## 4 Results
### Results of Correlation Analysis
This may be related to the frequency with which the mathematics teacher uses the homework release function. The mathematics teacher will check the homework report promptly and re-release exercises that students are prone to make mistakes on for students to practice, leading to a reduced frequency of students' autonomous similar problem exercises, but still achieving the learning effects. The improvement in chemistry subject scores is positively correlated with each student's autonomous practice data; the similar problem exercises, especially, show a relatively strong correlation. Looking back, the chemistry subject score improvement, ranking second, the teacher's frequency of using homework functions is not high, and it is speculated that the reason for this is the high frequency of students' autonomous similar problem and personalized problem exercises. Physics and biology both show a positive correlation in similar problem exercises, proving that the frequency of students' self-directed gap-checking exercises affects their score improvement. Typical Student Improvement Ranking: In order to further investigate the correlation between students' self-directed learning and their learning outcomes under personalized learning paths, the researchers selected typical students for analysis. Researchers will rank the students in the intervention class for mathematics, physics, chemistry, and biology according to their improvement in rank, select the top five students who made the greatest progress and the five students who experienced the greatest regression in each subject, and analyze their autonomous learning data to explore its correlation. The individual's autonomous practice data includes the number of similar problems completed and the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Subject** & **Homework Completion Rate** & \
\begin{tabular}{c} **Similar Questions** \\ **Completed Count** \\ \end{tabular} &
\begin{tabular}{c} **Personalized Exercises** \\ **Completed Count** \\ \end{tabular} \\ \hline M & Y = 0.0031 \(\times\) X + 9.663 & Y = \(-\)0.5400 \(\times\) X + 30.81 & Y = 0.0167 \(\times\) X + 4.917 \\ P & Y = \(-\)0.1662 \(\times\) X + 95.13 & Y = 1.277 \(\times\) X + 21.50 & Y = \(-\)0.0298 \(\times\) X + 3.857 \\ C & Y = 0.4216 \(\times\) X + 95.66 & Y = 4.283 \(\times\) X + 5.579 & Y = 2.174 \(\times\) X\(-\)2.325 \\ B & Y = \(-\)0.2373 \(\times\) X + 94.84 & Y = 1.306 \(\times\) X + 35.34 & Y = \(-\)0.4493 \(\times\) X + 21.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Class 1-6 students’ self-directed learning data table. number of personalized problems completed. The analysis results are shown in Table 4 below. The data in the table show that for mathematics, physics, and chemistry, the slope of the function corresponding to the degree of improvement of the typical students is positive, proving that there is a positive correlation. For the biology subject, the slope of the function corresponding to the degree of improvement of the typical students is negative with the number of personalized exercises, showing no correlation, but it is positively correlated with the number of similar problem exercises. Combining this with the \"Results of correlation analysis of Class student self-directed learning,\" it is concluded that practice with similar problems helps to improve the scores in mathematics, physics, chemistry, and biology. The two subjects with greater progress in scores, mathematics and chemistry, also show a positive impact in terms of homework completion rate and the number of personalized exercises practiced. This also answers the second question raised above: the quantity of students' autonomous practice with similar problems is the reason for the difference in effectiveness of the precision teaching model in teaching mathematics, physics, chemistry, and biology. The homework completion rate and the number of personalized exercises only affect the two subjects of mathematics and chemistry.