A Study on the Optimisation of Personalised English Learning Paths Based on Big Data Analysis and Its Impact on Teaching Efficiency

The sparsity measure is shown in equation (1): (1) Sparsity=1−rm×n $$Sparsity = 1 - \frac{r}{{m \times n}}$$

Equation (2) was used to calculate the user’s predictive score:: (2) P(u,i)=∑jeLsimc(i,j)×R(u,j)|L| $$P(u,i) = \frac{{\sum\limits_{jeL} s i{m_c}(i,j) \times R(u,j)}}{{|L|}}$$

The core of the user-based collaborative filtering algorithm (UCF) is to find the set of similar neighboring users by calculating the user similarity and finding the set of similar neighboring users according to the similarity level [21]. The specific implementation steps are as follows: 1)

Construct a user similarity matrix. The Pearson correlation coefficient is utilized to calculate the similarity between users and users, with full consideration of the user’s scoring habits, and the calculation formula is: (3) simuser(u,v)=∑i∈Bvv(Ru,i−Ru¯)⋅(Rv,i−Rv¯)∑i∈Bvv(Ru,i−Ru¯)2∑i∈Bvv(Rv,i−Rv¯)2 $$si{m_{user}}(u,v) = \frac{{\sum\limits_{i \in {B_{vv}}} {({R_{u,i}} - \overline {{R_u}} )} \cdot ({R_{v,i}} - \overline {{R_v}} )}}{{\sqrt {\sum\limits_{i \in {B_{vv}}} {{{({R_{u,i}} - \overline {{R_u}} )}^2}} } \sqrt {\sum\limits_{i \in {B_{vv}}} {{{({R_{v,i}} - \overline {{R_v}} )}^2}} } }}$$

The steps to implement the item-based collaborative filtering algorithm (ICF) are as follows: 1)

Construct the item similarity matrix. In this paper, we use the optimized scoring matrix to calculate the cosine similarity between items and items, which is calculated as: (5) simbook(i,j)=∑u=1mRu,i×Ru,j∑u=1m(Ru,i)2×∑u=1m(Ru,j)2 $$si{m_{book}}(i,j) = \frac{{\sum\limits_{u = 1}^m {{R_{u,i}}} \times {R_{u,j}}}}{{\sqrt {\sum\limits_{u = 1}^m {{{({R_{u,i}})}^2}} } \times \sqrt {\sum\limits_{u = 1}^m {{{({R_{u,j}})}^2}} } }}$$

No recommendation algorithm can be applied to all datasets. UCF, which calculates user similarity to make recommendations, has higher accuracy in datasets where the number of users is larger than the number of items; on the contrary, ICF is more suitable for datasets where the number of items is larger than the number of users. The more common learning path optimization generally adopts hybrid collaborative filtering recommendation (UCF-ICF), which fuses the predictive scores of user-based collaborative filtering algorithm and item-based collaborative filtering algorithm by introducing the weighting coefficients β and then unfolds the final recommendation based on the fused predictive scores, and the formula computes the fused predictive scores: (7) P(u,i)=βPuser(u,i)+(1−β)Pbook(u,i) $$P(u,i) = \beta {P_{user}}(u,i) + (1 - \beta ){P_{book}}(u,i)$$

By using existing learner data and cognitive characteristics, it aids in inferring the learning preferences and needs of new learners, thus enabling the generation of personalized learning paths and improving the applicability and universality of the system. The schematic diagram of learner portrait modeling is shown in Figure 1.

Figure 1.

Schematic diagram of learner portrait modeling

1)

Quantitative representation based on Felder-Silverman learning styles

Learning style refers to the preferences, habits and tendencies shown by individuals in the learning process, as well as their attitudes and behavioral approaches to learning activities. Learning styles cover the characteristics of how individuals approach learning tasks, how they acquire and process information, and how they choose and utilize learning environments.

The different learning styles are specifically active, perceptive, visual and sequential. The behavioral patterns to the right of the learning styles are contemplative, intuitive, verbal, and integrative. For each learner, the sequence of learning behaviors can be represented as x₁, x₂, ⋯, x_n, where x_i represents the i th learning behavior pattern. Learning styles can be represented as S₁, S₂, ⋯, S_n, where S_i represents the i th learning behavior pattern. The quantitative Kan values of the patterns can be divided into three levels, which are expressed as low L, medium M, and high H. According to the above definition, the three levels are satisfied as shown in Equation (8): (8) maxSi(f(x)i|xi(L),xi(M),xi(H))=12πσe−(a−μ)22σ2∑i=1n∫LHf(x)idx $$\max {S_i}(f{(x)_i}|{x_i}(L),{x_i}(M),{x_i}(H)) = \frac{{\frac{1}{{\sqrt {2\pi \sigma } }}{e^{ - \frac{{{{(a - \mu )}^2}}}{{2{\sigma ^2}}}}}}}{{\sum\limits_{i = 1}^n {\int_L^H f } {{(x)}_i}\:dx}}$$

Where the column vector α, γ represents the eigenvalues in the dimensions of basic attributes and learning outcomes, respectively, and β is composed of the eigenvalues λ₁, λ₂ of the two-dimensional matrix consisting of its matrix learning features β₁ and learning overhead β₂, and satisfies Equation (10): (10) {β1=[Are,Sty]⊤β2=[Exp,Tim]⊤ $$\left\{ {\begin{array}{*{20}{l}} {{\beta _1} = {{\left[ {Are,Sty} \right]}^ \top }} \\ {{\beta _2} = {{\left[ {Exp,Tim} \right]}^ \top }} \end{array}} \right.$$

Then, in this paper, take the use of the K-means algorithm sequentially for each ID in the learner collection corresponding to the eigenvalues to find its spatial distance, calculate its maximum similarity, and respectively, the basic vectors in the above as the center vector to obtain the center clusters. Such as age. Age as the clustering center vector in the learner collection library Stu to get the cluster vector can be expressed as equation (11): (11) NAge={ID1(α1⊤,β1⊤,γ1⊤),ID2(α2⊤,β2⊤,γ2⊤),⋯,IDn} $${N_{Age}} = \left\{ {\begin{array}{*{20}{c}} {I{D_1}\left( {\alpha _1^ \top ,\beta _1^ \top ,\gamma _1^ \top } \right),I{D_2}\left( {\alpha _2^ \top ,\beta _2^ \top ,\gamma _2^ \top } \right), \cdots ,I{D_n}} \end{array}} \right\}$$

In the K-means algorithm, the distance between two eigenvalues obtained at the end is usually calculated using the Euclidean distance. Euclidean distance is one of the most commonly used distance measures for continuous eigenvalues in the feature space. The Euclidean distance d between two eigenvalues x and y can be expressed as equation (12): (12) d(x,y)=∑i=1n(xi−yi)2 $$d(x,y) = \sqrt {\sum\limits_{i = 1}^n {{{\left( {{x_i} - {y_i}} \right)}^2}} }$$

The core idea of collaborative filtering is to discover the correlation between learners and knowledge points based on learner behavior data, such as learners’ behaviors of studying, reviewing, and quizzing on knowledge points, in order to make personalized recommendations. The key to collaborative filtering is to use user behavior data to discover the correlation between users and items without explicitly characterizing users or items beforehand. 1)

Similarity calculation

The first layer represents three different learners, the second layer is five different unlearned knowledge points, the third layer is the knowledge point fusion coding layer, and the fourth layer is the learner relationship embedding layer [22]. In the knowledge point fusion coding layer, this paper calculates the fusion coding of knowledge points by weighted summation of all learner embedding vectors and knowledge point embedding vectors, where the weighted summation is used in order to take into account the different degrees of importance of different learners for the same knowledge point. Specifically, Equation (13) can be used to calculate the fusion coding of knowledge points: (13) Fj=∑i=1mαij⋅σ(Ui⋅Pj⊤) $${F_j} = \sum\limits_{i = 1}^m {{\alpha _{ij}}} \cdot \sigma ({U_i} \cdot P_j^ \top )$$

Where F_j denotes the fusion coding of the knowledge point j , α_ij is the weight of the learner i on the knowledge point j , U_i is the embedding vector of the learner i and P_j is the embedding vector of the knowledge point j . The formula represents the fusion of the information of knowledge points through the learner’s embedding and the corresponding weights to get a comprehensive fusion coding of knowledge points. The formula σ(x)=11+e−x $$\sigma (x) = \frac{1}{{1 + {e^{ - x}}}}$$ represents the fusion of the information of knowledge points through the learner’s embedding and the corresponding weights to get a comprehensive fusion coding of knowledge points.

After fusing the knowledge points in the fourth layer to encode the learner vectors, this paper uses a similarity matrix to represent the similarity between the learners and adjusts the encoding of the target learner by weighted summing the encodings of the similar learners, where S_ij denotes the similarity between Learner i and Learner j. Then, this similarity matrix is used to compute the new encoding of the learner vectors as in (14): (14) Ui′=∑j=1mSij⋅Uj+∑k=1nγik⋅Fk $${U'_i} = \sum\limits_{j = 1}^m {{S_{ij}}} \cdot {U_j} + \sum\limits_{k = 1}^n {{\gamma _{ik}}} \cdot {F_k}$$ 2)

Generation of Predictive Rating Matrix

In the field of education and teaching, collaborative filtering-based methods can be applied to predict learners’ ratings of unevaluated knowledge points in order to generate a personalized list of knowledge point recommendations.

Suppose, the sequence of learner feature vectors can be denoted as U₁, U₂, ⋯, U_N and the sequence of knowledge point feature vectors can be denoted as P₁, P₂, ⋯, P_N. Then learner i rating of knowledge point j can be predicted by the similarity between learner i feature vector U_i and knowledge point j feature vector P_j. This can be represented by equation (15): (15) r^ij=(∑k=1NUik⋅Pjk⊤+b)⋅E $${\hat r_{ij}} = \left( {\sum\limits_{k = 1}^N {{U_{ik}}} \cdot P_{jk}^ \top + b} \right) \cdot E$$

Where b is the bias term and E is the unit matrix that recommends only the most suitable knowledge points to the learner each time.

The cognitive level diagnostic layer focuses on analyzing and decomposing the eigenvalues of learners and knowledge points to assess learners’ mastery of specific aspects of different knowledge points [23]. This part is divided into two main parts: static parameter estimation and dynamic data analysis.

The static parameter estimation component allows for a fine-grained assessment of the student’s level of competence in each skill and permits the consideration of interactions between the requisites. On the other hand, the dynamic data analysis part compensates for the lack of consideration of the dynamics of learning by using the feature interaction matrix as parameter input and the learner’s personalized learning path as the output of the model training. 1)

Static parameter estimation

In the static parameter estimation part, U_ik denotes the learner’s i response to the test k, and P_kj denotes the test k ’s examination of the knowledge point j, so the learner i ’s mastery of the knowledge point j can be expressed as a score matrix y_ij , as in Equation (16):

Knowledge point or not. Assuming that the topic j belongs to the knowledge point , then the elements in the score matrix y_ij can be defined as a binary variable indicating whether the student S_ij has mastered the knowledge point j or not, as in Equation (17): (17) Sij={1,if∑kUik×Pkj>bij0,otherwise $${S_{ij}} = \left\{ {\begin{array}{*{20}{l}} {1,}&{if\:\sum\limits_k {{U_{ik}}} \times {P_{kj}} > {b_{ij}}} \\ {0,}&{otherwise} \end{array}} \right.$$

The static parameter estimation feature parameter b_ij is introduced here as the a priori information for the dynamic data analysis, which is defined as Equation (18): (18) bij=bi+bj=1J×∑j=1JSij+1I×∑i=1ISij $${b_{ij}} = {b_i} + {b_j} = \frac{1}{J} \times \sum\limits_{j = 1}^J {{S_{ij}}} + \frac{1}{I} \times \sum\limits_{i = 1}^I {{S_{ij}}}$$

Therefore, it is only necessary to filter out the first q knowledge points where the proportion of knowledge points mastered by that learner is less than the proportion of knowledge points mastered by all learners, i.e., the set of explicitly weak knowledge points of that learner, which can be expressed as Equation (19): (19) Xij(i)={j:Sij=1,rank(bi∑i=1ISij)<1q⋅∑i=1qSij} $${X_{ij}}\left( i \right) = \left\{ {j:{S_{ij}} = 1,rank\left( {\frac{{{b_i}}}{{\sum\limits_{i = 1}^I {{S_{ij}}} }}} \right) < \frac{1}{q} \cdot \sum\limits_{i = 1}^q {{S_{ij}}} } \right\}$$ 2)

Dynamic parameter analysis

In the process of dynamic learning data analysis, the learner-knowledge point feature vector interaction matrix output from the neural collaborative filtering layer is weighted and averaged for each of its components in turn to obtain the interaction matrix, which can be expressed as Equation (20):

Next, this paper performs an SVD decomposition of the interaction matrix R_ij because each learner is associated with a vector of real values of potential features of the knowledge point, modeling the bidirectional interaction between the learner and the potential factors of the knowledge point, and each dimension in the potential space is independent of each other, so that the same weights can be used to linearly combine them. Thus, the learner low-dimensional feature projection matrix R_i and the learner behavioral data latent feature matrix R_j can be expressed as Equation (21): (21) Rij=∑k=1KRik⋅Rjk+ϵ $${{R}_{ij}}=\sum\limits_{k=1}^{K}{{{R}_{ik}}}\cdot {{R}_{jk}}+\epsilon$$

Thus, the set of implicitly weak knowledge points Z_ij can be represented as a quintuple, satisfying equation (22): (22) Zij(i)={j:maxθ∑(i,j)∈ℝlogp(zi,j|bi,j,Ri,Rj,σR,θk)} $${Z_{ij}}(i) = \left\{ {j:{{\max }_\theta }\sum\limits_{(i,j) \in \mathbb{R}} {\log } p\left( {{z_{i,j}}\:|{b_{i,j}},{R_i},{R_j},{\sigma _R},{\theta _k}} \right)} \right\}$$

Based on the above analysis, the basic characteristic parameter information of this learner has been obtained through the neural synergistic filtering layer and the cognitive level diagnostic layer, so the fuzzy cognitive diagnostic result of the learner under the model can be expressed as Equation (23) (23) yui^=ρ⋅Xij+(1−ρ)⋅Zij $$\widehat {{y_{ui}}} = \rho \cdot {X_{ij}} + (1 - \rho ) \cdot {Z_{ij}}$$

Since X_ij is a static covariate that remains almost unchanged during the learning session interaction recording, only the dynamic parameter estimation part, i.e., Z_ij, needs to be updated and self-repairing. its final objective function can be expressed as Eq. (24): (24) L=12∑(i,j)∈Rij(Rij−β1⊤β2)2+λ12‖β1‖2+λ22‖β2‖2 $$\mathcal{L} = \frac{1}{2}\sum\limits_{(i,j) \in {R_{ij}}} {{{({R_{ij}} - \beta _1^ \top {\beta _2})}^2}} + \frac{{{\lambda _1}}}{2}{\left\| {{\beta _1}} \right\|^2} + \frac{{{\lambda _2}}}{2}{\left\| {{\beta _2}} \right\|^2}$$

Where λ₁ and λ₂ are the regularization parameters. Using stochastic gradient descent, the learner feature matrix β₁ and the knowledge point feature matrix β₂ can be updated. Specifically, for each (i, j), the updating formula is as in (25), (26): (25) β1i↔β1i+α(eijβ2j−λ1β1i) $${\beta _{1i}} \leftrightarrow {\beta _{1i}} + \alpha ({e_{ij}}{\beta _{2j}} - {\lambda _1}{\beta _{1i}})$$ (26) β2j↔β2j+α(eijβ1i−λ2β2j) $${\beta _{2j}} \leftrightarrow {\beta _{2j}} + \alpha ({e_{ij}}{\beta _{1i}} - {\lambda _2}{\beta _{2j}})$$

The gradient descent method is also used to maximize the log-likelihood function to achieve the update of model parameters. For each model parameter θ_k, its gradient can be calculated by back propagation algorithm as in equation (27): (27) ∂Z∂θk=∂Z∂p^∂p^∂θk $$\frac{{\partial Z}}{{\partial {\theta _k}}}\: = \:\frac{{\partial Z}}{{\partial \hat p}}\:\frac{{\partial \hat p}}{{\partial {\theta _k}}}$$

Where Z is the likelihood function in Eq. (22), p^ $$\hat p$$ is the probability predicted by the model, and θ_k is the k h model parameter. The backpropagation algorithm can recursively compute ∂Z∂p^ $$\frac{{\partial Z}}{{\partial \hat p}}$$ and ∂p^∂θk $$\frac{{\partial \hat p}}{{\partial {\theta _k}}}$$ and multiply them by the chain rule to obtain ∂Z∂θk $$\frac{\partial }{Z}\partial {\theta _k}$$ .

The model parameters can then be updated based on the gradient using gradient descent as in Eq. (28): (28) θk=θk−η∂L∂θk $${\theta _k} = {\theta _k} - \eta \frac{{\partial L}}{{\partial {\theta _k}}}$$

Where η is the learning rate and controls the size of the update step. By continuously iterating the above update process, the optimal model parameters can be obtained.

1)

Data sets

In order to verify the generality and generalization ability of the model, public datasets are used for model performance comparison experiments, with data specifications of 100K and 1M, respectively.

The experimental results of the performance comparison of each model on the public datasets are shown in Fig. 2, where (a) and (b) are the experimental results on the ML-100K dataset and ML-1M dataset, respectively. On both ML-100K and ML-1M datasets, it can be seen that the model of this paper outperforms the comparison model in both HR@10 and NDCG@10 performance results.

The model in this paper achieves the best NDCG@K metric performance. The NDCG@K scores on the two datasets are 0.352 and 0.465, respectively. The OPB-MLP model on the ML-100K dataset has the next best results. The fact that OPB-MLP outperforms DCRM-A is just a side-effect of the importance of the outer product for recommendation performance improvement, and the reason why OPB-MLP is not as good as the model in this paper may lie in the lack of the model’s ability to capture nonlinearities of the interaction function due to the use of the MLP. It can be demonstrated that the improved convolutional architecture is used in the process of expressing predictive neural networks in the expression prediction process. However, on the ML-1M dataset, DCRM-A outperforms the OPB-MLP model probably because a large amount of data is more conducive to the extraction of information by the DCRM-A model, and DR-CNN works best because of the improvements made on the basis of DCRM-A. In addition, the DCRM-A model outperforms GMF and MLP on both datasets because DCRM-A combines the advantages of both and introduces an attention mechanism and feedback information.

Figure 2.

Different recommendation algorithms are compared in each data set

The change of experimental parameters has an important impact on the model performance, so that the experimental analysis will be carried out next for the main parameters. The main focus is to analyze the effect of changes in K value and the number of feature maps on experimental results. Firstly, the results of different Top-K experiments are analyzed.

Top-K usually refers to selecting the top K items with the highest ratings from all possible recommendation items as the user’s recommendation result. In order to fully validate the performance of the collaborative filtering model after fusion of neural networks, therefore, experimental comparisons will be conducted next on the larger dataset, ML-1M dataset, for HR@K and NDCG@K under different K values. Figure 3 shows the comparison results, where (a) and (b) are the HR@K and NDCG@K metrics, respectively.

It can be observed that the length of the recommendation list has an impact on the accuracy of the model prediction, and as the recommendation list length increases, the recommendation performance of both models improves. Secondly, it is easy to conclude from the figure that this paper’s model is significantly better than the mainstream hybrid collaborative filtering recommendation algorithm (UCF-ICF), and the HR@K and NDCG@K indexes of this paper’s model when the list length is 10 are 0.656 and 0.402, respectively, which are higher than those of the traditional hybrid collaborative filtering algorithm (0.625 and 0.371). It is further demonstrated that the proposed collaborative filtering algorithm using neural networks to enhance the performance of this chapter helps to better model user and item characteristics, and improve the quality of system recommendations. Algorithm performance helps to better model user and item characteristics and improve the quality of system recommendations.

Figure 3.

Different types of performance comparison diagrams

Then, the effect of the number of feature maps on the model performance is verified. Experiments are conducted on the larger dataset ML-1M dataset for different numbers of feature maps to verify the performance of the model in this paper. Fig. 4 shows the results of the analysis of the effect of the change of feature maps on the HR@K and NDCG@K of the model under different epochs; (a) and (b) are the HR@K and NDCG@K metrics, respectively.

All the curves are growing steadily, and although there are some slight differences in the convergence curves with the increase in the number of training times, similar performance is finally achieved, which indicates that increasing the number of neural network parameters under the model framework of this paper does not lead to overfitting.

Figure 4.

Comparison of characteristics of different quantitative characteristics

The purpose of designing a comparative experiment of precision teaching intervention is to apply the designed precision teaching intervention based on learner profiling to the actual teaching process and to test the effect of this paper’s Learning Path Optimization Model intervention strategy on the enhancement of students’ English learning.

Two classes taught by the same teacher were selected as the experimental class and the control class, both with 50 students. The experimental class used this model to optimize students’ personalized learning paths in the course learning arrangement. The control class was taught using the traditional teaching model. Other than that, the teaching conditions of both classes are the same.

According to the curriculum standard and the requirements of the teaching and research group, a simple understanding of the unit grammar knowledge, focusing on the learning of words, phrases, and parts of speech content, the experiment focused on the teaching of English reading. A teaching quality test was conducted before the beginning of the semester and after the end of the semester as the pre-test and post-test of the experiment.

The pre- and post-intervention scores of the experimental and control classes were analyzed descriptively, and the results are shown in Table 1. The mean pre-test scores of the experimental class were 56.36 points, and the mean post-test scores were 82.19 points, an improvement of 25.83 points. The pre-test and post-test scores of the control class were 56.94 and 58.43, respectively, and the difference between the pre-test and post-test was extremely small, only 1.49 points. It can be seen that the difference between the pre-test scores of the experimental class and the control class is not large, but the experimental class’s learning achievement progress is significantly larger than that of the control class. It can be preliminarily concluded that the learning path optimization model designed in this paper has a great positive impact on students’ English learning.

Figure 5 shows the distribution of grades before and after the experimental and control classes. It can be seen that the final grade of the experimental class has significantly improved compared to the pre-semester, and students in all segments have made significant progress. There were no students above 80 points in the pre-test, while there were as many as 11 students above 90 points at the end of the semester. There is no significant trend of student movement in all segments of the control class.

Figure 5.

The experimental class and the comparison of the grade are distributed

In summary, it can be concluded that the learning path optimization model designed in this paper realizes the accurate judgment of learners’ cognitive characteristics and dynamic learning behavior state and can design scientific and reasonable learning paths on the basis of this in order to help improve the efficiency of teaching and learning effects.