Motivation and Feelings of Competence Among University Students in Introductory Physics

This work analyzes whether there is a cognitive bias between the ideal perception of the skills and the real performance in an introductory physics class, and additionally, whether predictions of students’ performance are related to various motivational variables. We examined through a validated survey and network analysis the relationship between several motivational aspects and volitional variables with the accuracy of their predictions. The results show that the students present a motivational bias when students’ desires were considered, mainly in the students with low academic performance. Finally, it is necessary to explore the development of specific interventions that target the motivations of students, in order to be effective, and to reduce the gap between expected and actual grades, increasing students’ metacognitive skills and thus their academic performance.


Introduction
Students who are starting their studies in scientific-technical degrees usually have a low opinion of their skills and competences in subjects such as Physics or Calculus. They also underestimate the study effort needed to pass the subject. (Blackmore et al., 2021;Fakcharoenphol et al., 2015;Galloway et al., 2013;Morphew, 2021). It is usual for students to fail in the first exams and choose to drop the course that creates a feeling of frustration (King, 2015;Lippert, 2020;Morphew, 2021). Additionally, the syllabus for high school physics in Spain does not completely overlap with physics syllabus for engineering found in first year of university, which is more oriented toward applied physics. So for most students, the discipline-specific physics course is their first encounter with those topics (Seidel et al., 2008). On the other hand, many students view introductory physics as a required hurdle and focus strictly on how to do exam-like problems. (Buckman et al., 1975).
One of the possible causes of this academic failure may be because they have an illusion of competence. This cognitive bias is known as the Dunning-Kruger effect (Kruger & Dunning, 2002). This effect shows that low-performing students often overestimate their performance on exams and tend to be overconfident. Moreover, this lack of conscience regarding their low performance implies that these students do not usually try to improve that performance. That is, these students do not have metacognitive self-awareness of their low performance, nor how to get it (Dunning et al., 2004). However, the students are able to improve their self-perception with metacognitive training (Kruger & Dunning, 2002).
Several studies have focused on the clinical and non-academic field in order to find out the psychological and social mechanisms of this effect, ranging from humor, logical reasoning, and grammar (Kruger & Dunning, 1999), geography (Ehrlinger & Dunning, 2003), to criticism of science (Caputo & Dunning, 2005). Even so, several studies have criticized that this cognitive bias actually consists of a statistical trap based on unreliable measurements (Krueger & Mueller, 2002), but they have subsequently been refuted conclusively (Freund & Kasten, 2012).
In the field of educational psychology, various empirical studies have been carried out (Bol et al., 2005;Nietfeld et al., 2006), although only a few have focused on the field of STEM (Science, Technology, Engineering and Mathematics) education. The presence of the effect has been corroborated, for example, in the reading comprehension (Huff & Nietfeld, 2009); biology (Rachmatullah & Ha, 2019), chemistry (Pazicni & Bauer, 2014), or mathematics course (Labuhn et al., 2010), but, as far as we know, there have been no studies applied to computer engineering. In all these studies, the variables on which the presence of cognitive bias depends have not been fully identified.
As a result, several studies have focused on finding the causes of this cognitive bias, indicating some of them the origin in the motivation and self-image protection (Blanton et al., 2001), while others are inclined toward a limited information processing (Chambers & Windschitl, 2004), or previously conceived beliefs about their skill and knowledge (Critcher & Dunning, 2009). A recent work suggests that metacognitive judgments may be affected by motivation, for example, when students make predictions about their performance on a forthcoming exam, they may, explicitly or implicitly, take into account the desired grades (Bol et al., 2005). That is, student motivations can influence performance predictions (Saenz et al., 2017). Since academic performance is an important aspect of students' self-concept (Elliot & McGregor, 2001) and self-esteem (Crocker et al., 2003), it is interesting to study how motivational variables play a role in self-assessments.

The Present Study
In view of all this, the following objectives are proposed in this paper: 1. To analyze whether there is a cognitive bias between the ideal perception of the skills and the real performance in the exams (Dunning-Kruger effect) in the first year of the Computer Engineering Degree course. 2. To determine whether predictions of students' performance are related to various motivational variables.

Participants
In this study, the students were enrolled in the first course at the Pablo de Olavide University (Spain) during the 2017-2018 academic year. The students attended the face-to-face Fundamentals of Physics subject in the Degree in Computer Engineering. The students were invited to participate in the study during the final semester exam. Only 72 of the 82 enrolled students (76.8%) signed the informed consent; a total of 57 males (90.4%) and 6 females (9.6%) participated in the study, a usual ratio in Computer Science degree. The consent form did not include any specific reference that it was a Dunning-Kruger effect study. The study was approved by the Ethics Committee of the University Pablo Olavide.

Procedure
The students were informed of the details of the survey and the objective of the study. The instrument was applied before beginning the exam at the end of the semester. The exam grade used a scale from 0 up to 10 points. Participation was voluntary and anonymous, although students who did participate were rewarded with a 0.1 point incentive for completing the questionnaire, which in other studies has been found not to affect the results (Ehrlinger et al., 2008). For all calculations, the statistical package R version 3.4.3 was used for quantitative analysis of the results including questionnaire reliability and validity statistics, t test for paired samples, and multiple regression. Additionally, to assess the connection between predicted performance and selected variables, a network analysis (NA) approach was applied. NA does not involve any a priori assumption and accepts simultaneous testing of reciprocal interplay between variables. Consequently, this methodological approach overcomes the need to hypothesize about causal latent variables (Borsboom, 2017).
NA was performed using the qgraph package of R (Epskamp et al., 2012). Variables are represented in a network by nodes related through edges between the nodes. Networks are based on partial correlations associating two nodes while considering the influence of all other variables. The edge is colored blue/green or red corresponding to positive or negative correlation/covariance between variables, respectively. And the magnitude of the association is characterized by the thickness of the edge. In a network, the relative importance of a node in the context of the other nodes is provided by centrality indices. There are three different and highly related measures of centrality implemented in qgraph package: strength (referring to the sum of the strength of all connections between a specific node and all other nodes), closeness (referring to the average distance involving indirect connections from that node to all the other nodes), and betweenness (referring to the relevance of that node in the average pathway between two other nodes) (McNally, 2016).
In order to facilitate the interpretation of the results, a "least absolute shrinkage and selection operator" (LASSO) regularization was applied making small correlations automatically reduce to zero and retaining only significant associations (Friedman et al., 2008). In this analysis, extended Bayesian information criterion (EBIC) was set to 0.5 enhancing both the accuracy and interpretability of network through the degree of regularization/penalty applied to sparse correlations (Chen & Chen, 2008).
Finally, in order to estimate the accuracy of the network, confidence intervals (95% CI) on the edge-weights were calculated for their estimates (Epskamp et al., 2018). CI were obtained through non-parametric bootstrapping (nboots = 2500) creating new plausible datasets with resampled observations in the data. Network stability analysis was performed using the bootnet package.

Questionnaire
The instrument was an in-house questionnaire and measures the expected exam grade in the subject for each student with respect to the rest of his or her classmates, following a common procedure in other studies (Pazicni & Bauer, 2014), as well as various motivational parameters about the student's desires with respect to his or her assessment (Saenz et al., 2017). The questionnaire consisted of 5 items.
The first item (Expected Grade) was related to the Dunning-Kruger effect and was concerned with measuring the test scores self-predicted by the students, "What grade (out of 10.00) do you think you will score on this exam?". The second item (Expected Range) was associated to the test scores self-predicted by the students relative to the grades of their peers, "Please indicate what range in % you expect your grade to be in with respect to the rest of the class (mark with an X)," where it can take values from 1 ("I'm at the very bottom"), up to 99 ("I'm at the very top") in increments of 10, and going through 50 ("I'm exactly at the average"). It is important to note that students were not aware of the performance of their peers.
Items 3 and 4 explored how students' motivations influence their expected grades. Item 3 studied positive motivation through the ideal grades that students would like to receive (Ideal Grade), "What is your ideal grade (out of 10.00) considering your individual efforts for this exam?". And item 4 (Minimum Grade) considered negative motivation through the minimum grades that students would be happy with, "What is the lowest grade (out of 10.00) you would be happy with on this test." Finally, item 5 (Effort) explored students' volition through an overall score of their level of preparation, "Compared to your prior efforts, how much did you prepare for this exam?", using a Likert 5-point scale (1: Not at all; 5: More than ever).
A test of the validity of the questionnaire was performed using exploratory factor analysis (EFA). Previously, Bartlett's test of sphericity (χ 2 = 196; p = < 0.001) revealed the existence of high dependence among the five items (Hair et al., 2010). Additionally, the value of the Kaiser-Meyer-Olkin index was > 0.6; and therefore, it was determined that data were adequate to carry out a factor analysis (Hair et al., 2010).
EFA using maximum likelihood (ML) with oblimin rotation and parallel analysis was performed. The exploratory factor analysis revealed a unique factor that explained 61.5% of the total variance with an eigenvalue of 3.08. As observed in Table 1, items showed factor loadings varying between 0.335 (Effort) and 0.938 (Expected grade), being the lowest factor loading minimally accepted (Hair et al., 2010). Finally, Cattell's scree plot (see Fig. 1) showed that the one-factor model was satisfactory to represent the data. The items seem to be measuring the same construct.
A test of the reliability of the questionnaire was performed using a Cronbach alpha (see Table 2).
The measurement of reliability using Cronbach's alpha assumes that the items (measured in Likert-type scale) measure the same construct, and are highly correlated (Zinbarg et al., 2005). The closer the alpha value is to 1, the greater the internal consistency of the scale items. In the first instance, the value obtained (0.392) highlighted the limitations of the unstandardized Cronbach's alpha coefficient, as it is affected by items measured with very different scales (see item 2). A satisfactory standardized Cronbach's alpha coefficient was obtained (0.862). Additionally, McDonald's omega coefficient was used, which uses factorial loads making the calculations more stable (Gerbing & Anderson, 1988), and does not depend on the number of items. A reliability value of the McDonald's omega   (Katz, 2003) is normally accepted, so our value of 0.88 strengthens the reliability of the questionnaire used. Guttman's λ 6 was also used as an additional measure of reliability. This coefficient uses the amount of an item's variance which is predictable by all of the other variable, so our value of 0.882 indicates acceptable reliability coefficients. Finally, average inter-item correlation compares correlations between all pairs of items that test the same construct by calculating the mean of all paired correlations. The obtained value exceeds 0.30, thus the construct validity was satisfied (Robinson et al., 1991).

Analyses and Results
First, we have focused on analyzing the trend of these data in a graph, and then perform a statistical analysis of them. Figure 2 shows the expected grades of the students against the actual grades obtained in the exam. It is important to note that students' grades have not been moderated inflating the average grade. If all students had perfect knowledge of their exam performance, the relationship between predicted and actual grades would be a straight line of 45° slope shown in Fig. 2. However, it can be seen how the relationship follows the usual tendency in studies of the Dunning-Kruger effect. Thus, lower performing students tend to overestimate their grades, and higher performing students tend to underestimate them, as seen by comparing the solid line related to the ideal line trend, and the dashed line that represents the trend of the relationship between the two variables.
On the other hand, students have been divided into quartiles according to their performance, from the lowest 25% (Q1) to the highest 25% (Q4) (see Fig. 3). Students in the lower quartile tend to overestimate their performance on the exam, their actual performance being around the 10th percentile, but they perceive their performance to be around the 26th percentile. However, students in the higher performance quartiles underestimate their exam grades, so students in the higher quartile underestimate their performance on the exam, so that students in the 90th percentile perceive their performance around the 69th percentile.
As explained above, students were ranked according to their performance based on actual grades. Differences between actual and perceived scores were analyzed using a t test for paired samples. The difference between the two grades was significant for both the worst quartile Q1, t = 11.170 and p < 0.001, and for the next quartile Q2, t = 6.180 and p < 0.0001. On the other hand, these differences disappear in students with higher performance, t = 0.848 and p = 0.410 for Q3, and t = 1.275 and p = 0.222 for Q4 (see Table 3).
In contrast, in the case of female students, as they are still a small percentage of computer engineering students (Buse et al., 2017), the sample is very small, which makes it impossible to obtain reliable statistical parameters.
This study has gone a step further by examining the contribution of students' motivations to their predictions about their exam grades. The students' desired grades were  obtained through the questions described above where they had to indicate the grade they would like to receive (Ideal Grade), along with the lowest grade they would not mind obtaining (Minimum Grade). In this way, both the positive motivation was evaluated, measuring the desired grade they expected to obtain given their own circumstances and preparation (Ideal Grade), and the negative motivation through the lowest grade with which the students would be happy.
To compare whether other types of variables might be related to the desired or expected grades, students were also asked to indicate the effort they made prior to the examination. In this way, it is possible to analyze whether this information is not associated with the perceived grades, which would reinforce the presence of desires and motivations in the metacognitive processes of the students.
Thus, all variables, both motivational (Ideal Grade and Minimum Grade) and volitional (Effort), were used as data in a multiple regression analysis to determine how they could predict the grades (absolute prediction) or range (relative prediction) perceived by students. The multiple regression analysis was preceded by an analysis of multicollinearity computing a variance inflation factor (VIF) for each independent variable. In the presence of multicollinearity, regression estimates are unstable and have high standard errors. VIF measures how much the variance of a regression coefficient is inflated due to multicollinearity in the model. The obtained VIF values for the predictor variables (see Table 4) was low (VIF < 4) indicating lack of multi-collinearity (Hair et al., 2010).
Regarding multiple regression analysis, it was obtained that all the variables together constituted a good predictive model (F = 48.38, R 2 = 0.711, p < 0.001) of the grades perceived by the students. Among these variables, volitional one was not a good predictor of the model (Effort: β = − 0.027, t = − 0.202, p = 0.840) (see Table 4). On the other hand, the variables that reflect the student's motivation were very good predictors of the behavior of the expected grades (Ideal Grade: β = 0.360, t = 4.266, p < 0.001; Minimum Grade: β = 0.545, t = 5.176, p < 0.001). In addition, the co-linearity between both explanatory variables of the model was checked, obtaining a condition index lower than 20, so that both variables are associated with a weak co-linearity (Goldstein, 1993).
However, due to the significance of Minimum Grade, and the non-significance of Ideal Grade, further information is needed to produce a complete understanding of these findings. We examined the predictions corresponding to the low performance quartile (Q1) and the high-performance quartile (Q4), and the items related to them. All items were entered into a multiple regression including the volitional item (Effort) and the motivational items (Minimum and Ideal Grade). At the end, we obtained a significant model for Q1 (F = 12.92, R 2 = 0.779, p < 0.001) and Q4 (F = 18.09, R 2 = 0.819, p < 0.001). Results showed that different motivational items were associated with grade predictions for Q1 and Q4 (see Table 5).
Thus, it appears that high-performing students were strongly associated with Ideal Grade, and low-performing students with Minimum Grade.
Regarding network analysis, the network structure of Ideal Grade, Minimum Grade, Effort, Predicted Grade, and Predicted Range is reported in Fig. 4 (Network-1). Network analysis allows us to examine how the items relate to each other and can reveal important structural relationships that regression cannot reveal. Green lines indicate positive correlations while red lines represented negative correlation.
The network centrality indices are plotted in Fig. 5. Expected grade and Expected range had the highest strength value (or degree) and higher closeness values because they have strong associations to the nodes nearby. Expected grade and Expected range play an essential role in the network and their activation influences strongly to the other nodes. However, Ideal Grade had the highest betweenness values acting as the bridge between the communities of nodes.
Central stability is shown in Fig. 6. The percentage of the students included in the calculation of the centrality indices was decreased obtaining the correlation between the indices from the subsample and the indices from the original entire sample of students. The centrality indices become unstable when the correlation goes below 0.7 for the subset sample involving 50% of the original sample. Strength and closeness had the highest stability; therefore, both indices are interpretable with some care, while betweenness is not (Epskamp et al., 2018).
The bootstrapped confidence intervals of estimated edge-weights are reported in Fig. 7. The resulting plots reveal sizable bootstrapped CIs around the estimated edge-weights, indicating that many edge-weights likely do not significantly differ from one-another. The generally large bootstrapped CIs imply that interpreting the order of most edges in the network should be done with care. The edges Expected grade-Expected range, Expected range-Minimum grade, and Effort-Expected range are reliably the three strongest edges since their bootstrapped CIs do not overlap with most other of the bootstrapped CIs. The network should be interpreted with caution due to the shape of Cis obtained for the edge weights.
Additionally, since there is a positive correlation between motivation as a key factor in academic performance (Pintrich & de Groot, 1990), the relationship between the  4 Partial correlation network constructed using EBIC-glasso depicting the association between volitional and motivational variables, and predictions (Network-1). Red lines denote negative associations between variables; green lines imply positive associations between variables; thicker lines reveal stronger associations     Fig. 8 (Network-2).
Expected grade and Expected range had also the highest strength value (or degree) as in Network-1 (Fig. 9). Ideal grade had the highest closeness value showing a strong association to the nodes nearby, and it had also the highest betweenness value acting as the bridge between the communities of nodes.
Central stability is shown in Fig. 10 for Network-2. Strength had the highest stability; therefore, it is interpretable with some care, while betweenness and closeness are not.
The bootstrapped confidence intervals of estimated edge-weights are reported in Fig. 11 for Network-2. The edges Expected grade-Expected range, Expected range-Minimum grade, and Effort-Previous grade, and Ideal grade-Previous grade are reliably the five strongest edges since their bootstrapped CIs do not overlap with most other of the bootstrapped CIs. The network should be interpreted with caution due to the shape of Cis obtained for the edge weights.

Discussion and Conclusions
The objectives of the present study were, firstly, to analyze whether there was a cognitive bias between the ideal perception of knowledge and performance, and the real ones presented in the exam scores, and secondly, to determine whether the predictions of the students' performance were related to different motivational variables.
With respect to the first objective, the results showed that there is indeed a cognitive bias between the ideal perception of their performance and the actual performance. It is noted that lower performing students tend to overestimate their grades, while better performing students tend to underestimate their performance. This result coincides with previous work in which similar results have been obtained (de Bruin et al., 2017).
The results also show that the students' perception of their academic performance does not seem to be influenced by the feedback provided by professors (Previous Grade variable). Students who held illusions of competence on the midterm exam tended to do so throughout the end of semester, failing feedback to obtain a more accurate self-awareness of their low performance. That is, as Critcher and Dunning, (2009) point out, cognitive bias on performance is not affected by students' concrete experiences of similar exams, but by preconceived notions of their ability. There are different forms of intervention that can reduce this effect and that can serve as metacognitive training, such as home activities or group work, with the aim of achieving a better perception of one's own performance (de Bruin et al., 2017).
Network analysis found that performance variables had relatively few connections to motivation and prediction variables. However, performance variables are positively interconnected, not being affected by feedback. Students who overestimate their ability are not likely to change their study methods. The overestimation effect might obscure the student's perception about the effort required to pass the exam (Boekaerts & Rozendaal, 2010). This connection can be observed in the negative association between volition and performance variables. The more effort students reported expending, lower the grade earned on the midterm exam.
Regarding the second objective, volition is, however, positively related to motivational variables. Students report a high Ideal grade according to the effort expended. However, there is no connection between Effort and Minimum grade variable. Students who have studied more than they usually do have based their grade prediction on optimistic form of motivation.
Our findings show that there is a strong relationship between students' motivation (Ideal grade and Minimum grade), and the grades that students expect to obtain in the exam (Expected grade and Expected range). Thus, motivational variables play a more important role than academic variables in predicting future student performance. The network structures illustrate that the nodes that are specifically strongly paired are Ideal grade-Expected grade, and Minimum grade-Expected range. The optimistic form of motivation materialized in students' optimal level of performance (Ideal grade) is directly connected to their predicted grade. Their desires are directly connected with their metacognitive judgments. On the other hand, the pessimistic form of motivation expressed by students' lowest level of performance (Minimum grade) is connected to their perceived performance ranking. The minimum grade that students desire to earn is strongly related to student's ranking indicating their position among peers within the same class.
Additionally, multiple regression provides us with additional information about the Dunning-Kruger effect. The relationship between desired grades and expected grades has been corroborated. If we focused on students' ranking, Minimum Grade variable was a significant unique predictor for low-performing students (Q1), capturing the students' fear of failure and establishing the floor of students' desire. Those low-performing students may be content with just a passing grade. On the other hand, high-performing students (Q4) tend to focus on Ideal Grades, as a possible desire of self-improvement.
Recent works also suggests that individuals with stronger cognitive knowledge may simply be more cautious in their self-evaluations, providing lower judgements of their performance (de Carvalho Filho, 2009), or they may avoid appearing too competent for social reasons (Schunk & Pajares, 2004). On the other hand, students who underestimate their performance ability report tend to be more conformist (Bol et al., 2005).
However, this study presents limitations as the questionnaire had to have few questions because students had to fill it before the semester exam. Therefore, more information is needed to achieve a complete understanding of the results of this work, which precisely identifies the variables that influence both students' perception of their competences and their academic performance. Limitations of the study include a small sample size in which the gender distribution is not equal. There are recent studies that have looked at how gender influences the Dunning-Kruger effect. However, the results are inconclusive, as the results range from no gender difference (Pirttinen et al., 2020;Rachmatullah & Ha, 2019) to women having a greater effect than men (Harrington et al., 2018;Pazicni & Bauer, 2014). Future studies on the Dunning-Kruger effect may focus on gender differences and the influence of the mentioned variables, as well as on cognitive bias. On the other hand, students who participated in this study were not split into several streams. The influence of stream on the Dunning-Kruger effect has been scarcely studied in the literature (Bewes & Sharma, 2006;Muller & Sharma, 2012). The main results indicate that the students in advanced streams showed evidence of smaller bias or were better calibrated. Finally, it is necessary to explore the development of specific interventions that target the motivations of students, in order to be effective, and to reduce the gap between expected and actual grades.