## 4 Research Methodology
### Model Cross Validity  
To further validate the stability of the model, this study verified the invariance of the two cohorts by gender, the two cohorts with or without participation in after-school services, and the three cohorts by grade level, respectively. This includes measuring the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**SIE**} & \multirow{2}{*}{\
\begin{tabular}{c} **Point** \\ **Estimate** \\ \end{tabular} } & \multicolumn{2}{c}{
\begin{tabular}{c} **Product of** \\ **Coefficients** \\ \end{tabular} } & \multicolumn{2}{c}{**Bias Corrected 95\% CI**} & \multicolumn{2}{c}{**Percentile 95\% CI**} \\ \cline{3-8}  & & **SE** & **Z** & **Lower** & **Upper** & **Lower** & **Upper** \\ \hline Indirect effects & & & & & & & \\ Experential learning in & & & & & & & \\ mathematics \(\rightarrow\) & 0.529 & 0.105 & 5.038 & 0.365 & 0.788 & 0.365 & 0.788 \\ mathematical self-efficacy & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mediating effects of the structural model. factor loadings, structural path coefficients, and factor covariances of the model, and if they do not differ from each other, the model can be said to have considerable stability, i.e., cross validity. In this study, using SPSS 26.0, the sample was divided into the following groups: 1. boys and girls; 2. those who participated in after-school services and those who did not; and 3. grade 2, grade 3, and grade 6. The analysis was then conducted separately for the different cohorts using Amos 24.0. As can be seen from Table 6, under the assumption that the structural model is correct, the two groups of boys and girls were compared. 1. The factor loadings were set as equal between the two groups, and the model structure had a total of 26 factor loadings that were set as equal (DF = 26), with a cardinality (CMIN) increase of 27.082, and the \(p\)-value of the test result was 0.405, which did not reach the significance level (\(p<2\). In addition to maintaining the limitations of the measurement model, three additional structural path coefficients were set, and the chi-squared value (CMIN) increased by 2.695, with a \(p\)-value of 0.000, reaching a significance level of \(p<0.05\), indicating that the three structural path coefficients were not equal. The coefficients are not all equal, but the \(\Delta\)CFI is 0.000, which does not reach the standard indicator of 0.05 and meets the criterion of \(\Delta\)CFI \(\leq 0.05\), so these three conformational path coefficients are in the acceptable range. 2. In addition to maintaining the structural model path coefficients, the settings of the variance and covariance matrices are increased by one more, and the chi-squared value (CMIN) increased by 0.498, with a check result of \(p=0.000\), reaching a significance level of \(p<0.05\), but the \(\Delta\)CFI is 0.000, not reaching the standard indicator of 0.05, and meeting the criterion of \(\Delta\)CFI \(\leq 0.05\), indicating that it is acceptable for this one variance and covariance to be set equal, so this one variance and covariance are equal. As can be seen from Table 7, under the assumption that the structural model is correct, there is a comparison of the two clusters of participants in after-school services with those not participating in after-school services. 1. Setting the factor loadings equal between the two clusters, the model structure has a total of 20 factor loadings that are set as equal (DF = 20) and the chi-squared value (CMIN) increases by 23.519, with a check result \(p\)-value of 0.264, which does not reach the significant level. In addition to maintaining the limitations of the measurement model, six additional structural path coefficients were set