## 3 Proposed Method
### Model Architecture
#### 3.2.1 Using CNN to Extract Spatial Features of Students' Learning Sequences  
Because CKT has significant advantages in modeling students' personalized learning interaction process [14], DKT-STDRL borrows the CKT model structure. Different from CKT, this component of the DKT-STDRL model needs only to extract the spatial features of students' learning sequences without predicting the performance of the next time step. The steps are as follows:  
1. Convert the input to an embedded matrix. In order to make the model better extract spatial features of students' learning process, CKT uses the embedding matrix to represent the given students' exercise history \(\mathbf{U}_{k}\). \(k\) is the max sequence length. Specifically, \(s_{t}\) is randomly initialized as an n-dimensional vector \(\mathbf{s}_{t}\) by embedding; \(n\) is far less than the total counts of knowledge points in the whole dataset \(M\). Then, according to whether the answer result is correct or not, the same n-dimensional zero vector is spliced on the right or left side of \(\mathbf{s}_{t}\), so that \(\mathbf{U}_{k}\) is transformed into a \(k\times 2n\) dimensional matrix through embedding [14];
2. Calculate the prior knowledge of different students. Because the learners' prior knowledge varies with each individual and can be reflected by students' past exercise performance and correct ratio on relevant skills [31], CKT obtains the prior knowledge of different students by calculating values for historical relevant performance (HRP) and concept-wise percent correct (CPC) [14];
3. Calculate the learning rate of different students. Because the learning efficiency of different students is different, and the continuous exercise performance of students over a period can reflect the learning efficiency of different students in different learning stages, CKT extracts the learning rate of different students through multi-layer CNN. Through the above process, the spatial learning features of students' learning sequences can be obtained. The process is visualized by the formula as follows:  
\[\mathbf{e}_{t}= \begin{cases}[\mathbf{s}_{t}\oplus\mathbf{0}],&\text{if }r_{t}=1\\ [\mathbf{0}\oplus\mathbf{s}_{t}],&\text{if }r_{t}=0\end{cases}  \tag{1}\]  
\[\mathbf{F}_{LIS}=(\mathbf{e}_{1},\mathbf{e}_{2},\ldots,\mathbf{e}_{k})^{\top} \tag{2}\]  
\[\left\{ \begin{array}{l}\mathbf{relation}(j)=\text{ Masking }(\mathbf{s}_{j}\cdot\mathbf{s}_{t}),j \in(t,k)\\ \mathbf{weight}_{t}(j)=\text{ Softmax }(\mathbf{relation}_{t}(j)),j\in(1,k)\end{array} \right. \tag{3}\]  
\[\mathbf{F}_{HRP}(t)=\sum_{j=1}^{t-1}\mathbf{weight}_{t}(j)\mathbf{e}_{j} \tag{4}\]  
\[\mathbf{F}_{CPC_{l}}(m)=\frac{\sum_{j=0}^{t-1}r_{j}^{m}==1}{\text{total}(s^{m})} \tag{5}\]  
\[\left\{ \begin{array}{l}\mathbf{F}_{\text{\tiny{IPF}}}=\mathbf{F}_{LIS}\oplus\mathbf{ F}_{HRP}\oplus\mathbf{F}_{CPC}\\ \mathbf{F}_{ILA}=\left(\mathbf{F}_{\text{\tiny{IPF}}}*\mathbf{W}_{3}+\mathbf{b}_{3} \right)\otimes\sigma\big{(}\mathbf{F}_{\text{\tiny{IPF}}}*\mathbf{W}_{4}+\mathbf{b}_{4} \big{)}\\ \end{array} \right. \tag{6}\]  
\[\mathbf{F}_{KS}=\left\{GLU(\text{Conv }1d(\mathbf{F}_{ILA}))\right\}_{3} \tag{7}\]where \(\mathbf{F}_{LIS}\) is the embedded matrix expression of the learning interaction sequence \(\mathbf{U}_{k}\). _Masking_ is an operation used to exclude subsequences. \(\mathbf{weight}\) is used to estimate the resemblance between the present question and the previous question, and to then comprehensively analyze the impact of previous exercise performance on the current exercise. \(\mathbf{F}_{HRP}\) is a historically related performance feature. \(m\) is the number of the knowledge point. \(\mathbf{F}_{CPC_{t}}(m)\) indicates the correct rate when the student answers the question of a knowledge point at a certain time. \(\mathbf{F}_{JPF}\) is the prior knowledge feature obtained by connecting \(\mathbf{F}_{LIS}\), \(\mathbf{F}_{HRP}\), and \(\mathbf{F}_{CPC}\).