Motivational Profiles of High Achievers in Mathematics: Relations with Metacognitive Processes and Achievement Emotions

Abstract

The current study aimed to explore alternative motivational profiles of high achievers in Mathematics, within the framework of the Situated Expectancy-Value Theory. Furthermore, it aimed to examine the profiles’ potential differences in relation to self-reported metacognitive processes, such as metacognitive awareness and experiences, and achievement emotions related to Mathematics. A comprehensive evaluation in Mathematics was conducted on a total of 492 ninth-graders, including students from regular junior high schools, experimental junior high schools, and an academically advanced summer program. The assessment involved a battery of school-type mathematical tasks, resulting in the identification of 141 high achievers. Cluster analysis, based on students’ expectancies for success, subjective value, and perceived cost in relation to Mathematics, revealed five motivational profiles labeled as follows: Cluster 1: Higher Motivation; Cluster 2: Higher Expectancies, Value, and Cost; Cluster 3: Lower Expectancies; Cluster 4: Lower Value; Cluster 5: Lower Motivation. Differences were found among the five profiles in terms of students’ reported metacognitive awareness and their emotions of enjoyment, pride, anxiety, shame, and boredom toward Mathematics. Students with the Higher Motivation profile appeared to be the most adaptive across all of the examined variables, while students with the Lower Motivation profile reported less favorable levels of motivational and affective variables than most others. However, high achievers did not differ significantly regarding their metacognitive accuracy. Examination of the gender distribution within the clusters did not reveal any differences in gender representation.

1. Introduction

Τhe development of students’ mathematical competence is among the core aims of school curriculums. Traditionally, enhanced cognitive skills had been considered as the underlying factor that ensured high Mathematics achievement [1,2]; however, it soon became clear that such skills were not sufficient [3]. Educational research has gradually revealed a wide range of factors relative to high achievement in Mathematics, highlighting, among others, the significant role of motivational beliefs, metacognitive processes, and, more recently, achievement emotions [4,5,6,7,8].

Moreover, during the last years, STEM education has become the focus area of many education systems worldwide [9], triggering an increasing interest in Mathematics as its foundational component. This has facilitated the advancement in Mathematics research, leading to several applications in educational contexts that would enhance students’ achievement and STEM choices [10,11,12,13]. However, there is still a need for further investigation toward a better understanding of high achievement. For example, typical studies in the area usually examine independently the various factors relating to achievement in Mathematics, while holistic approaches are limited. Also, continuous developments in educational research, especially in the fields of achievement motivation and emotions [14,15,16], require multiple research approaches and further enrichment of the relevant results.

At the same time, a significant part of research related to Mathematics achievement has focused on low achievers and ways to enhance their performance. However, studies on students who excel in Mathematics are less frequent; these students are often considered to require minimal support in academic settings due to the widespread notion that their abilities suffice for overcoming potential academic challenges [17]. Moreover, high achievers are usually treated in schools as a homogenous group [18], and their unique characteristics and needs are often overlooked [19]. Person-centered methodologies [20,21,22] could highlight the diversity of these students, but studies utilizing them for this purpose are still generally lacking.

Given these considerations, this study aimed to explore the motivational profiles of high achievers in Mathematics via a person-centered approach and investigate their potential differences in relation to metacognitive processes and achievement emotions.

1.1. High Achievers in Mathematics: Motivational Beliefs, Metacognitive Processes, and Achievement Emotions

1.1.1. Motivational Beliefs

Researchers have been for long investigating the factors that are associated with Mathematics performance, including motivational beliefs. These beliefs are closely related to performance on tasks with specific criteria of completion, such as mathematical exercises. Overall, research is constantly highlighting motivational beliefs as strong predictors of Mathematics performance [23,24,25,26].

Among the several theoretical frameworks describing achievement motivation, Expectancy-Value Theory (EVT) has evolved over the years to take into account both ability beliefs and perceived task value, examined within a specific context (thus being renamed as “Situated EVT”—SEVT—after 2020) [5,27]. According to the theory, students’ expectancies for success, defined as individuals’ beliefs regarding their anticipated performance on an upcoming task, are a decisive factor for engaging with this task. Expectancies for success, even if conceptually distinct, are empirically proximal to academic self-concept and to self-efficacy, which is defined as one’s beliefs in successfully carrying out the actions needed to complete a task [28,29]. Moreover, the theory describes a task’s subjective value in relation to the task’s qualitative characteristics and the ways these characteristics drive engagement with the task. More specifically, SEVT identifies four dimensions of value: (i) intrinsic value, referring to one’s enjoyment while engaging or planning to engage with a task, which is similar to the concept of interest [5,30], and intrinsic motivation of Self-Determination Theory [31,32], (ii) attainment value, describing the importance of dealing with a task in order to preserve one’s own identity, (iii) utility value, representing the usefulness of engaging with a task as a means for the fulfillment of other goals, and (iv) perceived cost, including the effort needed to engage with a task, the time to be invested (opportunity cost), and the emotional and psychological consequences of engaging with a task (e.g., disappointment after a possible failure). In contrast with the first three dimensions of value, perceived cost refers to the negative aspects of engaging with a task, and thus, it has a negative impact on the overall achievement motivation. For this reason, cost is often studied independently from the rest dimensions of value, providing more nuanced insights regarding students’ achievement motivation [21,33,34,35,36]. Finally, the theory takes into account a series of contextual factors, which also affect motivational beliefs, such as a family’s socio-economic status and socio-emotional climate, parent’s general beliefs, societal stereotypes, etc. [5].

Situated Expectancy-Value Theory is especially prevalent among studies on Mathematics achievement; students’ perceptions of competence in a subject of particular difficulty, such as Mathematics, as well as the varying value they could attribute to this subject, are catalyzing factors for students’ achievement. When examining the motivation of high achievers in Mathematics through the lens of SEVT, consistent findings show that these students tend to have higher levels of expectancy for success, and they generally assign greater value to Mathematics in comparison to their peers [24,37,38]. For example, in a study of German students from fifth to twelfth grade, Gaspard and colleagues [38] found that students’ grades in Mathematics were positively predicting their expectancies for success. Moreover, the study found that high achievers attributed greater overall value to Mathematics than low achievers, but their lead was less eminent than the one found regarding their expectancies for success. The researchers also noticed that utility value had the weakest correlation with achievement in Mathematics compared to the other value facets.

SEVT also stresses the role of social stereotypes in the formation of students’ expectancies for success and subjective task value [5,39]. Among others, it predicts that male students are expected to hold more adaptive motivational beliefs related to domains traditionally perceived as male-dominated, such as Mathematics. Indeed, the majority of empirical findings report expectancies for success and at least some dimensions of perceived task value to be lower for female students [40,41,42,43,44]. For example, Guo and colleagues [40] studied three cohorts of eighth-grade students from Hong Kong and found a predictive direct effect of gender on motivational beliefs. More specifically, male students had higher math self-concept and intrinsic value for Mathematics than female students, even if no significant differences were found for utility value. Brown and Putwain [45] reported similar results for English adolescents, both for expectancies for success and a combined measure of all three dimensions of subjective task value. This pattern favoring boys persisted even when considering only high achievers in Mathematics [46,47]. For example, Preckel and colleagues [46] compared academically gifted and average-ability German students, finding, in both cases, significantly higher expectancies for success and perceived value of Mathematics for boys than for girls, with the academically gifted sample showing the largest gender differences.

1.1.2. Metacognitive Processes

Metacognition refers to an individual’s awareness of one’s cognitive processes and the ways they can be regulated [48,49,50]. It takes various forms, the major of which are metacognitive knowledge, metacognitive skills, and metacognitive experiences [48,49,51,52]. Metacognitive knowledge is declarative knowledge about cognitive tasks, processes, and strategies stored in one’s memory. It also comprises procedural knowledge on how to apply specific cognitive strategies, as well as conditional knowledge on when to apply such strategies. Metacognitive skills are abilities that facilitate monitoring, control, and evaluation of cognitive processes. Such skills play an important role in self-regulating learning, enabling individuals to acquire new knowledge and master new skills more strategically. Finally, metacognitive experiences refer to any kind of cognitive or affective experiences one might have (feelings, judgments, or estimates), which are also related to one’s learning [50,53,54]. For example, depending on the level of feeling of certainty, a student might opt to revise the learning material once more before the exams.

Metacognition has been found to contribute significantly to Mathematics achievement, especially in the domain of problem-solving [8]. In their review of the relationship between metacognition and Mathematics education, Schneider and Artelt [55] presented previous research suggesting a 15–20% shared variance between metacognition and Mathematics performance. Moreover, they acknowledged a significant impact of declarative metacognitive knowledge in Mathematics performance, even when controlling for cognitive abilities. In a more recent review, Desoete and de Craene [56] highlighted that teaching metacognition is necessary to foster the development and improvement of mathematical skills. In addition to metacognitive knowledge and metacognitive skills, metacognitive experiences [57] have also been studied in relation to students’ academic performance. For example, Tay and colleagues [52] examined 14-year-old students and found that their feeling of difficulty while solving mathematical tasks was a stronger predictor of their performance than their metacognitive knowledge or skills.

Research on mathematically talented children has identified advanced metacognitive skills already from the early years of primary school [58]. During adolescence, high achievers in Mathematics continue to show enhanced metacognitive skills compared to their peers; they make better use of cognitive strategies than relying on trial and error, or they transfer appropriate strategies more easily among similar problems [59,60]. Various studies have also addressed the metacognitive experiences of high-ability students [61]. For example, high academic achievers are often found to have weakened feelings of certainty for their performance in comparison to low achievers in general [62,63,64], with the exception of mathematical tasks, where overconfidence is usually the norm for all achievement levels [65,66,67]. However, high achievers in Mathematics seem to be better calibrated than low achievers, estimating their performance with greater accuracy (i.e., the extent of deviation of feeling of certainty from actual performance) [65,66,67,68].

1.1.3. Achievement Emotions

The relationship between affect and achievement in Mathematics has been traditionally studied mostly for Mathematics anxiety [69,70,71,72,73]. However, during the last years, the research focus has taken a shift toward a more detailed investigation of other discrete emotions arising in achievement settings, such as enjoyment, shame, hope, boredom, etc. [74,75,76]. This change has also been facilitated by theoretical advances in the field, with Pekrun’s Control-Value Theory—CVT—[77,78] emerging as one of the most established theoretical frameworks describing emotions related to achievement activities or their results. According to CVT, students’ beliefs regarding their control over an achievement activity (e.g., successfully passing the academic year final exams), combined with the value they pose on the activity per se, are decisive factors both for the type and the intensity of the arising emotion in relation to this activity. Apart from the typical distinction between positive and negative, CVT distinguishes emotions based on the degree of activation they elicit. Emotions such as pleasure, hope, gratitude, anger, anxiety, and shame are capable of driving a person to take action in relation to an activity. On the other hand, emotions like relief, satisfaction, disappointment, despair, and boredom reduce an individual’s activation, for example, by inhibiting a student’s engagement with a specific mathematical task.

Lots of studies have investigated the interaction between achievement emotions and academic achievement, including performance in school Mathematics. In their review, Goetz and Hall [79] reported statistically significant, even if moderate, negative correlations between anxiety and student academic achievement, ranging in average between −0.20 and −0.25. These outcomes were based on three meta-analyses, which included studies on Mathematics performance [70,80,81]. When other discrete emotions like enjoyment, pride, anxiety, anger, boredom, etc., were considered, their mean correlations to Mathematics performance were also found to be close to |r| = 0.25 [79]. Overall, the results are consistent regarding the emotion’s valence and its relation to achievement: positive activating emotions (e.g., enjoyment, pride, and hope) are positively related to achievement, while negative deactivating emotions (e.g., disappointment and boredom) are negatively related [78].

Most studies indicate, on average, more positive and less negative emotions for high achievers. For example, Roos et al. [82] found lower levels of both state and trait anxiety in relation to Mathematics for ninth- and tenth-grade high achievers, compared to their low-achieving peers. In the study of van der Beek and colleagues [83], including ninth-grade Dutch students, the researchers found that high achievers had enhanced self-concept, higher levels of enjoyment, and lower levels of anxiety in comparison to their peers. Also, Goetz and colleagues [84] investigated the emotional experiences of German students in early adolescence before, during, and after taking a Mathematics test, considering their cognitive abilities. They found increased enjoyment for students at the upper quartile of abstract reasoning abilities, who also scored higher on the math test than the rest of the students. Moreover, high achievers reported lower levels of anxiety and anger than their low-achieving peers. However, when the emotion of boredom was considered, student reports were similar, regardless of achievement level.

Boredom usually stands out from the rest of the achievement emotions, with studies demonstrating contradictory results. Some findings show that high achievers exhibit lower levels of boredom compared to low achievers [85], while others find it similar [84,86] or even higher [87]. Trying to investigate this discrepancy, Preckel and colleagues [88] evaluated the reasons behind the manifestation of boredom by distinguishing between boredom arising in environments with low academic challenges and boredom arising in environments with excessively high academic challenges. For gifted students who transferred to a more academically demanding environment, the researchers found that boredom arising from low academic demands decreased during the transition, while boredom arising from higher academic demands actually increased. As a result, the overall reported level of boredom did not differ significantly from the boredom experienced by students in regular classes with typical achievement levels.

The relationship between achievement emotions and achievement motivation has also been investigated. Increased motivation is often accompanied by more positive activating and fewer negative deactivating emotions, as predicted by theory [77,78] and demonstrated in a series of empirical studies. In one of them, Peixoto and colleagues [89] examined a series of achievement emotions of Portuguese adolescents during class and during a test in Mathematics. In both cases, the correlations of achievement emotions with academic self-concept and perceived value of Mathematics were found to be positive for positive emotions and negative for the negative ones (except for the emotion of relief). In another study, Goetz and colleagues [90] investigated the academic self-concept of German adolescents and its relation to enjoyment, pride, anxiety, anger, and boredom. The researchers found significant correlations with all types of emotions, which were stronger for Mathematics than for the English or German Language, implying the importance of considering emotions when studying Mathematics achievement.

1.1.4. Studying High Achievers with Person-Centered Approaches

Person-centered approaches usually provide a more fine-grained depiction of student profiles, since they do not examine variables in isolation like traditional methods but seek to classify individuals into meaningful groups [20,22,91]. For example, Andersen and Cross [92] used latent class analysis to investigate the motivational profiles of a large representative sample of ninth-graders from the US, evaluating their academic self-concept, their interest in Mathematics, their perceived importance of the subject, as well as its utility value. Their analysis revealed four distinct motivational profiles, with 15% of the high achievers belonging to the low motivation cluster. In another study, Conley [91], using EVT theory, found seven motivational student profiles, two of which were characterized by high expectancies for success, high value of Mathematics, and also high cost, differing only in terms of their members’ achievement goals. Similarly, Watt and colleagues [21] used the EVT framework to examine the motivational profiles of Australian adolescents. The researchers identified three motivational profiles, including a low motivation cluster comprising one out of five high achievers. Moreover, it was found that high perceived cost was associated with weakened psychological and emotional wellbeing, especially for students with high expectancies for success and high perceived value of Mathematics. Gonida and colleagues [20], using latent class analysis in the framework of achievement goal theory, found a similar percentage (17%) of high achievers with all goal orientations low (both performance and mastery goals, approach and avoidance), characterized also by maladaptive help-seeking beliefs and low intention to seek instrumental academic help.

These studies suggest that person-centered approaches may provide richer insights regarding high achievers’ motivation, as they shift the focus from the average high achiever to alternative profiles. Moreover, being aware of the diverse student profiles could also inform instructional practice. In a recent review of studies on differentiated instruction [93], among the main conclusions was high achievers’ positive perception of instruction that was tailored to their specific needs and interests. However, most teachers do not differentiate their instruction [93], also due to beliefs that “gifted children are already good, small in number and do not need a different education” [94]. Further research highlighting the heterogeneity of high achievers could facilitate raising awareness of their specific but usually overlooked educational needs (see also [20]).

1.2. The Present Study

The present study focused on Mathematics achievement during adolescence and adopted a person-centered approach to examine high achievers’ motivational beliefs, using the framework of the Situated Expectancy-Value Theory. The identified profiles would be associated with metacognitive processes (metacognitive awareness, perceived task difficulty and certainty for the solution provided to each task, metacognitive accuracy) and achievement emotions. Moreover, as gender effects have been discussed in SEVT and supported by prior research in relation to Mathematics achievement and motivational beliefs, an additional aim of the present study was to examine the composition of the resulting motivational profiles in terms of gender. Thus, the main research aims of the study were: (i) the identification of different motivational profiles of high achievers in Mathematics based on their expectancies of success, the value they assigned to this school subject, and their perceived cost, (ii) the investigation of how the different motivational profiles are associated with experienced emotions and metacognitive processes, and (iii) the examination of gender distribution within the resulting profiles.

In accordance with these aims and based on prior empirical evidence, the hypotheses of the study were the following:

Hypothesis 1.

Students with adaptive motivational profiles (i.e., high motivational beliefs such as high expectancies of success, high assigned value on Mathematics, and low cost) would be more likely to report higher metacognitive awareness, have better metacognitive accuracy, and report more positive and less negative achievement emotions [77,89,90];

Hypothesis 2.

Clusters of high achievers with a maladaptive motivational profile (i.e., poor motivational beliefs such as low expectancies of success, low assigned value on Mathematics, and high cost) were expected to emerge [20,21,92];

Hypothesis 3.

Male students were expected to have more adaptive motivational profiles in relation to Mathematics [46,47,95].

2. Materials and Methods

2.1. Participants

The initial study sample included 492 (56% female) 9th-graders from (i) 10 regular junior high schools (297 students), (ii) 3 experimental junior high schools for academically advanced students (138 students), and (iii) a summer program for academically talented adolescents (57 students). The final sample of the study consisted of 141 high achievers, who were identified from the initial sample via an ad hoc procedure, described in Section 3.

2.2. Measures

2.2.1. Achievement in Mathematics

Participants’ Mathematics achievement was assessed with a battery of school-type mathematical tasks. These tasks were based on the 9th grade’s curriculum and were carefully selected after an initial iterative evaluating procedure, including experienced professional teachers and students. The selected tasks were further tested in a pilot study with 114 ninth-graders, attending a public regular junior high school in an urban setting. The pilot results showed that the distribution of Mathematics performance was skewed to the right, indicating a floor effect. Thus, it was necessary to decrease the overall difficulty of the tasks to ensure better alignment to the students’ academic level. The professional teachers agreed on the use of the shortlisted seven mathematical tasks, which were of gradual difficulty and included linear and quadratic equations, factorization of algebraic expressions, as well as root calculation.

2.2.2. Motivational Beliefs

Participants’ motivational beliefs for school Mathematics were assessed via the Expectancy-Value-Cost scale of Kosovich and colleagues [34]. The scale consists of 10 items measuring 3 motivational variables: (i) students’ expectancies for successfully fulfilling the requirements of their Mathematics class (3 items, e.g., “I am confident that I can understand the material in my math class”), (ii) the aggregated value that students attribute to the subject of Mathematics (intrinsic, attainment and utility value, 3 items, e.g., “I think my math class is useful”), and (iii) the perceived cost of working on school Mathematics (4 items, e.g., “My math classwork requires too much time”). Students provided each of their answers in a 5-point Likert scale, ranging from 1 (absolute disagreement) to 5 (absolute agreement).

2.2.3. Metacognitive Processes

The metacognitive processes examined were students’ metacognitive awareness and metacognitive experiences. Specifically, participants were asked to complete the jr Metacognitive Awareness Inventory—jr MAI [96], which has two subscales: (i) knowledge of cognition (9 items, e.g., “I know when I understand something”) and (ii) regulation of cognition (9 items, e.g., “I occasionally check to make sure I’ll get my work done on time”). Students responded with a 5-point Likert scale, ranging from 1 (absolute disagreement) to 5 (absolute agreement). In the end, the two subscales were combined into a single measure of metacognitive awareness.

Perceived task difficulty and students’ certainty for their provided solution were also measured with Likert scales. Participants assessed each mathematical task in a range from “not difficult at all” (1) to “very difficult” (5) and their level of certainty from “not certain at all” (1) to “very certain” (5). Students’ reported certainty was also used to calculate their performance calibration as a measure of students’ metacognitive accuracy on estimating their own performance, described in the Section 3.

2.2.4. Achievement Emotions

Five achievement emotions in relation to Mathematics were measured with the Achievement Emotions Questionnaire—AEQ [97], an instrument which was developed based on Pekrun’s Control-Value Theory [77,78]. The classroom-related part of AEQ was designed to evaluate emotions experienced before, during, and after a typical school lesson. Specifically, 53 items were used to measure 2 positive and 3 negative emotions, namely enjoyment (10 items, e.g., “I enjoy being in class”), pride (9 items, e.g., “I am proud of the contributions I have made in class”), anxiety (12 items, e.g., “I worry the others will understand more than me”), shame (11 items, e.g., “I’d rather not tell anyone when I don’t understand something in class”), and boredom (11 items, e.g., “I get so bored I have problems staying alert”). Students’ responses were recorded in a 5-point Likert scale, with (1) indicating absolute disagreement and (5) absolute agreement.

2.3. Procedure

For the implementation of the study, special permission was requested and consequently granted from the National Institute of Educational Policy. Students participated voluntarily after being informed about the aims of the study and the right to withdraw at any part of the procedure, and they were also ensured for the anonymity of the data they would provide [98]. Parental consent was also obtained via signed forms with details of the study. Students were initially assessed in Mathematics during a typical school hour (45 min). During another school hour, they completed the Likert-type questionnaires for the evaluation of the motivational, metacognitive, and affective variables under examination.

3. Results

3.1. Psychometric Properties of the Scales

The validity of each scale was tested through Principal Component Analysis (PCA) with varimax rotation using SPSS version 23. The number of principal components for each scale was identified using Cattell’s criterion (scree plot) [99]. The PCA revealed three components for the Expectancy-Value-Cost scale, with all items loading to the corresponding component, as expected. The analysis for the jr Metacognitive Awareness Inventory revealed two components, approximating the two subscales of knowledge and regulation of cognition. However, following the suggestion of the jr MAI authors [96], the two scales were combined into a single measure of metacognitive awareness, which would suffice for the aims of the study. The PCA for the Achievement Emotions Questionnaire was applied to each one of the five emotion subscales, and the scree plot indicated one component in each case. These results confirmed the theoretical structure of all used scales.

The reliability of each scale was assessed with Cronbach alpha, and the results were equally satisfying: (i) a_exp = 0.86, a_val = 0.84, a_cost = 0.74 for the EVC subscales, (ii) a_met = 0.84 for jr MAI, and (iii) a_enj = 0.92, a_pride = 0.85, a_bor = 0.95, a_shame = 0.92, a_anx = 0.90 for the AEQ subscales. Moreover, the examination of the correlations between each item and the total score of the corresponding sub-scale revealed only three items from the jr MAI scale with a correlation under the threshold of 0.30 [100]; however, their potential exemption from the subsequent analyses would not impact the overall reliability of the scale, since further analyses showed that the corresponding Cronbach alpha would remain at similar levels.

3.2. Identification of High Achievers

In order to enhance the accuracy of students’ Mathematics performance assessment, their grading on the mathematical tasks was Rasch-analysed [101,102] with jMetrik software, version 4.1.1 [103], and thus their aggregated grades were transformed to better estimators of Mathematics performance. Based on these performance indicators, students were ranked accordingly in order to facilitate the identification of high achievers in the sample. The participants were classified into four distinct performance categories with the JASP software, version 0.14 [104]. The criterion used was one standard deviation above and below the mean performance of students attending regular junior high schools (N = 297), who better represented the general population among study participants (Figure 1). High achievement was defined as performance at least one standard deviation above the mean, resulting in approximately 10% of this sub-sample being considered as high achievers (

Table 1. Distribution of Students Attending Regular Junior High Schools in Achievement Categories.

Category	Achievement	Ν	%
1	High	31	10.4
2	Average to High	104	35.0
3	Average to Low	124	41.8
4	Low	38	12.8
Total		297	100.0

). Subsequently, the high achievement threshold was used to identify high achievers in the total sample (N = 497), resulting in 141 high-achieving students. This sub-sample comprised 31 students from the regular junior high schools, 80 students from the experimental junior high schools, and 30 students from the academic summer program. The subsequent analyses were based on this sub-sample, representing high achievers in Mathematics.

3.3. Performance Calibration

Preliminary analyses also included the calculation of participants’ metacognitive accuracy, which was one of the metacognitive measures included in the study.

The accuracy of students’ estimation of their Mathematics performance was assessed with the Absolute Accuracy Index [105]. Students’ reported certainty for their provided solution in each task (c_i), along with students’ actual performance at the task (p_i), were used for the calculation of the index according to the formula:

$$\frac{1}{{\rm N}}\sum _{i=1}^{{\rm N}}{\left(c_i-p_i\right)}^2,$$

where N is the number of Mathematics tasks (N = 7).

3.4. Motivational Profiles of High Achievers

The motivational profiles of the 141 high achievers in Mathematics were investigated via cluster analysis with k-means, using JASP 0.14 [104]. This method facilitated the classification of students according to their expectancies for success, the value they attributed to Mathematics, and their perceived cost of dealing with this subject. Since k-means algorithms require an a priori specification of the number of clusters to be formed, potential models with different numbers of clusters were compared with each other using the Bayesian Information Criterion (BIC). This index was minimized for the classification of students in five clusters (Figure 2). The selected model explained 66% of the variance (R²) of the motivational variables, above the 50% indicative threshold [106]. The five clusters are depicted in Figure 3, and the centroids of each one are shown in

Table 2. Cluster Sizes and Respective Means of the Three Motivational Variables.

Cluster	1	2	3	4	5
Relative Size	27%	17.7%	34%	12.8%	8.5%
Expectancy	4.95 (0.93)	4.75 (0.64)	3.85 (−0.64)	4.28 (−0.04)	3.17 (−1.63)
Value	4.52 (0.73)	4.41 (0.59)	3.97 (−0.02)	2.78 (−1.62)	3.19 (−1.05)
Cost	1.54 (−0.87)	2.62 (0.62)	2.05 (−0.17)	2.19 (0.03)	3.69 (2.08)

Note. z-scores are indicated in parentheses.

. Their corresponding z-scores represent the deviation of each cluster’s mean from the total mean (z-score = 0) of the 141 high achievers, for each one of the three motivational variables. The difference, measured in standard deviations, could be interpreted as the effect size of a student’s participation in the cluster [107]. Thus, a value close to 0.2 indicates a small effect, a value around 0.5 indicates a moderate effect, whereas a value of 0.8 or above implies a large effect, according to the definition of Cohen’s d effect size [108]. The majority of these differences for each motivational variable, presented in Table 2, indicate a moderate to large deviation from the total mean value of each variable in the high-achieving sample. Moreover, the five clusters seem to differ significantly from each other at least in one of the three variables (z-score differences at least 0.8), which further supports the internal validity of the resulting classification.

Students belonging to Cluster 1 represented 27% of high achievers. They were characterized by significantly higher expectancies for success, attributed higher value to Mathematics and perceived lower cost compared to the rest of high achievers. Students in Cluster 2, which represented 17.7% of the sample, did not differ significantly from the students in Cluster 1 in terms of expectancies for success and value attributed to Mathematics, since they also reported quite high values for both of these variables. Cluster 3 was the most populated, representing 34% of high achievers. Its members stood out due to their lower expectancies for success; however, the value and cost they reported did not deviate significantly from the corresponding mean values of the high achieving sample. Students who belonged to Cluster 4, representing 12.8% of the sample, differed from the rest only regarding their beliefs on the value of Mathematics, considering it much lower than the average high achiever. Finally, Cluster 5 included students who held significantly lower expectancies for success and value of Mathematics, and at the same time, they perceived significantly higher cost in comparison to the corresponding means of the sample, making them the least motivated group among high achievers.

With the emphasis given to the most prominent characteristics of each cluster’s members, the five clusters were labeled as follows: Cluster 1—Higher Motivation; Cluster 2—Higher Expectancies, Value, and Cost; Cluster 3—Lower Expectancies; Cluster 4—Lower Value; Cluster 5—Lower Motivation (Figure 3). It is noted that these labels are not absolute but relative to the corresponding mean of the total high-achieving sample for each one of the three motivational variables.

Next, MANOVA with JASP [104] was applied in order to test whether the five clusters significantly differed in terms of the metacognitive processes (metacognitive awareness, perceived task difficulty, perceived certainty and accuracy) and the achievement emotions of their members. Significant differences were found in relation to metacognitive processes, V_pilai’s = 0.345, F (16, 540) = 3.187, p < 0.001, partial η² = 0.086. After univariate analyses with Bonferroni correction, these differences were traced to students’ self-reported metacognitive awareness, F (4, 135)= 9.102, p < 0.001, partial η² = 0.212, and their perceived certainty for the solution provided, F (4, 135)= 4.332, p = 0.003, partial η² = 0.114, but not to their perceived task difficulty or to their metacognitive accuracy. Post hoc tests revealed similar levels of metacognitive awareness for students belonging to the Higher Motivation and the Higher Expectancies, Value, and Cost clusters (Cluster 1 and Cluster 2). The rest of the clusters (Clusters 3, 4, and 5) were found to have lower metacognitive awareness compared to the Higher Motivation cluster, but nonsignificant differences were found among them. Students in the Higher Motivation cluster also reported being more certain about their provided solutions compared to the students in the Lower Expectancies and Lower Motivation clusters. Cluster means are summarized in

Table 3. Comparison of the Means of Metacognitive and Emotional Variables for each cluster.

Cluster	(1)	Higher Motivation	(2)	Higher Expectancies, Value & Cost	(3)	Lower Expectancies	(4)	LowerValue	(5)	Lower Motivation
	Mean (z)	sd	Mean (z)	sd	Mean (z)	sd	Mean (z)	sd	Mean (z)	sd	F	ηp2
Metacognitive Awareness	3.93 (0.59) a	0.37	3.84 (0.39) a,b	0.44	3.49 (−0.40) c	0.30	3.48 (−0.43) c	0.58	3.49 (−0.43) b,c	0.42	9.102 ***	0.212
Perceived Difficulty	1.38 (−0.44)	0.36	1.63 (0.16)	0.41	1.66 (0.23)	0.45	1.57 (0.02)	0.32	1.64 (0.18)	0.48	2.852 *	0.078
Perceived Certainty	4.83 (0.54) a	0.28	4.55 (−0.06) a,b	0.43	4.48 (−0.21) b	0.57	4.51 (−0.15) a,b	0.37	4.37 (−0.45) b	0.47	4.332 **	0.114
Metacognitive Accuracy	0.34 (−0.31)	0.53	0.66 (0.09)	0.75	0.70 (0.14)	1.06	0.66 (0.08)	0.74	0.70 (0.13)	0.66	1.263	0.036
Enjoyment	4.06 (0.77) a	0.54	3.40 (−0.10) c,b	0.78	3.45 (−0.04) b	0.51	2.88 (−0.79) c	0.66	2.81 (−0.89) c	0.93	15.545 ***	0.314
Pride	4.09 (0.76) a	0.53	3.69 (0.23) a,b	0.62	3.34 (−0.23) c,b	0.46	2.98 (−0.71) c	1.01	2.85 (−0.88) c	0.83	15.834 ***	0.318
Boredom	1.73 (−0.47) c	0.89	2.27 (0.11) b,c	1.07	1.95 (−0.23) c	0.59	2.88 (0.76) a,b	0.80	3.15 (1.04) a	0.91	11.161 ***	0.247
Anxiety	1.42 (−0.73) c	0.36	2.04 (0.14) b	0.77	2.11 (0.24) b	0.59	1.80 (−0.19) b,c	0.54	2.85 (1.29) a	0.82	15.770 ***	0.317
Shame	1.51 (−0.52) c	0.56	1.96 (0.05) b,c	0.84	2.06 (0.17) b	0.71	1.74 (−0.23) b,c	0.58	2.89 (1.21) a	1.04	9.228 ***	0.213

Note. Means in the same raw with different exponent differ significantly at p < 0.05 level. * p < 0.05, ** p < 0.01, *** p < 0.001.

.

Significant differences were also found among the five clusters regarding the emotions under examination of high achievers, V_pilai’s = 0.735, F (20, 540) = 6.076, p < 0.001, partial η² = 0.184. Univariate analyses with Bonferroni correction indicated stronger differences for Enjoyment, F (4, 136) = 15.545, p < 0.001, partial η² = 0.314, Pride, F (4, 136) = 15.834, p < 0.001, partial η² = 0.318, and Anxiety, F (4, 136) = 15.770, p < 0.001, partial η² = 0.317. Smaller differences, but still with large effect sizes, were found for Boredom, F (4, 136) = 11.161, p < 0.001, partial η² = 0.247 and Shame, F (4, 136) = 9.228, p < 0.001, partial η² = 0.213. Post hoc comparisons showed that students in the Higher Motivation cluster (Cluster 1) reported significantly more positive emotions (enjoyment and pride) and less negative (boredom, anxiety, and shame) than the students of the Lower Motivation cluster (Cluster 5). Students in the Higher Expectancies, Value, and Cost cluster (Cluster 2) and Lower Expectancies cluster (Cluster 3) did not differ in terms of their achievement emotions, while some differences were found in the other combinations of cluster emotions comparison, as can be seen in Table 3.

Finally, the examination of gender distribution in each cluster did not reveal any significant deviation from the gender ratio in the total sample of high achievers, X² (4) = 1.461, p = 0.833.

4. Discussion

Based on the Situated Expectancy-Value Theory, the present study aimed to investigate alternative motivational profiles of high achievers in Mathematics. Also, it intended to explore potential differences among them in relation to metacognitive variables and a series of positive and negative achievement emotions. Using cluster analysis, five distinct motivational profiles of the identified high achievers emerged, labeled according to their deviation from the average high achiever: (1) Higher Motivation, (2) Higher Expectancies, Value, and Cost, (3) Lower Expectancies, (4) Lower Value, and (5) Lower Motivation.

The results showed that almost one out of four high achievers had the most adaptive profile of Higher Motivation (Cluster 1). Students with this profile reported higher expectancies for success, attributed greater value to Mathematics, and perceived less cost when dealing with this school subject compared to the average high achiever of the sample. Students in this cluster also showed increased metacognitive awareness and felt more certain about their solutions in the given mathematical tasks. Moreover, they were experiencing, on average, greater enjoyment and pride for their performance in Mathematics compared to their high-achieving peers while feeling less anxiety, shame, and boredom. These results confirmed Hypothesis 1 and are in line with the predictions of the Control-Value Theory, which poses that increased motivation is accompanied by more positive activating and fewer negative deactivating emotions [77,78]. The results are also in accordance with numerous empirical studies, which indicate strong correlations between motivational variables, such as achievement goals and perceived task value, and achievement emotions, especially in classroom settings [89,109,110].

Students with the Higher Expectancies, Value, and Cost profile (Cluster 2) represented almost one-fifth of the sample. They shared a lot in common with students with the Higher Motivation profile (Cluster 1) despite the increased perceived cost of dealing with Mathematics. Specifically, students of Cluster 2 reported similar levels of metacognitive processes and most achievement emotions with students from Cluster 1, with the exception of significantly lower enjoyment and increased anxiety. Probably, high performance in Mathematics for many students in this cluster was the outcome of excess effort and time invested, and thus, it came at a high cost. This finding highlights the importance of evaluating cost separately rather than combining it with the positive dimensions of value into an aggregated score. Combining the positive and the negative dimensions of this cluster’s value would cancel each other out, resulting in an overall value typical for the high-achieving students in the sample (Figure 3). Such aggregation would make it difficult to trace the reasons behind the increased anxiety that was observed in students belonging to this cluster, an emotion often associated with high perceived cost [111]. Conley [91] also found clusters of students with differences in reported affective variables when the perceived cost of Mathematics was considered independently, while they shared similar levels of overall value attributed to Mathematics. The elevated cost experienced by students in Cluster 2 might also explain their moderate enjoyment of math class; their enjoyment did not differ significantly from the enjoyment levels of students in the Lower Motivation cluster (Cluster 5), despite Cluster 2 students’ increased interest in the subject (Table 3).

The Lower Expectancies cluster (Cluster 3) was the largest one, representing one-third of high achievers. With the exception of their expectancies for success, the other two motivational variables (value and cost) were close to the corresponding means of high achievers. Their reported metacognitive processes and achievement emotions were also around the means of the high-achieving sample, making this profile the most typical for high achievers.

The remaining two clusters were the least populated but of particular interest. Students within the Lower Value cluster (Cluster 4), while not differing from the average high achiever in terms of expectancies for success and perceived cost, attributed moderate value to Mathematics on average. This finding might explain the moderate levels of enjoyment and pride of students with this specific profile. Students belonging to this cluster might have had more interest in other school subjects, and their high achievement could be merely the result of their quest for a high-grade point average in school. More importantly, this finding suggests that the value of Mathematics could be questioned also by students who excel in this school subject, a counterintuitive result given the consistently reported positive correlation between achievement in Mathematics and its perceived value [24,38,112]. However, this finding highlights the importance of using person-centered approaches when more refined insights into individual variation are sought.

Moderate value was attributed to Mathematics also by students in the Lower Motivation cluster (Cluster 5), who were additionally characterized by moderate expectancies for success and increased cost. High achievement for students with this profile seems to come along with significant effort and low to moderate overall motivation. Metacognitive awareness and perceived certainty were also below the average levels of high achievers, while their positive emotions were below, and their negative emotions were above the average high achiever’s respective levels. These results make this profile the least adaptive among the five emerged profiles of high achievers and confirmed Hypothesis 2 for a cluster of high achievers with overall low achievement motivation.

Studies which have examined the psychological profiles of high achievers have primarily focused on their academic self-concept [113,114,115] and their achievement goals [20]. However, there are studies that investigated the psychological profiles of typical adolescents, regardless of their achievement level in Mathematics, which considered a range of motivational and affective variables, and resulted in outcomes similar to this study’s. Among these common outcomes were the increased levels of positive emotions and decreased levels of negative emotions characterizing students with high motivational profiles [21,91,116]. Moreover, at least two studies [21,91] identified clusters similar to the Higher Expectancies, Value, and Cost cluster of the current study. These results highlight that high cost could co-exist with high expectancies and value for Mathematics.

The results of the study also confirmed that high-achieving students are well- calibrated regarding the estimations of their performance, a finding that has been consistently documented in a series of studies [65,66,67,68]. Moreover, no significant differences in terms of metacognitive accuracy were found among students with different motivational profiles, suggesting that accurate performance estimation is a well-developed skill of high achievers, independent of their motivational beliefs, such as the ones examined in the present study.

Moreover, it is worth noting the balanced gender ratio in each one of the five clusters, which followed the gender distribution in the initial high achieving sample. This was also the case for the cluster of Lower Expectancies, despite the anticipation of over-representation of female students (Hypothesis 3), grounded on the predictions of the Situated Expectancy-Value Theory [5,27]. SEVT emphasizes the influence of social stereotypes on students’ motivational beliefs, suggesting that, theoretically, more girls would be expected to show reduced expectancies for success in a domain often perceived as male-dominated. This result is also contradictory to the majority of empirical findings, indicating low expectancies for success among female students [40,41,42,43,44], even for the ones scoring high in Mathematics [46,47]. While this phenomenon is not common, it is in accordance with some studies that have noted the gradual closing of the gender gap in expectancies for success throughout school grades, which, during adolescence, might even become negligible [95,117,118].

5. Limitations and Future Research

The present study provided further insights into the diversity of high achievers’ motivational profiles, but it also had certain limitations which should be acknowledged. Among them is the evaluation of performance in Mathematics that was conducted using ad hoc mathematical tasks, since the focus was on high achievement within a school setting rather than using standardized procedures. Moreover, despite the carefully designed process of shortlisting these tasks to guarantee their representativeness for the grade’s academic level, the final selection did not cover the whole curriculum of the ninth grade, evaluating Mathematics performance mostly in algebra. However, the resulting performance distribution of the students attending regular junior high schools resembled a Gaussian distribution.

Another limitation concerned the use of self-report instruments to measure the psychological variables of the study. Such methods lack the accuracy levels of more direct ways of measuring, for example, an emotion the moment it arises [119]. Also, despite the thorough validation of the AEQ tool used in the study [78,120,121], it is important to consider that the emotions identified and reported by students often depend on their willingness to disclose these emotions, which could be affected by students’ social stereotypical beliefs [122]. Similarly, students do not always provide an entirely accurate depiction of their utilization of metacognitive strategies when they use self-reporting questionnaires [123]. This discussion highlights the benefits of the combined use of qualitative and quantitative approaches.

Future studies could also investigate whether the profiles found among high achievers would remain consistent throughout the whole period of secondary education or if they would vary according to students’ school grades and the classroom environment, which is likely to be different among the school years. It would be of particular interest to examine a possible fluctuation of motivational profiles, especially during the critical stage of transition from primary to secondary school, when significant changes to students’ ability beliefs are taking place [124,125,126]. Moreover, when investigating students’ motivational profiles, it is worth considering contextual variables affecting student engagement with Mathematics. Such variables could include teachers’ expectations for their students’ success, as well as teachers’ own motivational beliefs, emotions, or instructional style [5,127]. Families’ SES could also be considered, given its impact on student motivation [5,40,128]. Finally, future studies could investigate further the nature of perceived cost when dealing with Mathematics, given its significant role in the present study as well. For example, it could be interesting to know more about students’ specific conceptualization of cost, i.e., whether it is related mostly to increased effort, missing opportunities, or it is predominantly rooted in psychological factors [14]. Such approaches could provide further insights while evaluating the motivational profiles of high-achieving students in Mathematics.

6. Conclusions and Implications for Educational Practice

The present study contributes to the existing literature on high-achieving students in Mathematics by highlighting their alternative motivational profiles, which, in turn, are related to different metacognitive and affective characteristics. Contrary to the commonly held belief that high achievers comprise a homogenous group without significant challenges or needs for special support within academic settings [17,19], this study provided a more nuanced psychological depiction using a person-centered approach. The results showed that a considerable percentage of high achievers, despite their strong school track record, held counterintuitive motivational beliefs regarding Mathematics. Such beliefs included moderate expectancies of success, attributions of low value to Mathematics, increased perceived cost, or a combination of them. These outcomes were in line with previous research, which raised similar concerns for sub-groups of high achievers with non-adaptive motivational beliefs [20,113,114,115].

Moreover, the motivational profiles of high achievers were found to be closely aligned with their math-related achievement emotions. Students with the most adaptive motivational profile reported increased levels of enjoyment and pride and decreased levels of anxiety, shame, and boredom compared to the average high achiever of the sample. The opposite was observed for students with the least adaptive profile, thus replicating previous findings that indicated similar associations between motivational beliefs and achievement emotions [83,89,90,129]. Participants with higher motivational beliefs also reported enhanced metacognitive awareness and higher certainty for their Mathematics performance; however, all high achievers showed similarly high levels of metacognitive accuracy, regardless of their level of motivation.

These findings could have particular implications for educational practice. For example, they suggest that motivational interventions would be beneficial not only for low achievers in Mathematics but for high achievers as well. This might be applicable to the low value attributed to Mathematics by a considerable proportion of study participants, which could be successfully addressed even with short interventions that promote the utility component of the value of Mathematics. This has been shown in a series of studies targeting students [10,130,131] or engaging their parents as well [11,132]. Such interventions focusing precisely on utility value are the most common, as it is considered the most easily manipulable value dimension [133]. However, other interventions targeting attainment value, perceived cost, or a combination of value dimensions could also deliver favorable results [134]. These interventions are not yet commonly used in the context of Mathematics, but they have been already applied to other STEM subjects. For example, Johnson and Sinatra [135] found that the participants who adopted the perspective of a fictional student in a narrative, who believed in the importance of doing well in a biology course (attainment value), performed better on a post-test than those in the study’s control group. Also, Rosenzweig et al. [136] tried successfully to reduce students’ perceived costs related to college physics coursework by altering their attributions about challenges they faced in the course. The results of our study also underscore the importance of identifying high achievers who perceive a high cost when engage with Mathematics. This could help to timely address its causes before the increased cost potentially affects the overall total motivation and disengage students from their academic pursuits, as indicated in the longitudinal study of Tuominen-Soini and Salmela-Aro [111].

Another finding that could inform educational practice is the association between adaptive motivational profiles and adaptive emotions. Overall, higher expectancies for success in Mathematics are associated with enhanced perceived control when engaging with a mathematical task, and according to Control-Value Theory [78], this situation could potentially trigger positive emotions. Such associations were also found to be strong in previous research [90,137] and indicate a way for teachers to promote positive learning experiences through enhancing their students’ academic self-beliefs.

In conclusion, a better understanding of high achievers’ diverse motivational profiles and of the associated metacognitive and emotional differences could significantly contribute to the development of a more inclusive and supportive learning environment for high achievers, who are frequently overlooked in a regular class.