STUDENTS' MOTIVATION IN THE MATHEMATICS ... - Springer Link

Marilena Pantziara and George N. Philippou

STUDENTS’ MOTIVATION IN THE MATHEMATICS CLASSROOM. REVEALING CAUSES AND CONSEQUENCES Received: 18 March 2012; Accepted: 14 November 2013

ABSTRACT. Students’ affective domain has been popular in the mathematics education community in an ongoing attempt to understand students’ learning behavior. Specifically, enhancing students’ motivation in the mathematics classroom is an important issue for teachers and researchers, due to its relation to students’ behavior and achievement. This paper utilized achievement goal theory—an important theoretical prospect on students’ motivation in school settings—to investigate the existence of a model presenting the relation between motivation and other affective constructs and students’ performance in mathematics. In this regard, two types of tests were administered to 321 sixth grade students measuring their motivation and other affective constructs and their performance in mathematics. Using structural equation modeling, we examined the associations among the affective constructs, motivation, and the extent to which these constructs influence students’ performance and interest in mathematics. The data revealed that students’ performance and their interest in mathematics were influenced by fear of failure, selfefficacy beliefs, and achievement goals. We discuss these findings in terms of teaching implications in the mathematics classroom. KEY WORDS: achievement goals, structural equation modeling, students’ performance

INTRODUCTION The importance of students’ motivation is reflected in the vast amount of related research in the teaching and learning process (Stipek, Salmon, Givvin, Kazemi, Saxe & MacGyvers, 1998; Elliot, Henrly Sell & Maier, 2005; Friedel, Cortina, Turner & Midgley, 2007). Researchers, mainly in the Educational Psychology domain, examine the role of motivation in the learning and teaching context in an attempt to understand why some students seem to learn and succeed in school contexts, while other students seem to struggle to obtain analogous outcomes (Pintrich, 2003). In this respect, a considerable body of research highlights not only the various cognitive, motivational, and regulatory constructs, but also the importance of the social context and the interactions with other constructs in the development of students’ motivation (Elliot, 1999; Liem, Lau & Nie, 2008; Kaplan & Maehr, 2007; Zusho, Pintrich & Cortina, 2005). However, most of these studies were focused on middle school and college students. International Journal of Science and Mathematics Education (2015) 13(Suppl 2): S385YS411 # National Science Council, Taiwan 2014

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Accordingly, the importance of student motivation is reflected in the realm of mathematics education which treats motivation as a desirable outcome and a means to enhance understanding. Researches in the field support not only the close association between cognitive, motivational, and affective factors in students’ learning (Hannula, 2006; Op’t Eynde, De Corte & Verschaffel, 2006), but also the importance of the specific classroom context and the students’ sociohistorical context in the formation of students’ affect. Yet, further research is needed on how these constructs interact and relate with each other (Hannula, 2006). In this respect, the main objective of the present study was to investigate the relationships among elementary (sixth grade) students’ affective constructs, their achievement goals, and their mathematical behavior. The study was conducted in a specific context and students’ mathematical performance was measured based on an explicitly designed test.

MOTIVATION The term motivation derives from the Latin verb movere which means to move. Thus, in Educational Psychology, motivational theories seek to investigate what gets individuals to move towards what activities and to describe the characteristics of these activities (Pintrich, 2003). Motivation is characterized as a complex and multidimensional construct (Zhu & Leung, 2010). Pintrich (2003) refers to five frameworks of social–cognitive constructs that have recently been used in research on students’ motivation in the classroom context. The framework of selfbeliefs (e.g. self-efficacy) asserts that when students believe that they can do well in the educational settings, they feel confident, tend to try hard, persist more, and perform better (Bandura, 1997). Attribution and control beliefs (Weiner, 1986) support that students who believe that they have more personal control of their learning are expected to do better and achieve at higher levels than students who do not feel in control of their learning behavior. Interest and intrinsic motivation is another theoretical framework that expands the traditional distinction between students’ intrinsic and extrinsic motivation to a more complex differentiation of extrinsic motivation. These different types of interest were found to lead students to different learning outcomes (Ryan & Deci, 2000). Expectancyvalue theory (Eccles & Wigfield, 1995) supports the view that students are motivated when they judge a task as somehow important. Research under this framework revealed that task valued beliefs seem to influence

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students’ choices, like the enrollment in future courses. Finally, achievement goal theory focuses on goals, as the reasons and purposes for engaging in achievement tasks (Elliot, 1999). The current study investigates how various affective constructs, derived from self-beliefs, interest and intrinsic motivation, and achievement goal theory, interact and are related to students’ mathematical performance. Achievement Goal Theory and the Hierarchical Model of Achievement Motivation Achievement goal theory has been developed during the last two decades as an important theoretical prospect on students’ motivation in school settings, as it provides an extensive research framework on motivational orientations that contributes to students’ adaptive and maladaptive learning behavior. This theory defines motivation in terms of quality, focusing on how students think about themselves, their tasks, and their performance, while it investigates both individual and social factors that may influence students’ motivation (Elliot, 1999). Furthermore, one of its strengths in understanding students’ motivation is that it explicitly investigates the role of the teacher and the instructional context in identifying students’ personal goal orientation (Anderman, Patrick, Hruda & Linnenbrink, 2002). Achievement goal theory defines why and how students attempt to achieve various objectives (Kaplan & Maehr, 2007). Achievement goals are defined as the competence-relevant purposes or aims that individuals attempt to achieve—committed to serve—and these different purposes or aims are associated with different quality of engagement in schoolwork as well as with different cognitive, affective, and behavior consequences (Elliot et al., 2005; Kaplan, Middleton, Urdan & Midgley, 2002). Elliot et al. (Elliot & Church, 1997; Elliot, 1999) proposed a hierarchical achievement goal framework in which students’ goal orientations are presumed to mediate the relation between individual and social variables and selected achievement and motivational variables. Particularly, they grouped antecedent variables in categories labeled as competence-based variables (e.g. need for achievement), self-based variables (e.g. self-esteem), relationally based variables (e.g. fear of rejection), demographic variables (e.g. gender), and neurophysiological predispositions (e.g. sensitivity) and presume that these variables prompt the adoption of specific achievement goals. The variability in the motivational roots of achievement goals is believed to be translated into difference in the effect that achievement goals have on achievement

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behavior and outcomes (Kaplan & Maehr, 2007). Many studies have validated the model mostly with secondary and college students in different social contexts (Cury, Elliot, Da Fonseca & Moller, 2006; Elliot & Church, 1997; Zusho et al., 2005). In this study, we focus on three types of goals that have been identified: mastery goals, performance-approach goals, and performance-avoidance goals. Mastery goals refer to an individual’s purpose of developing personal competence and growth (Kaplan & Maehr, 2007). Mastery goals are believed to produce a steady empirical pattern due to the alignment of the primary sources of mastery goals with the focus inherent in its representation (Elliot, 1999). Since mastery goals focus on the development of competence, it is believed to lead to positive outcomes. Earlier research has documented that students’ engagement with mastery goals evokes positive processes like effort, expenditure, persistence, task absorption, self-efficacy, selfregulated learning as well as positive affect and well-being (Elliot et al., 2005; Kaplan & Maehr, 2007; Liem et al., 2008; Wolters & Rosenthal, 2000; Zusho et al., 2005). However, research revealed that mastery goals do not provide a consistent facilitator of students’ achievement. Some studies found positive relationship between students’ mastery goals and their achievement (Anderman & Midgley, 1997; Pantziara & Philippou, 2006), while other studies found that mastery goals did not have a direct effect on students’ performance (Cury et al., 2006; Zusho et al., 2005). Performance-approach goals refer to the purpose of demonstrating competence, focusing on attempts to create an impression of high ability often through the comparison with others’ ability (Kaplan & Maehr, 2007). The model holds the view of a complex empirical pattern associated with performance-approach goals since the focus of these goals can be aligned or not with their motivational roots. When the antecedents of these goals are congruent (e.g. need for achievement), then the adoption of performance-approach goals is expected to lead to positive outcomes. Research (Elliot, 1999; Elliot & Church 1997; Zusho et al., 2005) found that performance-approach goals are associated with positive processes like persistence, positive affect, and graded performance. In the case of incongruent antecedents (e.g. fear of failure), the adoption of these goals represents approach in order to avoid something and it is likely to lead to both positive and negative outcomes (Elliot, 1999; Midgley, Kaplan & Middleton, 2001). Performance-avoidance goals focus on the possibility of failure and on the attempt to avoid it (Kaplan & Maehr, 2007, p. 144). The same pattern with mastery goals


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underpins performance-avoidance goals which are expected to lead to negative processes since many of the antecedents (e.g. fear of failure) are oriented in a negative direction (Elliot, 1999). Performance-avoidance goals were found to be related to low efficacy, low grades, and avoidance of help-seeking and self-handicapping strategies (Urdan, Ryan, Anderman & Gheen, 2002). Achievement goals have been consistently studied and incorporated in various models (Friedel et al., 2007; Kaplan & Maehr, 1999; Liem et al., 2008). Researchers investigated variables that prompt the adoption of these goals and their relation to students’ psychological, social, and behavioral functioning and also to their achievement. In the present study, we examine whether students during their mathematics classrooms identify their goals and therefore distinguish and adopt the three different kinds of achievement goals. Moreover, we investigate the role of each of these three achievement goals as mediators of students’ beliefs and motives in their mathematical behavior. Fear of Failure and Achievement Goals Among the numerous antecedent variables identified by researches to orient students towards specific achievement goals, we focus on fear of failure and self-efficacy beliefs, which we expect that students could deal with. McGregor & Elliot (2005) state that “fear of failure may be interpreted as ‘a self-evaluative’ framework that influences how the individual defines, orientates to, and experiences failure in achievement situations” (p. 219). More explicitly, an individual with a high fear of failure perceptually and cognitively orientates to failurerelevant information and thus encounters anxiety prior to and during task engagement. He/she seeks to avoid failure by avoiding the situation, by quitting or withdrawing effort, or by trying hard to succeed and thus avoid failure. The core emotion of fear of failure is most likely shame, a devastating emotion that entails a sense of one’s global incompetence. Fear of failure is hypothesized to prompt the adoption of performance-avoidance goals that focus on the avoidance of negative consequences. However, fear of failure is also posited to lead to performance-approach goals, since the desire to avoid failure is strategically served by striving to attain success (Elliot, 1999; Elliot & Church, 1997). In the present study, we investigate fear of failure

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as an antecedent variable that sets off the adoption of specific kind of achievement goals and particularly performance-approach and performance-avoidance goals. Self-Efficacy Beliefs and Achievement Goals In the hierarchical model by Elliot (1999) and the review of research by Kaplan & Maehr (2007), it is assumed that individuals hold a certain cognitive–affective system of beliefs about themselves; this system exerts a direct effect on student’s achievement goals which in turn serve as proximal ancestors to achievement-related outcomes. This system includes, among other constructs, competence perceptions and selfefficacy beliefs. Cury et al. (2006) define perceptions of competence as individuals’ beliefs about their ability to accomplish a task or not, in competence-relevant settings. Academic self-efficacy refers to individuals’ belief that they can successfully achieve at a specific level on an academic task or attain an academic goal (Bandura, 1997). In education, the boundaries between competence perceptions and self-efficacy beliefs have gradually been blurred (Bouffard & Couture, 2003). Initially perceived competence has been applied as a more general concept, referring to a more general range of operations, encompassing a variety of tasks while self-efficacy beliefs were more depended on specific tasks. In educational research, the two concepts are often used interchangeably while in areas other than education each construct is used in their stern sense. Researchers (Cury et al., 2006; Elliot, 1999; Elliot & Church, 1997) support the view that individuals with high competence perceptions are expected to adopt approach goals, mastery, and performance, whereas individuals with low competence perceptions are expected to orient toward the adoption of performance-avoidance goals. Cury et al. (2006) found that perceived competence was a significant predictor for approach goals and a negative predictor of avoidance goals (Cury et al., 2006). Self-efficacy beliefs positively correlate with students’ persistence, effort, interest, and achievement (Chen & Zimmerman, 2007; Nicolaou & Philippou, 2007). Self-efficacy has been consistently found to be related to mastery goals (Friedel et al., 2007; Kaplan & Maehr, 1999; Pantziara & Philippou, 2006), whereas the relation between performance goals and self-efficacy has been less consistent. Some studies have found positive relation (Liem et al., 2008), other studies have found negative association (Schunk 1996), and some studies found moderate relation between the


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two constructs (Bråten, Samuelstuen & Strømsø, 2004; Friedel et al., 2007). Differences in studies also refer to the placement of self-efficacy beliefs in the various models related to achievement goals. Some studies (Liem et al., 2008) incorporate in their model self-efficacy beliefs as antecedents of achievement goals, whereas in other studies (Bandalos, Finney & Geske, 2003; Friedel et al., 2007), self-efficacy beliefs are conceived as consequences of achievement goals. This bidirectional relation is in accordance with Bandura’s (1997) notion of reciprocal determinism, stating that how individuals interpret the results of their achievement informs and modifies their belief system, while their self-efficacy beliefs inform and modify their achievement. In the present study, in accord with the hierarchal model of achievement motivation (Elliot, 1999), we examine self-efficacy as an antecedent variable that may prompt the adoption of mastery and performance goals. Achievement Goals and Interest Research using the trichotomous achievement goal framework (Elliot & Church, 1997; Zusho et al., 2005) has focused on the consequences of adopting different achievement goals. One of these positive outcomes refers to interest. We conceptualize interest according to self-determination theory (Ryan & Deci, 2000) which argues that interest or intrinsic motivation is the engagement in an activity for its own innate satisfactions and enjoyment rather than for some separable consequences. Intrinsic motivation has emerged as an important issue for educators because it results in highquality learning. In this respect, self-determination theory investigates factors and forces that stimulate or undermine interest. One of the most consistent patterns emerged from the achievement goal research is the positive influence of mastery goals to interest (Cury et al., 2006; Pantziara & Philippou, 2006; Zusho et al., 2005). At odd, performance-avoidance goals are usually found to have negative effect on students’ interest (Cury et al., 2006; Zusho et al., 2005). In the case of performance-approach goals, some studies have found positive effect (Pantziara & Philippou, 2006; Zusho et al., 2005), and some others found null effect (Cury et al., 2006). In this study, we investigate interest as an outcome variable that in many ways serves as an indicator of adoptive behavior in the mathematics classroom.

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Achievement Goals and Students’ Age The hierarchical model of achievement motivation and other models based on achievement goal theory have been investigated regularly in secondary and college students (Bandalos et al., 2003; Cury et al., 2006; Elliot & Church, 1997; Zusho et al., 2005). Some studies referred to elementary students’ achievement goals included mastery and performance-approach goals but not performance-avoidance goals (Anderman & Midgley, 1997; Friedel et al., 2007), while Midgley, Kaplan, Middleton, Maehr, Urdan, Anderman, Anderman et al. (1998) validated the existence of the three types of achievement goals in a wide range of grade levels including the elementary one. Anderman & Midgley (1997) found that students in the last grade of elementary school were more oriented to mastery goals and felt more academically competent, while the same students in middle school gave greater emphasis on performance goals. This was rendered to the policies and practices in elementary and middle school with middle school to emphasize performance goals more.

MOTIVATION AND MATHEMATICS EDUCATION The importance of motivation in mathematics education has been well documented (Hannula, 2006; Pantziara & Philippou, 2006). Hannula (2006) perceives motivation not as another dimension of affect but as another perspective that illuminates some aspects of affect. He considers motivation as a potential to direct behavior that is built into the system which controls emotion. He states that motivation can be indirectly observed through cognition, emotion, and behavior. In this respect, a student’s desire to get involved in a mathematical task may be driven by his belief about the importance of the task (cognition), his anger for the failure to solve the task (emotion), or his persistence to solve the task (behavior). Hannula (2006) suggests the integration of motivation into theories of self-regulation established on needs and goals. Particularly, he views motivation in terms of needs and goals, considering needs as more general than goals influenced by students’ beliefs of themselves, as learners of mathematics. In line with Elliot’s model (1999), Hannula (2004) has found that different dominating needs lead to the adoption of different primary goals and therefore to different behavior in mathematical situations. Moreover, based on the theory of self-regulation, he


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suggests that a change in motivation emerges from a desired goal and ones’ beliefs like self-efficacy beliefs. Op’t Eynde et al. (2006) in line with social–cognitive theories incorporated the issues of motivation in the socio-constructivist perspective on learning, recognizing the close association between (meta) cognitive, motivational, and affective factors in students’ learning. Particularly, they consider students’ learning as a form of engagement that allows them to actualize their identity through the involvement in activities situated in a specific context. Accordingly, students’ identity reveals their needs, their values, and their beliefs which are expressed through their emotions. These affective constructs are related to the development of specific motivation. In line with the framework of social cognitive models, this theoretical perspective states that the specific classroom-context and students’ sociohistorical context is in close interaction with students’ affective constructs, learning, and problem solving.

THE PRESENT STUDY While researchers in mathematics education identified patterns of students’ behavior like negative emotional disposition with fear, anger, boredom, and low perceived competence (Di Martino & Zan, 2010), further research is needed on how different elements of affect and motivation relate and interact to and with each other (Hannula, 2006). Moreover the effect of culture on learning and motivation has been well recognized (Zhu & Leung, 2010). The researchers argue that it is important to consider cultural values, norms, and practices in the investigation of students’ motivation. The purpose of the present study was to test a model of relations among sixth grade students’ affective constructs, motivation, and performance in mathematics based on achievement goal framework. The study was conducted in Cyprus—a different social context, where there is no grading for students in elementary school (grades 1—6), a practice that may affect students’ goals. Additionally, we introduce a different way to measure students’ performance in a specific mathematical concept. Our test focuses on fraction understanding, examining student’s performance in progressively demanding activities reflecting the characteristics of the three stages proposed by Sfard (1991). Explicitly in the model shown in Fig. 1, students’ achievement goals mediate the relation between self-efficacy beliefs and fear of failure on

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MARILENA PANTZIARA AND GEORGE N. PHILIPPOU Interest

Self-efficacy beliefs Achievement goals

Mathematical performance

Fear of failure

Figure 1. A model of students’ motivational constructs and outcomes

one side and students’ mathematics performance and interest on the other. Specifically, we assume that students’ self-efficacy beliefs will influence mastery goals and performance-approach goals, while students’ fear of failure may affect both performance-approach and performanceavoidance goals. Furthermore, we support the view that students’ mastery goals will have a direct effect on their mathematical performance and their interest in mathematics due to the absence of grading in this particular school level. Last, we argue that the influence of students’ performance-approach goals on their performance and interest in mathematics will depend on their antecedents (self-efficacy beliefs and/or fear of failure). The investigation of a model including different affective constructs, their relationship, and interaction with each other will provide researchers and teachers with insights of students’ behavior in the elementary mathematics classroom enabling them to explain, predict, and alternate this behavior. The study aims to expand the existing results on students’ motivation in mathematics education (Di Martino & Zan, 2010; Hannula, 2004; Stipek et al., 1998) since it is crucial to keep exploring factors that may develop students’ motivation informing teachers of how to keep students motivated to learn and thus improving the quality of teaching and learning of mathematics.

RESEARCH QUESTIONS This study is a part of a larger study that investigates students’ motivational constructs as well as variables from the classroom and social context that may develop students’ mathematical performance and interest. The purpose of the present study was as follows: 1. To test, in the Cyprus social context and for the specific age group, the validity of the measures for the six factors: fear of failure, selfefficacy, interest, mastery goals, performance-approach goals, and performance-avoidance goals


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2. To investigate the relation between students’ affective constructs (fear of failure, self-efficacy, achievement goals, and interest) 3. To confirm the existence of a model that presents the relations between students’ fear of failure, self-efficacy, achievement goals, mathematical performance, and interest

METHODS Participants and Setting Participants were 321 sixth grade students, 136 males and 185 females, from 15 intact classes in Cyprus. The participants completed a questionnaire concerning their motivation in mathematics and a test assessing their performance in mathematics comprised of fraction items. The collection of the data took place in the mid of the second semester of the school year. The collection period (mid of the second semester) was chosen so as to allow the evolution of certain motivational constructs and achievement goals with the specific mathematics teacher in each class. Thus, taking into account that achievement goal theory suggests that apart from students’ individual characteristics, the environment (e.g. classroom context, teachers’ practices) may influence their needs and goals. The questionnaire of motivation was administrated to students in their classrooms by the researchers. The completion of the questionnaire lasted for approximately 45 min. After a week, the same students completed a test on fractions of about 50 min of duration. Students who were absent from either of the two administrations were excluded from the study. Measures Measuring Students’ Affective Constructs. The questionnaire measuring students’ affective constructs was developed for the needs of the study and was comprised of 35 Likert-type five-point items (1—strongly disagree, 5—strongly agree). The six subscales comprising the questionnaire were (a) mastery goals (five items), (b) performance-approach goals (five items), (c) performance-avoidance goals (four items), (d) selfefficacy beliefs (five items), (e) fear of failure (nine items), and (f) interest (seven items). All items for the first four subscales were adopted from the patterns of adaptive learning scales (PALS revised) (Midgley, Maehr, Hruda, Anderman, Anderman, Freeman, Gheen et al., 2000). We used PALS instead of other scales because the development and validation of these

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scales involved primary students. We rephrased each of the scales’ items to appraise mathematics. The mastery goal orientation scale was conceptualized as a student’s focus on learning and understanding (e.g. “One of my goals in class is to learn as much mathematics as I can”). The performance-approach scale was defined as the tendency to focus on looking superior in comparison to other students (e.g. “One of my goals is to show others that I’m good at my mathematics work”). Performance-avoidance goals were defined in terms of the student’s focus on not looking dumb or stupid in relation to the other students (e.g. “It’s important to me that I don’t look stupid in mathematics class”). Selfefficacy beliefs were defined in terms of students’ perceptions of their competence to do their mathematics work (e.g. “I’m certain I can master the mathematics skills taught in class this year”). Students’ fear of failure was assessed using nine items adopted from the Herman’s fear of failure scale (Thrash & Elliot, 2003) (e.g. “I often avoid a task because I am afraid that I will make mistakes”). The researchers stated that the short form compared with the full scale, correlated strongly, covered the same content universe, and displayed internal consistency and predictive validity. The item scales were adjusted to assess students’ fear for mathematics. Finally, we used seven items from the Elliot & Church’s (1997) scale to measure students’ interest in achievement tasks. Interest was defined in terms of personal interest as well as appeal in the task while the items of the scale were adjusted to the age of students; instead of referring to class, they were referring to mathematics lessons (e.g. “I think mathematics lessons are interesting”). These 35 items were randomly spread throughout the questionnaire, to avoid the formation of possible reaction patterns. Measuring Students’ Mathematical Performance. In order to ensure that the test would be progressively demanding according to specific characteristics, we developed a three-dimensional test, each dimension corresponding to one of the levels of conceptual understanding—interiorization, condensation, and reification—proposed by Sfard (1991). In order to assess students’ level of understanding, we consider their performance in a single concept (the fraction concept) and investigate their achievement according to the characteristic of Sfard’s three stages. The choice of fractions was motivated by the fact that many students face a lot of difficulties in learning fractions and fail to understand them conceptually, remaining at the procedural, superficial knowledge level (Charalambous & Pitta-Pantazi, 2007). Most of the tasks comprising the test were adopted from published research (e.g. Lamon, 1999), assessing students’ understanding of fraction


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as part of a whole, as measure, fraction equivalence, fraction comparison, and addition of fractions with common and non-common denominators. We chose to investigate fraction as part of whole and measure, since in Kieren’s (1993) theoretical model of the fraction concept, the part–whole subconstruct is considered fundamental for developing understanding of the concept of fraction, while the measure subconstruct is considered necessary for developing proficiency in additive operations on fractions (Charalambous & Pitta-Pantazi, 2007). In Cyprus, fraction as part–whole subconstruct dominates mathematics textbooks and instruction from grade 1, while the measure subcontract appears less frequently. Cyprus mathematics textbooks for grades 5 and 6 give emphasis to equivalence, comparison, and addition of fractions. The tasks were further developed to correspond to the characteristics and difficulty of each of Sfards’ stages. In Sfards’ model, the ability of a learner to develop a mathematical concept is a gradual progress leading to reified objects whose structure gives a conceptual growth focusing on the properties of the objects. Sfard (1991) distinguished three hierarchical stages, which correspond to three degrees of structuralization. The test comprised of 21 tasks in seven triads; each triad of tasks aimed to measure understanding at each of the three Sfard’s stages (for details see Pantziara & Philippou, 2012). Students’ performance in the test was based on their total score on the fraction test; each of the tasks was graded with 0 (wrong) or 1 (correct). Data Analysis To test the validity of the measure for the six factors in the specific context, we used confirmatory factor analysis using maximum likelihood estimates in the structural equation model framework with latent variables. Exploratory factor analysis took place in a previous phase of this study (Pantziara & Philippou, 2006). The model referring to the affective constructs also reveals the relations between students’ affective constructs and motivation (fear of failure, self-efficacy, achievement goals, and students’ interest). To measure students’ achievement in the fraction test, two types of analysis on students’ responses were performed, Rasch model and cluster analysis. The first analysis was used to examine the reliability of the test as a whole and to specify the sequence of items by difficulty and the second analysis was used to search for groups of students in each of the difficulty levels. More details on the analyses can be found in Pantziara & Philippou (2012). In this study, we used the results of the Rasch model.

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Finally, structural equation modeling using maximum likelihood estimates (Hu & Bentler, 1999) was applied to assess the fit of the proposed model presented in Section 2, suggesting that the relation of students’ fear of failure and self-efficacy beliefs to mathematical performance and interest is mediated through achievement goals.

FINDINGS Factor Analysis of Students’ Affective Constructs To examine whether the factor structure yields the six affective constructs, confirmatory factor analysis was conducted using EQS (Hu & Bentler, 1999). By maximum likelihood estimation method, three types of fit indices were used to assess the overall fit of the model: the chisquare index, the comparative fit index (CFI), and the root mean square error of approximation (RMSEA). The chi-square index provides an asymptotically valid significance test of model fit. The CFI estimates the relative fit of the target model in comparison to a baseline model where all of the variables in the model are uncorrelated (Hu & Bentler, 1999). The values of the CFI range from 0 to 1, with values greater than 0.95 indicating an acceptable model fit. Finally, the RMSEA is an index that takes the model complexity into account; an RMSEA of 0.08 or less is considered to be an acceptable fit (Browne & Cudeck, 1993). For the identification of the six factors including the reduction of raw scores to a limited number of representative scores, we followed an approach suggested by proponents of structural equation modeling (SEM) (Hu & Bentler, 1999). One item from each of the factors mastery goals, fear of failure, and interest was deleted because of low loading. Some variables for the factors performance-approach goals, interest, self-efficacy, and fear of failure were grouped together since they correlated highly. Then, a six-factor model was tested (see Fig. 2) with items to load only on their respective latent variables. All factors were confirmed except from the factor performance-avoidance goal. The confirmation of the factors reveals the validity of the scales since all scales were spread throughout the questionnaire and students were able to distinct them expressing what they felt and believed. The failure to confirm performance-avoidance factor, both in an earlier exploratory analysis (Pantziara & Philippou, 2006) and the current analysis, may be due to the different social context that the study was conducted. Figure 2 presents the model with the five factors and their correlations. All indices (x2 = 343.487, df = 198, p G 0.000; CFI = 0.931 and RMSEA =


Interest

.715 .827 .664 .862

.698

Performance goals

.595 .751 .831 .700

.105

.621 Efficacy-beliefs

.545 .462 .728

.829 .488 .573

.-459

p.2.8 p.3.14

s.1.4 s.2.10 s.3.16

m.3.13 m.4.19 m.5.25

-.420

f.2.11

.510 .519

Fear of failure

p.1.2

m.2.7

.452

.-615

i.4.24

s.65

.627

Mastery goals

i.2.12

p.64

.347 .162

i.1.6

i.5.6.7

.192

.595

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f.6.30

.613 .650

f.7.32

.652

f.8.34

.399

f.13 f.49

Figure 2. The factor model of students’ affective constructs with factor parameter estimates

0.049) signified good fit for the proposed structure of the scale. The five factors’ items with their loadings together with Cronbach’s alpha for each factor appear in the Table 3 of Appendix. Loadings for fear of failure scale ranged from 0.399 to 0.652 (Fig. 2), while for self-efficacy beliefs, they ranged from 0.462 to 0.728. Loadings for goals ranged from 0.452 to 0.829 for mastery goals, from 0.595 to 0.831 for performance-approach goals, and from 0.664 to 0.862 for interest.

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TABLE 1 Mean scores for each factor Factor

Mean

SD

Fear of failure Self-efficacy beliefs Mastery goals Performance-approach goals Interest

2.43 4.03 4.61 3.02 3.84

0.80 0.69 0.56 1.08 0.97

Table 1 presents mean scores and standard deviations for all variables referred to students’ affective constructs. At first glance, the specific sample of students seems to have a positive view of mathematics. Specifically, they had high self-efficacy beliefs, mastery goals, and interest while they indicated moderate fear of failure. Students in this age seem to get involved in mathematics more for mastery reasons and lesser for the demonstration of their mathematical ability in relation to other students. Correlations The correlations among the SEM latent variables as they emerged from the confirmatory factor analysis are shown in Table 2. As it was expected, mastery and performance-approach goals were found to correlate significantly, but this correlation was low (0.105) due to the different orientation that each of them represent (mastery focus and demonstration of ability). The study once again confirms the positive processes

TABLE 2 Correlations between factors Correlations Factors

1

2

3

4

5

1. 2. 3. 4. 5.

– 0.105 −0.420 0.627 0.595

– 0.162 0.347 0.192

– −0.615 −0.459

– 0.698

–

Mastery goals Performance-approach goals Fear of failure Self-efficacy beliefs Interest

All correlations are significant at the 0.05 level


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associated with mastery goals. The correlation of mastery goals with fear of failure was found negative and statistically significant (−0.420), whereas mastery goals were highly correlated with students’ interest for mathematics (0.595) and with students’ self-efficacy beliefs (0.627). Table 2 also shows that performance-approach goals were significantly and positively correlated both with fear of failure (0.162) and with selfefficacy beliefs (0.347). Students’ interest in mathematics was highly correlated with their self-efficacy beliefs (0.698) and also negatively correlated with fear of failure (−0.459). The Structural Equation Model To investigate possible relations between different students’ affective constructs, motivation, and their mathematical behavior, we applied path analysis (SEM). In the first level, we examined the relations between students’ self-efficacy and fear of failure; in the second level, between students’ adoption of achievement goals; and in the third level, we examined consequences of achievement goals on students’ performance and interest in mathematics (see Fig. 1). As presented by the iterative summary of the initial analysis, the data did not fit the model well (x2 = 99.45, df = 6, CFI = 0.738, and RMSEA = 0.223). Subsequent model tests suggested that the model fit indices could be improved by (1) adding a direct path between fear of failure and students’ mathematical performance and (2) adding a direct path between students’ self-efficacy beliefs and students’ interest. Including these modifications, the data had very good fit to the model (x2 = 8.219, df = 5, CFI = 0.991, and RMSEA = 0.045). The model has been cross-validated using two subsamples of the original sample. Cases in each subsample have been randomly selected (N1 = 160 and N2 = 161). The corresponding fit indices were satisfactory for both models: model 1 (x2 = 8.689, df = 5, CFI = 0.980, RMSEA = 0.068) and model 2 (x2 = 5.820, df = 5, CFI = 0.995, RMSEA = 0.033). Figure 3 depicts the model including standardized regression coefficients. The path coefficients in Fig. 3 suggest that students with high fear of failure had low mastery goals (β = −0.158) and the more fear of failure students stated the more their performance-approach goals were (β = 0.287). Students’ fear of failure was also found to directly and negatively influence their mathematical performance (β = −0.340). Regarding students’ self-efficacy beliefs, students who believed that they could do well in mathematics had also high mastery goals (β = 0.367) and performance-approach goals (β = 0.377). Thus, the results

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MARILENA PANTZIARA AND GEORGE N. PHILIPPOU .808

.890

Fear of failure

-.420

-.158

Self-efficacy beliefs

.018 .421

.287

.367

.377

Interest for mathematics

.263

Mastery goals

-.340

.116 Mathematical performance

Performance approach goals

.931

.920

Figure 3. Final model parameter estimates

validated the ambiguity of the roots of performance goals based on both negative and positive causes (fear of failure and self-efficacy, respectively). Students’ self-efficacy beliefs were also found to indirectly influence their performance in mathematics through their achievement goals and directly influence their interest in mathematics (β = 0.421). The model shows that students who stated high mastery goals also had high mathematical performance (β = 0.116). Students’ performanceapproach goals did not influence their performance in mathematics. This may be due to the fact that students do not receive any grades in the primary school, something that may lead students to focus on their mastery and personal development rather than on the demonstration of their performance in comparison to fellow students. The model suggests that mastery goals (β = 0.263) and performance-approach goals (β = 0.018) mediate the effect of self-efficacy beliefs and fear of failure on students’ interest in mathematics. DISCUSSION The outcomes of the present study tend to confirm some of the earlier findings, contradict others, and somehow complement existing knowledge about students’ motives to learn mathematics. In this section, we summarize our findings in view of prior research, indicate possible implications for teachers, and propose considerations for future research. Regarding the first aim of the study, our analysis confirmed the validity of five out of the six constructs in the specific context. This is in line with findings in other studies (Midgley et al., 1998; Friedel et al., 2007; Zusho et al., 2005). However, contrary to other studies


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(Cury et al., 2006; Elliot & Church, 1997; Zusho et al., 2005), performance-avoidance goals failed to be confirmed. Students’ inability to distinguish among the performance-approach and the performance-avoidance goals may be due to their age (elementary grades); in most of the other studies (Zusho et al., 2005; Cury et al., 2006), participants came from middle and upper grades, or university, while some studies referring to elementary students measured only mastery and performance-approach goals (Anderman & Midgley, 1997; Friedel et al., 2007). Since Midgley et al. (1998) validated the existence of the three types of achievement goals also in elementary level, this result may highlight the impact of the different social context on students’ motivation, as other studies have demonstrated (Zhu & Leung, 2010). In general, the results of the present study indicate a positive outlook of these students’ motivation. On the average, students reported low levels of fear of failure and endorsed mastery goals rather than performanceapproach goals; this is consistent with previous studies referring to similar age students (Anderman & Midgley, 1997; Di Martino & Zan, 2010; Friedel et al., 2007). They also reported high levels of self-efficacy beliefs and interest in mathematics. One possible reason for these outcomes may be the absence of grading in the Cyprus educational system before grade 7. These findings give a more positive picture about students’ affect than studies referring to upper grades students (e.g. Zusho et al., 2005). Numerous studies have documented a decline in students’ affect in mathematics as students get older (Anderman & Midgley, 1997; Di Martino & Zan, 2010). Regarding the second research target, the relationship between students’ affective constructs (fear of failure, self-efficacy, achievement goals and interest), the results of the study revealed the positive effect of the adoption of mastery goals. Specifically, a significantly positive correlation was found between mastery goals and interest, in line with the results of other studies (Cury et al., 2006; Zusho et al., 2005). The study showed that mastery goals highly correlated with self-efficacy beliefs, similar to the results of previous studies (Friedel et al., 2007; Bråten et al., 2004; Wolters & Rosenthal, 2000). In addition, mastery goals were found to be negatively correlated to fear of failure. Other studies (Elliot & Church, 1997; Zusho et al., 2005) stated null correlation. The results regarding performance-approach goals were somehow inconsistent; that may be due to the absence of the approachavoidance distinction (Kaplan et al., 2002). Specifically, the perfor-

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mance-approach goals were found to be positively correlated with the fear of failure, another finding congruent with earlier findings (Elliot & Church, 1997; Zusho et al., 2005). Performance-approach goals were also positively correlated with self-efficacy beliefs. Some studies found similar results (Liem et al., 2008) while others found no association between these two affective constructs (Friedel et al., 2007; Kaplan & Maehr, 1999). Performance-approach goals were also positively correlated with students’ interest, in contrast with the results of other studies (Cury et al., 2006; Zusho et al., 2005) in which no correlation was detected. The positive correlation between performance-approach goals and interest in this study may be due to the correlation detected between performance-approach goals and selfefficacy beliefs. Elliot (1999) states that individuals with high competence perceptions are expected to orient toward success and adopt performance-approach goals; thus, the root (self-efficacy beliefs) that orients towards performance-approach goals is positive leading to adaptive behavior. The present study detected a low correlation between mastery and performance-approach goals, similar to other studies (Anderman & Midgley, 1997; Midgley et al., 1998). This should be expected since these two orientations are associated with different purposes and beliefs about intelligence and ability. The third aim of the study was to propose and empirically validate a model presenting relations between students’ affective constructs, motivation, and their mathematical performance in elementary grades while other studies confirmed models based on achievement goal theory with middle school and college students (Bandalos et al., 2003; Elliot & Church, 1997; Zusho et al., 2005). The presence of this model does not disprove alternative models with similar affective constructs in elementary grades. The model in the present study restricted in two goals, mastery goals and performance-approach goals, since the performanceavoidance goals failed to be confirmed in this specific social context. Based on this model, the results of the present study shed some light on students’ affective constructs when they are engaged in the learning of mathematics. Specifically, the results revealed the negative effect that fear of failure has in the mathematics learning. Students’ fear of failure was found to negatively influence the adoption of mastery goals, positively influence the adoption of performance-approach goals, and also directly and negatively influence their mathematics performance. The effect of fear of failure on mastery and performance-approach goals supports the results


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of other studies involving college students (Elliot & Church, 1997; Zusho et al., 2005). Repetitive failure in mathematics or the lack of understanding has been identified as some of the causes that develop students’ fear of failure in mathematics (Di Martino & Zan, 2010). Teachers of mathematics should be aware of the causes and the negative consequences of students’ fear of failure in order to explain, predict, and change students’ behavior in the classroom. This study demonstrated that self-efficacy beliefs had direct positive effect on mastery and performance-approach goals in line with the findings by Liem et al. (2008). In reverse, other studies (Bandalos et al., 2003; Friedel et al., 2007) found that mastery and performance-approach goals had direct effect on self-efficacy beliefs. This bidirectional relationship may be due to the notion of reciprocal determinism (Bandura, 1997) which states that a person’s behavior both influences and it is influenced by various factors. Furthermore, self-efficacy beliefs directly influenced students’ interest in mathematics. We suggest that teachers should give students the opportunity to experience success in mathematical tasks developing their confidence in a logical sense. The results of this study indicate the positive consequences of the adoption of mastery goals stated in the present model as well as in other studies (Elliot & Church, 1997; Liem et al., 2008; Zusho et al., 2005). Specifically, it is found that students’ mastery goals influence their interest and their performance in mathematics while other studies (Cury et al., 2006; Elliot & Church, 1997) found that performance-approach goals and not mastery goals influence graded performance. This difference may be due to two causes. First, the sample of the present study consisted of elementary students who are rarely exposed to graded assessment. As Elliot et al. (2005) state, students in the middle grades through college are exposed to graded assessment that orients them to external evaluative considerations, something that guides them to the adoption of performance-approach goals. A second reason is the fact that we measured students’ performance through a test specifically designed to assess conceptual understanding of the fraction concept (Sfard, 1991); that has probably introduced a bias in favor of mastery goals and consequently the prediction of mathematics performance from mastery goals. Elliot et al. (2005) state the advantage of performance-approach goals over mastery goals is most likely to be observed among other criteria when students’ performance is evaluated using a normative standard and when superficial processing is required for the completion of the

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test. In this regard, we underline the importance of teaching and assessing for conceptual understanding. The results of the present study support not only the value of motivation in students’ mathematics learning, but also its multifaceted nature (Hannula, 2006; Op’t Eynde et al., 2006; Zhu & Leung, 2010). Our conclusions deal with the positive effect of self-efficacy beliefs and mastery goals on students’ achievement and interest. Teachers of mathematics have a key role in the development and change of students’ motivation. Since research demonstrates that students’ affect is unstable in school years (Di Martino & Zan, 2010; Op’t Eynde et al., 2006), the adoption of teaching practices for the development of positive affect in mathematics becomes vital. Achievement goal researchers state that if the achievement setting is strong enough, it alone can set up situation-specific concerns that lead to goal preferences for the individual, either in the absence of a priori tendencies or by overwhelming such tendencies (Elliot, 1999). However, we argue that teachers’ role in the mathematics classroom is not straightforward as far as it concerns goals. Initially, the question of “what kind of goals do teachers want their students to hold?” must be answered. Accumulated research has consistently shown the positive effects of mastery goals, although existing evidence is not sufficient to advice teachers to discourage students from adopting performanceapproach goals, especially in specific contexts and for students in upper grades. Similar to Zhu & Leung’s (2010) argument regarding intrinsic and extrinsic motivation, mastery and performance-approach goals may not necessarily be “polar opposite” but under some conditions or in combination with other affective constructs to have various effects on students’ behavior and achievement. Even thought, the results of numerous studies (e.g. Elliot et al., 2005; Kaplan & Maehr, 2007) suggest that teachers should work for the development of students’ mastery goals due to their overall importance and value on students’ learning and behavior; more research is needed to clarify the affects of the combination of these two affective constructs under different social conditions and with different age students. Finally, there are some limitations to this study. First, the data were gathered once and therefore no inference of bidirectionality can be investigated. It will be interesting to investigate this model through a longitudinal study, selecting data from a group of students in successive measurements over a time period. Secondly, in this study, we investigated students’ performance in one mathematical topic, this of fractions. Accordingly, we acknowledge that motivational patterns of these students may differ from a sample that its mathematical performance is measured


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through a more inclusive in terms of topics test. Thirdly, we are aware that students may not have only one achievement goal but may have diverse levels of different achievement goals. Consequently, it will be important to investigate this model to other samples gathering also qualitative data. In addition, an interesting avenue for further research involves the observation of teachers’ practices in the classroom in which students are characterized by certain achievement goals and high mathematical performance. Teachers of mathematics must be alert to the apparent importance of students’ affective domain and to their practices in teaching so as students enjoy learning mathematics, develop their performance, and continue their educational career studying mathematics. APPENDIX

TABLE 3 Factor loadings of the five factors against the statements associated with students’ affective constructs Items F1: Interest (Cronbach’ s alpha=0.844) Ι.1.6. I think mathematics lessons are interesting. Ι.2.12. I am enjoying mathematics lessons very much. Ι.4.24. I think mathematics lessons are fan. Ι.5.29, Ι.1.6, and Ι.7.33. I think mathematics lessons are boring. I’m glad I have mathematics lessons. I don’t like mathematics lessons. F2: Performance-approach goals (Cronbach’ s alpha=0.810) P.1.2. It’s important to me that other students in my class think I am good at my mathematics work. P.2.8. One of my goals is to show others that I’m good at my mathematics work. P.3.14. One of my goals is to show others that mathematics work is easy for me. P.4.20 and P.5.26. One of my goals is to look smart in comparison to the other students in my mathematics class. It’s important to me that I look smart compared to others in my mathematics class. F3: Self-efficacy beliefs (Cronbach’ s alpha=0.655) S.1.4. I’m certain I can master the mathematics skills taught in class this year. S.2.10. I’m certain I can figure out how to do the most difficult mathematics work. S.3.16. I can do almost all the mathematics work in class if I don’t give up. S.4.22, S.5.27. Even if the mathematics work is hard, I can learn it. I can do even the hardest mathematics work in this class if I try.

Loadings

0.715 0.827 0.664 0.862

0.595 0.751 0.831 0.700

0.621 0.545 0.462 0.728

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F4: Mastery goals (Cronbach’ s alpha=0.683) Μ.2.7. One of my goals in class is to learn as much mathematics as I can. Μ.3.13. It’s important to me that I learn a lot of new mathematics concepts this year. Μ.4.19. It’s important to me that I thoroughly understand my mathematics work. Μ.5.25. It’s important to me that I improve my mathematics skills this year. F5: Fear of failure (Cronbach’ s alpha=0.725) F.1.5, F.3.17. When I start doing poorly on a mathematics task, I feel like giving up. When I fail at a mathematics task, I am even more certain that I lack the ability to perform the task. F.2.11. If given a choice, I have a tendency to select a relatively easy mathematics task rather than risk failure. F.6.30. Sometimes I think it is better not to have tried at all, than to have tried and failed. F.7.32. When I am tackling a challenging task, I find that I am reminded of my previous failures. F.8.34. I often avoid a task because I am afraid that I will make mistakes. F.4.23 and F.9.35. I often find that I am well prepared for success on a task, but I do not perform the task well under pressure. I find that I can learn to perform a task very well, but I crack under the pressure of the situation and often do not perform anywhere close to my potential.

0.452 0.829 0.488 0.573 0.652

0.510 0.619 0.613 0.650 0.399

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Zusho, A., Pintrich, P. & Cortina, K. (2005). Motives, goals and adaptive patterns of performance in Asian American and Anglo American students. Learning and Individual Differences, 1(2), 141–158. Marilena Pantziara Cyprus Pedagogical Institute 40 Macedonia Avenue, 2238 Latsia, P.O. Box 12720, 2252 Nicosia, Cyprus E-mail: [email protected] George N. Philippou University of Nicosia Nicosia, Cyprus E-mail: [email protected]