The Relationships between Parental Involvement, Teacher Support, and Mathematics Performance: Mediating Roles of Academic Self-Efficacy and Academic Buoyancy

The Relationships between Parental Involvement, Teacher Support, and Mathematics Performance: Mediating Roles of Academic Self-Efficacy and Academic Buoyancy | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The Relationships between Parental Involvement, Teacher Support, and Mathematics Performance: Mediating Roles of Academic Self-Efficacy and Academic Buoyancy İlhan İlter, Nuri Can Aksoy, Mehmet Ceylan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6903576/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 04 Nov, 2025 Read the published version in BMC Psychology → Version 1 posted 13 You are reading this latest preprint version Abstract Background Given its multifaceted role in fostering students’ academic outcomes, parental involvement remains a critical area of investigation in educational research. Parental involvement not only complements teachers' efforts and supportive behaviours, but also has the potential to increase students' academic engagement and academic self-efficacy, both of which are among the key determinants of mathematics achievement. Methods This study examined the relationships between perceived teacher support (TS), parental involvement (PI), academic self-efficacy (ASE), academic buoyancy (AB), and mathematics performance (MP) among middle school students. More specifically, it investigated the mediating effects of ASE and AB in the relationships between PI and TS with MP. Participants included 363 middle school students from Türkiye. Data were analysed using Partial Least Squares (PLS) method and structural equation modeling (SEM). Results The findings revealed that both TS and PI have direct positive effects on students’ MP. TS and PI demonstrated significant indirect effects on MP through the mediating roles of ASE and AB. Furthermore, both ASE and AB emerged as significant positive predictors of MP. Conclusion The study indicates that students who are supported by teachers and parents tend to perform better in math, both directly and indirectly through increased academic self-efficacy and academic buoyancy. Results extend our understanding by providing important insights into the critical role of the complex interplay between external support mechanisms and internal psychological resources in enhancing mathematics performance. Teacher support parental involvement academic self-efficacy academic buoyancy mathematics performance Figures Figure 1 Figure 2 Introduction The issue of enhancing students' mathematical performance remains a persistent challenge for both educators and policymakers, as well as parents. Despite consistent interest and efforts to improve students’ math performance and achievement, the recent PISA report indicated that between 2018 and 2022; the average performance in mathematics across OECD countries dropped by a record 15 points (OECD, 2023 ). Nevertheless, in the four weeks preceding the PISA assessment, an average of 10% of students across the OECD countries reported that they did not feel safe at school, while 30% 30% indicated being distracted by digital devices. These findings support the notion that learning mathematics extends beyond cognitive challenges and entails a complex interplay of emotional and motivational factors (Pekrun & Loderer, 2020 ; Živković et al., 2023 ). Factors such as emotional security, motivation, and focus may be crucial in shaping students' mathematical performance. Addressing the challenges in promoting and teaching mathematics learning requires a deeper understanding of how both external support, such as teacher support and parental involvement, and internal non-cognitive attributes including academic self-efficacy, grit, academic motivation, academic buoyancy, contribute to students' performance and achievement. Non-cognitive factors are critical, as they encompass intrinsic motivations and behaviours that enable students to make the most of their potential (Chamorro-Premuzic & Furnham, 2004 ). These intrinsic attributes not only influence students' academic performance but also play a pivotal role in shaping their long term educational outcomes (Farrington et al., 2012 ). This study builds upon the importance of mathematics education by examining how social and motivational factors influence middle school students' mathematics performance. In particular, it investigates how academic buoyancy and academic self-efficacy mediate the links between math performance, teacher support, and parental involvement. By focusing on these mediators, the study aims to reveal how external factors, such as social support, interact with students’ internal non-cognitive attributes to influence their performance in mathematics. The study aims to provide a more thorough understanding of the intricate aspects impacting mathematical achievement by utilizing a multiple mediation approach. This study aims to contribute to the identification of non-cognitive factors that support students’ mathematics performance and achievement, while also examining the role of external support provided by teachers and parents. Conceptual framework Academic self-efficacy and mathematics performance According to the Social Cognitive Theory (SCT), both external social influences and self-regulatory factors are crucial in motivating and controlling behaviour (Bandura, 2012). One important component of these self-regulatory elements is self-efficacy, which is a person's assessment of their capacity to plan and carry out the actions or tasks required to accomplish certain goals (Bandura, 1997). In educational settings, self-efficacy is often conceptualized as academic self-efficacy (Bandura, 2012; Schunk & DiBenedetto, 2022), which is defined as students’ belief in their own ability to succeed in academic tasks (Bandura, 1997; Zimmerman et al., 1992). Academic self-efficacy refers to students' perseverance and determination towards their abilities in achieving academic achievement, as well as their beliefs about fulfilling certain academic tasks (Bandura, 1997). Academic self-efficacy refers to a person's belief in their ability to complete required tasks (Zakariya, 2022) and their confidence when faced with mathematical challenges (Yu et al., 2023). Research shows that academic self-efficacy has a significant effect on shaping students' academic behaviours and outcomes. A strong feeling of academic self-efficacy not only increases the likelihood that students will set higher goals and exert more effort, but it also fosters resilience when they encounter challenges (Usher & Pajares, 2008; Zimmerman & Kitsantas, 2005). A belief that one can succeed has been related to increased academic performance across a range of content areas in several studies, and the extensive literature highlights its significant impact on achievement and test performance (Camelo-Lavadores et al., 2017; Honicke & Broadbent, 2016; Roick & Ringeisen, 2018; Zysberg & Schfwabsky, 2020). In mathematics, academic self-efficacy refers to a person's belief in their ability to effectively complete mathematics activities (Yang et al., 2021) and their confidence when faced with mathematical challenges (Yu et al., 2023). In their study, Lera et al. (2023) used a multilevel structural equation model to better understand the effect of self-efficacy in mathematics education. They found that self-efficacy also predicted math performance at the class level. However, while academic self-efficacy is recognized as a primary predictor of mathematics performance, it remains unclear whether teacher support and parental involvement influence middle school students' mathematics performance through their interaction with academic self-efficacy. This gap in understanding highlights the necessity of exploring how the interplay between these factors gives a more comprehensive picture of middle school students’ achievement in mathematics. Academic buoyancy and math performance Academic buoyancy is the capacity of students' ability to effectively deal with daily academic challenges or difficulties such as low grades, difficult tasks, or classroom distractions (Martin & Marsh, 2008). It not only acts as a buffer against academic stress but also promotes a positive attitude towards learning, which is crucial for long-term academic achievement (Martin, 2013). In the context of mathematics, the broad range of topics and the reasoning required during the learning process pose unique challenges to maintaining consistent motivation and effort. When students struggle to make progress, they may experience setbacks that hinder their academic performance. The ability to adapt to such challenges in mathematics is referred to as mathematical buoyancy (Hoon et al., 2024). Strengthening mathematical academic buoyancy is crucial for overcoming unexpected challenges (e.g., anxiety, lack of motivation, and unstable academic development), all of which can negatively affect students' math performance (Ang et al., 2022). By encouraging students' interest and persistence in pursuing goals for the future, mathematical buoyancy aims to improve achievement and reduce math-related anxiety. Previous studies showed that academic buoyancy, particularly in mathematics, significantly impacts academic performance and achievement (Colmar et al., 2019; Datu & Yang, 2021) with academic self-efficacy mediating this relationship (Weißenfels et al., 2023) and remaining significant even when controlling for gender (Lei et al., 2022). Overall, developing mathematical buoyancy is crucial for enhancing students' motivation, engagement, and performance in the school setting (Collie et al., 2024; Colmar et al., 2019). Teacher support in mathematics performance One important resource that has a significant effect on how students’ academic performance develops is teacher supportive behaviours (Kim, et al., 2018). Teacher support enhances students' academic achievement (Glozah & Pevalin, 2014) and is positively correlated with math self-concept while negatively correlated with math anxiety (Yıldırım & Yıldırım, 2019). Research indicates that teacher support has a direct impact on three components of math engagement: emotional, behavioural, and cognitive (Liu et al., 2017). Positive teacher supportive behaviours not only boost students' autonomy in their math abilities but also enhance their overall engagement, promoting improved academic outcomes (Yang et al., 2021). Furthermore, close relationships with teachers positively predict school engagement, while conflict with teachers negatively predicts it (Engels et al., 2021). However, some research suggests that teacher support does not directly influence students' math performance. In their study, Chang et al. (2023) concluded that although teacher support significantly enhances students’ math self-efficacy but did not directly extend to their math performance. The study further emphasizes that the role of teacher support varies across different demographic groups and may influence math achievement through indirect pathways. Parent involvement in students’ learning mathematics Given its multifaceted role in fostering students’ academic outcomes, parental involvement remains a critical area of investigation in educational research. It not only complements teacher efforts but also enhances academic engagement, and academic self-efficacy among students, both of which are key predictors of math achievement (Fan & Williams, 2010; Hill & Tyson, 2009). Ginsburg et al. (2008) identified three types of parental involvement in mathematics subject areas; (1) parent involvement in completing homework assignments, (2) parent reengagement with mathematics learning, and (3) math talks during mathematics activities. These practices directly contribute to students' academic progress by fostering a supportive learning environment in the home (Gonzalez-DeHass et al., 2005; Hill & Tyson, 2009). Previous research indicates that parental involvement in their children's learning strengthens school efforts and is connected to increased positive academic outcomes, including higher academic achievement and better school attendance (Jeynes, 2007). Students who have received parental support tend to demonstrate higher math performance than those who have not. However, some research suggests that sustaining this positive effect requires consistent involvement and high-quality parent-child interactions (Ma et al., 2021; Morkoyunlu &Konyalıoğlu, 2020). Moreover, not all kinds of involvement are positively related to academic achievements, and the effects may vary across different ethnicities (Boonk et al., 2018). Considering mathematics education, parental involvement can have both benefits and drawbacks, particularly when parents experience math anxiety, which can inadvertently hinder students’ learning (Maloney et al., 2015; Oh et al., 2022). Building on these findings, the present study investigates the mediating roles of academic buoyancy and academic self-efficacy to better understand how perceived teacher support and parental involvement influence middle school students' math performance. Mediating effect of academic self-efficacy Academic self-efficacy beliefs reflect one's confidence and capacity in their ability to carry out certain academic tasks across various subject areas, such as mathematics, science, and foreign languages. In challenging subjects like mathematics, the notion of academic self-efficacy is one of the strongest predictors of performance of an activity and school success (Han & Wang, 2021). According to Martin and Rimm-Kaufman's (2015) research, students’ self-efficacy has a direct effect on their mathematics learning process and significantly shapes their achievement outcomes. However, academic self-efficacy is not solely an individual characteristic; it is significantly influenced by the social support students receive from their environment (Bandura, 1997). Parents who are involved in their children's studies put more effort into involving their children in the educational process. Parental involvement in children's education fosters positive outcomes, including academic achievement. Research indicates that it serves as a key source of external support, enhancing students' academic self-efficacy in mathematics (Cheung & Pomerantz, 2011). Moreover, emotional support from parents not only motivates students to overcome challenges in mathematics but also promotes their autonomy, contributing to their overall academic success. As Fan & Williams (2010) noted, parental support often begins early in a child’s education, as parents are typically their first educators, shaping their aspirations and expectations (Metheny & McWhirter, 2013). This is because high-quality parent-child relationships foster positive educational outcomes, including enhanced self-efficacy and academic engagement (Fan & Williams, 2010). When teachers provide structure by managing student-centred activities and providing positive feedback that improves students' self-efficacy perceptions, this positively influences their academic performance. Kim et al. (2018) found that students who received more support from their teachers had a stronger sense of academic self-efficacy, which in turn, led to improved overall achievement. Collie et al. (2016) concluded that high-quality relationships with teachers and parents— characterised by positive interactions, fair treatment and a sense of belonging —often result in students with high self-efficacy and greater academic engagement. Academic self-efficacy can serve as a mediator in the relationships between parental involvement, teacher support, and mathematics achievement. When parents and teachers are actively involved in mathematics learning, they provide essential support that enhances students' academic self-efficacy. Consequently, this strengthened self-efficacy boosts students' confidence and motivation, ultimately leading to improved performance in mathematics. Mediating effect of academic buoyancy Students who overcome minor academic difficulties and challenges with academic vitality demonstrate school satisfaction and effective classroom behaviours (Hoferichter et al., 2021). It is documented in the literature that academic buoyancy is positively related to students' performance in the classroom (Datu & Yang, 2021; Martin & Marsh, 2009; Yun et al., 2018). Research has shown that academic buoyancy encourages students to adopt effective learning strategies (Collie et al., 2016) and supports both their emotional and behavioural engagement in school settings (Datu et al., 2018), self-regulation, and higher academic achievement (Miller et al., 2013). Martin and Marsh (2008) highlight the importance of adopting a domain-specific approach when studying academic buoyancy. In the field of mathematics, some studies have indicated a positive relationship between academic buoyancy and academic achievement in the subject (Martin & Marsh, 2008; Weißenfels et al., 2023). Academic buoyancy is influenced by an individual's social environment, as environmental factors significantly affect students’ thoughts and actions (Martin, & Marsh, 2008). For students to cultivate academic buoyancy, appropriate support is essential. The home environment, with parental involvement (Chen & Mok, 2023), and the school environment, with teacher support, can greatly influence a child's development. Hejazi and Abbasi (2021) noted that students’ academic buoyancy can improve when teachers help guide them toward optimal personal goals and parents can foster a supportive family environment. Consequently, parental involvement and teacher support are thus crucial in enhancing academic buoyancy, ultimately helping students improve their academic performance, including in subjects like mathematics. In this respect, academic buoyancy may mediate the relationship between parent involvement, teacher support, and math performance. The Current Study Previous research has highlighted the impact of internal factors, such as academic self-efficacy and academic buoyancy on middle school students’ mathematics performance (Hoon et al., 2024; Lei, 2024; Liu et al., 2017). Similarly, external factors such as parental involvement and teacher support have been shown to significantly influence students' mathematics outcomes (Rameli et al., 2024; Silinskas & Kikas, 2017). A literature review of 137 studies on children aged 6-16 years concluded that most of the indicators of parental involvement are related to children's mathematical achievement, performance and skills. In addition, a positive generalization about parental involvement has the potential to erroneously hide negative aspects (Fiskerstrand, 2022). Moreover, the effect of parental involvement can vary across ethnic and cultural groups (Boonk et al., 2018). Despite these findings, little attention has been given to exploring how these factors interact through mediating variables, particularly in middle-school students (Weißenfels et al., 2023). This gap is noteworthy, as academic buoyancy and academic self-efficacy are likely to play a particularly significant role in shaping math performance during the middle school years for several developmental and contextual reasons. First, this period may be largely due to encountering formal examinations which can create new academic pressures for the first time at the school. Second, middle school is characterized by a grading system that brings increased performance pressures, often accompanied by higher expectations from both parents and teachers. Third, students begin to encounter increasingly complex and abstract mathematical concepts during these years. Finally, this period often coincides with developmental changes, including the transition to high school and adolescence. These combined challenges can lead to declines in students’ math performance. Although cognitive factors have been the main focus of many studies in the existing literature, researchers emphasize the need to consider motivational, emotional and social factors alongside cognitive factors to gain a comprehensive understanding of mathematics achievement (Ramirez et al., 2013; Weißenfels et al., 2023; Živković et al., 2022). To address a notable gap in the literature, the present study investigates a multifaceted mediation model that examines how external factors (i.e., parental involvement and perceived teacher support) predict middle school students’ math performance, directly and indirectly through the mediating roles of academic buoyancy and academic self-efficacy which function as internal mechanisms. By exploring these relationships, the study seeks to understand how external factors interact with these internal factors to influence students’ mathematics performance. Specifically, it aims to contribute to the literature in two key ways: (1) It broadens the understanding of how interactions between external influences (i.e., such as parental involvement and teacher support) and internal attributes such as academic self-efficacy and academic buoyancy affect mathematics performance, thereby supporting improvements in mathematics learning and achievement. (2) It sheds light on the mechanisms linking these external factors to mathematics performance through academic self-efficacy and academic buoyancy, offering insights into the key drivers of middle school students' math performance. The Hypotheses Based on the literature, the default model (see Figure 1) developed in this study aims to provide a comprehensive understanding of the intrinsic and extrinsic factors influencing middle school students' mathematics performance, as well as to examine the interactions among these factors. For this purpose, the following hypotheses (H) were formulated: H 1 . Teacher support is positively associated with math performance. H 2 . Parent involvement is positively associated with math performance. H 3 . Academic self-efficacy is positively associated with math performance. H 4 . Academic buoyancy is positively associated with math performance. H 5 . Academic self-efficacy mediates the relationship between perceived teacher support and math performance. H 6 . Academic self-efficacy mediates the relationship between parental involvement and math performance. H 7 . Academic buoyancy mediates the relationship between perceived teacher support and math performance. H 8 . Academic buoyancy mediates the relationship between parental involvement and math performance. Methods Participants The study used a cross-sectional design and participants were selected through convenience sampling among sixth grade students enrolled in two private and one public middle schools in Turkey. Data were collected using a 35-item questionnaire administered to students with the permission of school principals and teachers in selected schools under the supervision of the second author of the present study. Of the 400 distributed questionnaires, 52 were excluded due to incomplete responses or evidence of inattentive responding. As a result, the final sample consisted of 363 students, including 185 boys (51.0%) and 178 girls (49.0%). Participants’ ages ranged from 11 to 12 years, with a mean age of 12.50 (SD = 0.67). Measures The Interpersonal Behaviours Questionnaire, developed by Rocchi et al. (2016) and translated into Turkish by the first author, was used to measure perceived teacher supportive behaviours. The scale consists of three subscales—competence support, autonomy support, and relatedness support—with a total of 12 items. Responses to items were rated on a 7-point Likert-type scale (1=strongly disagree, 7=strongly agree). Higher scores indicate greater perceived teacher support. Sample items include (1) “My mathematics teacher gives me the freedom to make my own choices.”, (2) “My mathematics teacher supports me in developing my skills.” The scale demonstrated high reliability in this study, with a Cronbach's Alpha of 0.93. The Academic Self-Efficacy Scale , originally developed by Lee et al. (2010), was adapted from the relevant dimension of the Motivational Strategies for Learning Questionnaire . The scale consists of 6 items, which were specially adapted to mathematics by the authors of this study. Sample items include (1) "Compared to other students in this class, I think I know more about mathematics topics," and (2) "Compared to other students in this class, I think I am a good student in mathematics." Items were rated on a 5-point Likert scale (1= not at all true for me, 5= very true for me), with higher scores indicating greater academic self-efficacy in mathematics. Parental Involvement Scale, developed by Cheung and Pomerantz (2011), was used to measure students’ perceptions of their parents’ involvement in mathematics learning. The scale consists of one-dimension with ten items, covering a range of parental involvement practices in math. For this study, all items were adapted to specifically reflect mathematics-related activities. Sample items include (1) “My parents are in contact with my math teacher at school” and (2) “My parents buy extra math workbooks or supplementary materials for me.” Students responded on a 5-point Likert scale ranging from 1 (not at all true) to 5 (very true). High scores indicate higher perceived parental involvement in students’ mathematics learning. Academic Buoyancy Scale, developed by Martin and Marsh (2008), was used to measure participants' academic buoyancy in mathematics. The scale consists of 4 items, rated on a 5-point Likert scale (1=strongly disagree, 5= strongly agree). Sample items include “I am good at dealing with setbacks in math class (e.g., poor grades, negative feedback, getting a question wrong).” For this study, the original items were adapted into Turkish by the researchers to specifically address mathematics learning. The four items evaluate students’ ability to cope with setbacks, challenges, and stress related to mathematics. Higher scores indicate greater mathematics academic buoyancy. Math performance was measured using scores from a standardized national mathematics exam administered by the Turkish Ministry of National Education in the spring term of the 2023-2024 school year. Exam scores were collected from participants’ official school records at the time of the survey administration. Participants' mathematics exam scores ranged from 8 to 100 (M = 64.15, SD = 27.39). A higher exam score shows higher levels of mathematics achievement. Some studies showed that standardized mathematics test scores are a reliable measure of students’ math performance (Yang et al., 2021). Validity and reliability studies of the instruments In this study, the Academic Self-Efficacy Scale (ASES), Parental Involvement Scale (PIS), and Academic Buoyancy Scale (ABS) were initially translated into Turkish by linguistic experts. A rigorous back-translation method was applied to ensure accuracy and cultural appropriateness in the translation process. Expert reviewers specializing in translation and applied linguistics carefully examined the initial and back-translated versions. Throughout the process, special sensitivity was shown to cultural nuances, contextual changes, and cultural expressions of emotions. Pilot research with 50 middle school students was conducted to evaluate the clarity, cultural compatibility, and dependability of the items. Considering the feedback, revisions were made to better align the items with Turkish cultural norms and ensure the constructs were effectively captured. Subsequently, a preliminary study was conducted with 200 middle school students who were not included in the main sample to examine the psychometric properties of the finalized Turkish versions of the scales. Reliability was assessed using Cronbach’s alpha and composite reliability (CR), while construct validity was evaluated through factor loadings and average variance extracted (AVE). Academic Self-Efficacy Scale. CFA was performed to assess the overall goodness-of-fit of all the constructs to determine the validity of the scale. The goodness-of-fit indices for the scale were as follows: (χ²/df = 2.408, GFI = 0.96, TLI = 0.95, CFI = 0.96, RMSEA = 0.064). The factor loadings of all items ranged between 0.48 and 0.82. Further reliability analyses yielded a Cronbach’s alpha of 0.87, AVE of 0.58, and a CR of 0.84, demonstrating satisfactory internal consistency and construct reliability (Hair et al., 2019). Parental Involvement Scale , CFA was performed to evaluate the goodness-of-fit of the model, yielding good indices (χ²/df = 2.67, RMSEA = 0.048, GFI = 0.95, CFI = 0.96, TLI = 0.94). The factor loadings for the items ranged from 0.62 to 0.74. The Cronbach's alpha was 0.89, and AVE and CR values were 0.56 and 0.82, respectively. Academic Buoyancy Scale. The fit indices of the CFA model indicated that the model was a good fit for the data (χ²/df = 3.71, RMSEA = 0.061, GFI = 0.93, CFI = 0.94, TLI = 0.93). Factor loadings for each item were greater than 0.40, with items loading strongly onto their respective factors. The Cronbach’s alpha for the scale was 0.91, the AVE was 0.57 and the CR value was 0.83. Data Collection Following ethical approval, informed consent forms were distributed to students who voluntarily agreed to participate in the study. During survey administration, the second author clarified that the instruments were not mathematics tests or exams, emphasized that there were no right or wrong answers, and assured students that their responses would not affect their mathematics course grades. Participants were encouraged to respond honestly and were informed that their answers would remain confidential and analyzed in aggregate. It was also explicitly stated that their teachers would not have access to individual responses. The survey took approximately 30 minutes to complete, and data were collected in June 2024. Data Analysis The SmartPLS 4.0 software was used, employing partial least squares structural equation modelling (PLS-SEM) as an analytical approach in order to test the hypothesized model. PLS-SEM is particularly well suited for testing complex models with small sample sizes and is less restrictive in terms of data normality assumptions (Hair et al., 2017; Ringle et al., 2022). The PLS-SEM procedure follows a two-step process: first, the measurement model is evaluated to assess the reliability and validity of the constructs, and second, the structural model is analyzed to test the hypothesized relationships (Henseler et al., 2009). Before SEM analysis, firstly, the validity and reliability of the measurement model were evaluated by examining outer loadings, CR, CA, and AVE coefficients. The results confirmed that the measurement model was reliable and valid, as indicated by Composite Reliability (CR) and Cronbach's Alpha (CA) values exceeding 0.7 and Average Variance Extracted (AVE) values higher than 0.5 (Hair et al., 2019). Structural equation modeling (SEM) was used to investigate the direct effects of parental involvement and teacher support on mathematics performance as well as the indirect effects through academic self-efficacy and academic buoyancy. To examine the hypothesized indirect effect of mediating variables, bias-corrected bootstrap analysis with a 95% confidence interval using 5000 bootstrap re-samples was used. Results Preliminary Analysis As presented in Table 1, perceived teacher support (TS) showed significant and positive correlations with academic self-efficacy (ASE; r = 0.457, p < 0.01), academic buoyancy (AB; r = 0.382, p < 0.01), and math performance (MP; r = .402, p < 0.01). Similarly, parental involvement (PI) was positively associated with ASE (r = 0.448, p < 0.01), AB (r = 0.349, p < 0.01), and MP (r = 0.332, p < 0.01). Furthermore, both ASE (r = 0.570, p < .01) and AB (r = 0.554, p < 0.01) showed significant and positive correlations with MP. Finally, there is a positive correlation between PI and TS (r = 0.359, p < 0.01). Table 1 . Means, Standard Deviations and Correlations between Variables 1 2 3 4 5 1. TS 0.359** 0.457** 0.382** 0.402** 2. PI 0.448** 0.349** 0.332** 3. ASE 0.570** 0.554** 4. AB 0.484** 5. MP M 5.13 3.53 3.58 3.35 68.55 SD 1.45 0.768 0.823 0.813 26.03 Note: ** p < 0.001, Abbreviations: TS= Teacher support, PI= Parent involvement, ASE= Academic efficacy, AB= Academic buoyancy, MP= Math performance Measurement Model The measurement model (Table 2) was tested to verify the theoretical appropriateness of the model developed based on the theoretical framework. To validate the measurement model, CA coefficients as well as convergent and discriminant validity values were evaluated. Factor loadings were examined to determine the validity and reliability indicators of the model. Factor loadings of 0.70 or higher and AVE values above 0.50 indicate an acceptable level of convergent validity (Hair et al., 2017). An initial analysis was conducted to identify any items that had factor loadings below the threshold of 0.70. All factor loadings within the measurement model exceeding 0.70 ranged from 0.708 to 0.823. The CA ranged from 0.786 to 0.831, while the CR ranged from 0.785 to 0.855. CA and CR coefficients surpassed the recommended threshold of 0.70, indicating robust reliability for all latent constructs (Hair et al., 2017) and demonstrating good internal consistency. The AVE for all constructs ranged from 0.547 to 0.601, confirming both the reliability and convergent validity of the latent constructs within the model (Hair et al., 2021). These findings demonstrate that the measurement model meets the established criteria. Table 2 . Outer Loadings, Internal Consistency and Average Variance Extracted Constructs Items Loadings rho_A CA CR AVE Academic buoyancy (AB) AB1 0.723 0.823 0.787 0.785 0.551 AB2 0.759 AB3 0.745 AB4 0.728 Teacher Support (TS) TS1 0.734 0.832 0.789 0.823 0.601 TS2 0.789 TS3 0.712 TS4 0.778 TS5 0.796 TS6 0.709 TS7 0.733 TS8 0.745 TS9 0.751 TS10 0.833 TS11 0.744 TS12 0.777 Academic self-efficacy (ASE) ASE1 0.708 0.886 0.786 0.855 0.578 ASE2 0.765 ASE3 0.807 ASE4 0.711 ASE5 0.745 ASE6 0.766 ASE7 0.823 ASE8 0.755 ASE9 0.788 Parental involvement (PI) PI1 0.724 0.932 0.831 0.804 0.547 PI2 0.745 PI3 0.767 PI4 0.786 PI5 0.790 PI6 0.733 PI7 0.790 PI8 0.821 PI9 0.713 PI10 0.747 As seen in Table 2, the rho_A values ranged from 0.823 to 0.932, exceeding the 0.70 threshold and indicating reliable constructs (Hair et al., 2019). Further, the Heterotrait-Monotrait Ratio (HTMT) criteria for discriminant validity evaluations were examined. The HTML values ranged from 0.55 to 0.82, all of which were below the threshold of 0.85, indicating that the discriminant validity was established (Henseler et al., 2014). Variance inflation factor (VIF) values were examined to determine whether there is indicator collinearity. VIF values are less than 3 for the indicators in each factor (Hair et al., 2021) and range between 1.256 and 2.411. These values indicate that there is no collinearity problem among the main variables (Hair et al., 2019). Hypotheses testing results After testing the measurement model, the PLS algorithm was used to test the assumed structural model. The results of the structural model and hypothesis testing are presented in Table 3 and Figure 2. Table 3. Hypotheses Test Results Original sample Sample Mean t p Decision H1 TS—> MP 0.151 0.152 3.342 0.001 Supported H2 PI—> MP 0.093 0.096 2.059 0.040 Supported H3 ASE—> MP 0.252 0.252 4.067 0.000 Supported H4 AB—> MP 0.334 0.332 5.975 0.000 Supported H5 TS —> ASE —> MP 0.084 0.022 3.853 0.000 Supported H6 PI—> ASE —> MP 0.962 0.098 3.498 0.000 Supported H7 TS —> AB —> MP 0.108 0.107 4.790 0.000 Supported H8 PI—> AB—> MP 0.987 0.100 3.723 0.000 Supported As seen in Figure 2, both perceived TS (β = 0.151, p = 0.01) and PI (β = 0.093, p = 0.040) had positive and statistically significant direct effects on MP, thereby supporting hypotheses H1 and H2. In addition, TS (β = 0.333, p = 0.00; β = 0.324, p = 0.01) and PI (β = 0.384, p = 0.00; β = 0.294, p = 0.05) also positively and significantly predicted both ASE and AB, respectively. Both ASE (β = 0.334, p = 0.00) and AB (β = 0.252, p = 0.05) had significant positive effects on MP, providing support for hypotheses H3 and H4. Further analysis revealed significant indirect effects. ASE mediated the relationship between TS and MP (β = 0.084, p = 0.00), as well as between PI and MP (β = 0.096, p = 0.00), thereby supporting hypotheses H5 and H6. Similarly, AB significantly mediated the relationship between TS and MP (β = 0.108, p = 0.00), and between PI and MP (β = 0.098, p = 0.00), confirming hypotheses H7 and H8. Overall, the model demonstrated that TS and PI influence MP not only through direct pathways but also indirectly through the mediation of students’ ASE and AB. Discussion The present study aimed to investigate the associations between perceived teacher support (TS) and parent involvement (PI) on middle school students’ math performance (MP), both directly and indirectly through academic self-efficacy (ASE), and academic buoyancy (AB). The findings revealed that all four variables had significant and positive direct effects on MP. Moreover, ASE and AB played mediating roles in the relationship between both TS and PI with the students’ MP. Prior research has consistently emphasized the central role of supportive teacher-student relationships in fostering students’ academic engagement and achievement, particularly in challenging subjects such as mathematics (Chen & Leung, 2023 ; Wentzel, 2002 ). Middle school students, who are navigating increased academic pressure and a more abstract mathematics curriculum, may particularly benefit from emotionally responsive and academically supportive teachers (Yang et al., 2023 ). Consistent with this literature, the present study found that perceived TS was a significant direct predictor of MP. This suggests that students who perceive their teachers as supporting autonomy, competence, and emotional well-being tend to perform and achieve better outcomes in mathematics (Jung et al., 2023 ; Yu & Singh, 2016 ). Beyond this direct effect, TS also indirectly predicted MP through its positive associations with both ASE and AB, suggesting TS not only contributes to students’ mathematics performance by offering instructional support, but also by enhancing their academic self-efficacy in mathematics—that is, their belief in their ability to succeed in math—and by fostering their academic buoyancy, or their capacity to recover from routine academic setbacks encountered in math learning. This aligns with Bandura’s ( 1997 ) socio-cognitive perspective, which posits that supportive social contexts enhance students’ motivational beliefs and academic performance. Specifically, students who receive more social support from their teachers are more likely to acquire mathematical knowledge, build self-confidence (i.e., competence), develop an interest in mathematics (Wu et al., 2022 ), form positive evaluations of the value of mathematics (Pekrun et al., 2017 ).Such support also helps students perceive mistakes as learning opportunities rather than as sources of judgment, thereby reducing math anxiety. Teacher support thus not only motivates students to engage with learning but also increases their self-efficacy beliefs (Liu et al., 2017 ), which in turn reduces setbacks or challenges encountered in mathematics learning (Li et al., 2023 ). The current findings indicate that TS serves as a dual mechanism, encompassing both the instructional quality provided by teachers and relational and motivational support that foster students’ ASE and AB, which in turn contribute positively to their mathematics performance. The present study extends existing literature on the role of PI on students' MP. Active PI is associated with improved children’s learning and classroom performance (Jeynes, 2007 ). A recent meta-analysis also confirmed the significantly positive influence of PI on students' MP (Wang & Wei, 2024 ). When parents actively engage in their children’s learning processes, students are more likely to achieve higher levels of achievement in mathematics (Fiskerstrand, 2022 ). At the middle-school level, where the mathematics curriculum becomes increasingly abstract and includes more specialized topics requiring advanced reasoning and problem-solving skills, PI may provide students with the emotional support and structured guidance needed to persist through academic challenges. Consistent with this body of research, the present findings revealed that PI not only directly supports students’ MP but also has an indirect effect on MP through its positive associations with ASE and AB. Several studies indicate that higher levels of PI enhance students' ASE, which in turn contributes to improved academic performance (Weiser & Riggio, 2010 ; You et al., 2015 ). Similarly, encouraging active and supportive PI can significantly enhance AB, which fosters students’ ability to cope with everyday academic setbacks and challenges and thus support their MP (Chen & Mok, 2023 ). Parents who demonstrate consistent involvement in their children's education may reinforce students’ confidence in their academic capabilities and promote a resilient attitude toward learning challenges (Gu et al., 2023 ; Morkoyunlu & Konyalıoğlu, 2020 ). In this context, parental involvement can serve as an external resource that creates a positive impact on students' mathematics performance by increasing their self-efficacy beliefs in mathematics and helping them cope with math anxiety or typical difficulties. Because emotional and autonomy support from parents can create an environment that enables them to guide students to pursue mathematical challenges, take risks, and strive for learning and performance with increased self-confidence, ultimately increasing mathematics achievement (Oh et al., 2022 ). These findings underscore the multifaceted role of PI in supporting math performance, both directly and through academic self-efficacy and buoyancy, and suggest that fostering meaningful and sustained parental involvement may therefore be an effective strategy for increasing students' academic self-efficacy and buoyancy in mathematics. Importantly, this support must be perceived by students themselves, as their sense of being supported plays a critical role in the development of ASE and AB. As with external factors, the present study also highlights the internal factors, such as ASE and AB, in relation to middle school students’ MP, revealing both significant direct and mediated effects. This finding aligns with established theoretical perspectives. Students with higher ASE, characterized by strong confidence in their ability and effort to succeed, tend to achieve better MP, consistent with Bandura’s ( 1997 ) assertion that self-efficacy is a key determinant of academic success. Similarly, students with higher AB are better able to cope with everyday academic setbacks, pressures, and daily challenges and tend to maintain engagement and persistence, leading to improved performance in demanding subjects such as mathematics and science (Colmar et al., 2019 ; Kul et al., 2024 ; Martin & Marsh, 2008 ; Weißenfels et al., 2023 ). Beyond their direct contributions, both ASE and AB served as mediators between external support (TS and PI) and students’ MP. This suggests that supportive environments foster students’ confidence in their academic abilities and their buoyancy in everyday academic challenges. When students perceive encouragement and involvement from teachers and parents, they are more likely to believe in their capabilities and persist through academic difficulties. Thus, ASE and AB function as a key psychological mechanism, which translates external support into sustained engagement and improved academic performance in mathematics. Conclusion This study supports our understanding of the relationship between middle school students' mathematics performance and the non-cognitive psychological factors of academic self-efficacy and academic buoyancy. The findings suggest that these internal psychological resources positively affect students' mathematics performance in both direct and indirect ways. The study indicates that students who are supported by teachers and parents tend to perform better in math, both directly and indirectly through increased academic self-efficacy and academic buoyancy. These intrinsic psychological factors seem to positively mediate the effect of external factors by emphasizing their transformative function on the performance of external social resources in mathematics. Moreover, teacher support and parental involvement appear to play a dual role not only in directly improving mathematics performance but also in shaping students' mathematics performance by supporting students' academic self-efficacy and academic vitality. Taken together, these findings provide empirical evidence that students can acquire meaningful outcomes in mathematics learning when they receive support from both their teachers and their parents. Overall, the study emphasizes the importance of holistic approaches, including both intrinsic and extrinsic factors, in mathematics education, sheds light on the dynamic interaction between these factors, and emphasizes the importance of both to support students' mathematics achievement. Limitations and Future Studies This study has some limitations. First, our understanding regarding students' parent involvement and perceptions about their teacher support is based solely on their self-reported. This may not fully capture the extent of support provided by parents and teachers. Future studies may incorporate multi-sources approaches, including data collected from parents, teachers, and direct observations, to provide a more comprehensive understanding. Second, the sample size was relatively small and drawn from a single urban region Türkiye. Findings may not be generalizable to broader, more diverse populations. Future research should include larger and more diverse samples to improve the external validity. Third, the use of a cross-sectional design restricts the ability to establish causal relationships among the variables. Longitudinal designs are recommended to explore how these relationships evolve and influence mathematics performance over time. Implications for Practice This study offers practical insights for teachers and parents aiming to improve middle school students’ math performance. Teachers are encouraged to provide opportunities for student decision-making, support their choices, and show genuine interest, as these practices can positively strengthen students’ perceptions of teacher support and boost their performance. Additionally, parents can enhance their involvement by maintaining regular communication with mathematics teachers to monitor their children’s progress and engage in meaningful conversations with them about math-related challenges and achievements. Supporting students’ self-efficacy and academic buoyancy in mathematics is essential, as these internal resources play a critical role in enhancing students’ learning experiences and enabling them to overcome the difficulties inherent in challenging subjects such as mathematics. Feeling supported in coping with such challenges can motivate students to persist and succeed in mathematics and increase their success. Ultimately, coordinated efforts between parents and teachers—characterized by consistent encouragement, targeted support, and open communication—can foster students’ academic buoyancy, strengthen their self-efficacy, and improve their overall mathematics performance. Abbreviations TS: Teacher Support, PI: Parental Involvement, ASE: Academic Self-Efficacy, AB: Academic Buoyancy, MP: Mathematics Performance, PLS: Partial Least Squares, SEM: Structural Equation Modelling, OECD: Organisation For Economic Co-Operation And Development, ASES: Academic Self-Efficacy Scale, PIS: Parental Involvement Scale, ABS: Academic Buoyancy Scale, PLS-SEM: Partial Least Squares Structural Equation Modelling, CR: Composite Reliability, AVE: Average Variance Extracted, HTMT: Heterotrait-Monotrait Ratio, VIF: Variance Inflation Factor. Declarations Ethics approval and consent to participate This study was conducted in Türkiye with the participation of middle school students. Ethical approval was obtained from the Ethics Committee of Hasan Kalyoncu University (Approval No: E-97105791-050.04-57875). All legal and ethical requirements were fulfilled, and written informed consent was obtained from parents or legal guardians in accordance with the Declaration of Helsinki. Consent for publication Not applicable Availability of data and materials The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request. Competing interests The authors have no competing interests. Funding The author acknowledge that they received no external funding in support of this research. Authors’ contributions İİ, NCA, and MC: Writing, Review & Editing. İİ: Resources, Conceptualization, Methodology, Investigation, Visualization and analysis. 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Multidimensional aspects of parental involvement in Korean adolescents’ schooling: a mediating role of general and domain-specific self-efficacy. Educational Psychology, 36 (5), 916–934. https://doi.org/10.1080/01443410.2015.1025705 Yu, W., Zhou, S., & Zhou, Y. (2023). Measuring mathematics self-efficacy: Multitrait-multimethod comparison. Frontiers in Psychology, 14 https://doi.org/10.3389/fpsyg.2023.1108536 Yu, R., & Singh, K. (2016). Teacher support, instructional practices, student motivation, and mathematics achievement in high school. The Journal of Educational Research, 1 –14. https://doi.org/10.1080/00220671.2016.1204260. Yun, S., Hiver, P., & Al-Hoorie, A. H. (2018). Academic buoyancy: Exploring learners’ everyday resilience in the language classroom. Studies in Second Language Acquisition, 40 (4), 805-830. https://doi.org/10.1017/S0272263118000037 Zakariya, Y. F. (2022). Improving students’ mathematics self-efficacy: A systematic review of intervention studies. Frontiers in Psychology, 13 . https://doi.org/10.3389/fpsyg.2022.986622 Zimmerman, B. J., & Kitsantas, A. (2005). Homework practices and academic achievement: The mediating role of self-efficacy and perceived responsibility beliefs. Contemporary Educational Psychology, 30 (4), 397-417 https://doi.org/10.1016/j.cedpsych.2005.05.003 Zimmerman, B. J., Bandura, A., & Martinez-Pons, M. (1992). Self-motivation for academic attainment: The role of self-efficacy beliefs and personal goal setting. American Educational Research Journal, 29 (3), 663-676 https://doi.org/10.3102/00028312029003663 Živković, M., Pellizzoni, S., Doz, E., Cuder, A., Mammarella, I., & Passolunghi, M. C. (2023). Math self-efficacy or anxiety? The role of emotional and motivational contribution in math performance. Social Psychology of Education, 26 (3), 579-601. https://doi.org/10.1007/s11218-023-09760-8 Živković, M., Pellizzoni, S., Mammarella, I. C., & Passolunghi, M. C. (2022). Executive functions, math anxiety and math performance in middle school students. British Journal of Developmental Psychology, 40 (3), 438-452. https://doi.org/10.1111/bjdp.12412 Zysberg, L., & Schwabsky, N. (2020). School climate, academic self-efficacy and student achievement. Educational Psychology, 41 (4), 467-482. https://doi.org/10.1080/01443410.2020.1813690 Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6903576","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":490082888,"identity":"0756fa93-55f1-4527-8d00-e0e049a22e20","order_by":0,"name":"İlhan İlter","email":"","orcid":"","institution":"Kahramanmaraş Sütçü İmam University","correspondingAuthor":false,"prefix":"","firstName":"İlhan","middleName":"","lastName":"İlter","suffix":""},{"id":490082891,"identity":"4d556eb3-e832-45d6-90e0-7ba4fb9c4601","order_by":1,"name":"Nuri Can Aksoy","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6ElEQVRIie3RPQrCMBiA4a8I7SLWMVn0ChGhIgi9SkqhU1ylg6AunYqzuYUuuhYEuxTnjBahsy4dxUZxjRkF80J+hjwkEACT6XejzWidrKXcZ/rEjj7kvWqQtqdHRs6xusA88l3E6pIn0OsI6lxjBRmn0YjAiQV8Mz0MdgkMsaDWqlAQIsBDYMeUiOkelwkEW0lULyPCqRE8Yp8IVkmy0CBtD1kJs7aC2bh5WHPdN1KwGQrWUcDTysObMxrwolxxJcnzPbrVoe86YYXT2aTfycPsriKvqJy6rxk1Q+snZW6medBkMpn+ridhi1EQZxz1dwAAAABJRU5ErkJggg==","orcid":"","institution":"Hasan Kalyoncu University","correspondingAuthor":true,"prefix":"","firstName":"Nuri","middleName":"Can","lastName":"Aksoy","suffix":""},{"id":490082892,"identity":"46d75baa-0c2f-4e5e-8806-b5ef00c8fa87","order_by":2,"name":"Mehmet Ceylan","email":"","orcid":"","institution":"Hasan Kalyoncu University","correspondingAuthor":false,"prefix":"","firstName":"Mehmet","middleName":"","lastName":"Ceylan","suffix":""}],"badges":[],"createdAt":"2025-06-16 08:53:28","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6903576/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6903576/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1186/s40359-025-03547-6","type":"published","date":"2025-11-04T15:57:28+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":87693786,"identity":"9a2af1a6-603b-4d98-94e8-5dbd58fe6c45","added_by":"auto","created_at":"2025-07-28 05:34:58","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":91420,"visible":true,"origin":"","legend":"\u003cp\u003eThe default model\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6903576/v1/1f5df361a915d6a06bb3f949.png"},{"id":87693789,"identity":"b047dbd3-d1de-4ff9-96ed-dccb0a576e67","added_by":"auto","created_at":"2025-07-28 05:34:58","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":132806,"visible":true,"origin":"","legend":"\u003cp\u003eHypothesized model results\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6903576/v1/334ccc1ad675e7d5ba79f4e4.png"},{"id":95564227,"identity":"bfea072f-2535-4e8f-b129-eb1896399d70","added_by":"auto","created_at":"2025-11-10 16:09:06","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1100535,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6903576/v1/36d2beb2-ac2b-4357-bcae-ab9a96fbcdd1.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The Relationships between Parental Involvement, Teacher Support, and Mathematics Performance: Mediating Roles of Academic Self-Efficacy and Academic Buoyancy","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe issue of enhancing students' mathematical performance remains a persistent challenge for both educators and policymakers, as well as parents. Despite consistent interest and efforts to improve students\u0026rsquo; math performance and achievement, the recent PISA report indicated that between 2018 and 2022; the average performance in mathematics across OECD countries dropped by a record 15 points (OECD, \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Nevertheless, in the four weeks preceding the PISA assessment, an average of 10% of students across the OECD countries reported that they did not feel safe at school, while 30% 30% indicated being distracted by digital devices. These findings support the notion that learning mathematics extends beyond cognitive challenges and entails a complex interplay of emotional and motivational factors (Pekrun \u0026amp; Loderer, \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Živković et al., \u003cspan citationid=\"CR81\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eFactors such as emotional security, motivation, and focus may be crucial in shaping students' mathematical performance. Addressing the challenges in promoting and teaching mathematics learning requires a deeper understanding of how both external support, such as teacher support and parental involvement, and internal non-cognitive attributes including academic self-efficacy, grit, academic motivation, academic buoyancy, contribute to students' performance and achievement. Non-cognitive factors are critical, as they encompass intrinsic motivations and behaviours that enable students to make the most of their potential (Chamorro-Premuzic \u0026amp; Furnham, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). These intrinsic attributes not only influence students' academic performance but also play a pivotal role in shaping their long term educational outcomes (Farrington et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2012\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eThis study builds upon the importance of mathematics education by examining how social and motivational factors influence middle school students' mathematics performance. In particular, it investigates how academic buoyancy and academic self-efficacy mediate the links between math performance, teacher support, and parental involvement. By focusing on these mediators, the study aims to reveal how external factors, such as social support, interact with students\u0026rsquo; internal non-cognitive attributes to influence their performance in mathematics. The study aims to provide a more thorough understanding of the intricate aspects impacting mathematical achievement by utilizing a multiple mediation approach. This study aims to contribute to the identification of non-cognitive factors that support students\u0026rsquo; mathematics performance and achievement, while also examining the role of external support provided by teachers and parents.\u003c/p\u003e"},{"header":"Conceptual framework","content":"\u003cp\u003e\u003cstrong\u003eAcademic self-efficacy and mathematics performance\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAccording to the Social Cognitive Theory (SCT), both external social influences and self-regulatory factors are crucial in motivating and controlling behaviour (Bandura, 2012). One important component of these self-regulatory elements is self-efficacy, which is a person\u0026apos;s assessment of their capacity to plan and carry out the actions or tasks required to accomplish certain goals (Bandura, 1997). In educational settings, self-efficacy is often conceptualized as academic self-efficacy (Bandura, 2012; Schunk \u0026amp; DiBenedetto, 2022), which is defined as students\u0026rsquo; belief in their own ability to succeed in academic tasks (Bandura, 1997; Zimmerman et al., 1992). Academic self-efficacy refers to students\u0026apos; perseverance and determination towards their abilities in achieving academic achievement, as well as their beliefs about fulfilling certain academic tasks (Bandura, 1997). Academic self-efficacy refers to a person\u0026apos;s belief in their ability to complete required tasks (Zakariya, 2022) and their confidence when faced with mathematical challenges (Yu et al., 2023).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eResearch shows that academic self-efficacy has a significant effect on shaping students\u0026apos; academic behaviours and outcomes. A strong feeling of academic self-efficacy not only increases the likelihood that students will set higher goals and exert more effort, but it also fosters resilience when they encounter challenges (Usher \u0026amp; Pajares, 2008; Zimmerman \u0026amp; Kitsantas, 2005). A belief that one can succeed has been related to increased academic performance across a range of content areas in several studies, and the extensive literature highlights its significant impact on achievement and test performance (Camelo-Lavadores et al., 2017; Honicke \u0026amp; Broadbent, 2016; Roick \u0026amp; Ringeisen, 2018; Zysberg \u0026amp; Schfwabsky, 2020).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn mathematics, academic self-efficacy refers to a person\u0026apos;s belief in their ability to effectively complete mathematics activities (Yang et al., 2021) and their confidence when faced with mathematical challenges (Yu et al., 2023). In their study, Lera et al. (2023) used a multilevel structural equation model to better understand the effect of self-efficacy in mathematics education. They found that self-efficacy also predicted math performance at the class level. However, while academic self-efficacy is recognized as a primary predictor of mathematics performance, it remains unclear whether teacher support and parental involvement influence middle school students\u0026apos; mathematics performance through their interaction with academic self-efficacy. This gap in understanding highlights the necessity of exploring how the interplay between these factors gives a more comprehensive picture of middle school students\u0026rsquo; achievement in mathematics.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcademic buoyancy and math performance\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAcademic buoyancy is the capacity of students\u0026apos; ability to effectively deal with daily academic challenges or difficulties such as low grades, difficult tasks, or classroom distractions (Martin \u0026amp; Marsh, 2008). It not only acts as a buffer against academic stress but also promotes a positive attitude towards learning, which is crucial for long-term academic achievement (Martin, 2013). In the context of mathematics, the broad range of topics and the reasoning required during the learning process pose unique challenges to maintaining consistent motivation and effort. \u0026nbsp;When students struggle to make progress, they may experience setbacks that hinder their academic performance. The ability to adapt to such challenges in mathematics is referred to as mathematical buoyancy (Hoon et al., 2024). Strengthening mathematical academic buoyancy is crucial for overcoming unexpected challenges (e.g., anxiety, lack of motivation, and unstable academic development), all of which can negatively affect students\u0026apos; math performance (Ang et al., 2022). By encouraging students\u0026apos; interest and persistence in pursuing goals for the future, mathematical buoyancy aims to improve achievement and reduce math-related anxiety. Previous studies showed that academic buoyancy, particularly in mathematics, significantly impacts academic performance and achievement (Colmar et al., 2019; Datu \u0026amp; Yang, 2021) with academic self-efficacy mediating this relationship (Wei\u0026szlig;enfels et al., 2023) and remaining significant even when controlling for gender (Lei et al., 2022). Overall, developing mathematical buoyancy is crucial for enhancing students\u0026apos; motivation, engagement, and performance in the school setting (Collie et al., 2024; Colmar et al., 2019).\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTeacher support in mathematics performance\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOne important resource that has a significant effect on how students\u0026rsquo; academic performance develops is teacher supportive behaviours (Kim, et al., 2018). Teacher support enhances students\u0026apos; academic achievement (Glozah \u0026amp; Pevalin, 2014) and is positively correlated with math self-concept while negatively correlated with math anxiety (Yıldırım \u0026amp; Yıldırım, 2019). Research indicates that teacher support has a direct impact on three components of math engagement: emotional, behavioural, and cognitive (Liu et al., 2017). Positive teacher supportive behaviours not only boost students\u0026apos; autonomy in their math abilities but also enhance their overall engagement, promoting improved academic outcomes (Yang et al., 2021). Furthermore, close relationships with teachers positively predict school engagement, while conflict with teachers negatively predicts it (Engels et al., 2021). However, some research suggests that teacher support does not directly influence students\u0026apos; math performance. In their study, Chang et al. (2023) concluded that although teacher support significantly enhances students\u0026rsquo; math self-efficacy but did not directly extend to their math performance. The study further emphasizes that the role of teacher support varies across different demographic groups and may influence math achievement through indirect pathways.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eParent\u0026nbsp;involvement in students\u0026rsquo; learning mathematics\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eGiven its multifaceted role in fostering students\u0026rsquo; academic outcomes, parental involvement remains a critical area of investigation in educational research. It not only complements teacher efforts but also enhances academic engagement, and academic self-efficacy among students, both of which are key predictors of math achievement (Fan \u0026amp; Williams, 2010; Hill \u0026amp; Tyson, 2009). Ginsburg et al. (2008) identified three types of parental involvement in mathematics subject areas; (1) parent involvement in completing homework assignments, (2) parent reengagement with mathematics learning, and (3) math talks during mathematics activities. These practices directly contribute to students\u0026apos; academic progress by fostering a supportive learning environment in the home (Gonzalez-DeHass et al., 2005; Hill \u0026amp; Tyson, 2009). Previous research indicates that parental involvement in their children\u0026apos;s learning strengthens school efforts and is connected to increased positive academic outcomes, including higher academic achievement and better school attendance (Jeynes, 2007). Students who have received parental support tend to demonstrate higher math performance than those who have not. However, some research suggests that sustaining this positive effect requires consistent involvement and high-quality parent-child interactions (Ma et al., 2021; Morkoyunlu \u0026amp;Konyalıoğlu, 2020). Moreover, not all kinds of involvement are positively related to academic achievements, and the effects may vary across different ethnicities (Boonk et al., 2018). Considering mathematics education, parental involvement can have both benefits and drawbacks, particularly when parents experience math anxiety, which can inadvertently hinder students\u0026rsquo; learning (Maloney et al., 2015; Oh et al., 2022). Building on these findings, the present study investigates the mediating roles of academic buoyancy and academic self-efficacy to better understand how perceived teacher support and parental involvement influence middle school students\u0026apos; math performance.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMediating effect of academic self-efficacy\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAcademic self-efficacy beliefs reflect one\u0026apos;s confidence and capacity in their ability to carry out certain academic tasks across various subject areas, such as mathematics, science, and foreign languages. In challenging subjects like mathematics, the notion of academic self-efficacy is one of the strongest predictors of performance of an activity and school success (Han \u0026amp; Wang, 2021). According to Martin and Rimm-Kaufman\u0026apos;s (2015) research, students\u0026rsquo; self-efficacy has a direct effect on their mathematics learning process and significantly shapes their achievement outcomes. However, academic self-efficacy is not solely an individual characteristic; it is significantly influenced by the social support students receive from their environment (Bandura, 1997). Parents who are involved in their children\u0026apos;s studies put more effort into involving their children in the educational process. Parental involvement in children\u0026apos;s education fosters positive outcomes, including academic achievement. Research indicates that it serves as a key source of external support, enhancing students\u0026apos; academic self-efficacy in mathematics (Cheung \u0026amp; Pomerantz, 2011). Moreover, emotional support from parents not only motivates students to overcome challenges in mathematics but also promotes their autonomy, contributing to their overall academic success.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAs Fan \u0026amp; Williams (2010) noted, parental support often begins early in a child\u0026rsquo;s education, as parents are typically their first educators, shaping their aspirations and expectations (Metheny \u0026amp; McWhirter, 2013). This is because high-quality parent-child relationships foster positive educational outcomes, including enhanced self-efficacy and academic engagement (Fan \u0026amp; Williams, 2010). When teachers provide structure by managing student-centred activities and providing positive feedback that improves students\u0026apos; self-efficacy perceptions, this positively influences their academic performance. Kim et al. (2018) found that students who received more support from their teachers had a stronger sense of academic self-efficacy, which in turn, led to improved overall achievement. Collie et al. (2016) concluded that high-quality relationships with teachers and parents\u0026mdash; characterised by positive interactions, fair treatment and a sense of belonging \u0026mdash;often result in students with high self-efficacy and greater academic engagement. Academic self-efficacy can serve as a mediator in the relationships between parental involvement, teacher support, and mathematics achievement. When parents and teachers are actively involved in mathematics learning, they provide essential support that enhances students\u0026apos; academic self-efficacy. Consequently, this strengthened self-efficacy boosts students\u0026apos; confidence and motivation, ultimately leading to improved performance in mathematics.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMediating effect of academic buoyancy\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eStudents who overcome minor academic difficulties and challenges with academic vitality demonstrate school satisfaction and effective classroom behaviours (Hoferichter et al., 2021). It is documented in the literature that academic buoyancy is positively related to students\u0026apos; performance in the classroom (Datu \u0026amp; Yang, 2021; Martin \u0026amp; Marsh, 2009; Yun et al., 2018). Research has shown that academic buoyancy encourages students to adopt effective learning strategies (Collie et al., 2016) and supports both their emotional and behavioural engagement in school settings (Datu et al., 2018), self-regulation, and higher academic achievement (Miller et al., 2013). Martin and Marsh (2008) highlight the importance of adopting a domain-specific approach when studying academic buoyancy. In the field of mathematics, some studies have indicated a positive relationship between academic buoyancy and academic achievement in the subject (Martin \u0026amp; Marsh, 2008; Wei\u0026szlig;enfels et al., 2023). Academic buoyancy is influenced by an individual\u0026apos;s social environment, as environmental factors significantly affect students\u0026rsquo; thoughts and actions (Martin, \u0026amp; Marsh, 2008). For students to cultivate academic buoyancy, appropriate support is essential. The home environment, with parental involvement (Chen \u0026amp; Mok, 2023), and the school environment, with teacher support, can greatly influence a child\u0026apos;s development. Hejazi and Abbasi (2021) noted that students\u0026rsquo; academic buoyancy can improve when teachers help guide them toward optimal personal goals and parents can foster a supportive family environment. Consequently, parental involvement and teacher support are thus crucial in enhancing academic buoyancy, ultimately helping students improve their academic performance, including in subjects like mathematics. In this respect, academic buoyancy may mediate the relationship between parent involvement, teacher support, and math performance. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThe Current Study\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ePrevious research has highlighted the impact of internal factors, such as academic self-efficacy and academic buoyancy on middle school students\u0026rsquo; mathematics performance (Hoon et al., 2024; Lei, 2024; Liu et al., 2017). Similarly, external factors such as parental involvement and teacher support have been shown to significantly influence students\u0026apos; mathematics outcomes (Rameli et al., 2024; Silinskas \u0026amp; Kikas, 2017). A literature review of 137 studies on children aged 6-16 years concluded that most of the indicators of parental involvement are related to children\u0026apos;s mathematical achievement, performance and skills. In addition, a positive generalization about parental involvement has the potential to erroneously hide negative aspects (Fiskerstrand, 2022). Moreover, the effect of parental involvement can vary across ethnic and cultural groups (Boonk et al., 2018). Despite these findings, little attention has been given to exploring how these factors interact through mediating variables, particularly in middle-school students (Wei\u0026szlig;enfels et al., 2023). This gap is noteworthy, as academic buoyancy and academic self-efficacy are likely to play a particularly significant role in shaping math performance during the middle school years for several developmental and contextual reasons. First, this period may be largely due to encountering formal examinations which can create new academic pressures for the first time at the school. Second, middle school is characterized by a grading system that brings increased performance pressures, often accompanied by higher expectations from both parents and teachers. Third, students begin to encounter increasingly complex and abstract mathematical concepts during these years. Finally, this period often coincides with developmental changes, including the transition to high school and adolescence. These combined challenges can lead to declines in students\u0026rsquo; math performance. Although cognitive factors have been the main focus of many studies in the existing literature, researchers emphasize the need to consider motivational, emotional and social factors alongside cognitive factors to gain a comprehensive understanding of mathematics achievement (Ramirez et al., 2013; Wei\u0026szlig;enfels et al., 2023; Živković et al., 2022).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTo address a notable gap in the literature, the present study investigates a multifaceted mediation model that examines how external factors (i.e., parental involvement and perceived teacher support) predict middle school students\u0026rsquo; math performance, directly and indirectly through the mediating roles of academic buoyancy and academic self-efficacy which function as internal mechanisms. By exploring these relationships, the study seeks to understand how external factors interact with these internal factors to influence students\u0026rsquo; mathematics performance. Specifically, it aims to contribute to the literature in two key ways: (1) It broadens the understanding of how interactions between external influences (i.e., such as parental involvement and teacher support) and internal attributes such as academic self-efficacy and academic buoyancy affect mathematics performance, thereby supporting improvements in mathematics learning and achievement. (2) It sheds light on the mechanisms linking these external factors to mathematics performance through academic self-efficacy and academic buoyancy, offering insights into the key drivers of middle school students\u0026apos; math performance.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThe Hypotheses\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eBased on the literature, the default model (see Figure 1) developed in this study aims to provide a comprehensive understanding of the intrinsic and extrinsic factors influencing middle school students\u0026apos; mathematics performance, as well as to examine the interactions among these factors. For this purpose, the following hypotheses (H) were formulated:\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e1\u003c/sub\u003e. Teacher support is positively associated with math performance.\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e2\u003c/sub\u003e. Parent involvement is positively associated with math performance.\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e3\u003c/sub\u003e. Academic self-efficacy is positively associated with math performance.\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e4\u003c/sub\u003e. Academic buoyancy is positively associated with math performance.\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e5\u003c/sub\u003e. Academic self-efficacy mediates the relationship between perceived teacher support and math performance.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e6\u003c/sub\u003e. Academic self-efficacy mediates the relationship between parental involvement and math performance.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e7\u003c/sub\u003e. Academic buoyancy mediates the relationship between perceived teacher support and math performance.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eH\u003csub\u003e8\u003c/sub\u003e. Academic buoyancy mediates the relationship between parental involvement and math performance.\u0026nbsp;\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003e\u003cstrong\u003eParticipants\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe study used a cross-sectional design and participants were selected through convenience sampling among sixth grade students enrolled in two private and one public middle schools in Turkey. Data were collected using a 35-item questionnaire administered to students with the permission of school principals and teachers in selected schools under the supervision of the second author of the present study. Of the 400 distributed questionnaires, 52 were excluded due to incomplete responses or evidence of inattentive responding. As a result, the final sample consisted of 363 students, including 185 boys (51.0%) and 178 girls (49.0%). Participants\u0026rsquo; ages ranged from 11 to 12 years, with a mean age of 12.50 (SD = 0.67).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeasures\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThe Interpersonal Behaviours Questionnaire,\u0026nbsp;\u003c/strong\u003edeveloped by Rocchi et al. (2016) and translated into Turkish by the first author, was used to measure perceived teacher supportive behaviours. The scale consists of three subscales\u0026mdash;competence support, autonomy support, and relatedness support\u0026mdash;with a total of 12 items. Responses to items were rated on a 7-point Likert-type scale (1=strongly disagree, 7=strongly agree). Higher scores indicate greater perceived teacher support. Sample items include (1) \u0026ldquo;My mathematics teacher gives me the freedom to make my own choices.\u0026rdquo;, (2) \u0026ldquo;My mathematics teacher supports me in developing my skills.\u0026rdquo; The scale demonstrated high reliability in this study, with a Cronbach\u0026apos;s Alpha of 0.93.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThe Academic Self-Efficacy Scale\u003c/strong\u003e, originally developed by Lee et al. (2010), was adapted from the relevant dimension of the \u003cem\u003eMotivational Strategies for Learning Questionnaire\u003c/em\u003e. The scale consists of 6 items, which were specially adapted to mathematics by the authors of this study. Sample items include (1) \u0026quot;Compared to other students in this class, I think I know more about mathematics topics,\u0026quot; and (2) \u0026quot;Compared to other students in this class, I think I am a good student in mathematics.\u0026quot; Items were rated on a 5-point Likert scale (1= not at all true for me, 5= very true for me), with higher scores indicating greater academic self-efficacy in mathematics.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eParental Involvement Scale,\u0026nbsp;\u003c/strong\u003edeveloped by Cheung and Pomerantz (2011), was used to measure students\u0026rsquo; perceptions of their parents\u0026rsquo; involvement in mathematics learning. The scale consists of one-dimension with ten items, covering a range of parental involvement practices in math. For this study, all items were adapted to specifically reflect mathematics-related activities. Sample items include (1) \u0026ldquo;My parents are in contact with my math teacher at school\u0026rdquo; and (2) \u0026ldquo;My parents buy extra math workbooks or supplementary materials for me.\u0026rdquo; Students responded on a 5-point Likert scale ranging from 1 (not at all true) to 5 (very true). High scores indicate higher perceived parental involvement in students\u0026rsquo; mathematics learning.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcademic Buoyancy Scale,\u0026nbsp;\u003c/strong\u003edeveloped by Martin and Marsh (2008), was used to measure participants\u0026apos; academic buoyancy in mathematics. The scale consists of 4 items, rated on a 5-point Likert scale (1=strongly disagree, 5= strongly agree). Sample items include \u0026ldquo;I am good at dealing with setbacks in math class (e.g., poor grades, negative feedback, getting a question wrong).\u0026rdquo; For this study, the original items were adapted into Turkish by the researchers to specifically address mathematics learning. The four items evaluate students\u0026rsquo; ability to cope with setbacks, challenges, and stress related to mathematics. Higher scores indicate greater mathematics academic buoyancy.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMath performance\u0026nbsp;\u003c/strong\u003ewas measured using scores from a standardized national mathematics exam administered by the Turkish Ministry of National Education in the spring term of the 2023-2024 school year. Exam scores were collected from participants\u0026rsquo; official school records at the time of the survey administration. Participants\u0026apos; mathematics exam scores ranged from 8 to 100 (M = 64.15, SD = 27.39). A higher exam score shows higher levels of mathematics achievement. Some studies showed that standardized mathematics test scores are a reliable measure of students\u0026rsquo; math performance (Yang et al., 2021).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eValidity and reliability studies of the instruments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn this study, the Academic Self-Efficacy Scale (ASES), Parental Involvement Scale (PIS), and Academic Buoyancy Scale (ABS) were initially translated into Turkish by linguistic experts. A rigorous back-translation method was applied to ensure accuracy and cultural appropriateness in the translation process. Expert reviewers specializing in translation and applied linguistics carefully examined the initial and back-translated versions. Throughout the process, special sensitivity was shown to cultural nuances, contextual changes, and cultural expressions of emotions. Pilot research with 50 middle school students was conducted to evaluate the clarity, cultural compatibility, and dependability of the items. Considering the feedback, revisions were made to better align the items with Turkish cultural norms and ensure the constructs were effectively captured. Subsequently, a preliminary study was conducted with 200 middle school students who were not included in the main sample to examine the psychometric properties of the finalized Turkish versions of the scales. Reliability was assessed using Cronbach\u0026rsquo;s alpha and composite reliability (CR), while construct validity was evaluated through factor loadings and average variance extracted (AVE).\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eAcademic Self-Efficacy Scale.\u0026nbsp;\u003c/em\u003eCFA was performed to assess the overall goodness-of-fit of all the constructs to determine the validity of the scale. The goodness-of-fit indices for the scale were as follows: (\u0026chi;\u0026sup2;/df = 2.408, GFI = 0.96, TLI = 0.95, CFI = 0.96, RMSEA = 0.064). The factor loadings of all items ranged between 0.48 and 0.82. Further reliability analyses yielded a Cronbach\u0026rsquo;s alpha of 0.87, AVE of 0.58, and a CR of 0.84, demonstrating satisfactory internal consistency and construct reliability (Hair et al., 2019).\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eParental Involvement Scale\u003c/em\u003e, CFA was performed to evaluate the goodness-of-fit of the model, yielding good indices (\u0026chi;\u0026sup2;/df = 2.67, RMSEA = 0.048, GFI = 0.95, CFI = 0.96, TLI = 0.94). The factor loadings for the items ranged from 0.62 to 0.74. The Cronbach\u0026apos;s alpha was 0.89, and AVE and CR values were 0.56 and 0.82, respectively.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eAcademic Buoyancy Scale.\u003c/em\u003e The fit indices of the CFA model indicated that the model was a good fit for the data (\u0026chi;\u0026sup2;/df = 3.71, RMSEA = 0.061, GFI = 0.93, CFI = 0.94, TLI = 0.93). Factor loadings for each item were greater than 0.40, with items loading strongly onto their respective factors. The Cronbach\u0026rsquo;s alpha for the scale was 0.91, the AVE was 0.57 and the CR value was 0.83.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Collection\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eFollowing ethical approval, informed consent forms were distributed to students who voluntarily agreed to participate in the study. During survey administration, the second author clarified that the instruments were not mathematics tests or exams, emphasized that there were no right or wrong answers, and assured students that their responses would not affect their mathematics course grades. Participants were encouraged to respond honestly and were informed that their answers would remain confidential and analyzed in aggregate. It was also explicitly stated that their teachers would not have access to individual responses. The survey took approximately 30 minutes to complete, and data were collected in June 2024.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Analysis\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe SmartPLS 4.0 software was used, employing partial least squares structural equation modelling (PLS-SEM) as an analytical approach in order to test the hypothesized model. PLS-SEM is particularly well suited for testing complex models with small sample sizes and is less restrictive in terms of data normality assumptions (Hair et al., 2017; Ringle et al., 2022). The PLS-SEM procedure follows a two-step process: first, the measurement model is evaluated to assess the reliability and validity of the constructs, and second, the structural model is analyzed to test the hypothesized relationships (Henseler et al., 2009).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eBefore SEM analysis, firstly, the validity and reliability of the measurement model were evaluated by examining outer loadings, CR, CA, and AVE coefficients. The results confirmed that the measurement model was reliable and valid, as indicated by Composite Reliability (CR) and Cronbach\u0026apos;s Alpha (CA) values exceeding 0.7 and Average Variance Extracted (AVE) values higher than 0.5 (Hair et al., 2019). Structural equation modeling (SEM) was used to investigate the direct effects of parental involvement and teacher support on mathematics performance as well as the indirect effects through academic self-efficacy and academic buoyancy. To examine the hypothesized indirect effect of mediating variables, bias-corrected bootstrap analysis with a 95% confidence interval using 5000 bootstrap re-samples was used.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003e\u003cstrong\u003ePreliminary Analysis\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAs presented in Table 1, perceived teacher support (TS) showed significant and positive correlations with academic self-efficacy (ASE; r = 0.457, p \u0026lt; 0.01), academic buoyancy (AB; r = 0.382, p \u0026lt; 0.01), and math performance (MP; r = .402, p \u0026lt; 0.01). Similarly, parental involvement (PI) was positively associated with ASE (r = 0.448, p \u0026lt; 0.01), AB (r = 0.349, p \u0026lt; 0.01), and MP (r = 0.332, p \u0026lt; 0.01). Furthermore, both ASE (r = 0.570, p \u0026lt; .01) and AB (r = 0.554, p \u0026lt; 0.01) showed significant and positive correlations with MP. Finally, there is a positive correlation between PI and TS (r = 0.359, p \u0026lt; 0.01).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1\u003c/strong\u003e.\u003cem\u003e\u0026nbsp;\u003c/em\u003eMeans, Standard Deviations and Correlations between Variables\u0026nbsp;\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"597\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e1\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e2\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e3\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e4\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e5\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e1. \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;TS\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.359**\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.457**\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.382**\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.402**\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e2. \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;PI\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.448**\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.349**\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.332**\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e3. \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;ASE\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.570**\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.554**\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e4. \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;AB\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.484**\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003e5. \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp;MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003eM\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e5.13\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e3.53\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e3.58\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e3.35\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e68.55\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 99px;\"\u003eSD\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 54px;\"\u003e1.45\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.768\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.823\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e0.813\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 111px;\"\u003e26.03\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eNote: ** \u003cem\u003ep \u0026lt; 0.001, Abbreviations: TS= Teacher support, PI= Parent involvement, ASE= Academic efficacy, AB= Academic buoyancy, MP= Math performance\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMeasurement Model\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe measurement model (Table 2) was tested to verify the theoretical appropriateness of the model developed based on the theoretical framework. To validate the measurement model, CA coefficients as well as convergent and discriminant validity values were evaluated. Factor loadings were examined to determine the validity and reliability indicators of the model. Factor loadings of 0.70 or higher and AVE values above 0.50 indicate an acceptable level of convergent validity (Hair et al., 2017). An initial analysis was conducted to identify any items that had factor loadings below the threshold of 0.70. All factor loadings within the measurement model exceeding 0.70 ranged from 0.708 to 0.823. The CA ranged from 0.786 to 0.831, while the CR ranged from 0.785 to 0.855. CA and CR coefficients surpassed the recommended threshold of 0.70, indicating robust reliability for all latent constructs (Hair et al., 2017) and demonstrating good internal consistency. The AVE for all constructs ranged from 0.547 to 0.601, confirming both the reliability and convergent validity of the latent constructs within the model (Hair et al., 2021). These findings demonstrate that the measurement model meets the established criteria.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 2\u003cem\u003e.\u0026nbsp;\u003c/em\u003e\u003c/strong\u003eOuter Loadings, Internal Consistency and Average Variance Extracted\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"596\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 143px;\"\u003eConstructs\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eItems\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003eLoadings\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 72px;\"\u003erho_A\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eCA\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 70px;\"\u003eCR\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 70px;\"\u003eAVE\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"4\" valign=\"top\" style=\"width: 143px;\"\u003eAcademic buoyancy (AB)\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eAB1\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.723\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"4\" valign=\"top\" style=\"width: 72px;\"\u003e0.823\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"4\" valign=\"top\" style=\"width: 76px;\"\u003e0.787\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"4\" valign=\"top\" style=\"width: 70px;\"\u003e0.785\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"4\" valign=\"top\" style=\"width: 70px;\"\u003e0.551\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eAB2\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.759\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eAB3\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.745\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eAB4\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.728\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"12\" valign=\"top\" style=\"width: 143px;\"\u003eTeacher Support (TS)\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS1\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.734\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"12\" valign=\"top\" style=\"width: 72px;\"\u003e0.832\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"12\" valign=\"top\" style=\"width: 76px;\"\u003e0.789\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"12\" valign=\"top\" style=\"width: 70px;\"\u003e0.823\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"12\" valign=\"top\" style=\"width: 70px;\"\u003e0.601\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS2\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.789\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS3\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.712\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS4\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.778\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS5\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.796\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS6\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.709\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS7\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.733\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS8\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.745\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS9\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.751\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS10\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.833\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS11\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.744\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eTS12\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.777\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"9\" valign=\"top\" style=\"width: 143px;\"\u003eAcademic self-efficacy (ASE)\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE1\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.708\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"9\" valign=\"top\" style=\"width: 72px;\"\u003e0.886\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"9\" valign=\"top\" style=\"width: 76px;\"\u003e0.786\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"9\" valign=\"top\" style=\"width: 70px;\"\u003e0.855\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"9\" valign=\"top\" style=\"width: 70px;\"\u003e0.578\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE2\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.765\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE3\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.807\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE4\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.711\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE5\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.745\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE6\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.766\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE7\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.823\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE8\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.755\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eASE9\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.788\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd rowspan=\"10\" valign=\"top\" style=\"width: 143px;\"\u003eParental involvement (PI)\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI1\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.724\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"10\" valign=\"top\" style=\"width: 72px;\"\u003e0.932\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"10\" valign=\"top\" style=\"width: 76px;\"\u003e0.831\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"10\" valign=\"top\" style=\"width: 70px;\"\u003e0.804\u003cbr\u003e\u003c/td\u003e\n \u003ctd rowspan=\"10\" valign=\"top\" style=\"width: 70px;\"\u003e0.547\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI2\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.745\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI3\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.767\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI4\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.786\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI5\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.790\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI6\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.733\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI7\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.790\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI8\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.821\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI9\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.713\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003ePI10\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 88px;\"\u003e0.747\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;As seen in Table 2, the rho_A values ranged from 0.823 to 0.932, exceeding the 0.70 threshold and indicating reliable constructs \u0026nbsp;(Hair et al., 2019). Further, the Heterotrait-Monotrait Ratio (HTMT) criteria for discriminant validity evaluations were examined. The HTML values ranged from 0.55 to 0.82, all of which were below the threshold of 0.85, indicating that the discriminant validity was established (Henseler et al., 2014). Variance inflation factor (VIF) values were examined to determine whether there is indicator collinearity. VIF values are less than 3 for the indicators in each factor (Hair et al., 2021) and range between 1.256 and 2.411. These values indicate that there is no collinearity problem among the main variables (Hair et al., 2019).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eHypotheses testing results\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAfter testing the measurement model, the PLS algorithm was used to test the assumed structural model. The results of the structural model and hypothesis testing are presented in Table 3 and Figure 2.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 3.\u0026nbsp;\u003c/strong\u003eHypotheses Test Results\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"612\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"2\" valign=\"top\" style=\"width: 218px;\"\u003e\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003eOriginal sample\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003eSample Mean\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003et\u0026nbsp;\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003ep\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eDecision\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH1\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003eTS\u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.151\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.152\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e3.342\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.001\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH2\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003ePI\u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.093\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.096\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e2.059\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.040\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH3\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003eASE\u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.252\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.252\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e4.067\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.000\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH4\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003eAB\u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.334\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.332\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e5.975\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.000\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH5\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003eTS \u0026mdash;\u0026gt; ASE \u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.084\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.022\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e3.853\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.000\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH6\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003ePI\u0026mdash;\u0026gt; ASE \u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.962\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.098\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e3.498\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.000\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH7\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003eTS \u0026mdash;\u0026gt; AB \u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.108\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.107\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e4.790\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.000\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003eH8\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 161px;\"\u003ePI\u0026mdash;\u0026gt; AB\u0026mdash;\u0026gt; MP\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e0.987\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 66px;\"\u003e0.100\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 59px;\"\u003e3.723\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e0.000\u003cbr\u003e\u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 127px;\"\u003eSupported\u003cbr\u003e\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eAs seen in Figure 2, both perceived TS (\u0026beta; = 0.151, p = 0.01) and PI (\u0026beta; = 0.093, p = 0.040) had positive and statistically significant direct effects on MP, thereby supporting hypotheses H1 and H2. In addition, TS (\u0026beta; = 0.333, p = 0.00; \u0026beta; = 0.324, p = 0.01) and PI (\u0026beta; = 0.384, p = 0.00; \u0026beta; = 0.294, p = 0.05) also positively and significantly predicted both ASE and AB, respectively. Both ASE (\u0026beta; = 0.334, p = 0.00) and AB (\u0026beta; = 0.252, p = 0.05) had significant positive effects on MP, providing support for hypotheses H3 and H4. Further analysis revealed significant indirect effects. ASE mediated the relationship between TS and MP (\u0026beta; = 0.084, p = 0.00), as well as between PI and MP (\u0026beta; = 0.096, p = 0.00), thereby supporting hypotheses H5 and H6. Similarly, AB significantly mediated the relationship between TS and MP (\u0026beta; = 0.108, p = 0.00), and between PI and MP (\u0026beta; = 0.098, p = 0.00), confirming hypotheses H7 and H8. Overall, the model demonstrated that TS and PI influence MP not only through direct pathways but also indirectly through the mediation of students\u0026rsquo; ASE and AB.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe present study aimed to investigate the associations between perceived teacher support (TS) and parent involvement (PI) on middle school students\u0026rsquo; math performance (MP), both directly and indirectly through academic self-efficacy (ASE), and academic buoyancy (AB). The findings revealed that all four variables had significant and positive direct effects on MP. Moreover, ASE and AB played mediating roles in the relationship between both TS and PI with the students\u0026rsquo; MP.\u003c/p\u003e\u003cp\u003ePrior research has consistently emphasized the central role of supportive teacher-student relationships in fostering students\u0026rsquo; academic engagement and achievement, particularly in challenging subjects such as mathematics (Chen \u0026amp; Leung, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Wentzel, \u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). Middle school students, who are navigating increased academic pressure and a more abstract mathematics curriculum, may particularly benefit from emotionally responsive and academically supportive teachers (Yang et al., \u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Consistent with this literature, the present study found that perceived TS was a significant direct predictor of MP. This suggests that students who perceive their teachers as supporting autonomy, competence, and emotional well-being tend to perform and achieve better outcomes in mathematics (Jung et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Yu \u0026amp; Singh, \u003cspan citationid=\"CR76\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). Beyond this direct effect, TS also indirectly predicted MP through its positive associations with both ASE and AB, suggesting TS not only contributes to students\u0026rsquo; mathematics performance by offering instructional support, but also by enhancing their academic self-efficacy in mathematics\u0026mdash;that is, their belief in their ability to succeed in math\u0026mdash;and by fostering their academic buoyancy, or their capacity to recover from routine academic setbacks encountered in math learning. This aligns with Bandura\u0026rsquo;s (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1997\u003c/span\u003e) socio-cognitive perspective, which posits that supportive social contexts enhance students\u0026rsquo; motivational beliefs and academic performance. Specifically, students who receive more social support from their teachers are more likely to acquire mathematical knowledge, build self-confidence (i.e., competence), develop an interest in mathematics (Wu et al., \u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), form positive evaluations of the value of mathematics (Pekrun et al., \u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e2017\u003c/span\u003e).Such support also helps students perceive mistakes as learning opportunities rather than as sources of judgment, thereby reducing math anxiety. Teacher support thus not only motivates students to engage with learning but also increases their self-efficacy beliefs (Liu et al., \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), which in turn reduces setbacks or challenges encountered in mathematics learning (Li et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). The current findings indicate that TS serves as a dual mechanism, encompassing both the instructional quality provided by teachers and relational and motivational support that foster students\u0026rsquo; ASE and AB, which in turn contribute positively to their mathematics performance.\u003c/p\u003e\u003cp\u003eThe present study extends existing literature on the role of PI on students' MP. Active PI is associated with improved children\u0026rsquo;s learning and classroom performance (Jeynes, \u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). A recent meta-analysis also confirmed the significantly positive influence of PI on students' MP (Wang \u0026amp; Wei, \u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). When parents actively engage in their children\u0026rsquo;s learning processes, students are more likely to achieve higher levels of achievement in mathematics (Fiskerstrand, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). At the middle-school level, where the mathematics curriculum becomes increasingly abstract and includes more specialized topics requiring advanced reasoning and problem-solving skills, PI may provide students with the emotional support and structured guidance needed to persist through academic challenges. Consistent with this body of research, the present findings revealed that PI not only directly supports students\u0026rsquo; MP but also has an indirect effect on MP through its positive associations with ASE and AB. Several studies indicate that higher levels of PI enhance students' ASE, which in turn contributes to improved academic performance (Weiser \u0026amp; Riggio, \u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; You et al., \u003cspan citationid=\"CR74\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). Similarly, encouraging active and supportive PI can significantly enhance AB, which fosters students\u0026rsquo; ability to cope with everyday academic setbacks and challenges and thus support their MP (Chen \u0026amp; Mok, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Parents who demonstrate consistent involvement in their children's education may reinforce students\u0026rsquo; confidence in their academic capabilities and promote a resilient attitude toward learning challenges (Gu et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Morkoyunlu \u0026amp; Konyalıoğlu, \u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). In this context, parental involvement can serve as an external resource that creates a positive impact on students' mathematics performance by increasing their self-efficacy beliefs in mathematics and helping them cope with math anxiety or typical difficulties. Because emotional and autonomy support from parents can create an environment that enables them to guide students to pursue mathematical challenges, take risks, and strive for learning and performance with increased self-confidence, ultimately increasing mathematics achievement (Oh et al., \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). These findings underscore the multifaceted role of PI in supporting math performance, both directly and through academic self-efficacy and buoyancy, and suggest that fostering meaningful and sustained parental involvement may therefore be an effective strategy for increasing students' academic self-efficacy and buoyancy in mathematics. Importantly, this support must be perceived by students themselves, as their sense of being supported plays a critical role in the development of ASE and AB.\u003c/p\u003e\u003cp\u003eAs with external factors, the present study also highlights the internal factors, such as ASE and AB, in relation to middle school students\u0026rsquo; MP, revealing both significant direct and mediated effects. This finding aligns with established theoretical perspectives. Students with higher ASE, characterized by strong confidence in their ability and effort to succeed, tend to achieve better MP, consistent with Bandura\u0026rsquo;s (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1997\u003c/span\u003e) assertion that self-efficacy is a key determinant of academic success. Similarly, students with higher AB are better able to cope with everyday academic setbacks, pressures, and daily challenges and tend to maintain engagement and persistence, leading to improved performance in demanding subjects such as mathematics and science (Colmar et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Kul et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; Martin \u0026amp; Marsh, \u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Wei\u0026szlig;enfels et al., \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Beyond their direct contributions, both ASE and AB served as mediators between external support (TS and PI) and students\u0026rsquo; MP. This suggests that supportive environments foster students\u0026rsquo; confidence in their academic abilities and their buoyancy in everyday academic challenges. When students perceive encouragement and involvement from teachers and parents, they are more likely to believe in their capabilities and persist through academic difficulties. Thus, ASE and AB function as a key psychological mechanism, which translates external support into sustained engagement and improved academic performance in mathematics.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThis study supports our understanding of the relationship between middle school students' mathematics performance and the non-cognitive psychological factors of academic self-efficacy and academic buoyancy. The findings suggest that these internal psychological resources positively affect students' mathematics performance in both direct and indirect ways. The study indicates that students who are supported by teachers and parents tend to perform better in math, both directly and indirectly through increased academic self-efficacy and academic buoyancy. These intrinsic psychological factors seem to positively mediate the effect of external factors by emphasizing their transformative function on the performance of external social resources in mathematics. Moreover, teacher support and parental involvement appear to play a dual role not only in directly improving mathematics performance but also in shaping students' mathematics performance by supporting students' academic self-efficacy and academic vitality. Taken together, these findings provide empirical evidence that students can acquire meaningful outcomes in mathematics learning when they receive support from both their teachers and their parents. Overall, the study emphasizes the importance of holistic approaches, including both intrinsic and extrinsic factors, in mathematics education, sheds light on the dynamic interaction between these factors, and emphasizes the importance of both to support students' mathematics achievement.\u003c/p\u003e\u003cdiv id=\"Sec24\" class=\"Section2\"\u003e\u003ch2\u003eLimitations and Future Studies\u003c/h2\u003e\u003cp\u003eThis study has some limitations. First, our understanding regarding students' parent involvement and perceptions about their teacher support is based solely on their self-reported. This may not fully capture the extent of support provided by parents and teachers. Future studies may incorporate multi-sources approaches, including data collected from parents, teachers, and direct observations, to provide a more comprehensive understanding. Second, the sample size was relatively small and drawn from a single urban region T\u0026uuml;rkiye. Findings may not be generalizable to broader, more diverse populations. Future research should include larger and more diverse samples to improve the external validity. Third, the use of a cross-sectional design restricts the ability to establish causal relationships among the variables. Longitudinal designs are recommended to explore how these relationships evolve and influence mathematics performance over time.\u003c/p\u003e\u003cdiv id=\"Sec25\" class=\"Section3\"\u003e\u003ch2\u003eImplications for Practice\u003c/h2\u003e\u003cp\u003eThis study offers practical insights for teachers and parents aiming to improve middle school students\u0026rsquo; math performance. Teachers are encouraged to provide opportunities for student decision-making, support their choices, and show genuine interest, as these practices can positively strengthen students\u0026rsquo; perceptions of teacher support and boost their performance. Additionally, parents can enhance their involvement by maintaining regular communication with mathematics teachers to monitor their children\u0026rsquo;s progress and engage in meaningful conversations with them about math-related challenges and achievements. Supporting students\u0026rsquo; self-efficacy and academic buoyancy in mathematics is essential, as these internal resources play a critical role in enhancing students\u0026rsquo; learning experiences and enabling them to overcome the difficulties inherent in challenging subjects such as mathematics. Feeling supported in coping with such challenges can motivate students to persist and succeed in mathematics and increase their success. Ultimately, coordinated efforts between parents and teachers\u0026mdash;characterized by consistent encouragement, targeted support, and open communication\u0026mdash;can foster students\u0026rsquo; academic buoyancy, strengthen their self-efficacy, and improve their overall mathematics performance.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e"},{"header":"Abbreviations","content":"\u003cp\u003eTS: Teacher Support, PI: Parental Involvement, ASE: Academic Self-Efficacy, AB: Academic Buoyancy, MP: Mathematics Performance, PLS: Partial Least Squares, SEM: Structural Equation Modelling, OECD: Organisation For Economic Co-Operation And Development, ASES: Academic Self-Efficacy Scale, PIS: Parental Involvement Scale, ABS: Academic Buoyancy Scale, PLS-SEM: Partial Least Squares Structural Equation Modelling, CR: Composite Reliability, AVE: Average Variance Extracted, HTMT: Heterotrait-Monotrait Ratio, VIF: Variance Inflation Factor.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cem\u003eEthics approval and consent to participate\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThis study was conducted in T\u0026uuml;rkiye with the participation of middle school students. Ethical approval was obtained from the Ethics Committee of Hasan Kalyoncu University (Approval No: E-97105791-050.04-57875). All legal and ethical requirements were fulfilled, and written informed consent was obtained from parents or legal guardians in accordance with the Declaration of Helsinki.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eConsent for publication\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eAvailability of data and materials\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eCompeting interests\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no competing interests.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eFunding\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe author acknowledge that they received no external funding in support of this research.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eAuthors\u0026rsquo; contributions\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eİİ, NCA, and MC: Writing, Review \u0026amp; Editing. İİ: Resources, Conceptualization, Methodology, Investigation, Visualization and analysis. NCA: Conceptualization, Data collection Investigation, MC: Writing- Original draft preparation. All authors read and approved the final manuscript\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eAcknowledgements\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAng, W. H. D., Shorey, S., Lopez, V., Chew, H. S. J., \u0026amp; Lau, Y. (2022). Generation Z undergraduate students\u0026rsquo; resilience during the COVID-19 pandemic: A qualitative study. \u003cem\u003eCurrent Psychology, 41\u003c/em\u003e(11), 8132-8146. https://doi.org/10.1007/s12144-021-01830-4\u003c/li\u003e\n\u003cli\u003eBandura, A. (1997). \u003cem\u003eSelf-efficacy: The exercise of control.\u003c/em\u003e W.H. Freeman.\u003c/li\u003e\n\u003cli\u003eBandura, A. (2012). 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The role of emotional and motivational contribution in math performance. \u003cem\u003eSocial Psychology of Education, 26\u003c/em\u003e(3), 579-601. https://doi.org/10.1007/s11218-023-09760-8\u003c/li\u003e\n\u003cli\u003eŽivković, M., Pellizzoni, S., Mammarella, I. C., \u0026amp; Passolunghi, M. C. (2022). Executive functions, math anxiety and math performance in middle school students. \u003cem\u003eBritish Journal of Developmental Psychology, 40\u003c/em\u003e(3), 438-452. https://doi.org/10.1111/bjdp.12412\u003c/li\u003e\n\u003cli\u003eZysberg, L., \u0026amp; Schwabsky, N. (2020). School climate, academic self-efficacy and student achievement. \u003cem\u003eEducational Psychology, 41\u003c/em\u003e(4), 467-482. https://doi.org/10.1080/01443410.2020.1813690\u003cu\u003e \u003c/u\u003e\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"bmc-psychology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"psyo","sideBox":"Learn more about [BMC Psychology](http://bmcpsychology.biomedcentral.com/)","snPcode":"","submissionUrl":"","title":"BMC Psychology","twitterHandle":"BMC_series","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Teacher support, parental involvement, academic self-efficacy, academic buoyancy, mathematics performance","lastPublishedDoi":"10.21203/rs.3.rs-6903576/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6903576/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003ch2\u003eBackground\u003c/h2\u003e\u003cp\u003eGiven its multifaceted role in fostering students\u0026rsquo; academic outcomes, parental involvement remains a critical area of investigation in educational research. Parental involvement not only complements teachers' efforts and supportive behaviours, but also has the potential to increase students' academic engagement and academic self-efficacy, both of which are among the key determinants of mathematics achievement.\u003c/p\u003e\u003ch2\u003eMethods\u003c/h2\u003e\u003cp\u003eThis study examined the relationships between perceived teacher support (TS), parental involvement (PI), academic self-efficacy (ASE), academic buoyancy (AB), and mathematics performance (MP) among middle school students. More specifically, it investigated the mediating effects of ASE and AB in the relationships between PI and TS with MP. Participants included 363 middle school students from T\u0026uuml;rkiye. Data were analysed using Partial Least Squares (PLS) method and structural equation modeling (SEM).\u003c/p\u003e\u003ch2\u003eResults\u003c/h2\u003e\u003cp\u003eThe findings revealed that both TS and PI have direct positive effects on students\u0026rsquo; MP. TS and PI demonstrated significant indirect effects on MP through the mediating roles of ASE and AB. Furthermore, both ASE and AB emerged as significant positive predictors of MP.\u003c/p\u003e\u003ch2\u003eConclusion\u003c/h2\u003e\u003cp\u003eThe study indicates that students who are supported by teachers and parents tend to perform better in math, both directly and indirectly through increased academic self-efficacy and academic buoyancy. Results extend our understanding by providing important insights into the critical role of the complex interplay between external support mechanisms and internal psychological resources in enhancing mathematics performance.\u003c/p\u003e","manuscriptTitle":"The Relationships between Parental Involvement, Teacher Support, and Mathematics Performance: Mediating Roles of Academic Self-Efficacy and Academic Buoyancy","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-07-28 05:34:53","doi":"10.21203/rs.3.rs-6903576/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-08-08T09:16:58+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-08-05T03:34:17+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-30T09:29:09+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"324143563939208792338342955885211334243","date":"2025-07-30T08:41:51+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"199977815889701307646607820049195313502","date":"2025-07-26T14:00:49+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-25T08:18:41+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"131782013486519130989850905848795178836","date":"2025-07-25T03:55:54+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"7203803605887839325849606288375916380","date":"2025-07-23T04:20:50+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-07-23T04:12:44+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-07-16T15:24:07+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-06-26T14:28:27+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-06-23T21:38:39+00:00","index":"","fulltext":""},{"type":"submitted","content":"BMC Psychology","date":"2025-06-23T21:35:15+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"bmc-psychology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"psyo","sideBox":"Learn more about [BMC Psychology](http://bmcpsychology.biomedcentral.com/)","snPcode":"","submissionUrl":"","title":"BMC Psychology","twitterHandle":"BMC_series","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1e14af2c-bacc-423f-84c7-49ce880a2c3f","owner":[],"postedDate":"July 28th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-11-10T16:06:33+00:00","versionOfRecord":{"articleIdentity":"rs-6903576","link":"https://doi.org/10.1186/s40359-025-03547-6","journal":{"identity":"bmc-psychology","isVorOnly":false,"title":"BMC Psychology"},"publishedOn":"2025-11-04 15:57:28","publishedOnDateReadable":"November 4th, 2025"},"versionCreatedAt":"2025-07-28 05:34:53","video":"","vorDoi":"10.1186/s40359-025-03547-6","vorDoiUrl":"https://doi.org/10.1186/s40359-025-03547-6","workflowStages":[]},"version":"v1","identity":"rs-6903576","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6903576","identity":"rs-6903576","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}