The Role of Data-Driven Instructional Models in the Development of Students’ Critical Thinking Skills in Core Literacy Education

Through the qualitative analysis of classroom observation data and student interview data, the key elements and their characteristics that support the development of higher-order thinking in high school language are analyzed from the real classroom context, and the theoretical model of the data-driven teaching mode to support the development of student’s critical thinking ability is constructed as shown in Fig. 1, combining with the previous understanding of the structure and function of the technology-enriched classroom environment.

Figure 1.

The theoretical model of students’ critical thinking ability development

The data-driven teaching model acts on the process of students’ critical thinking ability development with three dimensions of rich physical space, rich activity space and rich psychological space, and supports students’ critical thinking ability development. The theoretical model is explained below in terms of the interrelationships between three-dimensional spatial elements and their role in the development of higher-order thinking in high school language.

Hypothesis 1: The data-driven instructional model activity space element significantly promotes the development of student’s critical thinking skills in core literacy education.

Hypothesis 2: The physical space element of the data-driven instructional model significantly promotes the development of students’ critical thinking skills in core literacy education.

Hypothesis 3: The data-driven instructional model mental space element significantly promotes the development of student’s critical thinking skills in core literacy education.

Hypothesis 4: The data-driven instructional model shows a correlation with the development of students’ critical thinking skills.

In the pre-survey phase of this study, one prestigious high school, one high-quality high school, and one average high school were selected, and there was a large gap between the student population of the three high schools. The pre-survey included three grades of students, and one class was randomly selected for each grade. The “pre-survey” used on-site organization of the questionnaire mode. The two surveys issued 320 questionnaires and recovered 300 valid questionnaires, with a recovery rate of 93.8%.Among them, 135 were girls and 165 were boys.

Where: b₀ is a constant term that represents the estimated value of y when all the respective variables are zero; b₁,b₂,⋯,b_k is the sample partial regression coefficient, or known as the partial regression coefficient of y corresponding to x₁,x₂,……x_k, which represents the average change in y for each unit change in x₁,x₂,……x_k when the other independent variables are unchanged; ε is known as the random variable of the error term, which is the variability that is contained inside the y but cannot be explained by the linear relationship of the k independent variables [23].

When the original information establishes the multivariate regression model, in order to ensure that the regression model has excellent explanatory ability and predictive effect, the theoretical conditions that should be satisfied are: (1)

Linearity: the relationship between the dependent variable and the independent variable is linear. That is to say, the independent variable has a significant effect on the dependent variable and has a close linear correlation.

Estimation of biased regression coefficients:

One of the purposes of regression analysis is to develop regression equations that enable researchers to predict the value of the dependent variable based on the known independent variables. The estimation of the partial regression coefficients for a multiple regression model, like the same linear regression equation, is also done by solving the parameters by the least squares method, provided that the sum of squared errors (∑e²) is required to be minimized. Taking the bilinear regression model as an example, the standard set of equations for solving the regression parameters is: (2) y=b0+b1x1+b2x2,(k=2) \[y={{b}_{0}}+{{b}_{1}}{{x}_{1}}+{{b}_{2}}{{x}_{2}},(k=2)\] (3) B=[ b0b1b2 ],X=(1x11x12.........1xn1xn2) \[B=\left[ \begin{matrix} {{b}_{0}} \\ {{b}_{1}} \\ {{b}_{2}} \\ \end{matrix} \right],X=\left( \begin{matrix} 1 & {{x}_{11}} & {{x}_{12}} \\ ... & ... & ... \\ 1 & {{x}_{n1}} & {{x}_{n2}} \\ \end{matrix} \right)\]

The formula to derive b₀,b₁b₂ is as follows: (4) { ∑ y=nb0+b1∑ x1+b2∑ x2∑ x1y=b0∑ x1+b1∑ x12+b2∑ x1x2∑ x2y=b0∑ x2+b1∑ x1x2+b2∑ x22 \[\left\{ \begin{array}{*{35}{l}} \sum{y}=n{{b}_{0}}+{{b}_{1}}\sum{{{x}_{1}}}+{{b}_{2}}\sum{{{x}_{2}}} \\ \sum{{{x}_{1}}}y={{b}_{0}}\sum{{{x}_{1}}}+{{b}_{1}}\sum{x_{1}^{2}}+{{b}_{2}}\sum{{{x}_{1}}}{{x}_{2}} \\ \sum{{{x}_{2}}}y={{b}_{0}}\sum{{{x}_{2}}}+{{b}_{1}}\sum{{{x}_{1}}}{{x}_{2}}+{{b}_{2}}\sum{x_{2}^{2}} \\ \end{array} \right.\]

Solve this equation to find the value of b₀,b₁,b₂. The following matrix method can also be used to find b=(x′x)^–1·x′y i.e: (5) [ b0b1b2 ]=[ n∑ x1∑ x2∑ x1∑ x12∑ x1x2∑ x2∑ x1x2∑ x22 ]−1[ ∑ y∑ x1y∑ x2y ] \[\left[ \begin{matrix} {{b}_{0}} \\ {{b}_{1}} \\ {{b}_{2}} \\ \end{matrix} \right]={{\left[ \begin{matrix} n & \sum{{{x}_{1}}} & \sum{{{x}_{2}}} \\ \sum{{{x}_{1}}} & \sum{x_{1}^{2}} & \sum{{{x}_{1}}}{{x}_{2}} \\ \sum{{{x}_{2}}} & \sum{{{x}_{1}}}{{x}_{2}} & \sum{x_{2}^{2}} \\ \end{matrix} \right]}^{-1}}\left[ \begin{matrix} \sum{y} \\ \sum{{{x}_{1}}}y \\ \sum{{{x}_{2}}}y \\ \end{matrix} \right]\]

The closer r² is to 1, the better the model fits.

The test of significance of the equation aims to make an inference about whether the linear relationship between the explanatory variables and the explanatory variables in the model holds significantly in the aggregate.

The original hypothesis b₁=0,b₂,⋯,b_k = 0 and the alternative hypothesis b_i are not all 0. According to the definition of mathematical statistics, under the condition that the original hypothesis holds, the statistics are as follows: (7) F=ESS / kRSS / (n−k−1) \[F=\frac{ESS\;/\;k}{RSS\;/\;(n-k-1)}\]

Obey the F distribution with degree of freedom (k,n–k–1). Given the significance level α, the critical value Fα(k,n–k–1), the value of the statistic F is derived from the sample, and the original hypothesis can be rejected or accepted by either F>Fα(k,n–k–1) or F≤Fα(k,n–k–1) to determine whether the overall linear relationship of the original equation holds significantly.

Hypothesis test for biased regression coefficients: the t test. In such a test, we need to make a statistically significant (i.e., with a certain level of confidence) test of whether a given (overall) parameter in the model satisfies the dummy original hypothesis b_i = a_i, where a_i is some given known number. In particular, when a_i=0, it is called a significance test for the parameter. If the original hypothesis is rejected, it means that the explanatory variable x_i has a significant linear effect on the explanatory variable Y, and the estimate b^i\[{{\hat{b}}_{i}}\] will dare to be used. Conversely, it means that the explanatory variable x_i does not have a significant linear effect on the explanatory variable Y and the estimate b^i\[{{\hat{b}}_{i}}\] would not be meaningful to us.

Since the model parameter b_i obeys the following normal distribution: b^i~N(bi,σ2cii)${{\hat{b}}_{i}}\tilde{\ }N({{b}_{i}},{{\sigma }^{2}}{{c}_{ii}})$ , where c_i denotes the ith element on the main diagonal of the matrix (X′X)−1 and σ² is the variance of the random error term, it is replaced by its estimator in the actual calculation: (8) σ^2=∑ εi2n−k−1=ε′εn−k−1 \[{{\hat{\sigma }}^{2}}=\frac{\sum{\varepsilon _{i}^{2}}}{n-k-1}=\frac{{\varepsilon }'\varepsilon }{n-k-1}\]

A statistic can be constructed as follows: (9) t=b^i−biSi=b^i−biciiε′εn−k−1 \[t=\frac{{{{\hat{b}}}_{i}}-{{b}_{i}}}{{{S}_{i}}}=\frac{{{{\hat{b}}}_{i}}-{{b}_{i}}}{\sqrt{{{c}_{ii}}\frac{{{\varepsilon }^{\prime }}\varepsilon }{n-k-1}}}\]

Given the significance level α, the critical value t_a/2(n–k–1) is obtained, whereupon the original hypothesis can be rejected or accepted based on: |t|>t_a/2(n–k–1) or |t|≤t_a/2(n–k–1), thus determining whether the corresponding explanatory variables should be included in the model.

The principle of correlation analysis is the process of analyzing the degree of correlation between two or more variables and expressing the results of the calculations through appropriate indicators. Variable selection generally consists of two parts: rules and query mechanisms. By rules, we mean the rules for evaluating the joint correlation between variables in a set of variables [24]. The main purpose of the query mechanism is to delete or add variables to the set of variables. Evaluation rules, also known as evaluation criteria, play a decisive role in variable selection. Both parts and feature construction are inseparable from correlation. Correlation is a general concept used to describe the degree of data closeness between variables and the amount of mutual information contained, including asymmetric causal and driving relationships. The basic idea of feature selection based on correlation analysis is to select effective variables with higher correlation with the prediction target, i.e., feature variables contributing to the performance of the prediction model, by analyzing the correlation strength relationship between variables.

Commonly used correlation coefficients: 1)

Pearson’s correlation coefficient

Define the aggregate of the two-dimensional variable X,Y to be (X,Y)^T,(x₁,y₁)^T,(x₂,y₂)^T,⋯,(x_n,y_n)^T is an experimental sample of the variable X,Y, which is also an observation, to obtain the observation matrix M: (10) M=[ x1x2⋯xny1y2⋯yn ]T \[M={{\left[ \begin{matrix} {{x}_{1}} & {{x}_{2}} & \cdots & {{x}_{n}} \\ {{y}_{1}} & {{y}_{2}} & \cdots & {{y}_{n}} \\ \end{matrix} \right]}^{T}}\]

The means of variable X and variable Y were calculated separately: (11) x¯=1n∑i=1nxi \[\bar{x}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}\] (12) y¯=1n∑i=1nyi \[\bar{y}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{y}_{i}}}\]

Equation (11) represents the mean of observed data for variable X and equation (12) is the mean of observed data for variable Y. The data variance is further calculated: (13) Sxx=1n−1∑i=1n(xi−x¯)2 \[{{S}_{xx}}=\frac{1}{n-1}\sum\limits_{i=1}^{n}{{{({{x}_{i}}-\bar{x})}^{2}}}\] (14) Syy=1n−1∑i=1n(yi−y¯)2 \[{{S}_{yy}}=\frac{1}{n-1}\sum\limits_{i=1}^{n}{{{({{y}_{i}}-\bar{y})}^{2}}}\] (15) Sxy=1n−1∑i=1n(xi−x¯)(yi−y¯) \[{{S}_{xy}}=\frac{1}{n-1}\sum\limits_{i=1}^{n}{({{x}_{i}}-\bar{x})}({{y}_{i}}-\bar{y})\]

Equations (13), (14), and (15) compute the covariance of the observations for variable X, variable Y, and two-dimensional variable X,Y, respectively, defining S as the covariance matrix of the observations, then: (16) S=[ SxxSxySyxSyy ] \[S=\left[ \begin{matrix} {{S}_{xx}} & {{S}_{xy}} \\ {{S}_{yx}} & {{S}_{yy}} \\ \end{matrix} \right]\]

In matrix S, there is S_xy=S_x, so S The diagonal elements are equal and it is a symmetric matrix. Moreover, according to Schwarz inequality, there is: (17) Sxy2≤SxxSyy \[S_{xy}^{2}\le {{S}_{xx}}{{S}_{yy}}\]

Therefore the matrix S is a positive definite matrix. Define the correlation coefficient of the observations of X,Y is given by: (18) rxy=SxySxxSyy \[{{r}_{xy}}=\frac{{{S}_{xy}}}{\sqrt{{{S}_{xx}}}\sqrt{{{S}_{yy}}}}\]

|r_xy|≤1 holds according to Schwarz’s inequality. r_xy is used to measure the degree of linear correlation of the variable X,Y. Let the overall two-dimensional variable X,Y be (X,Y)^T, then the correlation coefficient of (X,Y)^T is calculated as: (19) rXY=Cov(X,Y)Var(X)Var(Y) \[{{r}_{XY}}=\frac{Cov(X,Y)}{\sqrt{Var(X)}\sqrt{Var(Y)}}\]

Where Var(X),Var(Y) is the variance of variable X and variable Y, respectively, and Cov(X,Y) is the covariance of the two-dimensional aggregate (X,Y)^T. The range of earson’s correlation coefficient is rϵ[–1,1], and the absolute value of Pearson’s coefficient is usually used to determine the strength of correlation between two variables, the rules are as follows: r>0 means positive correlation, r<0 means negative correlation; |r|=0 means there is no linear relationship; and |r|=1 means complete linear correlation.

In short, the absolute value of the Pearson coefficient is close to 1, which proves that the correlation between the two variables is stronger; on the contrary, the smaller the absolute value is, the weaker the correlation between the variables is, and the stronger the independence is. Meanwhile, it should be noted that the use of Pearson’s correlation coefficient needs to follow the following principles.

Where the covariance dCor(X,Y) is defined as: (21) dCov(X,Y)=1n2∑k,l=1nXklYkl \[dCov(X,Y)=\frac{1}{{{n}^{2}}}\sum\limits_{k,l=1}^{n}{{{X}_{kl}}}{{Y}_{kl}}\]

The relevant term in the above equation is defined as: (22) Xkl=kl−x¯k⋅−x¯⋅l+x¯ \[{{X}_{kl}}{{=}_{kl}}-{{\bar{x}}_{k\cdot }}-{{\bar{x}}_{\cdot l}}+\bar{x}\] (23) x¯k•=1n∑i=1nxkl x¯⋅l=1n∑k=1nxkl x¯=1n2∑k,l=1nxkl \[{{\bar{x}}_{k\bullet }}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{kl}}}\quad {{\bar{x}}_{\cdot l}}=\frac{1}{n}\sum\limits_{k=1}^{n}{{{x}_{kl}}}\quad \bar{x}=\frac{1}{{{n}^{2}}}\sum\limits_{k,l=1}^{n}{{{x}_{kl}}}\]

The term of interest for the random variable Y is defined above. The distance correlation coefficient is an improved method of Pearson’s correlation coefficient. When Pearson’s correlation coefficient is 0, we can not conclude that the two variables are independent because Pearson can not measure the nonlinear relationship, but if the distance correlation coefficient is 0, it can be confirmed that the two variables are independent of each other.

Where H(X) is the information entropy of X, I(x) stands for self-information or informativeness and denotes the entropy contributed by a single piece of information, and E is the expectation function.

For random variables x and y containing finite values, if x and y are signals from two sources that are not independent of each other, and their joint distribution is set to be p(x,y) and their marginal distributions to be p(x) and p(y), then the joint information entropy of x and y is: (25) H(X,Y)=−∑x,yp(x,y)logp(x,y) \[H(X,Y)=-\sum\limits_{x,y}{p}(x,y)\log p(x,y)\]

Among them: (26) p(x,y)=p(X=x,Y=y)p(x)=p(X=x)p(y)=p(Y=y) \[p(x,y)=p(X=x,Y=y)p(x)=p(X=x)p(y)=p(Y=y)\]

The conditional entropy of X in the Y condition is then expressed as: (27) H(X|Y)=−∑yp(y)H(X|Y=y)=−∑x∑yp(y)p(x|y)logp(x|y)=−∑x,yp(x,y)logp(x|y) \[\begin{align} & H(X|Y)=-\sum\limits_{y}{p}\left( y \right)H(X|Y=y) \\ & =-\sum\limits_{x}{\sum\limits_{y}{p}}(y)p(x|y)\log p(x|y) \\ & =-\sum\limits_{x,y}{p}(x,y)\log p(x|y) \\ \end{align}\]

Mutual information in information theory denotes the interdependent inclusion of two variables X and Y, i.e., the amount of information that Y contains about. For two discrete random variables X and Y, the mutual information I(X;Y) between them is defined as: (28) I(X;Y)=Ex,y(SI(x,y))=∑x,yp(x,y)logp(x,y)p(x)p(y) \[I(X;Y)={{E}_{x,y}}(SI(x,y))=\sum\limits_{x,y}{p}(x,y)\log \frac{p(x,y)}{p(x)p(y)}\]

Where p(x,y) denotes the joint probability distribution of X and Y, and SI(x,y) is the mutual information between the sample points emanating from X and Y. For two continuous random variables X and Y, I(X;Y) is defined as: (29) I(X;Y)=∬p(x,y)logp(x,y)p(x)p(y)dxdy \[I(X;Y)=\iint{p}(x,y)\log \frac{p(x,y)}{p(x)p(y)}dxdy\]

Mutual information can also be represented by information entropy with the following formula: (30) I(X;Y)=H(X)−H(Y|X)=H(Y)−H(Y|X)=H(X)+H(Y)−H(X,Y) \[\begin{align} & I(X;Y)=H(X)-H(Y|X) \\ & =H(Y)-H(Y|X) \\ & =H(X)+H(Y)-H(X,Y) \end{align}\]

The overall profile of the study population is shown in Table 2. Among them, 131 students belonging to quality schools (43.67%), 55% of the total number of females, 51.67% of the total number of students from urban areas, and 198 students from arts and sciences (66%), which is in line with the thinking characteristics of the development of critical thinking skills.

The students’ CTDI-CV scores for each dimension and the percentage of critical thinking tendencies are shown in Table 3. The total score of students’ CTDI-CV is (293.14±30.6), which is positive critical thinking. The highest score in each dimension is curiosity, the lowest is self-confidence, and there are a few people who have positive and strong critical thinking in each dimension, in which the proportion of the number of people in the curiosity dimension (22.4%) is higher than the other dimensions.

In order to examine whether there is a significant difference between the student’s critical thinking in terms of gender, an independent samples t-test was conducted on the students’ total critical thinking scores and their sub-dimension scores, and the results of the t-test are shown in Table 4. The table shows that although the mean of the total critical thinking scores of male students (266.19) is higher than that of female students (264.28), it does not reach a significant difference (P=0.598>0.05), so there is no significant difference between the total scores of the student’s critical thinking in terms of gender but there is a significant difference between the sub-dimensions of truth-seeking, self-confidence, and cognitive maturity, where the scores of male students are significantly higher than those of female students in the sub-dimensions of truth-seeking and self-confidence. Aspect boys scored significantly higher than girls, while in the cognitive maturity sub-dimension, girls scored significantly higher than boys.

Similarly, to examine whether there is a significant difference between students in terms of arts and sciences, we conducted an independent samples t-test. The differences in college students’ critical thinking in terms of majors are shown in Table 5, which shows that there is a significant difference between the students’ total scores of critical thinking in terms of majors (P=0.018<0.05). Also, there is a significant difference between the two subdimensions of truth-seeking and analytical ability, where the science students’ Critical Thinking is significantly higher than that of Arts students, and Science students’ scores in the two sub-dimensions of truth-seeking and emancipation are also significantly higher than that of Arts students.