Strategies for Sharing and Utilizing Internet-Based Curriculum Resources in Teaching Higher Mathematics

The construction of a teaching resource base for higher mathematics aims to create a resource-rich, up-to-date, and interactive environment for online teaching and learning. From the higher mathematics teaching resource management and full utilization, to the cooperative learning between teachers and students online, and then to the students’ own knowledge reconstruction of the three processes of the construction of higher mathematics teaching resource library, higher mathematics teaching resource library can be regarded as a platform for refining and integrating the knowledge and teaching materials [21]. The structural framework for the realization of a higher mathematics teaching resource base is shown in Figure 1.

Figure 1.

Teaching repository framework

This Advanced Mathematics Teaching Resource Library mainly consists of four major functional submodules: shared resource library sub-module, fine course management sub-module, network teaching management sub-module, and system management sub-module. Each sub-module consists of several functional modules, thus implementing all the functional modules of the higher mathematics teaching resource library. The functional composition of the teaching resource library is shown in Figure 2. Shared Resource Library Submodule: used to store uploaded media resources, animation resources, courseware resources, and facilitate resource retrieval.

Figure 2.

The function of the teaching resource library is composed

The media resources mainly contain course video, external video, streaming video materials, etc. The file formats are: AVI, RMVB, ASF, FLV and other network streaming media formats. It can be nested in the webpage and downloaded while playing. Animation resources include the demonstration animation of a knowledge point in an advanced mathematics course, the demonstration animation of practical training guidance, and the document converted by the Falshpaper tool, etc. The file formats are: SWF, GIF and other common animation file formats. Courseware resources include courseware commonly used in teaching, file format: PPT, PPTX and other slide format files. Resource retrieval is a must-have function for a resource library with a huge amount of data. It allows users to easily and quickly find the resources they need. Fine Courses Module: It is used to store and manage the national, provincial and school-level fine courses over the years, and at the same time, it can also release the existing courses on the platform into fine courses through evaluation. Fine course template function, according to all kinds of fine course evaluation standards, the columns of fine courses into a template, teachers in the release of fine courses in accordance with the template columns to fill in the content can be. Fine Courses management function, the existing fine courses for editing, deleting, adding and modifying the content, etc., at the same time, part of the external link “China’s Fine Courses Network” resources for teachers and students to study for reference. Fine course evaluation function, teachers and student users can evaluate and score a course on the platform according to the fine course evaluation standard, and the administrator will decide whether to release it as a fine course according to the evaluation. At the same time, the teaching administration can also organize experts to evaluate the courses on the platform. Network Teaching Module: Including the creation and management of internal and external courses, which are used for teaching and students’ daily learning. Internal course management: Teachers can directly create courses on the platform through this function, and upload teaching plans, teaching logs, student lists, lesson plans, courseware, classroom exercises, extracurricular assignments, videos of the teaching process, etc. Meanwhile, they can delete the original courses and edit the contents of the original courses. External course management is similar to internal courses, but the difference is that the resources of internal courses are directly uploaded to the platform’s shared resource library, while external course resources directly call external resources via URL. Interactive module: Teachers can create polls, forums, and Q&A within the course through this function to fully interact with students. System management module: including specialty management, class management, user management, and system maintenance. Specialties, classes and users all belong to the basic data. Specialties and classes are all the specialties and class users opened by the school, including student users, teacher users and administrators at all levels. The basic data can either be imported directly from the original teaching management information system or you can become an official user of the platform by registering with the basic data administrator. System maintenance function, the system administrator’s daily maintenance of the system. Including analysis and statistics of the system’s operation status, as well as analysis and statistics of the operation log. The functional relationship of each submodule is not independent, but interconnected and interdependent. For example, if an internal course is created in the online teaching module, its uploaded resources will be stored in the shared resources module. Teachers use the Interactive module to teach using the Fine Mathematics program.

The weight calculation part is mainly to calculate the correlation weights between exercises and related knowledge points. Firstly, we need to calculate the embedding of the input information, the input exercise q with the number of times to do the problem attempt_count, the request hint first _ action and the difficulty of the exercise diffculty, the type of exercise type through the embedding matrix to obtain the embedding vectors q_t, count_t, hint_t, diff_t, and type_t, respectively. q_t and these features in different combinations of the feature splicing to construct a combination of features.

The influence of different features on students’ answer results varies from person to person, and the attention mechanism can assign corresponding weights to the elements according to their correlation. Therefore, this section applies the attention mechanism to calculate the importance degree of each feature of the input model, and then assigns weights to them, realizing the purpose of measuring the importance degree of different features by focusing the attention on the features that have a greater impact on the current students’ answering results. The feature vectors are first integrated into the feature vector matrix U = (u₁, u₂,⋯, u_n). The query, key and value matrices W^Q, W^K and W^V are defined, and the feature vector U is linearly transformed by W^Q, W^K and W^V to obtain the corresponding query, key and value vectors, i.e., Q, K and V, respectively. 1Q=UWQ 2K=UWK 3V=UWV

Then the vectors corresponding to Q and K are dot-producted and normalized by Softmax layer and multiplied with the corresponding V to obtain the attention weight, as shown in Eq. (4). 4Attention(Q,K,V)=∑ Softmax(QKTd)V

After the above process, the different features obtain the corresponding attention weights, and the combined feature c_t is obtained by calculating the weighted sum of all feature vectors. Then the inner product is calculated for each memory slot of the c_t and M^k matrices and the result of the inner product is passed through the Softmax activation function to obtain the correlation weight w_t, which represents the correlation between the exercises and the potential knowledge concepts in the M^k matrix as shown in Equation (5). Compared with the original DKVMN, this model takes into account not only the exercise labels but also the effects of multiple features in the input. 5wt(i)=Softmax(ctMk(i))

For each exercise, the weighted sum r_t of the knowledge states related to the exercise in the M^v matrix is calculated by the relevant weights w_t, which is used to represent the overall mastery level of students on the current exercise, as shown in the following equation. 6rt=∑i=1Nwt(i)Mtv(i)

The amount of knowledge memorized by students is not fixed. Considering that every learner experiences forgetting after learning, the degree of memory retention R_t is introduced into the model, and r_t is denoted as the degree of mastery that follows the forgetting law. Specifically, the reading vector re_t based on the forgetting law is obtained by multiplying the corresponding positional elements of readings r_t and R_t.

Then considering the influence of students’ learning behaviors and exercise characteristics on students’ answer results, re_t is spliced with the combined feature vector c_t, and the summary vector f_t is obtained through a fully connected layer with Tanh an activation function, as shown in Equation (7). In contrast, the model in this chapter f contains more information, including students’ knowledge status, behavioral characteristics, and exercise characteristics. 7ft=Tanh(w1T[ ret,ct ]+b1)

Finally, in order to predict whether the student can answer the current exercise correctly, f_i is passed through another fully connected layer and activated using the Sigmoid function, which in turn obtains the probability p_t that the student will answer Exercise q correctly, as shown in the following equation. p_t represents the performance of the student in answering the question, and if p_t is higher than 0.5 then the student is considered to have answered the question correctly. 8pt=Sigmoid(w2Tft+b2)

After the students answer the exercises, the writing process realizes the dynamic update of the students’ knowledge status in the M^v matrix according to the students’ answer results. The time used by the student to answer the question can reflect the student’s mastery of the knowledge concept to a certain extent, so in the process of updating the knowledge state matrix, the duration of answering the question is taken into account ms_first_response. Firstly, the combination of exercise and answer results feature (q_t, y_t) is passed through the embedding matrix A to get the embedding vector ar_t, and at the same time, the discretized time is passed through the embedding matrix B to get the embedding vector time_t, and then y_t and time_t are concatenated to v_t for updating the student’s knowledge status.

According to the working principle of bi-directional long and short-term memory network, the knowledge state update process consists of two parts: erasure and update. Before updating the value matrix, it is necessary to delete the invalid content in the matrix, i.e., to remove the information of the long interval exercises in the matrix through the erasure vector e_t, and at the same time to update the memory value of the current time M^v matrix according to e_t and the related weights, and the related equations are shown as follows. 9et=Sigmoid(ETvt+be) 10M¯tv(i)=Mt−1v(i)[ 1−wt(i)et ]

The new exercise information is then added to the VALUE matrix by adding vector a_t and updating the memory values of the VALUE matrix according to a_t with the associated weights as shown in Eqs. (11) and (12). 11at=Tanh(DTvt+ba)T 12Mtv(i)=M¯t−1v(i)+wt(i)at

In the process of training the DKVMN-FMF model, the training method of the original model is still used to train each parameter and embedding matrix, etc. by minimizing the standard cross-entropy loss between the predicted value and the true value, i.e., between p_t and r_t, as shown in Equation (13). 13L=−∑t(rtlogpt+(1−rt)log(1−pt))

In the personalized knowledge tracking model, this section introduces three time-dependent features to augment the deep knowledge tracking model to address learners’ forgetfulness during the learning process. In addition, this section uses the learner’s personality to model the differences between learners to more accurately capture the learner’s dynamically evolving knowledge level. After obtaining the predicted knowledge level y^aux of the learner, we further use it as part of the learner portrait [22–23]. We use an attentional mechanism to adaptively fuse the learner’s knowledge level y^aux with the learner’s sequential behavior u_b and the learner’s personality representation u_p based on the contextual information to generate the learner’s portrait U. The goal of this section is to generate a course recommendation score y^pri from the learner’s portrait U and course description C, and to generate a list of TOP-N course recommendations based on the ranked recommendation scores. The knowledge-enhanced multi-task learning course recommendation model is shown in Fig. 4.

Figure 4.

A multi-task learning course recommendation model

In this section, the implementation details of the knowledge-enhanced multi-task learning course recommendation model proposed in this section are unfolded. An example of the candidate course is shown in Fig. 5, which contains four parts, namely: 1)

Input layer, which contains the features we use.

Figure 5.

Candidate course examples

The knowledge-enhanced multi-task learning course recommendation model proposed in this section has the same parts in the input and embedding layers and the enhanced knowledge tracking task as the corresponding parts of the personalized knowledge tracking model proposed in Chapter 3, and thus will not be repeated here. Next, the course recommendation task is described in detail.

In the course recommendation task, this chapter generates a recommendation score y^pri based on the learner’s portrait U and the course’s representation C. With respect to the learner’s portrait U, the knowledge-augmented personalized course recommendation model adaptively incorporates three relevant features based on a specific context: 1)

The learner’s current knowledge-level features about the target course y^auxck+1 , which are captured by an enhanced knowledge-tracking task.

For a learner’s sequential behavioral representation u_b, not all courses taken by the learner are important, so this chapter first designs a gating mechanism that accepts each course e_ci as an input and controls how much information passes through to the next layer, and later uses the attention mechanism to adaptively compute a vector of behavioral representations of the learner using the history of behaviors associated with the target course. 14e¯ci=σ(wceci+bc)⊙tanh(wcceci+bcceci) 15αj=exp(e¯cjWeck+1)∑p=1kexp(e¯cpWeck+1) 16ub=∑j=1kαje¯cj

For the learner’s personality, we use a gating mechanism to adapt the original features considering that different dimensions have different priorities. To help the model can quickly learn the features related to the learner’s personality, we further use batch regularization: 17up*=BatchNormalization(σ(wpep*+bp)⊙tanh(wppep*+bpp)) where p* is an abbreviation for p_{(O, C, E, A, N)}. w_p, w_pp and b_p, b_pp are trainable weights and bias terms, respectively. We further adaptively fuse the three features based on a specific context to generate a learner’s portrait: 18U=α*up+β*ub+γ*y^auxck+1 where α, β, γ denotes the adaptive weights of the relevant features, respectively, and these three weights are obtained by means of the softmax function, i.e., α + β + γ = 1 and α, β, γ ∈ [0,1].

For course representation C, this section designs a rule-based course navigation algorithm that relies on expert knowledge to select candidate courses for recommendation. The use of candidate courses not only ensures a logical knowledge structure but also reduces the search space. A quiz exists to assess the learner’s learning outcomes after the learner has completed a series of courses. When constructing a candidate set of courses, this section selects from two types of courses: (1) courses that the learner has already taken. (2) Courses that have the same quiz as the last course in the learner’s learning sequence. For the last interaction k in the learning sequence. After which the course representation C is represented by the embedding vector of the candidate set of courses.

In this section, the factorization machine is used to compute the predicted recommendation scores y^pri : 19y^pri=∑i=1dwixir+∑i=1d∑j=i+1d〈 zi,zj 〉xirxjr+b where x^r = U ⊕ C, b denote bias terms. w is the weight of the linear regression, and 〈z_i,z_j〉 is the weight of the pairwise interaction between xir and xjr .

This section uses the pairwise Bayesian personalized ranking loss function to optimize the model parameters. It assumes that interactions that can be observed should be assigned higher prediction scores than interactions that cannot be observed: 20LR=∑(u,i,j)∈O−lnσ(y^prii−y^prij) where O = {(u, i, j)|(u, i) ∈ R⁺, (u, j) ∈ R⁻} denotes pairs of training data. R⁺ and R⁻ denote observable and unobservable interactions, respectively.

For the knowledge-enhanced multi-task learning course recommendation model training objective, the loss function of the course recommendation task and the loss function of the knowledge tracking task are linearly combined to jointly learn the two tasks in an end-to-end manner: 21L=LR+λLK where λ is the control parameter. The statistical details of the data set are shown in Table 2.

The mastery of the 124 students who participated in the quiz on the knowledge point k1 (operation of fractions) is shown in Figure 7. As can be seen from the figure, the number of failures who did not master the knowledge point was 8, the number of those who mastered the level of excellence was 68, and the level of mastery of 48 students was good, and on the whole, the level of mastery of all the students for the knowledge point k1 was high, and basically reached the academic level.

Figure 7.

Students have a sense of knowledge at k1

The students’ mastery of knowledge point k2 (Properties and Applications of Proportions) is shown in Figure 8. For knowledge point k2, 29 students achieved an excellent level of mastery, 78 students had a good level of mastery, and 17 students failed and did not master the knowledge point. The application of proportions involves mathematical modeling. The question types are more novel and flexible, and the difficulty has increased compared to the knowledge point k1.

Figure 8.

Students have a sense of knowledge at k2

The students’ mastery of knowledge point k3 (space cube) is shown in Figure 9. For the knowledge point k3, 12 students had an excellent mastery level, 95 students had a good mastery level, and 17 students failed and did not master the knowledge point. The knowledge point related to space cube is a difficult and key point in the advanced mathematics course, students’ first contact with spatial geometry, it is difficult to learn the knowledge points such as three-view drawing and unfolding diagram, and the spatial imagination ability needs to be further cultivated. In the study of this knowledge point, the number of failures is high and the number of excellencies is low.

Figure 9.

Students have a sense of knowledge at k3

Students’ mastery of knowledge point k4 is shown in Figure 10. For the knowledge point k4, 41 students achieved an excellent level of mastery, 72 students had a good level of mastery, and 11 students failed and did not master the knowledge point. The knowledge point of quadratic equations is generally examined in the context of real-life scenarios, which requires students to establish the concepts of variables and unknowns, analyze and establish equations from the meaning of the problem and solve it, and is the key knowledge required by the syllabus of higher mathematics.

Figure 10.

Students have a sense of knowledge at k4

Students’ mastery of knowledge point k5 (Preliminary knowledge of statistics) is shown in Figure 11. For knowledge point K5, 62 students mastered it at an excellent level, 57 students mastered it at a good level, and 5 students failed to master the knowledge point. The statistical knowledge point mainly examines the proficiency in three types of statistical graphs and simple calculations. Students only need to have a firm grasp of the definitions of three kinds of statistical graphs, so that they can learn by example and achieve excellent learning results.

Figure 11.

Students have a sense of knowledge at k5

In summary, comparing the mastery of the five knowledge points, the overall mastery level of students on knowledge point k1 and knowledge point k5 is higher, the number of outstanding is higher, and the number of failures is lower, the overall score distribution is more concentrated, indicating that the use of this paper’s method of teaching on these two knowledge points is very correct and fruitful, but it is still necessary to pay attention to additional attention to poorer students, appropriate personalized tutoring to help them improve. In contrast, students’ mastery of knowledge points k2 and k4 declined, but overall they are still at an excellent level. In addition, their mastery is more dispersed.

To analyze the differences in students’ mastery levels of knowledge points at the class level, all 124 students who participated in this quiz came from three classes, of which 40 were from class A, 42 were from class B, and 42 were from class C. The mastery of knowledge points in different classes is shown in Figure 12. According to the figure, the mastery level of class C for the knowledge points k1, k2, k4 and k5 is better than that of the other two classes, and the overall performance of the students in class C is very good. The mastery level of the students in class A for the knowledge points k2, k3 and k4 is the lowest among the three classes, which indicates that the students in class A need to strengthen the practice and review and consolidate the three knowledge points in order to improve the mastery level of the knowledge points. In addition, students in class B have a medium level of mastery of the knowledge points k1, k2, k4 and k5, but their mastery of the most difficult knowledge point k3 is significantly higher than that of the other classes.

Figure 12.

Different class knowledge points master the situation

For two students with the same test scores or the same ranking, their mastery of various knowledge points is likely to be quite different. Therefore, the total score or ranking of a test alone is not enough to make a comprehensive evaluation of a student’s academic performance at a certain stage. It is only through personalized evaluation and guidance for each student’s specific learning situation that the evaluation and guidance function of the test can be brought into full play. Both Student #40 and Student #90 scored a total of 24 points on this quiz. A comparison of the students’ individual knowledge acquisition is shown in Figure 13. From the figure, we can see the difference between the two people with the same score within the knowledge structure, student #40 has an excellent level of mastery of knowledge points k1 and k2, good mastery of knowledge points k3 and k5, but poor mastery of knowledge point k4. In contrast, student #90 has good mastery of point k4, excellent mastery of points k1 and k5, but poor mastery of point k3.

Figure 13.

The students’ knowledge points are compared