## Introduction

### Background

According to the National Council of Teachers of Mathematics (2000), “teachers must help every student develop conceptual and procedural understandings of number, operations, geometry, measurement, statistics, probability, functions, and algebra and the connections among ideas… and to develop the self-confidence and interest to do so” (p. 21). Besides, teachers, who demonstrate confidence in mathematics teaching strategies, have an ability to influence their students’ confidence and use their beliefs in their own competence to develop the required students’ outcomes in mathematics (Kazempour, 2008). There is also a relationship between teacher self-efficacy that guides students’ behavior and the actions chosen by students to achieve the required goals (Ashton & Webb, 1986; Gibson & Dembo, 1984). Different research studies indicate that the teachers’ efficacy influences students’ achievements by increasing students motivation and self-efficacy in order to accomplish various academic goals at a high level (Haverback & Parault, 2008; Swackhamer, Koellner, Basile, & Kimbrough, 2009; Turner, Cruz, & Papakonstantinou, 2004; Woolfolk Hoy & Spero, 2005). Thus, the understanding of the best methods to promote self-efficacy on pre-service mathematics teachers is important (U.S. Department of Education, 2008). Such importance can be explained by the possible impact of efficacy beliefs on teachers’ effectiveness, attitudes, and behaviors. The challenges that pre-service mathematics teachers could encounter may be partly explained by the lack of confidence in a teaching process (Aemi, 2008).

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The most important period for the lifelong development of a teachers’ self-efficacy entails the years of training (Bandura, 1997; Tschannen-Moran & Hoy, 2001). Self-efficacy in the first year of training is essential because, once a feeling of efficacy is developed, it cannot be changed (Bandura, 1997; Tschannen-Moran & Hoy, 2001; Woolfolk Hoy & Spero, 2005). Teachers can improve their performance and self-efficacy through training and experience. Teacher learning and development are two crucial tasks that cannot be neglected by researchers and have to be evaluated in order to support the idea of lifelong learning (Hammerness et al., 2012). Cakiroglu (2000) noted that pre-service elementary teachers should take part in “a mathematics methods course in order to increase mathematics teacher efficacy” (p.92). In addition, when pre-service teachers explore hands-on instructions in a variety of contexts with multiple levels of complexity and challenges in order to provide more mastery experiences with quality performance feedback from their instructions, they get a number of chances to improve their efficacy. When teachers learn and apply some key strategies, their confidence may be improved as well (Albayrak & Unal, 2011).

The research study that is related to mathematics pre-service elementary teachers and self-efficacy has certain limitations. The results show that self-efficacy is one of the factors that predict the pre-service teachers’ behaviors, attitudes, and effectiveness in the classroom context (Albayrak & Unal, 2011; Chan, 2005; Haverback & Parault, 2008). A part of the existing literature demonstrates that pre-service teachers, who are conscious of their self-efficacy and teaching efficacy, work effectively and efficiently (Briley, 2012). Such teachers endeavor to spend more time on their work in order to increase their chances of success (Onen & Kaygisiz, 2013). In contrast, the mathematics pre-teachers with low self-efficacy are likely to show fewer efforts and commitment to their work (Tschannen-Moran & Hoy, 2007). Several studies have demonstrated that a mathematics methodology course increases pre-service confidence (Briley, 2012; Swars, Hart, Smith, Smith, & Tolar, 2007). Despite these positive effects, only a few studies have examined the impact of various teaching programs on the attitudes and beliefs of the future educators (Albayrak & Unal, 2011; Fajet, Bello, Leftwich, Mesler, & Shaver, 2004).

The current study on the effects of the mathematics methodology course on self-efficacy is mostly based on quantitatively oriented designs that utilize survey approaches (Tschannen-Moran, Woolfolk Hoy, & Hoy, 2001). By understanding the need for steady and strong self-efficacy, researchers suggest that it is important to develop mathematics pre-service teachers’ self-efficacy by using case studies or qualitative data (Tschannen-Moran et al., 2001).

Most of the investigations about self-efficacy in mathematics teaching concentrate on in-service teachers (Gibson & Dembo, 1984). However, there is a need to understand how to promote the pre-service teachers’ perceptions of their skills, competence, and beliefs about teaching mathematics (Ball & Bass, 2003; Musser, Peterson, & Burger, 2008; Prado, Hill, Phelps, & Friedland, 2007). Consequently, future researchers should develop more studies about mathematics pre-service teachers’ self-efficacy beliefs (Tschannen-Moran et al., 2001). This study helps to examine the impact of mathematics methodology courses on pre-service teachers’ self-efficacy and clarify the possible factors responsible for such teaching efficacy.

Self-efficacy is a self-belief of someone’s ability to complete a task successfully (Bandura, 1986). Therefore, this term can be applied to different areas of human activities. This concept is associated with specific examples such as teacher self-efficacy that can be described as a degree to which an educator could believe in their ability to promote students’ learning or their cognitive skills development. This attribute is important for educators who may represent different fields of education. Additionally, this term can be applied to a particular task or subject such as mathematics. For instance, mathematics self-efficacy can be viewed as an “individual’s belief about his/her capacity to solve mathematical problems” (Marshall, 2007, p. 180). Finally, it is important to mention mathematics teaching self-efficacy, which can be regarded as a teachers’ perception of their ability to promote learners’ acquisition of mathematical knowledge and skills. These terms are vital for identifying the purpose of the study, clarifying the significance of the study, and examining the main research questions.

### Purpose of the Study

There are two main purposes of the study under consideration. The first purpose is to examine the impact of mathematics methodology courses on pre-service early childhood and special educators’ self-efficacy and beliefs. The second purpose is to analyze the possible factors responsible for the pre- service teachers’ beliefs and determine the levels of self-efficacy of pre-service teachers’ regarding their skills in a mathematics methodology course.

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### Problem Statement

A critical element of training future teachers is a need to understand and improve the level of teachers’ self-efficacy in teaching mathematics. While the topic of teachers’ self-efficacy has been discussed in educational research for the last couple of years (Ashton & Webb, 1986; Gibson & Dembo, 1984; Guskey & Passaro, 1994; Swars, 2005), only several researchers have investigated the ways of how mathematics methodology courses can affect self-efficacy of future educators (Cakiroglu, 2000; Huinker & Madison, 1997).

A number of current researchers, who focus on the discussions of mathematics teaching self-efficacy, underline the importance of using surveys as the main research method (Albayrak & Unal, 2011; Arslan & Yavuz, 2012; Berna & Gunhan, 2011; Cakiroglu, 2008; Isiksal, 2005; Savran-Gencer & Cakiroglu, 2007) instead of paying attention to case studies or various qualitative research methods (Charalambous & Philippou, 2003; Esterly, 2003; Woolfolk Hoy & Spero, 2005). As a result, future research about mathematics teachers’ efficacy should include interviews and case studies to be aware of the impact of mathematics methodology courses on pre-service teachers’ self-efficacy beliefs. This study can add the information about the connection between methodology courses and self-efficacy levels to the existing body of research. In particular, it is necessary to prove that participation in mathematic methodology courses may influence pre-service teachers’ perceptions of their skills.

### Research Questions

- Are there any differences in self-efficacy between pre-service teachers, who had content pedagogy courses, and teachers, who had one or two mathematics methods courses?
- Are there any differences in self-efficacy for pre-service teachers between those pre-service teachers, who had one content pedagogy mathematics course, and those pre-service teachers, who had two content pedagogy mathematics courses?
- How does self-efficacy vary among pre-service teachers, who had one methods course, and those, who had two methods courses?
- What is the impact of mathematics methodology courses on pre-service teachers’ self-efficacy?
- Are there any differences in self-efficacy for pre-service teachers of different genders?
- What are pre-service teachers’ perceptions of their skills, competencies, and abilities to teach mathematics?
- What aspects of mathematics methods courses influence the self-efficacy beliefs of future teachers of mathematics?

These questions are related to both qualitative and quantitative research designs. The first five questions aim at discussing the relationships between variables. The sixth and seventh questions are important for understanding the experiences of pre-service teachers in their teaching training programs.

### Theoretical Framework

Teacher efficacy is a theory that comes as a result of self-efficacy discussions developed by Bandura (1986). There are two possible dimensions that could be applied to Bandura’s theory: efficacy expectations and outcome expectations. The expectations of efficacy include the beliefs of individuals in their capabilities toward a successful achievement of a given behavior. Outcome expectations are people’s estimations of performed behavior and the actual causes with specific effects (Bandura, 1986). This expectancy includes teachers’ beliefs about their effective teaching processes that can cause a positive learning outcome in a student without regard to external teaching and learning factors such as culture or race (Enochs, Smith, & Huinker, 2000; Swars et al., 2007). The efficacy of a mathematics teacher will be defined in terms of the teachers’ “judgment of his or her capabilities to bring about desired outcomes of student engagement and learning, even among those students who may be difficult or unmotivated” (Tschannen-Moran et al., 2001, p. 783).

### Significance of the Study

Overall, the results of this study could help to explore the self-efficacy level of pre-service teachers after their participation in mathematics methodology courses. A methodology course can be viewed as an independent variable that shapes the beliefs of pre-service teachers about their competence, skills, and the abilities to improve the learning of children behavior in the classroom (Dembo & Gibson, 1985). The teaching challenges that pre-service mathematics teachers encounter can be partly explained by the lack of confidence in their skills (Enochs et al., 2000). Therefore, certain recommendations should be made about the design of methodology courses to raise the level of pre-service teachers’ confidence. The changes in teachers’ self-efficacy after participation in methodology courses can be viewed as one of the criteria to provide sufficient support to future teachers (Woolfolk Hoy & Spero, 2005). Hence, the results of this study can play an active role in helping educators develop students’ confidence and beliefs in their abilities to excel at teaching mathematics. This information is of immense importance in the design of educational programs that are offered to future teachers, who need to develop various capacities including guiding students to academic excellence and coping with possible challenges. Additionally, the findings of this study may be instrumental in assisting educators and policy makers with understanding how to develop an appropriate learning environment and the opportunities for a customized professional growth in the field of mathematics. Moreover, the data and findings could add the value to the limited mixed method data that exists on the pre-service self-efficacy research.

### Methods

In this study, interviews and surveys are used to gather the required portion of information about the influence of mathematics methodology courses on pre-service early childhood teachers’ self-efficacy and determine the level of prospective teacher self-efficacy. This mixed methods study involves the investigations of the Western University of Pennsylvania. The reason to use mixed methods is to allow the researcher to investigate the strengths and weaknesses of each method. Ivankova, Creswell, and Stick (2006) admit that such mixed-method approach is expected to produce a strong analysis. This study contains three phases during which data collection is possible. The first phase of the study is a pre-survey that aims at collecting the information to measure the level of the pre-service early childhood teachers’ self-efficacy before the mathematics method course occurs. The survey consists of 21 Likert skills questions developed by Enochs et al. (2000). Permission of the authors should be granted to use the survey in the study. The modification of additional demographic data collection is required as well at this phase. The second phase of the study includes the post surveys where the same instrument is used, and the same procedures are followed to distribute the post-survey. The third phase of the study introduces follow-up qualitative interviews with six participants. The interviewees could be randomly selected among the participants, who demonstrate their willingness to participate in this kind of interview. It is planned to develop 14 open-ended questions in order to provide a broader understanding of teacher candidates’ perception about the mathematics methods course.

### Delimitations of the Study

The sample of this study includes college students attending the Western University of Pennsylvania only. The focus of the study is on the experiences of participants studying early childhood and special education and currently taking mathematics methodology courses. The study examines self-efficacy of pre-service teachers when they are involved in planning and delivering mathematics lessons to young children.

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### Summary

The introductory chapter explains the main statements and goals of the project. The descriptive study is used to explore the impact of mathematics methodology courses on pre-service early childhood and special educators’ self-efficacy and beliefs. It also examines the possible factors responsible for the teachers’ teaching efficacy beliefs. In addition, it attempts to determine the level of self-efficacy among pre-service teachers’ regarding their skills in the mathematics methodology course. The second chapter presents a review of the literature as it relates to teacher self-efficacy. The review of the literature also provides an explanation of the theory of self -efficacy and teacher efficacy. It also outlines a summary of the four sources of developing a strong sense of self -efficacy in mathematics pre-service teachers which include mastery, social experiences, social persuasion, and physiological conditions. The third chapter provides the information that is relevant to the research methodology. The study has a mixed methods design and utilizes both quantitative and qualitative data collection techniques to examine pre-service early childhood and special educator’ self-efficacy and beliefs. The chapter provides the description of the subject selection for the sample, the data analysis, and the discussion of the results. The fourth chapter describes the results of the study. The final chapter of the dissertation provides a discussion of the findings, an overview of the details, the summary of research, and the recommendations for a further research project.

### Definition of Terms

Self-efficacy: The self-belief of one’s ability to complete a task successfully (Bandura, 1986). Teaching efficacy is a type of self-efficacy. Therefore, the terms of teaching efficacy and self-efficacy are interchangeable in this study (Tschannen-Moran & Hoy, 2001).

Teacher efficacy: The judgment and belief of a teacher that their abilities and strategies will bring the desired results in all students’ learning processes and engagement (Tschannen-Moran & Hoy, 2001).

Mathematics teaching efficacy: A teacher’s perception of their ability to promote learners’ acquisition of mathematical knowledge or skills (Enochs et al., 2000)

Pre-service teachers education: Students are enrolled in a teacher-training program.

Early childhood/special education program: A major that prepares students to be teachers of all learners from “preK through grade 4” as well as “special education preK to grade 8” (Indiana University of Pennsylvania, 2013, p. 67).

Self-fulfilling prophecies: Teacher’s assumption about his/her students’ abilities may influence how well his/her students perform and achieve (Rosenthal & Jacobson, 1968).

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Pedagogical content knowledge: A combination of knowledge of math content, knowledge of pedagogical, and knowledge of how children should learn (Shulman, 1986). The knowledge and understanding of what should be taught, how to teach, and who could be taught (Shulman, 1986).

## Literature Review

### Introduction

Chapter 2 presents a review of research studies that investigate teacher self-efficacy beliefs and the peculiarities of teaching mathematics process. Specifically, the chapter focuses on the literature taken from the following areas: Bandura’s theory of social learning, self-efficacy, self-efficacy sources, self-efficacy in the pre-service teacher context, teaching efficacy, the summaries of research studies on teacher self-efficacy, the role of the affective domain in mathematics teachers’ performance, the importance of mathematics methodology courses, and the effectiveness of mathematics teaching.

### Self-Efficacy

Self-efficacy is one of the fundamental beliefs of the social learning theory (Bandura, 1977). This concept is incorporated into contemporary teacher-education programs research to increase teaching confidence (Bray-Clark & Bates, 2003). Self-efficacy refers to “the beliefs in one’s capability to organize and execute the courses of action, which are required to produce given attainments” (Bandura, 1997, p. 3). Also, self-efficacy refers to one’s confidence in their ability to execute the action:

Self-efficacy reflects an individual’s confidence in his/her ability to perform the behavior required to produce specific outcome and it’s thought to directly impact the choice to engage in a task, as well as the effort that will be expended and the persistence that will be exhibited. (Kinzie, Delcourt, & Powers, 1994, p. 747)

According to Bandura (1997), people’s choices to handle or avoid challenges depend on their level of self-efficacy. Self-efficacy affects thinking processes and enhances the level of cognitive functioning (Bong & Clark, 1999). Self-efficacy is able to “mobilize the motivation, cognitive resources, and courses of action needed to meet given situational demands” (Wood & Bandura, 1989, p. 408).

Bray-Clark and Bates (2003) posit that self-efficacy “is a task-specific belief that regulates choice, effort, and persistence in the face of obstacles and in concert with the emotional state of an individual” (p. 14). Self-efficacy influences pre-service teachers’ response, persistence, and effort when learning subject matter and learning to teach. Albayrak and Unal (2011) argue that efficacy beliefs “govern how people think, feel, motivate themselves and behave, and determine whether coping behavior is initiated, how much effort is expended, [and] how long the behavior is sustained when faced with obstacles and unfavorable experiences” (p. 183). Additionally, these authors note that individuals must demonstrate the necessity of knowledge, skills, and efficacy beliefs in order to develop the capacity and perform specific actions efficiently. Following this explanation, Berna and Gunhan (2011) acknowledge that individuals with a strong sense of self-efficacy beliefs may show more efforts when they learn subject matter and the ways of how to teach and continue demonstrating confidence and faithfulness as they achieve the skills necessary to overcome the obstacles (Smith, 2008).

Self-efficacy is also one of the motivational concepts (Zimmerman, 2000). The beliefs contained in this construct not only affect people’s judgments and perceptions, but they also shape how an individual can perform in a given scenario (Hinton, Flores, Burton, & Curtis, 2015; Pajares & Graham, 1999; Phan, 2012;). This research study seeks to connect self-efficacy and the relevant theoretical perspectives as explained under the theories of expectancy-value and self-concept in the available literature. According to the expectancy-value theory, “individuals will be motivated to engage in tasks when they value the outcome expected; they will be less predisposed to perform tasks whose outcomes they do not value” (Pajares, 1996, p. 558). This statement implies that positive self-efficacy beliefs will lead to a positive contribution to the expectation for an action. For instance, if a mathematics pre-service teacher tends to have high-level confidence about their activity, it is most likely that the teacher has high expectations for the success of the activity.

The available literature demonstrates that the self-efficacy belief has two components outcome expectancy and efficacy expectation (Bray-Clark & Bates, 2003). On the one hand, the efficacy expectancy refers to the belief that an individual has the capability to get the required portion of knowledge through learning. On the other hand, the outcome expectancy underscores the belief that the accomplished task could result in the desired outcomes (Bandura, 1986). Most people develop self-efficacy through observational learning and experiences in social settings to develop one’s personality (Czerniak & Schriver, 1994). The experiences of people provide them with an opportunity to develop self-efficacy. Abilities, attitudes, and cognitive skills make up self-efficacy, which plays an important role in people’s perception of situations and responses to these different situations (Bandura, 1986; Kranzler & Pajares, 1997; Swars, 2005). In practice, people believe in their abilities and take chances in accomplishing tasks based on self-efficacy (Grossman & McDonald, 2008). Such individuals trust themselves and believe that they could achieve realistic results when they focus on doing something (Hall & Ponton, 2005). Conversely, people, who possess low self-efficacy, have little belief in their abilities and remain doubtful about their ability to achieve positive outcomes very often (Pendergrast, Garvis, & Keogh, 2011). Therefore, their efforts and determination could fall below the standards and force them to get undesired results.

### In-Service Teachers and Teacher Efficacy

Any teachers’ training program should consider teacher efficacy as one of the most important factors that could affect pre-service teachers’ readiness to teach. “Teacher’s sense of efficacy is a mediating cognitive process that significantly influences teacher motivation, professional duration, [and] teacher adjustment” (Jerkins, 2001, p. 6). Teacher efficacy is a construct that is developed from the self-efficacy theory. With reference to Bandura’s theoretical framework, it is possible to say how important “teachers’ beliefs about their own capacities as teachers” could actually be (p. 202). A number of other studies on self-efficacy and teachers’ self-efficacy may be used to prove the same position and develop the chosen concept including the projects of Ashton, Webb, and Doda (1982a), Ashton and Webb (1986), and Tschannen-Moran and Hoy (2001). Besides, Tschannen-Moran and Hoy (2001) underline that teachers, who “express confidence in their ability to teach difficult or unmotivated students evidence a belief that reinforcement of teaching activities lies within the teacher’s control” (p. 784).

The teachers’ efficacy belief is an adaptive and forceful factor (Kim, Sihn, & Mitchell, 2014). “Few would argue that the beliefs teachers hold influence their perceptions and judgments, which, in turn, affect their behavior in the classroom, or that understanding the belief structures of teachers and teacher candidates is essential to improving their professional preparation and teaching practices” (Pajares, 1992, p. 307). In the literature, the opinion that teachers’ beliefs shape students’ learning and achievements is held (Albayrak & Unal, 2011). Besides, the capacity of teachers to perform particular teaching tasks successfully in their current teaching conditions depends on teacher efficacy (Lampert, 1990; Steele & Wildman, 1997).

Teachers’ efficacy beliefs form strong determinants of the extent to which they can accomplish various tasks (Pajares, 1996; 2002). Most cognitive domains are related to the idea of how individuals perceive their self-efficacy (Artistico, Cervone, & Pezzuti, 2003). For example, Chan posits that “educators often come to interpret the positive association between teacher self-efficacy and teacher performance or student achievement to mean that a higher sense of teacher self-efficacy would lead to better teacher performance and higher student achievement” (p. 85). At the same time, such issues as motivation and achievement tend to decrease as the individual’s level of self-efficacy decrease. Artistico et al. (2003) explain that an individual’s sense of self-efficacy could be used to predict how well people, regardless of their age, can accomplish problem-solving tasks. This study also focuses on the abilities of participants with higher self-efficacy levels and the possibilities to achieve more in comparison with those people, who have lower levels of self-efficacy.

In teacher efficacy research, it is evident that classroom activities implemented by a teacher influence the students’ learning outcomes (Bray-Clark & Bates, 2003; Isiksal, 2005). Consequently, as documented in various teacher-education studies, the concept of teacher efficacy has two types of beliefs. The first one is personal teaching efficacy. The second one is teaching outcome expectancy. The second type states that teachers believe that effective teaching can affect the students’ learning and grade achievement (Albayrak & Unal, 2011; Cohen, 1988; Lee, 2010). Bandura (1986) called for a distinction between those two dimensions of teaching efficacies because a teacher could assume that student learning originated from effective teaching while being uncertain of their essential capabilities for the successful delivery of lessons. The concept of teacher efficacy focuses on the factors that enhance their confidence and enable them to achieve the goals and objectives associated with classroom instructions and management, reflective teaching, student motivation and engagement, and stakeholder engagement in an educational process (Kazempour, 2008).

The investigation of the influence of self-efficacy on teaching has been a leading concern in several educational studies (Battista, 1994; Bray-Clark & Bates, 2003; Charalambous & Philippou, 2003; Czerniak, 1990; Gavora, 2011; Woolfolk Hoy & Spero, 2005). Most of these studies relate the concept of self-efficacy belief with the teacher efficacy belief to demonstrate how the latter enhances the student learning outcomes in school. Albayrak and Unal (2011) acknowledged that:

Teachers who believe student learning can be influenced by effective teaching outcomes expectancy beliefs and who also have confidence in their teaching abilities self-efficacy beliefs should persist longer, provide a greater academic focus in the classroom, and exhibit different types of feedback than teachers who have lower expectations concerning their ability to influence student learning. (p. 184)

Other studies indicate that teachers with positive teaching efficacy beliefs can be engaged in risk-taking behaviors such as trying harder with math problems or strategies they usually avoid (Arslan & Yavuz, 2012; Berna & Gunhan, 2011). Teachers with high teaching efficacy employ inquiry and student-centered strategies for efficiency and effectiveness. They demonstrate their personal beliefs of having the capacity to influence students’ achievements and motivation (Ashton & Webb, 1986; Savran-Gencer & Cakiroglu, 2007). In their investigations, Kim et al. (2014) acknowledge that:

Students’ development of mathematical proficiency is related to teachers’ efficacy in teaching mathematics and highly effective teachers have a positive effect on the student learning outcomes because effectiveness influences the teachers’ determination for a task, willingness to take risks, and the adoption of new ideas in their teaching. (p. 2)

In-service teachers and pre-service teachers with high self-efficacy are more willing to adapt and use several of instructional strategies at the same time (Riggs & Enochs, 1990). Teacher efficacy is shown through the use of various instructional and student-centered approaches. A diversity of instructional approaches means that the teacher does not use the same teaching methods from the first day to the last (Hofer & Pintrich, 1997). Turner (2010) mentions that teacher self-efficacy has a positive association with the willingness or eagerness of a teacher to start new teaching ideas and use them as variations in teaching strategies. Such teachers play the role of supervisors and mentors, who train students on how to acquire information and use it as knowledge (Cady & Rearden, 2007; Charalambous Philippou, & Kyriakides, 2008). In their turn, students tend to work in groups to acquire and synthesize knowledge in order to approach their teachers only when they experience a significant setback or challenge (Czerniak, 1990). In contrast, teacher-centered learning entails a situation whereby teachers control all class activities and allow little room for student contribution (Hoffman, 2010). Pre-service teachers with low self-efficacy tend to use teacher-centered learning more than student-centered learning (Guskey & Passaro, 1994). Swars’ investigations (2005) prove that collaboration for that argument by observing that teachers with a high perception of self-efficacy “are more likely to use inquiry and student-centered teaching strategies, while teachers with a low sense of self-efficacy are more likely to use teacher-directed strategies such as lecture and reading from the text” (p. 2). As such, it is common to find teachers with a low level of self-efficacy in classroom contexts using a traditional or teacher-directed method or technique, who are different from highly effective teachers with the intentions to build confidence among students, use student groups, and allow learners to navigate throughout their learning process for optimal comprehension generously.

Teaching efficacy predicts “the percentage of goal achieved, amount of teacher change, improved student performance, and continuation of both project methods and material” (Dembo & Gibon, 1985 p. 173). The views, perceptions, and beliefs held by teachers affect their abilities to teach and manage learning activities effectively in the classroom. It also affects students’ achievements (Guskey & Passaro, 1994). Additionally, it is related to the behavior of teachers in the classroom. For example, Tschannen-Moran and Hoy (2001) note that teachers with a strong sense of efficacy could exhibit greater levels of organization in the classroom and remain to be more open to new ideas and experiments with new methods.

Teacher efficacy could have a certain impact on teachers and students in the same classroom and at the same time. The quality of student’s performance in course work, students’ attitudes to their tasks and responsibilities, and the possibilities to promote their social, mental, and cultural growth are under a considerable influence of teacher efficacy (Broadbent, Boyle, & Brady, 2009; Granger et al., 2012; Guskey & Passaro, 1994; Hackett & Betz, 1989; Hofer & Pintrich 1997; Lampert, 1990; Marshall, 2007; Pajares & Graham, 1999; Rimm-Kaufman & Sawyer, 2004;). It is important to assess various self-efficacy beliefs during training processes because teacher efficacy has a certain power over teachers’ actions and decisions.

### Self-Efficacy Factors

Bandura (1977) identified four major sources that could contribute to the development of self-efficacy beliefs. Tschannen-Moran and Hoy (2007) investigated “both theoretical and practical importance to understand the sources teachers taps when making judgments about their capability for instruction” (p. 953). Regarding the attention of different researchers to these four sources, it is necessary to understand their essence and possible effects on teaching strategies, and the possible development and improvement of self-efficacy among pre-service teachers.

### Performance Accomplishment or Mastery Experiences

This factor may be referred to previous task experiences. It is one of the greatest contributors and influential sources of efficacy information because of the individuals, who have success in a task and are likely to perform outstandingly in similar tasks in the future (Charalambous & Philippou, 2003). However, not all successful experiences reinforce efficacy. “Successes raise self-efficacy appraisals; repeated failures lower them, especially if the failures occur early in the course of events and do not reflect lack of effort or adverse external circumstances” (Bandura, 1986, p. 399).

An individual’s sense of self-efficacy cannot be reinforced when success is attained through unbalanced external assistance or being exposed to an easy and unchallenged task (Bray-Clark & Bates, 2003). Successful completion of a task strengthens one’s sense of self-efficacy and allows individuals to believe that they have the required portion of skills to fulfill each task. However, there is a threat that the level of self-efficacy could be weakened in case a person cannot complete a task or understand the essence of a challenge (Enochs et al., 2000). Pre-service teachers gain and use much-needed teaching experience in the methods courses. Long-term exposure to different levels of complexities in the methods classes often enhances teacher efficacy. Hackett and Betz (1989) explained that mastery experiences allowed pre-service teachers to develop a stable sense of efficacy.

### Vicarious Experiences

As identified by Woolfolk Hoy and Spero (2005), vicarious experiences are usually modeled by someone else. It also refers to such methods as observation or participation. Research indicates that vicarious experiences may modify efficacy beliefs, expectations, or judgments about self-competence through comparison with the achievement of others (Berna & Gunhan, 2011). This aspect implies that watching peers with more or less the same capabilities can influence the observer’s self-efficacy beliefs (Charalambous & Philippou, 2003). For example, when pre-service teachers help to understand or model a specific task, others can be motivated that they can take the same steps with each. In this situation, peers and teachers can be used as a mastery experience. “Degree to which the observer identifies with the model moderates the efficacy effect on the observer” (Woolfolk Hoy & Spero, 2005, p. 3). This assertion means that the more directly an observer relates to knowledgeable, motivated, and credible individuals, the stronger the outcome of efficacy can be. Therefore, when pre-service teachers watch other experienced teachers complete their tasks successfully, they could also want to trust their abilities and work hard to achieve the same outcomes. According to Battista (1994), when people see others with whom they have similar characteristics to succeed through sustained effort, they raise their own beliefs and try to find out the same capabilities and chances to succeed. One of the main problems in teaching is teachers’ intentions to move forward too rapidly neglecting the development of possible difficulties among students (Ediger, 2012). Mathematics pre-service self-efficacy may be improved by means of teaching programs or observations of the actions of other people (Bray-Clark & Bates, 2003). Thus, training programs could increase pre-service teachers ‘self-efficacy by encouraging the cooperative learning experiences and positive feedback. In their turn, pre-services teachers could explain the mathematics concepts to teach others as a good strategy to improve the teachers’ sense of self-efficacy.

### Verbal or Social Persuasion

Verbal or social persuasion provides a further opportunity for reinforcing the beliefs or expectations of an individual, particularly in the context whereby significant others express confidence and faith in the capabilities demonstrated by the individual (Charalambous & Philippou, 2003). This assertion could also be used when encouragement is provided effectively and realistically by real experiences (Berna & Gunhan, 2011; Bursal & Paznokas, 2006; Phelps, 2010). Woolfolk Hoy and Spero (2005) added that verbal or social persuasion could entail “a pep talk or specific performance feedback from a supervisor or a colleague or it may involve the general chatter in the teachers’ lounge or in the media about the ability of teachers to influence students” (p. 3). Such explanation could be used to prove that individuals are more likely to do the task when they are persuaded that they can succeed. Social persuasion is a major source of self-efficacy in removing past hindrances responsible for encouraging self-doubt and disorder. Additionally, it influences the credibility, trustworthiness, and expertise of convincing individuals (Woolfolk Hoy & Spero, 2005). Any feedback should be positive, corrective and immediate in order to lead to the required portion of successful results (Coulter & Grossen, 1997; O’Reilly et al., 1992; O’Reilly, Renzaglia, & Lee, 1994). In training programs, pre-service teachers are exposed to corrective and positive feedbacks, which influence them to raise their confidence so that they can also excel (Enochs et al., 2000; Hackett & Betz, 1989). Feedback can be used as a clarification for their success or their failure. Butler and Winne (1995) also explain that it is important to know how pre-service teachers could benefit from the feedbacks they got and what steps should be taking to succeed in self-regulating learning.

### Physiological or Affective States

Physiological or affective states underscore how positive feelings such as relaxation and confidence or negative mood or feelings such as anxiety, fear, tension, or fatigue can affect people’s decisions (Charalambous & Philippou, 2003). Battista (1994) argues that emotional reactions and responses to situations influence the development of self-efficacy. This assertion implies that emotions, moods, stress, and physical reactions have effects on a person’s opinion of his/her abilities in a given situation (Hall & Goetz, 2013). However, the actual awareness of a physical or emotional reaction is not a very significant aspect of the relationships that exist between psychological responses and the development of self-efficacy. On the contrary, the most significant factor is the perception and interpretation that a person uses to reduce stress and elevate mood during different challenging and difficult tasks (Ashton & Webb, 1986; Battista, 1994; Cakiroglu, 2008). The ways educators use to reduce stressful situations and minimize anxiety including the creation of a friendly environment and communication with right people play an important role in the emotional experiences of teachers and their student (Evans & Tribble, 1986).

### Characteristics Associated with Pre-Service

#### Low Self-Efficacy Characteristics

Pre-service teachers with a negative sense of self-efficacy display characteristics that function to disrupt or affect students’ educational outcomes and achievements. Tschannen-Moran and Hoy (2007) state that:

According to social-cognitive theory, teachers who do not expect to be successful with certain pupils are likely to put forth less effort in preparation and delivery of instruction, and to give up easily at the first sign of difficulty, even if they actually know of strategies that could assist these pupils if applied. Self-efficacy beliefs can thus become self-fulfilling prophecies, validating beliefs of either capability or incapacity. (p. 80)

Social-cognitive theory helps to test and underline the essence of self-efficacy performance among teachers (Harrison, Rainer, Hochwarter, & Thompson, 1997; Henson, 2002). Teachers with a low sense of self-efficacy are often less motivated and tend to keep away from demanding and difficult tasks. In most cases, they believe that “the more people distrust their self-efficacy, the more they shy away from activities and products requiring higher cognitive skills” (Bandura, 1997, p. 460). They try to use various teacher-directed strategies in order to make sure they could meet the expectations of an organization (Swars, 2008).

The practice becomes routine among teachers who avoid difficult tasks, which undermines their ability to acquire the required skills and solve various challenges. Such teachers try to concentrate on their limitations, failures, and negative outcomes (Bates, Latham, & Kim, 2011; Charalambous et al., 2008). Pre-service teachers are poorly confident when the time to teach mathematics and use new ideas or strategies comes (Tschannen-Moran et al., 1998). They lack the ability to come back and start planning again in order to achieve the required portion of success. Finally, they lose confidence in their personal abilities and stop working on tasks that they think they cannot cope with. They try to ignore such practices by any possible means (Albayrak & Unal, 2011). Self-efficacy beliefs are influenced by an individual’s previous performance (Bandura, 1997). Therefore, it suffices to conclude that past unsuccessful experiences of pre-service teachers may form the reasons for why teachers develop low-efficacy beliefs and internalize negative behaviors (Cone, 2009).

#### High Self-Efficacy Characteristic

Mathematics pre-service teachers with high levels of self-efficacy have a number of characteristics that make them differ from other pre-service teachers. For example, such teachers are usually more motivated in order “to initiate a task, attempt new strategies, or try hard to succeed” (Tschannen-Moran, 1998, p. 212). They are expected to be more successful and aware of how to deal with different barriers. They try to identify different tasks that must be mastered and involve as many people as possible in completing the required goals (Bates et al., 2011; Cone, 2009). When mathematics pre-service teachers are faced with new teaching ideas, particularly techniques that are difficult, high self-efficacy helps them master a problem successfully at the course level and in the future. This group of teachers is more willing to change or adapt their teaching practices that are towered over the inclusive practices. For example, pre-service teachers with high self-efficacy want to use manipulative techniques and work with future students, who encounter difficulties (Enochs et al., 2000). As soon as pre-service teachers enrich their experiences, they get all chances to change their efficacy beliefs and improve their methods in mathematics teaching (Huinker & Madison, 1997; Kagan, 1992).

Accordingly, they develop the prerequisite skills that boost their confidence in the related tasks or other challenging mathematics problems. Pre-service teachers with high levels of self-efficacy make other pre-service teachers develop firm interests in the activities they undertake (Charambous et al., 2008; Grossman & Mcdonald, 2008). Besides, such teachers try to stay motivated all the time in order to write various lesson plans and focus on learning as something exploratory not routine and to underline the role of teachers in learning and problem-solving processes (Clift & Brady, 2005).

Pre-service teachers with high levels of self-efficacy are likely to develop and form a firm sense of commitment to their activities. Commitment allows mathematics pre-service teachers to acquire new skills since they are often ready to learn approaches and strategies for tackling mathematics problems and challenges students face (Esterly, 2003; Riggs & Enochs, 1990: and Czerniak & Schriver, 1994). Pre-service teachers with high levels of self-efficacy are proactive and self-organizing. In most cases, they are more willing to adapt various strategies and curriculum ideas using a variety of materials and approaches (Battista, 1994). Latham and Locke (2006) stated that “people with high self-efficacy set their goals high, because they are not satisfied with less” (p. 332). Therefore, they work harder in order to improve their teaching skills and change the existing teaching practices. They also usually use the inquiry and students centered strategies (Swars, 2005).

Mathematics pre-service teachers with positive self-efficacy work harder or acquire better skills required to solve future challenges when they face negative outcomes (Hall & Ponton, 2005). Self-efficacy enables mathematics pre-service teachers to create strategies that enhance performance and provide the desired feedback for positive results (Cone, 2009). Since self-efficacy affects their choice and their instructional decisions, mathematics pre-service teachers with a high level of self-efficacy can create new strategies to solve different class problems (Ashton, Webb & Doda, 1982b). They should have some alternatives to develop a strong sense of success (Tatar & Buldur, 2013). In teaching practice, the best results are often attributable to the sourcing and utilization of several strategies (Charalambous et al., 2008). Thus it is important to promote their self-efficacy beliefs during their training.

#### Pedagogical Content Knowledge

There are seven types of knowledge required for teachers including general pedagogical knowledge, knowledge of learners, knowledge of educational context, knowledge of educational ends, content knowledge, pedagogical content knowledge, and curriculum knowledge (Shulman, 1987). The study focused on pedagogical content knowledge to deepen the pre-service understanding of elementary mathematics and pedagogy.

Through the work of Shulman (1987), the idea of pedagogical content was introduced as:

The most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations – in a word, the most useful ways of representing and formulating the subject that make it comprehensible to others… Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons. (p. 9)

Content knowledge is the amount of actual knowledge and the ways of how it could be organized in the mind of a teacher (Shulman, 1986). In its turn, pedagogical knowledge is the type of knowledge that entails the strategies and principles of classroom management and organization in education (Shulman, 1987; 2004). All learners need a teacher who has an experience in their subject and knows how to teach the subject. Teachers’ training programs should create the connection between the knowledge of the subject and the knowledge of teaching and learning to prepare pre-service teachers in a proper way (Cheong, 2010; Shulman, 1989). Ball and Bass (2000) emphasized that pre-service teachers should have knowledge that could go beyond the common knowledge of mathematics. Thus, teacher-training programs should consider the idea of improvement of their mathematics courses to create a focus on how to teach mathematics and increase their students’ sense of self-efficacy (Palmer, 2001). One of the most important characteristics of an effective program to enhance pre-service teachers’ high self-efficacy is the provision of sufficient content and pedagogical knowledge (Guskey, 2003)

Teachers with strong pedagogical content knowledge are more confident than other teachers who have weaker pedagogical content knowledge (Van Driel, Veal & Janseen, 2001). Additionally, teachers, who are less confident in their abilities to teach, may affect the development of their knowledge of the subject (Van Driel et al., 2001). Mixing mathematical and pedagogical tasks could enhance mathematics knowledge among pre-service teachers (Steele & Hillen, 2012). When pre-service teachers learn the mathematics content from different aspects and clarify how they can teach students, they get all chances to increase their self-efficacy. Mastery experiences may develop from teacher knowledge about the subject and pedagogical knowledge in the subject (Yilmaz, 2011). This type of knowledge helps to prepare teachers to teach the desired content (Shulman, 1989). Yilmaz’s investigation (2011) shows that there is a certain relationship between teaching efficacy and pedagogical content knowledge since self-efficacy levels increased by the content-specific knowledge through pedagogical emphasis (Palmer, 2001).

Pedagogical content knowledge could also increase pre-service teachers’ self-efficacy level (Yilmaz, 2011). Notably, it enables pre-service teachers to formulate and represent the subject to students in a comprehensible manner. Furthermore, it enables teachers to comprehend what makes certain concepts hard or easy for children to understand (Graeber, 1999). Taking into consideration this statement, it is possible to say that pre-service teachers can take the subject matter and transform their understanding into explanations that students can comprehend in their own ways (Shulman, 1986). According to Shulman (1984), teachers must involve themselves in deliberation, debate and decision making in order to understand how to teach because it is the only way in which sufficient knowledge or pedagogical content knowledge can be developed. In this regard, deliberation, debate and decision participation improves teachers’ self-efficacy regarding the content delivery.

Ball and Bass (2000) explain that teachers “need mathematical knowledge in ways that equip them to navigate these complex mathematical transactions flexibly and sensitively with diverse students” (p.49). Pre-service teachers may not have an opportunity to work with students due to the difficulty of placing all pre-service teachers in field experiments. However, there are some strategies that could help to connect early childhood/special education program and theories with practice through case-based approaches, collaborative learning environments, or role playing (Ewing, Smith & Horsley, 2003). The usage of any of these techniques allows pre-service teachers to experience children reasoning, apply what they have learned, experience the complexity of teaching and learning, and provide reflective practice (Ewing et al., 2003).

Teachers with higher confidence in their content area should have a high self-efficacy level to teach mathematics because the mathematics knowledge that pre-service teachers need to have differs from the knowledge that is required for other majors (Muijs and Reynolad, 2002). Thus, if pre-service teachers want to demonstrate effective practices, they need to know the subject they teach and consider the approaches on how to teach students about the concepts and their importance (Shulman, 1986). For example, a fraction definition is the division of one whole number by another. However, pre-service teachers need to know more than that part of a whole. Pre-service teachers should know about common fraction misconceptions and mistakes, logical reasoning behind fractions, more than one method to solve fractions, and the information about how fractions apply to daily life and how to connect different representation (e.g. pictures or manipulative and spoken language) (Beckmann, 2014). Pre-service teachers “must know more than just how to carry out basic mathematical procedures; they must be able to explain way” (Beckmann, 2014, p. xvi).

#### Summary of Research Studies on Teacher Self-Efficacy

Taking the conclusions made in the frames of the social learning theory into consideration, it is possible to say that the role of methodology courses in teacher training is important because it enhances the development of self- efficacy beliefs and reinforces the pre-service teachers’ ability to deal with learning and promote desired behaviors (Darling-Hammond, 2000; Hart, 2002; Huinker & Madison, 1997). “Teachers who have had more preparation for teaching are more confident and successful with students than those who have had little or none” (Darling-Hammond, 2000, p. 166). However, some research results revealed that pre-service teachers’ attitudes were not affected, and there was no significant change or decrease on pre-service teachers’ self-efficacy after completing the method course (Plourde, 2002; Yilmaz & Çavaş, 2008). These researchers explained that the negative impact of the method courses could be proved by the fact that pre-service prior experiences before the program and during the program could not be defined as perfect. Teachers have to improve their skills and make decisions on their own. There are several method courses that could be used to describe the actions of teachers in such situations.

The analysis of the literature and the evaluation of the material available prove that self-efficacy as a concept has been studied from many different perspectives (Alsup, 2004; Bleicher, 2004). It was at the core of teacher-education studies during several decades as one of the most fundamental aspects for influencing the behaviors, attitudes, and effectiveness of teachers (Albayrak & Unal, 2011). However, very little attention was paid on pre-service teacher self-efficacy toward teaching mathematics. There were the studies that demonstrated that mathematics methods courses could improve the pre-service teachers’ self-efficacy in many ways. Wenner (2001) explored self-efficacy beliefs of elementary pre-service teachers regarding mathematics concepts and teachers’ preparation programs and compared pre-service and in-service teachers’ self-efficacy in teaching mathematics. Wenners (2001) also reported that, by the end of the program, their self-efficacy increased. Therefore, there are the chances to change the level of self-efficacy considerably. In general, it turns out to be that pre-service teachers may have higher self-efficacy in comparison to other in-service teachers, and pre-service and in-service teachers’ self-efficacy levels in teaching mathematics should be higher than their self-efficacy level in teaching. The differences between in-service and pre-service teachers’ self-efficacy have to be investigated and improved because of the current lack of support and mentoring activities when teachers have to enter the field. Besides, pre-services teachers have to think about the support they could provide their students with in order to improve the classroom activities (McClain & Cobb, 2001; Marzano, 2003). Therefore, it is important to think about the situations when pre-service awareness about self-efficacy, as well as content knowledge, could be increased and take the steps in order to achieve the required results and promote support as the best means to explain new material and underline the expectations (Wilkins, 2008). For example, Swars et al. (2007) examined the mathematics beliefs and the content knowledge of elementary pre-service teachers. Their results show that pre-service teachers’ self-efficacy could be increased after taking two mathematics methods courses.

The investigation of Susan (2008) in a similar study helped to understand that mathematics elementary pre-service teachers could always improve their self-efficacy level and demonstrate how their teaching effectiveness could be changed. Four elementary pre-service teachers enrolled in mathematics methods course participated in that study. Pre-/post-test analysis of the MTEBI and the interview analysis revealed that past experiences in mathematics affected their self-efficacy. It was also found that pre-service teachers with high self-efficacy were more comfortable when they used manipulative behavior in their teaching. It was suggested to examine the pre-service elementary teachers’ past experiences in order to improve their self-efficacy.

Esterly (2003) used both quantitative and qualitative research methodologies to investigate the mathematics teaching efficacy and epistemological beliefs of middle school mathematics pre-service teachers with a view to expand knowledge on the concept and to consider students’ performance outcomes. The study proved that teachers with a low efficacy score could employ teacher-directed approaches to teaching mathematics in classrooms with two of the foremost outcomes associated with those approaches, low student performance in mathematics, and diminished student participation. The study could also be used to explain that teachers’ sense of efficacy for an effective usage of instructional strategies could be influenced by instructional time management and teacher-related skills in effectively addressing students’ mathematics performance levels to a large extent (Esterly, 2003). Besides, teachers’ beliefs about their capabilities in the core areas of student engagement, classroom management, and the use of instructional strategies influence their evaluation of their competence to teach should be taken into consideration. Consequently, the study concluded that new interventions’ design to enhance teacher efficacy in teaching mathematics might have long-term effects on maximizing students’ confidence, intensity, and achievement in mathematics.

A study conducted by Isiksal (2005) examined the effects of the gender issues and the importance of years in a program on the performance and mathematical self-efficacy beliefs. In this study, 145 pre-service mathematics teachers from Turkey schools were invited (Isiksal, 2005). It was found that there were substantial statistical effects of gender and year in a program on both pre-service teachers’ performance and self-efficacy scores. Specifically, the study explained that female pre-service teachers scored substantially higher results than their male counterparts in performing their mathematics self-efficacy. Still, no noteworthy variation was discovered between the two groups with respect to mathematics self-efficacy scores, and that senior pre-service teachers scored highly on performance, as well as mathematics self-efficacy scores, compared to newer students enrolled in the education program.

In a study conducted by Savran-Gencer and Cakiroglu (2007), the task was to investigate the efficacy beliefs of the Turkish pre-service teachers and their classroom management. It was found that pre-service science teachers usually expressed positive efficacy beliefs. Moreover, the study concluded that their teaching practices were the most important factors that affected their teaching efficacy.

In another quantitative study developed by Albayrak and Unal (2011), the exploration of the effects of mathematics methodology courses on elementary pre-service teachers’ mathematics teaching efficacy beliefs in Turkey was offered. It was explained that when students and teachers attended a mathematics methodology course with positive attitudes and goals, a number of positive changes in the teaching efficacy beliefs among elementary pre-service teachers could be observed.

Arslan and Yavuz (2012) conducted a study at the Istanbul University in Turkey with a view to not only establish prospective (pre-service) teachers’ self-efficacy beliefs about mathematical literacy but also investigate those beliefs against a set of variables that included a teaching department and gender factor. Several conclusions were made in the study. First, it was proved that mathematical literacy self-efficacy beliefs of prospective mathematics and physics teachers were below average (Arslan & Yavuz, 2012). Besides, prospective teachers’ mathematics literacy self-efficacy beliefs did not differ in regards to the department factor. Finally, a gender issue had its role in the study but did not mediate or influence the mathematical literacy self-efficacy beliefs of prospective mathematics and physics teachers (Arslan & Yavuz, 2012).

Informed by the need to explain ways through which teacher efficacy can be enhanced, Kim et al. (2014) conducted a study to investigate “the South Korean elementary teachers’ mathematics teaching efficacy beliefs and the factors that increase the efficacy beliefs demonstrated by teachers” (p. 1). According to the conclusions made by those authors, it was evident that teachers, who believed that teaching could influence student learning (teacher efficacy) and who demonstrated high self-efficacy may provide “a greater academic focus in the classroom and offer diverse feedback according to the students’ academic backgrounds more than teachers who have low mathematics teaching efficacy beliefs” (Kim et al., 2014, p. 3).

Wilkins and Brand (2004) examined the effect of mathematics method courses on elementary pre-service teachers’ beliefs. In that study, a sample of eighty- nine pre-service teachers administrated mathematics beliefs instruments. Wilkins and Brand (2004) found a significant positive relationship between mathematics methodology courses and self-efficacy. That relationship suggested that mathematics teacher self-efficacy could be increased after the end of the course if the same methods and direction were chosen.

Incikabi (2013) surveyed 200 pre-service teachers’ self-efficacy cases. During the mathematics methods courses, the participants were introduced to the play-generated curriculum where teachers demonstrated play experiences that improved mathematics learning concepts and skills. Incikabi (2013) used pre- and post-experimental design to measure the level of self-efficacy. As a result, mathematics self-efficacy was increased at the end of course. The changes because of the received experiments and participating in the activity were stated. The conclusions were clear and information. Incikabi (2013) proved that training offered by pre-service teachers could be improved and strengthened in order to change teachers’ self-efficacy beliefs. The previous experiences also mentioned in the study could be used to improve their teaching self-efficacy.

#### Research-Practice Connection

As suggested by Bandura (1986), it is possible that the theory of social cognition could offer a reasonable explanation of the sources of such self-efficacy beliefs among teachers including their mastery experiences, verbal or social persuasions, their vicarious experiences, and physiological arousal. Arguably, the most important factor in developing self-efficacy is the mastery experiences (Arslan & Yavuz, 2012; Bandura, 1997; Hoffman, 2010). Mathematics methodology courses can improve the vicarious experiences through collaborative training, peer interaction, and observations of other pre-service teachers (Bray-Clark & Bates, 2003). It is possible that pre-service mathematics teachers observe their peers’ successful teaching behavior and try to take the steps in order to change their own self-efficacy levels or attitude to the situations (Bandura, 1993). Educators should concentrate on the development of the mastery experiences by demonstrating their teaching abilities and relations that could be offered to their peers. Thus, peer groups and classmates should be stated as important sources of self-efficacy. Lin, Gorrell, and Taylor (2002) also offer to consider parents as another important group for cooperation in order to improve teachers’ self-efficacy.

Pre-service teachers enter their programs with wide past experiences. While mastery experiences are developed from the training programs in which pre-service mathematics teachers enroll, the use of manipulative, technological, and cooperative learning, and discourses on practical experiences in the teaching of mathematics reinforce their mastery experiences and increase teachers’ self-efficacy (Kazemi, Lampert, & Ghousseini, 2007; Woolfolk Hoy & Spero, 2005). In their turn, Hawley and Valli introduce their own explanation of teacher efficacy and the possibilities to promote changes. The say that “teacher efficacy is enhanced when teachers have opportunities to observe new strategies, practice them, engage in peer coaching, acclimate students to new ways of learning, and use new teaching and learning strategies regularly and appropriately” (Hawley & Valli, 1999, p. 130). Training programs can provide the mastery experiences by using mini-lessons. Pre-service teachers introduce teaching a specific mathematics concept or skill to peers, receive feedback from instructors and peers, learn by doing, and reflect upon their practices. When pre-service teachers implement new practices and observe positive results, they can apply them in the future (Guskey, 2003)

Another way to promote their mastery experiences is to use classroom simulations in order to increase pre-service self-efficacy (Bray-Clark & Bates, 2003). Simulation is a strategy that allows pre-service teachers to take on the role of a teacher and develop their skills and knowledge in order to help students (Bray-Clark & Bates, 2003). In addition to simulations, role-play techniques are used. Any role-play method is a technique to solve a problem with peers followed by a deep discussion about learning and behavior management within primary classrooms.

Another way to experience teaching is video. This method is appropriate due to “its closeness to the complex reality of the classroom, teachers’ past experiences and beliefs about what is possible and not possible in teaching may turn even video into an artificial representation of teaching that teachers can easily dismiss” (Santagata & Guarino, 2011, p. 143). Teaching videos followed up by a discussion is also another kind of experience which connects theory and practice (Lampert & Ball, 1998). Teaching videos have been regarded as an effective means of deepening pre-service teachers’ learning (Linares & Valls, 2010). Video observations are more effective than different aspects of field experiments.

Verbal persuasion is another source that develops mathematics self-efficacy. Instructor and peer feedback, collaborative teaching, and peer observations provide a good portion of the verbal support to mathematics pre-service teachers. Feedbacks encourage pre-service teachers to develop their strengths and other areas they need to improve. If they don’t receive feedback on the new practices, they may ignore such new practices and chose those ideas they find appropriate for themselves (Guskey, 2003).

Concerning physiological states, available research indicates that a positive correlation between the feelings of satisfaction could be developed by teachers in their simulated teaching experiences, and the satisfaction could be derived from the achievement of actual teaching (Tschannen-Moran & Hoy, 2007). Stress could be decreased among pre-service teachers in case teachers try to increase the level of mathematics knowledge and use manipulative, technological, and cooperative learning in the teaching of mathematics. These factors influence teachers’ self-efficacy and their beliefs in a manner that affects their capability to accomplish their desired tasks or objectives.

The available literature shows that teacher efficacy influences students’ learning outcomes, motivation, and attitudes toward the learning of different subjects. It changes the students’ beliefs, attitudes, and learning priorities toward their behavior in the classroom (Boud, 2012; Rimm-Kaufman, 2004). Many social learning theories support this concept and observe “understanding the belief structures of teachers and teacher candidates is essential to improving their professional preparation and teaching practices” (Pajares, 1992, p. 307). This observation implies that it is important for researchers in the field of education to consider how these factors could influence the efficacy of a teacher with a view to determining what is needed to assist teachers (especially teachers of mathematics) and gain a greater sense of teacher efficacy.

The investigation on the various methodology courses offered at mathematics teacher-training institutions gains the importance as educators and other relevant stakeholders realize that teacher quality is tied to the students’ educational outcomes (Arslan & Yavuz, 2012; Haverback & Parault, 2008; Isiksal, 2005; Kim et al., 2014; Lancaster & Bain, 2010). Within the context of mathematics, teaching efficacy focuses on two fundamental areas that relate to the two aspects of teacher efficacy: mathematics teaching efficacy and teaching outcome expectancy. Few studies have been performed to analyze mathematics teaching efficacy among early childhood pre-service teachers. However, available studies have shown a considerable increase in mathematics teaching efficacy is possible after completion of one of the methods courses or a number of methods courses offered by the institution (Huinker & Madison, 1997; Cakiroglu, 2000). Moreover, such improvements could also be observed after completing content in mathematics courses (Swars et al., 2007)

Curriculum design for pre-service teachers should focus on developing self-efficacy. The teacher-education programs are important in order to increase mathematics pre-service teachers’ self-efficacy beliefs, which are critical in the field of education due to their abilities to influence teaching experiences as well as teacher-student interactions (Bray-Clark & Bates, 2003; Kim et al., 2014). Research has demonstrated that mathematics pre-service teachers, who were exposed to social and verbal persuasion by observing their peers and learning new materials, should have a higher chance to increase their efficacy in comparison to other pre-service teachers (Turner et al., 2000). Charalambous and Philippou (2003) gave evidence for the assumption that was carefully designed for the existing teaching practice programs and could result in positive shifts in the components of the affective domain. In addition, the authors stated that it was possible to modify student-teachers’ efficacy beliefs since their self-efficacy was changeable as with experienced in-service teachers.

#### Self-Efficacy and Attitudes toward Mathematics

Mathematics self-efficacy is “a situational or problem-specific assessment of an individual’s confidence in her or his ability to perform or accomplish a particular mathematical task or problem successfully” (Hackett & Betz, 1989, p. 8). Self-efficacy is also a strong influence for individual or pre-service teachers’ attitudes toward mathematics (Hackett & Betz, 1989). Hembree (1990) compared elementary pre-service teachers’ attitudes in mathematics to students in other majors. The results of that investigations proved that elementary pre-service teachers could have negative attitudes to more than other majors. Indeed, some pre-service teachers confirmed their dislikes for subjects that they were supposed to teach once they got their profession (Bates et al., 2011). Students observed their teachers’ self-efficacy as the subjects and tried to adopt the same behaviors toward the subjects. Often, the attitudes and judgments of teachers concerning their abilities could have a direct impact on the attitudes and outcomes of their students toward their subjects (Hackett & Betz, 2009).

Self-efficacy theories propose that self-efficacy beliefs determine the behaviors of people through the development of attitudes toward their capabilities (Bates et al., 2011; Cone, 2009). Accordingly, when mathematics pre-service teachers develop an attitude toward their abilities, they tend to determine what they can or cannot do with their knowledge and skills (Lampert, 1990; Steele & Widman, 1997). Self-efficacy is a fundamental factor in human competence since it mediates between the beliefs, behaviors, and abilities (Ashton et al., 1982a; Bandura, 1977; Bursal, 2007).

Teacher self-efficacy implies a function of the level of comfort that an individual has with the content taught in the classroom environment and the performance of students. Pre-service teachers with a high level of self-efficacy while handling a reading lesson may show low self-efficacy in teaching mathematics (Arslan & Yavuz, 2012; Brown, 2012; Kim et al., 2014). Such conclusion helps to clarify that the evaluation of the impact of the efficacy of math pre- service teachers should be done with the factors affecting mathematics self-efficacy. Some of the factors identified previously include the performance accomplishment, vicarious experiences, verbal persuasion, and physiological states (Albayrak & Unal, 2011; Briley, 2012; Cakiroglu, 2000; Gresham, 2008; Stipek, Givvin, Salmon, & MacGyvers, 2001). Watson (1987) indicated that there was a high correlation between teacher training programs and pre-service teachers’ attitudes and beliefs toward mathematics. Studies also indicated that teachers with a low opinion of and negative attitudes about math could end up using traditional instructional methods that were essentially in the teacher-directed environment (Brown, 2012; Swars, 2005). Pre-service teachers often preconceived beliefs concerning mathematics and their teaching and learning abilities in the subject (Cakiroglu, 2008). A number of pre-service teachers have different views about mathematics, and most of them originate from their experiences as students. According to Cakiroglu (2008), pre-service teachers also noted that exposure to the creation of new strategies in mathematics methods courses could influence the already achieved level of knowledge in the field of mathematics, as well as the level of teachers’ efficacy. It was important for elementary pre-service teachers to take part in “a mathematics methods course in order to increase mathematics teacher efficacy” (Cakiroglu, 2000, p.92). In addition, pre-service teachers gained and used much-needed teaching experience considering that long-term exposure to different levels of complexities in the methodology class often enhances mathematics teaching efficacy and high-level delivery of lesson content.

#### Conclusion

Educational success depends on the level of knowledge gained in the chosen field and the abilities of teachers to share their knowledge. Mathematics pre-service teachers’ efficacy beliefs influence teachers’ perceptions and judgments, which affect their decisions, behaviors, and practices.

Limited research has explored pre-service teachers’ self-efficacy in general or toward mathematics during the enrollment in teacher education program (Hoy&Woolfolk, 2000). The characteristics associated with pre-service teachers with high levels of self-efficacy are comfortable and more confident in their ability to teach mathematics and develop a strong sense of commitment and motivation to their activities. Teachers want to adapt various strategies and curriculum ideas more in case they are motivated to teach math and develop professional qualities obtained. Self-efficacy has been identified as one way to assist pre-service teachers in obtaining those qualities.

Therefore, the chosen mixed methods study aims to explore the impact of mathematics methodology courses in pre-service early childhood and special educator’ self-efficacy and beliefs. The study also examines the possible factors responsible for the teachers’ self-efficacy beliefs and attempts to determine the levels of self-efficacy of pre-service teachers. It is also necessary to take into consideration the skills of teachers that could be used in the mathematics field and developed during the methodology courses.

In general, Chapter Two provides a clear overview of self-efficacy and its impact on pre-service teachers. The review is divided into several major sections. In one of the sections, it is possible to find out the basics of a self-efficacy theory and teaching efficacy. There is also a summary of the four sources of developing a strong sense of self-efficacy in mathematics pre-service teachers, which include mastery, social experiences, social persuasion and physiological states. Previous self-efficacy research about the teachers and pre-service teachers’ self-efficacy is used in the study. Finally, the information that is relevant to the methodology of the research is mentioned to get the reader prepared for the next chapter.

## Methodology

The purpose of the study is the examination of the impact of mathematics methodology courses on pre-service early childhood and special educator’ self-efficacy and beliefs. Additionally, it is necessary to identify and evaluate the possible factors responsible for the pre-service teachers’ beliefs and determine the levels of self-efficacy of pre-service teachers’ regarding their skills in mathematics methodology courses.

Both quantitative and qualitative methods of data collection and analysis were employed throughout the investigation. The study was conducted with students from the Western University of Pennsylvania. The study was conducted in two phases to gather accurate information regarding students’ efficacy beliefs and various learning aspects during different stages of the mathematics-methodology course. The specific methodology, research strategies, and approaches discussed in this chapter were used to provide a research framework to answer the following research questions:

- Are there any differences in self-efficacy between pre-service teachers, who had content pedagogy courses, and teachers, who had one or two mathematics methods courses?
- Are there any differences in self-efficacy for pre-service teachers between those pre-service teachers, who had one content pedagogy mathematics course, and those pre-service teachers, who had two content pedagogy mathematics courses?
- How does self-efficacy vary among pre-service teachers, who had one methods course, and those, who had two methods courses?
- What is the impact of mathematics methodology courses on pre-service teachers’ self-efficacy?
- Are there any differences in self-efficacy for pre-service teachers of different genders?
- What are pre-service teachers’ perceptions of their skills, competencies, and abilities to teach mathematics?
- What aspects of mathematics methods courses influence the self-efficacy beliefs of future teachers of mathematics?

All these questions were related to both qualitative and quantitative research designs. The connection between the variables of the study could be defined by answering the first three questions in the list. Other questions were aimed at explaining the experiences of pre-services teachers that could be observed during the offered teaching training programs.

### Research Procedures and Methodology

The study adopted a mixed methods research approach in its attempt to examine not only pre-service early childhood teachers’ sense of self-efficacy in relation to teaching mathematics but also the ways of how experiences in teaching mathematics could affect self-efficacy. The advantages of the mixed methods research design include the possibilities of using quantitative and qualitative research methods sequentially in order to investigate a certain phenomenon in detail (Creswell & Clark, 2011; Lampert, 2001). Mixed methods studies were discussed during the paradigm debate period when the effectiveness of and the connection between qualitative and quantitative research methods had to be investigated (Creswell & Clark, 2011). The importance of using those approaches in educational research was discussed due to their ability to yield informative and valuable data in generating a cumulative body of knowledge (Creswell & Clark, 2011; Creswell, 2007).

The study employed quantitative methods in the form of a survey design technique in order to find out the impact of mathematics methodology courses on pre-service teachers’ self-efficacy and identify the differences between self-efficacy levels of the pre-service teachers, who have to cooperate two types of students: those, who completed one mathematics methodology course, and those, who completed two mathematics methodology courses.

Quantitative methods provided a numeric representation of opinions and perspectives (Creswell, 2009). According to Gay and Airasian (2000), quantitative data is used to describe certain conditions, and quantitative data is used to analyze the link between various variables. Thus, the current study utilized quantitative methods to work with the descriptive quantitative data. It is important to note that the use of quantitative descriptive data allowed the researcher to assess the current condition of beliefs held by mathematics teachers in relation to self-efficacy (Creswell & Clark, 2011; Gay & Airasian, 2000). The quantitative research approach was used to investigate the level of self-efficacy of pre-service teachers and the relationship between the components of mathematics methodology courses and its impact on teachers’ self-efficacy related to mathematics.

The qualitative data was used in order to explore pre-service teachers’ perceptions of their skills, competence, and the ability to teach mathematics and determine the aspects of mathematics methods courses that could affect the self-efficacy beliefs of pre-service educators. The qualitative research approach was important in determining the opinions of the participants in relation to variations in the methodologies that could be used to teach mathematics. The data were analyzed to explore the relationship between learning the components of mathematics methodology courses and its impact on teachers’ mathematics self-efficacy (Bandura, 1993; Hall & Ponton, 2005). The qualitative and quantitative research findings offered the information on teaching methods that pre-service teachers perceived as providing the best results (Johnson & Onwuegbuzie, 2004). The argument for using the mixed methods research approach was the necessity to assist the researcher in discussing the relationships between various variables of interest through the quantitative means and of understanding the experiences of pre-service teachers using the qualitative means (Brown, 2012; Creswell, 2009; Creswell & Clark, 2011).

The mixed methods helped to describe a specific class of research when it is possible to combine different research techniques and concepts in a single study to achieve the desired result (Creswell, 2009; Creswell & Clark, 2011; Hall & Ponton, 2005; Johnson & Onwuegbuzie, 2004). The mixed method approach was used in this research. Its focus was on a survey and face-to-face interviews with the participants. The quantitative data was collected and analyzed to increase the quality of research (Creswell, 2009; Creswell & Clark, 2011; Johnson & Onwuegbuzie, 2004). Thus, the stages of the quantitative survey were taken before the administration of the qualitative interviews.

### Selection of Research Participants

The Institutional Review Board approval was obtained prior to data collection. The participants were chosen for this study by means of purposeful sampling. It included all students in Math151, Math 152, Math 320 and Math 330 during the spring semester of 2016 at the Western University of Pennsylvania. All students from the chosen courses were invited to complete a pre-/post-self-efficacy survey. The same students were also invited to be interviewed voluntarily. Six participants (identified with pseudonyms) were randomly selected from the volunteers to participate in the one to one interview. This sample included at least one participant from each course. Pre-service teachers completed an informed consent form prior to participating in the surveys or interviews.

### Purposeful Sampling

A purposeful sampling approach was used in this research (Creswell & Clark, 2011). Purposeful sampling enables the researcher to focus on the population with the characteristics that address the research needs. According to this technique, the participants were selected according to certain inclusion and exclusion criteria (Creswell, 2009). Following the inclusion criteria, only students with the majors in early childhood /special education, who were enrolled in mathematics-methodology courses at the Western University of Pennsylvania, were invited to participate in the study during spring 2016. Those future pre-service teachers, who were not taking mathematics-methodology courses, were excluded from the participation in the study. The chosen approach allowed the researcher to focus on studying how students, who were enrolled in mathematics-methodology courses, would evaluate their self-efficacy in relation to teaching mathematics.

### Population Characteristics

While focusing on the specific population’s characteristics, it is important to note that approximately 80 to 100 students of the Western University of Pennsylvania studying in the mathematics methodology courses were invited to participate in the study because they fit the inclusion criteria. While the age was not an inclusion factor for this study, most of the participants were between 18 and 25 years old. The participants involved in the study were at various stages in their teacher preparation: the first, second, third, or fourth semester of taking mathematics methods courses.

### Study Site

The site selected for the study is the Western University of Pennsylvania. The university serves almost 15,000 undergraduates and about 2,500 graduate students. Students majoring in early childhood and special education are required to take four courses in mathematics methodology. The mathematics methodology courses at this public university are Mathematics for Early Childhood Education (Math 320), Teaching Mathematics in Elementary School (Math 330), Elements of Math I (Math 151), and Elements of Math II (Math 152). Math 320 and Math 330 are the traditional mathematics methods courses. These courses are not required being taken sequentially. Math 151 and 152 require to be taken sequentially. Math 151 and 152 are prerequisites for Math 320 and Math 330.

The description of the courses descriptions should help to get a clear idea of how the study was developed and what kind of information was gathered and introduced to the participants.

### Instruments for Data Collection

Quantitative and qualitative instruments were used in this study to provide complete information regarding pre-service teachers’ self-efficacy beliefs and experiences in teaching mathematics. The quantitative data was gathered with the help of the Mathematics Teaching Efficacy Belief Instrument (MTEBI) that was developed by Enochs, Smith, and Huinker. This instrument was used during the first phase of the survey, including two stages of the quantitative research. The qualitative data were obtained through conducting face-to-face interviews with the participants that were based on the use of a questionnaire developed by the researcher to address the important aspects associated with the investigation.

### The Mathematics Teaching Efficacy Belief Instrument (MTEBI)

Quantitative research was used to test objective theories through the examination of the relationships between the measurable variables. The quantitative data were used to provide the broader understanding of the research problem through the data collected from the participants (Creswell, 2009). Following Creswell and Clark, the quantitative approach added strengths to the research such as 1) a possibility of using the numerical data; 2) independence of the research results; and 3) a possibility of focusing on deductive and explanatory statistical analysis (Creswell & Clark, 2011; Johnson & Onwuegbuzie, 2004). The quantitative approach was used in this research to provide a numerical array of the elementary mathematics teachers’ opinions through a study of a sample of the population (Creswell & Clark, 2011; Hackett & Betz, 2009; Johnson & Onwuegbuzie, 2004). It is important to state that the quantitative approach is effective because it provides the most accurate data regarding the level of participant’s self-efficacy.

The first part of the quantitative phase included Likert-type questions partially derived from an established questionnaire (see Appendix A). The researcher added demographic questions about the students enrolled in the mathematics courses. Students were asked if they had taken mathematics methodology courses. If so, they indicated in which courses. The demographic information about gender information allowed the researcher to explore relationships or patterns of gender to self-efficacy scores. To identify students for the pre- and post- survey, students were asked to indicate the last four cell phones numbers.

The Mathematics Teaching Efficacy Belief Instrument (MTEBI) was used in research to determine students’ self-efficacy ratings regarding their success in teaching mathematics at the beginning and final stages of taking the mathematics methodology course (Enochs et al., 2000). Thus, the MTEBI scale was used to determine the possible development in the pre-service teachers’ beliefs regarding their success in teaching mathematics. The validity and reliability of the MTEBI were supported in many studies. The MTEBI was often used in the studies involving the assessment of teachers’ self-efficacy (Bandura, 1993; Enochs et al., 2000). The MTEBI is a modification of the Science Teaching Efficacy Belief Instrument (STEBI-B) developed by Enochs and Riggs in 1990 (Bleicher, 2004). The instrument employs a five-point Likert scale and includes 21 items divided into two subscales (Enochs et al., 2000). The first subscale is the Personal Mathematics Teaching Efficacy (PMTE) subscale with 13 items in it. The second subscale is the Mathematics Teaching Outcome Expectancy (MTOE) subscale with 8 items in it (Enochs et al., 2000). The two subscales of the MTEBI were adapted to become understandable for teachers in order to guarantee the accuracy of the answers regarding self-efficacy beliefs at the different stages of the survey (Enochs et al., 2000). The reliability analysis helped to find out that the alpha coefficient was 0.88 for the first subscale and 0.75 for the second subscale (Enochs et al., 2000). As a result, it was possible to speak about a high level of the instrument’s reliability. Furthermore, the confirmatory factor analysis indicated the high level of the MTEBI’s validity.

### Interview Questionnaire

The interview questionnaire was designed to conduct the follow-up interviews and gather the information regarding the participants’ opinions on taking the mathematics methodology courses, personal experiences of the participants concerning the courses, and the reasons for the low or high levels of self-efficacy. Approximately, the questions were piloted with three graduate students with whom the researcher had a direct contact. Students had the option to participate in the pilot study or not. Following the pilot of the interview questions, the researcher received feedbacks and made the required changes. The analysis of feedbacks proved that some questions had to be omitted or modified. The pilot study resulted in gathering the participants’ suggestions to promote certain changes in the interview questions and to make sure that the interview questions addressed the research question.

Fourteen open-ended questions were developed by the researcher and included in the interview questionnaire (Appendix B). The interview questions were constructed in correspondence with the research questions so that the participants could provide more detailed information, express their opinions about their personal experiences of the subjects, and make it possible to focus on cross-case comparisons (Creswell & Clark, 2011; Johnson & Onwuegbuzie, 2004). The questions were focused on the changes in the pre-teachers’ experience, the helpfulness of the course, and the possible improvements in pre-teachers’ strategies and the vision of their success. Thus, the questions helped to establish the relationship between the mathematics methodology class and the pre-service teachers’ visions of self-efficacy. All participants responded to the same questions. To contribute to the reliability and validity of the instrument used for conducting interviews, the questions were revised according to experts’ commentaries.

### Data-Collection Methods

The study included both quantitative and qualitative methodologies. The participation was strictly voluntary. The Institutional Review Board approval was obtained prior to data collection. The researcher obtained the approval to conduct the study from a chairperson of the Mathematics Department at the university. The permission from the faculty members, who taught Mathematics during the Early Childhood Education course (Math 320), Teaching Math in Elementary School course (Math 330), Elements of Math I course (151), and Elements of Math II course (152) to distribute a survey in their classes, were obtained. The study was utilized in the way that any suspicion of respondent coercion by a professor or a principal investigator could be eliminated. As such, the coordinator provided a research assistant, who helped in the distribution of the research materials and the collection of data, with the required portion of information. The data collection period of this study included a spring 2016 academic semester and was determined in consultation with the instructors. The students were informed that the participation in the study was voluntary, and that it had no bearing on enrollment or their course grades. The researcher organized an appropriate schedule for the data distribution in cooperation with the Mathematics Program coordinator and faculty members at the Western University of Pennsylvania. As such, the course instructor was not present during the recruitment process, the data collection process, and the distribution of the research materials.

### Phase One

A cover letter and a questionnaire were administrated in the classroom setting to the participants in the study. The letter informed the students of the purpose of the study and their rights (Appendix C). In addition, the cover letter stated that the principle of confidentiality was going to be upheld throughout the entire research process. The participation in the study was voluntary regardless of the survey instruments that were provided to the respondents. Caution measurements were taken to make sure that the identity of the subjects remained anonymous to their peers, as well as to the researcher. The students, who were unwilling to participate in the survey, were given a math activity as an alternative. The pre- and post-surveys took 20 to 30 minutes to complete. Research involved data from the same participant on multiple occurrences, so the pre- and post-surveys needed to be labeled with an identifier.

Confidentiality was maintained by labeling the survey with the last seven cell phone numbers of the participants as the way to match the data in the pre- and post-survey. Once the pre- and post-surveys were conducted, the data was matched using the last four cell phone numbers’ of the participants. Any identifiers were removed from the data. Next, the last four numbers in the participants’ cell phone was securely discarded. To prevent any disclosure of such information, only the researcher had access to the information that linked participants to their responses. Students placed their own surveys in an envelope near the entrance to the classroom.

### Phase Two

A representative of the research team followed the same pre-survey procedures to collect the post-survey data. The list of the last four numbers of the students’ cell phones were securely discarded and removed from the data soon after pre- and post-data was entered.

### Phase Three

The next phase of data collection included face-to-face follow-up interviews. All students were issued with response papers and were asked to drop them in an envelope near to the classroom entrance. The selection of the interview participants was made through the issuance of response papers. All subjects, willing or unwilling to be interviewed, deposited their response papers in the box. Such step was taken to uphold and maintain confidentiality.

The students, who agreed to be interviewed, provided their names and telephone contacts, as well as their email addresses. Those students, who did not agree to be interviewed, deposited blank response paper. The researcher selected six participants from those who expressed their interest in participating in the interview randomly. The sample included the participants who took the methods classes Math 320 or Math 330. The potential participants were contacted individually either through email or telephone calls to schedule a time. The place of the interview for all participants was scheduled individually. The same manner as the survey was developed, cover letters were issued to the participants in the face-to-face interviews that explained the purpose of the interview and their rights (Appendix D). Copies of voluntary consent letters were provided to the participants.

The interview participants were informed well in advance that the interview would take 30 minutes and would be digitally recorded. The identities of those, who responded to the interview questions, were kept confidential by the use of pseudonyms that could make the identification of the participants during the discussion of the research findings possible. The researcher stored personal information of the participants and data safely in a locked cabinet in the researcher’s home. The confidentiality and anonymity of the participants were guaranteed (Turner, 2010). The participants were offered a $20 gift card as a kind of appreciation for their participation, time, and effort.

### Analysis Methods

The results of the quantitative survey were analyzed with the help of a statistical tool known as SPSS (Creswell, 2007; Creswell, 2009). The program allowed the researcher to conduct the complete descriptive and inferential statistical analyses. The use of the computer software was necessary to present the most accurate results, state the relationship between the variables, and make conclusions about the observed correlations (Cohen, 1988; Creswell, 2007; Creswell, 2009). The analysis of covariance (ANCOVA) and the paired t-test were used to determine if the differences between the courses Math 151, Math 152, Math 320, and Math 330 existed when controlling for pre-survey.

The analysis of the qualitative data was based on the process of coding and identifying thematic patterns in the participants’ answers to the interview questions. First, the researcher focused on the transcribing process of the recorded interviews and reviewed the data in order to pay attention to all the details mentioned by the participants (Cohen, 1988; Creswell, 2007; Creswell, 2009). The material was read several times to code different segments of the qualitative data according to the determined themes (Turner, 2010). After coding the data, the researcher organized the qualitative material in regards to the themes raised by the participants (Turner, 2010). The emergent themes and the main characteristics of the qualitative data were identified as a result of the qualitative data analysis stage.

### Ethical Issues

Ensuring confidentiality is necessary in order to encourage the participation of people and promote honest responses (Creswell, 2007). Therefore, the researcher sought permission from the individuals to participate in the study. The participants were informed of the precautions that would be taken to protect their confidentiality. The participation in the study was voluntary, and the participants were free to withdraw it any time. The participation or non-participation in the study was entirely voluntary and had no bearing on enrollment or students’ grades.

### Chapter Summary

The purpose of the study was to examine the impact of mathematics methodology courses on pre-service teachers’ self-efficacy and beliefs. It also examined the possible factors responsible for the pre-service teachers’ teaching efficacy beliefs. In addition, an attempt to determine the levels of the self-efficacy of pre-service teachers’ regarding their skills in the mathematics methods courses was made. Quantitative and qualitative methods of data collection were employed throughout the investigation to investigate the visions of the students of the Western University of Pennsylvania. The quantitative research method consisted of two stages during which the participants completed the surveys with the help of the Mathematics Teaching Efficacy Belief Instrument (MTEBI). The qualitative research method involved those students, who agreed to participate in the follow-up interviews. To explore the teachers’ self-efficacy beliefs, the researcher asked six participants to answer the open-ended interview questions prepared in correspondence with the research topic. The quantitative data was analyzed with the help of the t-test and appropriate SPSS software to calculate the statistical variances.

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