the final classification results. The membership function can be built in a variety of ways. The dominant one uses the distance between the sample point and the class center to calculate membership. Given the sample {A1, …, An}, the sample dimension is m, A0 denotes the class centroid, and ε is the class radius:(12)A0=1n∑j=1nAj,ε=maxAj−A0,j=1,2,…,n.The expression of the membership of the class center is as follows:(13)Mj=1−Aj−A0ε+D.


## 3.1. Ideas of Teaching Quality Evaluation

The idea of the proposed TQE method is shown in Figure3. First, the teaching related data were obtained through survey questionnaires, system scoring, and various platform data. After the raw data are obtained, the data need to be preprocessed, including the elimination of invalid data or noisy data. The preprocessed TQE data are then separated into training and testing samples. Thirdly, the classification model is trained using the training samples. Then input the test sample to get the classification results of the test sample.Figure 3
Ideas of teaching quality evaluation.The TQE data are mainly collected from the following aspects, as shown in Table1. There are 5 primary indicators and 18 secondary indicators in total. The data of each index is quantified into specific values in order to start the calculation; therefore, the value interval of each index is [0, 100]. The scores of the secondary indicators are divided into 5 levels, namely, excellent, good, moderate, pass, and fail. The specific values corresponding to yes are [90, 100], good are [80, 89], medium are [70, 70], pass are [60, 69], and fail are [0, 59].Table 1
Data collection indicators.
First-level indicatorSecondary indicatorsTeaching ability1. Putonghua level2. Reasonable course schedule3. Board book combined with information technology toolsTeaching method4. Cultivate students’ creative thinking5 Classroom content is tightly arranged6. Rich teaching toolsTeaching attitude7. Prepare lessons carefully8. Patiently counseling and answering questions, timely correction of homework9. Require strict classroom discipline10. Interaction with studentsTeaching content11. The teaching content is vivid and interesting12. Peal teaching content with depth and breadth13. The teaching content is cutting-edgeTeaching effect14. Active classroom atmosphere15. Students are basically able to grasp what the teacher teaches in class16. Enlighten students’ thinking and improve their problem-solving skills17. Improve students’ ability to work together as a team18. Improve students  enthusiasm to participate in classroom interaction


## 3.2. Fuzzy Support Vector Machine

In supervised learning, the performance of many learning algorithms is very similar, so it is not the choice of learning algorithm A or learning algorithm B that is important, but rather the level of algorithmic dependence on which the performance of a large amount of data is usually applied to these algorithms. This is reflected in the choice of algorithm features and the choice of regularization parameters. One of the most widely used in industry and academia is Support Vector Machine (SVM) [11–13]. Compared to logistic regression and neural networks, SVMs offer a more powerful way in learning complex nonlinear equations. In machine learning [14–16], SVM is supervised learning model. A supervised learning model is one in which the data is labeled with a classification beforehand so that the machine knows which classification the data belongs to. Unsupervised learning means that the data is not labeled with a classification, because either prior knowledge is not available or the cost of labeling is high. In summary, the machine can be used for preprocessing related tasks. For example, data clustering analysis makes subsequent manual analysis of each class easier. As a supervised learning model, SVM is capable of pattern recognition, classification, and regression analysis. FSVM introduces a membership strategy based on the classic SVM [17, 18]. Membership is the soul of the FSVM algorithm. Membership can give different weights according to the size of each sample in the classification task. Good samples with large positive effects are given large weights; otherwise small weights are given.


### 3.2.1. Fuzzy Set Concept

The analysis of many problems cannot be defined by absolute yes or no, belonging or not belonging, etc. conclusions. Therefore, fuzzy concepts are proposed by scholars to describe some specific problems. Fuzzy mathematics is mainly used to solve fuzzy class problems. Combining fuzzy mathematics with other disciplines, it has been successfully used in pattern recognition, classification, and other fields [19, 20]. For example, fuzzy algorithm is one of the typical applications. Before there is no fuzzy concept, the relationship between a sample and a set can only belong or not belong. After the fuzzy concept appears, the relationship between the sample x and the set W can be expressed as follows:(1)RWx=1,x∈W,0,x∉W.The individual data in the set can be represented by fuzzy numbers, so that the fuzzy sets evolve from the traditional 0 or 1 to all numbers between 0 and 1.SupposeW is a subset of the set ψ and ψ takes a range of values 0,1. The higher the degree to which x belongs to W, the larger the value of pWx. If x∉W, then pWx=0. Call pW,W a fuzzy set and PW a fuzzy membership of the set W.


### 3.2.2. Fuzzy Support Vector Machine

The training set isM=A1,B1,M1,…,An,Bn,Mn, where Aj∈Rm, Bj∈−1,1, 0≤Mj≤1,j=1,2,…,n. Mj is the fuzzy membership of the training point Aj,Bj after fuzzification. The significance of fuzzy membership Mj represents the membership of the training points Aj,Bj in a specific class. Parameter ηj represents the proportion of wrong samples in the samples divided by SVM. Therefore, Mjηj can be used to measure the degree of correctness of the SVM to classify the data with different membership degrees.When the input test data is linearly divisible, the mathematical expression of the classification model to find the optimal hyperplane is given by(2)mink,b,η12k2+C∑j=1nMjηj,st⋅Bjk⋅A+b+ηj≥1,j=1,2,…,n,ηj≥0,j=1,2,…,n,where C represents the penalty parameter, η=η1,…,ηn′. To solve the quadratic programming problem shown in equation (2), construct the Lagrangian equation shown in the following equation:(3)Lk,b,η,x,y=12k2+C∑j=1nMjηj−∑j=1nxjBjk⋅A+b+ηj−1−∑j=1nyjηj.By the definition of Wolfe dual, the Lagrangian function onk,b,η is found to be minimal, i.e.:(4)∇kLk,b,η,x,y=k−∑j=1nxjBjAj=0,(5)∇bLk,b,η,x,y=−∑j=1nxjBj=0,(6)∇ηjLk,b,η,x,y=MjC−xj−yj=0,j=1,2,…,n.Substituting equations (4) and (5) into equation (6) and maximizing for x, we obtain the following equation:(7)minx∑j=1nxj−12∑i=1n∑j=1nBjxjxiAi,Aj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.To find the minimal value ofx, equation (7) is transformed into the following equation:(8)minx12∑i=1n∑j=1nBjxjxiAi,Aj−∑j=1nxj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.Solving equation (8) yields the optimal solution x∗=x1∗,...,xn∗′, and the expression for the fuzzy optimal hyperplane is obtained as follows:(9)fuzA=sgnk∗⋅A+b∗,A∈Rm,where k∗=∑j=1nxj∗BjAj , b∗=Bi−∑j=1nBixjAj⋅Ai,i∈i|0<xi∗<MiC.The training dataAi corresponding to x∗>0 in the set x∗=x1∗,...,xn∗′ is called the support vector. The support vectors are divided into two types. One is the correct test samples with the value of 0<xi∗<MiC. The other class is the incorrect test samples with the value of xi∗>MiC. The major difference between FSVM and SVM is that the former has more fuzzy membership than the latter.When dealing with nonlinear data, the kernel functionΠxi,xj can be introduced in FSVM, and then the objective function formula for solving the problem is as follows:(10)minx12∑i=1n∑j=1nBjxjxiΠAi,Aj−∑j=1nxj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.Solving equation (10) yields the fuzzy optimal classification function as follows:(11)fuzA=sgnxj∗BjΠA,Aj+b∗,A∈Rm,where b∗=Bi−∑j=1nBixjAj⋅Bi,i∈i|0<xi∗<MiC.


### 3.2.3. Fuzzy Membership

Fuzzy membership is the probability that a sample will be classified into some class in the range of 0 to 1. 0 is assigned to determine the position that does not belong to the specified set and 1 is assigned to determine those values that belong to the specified set. Probabilities in the entire range between 0 and 1 are assigned to possible members of some class. The core of FSVM is the fuzzy membership function. The fuzzy membership is mainly used to assign weights to the samples. Different approaches to designing the membership function have a significant impact on the algorithm’s ease of implementation and the final classification results. The membership function can be built in a variety of ways. The dominant one uses the distance between the sample point and the class center to calculate membership. Given the sample {A1, …, An}, the sample dimension is m, A0 denotes the class centroid, and ε is the class radius:(12)A0=1n∑j=1nAj,ε=maxAj−A0,j=1,2,…,n.The expression of the membership of the class center is as follows:(13)Mj=1−Aj−A0ε+D.


## 3.2.1. Fuzzy Set Concept

The analysis of many problems cannot be defined by absolute yes or no, belonging or not belonging, etc. conclusions. Therefore, fuzzy concepts are proposed by scholars to describe some specific problems. Fuzzy mathematics is mainly used to solve fuzzy class problems. Combining fuzzy mathematics with other disciplines, it has been successfully used in pattern recognition, classification, and other fields [19, 20]. For example, fuzzy algorithm is one of the typical applications. Before there is no fuzzy concept, the relationship between a sample and a set can only belong or not belong. After the fuzzy concept appears, the relationship between the sample x and the set W can be expressed as follows:(1)RWx=1,x∈W,0,x∉W.The individual data in the set can be represented by fuzzy numbers, so that the fuzzy sets evolve from the traditional 0 or 1 to all numbers between 0 and 1.SupposeW is a subset of the set ψ and ψ takes a range of values 0,1. The higher the degree to which x belongs to W, the larger the value of pWx. If x∉W, then pWx=0. Call pW,W a fuzzy set and PW a fuzzy membership of the set W.


## 3.2.2. Fuzzy Support Vector Machine

The training set isM=A1,B1,M1,…,An,Bn,Mn, where Aj∈Rm, Bj∈−1,1, 0≤Mj≤1,j=1,2,…,n. Mj is the fuzzy membership of the training point Aj,Bj after fuzzification. The significance of fuzzy membership Mj represents the membership of the training points Aj,Bj in a specific class. Parameter ηj represents the proportion of wrong samples in the samples divided by SVM. Therefore, Mjηj can be used to measure the degree of correctness of the SVM to classify the data with different membership degrees.When the input test data is linearly divisible, the mathematical expression of the classification model to find the optimal hyperplane is given by(2)mink,b,η12k2+C∑j=1nMjηj,st⋅Bjk⋅A+b+ηj≥1,j=1,2,…,n,ηj≥0,j=1,2,…,n,where C represents the penalty parameter, η=η1,…,ηn′. To solve the quadratic programming problem shown in equation (2), construct the Lagrangian equation shown in the following equation:(3)Lk,b,η,x,y=12k2+C∑j=1nMjηj−∑j=1nxjBjk⋅A+b+ηj−1−∑j=1nyjηj.By the definition of Wolfe dual, the Lagrangian function onk,b,η is found to be minimal, i.e.:(4)∇kLk,b,η,x,y=k−∑j=1nxjBjAj=0,(5)∇bLk,b,η,x,y=−∑j=1nxjBj=0,(6)∇ηjLk,b,η,x,y=MjC−xj−yj=0,j=1,2,…,n.Substituting equations (4) and (5) into equation (6) and maximizing for x, we obtain the following equation:(7)minx∑j=1nxj−12∑i=1n∑j=1nBjxjxiAi,Aj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.To find the minimal value ofx, equation (7) is transformed into the following equation:(8)minx12∑i=1n∑j=1nBjxjxiAi,Aj−∑j=1nxj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.Solving equation (8) yields the optimal solution x∗=x1∗,...,xn∗′, and the expression for the fuzzy optimal hyperplane is obtained as follows:(9)fuzA=sgnk∗⋅A+b∗,A∈Rm,where k∗=∑j=1nxj∗BjAj , b∗=Bi−∑j=1nBixjAj⋅Ai,i∈i|0<xi∗<MiC.The training dataAi corresponding to x∗>0 in the set x∗=x1∗,...,xn∗′ is called the support vector. The support vectors are divided into two types. One is the correct test samples with the value of 0<xi∗<MiC. The other class is the incorrect test samples with the value of xi∗>MiC. The major difference between FSVM and SVM is that the former has more fuzzy membership than the latter.When dealing with nonlinear data, the kernel functionΠxi,xj can be introduced in FSVM, and then the objective function formula for solving the problem is as follows:(10)minx12∑i=1n∑j=1nBjxjxiΠAi,Aj−∑j=1nxj,st⋅∑j=1nBjxj=0,0≤xj≤MjC,j=1,2,…,n.Solving equation (10) yields the fuzzy optimal classification function as follows:(11)fuzA=sgnxj∗BjΠA,Aj+b∗,A∈Rm,where b∗=Bi−∑j=1nBixjAj⋅Bi,i∈i|0<xi∗<MiC.


## 3.2.3. Fuzzy Membership

Fuzzy membership is the probability that a sample will be classified into some class in the range of 0 to 1. 0 is assigned to determine the position that does not belong to the specified set and 1 is assigned to determine those values that belong to the specified set. Probabilities in the entire range between 0 and 1 are assigned to possible members of some class. The core of FSVM is the fuzzy membership function. The fuzzy membership is mainly used to assign weights to the samples. Different approaches to designing the membership function have a significant impact on the algorithm’s ease of implementation and the final classification results. The membership function can be built in a variety of ways. The dominant one uses the distance between the sample point and the class center to calculate membership. Given the sample {A1, …, An}, the sample dimension is m, A0 denotes the class centroid, and ε is the class radius:(12)A0=1n∑j=1nAj,ε=maxAj−A0,j=1,2,…,n.The expression of the membership of the class center is as follows:(13)Mj=1−Aj−A0ε+D.


## 4. Experiments

### 4.1. Teaching Process Data Collection

180 teachers’ data were gathered in five categories: teaching attitude, teaching ability, teaching method, teaching content, and teaching effect. Among these 180 teachers, 40 teachers had more theoretical courses, 80 teachers had more science and practical courses, and 60 teachers had more practical courses. The data were obtained from the teaching system and questionnaires. The questionnaires were administered to students, peers, and leaders. The questionnaires for each study subject were generally scored on site at the end of each class. The highest score was 100 and the low score was 0. There were 5 levels. The details of the questionnaire are shown in Table2. The total number of samples was 180, with 18 dimensions for each sample. The training sample size was 140, and the test sample size was 40. Sample examples are shown in Table 3. The data were preprocessed such as normalization to obtain the sample data results as shown in Table 4. Before training the data into a model, the main features of the data were extracted using principal component analysis to reduce the dimensionality of the data. The final data dimensionality was reduced to 8 dimensions.Table 2
Survey questionnaire.
Scoring items90–10080–8970–7960–690–591. Putonghua level2. Reasonable course schedule3. Board book combined with information technology tools4. Cultivate students’ creative thinking5 Classroom content is tightly arranged6. Rich teaching tools7. Prepare lessons carefully8. Patiently counseling and answering questions, timely correction of homework9. Require strict classroom discipline10. Interaction with students11. The teaching content is vivid and interesting1 Peal teaching content with depth and breadth13. The teaching content is cutting-edge14. Active classroom atmosphere15. Students are able to understand and master the content of the teacher’s lecture16. Enlighten students’ thinking and improve their problem-solving skills17. Improve students’ ability to work together as a team18. Improvement of thinking abilityTable 3
Sample example.
Number12345678…18S1788071809098947474S27578879870827068…93S39186867480669370…83S49288678388667085…82S56790656996709779…97……………………………S1807190907583636281…85Table 4
Example of preprocessed sample data.
Number12345678...18S10.77550.79590.71040.79930.90050.98370.93630.74000.7407S20.75260.77820.87190.98390.70200.81890.70170.6786...0.9323S30.90620.85850.86200.73620.80240.65540.92570.7004...0.8341S40.91810.88370.66500.83410.87960.65970.69740.8464...0.8199S50.67470.90190.64760.68950.95640.70300.97170.7893...0.9669.................................S1800.71430.90290.90150.75220.82710.63030.62160.8123...0.8510


### 4.2. Experimental Discussion

Several comparison models were used to validate the performance of the method used to verify the feasibility of the teaching evaluation model used in this paper. The main comparison models are SVM [21], Radial Basis Neural Network (RBFNN) [22], Random Forest (RF) [23], Iterative Dichotomiser 3 (ID3) [24], and K-Nearest Neighbor (KNN) [25], and nine teaching experts were asked to score 180 teachers during the experiment, and each expert scored 40. The final teaching quality score of each teacher was given by taking the average value. And according to the scores, the grade where they were located was determined as A to E. 90–100 was A, 80–89 was B, 70–79 was C, 60–69 was D, and 0–59 was E. The classification accuracy was used as the evaluation index for the performance of each algorithm. Table 5 displays the experimental results. All of the experimental data are mean values obtained by running the algorithm ten times.Table 5
TQE results.
AlgorithmSVMRBFNNRFID3KNNFSVMAccuracy0.75440.90770.83190.81990.85800.9412Std0.14880.05040.12530.16070.09040.0560Time (ms)94.549.148.933.890.036.9According to the experimental results in Table5, the FSVM algorithm used in this paper yielded the best evaluation result of 0.9412, followed by RBFNN with a result of 0.9077. The results obtained by the three algorithms RF, ID3, and KNN are all between 0.8 and 0.9. The worst result obtained by SVM is only 0.7544. The reason why FSVM can achieve excellent experimental results is the introduction of fuzzy membership, which improves the classification accuracy of the samples. On the other hand, the experimental results plot shown in Figure 4 shows that the stability of the results obtained by each algorithm is not the same. Among them, ID3 and SVM have the highest volatility, which indicates that these two algorithms are the most unstable. The next best is KNN, and the best is RBFNN and FSVM. In summary, the FSVM used in this paper not only achieves the optimal teaching evaluation results, but also has stable performance.Figure 4
Comparison of experimental accuracy.Figure5 compares the time required to train each of the algorithms under consideration. It is clear from the figure that KNN and SVM take the longest time, followed by RF and RBFNN, followed by FSVM. The shortest time spent among the algorithms is ID3, but the difference between the time spent by FSVM and ID3 is very small, with a difference of 3.1 ms. Therefore, it can be said that the FSVM algorithm has an advantage in terms of model training time.Figure 5
Algorithm time consumption comparison.


## 4.1. Teaching Process Data Collection

180 teachers’ data were gathered in five categories: teaching attitude, teaching ability, teaching method, teaching content, and teaching effect. Among these 180 teachers, 40 teachers had more theoretical courses, 80 teachers had more science and practical courses, and 60 teachers had more practical courses. The data were obtained from the teaching system and questionnaires. The questionnaires were administered to students, peers, and leaders. The questionnaires for each study subject were generally scored on site at the end of each class. The highest score was 100 and the low score was 0. There were 5 levels. The details of the questionnaire are shown in Table2. The total number of samples was 180, with 18 dimensions for each sample. The training sample size was 140, and the test sample size was 40. Sample examples are shown in Table 3. The data were preprocessed such as normalization to obtain the sample data results as shown in Table 4. Before training the data into a model, the main features of the data were extracted using principal component analysis to reduce the dimensionality of the data. The final data dimensionality was reduced to 8 dimensions.Table 2
Survey questionnaire.
Scoring items90–10080–8970–7960–690–591. Putonghua level2. Reasonable course schedule3. Board book combined with information technology tools4. Cultivate students’ creative thinking5 Classroom content is tightly arranged6. Rich teaching tools7. Prepare lessons carefully8. Patiently counseling and answering questions, timely correction of homework9. Require strict classroom discipline10. Interaction with students11. The teaching content is vivid and interesting1 Peal teaching content with depth and breadth13. The teaching content is cutting-edge14. Active classroom atmosphere15. Students are able to understand and master the content of the teacher’s lecture16. Enlighten students’ thinking and improve their problem-solving skills17. Improve students’ ability to work together as a team18. Improvement of thinking abilityTable 3
Sample example.
Number12345678…18S1788071809098947474S27578879870827068…93S39186867480669370…83S49288678388667085…82S56790656996709779…97……………………………S1807190907583636281…85Table 4
Example of preprocessed sample data.
Number12345678...18S10.77550.79590.71040.79930.90050.98370.93630.74000.7407S20.75260.77820.87190.98390.70200.81890.70170.6786...0.9323S30.90620.85850.86200.73620.80240.65540.92570.7004...0.8341S40.91810.88370.66500.83410.87960.65970.69740.8464...0.8199S50.67470.90190.64760.68950.95640.70300.97170.7893...0.9669.................................S1800.71430.90290.90150.75220.82710.63030.62160.8123...0.8510


## 4.2. Experimental Discussion

Several comparison models were used to validate the performance of the method used to verify the feasibility of the teaching evaluation model used in this paper. The main comparison models are SVM [21], Radial Basis Neural Network (RBFNN) [22], Random Forest (RF) [23], Iterative Dichotomiser 3 (ID3) [24], and K-Nearest Neighbor (KNN) [25], and nine teaching experts were asked to score 180 teachers during the experiment, and each expert scored 40. The final teaching quality score of each teacher was given by taking the average value. And according to the scores, the grade where they were located was determined as A to E. 90–100 was A, 80–89 was B, 70–79 was C, 60–69 was D, and 0–59 was E. The classification accuracy was used as the evaluation index for the performance of each algorithm. Table 5 displays the experimental results. All of the experimental data are mean values obtained by running the algorithm ten times.Table 5
TQE results.
AlgorithmSVMRBFNNRFID3KNNFSVMAccuracy0.75440.90770.83190.81990.85800.9412Std0.14880.05040.12530.16070.09040.0560Time (ms)94.549.148.933.890.036.9According to the experimental results in Table5, the FSVM algorithm used in this paper yielded the best evaluation result of 0.9412, followed by RBFNN with a result of 0.9077. The results obtained by the three algorithms RF, ID3, and KNN are all between 0.8 and 0.9. The worst result obtained by SVM is only 0.7544. The reason why FSVM can achieve excellent experimental results is the introduction of fuzzy membership, which improves the classification accuracy of the samples. On the other hand, the experimental results plot shown in Figure 4 shows that the stability of the results obtained by each algorithm is not the same. Among them, ID3 and SVM have the highest volatility, which indicates that these two algorithms are the most unstable. The next best is KNN, and the best is RBFNN and FSVM. In summary, the FSVM used in this paper not only achieves the optimal teaching evaluation results, but also has stable performance.Figure 4
Comparison of experimental accuracy.Figure5 compares the time required to train each of the algorithms under consideration. It is clear from the figure that KNN and SVM take the longest time, followed by RF and RBFNN, followed by FSVM. The shortest time spent among the algorithms is ID3, but the difference between the time spent by FSVM and ID3 is very small, with a difference of 3.1 ms. Therefore, it can be said that the FSVM algorithm has an advantage in terms of model training time.Figure 5
Algorithm time consumption comparison.


## 5. Conclusion

TQE’s objectivity and accuracy are critical for teachers and schools. The traditional method of manual evaluation is no longer applicable to the present situation. More objective and accurate teacher TQE methods need to be generated urgently. As a result, this paper proposes a fuzzy support vector machine-based TQE method. A fuzzy mathematical model is established by the teacher teaching evaluation index system, and this fuzzy mathematical model is used to analyze the teacher teaching evaluation process. When developing the fuzzy mathematical model, a multilevel fuzzy comprehensive evaluation model that takes into account many different categories and levels of factors for the actual evaluation problem is required. In the fuzzy mathematical model for teacher teaching evaluation process analysis, the evaluation hierarchy is established, the factors affecting the evaluation results and their scores are determined, membership relationships are established, a fuzzy evaluation matrix is built, the evaluation levels are determined, and the evaluation results are derived. The results of the algorithm run verified that the used evaluation method can achieve the expected results. However, there are still some places in this study that need to be remembered not improved. One reason is that the data set is on a small scale. The feasibility of the algorithm can be verified by a larger data set in the future, and the evaluation system can be clarified to make it more reasonable and improve its applicability. Second, the dimensions of data collection can be more extensive, so that the data information is more comprehensive. Third, the importance of data of each dimension is different, so the introduction of weighting mechanism can be considered subsequently. The work of TQE itself cannot be fully supported based on data but can only be said to be as objective and fair as possible. We will also continue to explore in this field and propose intelligent teaching evaluation methods that can be applied in practice.



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*Source: 7354972-2022-07-06.xml*