Introduction

The value of education lies in individuals’ development, which is based partly on their subjective well-being (SWB; Pring, 2010). The development of students’ SWB, which is the one important factor of non-cognitive ability, is thus an important outcome in education. For many years, however, research on educational effectiveness has considered academic achievement as the sole output factor in the assessment of educational processes (Teddlie & Reynolds, 2000). In particular, school effectiveness research has mainly focused on cognitive outcomes and especially on language, mathematics, or science achievement (Van Damme, De Fraine, Van Landeghem, Opdenakker, & Onghena, 2002). According to data from the China National Knowledge Infrastructure (CNKI), which is the largest index database in China, there is only a small amount of literature involving students’ non-cognitive outcomes. Furthermore, a very widespread phenomenon exists in China wherein elementary school students consistently achieve mathematics scores above 90% but are nonetheless given the lowest ranking in their grades. As this phenomenon becomes more and more serious, non-cognitive outcomes are increasingly being seen as important and as educational aims in themselves (Opdenakker & Van Damme, 2000).

Affective issues play a central role in mathematics learning and instruction. When teachers talk about their mathematics classes, they often mention that they equate students’ enthusiasm or hostility with their mathematical cognitive achievements. Similarly, students might also equate their passion for and like or dislike of mathematics with their mathematics cognitive outcomes. If research on learning and instruction is to maximize its impact on students and teachers, affective issues need to occupy a more central position in the minds of researchers (Lindqvist & Vestman, 2009; McLeod, 1992).

Students in Shanghai ranked at the top in mathematics on the past two assessments of the Program for International Student Assessment (PISA). However, it is also documented that even though Asian students in general, and Chinese students in particular, ranked at the top in mathematics performance (Organisation for Economic Co-operation and Development (OECD), 2013), there are few available school effectiveness studies concerned with non-cognitive outcomes (e.g. Konu, Lintonen, & Autio, 2002; Opdenakker & Van Damme, 2000). In recent years, the educational research community has started to pay attention to students’ SWB (Liu, Tian, Huebner, Zheng, & Li, 2015; Proctor, Linley, & Maltby, 2009). So far, we know little about students’ SWB in Shanghai. The purpose of this study is to investigate the mathematical achievement and SWB of elementary and middle school students in Shanghai, and analyze how students’ SWB affects their mathematics achievement.

Theoretical Bases

Students’ SWB

Diener, Wirtz, Tov, Chu, Choi, Oishi and Biswas-Diener (2009) and Diener, Lucas and Scollon (2013) identified SWB as a broad term which refers to individuals’ experiences, including feelings of happiness, low negative affect, and high life satisfaction. This definition has been used in most research on SWB. Based on this definition, most researchers have attempted to measure SWB in terms of cognitive and affective aspects (Calaguas, 2017; Diener, Oishi, & Lucas, 2003; Schimmack, Schupp, & Wagner, 2008). Researchers in prior studies used open-ended questions to investigate individuals’ SWB. Some surveys, like the World Values Survey (WVS, 2009), query respondents about their life satisfaction, asking, “All things considered, how satisfied are you with your life these days?” The Gallup World Poll lets respondents choose a number that represents their life satisfaction (1 meaning you are “completely dissatisfied” and 10 meaning you are “completely satisfied”; Fleche, Smith, & Sorsa, 2011).

Based on the studies of adults’ SWB, the study of students’ SWB in school was introduced. Suldo, Hearon, Bander, McCullough, Garofano, Roth and Tan (2015) claimed that students’ SWB indicates their satisfaction with and affect towards themselves. In addition to this, Tian, Du and Huebner (2015) have proposed a tripartite model for students’ SWB, which includes school satisfaction and positive and negative affect in school. Furthermore, most previous studies have only investigated students’ SWB from the perspectives of students’ health, their positive attitude towards school, their satisfaction with their life, and their social relationships in school (Lovat, Toomey, & Clement, 2010; OECD, 2017).

However, the research of Opdenakker and Damme (2000) focused on the effect of more detailed aspects of SWB on students’ learning. Their descriptions of SWB in school can be grouped into two categories: (1) the students’ affect for their studies as well as for school, such as academic self-concept, attitude to homework, motivation towards learning tasks, attentiveness in the classroom, interest in learning tasks, and well-being at the school, and (2) interpersonal relationships, like relationships with teachers and social integration within the class.

This study therefore uses the questionnaire developed by Opdenakker and Damme (2000) to investigate students’ SWB in Shanghai. Opdenakker, Van Damme, De Fraine, Van Landeghem and Onghena (2002), who designed the LOSO project, proposed a comprehensive concept suggesting that SWB is the sole non-cognitive outcome in school effectiveness research. In school, students’ SWB involves a satisfactory school life, which includes the following eight scales: academic self-concept (ACSELFC), motivation towards learning tasks (MOTLT), interest in learning tasks (INTERLT), attentiveness in the classroom (ATTENTCL), well-being at the school (WELLBS), relationship with teachers (RELTEACH), social integration in the class (SOCINTCL), and attitude to homework (ATHOMEW). They chose first- and second-grade students as the research subjects.

SWB is an indicator of school effectiveness and the ultimate aim of students’ development (Baker & Maupin, 2009; Park, 2004). However, most studies have investigated the definition of and the situation surrounding students’ SWB. To date, much less research has been carried out examining the relationship between students’ SWB and their academic achievement. Thus, the question arises: What is the relationship between students’ SWB and their mathematics performance? How does students’ SWB affect their mathematics performance?

Students’ SWB and Mathematics Performance

In the past several decades, a number of studies focused solely on students’ mathematics achievement and a variety of influencing factors such as anxiety, academic self-concept, and self-efficacy. Although there are some previous research about the relationship between students’ academic achievement and their SWB (Abadi, Tabbodi, & Rahgozar, 2013; Steinmayr, Crede, McElvany, & Wirthwein, 2016), there has been no direct research involving how students’ SWB affects their mathematics performance. We therefore reviewed prior research on the relationships between each of the scales of SWB and students’ mathematics learning.

With respect to the first scale, students’ academic self-concept is related to their achievement significantly (Marsh & Craven, 2006). There is also a significant relationship between students’ mathematical self-concept and their mathematics achievement (Chmielewski, Dumont, & Trautwein, 2013; Huang, 2011). Higher school achievement, in turn, contributes to developing students’ positive self-evaluation and self-cognition. For example, the results of PISA 2012 indicated that in all of the Western countries, greater self-concept tends to make less of a difference in impacting the performance of the lowest-achieving students than that of the highest-achieving students.

For the second, third, and fourth scales, numerous studies have clearly demonstrated that students’ learning motivation (Lim & Chapman, 2013), learning interest (Hidi & Anderson, 1992), and attitude to homework (Cooper, Robinson, & Patall, 2006) are significantly related to students’ mathematics achievement. Previous studies demonstrated that motivation has positive effects on students’ mathematics performance (Ruzek, Domina, Conley, Duncan, & Karabenick, 2015; Schiefele & Csikszentmihalyi, 1995). Furthermore, learning interest also is an important affective variable of mathematics learning (Kankia, 2008; Schiefele & Csikszentmihalyi, 1995). And students’ attitude to their homework has two main goals: improving learning processes and teaching students how to study independently (Simons, 1989). Students with a positive attitude to homework are more likely to do well on these two aspects (Planchard, Daniel, Maroo, Mishra, & Mclean, 2015).

For the fifth scale, attentiveness in the classroom, as an outcome of student learning, should be the most important variable considered. Researchers considered attentiveness in the classroom and school as an important educational objective (Elmore & Others, 1990) and suggested that it contributes to the development of students’ deep learning (Lau, Liem, & Nie, 2008) and social and cognitive abilities (Finn, 1993). Although student attentiveness in the classroom is very important, we found few existing studies in the ERIC index addressing students’ attentiveness in the classroom in direct relation to their mathematics performance. Thus, also little is known about the relationship between attentiveness in the classroom and student mathematics achievement.

With reference to the last three scales, previous studies also approved that students’ well-being at the school (Abadi et al., 2013; McDonald, 2012; Steinmayr et al., 2016) and the relationship between students and their teachers or peers (e.g. Eryilmaz, 2015; Valiente, Lemery-Chalfant, Swanson, & Reiser, 2008) are related to their achievement positively. Samdal, Wold and Bronis (1999) found that well-being at the school is positively related to student achievement. These findings imply that interventions which enhance students’ satisfaction with school are likely to improve their achievement as well. And the study of Lilienfeld, Turner and Jacob (1996) indicated that perceived teacher enthusiasm and supportive achievement-contingent feedback from the teacher were positively related to individual levels of academic enjoyment. Moreover, Wentzel (2005) also found that peer relationships can affect students’ learning in a non-linear way, especially for students with low achievement.

Hence, we can formulate the hypothesis that students’ SWB is related to their mathematical performance and then the SWB affects their mathematical performance in some way. Although there is mutual interaction between students’ SWB and their mathematical performance which literature review shows, studying how students’ SWB affects their performance on mathematics will be the foundation of understanding the mutual relation and impact between these two aspects. Thus, this study will analyze how students’ SWB affects their mathematics based on the correlation analysis.

Besides, most studies were done within a learning stage, like within elementary school or middle school. To date, much less research has been done examining the differences of students’ SWB between elementary and middle school. Hence, the differences of students’ SWB between elementary and middle school will be analyzed in this study.

Differences in Different Learning Stages

The transition from elementary to middle school is an important time for students (Odegaard & Heath, 1992; Symonds & Hargreaves, 2016). The change in grade level and environment can cause instability and academic failure (Wigfield & Eccles, 1994). Such changes can decrease students’ well-being at the school and their positive attitude towards knowledge and can inversely increase their negative reaction to teachers (Salmela-Aro, 2009; Symonds & Hargreaves, 2016). Ultimately, students’ self-efficacy and self-esteem may be decreased, because of the changing of their learning environment and the relationships with their peers, teachers, or school (Malmberg & Little, 2007; Vasalampi, Salmela-Aro, & Nurmi, 2010).

In order to understand what kinds of factors can affect students’ development as they transition from elementary to middle school and from middle to high school, using the person-environment fit and self-determination theory (Deci & Ryan, 2002), Eccles and Midgley (1989) proposed the stage-environment fit theory (SEFT) and used it to study how the transition change affected students’ development (Eccles et al., 1993). Specifically, Eccles, Roeser, Wigfield and Freedman-Doan (1999) built a framework which can be used to measure how school affects students’ development. This framework includes a series of hierarchically ordered, interdependent levels of organization beginning at the most basic level of the classroom and school as an organizational system embedded in a larger cultural system. There are five aspects to this framework: (1) schools are multiple-level systems composed of regulatory processes (e.g. organizational, interpersonal, and instructional in nature); (2) these processes are interrelated across levels of analysis; (3) these processes are generally dynamic, sometimes being worked out each day between the various social actors (like teachers and students); (4) these processes will be changed when students transition from elementary to middle school; and (5) these processes control students’ development in cognition, social integration, and behavior.

Therefore, students’ lives and learning environments (such as the school environment, school structure, the relationship between teacher and student, individual psychology, and even parental involvement) will be changed when they transition from elementary to middle school. However, there are few studies which pay attention to whether students’ SWB is different from elementary to middle school and what factors influence the change in students’ SWB. In this paper, results from two studies are reported to answer the following two questions: (1) What is the difference in Shanghai students’ SWB between Grade 5 (last year of elementary school) and Grade 6 (first year of middle school)? (2) Is the students’ mathematical performance related to their SWB? If yes, how does their SWB affect mathematical performance? Study 1 is designed to answer the first research question, and study 2 is designed to answer the second research question.

Study 1

Method

Subjects

A total of 4475 Shanghai students in Grades 3 – 9 participated in study 1 in the spring of 2013. Table 1 shows the subjects in each grade. The participants consisted of 2226 elementary school students and 2249 middle school students. The total number of valid questionnaires in elementary school (from Grades 3 to 5) was 2149, making the return rate 97%. The total number of valid returns in middle school (from Grades 6 to 9) was 2180, for a return rate of 97%. The percentages of boys and girls were each around 50%.

Table 1 Subjects of study 1
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Elementary and middle school students were chosen as the research subjects for several reasons. Firstly, from elementary school to middle school, there is a great change in the kind of knowledge that students develop, moving from arithmetic to algebra, from constancy to variance, and from simple and intuitive experimental geometry to the demonstrative geometry of abstract logical reasoning. These changes lead to higher requirements for new middle students in their cognitive and mental levels and require students to understand more complex knowledge than they do in elementary school (Van de Walle, 1998), such as the real number system, equations, and variance. Secondly, according to Piaget’s cognitive development theory, the students’ cognitive level may change when they transit to middle school from elementary school. Finally, the way of teaching changes from elementary school to middle school. Shaped by different content, requirement, and assessment formats, the teaching methods of middle school teachers are different from those of elementary school teachers (Wang, 2009). And students’ academic environment and social interaction would be changed from elementary school to middle school (Deci & Ryan, 2002; Eccles et al., 1999; Eccles & Midgley, 1989).

Measures of Students’ SWB

We translated the SWB questionnaire developed by Opdenakker and Van Damme (1998) into Chinese, after which it was checked by three professors from two different universities. One professor is Chinese-American and teaching at an American university, and the other two professors are from a Chinese university. The questionnaire consists of 61 items belonging to the eight factors previously mentioned. The scale used is a 5-point Likert-type scale: 1—strong disagreement, 2—disagreement, 3—not sure, 4—agreement, and 5—strong agreement. Some of the questionnaire items are reverse questions, signified by the use of (-).

In the Chinese version, the SWB’s overall scale reliability coefficient is 0.97 (with scales ranging from 0.60 to 0.88) for elementary school and 0.96 (with scales ranging from 0.70 to 0.88) for middle school data. Following this, principal component analyses were conducted to examine the underlying factor structure of the SWB scores using the data of elementary and middle school. For the data of elementary school, the screen plot displayed that the magnitudes of the eigenvalues for the first eight factors were relatively large (18.769, 3.306, 2.476, 1.945, 1.426, 1.306, 1.153, 1.048) and descended below 1. And the value of Kaiser-Meyer-Olkin (KMO) was 0.98 (p < .001). For the data of middle school, the screen plot displayed that the magnitudes of the eigenvalues for the first eight factors were relatively large (19.542, 3.309, 2.944, 2.176, 1.861, 1.332, 1.255, 1.154) and descended below 1. And the value of KMO was 0.97 (p < .001). And the relationship among each item within each factor was thus strong for both the elementary and middle school data, making this questionnaire very suitable for factor analyses (Spicer, 2004).

Based on these results, we performed exploratory factor analysis five times for both elementary and middle school samples in order to examine the items and scales of the pilot SWB questionnaire. After each analysis, we deleted several items and revised the scales of the SWB questionnaire according to the following principles: first, some items in a given scale had low correlations with the other items; second, some items contributed to the decline of the reliability of the SWB questionnaire; third, previous studies have showed that motivation towards learning tasks is closely and significantly related to attitude towards homework (DeNeve & Cooper, 1998; Keith & Cool, 1992) and students’ motivation towards learning is reflected when they actively complete homework (Singh, Granville, & Dika, 2002). We therefore removed the two items related to attitude to homework (ATHOMEW) and considered the rest of the three items of ATHOMEW and the two items of motivation towards learning tasks (MOTLT) as part of the motivation towards learning tasks (MOTLT) scale, resulting in only seven scales of SWB. In total, we deleted 27 items, resulting in a new SWB questionnaire containing seven scales totaling 34 items.

Subsequently, confirmatory factor analysis (CFA) was used to confirm the validity of this new SWB questionnaire. Based on the results from the elementary and middle school data, the new SWB questionnaire was suitable for these two learning stages. We then considered these two portions of data as a whole and randomly chose 2150 subjects to confirm the validity of the seven scales of the new questionnaire. There were 34 observed variables with seven latent variables, as mentioned above. Each of the seven exogenous latent variables was measured through its associated questionnaire items. The model assumed that each of the seven scales measured a different but correlated aspect of student SWB in school. AMOS 20.0 was used, and model parameters were estimated by the maximum-likelihood estimation procedure.

Absolute and incremental fit indices were used to evaluate model fit (Wang, Young, Wilhite, & Marczyk, 2011). The goodness-of-fit (GOF) indices used were chi square/degrees of freedom, root-mean-square error of approximation (RMSEA), unadjusted GOF index (GFI), and average GOF index (AGFI). The values were chi square/degrees of freedom = 4.97, RMSEA = .043, GFI = .933, and AGFI = .921. The incremental GOF indices used were comparative fit index (CFI) and standardized root-mean-square error residual (SRMR). The resulting values were SRMR = .03 and CFI = .93. According to the requirements of structural equation modeling (SEM), these indices showed that this model has a good fit for the full data. The CFA model results provided a reasonably close fit to the data and confirmed the seven-factor model of SWB. Hence, the new SWB questionnaire consisting of seven scales comprised of 34 items was ready to use in investigating Shanghai students’ SWB.

In study 1, we collected students’ scores on an external mathematics assessment in a district of Shanghai. However, because this assessment was a proficiency test rather than a selective test, most students’ grades were very high. The mathematical achievement of elementary and middle school students as measured by this assessment did not represent a normal distribution; thus, this data could not be used to analyze the relationship between students’ SWB and their mathematics achievement.

Results

All seven scales were tested by analysis of variance (ANOVA). Results of the ANOVA (see Table 2) indicated that there were significant differences (α = 0.05) between elementary and middle school students on all seven scales. Compared to the mean scores of elementary school students, the mean scores of middle school students were significantly lower on all seven scales.

Table 2 Differences between elementary and middle school based on results from ANOVA
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Based on this result, we conducted a further comparative study of elementary and middle school students’ mean SWB scores. Figure 1 shows the students’ mean SWB scores in elementary and middle school. Multivariate ANOVA was used to describe the differences in SWB scores. These students’ mean scores on the seven scales exhibited the same trend from Grade 3 to Grade 5 with no significant differences, and across grade levels in middle school, the scores of SWB were not significantly different. However, it is worth noting that both in elementary and middle school, the mean scores of academic self-concept were the lowest (elementary school is from 3.98 to 4.02 and middle school is from 3.26 to 3.31).

Fig. 1
figure 1

SWB’s mean scores of elementary and middle school samples

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Throughout study 1, we revised the SWB instrument and returned initial findings about the students’ SWB in elementary and middle school. There are two main reasons why a study 2 was necessary: firstly, study 2 confirmed the validity of the new SWB instrument that was developed during study 1, and secondly, we investigated the students’ mathematics achievement by the questions of mathematical problem-solving and their SWB in order to analyze how students’ SWB related to their mathematics achievement.

Study 2

Based on the analysis of the data from study 1, we collected data again using the new SWB questionnaire comprised of seven scales and 34 items in study 2. Furthermore, because the achievement data in study 1 did not represent a normal distribution, we collected students’ mathematical achievement in study 2 by examining their problem-solving ability.

Method

Subjects

We chose the district in which the students’ academic performance in Grades 5 and 6 was located at the middle level of all the districts of Shanghai as our new subjects; the subjects from Grade 5 represent elementary school students, while the subjects from Grade 6 represent middle school students. Table 3 shows the subjects in each grade and school.

Table 3 Subjects of Grade 5 and Grade 6
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The subjects were selected randomly. And the participants consisted of 500 Grade 5 students who came from seven elementary schools and 750 Grade 6 students who also came from seven middle schools. The total number of valid questionnaires in Grade 5 was 464, making the return rate 92.8%. The total number of valid returns in Grade 6 was 698, for a return rate of 93.1%. The percentages of boys and girls were each around 50%.

Measures of Students’ SWB

We collected data again from Grade 5 and Grade 6 students with the new SWB questionnaire. The SWB’s overall scale reliability coefficient for the Grade 5 sample was 0.90, with scales ranging from 0.50 to 0.84. Principal axis analyses were used to confirm the underlying factor structure of the SWB scores. The screen plot displayed that the magnitudes of the eigenvalues for the first seven factors were relatively large (9.194, 2.499, 1.948, 1.505, 1.334, 1.150, 1.126) and the value of KMO was 0.92 (p < .001). For the Grade 6 sample, the SWB’s overall scale reliability coefficient was 0.90, with scales ranging from 0.50 to 0.83. The screen plot displayed that the magnitudes of the eigenvalues for the first seven factors were relatively large (9.043, 2.465, 1.779, 1.574, 1.359, 1.163, 1.025) and the value of KMO was 0.92 (p < .001).

Measures of Students’ Mathematics Achievement

Since students’ mathematics achievement consisted of a sharply negatively skewed distribution, we decided to consider students’ performance on mathematical problem-solving tasks as a measure of their mathematics achievement. We chose four non-routine mathematical problems developed by Cai (2000) to measure students’ mathematical achievement. Each of the four problems involves different content areas such as ratio and proportion, number sense, pattern recognition, and pre-algebra. Students were asked not only to provide their answer but also to explain their thinking process.

We used Cai (1995) general scoring rubric to reflect on three interrelated performance dimensions: mathematical conceptual and procedural knowledge, strategic knowledge, and communication. Based on consideration of these three aspects, each student response was scored at each of five levels (0 – 4). According to the scoring rubric of Cai (1995), a score of 4 means that a student’s explanation or solution process shows a correct and complete understanding of the problem; a score of 3 means that a student’s explanation or solution process is basically correct and complete except for a minor error, omission, or ambiguity; a score of 2 means that the student’s explanation or solution process, although it might express their understanding of the problem, is incomplete; a score of 1 means that the student’s understanding of the problem is very limited; and finally, a score of 0 means that there is no answer or that the student’s answer shows no understanding of the problem.

Although this scoring rubric was tested by Cai (2000), we still needed to confirm its reliability with respect to the present study due to our use of different subjects and due to the number of mathematical problems. For this study, the reliability estimates (the Cronbach alpha coefficients) were 0.76 for Grade 5 and 0.84 for Grade 6.

Inter-rater Agreement of Students’ Mathematical Achievement

To ensure inter-rater reliability of the four problems used to measure students’ mathematical achievement, we randomly chose 100 students’ test papers from Grade 5 and Grade 6, respectively, and invited two people who know this study and the scoring rubric to score the students’ mathematical achievement. The correlation coefficients show that these two data sets have strong correlations on each question as well as on the total test score (t > 0.90, p < .001).

Results

Students’ SWB

Both Grade 5 and Grade 6 students had high mean scores of SWB, with the academic self-concept of both samples being the lowest of all the scales of SWB. That means most of the students had very low confidence in their learning capacity, which was the main reason for the students’ low academic self-concept, because the students have got the lowest mean scores on the item “I think that I am good at learning,” which is in the factor of academic self-concept. Furthermore, students’ attentiveness in the classroom was also lower than the other five scales of SWB. Most of the students in both Grade 5 and Grade 6 expressed they have been distracted in class easily.

Comparing the Grade 5 and Grade 6 students’ SWB (Table 4), we found that the Grade 5 students’ SWB was significantly higher than that of the Grade 6 students. The differences between Grade 5 and Grade 6 students on SWB also verify the results of study 1. Table 4 shows the mean scores for each variable of SWB for the Grade 5 and 6 students. The mean scores for the Grade 5 sample are significantly larger than those for the Grade 6 sample on the variables of academic self-concept (t = 3.56, p < .001), attentiveness in the classroom (t = 4.69, p < .001), interest in learning tasks (t = 7.34, p < .001), motivation towards learning tasks (t = 4.79, p < .001), relationship with teachers (t = 6.70, p < .001), social integration in the class (t = 3.14, p < .01), and well-being at the school (t = 7.05, p < .001).

Table 4 Mean scores of Grade 5 and Grade 6 students on each scale of SWB
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Relationship Between Students’ SWB and Their Mathematics Achievement

To examine the relationship between SWB and students’ mathematical achievement, the correlation coefficients between SWB scores and students’ problem-solving scores were calculated and examined (Table 5). For both samples, the scores on most of the SWB factors are moderately correlated with students’ mathematics scores. In particular, for the Grade 5 students, three scales of SWB scores are significantly correlated with students’ mathematics scores: The correlation coefficients are 0.15 (p < .01) between academic self-concept and mathematics score, 0.18 (p < .01) between attentiveness in the classroom and mathematics score, and 0.16 (p < .01) between relationship with teachers and mathematics score. For the Grade 6 students, all the scales of SWB are significantly correlated with students’ mathematics scores. Nevertheless, the Grade 6 students’ single mean mathematics score also has a stronger correlation with the abovementioned three scales of SWB than the other scales.

Table 5 Correlation coefficients between students’ mathematical achievement and their SWB
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In addition, the highest and lowest mathematical achievements were also differentiated by comparing the mean scores of students’ SWB. The mean mathematical achievement scores of the top 10% of students are significantly higher than the mean mathematical achievement scores of the bottom 10% for both the Grade 5 sample (t = 45.67, p < .001) and the Grade 6 sample (t = 170.08, p < .001).

The top 10% of students on mathematics score demonstrated a significantly higher SWB mean score than the bottom 10% of students for both the Grade 5 sample (t = 2.20, p < .05) and the Grade 6 sample (t = 6.20, p < .001). Table 6 shows each scale’s mean scores of students with the highest and lowest mathematical achievements.

Table 6 Mean SWB scores of students with the highest and lowest mathematics scores
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For the Grade 5 sample, the mean scores of the highest mathematical performance students on academic self-concept (t = 3.78, p < .001), relationship with teachers (t = 2.79, p < .01), and attentiveness in the classroom (t = 2.22, p < .05) are significantly higher than the mean scores on these scales of the lowest mathematical performance students. On the other scales of SWB, there are no significant differences between the two groups. This means that, compared to the student who has the lowest mean score in mathematics, the student who has the highest mean score in mathematics has more confidence in learning, is less easily distracted in class, and gets along better with most of the teachers.

For the Grade 6 sample, on the other hand, the mean scores of the highest mathematical performance students on all the scales of SWB are significantly higher than the mean scores on these scales of the lowest mathematical performance students. In particular, there are very significant differences between the mean scores of the highest mathematical performance students and the lowest mathematical performance students on attentiveness in the classroom (t = 7.16, p < .001), academic self-concept (t = 6.58, p < .001), and relationship with teachers (t = 4.07, p < .001). Furthermore, the mean scores of the highest mathematical performance students are significantly higher than the mean scores of the lowest mathematical performance students on the other four scales—interest in learning tasks (t = 3.88, p < .001), well-being at the school (t = 2.91, p < .01), social integration in the class (t = 2.60, p < .05), and motivation towards learning tasks (t = 2.45, p < .05).

For each scale, stepwise regression analyses were conducted to examine how well the seven SWB scales as a whole predict student achievement in mathematics (Table 7). For the Grade 5 sample, the results of the regression analyses suggest that four scales of SWB significantly (p < .01) contribute to student performance in mathematics. However, just 7% of the variability of student performance in mathematics is accounted for by the variability of these four scales of SWB. The results also showed that attentiveness in the classroom, as the independent variable, is the most important predictor of student performance in mathematics. Nevertheless, for the Grade 5 sample, the seven SWB scales as a whole failed to predict student performance in mathematics. There are two possible reasons for this result. First, this result might be related to the sharply negatively skewed distribution of the students’ SWB scores. The second possible reason is that the students’ mathematics performance in elementary school can be only partially predicted by their SWB.

Table 7 Multiple regression analysis with students’ mathematical achievement as dependent variable
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Furthermore, for the Grade 6 sample, the regression analyses showed that all seven SWB scales’ roles as a whole contribute significantly to student mathematical performance (p < .001). The coefficient of determination (R 2) for the regression analysis using the seven SWB scores as independent variables was 0.143. The stepwise regression analyses results showed that the three independent variables of SWB—academic self-concept, attentiveness in the classroom, and relationship with teachers—are the most important predictors of Grade 6 students’ mathematical performance (R 2 = 0.15) and that the other four scales can be removed from the regression model without not much loss in predictive power.

Discussion

The results of study 1 and study 2 indicated that for those students at two different learning stages (e.g. elementary and middle school), all seven scales of SWB showed significant differences. However, there is no significant difference on the seven scales of SWB within the same learning stage, regardless of the specific learning stage. The stage-environment fit theory (SEFT) can be used to explain the part of the reasons for these differences. According to SEFT, school, class, teachers, peers, and other environmental characteristics are changed during the transition period from elementary to middle school. These changes are proposed to affect the overall “fit” between adolescents and their school environment (Eccles & Midgley, 1989). First of all, both study 1 and study 2 found that academic self-concept, motivation towards learning tasks, and interest in learning tasks of middle school students are lower than they are for elementary school students. These findings also support the argument that after students’ transition from elementary to middle school, their learning motivation and interest declines (Symonds & Galton, 2014; Wigfield, Byrnes, & Eccles, 2006).

Secondly, we also found that the relationship between students and teachers or peers declined when students moved to middle school from elementary school. This result aligns well with results of previous studies which have examined the interactions between students and their school environment (Symonds & Hargreaves, 2016; Wigfield et al., 2006). Although middle school teachers may care about students or provide help just as elementary school teachers do, our study shows that students expressed more nervousness or unease when in the presence of their teachers (“I feel at ease with most of the teachers,” mean score is 3.72; “I get on well with most of the teachers,” mean score is 3.87). Therefore, future research should focus not only on students’ feelings but also on the interactive process between teachers and students in middle school.

Thirdly, for both samples, the mean scores of the students in both elementary school and middle school on academic self-concept and attentiveness in the classroom are very low compared to the other scales of SWB. This means that students in Shanghai lack confidence in their learning. Both study 1 and study 2 support the result of PISA 2012 (OECD, 2013), which reported that Shanghai students’ average mathematics self-concept was lower than that for almost 20 other countries or districts. However, since we still do not know the reasons for this result, future studies are needed to explore it further. Study 2 extended earlier work (Marks, 2000; Newmann, 1992) to examine how students’ engagement in their classroom affects their learning, with a particular focus on students’ attentiveness in the classroom. However, this study does not investigate what factors affect students’ attentiveness and how students’ attentiveness in the classroom influences their mathematics performance. Therefore, the findings may serve as a basis for future studies to investigate and analyze the reasons and mechanisms which affect students’ attentiveness in the classroom and, consequently, their learning of mathematics.

Besides, the findings of both study 1 and study 2 also suggest that students perceive a decline in the emotional support provided by their school and in their sense of belonging in their school (Roeser, Peck, & Nasir, 2006; Wigfield et al., 2006). The results of these studies may serve as an effective way or viewpoint for future studies to research the transition problems students face during the move to middle school from elementary school or from middle school to high school, and help students successfully adapt to new learning environments.

Study 2 not only investigated students’ SWB in school but also examined the relationship between students’ mathematical achievement and their SWB. For both grades, the scales of SWB were closely related to students’ mathematical achievement. The contribution of all seven SWB scales towards Grade 6 students’ mathematics performance was statistically significant. Furthermore, only three scales of SWB were found to contribute significantly to Grade 5 students’ mathematical performance.

In determining which scales of SWB are the most important predictors of students’ mathematical performance, the results of study 2 indicate that three of the scales of students’ SWB in school—academic self-concept, attentiveness in the classroom, and relationship with teachers—are the most important predictors. Furthermore, we found that students with high and low mathematics scores have different SWB scores. The SWB of the students with the top 10% mathematics scores was significantly higher than that of the students with the bottom 10% mathematics scores. Based on the correlation between students’ SWB and their mathematical performance and the regression results, we could call this “the good circle” (Mortimore, 1991), according to which high SWB in school contributes towards good mathematical achievement, which in turn contributes to students’ self-efficacy, positive affection and motivation, and attitude to mathematics (Cai & Merlino, 2011). Therefore, when trying to enhance achievement among students who are not performing very well in mathematics, a possible approach may be to raise their SWB in school.

The abovementioned results corroborate the results of Marsh and Craven (2006) and provide further support for improving students’ academic self-concept to enhance their mathematics achievement. We also found that the other scales of SWB were significantly related to students’ mathematics achievement, similar to previous research findings (Roorda, Koomen, Spilt, & Oort, 2011; Steinmayr et al., 2016). Meanwhile, the positive effect size on mathematics achievement of Grade 6 students was more than that of Grade 5 students. Based on the findings of this study, it contributes to provide some potential teaching opportunities and views for teachers, especially for the teachers who teach the students when they transition from elementary school to middle school.

Furthermore, in order to improve students’ dispositions or emotion towards mathematics, teachers could help students experience the success of solving mathematical problems and see the importance of learning mathematics. The study could also contribute to teachers developing better strategies to engage reluctant learners and win over complacent student peer cultures (Cai & Merlino, 2011). Besides, teachers can know students in various aspects and develop their positive SWB through cooperating with other colleagues, because teaching staff cooperation in relation to teaching methods and pupil counseling has a significant positive effect on students’ SWB (Opdenakker & Van Damme, 2010).

It should be indicated that in study 2, only about 15% of the variance in students’ mathematics achievement can be explained by the variability of SWB. This implies that the higher the SWB students have, the better their performance in school mathematics; however, students’ SWB is not the only influence on their mathematical learning. Although students’ SWB is neither a necessary nor a sufficient condition in itself for students’ academic success in mathematics, students’ mathematical SWB represents for us a new way to think about affective development in mathematics education (Bishop, 2012).

Limitations and Future Studies

There were two limitations present in this study, though efforts were made to minimize them. First, the number of subjects needs to be increased, especially in a future study. Although the students included in the studies reported here were drawn from two districts of Shanghai, they represented subgroups that are far from homogeneous in school level or quality. Thus, it will be important for future studies to include participants from other districts and to ensure that various socioeconomic and school groups are adequately represented in the sample. Second, because of limited time and subjects, this study aimed to identify the changes or differences in the SWB of students who transitioned between only two learning stages—specifically, elementary school to middle school. Future studies ought to track students across multiple learning stages.

Since this study focused on students’ SWB in school and analyzed the relationship between students’ SWB and their mathematical performance, future studies should consider students’ SWB in mathematics based on the concrete characteristics of mathematics learning. Although this kind of research has already been carried out (for example, the study of Clarkson, Bishop and Seah (2013) researched the “Stages of the construct ‘mathematical well-being’”), we still know little about how students’ SWB affects their mathematics learning. These findings indicate a need for further research using the scales of SWB in school in order to better understand the definition of students’ SWB and how these factors affect students’ mathematical well-being.