Associations Between School Climate and Student Mathematics Achievement: A Multilevel Structural Equation Modeling Approach

Abstract

Students’ academic achievement is related to various factors. While extensive research has examined students’ confidence and perceived difficulty in mathematics, relatively few studies have comprehensively examined the associations between school climate and student achievement in mathematics within a multilevel framework. This study aims to investigate the associations between school climate-related factors and students’ mathematics achievement using multilevel structural equation modeling. This study used U.S. data from the 2019 Trends in International Mathematics and Science Study and analyzed 7589 eighth-grade students across 235 schools, focusing on nine school climate-related factors. The student-level factors were school belonging and safety, mathematics class climate, and bullying, while the school-level factors were mathematics resources, school discipline and safety, teacher characteristics, parental characteristics, student characteristics, and school support. The intraclass correlation coefficients for mathematics scores ranged from 0.340 to 0.351, indicating the importance of school-level factors. This study found that, at the student level, school belonging and safety and mathematics class climate were positively associated with students’ mathematics achievement, whereas bullying was negatively associated with both mathematics achievement and the aforementioned factors. At the school level, shortages of resources and school discipline and safety were negatively associated with students’ mathematics achievement. The findings highlight the importance of school safety, classroom climates, bullying prevention, and resources for mathematics in school.

1. Introduction

Mathematics provides the basis for other STEM fields, and proficiency in mathematics is closely associated with students’ academic success as well as their preparation for college and future careers (Claessens & Engel, 2013; Cogan et al., 2018; Just & Siller, 2022). Thus, many studies have examined various factors associated with mathematics achievement, confidence or self-efficacy in mathematics, socioeconomic status (SES), and school climate. Specifically, students’ confidence or self-efficacy in mathematics has been found to be positively related to their mathematics achievement across many countries, for example, the U.S. and Singapore (Ker, 2016), South Korea (House & Telese, 2016), Morocco (Chatri et al., 2021), Hong Kong and Singapore (Chen, 2014), and Malaysia and Singapore (Ghagar et al., 2011). SES-related factors, such as the number of books, the possession of computers, and parents’ educational backgrounds, were also positively associated with students’ mathematics scores (e.g., Takashiro, 2017).

Additionally, some studies have examined students’ mathematics achievement through school climate, a five-dimensional framework, including school safety, relationships, teaching and learning, institutional environment (resources and supplies), and improvement efforts (Thapa et al., 2013). Shindler et al. (2016) found a strong correlation (r = 0.7) between school climate and students’ achievement. Other studies have identified similar results. Studies reported a positive relationship between school climate and students’ academic achievement (e.g., Bear et al., 2014, 2015; Benbenishty et al., 2016; Daily et al., 2019; Sakız, 2017; Thapa et al., 2013).

However, these studies focused on either school-level or student-level factors and did not address the hierarchical structure of students nested within the school. In contrast, other studies explored the relationship between school climate and student outcomes in multilevel frameworks, such as hierarchical linear modeling (HLM). For example, Ghagar et al. (2011) applied HLM with the 2003 Trends in International Mathematics and Science Study (TIMSS) data on eighth-grade students in Malaysia and Singapore and found that the school climate, as perceived by school principals, was the most important factor in students’ mathematics scores. W. Wang et al. (2014) also utilized HLM and highlighted that school climate was positively associated with fifth-grade students’ academic achievement, identifying both direct and indirect associations between school climate and students’ academic achievement.

1.1. Methodological Considerations

A multilevel framework can provide deeper insights into contextual patterns by examining how school factors are associated with student-level academic outcomes. Failing to account for this structure can lead to misinterpretation of results due to ecological or atomistic fallacies, in which inferences are made at inappropriate levels of analysis (Hox et al., 2017). Thus, incorporating a multilevel approach helps capture the nested nature of educational data and investigate the interactions between student- and school-level factors.

However, HLM has several limitations that can affect the accuracy and interpretability of multilevel analyses. First, HLM typically relies on observed variables or scale means, which can result in measurement errors. The unreliability of observed variables may reduce the power of regression estimates. Second, it has constraints in addressing mediation pathways within a multilevel framework, as it typically requires a multi-step estimation process, which can introduce bias. Third, assessing overall model fit is challenging in the HLM framework (Preacher et al., 2010).

Multilevel structural equation modeling (MSEM) has several advantages that address the above limitations (Preacher et al., 2010). MSEM employs latent variables at both the student and school levels. This feature of MSEM helps to reduce measurement error and improve the precision of factor modeling. It also allows indirect associations to be examined within a multilevel framework by estimating all paths simultaneously, which avoids the need for multi-step estimation procedures. Moreover, MSEM provides fit indices in order to evaluate overall model adequacy (Preacher et al., 2010).

Bayesian structural equation modeling (BSEM) can be used as an alternative because it is flexible in parameter estimation and can effectively handle complex models, particularly when the sample size is relatively small (B. Muthén & Asparouhov, 2012).

Given the relatively large sample size, this study employed the MSEM approach because it is well-suited for modeling hierarchical data using maximum likelihood estimation. With the methodological approach established, it is important to define the key factors in this study. In particular, school climate is a multi-faceted concept that requires a comprehensive understanding from both student and school perspectives.

1.2. School Climate at the Student and School Levels

School climate is a multi-faceted concept that requires a comprehensive understanding from various perspectives. Since there is no unified definition of school climate (Cohen et al., 2009; Kutsyuruba et al., 2015; Thapa et al., 2013), previous studies have used different definitions and measures. Berkowitz et al. (2017) also pointed out that the definition and measurement of school climate lacks clarity and consistency. As a result, the term ‘school climate’ is often used to cover a wide range of aspects within the school environment, such as safety, interpersonal relationships, and perceptions of teaching and learning (Cohen et al., 2009; Gase et al., 2017; M.-T. Wang & Degol, 2016). Some researchers defined school climate as the overall quality and character of life within a school that reflects teaching practices, interpersonal relationships, norms, values, goals, organizational structures, and expectations that foster a sense of safety (Cohen et al., 2009; National School Climate Council, 2007). Other researchers emphasize that school climate stems from the quality of interpersonal relationships among students, school personnel, parents, and administrators (Kutsyuruba et al., 2015).

According to Thapa et al. (2013), the school climate includes the social, emotional, ethical, civic, and academic experiences of students, parents, and school staff in the school environment. They identified five dimensions of school climate: school safety, relationships, teaching and learning, institutional environment, and school improvement processes. Similarly, the National School Climate Center (NSCC, n.d.) proposed five dimensions of school climate, which include the first four dimensions identified by Thapa et al. (2013), along with leadership and efficacy as the fifth dimension.

The current study drew nine relevant factors from the TIMSS 2019 data (Mullis et al., 2020) based on the five dimensions from Thapa et al. (2013) and National School Climate Center (NSCC, n.d.). The nine factors consisted of three for the student level and six for the school level. Student-level factors included bullying, school belonging and safety, and mathematics class climate. These three student-level factors correspond to the dimensions of school safety and interpersonal relationships. School-level factors included school resources; school support; discipline and safety; and principals’ perceptions of parental, teacher, and student characteristics. The discipline and safety factor corresponds to the school safety dimension at the school level, while school resources and support correspond to the institutional environment dimension. Principals’ perceptions of parental, teacher, and student characteristics correspond to the leadership and efficacy dimension.

These student- and school-level factors were utilized to specify a structural equation model to investigate the association between school climate and student mathematics achievement (see Figure 1). The direct and indirect associations between school climate and mathematics achievement are theoretically based on ecological system theory (Bronfenbrenner, 1979). This study conceptualizes students’ mathematics achievement as a developmental outcome shaped by interconnected environmental systems, especially the microsystem where individuals are directly related to their family members, teachers, peers, and schools. The following section reviews prior studies that provide support for the proposed model.

1.2.1. Student-Level Factors

• Bullying

Bullying is a widespread form of violence within schools, where students experience it either as victims, bullies, or both (Mohtar et al., 2019; Yang & Salmivalli, 2013). Bullying includes various forms, such as verbal abuse, threats, physical assaults, language, and criticisms, and has been mostly shown to have a negative association with students’ academic achievement (Al-Raqqad et al., 2017; Konishi et al., 2010; Topçu et al., 2016) and students’ commitment to schoolwork (Thapa et al., 2013). Specifically, bullying demonstrated a negative relationship with mathematics achievement at school (Konishi et al., 2010). In another study using TIMSS 2011 eighth-grade data, bullying showed a significant negative association with mathematics achievement for students in Turkey, but not in Korea (Topçu et al., 2016). Bullying also showed an indirect association with mathematics achievement through students’ sense of school belonging, where bullying had a strong negative association with school belonging (Konishi et al., 2010; Ren et al., 2025). In addition, bullying was negatively associated with class climate (Thornberg et al., 2022). Based on these findings, this study examined another indirect association between bullying and mathematics achievement through mathematics class climate.

2.

School Belonging and Safety

School belonging and safety reflect students’ perception of being respected, accepted, and supported by others at school (Goodenow, 1993), along with their sense of physical and emotional safety. This factor is important for academic achievement as multiple studies across contexts have shown that positive school belonging and safety are associated with higher academic performance. For example, perceptions of safety at school were positively associated with both mathematics and reading achievement (Kwong & Davis, 2015). In the U.S., initiatives aimed at improving school belonging and safety resulted in a 28% reduction in suspensions and strengthened students’ perceptions of safety and academic proficiency (Huguley et al., 2020).

3.

Mathematics Class Climate

Marder et al. (2023) found that disruptive behavior in mathematics classrooms was negatively related to students’ mathematics achievement. This study examined the mathematics class climate using the TIMSS 2019 questionnaire on the overall environment and discipline within the mathematics classroom, including student conduct, noise levels, respect for the teacher, and adherence to classroom rules (Mullis et al., 2020).

1.2.2. School-Level Factors

• School Resources for Mathematics Instruction

After discussing the student-level factors, it is also critical to consider school-level factors that may influence mathematics achievement. Afana et al. (2013) indicated that the association between school resources and students’ mathematics achievement varied across different educational systems. They reported that shortages in computer hardware and software were significantly related to lower achievement in some school contexts but not others. In this study, school resources for mathematics instruction refer to the educational tools and personnel available to support teaching and learning in mathematics. This factor includes mathematics teachers, computer software for mathematics instruction, library resources relevant to mathematics, calculators, and materials for understanding quantities or procedures (Mullis et al., 2020).

2.

School Discipline and Safety

Gase et al. (2017) found that for students in grades six through twelve, school safety was significantly associated with students’ grade point averages at the individual level, but this association was not statistically significant at the school level. Another study examining absenteeism and academic achievement among K–3 students using HLM reported that chronic absenteeism at the school level was negatively associated with student achievement (May et al., 2025). In this study, school discipline and safety encompass various issues identified in the TIMSS 2019 questionnaire, including tardiness, absenteeism, classroom disruptions, cheating, profanity, vandalism, theft, and various forms of abuse and intimidation among students, as well as towards teachers and staff (Mullis et al., 2020).

3.

Parental, Teacher, and Student Characteristics

Researchers have emphasized the importance of studying parents’, school staff’s, and students’ perceptions of school climate (Berkowitz et al., 2017; Thapa et al., 2013), as well. A review study on the relationship between parental involvement and academic achievement found that parental expectations, valuing academic achievement, and academic encouragement and support were positively related to academic achievement in middle, high school, and beyond (Boonk et al., 2018). Rogers et al. (2009) found that the relationship between parental involvement and academic achievement was indirect through fifth- and sixth-grade children’s academic competencies in Canada. Similarly, another study reported that parental expectations played a significant role in the academic achievement of 780 primary school students in Hong Kong (Phillipson & Phillipson, 2012). Wilder (2014) reviewed nine studies on parental involvement in student academic achievement, defining parental involvement as parental expectations for their children’s academic success, and reported a strong positive relationship between the two. Utilizing TIMSS 2015 data from the United Arab Emirates, Badri (2019) examined the relationships between school-level factors, including teachers, parents, and students’ characteristics, and students’ mathematics and science achievement for fourth-grade students. In their study, teachers’ characteristics were not directly associated with students’ achievement, but they showed a significant and positive indirect association through students’ characteristics. Also, parental characteristics were positively and significantly associated with students’ achievement both directly and indirectly, with student characteristics involved in the indirect associations (Badri, 2019).

For the current study, the characteristics of parents, teachers, and students were drawn from the TIMSS 2019 questionnaire. Teacher characteristics in this study include teachers’ comprehension of and alignment with the school’s curricular goals, their expectations for student achievement, and their ability to inspire students. Parental characteristics include parents’ involvement in school activities, expectations, and support for their children’s academic achievement. Student characteristics in this study encompass students’ motivation to do well in school, their ability to meet the school’s academic goals, and their respect for peers who excel academically. Based on the findings of Badri (2019), the present study examines indirect associations between parental and teacher characteristics and mathematics achievement through student characteristics.

4.

School Support

A meta-analysis of 46 studies by Lynch et al. (2025) found that teachers’ professional development was positively associated with students’ mathematics achievement, and similarly, Casing and Casing (2024) determined a positive association of after-school programs for sixth- and eighth-grade students from the U.S and mathematics achievement. On the other hand, Mori (2012) studied supplementary tutoring programs and found no significant association with academic achievement among 15-year-old students in the U.S. and Japan. Although the school support questionnaires in TIMSS 2019 encompass both mathematics and science, the present study included them to investigate their potential associations with students’ mathematics achievement. The school support factor includes eight items: career guidance, initiatives to promote student engagement and achievement, professional development for teachers, supplementary lessons, specialized activities for students, targeted educational goals, additional teacher involvement to encourage continued study in the field, and extra time working with students.

1.3. Research Objectives and Questions

Research applying MSEM to examine the relationship between school climate and students’ mathematics achievement is limited. Building on prior findings and addressing gaps in the literature, this study investigates the association between school climate and the mathematics achievement of U.S. eighth-grade students using nested educational data, where students are nested within schools. The school climate factors examined in this study align with most of the five dimensions outlined by Thapa et al. (2013) and National School Climate Center (NSCC, n.d.). This study focuses on eighth-grade students because their growing needs for autonomy and relatedness (M.-T. Wang & Degol, 2016) make them particularly sensitive to school climate factors. Furthermore, to address the need for studies that incorporate multiple perspectives (Berkowitz et al., 2017; Cohen et al., 2009), this study includes not only students’ responses on student-level data but also principals’ perspectives on school-level data. This approach provides a more comprehensive understanding of how school climate is associated with students’ mathematics achievement. The research questions are as follows:

RQ1. How are student-level factors—bullying, sense of school belonging and safety, and mathematics class climate—directly associated with students’ mathematics achievement?

RQ2. How is bullying indirectly associated with students’ mathematics achievement through students’ sense of school belonging and safety, as well as mathematics class climate?

RQ3. How are school-level factors—shortage of school resources for mathematics, school discipline and safety; teacher, parental, and student characteristics; and school support—directly associated with students’ mathematics achievement?

RQ4. How are teacher and parental characteristics indirectly associated with students’ mathematics achievement through students’ academic and motivational characteristics?

2. Materials and Methods

2.1. Data

TIMSS is an international large-scale assessment study (Fishbein et al., 2021; Mullis et al., 2020). It has been administered by the National Center for Education Statistics (NCES) and organized by the International Association for the Evaluation of Educational Achievement (IEA) every four years since 1995 to provide mathematics and science achievement for fourth- and eighth-grade students. TIMSS has a nested data structure that includes variables at the student and school levels. This study utilized TIMSS 2019 data of the U.S. eighth-grade students. Ten schools with eight or fewer students were excluded from the analyses because of the lack of within-cluster variation1. A total of 7589 students from 235 schools were included in the analysis. The average number of students in schools was 32 (range 9 to 74). The missing rates for the variables used in MSEM ranged from 2.0% to 4.1% for the variables at the student level and from 0.9% to 2.3% for the variables at the school level. There was no missing value for students’ mathematics scores. As missing rates were relatively low (e.g., less than 10%) and there was no missing data on the outcome variable, the missing-at-random assumption underlying FIML was considered reasonable (Dong & Peng, 2013; Enders, 2001).

In addition to these variables, some sociodemographic variables were examined.

Table 1. Sociodemographic Characteristics of U.S. Eighth Grade Students.

Characteristic	N	%
Gender
Female	3837	50.56
Male	3750	49.41
Omitted/Blank	2	0.03
Race/Ethnicity
White	3314	43.67
Hispanic	2225	29.32
Black	974	12.83
Two or more races	444	5.85
Asian	417	5.49
Other	115	1.52
Omitted/Blank	100	1.32
Parents’ Highest Education Level
University or Higher	3111	40.99
Post-secondary but not University	732	9.65
Upper Secondary	1311	17.28
Lower Secondary	344	4.53
Some Primary, Lower Secondary or No School	169	2.23
Don’t Know	1750	23.06
Omitted/Blank	172	2.27
Books
None or very few (0–10 books)	1646	21.69
Enough to fill one shelf (11–25 books)	1670	22.01
Enough to fill one bookcase (26–100 books)	1996	26.30
Enough to fill two bookcases (101–200 books)	1129	14.88
Enough to fill three or more bookcases (more than 200)	1006	13.26
Omitted/Blank	142	1.87
How often speak English at home
Always	5356	70.58
Almost always	1359	17.91
Sometimes	675	8.89
Never	87	1.15
Omitted/Blank	112	1.48
Language spoken at home
Spanish	1700	22.40
Other	823	10.84
Omitted/Blank	5066	66.75
	M (SD)	Range
Age	14.21 (0.46)	[9.42, 17.67]

Note. N = 7589.

summarizes the sociodemographic characteristics of U.S. eighth-grade students included in this study. Missing data are indicated as omitted or blank. Approximately 50.6% of the respondents were female. White students constituted the largest racial/ethnic group (43.7%), followed by Hispanic (29.3%) and Black (12.8%) students. About half of the students reported that at least one parent had attained postsecondary education or higher. Regarding books owned at home, 54.4% of students reported having enough books to fill at least one bookcase (i.e., 26 or more books). Most students (70.6%) indicated that English was always spoken at home, while Spanish was the most commonly spoken non-English language (22.4%). The average age of eighth-grade students was 14 years, with a range from 9 to 17 years.

2.2. Measures

Both student-level and school-level school climate factors in TIMSS were included in the MSEM framework in this study (see Figure 1). Student-level factors consisted of school belonging and safety, mathematics class climate, and bullying. All student-level items were extracted from the TIMSS 2019 student questionnaires, and each student completed a student questionnaire (Mullis et al., 2020). Eighth-grade students were given 45 min for each of the two sections of the TIMSS assessment, which included both mathematics and science items. They then had 30 min to complete the student questionnaire, and extra time was allowed for this questionnaire when needed (Johansone, 2020). As shown in Figure 1, this study examined whether student-level factors are associated with students’ mathematics achievement and whether bullying was indirectly associated with academic success through the other two factors included in the school climate model.

School-level factors included the school resources for mathematics instruction, school discipline and safety, teacher characteristics, parental characteristics, student characteristics, and school support for mathematics. All school-level items were drawn from the TIMSS 2019 school questionnaire, which was completed by the school principal (Mullis et al., 2020). According to the TIMSS questionnaire, school principals completed it in approximately 30 min. The principal provided school-wide evaluations based on their overall impressions of the school environment, including characteristics of teachers, students, and parents. All items are listed in

Table A1. Items Used in This Study in TIMSS 2019.

Item	School belonging and safety
BSBG13A	I like being in school
BSBG13B	I feel safe when I am at school
BSBG13C	I feel like I belong at this school
BSBG13D	Teachers at my school are fair to me
BSBG13E	I am proud to go to this school
	Mathematics Class Climate
BSBM18A	Students don’t listen to what the teacher says
BSBM18B	There is disruptive noise
BSBM18C	It is too disorderly for students to work well
BSBM18D	My teacher has to wait a long time for students to quiet down
BSBM18E	Students interrupt the teacher
BSBM18F	My teacher has to keep telling us to follow the classroom rules
	Bullying
BSBG14A	Said mean things about my physical appearance (e.g., my hair, my size)
BSBG14B	Spread lies about me
BSBG14C	Shared my secrets with others
BSBG14D	Refused to talk to me
	Shortage of School Resources for Mathematics Instruction
BCBG13BA	Teachers with a specialization in mathematics
BCBG13BB	Computer software/applications for mathematics instruction
BCBG13BC	Library resources relevant to mathematics instruction
BCBG13BD	Calculators for mathematics instruction
BCBG13BE	Concrete objects or materials to help students understand quantities or procedures
	School Discipline and Safety
BCBG16A	Arriving late at school
BCBG16B	Absenteeism (i.e., unjustified absences)
BCBG16C	Classroom disturbance
BCBG16D	Cheating
BCBG16E	Profanity
BCBG16F	Vandalism
BCBG16G	Theft
BCBG16H	Intimidation or verbal abuse among students (including texting, emailing, etc.)
BCBG16I	Physical injury to other students
BCBG16J	Intimidation or verbal abuse of teachers or staff (including texting, emailing, etc.)
BCBG16K	Physical injury to teachers or staff
	Teacher characteristics
BCBG14A	Teachers’ understanding of the school’s curricular goals
BCBG14B	Teachers’ degree of success in implementing the school’s curriculum
BCBG14C	Teachers’ expectations for student achievement
BCBG14D	Teachers’ ability to inspire students
	Parental characteristics
BCBG14E	Parental involvement in school activities
BCBG14F	Parental commitment to ensure that students are ready to learn
BCBG14G	Parental expectations for student achievement
BCBG14H	Parental support for student achievement
	Student characteristics
BCBG14I	Students’ desire to do well in school
BCBG14J	Students’ ability to reach school’s academic goals
BCBG14K	Students’ respect for classmates who excel academically
	School support for mathematics
BCBG15A	The school provides students with information about career options in mathematics and science
BCBG15B	The school has initiatives to promote student interest in mathematics and science (e.g., student clubs, competitions)
BCBG15C	The school promotes professional development for teachers of mathematics and science
BCBG15D	The school provides extra lessons to help students excel in mathematics and science
BCBG15E	The school provides special activities in mathematics and science for interested students
BCBG15F	The school has a specific goal to improve mathematics and science education
BCBG15G	The school encourages students to continue studying mathematics and science in the future
BCBG15H	Mathematics and science teachers in this school spend extra time working with students interested in mathematics and science

Note. Bold text indicates factor names; items are listed under each factor.

, and descriptive statistics, including sample sizes, means, standard deviations, skewness, and kurtosis, are presented in

Table A2. Descriptive Statistics of All Items.

Factor	Item	n	M	SD	Skewness	Kurtosis
Student level
School belonging and safety	BSBG13A	7435	2.746	0.930	−0.479	−0.578
	BSBG13B	7419	3.113	0.872	−0.767	−0.112
	BSBG13C	7329	3.032	0.958	−0.735	−0.417
	BSBG13D	7405	3.229	0.877	−0.972	0.155
	BSBG13E	7415	3.036	0.975	−0.744	−0.466
Mathematics Class Climate	BSBM18A	7344	2.448	0.959	−0.136	−0.986
	BSBM18B	7341	2.394	1.012	−0.044	−1.154
	BSBM18C	7271	3.075	0.974	−0.749	−0.514
	BSBM18D	7314	2.771	1.028	−0.439	−0.933
	BSBM18E	7312	2.642	1.054	−0.310	−1.106
	BSBM18F	7324	2.961	1.068	−0.662	−0.842
Bullying	BSBG14A	7385	1.929	1.075	0.830	−0.658
	BSBG14B	7395	1.784	0.948	1.001	−0.052
	BSBG14C	7376	1.703	0.916	1.145	0.305
	BSBG14D	7350	1.666	0.923	1.256	0.533
Mathematics Scores (z-score)	BSMMAT01	7589	0.002	1.000	−0.145	−0.321
	BSMMAT02	7589	0.001	1.000	−0.145	−0.318
	BSMMAT03	7589	0.002	0.999	−0.153	−0.307
	BSMMAT04	7589	0.001	0.999	−0.155	−0.306
	BSMMAT05	7589	0.001	0.999	−0.144	−0.345
School level
Shortage of School Resources for Mathematics Instruction	BCBG13BA	7430	1.636	0.905	1.232	0.403
	BCBG13BB	7464	1.721	0.833	0.933	0.060
	BCBG13BC	7427	1.848	0.862	0.850	0.083
	BCBG13BD	7489	1.499	0.812	1.549	1.474
	BCBG13BE	7452	1.671	0.837	1.217	0.895
School Discipline and Safety	BCBG16A	7517	2.103	0.726	0.464	0.265
	BCBG16B	7517	2.192	0.774	0.537	0.146
	BCBG16C	7479	2.051	0.751	0.516	0.208
	BCBG16D	7517	1.520	0.580	0.590	−0.619
	BCBG16E	7517	2.062	0.812	0.429	−0.305
	BCBG16F	7517	1.422	0.540	0.771	−0.535
	BCBG16G	7517	1.396	0.552	1.217	1.692
	BCBG16H	7517	2.162	0.735	0.330	−0.029
	BCBG16I	7517	1.478	0.618	1.149	1.297
	BCBG16J	7468	1.436	0.653	1.538	2.383
	BCBG16K	7517	1.067	0.266	4.055	16.903
Teacher characteristics	BCBG14A	7517	4.175	0.684	−0.427	−0.128
	BCBG14B	7478	3.873	0.741	−0.247	−0.245
	BCBG14C	7517	3.976	0.723	−0.610	1.137
	BCBG14D	7468	3.703	0.795	−0.350	0.226
Parental characteristics	BCBG14E	7517	3.134	1.039	0.092	−0.635
	BCBG14F	7480	3.250	0.943	−0.131	−0.205
	BCBG14G	7480	3.734	0.823	−0.147	−0.562
	BCBG14H	7517	3.435	0.884	−0.007	−0.506
Student characteristics	BCBG14I	7517	3.619	0.713	0.057	−0.328
	BCBG14J	7517	3.721	0.694	−0.055	−0.256
	BCBG14K	7480	3.727	0.745	−0.377	0.459
School support for mathematics	BCBG15A	7445	3.315	0.619	−0.329	−0.663
	BCBG15B	7482	3.289	0.747	−0.929	0.663
	BCBG15C	7482	3.564	0.580	−1.210	1.822
	BCBG15D	7445	3.187	0.722	−0.527	−0.179
	BCBG15E	7482	3.216	0.734	−0.629	−0.042
	BCBG15F	7417	3.467	0.689	−1.197	1.161
	BCBG15G	7482	3.468	0.638	−0.793	−0.411
	BCBG15H	7445	3.246	0.746	−0.880	0.686

. None of the items exhibited high levels of skewness or kurtosis (Curran et al., 1996).

2.2.1. Student Level

The factor of school belonging and safety was measured using five items that assess whether students like being in school, feel safe at school, feel a sense of belonging at school, believe teachers treat students fairly, and feel proud to attend their school. Mathematics class climate was measured by six items, including students not listening to what the teacher says, disruptive noise, classrooms being too disorderly to work, long waiting times for students to quiet down, students interrupting the teacher, and the need to repeatedly remind students of the rules. Bullying consisted of four items assessing how often other students at school engaged in the following behaviors: saying mean things about students’ physical appearance, spreading lies about the students, sharing the students’ secrets with others, and refusing to talk to other students. All items for the three factors were measured with a 4-point Likert scale. The items in the school belonging and safety factor ranged from ‘agree a lot (1)’ to ‘disagree a lot (4)’, and mathematics class climate ranged from ‘every or almost every lesson (1)’ to ‘never (4)’, and bullying ranged from ‘At least once a week (1)’ to ‘never (4).

2.2.2. School Level

The shortage of school resources for mathematics instruction was measured by five items, including teachers specialized in mathematics, computer software and applications for mathematics instruction, library resources relevant to mathematics, calculators for mathematics instruction, and materials designed to help students understand quantities. The questions were answered with a 4-point Likert scale ranging from ‘not at all (1)’ to ‘a lot (4)’.

School discipline and safety were measured using 11 items. Principals were asked to indicate the extent to which each of the following was a problem among eighth-grade students in their school: arriving late to school, absenteeism, classroom disturbance, cheating, use of profanity, vandalism, theft, intimidation or verbal abuse among students, physical injury to other students, intimidation or verbal abuse among students and toward teachers or staff, and physical injury to teachers or staff. All items were measured by a 4-point Likert scale from ‘not a problem (1)’ to ‘serious problem (4)’.

The three characteristics-related factors (i.e., teacher, parent, student) correspond to the following question: “How would you characterize each question within your school?” Specifically, the teacher characteristics factor consists of four items: teachers’ understanding of the school’s curricular goals, their success in implementing the curriculum, their expectations for student achievement, and their ability to inspire students. The parental characteristics factor includes four items: parental involvement in school activities, commitment to ensuring that students are ready to learn, expectations for student achievement, and support for student achievement. Student characteristics consist of three items: students’ desire to do well in school, ability to reach the school’s academic goals, and respect for classmates who excel academically. All items were answered by a 5-point Likert scale from ‘very high (1)’ to ‘very low (5)’.

School support was measured using eight items assessing agreement with statements about mathematics and science education within the school. The items include providing students with information about career options in mathematics and science, promoting student interest through clubs and competitions, supporting professional development for teachers, offering extra lessons and special activities, setting explicit goals to improve mathematics and science education, encouraging students to pursue these subjects in the future, and fostering additional teacher–student engagement for those with particular interest in mathematics and science. All items were measured on a 4-point Likert scale, ranging from ‘agree a lot (1)’ to ‘disagree a lot (4)’.

2.2.3. Mathematics Achievement

In TIMSS, students completed a mathematics assessment that covered four domains: number, algebra, geometry, and data and probability (Fishbein et al., 2021). Five mathematics scores in TIMSS (i.e., BSMMAT01-BSMMAT05) were used for a mathematics achievement outcome. These five scores are known as plausible values, which represent the best available estimates of student mathematics achievement and are recommended for use as outcome variables in research studies (Fishbein et al., 2021; Foy et al., 2020; von Davier, 2020). The mean values of the five scores range from 519.9 to 521.2, and standardized scores of each variable were calculated and utilized for the analyses (see Table A2).

2.3. Statistical Analyses

The R package of Dire version 2.2.0 (Bailey et al., 2023a) and EdSurvey version 4.0.7 (Bailey et al., 2023b) were utilized in R version 4.4.0 (R Core Team, 2024) to download the TIMSS 2019 U.S. data. The multilevel confirmatory factor analysis (MCFA) was conducted first to evaluate the construct validity of the TIMSS measures used in this study and to confirm that each predictor was appropriately measured before testing the mediation model using MSEM (Hox et al., 2017). McDonald’s omega (McDonald, 1970) was computed based on the standardized factor loadings to assess internal consistency because omega is considered a more general and robust reliability coefficient than Cronbach’s alpha (Padilla & Divers, 2016). Values of 0.70 or higher are typically considered indicative of acceptable internal consistency, whereas values of 0.90 or higher are often considered excellent. Then, the MSEM was applied to investigate the climate factors associated with students’ mathematics achievement at both school and student levels, including indirect associations. Mplus version 8.11 (L. K. Muthén & Muthén, 1998–2017) was used to conduct MCFA and MSEM using maximum likelihood estimation with robust standard error (MLR). We initially specified the Likert-scale predictors as categorical, but the model did not converge in Mplus. Previous research found that treating Likert-scale variables with five or more categories as continuous generally yields minimal bias (Rhemtulla et al., 2012). Although some items had four categories in our study, we chose MLR due to the model complexity and large sample size, and we note the implication from Li (2016) that MLR often provides less biased standard errors and good recovery of interfactor correlations, even though it can attenuate factor loadings when applied to ordinal data.

To account for the complex sampling design of TIMSS, both student- and school-level sampling weights were applied in the analysis. The weight variables are defined as the reciprocal of the probability of selection at each level, with adjustments to account for nonparticipation (LaRoche et al., 2020). Specifically, the total student weight (TOTWGT) was designed for student-level analysis, and the school-level weight (SCHWGT) was designed for school-level analysis. The TOTWGT reflects overall student sample weight and ensures that student-level estimates are representative of the population. The SCHWGT is intended for analyses where schools are the unit of analysis and ensures appropriate representation of schools in school-level estimates (Fishbein et al., 2021). The names of these variables are from publicly available TIMSS data.

To handle missing data, full information maximum likelihood (FIML) estimation, implemented in Mplus 8.11, was employed (Enders, 2022; L. K. Muthén & Muthén, 1998–2017). FIML is based on maximum likelihood estimation and utilizes all available information in the data. Specifically, a likelihood function is computed based on the observed responses of each participant, and the total likelihood is maximized across the sample (Enders, 2022). FIML has some limitations, such as the potential to underestimate standard errors under nonnormality. However, it is generally recommended and widely used in structural equation modeling because it produces efficient and unbiased parameter estimates under both missing completely at random and missing at random conditions (Enders, 2001; Enders & Bandalos, 2001; Lim & Cheung, 2022).

To evaluate the MSEM model, the comparative fit index (CFI; Bentler, 1990), the Tucker–Lewis index (TLI; Tucker & Lewis, 1973), the root mean square error of approximation (RMSEA; Steiger & Lind, 1980), and the standardized root mean square residual (SRMR; Bentler, 1995) were utilized. Values greater than 0.90 and 0.95 are considered as an acceptable and excellent fit to the data for CFI and TLI (Hu & Bentler, 1999; Marsh et al., 2004). The value of RMSEA, which is smaller than 0.08 and 0.05, is considered acceptable and a good fit for the data (Browne & Cudeck, 1992; Hu & Bentler, 1999). For SRMR, a value below 0.08 was suggested as a good fit (Hu & Bentler, 1999). Regarding between-level SRMR, the literature has noted additional complexity due to finite sample sizes and reported that the conventional cutoff value (0.08) may be overly strict (Asparouhov & Muthén, 2018). The parameter estimates of each model were examined after assessing the model fit. After fitting the MSEM model, Monte Carlo confidence intervals for the indirect effects were calculated using the RMediation package version 1.2.2 in R (Preacher & Selig, 2012; Tofighi & MacKinnon, 2011).

3. Results

The intraclass correlation coefficients for mathematics achievement ranged from 0.340 to 0.351, indicating that 34–35% of the variance in students’ mathematics achievement was due to differences between schools. This supports the use of a multilevel modeling framework for examining potential associations between school-level factors and students’ mathematics achievement. MCFA was conducted first, and its model fits were presented in

Table 2. Model fit for MSEM Model.

Model	${\mathit{\chi }}^2$ (df)	CFI	TLI	RMSEA	SRMRwithin	SRMRbetween
MCFA	2393.58 *** (709)	0.973	0.970	0.018	0.032	0.097
MSEM	3060.06 *** (887)	0.964	0.961	0.018	0.055	0.091

Note. *** p < 0.001.

. The model fit results of MCFA showed an acceptable or excellent fit (i.e., CFI = 0.973, TLI = 0.970, RMSEA = 0.018, within-level SRMR = 0.032). The between-level SRMR (0.097) was slightly lifted, but the between-level SRMR value should be interpreted with caution rather than as evidence of model misspecification, as it may exceed the conventional cut-off value (Asparouhov & Muthén, 2018). All standardized factor loadings of the MCFA model were above 0.4 and mostly above 0.6 (see

Table A3. Standardized Factor Loadings in MCFA model and Internal Consistency Estimates (ω).

Factor	Item	Factor Loading	SE
Student level
School belonging and safety	BSBG13A	0.648	0.015
$(\omega$ = 0.830)	BSBG13B	0.661	0.016
	BSBG13C	0.784	0.011
	BSBG13D	0.592	0.023
	BSBG13E	0.816	0.009
Mathematics Class Climate	BSBM18A	0.768	0.012
$(\omega$ = 0.915)	BSBM18B	0.803	0.009
	BSBM18C	0.777	0.011
	BSBM18D	0.844	0.009
	BSBM18E	0.823	0.010
	BSBM18F	0.788	0.012
Bullying	BSBG14A	0.666	0.015
$(\omega$ = 0.788)	BSBG14B	0.809	0.011
	BSBG14C	0.669	0.018
	BSBG14D	0.626	0.019
Mathematics Scores	BSMMAT01	0.964	0.002
$(\omega$ = 0.985)	BSMMAT02	0.964	0.002
	BSMMAT03	0.966	0.002
	BSMMAT04	0.964	0.002
	BSMMAT05	0.964	0.002
School level
Shortage of School Resources for   Mathematics Instruction	BCBG13BA	0.711	0.055
$(\omega$ = 0.924)	BCBG13BB	0.909	0.040
	BCBG13BC	0.841	0.071
	BCBG13BD	0.815	0.065
	BCBG13BE	0.921	0.045
School Discipline and Safety	BCBG16A	0.676	0.081
$(\omega$ = 0.904)	BCBG16B	0.691	0.074
	BCBG16C	0.782	0.056
	BCBG16D	0.434	0.084
	BCBG16E	0.777	0.045
	BCBG16F	0.595	0.071
	BCBG16G	0.595	0.080
	BCBG16H	0.766	0.043
	BCBG16I	0.758	0.050
	BCBG16J	0.781	0.044
	BCBG16K	0.580	0.094
Teacher characteristics	BCBG14A	0.741	0.059
$(\omega$ = 0.885)	BCBG14B	0.877	0.026
	BCBG14C	0.811	0.037
	BCBG14D	0.811	0.037
Parental characteristics	BCBG14E	0.880	0.041
$(\omega$ = 0.932)	BCBG14F	0.946	0.022
	BCBG14G	0.837	0.045
	BCBG14H	0.852	0.054
Student characteristics	BCBG14I	0.856	0.041
$(\omega$ = 0.769)	BCBG14J	0.691	0.066
	BCBG14K	0.618	0.096
School support for mathematics	BCBG15A	0.638	0.083
$(\omega$ = 0.846)	BCBG15B	0.566	0.097
	BCBG15C	0.628	0.070
	BCBG15D	0.761	0.056
	BCBG15E	0.650	0.081
	BCBG15F	0.590	0.073
	BCBG15G	0.492	0.111
	BCBG15H	0.759	0.051

for details). All factors demonstrated acceptable to excellent reliability. Specifically, McDonald’s omega coefficients ranged from 0.77 for student characteristics to 0.99 for mathematics scores (see Table A3 for details). This indicated that the observed variables loaded strongly and consistently on their respective latent constructs and therefore supported the quality of the measurement model. Thus, MSEM was conducted to examine the research questions mentioned above. The model fit indices indicated a good fit (i.e., CFI = 0.964, TLI = 0.961, RMSEA = 0.018, within-level SRMR = 0.055), except for the between-level SRMR (0.091), as demonstrated in Table 2. Additionally, most standardized factor loadings were above 0.6, indicating strong relations between the observed variables and their corresponding factors in the model (see

Table A4. Standardized Factor Loadings of Each Item in MSEM model.

Factor	Item	Factor Loading	SE
Student level
School belonging and safety	BSBG13A	0.648	0.016
	BSBG13B	0.658	0.016
	BSBG13C	0.785	0.011
	BSBG13D	0.590	0.022
	BSBG13E	0.818	0.009
Mathematics Class Climate	BSBM18A	0.768	0.012
	BSBM18B	0.804	0.009
	BSBM18C	0.775	0.011
	BSBM18D	0.844	0.009
	BSBM18E	0.823	0.010
	BSBM18F	0.787	0.012
Bullying	BSBG14A	0.665	0.015
	BSBG14B	0.807	0.011
	BSBG14C	0.669	0.018
	BSBG14D	0.625	0.019
Mathematics Scores	BSMMAT01	0.940	0.003
	BSMMAT02	0.939	0.003
	BSMMAT03	0.942	0.003
	BSMMAT04	0.939	0.004
	BSMMAT05	0.939	0.003
School level
Shortage of School Resources for   Mathematics Instruction	BCBG13BA	0.716	0.053
	BCBG13BB	0.912	0.039
	BCBG13BC	0.840	0.073
	BCBG13BD	0.813	0.066
	BCBG13BE	0.919	0.046
School Discipline and Safety	BCBG16A	0.678	0.081
	BCBG16B	0.693	0.073
	BCBG16C	0.783	0.056
	BCBG16D	0.433	0.085
	BCBG16E	0.776	0.044
	BCBG16F	0.594	0.071
	BCBG16G	0.597	0.077
	BCBG16H	0.765	0.043
	BCBG16I	0.759	0.050
	BCBG16J	0.779	0.043
	BCBG16K	0.579	0.093
Teacher characteristics	BCBG14A	0.730	0.063
	BCBG14B	0.875	0.028
	BCBG14C	0.811	0.040
	BCBG14D	0.814	0.037
Parental characteristics	BCBG14E	0.882	0.041
	BCBG14F	0.947	0.022
	BCBG14G	0.837	0.042
	BCBG14H	0.851	0.052
Student characteristics	BCBG14I	0.859	0.045
	BCBG14J	0.685	0.064
	BCBG14K	0.624	0.087
School support for mathematics	BCBG15A	0.636	0.083
	BCBG15B	0.570	0.098
	BCBG15C	0.632	0.070
	BCBG15D	0.765	0.054
	BCBG15E	0.653	0.078
	BCBG15F	0.598	0.070
	BCBG15G	0.492	0.111
	BCBG15H	0.746	0.049
Mathematics Scores	BSMMAT01	1.000	0.000
	BSMMAT02	1.000	0.000
	BSMMAT03	1.000	0.001
	BSMMAT04	1.000	0.001
	BSMMAT05	1.000	0.001

for details).

To further assess model fit, modification indices were examined. The largest modification index suggested allowing a residual covariance between two mathematics class climate variables (i.e., BSBM18A, BSBM18B) at the within-student level, whereas all remaining indices were substantially smaller. The model fit from the modified model, which added residual covariance between those two variables, did not meaningfully improve the model fit, including the between-level SRMR, and it was not theoretically supported. Thus, this modification was not retained in the final model.

The standardized path coefficients and their standard errors from the MSEM are presented in

Table 3. Standardized Path Coefficients of Factors for the MSEM model.

Paths	b*	95% CI	SE	p
Student level
School belonging and safety→ Mathematics Achievement	0.192 ***	[0.144, 0.240]	0.025	<0.001
Bullying → Mathematics Achievement	−0.087 **	[−0.139, −0.034]	0.027	0.001
Mathematics Class Climate → Mathematics Achievement	0.090 *	[0.018, 0.162]	0.037	0.014
Bullying → School belonging and safety	−0.376 ***	[−0.425, −0.327]	0.025	<0.001
Bullying → Mathematics Class Climate	−0.291 ***	[−0.340, −0.241]	0.025	<0.001
Indirect Paths
Bullying → School belonging and safety → Mathematics Achievement	−0.072 ***	[−0.093, −0.053]	0.011	<0.001
Bullying → Mathematics Class Climate → Mathematics Achievement	−0.026 *	[−0.050, −0.005]	0.011	0.020
School level
Shortage of School Resources → Mathematics Achievement	−0.320 *	[−0.566, −0.073]	0.126	0.011
School Discipline and Safety → Mathematics Achievement	−0.183 *	[−0.347, −0.018]	0.084	0.030
Teacher Characteristics → Mathematics Achievement	−0.182	[−0.573, 0.209]	0.199	0.362
Parent Characteristics → Mathematics Achievement	0.130	[−0.104, 0.363]	0.119	0.276
Student Characteristics → Mathematics Achievement	0.366	[−0.003, 0.735]	0.188	0.052
School Support for Mathematics → Mathematics Achievement	−0.023	[−0.209, 0.163]	0.095	0.812
Teacher Characteristics → Student Characteristics	0.763 ***	[0.535, 0.991]	0.116	<0.001
Parent Characteristics → Student Characteristics	0.173	[−0.097, 0.444]	0.138	0.209
Indirect Paths
Teacher Characteristics → Student Characteristics → Mathematics Achievement	0.279	[−0.002, 0.609]	0.160	0.080
Parent Characteristics → Student Characteristics → Mathematics Achievement	0.063	[−0.043, 0.203]	0.055	0.252

Note. * p < 0.05, ** p < 0.01, *** p < 0.001.

. The same results, ordered by coefficient magnitude (i.e., effect size), are provided in Supplementary Table S1. In Figure 1, thicker arrows indicate significant paths among the school climate factors and students’ mathematics achievement; a version of the figure displaying only significant paths is presented in Supplementary Figure S1. For within-level (i.e., student-level), the school belonging and safety showed a positive and statistically significant association with students’ mathematics achievement (b* = 0.192, SE = 0.025, 95% CI [0.144, 0.240], p < 0.001), as well as mathematics class climate (b* = 0.090, SE = 0.037, 95% CI [0.018, 0.162], p = 0.014). Bullying showed a negative and significant association with students’ mathematics achievement (b* = −0.087, SE = 0.027, 95% CI [−0.139, −0.034], p = 0.001). The two paths from bullying to school belonging and safety (b* = −0.376, SE = 0.025, 95% CI [−0.425, −0.327], p < 0.001) and to mathematics class climate (b* = −0.291, SE = 0.025, 95% CI [−0.340, −0.241], p < 0.001) were negative and statistically significant. Regarding indirect associations, bullying was negatively associated with students’ mathematics achievement through school belonging and safety (b* = −0.072, SE = 0.011, 95% CI [−0.093, −0.053], p < 0.001) and mathematics class climate (b* = −0.026, SE = 0.011, 95% CI [−0.050, −0.005], p = 0.020). This indicated that being bullied by other students was significantly associated with mathematics achievement both directly and indirectly through school and mathematics class climate. The total combined standardized association was −0.185.

For between-level (i.e., school-level) factors, both the shortage of school resources for mathematics instruction (b* = −0.320, SE = 0.126, 95% CI [−0.566, −0.073], p = 0.011) and school discipline and safety (b* = −0.183, SE = 0.084, 95% CI [−0.347, −0.018], p = 0.030) showed significantly negative relationships with students’ mathematics achievement. Teacher, parental, and student characteristics, as well as school support for mathematics, did not show significant direct associations with students’ mathematics achievement (ps > 0.05). However, teachers’ characteristics were significantly positively associated with student characteristics (b* = 0.763, SE = 0.116, 95% CI [0.535, 0.991], p < 0.001) while parent characteristics were not significantly related to students’ characteristics (b* = 0.173, SE = 0.138, 95% CI [−0.097, 0.444], p = 0.209). School support was not significantly associated with students’ mathematics achievement (p > 0.05). Neither the indirect associations between teacher characteristics and mathematics achievement via student characteristics nor the indirect association between parent characteristics and mathematics achievement via student characteristics was statistically significant (p > 0.05). Because all structural paths were estimated simultaneously within a single, theory-driven MSEM, formal corrections for multiple comparisons were not applied in this study.

The correlations between factors at the school level are presented in

Table 4. Correlation Coefficients of School-Level Factors for MSEM.

	1	2	3	4
Shortage of School Resources
2. School Discipline and Safety	−0.027
3. Teacher Characteristics	−0.390 ***	−0.540 ***
4. Parent Characteristics	0.109	−0.528 ***	0.419 *
5. School Support	−0.320 ***	−0.162	0.409 ***	0.261

Note. * p < 0.05, *** p < 0.001.

. The school discipline and safety factor showed a negative correlation with both teacher characteristics (r = −0.540, p < 0.001) and parent characteristics (r = −0.528, p < 0.001). Moreover, the shortage of school resources for mathematics was negatively correlated with both teacher characteristics (r = −0.390, p < 0.001) and school support for mathematics (r = −0.320, p < 0.001). Conversely, school support for mathematics had a strong positive correlation with teacher characteristics (r = 0.409, p < 0.001), and teacher characteristics were positively correlated with parent characteristics (r = 0.419, p = 0.021). Overall, these correlations indicate that the school-level factors are interrelated. This pattern highlights the importance of modeling school-level factors simultaneously in the MSEM to account for their shared variance and to assess the unique associations of each factor with the outcome.

4. Discussion

This study applied MSEM to investigate the association between multiple aspects of school climate factors and students’ mathematics achievement at both the student and school levels. The MSEM approach allows for the examination of complex models of latent constructs in a hierarchical data structure and explicitly accounts for measurement errors by modeling latent factors.

4.1. Key Findings

This study finds that a sense of school belonging and safety is positively related to students’ mathematics performance. This is consistent with the findings from Topçu et al. (2016) for Korean students. They reported that Korean students who felt safe and had a sense of belonging at school showed higher academic performance in science and mathematics. However, they found a negative relationship for Turkish students and explained that this negative association may stem from Turkish students’ negative perceptions of school as a pressure-laden environment, where teachers are often viewed as judges.

In the present study, the positive association observed between school belonging and safety and mathematics achievement suggests that school belonging may reflect supportive school environments and higher levels of academic engagement, which in turn may facilitate students’ performance in mathematics. In addition, a well-managed mathematics classroom with minimal disruptive noise and orderly student behavior improved students’ mathematics achievement. Conversely, being bullied shows significant negative direct and indirect associations with students’ mathematics scores through the two other student-level factors, school belonging and safety and mathematics class climate. Consistent with the previous studies that reported a negative relationship between bullying and academic achievement directly (Al-Raqqad et al., 2017; Konishi et al., 2010; Topçu et al., 2016) and indirectly (Ren et al., 2025), this study also showed direct and indirect associations between bullying and students’ mathematics achievement, suggesting that negative peer experiences may undermine students’ academic performance.

Although some associations demonstrated moderate effect sizes (e.g., bullying and school belonging and safety, b* = −0.376), several significant paths, such as mathematics class climate and mathematics achievement (b* = 0.090) and bullying and mathematics achievement (b* = −0.087), indicate small effect sizes (Cohen, 1988). However, Kraft (2020) proposed empirical benchmarks in which effect sizes from 0.05 to less than 0.20 were considered medium and emphasized that even small effects can be practically meaningful when they are observed in large-scale educational studies (Kraft, 2020). Therefore, the significant paths identified in this study may be considered meaningful.

At the school level, the shortage of school resources, such as computer software, calculators, and library resources, showed a negative association with students’ performance in mathematics. In addition, more issues in school discipline and safety—absenteeism, being late at school, cheating, theft, and physical or verbal abuse—are also negatively related to students’ achievement in mathematics. These findings are consistent with previous studies reporting that absenteeism was negatively related to students’ achievement at the student level (Smerillo et al., 2018) and school level (May et al., 2025). However, this study extends the literature by examining school safety and discipline at the school level and highlights that school-wide discipline issues, not limited to absenteeism, may be related to lower levels of students’ mathematics performance.

In this study, teacher, parental, and student characteristics are not significantly associated with students’ mathematics achievement, which is inconsistent with a previous study (Badri, 2019). In addition, school support was not significantly associated with students’ mathematics achievement in this study. Several considerations may help explain these statistically non-significant associations. First, these school-level factors are interrelated, and their unique associations may be attenuated when shared variance is modeled simultaneously. Second, the number of schools available for school-level estimation was modest, which may have limited the statistical power to detect smaller unique associations. Third, school-level factors were assessed solely based on school principals’ responses, which may capture general perceptions of school climate but may be less sensitive to distinctions among specific factors. These results are consistent with those of Topçu et al. (2016), who reported that parental involvement did not yield a significant association with Turkish students’ mathematics achievement but was different from the results for Korean students. Although cross-national differences were not examined in the present study, these findings from prior studies suggest that the associations between school climate and student outcomes may vary across broader contexts. As M.-T. Wang and Degol (2016) noted, students’ development is shaped by multiple interacting contexts, including school practices and interpersonal relationships, which may alter how school climate operates. This implies that cultural and contextual factors may shape the ways in which school climate relates to students’ academic achievement. In addition, teacher characteristics and student characteristics were strongly associated at the school level in this study, although they are theoretically distinct factors. This association likely reflects the shared schools’ academic climate and some degree of conceptual proximity between the factors. It also may partly arise from the fact that both factors were reported by school principals, who have more direct and routine exposure to teachers and students than to parents.

Lastly, the correlations among school-level factors highlight the connected nature of school climate factors. The strong negative correlations between school discipline and safety and both teacher characteristics (r = −0.540) and parent characteristics (r = −0.528) indicate that schools with more discipline issues are perceived to have less favorable teachers’ instructional practices and expectations, as well as lower levels of parental involvement and support. Additionally, greater shortages in school resources for mathematics instruction tended to have lower perceived quality of teachers’ instructional practices and expectations (r = −0.390) and lower levels of school support for mathematics and science instruction (r = −0.320). More favorable perceptions of teachers’ instructional practices and expectations are moderately associated with higher levels of parental involvement and support (r = 0.419) and higher levels of school support for mathematics and science (r = 0.409).

4.2. Practical Implications

The findings of this study suggest that students’ mathematics achievement is associated with multiple aspects of school climate, including perceptions of safety, school belonging, classroom order, and bullying. This demonstrates the potential importance of creating school and classroom environments in which students feel supported, respected, and are able to focus on learning. Additionally, anti-bullying programs that emphasize peer relationships and a sense of belonging within classrooms may be beneficial for schools to improve students’ academic achievement.

At the school level, limited resources for teaching mathematics—such as qualified teachers, calculators, or learning materials—as well as discipline problems, may be negatively associated with students’ ability to learn mathematics. Therefore, providing instructional technology and software that supports mathematics teaching and learning is important. Schools need to ensure students’ access to mathematics-related library resources that complement classroom instruction. Our findings also show that shortages of school resources are negatively related to both school support and perceptions of discipline and safety. This suggests that resource availability and schoolwide conditions may be relevant contextual factors when considering efforts to support students’ learning in mathematics. In addition, how schools approach discipline may be relevant to students’ academic achievement. Reducing instructional disruptions caused by lateness, absenteeism, and classroom disturbances, and promoting a school environment that minimizes verbal aggression and intimidation among students may be helpful in improving students’ academic achievement. Overall, this research highlights school climate as a contextual factor associated with students’ academic achievement.

4.3. Limitations and Future Directions

While this study explores various factors associated with the students’ mathematics achievement in the MSEM framework, the limitations and future study directions should be discussed. First, as a cross-sectional study, this research investigates associations among multiple climate factors, including indirect associations with mathematics achievement, at one point in time. The findings of this study further highlight the need for longitudinal research. Although some school-level factors did not show significant associations with students’ mathematics achievement in the current cross-sectional analyses, substantial between-level variances suggest that school climate may influence achievement through cumulative or delayed processes over time. As climate factors may be differently associated with students’ mathematics achievement over time (Thapa et al., 2013), longitudinal designs may help to examine how school climate-related changes are related to students’ academic achievement.

In addition, researchers have raised concerns about cross-sectional studies that apply mediation analyses because those designs do not allow for the passage of time between variables. This poses an issue because mediation occurs over time, potentially inducing bias and misrepresenting the nature of indirect association (e.g., Cole & Maxwell, 2003; Maxwell et al., 2011; Selig & Preacher, 2009). Although this study examines indirect associations with students’ mathematics achievement, these associations do not imply causal relationships. Future studies that apply longitudinal data are needed to more rigorously assess the timing and directionality of indirect associations (Caemmerer et al., 2024; Cole & Maxwell, 2003; Preacher, 2015). Moreover, future longitudinal studies could also identify when and how school climate interventions are most effective.

Second, this study focuses on eighth-grade students’ achievement in mathematics. Future research could extend this work to other subjects or overall academic achievement, as many of the school climate factors examined here—such as school belonging and safety, bullying, school discipline, teacher, parent, and student characteristics, and school support—may also be associated with performance in other domains. Additionally, future research could examine fourth-grade students, for whom TIMSS data are already available, to allow early identification of relevant factors, so that schools can address potential challenges and support positive conditions at earlier stages. Future studies at higher grade levels, such as high school, are also warranted, given evidence that the associations between school climate factors and students’ academic achievement vary in magnitude between middle school and high school students (Daily et al., 2019). Extending these analyses to other grade levels may identify developmental periods when school climate interventions are most strongly associated with student outcomes.

Third, this study used U.S. TIMSS data. The results of this study should be interpreted with caution when applying to other countries, as previous studies have reported that the factors related to mathematics achievement vary across countries (e.g., Topçu et al., 2016; X. S. Wang et al., 2023). Future studies on how school climates vary across cultural contexts and how these differences are related to students’ mathematics achievement are necessary. In addition, variations may exist within the U.S. across different regions or school contexts. Although analyses of within-U.S. variation across regions, urbanicity, and school types were not conducted in this study because this information was not available in the publicly accessible TIMSS international database, future studies examining these differences would provide more targeted implications for practice.

Fourth, school-level factors in this study were solely based on principals’ self-reports in the TIMSS school questionnaire. While this is meaningful because studies incorporating school principals’ perceptions are limited, the findings may primarily capture principals’ perspectives rather than those of teachers, parents, or students. Although we did not find direct relationships between teacher, parental, and student characteristics and students’ mathematics achievement, the results might differ if these factors were reported by teachers, parents, and students themselves, as prior research suggested that different raters may provide distinct perspectives on school climate and related factors (e.g., M.-T. Wang & Degol, 2016). Therefore, further research using multiple informants and diverse measurement methods is needed to develop strategies that address the needs of different stakeholders within schools.

Lastly, the number of schools is acceptable for multilevel modeling, but statistical power at the school level may be limited in detecting unique associations, especially given the simultaneous inclusion of multiple school-level predictors in the MSEM. Therefore, associations with statistically nonsignificant results should be interpreted with caution. Future research using larger numbers of schools or longitudinal designs may provide greater power to detect and clarify school-level associations.

5. Conclusions

This study examines various school climate factors in relation to eighth-grade students’ mathematics using the U.S. TIMSS data. Unlike prior studies that examined school climate factors at a single level or that used HLM, this study uses MSEM to provide a more nuanced understanding of how climate factors relate to mathematics achievement. The research finds that a stronger sense of school belonging and safety, a positive mathematics class climate, and lower levels of bullying are associated with higher mathematics achievement at the student level, whereas a shortage of resources for mathematics instruction and frequent discipline or safety incidents are negatively associated with mathematics achievement at the school level. These findings highlight the importance of fostering a safe and orderly learning environment with adequate instructional resources in students’ mathematics learning.