School psychological capital refers to an individual’s positive psychological state or psychological resources that arise from endorsing four psychological capacities: hope, optimism, resilience, and self-efficacy (Luthans et al., 2004). Since PsyCap was first introduced from the industrial-organizational context to the educational domain, this construct has been hotly discussed (Carmona-Halty et al., 2019; Kang et al., 2025; Vîrgă et al., 2022). However, the measurement of school PsyCap was controversial, given that this construct was established based on the synchronicity of the academic community (Kang et al., 2021). For example, the unidimensional model regarded school PsyCap as an omnibus construct, and the value of this construct was calculated by summing up and averaging the value of each item (Li et al., 2014). In comparison, the four-factor model treats the four individual components as observational variables (Carmona-Halty et al., 2021; Datu and Valdez, 2016). Recently, the hierarchical model (or second-order model) that views both school PsyCap and its four components as latent variables has begun to gain support from scholars (Datu et al., 2018; Kang et al., 2021; King et al., 2020; King and Caleon, 2021).

Hope, self-efficacy, resilience, and optimism or the “HERO” within (e.g., see Luthans, 2012) were the four psychological resources that constitute PsyCap, each of which has been extensively discussed in educational literature. Hope pertains to the goal-related thinking orientation characterized by three components: goals, pathway, and agency (Snyder, 2000). Existing studies have shown that hope positively predicted key learning-related outcomes, including academic performance (Buckelew et al., 2008; Day et al., 2010; Marques et al., 2011; Snyder et al., 2002), emotional well-being (Ciarrochi et al., 2007), behavioral engagement (Chen et al., 2020; van Ryzin, 2011), and psychological adjustment (Du and King, 2013). Self-efficacy (or academic confidence) refers to an individual’s judgment of his or her ability to complete challenging tasks (Urdan and Schoenfelder, 2006). A considerable number of studies showed that self-efficacy is positively correlated with different indices of optimal functioning such as motivation and learning (Linnenbrink and Pintrich, 2003; Van Dinther et al., 2011), academic performance, and learning-related emotions (Putwain et al., 2013), course satisfaction (Doménech-Betoret et al., 2017), mental well-being (Andretta and McKay, 2020), and engagement (Lin, 2021; Simón et al., 2024). Besides, self-efficacy has also been found to positively correlate with life satisfaction and moderate the relationship between stress and life satisfaction (Moksnes et al., 2019). Resilience is an individual’s ability to sustain and bounce back when facing problems and adversities (Martin and Marsh, 2006). Those high in resilience would see adversities as challenges rather than threats (Symes et al., 2015). The positive effect of resilience on crucial indices of school functioning has also been extensively verified (Allan et al., 2014; Ayala and Manzano, 2018; Kotzé and Kleynhans, 2013). For example, García-Izquierdo et al. (2018) documented that resilience positively correlated with psychological health and moderated between burnout and psychological health in a sample of sophomore nursing students. Moreover, the predictive effects of resilience on academic performance (Ayyash-Abdo et al., 2016; Cai and Meng, 2025), academic satisfaction (Meneghel et al., 2019), emotional engagement (Kim et al., 2021), and psychological well-being (Voon et al., 2021) were also confirmed. Optimism pertains to an individual’s expectation of positive outcomes in their endeavors. Researchers have found that optimism positively affects academic achievement and psychological well-being. For example, in a study among high school students, Hoy et al. (2006) documented that academic optimism was positively correlated with student achievement. Taking undergraduate students as participants, Vizoso et al. (2019) reported that optimism positively affected academic performance by mitigating academic burnout. In another study with undergraduate women, Augusto-Landa et al. (2011) observed that optimism was positively associated with all dimensions of psychological well-being. Similarly, the predictive effects of optimism on engagement (Geers et al., 2009) and psychological well-being (Krok, 2015) were also confirmed.

The predictive effect of school PsyCap has also been recognized within educational contexts. For example, Siu et al. (2014) documented that PsyCap predicts school engagement directly or indirectly via intrinsic motivation among university students in Hong Kong. Similarly, in a study conducted among Filipino high school students, school PsyCap positively correlates with academic motivation, engagement, achievement, and well-being (Datu et al., 2018; Datu and Valdez, 2016). In line with these findings, research involving Chilean high school students supported the positive association between PsyCap and academic performance (Carmona-Halty et al., 2019, 2021; Carmona–Halty et al., 2019).

Based on the above review, it is evident that both school PsyCap and its four components demonstrate a positive correlation with academic engagement. However, relatively few studies have been conducted to explore the mechanism between school PsyCap and academic engagement, especially in a Mainland Chinese context. An exception was the work of King et al. (2020), who demonstrated that positive emotions partially mediate the relationship between school PsyCap and academic engagement in a sample of 71 Hong Kong secondary school students. One key limitation of this study was that it ignored the domain-specificity of emotions and academic engagement in the school context (Goetz et al., 2007; Green et al., 2007). In addition, both positive and negative achievement emotions would be experienced in the subject-related learning process (e.g., Lichtenfeld et al., 2012; Pekrun, 2006; Raccanello et al., 2013), however, only positive emotions have been shown to mediate the relationship between school PsyCap and academic engagement in their study (King et al., 2020). The present study aimed to extend this research using mathematics-related data to investigate the emotional connections (both positive and negative achievement emotions) between school PsyCap and behavioral engagement in the context of mathematics classrooms.

Achievement emotions, which refer to discipline-related emotions that are closely related to achievement activities or achievement outcomes (Pekrun, 2006), have drawn a growing number of educational researchers’ attention as their ubiquity and their considerable influence on learning and achievement (Pekrun and Stephens, 2010; Putwain et al., 2022; Villavicencio and Bernardo, 2016). For example, King and Gaerlan (2014) investigated the consequence of achievement emotions. They found that achievement emotions had a predictive effect on engagement and, thus, academic achievement in a sample of Filipino university students. Furthermore, the beneficial effects of positive emotions on self-efficacy, self-regulation, achievement (Villavicencio and Bernardo, 2016), intrinsic motivation and engagement (Oriol et al., 2016), coping strategies (Vierhaus et al., 2016), and learning satisfaction (Lee et al., 2021) were also confirmed.

Through a series of qualitative and quantitative studies, Pekrun et al. (2002) identified nine emotions that students most commonly experienced (i.e., enjoyment, pride, hope, relief, anxiety, boredom, hopelessness, anger, and shame) in their learning experiences. Valence, activation, and object focus are the three dimensions for assessing achievement emotions (Pekrun, 2006; Pekrun et al., 2002). According to this classification criteria, every achievement emotion can be concluded by three dimensions: valence–positive versus negative, activation–activated versus deactivated, and object focus–activity-related versus outcome-related. Concerning these three dimensions, enjoyment, for example, can be described as a positive activity-related emotion that facilitates activation.

According to the control-value theory (CVT) of achievement emotions (Pekrun, 2006), students’ achievement-related controllability and value that are experienced at school are the two most essential antecedents of their achievement emotions (Pekrun, 2006; Pekrun and Stephens, 2010). The learning environment, including environmental, situational, and individual information, plays a vital role in the arousal of achievement emotions (Forsblom et al., 2021). Previous studies concentrated mainly on the influence of environmental and situational factors on achievement emotions (e.g., Clem et al., 2021; Forsblom et al., 2021; Westphal et al., 2018), however, the impact of individual factors on achievement emotions was relatively unexplored (for exceptions, see Kang and Wu, 2021).

As one subtype of academic engagement (including behavioral, affective, and cognitive) (e.g., Archambault et al., 2009; Ben-Eliyahu et al., 2018), behavioral engagement refers to students’ intensity of involvement within both academic or non-academic activities in the school context (Ben-Eliyahu et al., 2018; Chen et al., 2020). These three subtypes differ in connotation but are concurrently interrelated (Fredricks et al., 2004). Among them, behavioral engagement has been the most tremendously studied in connection with achievement-related outcomes, partly because students’ behavioral engagement can be observed in learning-related situations regarding participation and interest in academic tasks, following teacher instructions, and persistence and inhibition toward academic tasks (Chen et al., 2020; Darensbourg and Blake, 2013; Gregory et al., 2014). In addition, behavioral engagement demonstrated a more substantial predictive effect on school achievement than the other two subtypes of academic engagement. In a sample of high school students in Quebec, Archambault et al. (2009) found that only behavioral engagement was significantly correlated with dropout, while affective and cognitive engagement was not. During the primary grades, Ladd and Dinella (2009) found that the predictive contribution of behavioral engagement to long-term achievement was more potent than emotional engagement. Similarly, Wang et al. (2012) compared the contribution of these three types of engagement to academic performance and educational aspiration in a sample of African-American and European-American students in grades seven through eleven. They found that emotional engagement can only contribute to academic performance via behavioral or cognitive engagement. Furthermore, the present study was concerned with the participation dimension of academic engagement in the context of the mathematics classroom, so the key constructs referring to classroom activities–namely, behavioral engagement were chosen (e.g., Hospel et al., 2016).

Behavioral engagement was believed to affect a diverse set of desired learning-related outcomes positively. For instance, Wang and Pomerantz (2009) documented that students’ continuous behavioral engagement contributed to their school performance in a sample of American and Chinese secondary school students. In the same vein, the robust predictions of behavioral engagement on school graduation (Kieffer et al., 2013), interpersonal relationships (i.e., peer relation and teacher-student relations) (Engels et al., 2016), and student retention (Sengupta and Williams, 2021) were also confirmed. Given the self-determination theory, students’ school engagement was seen as manifesting their inner motivational resources (Reeve and Halusic, 2009; Skinner et al., 2009; Skinner and Pitzer, 2012). It plays a positive role in predicting various achievement-related outcomes.

To explore the emotional connection between school PsyCap and behavioral engagement, the present study examined how school PsyCap influences enjoyment and anxiety and how these two achievement emotions, in turn, affect behavioral engagement in a sample of Chinese secondary school students in the mathematics classroom context (see Figure 1 for the mediational model). Building on the literature review, the present study posited that variability in students’ school PsyCap would lead to differential experiences of achievement emotions in the learning environment. For example, in a sample of Chinese secondary school students, Kang and Wu (2021) found that students with higher levels of school PsyCap tend to experience more positive achievement emotions (i.e., enjoyment, hope, and pride) and fewer negative emotions (i.e., anxiety, anger, shame, boredom, and hopelessness) in mathematics learning. Furthermore, the significant association between achievement emotions and academic engagement has also been demonstrated in a large body of literature (e.g., Carmona-Halty et al., 2021; King et al., 2020; King and Gaerlan, 2014; Oriol et al., 2016). Collectively, the present study posited that achievement emotions (i.e., enjoyment and anxiety) mediate the association between school PsyCap and behavioral engagement. The present study contributes to literature mainly by testing the following three hypotheses:

One thousand one hundred thirty-five participants from two secondary schools took part in the present study, comprising seventh graders (N = 557) and eighth graders (N = 578). The two secondary schools are in two towns in Xishuangbanna Dai Autonomous Prefecture, the extreme southwestern border of Yunnan Province, China. The mean age of participants was 13.15 years (SD = 0.75), with 574 (50.6%) males and 561 (49.4%) females. When students were invited to participate in this survey, permission was sought from students, administrators, teachers, and parents. After these consents were obtained, participants completed the questionnaire under the guidance of their headteacher. Throughout the data collection process, students were informed that their participation was voluntary, and refusal to participate would not affect their academic standing or relationships within the school.

This empirical study was based on participants’ self-reported data. Therefore, it is necessary to address common method bias before conducting the data analysis. To eliminate the potential common method bias, Harman’s single-factor test was employed (Podsakoff et al., 2003). The single-factor confirmatory factor analysis yielded a poor model fit: χ² (324) = 2695.213, p < 0.001, RMSEA = 0.083, CFI = 0.750, TLI = 0.730, SRMR = 0.070. The findings demonstrate that common method bias was not expected to substantially impact the data analysis results.

Table 1 presents the descriptive statistics and correlation coefficients among the study variables. In the analysis, school PsyCap was designated as the independent variable, while mathematics enjoyment and anxiety were viewed as mediators and behavioral engagement in mathematics classes as the dependent variable. Besides, gender and prior mathematics achievement were included as covariates.

Table 2 presents the goodness of fit indexes from CFAs of the study variables. Traditionally, the excellent and adequate cutoff values for model fit indexes were: (1) comparative fit index (CFI) and Tucker-Lewis index (TLI) are greater than or equal to 0.95 and 0.90, respectively; (2) the root mean square error of approximation (RMSEA) are less than or equal to 0.06 and 0.08, respectively; and (3) the values of standardized root mean square residual (SRMR) are less than or equal to 0.08 and 0.10, respectively (Chen, 2007; Hu and Bentler, 1999). Accordingly, the results indicated that a good model fit for all latent constructs was supported based on the conventional criterion of cutoff values.

First, the associations among school PsyCap, behavioral engagement, and two covariates (i.e., gender and prior mathematics achievement) were tested. Results indicated good model fits, with χ² (182) = 656.533, CFI = 0.930, TLI = 0.920, RMSEA = 0.051 and SRMR = 0.047. Significant associations between school PsyCap (β = 0.713, SE = 0.023 and p < 0.001) and behavioral engagement as well as gender (β = 0.122, SE = 0.028 and p < 0.001) and behavioral engagement were identified.

Second, the two mediators (i.e., academic enjoyment and anxiety) were added into the model. Results showed that the mediation model also has good model fits, with χ² (362) = 1197.468, CFI = 0.914, TLI = 0.904, RMSEA = 0.048 and SRMR = 0.057 (Figure 1). Path one (school PsyCap → academic enjoyment → behavioral engagement) was significant (β = 0.339, SE = 0.030 and p < 0.001) for the 95% bootstrap bias-corrected confidence interval (BCa 95% CIs [0.285, 0.403]) did not contain zero. Path two (school PsyCap → academic anxiety → behavioral engagement) was also significant (β = −0.034, SE = 0.013 and p < 0.05) as (BCa 95% CIs [−0.065, −0.013]) did not contain zero. The significance of Path one and Path two indicated that academic enjoyment and anxiety mediate the relationship between school PsyCap and behavioral engagement. The association between school PsyCap and behavioral engagement, after taking mediators into consideration, decreased from β = 0.713 and p < 0.001 to β = 0.402 and p < 0.001, which revealed that academic enjoyment and anxiety act as partial mediators. Altogether, the indirect effect accounted for 48.1% of the link between school PsyCap and behavioral engagement, with a stronger link via academic enjoyment (43.7%) than via academic anxiety (4.4%), which could also be drawn from the small mediation of academic anxiety (β = 0.034).

Regarding the observed variables, results indicated prior mathematics achievement related positively to academic enjoyment (β = 0.163 and p < 0.001) and their behavioral engagement (β = −0.055 and p < 0.05) and negatively to academic anxiety (β = −0.138 and p < 0.001). Furthermore, the significant negative correlation between gender (male = 1 and female = 2) and academic enjoyment (β = −0.062 and p < 0.05) suggested that female students were less likely to take delight in mathematics lessons. While the significant positive correlation between gender and academic anxiety (β = 0.282 and p < 0.001) and behavioral engagement (β = 0.097 and p < 0.01) meant that female students were more engaged and anxious in mathematics classes.

The present study simultaneously considered the antecedents and consequences of achievement emotions, thereby providing empirical support for the control-value theory and establishing an integrative framework for the comprehensive analysis of achievement emotions. The content of the CVT comprises two parts: on the one hand, control and value appraisals are the proximal antecedents of achievement emotions; other proximal antecedents of achievement emotions (e.g., school PsyCap) influence achievement emotions indirectly through control and value appraisals (Kang et al., 2025; Pekrun, 2006, 2019). On the other hand, achievement emotions have a significant impact on academic outcomes, including academic engagement, learning motivation, and academic performance (Earl et al., 2024; Kang and Wu, 2022a; Muwonge et al., 2018). In contrast to prior research that has primarily focused on either the antecedents or the consequences of achievement emotions (Kang and Wu, 2022b; Wu and Yu, 2022), the present study adopted an integrative approach by examining both aspects concurrently. This study proposed the “school PsyCap → achievement emotions → academic engagement” model, thereby facilitating a more comprehensive perspective for understanding achievement emotions.

This study contributes to the body of literature (Carmona–Halty et al., 2019; Datu et al., 2018; Datu and Valdez, 2016; Kang et al., 2021; King et al., 2020; Siu et al., 2014), suggesting that school PsyCap is adaptive for students’ behavioral engagement via the emotional mediators of academic enjoyment and anxiety. Teachers can implement a variety of measures to improve the level of their students’ school PsyCap. Given that the four higher-order core factors of school PsyCap (i.e., hope, self-efficacy, resilience, and optimism) are open to development, intervention training programs such as expanding students’ personal resources to facilitate them to overcome learning obstacles, setting up stepwise techniques and timelines to help students achieve their learning goals, identifying learning obstacles in advance and proposing corresponding solutions to increase students’ expectations for academic success are the effective ways for teachers to develop their students’ school PsyCap (see Luthans et al., 2010). Second, teachers are recommended to create a cooperative classroom atmosphere and to give full play to the roles of study groups and peer leaders as students’ self-confidence can be effectively boosted (Luthans et al., 2014), and thus the level of their school PsyCap was raised. Third, group-based acceptance and commitment therapy, which emphasizes mindfulness, acceptance, values, and cognitive defusion, has also been shown to boost adolescent students’ PsyCap (Fang and Ding, 2020). This indicates that teachers can help by creating a therapeutic classroom climate in which students’ PsyCap level is improved. Fourth, compared to their male counterparts, female students exhibited greater behavioral engagement in the mathematics classrooms. However, female mathematics learners experienced lower levels of enjoyment and higher levels of anxiety, suggesting that mathematics educators should provide greater support and care for female students to help mitigate their negative emotional experiences. Fifth, the mediating influence of academic enjoyment in the relationship between school PsyCap and behavioral engagement was remarkably pronounced than that of academic anxiety. Mathematics educators are suggested to prioritize strategies aimed at fostering students’ positive emotional experiences in learning mathematics, rather than exclusively focusing on the reduction or remediation of negative emotions. This proposition is consistent with the principles of the positive psychology movement within educational contexts. Mathematics educators could significantly enhance students’ positive emotional experiences by providing increased attention and care, fostering harmonious teacher-student relationships, and offering timely academic and emotional support. Such interventions are likely to exert a substantial positive influence on academic outcomes, such as academic engagement.

This empirical study investigated the emotional connections of school PsyCap with behavioral engagement in mathematics classrooms of Chinese secondary school students, covering both positive and negative achievement emotions. Nevertheless, four potential limitations need to be acknowledged. First, only gender and prior academic achievement were controlled for in the present study, and it is suggested that more possible covariates should be considered in future studies. Family obligations (King and Ganotice, 2015), interpersonal relationships (Collie et al., 2016), family socioeconomic status (Chen et al., 2021) and classroom social climate (Mohammad Hosseini et al., 2022), for example, have also been shown to have an impact on the levels of behavioral engagement, indicating that future studies could deepen the understanding of the linkage between school PsyCap and behavioral engagement by way of controlling more antecedents of behavioral engagement. Second, a rigorous causal relationship between school PsyCap and behavioral engagement could not be determined because of the cross-sectional design of the present study. A longitudinal design is suggested to establish causality between these two constructs. Third, the data in the present study were all self-reported by the participants. Although common method bias has been addressed, future research is recommended to collect data from a broader range of resources, such as participants’ parents, teachers, and peers, to overcome the potential response bias effect (Conway and Lance, 2010). Fourth, although the Cronbach’s alpha values for academic anxiety, optimism, and self-efficacy were acceptable, they were not considered good. Thus, future research is recommended to undertake item refinement to enhance the reliability of these scales.