## Abstract

A mixed methods research study was conducted to explore the impact of mathematics methods courses and content pedagogy courses on pre-service Early Childhood and Special Educators’ self-efficacy and beliefs. The purpose was to examine the possible factors responsible for the pre-service teachers’ beliefs, and to determine the levels of self-efficacy of pre-service teachers regarding their skills in the mathematics content pedagogy and mathematics methods courses. The quantitative sample included 164 Early Childhood with Special Education pre-service teachers who were given MTEPI instrument at the beginning and at the end of the courses. Qualitative data were collected in the form of interviews in which six pre-service teachers who were then currently enrolled in mathematics methods courses discussed their perceptions of the topic.

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The findings of the quantitative and qualitative data indicated a significant difference in self-efficacy from the pre-instrument at the beginning of the mathematics methods courses to the post-instrument at the end of those courses. However, the quantitative data revealed a decrease in pre-service teachers’ self-efficacy after the content pedagogy courses. On the other hand, the results of the qualitative data revealed that pre-service teachers believed the content pedagogy courses increased their self-efficacy. The qualitative data also revealed that the growth in self-efficacy related consistently to mastery experiences, vicarious experience, verbal persuasion and physiological states. Moreover, it was found that all the participants after taking the mathematics methods courses seemed to be more confident in overcoming the challenges and implementing new educational practices, which are related to some of the characteristics associated with high self-efficacy. Thus, the data gathered in this study indicated that pre-service teachers need active involvement opportunities, modeling opportunities, effective feedback, and supportive environments to improve their self-efficacy.

## Introduction

The definition of the term “Self-efficacy” is a “belief in one’s capabilities to organize and execute the courses of action required to produce given attainments” (Bandura, 1997, p.3). College students, pre-service teachers, and teachers could each face concepts of self-efficacy. An individual can have different types of self-efficacy concerning different academic subjects, such as reading and mathematics (Bandura, 1986). A pre-service teacher who demonstrates a high level of self-efficacy while handling a reading lesson, for example, may show lower self-efficacy in teaching mathematics (Arslan & Yavuz, 2012; Brown, 2012; Kim, Sihn & Mitchell, 2014). The available research on self-efficacy demonstrates that the concept has been studied from many perspectives (Alsup, 2004; Bleicher, 2004). In fact, it has been at the core of educational studies for several decades as one of the aspects for influencing the behaviors, attitudes, and effectiveness of teachers and their students (Albayrak & Unal, 2011; Tschannen-Moran & Hoy, 2001). It is also one of the factors that predicts a teacher’s persistence in the field of education (Tschannen-Moran, & Hoy, 2001). Accordingly, when pre-service teachers or teachers develop an attitude about their abilities, they tend to determine what they can do or not do with their knowledge and skills (Lampert, 1990; Steele & Widman, 1997).

Because of the effect of self-efficacy on behaviors and attitudes, self-efficacy theory is of interest to all who are concerned with learning and teaching (Bandura, 1997). Most of the research about self-efficacy in the field of education focuses on teaching mathematics as in-service teachers (Gibson & Dembo, 1984). Therefore, a need exists 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) before they become in-service teachers. Consequently, it is important for researchers to investigate pre-service teachers’ self-efficacy beliefs regarding the teaching of mathematics (Tschannen-Moran & Hoy, 2001). This study helps to examine the impact of mathematics methods courses on the pre-service teachers’ self-efficacy and the possible factors responsible for their self-efficacy in teaching effectively.

The first years of learning how to teach are important for the development of a teacher’s self-efficacy (Bandura, 1997; Tchannen-Moran, Hoy, & Hoy, 2001). For this reason, pre-service teachers’ knowledge and beliefs are two important aspects that cannot be neglected by researchers and have to be evaluated in order to support the idea of lifelong-learning self-efficacy (Briley, 2012). Consequently, it is important to increase self-efficacy because it could predict pre-service teachers’ behaviors, attitudes, and effectiveness in the classroom context (Albayrak & Unal, 2011; Haverback & Parault, 2008). In contrast, mathematics pre-teachers teachers with low self-efficacy are likely to show less effort and commitment in the classroom (Tschmannen-Moran, Hoy, & Hoy, 2007). 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) Consequently, when pre-service teachers learn and apply some key teaching strategies, their self-efficacy may improve (Albayrak & Unal, 2011).

### Purpose of the Study

The purpose of this study is to examine the impact of mathematics methods courses on pre-service Early Childhood and Special Educators’ self-efficacy and beliefs. Additionally, the purpose is to examine the possible factors responsible for the pre-service teachers’ beliefs, and to determine the levels of self-efficacy of pre-service teachers regarding their skills in the mathematics methods courses.

### The Statement of the Problem

According to the National Council of Teachers of Mathematics (NCTM, 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). Teachers who demonstrate confidence in their ability to teach mathematics 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 teachers’ self-efficacy that guides students’ behavior and the actions as chosen by students to achieve the required goals (Ashton & Webb, 1986; Gibson & Dembo, 1984; Muijs and Reynolad, 2002). Different research studies indicate that teachers’ self-efficacy influences students’ achievements by increasing students’ motivation and self-efficacy (Haverback & Parault, 2008; Hoy & Spero, 2005; Turner, Cruz & Papakonstantinou, 2004). Thus, there is an important need to determine the best methodology to promote pre-service teachers’ self-efficacy. This importance can be explained by the possible impact of self-efficacy beliefs regarding mathematics on pre-service teachers’ effectiveness, attitudes, and behaviors (Swars, 2008).

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### Definition of Terms

Early-childhood/special education program: An educational major that prepares students to be teachers of learners from Pre-K through Grade 4, as well as special education students Pre-K to Grade 8 (Indiana University of Pennsylvania, 2013, p. 67).

Mathematics teaching efficacy: Teacher’s conception of their ability to promote learners’ achievement in mathematics (Enochs, Smith, & Huinker, 2000).

Pedagogical content knowledge: The combination of the knowledge of mathematical content, knowledge of pedagogy, and knowledge of how children learn (Shulman, 1986).

Pre-service teachers: Students enrolled in an education program at an institution of higher learning.

Self-efficacy: “beliefs in one’s capabilities to organize and execute the courses of action required to produce given attainments” (Bandura, 1997, p.3). Teacher self-efficacy or teacher efficacy is considered a type of self-efficacy. Therefore, the researcher used “teacher efficacy” interchangeably with “self-efficacy” in this study (Tschannen-Moran & Hoy, 2001).

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

Teacher efficacy: A “judgment of a teacher’scapabilities to bring about desired outcomes of student engagement and learning, even among those students who may be difficult or unmotivated” (Tschannen-Moran & Hoy, 2001, p.783). Teaching efficacy is considered a type of self-efficacy. Therefore, the researcher used “teacher efficacy” interchangeably with “self-efficacy” in this study.

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### Re**search Questions**

Seven research questions guide this study:

- Are there differences in self-efficacy between pre-service teachers who have had content pedagogy courses and those who have had mathematics methods courses?
- Are there differences in self-efficacy of pre-service teachers between those pre-service teachers who have had one content pedagogy mathematics course and those pre-service teachers who have had two content pedagogy mathematics courses?
- How does self-efficacy vary among pre-service teachers who have had one methods course and those who have had two methods courses?
- What is the impact of mathematics methods courses on pre-service teachers’ self-efficacy?
- Based on gender, are there differences in self-efficacy of pre-service teachers?
- What are pre-service teachers’ perceptions of their skills, competence, and ability 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 are aimed at investigating the relationships among variables, whereas the sixth and seventh questions are important for understanding the experiences of pre-service teachers during their teaching education program.

### Theoretical Position

The conceptual framework in this mixed methods research provided a basis for understanding pre-service teachers’ self-efficacy regarding teaching mathematics and examining the factors in relation to past practices to improve their self-efficacy. Self-efficacy beliefs determine the behaviors of people through the development of attitudes toward their capabilities (Bates, Latham & Kim, 2011; Cone, 2009). It refers to the “beliefs in one’s capabilities to organize and execute the courses of action required to produce given attainments” (Bandura, 1997, p.3). This term can be applied to different areas of activities, specifically teaching. This concept is associated with pre-service teachers’ self-efficacy that can be described as a degree to which educators could believe in their ability to promote students’ learning or their cognitive skills development.

This attribute is important for educators in different content areas, but specifically toward teaching mathematics. The theoretical perspectives of this study also revealed more specific detailed information about the developments of pre-service teachers’ self-efficacy beliefs after the mathematics methods courses. It was critical to acknowledge four factors that could impact pre-service teacher’s self-efficacy: mastery experiences, vicarious experiences, social persuasion, and physiological states (Bandura, 1986). As Bandura’s theory indicates, pre-service teachers’ success is related to these four factors in the teaching and learning environment. Therefore, this theory offers the perspective that self-efficacy is an important factor in how pre-service teachers learn as they prepare to become classroom teachers.

### Significance of the Study

Over the years, a number of studies have investigated the topic of teachers’ self-efficacy and it has been discussed in educational research (Ashton & Webb, 1986; Gibson & Dembo, 1984; Guskey & Passaro, 1994; Swars, 2005). However, important gaps in the literature remain. These include: 1) a limited amount of research studies examining pre-service teachers’ mathematics self-efficacy; 2) a limited number of research studies examining how mathematics methods courses can affect self-efficacy of future educators; and 3) a limited amount of qualitative research studies examining the effect of the mathematics methods courses on self-efficacy. Therefore, the results of this study could help to explore the self-efficacy level of pre-service teachers after their participation in mathematics methods courses by using a mixed methods design. Mathematics methods courses can be viewed as an independent variable that shapes the beliefs of pre-service teachers about their competence, skills, and the abilities to improve children’s learning behavior (Dembo & Gibson, 1985). The teaching challenges that pre-service teachers encounter can be partly explained by the lack of confidence in their ability to teach (Enochs, Smith, & Huinker, 2000). This study will be of great significance to teacher education programs on developing an appropriate learning environment and the opportunities for customized professional growth in pre-service teachers’ self-efficacy.

### Methods

In this study, the researcher utilized interviews and data collection instruments to gain information regarding the influence of the mathematics methods courses on pre-service Early Childhood and Special Educators’ self-efficacy and to determine the level of the prospective pre-service teachers’ self-efficacy. It was a mixed methods study involving a western Pennsylvanian university. The reason behind mixing methods is to allow the researcher to investigate while using the strengths of each method. This study consisted of three phases of data collection. The first phase of the study was the pre-instrument that measured the level of the pre-service early childhood teachers’ self-efficacy before the mathematics methods courses. The instrument consisted of 21 Likert scale questions developed by Enochs, Smith, and Huinker (2000) (see Appendix A).

Permission was granted to use the instrument in the study from the authors (see Appendix B). The instrument has been modified to collect additional demographic data. The second phase of the study was the post instruments using the same instrument as the first phase, and the same procedures were followed to distribute the post-instrument. The third phase of the study consisted of follow-up qualitative interviews with six students who were taking MATH 320 and MATH 330 because the study focused on examining the influence of the mathematics methods courses on pre-service teachers’ self-efficacy. The interviewees were randomly selected from the participants who indicated a willingness to participate in a follow-up interview. The interview questions consisted of 16 open-ended questions in order to provide a broader understanding of pre-service teachers’ perceptions about the mathematics methods courses (see Appendix C).

### Delimitations of the Study

The sample of this study included only college students attending one western Pennsylvanian university who were majoring in Early Childhood Education with Special Education and were enrolled in one of the following Early Childhood Education courses: Elements of Math I (MATH 151), Elements of Math II (MATH 152), Mathematics for Early Childhood (MATH 320), or Teaching Math in Elementary School (MATH 330). The focus of the study was on the experiences of the participants.

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### Summary

The descriptive study is used to explore the impact of mathematics methods courses on pre-service Early Childhood and Special Education pre-service teachers’ self-efficacy. It also examined the possible factors responsible for developing the teaching efficacy beliefs. In addition, it attempted to determine the levels of the self-efficacy of pre-service teachers’ regarding their own skills in the mathematics methods courses. The second chapter presents a review of the literature as it relates to pre-service teachers’ self-efficacy. The review of the literature also provides an explanation of the theory of self-efficacy. It also outlines a summary of the four sources for developing a strong sense of self-efficacy in mathematics pre-service teachers, which include mastery experiences, social experiences, social persuasion, and physiological states. The third chapter provides the information relevant to the methods of the research. The study is a mixed methods design, utilizing both quantitative and qualitative data collection techniques to examine pre-service Early Childhood and Special Educators’ self-efficacy. The chapter provides a description of the subject selection for the sample, an analysis of the collected data, and a discussion of the results. The fourth chapter describes the results of the study. The fifth and final chapter of the dissertation provides a discussion of the findings, an overview, and summary of the research, as well as recommendations for future research.

## Literature Review

The purpose of the study is to examine the impact of mathematics methods courses on pre-service Early Childhood and Special Educators’ self-efficacy and beliefs regarding teaching mathematics to children. An additional purpose of this study is to examine the possible factors responsible for the pre-service teachers’ beliefs and to determine their levels of self-efficacy with regard to teaching mathematics. This chapter begins with a literature review about self-efficacy, self-efficacy factors, and pre-service teachers’ characteristics and belief systems. A brief literature review is also included about pedagogical content knowledge. The final section is a review of research—practice connections and conclusions.

### Self-Efficacy

Self-efficacy is one of the fundamental beliefs of the social-learning theory (Bandura, 1997). This concept is incorporated into education research to understand how physiological variables could influence students’ and teachers’ performance (Bray-Clark & Bates, 2003). Self-efficacy refers to “the belief in one’s capability to organize and execute the courses of action, which are required to produce given attainments” (Bandura, 1997, p. 3). Kinzie, Delcourt, and Powers (1994) reported that “self-efficacy reflects an individual’s confidence in his/her ability to perform the behavior required to produce specific outcomes and it is 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” (p. 747). A self-efficacy belief has two components: outcome expectancy and efficacy expectation (Bray-Clark & Bates, 2003). The efficacy expectancy (personal mathematics teaching efficacy) refers to “conviction that one can successfully execute the behavior required to produce the outcome.” On the other hand, outcome expectancy (in this study, mathematics teaching outcome expectancy) is “a person’s estimate that a given behavior will lead to certain outcomes” (Bandura, 1977, p.193).

Most people develop self-efficacy through observational learning and experiences in social settings while developing their personality (Czerniak & Schriver, 1994). That is, people’s experiences provide them with an opportunity to develop self-efficacy. Abilities, attitudes, and cognitive skills make up self-efficacy, which play 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 thus they 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 unconfident in their ability to achieve positive outcomes (Pendergrast, Garvis, & Keogh, 2011).

According to Bandura (1997), people’s choices to handle or avoid challenges depend on their level of self-efficacy. Self-efficacy affects thinking and can enhance the level of cognitive performance (Pajares, 1996). A person’s 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) purported 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). Therefore, it can be surmised that self-efficacy influences pre-service teachers’ responses, persistence, and efforts when learning to teach. Albayrak and Unal (2011) state 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 noted that individuals must demonstrate the necessary knowledge, skills, and self-efficacy beliefs to develop the capacity to perform specific actions efficiently. Following this explanation, Berna and Gunhan (2011) acknowledged that individuals with a strong sense of self-efficacy beliefs may show more effort when they learn the subject matter and the ways of teaching.

According to Bandura (1994),” self-efficacy beliefs determine how people feel, think, motivate themselves and behave. Such beliefs produce these diverse effects through four major processes: cognitive, motivational, affective and selection processes” (p. 71). It not only affects people’s judgments and perceptions, but also shapes how they perform in a given scenario (Hinton, Flores, Burton, & Curtis, 2015; Pajares & Graham, 1999; Phan, 2012). Self-efficacy theory can be applied to almost everyone (Bandura, 1997). Therefore, this research study seeks to connect self-efficacy and relevant practical applications for developing self-efficacy in the literature to pre-service teachers when preparing to teach mathematics concepts to children.

### In-Service Teachers and Teacher Efficacy

Teacher efficacy is a construct developed from the self-efficacy theory. With reference to Bandura’s theoretical framework, Tschannen-Moran and Hoy (2001) defined teacher efficacy as “a judgment of his/her capabilities to bring about desired outcomes of student engagement and learning, even among those students who may be difficult or unmotivated” (p. 783). Teaching efficacy predicts “the percentage of goal achieved, amount of teacher change, improved student performance, and continuation of both project methods and material” (Gibson & Dembo, 1984, p. 173). The views, perceptions, and beliefs held by teachers affect their ability to teach and manage learning activities effectively in the classroom. For example, Tschannen-Moran and Hoy (2001) noted that teachers with high self-efficacy are more organized in the classroom and more inclined to adapt new methods. The quality of students’ performance and students’ attitudes toward their tasks are under a considerable influence of their teacher’s self-efficacy (Granger, Bevis, Saka, Southerland, Sampson, & Tate 2012; Guskey & Passaro, 1994; Hackett & Betz, 1989; Hofer & Pintrich, 1997; Lampert, 1990; Marshall, 2007; Pajares & Graham, 1999; Rimm-Kaufman & Sawyer, 2004).

A teacher education program at an institution of higher learning should consider teacher efficacy as one of the important factors that could affect pre-service teachers’ readiness to teach (Jerkins, 2001; Tschannen-Moran & Hoy, 2007). It is important to consider that “teacher’s sense of efficacy is a mediating cognitive process that significantly influences teacher motivation, professional duration, and teacher adjustment” (Jerkins, 2001, p. 6). Teachers’ beliefs shape the students’ learning and achievements (Albayrak & Unal, 2011; Bandura, 1977; Dembo & Gibson, 1984; Pajares, 1992; Muijs & Reynolad, 2002; Tschannen-Moran, Hoy, & Hoy, 1998). Besides, the ability of teachers to perform particular teaching tasks successfully in their current teaching conditions depends on teachers’ efficacy (Lampert, 1990; Steele & Wildman, 1997).

Teachers’ efficacy beliefs are strong determinants of the extent to which they can accomplish various tasks (Pajares, 1996). Indeed, teachers’” beliefs 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).

The investigation of the influence of self-efficacy on teaching has been a leading concern for several educational studies (Battista, 1994; Bray-Clark & Bates, 2003; Charalambous & Philippou, 2003; Czerniak, 1990; Gavora, 2011; Hoy & Spero, 2005). Most of these studies relate the concept of self-efficacy belief with the teacher efficacy belief to demonstrate how teacher efficacy 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).

Indeed, other studies indicated that teachers with positive teaching efficacy beliefs can be engaged in risk-taking behaviors such as trying harder with mathematics 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 a personal belief that they have the capacity to influence student achievement and motivation (Ashton & Webb, 1986; Savran-Gencer & Cakiroglu, 2007). In their investigations, Kim, Sihn, and Mitchell (2014) acknowledged 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). Tschannen-Moran and Hoy (2001) asserted that teachers could assume that student learning originates 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 instruction and management, reflective teaching, student motivation and engagement, and stakeholder engagement in the educational process (Kazempour, 2008).

Teachers with high self-efficacy are more willing to adapt and use several instructional strategies (Riggs & Enochs, 1990). Teacher efficacy is shown through the use of various instructional and student-centered approaches (Tschannen-Moran & Hoy, 2001). A diverse range 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, Cruz and Papakonstantinou (2004) mentioned that teachers’ self-efficacy has a positive association with the willingness of a teacher to implement new teaching ideas. 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). Accordingly, students tend to work in groups to acquire knowledge, and they approach the teacher only when they experience a significant setback or challenge (Czerniak, 1990).

In contrast, teacher-centered learning entails a situation whereby the teacher controls all class activities and allows little room for student contribution (Hoffman, 2010). Teachers with low self-efficacy tend to use teacher-centered learning more than student-centered learning (Guskey & Passaro, 1994). In fact, Swars’ study (2005) proved 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 and technique, which is different from highly effective teachers who tend to build confidence among students, use student groups, and generously allow the learners to explore through their learning process for optimal comprehension (Muijs and Reynolds, 2002). Thus, it is important to assess various self-efficacy beliefs during education processes because teacher efficacy has a certain power over teachers’ actions and decisions and consequently, student performance.

### Self-Efficacy Factors

Bandura (1977) identified four major sources (mastery experiences, vicarious experiences, social persuasion, and physiological state) that contribute to the growth of self-efficacy beliefs. The four factors are:. Tschannen-Moran and Hoy (2007) investigated “it is of both theoretical and practical importance to understand the sources teachers tap when making judgments about their capability for instruction” (p. 953). It is necessary to understand the four sources’ possible effects on teaching strategies, and the possible development and improvement of self-efficacy among pre-service teachers.

### Performance Accomplishment or Mastery Experiences

“Performance accomplishment” refers to previous task experiences (Bandura, 1986) that could be constructed by doing a task (Bandura, 1986). Pre-service teachers’ mastery experiences could be developed by practicing teaching. It is considered the greatest contributor and influential source of efficacy information because pre-service teachers who have success in a task are likely to perform successfully in similar tasks in the future (Charalambous & Philippou, 2003). However, not all successful experiences reinforce self-efficacy. For instance, 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, which allows pre-service teachers to believe that they have the required skills to teach mathematics. However, there is a concern that the level of self-efficacy could be weakened in case a person cannot complete a task or understand the task. (Enochs, Smith, & Huinker, 2000).

### Vicarious Experiences

As identified by Hoy and Spero (2005) vicarious experiences are those that are usually modeled by someone else. The term also refers to such methods as observation or participation. Research indicates that vicarious experiences may modify self-efficacy beliefs, expectations, or judgments about self-competence through comparison with the achievement of others (Berna & Gunhan, 2011). According to Schunk and Zimmerman (2007) “observing similar others succeed at a task, such as reading aloud in front of the class, may raise observers’ self-efficacy” (p.10). This aspect implies that watching a peer with 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 specific tasks, others can be motivated that they can do the same. Tschannen-Moran, Hoy, & Hoy (1998) stated that the more closely the pre-service teacher identifies with the cooperating teacher as a vicarious model, the stronger the impact on the pre-service teacher’s efficacy formation.

This means that self-efficacy would be stronger if an observer relates to others who display confidence and self-efficacy. Therefore, when pre-service teachers watch other experienced teachers complete their tasks successfully, they will also want to trust their abilities and work hard to achieve the same outcomes. Also, Battista (1994) stated that when people see others with similar characteristics succeed, they raise their own beliefs and try to find out the same capabilities and chances to succeed. Pre-service teachers’ self-efficacy may be improved by observations of the actions of other people (Bray-Clark & Bates, 2003). Thus, teacher education programs could increase pre-service teachers’ self-efficacy by encouraging cooperative learning experiences in actual classroom settings and encourage positive feedback from the cooperating teacher. In turn, pre-services teachers could explain mathematical concepts to children as a good strategy to improve the pre-service teachers’ sense of self-efficacy.

### Verbal or Social Persuasion

Verbal or social persuasion provides a further opportunity for reinforcing the beliefs of an individual, particularly in the context in which others provide information or explanation in the skills performed by the individual (Charalambous & Philippou, 2003). This assertion could be used when encouragement is provided effectively and realistically by actual experiences (Berna & Gunhan, 2011; Bursal & Paznokas, 2006; Phelps, 2010; O’Reilly, Renzaglia, & Lee, 1994). 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 demonstrate that individuals are more likely to do the task when they are persuaded that they can succeed. Social persuasion is important in removing past hindrances responsible for encouraging self-doubt and disorder. Additionally, it influences the credibility, trustworthiness, and expertise of convincing individuals (Hoy & Spero, 2005). Any feedback should be positive, corrective, and immediate in order to gain successful results (Coulter & Grossen, 1997; O’Reilly, Renzaglia, & Lee, 1994). In teacher education programs, pre-service teachers are exposed to corrective and positive feedback, which raises their confidence so that they can also succeed (Enochs, Smith & Huinker, 2000; Hackett & Betz, 1989).

### Emotional and Physiological States

Physiological states emphasize how positive feelings such as relaxation and confidence or negative mood or feelings such as anxiety, fear, and fatigue affect people’s decisions (Charalambous & Philippou, 2003). According to Bandura (1997), self-efficacy could be developed by “enhanced physical status, which reduces stress levels and negative emotional proclivities, and corrects misinterpretations of bodily states” (p. 106**). **In pre-service teachers’ cases, it refers to the feeling or the mood after various teaching experiences in the mathematics methods courses.

The levels of arousal, fatigue, stress, anxiety, tension, pain, and mood conditions are signals that change the levels of self-efficacy (Bandura, 1986). Battista (1994) argues that emotional reactions and responses to situations influence the development of self-efficacy. This concept implies that emotions, moods, stress, and physical reactions affect a person’s opinion of his/her abilities in a given situation (Hall & Goetz, 2013). On the contrary, the most significant factor is the perception and interpretation that a person uses to reduce stress and improve mood during different challenging and difficult tasks (Ashton & Webb, 1986; Battista, 1994; Cakiroglu, 2008). Chong and Kong (2012) stated that to improve pre-service teachers’ self-efficacy, the teacher educator should use instructional practices that provide a supportive environment. They found that group activities and collaborative work could reduce stress. The strategies or instructional practices teacher educators use to reduce stressful situations and minimize anxiety, including the creation of a friendly environment and communication, play an important role in the growth of pre-service teachers’ self-efficcy (Chong & Kong, 2012; Turner, Cruz, & Papakonstantinou, 2004). It should be noted that comfortable classroom environments promote positive mood (Ryan, & Patrick, 2001). Educators could support pre-service teachers’ positive mood through instructor encouragements and working with groups (Ashton & Webb, 1986).

### Characteristics Associated with Pre-Service Teachers

#### Low Self-Efficacy Characteristics

Pre-service teachers with a low sense of self-efficacy display characteristics that affect the student educational outcomes and achievements. Tschannen-Moran and Hoy (2007) posit:

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).

Pre-service teachers with a low sense of self-efficacy are often less motivated and tend to keep away from demanding and difficult tasks (Tschannen-Moran, Hoy, & Hoy, 1998). 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). Thus, they tend to use the teacher-directed strategies instead of the child-centered strategies (Swars, 2005). The practice becomes routine among pre-service teachers who avoid difficult tasks which undermines their ability to acquire the required skills to teach mathematics (Clift & Brady, 2005). Such pre-service teachers try to concentrate on their limitations, failures, and negative outcomes (Bates, Latham, & Kim, 2011; Charalambous, Philippou, & Kyriakides 2008). Pre-service teachers are less confident when it is time to teach and use new strategies, and strategies that they think they will not master (Tschannen-Moran, Hoy, & Hoy, 1998). Self-efficacy beliefs are influenced by pre-service teachers’ previous performance (Bandura, 1997). Therefore, unsuccessful experiences of pre-service teachers may form the reasons for why teachers develop low-efficacy beliefs indicated above (Cone, 2009).

#### High Self-Efficacy Characteristics

Pre-service teachers with high self-efficacy have a number of characteristics that distinguish them from other pre-service teachers. Some of these characteristics are listed below:

- They are usually more motivated in order “to initiate a task, attempt new strategies, or try hard to succeed” (Tschannen-Moran, Hoy, & Hoy 1998, p. 212). Besides, such pre-service teachers try to stay motivated all the time in order to write various lesson plans and focus on learning as something exploratory, not routine (Clift & Brady, 2005).
- They are expected to be more successful and aware of how to deal with different circumstances (Tschannen-Moran, Hoy, & Hoy, 1998). They are more willing to master a problem successfully at the course level and in the future. 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).
- They are more willing to change or adapt their teaching practices to include all students on all levels at the same time. For example, pre-service teachers with high self-efficacy want to use manipulative techniques and work with future students who encounter difficulties (Enochs, Smith, and Huinker, 2000; Turner, Cruz, Papakonstantinou, 2004). As soon as pre-service teachers enrich their experiences, they get all chances to improve their methods in teaching (Huinker & Madison, 1997; Kagan, 1992).
- They are more likely committted to their activities (Enochs, Smith & Huinker, 2000). Such commitment allows pre-service teachers to acquire new skills since they are often ready to learn approaches and strategies for tackling mathematics problems and challenge students (Czerniak & Schriver, 1994; Esterly, 2003; Riggs & Enochs, 1990).
- They usually use the more child-centered or constructivist learning environments (Swars, 2005). They try to identify different tasks that must be mastered and co-constructed by the teachers and children in completing the required goals (Bates, Latham & Kim, 2011; Cone, 2009).
- They can create new strategies to solve different class problems since self-efficacy affects their choices and their instructional decisions, mathematics (Ashton, Webb & Doda, 1982b).

### Pedagogical Content Knowledge

There are seven kinds of knowledge required of teachers. They should be knowledgeable about: 1) general pedagogical, 2) learners, 3) educational context, 4) educational ends, 5) content knowledge, 6) pedagogical content, and 7) curriculum (Shulman, 1987). This study focused on pedagogical content knowledge because the study was conducted with courses that are designed to deepen the pre-service teachers’ understanding of elementary mathematics and pedagogy.

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

The most useful form 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). On the other hand, pedagogical knowledge is knowledge that entails the strategies and principles of classroom management and organization in education (Shulman, 1987). All learners need teachers who have expertise in their subject and who also knows how to teach the subject to their students. Teacher education programs should create the connection between the knowledge of the subject and the knowledge of teaching and learning to prepare pre-service teachers effectively (Cheong, 2010; Huinker & Madison, 1997; Shulman, 1989; Swars, 2005)

Pre-service teachers should have knowledge that goes beyond the common knowledge in mathematics (Ball & Bass, 2000). Thus, teacher education programs should consider improving their mathematics courses that focus on how to teach mathematics to increase their future students’ sense of self-efficacy (Palmer, 2001). One of the most important characteristics of an effective program to enhance pre-service teachers’ self- efficacy is the provision of necessary content and pedagogical knowledge (Guskey & Passaro, 1994).

Teachers with strong pedagogical content knowledge are more confident than other teachers with weaker pedagogical content knowledge (Van Driel, Veal & Janseen, 2001). Additionally, teachers who are less confident about their ability to teach may affect the development of their own knowledge of the subject (Van Driel, Veal & Janseen, 2001). Mixing mathematical and pedagogical tasks could enhance mathematics knowledge among pre-service teachers (Shulman, 1987). 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 (Swackhamer, Koellner, Basile, & Kimbrough, 2009).

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 subjects (Shulman, 1989). Yilmaz’s investigation (2011) showed that there is a positive link between teaching self-efficacy and pedagogical content knowledge (Palmer, 2001). Indeed, according to Schunk and Zimmerman (2007), “high self-efficacy will not produce competent performance when requisite knowledge and skills are lacking” (p.3). Also, the findings of a study done by Palmer (2001) showed that pre-service teachers’ self-efficacy levels increased by the content-specific knowledge through pedagogical emphasis. Pedagogical content knowledge increases 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). In this regard, 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 discussion, debate, and decision-making on how to teach because it is the only way in which sufficient knowledge or pedagogical content knowledge can develop. On this matter, discussion, debate, and decision participation improves teachers’ self-efficacy concerning content delivery.

Ball and Bass (2000) explained 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 experiences. However, there are some strategies that could help teacher education programs connect theory to practice through case-based approach, collaborative learning environments, or role playing (Ewing, Smith & Horsley, 2003). The usage of any of these techniques allows pre-service teachers to experience the children’s reasoning, apply what they have learned, experience the complexity of teaching and learning, and provide reflective practice (Ewing, Smith & Horsley, 2003).

Pre-service teachers need to have mathematics knowledge beyond the knowledge that is required for other majors (Ball & Bas, 2000). Thus, if pre-service teachers want to demonstrate effective practices, they need to know the subject they teach and how to teach it (Shulman, 1986). For example, 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 representations (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 the way” (Beckmann, 2014, p. xvi). Indeed, they need to know how to teach mathematics problems in different ways and figure out how and where students usually could make mistakes (Ball & Bass, 2000). In sum, pre-service teachers or teachers need sufficient knowledge and higher self-efficacy to be highly effective in teaching mathematics (Muijs & Reynolds, 2002).

Self-efficacy, as well as content knowledge, could be increased through mathematics methods courses (Wilkins, 2008; Hinton, Flores, Burton, & Curtis, 2015). For example, the results of Swars, Hart, Smith, Smith, and Tolar (2007) study showed that pre-service teachers’ self-efficacy increased after taking the first mathematics methods courses. After the second mathematics course that had field experiences, their self-efficacy did not increase. They explained the decreasing in their self-efficacy as related to the unrealistic expectation about teaching mathematics in the field experiences. This self-efficacy level was likely due to the difficulty of putting into practice the concepts they had learned in their mathematics methods course. The researchers concluded from their findings that there is a strong relationship between content knowledge and self-efficacy in the classroom setting.

### Summary of Research Studies on Pre-Service Teacher’ Self-Efficacy

The research on the effect of various methods courses on self-efficacy provided at teacher education programs is gaining importance as more educators and other relevant stakeholders realize that teacher quality is tied to the teachers’ skills and beliefs (Arslan & Yavuz, 2012; Haverback & Parault, 2008; Isiksal, 2005; Kim, Sihn, & Mitchell, 2014; Lancaster & Bain, 2010). Few studies have been performed to analyze mathematics-teaching efficacy among early childhood pre-service teachers. However, available studies have shown that mathematics methods and content courses could increase pre-service teachers’ self-efficacy (Cakiroglu, 2000, Huinker & Madison, 1997).

Moreover, such improvements were evident after completing content in mathematics courses (Hinton, Flores, Burton, & Curtis, 2015; Swars, Hart, Smith, & Tolar, 2007) Therefore, the role of methods courses in teacher education program is important because it enhances the development of self-efficacy beliefs and reinforces the pre-service teachers’ ability to improve their teaching practice (Darling-Hammond, 2000; Hart, 2002; Huinker & Madison, 1997). Despite that, few researchers have focused specifically on pre-service teachers in early childhood classrooms. It is important for teacher education programs to research pre-service teachers’ self-efficacy because “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). This concept applies to pre-service teachers on many levels. The following section will provide an overview of what is known about self-efficacy improvement after taking mathematics methods courses.

Wenner (2001) explored self-efficacy beliefs of elementary pre-service teachers regarding mathematics concepts and teacher education programs and compared pre-service and in-service teachers’ self-efficacy in teaching mathematics. Wenner reported that by the end of the program, their self-efficacy increased. Therefore, the chances to change the level of self-efficacy are considerable with effective instruction in the teacher education program. In general, it turns out to be that pre-service teachers had higher self-efficacy levels in comparison to other in-service teachers. Their results indicated that the lack of support and mentoring activities could hinder teachers and pre-service teachers’ self-efficacy.

Using both quantitative and qualitative research methodologies, Esterly (2003) investigated the mathematics teaching self-efficacy and epistemological beliefs of elementary pre-service and novice teachers with a view to expand knowledge on the concept and to consider students’ performance outcomes. Esterly’s data were collected at three points: after the first methods course, after the second methods course, and after the students’ teaching experiences. The findings showed that pre-service teachers’ sense of efficacy increased after their methods courses, but slightly decreased after their student teaching experiences in a classroom setting. Also, the study found that there was no direct link between mathematics content knowledge and mathematics teaching efficacy. Their self-efficacy was high even though their performances were low in the test (regarding their knowledge in the mathematics content knowledge).

Play-generated curriculum is an example of a curriculum that can increase pre-service teacher’ self-efficacy regarding mathematics (Incikabi, 2013). Incikabi investigated the self-efficacy of 37 pre-service teachers. During mathematics methods courses, the participants were introduced to the play-generated curriculum where pre-service teachers demonstrated play experiences that improved mathematics learning concepts and skills. In the play-generated curriculum, pre-service teachers were introduced to the benefit of play for children’s learning and provided teachers roles in the free play or the guided play. Incikabi used pre and post-experimental design to measure the level of self-efficacy. As a result, mathematics self-efficacy increased at the end of the course. The changes were due to the planned experiments and participation in the activities which increased teachers’ self-efficacy. Incikabi (2013) showed that pre-service teachers’ self-efficacy could improve after taking mathematics methods courses.

Another study conducted by Huinker and Madison (1997) attempted to find the effects of mathematics courses on pre-service teachers’ self efficacy. Sixty-two pre-service teachers were surveyed using the Science Teaching Efficacy Beliefs Instrument and the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI). They also conducted interviews with 22 pre-service teachers at the beginning and the end of the methods courses. More in-depth interviews were conducted with pre-service teachers who showed more changes in their self-efficacy. This methods course comprised science, mathematics, and social studies. In the course, the constructivist philosophy/approach was used with the teachers and pre-service teachers, constructing their own learning through hands-on activities. The study showed increases in both groups’ self-efficacy after taking the methods courses in mathematics and science.

The investigation of Swars (2005) in a similar study helped to understand the characteristics of pre-service teachers with high or low self-efficacy regarding elementary pre-service teachers’ self-efficacy. It also demonstrated how their teaching self-efficacy could be changed. Four elementary pre-service teachers who enrolled in mathematics methods courses participated in the study. Pre and 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 techniques in their teaching. It suggested examining the pre-service elementary teachers’ past experiences in order to improve their self-efficacy.

Informed by the need to explain ways through which teacher efficacy can be enhanced, Kim, Sihn, and Mitchell (2014) conducted a study to investigate South Korean elementary teachers’ mathematics teaching efficacy beliefs and the factors that increase these beliefs demonstrated by teachers (p. 1). In this study, it was evident that teachers who believed that teaching can 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, Sihn, & Mitchell, 2014, p. 3).

To investigate the best instructional practices to develop pre-service teachers’ self-efficacy regarding mathematics and science, Wilkins and Brand (2004) examined elementary pre-service teachers’ self-efficacy regarding these content areas after enrolling in mathematics and science methods courses. In their study, a sample of 89 pre-service teachers were given the Mathematics Teaching Efficacy Beliefs Instruments. Wilkins and Brand (2004) found a relationship between mathematics methods courses and the teachers’ self-efficacy. This relationship suggested that the mathematics teachers’ self-efficacy increased after having taken the courses. Also, they examined the best pedagogical practices in the courses that affected pre-service teachers’ self-efficacy. For example, the activities where pre-service teachers constructed their own learning were examples of the four factors (mastery experiences, social persuasion, vicarious experiences, and physiological states) leading to self-efficacy.

The effects of gender and the importance of years in a teacher education program on the performance and mathematical self-efficacy beliefs of mathematics teachers was examined by Isiksal (2005). In this study, 145 pre-service mathematics teachers from Turkish schools were invite to participate. It was found that there were substantial statistical effects of gender and years in the teacher education program on both pre-service teachers’ performance and self-efficacy scores. Specifically, the study explained that female pre-service teachers scored higher than their male colleagues in performing habits of self-efficacy in their teaching. Still, no noteworthy variation was discovered between the two groups with respect to mathematics self-efficacy scores. Pre-service teachers who have had more courses scored higher in performance and mathematics self-efficacy scores, compared to pre-service teachers who have had fewer courses.

Conversely, Bursal’s (2007) findings contradict Isiksal’s findings when evaluating mathematics teaching self-efficacy beliefs held by pre-service elementary teachers. The findings showed that gender significantly influenced the beliefs. Males’ mathematics self-efficacy scores were remarkably higher when compared to females’ mathematics self-efficacy scores. Number of years in the teacher education program was also found to have a significant influence on mathematics self-efficacy scores since the senior pre-service teachers scored higher than the ones in the junior levels. Thus, further research is needed to study the effect of gender on pre-service teachers’ self-efficacy regarding mathematics.

### Research-Practice Connection

The available literature shows that teacher self-efficacy influences students’ learning outcomes, motivation, and attitudes toward the learning of teaching mathematics. Tschannen-Moran, Hoy, and Hoy (1998) pointed out that self-efficacy is “related to teachers’ behavior in the classroom. It affects the effort they put into teaching, the goals they set, and their level of aspiration” (pp. 222-223). It also changes the students’ beliefs, attitudes, and learning priorities toward their behavior in the classroom (Boud, 2012; Rimm-Kaufman, 2004). Most social learning theories support “that understanding the belief structures of teachers and teacher candidates is essential to improving their professional preparation and teaching practices” (Pajares, 1992, p. 307). According to Briley (2012), pre-service teachers who have successfully taken elementary mathematics methods courses before their first pre-service teaching year have high teaching efficacy. Thus, teacher education programs for preparing students to become educators should provide field experience opportunities which could influence pre-service teachers’ self-efficacy.

This observation implies that it is important for researchers in education to consider which factors influence pre-service teachers’ self-efficacy with a conceptual understanding of what is needed to assist them. As suggested by Bandura (1986), the sources of such self-efficacy beliefs in pre-service teachers include their mastery experiences, verbal or social persuasions, their vicarious experiences, and physiological state. However, the most important factor in developing self-efficacy is the mastery experiences (Arslan & Yavuz, 2012; Bandura, 1998; Hoffman, 2010)

Pre-service teachers enter their teacher education programs with a vast array of past experiences. The use of manipulatives, technology, cooperative learning, and classroom discussions strengthens and enhances mastery experiences. Practical experiences in the teaching of mathematics reinforce their mastery experiences and increase teachers’ self-efficacy (Hoy & Spero, 2005; Kazemi, Lampert, & Ghousseini, 2007). One of the ways to provide mastery experiences are videos that show teachers demonstrating teaching in a classroom environment. 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 though the video is an artificial representation of teaching which some pre-service teachers can easily dismiss” (Santagata & Guarino, 2011, p. 143). Teaching videos, however, followed by a discussion are also a kind of mastery experience which connects theory and practice (Lampert & Ball, 1998).

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 practice teaching and develop their skills and knowledge in order to help students (Bray-Clark & Bates, 2003). In addition to simulations, role-play techniques could be used to increase self-efficacy. A role-playing method is a technique to read an article about teaching with peers followed by a deep discussion about learning and behavior management (Bedient & Fox, 1999)

With mini-lessons or lesson-study approach, pre-service teachers could receive all the factors that promote their self-efficacy: mastery experiences, vicarious experiences, verbal persuasion, and physiological states (Chong, Kong, 2012). In mini-lessons, groups of pre-service teachers introduced teaching a specific mathematics concept or skill to their peers, receiving feedback, receiving practical experiences, observing others and reflecting upon their practices (Chong, Kong, 2012). All the above practices involve practical experiences, observing others, and receiving feedback in a supporting and encouraging environment. Thus, it could create all the four factors that affect self-efficacy as previously mentioned.

Mathematics methods courses can improve pre-service teachers’ vicarious experiences through collaborative training, peer interaction, and observations of other pre-service teachers (Bray-Clark & Bates, 2003). Teacher efficacy could be improved by giving pre-service teachers the chance to observe strategies, practice them with their peers, and use them regularly (Bandura, 1993; Chong and Kong, 2012; Swars, 2005). It is possible that pre-service teachers observe their peers’ successful teaching behavior and try to take the same steps in order to change their own self-efficacy levels or attitude to the situations (Bandura, 1993). Teacher educators should concentrate on the development of vicarious experiences by demonstrating their own teaching abilities and relationship-building skills that could be shared with others. Thus, peer groups and classmates become important sources of self-efficacy for pre-service teachers.

Another source that develops mathematics self-efficacy is verbal persuasion (Bandura, 1993). Feedback, collaborative teaching, and observations provide a good portion of the verbal support for mathematics pre-service teachers. Feedback encourages pre-service teachers to develop their strengths and other areas in which they need to improve (Tschannen-Moran, Hoy & Hoy, 1998). If they do not receive feedback on their new practices, they may ignore them and choose those ideas they find appropriate for themselves (Guskey & Passaro, 1994). When pre-service teachers implement new practices and observe positive results, they can apply them in the future (Guskey & Passaro, 1994).

Concerning physiological states, Wadlington, Slaton, and Partridge (1998), suggested that stress could be decreased for pre-service teachers by increasing teachers’ knowledge of mathematics and the use of manipulatives, and teacher-supported cooperative learning in mathematics methods courses. Teacher educators could support their pre-service teachers through caring and understanding (Ryan, & Patrick, 2001), supporting the relationship between pre-service teachers to provide a good environment that reduces their stress (Ryan, & Patrick, 2001). This relationship could be provided through collaborative learning by implementing discussion, group activity, mini lesson, role-play, etc. (Chong & Kong, 2012).

Swars, Hart, Smith, Smith, and Tolar (2007) insisted that the curriculum design for pre-service teachers should focus on developing skills, knowledge and beliefs. Teacher education programs are important for increasing pre-service teachers’ self-efficacy beliefs which are critical to their overall education due to their ability to influence teaching experiences (Bray-Clark & Bates, 2003; Kim, Sihn, & Mitchell, 2014). Research has demonstrated that exposure to social and verbal persuasion by observing peers and learning new materials had a higher chance of increasing pre-service teachers self-efficacy (Turner, Cruz and Papakonstantinou, 2000; Butler & Winne, 1995). Charalambous and Philippou (2003) suggested that it is possible to modify the student teachers’ efficacy beliefs since their self-efficacy is changeable as seen with experienced in-service teachers.

### Conclusion

Self-efficacy strongly influences pre-service teachers’ attitudes toward mathematics (Hackett & Betz, 1989). Pre-service teachers’ self-efficacy toward mathematics influence their prospective students’ self-efficacy and outcomes (Albayrak & Unal, 2011; Cohen, 1988; Lee, 2010). Self-efficacy beliefs influence pre-service teachers’ decisions, behaviors, and practices (Tschannen-Moran & Hoy, 2001). Thus, success in mathematics teaching is not only based on one’s knowledge of mathematics, but also on one’s beliefs about his or her ability to teach (Shulman, 1987).

By studying the impact of the mathematics methods courses on pre-service teachers’ self-efficacy, one can use the findings to construct more effective courses to increase self-efficacy. The factors identified by Bandura’s self-efficacy theory – performance accomplishments, vicarious experiences, verbal persuasions, and physiological states – are important to developing pre-service teachers’ self-efficacy (Albayrak & Unal, 2011; Briley, 2012; Cakiroglu, 2000; Gresham, 2008; Stipek, Givvin, Salmon, & MacGyvers, 2001). Pre-service teachers often have preconceived beliefs concerning mathematics and their teaching and learning abilities in this subject (Cakiroglu, 2000). A number of these pre-service teachers have different views about mathematics, most of which originated from their experiences as students. Cakiroglu noted that it was important for elementary pre-service teachers to “take part in a mathematics methods course in order to increase mathematics teacher efficacy“ (p.92). In addition, long-term exposure to different levels of complexities in the methods courses often enhances mathematics teaching efficacy and high-level delivery of lesson content.

Limited research has explored pre-service teachers’ self-efficacy in general or toward mathematics during enrollment in teacher education programs (Tschannen-Moran & Hoy, 2001). Pre-service teachers with high levels of self-efficacy are more confident in their ability to teach mathematics, more willing to develop and form a strong sense of commitment and motivation to their activities, and more likely to adapt various strategies and curriculum ideas. Self-efficacy has been identified as the one way to assist pre-service teachers in improving their classroom teaching abilities. Therefore, the mixed methods of this study aim to explore the impact of mathematics methods courses in pre-service Early Childhood and Special Educator’ self-efficacy and beliefs. The study also examines the possible factors responsible for the teachers’ teaching efficacy beliefs and attempts to determine the levels of self-efficacy of pre-service teachers regarding their skills in mathematics methods courses. Chapter Two provides an overview of the self-efficacy and its impact on pre-service teachers. The first section provides a self-efficacy theory and teaching efficacy. It also outlines a summary of the four sources in developing a strong sense of self-efficacy in mathematics pre-service teachers, which are: mastery, social experiences, social persuasion, and pschological states. Previous self-efficacy research about the teachers and pre-service teachers’ self-efficacy is also provided. The third chapter, then, provides the information relevant to the methods of the research.

## Methods

The purpose of this study was to examine the impact of mathematics methods courses on pre-service Early Childhood and Special Educators’ self-efficacy and beliefs. Additionally, the purpose was to examine the possible factors responsible for the pre-service teachers’ beliefs, and to determine the levels of self-efficacy of pre-service teachers regarding their skills in the mathematics methods courses.

Both quantitative and qualitative methods of data collection and analysis were employed throughout the investigation. The study was conducted with students from a western Pennsylvanian university. The study was conducted in three phases to gather accurate information regarding pre-service teachers’ self-efficacy beliefs during different stages of learning in the mathematics methods courses. The specific methods, research strategies, and approaches discussed in this chapter were used to provide a research framework to answer the following seven research questions:

- Are there differences in self-efficacy between pre-service teachers who have had content pedagogy courses and those who have had mathematics methods courses?
- Are there differences in self-efficacy for pre-service teachers between those pre-service teachers who have had one content pedagogy mathematics course and those pre-service teachers who have had two content pedagogy mathematics courses?
- How does self-efficacy vary among pre-service teachers who have had one methods course and those who have had two methods courses?
- What is the impact of mathematics methods courses on pre-service teacher’s self-efficacy?
- Based on gender, are there differences in self-efficacy of pre-service teachers?
- What are pre-service teachers’ perceptions of their skills, competence, and ability 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 are aimed at discussing the relationships between variables, whereas the last two questions are important for understanding the experiences of pre-service teachers during their teacher education program.

### Research Procedures and Methods

The study adopted a mixed-methods research approach in its attempt to examine not only pre-service Early Childhood and Special Educators’ self-efficacy and beliefs in relation to teaching mathematics, but also how their practical experiences in the mathematics methods courses could affect self-efficacy. The advantages of mixed-methods research are in the possibilities of using quantitative and qualitative research methods sequentially in order to research a certain topic in detail (Creswell, 2009). The importance of using these approaches in educational research were discussed due to their ability to yield informative and valuable data in generating a cumulative body of knowledge (Creswell, 2009).

This study employed quantitative methods in the form of an instrument-design technique to: (a) find the impact of mathematics methods courses on pre-service teachers’ self-efficacy; (b) find differences in self-efficacy between pre-service teachers who completed one pedagogy course and those who completed two pedagogy courses; (c) find differences in the pre-service teachers’ self-efficacy between those who completed one mathematics methods course and those who completed two mathematics methods courses; (d) find differences in the self-efficacy between pre-service teachers who completed the pedagogy courses and those who completed one or two mathematics methods courses; and (e) find differences in the pre-service teachers’ self-efficacy based on gender.

Quantitative methods provide a numeric representation of opinions or perspectives (Creswell, 2009). According to Gay and Airasian (2000), quantitative data are used to describe certain conditions, and to analyze the changes 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 pre-service teachers in relation to self-efficacy (Gay & Airasian, 2000).

The qualitative data were used to (a) explore pre-service teachers’ perceptions of their skills, competence, and ability to teach mathematics; and (b) determine what aspects of mathematics methods courses affected the self-efficacy beliefs of pre-service teachers. The qualitative research approach was important in determining the opinions of the participants in relation to variations in the methodologies used to teach mathematics. The data were analyzed to explore the relationship between learning the components of mathematics methods courses and its impact on pre-service teachers’ self-efficacy. The qualitative and quantitative research findings provided information on the teaching methods that pre-service teachers perceive as providing the best result. The mixed-methods research approach was necessary in assisting the researcher with discussing the relationships between various variables of interest through quantitative means, and understanding the experiences of pre-service teachers using qualitative means (Brown, 2012; Creswell, 2009).

The mixed methods being used are described as the combination of different research techniques and concepts in a single study to achieve the desired result (Creswell, 2009). In this research, the mixed-methods approach focused on an instrument and face-to-face interviews with the participants.

### Selection of Research Participants

Participants were chosen for this study by purposeful sampling. The sample included all students in MATH 151, MATH 152, MATH 320, and MATH 330 during the Spring semester of 2016 at a western Pennsylvanian university. All students in the participating courses were invited to complete pre- and post-instruments. Students in MATH 320 and MATH 330 were invited to volunteer to be interviewed. Content pedagogy courses were not part of the data collection process because the research focused on exploring the mathematics methods courses.

Six participants (identified with researcher-generated ID numbers) were randomly selected from the volunteers to participate in the one-on-one interviews. Pre-service teachers completed an informed consent form prior to participating in the instruments or interviews. Institutional Review Board approval was obtained prior to data collection.

### Sampling Procedure

A purposeful sampling approach was used in the quantitative phase. Purposeful sampling enables the researcher to focus on the population with the characteristics that address the research needs. According to this technique, participants were selected according to certain inclusion and exclusion criteria (Creswell, 2009). Following the inclusion criteria, all students who were enrolled in mathematics-methods courses and content pedagogy courses were invited to participate in the study during Spring 2016. All students who were not majoring in Early Childhood /Special Education were excluded after the data collection. The purposeful sampling approach was used in the qualitative phase to target only pre-service teaches in mathematics methods courses. Content pedagogy courses were not part of the data collection process because the researcher was focusing on exploring the mathematics methods courses. This approach allowed the researcher to focus on studying how students who were enrolled in mathematics methods courses would evaluate their self-efficacy in relation to teaching mathematics.

### Sample Characteristics

While focusing on the specific population’s characteristics, it is important to note that approximately 164 students were of a western studying in the mathematics methods courses and content pedagogy courses at the western Pennsylvanian university. Students were invited to participate in the study because they fit the inclusion criteria. While age was not an inclusion factor for this study, most of the participants were between ages 18 and 25. The participants involved in the study were at various stages in their teacher preparation: first, second, third, or fourth semester of taking mathematics courses.

### Study Site

The site selected for the study is a western Pennsylvanian university. Students majoring in Early Childhood and Special Education are required to take four courses in mathematics methods. The mathematics methods courses and the content pedagogy courses at this public university are Elements of Math I (MATH 151), Elements of Math II (MATH 152) Mathematics for Early Childhood Education (MATH 320), and Teaching Mathematics in Elementary School (MATH 330). MATH 151 and MATH 152 are required to be taken sequentially. MATH 151 and 152 are prerequisites for MATH 320 and MATH 330.Table 1 provides information about MATH 151 and MATH 152 contents

*Table 1. Content Pedagogy Courses Descriptions.*

On the other hand, MATH 320 and MATH 330 are traditional mathematics methods courses. These courses are not required to be taken sequentially. Table 2 provides information about MATH 320 and MATH 330 contents.

*Table 2. Mathematics Methods Courses Description.*

### 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 self-efficacy instrument used to measure self-efficacy levels for this study was the Mathematics Teaching Efficacy Belief Instrument. (Appendix A). Permission was granted to use the instrument in the study from the authors (Appendix B). This instrument was used during the first and second phase of the data-gathering process. The qualitative data were obtained in the third phase through face-to-face interviews with the participants that were based on a questionnaire developed by the researcher to address the important aspects associated with the investigation (Appendix B).

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

The quantitative data were used to provide broader understanding of the research problem through the data collected from the participants (Creswell, 2009). It strengthened the research by providing: 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, 2009). 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, 2009). It is important to note 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 methods courses. If so, they indicated those courses. The demographic information about gender information allowed the researcher to explore relationships or patterns of gender to self-efficacy scores. To match students for the pre- and post- instrument, students were asked to indicate the last four digits of their cell phone numbers.

The Mathematics Teaching Efficacy Belief Instrument (MTEBI) developed by Enochs, Smith, and Huinker (2000) was used in the research to determine the pre-service teachers’ self-efficacy rating regarding their success in teaching mathematics at the beginning and final stages of taking the mathematics methods courses. Permission was granted to use the survey in the study from the authors prior to the study (see Appendix B). The validity and reliability of the MTEBI is supported by many studies (Huinke & Madison, 1997; Swars, 2005). The MTEBI has often been used in studies involving the assessment of pre-service teachers’ self-efficacy regarding mathematic methods courses (Enochs Smith, & Huinker, 2000). The MTEBI is a modification of the “Science Teaching Efficacy Belief Instrument” (STEBI) developed by Enochs and Riggs (1990).

The instrument employs a five-point Likert scale with 21 items divided into two subscales (Enochs, Smith, & Huinker, 2000). The first subscale is the “Personal Mathematics Teaching Efficacy” (PMTE) subscale, which has 13 items (2, 3, 5, 6, 8, 11, 15, 16, 17, 18, 19, 20, and 21). The second is the “Mathematics Teaching Outcome Expectancy” (MTOE) subscale, which includes eight items (1, 4, 7, 9, 10, 12, 13, and 14) (Enochs, Smith, & Huinker, 2000). The two subscales of the MTEBI were adapted to become understandable for pre-service teachers in order to guarantee the accuracy of the answers regarding self-efficacy beliefs at the different stages of the instrument. Enochs and the group of researchers examined the instruments with 324 elementary pre-service teachers before and after taking the mathematics method courses in multiple universities. They conducted the reliability analysis, finding that the alpha coefficient was 0.88 for (PMTE) and 0.75 for (MTOE) (Enochs Smith, & Huinker, 2000). As a result, it is possible to speak about the high level of the instrument’s reliability. Furthermore, the confirmatory factor analysis indicated the high level of the MTEBI’s validity.

#### The 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-methods courses, the personal experiences about the courses from the participants, and the reason for the low or high self-efficacy. The questions were piloted with three graduate students with whom the researcher had direct contact. Students had the option to participate in the pilot study. Following the pilot of the interview questions, the researcher received feedback and made changes as needed. Through the feedback, there were questions that needed 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 questions.

Sixteen open-ended questions were developed by the researcher and included in the interview questionnaire (Appendix C). The interview questions were constructed to correspond with the research questions so that the participants could provide more detailed information, express their opinions, and offer an understanding of their personal experiences of the subjects after taking the mathematics method courses (Creswell, 2009). The interview questions investigated observed changes in the pre-service teachers’ experiences, the helpfulness of the courses, and the possible changes in pre-teachers’ strategies and vision of their success. Thus, the questions helped to establish the relationship between the mathematics methods courses and the pre-service teachers’ visions of self-efficacy.

### Data-Collection Methods

The study involved the collection of both qualitative and quantitative data, conducted in three phases: pre-instrument, post-instrument, and interview. Institutional Review Board approval was obtained prior to data collection. The researcher obtained the approval to conduct the study from the chairperson of the Mathematics Department at the university (see Appendix D). Permission was obtained from the faculty members who taught the Elements of Math I (MATH 151), and Elements of Math II (MATH 152), Mathematics for Early Childhood Education course (MATH 320) and Teaching Math in Elementary School course (MATH 330), to distribute the instruments during class time on two separate occasions (once for the pre-instrument and once for the post-instrument) was obtained (see Appendix E). The data collection period of this study was during the Spring 2016 academic semester, determined in consultation with the instructors. The researcher organized the appropriate schedule for the data distribution in cooperation with the faculty members at a western Pennsylvanian university. The study was utilized in such a way that any suspicion of respondent coercion by the professor or the principal investigator was eliminated. As such, a representative of the researcher, a faculty member who was not the teacher of the mathematics courses, administrated the instrument in each mathematics class to the undergraduate students. The researcher provided a script for the researcher’s representative to read in the classes.

#### First Phase (Quantitative)

The first phase of the study involved administrating the pre-instrument within the first three weeks of the Spring 2016 semester. A cover letter and questionnaire were given by the researcher’s representative to the participants in the study. The letter informed the students of the purpose of the study and their rights (see Appendix F). In addition, the cover letter stated how the principle of confidentiality was going to be upheld throughout the entire research process. The students were informed that participating in the study was voluntary and that it had no bearing on enrollment in the course or their course grade. Caution was taken to make sure that the identity of the subjects remained anonymous to their peers, as well as confidential to the researcher. The students who were unwilling to participate in the instrument were given a mathematics activity as an alternative activity. The pre-instruments took 10-15 minutes to complete.

The research involved data from the same participants on multiple occurrences, so the pre- and post-instruments needed to be labeled with an identifier. Confidentiality was maintained by labeling the instrument with the last four numbers of the participant’s cell phone number as a way to match the data in the pre- and post-instruments. Students placed their own instrument in an envelope near the entrance to the classroom. Also, MATH 330 and MATH 320 students were issued response papers and were asked to drop them in an envelope at the classroom entrance. The selection of the interview participants was done through the issuance of response papers. All subjects, willing or unwilling to be interviewed, deposited their response papers in the envelope. This was done to uphold and maintain confidentiality. The students who agreed to be interviewed, provided their names and telephone numbers, as well as their email addresses, while those who did not agree to be interviewed deposited blank response papers.

#### Second Phase (Quantitative)

The next phase involved administrating the post-instrument using the same instruments within the last two weeks of Spring 2016. The researcher’s representative followed the same pre-instrument procedures to collect the post-instrument. Once the pre- and post-instruments were collected, the data were matched using the last four digits of the participants’ cell phone numbers. Any identifiers were removed from the data. Next, the last four digits of the participants’ cell phone numbers were securely discarded. To prevent any disclosure of such information, only the researcher had access to the information that linked participants to their responses.

#### Third Phase (Qualitative)

Participants for the third phase of the study were identified with students only in MATH 320 and MATH 330 who volunteered after the pre- and post-instruments were given. Students in MATH 151 and MATH 152 were not included because this study focused to examine the impact of mathematics methods courses on pre-service teachers’ self-efficacy. Respondents were asked if they would like to participate in the qualitative phase. The students who agreed to be interviewed provided their names and telephone contacts, as well as their email addresses. The researcher randomly selected six participants from those who expressed interest in participating in the interview. At the end of the semester, potential participants were contacted individually either through email or telephone to schedule an interview time. The place of the interview for all participants was scheduled individually. In the same manner as the instrument, cover letters were issued to the participants in the face-to-face interviews that explained the purpose of the interview and their rights (see Appendix G). Copies of voluntary consent letters were provided to the participants.

The interview participants were informed well in advance that the interview would be digitally recorded and would take 10-15 minutes. The identities of those who responded to the interview were kept confidential by the use of pseudonyms to identify the participants during the discussion of the research findings. The researcher stored the personal information 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 appreciation for their time and effort in follow-up interview.

### Analysis Methods

The results of the quantitative instrument were analyzed with the help of SPSS. This 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 and to state the relationship between the variables to make conclusions about the observed correlations (Cohen, 1988; 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-instrument.

The analyses of the qualitative data were based on the process of coding and identifying thematic patterns in the participants’ answers to the interview questions. First, the researcher focused on transcribing. Software called Google Doc was used to transcribe the audio. Then the researcher revised the interview in order to pay attention to all the details mentioned by the participants (Cohen, 1988; Creswell, 2009). Once transcribed, a colleague checked the transcriptions for accuracy.

The material was read several times and analyzed in order to determine different and similar parts (Turner, 2010). Two kinds of coding were used: open and axial coding to identify common concepts in order to proceed to themes about the courses and factors effecting pre-service teachers’ self-efficacy (Strauss & Corbin, 1998). These coding strategies are described in the following chapters.

### Ethical Issues

Ensuring confidentiality is necessary in order to encourage the participation of people and ensure honest responses (Creswell, 2009). Therefore, the researcher sought permission from individuals to participate in the study. The participants were informed of the precautions that would be taken to protect their confidentiality. Their participation in the study was voluntary, and the participants were free to withdraw at any time. The participation or non-participation in this particular study was entirely voluntary and had no bearing on enrollment or students’ grades.

### Chapter Summary

The purpose of this study was to examine the impact of mathematics methods courses on pre-service teachers’ self-efficacy and beliefs. It also sought to examine the possible factors responsible for the pre-service teachers’ self-efficacy beliefs. In addition, it attempted to determine the levels of the self-efficacy of pre-service teachers regarding their skills in the mathematics methods courses. Quantitative and qualitative methods of data collection were employed throughout the investigation to examine the visions of the students of the western Pennsylvanian university. The quantitative research method was realized in two stages during which the participants completed the instruments with the help of the Mathematics Teaching Efficacy Belief Instrument (MTEBI).

The qualitative research method involved those pre-service teachers who agreed to participate in the follow-up interviews. To explore the pre-service teachers’ self-efficacy beliefs, the researcher asked six participants to answer the open-ended questions prepared in correspondence with the topic of the research. The quantitative data were analyzed using the paired T-test and ANCOVA with the help of the appropriate SPSS software to calculate the statistical variances. The qualitative data that were the result of this study will be described in detail in Chapter Four. The qualitative data were analyzed with the focus on coding the participants’ responses and identifying the themes regarding the participants’ self-efficacy beliefs and visions of their success in teaching after completing the mathematics-methods courses.

## Data Analysis

In this chapter, the results of the data analysis are presented. One purpose of this mixed-methods research study was to examine the impact of mathematics methods courses on pre-service Early Childhood and Special Educators’ self-efficacy and beliefs. It also aimed to examine the possible factors responsible for the pre-service teachers’ beliefs, and to determine the levels of self-efficacy of pre-service teachers’ regarding their skills in the mathematics methods courses. This study addressed seven research questions:

- Are there differences in self-efficacy between pre-service teachers who have had content pedagogy courses and those who have had mathematics methods courses?
- Are there differences in self-efficacy for pre-service teachers between those pre-service teachers who have had one content pedagogy mathematics course and those pre-service teachers who have had two content pedagogy mathematics courses?
- How does self-efficacy vary among pre-service teachers who have had one methods course and those who have had two methods courses?
- What is the impact of mathematics methods courses on pre-service teacher’s self-efficacy?
- Based on gender, are there differences in self-efficacy of pre-service teachers?
- What are pre-service teachers’ perceptions of their skills, competence, and ability to teach mathematics?
- What aspects of mathematics methods courses influence the self-efficacy beliefs of future teachers of mathematics?

The first five questions addressed the data that were collected with the METBI (Mathematics Teaching Efficacy Belief Instrument) aimed at discussing the relationships between variables, whereas the sixth and seventh questions are important for understanding the experiences of pre-service teachers during their teacher education program.

The study was conducted in three phases during the Spring 2016 semester at a western Pennsylvanian university. In Phases One and Two, the quantitative data were obtained via an existing instrument known as the MTEBI developed by Enochs, Smith, and Huinker (2000). This instrument has 21 items. In the first phase, the students who were enrolled in MATH 151, MATH 152, MATH 320, and MATH 330 courses were given the METBI pre-instrument. At the end of the semester, the post-instrument of the METBI was administered. Twenty minutes were allocated to the classes to administer the METBI pre- and post-instrument questionnaire.

In the Phase Three, the qualitative data were collected through one-on-one interviews conducted using a 16-question interview protocol (Appendix C). This interview was designed to provide participants with an opportunity to elaborate on their previously-recorded responses. Six participants from the methods courses MATH 320 and MATH 330 were randomly-selected based on their willingness to participate.

### Quantitative Description of Sample

One hundred sixty-four pre- and post-instruments were returned. However, only 157 of the returned instruments were completed. The incomplete instruments were removed from the database. Only students who had completed the pre-instrument and post-instrument were included in this analysis. An additional 25 instruments, which belonged to non-early childhood educators, were eliminated.

The numbers of students from each course who completed both the pre and post- instruments were 132 participants: 18 students were from MATH 151, 49 were from were MATH 152, 22 were from MATH 320, and 43 were from MATH 330.

### Qualitative Description of Sample

Six pre-service teachers who were enrolled in the courses MATH 320 and MATH 330 were randomly selected to participate in the interview (Table 3). These participants were randomly selected for the interviews from pre-service teachers in Phase One and Two who indicated their interest in participating in Phase Three of the research study. It was a coincidence that all the participants who volunteered for the interview were female. Each interview was digitally recorded and then transcribed prior to analysis. Table 3 provides information on the six participants’ courses.

*Table 3. Demographic Qualitative Data.*

### Quantitative Data Analysis

An important purpose of this study was to investigate whether pre-service teachers’ self-efficacy changed during their mathematics courses of their teacher education program. To track these changes, the researcher utilized data collected from the pre- and post-instruments. The quantitative data were gathered twice throughout the study: (a) Phase One in the beginning of the semester, and (b) Phase Two at the end of the course. The dependent variables included: (a) the self-efficacy pre-test (pre-instrument), and (b) the self-efficacy post-test (post-instrument). To compare the participants’ self-efficacy of before and after the courses, it would require a paired T-Test and the ANCOVA. The ANCOVAs compared post-instruments for the courses after removing the effects of pre-instruments measures on the post-instruments measures. While paired sample t-tests were performed to measure the impact of the courses in self-efficacy. Both were employed because they are commonly used for two-assessment time points.

Quantitative analyses were conducted for research questions 1-5. These results are addressed in greater detail below.

#### Research Question 1

**Are there differences in self-efficacy between pre-service teachers who have had content pedagogy courses and those who have had one or two mathematics methods courses?**

To answer this question, analysis of covariance (ANCOVA) was used to determine if there was a significant difference between the post-instruments scores for pre-service teachers who have had content pedagogy courses, and post- instruments scores for pre-service teaches who have had one or two mathematics methods courses.

**Variables**. The independent variables included the content pedagogy courses and the mathematics methods courses. The dependent variable was the pre-service teachers’ post-instrument scores, and the covariate was the pre-instrument scores.**Null hypothesis**. There is no significant difference in self-efficacy between pre-service teachers who have had content pedagogy courses and who have had one or two mathematics methods courses.**Alternative hypothesis**. There is a significant difference in self-efficacy between pre-service teachers who have had content pedagogy courses and those who have had one or two mathematics methods courses.**Summary of Findings for Research Question 1**. An ANCOVA was performed to examine pre-service teachers’ self-efficacy in content pedagogy courses and mathematics methods courses. The confidence intervals are presented in Table 4, and the results of the ANCOVA are presented in Table 5 with the means and standard deviations.

*Table 4. Descriptive Statistics for the Mathematics Methods Courses and the Content Pedagogy Courses.*

The pre-service teachers’ group consisting of one or two content pedagogy courses showed a lower mean (74.5) compared to the one or two mathematics methods courses group (85.8). The ANCOVA results suggested that there was a significant difference in self-efficacy [F (1,129)= 29.878, p=0.00] between the pre-service teachers who have had content pedagogy courses and those who have had mathematics methods courses while adjusting pre-instrument. When the two groups have values greater than.05, it means that the two groups have equal variances according to the results from the Levene’s test for equality of the variances (F=1.061 with p =.305) (see Table 5). Therefore, the conclusion is that there is a statistical difference in the self-efficacy between pre-service teachers who have had content pedagogy courses and those who have had mathematics methods courses accounting for the pre-intervention condition of the two groups while adjusting for pre-self efficacy.

*Table 5. ANCOVA Analysis of Pre-service Teachers’ Self-efficacy Comparing Content Pedagogy Courses and Two Mathematics Methods Courses.*

#### Research Question 2

**Are there differences in self-efficacy for pre-service teachers between those pre-service teachers who have had one content pedagogy mathematics course and those pre-service teachers who have had two content pedagogy mathematics courses?**

To answer this question, analysis of covariance (ANCOVA) was used to determine if there was a significant difference between the post-instruments scores for pre-service teachers who have had one content pedagogy course and post- instruments scores for pre-service teaches who have had two content pedagogy courses.

**Variables**. The independent variables were the content pedagogy courses and the mathematics methods courses. The dependent variable was the pre-service teachers’ post-instrument scores, and the covariate was the pre-instrument scores.**Null hypothesis**. There is no significant difference in self-efficacy between pre-service teachers who have had one content pedagogy course and those who have had two content pedagogy courses.**Alternative hypothesis**. There is a significant difference in self-efficacy between pre-service teachers who have had one content pedagogy course and those who have had two content pedagogy courses.**Summary of Findings for Research Question 2**. The means, standard deviations, and confidence intervals are presented in Table 6 and the results of the ANCOVA are presented in Table 7.

*Table 6. Descriptive Statistics for Content Pedagogy.*

By looking at Table 6, it was found that the mean of pre-service teachers with one content pedagogy course is slightly lower (70.0) compared to that of the pre-service teachers with two content pedagogy courses (79.1). The ANCOVA results in Table7 suggested that there was a significant difference in self-efficacy [F (2,128)= 15.587, p= 0.00] between those teachers who have had one content pedagogy mathematics course and those teachers who have had two content pedagogy mathematics courses.

*Table 7. ANCOVA Analysis of Pre-service Teachers’ Self-efficacy Comparing One Content Pedagogy Course and Two Content Pedagogy Courses.*

Additionally, the researcher wondered about the self-efficacy impact of the content pedagogy courses and performed another analysis to evaluate the impact of the mathematics content pedagogy courses on pre-service teachers’ self-efficacy. Paired sample t-tests were performed for the content pedagogy courses to determine if there was a change in self-efficacy to teach mathematics as measured by the pre- and post- scores on the MTEBI instrument.

Table 8 indicates the descriptive statistics of the pre-service teachers’ self-efficacy and showed that there is a change in the participants’ self-efficacy between pre- and post-instrument scores for pre-service teachers in the mathematics content pedagogy courses. Table 8 shows that pre-service teachers showed lower means on self-efficacy after the mathematics content pedagogy courses (M = 72.1) than before the mathematics content pedagogy courses (M = 77.5).

Table 8. *Paired Samples Descriptive Statistics for Question 2.*

In addition, the paired T-Test results in Table 9 indicate that there were differences between the pre-test and post-test instrument scores in self-efficacy among the respondents (t=2.43, p=. 018) at the 95% confidence interval. Therefore, these results indicate that pre-service teachers’ self-efficacy decreased between the beginning and the end of mathematics content pedagogy courses.

*Table 9. Paired Sample T-Test Results of Pre-service Teachers ’self-efficacy after the Content Pedagogy Courses.*

Therefore, it is established that there is a difference in self-efficacy between pre-service teachers who have had one content pedagogy course and those who have had two content pedagogy courses. However, the result also indicated a slight decrease in pre-service teachers’ self-efficacy after taking the mathematics content pedagogy courses.

#### Research Question 3

In order to answer this question, 21 ANCOVAs were performed for each item of the self-efficacy instruments to determine statistically-significant differences between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses on the pre- and post-self-efficacy variable controlling for the pre-instrument.

Dependent variables were measured by the MTEBI instrument. The instrument employs a five-point Likert scale and includes twenty-one items divided into two subscales (Enochs, Smith, & Huinker, 2000). The first subscale is the Personal Mathematics Teaching Efficacy (PMTE) subscale, which has 13 items (2, 3, 5, 6, 8, 11, 15, 16, 17, 18, 19, 20, and 21). The second is the Mathematics Teaching Outcome Expectancy (MTOE) subscale, which includes eight items (1, 4, 7, 9, 10, 12, 13, and 14). PMTE is the pre-service teachers own belief about their ability to teach mathematics. MTOE is pre-service teachers’ expectation that their teaching will lead students to learn mathematics (Bandura, 1977).

**The following items are included in the MTEBI instrument and are dependent variables**.

- When a student does better than usual in mathematics, it is often because the teacher exerted a little extra effort.
- I will continually find better ways to teach mathematics.
- Even if I try very hard, I will not teach mathematics as well as I will most subjects.
- When the mathematics grades of students improve, it is often due to their teacher having found a more effective teaching approach.
- I know how to teach mathematics concepts effectively.
- “I will not be very effective in monitoring mathematics activities.”
- If students are underachieving in mathematics, it is most likely due to ineffective mathematics teaching.
- I will generally teach mathematics ineffectively.
- The inadequacy of a student’s mathematic background can be overcome by good teaching.
- When a low-achieving child progresses in mathematics, it is usually due to extra attention given by the teacher.
- I understand mathematics concepts well enough to be effective in teaching elementary mathematics.
- The teacher is generally responsible for the achievement of students in mathematics.
- Students’ achievement in mathematics is directly related to their teacher’s effectiveness in mathematics teaching.
- If parents comment that their child is showing more interest in mathematics at school, it is probably due to the performance of the child’s teacher.
- I will find it difficult to use manipulatives to explain to students why mathematics works.
- I will typically be able to answer students’ questions.
- I wonder if I will have the necessary skills to teach mathematics.
- Given a choice, I will not invite the principal to evaluate my mathematics teaching.
- When a student has difficulty understanding a mathematics concept, I will usually be at a loss as to how to help the student understand it better.
- When teaching mathematics, I will usually welcome students’ questions.
- I do not know what to do to turn students on to mathematics.

**Independent variable**. The independent variables included the mathematics methods courses**Null hypothesis**. There is no significant difference in self-efficacy between pre-service teachers who have had one methods course and those who have had two methods courses.**Alternative hypothesis**. There is a significant difference in self-efficacy between pre-service teachers who have had one methods course and those who have had two methods courses.

In order to answer Question Three, 21 separate ANCOVA statistical analyses were conducted to determine differences between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses on the pre and post self-efficacy variable controlling for the pre-instrument. The means, standard deviations and confidence intervals for each item in the instrument are presented in Table 10.

*Table 10. Descriptive Statistics for Each Item in the Instrument.*

#### ANCOVA Results for Each Item in the Instrument

**Item 1**. When a student does better than usual in mathematics, it is often because the teacher exerted a little extra effort.

Table 11 shows the ANCOVA output comparing differences in Item 1 between pre-service teachers who have had one mathematics methods course (M = 3.7), and those who have had two mathematics courses (M = 3.8). The ANCOVA results suggested that there was no significant difference in Item 1 [F (2,128) = 4.1099, p= 0.272] between the pre-service teachers who have had one mathematics course and those who have had two mathematics method courses while adjusting pre-item 1. The conclusion is that there is no statistically- significant difference in Item 1 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 11. ANCOVA Result for Item 1 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 2**. I will continually find better ways to teach mathematics.

Table 12 shows the ANCOVA output comparing differences in Item 2 between pre-service teachers who have had one mathematics methods course (M = 3.2), and those who have had two mathematics methods courses (4.7). The ANCOVA results suggested that there was a significant difference in Item 2 [F (2,128) = 4.342, p=0.015] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses while adjusting pre-item 2. The conclusion is that there is a statistically-significant difference in Item 2 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 12. ANCOVA Result for Item 2 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 3**. Even if I try very hard, I will not teach mathematics as well as I will most subjects.

Table 13 shows the ANCOVA output comparing differences in Item 3 between pre-service teachers who have had one mathematics methods course (M=2.6), and those who have had two mathematics methods courses (M=3.5). The ANCOVA results suggested that there was a significant difference in Item 3 [F (2,128) = 4.923, p=0.009] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses while adjusting pre-item 3. The conclusion is that there is a statistically-significant difference in Item 3 of the self-efficacy instrument between pre-service teachers who have had one mathematics course and those who have had one two mathematics methods courses.

*Table 13. ANCOVA Result for Item 3 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 4**. When the mathematics grades of students improve, it is often due to their teacher having found a more effective teaching approach.

Table 14 shows the ANCOVA output comparing differences in Item 4 between pre-service teachers who have had one mathematic methods course (M=4.1), and those who have had two mathematics methods courses (M= 4.0). The ANCOVA results suggested that there was no significant difference in Item 4 [F (2,128) =.495, p=0.087] between the pre-service teachers who have had one mathematics course and those who have had two mathematics method courses while adjusting pre-item 4. The conclusion is that there is no statistical difference in Item 4 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 14. ANCOVA Result for Item 4 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 5**. I know how to teach mathematics concepts effectively.

Table 15 shows the ANCOVA output comparing differences in Item 5 between pre-service teachers who have had one mathematic methods course (M=3.3), and those who have had two mathematics methods courses (M=4.5). The ANCOVA results suggested that there was a significant difference in Item 5 [F (2,128) =.493, p=0.005] between the pre-service teachers who have had one mathematics course and those who have had two mathematics method courses while adjusting pre-item 5 status. The conclusion is that there is a statistical difference in Item 5 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 15. ANCOVA Result for Item 5 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 6**. I will not be very effective in monitoring mathematics activities.

Table 16 shows the ANCOVA output comparing differences in Item 6 between one mathematics pre-service teachers who have had one mathematic methods course (M=3.2), and those who have had two mathematics methods courses (M=4.3). The ANCOVA results suggested that there was a significant difference in Item 6 [F (2,128) = 4.714, p=0.011] between the pre-service teachers who have had one mathematics course and those who have had two mathematics method courses while adjusting pre-item 6. The conclusion is that there is a statistical difference in Item 6 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses accounting for of the two groups while adjusting for pre-item 6.

*Table 16. ANCOVA Result for Item 6 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 7**. If students are underachieving in mathematics, it is most likely due to ineffective mathematics teaching.

Table 17 shows the ANCOVA output comparing differences in Item 7 between one mathematics pre-service teachers who have had one mathematic methods course (M=3.1), and those who have had two mathematics methods courses (M=3.3). The ANCOVA results suggested that there was no significant difference in Item 7 [F (2,128) = 2.780, p=0.066] between the pre-service teachers who have had one mathematics course and those who have had two mathematics method courses while adjusting pre-item 7. The conclusion is that there is no statistical difference in Item 7 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 17. ANCOVA Result for Item 7 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 8**. I will generally teach mathematics ineffectively.

Table 18 shows the ANCOVA output comparing differences in Item 8 between pre-service teachers who have had one mathematic methods course (M=4.3), and those who have had two mathematics methods courses (M=4.3). The ANCOVA results suggested that there was no significant difference in Item 8 [F (2,128) = 2.598, p=0.078] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 8. The conclusion is that there is no statistical difference in Item 8 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 18. ANCOVA Result for Item 8 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 9**. The inadequacy of a student’s mathematic background can be overcome by good teaching.

Table 19 shows the ANCOVA output comparing differences in Item 9 between pre-service teachers who have had one mathematics methods course (M=4.0), and those who have had two mathematics methods courses (M=4.2). The ANCOVA results suggested that there was no significant difference in Item 9 [F (2,128) =. 846, p=0.431] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 9. The conclusion is that there is no statistical difference in Item 9 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 19. ANCOVA Result for Item 9 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 10**. When a low-achieving child progresses in mathematics, it is usually due to extra attention given by the teacher.

Table 20 shows the ANCOVA output comparing differences in Item 10 between pre-service teachers who have had one mathematics methods course (M=4.1), and those who have had two mathematics methods courses (M=4.0). The ANCOVA results suggested that there was no significant difference in Item 10 [F (2,128) =2.207, p=0.114] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 10. The conclusion is that there is no statistical difference in Item 10 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 20. ANCOVA Result for Item 10 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 11** I understand mathematics concepts well enough to be effective in teaching elementary mathematics.

Table 21 shows the ANCOVA output comparing differences in Item 11 between pre-service teachers who have had one mathematics methods course (M=3.5), and those who have had two mathematics methods courses (M=4.7). The ANCOVA results suggested that there was a significant difference in Item 11 [F (2,128) = 3.995, p=0.021] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 11. The conclusion is that there was a statistical difference of Item 11 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses accounting.

*Table 21. ANCOVA Result for Item 11 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 12**. The teacher is generally responsible for the achievement of students in mathematics.

Table 22 shows the ANCOVA output comparing differences in Item 12 between pre-service teachers who have had one mathematics method course (M=3.0), and those who have had two mathematics methods courses (4.8). The ANCOVA results suggested that there was a significant difference in Item 12 [F (2,128) = 3.204, p=0.044] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 12. The conclusion is that there was a statistical difference of Item 12 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 22. ANCOVA Result for Item 12 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 13**. Students’ achievement in mathematics is directly related to their teacher’s effectiveness in mathematics teaching.

Table 23 shows the ANCOVA output comparing differences in Item 13 between pre-service teachers who have had one mathematics methods course (M=3.7), and those who have had two mathematics methods courses (M=4.9). The ANCOVA results suggested that there was a significant difference in Item 13 [F (2,128) = 4.022, p=0.020] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 13. The conclusion is that there was a statistical difference in Item 13 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 23. ANCOVA Result for Item 13 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 14**. If parents comment that their child is showing more interest in mathematics at school, it is probably due to the performance of the child’s teacher.

Table 24 shows the ANCOVA output comparing differences in Item 14 between pre-service teachers who have had one mathematics methods course (M=3.9), and those who have had two mathematics courses (M=3.8). The ANCOVA results suggested that there was no significant difference in Item 14 [F (2,128) =. 759, p=0.470] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 14. The conclusion is that there was no statistical difference in Item 14 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 24. ANCOVA Result for Item 14 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 15**. I will find it difficult to use manipulative to explain to students why mathematic works.

Table 25 shows the ANCOVA output comparing differences in Item 15 between pre-service teachers who have had one mathematics methods course (M=3.5), and those who have had two mathematics courses (M=4.3). The ANCOVA results suggested that there was a significant difference in Item 15 [F (2,128) = 7.978, p=0.001] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 15. The conclusion is that there was a statistical difference in Item 15 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 25. ANCOVA Result for Item 15 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 16**. I will typically be able to answer students’ questions.

Table 26 shows the ANCOVA output comparing differences in Item 16 between pre-service teachers who have had one mathematics methods course (M=3.4), and those who have had two mathematics methods courses (M=3.5). The ANCOVA results suggested that there was no significant difference in item 16 [F (2,128) =1.942, p=0.148] between the pre-service teachers who have had one mathematics course and those who have had two mathematics method courses while adjusting pre-item 16. The conclusion is that there was no statistical difference in Item 16 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 26. ANCOVA Result for Item 16 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 17**. I wonder if I will have the necessary skills to teach mathematics.

Table 27 shows the ANCOVA output comparing differences in Item 17 between pre-service teachers who have had one mathematics methods course (M=3.4), and those who have had two mathematics methods courses (M=4.2). The ANCOVA results suggested that there was a significant difference in Item 17 [F (2,128) = 4.457, p=0.013] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 17. The conclusion is that there was a statistical difference in Item 17 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 27. ANCOVA Result for Item 17 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 18**. Given a choice, I will not invite the principal to evaluate my mathematic teaching.

Table 28 shows the ANCOVA output comparing differences in Item 18 between pre-service teachers who have had one mathematics methods course (M=3.3), and those who have had two mathematics methods courses (M=4.0). The ANCOVA results suggested that there was a significant difference in Item 18 [F (2,128) = 4.457, p=0.027] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 18. The conclusion is that there was a statistical difference in Item 18 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 28. ANCOVA Result for Item 18 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 19**. When a student has difficulty understanding a mathematics concept, I will usually be at a loss as to how to help the student understand it better.

Table 29 shows the ANCOVA output comparing differences in Item 19 between pre-service teachers who have had one mathematics methods course (M=3.4), and those who have had two mathematics methods courses (M=4.2). The ANCOVA results suggested that there was a significant difference in Item 19 [F (2,128) =1.651, p=0.018] between the pre-service teachers who have had one mathematics methods course and those who have had two mathematics method courses while adjusting pre-item 19. The conclusion is that there was a statistical difference in Item 19 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 29. ANCOVA Result for Item 19 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 20**. When teaching mathematics, I will usually welcome student questions.

Table 30 shows the ANCOVA output comparing differences in Item 20 between pre-service teachers who have had one mathematics methods course (M=4.4), and those who have had two mathematics methods courses (M=4.6). The ANCOVA results suggested that there was no significant difference in Item 20 [F (2,128) =2.853, p=0.061] between the pre-service teachers who have had one mathematics course and those who have had two mathematics method courses while adjusting pre-item 20. The conclusion is that there was no statistical difference in Item 20 of the self-efficacy instrument between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 30. ANCOVA Result for Item 20 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

**Item 21**. I do not know what to do to turn students on to mathematics.

Table 31 shows the ANCOVA output comparing differences in Item 21 between pre-service teachers who have had one mathematics methods course (M=3.3), and those who have had two mathematics methods courses (M=4.4). The ANCOVA results suggested that there was a significant difference in Item 21 [F (2,128) = 4.841, p=0.009] between the pre-service teachers who have had one mathematics methods course and those who have had one or two mathematics method courses while adjusting pre-item 21. The conclusion is that there was a statistical difference in Item 21 of the self-efficacy instrument between pre-service who teachers have had one mathematics methods course and those who have had two mathematics methods courses.

*Table 31. ANCOVA Result for Item 21 Comparing One Mathematics Methods Course and Two Mathematics Methods Courses.*

#### Research Question 4

**What is the impact of mathematics methods courses on pre-service teacher’s self-efficacy?**

To evaluate the pre-service teachers’ self-efficacy after taking the mathematics methods courses, a paired T-Test was conducted. A paired sample t-test was performed for the mathematics methods courses to determine if there was a change in self-efficacy measured by the pre- and post-treatment scores on the MTBI instrument for each course.

**Variables**. The independent variable was the mathematics methods courses groups, the dependent variable was the pre-service teachers’ post-instruments scores, and the covariate was the pre-instruments scores.**Null hypothesis**. Mathematics methods courses have a significant impact on pre-service teachers’ self-efficacy.**Alternative hypothesis**. Mathematics methods courses have no significant impact on pre-service teachers’ self-efficacy.**Summary of Findings for Research Question 4**. Table 32 indicates the descriptive statistics of the pre-service teachers’ self-efficacy, and shows that there was a change in the participants’ self-efficacy between pre- and post-instrument scores for pre-service teachers in the mathematics methods courses.

The descriptive statistics showed that the pre-service teachers had higher means on the self-efficacy after the mathematics methods courses (M = 89) than before the mathematics methods courses (M = 80).

*Table 32. Descriptive Statistics for the Pre- and Post Instrument.*

In addition, the Paired T-Test results in Table 33 indicated that there was a statistically significant improvement (t= -4.553, p=. 000). Therefore, one can conclude that there are significant differences of pre-service teachers’ self-efficacy before and after the mathematics methods courses, indicating an improvement in self-efficacy.

*Table 33. Paired Sample T-Test Results of Pre-service Teachers ’self-efficacy after the Content Pedagogy Courses.*

#### Research Question 5

**Based on gender, are there differences in self-efficacy of pre-service teachers?**

To answer this question, analysis of covariance (ANCOVA) was used to determine if there was a significant difference between the post-instruments scores between gender groups.

**Variables**. The independent variable was gender groups, the dependent variable was the pre-service teachers’ post-instruments scores, and the covariate was the pre-instruments scores.**Null hypothesis**. There is no significant difference in self-efficacy between pre-service teachers who have had content pedagogy courses and who have had one or two mathematics methods courses.**Alternative hypothesis**. There is a significant difference in self-efficacy between pre-service teachers who have had content pedagogy courses and who have had one or two mathematics methods courses.

**Summary of Findings for Research Questions 5**. In order to test whether there is a difference between male and female pre-service teachers, an ANCOVA was used to determine if there were any significant differences between the mean gain score of the male and female groups. To evaluate whether the group size was a problem, the researcher ran Levene’s test for homogeneity of variance at the same time that the ANCOVA was run. When the two groups have a value greater than.05, it means they have equal variances according to the results of the Levene’s test for equality of the variances as seen in Table 34 (F=0.187 with p-value =.608). A value greater than.05 means the variance.

*Table 34. Levene’s Test of Equality Gender.*

Results from the ANCOVA analyses in Table 35 (F= 1.551, p=. 215) suggested that gender was not a significant influence on self-efficacy. Therefore, the conclusion is that there is no difference in the self-efficacy based on gender between pre-service teachers. A major limitation of the current analysis is the extremely small sample size of the male group of participants.

*Table 35. ANCOVA Analysis of Pre-service Teachers’ Self-efficacy Comparing Gender Groups.*

### Qualitative Data Analysis

Data analysis for the qualitative data began with the transcription of the data. The length for each interview ranged from 10-15 minutes and all were audio-recorded. All interviews were transcribed by using Google Docs software without identifiers, ensuring the anonymity of the participants. After transcribing, the researcher identified common themes among the interviews.

To answer the Research Questions 6 (What are pre-service teachers’ perceptions of their skills, competence, and ability to teach mathematics?) and 7 (What aspects of mathematics methods courses influence the self-efficacy beliefs of future teachers of mathematics?), two types of coding (open and axial) were used to identify themes from the interviews.

According to Corbin and Strauss (1998), open coding is the process of “dividing the data into blocks so as to be able to outline the concepts that will represent the raw data. At the same time, one is qualifying those concepts in terms of their properties and dimensions” (p. 195). The process of axial coding is also described as a “process of relating categories to their subcategories” (Strauss & Corbin, 1998, p. 123). When it came to the open coding process, the researcher examined and compared the transcript several times to determine similarities and differences. The important sentences were highlighted using the Microsoft Word program to distinguish the main idea of the interview. The important sentences were identified as either frequently occurring or being significant to the researcher. Then the significant parts of it were grouped together using a code that represents a concept associated with the study. The labels of the codes were experiences, receiving feedback, modeling, concern, positive change, content courses, positive environment, new strategies, coping, and confidence. The axial coding followed open coding, by clarifying and expanding the relationships between these different labels. These themes are addressed in greater detail below.

#### Research Question 6

**Theme 1: Growing self-efficacy**. The first theme, which addressed the first research question, has to do with the pre-service teachers’ perceptions of their skills. All the participants felt their self-efficacy increased after the courses. The pre-service teachers mainly assessed their self-efficacy as rather high; most respondents assessed it as 8 or 9 on the scale from 1 to 10. In particular, pre-service teachers 1, 2 and 3 gave themselves the mark of 8 on a 10-point scale, pre-service teacher 4 gave him/herself a 9, pre-service teacher 5 gave him/herself a 7, and pre-service teacher 6 ranked him/herself as being between 8 and 9 (Table 36).

*Table 36. Pre-Service Teachers’ SELF-RATING on Their Level of Self-Efficacy.*

Pre-service teachers emphasized that the mathematics methods courses provided them with the knowledge of techniques which may be used to teach mathematics, as well as with some practical experience of teaching, even if they only practiced with their colleagues. However, they did not give themselves a 10 as they felt they needed more practice and field experiences to develop their skills. One of the participants explained that she needed the field experiences in the mathematic methods courses because teaching each other is different than teaching children. Following are some quotes from the participants’ interviews supporting these claims:

“I think I am about an eight now. Because I have different tools and I know the techniques that are useful if a student doesn’t learn a concept one way, I will have another way and a different perspective to offer the same idea. And before, I could only think of it the way I learned it. And now I can see it from different views.

(Not a ten because) I just don’t have any real practical experience teaching Math. I’m still a pre service teacher. So hopefully it will increase to a ten.”

“I would probably say, like, a seven. I don’t know. I feel like I still kind of need more practice.”

“If there was an opportunity to add a field placement with the Math methods classes, I think it would be a helpful tool. Logistically, I don’t know how that would work out. I think that’s everything. Teaching it to our colleagues like our peers is helpful. It gets us to experience in front of people teaching, but it’s very different to teach a skill to twenty-year-old than it is to teach it to eight-year old. We do get field experience. It’s just not always in math, which is kind of like different.”

When asked about their ability to teach before and after the courses, many of the pre-service teachers were not confident of their knowledge of mathematics prior to taking the courses. Most of them stated that they knew mathematics well, but were not prepared to explain the very basic concepts of mathematics to their peers. On the other hand, after taking the courses, the pre-service teachers became significantly more confident when it came to assessing their knowledge of mathematics. The respondents stated that the courses allowed them to remember not only the basics but also more advanced concepts and notions of mathematics, and that explaining them allowed for a better understanding of these concepts, being very beneficial to those who did the explaining. All six pre-service teachers indicated that they were much more confident in their math skills compared to what they had felt before taking the coursework. Some quotes from the participants’ interviews regarding their experiences before and after the course appear below:

“Before going into it, I definitely probably wasn’t very confident in my skills. Now after doing MATH 320 I feel like I could get up in front of a classroom and actually teach a lesson.”

“Before, I was very nervous. I wasn’t sure if I was going to be able to do it because like I said, I have always been terrible at math. But I think Dr.X really gave us good examples from her experiences and provided us with a list of multiple activities that we can use.”

**Theme 2: The benefit of the content pedagogy courses on pre-Service teachers.** Most of the pre-service teachers indicated that the content pedagogy courses were beneficial for bringing them back to basic material and contributed to their understanding of what their students usually did not understand. Moreover, they indicated that these courses were important to learning about the conceptual knowledge of students learning mathematics. Also, they explained that in content pedagogy courses, they not only learned how to teach, but also, learned mathematics from students’ perspective. Being able to understand mathematics from the students’ perspective had helped them become better at understanding ways to address students’ mathematical struggles. An exception was with one interviewee who did not have a positive experience with content courses because shehe struggled with the mathematical content itself.

The following are quotes from the participants’ interviews supporting these claims regarding the importance of the content pedagogy courses:

“I feel like MATH 151 and 152 really helped me become more confident with the content itself, because they’re more learning what you’re going to teach and not necessarily how to teach it.”

“I think MATH 151 and 152 helped me see from my students’ perspective, see what they might struggle with, what issues they could have.”

“My experience in MATH 151 and 152, I didn’t like them very much. I did better in MATH 151 than I did in MATH 152. MATH 152 has a focus on geometry, geometry was an area that overall I personally struggle in. I sort of felt like those two classes were kind of almost intimidating.”

**Theme 3:Mathematics methods courses increased self-efficacy more than the content pedagogy courses**. When asked about which course had a strong effect on their self-efficacy, all pre-service teachers agreed that mathematics methods courses increased their self-efficacy more than the content pedagogy courses. They explained that mathematics methods courses provided more methods like hands-on activities and improved their teaching skills more than the content pedagogy courses. According to the participants, mathematics methods courses provided them with various techniques and hands-on tools to help them in the future. MATH 320 was mentioned more than any other course as most beneficial but many students also included MATH 330. One pre-service teacher mentioned that mathematics methods courses were more beneficial because they adopt a child-centered approach. Others stated they practiced many methods in mathematics. Even the pre-service teacher who struggled with the content courses had positive experiences with the mathematics methods courses. The following are quotes from the participants’ interviews supporting these claims:

“I haven’t taken MATH 330 yet. So far in MATH 320 I have developed a lot of skills that are helpful to give young children conceptual understanding of basic math concepts, and help me gain perspective into counting.”

“I would say MATH 330 had the biggest effect because I was worried that I wouldn’t be able to teach my students properly how to do. Especially with the more complex stuff that we’re learning in MATH 330, I feel like it helped me get the tools to teach my students how to learn the things. It helped me feel more comfortable teaching my subjects.”

“I would say MATH 320 had the most effect on developing- That’s when I really started to learn like these are the technique you can use and we actually started applying them. We practiced them with each other, we practiced them for a grade to see if we could teach. We were actually supposed to go into a class, but never did, so I think that would have been helpful, but yeah, that never happened.”

“Definitely the one that I’m taking now increased it. I’ve never really been good at math but when I do hands on activities like we are doing in MATH 320, I feel better about it and I know that I think I could teach it.”

“I was very anxious going into MATH 320 this year because I was kind of thinking, like, “Great. It’s just going to be another extension on top of those and will be even harder.” It’s totally different because it’s actually teaching the math. We’re doing everything at a pre-kindergarten to first grade level. We do a lot of hands on stuff, and a lot of manipulatives and activities in class and stuff. We would play games. It’s very, like, child-centered.”

“I think that MATH 320 and 330 have definitely helped me feel more comfortable teaching. They all build off of each other, which is nice. 152 is a continuum of 151. It’s the next step up. Then when she gets into the methods classes of 320 and 330, it’s really looking at different strategies to teach it, and how to do it yourself.”

**Theme 4: Confidence growth**. All the participants expressed positive confidence after the mathematics methods courses. Most of the participants believed that the courses contributed to their confidence. The following are quotes from the participants’ interviews supporting these claims:

“I feel confident in my ability that I can teach them. If there is a topic that I am unfamiliar with or I am unsure how to teach, I know that there are resources out there that I can go look at to learn more myself so I can better help my students learn the skills that they need. I definitely feel a hundred times more confident now than I did in freshman year.”

“I feel like the course has definitely helped me feel more confident in my ability to teach mathematics.”

“I just feel like I’ve taken a lot from the course and have definitely become more confident and kind of more motivated to be like, I can do this one day when in my own classroom when I am in the real world.”

“Having them taught the way they are taught and by the people that they are taught by makes me as someone who is getting ready to go into the field feel a lot more confident.”

“I feel confident now to communicate the concepts to students.”

“I think it helped me out because I’m a lot more confident now. By using my methods, it just helped me out a lot to show me that I’m able to do it.”

**Theme 5: Willing to use new strategies.** Willingness to use new strategies is one of the characteristics of high self-efficacy that was mentioned in the interviews. When asked about their willingness to use the new strategies they have learned, all of the participants appreciated all the new strategies they had in the courses and will use it in the future. Two of the participants used some of the strategies they have learned in tutoring. The following are quotes from the participants’ interviews supporting these claims:

“I will use the new strategies. They are the most useful teaching strategies that I learned.”

“I actually used a few of the new strategies. I used to tutor some students, so I used all the strategies. “

“I will definitely use everything I learned.”

**Theme 6. Coping with difficulties**. Coping with difficulties was another characteristic of high self-efficacy that was explored in the interview. When asked about their ability to cope with difficult tasks or negative experiences, they mentioned some of the ways to cope with difficult concepts in the courses. For example, participants reported that during times of difficulty they would communicate with their peers or go back to figure out what the problem was to overcome the challenges. The following are quotes from the participants’ interviews supporting these claims:

“We do our lessons, sometimes it goes bad. We can turn it around and can positively affect our self-efficacy”

“Even if I don’t know the way to fix a problem myself, I can communicate it to someone else.”

“I’ve had negative experiences. I don’t think any of them decreased my self-efficacy. I think they helped me feel more prepared to do it. It might not work, but I can always improve on it. I always went back and looked and see what exactly didn’t work and see how I can change that and see if I can make it better. And I think that’s what helped me the most.”

#### Research Question 7

**Theme 1: Mastery experiences**. Mastery experience is the experience when pre-service teachers actually practice teaching in the courses. Participants provided examples for some of the many experiences they had in the courses that they felt beneficial for improving their self-efficacy. They all acknowledged that hands-on activities, working in small groups, and mini-lessons were examples to construct mastery experiences. Most of the pre-service teachers valued manipulative tasks and tools that facilitated their self-efficacy. Two of the participants mentioned teaching mathematics to their professor that acted as a school student. Hands-on opportunities and activities were outlined as beneficial by most of the respondents. One of the participants mentioned that they were provided with educational articles to be discussed and used to teach each other. They also mentioned teaching each other while other pre-service teachers enjoyed watching videos of other students teaching. One of the participants indicated that struggling in the mini-lessons helped her be aware of some of the challenges she could be facing as a teacher in the future. The following are quotes from the participants’ interviews supporting these claims:

“She always provides us with lots of examples and lots of research-based articles that are things that people are doing in the field now. Then we have to talk about them, and implement them, and do lessons with the college kids as if they are second graders, which really get us thinking. We are spending class periods just doing different activities that you could do with kids.”

“You had to actually do the activity with them like you were the teacher and they were the students. So you practiced mini teaching in your little group.”

“It’s like co-teaching pairs, and then you and your partner have to work together to come up with how to teach this concept using an idea from the textbook and manipulatives that are available in the classroom.”

“We do a lot of group activities, like I said. We are either teaching each other or we’re teaching a lesson in the front of the classroom to the whole entire class, so we are playing the role of the teacher.”

“I think it helped me see what struggles I could come in to. We were doing addition, subtraction with base ten blocks and I could see what struggles the teacher had. If there weren’t enough blocks for everybody then those things could impact the course. It helped me feel like it’s okay if not everything works all at once and that’s why you get to do it over and over again and improve upon it. Not to sweat so much if it doesn’t work the first time because it might work another time after I see what works and what doesn’t work. It helped me see what may and may not work and what I can do to fix things.”

**Theme 2: Vicarious experience.** Vicarious experience is the experience of observing other pre-service teachers or professor performing teaching. When discussing vicarious experiences with pre-service teachers, they all stated that modeling activities positively influenced their self-efficacy. They mentioned that peer observation was useful for acquiring new knowledge, skills, and mathematics problem-solving from a new perspective. Also, it encourages them to try what they have observed. Some participants further mentioned that working with groups is similar to co-teaching which increased their self-efficacy. In the groups, they planned lessons, examined students’ work, watched each other teach, and provided feedback. One of the participants mentioned that she felt comfortable when she compared her skills to others. One of the most interesting comments occurred when she referred that she felt confident when she saw others struggle with the same activity with which she struggled. To summarize, many participants highlighted some instructional strategies to construct vicarious experiences. They all acknowledged that seeing their peers or professor use manipulative tools, repeating concepts, and teaching new concepts to a partner and/or small group helped enhance their own self-efficacy. The following are quotes from the participants’ interviews supporting these claims:

“There was a great deal of peer interaction and group work. We would have individual assignments that we would bring back and teach each other about.”

“My professor really gave us good examples from her experiences and provided us with a list of multiple activities that we can use”.

“We were given materials and teacher partner. Just given these materials, do whatever you want to teach your partner how to find the area or perimeter of this, so just pretend they’re an elementary school student and teach them. So we do that and then we pair up and talk.”

“Okay, this is what I might have done differently. This was a little bit confusing, you could change that.” And then our professor has the same materials and she models for us, “This is how I would have done it.”

“I’d say for that one definitely peer interaction. Just getting ideas from each other, and then the professor modeling, like, this is how I would do it, so go ahead and try, and then put your own spin on it. The professor modeling, and then the peer interactions helped me a lot. We sit in pairs, so we have two people that sit together and work together every class period”

“The small group instruction and seeing what my peers came up with and discussing our ideas with that because we’re all in the same boat, we’re all in the same area of our learning. Seeing what they’re doing or what they’re struggling with, I think that helped me the most to see, oh, I’m not the only one that’s feeling this way, or oh, that’s a good idea. I can definitely use that. So I would say small group instruction.”

**Theme 3: Social persuasion**. Social persuasion comes from experiences where students receive verbal or written feedback during pre-service teachers’ learning-to-teach activities. The component labeled as social persuasion was seen as being effective when receiving feedback from the instructor or peers on performance. Many of the interviewed pre-service teachers indicated that explanations and positive feedback given by their professors increased their self-efficacy. Also, they explained that their instructor provided them with an explanation of how they could improve their teaching and suggest strategies for future student engagement. Four interviewees mentioned that the task of teaching mini-lessons to other students promoted self-efficacy alongside with the feedback given by the professor. The following are quotes from the participants’ interviews supporting these claims:

“And my professor will give us feedback on maybe something we could change or make better. Or maybe if it was just really, really good.”

“There she’ll kind of tell us, like, Oh, you really should’ve incorporated this or Next time you do this try to do this. She’ll kind of be like, I’m not really sure what you’re explaining. Can you repeat that or can you explain it in a different way?”

“The professor and our peers give us feedback, and I say that’s probably the most helpful, because they can say, “Okay, that was good, but here’s what I would have done if I were to do it. Here’s another way you could do it.”

“Knowing that I can teach a lesson, have these lessons planned, and get good grades on them it makes me feel more confident in doing it. My professor usually writes how we could improve our lessons. Different questions that we could ask our students instead of just saying, “Oh good job”, “Great job.”

**Theme 4: Supportive and inviting environment**. Physiological factors such as pre-service teachers’ feelings and moods during or after the mathematics methods courses affected interviewees’ self-efficacy. Negative moods decreased pre-service teachers’ beliefs about their ability to teach, whereas positive moods improved their beliefs about their ability to teach. Most of the pre-service teachers interviewed in study indicated that their physiological state had a critical influence on their self-efficacy, as well as on their ability to successfully participate in the teaching and learning activities. Participants provided examples on how the physiological conditions could be positive supportive environment, collaborative environment, teacher encouragement, and teachers’ positive attitudes. Most of the participants had a positive mood, which was affected by a supportive and inviting environment.

Pre-service teachers in the study cited a supportive and inviting environment in reflecting on the prospect of their physiological state, and how it had enhanced their self-efficacy. Participants provided examples on how the physiological condition could be controlled by supportive and collaborative environments. Many participants highlighted how their professors’ positive attitudes helped increase their self-efficacy. They mentioned that a high-level of responsibility for the support and growth of pre-service teachers’ self-efficacy is held by their student-faculty relationship. This relationship encourages them to participate and to communicate which increased their self-efficacy. Also, the support from their peers and their professors created a positive environment, which limited any negative moods. The interviewees stated that the learning environment was inviting because they had discussions, mathematical games, and beneficial feedback. One of the respondents mentioned how their teachers’ feedback of support rather than a formal evaluation created an inviting environment. Pre-service teachers’ interview responses highlighted some of these experiences. The following are quotes from the participants’ interviews regarding their classroom experiences that controlled or affected physiological states:

“She really has everybody communicate. I feel like that’s really helpful, especially in math, I feel like a lot of people are scared to share their answer because they think they’re going to be wrong, and even if they’re right they still don’t want to share their answer because they don’t want to be the only one that’s right. Being that she makes everyone kind of participate, I feel like that really helps.”

“We do activities that seem like fun games that actually have great value in learning math. She helps us to be relaxed but also engaged and open-minded to the new concepts and the instructional methods that we’re learning.”

“Our instructor always starts off with a little activity at the beginning to try to get us into the math mindset. I think that does help with getting you into the correct mood and not just throwing you in immediately and floundering around.”

“We start out every class by sharing ideas, and I think that really helps because you don’t necessarily shut anyone down for their ideas, whether you think it’s right or wrong. It’s a very open environment for sharing and trying new things, and experimenting.”

“I definitely could see how if your teacher is a more negative person, how then you would feel negative about your abilities to teach.”

“There’s always class discussions that if there is a disagreement about a problem, it’s not I am right, you’re wrong. It’s okay, let’s talk about this. How can we…? My wording could be changed. How can I change it? It’s a very inviting environment to learn in, because you feel like not only are you still learning, but the professors are also learning with you in a way.”

“Then she’s always there to support us.”

### Chapter Summary

This chapter presented the data and analyses for the demographic variables and research questions. This chapter also provided a description of the sample, the research questions, and quantitative and qualitative results. The first five questions were analyzed using One-way Analysis of Covariance (ANCOVA) and Paired T-Test. For the qualitative data, responses were examined for themes in which the transcription was analyzed for themes. Each theme was explored, and examples of pre-service teachers’ comments were given. The quantitative data showed that the results from the pre-instrument and post-instrument were a statistically-significant improvement in pre-service teachers’ self-efficacy while the interviews with the participants created a picture of a noticeable increase in pre-service teachers’ self-efficacy. Chapter Five presents a summary and discussion of the findings, implications for practice, and recommendations for future research.

## Findings, Research, Implications, and Conclusion

In this final chapter, a summary and discussion of the results, limitations of the study, implications for practice, and the recommendations for future research are presented.

### Summary of the Study

The researcher investigated how pre-service teachers’ self-efficacy for the teaching profession changed during their mathematics method courses. By examining the changes of pre-service teachers’ self-efficacy, teacher educators might provide proper instructional practices to help increase pre-service teachers’ self-efficacy. In this study, it was found that the pre-service teachers’ self-efficacy did significantly change after taking the mathematics methods courses. Their self-efficacy in the mathematics methods courses increased significantly from the start of their teacher education program to the end of their training. The study was aimed to examine the impact of mathematics methods courses on pre-service Early Childhood and Special Educators’ self-efficacy and beliefs. Additionally, the purpose was to examine the possible factors responsible for the pre-service teachers’ beliefs, and to determine the levels of self-efficacy of pre-service teachers’ regarding their skills in the mathematics methods courses.

### The Quantitative Findings

In this section, quantitative findings are discussed for the first five research questions.

#### Research Question 1

**Are there Differences in Self-Efficacy Between Pre-Service Teachers Who Have Had Content Pedagogy Courses and those Who Have Had Mathematics Methods Courses?**

ANCOVA tests showed a significant difference in self-efficacy beliefs between pre-service teachers exposed to content pedagogy courses and those exposed to one or two mathematics methods courses. The mean self-efficacy score of pre-service teachers exposed to content pedagogy courses was lower (74.5) compared with those exposed to mathematics methods courses (85.8). This finding is a good indicator of the fact that mathematics methods courses may have more beneficial effects in developing the self-efficacy of pre-service teachers than general content pedagogy courses. The pre-service teachers who were interviewed explained that content pedagogy courses are important because they provide the theory behind how students learn mathematics, but the mathematics methods courses provide more practical experiences for pre-service teachers to teach mathematics. For example, one of the participants has explained the reason of the differences between mathematics methods courses and mathematics content pedagogy courses. as expressed in the following response:

“I think that MATH 320 and 330 have definitely helped me feel more comfortable teaching. They all build off of each other, which is nice. 152 is a continuum of 151. It’s the next step up. Then when she gets into the methods classes of 320 and 330, it’s really looking at different strategies to teach it, and how to do it yourself.”

Although no study has specifically evaluated how content pedagogy courses and mathematics methods courses compare in developing the self-efficacy of pre-service teachers, Lancaster and Bain (2010) found statistically-significant gains in self-efficacy among pre-service elementary education teachers exposed to a 13-week required undergraduate course. They stated that the exposure to more courses and more preparation increased their self-efficacy. These results suggest that cumulative educational training in the program is an effective way to increase self-efficacy. Research from Darling-Hammond (2000) also supports the concepts that pre-service teachers who have had more preparation on teaching are more likely to have higher self-efficacy.

#### Research Question 2

**Are there Differences in self-efficacy for pre-service teachers between those pre-service teachers who have had one content pedagogy mathematics course and those Pre-service teachers who have had two content pedagogy mathematics courses?**

The second research question addressed the differences in self-efficacy between pre-service teachers exposed to one content pedagogy mathematics course and pre-service teachers exposed to two content pedagogy courses. ANCOVA tests showed that pre-service teachers exposed to one content pedagogy course had a lower self-efficacy mean score (70.0) than those exposed to two content pedagogy courses (79.1). While this question did not ask for the impact of the content pedagogy courses on their self-efficacy, the Paired T-test findings indicated that there was a decrease in pre-service teachers’ self-efficacy after completing the content pedagogy courses. This finding contradicted previous information obtained from the literature about the importance of the content pedagogy courses to improve pre-service teachers’ self-efficacy (Guskey & Passaro, 1994; Muijs and Reynolad, 2002; Shulman, 1986; Swars, 2007; Van Driel, Veal & Janseen, 2001). Thus, more research is needed to investigate the effect of content pedagogy courses on pre-service teachers’ self-efficacy. The result that the pre-service teachers’ self-efficacy decreased after the content pedagogy courses could be due to the difficulty of the mathematics content in the courses. For example, one of the participants who struggled with the content pedagogy, as expressed in the following response:

“My experience in MATH 151 and 152, I didn’t like them very much. I did better in MATH 151 than I did in MATH 152. MATH 152 has a focus on geometry, geometry was an area that overall I personally struggle in. I sort of felt like those two classes were kind of almost intimidating.”

Also, it may be that self-efficacy itself may change over the course of the semester and one rating at the end of the semester is not sensitive enough to detect the changes. Another explanation is the self-efficacy instrument that was used is designed to predict pre-service beliefs about teaching. The content pedagogy courses were not focused on teaching methods or strategies. The content pedagogy courses afford experiences of examining mathematics concepts, but they lack the opportunity to practice teaching, so it may be insufficient to use the same instrument for both courses. Pre-service teachers may begin the content pedagogy courses excited about teaching, but when faced with difficult mathematics content without teaching practice, their self-efficacy in their teaching skills may decrease.

#### Research Question 3

The third research question addressed the variations in self-efficacy between pre-service teachers exposed to one mathematics methods course and those exposed to two methods courses. The Mathematics Teacher Efficacy Beliefs Instrument was used to collect data. The MTEBI instrument compared two uncorrelated components: The Personal Mathematics Teaching Efficacy (PMTE) and The Mathematics Teaching Outcome Expectancy (MTOE) between the pre-service teachers. The first subscale is the Personal Mathematics Teaching Efficacy (PMTE) subscale, which has 13 items (2, 3, 5, 6, 8, 11, 15, 16, 17, 18, 19, 20, and 21). The second is the Mathematics Teaching Outcome Expectancy (MTOE) subscale, which includes eight items (1, 4, 7, 9, 10, 12, 13, and 14) (Enochs, Smith, & Huinker, 2000). PMTE is the pre-service teachers’ own belief about their ability to teach. MTOE is pre-service teachers’ belief that their teaching will lead to the desired outcomes (Bandura, 1977). PMTE represents the question, “Can I teach mathematics?”, while MTOE represents the question, “If I teach mathematics, what will happen?”. Figure 1 represents the relationship between PMTE and MTOE (Bandura, 1977). PMTE will lead to teaching mathematics, while MTOE will lead to a specific outcome (Bandura, 1977). According to Bandura (1977), personal mathematics teaching efficacy comes before mathematics teaching outcome expectations.

Twenty-one ANOVAs were performed to explore the changes of each item between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses. The results indicate that there is no difference in items 1, 4, 7, 8, 9, 10, 16, and 20 of the self-efficacy instruments between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses. On the other hand, the results of items 2, 3, 5, 6, 11, 12, 13, 15, 17, 18,19, and 21 in the self-efficacy instruments showed statistically significant differences between pre-service teachers who have had one mathematics methods course and those who have had two mathematics methods courses. All these items were associated with the PTME scale except items 12 and 13, which were associated with the MTOE subscale.

Thus, pre-service teachers who completed one mathematics methods course have similar mathematics outcome expectancy but statistically lower personal mathematics teaching efficacy from pre-service teachers who have completed two mathematics methods courses. Two items in the MTOE: (12) The teacher is generally responsible for the achievement of students in mathematics, and (13) Students’ achievement in mathematics is directly related to their teacher’s effectiveness in mathematics teaching, were found to be different between the two groups of pre-service teachers. Pre-service teachers who have had two mathematics methods courses have higher mean scores. These scores may indicate higher personal mathematics teaching efficacy, than pre-service teachers who have had only one mathematics methods course. The study can conclude that both groups had high self-efficacy in all items, but the self-efficacy level for teaching was higher for those pre-service teachers who had two methods courses.

An explanation for the differences in means may be due to the additional experiences and instructional practices that benefitted pre-service teachers who had two mathematics methods courses rather than one. Pre-service teachers who have had two mathematics methods courses would have more experiences than pre-service teachers in the other group. Additionally, these pre-service teachers had more opportunities for in-class practice and experienced the entire mathematics methods instructional sequence for their program. The research has demonstrated that mathematics pre-service teachers who were exposed to social and verbal persuasion by observing their peers and learning new materials have a higher chance of increasing their efficacy than other pre-service teachers (Turner, Cruz, & Papakonstantinou, 2004). Therefore, the findings of this research question support the findings of research questions number one and two, which state that if pre-service teachers are exposed to more educational courses in their program, their self-efficacy will gradually increase. As mentioned previously in Question 1, pre-service teachers’ self-efficacy benefited from more training.

#### Research Question 4

**What is the impact of mathematics methods courses on pre-service teachers’ self-efficacy?**

The main purpose of this study was to examine the impact of mathematics methods courses on pre-service teachers’ self-efficacy and beliefs. In response to the fourth research question that assessed the impact of mathematics methods courses on pre-service teachers’ sense of self-efficacy, the study found significant increases in teachers’ self-efficacy upon exposure to mathematics methods courses. This finding is consistent with previous research studies documenting the positive effects of mathematics methods courses on pre-service teachers’ self-efficacy (Esterly, 2003; Huinker and Madison, 1997; Incikabi, 2013; Kim, Sihn, & Mitchell, 2014; Kim, Sihn, & Mitchell, 2014; Swars Hart, Smith, Smith, & Tolar, 2007; Swars, 2008; Wenner, 2001; Wilkins, 2008; Wilkins & Brand, 2004).

Also, a study by Wilkins and Brand (2004), has shown that “a positive relationship between participating in the mathematics methods course and changes in teacher beliefs and attitudes” is evident (p. 226). In another related study, Easterly (2003) found that mathematics self-efficacy, teacher efficacy, and mathematics teaching efficacy increased substantially among pre-service and novice elementary teachers exposed to a mathematics methods courses over a fixed period of time.

#### Research Question 5

**Are there differences in self-efficacy based on gender for pre-service teachers?**

The fifth research question assessed if there are significant differences in self-efficacy of pre-service teachers based on their gender. According to the ANCOVA test, there was no difference in self-efficacy between two genders (F= 1.551, p=. 215). The between-subjects effects test failed to reject the null hypothesis that there is no difference in self-efficacy between male and female. This finding may mean that gender does not mediate the difference in self-efficacy beliefs among pre-service teachers. The finding was consistent with Brusal’s (2010) finding which reveals that gender has no significant effect on mathematics teaching efficacy beliefs.

### Qualitative Findings

**In this section, qualitatve findings are discussed for the last two research questions.**

#### Research Question 6

The qualitative themes regarding the teachers’ perceptions of their skills, competence, and ability to teach mathematics are: 1) growing self-efficacy, 2) the benefit of the content pedagogy courses, 3) mathematics methods courses increased efficacy more than content pedagogy courses, 4) confidence growth, 5) willing to use new strategies, 6) and coping with difficulties. These themes provided insight into understanding the pre-service teachers’ perspectives about their skills, competency, and ability to teach mathematics.

**Growing self-efficacy**. Qualitative analysis supported the quantitative data, demonstrating that pre-service teachers felt their ability to teach mathematics improved after the mathematics methods courses. Interviews conducted with six pre-service teachers gave the following data on teachers’ self-perception about their ability to teach mathematics: three teachers said that their rating was eight, one gave a rating of nine, and one gave a rating of seven, while only one hesitated between eight and nine points. Such rating results (between seven and nine points) suggests that these courses taught respondents a great deal about effective practices, techniques, and tools. However, they also suggest that interviewed pre-service teachers did not have enough experience and confidence to give themselves a rating of ten. The fact that the majority of interviewees acknowledged that there was always room for improvement and growth suggests that more teaching experience will improve their self-efficacy.

Most of the pre-service teachers who took part in the current study thought that they had not been very skillful teachers of mathematics prior to taking the courses. On the other hand, it is apparent that after the courses, the participants became more confident in their ability to teach mathematics. It is very important to note that even those respondents who stated that they had forgotten the majority of the basic mathematics content and had doubts regarding their ability to teach properly became much more confident in their ability to teach mathematics after taking the mathematics methods courses as expressed in the following response:

“I was very anxious going into MATH 320 this year because I was kind of thinking, like, Great. It’s just going to be another extension on top of those [MATH 151 and 152] and will be even harder. It’s totally different because it’s actually teaching the math. We’re doing everything at a pre-kindergarten to first grade level. We do a lot of hands on stuff, and a lot of manipulative and activities in class and stuff. We would play games. It’s very, like, child-centered Therefore, mathematics methods courses have a major effect on pre-service teachers’ self-efficacy”.

These findings are closely aligned with the various studies done that found that pre-service teachers self-efficacy could increase after taking mathematics methods course (Easterly, 2003; Swars, Hart, Smith, & Tolar, 2007).

**The benefit of the content pedagogy courses**. Qualitative analysis, supported by the quantitative data, demonstrated the importance of the content pedagogy courses on the pre-service teachers as expressed by them. The majority of the pre-service teachers who participated in the study mentioned that content pedagogy courses affect their self-efficacy. These courses provide pre-service teachers with an understanding of the basic principles and simple application of ideas like adding, subtraction, and fraction. This finding was relatively consistent with the findings of Swars, Hart, Smith, and Tolar (2007). The interviewees’ perspectives about content pedagogy courses were beneficial for bringing them back to basic material and helped them realize how their future students might struggle. In the content pedagogy courses, they experienced learning mathematics as a student. It is important for pre-service teachers to both know the content they teach and know how to teach (Shulman, 1986).

**Mathematics Methods Courses Increased Efficacy more than Content Pedagogy Courses**. Qualitative analysis supported the quantitative data that mathematics methods courses affect pre-service teachers’ self–efficacy more than content pedagogy courses. All respondents suggested that mathematics methods courses were more effective than content pedagogy courses. They explained that methods courses have more practical experiences on teaching mathematics. In these courses, they applied the knowledge they have learned and practiced the necessary skills. In these courses they learned mathematics as learners and as teachers.

Pre-service teachers need to know the subject they teach and how to teach the concepts to students (Shulman, 1986). This finding is a good indicator of the fact that both content pedagogy courses and mathematics methods courses are important but mathematics methods courses may have more beneficial effects on the pre-service teachers’ self-efficacy development than content pedagogy courses. According to a study by Swackhhamer, Koellner, Basile and Kimbrough (2009), mathematics methods courses and content pedagogy courses are important for pre-service teachers’ self-efficacy. An explanation for this finding is that deference in self-efficacy levels between courses may be due to deferent instructional experiences or topics in the courses. Different instructional practices may differentiate pre-service teachers’ levels of self-efficacy. Unfortunately, there is not much research that compares the effects of the content pedagogy courses and mathematics courses on pre-service teachers’ self-efficacy.

**Confidence, willingness, and ability to cope**. The characteristics of most pre-service teachers that took part in the interview were very similar to characteristics found in earlier research that are associated with high self-efficacy, and on similar research about the characteristics associated with high self–efficacy regarding teaching mathematics (Swars, 2008). Awareness of these characteristics helps teacher educators to include various instructional activities to support pre-service teachers’ different self-efficacy levels. It was implied that participants in this study have high self-efficacy regarding teaching mathematics. Pre-service teachers were much more likely to cope with difficulties, more confident in their ability to teach, and to use new strategies. These characteristics are consistent with previous research focusing on a strong sense of efficacy (Moran & Hoy, 2007; Tschannen-Moran, Hoy, & Hoy, 1998). Also, some of the respondents believed in their ability to use all the new methods they have learned to teach all students with different backgrounds. Czernaik (1990) stated that teachers who possessed high self-efficacy were the most likely to apply new methods in their classrooms and manage to teach all students with different backgrounds. Similarly, participants described a variety of positive experiences and did not express any negative concerns about their ability to teach mathematics. These characteristics prove that mathematics methods courses have a positive effect on pre-service teachers’ beliefs and behaviors. It is therefore important for teacher education programs to identify pre-service teachers’ strengths and other characteristics to develop suitable instructional strategies to meet the needs of the pre-service teachers. Knowing pre-service teachers’ characteristics helps to determine their readiness to learn how to teach (Jerkins, 2001; Tschannen-Moran & Hoy, 2001).

#### Research Question 7

The qualitative data revealed four themes regarding the aspects that influence self-efficacy. These themes are mastery experiences, vicarious experiences, social persuasions and supportive and inviting environments which control teachers’ physiological states.

The results of this study support the self-efficacy theory with the four factors that develop self-efficacy (Banura, 1977). These factors are mastery experiences, vicarious experiences, verbal or social persuasion, and physiological states. The four factors provided pre-service teachers with the most efficient way of gaining higher levels of self-efficacy.

**Mastery experiences.** In this study, pre-service teachers were asked to provide specific examples of mastery experiences that influenced their self-efficacy. The experiences when pre-service teachers do something well and gain the sense of accomplishment as a result, their levels of self-efficacy are increased (Banura, 1977). This study’s findings support that pre-service teachers expressed the importance of instructional strategies in their courses to develop the mastery experiences in mathematics methods courses. Mastery experiences develop from activities such as mini-lessons, discussions, etc. Mini-lessons provided pre-service teachers with the experience of successfully teaching mathematical concepts and performing as teachers, which in turn increased their self-efficacy. In these courses, they experienced teaching by instructing their peers. The mathematics methods courses gave them an opportunity to perform when they were placed in the role of the teachers, which then increased their confidence regarding their ability to teach.

These results are consistent with previous research studies which stated that hands-on activities provide necessary mastery experiences. The teacher-role then acts as an important source of self-efficacy beliefs for pre-service teachers (Bray-Clark & Bates, 2003; Cone, 2009; Tenaw, 2013; Turner, Cruz, & Papakonstantinou, 2004). When asked about the most effective element in the courses, the pre-service teachers emphasized small group activities, peer interaction, partner assignments, and hands-on activities, allowing them to perform successfully through modeling, feedback, and an encouragement environment. Such results suggested that practical tasks were the most effective mastery experiences that helped develop self-efficacy in pre-service teachers. Pre-service teachers constructed their own knowledge through practice; as a result, this increased their self-efficacy. In these courses, pre-service teachers took a more active role in their own learning. When mathematics pre-service teachers enroll in such courses, the use of manipulative, cooperative learning experiences, and discourse reinforces this mastery experiences (Hoy & Spero, 2005; Kazemi, Lampert, & Ghousseini, 2007). In this way, the respondents learned certain strategies better and comprehended more successful methods of teaching.

**Vicarious experiences**. Pre-service teachers and instructors’ modeling occurred as another contextual factor affecting the pre-service teachers’ self-efficacy. Most of the participants mentioned that group learning and peer interaction were helpful in promoting their self-efficacy. This finding relates to a Bray-Clark and Bates (2003) study that mathematics methods courses could improve the vicarious experiences through collaborative training, peer interaction and observation of other pre-service teachers. By helping one another learn and teach, participants not only gained new knowledge and skills, but also understood that they are also capable of performing well. The findings from this research suggest that pre-service teachers’ self-efficacy levels are more likely to increase as a result of participating in mini-lessons in which they observe modeling of teaching, being able to interact successfully with the others and being encouraged by their professor.

**Social persuasion**. Pre-service teachers noted an important role of the instructor’s feedback during their learning and teaching process and how it increased their self-efficacy. They explained that receiving the feedback from their instructor identified specific trouble areas in their teaching, as well as strengths. According to Guskey and Passaro (1994), pre-service teachers should receive encouragement and corrective feedback in order to convince them that they could successively teach the subject. When instructors explained to students their strengths and weaknesses, it yielded better results (Coulter & Grossen, 1997; O’Reilly, Renzaglia, & Lee, 1994; Butler & Winne, 1995). It was a verbal/social persuasion to a certain extent, which clearly led to the improvement of self-efficacy among the respondents (Enochs, Smith & Huinker, 2000; Hackett & Betz, 2009).

**Supportive and inviting environment which control physiological states**. Physiological states, such as feeling and mood during or after the mathematics methods courses, affected interviewees’ self-efficacy. Most of the participants had a positive mood about math, which was affected by a supportive and inviting environment. The interviewees revealed that a supportive and inviting environment improved their self-efficacy. Most of the participants stated that their instructor provided a supportive learning classroom environment rather than performance evaluation which improved their self-efficacy. They explained that providing a supportive environment may have minimized their emotional stress. Based upon the overall responses in the interviews, the findings provided an example of how instructors can assess and modify their mathematics methods courses in terms of teachers’ self-efficacy beliefs by ensuring that the courses are able to not only provide hands-on use of manipulative and activity-based lessons, but also employ small-group activities, peer interactions and modeling during instruction. By implementing such educational practices, it creates a supportive relationship with the instructor and peers (O’Reilly, Renzaglia, & Lee, 1994). Based on previous research, the above teaching practices could provide a supportive environment to increase pre-service teachers’ self-efficacy (Chong & Kong, 2012; Turner, Cruz, & Papakonstantinou, 2004; Wadlington, Slaton, & Partridge, 1998).

### Limitations of the Study

Five limitations are noted regarding this study. First, respondents may feel obligated to impress the interviewer. Therefore, they may provide inaccurate information. To minimize this effect, the researcher used two assessment tools in the study: interviews and questionnaires (Turner, 2010). Second, the course length and the depth of content might not have been sufficient to develop significant changes in pre-service teachers’ self-efficacy. A longitudinal study of the same pre-service teachers throughout their time in the program, not just in the mathematics courses, could provide further understanding of how pre-service teachers develop their self- efficacy.

This process would entail a multi-year study that evaluated efficacy beliefs and actual ability at multiple points throughout the entire education program. Third, restricting analysis to only pre-service teachers enrolled in mathematics methods courses in one university limits the ability to generalize the results. Fourth, pre-service teachers enter the mathematical courses with different levels of self-efficacy. It is difficult to determine exactly if the prior experiences or the mathematics method courses affected their self-efficacy. It is important to include questions about previous experiences in the interview questions or the instrument. Finally, lack of changes on pre-service teachers’ self-efficacy between male and female may be attributed to unequal sample size of men and women involved in the study. The non- equivalent male and female sample is subject to threaten validity of the results.

### Future Research

The results of this study are limited by factors discussed above. Thus, a possibility of future research may involve a longitudinal study examining self-efficacy beliefs to determine how to maintain high levels of self-efficacy in a teacher education program.

Many participants in this study indicated that direct experience with students could increase self-efficacy. Some pre-service teachers identified the need for field experiences to increase their self-efficacy. Thus, a potential area for future research is comparing the self-efficacy of pre-service teachers in programs that provide field experiences in their curriculum to those that do not. Pre-service teachers underlined the need for actual teaching opportunities and teaching practicum in their program. Their concerns about field experiences contradicts with Lancaster and Bain’s (2010) findings, which showed no significant difference between courses with field experiences or courses without field experiences. However, further in-depth study may probably reveal the exact reason behind such contradiction.

A qualitative approach should be used to clarify the differences between pre-service teachers who have had one mathematics methods course and pre-service teachers who have had two mathematics methods courses on each of the items included in the two dimensions of self-efficacy (MTOE and PTME) on the MTEBI.

The small sample size for gender groups, female (127), and male (7) is one possible explanation for finding no statistical differences in measuring the means. A larger sample or equal size of the female and male groups might have led to statistically significant differences. Thus, another possibility of future research is examining the effects of gender on the pre-service teachers’ self-efficacy, in spite of the limited research on this topic.

#### Implications for Teacher Education Programs

The results of the current study have important implications for pre-service teacher education programs. From the literature review, the research indicates that self-efficacy and content knowledge can be used as an overall indicator of teachers’ readiness for the classroom (Jerkins, 2001; Tschannen-Moran & Hoy, 2001). The use of MTEBI measures may be beneficial to use during mathematics methods course to better meet pre-service teachers’ beliefs. Teacher education programs need to improve pre-service teachers’ self-efficacy by being aware of the factors that influence self-efficacy.

Educators and professional instructional designers may take interest in this study as it provides ways to construct the four factors identified by Bandura: mastery experiences, vicarious experiences, social persuasion and physiological states (Bandura, 1997). As shown in this study, pre-service teachers had high self-efficacy in teaching mathematics because they were more actively engaged. Pre-service teachers referenced a variety of other activities that they did that helped them create mastery experiences, have vicarious experiences, experience social persuasion, and be in supportive environment. Some of the activities they experienced in the courses were watching videos, playing games, working in groups, mini-lessons, collaborative works, and hands-on activities. In these instructional practices, they receive mastery experiences, vicarious experiences, and social persuasion in supportive environments.

There are many practical experiences that could add knowledge and skills for the pre-service teachers, which strengthens their confidence about their ability to teach. Most participants mentioned that by teaching each other, they obtained insight on what struggles they could be facing as teachers in the future. It then helps them learn the skills needed to overcome those challenges.

Also, teacher educators should be aware about the effect of the pre-service teachers’ physiological states on self-efficacy. Stress, anxiety, and negative moods affect pre-service teachers’ self-efficacy and behavior (Bandura, 1997). They can help their pre-service teachers minimize these physiological states by having a supportive and inviting environment. Also, the educator’s attitudes, feedback, practices could affect pre-service teachers’ self-efficacy. In sum, the teacher educator’s relationship with their pre-service teachers affects their moods and self-efficacy.

The results of this investigation are intended to help researchers and practitioners develop a better understanding of pre-service teachers’ self-efficacy beliefs and the factors that can affect or increase it. This investigation illustrates the importance of emphasizing proactive teaching approaches, teaching specific and practical classroom strategies, and providing pre-service teachers with the opportunity to practice, review, and receive feedback on their use of these strategies. These experiences may help pre-service teachers to report higher levels of self-efficacy (Huinker & Madison, 1997; Incikabi, 2013; Wilkins & Brand, 1990). In this way, since the mathematics methods courses cause high levels of self-efficacy, the teacher educator can use these instrument data to guide them on improving and providing adequate training, as well as ensuring that pre-service teachers are well prepared for the teaching profession. In addition, future education programs need to determine which instructional strategies or methods have the greatest effect on pre-service teachers’ efficacy. To conclude, it is important that teacher education programs should consider the self-efficacy beliefs of pre-service teachers and ensure that their courses provide the best preparation to better prepare them for their teaching careers.

## Conclusion

Pajares (1992) explained that the “beliefs teachers hold influence their perceptions and judgments, which, in turn, affect their behavior in the classroom, or the understanding that the belief structures of teachers and teacher candidates, is essential to improving their professional preparation and teaching practices” (p. 307). Self-efficacy is the best predictor for pre-service teachers’ behavior in the future (Swars, 2005). Thus, it is important to highlight the importance of mathematics methods courses on pre-service teachers’ self-efficacy. There are a number of studies that investigated self-efficacy among teachers (e.g., Gibson & Dembo, 1984; Tschannen-Moran & Hoy, 2001; Henson, 2002). However, few studies have explored this effect on mathematics pre-service teachers (e.g., Hoy & Isiksal, 2005; Kim, Sihn, & Mitchell, 2014; Swars, 2005; Swars, Hart, Smith, Smith, & Tolar, 2007).

Additionally, studies of mathematics pre-service teachers have often used quantitative approaches (e.g., Hoy & Isiksal, 2005; Kim, Sihn, & Mitchell, 2014; Swars, 2005; Swars, Hart, Smith, Smith, & Tolar, 2007). Through a mixed-methods approach, this study sought to gain broader understanding about the effects of mathematics methods courses on pre-service teachers’ self-efficacy regarding mathematics. By focusing on how mathematics methods courses influenced pre-service teachers’ self-efficacy, the results provide further confirmation for the benefits of the mathematics methods courses. As was discussed previously, this study examined the impact of mathematics methods courses on pre-service Early Childhood and Special Educators’ self-efficacy and beliefs. Additionally, the purpose was to examine the possible factors responsible for the pre-service teachers’ beliefs, and to determine the levels of self-efficacy of pre-service teachers’ regarding their skills in the mathematics methods courses.

Findings from this research indicated that the majority of pre-service teachers reported high self-efficacy (as indicated by both qualitative and quantitative data) after taking the mathematics methods courses. This result, which implies high level of self-efficacy after the mathematics methods courses, is consistent with the result of other studies (e.g., Darling-Hammond, 2000; Hart, 2002; Hunker & Madison, 1997). Pre-service teachers’ self-efficacy for teaching mathematics changed during their mathematics methods courses. It was noted that their self-efficacy in the mathematics methods courses changed and was found higher than in the mathematics content pedagogy courses. Also, the result of the qualitative component found that mathematics content pedagogy courses are important, but mathematics methods courses are more effective. The pre-service teachers who were interviewed explained that content pedagogy courses are important because they provide the theory behind how students learn mathematics, but the mathematics methods courses provide more practical experiences for pre-service teachers to teach mathematics. However, no known research has directly compared the effects of mathematics methods courses and content pedagogy courses; a potential study exists here.

Findings from this study aligned with research on self-efficacy in terms of developing self-efficacy (Bandura, 1997; Tschannen-Moran and Hoy, 2007). Considerable enhancement on pre-service teachers’ self-efficacy is possible when Bandura’s four factors (mastery experiences, vicarious experiences, social persuasion and psychological states) are integrated into the mathematics methods courses. The interview data revealed the importance of providing practical experiences, feedback, mini-lessons, and discussions that could enhance self-efficacy. When pre-service teachers construct their own knowledge and experiences about teaching mathematics through practice, review, and receiving immediate feedback, it improves their skills and self-efficacy (Swars, 2005).

According to Chong and Kong (2012), the four sources of self-efficacy would be created “through active engagement that includes meaningful discussions; planning classroom implementation; observing expert and experienced teachers; reviewing common student problems in learning; obtaining feedback in teaching through being observed; and engaging in reflective discussions about the learning processes” (p.3). When pre-service teachers learn how to teach through doing, observing, and receiving feedback in supportive environment, they could have more confidence about their ability to teach mathematics (Bandura, 1997; Tchannen-Moran, Hoy, & Hoy, 2001; Turner, Cruz, & Papakonstantinou, 2004).

Mastery experiences such as discussions or mini-lessons during courses provided them with examples of how they can successfully teach mathematical concepts, thus letting them believe that they are actually capable of teaching lessons to children at school could increase their level of self-efficacy (Jerkins, 2001; O’Reilly, Renzaglia, & Lee, 1994; Palmer, 20011). At the same time, vicarious experiences allowed the respondents to learn certain activities better, as well as to comprehend more ways of teaching different mathematical concepts to their learners. By helping one another to learn, the participants not only gained new knowledge and skills but also understood that they were capable of performing (Badura, 1986; O’Reilly, Renzaglia, & Lee, 1994; Swars, 2005). Moreover, social persuasion and physiological states are other factors that could be used to improve pre-service teachers’ self-efficacy (Bandura, 1997). These findings demonstrate that the providing pre-service teachers with mastery experiences, giving feedback modeling, and maintaining a supportive environment can lead to increased self-efficacy toward teaching mathematics.

Moreover, the results of qualitative data support a previous study by Marzano (2003), which provide some insight on the characteristics associated with high self-efficacy. For example, pre-service teachers tend to use new strategies and mini-lessons with their future students. They believe that they affect student achievement and motivation, as well as assist in adapting a class for a wide variety of backgrounds (Ashton, Webb, & Doda, 1982b; Tschannen-Moran, Hoy, & Hoy, 1998).

An interesting outcome of this study was that the progression of self-efficacy development depends on the cumulative effect of more mathematics methods and content pedagogy courses. For example, pre-service teachers exposed to two mathematics methods courses had higher self-efficacy than those exposed to one mathematics methods course. Pre-service teachers’ self-efficacy ranking from highest to the lowest: two methods courses, one methods course, two content courses, and one content course. Thus, the more time that was spent on the program, the more likely self-efficacy will increase. This result indicated that teacher education programs might have an effect on pre-service teachers’ self-efficacy. The interviews provided a broader understanding for pre-service teachers’ perceptions about the reason of the differences between mathematics methods courses and mathematics content pedagogy courses. It is evident that all pre-service teachers do benefit from mathematics methods courses. It also emphasizes the importance of making sure pre-service have the appropriate preparation and strategies to be able to develop their self-efficacy with teaching mathematics to young learners.

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