小学儿童数学焦虑的潜在类别转变及其父母教育卷入效应：3年纵向考察

doi:10.3724/SP.J.1041.2022.00355

摘要/Abstract

摘要：

研究采用潜在转变分析考察小学儿童数学焦虑的类别转变以及父母教育卷入在小学儿童数学焦虑类别转变中的作用。以1720名三、四年级儿童为被试, 对其数学焦虑和感知到的父母教育卷入进行3次追踪, 每次间隔1年。结果表明：(1)小学儿童数学焦虑存在低数学焦虑组、高数学评估焦虑组和高数学获得焦虑组3种不同类别; (2)随时间的推移, 高数学评估焦虑组倾向于向低数学焦虑组转变, 高数学获得焦虑组倾向于向高数学评估焦虑组转变, 而低数学焦虑组稳定性较强; (3)父亲/母亲教育卷入对儿童数学焦虑类别转变的预测作用, 因不同的数学焦虑类别而异。上述发现为深入理解数学焦虑的形成机制以及干预措施的制定提供了重要参考。

关键词: 小学儿童, 数学焦虑, 父母教育卷入, 潜在转变分析, 纵向研究

Abstract:

Mathematics anxiety is a sense of tense and anxious that an individual feels when solving the problems related to mathematics. This phenomenon has a considerable prevalence among children and youth, even in adults. Currently, most studies regard mathematics anxiety as a single-dimensional structure. However, mathematics anxiety is a multi-dimensional structure. For example, individuals with high mathematics learning anxiety are often associated with low mathematics achievement, while individuals with high mathematics evaluation anxiety do not necessarily lead to low mathematics achievement. And the dynamic developmental bio-psycho-social model holds that the interaction between individual factors and environmental factors makes the development of individuals’ mathematics anxiety heterogeneity. As individual factors and environmental factors are constantly developing and changing, the developmental trajectories of mathematics anxiety are dynamic. However, there were no studies has examined the individual heterogeneity of mathematics anxiety and the development and transitions of it from a longitudinal perspective. Moreover, parental educational involvement as one of important environmental factors might predict the transitions of mathematics anxiety over time. Thus, person-centered approach was used to solve these two problems in this study.
In this study, 1720 students of grade three and grade four in county primary schools were selected as participants. Children's Mathematics Anxiety Scale compiled by Chiu and Henry (1990) and revised by Geng and Chen (2005) and Questionnaire on Parental Involvement Behavior of Primary School Students compiled by Wu, Han, Wei, and Luo (2013) were used to measure children's mathematics anxiety and their perceived parental educational involvement separately three times over three years. Latent profile analysis and latent transition analysis were used to explore the possible subtypes of children's mathematics anxiety and the transitions between different subtypes over three waves in this study. Multiple logistic regressions were used to examine the effect of parental educational involvement in the latent transitions of different mathematics anxiety subtypes.
All data were analyzed by SPSS 22.0 and Mplus 8.0. Some valuable results were obtained as follows. (1)There were three different subgroups of mathematics anxiety in primary school children, including low mathematics anxiety group, high mathematics evaluation anxiety group and high mathematics acquisition anxiety group; (2)As time went by high mathematics evaluation anxiety group tended to change to low mathematics anxiety group, high mathematics acquisition anxiety group tended to change to high mathematics evaluation anxiety group, and low mathematics anxiety group were relatively stable; (3) Positive father involvement could promote the change of children's mathematics anxiety from high mathematics acquisition anxiety group to low mathematics anxiety group, which was mainly in girls. For girls, mother involvement was able to promote the change of their mathematics anxiety from high mathematics evaluation anxiety group to low mathematics anxiety group; however for boys, mother involvement was able to promote the change of their mathematics anxiety from high mathematics evaluation anxiety group to high mathematics acquisition anxiety group. For the low mathematics anxiety group, the positive effect of parental educational involvement was significant.
There was group heterogeneity in mathematics anxiety, and distinct subtypes of individuals may change over time, and parental educational involvement played different roles in different subgroups of children's mathematics anxiety. This study confirmed that the dynamic developmental bio-psycho-social model hypothesized that different individuals were affected by the different interaction of individual factors and environmental factors, and there were heterogeneity and dynamics in the developmental trajectories of individual mathematics anxiety. In view of this, parents or teachers should use different teaching methods for different subtypes of mathematics anxiety in mathematics learning. In addition, future researchers should consider individual heterogeneity of mathematics anxiety.

Key words: primary school children, mathematics anxiety, parental educational involvement, latent transition analysis, longitudinal study

中图分类号:

B844

司继伟, 郭凯玥, 赵晓萌, 张明亮, 李红霞, 黄碧娟, 徐艳丽. (2022). 小学儿童数学焦虑的潜在类别转变及其父母教育卷入效应：3年纵向考察. 心理学报, 54(4), 355-370.

SI Jiwei, GUO Kaiyue, ZHAO Xiaomeng, ZHANG Mingliang, LI Hongxia, HUANG Bijuan, XU Yanli. (2022). Transition of latent classes of children’s mathematics anxiety in primary school and the distinctive effects of parental educational involvement: A three-wave longitudinal study. Acta Psychologica Sinica, 54(4), 355-370.

图/表 10

参考文献 50

[1]	Bai, X. W., & Chen, Y. W. (2004). Measurement equivalence: Conception and test conditions. Advances in Psychological Science, 12(2), 231-239.
	[ 白新文, 陈毅文. (2004). 测量等价性的概念及其判定条件. 心理科学进展, 12(2), 231-239.]
[2]	Barger, M. M., Kim, E. M., Kuncel, N. R., & Pomerantz, E. M. (2019). The relation between parents' involvement in children's schooling and children's adjustment: A meta- analysis. Psychological Bulletin, 145(9), 855-980. doi: 10.1037/bul0000201 URL
[3]	Barroso, C., Ganley, C. M., McGraw, A. L., Geer, E. A., Hart, S. A., & Daucourt, M. C. (2020). A meta-analysis of the relation between math anxiety and math achievement. Psychological Bulletin. 147(2), 134-168. doi: 10.1037/bul0000307 URL
[4]	Bretherton, I. (2010). Fathers in attachment theory and research: A review. Early Child Development and Care, 180(1-2), 9-23. doi: 10.1080/03004430903414646 URL
[5]	Bronfenbrenner, U. (1979). The ecology of human development. Cambridge, MA: Havard University Press.
[6]	Cheung, S. S., & Pomerantz, E. M. (2011). Parents’ involvement in children’s learning in the United States and China: Implications for children’s academic and emotional adjustment. Child Development, 82(3), 932-950. doi: 10.1111/cdev.2011.82.issue-3 URL
[7]	Ching, H. H., Wu, H. X., & Chen, T. T. (2021). Maternal achievement-oriented psychological control: Implications for adolescent academic contingent self-esteem and mathematics anxiety. International Journal of Behavioral Development, 45(3), 193-203. doi: 10.1177/0165025420981638 URL
[8]	Chiu, L., & Henry, L. L. (1990). Development and validation of the mathematics anxiety scale for children. Measurement & Evaluation in Counseling & Development, 23(3), 121-127.
[9]	Collins, L. M., & Lanza, S. T. (2010). Latent class and latent transition analysis. Hoboken, NJ: Wiley.
[10]	Del Rio, M. F., Susperreguy, M. I., Strasser, K., & Salinas, V. (2017). Distinct influences of mothers and fathers on kindergartners' numeracy performance: The role of math anxiety, home numeracy practices, and numeracy expectations. Early Education and Development, 28(8), 939-955. doi: 10.1080/10409289.2017.1331662 URL
[11]	Demirtaş, A. S., & Uygun-Eryurt, T. (2020). Attachment to parents and math anxiety in early adolescence: Hope and perceived school climate as mediators. Current Psychological Research. doi: 10.1007/s12144-020-00964-1 doi: 10.1007/s12144-020-00964-1
[12]	Devine, A., Fawcett, K., Szücs, D., & Dowker, A. (2012). Gender differences in mathematics anxiety and the relation to mathematics performance while controlling for test anxiety. Behavioral and Brain Functions, 8, 33. doi: 10.1186/1744-9081-8-33 doi: 10.1186/1744-9081-8-33 pmid: 22769743
[13]	DiStefano, M., O’Brien, B., Storozuk, A., Ramirez, G., & Maloney, E. A. (2020). Exploring math anxious parents’ emotional experience surrounding math homework-help. International Journal of Educational Research, 99, 101526. doi: 10.1016/j.ijer.2019.101526 doi: 10.1016/j.ijer.2019.101526 URL
[14]	Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics anxiety: What have we learned in 60 years? Frontiers in Psychology, 7, 508. doi: 10.3389/fpsyg.2016.00508 doi: 10.3389/fpsyg.2016.00508 pmid: 27199789
[15]	Fan, X., Hambleton, R. K., & Zhang, M. (2019). Profiles of mathematics anxiety among 15-year-old students: A cross- cultural study using multi-group latent profile analysis. Frontiers in Psychology, 10, 1217. doi: 10.3389/fpsyg.2019.01217 doi: 10.3389/fpsyg.2019.01217 URL
[16]	Geng, L. N., & Chen, Y. H. (2005). The relationship between children’s mathematics anxiety and add-and-subtract cognitive strategy selection and implement. Psychological Development and Education, 21(4), 24-27.
	[ 耿柳娜, 陈英和. (2005). 数学焦虑对儿童加减法认知策略选择和执行的影响. 心理发展与教育, 21(4), 24-27.]
[17]	Harari, R. R., Vukovic, R. K., & Bailey, S. P. (2013). Mathematics anxiety in young children: An exploratory study. The Journal of Experimental Education, 81(4), 538-555. doi: 10.1080/00220973.2012.727888 URL
[18]	Hembree, R. (1990). The nature, effects and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21(1), 33-46. doi: 10.2307/749455 URL
[19]	Hickendorff, M., Edelsbrunner, P. A., McMullen, J., Schneider, M., & Trezise, K. (2018). Informative tools for characterizing individual differences in learning: Latent class, latent profile, and latent transition analysis. Learning and Individual Differences, 66, 4-15. doi: 10.1016/j.lindif.2017.11.001 URL
[20]	Hong, S., Yoo, S.-K., You, S., & Wu, C.-C. (2010). The reciprocal relationship between parental involvement and mathematics achievement: Autoregressive cross-lagged modeling. Journal of Experimental Education, 78(4), 419-439. doi: 10.1080/00220970903292926 URL
[21]	Hopko, D. R., Mahadevan, R., Bare, R. L., & Hunt, M. A. (2003). The abbreviated math anxiety scale (AMAS): Construction, validity, and reliability. Assessment, 10(2), 178-182. doi: 10.1177/1073191103010002008 URL
[22]	Jung, T., & Wickrama, A. S. (2008). An introduction to latent class growth analysis and growth mixture modeling. Social and Personality Psychology Compass, 2(1), 302-317. doi: 10.1111/spco.2008.2.issue-1 URL
[23]	Lerner, R. M. (2002). Concepts and theories of human development (3rd ed). London: Lawrence Erlbaum Associates.
[24]	Li, X., Han, M., Cohen, G. L., & Markus, H. R. (2021). Passion matters but not equally everywhere: Predicting achievement from interest, enjoyment, and efficacy in 59 societies. Proceedings of the National Academy of Sciences, 118(11), e2016964118. doi: 10.1073/pnas.2016964118 URL
[25]	Lital, D. C., & Orly, R. (2017). Mothers, intrinsic math motivation, arithmetic skills, and math anxiety in elementary school. Frontiers in Psychology, 8, 1939. doi: 10.3389/fpsyg.2017.01939 doi: 10.3389/fpsyg.2017.01939
[26]	Lubke, G. H., & Muthén, B. (2007). Investigating population heterogeneity with factor mixture models. Psychology Methods, 10(1), 21-39. doi: 10.1037/1082-989X.10.1.21 URL
[27]	Luo, L., Wu, Y. F., & Wei, W. (2014). Properties of parent- involved education with high quality. Journal of Beijing Normal University (Social Sciences), (1), 53-60.
	[ 罗良, 吴艺方, 韦唯. (2014). 高质量父母教育卷入的特征. 北京师范大学学报(社会科学版), (1), 53-60.]
[28]	Ma, M., Li, D., & Zhang, L. (2021). Longitudinal prediction of children's math anxiety from parent-child relationships. Learning and Individual Differences, 88(4), 102016. doi: 10.1016/j.lindif.2021.102016 URL
[29]	Maloney, E. A., Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2015). Intergenerational effects of parents' math anxiety on children's math achievement and anxiety. Psychological Science, 26(9), 1480-1488. doi: 10.1177/0956797615592630 pmid: 26253552
[30]	McKenna, J. S., & Nickols, S. Y. (1988). Planning for retirement security: What helps or hinders women in the middle years? Home Economics Research Journal, 17(2), 153-164. doi: 10.1177/1077727X8801700204 URL
[31]	Nylund, K. L. (2007). Latent transition analysis: Modeling extensions and an application to peer victimization (Unpublished doctorial dissertation). University of California, Los Angeles.
[32]	OECD. (2013). PISA 2012 assessment and analytical framework: Mathematics, reading, science, problem solving, and financial literacy. Paris: OECD Publishing.
[33]	Rolison, J. J., Morsanyi, K., & Peters, E. (2020). Understanding health risk comprehension: The role of math anxiety, subjective numeracy, and objective numeracy. Medical Decision Making, 40(2), 222-234. doi: 10.1177/0272989X20904725 URL
[34]	Rubinsten, O., Marciano, H., Levy, H. E., & Cohen, L. D. (2018). A framework for studying the heterogeneity of risk factors in math anxiety. Frontiers in Behavioral Neuroscience, 12, 291. doi: 10.3389/fnbeh.2018.00291 doi: 10.3389/fnbeh.2018.00291 pmid: 30559654
[35]	Sarason, I. G. (1978). Test anxiety scale:Concept and research. In C. D. Spielberger & I. G. Sarason (Eds.), Stress and anxiety: Vol. 5 (pp. 193-216). Washington D. C: Hemisphere Publishing Corp.
[36]	Si, J. W., Xu, Y. L., Feng, H. M., Xu, X. H., & Zhou, C. (2014). Differences of arithmetic strategy use in adults with different math anxieties: An ERP study. Acta Psychologica Sinica, 46(12), 1835-1849. doi: 10.3724/SP.J.1041.2014.01835 URL
	[ 司继伟, 徐艳丽, 封洪敏, 许晓华, 周超. (2014). 不同数学焦虑成人的算术策略运用差异:ERP研究. 心理学报, 46(12), 1835-1849.]
[37]	Szczygieł, M. (2020). When does math anxiety in parents and teachers predict math anxiety and math achievement in elementary school children? The role of gender and grade year. Social Psychology of Education, 23(4), 1023-1054. doi: 10.1007/s11218-020-09570-2 URL
[38]	Tomasetto, C., Mirisola, A., Galdi, S., & Cadinu, M. (2015). Parents’ math-gender stereotypes, children’s self-perception of ability, and children’s appraisal of parents’ evaluations in 6-year-olds. Contemporary Educational Psychology, 42, 186-198. doi: 10.1016/j.cedpsych.2015.06.007 URL
[39]	Tomasetto, C., Morsanyi, K., Guardabassi, V., & O’Connor, P. A. (2021). Math anxiety interferes with learning novel mathematics contents in early elementary school. Journal of Educational Psychology, 113(2), 315-329. doi: 10.1037/edu0000602 URL
[40]	Vanbinst, K., Bellon, E., & Dowker, A. (2020). Mathematics anxiety: An intergenerational approach. Frontiers in Psychology, 11, 1648. doi: 10.3389/fpsyg.2020.01648 doi: 10.3389/fpsyg.2020.01648 pmid: 32793047
[41]	Vukovic, R. K., Roberts, S. O., & Wright, G. L. (2013). From parental involvement to children's mathematical performance: The role of mathematics anxiety. Early Education and Development, 24(4), 446-467. doi: 10.1080/10409289.2012.693430 URL
[42]	Wang, B. Y., Zhang, M. Q., Zhang, J. T., & Hu, J. (2015). Describing the stage process of individual through transition matrix: Latent transition model. Psychological Research, 8(4), 36-43.
	[ 王碧瑶, 张敏强, 张洁婷, 胡俊. (2015). 基于转变矩阵描述的个体阶段性发展:潜在转变模型. 心理研究, 8(4), 36-43.]
[43]	Wang, C. K. (2001). Reliability and validity of test anxiety scale-Chinese version. Chinese Mental Health Journal, 15(2), 96-97.
	[ 王才康. (2001). 考试焦虑量表在大学生中的测试报告. 中国心理卫生杂志, 15(2), 96-97.]
[44]	Wang, M. T., & Degol, J. L. (2017). Gender gap in science, technology, engineering, and mathematics (STEM): Current knowledge, implications for practice, policy, and future directions. Educational Psychology Review, 29(1), 119-140. doi: 10.1007/s10648-015-9355-x URL
[45]	Wang, Z., Oh, W., Malanchini, M., & Borriello, G. A. (2020). The developmental trajectories of mathematics anxiety: Cognitive, personality, and environmental correlates. Contemporary Educational Psychology, 61, 101876. doi: 10.1016/j.cedpsych.2020.101876 doi: 10.1016/j.cedpsych.2020.101876 URL
[46]	Wang, Z., Shakeshaft, N., Schofield, K., & Malanchini, M. (2018). Anxiety is not enough to drive me away: A latent profile analysis on math anxiety and math motivation. PLoS ONE, 13(2), e0192072. doi: 10.1371/journal.pone.0192072 doi: 10.1371/journal.pone.0192072 URL
[47]	Wu, Y. F., Hang, X. H., Wei, W., & Luo, L. (2013). Theoretical model construction and testing of parental involvement in primary school children education. Journal of Beijing Normal University (Social Sciences), (1), 61-69.
	[ 吴艺方, 韩秀华, 韦唯, 罗良. (2013). 小学生父母教育卷入行为理论模型的建构与验证. 北京师范大学学报(社会科学版), (1), 61-69.]
[48]	Xiao, F., & Sun, L. (2021). Students' motivation and affection profiles and their relation to mathematics achievement, persistence, and behaviors. Frontiers in Psychology, 11, 533593. doi: 10.3389/fpsyg.2020.533593 doi: 10.3389/fpsyg.2020.533593 URL
[49]	Zhang, W. X., & Chen, G. H. (2009). Developmental contextualism: An instance of development system theories. Advances in Psychological Science, 17(4), 736-744.
	[ 张文新, 陈光辉. (2009). 发展情境论--一种新的发展系统理论. 心理科学进展, 17(4), 736-744.]
[50]	Zhou, H., & Long, L. R. (2004). Statistical remedies for common method biases. Advances in Psychological Science, 12(6), 942-950.
	[ 周浩, 龙立荣. (2004). 共同方法偏差的统计检验与控制方法. 心理科学进展, 12(6), 942-950.]

模型	S-Bχ²(df )	CFI	TLI	RMSEA	∆χ²(∆df )	p	∆CFI
父亲教育卷入
Model 1	8652.034 (3462)	0.931	0.925	0.030	-	-	-
Model 2	8805.099 (3510)	0.929	0.925	0.030	153.065 (48)	< 0.01	0.002
Model 3	9094.466 (3558)	0.926	0.922	0.030	289.367 (48)	< 0.01	0.003
母亲教育卷入
Model 1	7733.688 (3462)	0.926	0.920	0.027	-	-	-
Model 2	7923.238 (3510)	0.924	0.919	0.027	189.550 (48)	< 0.01	0.002
Model 3	8425.155 (3558)	0.916	0.912	0.028	501.917 (48)	< 0.01	0.008
数学焦虑
Model 1	3951.169 (1947)	0.956	0.951	0.024	-	-	-
Model 2	4032.857 (1983)	0.955	0.951	0.025	81.688 (36)	< 0.01	0.001
Model 3	4192.570 (2019)	0.952	0.949	0.025	159.713 (36)	< 0.01	0.003

模型	S-Bχ²(df )	CFI	TLI	RMSEA	∆χ²(∆df )	p	∆CFI
父亲教育卷入
Model 1	8652.034 (3462)	0.931	0.925	0.030	-	-	-
Model 2	8805.099 (3510)	0.929	0.925	0.030	153.065 (48)	< 0.01	0.002
Model 3	9094.466 (3558)	0.926	0.922	0.030	289.367 (48)	< 0.01	0.003
母亲教育卷入
Model 1	7733.688 (3462)	0.926	0.920	0.027	-	-	-
Model 2	7923.238 (3510)	0.924	0.919	0.027	189.550 (48)	< 0.01	0.002
Model 3	8425.155 (3558)	0.916	0.912	0.028	501.917 (48)	< 0.01	0.008
数学焦虑
Model 1	3951.169 (1947)	0.956	0.951	0.024	-	-	-
Model 2	4032.857 (1983)	0.955	0.951	0.025	81.688 (36)	< 0.01	0.001
Model 3	4192.570 (2019)	0.952	0.949	0.025	159.713 (36)	< 0.01	0.003

变量	T1父亲教育卷入	T2父亲教育卷入	T3父亲教育卷入	T1母亲教育卷入	T2母亲教育卷入	T3母亲教育卷入	T1儿童数学焦虑	T2儿童数学焦虑	T3儿童数学焦虑
性别
男	2.55 (0.71)	2.68 (0.79)	2.70 (0.73)	2.85 (0.61)	3.10 (0.63)	3.16 (0.63)	1.93 (0.62)	1.86 (0.67)	1.77 (0.68)
女	2.60 (0.69)	2.73 (0.74)	2.61 (0.73)	2.90 (0.60)	3.10 (0.62)	3.08 (0.64)	1.82 (0.60)	1.80 (0.62)	1.79 (0.61)
年级
三年级	2.52 (0.70)	2.59 (0.74)	2.67 (0.71)	2.78 (0.60)	3.01 (0.61)	3.14 (0.62)	1.99 (0.64)	1.92 (0.64)	1.83 (0.66)
四年级	2.62 (0.70)	2.82 (0.79)	2.66 (0.75)	2.96 (0.61)	3.19 (0.62)	3.11 (0.65)	1.78 (0.57)	1.75 (0.65)	1.73 (0.63)
性别(r)	0.04	0.03	-0.06*	0.04	-0.01	-0.06*	-0.09**	-0.05*	0.01
年级(r)	0.07**	0.15**	-0.01	0.14**	0.14**	-0.02	-0.17**	-0.14**	-0.08**
考试焦虑(r)	-0.13**	-0.14**	-0.14**	-0.10**	-0.11**	-0.09**	0.39**	0.29**	0.24**
T1数学成绩(r)	0.01	0.03	0.02	0.06*	0.03	0.02	-0.25**	-0.23**	-0.26**
T2数学成绩(r)	0.06*	0.08**	0.09**	0.10**	0.08**	0.09**	-0.26**	-0.24**	-0.31**
T3数学成绩(r)	0.09**	0.04	0.05*	0.10**	0.07**	0.06**	-0.26**	-0.27**	-0.33**

变量	T1父亲教育卷入	T2父亲教育卷入	T3父亲教育卷入	T1母亲教育卷入	T2母亲教育卷入	T3母亲教育卷入	T1儿童数学焦虑	T2儿童数学焦虑	T3儿童数学焦虑
性别
男	2.55 (0.71)	2.68 (0.79)	2.70 (0.73)	2.85 (0.61)	3.10 (0.63)	3.16 (0.63)	1.93 (0.62)	1.86 (0.67)	1.77 (0.68)
女	2.60 (0.69)	2.73 (0.74)	2.61 (0.73)	2.90 (0.60)	3.10 (0.62)	3.08 (0.64)	1.82 (0.60)	1.80 (0.62)	1.79 (0.61)
年级
三年级	2.52 (0.70)	2.59 (0.74)	2.67 (0.71)	2.78 (0.60)	3.01 (0.61)	3.14 (0.62)	1.99 (0.64)	1.92 (0.64)	1.83 (0.66)
四年级	2.62 (0.70)	2.82 (0.79)	2.66 (0.75)	2.96 (0.61)	3.19 (0.62)	3.11 (0.65)	1.78 (0.57)	1.75 (0.65)	1.73 (0.63)
性别(r)	0.04	0.03	-0.06*	0.04	-0.01	-0.06*	-0.09**	-0.05*	0.01
年级(r)	0.07**	0.15**	-0.01	0.14**	0.14**	-0.02	-0.17**	-0.14**	-0.08**
考试焦虑(r)	-0.13**	-0.14**	-0.14**	-0.10**	-0.11**	-0.09**	0.39**	0.29**	0.24**
T1数学成绩(r)	0.01	0.03	0.02	0.06*	0.03	0.02	-0.25**	-0.23**	-0.26**
T2数学成绩(r)	0.06*	0.08**	0.09**	0.10**	0.08**	0.09**	-0.26**	-0.24**	-0.31**
T3数学成绩(r)	0.09**	0.04	0.05*	0.10**	0.07**	0.06**	-0.26**	-0.27**	-0.33**

时间点	类别数目	AIC	BIC	aBIC	Entropy	LMRT (p)	BLRT (p)	所占比例(%)
T1	2	16547.51	16618.36	16577.06	0.90	< 0.01	< 0.01	71.3 / 28.7
	3	15381.52	15479.63	15422.44	0.86	< 0.01	< 0.01	49.2 / 35.2 / 15.6
	4	14698.64	14824.00	14750.93	0.88	< 0.01	< 0.01	45.8 / 32.2 / 4.4 / 17.6
	5	14480.28	14632.88	14543.93	0.83	>0.05	< 0.01	38.4 / 28.1 / 14.7 / 4.3 / 14.5
T2	2	16291.11	16361.96	16320.66	0.92	< 0.01	< 0.01	71.3 / 28.7
	3	15070.49	15168.59	15111.41	0.88	< 0.01	< 0.01	54.2 / 32.8 / 13.0
	4	14336.19	14461.55	14388.48	0.90	< 0.01	< 0.01	49.3 / 30.3 / 2.7 / 17.7
	5	14015.38	14167.98	14079.03	0.90	>0.05	< 0.01	46.6 / 9.6 / 14.1 / 2.2 / 27.5
T3	2	15736.23	15807.08	15765.78	0.94	< 0.01	< 0.01	76.3 / 23.7
	3	14499.57	14597.67	14540.49	0.93	< 0.05	< 0.01	68.1 / 24.7 / 7.2
	4	13692.59	13817.94	13744.87	0.90	< 0.05	< 0.01	53.3 / 27.6 / 15.2 / 3.9
	5	13374.71	13527.32	13438.36	0.88	>0.05	< 0.01	48.9 / 9.2 / 14.3 / 25.4 / 2.2