Research on the application of GLE teaching mode in English-medium colleges

The probability of the P_ij system from state E_i to another state through one transition is recorded as E_j, also known as moment. P is a transition matrix called P^(k)k; the secondary transition matrix is written as follows: (1) p=[p11p12⋯p1np21p22⋯p2n⋮⋮⋮pn1pn2⋯pnn] p = \left[{\matrix{{{p_{11}}} & {{p_{12}}} & \cdots & {{p_{1n}}} \cr {{p_{21}}} & {{p_{22}}} & \cdots & {{p_{2n}}} \cr \vdots & \vdots & {} & \vdots \cr {{p_{n1}}} & {{p_{n2}}} & \cdots & {{p_{nn}}} \cr}} \right]

The state transition matrix must be a probability matrix; so, the following conditions need to be met:

In the matrix, each element P_ij is a non-negative number, that is, (2) Pij0, i=1,2,⋯,n; j=1,2,⋯,n {P_{ij}}0,\quad i = 1,2, \cdots,n;\quad j = 1,2, \cdots,n

After a period of time, the k Markov chain tends to be stable after the state undergoes a sufficiently large sub-transition, and the stable state of the system can be obtained by the transition probability matrix.

In order to determine the primary transition matrix of the English teaching model, the probability of the relative state changes that of the stationary, increasing and decreasing English learning scale from 2012 to 2018 should be assumed first. Set the rate of change as follows: (4) c=(Ni−Ni−1)/Ni c = \left({{N_i} - {N_{i - 1}}} \right)/{N_i} where N_i is the number of students in the current year and N_i−1 is the number of students the previous year. If c ∈ [−0.1,0.1], the state is defined as stationary; if c > 0.1, the state is defined as increasing; if c < −0.1, the state is defined as decreasing. The scale of English learning in 2012 and 2013 was counted according to 0 people, s₁g state, 2 states and r₂ states, of which s → g₁g → g occurred, 1 occurred, g → r₁ occurred, 1 occurred, r → s and r₁ → r occurred. The first transition probability matrix of English learning scale is constructed as follows: (5) p=[01000.50.50.500.5] p = \left[{\matrix{0 & 1 & 0 \cr 0 & {0.5} & {0.5} \cr {0.5} & 0 & {0.5} \cr}} \right]

According to the idea of the least squares method, the constraints are introduced into the model at the same time, and a E Markov state transition probability is obtained with the minimum sum of squares of errors in the transition process of each stage in each state and the row condition and the non-negative condition as constraints. For the matrix optimisation model, assuming that the system has n, an incompatible state, the state can be expressed as follows: (6) {S1,S2,⋯,Sj,⋯,Sn} \left\{{{S_1},{S_2}, \cdots,{S_j}, \cdots,{S_n}} \right\}

After the k step transfer, the sum of squares of errors in the transfer process of each stage is written as follows: (7) E=∑j=1n∑k=1m(Sj(k)−∑i=1nSi(k−1)pij)2 E = \sum\limits_{j = 1}^n \sum\limits_{k = 1}^m {\left({S_j^{(k)} - \sum\limits_{i = 1}^n S_i^{(k - 1)}{p_{ij}}} \right)^2}

Based on this, the quadratic programming model is established as follows: (8) {minE=∑j=1n∑k=1m(Sj(k)−∑i=1nSi(k−1)pij)2s.t. ∑j=1npij=1, i=1,2,⋯,npij≥0, i=1,2,⋯,n \left\{{\matrix{{\min E = \sum\limits_{j = 1}^n \sum\limits_{k = 1}^m {{\left({S_j^{(k)} - \sum\limits_{i = 1}^n S_i^{(k - 1)}{p_{ij}}} \right)}^2}} \cr {{\rm{s}}{\rm{.t}}{\rm{.}}\;\sum\limits_{j = 1}^n {p_{ij}} = 1,\quad \quad i = 1,2, \cdots,n} \cr {{p_{ij}} \ge 0,\quad \quad i = 1,2, \cdots,n} \cr}} \right.

The purpose of this step is to find the membership function in the input space and the output space. Specifically, it is to cluster the training data, and each class corresponds to a fuzzy subset. The so-called clustering is to group the relatively close samples into one group. However, most of the input and output space division numbers (cluster numbers) are given in advance, which inevitably has a certain blindness, which directly affects the extraction quality of planning. Therefore, this paper adopts a density-based and grid-based clustering method to determine the initial cluster centre and the number of cluster centres, which can automatically divide the sample space and divide it in the sample space. Clustering learning time and other aspects have obvious advantages.

The transfer function of the first hidden layer node adopts the Gauss function, which is a kind of radial basis function, and the node output is written as follows: (9) fi,ki(xi)=e−(xi−cikP)2/σiki2 {{\rm{f}}_{\rm{i}}},{{\rm{k}}_{\rm{i}}}\left({{{\rm{x}}_{\rm{i}}}} \right) = {\rm{e}} - {\left({{{\rm{x}}_{\rm{i}}} - {{\rm{c}}_{\rm{i}}}{{\rm{k}}_{\rm{P}}}} \right)^2}/\sigma _{{\rm{iki}}}^2 where c_ik_iσ_iki are the centre and width of each membership function, respectively. The initial value is the initial cluster centre. Before giving the basic idea of determining the initial expected value, the concept of overlap should be given first. The so-called overlapping degree refers to the degree of overlapping of two fuzzy subsets, which is expressed by the maximum membership degree of the intersection of these two fuzzy subsets. The definition of its overlap is shown in Figure 2.

Fig. 2

As shown in Figure 2, the overlap is 0.45. In fuzzy control, the degree of overlap is an important factor that affects the control performance. If the overlap is too large or too small, the control effect will be reduced. Usually, it should be controlled at about 0.45. Based on this principle, this paper deduces the formula of the initial expected value.

Suppose the distance between any two adjacent cluster centres is as follows: (10) d=∥ci−ci−1∥ {\rm{d}} = \parallel {{\rm{c}}_{\rm{i}}} - {{\rm{c}}_{{\rm{i}} - 1}}\parallel

The radial basis function corresponding to the cluster centre is c_i: (11) fi(xp)=e−(xi−ci2/σ2 {{\rm{f}}_{\rm{i}}}\left({{{\rm{x}}_{\rm{p}}}} \right) = {\rm{e}} - \left({{{\rm{x}}_{\rm{i}}} - {{\rm{c}}_{\rm{i}}}^2/{\sigma ^2}} \right.

Expected value σ_i should be chosen so that the overlap of two adjacent fuzzy subsets is around 0.45. For this reason, let then, |x_i − c_i| = d/2, then, (12) 0.3679<fi(xi)<0.7788 0.3679 < {{\rm{f}}_{\rm{i}}}\left({{{\rm{x}}_{\rm{i}}}} \right) < 0.7788

At this time, the available public statements are: (13) 14<(xi−ci)2σi2=(d2)2σi2<1, σi=‖ci−ci−1‖γ, 1<γ<2 {1 \over 4} < {{{{\left({{x_i} - {c_i}} \right)}^2}} \over {\sigma _i^2}} = {{{{\left({{d \over 2}} \right)}^2}} \over {\sigma _i^2}} < 1,\quad \quad {\sigma _i} = {{\left\| {{c_i} - {c_{i - 1}}} \right\|} \over \gamma},\quad \quad 1 < \gamma < 2

Therefore, for the RBF network in this paper, the expected value is written as follows: (14) σiki=‖ciki−ciki−1‖γ, 1<γ<2 {\sigma _{i{k_i}}} = {{\left\| {{c_{i{k_i}}} - {c_{i{k_i} - 1}}} \right\|} \over \gamma},\quad \quad 1 < \gamma < 2

It is referred to here as the overlap coefficient γ.

In the forward propagation process, the input signal is processed one by one from the input layer and transmitted to the output layer, and the state of neurons in each layer only affects the state of neurons in the next layer. If the expected output cannot be obtained at the output layer, it will turn to back propagation and return the error of the output signal along the original connection path. By modifying the weights of neurons in each layer, the error signal is minimised. The output error evaluation function can be expressed as follows: (15) E=12∑j=1M(yj0−yj)2 {\rm{E}} = {1 \over 2}\sum\limits_{{\rm{j}} = 1}^{\rm{M}} {\left({{\rm{y}}_j^0 - {{\rm{y}}_{\rm{j}}}} \right)^2}

Among them, yj0 {\rm{y}}_j^0 is the target value, where y_j is the network output learning process. For each training sample, starting from the input node, calculate the output value of each node forward layer by layer, know the output node of the network and then start from the output node, calculated for all hidden layer nodes; the calculation formula can be expressed as follows: (16) −∂E∂yj=yj0−yj - {{\partial {\rm{E}}} \over {\partial {{\rm{y}}_{\rm{j}}}}} = {\rm{y}}_{\rm{j}}^0 - {{\rm{y}}_{\rm{j}}}

And adjust the weights on the connection so that the parameters of the membership function can be determined by learning from the measured data (learning samples), and the position and shape of the membership function can be adjusted.

English learning process c_iki and σ_iki the update is expressed by the following formula:

c_iki(t + 1) = c_iki(t) − ηδ E/δ c_iki + αΔC_iki(t), where Δc_ii = c_ii − c_{j j}(t − 1);

Among them, ΔE(t) = E(t) − E(t − 1). But how to determine the step size requires trial and error. If η is too small, the algorithm will converge slowly; if η is too large, it may diverge.

The heterogeneous encoding of multiple English words can be converted into a binary tree structure. All words correspond to the leaf nodes of the tree. The distance between the nodes where different word encodings are located on the tree represents the similarity between words, for example, for vocabulary w₁, w₂, a depth-based semantic similarity calculation method is given in the literature, and its formula can be expressed as follows: (17) (w1,w2)=0.9811×D+0.49770.1244×D+4.4612 \left({{w_1},{w_2}} \right) = {{0.9811 \times D + 0.4977} \over {0.1244 \times D + 4.4612}}

In the above formula, D is the w₁, w₂ depth of the nearest common node in the tree structure graph of the corresponding vocabulary code. This simple calculation formula converts the distance information into a similarity value between 0 and 1, which can well reflect the similarity. Encoding in a large number of words actually implements word embedding through the similarity relationship between multiple words. The model predicts the context word for each given word by maximising w_t. Its maximization formula can be expressed as follows: (18) LSG=1|V|∑t=1V∑−c≤i≤clogp(wt+iwt) {L_{{\rm{SG}}}} = {1 \over {|V|}}\sum\limits_{t = 1}^V \sum\limits_{- c \le i \le c} \log p\left({{\boldsymbol{w}_{t + i}}{w_t}} \right)

In the above formula, V is the corpus used to train the vocabulary vector; |V| is the total number of vocabulary; is w_t the vector corresponding to the c current central vocabulary; is the size of the context window; w_t+i is the w_t context w_t+i vocabulary vector p(w_t+iw_t).

After the vocabulary is vectorised, it is equivalent to establishing a mapping relationship between the vocabulary and the vector. Due to the natural operational properties between the vectors, the vocabulary vector forms a space. The similarity between words can be represented by the positional relationship of their corresponding word vectors in space. Use w_i, w_j to represent both the different words and the vectors corresponding to the words. The lexical similarity can be represented numerically by the inner product of its lexical vector wiTwj w_i^T{w_j} unit, and its geometric meaning is the cosine of the angle between the two vectors. Therefore, if it is used to S (w₁, w₂) express the similarity between words, its formula can be expressed as follows: (19) S(w1,w2)=cos(w1,w2) S\left({{w_1},{w_2}} \right) = \cos \left({{w_1},{w_2}} \right)

The more similar two vectors in the vocabulary vector space, the smaller the angle, the higher the similarity. For the word sense similarity based on knowledge ontology, the result is recorded as s₁, using the deep semantic similarity calculation method to calculate; for the word meaning similarity calculation based on vocabulary vector, the result is recorded as s₂, using the calculation formula of similarity between words to calculate. If it cannot be calculated, its value is recorded as 0. It is recorded S = (s₁, s₂) as the observation quantity, which is defined as the state set.

For each state S = (s₁, s₂), the reinforcement learning model F obtains the best action as A = (a₁, a₂), which is the sum according to the strategy A = F (s₁, s₂); so, the strategy under the best action can be expressed as follows: (20) V(S,A)=a1s1+a2s2 V(S,A) = {a_1}{s_1} + {a_2}{s_2}

The reinforcement model part, the input (s₁, s₂) is used as its state observation, the strategy is represented by a linear function, ɛ-the training is carried out through the greedy evaluation strategy, and the reward function is constructed with the difference between the action's income and the artificial value, and finally the learning result is obtained. When the learning model selects an action in the ɛ-greedy evaluation strategy, it will ɛ select a new action with 1 − ɛ a probability of.

For the data given above, the different environmental observations in the corresponding reinforcement learning model are calculated, where S = (s₁, s₂) and ɛ = 0.1, and the strategy obtained after training is as follows: (21) F(s1,s2)={a1=1.001, a2=0S∈S1a1=1.431, a2=−0.604S∈S2a1=0.729, a2=0.399S∈S3 F\left({{s_1},{s_2}} \right) = \left\{{\matrix{{{a_1} = 1.001,\quad {a_2} = 0} & {S \in {S_1}} \cr {{a_1} = 1.431,\quad {a_2} = - 0.604} & {S \in {S_2}} \cr {{a_1} = 0.729,\quad {a_2} = 0.399} & {S \in {S_3}} \cr}} \right.

When using this strategy, if the final result is a negative number, the similarity value is 0 because the similarity cannot be negative, which shows negative effects.

In order to verify the effectiveness of the GLE teaching model in this paper in terms of membership function learning, the RBF network designed for the problem needs to be a two-input single-output network, which represents the following relationship: y = f (x₁, x₂). Taking a representative surface with such a relationship as an example, a simulation experiment is carried out to illustrate the effectiveness of the GLE teaching model in this paper. Suppose the experimental class and the control class form a system, and this system can be described in the following way: (22) f(x1,x2)=64−((x1−0.6)2+(x2−0.5)2)/9−0.5, 0≤x1≤1, 0≤x2≤1 f\left({{x_1},{x_2}} \right) = \sqrt {64 - \left({{{\left({{x_1} - 0.6} \right)}^2} + {{\left({{x_2} - 0.5} \right)}^2}} \right)} /9 - 0.5,\quad \quad 0 \le {x_1} \le 1,\quad \quad 0 \le {x_2} \le 1

On this function, the learning samples are randomly changed. Assume that the distribution and size of the membership function of x₁ before learning is shown in Figure 3(a). After the learning adjustment of the GLE teaching mode, the final membership function of x₂ is obtained as shown in Figure 3(b). The relative error curve between the output of the system after learning and the theoretical output of the original nonlinear function is shown in Figure 4.

Fig. 3

Fig. 4

In the effectiveness experiment of membership function learning, the membership distribution in Figure 3(a) is between −2 and 1, while in Figure 3(a) after the teaching mode of this paper, the original membership distribution is analysed. It is automatically adjusted, and its membership degree is distributed between 0 and 1. Therefore, it shows that, using the GLE teaching mode proposed in this paper, after learning the learning samples of the experimental class and the control class, the membership function after the learning adjustment can be automatically obtained. And the distribution range of the membership function will also be changed; at the same time, it can be seen from Figure 4 that the relative error of the fuzzy system output curve to the original nonlinear analytic function curve is between 0 and 0.6, that is, within the allowable range. The overall trend of the curve is a downward trend, indicating that the use of the GLE teaching mode proposed in this paper will promote the relative error of the output curve of the fuzzy system to the original nonlinear analytic function curve to become smaller and smaller; so, the GLE teaching mode membership function learning is obviously effective.

According to the introduction of the teachers of the two classes, the observation of the author in the classroom, and the comparison of the reading scores of the students in the two classes the average scores of the students in the two classes in the reading part of the English final exam are similar, and the students’ conditions are basically similar. The author randomly selects class 1 to conduct task-based teaching experiments, namely the experimental class, while class 2 maintains the previous traditional teaching mode, that is, the control class. The number of students in the two classes was 59 (experimental class) and 61 (control class). According to the teaching plan, the weekly class hours of both classes are 4 periods, and the teaching order is normal in the experimental stage. First of all, two classes should be pre-tested, and the pre-test results are shown in Table 1.

Table 1 shows the basic statistics of the analysis variables: the number of samples covering the two groups (N), the sample mean score (mean), the standard deviation of the sample (std. deviation) and the standard error of the mean (std. error mean). As can be seen from Table 1, the average grades of the two classes are very close, which are 65.3765 and 64.2509, respectively, for control and experimental classes. Then the independent sample t-test for the English reading pre-test scores of the two classes is shown in Table 2.

Table 2 shows the results of the independent sample t-test. According to the relevant theoretical analysis of statistics, in the independent sample T test, the significance level of the homogeneity test is 0.079>0.05, indicating the homogeneity of variances. Therefore, the hypothesis of homogeneity of variances is correct corresponding to a row of T test results. The significance level is 0.687>0.05; so, under the 95% confidence level, this means that there is no significant difference in the English reading test scores of the two classes (that is, the English reading levels of the two classes are equivalent), that is, the two classes are parallel classes.

The teaching steps of the experimental class are divided into the following steps:

After 28 days of teaching how to read English for college students using the GLE teaching mode and the traditional teaching mode, the instructors will prepare and mark the questions in person. The two classes use the same set of test papers for reading tests. SPSS11.5 software performs statistics as shown in Table 3.

Table 3 shows that when the test questions and other conditions are consistent, the final reading scores of the two classes have improved, but the average scores of the two classes have a large gap, which is 67.2627 for the control class and 89.5371 for the experimental class. Compared with the control class, the average score was improved by 22 points. Then, the independent sample t-test for the English reading post-test scores of the two classes is shown in Table 4.

It can be seen from Table 4 that the significance of the homogeneity test is 0.051>0.05 in the independent sample T test of the post-experiment test at the end of the period, indicating the homogeneity of variance. Therefore, the hypothesis of homogeneity of variance is established, and the corresponding line of T test result is correct; the significance level is 0.000<0.05; so, under the 95% confidence level, there is a significant difference in the final grades of the two classes, which shows that the GLE teaching model proposed in this paper is more effective than the traditional mode.