First Year Engineering Students’ Difficulties with Math Courses- What Is the Starting Point for Academic Teachers?

Abstract

The discussion about first-year engineering students’ difficulties in mathematics is continuous in the fields of engineering, mathematics and higher education. The present study aimed to examine the initial barriers academic math teachers need to have in mind if they want to improve students’ performance in engineering math courses through appropriate teaching practices in order to face their initial interindividual differences. During the first phase of the study, we examined first year engineering students’ initial beliefs about the nature of mathematics, their self-efficacy beliefs about mathematics and their basic mathematical knowledge. The math school grade was used for their previous mathematical performance. Results indicated the predominant role of the previous mathematical knowledge and the important role of the formalistic disposition toward mathematics. The lack of experience of using mathematics for problem-solving situations within an engineering framework prevented students from recognizing and appreciating the value of mathematical courses during the engineering studies. The second phase of the study examined, through an interview with a group of students, their perceptions of the teaching practices which were introduced after their teacher attended a training program. The discussion concentrates on how academics can use teaching processes for equity and not equality in order to motivate their students.

1. Introduction

Pedagogical science has examined teaching and learning processes for centuries. There are no simple answers to the questions “how do we learn” and “how do teachers can enforce the students’ learning”. This is because education deals with specific differing purposes and contexts and with students as people, who are diverse in all aspects and ever changing [1]. We are always looking for appropriate and accountable teaching processes which can never be suitable for all students. Those processes respect the equality but not the equity, as they refer to the standard performance and the most preferable situations. Each special case is not a part of the “typical” or the “preferable”.

In the case of higher education, students bring with them different experiences, knowledge, backgrounds, motivations and expectations as a result of their previous school life. They are characterized by different cognitive and learning styles. Academic teachers need to take all those aspects into consideration, as their students experience the same teaching processes offered in different ways and consequently with different learning results. Student learning outcomes encompass a wide range of attitudes, beliefs, self-efficacy beliefs, knowledge and skills. Those abilities are both cognitive and affective, and they are indicative of how their learning experiences have supported students’ development as learners [2].

Teaching mathematics to engineering students in higher education has been discussed for many years by academics in the fields of engineering mathematics. Recently (2021), the “International Journal of Research in Undergraduate Mathematics Education” published a special issue with studies about the difficulties, the students’ performance and the intervention programs for the improvement of the teaching of mathematics in engineering courses. What is implemented is due to academics’ knowledge, beliefs and skills. Professional development is widely recognized as a necessary condition for the competitiveness of organizations [3]. Changes in higher education make the development of academics’ teaching skills a priority [4]. Insufficient professional development of academics creates a considerable risk to the quality of higher education. In our opinion, a part of this discussion needs to be related to issues concentrating on how we can take into consideration students’ interindividual differences in terms of their beliefs, self-efficacy beliefs, conceptions about mathematics and their performance. Their differences in affective and cognitive aspects necessitate a qualitative differentiation of teaching processes, otherwise equal opportunity in teaching is a “myth without real context”. The present study aimed to examine the initial students’ strengths and limitations that academic teachers need to take into consideration for the teaching of mathematics in first year engineering courses. Two main research questions were posed: (a) What mathematical knowledge, beliefs and self-efficacy beliefs do first year engineering students possess regarding mathematics, and what are the respective interrelations? (b) How can an introductory mathematics course enriched with alternative teaching methods contribute to students’ positive disposition toward mathematics at the beginning of their engineering studies? We relate our findings to the training that academics need in order to offer math courses characterized by quality education for students with a different starting point.

2. Theoretical Background

Mathematics is one of the core subjects in engineering education and forms the foundation of all disciplines in engineering. Mathematics education for engineering students has been a topic of discussion among university mathematics teachers, mathematics education researchers and university authorities for many years. Many universities report high drop-out rates in mathematics, which has prompted university departments to offer transition courses to progress students from school to university mathematics [5].

According to Tang, Chau, Lau and Ho [6], in Hong Kong, a significant number of students are admitted to engineering programs without sufficient mathematical background, and they are often unaware of how mathematically demanding their education could be [7]. Many students are unaware of the relationship between math courses and the course they have chosen to attend and the implementation of the knowledge in their future profession [8]. As Sazhin underlined many years ago, one cannot expect engineering students to perceive mathematics in the same way as professional mathematicians usually do [9]. They need to understand its applicability to real-life engineering problems. They try hard if they are aware that they need mathematical concepts in professional activities [10]. Kaspersen et al. [11] measured engineering students’ mathematical identities in relation to their grades in university mathematics courses. They found that there is a significant correlation with their self-efficacy beliefs as a part of their identity. Mathematics educators have pointed out that engineering students encounter various issues, including barriers to understanding math concepts and to believing that physical concepts are more closely associated with real-life experiences [6].

Song [12] found three reasons for engineering students’ lack of enthusiasm for mathematics: (a) it is a difficult subject, (b) the teaching is boring and (c) the teachers’ inability to attract them. Bengmark, Thunberg and Winberg [13] found that motivations, beliefs and perceptions regarding the nature of mathematics are important factors which explain the initial performance of students in the first year of their studies. It is not easy to engage first year engineering students in a course which is not directly related to their main interest. Rahman et al. [8] designed teaching strategies to support students’ development of mathematical knowledge and problem solving, communication and team working skills. Saiman, Wahyuningsih and Hamdani [14] underline the necessity of focusing on conceptual understanding by providing a contextual basis to connect the new knowledge to the previously established and to problem-solving context. They found that most engineering students feel that conceptual mathematics is less important than procedural mathematics for their future professional orientation. Kyle and Kahn [15] found that mathematics and statistics are often taught in schools as a collection of rules, procedures, theorems, definitions and applications. The most successful mathematics courses in engineering are thought to be those that have been well integrated in the engineering curriculum, facilitating contextual relevance of mathematical abstracts to engineering concepts [16], while passive lectures are criticized for many reasons.

Pedagogy underlines that active learning has a positive effect on students’ cognitive behavior [17]. It is not easy to engage students in a course if they have low self-efficacy beliefs about their abilities to face their respective previous difficulties. The method of peer support mechanisms for students was suggested many years ago [15]. Additionally, mathematics support centers have been established at universities in England (such as Manchester), which aim to offer supplementary online lectures, notes and videos for mathematics. Rensao [18] presents deliverable chosen mathematics episodes observed in a class of engineering in a calculus course and reveals the value of the instrumental strategies. In our opinion, the success of those attempts is affected by the students’ metacognitive knowledge of their difficulties, their desire to face them and their ability to activate self-regulatory strategies in order to overcome them.

Any educational reform or the implementation of any teaching innovation can be based on teachers’ knowledge, skills, concepts and dispositions. Appropriate lifelong learning training is an ongoing need for teachers in all levels of education, especially in the case of mathematics [19]. Unfortunately, some academics teach students without having much formal knowledge of how students learn [1]. The continual professional development of university teachers has received much attention in recent years [20]. In many European countries, academics are prepared for their role as researchers but not for their teaching duties [4]. In the Agenda for Higher Education [21], it was argued that too many higher education teachers have received little or no pedagogical training, and the systematic investment in this continuous professional development remains the exception.

Klingbeil et al. [22] proposed a model of the introduction of engineering students to mathematics. As they claimed, usually, the traditional approach to engineering mathematics education begins with the one year of freshman calculus as a prerequisite to the subsequent core engineering course. The motivation for their proposed model was the finding that only 42% of incoming freshmen who wish to pursue an engineering course at a specific university ever complete the required freshmen calculus program. Their proposed approach included the development of a novel freshman engineering mathematics course in order to increase students’ retention, motivation and success in engineering. By removing traditional math prerequisite requirements and moving core engineering courses earlier in the program, they proposed the restructuring of the engineering curriculum [23]. Many universities throughout the world are trying to determine appropriate training for academics in order to improve their teaching methods [24].

Santos et al. [3] conducted a systematic literature review on the main obstacles which are identified with respect to the academic’s difficulties to follow teaching innovations: (a) their unwillingness to move away from traditional teaching practices, (b) lack of formal procedures as evaluation criteria at universities, (c) lack of time and (d) lack of financial and institutional capacity to develop effective professional development schemes at the university level. Even if academics are aware of innovative teaching methods, they may lack the encouragement to implement them.

3. Methodology

The present study is a part of a project developed in a private university at Cyprus, with the purpose of facing the engineering students’ difficulties and the negative affective and cognitive performance in the math introductory courses. In this paper, we present two phases of the study. During the first phase, the students’ previous beliefs and self-efficacy beliefs were examined in relation to their initial mathematical skills. During the second phase, after a training, we concentrated on an academic who used alternative teaching and assessment methods.

The sample of the first phase of the study comprised 60 first year engineering students (the respective population was the same at the start of the specific semester). They attended an introductory, obligatory mathematical course during the first semester of their studies at the School of Engineering. The study was conducted during the fall semester of the academic year 2022–2023. The participation in the study was voluntary after their academic teachers explained them the purpose of the study. The sample consisted of 48 boys (80%) and 12 girls (20%), a ratio which is representative of the students’ gender at the School of Engineering. Those students were accepted for studies in the engineering programs at the specific school without any entrance exams, a policy which is used by many universities throughout the world. Only a high school leaving certificate from upper secondary education (either the general education or the technical education) was required.

Two research tools were constructed and used as the main tools for data collection during the first phase of the study, which examined the first research question: (a) a questionnaire about students’ beliefs and self-efficacy beliefs and (b) a test about their initial mathematical performance before attending the course. In the present study, we used data which derived from a 15-point Likert scale (1 = totally disagree and 5 = totally agree) including items on the students’ beliefs about mathematics and 7 items with the same Likert scale, on their self-efficacy beliefs about mathematics (all the items are presented at the next section, as part of the factor analysis). The test of mathematical performance consisted of 15 tasks which involved different mathematical concepts (patterns, problem solving with proportional and inversely proportional amounts, operations with negative numbers, solving equations and systems of equations, factorizing expressions, calculating the slope of a line, solving an inequality and constructing a simple geometrical proof), knowledge which was expected to have been acquired during the students’ school life in secondary education.

We analyzed our quantitative data by using SPSS version 25. Exploratory factor analysis was used in order to group the items of the questionnaire, and descriptive statistics was used in order to calculate the means and the standard deviations of the factors related to students’ beliefs and self-efficacy beliefs and their mean performance in mathematics. In order to examine the interrelations of the different cognitive and affective dimensions, the sample was divided into three groups (low, medium and high) by using cluster analysis based on the students’ initial mathematical performance in the test.

At the second phase of the study, 19 students who attended the specific course as a group (all the first-year engineering students were divided to groups of up to 20–25 students using a randomized approach) followed the teaching of the same concepts through supplementary learning materials which were introduced by their academic teacher after he attended a training program:

• (a). The solutions of many tasks were presented through on at the course platform in order to be available for study;

• (b). Students had to solve six online self-assessment quizzes throughout the 13-week course. Every week, the specific academic finished his lectures 15 min earlier, and he left the room after asking his students to start working on the online quiz either individually or in pairs by using their smart phones. The online quizzes were constructed through the e-learning platform of the course. Students were able to continue the quiz at home and to repeat it as many times as they wanted;

• (c). Students had peer tutoring hours as part of their timetable, offered by postgraduate students to whom they could pose their queries;

• (d). As expected, students were able to visit their teacher at his office during the office hours to pose their queries. The specific academic gave them the opportunity to arrange online meetings through the use of Zoom.

Although the sample during the second phase of the study was small, we used quantitative data about the number of times they worked on the quizzes in relation to their grade for the quiz and the time they spent working on it. We thus wanted to examine their persistence in working with the mathematical tasks. Additionally, we have analyzed the qualitative data about their conceptions and experiences through semi-structured interviews with six of them (3 boys and 3 girls with high, medium and low performance in the first midterm of the course).

Limitations: The participation, particularly in the first phase of the study, was voluntary (at least for the effort that devoted to the test or the concentration when completing the questionnaire, as the entire population indicated willingness to take part in the study) and without a reward. It was anticipated that the less motivated students may have been less willing to participate by spending the expected time. Additionally, the group of students which took part in the second phase of the study could be compared to the other groups due to the ethical rules regarding the other academics of mathematics who did not have any similar opportunity for training. The comparison of different teaching methods and the respective results are among our future aims, which are presented in the final section of the paper.

4. Results

We first subjected the students’ responses to exploratory factor analysis to examine the extent to which each part of the questionnaire reflected the constructs under examination. The analysis of the students’ responses to the 15 items of the first scale resulted in four factors with eigenvalues greater than 1 (KMO = 0.845, p < 0.05), which explained 72.75% of the total variance (Table 1). After the content analysis of the items, a description of each factor was used in order to examine further the students’ behavior toward each aspect. The items at the first factor expressed students’ beliefs about the factors of success in mathematics and mainly whether they depend on the person’s abilities and efforts or external factors. The second factor expressed their beliefs about the significance of mathematics in real-life situations. The third factor expressed a formalistic perspective about the nature of mathematics. The fourth factor expressed the experimental beliefs about the nature of mathematics.

The means (Table 2) of the four factors were used for further analyses. As was obvious, all the factor means were higher than 3. The highest mean was found in the case of the formalistic perspective of mathematics, where the small standard deviation indicated that participants had similar strong beliefs about the specific issue.

The respective analysis of the students’ responses to the seven items of the second scale resulted in two factors with eigenvalues greater than 1 (KMO = 0.783, p < 0.05) which explained the 73.56% of the total variance (Table 3). The content analysis of the items at each factor indicated that the first one expressed students’ self-efficacy beliefs about mathematics and the second a positive disposition toward mathematics.

The means of the factors in Table 4 indicated that the sample of the study had low self-efficacy beliefs about mathematics (mean = 2.56) and a negative disposition toward working in mathematics (mean = 2.32).

We examined the relationship of students’ school mathematical performance as revealed through their grade with their basic mathematical skills through the initial test which was used. Their previous school mathematical performance results indicated that 4 students had a grade of 19–20, 23 students 16–18, 20 students 13–15 and 11 students 10–12. It seems that based on the data derived by their assessment at high school, only 4 of them had high performance in mathematics, most of them (43) had medium performance and 11 of them had extremely low performance in mathematics. We could say that the grade for mathematics for 31 of the 60 students was under 15 (at a scale up to 20). Concerning their performance in the initial mathematical test which was used in the study, the minimum performance was 1.5 and the maximum 12.08 (the scale was 0–15). Based on their performance in the mathematical test, students were divided into three groups by using cluster analysis: 12 students had low performance in the test, 28 students had medium performance and 20 students had high performance.

The crosstabs analysis indicated that 33% of the students with medium performance in the test had math grade 13–15 in the last class of high school and 50% of them had a grade of 10–12. Obviously, the majority of the students with medium performance in the initial test were characterized as students with low performance in mathematics at high school (with grade 10–15). Similarly, 46.7% of the students with high performance in the test had at mathematics grade 16–18 in the last class of high school, and 20% had a grade of 19–20. All the students with grade 19–20 were part of the group with high performance in the test. Finally, 50% of the students with low performance in the test had, at the same time, low performance in mathematics in the last class of high school (10–12). It seems that the test which was used at the beginning of their studies in order to examine their initial mathematical performance confirmed their previous mathematical school performance, as characterized through the use of math grade at the end of high school.

The analysis of the variance of the three groups based on their mathematical performance in the test and their respective beliefs and self-efficacy beliefs indicated statistically significant differences only in the case of their beliefs about success in mathematics (F = 3.51, p < 0.05). Specifically, students with high performance in mathematics had the lowest mean concerning their belief about success in mathematics (2.83), and the students with low performance had the highest mean (3.27). We must highlight that students with low performance believe that success in mathematics depends on personal ability, which acts as a personal characteristic or as a talent.

The three groups with respect to the initial mathematical performance were examined concerning their beliefs about mathematics. The means for the three groups with respect to their beliefs are presented in Table 5, and their corresponding self-efficacy beliefs are presented in Table 6. Students with low performance in mathematics seemed to believe more than the other groups in the formalistic nature of mathematics and less in the experimental nature. Although the students with high performance in mathematics had lower mean for the formalistic perspective, it is even in this case too high (3.65), although they expressed a strong belief about an experimental perspective (3.66). It is impressive that students with high performance in mathematics do not recognize the implementation of mathematical concepts in real-life situations (3.11), while the respective mean, in the case of students with low performance, was 3.71.

In general, the findings at the first phase of the study revealed the interrelations of the previous students’ achievements and experiences with their beliefs and self-efficacy beliefs about mathematics. The specific university must face the difficulties in mathematics of the first-year engineering students with medium and low performance in mathematics, with a negative disposition toward mathematics and with a formalistic belief about the nature of mathematics. At the second phase of the study, we concentrated our attention on a group of students whose academic teacher attended a training program and used supplementary teaching materials, a teaching method which respected the difficulties which were revealed during the first phase of the study and which was based on differentiation.

The number of attempts made by students for each quiz are presented in Table 7. Additionally, in the first parenthesis, we present their average grade of each attempt at each quiz and in the second parenthesis the time (in minutes) they spent for solving it. Firstly, as shown, only in the first and second quizzes did all the students take part. In the third, there was one missing case, in the fourth there were two missing cases, in the fifth there were five missing cases and in the final quiz there were seven missing cases. Only at the second quiz was there a student who made five attempts to complete it. In all the other cases, the number of attempts was up to three. Secondly, in each case, the average grade became highest when the students continued to insist on solving the same quiz many times. In two cases, there were students who insisted on succeeding and scoring 100. Especially in the case of the second quiz, a student scored 100 at the fifth attempt. Additionally, the mean time which was spent was smaller in each new attempt as the students tried to again solve the tasks in which they were previously unsuccessful.

After the content analysis of the interviews, we concentrated our attention only on two aspects which are examined in the present paper: (a) the students’ initial conceptions about the attendance of math courses at the engineering studies (which are related to the first research question) and (b) their comments about the teaching processes of the specific course (which are related to the second research question). The students with medium and low performance in the midterm of the course would like to avoid attending a course on mathematics. They underlined the negative experiences they had during school life and their unwillingness to continue working on mathematics. A girl said that “I did not know that I would attend a compulsory math course when I chose to study engineering. We repeat things that we have done again at the school”. Their main concern was that they are not able to understand the implementation of the math knowledge in their future occupation as engineers. The two students with high performance expressed the certainty that they will be able “to understand the relation with real-life situations the following up semesters” of their studies, while a girl with medium performance expressed the suggestion to “add examples through videos or tutorials about the use of those concepts on engineering context”. It was impressive that all students had an extremely positive image of their teacher’s teaching processes. They distinguished him from the other academics with respect to his availability for the arrangement of extra meetings. They used the supplementary materials during their study for the midterms and the final exam. The students with high performance did not attend the peer tutoring meetings. The students with medium performance found them useful, although a boy said that “I would feel more comfortable and confident if the academic organized those meetings”. A boy with low performance was unable to spend time on attending them as he worked for economic reasons in order to be able to study, while another boy with low performance did not go due to his unwillingness to reveal his difficulties in mathematics: “it is difficult to ask someone explaining me concepts that I supposed to learn them during the first years of the secondary education”. All of them found the online quizzes useful for practice. They found useful the insistence of their teacher to start the first attempt at the last minutes of the lecture. However, there were different opinions about the work in pairs: “when I was not sure about something, it was helpful to share my ideas with someone else”, “there were students who took the advantage of the activity by cheating the answers”, “I prefer to spend time in order to work on a problem alone, before sharing any ideas. This was impossible to be done in 15 min. I preferred to solve the quiz alone, after the meeting. It is a useful tool for practice and a way to realize what you have not understood. However, I understand that our teacher tried to oblige all the students to try them by asking to start working in class”. Finally, a student with high performance in mathematics underlined the necessity of using different methods at each course. He said “it was something different and attractive. I would not like to be an imitation to all the courses because it will become a routine and not attractive”.

5. Discussion

5.1. Conclusions

Many previous studies identified first year engineering students’ difficulties in attending mathematical courses during their studies. The present study confirms previous findings that, often, students are not aware of how mathematically demanding their education could be [7], and this underlines the necessity of introducing mathematical knowledge in an engineering problem-solving context.

Students commence university engineering math courses with strong beliefs about the nature of mathematics and with established self-efficacy beliefs about their ability for mathematics. Results indicated that students view mathematics from a formalistic perspective of using symbols, rules and procedures [25], and they are not able to relate the mathematical concepts to problem-solving situations in the domain of engineering. Goold [26] indicated that students of engineering courses feel uncomfortable facing unknown situations, and they prefer the certainty of following rules and known processes. The emphasis on problem-solving situations can enable academics to introduce mathematical concepts through an experimental perspective and inquiry-based teaching processes. There is a problematic lack of connection between the mathematical knowledge which is related or not to the engineering and the real-life problem-solving situation [27]. The math academics of the specific courses need to contextualize mathematics with reference to engineering, and consequently they need to cooperate with the engineering academics.

Results indicated that math academics are frequently faced with participants’ low knowledge and skills in mathematics due to difficulties they faced during their school life. Academic teachers must take into consideration students’ different attitudes toward mathematics and their different self-efficacy beliefs about their success in mathematics. The use of the same teaching methods, which is common in higher education, such as lectures, exercises and midterm and final exams, acts as a barrier for students with medium and low performance and perpetuates the inequalities. We must train academics to develop alternative and attractive materials in order to fill the gap with the prerequisite mathematical knowledge and skills and encourage them to use innovative teaching processes in order to motivate their students to realize their difficulties and develop strategies to face them. During the first year of their studies, new students need to develop strategies of learning autonomy. They are insufficiently prepared to face this new situation [28], and the contribution of peer tutoring from other students and mentoring from their teachers can have a positive effect for some of them. The online self-evaluation quiz contributes to the understanding of the difficulties and the need for persistence and insistence during problem solving.

The expectations of math academics for first year university courses are high. The continuous discussions on the same learning issues indicated that there are no “teaching recipes” in order to face the previously structured barriers for some students. Undoubtedly, there are numerous universities throughout the world that accept students with low performance in mathematics, negative attitudes toward mathematics and low self-efficacy beliefs into engineering courses. The “easy way” is to ask secondary education to face the situation or to use the same teaching methods for all the students in order to offer equal opportunities and then choose those who were able to “survive”. The use of the same teaching methods with secondary education reproduces the same differences. The present study suggested the introduction of alternative materials for teaching (such as tutorials, peer tutoring and self-assessment quizzes) based on students’ difficulties, beliefs and dispositions. However, those materials cannot have the same impact on all students’ math performance and cannot be used by all the academics in the same way.

University academics frequently lack appropriate knowledge and skills to recognize and respect the interindividual differences. The teaching of mathematics in accordance with social justice requires recognition and cognitive diversity of new students in higher education. The introduction of any new teaching process or innovation is a prerequisite to appropriate training for teachers as part of their professional development. The present study is based on Skovsmose’s [29] main belief that critical mathematics education could mean something different for different groups of students. Undoubtedly, much research has addressed students who are at social risk. However, he successfully includes the students in not comfortable positions, such as university students and mainly engineering students who read and write the world with mathematics [30]. We believe that the inability to take into consideration the initial starting point with respect to the knowledge, skills, beliefs and dispositions may lead to future social inequality and the reproduction of the same lack of diversity. Academic teachers in higher education are not only researchers but also teachers who introduce young people to a science. Universities must continue working on the establishment of accountable capacity-building centers for teaching in order to improve their teachers’ professional development on learning and teaching issues. This attempt is a part of a lifelong learning process of academics as teachers.

5.2. Future Work

The present study is a part of a longitudinal project which aims to examine engineering students’ beliefs, self-efficacy beliefs about mathematics and their mathematical performance in relation to the respective cognitive and affective behavior toward the engineering courses throughout all the years of their studies. The starting point was to examine their initial conceptions, beliefs, skills and experiences. Future work must concentrate further on cognitive and learning characteristics of the students, their motivations and the difficulties they face in relating mathematical concepts to engineering concepts through problem-solving situations. We believe that the project-based teaching methods and the inquiry-based teaching processes of mathematics within the engineering framework must be used and examined further. A second perspective of the specific study is the academic teachers’ difficulties in improving students’ performance. The comparison of different teaching and assessment methods and the respective results must be one of the follow-up aims. Teaching models must be examined further regarding their accountability from a multidimensional perspective by taking into consideration the students, the teachers, the teaching materials and the curriculum.

Author Contributions

Methodology, E.T. and S.P.; Formal analysis, M.C.; Investigation, M.C.; Resources, S.P.; Writing—original draft, R.P.; Writing—review & editing, E.T.; Supervision, R.P.; Project administration, R.P. All authors have read and agreed to the published version of the manuscript.

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study. Before the participation at the study all the parents read a commitment of respecting their anonymity and a declaration that the research does not related with any biological intervention.

Data Availability Statement

Data can be shared directly by the authors if they obtain before in each case the permission of the University Council.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Tables

Table 1

Factor loading of the factors against the items associated with participants’ beliefs about mathematics.

Items	F1	F2	F3	F4
The hard work is a presupposition for success in Mathematics.	0.912
Clever people do not have to try hard in order to succeed in Mathematics	0.876
All people can improve their performance in Mathematics.	0.835
Those who solve a mathematical problem quickly are the cleverest in Mathematics.	0.724
Some people have an innate ability to learn Mathematics.	0.673
The success on Mathematics depends on the teacher.	0.554
Mathematics are related with everyday life.		0.719
Mathematical concepts are useful in our everyday life.		0.573
The symbols are essential in expressing thoughts in Mathematics.			817
In Mathematics you must know formulas by heart.			0.759
Mathematics is a set of rules.			0.570
Mathematical concepts are characterized by accuracy.			0.532
I can find many similarities between Mathematics and other fields.				0.742
In Mathematics there are not any absolute truths.				0.709
Mathematics is a human product and construction.				0.634

F1 = Success in Mathematics, F2 = Mathematics in Real Life, F3 = Formalistic Perspective of Mathematics, F4 = Experimental Perspective of Mathematics.

Table 2

Descriptive analysis of the factors which expressed students’ beliefs about mathematics.

Factors	Minimum	Maximum	Mean	SD
F1: Success in mathematics	2.2	4.2	3.04	0.44
F2: Mathematics in real life	1	5	3.52	1.05
F3: Formalistic perspective	1.5	5	3.75	0.55
F4: Experimental perspective	2	4.3	3.15	0.61

Table 3

Factor loadings of the factors against the items associated with participants’ self-efficacy beliefs about mathematics and attitudes.

Items	F1	F2
I feel that I can understand difficult mathematical concepts.	0.738
I am able to solve difficult math problems.	0.698
I feel confidence in learning mathematics.	0.657
I enjoy to learn new mathematical concepts.	0.511
I feel comfortable on solving mathematical problems.		0.725
I enjoy to solve new math problems.		0.708
The challenge of Maths is attractive to me.		0.539

F1 = Self-efficacy beliefs, F2 = Positive disposition toward mathematics.

Table 4

Descriptive analysis of the factors which expressed students’ self-efficacy beliefs about mathematics and attitudes.

Factors	Minimum	Maximum	Mean	SD
F1: Self-efficacy beliefs	1.13	4	2.56	0.78
F2: Positive disposition toward mathematics	0	4.2	2.32	0.86

Table 5

Students’ beliefs with respect to their mathematical performance.

	Success in Mathematics	Mathematics in Real Life	Formalistic Perspective	Experimental Perspective
Low performance	3.23	3.71	4.03	2.43
Medium performance	3.25	3.23	3.79	2.70
High performance	2.83	3.11	3.65	3.66

Table 6

Students’ self-efficacy beliefs and disposition with respect to their mathematical performance.

	Self-Efficacy Beliefs	Math Enjoyment
Low performance	2.29	2.25
Medium performance	3.26	2.30
High performance	2.84	3.02

Table 7

Frequency of the students’ attempts at the online quizzes, the average grade and the average time.

		Number of Attempts
		1	2	3	4	5
1st quiz	Frequency	7	5	7
	Mean grade	(55.8)	(68.2)	(80.5)
	Mean time	(20.2)	(15.6)	(12.8)
2nd quiz	Frequency	8	3	5	2	1
	Mean grade	(65.1)	(69.2)	(92.7)	(93.3)	(100)
	Mean time	(14.3)	(12.1)	(7.62)	(7.66)	(4.0)
3rd quiz	Frequency	12	3	3
	Mean grade	(67.7)	(65.0)	(96.6)
	Mean time	(9.66)	(8.66)	(8.66)
4th quiz	Frequency	9	5	3
	Mean grade	(65.8)	(72.5)	(95)
	Mean time	(12.7)	(12.8)	(8.75)
5th quiz	Frequency	5	3	6
	Mean grade	(53.2)	(53.3)	(90.0)
	Mean time	(12.0)	(12.7)	(11.6)
6th quiz	Frequency	4	5	3
	Mean grade	(61.6)	(90.0)	(100)
	Mean time	(10.0)	(7.25)	(5.0)

References

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