###### July 15, 2005

# Examining Perceptions of Mathematics Teaching Effectiveness Among Elementary Preservice Teachers With Differing Levels of Mathematics Teacher Efficacy

This study investigated perceptions of mathematics teaching effectiveness among elementary preservice teachers with high and low levels of mathematics teacher efficacy. Participants in this study included four elementary preservice teachers at a mid-sized university in the southeastern United States who had just completed a mathematics methods course. Data sources were the Mathematics Teaching Efficacy Beliefs Instrument and interviews. The interviews indicated that mathematics instructional strategies as well as past experiences with mathematics and their influences upon perceptions of teaching effectiveness were associated with mathematics teacher efficacy.

The National Council of Teachers of Mathematics (NCTM) has presented a vision of reform mathematics based upon constructivist approaches that has far-reaching implications for teacher practices in the mathematics classroom. Teachers are the crucial component to the success of the current reform movement in mathematics education (Battista, 1994). Teacher implementation of effective instructional practices in mathematics has been linked to teacher efficacy (Enon, 1995). Teacher efficacy is a significant predictor of mathematics instructional strategies, and highly efficacious teachers are more effective mathematics teachers than teachers with a lower sense of efficacy.

Teacher efficacy was derived from Bandura's (1977) conceptualization of self-efficacy, which is defined as individuals' judgments of their capabilities to accomplish certain levels of performance. Bandura asserted that self-efficacy beliefs govern most of human functioning and mediate how individuals, think, feel, motivate themselves, and behave. Using Bandura's theoretical framework, teacher efficacy is considered by many researchers to be a two-dimensional construct (Ashton, 1985: Dembo & Gibson, 1985; Enochs, Smith, & Huinker, 2000; Gibson & Dembo, 1984). The first factor, personal teacher efficacy, represents a teacher's belief in his or her skills and abilities to be an effective teacher. The second factor, teaching outcome expectancy, is a teacher's belief that effective teaching can bring about student learning regardless of external factors such as home environment, family background, and parental influences. Teacher efficacy was first investigated by the Rand Corporation (Dembo & Gibson, 1985). In the evaluation of education projects, it was found that teachers' sense of efficacy was positively related to the percentage of the project goals achieved, the amount of teacher changes, the continuity of project materials and methods, and the improvement of student performance.Teacher efficacy also has been correlated to such significant variables as classroom instructional strategies and willingness to embrace innovations. Inservice teachers, as well as preservice teachers, who have high teacher efficacy use a greater variety of instructional strategies (Riggs & Enochs, 1990; Wenta, 2000). Highly efficacious teachers are more likely to use inquiry and student- centered teaching strategies, while teachers with a low sense of efficacy are more likely to use teacher-directed strategies, such as lecture and reading from the text (Czcrnaik, 1990). In addition, teachers with high teaching efficacy are more likely to try new teaching strategies, particularly techniques that may be difficult to implement and involve risks such as sharing control with students (Hami, Czerniak, & Lumpe, 1996; Riggs & Enochs, 1990). The approaches to teaching and learning of highly efficacious teachers are more consistent with the vision of mathematics put forth by NCTM (2000).

Although there are many studies concerning teacher efficacy, there is limited research on mathematics teacher efficacy, specifically with elementary preservice teachers. In the few studies on mathematics teacher efficacy of elementary preservice teachers, it was found that preservice teachers' participation in a mathematics methods course corresponded to significant increases in mathematics teacher efficacy (Cakiroglu, 2000; Huinker & Madison, 1997; Wenta. 2000). Preservice teachers also reported that having exposure to reform approaches in a mathematic methods course influenced their levels of mathematics teacher efficacy (Cakiroglu, 2000). Mathematics teacher efficacy has also been linked to mathematics anxiety among elementary preservice teachers (Swars, 2004). Mathematics teacher efficacy and mathematics anxiety have a negative relationship, with highly efficacious elementary preservice mathematics teachers possessing, in general, lower levels of mathematics anxiety.

With limited research in the area of mathematics teacher efficacy of elementary preservice teachers and given the importance of teacher efficacy regarding instructional practices and willingness to embrace reform, further investigation should occur in this area. In order to facilitate the development of highly efficacious mathematics preservice teachers, an in-depth exploration should occur regarding preservice teachers' perceptions of mathematics teaching effectiveness. As previously discussed, teacher efficacy is considered by many researchers to be a two-dimensional construct, and the factor of personal teacher efficacy, which represents a teacher's belief in his or her skills and abilities to be an effective teacher, was explored in this study. Therefore, the purpose of this study was to investigate perceptions of effectiveness in teaching mathematics among preservice teachers with differing levels of mathematics teacher efficacy. More specifically, the study explored the commonalities and differences of elementary preservice teachers with high and low levels of mathematics teacher efficacy regarding their perceptions of their skills and abilities to teach mathematics.

Methodology

The study involved four elementary preservice teachers at a mid- sized university in the southeastern United States. The participants had just completed a 3-semester hour undergraduate mathematics methods course. The course provided extensive emphasis upon the reform vision of NCTM, and the participants were heavily involved in learning to provide effective instruction through the five process standards: (a) communication, (b) problem-solving, (c) connections, (d) representation, and (e) reasoning and proof (NCTM, 2000). The participants also had explicit instruction on a variety of effective mathematics teaching strategies such as manipulative and game usage. Furthermore, the participants were provided instruction on the integration of children's literature and technology in mathematics. Active participation was encouraged in the class, and small group work as well as whole group instruction was utilized. Much of the class time was devoted to hands-on learning of mathematics concepts. The participants also were involved in a clinical experience in order to have an opportunity to implement content learned in the course. The participants spent a total of 24 days in local elementary schools. During that time, the participants taught at least three mathematics lessons and worked each of the days with students who were struggling in mathematics.

During the last week of classes, the preservice teachers completed the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI). The MTEBI was used to determine the degree of mathematics teaching efficacy of the participants. The MTEBI consists of 21 items, 13 on the Personal Mathematics Teaching Efficacy subscale and 8 on the Mathematics Teaching Outcome Expectancy subscale (Enochs, Smith, & Huinker, 2000). The two subscales are consistent with the two-dimensional aspect of teacher efficacy. These items were used for this study with a slight modification of wording. The Personal Mathematics Teaching Efficacy subscale addresses the preservice teachers' beliefs in their individual capabilities to be effective mathematics teachers. The Mathematics Teaching Outcome Expectancy subscale addresses the preservice teachers' beliefs that effective teaching can bring about student learning of mathematics regardless of external factors. The instrument uses a Likert scale with five response categories including strongly agree, agree, uncertain, disagree, and strongly disagree. Possible scores on the Personal Mathematics Teaching Efficacy subscale range from 13 to 65; Mathematics Teaching Outcome Expectancy subscale scores range from 8 to 40. Reliability analysis produced an alpha coefficient of .88 for the Personal Mathematics Teaching Efficacy subscale and an alpha coefficient of .75 forme Mathematics Teaching Outcome Expectancy subscale (n = 324). Confirmatory factor analysis indicated that the two subscales are independent, adding to the construct validity of the MTEBI (Enochs, 2000).

An interview protocol (Appendix) was developed by the researchers based upon the Personal Mathematics Teaching Efficacy subscale of the MTEBI. The interview protocol was used to gather in-depth information on the participants' perceptions of their skills and abilities to teach mathematics effectively. Three experts in the field of mathematics education and research examined the protocol and offered suggestions for improvements, thus establishing content validity.

The participants in this study were the two that s\cored the highest on the MTEBI with the highest degree of mathematics teacher efficacy and the two that scored the lowest on the MTEBI with the lowest degree of mathematics teacher efficacy among the preservice teachers in the mathematics methods course. The four participants participated in semi-structured interviews within one week of completion of the mathematics methods course. The interviews were conducted at the researcher's office at the university with each interview lasting approximately 45 minutes. The researcher assured the preservice teachers of their confidentiality and obtained consent to tape-record the interviews. The data from the interviews were transcribed from the audiotapes.

In analyzing the data, grounded theory was used as prescribed by Strauss and Corbin (1998). Following the analytical procedures presented by Strauss and Corbin. the data from the interviews were first analyzed individually and then collectively. Micro-analysis was used, which is the detailed line-by-line analysis necessary to generate initial categories and suggest relationships among categories. Microanalysis involves very careful, often minute examination and interpretation of the data through the use of coding processes. Initially, open coding was used, which was concerned with identifying, naming, categorizing, and describing phenomena in the interviews. Asking questions making of comparison were important aspects of the open coding process. Data were initially broken down by asking simple questions such as what, where, how, when, and how much. Subsequently, data were compared and similar incidents were grouped together and given the same conceptual label or category. Axial coding processes were then used to develop connections between a category and its subcategories. Lastly, selective coding processes were used to integrate categories in order to build a framework with subsequent themes.

Results

Three themes emerged from the data related to perceptions of mathematics teaching effectiveness, which included past experiences with mathematics, influences upon perceptions of mathematics teaching effectiveness, and mathematics instructional strategies. For the purposes of this study, the preservice teachers with the lowest degrees of mathematics teacher efficacy are referred to as Lois and Beth. The preservice teachers with the highest degrees of mathematics teacher efficacy are referred to as Holly and Barbara.

Past Experiences with Mathematics

The two preservice teachers with the lowest degree of mathematics teaching efficacy. Lois and Beth, both had negative past experiences with mathematics. Beth expressed. "I have struggled as a math student, and that is what I considermy academic weakness." Lois also stated that "Math is my one academic weakness". In addition, both Lois and Beth expressed having feelings in school about the irrelevance of mathematics in their daily lives and in real world situations. Lois stated, "I remember being in high school thinking about when am I ever doing to need this and when will I ever need to know how to use this." Similarly, Beth stated. "I remember we used to ask a lot, why do we need to know this? It seemed we would never use it."

However, the two teachers with the highest degree of mathematics teacher efficacy. Holly and Barbara, expressed differing experiences concerning their past experiences with mathematics. Holly expressed positive experiences with mathematics in the past. She stated, "Math was always a better subject in school, especially in the lower grades." In contrast. Barbara indicated a struggle with mathematics in school and had to focus more on mathematics than other academic subjects. She asserted:

I struggled with math in elementary and high school when all my other subjects came naturally to me. I did not have to study that much in high school, but math was something I had to study more and really learn how to do.

Influences upon Perceptions of Mathematics Teaching Effectiveness

All four of the preservice teachers expressed the belief that they were confident in their abilities to be effective mathematics teachers. However, Lois and Beth, the two preservice teachers with the lowest degree of mathematics teacher efficacy, felt they could teach mathematics effectively, but it would take more time, work, and effort. Lois stated:

I believe I can. It would just take a little extra work. I would have to work more with preparing the lessons. I would be learning at the same time, so I could do it effectively. It would just take more time and more effort. I could do it; it would just take more work.

Similarly, Beth stated. "I believe I would have to put a lot of energy and effort into teaching mathematics. I would spend more time on my math lesson and conveying that information than I would with reading or language arts because that is something that has come easy to me." Both Lois and Beth expressed that the extra time, work, and effort spent on mathematics instruction were influenced by their past negative experiences with mathematics.

Barbara, who had the highest degree of mathematics teacher efficacy, expressed that she would be able to teach mathematics effectively because she struggled with mathematics in school. She felt that she was very comfortable teaching mathematics, but she was not comfortable learning mathematics. Barbara stated:

The reason I feel I can teach mathematics effectively is because I had such a struggle with it in elementary school and high school. Math was something I had to study more and learn how to do. I think that just being there and experiencing that can make me a good math teacher. I think with my students, when they ask questions, it will be one I have asked before which is why I feel that I will be good.

Holly, who also possessed the highest degree of mathematics teacher efficacy, also believed that she could teach mathematics effectively, but it was due to her comfort level with mathematics. Her experiences with mathematics in school were successful. Holly stated, "I believe I can teach mathematics effectively. Math was always a better subject."

Mathematics Instructional Strategies

All four of the preservice teachers indicated the most important teaching strategy in terms of motivating students to learn mathematics was to provide students with authentic mathematics activities. Since Lois and Beth, the two preservice teachers with the lowest degree of mathematics teacher efficacy, expressed past experiences with mathematics in schools where the subject seemed to lack usefulness, both of them felt it was important for students to know that the math they are learning can be used in a real world situations. Beth stated:

I would try to motivate students to learn mathematics by making it something that they will really use. Let's say you were trying to balance your checkbook. You wrote a check, and do not know how much money it was for. So, how do you figure it out? You have got your balance. You have got what you think you have. You know how much your bank says you have. That is an algebra problem, but it is put in a real world context so you would see how knowing how to do this would be useful. I would say make it authentic and show how you will use this in everyday life.

Similarly, both Holly and Barbara, the two preservice teachers with the highest degree of mathematics teacher efficacy, expressed the importance of students knowing the usefulness of mathematics. For example. Holly stated. ''For boys, especially maybe in upper elementary, use sports and statistics. Find something that's related to something they already know and enjoy and try to incorporate that." Barbara also expressed the importance of students' awareness of mathematics in their daily lives. She explained a mathematics activity where students could keep a log of how many hours they spent at the grocery store, watching television, and doing other activities. Students could then determine how much time they spent doing these activities and therefore realize the incorporation of mathematics in their daily lives.

Regarding the instructional strategy of using manipulatives as an aid to teaching and learning in the mathematics classroom. Lois and Beth expressed differing viewpoints. Lois did not embrace the use of manipulatives in mathematics as she stated, "When I was in elementary school, we did not use manipulatives. I am not really comfortable because I have not used them much." In contrast. Beth embraced the teaching strategy of manipulative usage and indicated that manipulatives are an authentic teaching tool that gives students a "hands-on experience".

Holly and Barbara, who had the highest levels of mathematics teacher efficacy, both expressed great enthusiasm for using manipulatives as a teaching and learning aid in the mathematics classroom. Barbara stated, "All of the math lessons I have done I have used them. I think they are a very good thing, and I will definitely use them to teach." Similarly, Holly expressed, "I think it is a more authentic learning experience when you are using a manipulative. It is a visual, and they can actually see something instead of just writing it on paper. I think it makes a big difference." Both of these preservice teachers felt that manipulative usage enhanced students' understandings of mathematics.

Conclusions

Past experiences with mathematics were associated with level of mathematics teacher efficacy and influential upon perceptions of mathematics teaching effectiveness among the preservice teachers. The preservice teachers with the lowest degree of mathematics teacher efficacy both reported negative experiences with mathematics in school. Bandura (1986) asserted that efficacy beliefs are primarily shaped as a result of an individual's previous performance and experiences. He asserted that individuals engage in tasks and activities, interpret the results of their actions, use the interpr\etations to develop beliefs about their capabilities to engage in subsequent tasks or activities, and act in relationship with the beliefs created. Typically, outcomes interpreted as successful raise efficacy and those interpreted as failures lower efficacy. The preservice teachers' experiences of failure with mathematics in school may have contributed to a lower sense of mathematics teacher efficacy. These past negative experiences led the preservice teachers to perceive that they would be effective mathematics teachers, but that teaching mathematics would take more time, work, and effort.

The preservice teachers with the highest degree of mathematics teacher efficacy had differing past experiences with mathematics. One of the preservice teachers with the highest level of mathematics teacher efficacy had positive experiences with mathematics in school. These successful experiences in school led to a comfort level with mathematics that translated into positive perceptions of mathematics teaching effectiveness. However, the other preservice teachers with a high degree of mathematics teacher efficacy had past negative experiences with mathematics in school. But, she viewed this struggle as a learner of mathematics to be an asset for effective teaching of mathematics. Her negative past experiences with mathematics had prepared her to be a better teacher of mathematics.

With reference to instructional strategies, the preservice teachers, regardless of level of mathematics teacher efficacy, indicated the importance of motivating students to learn mathematics through the use of "real world" experiences. Particularly, the two preservice teachers with the lowest degree of mathematics teacher efficacy expressed this viewpoint due to a perceived lack of relevance of mathematics when they were in school. NCTM (2000) asserted that teachers should focus upon mathematics content and processes that are worth the time and attention of students. The mathematics curriculum "should offer experiences that allow students to see that mathematics has powerful uses in modeling and predicting real-world phenomena'' (p. 16). Past research has indicated that highly efficacious teachers are more likely to use reform strategies in instruction (Hami. Czerniak, & Lumpe, 1996: Riggs & Enochs, 1990; Ross, 1992). However, regardless of level of mathematics teaching efficacy, all of the preservice teachers' viewpoints on the importance of "real world" situations in mathematics are consistent with the reform vision of NCTM.

In addition, regarding instructional strategies, the use of manipulatives was strongly embraced by the preservice teachers with the highest degree of mathematics teacher efficacy. The NCTM (2000) asserted the importance of teachers using "representations to model and interpret physical, social, and mathematical phenomena" (p. 70). The view of the preservice teachers with the highest degree of mathematics teacher efficacy in regard to mathematics manipulative usage is consistent with the reform vision of mathematics presented by NCTM. In contrast, one of the preservice teachers with the lowest degree of mathematics teacher efficacy expressed concerns about the use of manipulatives as a teaching and learning aid in the mathematics classroom. These findings are consistent with previous research, as studies have indicated a consistent relationship between teacher efficacy and classroom instructional strategies (Wertheim & Leyser, 2002) as well as willingness to embrace reform strategies (Hami, Czerniak, & Lumpe, 1996; Riggs & Enochs, 1990; Ross, 1992).

Studies have indicated that elementary preservice teachers' participation in a mathematics methods course corresponds to significant increases in mathematics teacher efficacy (Cakiroglu, 2000; Huinker & Madison, 1997; Wenta, 2000). Preservice teachers need positive experiences within mathematics methods courses in order to build efficaciousness towards teaching of mathematics. In addition, mathematics methods courses need to provide a self- awareness of past experiences with mathematics among preservice teachers, particularly negative experiences, in order to facilitate the building of mathematics teacher efficacy.

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Susan Lee Swars, Ph.D., Assistant Professor, Mathematics and Science Education, College of Education, Georgia State University.

Correspondence concerning this article should be addressed to Dr. Susan Lee Swars, Early Childhood Education, P.O. Box 3978, Atlanta, GA 30302-3978; Email: [email protected]

Appendix

Interview Protocol

1. Do you believe you can teach mathematics effectively? Why or why not?

2. What would you do to help low-achieving students in mathematics?

3. What is your level of understanding kindergarten through sixth grade mathematics concepts?

4. How comfortable do you feel using manipulatives in mathematics lessons? Why or why not?

5. What are some ways you would try to motivate students to learn mathematics?

Copyright Journal of Instructional Psychology Jun 2005