# Online Blended Learning in Small Private Online Course
## 5 Online Blended Learning Based on Network Experiment Courses
### Grade Aggregation Model Based on Ability Assessment  
(1) Problems in peer grading  
In the grade aggregation model, we must consider the reliability of the grade in order to improve the accuracy of grade aggregation. However, in the SPOC environment, the learners know each other and can communicate arbitrarily, and there is often collusion when reviewing submissions, meaning the law of large numbers cannot be used to model reliability characteristics. Therefore, accurately modeling the reliability characteristics of the graders is an important problem we have to solve. (2) Personalized ability estimation model  
We assess student abilities using the historical data of their watching videos and performing exercises as the basis for a reliability evaluation in order to establish a two-stage personalized ability evaluation model. First, we extract some characteristic values related to the students when they watch the video, including the total time for which the students watch the video, the number of times the video is forwarded, the number of videos watched, the length of forum posts, the total length of videos watched, the number of pauses, and the number of rewinds. Then we use the logistic regression method to fit the above features to the equation representative of the student's initial ability value. First, we define Equation (1):  
\[\mathrm{p(y=1\left|X;W\right\rangle=g\Big{(}w^{T}X\Big{)}=\frac{1}{1+e^{-W^{T}X}} \tag{1}\]  
Here, \(W\) represents the weight of each feature, and \(X\) represents the value of each feature, \(y\in\{0,1\}\) represents whether the feature exists. The maximum likelihood estimation method can be used to solve the parameter \(W\), the iteration process is Equation (2), and the \(\alpha\) is the expansion factor,  
\[\mathrm{WW-\alpha\sum_{i=1}^{n}\Big{(}y_{i}-g\Big{(}w^{T}X_{i}\Big{)}\Big{)}X_ {i}} \tag{2}\]  
In the second stage, we enter each student's answer sequence into the knowledge tracking model to evaluate their learning. As shown in Figure 7, the ability of each student is predicted using the output of the first stage as the S-node training personalized knowledge tracking model. In Figure 7, node S is a student node, node K is a knowledge level node, and node Q is a question answer node. \(\mathrm{P(}L_{0}\)) represents the initial probability of knowledge mastery, \(\mathrm{P(G)}\) represents the probability of a correct guess, \(\mathrm{P(S)}\) represents the probability of error, and \(\mathrm{P(T|S)}\) represents the probability of personalized learning. (3) Grade aggregation model  
In this work, we remodeled its reliability characteristics based on the ability of the graders. The first model we designed here is called PG6. Here, the reliability variable \(\tau_{v}\) follows a gamma distribution, wherein the hyperparameters \(\beta_{0}\), \(\eta_{0}\), \(\mu_{0}\) and \(\gamma_{0}\) are empirical values, and \(a_{v}\) is the shape parameter, the value of which represents the output result of the two-stage model we designed. \[\tau_{v}\sim\mathcal{G}(a_{v},\ \beta_{0}) \tag{3}\]  
\[b_{v}\sim\mathcal{N}(0,1/\eta_{0}) \tag{4}\]  
\[s_{u}\sim\mathcal{N}(\mu_{0},1/\gamma_{0}) \tag{5}\]  
\[z_{u}^{v}\sim\mathcal{N}(s_{u}+b_{v},1/\tau_{v}) \tag{6}\]  
where  
\(a_{v}\) represents the previous evaluation ability of each student grader (v). \(\tau_{v}\) represents the reliability distribution of the grader v.  
\(b_{v}\) represents the deviation of the grader v.  
\(s_{u}\) represents the true grade of submission (u). \(z_{u}^{v}\) represents the grade given by the grader v to the submission u.  
Figure 7: KT model of personalized learning probability. We also designed the PG7 model, which is similar to the PG6 model and in which the reliability variable \(\tau_{v}\) follows a Gaussian distribution. The mean parameter in the Gaussian distribution is \(a_{v}\). \[\tau_{v}\,\sim\,\mathcal{N}(a_{v},\,1/\beta_{0}) \tag{7}\]  
High-scoring students are believed to perform peer grading tasks more reliably.