# Exploring the evolving nature of three elementary preservice teachers' beliefs and practices: three parallel case studies.

The two preceding papers reported on the complexity of reforming mathematics education in one Midwestern state K-A elementary school. This paper focuses on teacher preparation for mathematics teaching and follows three preservice elementary teachers as they move through their university methods course and then into their practicum/student-teaching at another elementary school.Current research studies and documents from the National Council of Teachers of Mathematics (NCTM) address the importance of the teacher's role in creating learning opportunity in mathematical situations (Tsuruda, 1994; Bright, Bowman & Vacc, 1999; NCTM, 2000). Also, some researchers focus on understanding preservice teachers' beliefs and practices and how those beliefs and practices transform in the classroom context (Pourdavood & Harrington, 1998; Portnoy, Graham, Berk, Gutmann & Rusch, 1999). This increasing body of literature has led to some interesting insights.

Tsurda (1994) described his changing beliefs and how these changes affected his practices. He had been a middle school teacher for seventeen years, teaching in mostly a conventional style. Not all of Tsuruda's students passed his class, but he believed that more did so than students of other teachers in his building, thanks to his gregarious nature. Tsuruda was concerned, however, because even the students who were earning passing grades in his classes were not mathematically powerful. They could not apply problem-solving strategies or use critical thinking processes.

Most of my students would have done no better than the nationwide sample of students asked this question: There are 12S sheep and 5 dogs in a flock.

How old is the shepherd? Three out of four students across the nation responded with a numerical answer, the most common being that the shepherd's age is twenty-five. (Tsuruda, 1994, p. 2)

According to Tsuruda, there had to be a better way of teaching and learning mathematics so that students could critically think about what was being asked and develop a solution that not only made sense but that they could explain.

Tsuruda recognized two aspects inherent to teaching and learning once his beliefs had been perturbed by participating in national organizations such as the National Council of Teachers of Mathematics, dialogues with his colleagues, and actively listening, observing, and reflecting on his classroom mathematical activities and events. He identified these two aspects as form and spirit. Form is the method, such as small-group cooperative learning, use of technology, using manipulatives, and incorporating alternative assessment. Spirit is the teacher's belief on how the above methods can be used meaningfully for creating learning opportunities for students. Therefore, according to Tsuruda (1994), teachers may use the form of change without internalizing those changes. In this sense the spirit of change actually relates to individual transformation from one set of beliefs to another. It is a paradigm shift.

Cohen (1990) examined a second grade teacher's evolving beliefs and practices. His research findings, similar to Tsuruda's work, suggest that although a teacher may state what s/he believes, it may not be what is displayed in practice. "She [Mrs. O, the second grade teacher] eagerly embraced change, rather than resisting it...But, Mrs. O seemed to treat new mathematical topics as though they were a part of traditional school mathematics" (p.312). Cohen asserted that Mrs. O stated she believed she allowed for more students' understanding of mathematical ideas by revising the curriculum to allow for more hands-on lessons (form). However, in the actual classroom setting her beliefs were not consistent with the researcher's observations of the classroom mathematical practices (spirit). Cohen observed Mrs. O teach using manipulatives, but the teacher-controlled activities made it difficult for students to make sense of mathematical activities. Cohen attributed the inconsistency of Mrs. O's stated beliefs and her actual practices to a lack of content knowledge in mathematics and a lack of professional support.

Pourdavood and Harrington (1998) completed a study that encompassed form and spirit issues with secondary preservice teachers. Within this study preservice teachers were assessed as having either conventional or constructivist beliefs. Conventional method is synonymous with direct instruction, where the classroom is situated and centered on the teacher. Constructuvist method is synonymous with a problem-centered classroom where teachers and students are members of the mathematical community. A problem-centered classroom has three components: (1) a task which is presented by teacher or students, (2) small group interactions and dialogue among students, and (3) whole class discussions and debate (Wheatley & Reynolds, 1999).

When the secondary preservice teachers were placed in a field with teachers of similar or differing ideas, the researchers observed several interesting events. They examined classroom scenarios where preservice teachers believed they were constructivists; but when faced with a classroom in which rote learning took place, they reverted back to a conventional philosophy. Instead of making the form align with their stated philosophy, they allowed their philosophy to be influenced by the environment of the classroom.

In contrast, there were also some preservice teachers who fought to keep their practices in line with their beliefs. These students, when faced with the realities of the classroom, still were able to accomplish the goals of the curriculum but did it in a manner consistent with constructivism. They chose not to let their cooperating teacher's beliefs and the classroom/school environment change their ideas of how children best learn.

Portnoy, Graham, Berk, Gutmann & Rusch, (1999) found a similar occurrence with their study. Two students followed through student teaching were found either to fight to keep their pedagogy or change their pedagogy because of the classroom. Megan had constructivist's beliefs. After being in a classroom with primarily drill-and-practice lessons she struggled to maintain her constructivists' beliefs about how children best learn mathematics. Another student, Kate, who believed in a more traditional pedagogy, slowly began to see the benefits of a reform-minded classroom for students after her experience in a non-traditional classroom. Portnoy et al. (1999) asserted that this experience led her to search for better ways to motivate students.

In short, observations suggest that preservice teachers' beliefs and what they see and experience in the classroom and school may influence practices. This is an important statement within the scope of this research report.

Research Questions

This research report explores the extent preservice teachers' beliefs and practices are influenced and modified due to the college classroom experience, taking into account their prior beliefs about mathematics and mathematics instruction. It further explores how these preservice teachers' beliefs and practices are perturbed by the actual classroom in a practicum/student-teaching placement. Taking into account the preservice teacher's beliefs as well as the beliefs and practices of the cooperating teacher, the report addresses the following questions:

1. How do preservice teachers' beliefs evolve during the methods course instruction and how are these beliefs transformed within the context of the elementary classroom?

2. What influence do cooperating teachers' beliefs and practices have on the preservice teachers' beliefs and practices?

Setting and Participants

The school where these three preservice teachers did their practicum/student-teaching is located in an urban, inner-city community in-between an economically challenged urban area and an affluent suburban area. The student population is identified as almost exclusively African-American. The school is one of several elementary schools within a district that is in the bottom quartile when fourth grade standardized test scores and passage rates are compared throughout the state.

Grade-level teachers, those that do not teach specials (i.e., gym or art) are expected to teach only two to three content areas for two different grades. That is, teachers can teach first and second grade, third and fourth grade, or fifth and sixth grade. Each of these teachers only teaches specific content areas. For example, a teacher may be a third/fourth grade mathematics and science teacher, or a fifth/sixth grade language arts/civics teacher. No teachers in the building teach one specific grade above kindergarten exclusively throughout the day.

This research report involves two groups of participants: three cooperating teachers and three preservice teachers. The cooperating teachers' ages ranged from 27-42 and their years of teaching experience ranged from 5-17. There were two males and one female. All three were Caucasian teaching multiple grade levels between third and sixth in a teaming environment. All three volunteered for the research study after receiving information on what would be involved. The preservice teachers ranged in age from 23-39. There were one male and two females, all Caucasian. Two of the three were returning to college to receive licensure after previously attaining bachelors' degrees in nonteaching-related fields. One was a "traditional" student completing her degree requirements within the field of elementary education. All three preservice teachers were selected from the primary researcher's methods course after meeting the following criteria: (1) they were willing to participate, (2) they were able to articulate their be liefs concerning mathematics and mathematics education, and (3) they were at the end of their programs of study and taking the methods course in their final semester prior to student-teaching.

Data Collection and Analysis

Data sources included: the Teacher Belief and Attitude Survey in Mathematics, or T-BASM (Pourdavood, 1996); written responses to open-ended questions; classroom observations; field notes; document review from preservice teachers reflective assignments in their methods courses; and. transcribed audiotaped interviews with individual preservice teachers, whole group preservice teachers, and whole group inservice teachers.

The study procedure involved an initial phase while the three preservice teachers were in the methods course and an active phase after they were placed with cooperating teachers. The initial phase of the study started in the Spring Semester 1999 and lasted approximately 15 weeks. This phase included the responses to questionnaires, preservice teachers' reflection on their reading assignments, transcribed audiotaped interviews with both preservice and inservice teachers, and twice weekly preliminary observation of the cooperating teachers classrooms. The active phase of the study started in the Fall Semester 1999 and lasted 15 weeks. This phase included twice weekly classroom observations (both preservice and inservice teachers' mathematical activities), transcripts of audiotapes of one-on-one interviews with preservice teachers after classroom observation, transcripts of audiotapes of group dialogue after classroom observation (both preservice and inservice teachers), and preservice teachers' written reflect ion on their teaching.

Based on the multiple data sources, the researchers used Lincoln Guba's (1985) constant comparative method to make sense of the data. Based on emerging themes, several categories were identified such as: social norms of the classroom, socio-political norms within the classroom, school, district and state, time issues, and teachers' content knowledge.

Preservice Teachers' Stated Beliefs Prior to Student Teaching

At the beginning of the methods course, William (a preservice teacher participant) had devised a plan on how to teach in a manner different from how he had been taught. He believed that all subjects could be taught in conjunction with one another. That is, each subject could revolve around one particular concept. "Therefore, math, science, literature, etc. would all be taught in relation to one overall topic" (Written Reflection). Williams was also convinced that there was a better way to teach effectively within the classroom than rote memorization, as is evident by his responses to the initial T-BASM. William disagreed with the following statements: "Frequent drills on mathematics facts are essential in order for students to learn them" and "Effective mathematics teachers demonstrate the correct way to do a problem."

William's few traditional beliefs were replaced with constructivist beliefs regarding teaching and learning. According to constructivist epistemology, learning is built by individuals participating in and contributing to the activities of the mathematical classroom community (Cobb & Yackel, 1996). These stated changes were evident when exploring William's T-BASM responses, reflective papers he completed during the methods course, and ongoing conversations with the primary researcher.

William's initial responses on the T-BASM indicated he believed little consideration needed to be given to what children already knew when selecting the next topic to be taught. At the end of the methods course his idea changed significantly. His response during the second administration indicated significant consideration should be given to what children already knew before choosing a new concept to teach. William's beliefs regarding authority in the classroom also changed, according to his responses, from supporting a teacher-centered classroom to a student-centered classroom.

William at first asserted that it was necessary to know how to do mathematics in the traditional sense. He stated, "there is something to be said to be able to do a problem with paper and pencil" (Written Reflection). However, in his next Written Reflection William abandoned this idea. He indicated that problem-solving techniques needed to be taught so that students could use their creativity and not be restricted to one "correct" way to solve a problem.

In his own words, 'William described his teaching as "more of a constructivist approach" (Interview During the First Phase). He decided that student discovery of connections within mathematics would be more powerful than simply telling them how to do specific problems. "I realized that if I let them fool around with this stuff [manipulatives and concepts] for a while in a positive manner and in a directive manner that they would understand it on their own" (Interview During the First Phase).

Jackie (a preservice teacher participant) began the methods course unsure of her knowledge and ability within the realm of mathematics and mathematics teaching. According to her "Written Reflection" she learned mathematics from traditional teachers teaching in traditional classrooms. She expressed that she never understood why problems worked or did not work. This gave Jackie fear of teaching mathematics.

According to the responses on the T-BASM, Jackie did not know if allowing children to discuss their thinking helped them to make sense of mathematics. Before the methods course she offered a neutral response to this question. After the course, she agreed with the notion that children need to discuss their thinking. She also changed her response from neutral to strongly agree in regard to sacrificing coverage of the whole curriculum to allow children the time to explore some tasks thoroughly.

A significant change is observed in the evolution of her writings during the methods course. She described Tsuruda's (1994) Tell, Show, Practice, Test and Forget method of teaching as the style in which she learned mathematics. She revealed that she was never very good at that method. However, she never knew there was any other way to teach. "I spent most of my life thinking of math in one way" (Written Reflection). By the subsequent reflection paper Jackie had experienced enough in the methods course to state, "Slowly, my mind is making the transition from the behavioral approach (how I was taught) to the constructivist approach (how I wish I was [sic] taught!)." She confided that she thought the constructivist approach would be too time consuming. However, she believed that group work, allowing the children to help each other with abstract ideas, would help speed things along.

Toward the conclusion of the methods course Jackie was espousing the benefits of teaching for understanding. She stated in the last method course interview that she saw the benefit of teaching in a "hands-on" way. Her role, according to her, went from giving information to guiding the students to learn. Jackie's main goal was to keep the classroom managed so that cooperative learning in a constructivist sense could be accomplished.

Cassey (a preservice teacher participant) began the methods course searching for a pedagogy of teaching mathematics. She knew that the traditional way of teaching mathematics had not worked for her and she suspected that it was not working for many children. In her first Written Reflection she asked, "Where was Vygotsky when I was growing up?"

Although Cassey sought a different way to teach mathematics, she still held instead of help traditional beliefs about teaching. Initially, when she took the T-BASM at the beginning of the term, her responses to many questions served to indicate her traditional beliefs. For example, her response to the statement, "children learn mathematics best by figuring out for themselves the ways to find answers to simple word problems" was "strongly disagree." After course completion she answered this question as "strong agree." Another example is her response to the statement, "children can figure out ways to solve many mathematics problems without formal instruction." Her original response was "strongly disagree." At the completion of the methods course, she revised her answer to "strong agree."

In the methods course Cassey became aware of many concerns and issues that had previously not been a priority for her. The modification of answers to questions on the T-BASM indicates that Cassey began the methods course without adherence to a particular pedagogical stance for teaching mathematics. By completion of the methods course, Cassey said she believed in many components of constructivism. She stated that within her practicum/student-teaching class she planned on using cooperative learning and peer tutoring as a means to develop students' mathematical understanding.

Cassey began the methods course searching for a new pedagogy but without understanding all the issues involved with teaching mathematics. By the end of the methods course, she said she believed in the benefits of using a constructivist philosophy, including peer tutoring and manipulatives. Cassey said she had developed a belief in the constructivist pedagogy due to her experiences in the methods course.

Cooperating Teachers' Beliefs and Practices

One of the cooperating teacher participants, Mr. Hill, has set up his classroom in a manner consistent with the reform movement. Tables are around the room, with approximately 4-6 students around each table. His desk is in the front left of the room and sits low enough that even the smallest child could see over it. There is also a teacher workstation in the front middle of the classroom where the teacher can sit and work on an overhead. On Mr. Hill's walls are some posters depicting mathematics topics such as steps to complete arithmetic problems and fact families. Other posters depict differing units of measure in both English standard and metric.

Mr. Hill begins his class with a mathematics review. An example is a worksheet. Mr. Hill is concerned with passage rate on the state-mandated test and gives his students ample opportunities to work on basic skills and problems similar to those on the test.

At times Mr. Hill uses manipulatives in his teaching. He developed one lesson using Base-Ten Blocks for place value. However, although he uses manipulatives and the students work at tables, the classroom seems to be teacher-centered, with Mr. Hill owning most of the expertise and authority. Also at times he models the manipulatives in the front of the room rather than letting the students work with them at their desks.

Within the realm of learning, Mr. Hill agrees that students need frequent drills on mathematical facts in order to learn them. He disagrees with the notion that the goal of instruction is best achieved when the students routinely produce correct answers to problems. He agrees with the idea that a student's mathematical understanding comes from within and is unique to each individual. He disagrees that activities planned for students will be understood to mean different things due to personal goals and cultural experiences.

It was evident that Mr. Hill also valued questioning as a viable teaching technique. Within the class the primary researcher heard questions such as "What do we know?" and "What are we trying to figure out?" However, Mr. Hill usually did not offer answers, only further questions. The students were allowed to discuss answers after everyone had completed the assigned problems. They also could help those who were having trouble. Mr. Hill agreed on the T-BASM that students need to work together in cooperative groups where they may share their ideas and thinking with other group members. He also strongly agreed that to help make sense of mathematics, children need the opportunity to discuss their thinking. Mr. Hill frequently discussed his ideas and concerns with other teachers in the building. This was done informally in the hallway during the school day or at lunch.

Mr. Hill's conflicting styles are partially a product of the state-mandated tests and classroom management. His method of teaching is also closely related to the way he learned mathematics. He knows that basic skills need to be taught, as is evident from his agreement with the notion of frequent drilling of mathematical facts; but he is neutral in his beliefs about being firm and controlling. He prefers to be seen as a facilitator rather than a dispenser of knowledge.

Another cooperating teacher participant, Mrs. Masters, has a classroom set up in a more traditional manner than Mr. Hill's. There are desks in rows with her desk on the side of the room. The student desks face the front chalkboard and overhead screen. There are many examples of student work up on the walls as well as posters describing mathematical ideas such as multiplication, double and addition. There are five computers on the opposite wall and some manipulatives out on bookshelves.

Mrs. Masters' class begins with announcements. During one observed class, one announcement dealt with a packet of work the students had produced that Mrs. Masters wanted signed by their parents. Also, several students offered stories and asked both irrelevant and pertinent questions. It was evident that students feel free to talk and share in this non-threatening environment. Mrs. Masters' lessons start with preparing for the state-mandated test. Students spend some time during each class working on a concept in a manner directly related to the test. They are given questions Co answer in various ways, such as select the letter corresponding to the correct answer, or explain your answer in writing. After this is completed, the students work on various other assignments. Mrs. Masters reads a story related to mathematics and asks the students some leading questions. The students look at Mrs. Masters as the authority and she at times gives answers.

Mrs. Masters is in a transitional stage of how to teach and how students learn. Mrs. Masters' classroom is next to that of another teacher who has very different ideas of how to teach and how children learn. Mrs. S is a constructivist teacher, and Mrs. Masters has begun to be intrigued by Mrs. S's results with her students. Mrs. Masters has many conversations with Mrs. S about how to teach concepts. This is unusual because Mrs. S is a second-year teacher and Mrs. Masters is a 17-year teaching veteran.

It is evident that Mrs. Masters likes the way Mrs. S teaches, and Mrs. Masters has said she would like to have tables for the students instead of desks. "I would like to do more cooperative work but it is often difficult" (Written Reflection). Mrs. Masters has found the value of discussing experiences with peers, but she is still holding onto some traditional beliefs.

Mrs. Masters believes that frequent drilling is important. She also disagrees with the notion that children learn mathematics best by finding their own ways to solve simple problems. Mrs. Masters strongly agrees that teaching key words and short cuts helps students to learn to solve word problems. She agrees with the idea that teachers should use skill guides to teach so that each skill is measured. However, the clearest indication of her transitional phase is her quandary of whether to cover the entire curriculum or allow students to explore some ideas more fully at the curriculum's expense. Her answer to this question was neutral, indicating that she sees a problem with the curriculum being very broad but not very deep. She is not sure if it is acceptable for the students not to cover the entire curriculum for that particular grade. In fact, she agreed with the question regarding covering the entire curriculum as important.

Mrs. Masters is a traditional teacher who has begun to shift her philosophy. She is still teaching in more of an abstract manner but has seen that there are many problems with that method. An example of this is when the class was working on the topic of capacity. The students did the work looking at pictures of objects rather than using the actual objects. The question was posed as "Which holds more?" The students were shown a picture of a bucket and a picture of an eyedropper. The problem with this exercise was that both the bucket and eyedropper were pictured as the same size. The students had not had any experience with an eyedropper, so many of them answered the question wrong. If the students had been shown an actual bucket and an actual eyedropper, they would have had a better understanding of the question. To rectify this, Mrs. Masters brought in an eyedropper to the next class and the students let out a collective "oh, that's what it is."

Another cooperating teaching participant, Mr. Right, has his classroom set up in a traditional manner with desks in rows and his desk front and center facing the students. There are two computers on the side of the room and posters depicting Black History and class rules on the walls. There is a blackboard on the front wall where Mr. Right does examples of problems.

In Mr. Right's class students work individually on daily work. Also, Mr. Right is the sole voice of authority in the classroom. He strongly disagrees with the notion that children can learn effectively when the teacher does not tell them if their answers are right or wrong. As students complete work they bring it up for him to grade. Even when students did work on projects in-groups, Mr. Right was the "expert" from whose model the students checked their work. He best described his philosophy when he wrote:

The one thing that stays the same is the fact that I need to get as many of my students as I can to pass the sixth grade test, end of story. Fluff is cool and discovery activities are great but the bottom line is I get fifth graders with third grade skill and I have to get them ready for the sixth grade test. So my story is cover as much ground, with mastery, as possible. Don't turn the kids off in regards to math and hope near the end of sixth grade they "get it" and start to dig math. It happens but it takes nearly 2 years. (Written Reflection)

This traditional belief of mastery learning is interestingly juxtaposed with his ideas of mathematics teaching:

I like creating real life math problems and having the students solve them. Application is where it is, The kids hate it at first then they get into it and like it. This creates a situation where they want to know more. That is when it gets fun. (Written Reflection)

Mr. Right lets students communicate but only after he has checked their answers. He strongly agrees that children learn mathematics best by figuring for themselves the ways to find answers to simple word problems, but strongly disagrees with encouraging children to develop their own solutions even when they are inefficient.

Emerging Themes During Student Teaching

Many research studies, articles, and books have addressed specific ideas that have been highlighted within this research study. The following is a brief overview of the findings and observations of those writings, tied to the observations within this study during the three preservices teachers' practicum/student-teaching. The major areas that emerged as influencing preservice teacher's beliefs and practices within the study have been identified previously in this paper, and they are: social norms of the classroom; socio-political norms within the classroom, school, district, and state; time issues; and teachers' content knowledge. However, these issues did not work in isolation, but in a complex interplay.

Social Norms of the Classroom

Social norms of the classroom are the primary factors for creating learning opportunity for students. Social norms, according to Latz (1992), might be the most difficult issue for beginning teachers to understanding, and therefore the area that gives them the most concern. In a study conducted by Adams & Krockover (1997), nine of eleven preservice teachers had concerns about classroom social norms. Sturtevant, Castellani, Deal, Duling, Haid, Guth, and Tiss (1997) assert that the participants were uncertain about establishing classroom social norms.

Pajares (1992) offers an explanation for why classroom environment might be such a concern. He states that teachers may understand the why and how behind management of the social norms within the classroom, but they may be lacking in the when and under what circumstances to employ these techniques. Also, Calderhead and Robson (1991) indicate that preservice teachers may have a construct built in their mind of particular episodes in which to use differing techniques but not have the experience to know how or when to modify them. Winitzky and Kauchak (1995) found this to be the case in their research. Preservice teachers de-emphasized the role of college classes on development of management techniques and philosophies and mainly attributed the development to work in the field. However, these philosophies might not be strongly entrenched, as evident in the Jones and Veslind (1995) study. These researchers revealed that, "by the end of student-teaching preservice teachers experienced a conflict between their bel iefs in rules, their desire to be flexible and fair, and their desire to develop positive student relationships as a mechanism to promote student learning" (p. 313).

All three of the preservice teachers in this research study discussed the impact classroom social norms had upon their practices and ultimately upon the beliefs.

William (one of the preservice teacher participants) began his teaching experience by critically evaluating his cooperating teacher's technique for establishing classroom social norms. Initially he thought it was too unstructured. In his understanding it was easier to begin the year ultra structured to allow the student the opportunity to understand the environment of the classroom and what was expected of them. He thought that Mr. Right (his cooperating teacher) was not structured enough.

Further on in the experience, William decided that Mr. Right's style of making the children stand at the back of the room for misbehaving was not effective. He believed this stopped not only the misbehaving but also learning. Also, William noted that when he took over the classroom teaching he would not use that style, but explained that he could not introduce any management technique that his cooperating teacher would have to continue after he left. He introduced ringing a bell as a management technique.

Initially he confided in his Written Reflections and in-group interviews that the management style he chose was working. As the novelty wore off, though, the student behavior worsened, causing less instruction and learning within each class. Mr. Right intervened at times and also asked William to do more to control the noise. William was a bit disenchanted. He believed some talk was necessary for students to learn. This only served to show William that constructivist lessons would not be valued in such a classroom.

Classroom social norms were issues for Jackie (a preservice teacher participant) as well. When she began the experience she did not give much thought to how to manage the classroom. Jackie noted that the students, for the most part, behaved for Mr. Hill (her cooperating teacher). This changed when she decided to teach her first unconventional lesson. The room became chaotic and she immediately sent the students back to their desks. Although she did attribute the failure to lack of planning, the behavior was what made her end the lesson.

Jackie was able to design any management technique she wanted. She decided to give tickets to students who were acting "appropriately." These tickets were then used in a drawing at the end of each week for prizes such as pieces of candy or pencils. This seemed to work, but Mr. Hill was concerned that the students were not respecting Jackie because they did not know her limits. Mr. Hill encouraged Jackie to "get tough" with the kids to let them know her boundaries. Jackie changed her demeanor to become more rigid because she was afraid that if the students did not respect her as their teacher, her classroom time (a significant amount of the year) would be wasted and the students would fall behind on their standardized test preparation. Jackie confided that she was very uncomfortable with this change in style.

Immediately when things started to get out of control, Jackie sent the children back to their desks. She ended up teaching in a conventional manner when the cooperating teacher deemed the students were not acting "good." Ultimately, classroom management was one of the guiding factors in Jackie moving to a traditional method of teaching.

Classroom social norms had the greatest effect on Cassey's (a preservice teacher participant) teaching. Cassey began the experience not understanding that her cooperating teacher wanted her to continue many of the extrinsic motivational techniques to classroom management that the teacher employed. Cassey decided to try to develop her own techniques. This became too time consuming, and eventually Cassey stopped. Her management was reduced to turning off the lights or raising her voice when the students were getting "too loud."

Toward the end of the student teaching, Cassey's class was out of control and she decided that the only way to deal with student misbehavior was to put the students back into rows and teach from the book. Mrs. Masters (her cooperating teacher) commented that Cassey's lessons became very traditional. Cassey believed that this return to traditional teaching would in essence help the children. She did state that she missed the group-work, but did not know how to deal with the management issues.

Each of the cooperating teachers insisted that their preservice teachers maintain certain management techniques or levels. These cooperating teachers understood that eventually they would have to be in charge of the class again. They also set up other requirements to perpetuate the classroom environment as they had established it. These cooperating teachers' established social norms had an impact on the student-teachers' practices and eventually their beliefs.

The Socio-Political Nature of Education

Within this particular study some explicit and some not so explicit political forces acted on the student-teachers' practices and ultimately their beliefs. These were as simple as comments made to the preservice teachers or as complex as the implication of district mandated interventions to raise standardized test scores. In either instance, the preservice teachers were influenced.

Simon (1993) asserted various political realities in his study. He identified several issues revealing the influence of political patterns that affected the implementation of an innovative program. One such issue was administrator support of such programs. Simon stated "there is very little reward and often substantial risk for administrators who associate themselves with educational innovations" (p. 109). However, political influence goes beyond the administrators. Anderson and Piazza (1996) indicated "inservice and preservice teachers, who must be the agents of change, are products of the system they are trying to change" (p. 54). Anderson and Piazza also stated that using standardized tests frequently to assess speed and accuracy of computation makes certain that the teachers will not be likely to change the emphasis of their teaching. In short, the system perpetuates a traditional style of teaching. This is what was observed happening at the researched school in this study.

This school ranks near the bottom compared to other schools on the state mandated test results, The superintendent, according to Mr. Right, does not want the school to be opened any longer. Consequently, the district has implemented, and the administrators of this school ask teachers to practice a curriculum that encourages a more rote, traditional style of teaching. The school administration shows its bias toward the traditional style in everything from praising those who promote memorization of facts to checking lesson plans daily to make sure they contain measurable outcomes. Mosenthal & Ball (1992) asserted that "the teacher's role in orchestrating and fostering students' learning is uncertain and risky compared with traditional telling and showing" (p. 347). This makes change difficult for teachers at this elementary school and impossible for preservice teachers.

During the course of this study, Jackie and William were influenced by the socio-political nature of education within the elementary school and the district. In both instances, the cooperating teachers' interpretations of district-wide directives, such as designing interventions to raise mathematics test scores, led to a data-driven curriculum being favored over a problem-solving based curriculum.

Jackie's and William's cooperating teachers worked closely together. Jointly, they decided that the easiest way to implement interventions was to chart their students' progress by using timed arithmetic tests. Each day was devoted to completing review worksheets. These culminated with a weekly fact test. Unfortunately for Jackie and William, this left little time to teach lessons.

Time Issues

"In an era of accelerated growth in knowledge and of rapid changes in many life areas, teachers, like other professionals, face a problem of time" (Kremer-Hayon, 1995, p. 417). Kremer-Hayon goes on to state that educating students for the future, a future that is not known, places a heavy psychological burden on teachers. This burden is coupled with an environment in the classroom where the expectation is to achieve higher level goals." In this state of affairs, time becomes an important factor" (p. 417).

The district and state where this school is located judges high level goals within the classroom by evaluating student performance on the state mandated tests. It is obvious to them that if many students pass the test, there must be high levels of expectation and teaching going on in the classroom. The reality is that this only serves to prompt the teachers' coverage of a lot of material in a general sense. There is very little time to delve into the concrete meaning of mathematical concepts, only enough time for a traditional pedagogy that encourages rote memorization.

The problem of time was most experienced by William and Jackie. As stated earlier, their cooperating teachers are under a tremendous amount of pressure to have as many students pass the required state-mandated test as possible. The cooperating teachers believed that for this to occur the students needed as much review as possible. For example, Mr. Hill indicated that he could potentially create a study guide and just go over that throughout the year. He indicated that he would have as many students pass using that method as would pass if he employed an alternate pedagogy. However, he conceded that the students would have no conceptual understanding of the concepts presented. As a result of their belief in review, the cooperating teachers insisted that the preservice teachers cover review questions prior to teaching any lesson during the typical school day. This left on average ten to fifteen minutes for the preservice teachers to present a lesson.

Sturtevant et al. (1997) observed that time was one of the conditions that influenced "the teachers' beliefs about the value and efficacy of particular instructional strategies" (p. 4). Time management was also discussed in the Adams and Krockover (1997) study. All eleven preservice teachers in the study rated time management as a concern. Therefore, if time is in short supply, instructional strategies that can be accomplished quickly will be valued. This, of course, reinforces teachers' adoptions of a traditional philosophy of mathematics.

Another aspect of time management is content knowledge. When a teacher understands and can relay to students that mathematics is a whole rather than a group of disjointed parts, that teacher is more likely to teach in a reform-minded way which values the connective nature of mathematics (NCTM, 1989).

Content Knowledge

"The idea of a coherent subject matter means that knowledge of the subject matter is conceived as related concepts rather than as fragmented parts" (Mosenthal & Ball, 1992, p. 350). Barriers to reform exist and slow or stop the change process. One barrier is the preservice or inservice teacher's lack of specific content knowledge. "Teachers need deeper content knowledge and understanding of the connections between mathematical topics and about the connections of mathematics to real life" (Anderson & Piazza, 1996). Stoddart, Connell, Stofflett, & Peck (1993) attributed this unpreparedness to "deficiencies in the pedagogy practiced in traditional didactic...courses" (p. 229). As a result of lack of content knowledge, Anderson and Piazza (1996) insisted that "teachers [will rely] on textbooks" and that such reliance serves to "undermine their professional judgment about what constitutes appropriate teaching" (p. 54).

Chen and Ennis's (1995) study revealed that "enhancement of prospective teachers' pedagogical content knowledge should be emphasized in teacher preparation programs because it serves as a bridge linking the subject content knowledge with the curriculum delivered in classrooms" (p. 389). Unfortunately, most colleges and public schools focus on procedural knowledge development through drill and practice (Porter, 1989).

Within the context of this study, lack of content knowledge proved detrimental in the development of a teaching pedagogy for two of the three preservice teachers. More particularly, Jackie and Cassey both reported that lack of content knowledge adversely effected their lessons at times. This was evident in observations and interviews with the preservice and inservice teachers as well as their Written Reflections.

During the final interview, Jackie admitted that she taught some lessons conventionally. She stated that when she felt comfortable with the content she would try to employ student-centered activities. Jackie also indicated that she would use the book as the primary source of activity when she was less sure about the content.

Eventually, both lack of content knowledge and problems with classroom management affected Cassey's teaching methods. Initially, Cassey was permitted to rearrange the room to facilitate her teaching style. She explained that as the term progressed the students became "too comfortable" in the setting. During an informal conversation, Cassey indicated that she was beginning to become concerned because many of the students were turning in identical work. Also, she noticed that students were more apt to be off task in the small-group setting. These problems were intensified when Cassey encountered material that she indicated she had not seen since junior high school. Cassey abruptly ended group work in her classroom in desperation to have more control of the classroom and get the students to work independently. She placed the students back into rows and began lecturing from the book. Cassey did indicate that she "missed her groups" (Interview During the Second Phase); however, she ended up teaching in a tradition al manner.

Discussion

William's constructivist beliefs were actually strengthened by having to observe a traditional classroom. However, he practiced a traditional philosophy while in that classroom, and William attributed this to the cooperating teacher's requirements (i.e., because Mr. Right believed in a traditional classroom). William expressed that with the environment, social norms, and sociopolitical norms already established for this classroom, it was impossible for him to try to teach in a reform-minded way. He would have to spend the remaining time revamping the classroom and students so that he could actually implement constructivist methods.

Jackie's beliefs were changed during her practicum/student-teaching experience by her cooperating teacher's beliefs and practices. Mr. Hill believed that for the students to pass the state mandated test, they would need traditional instruction. Jackie at times practiced a traditional approach at Mr. Hill's request. She believed that there were appropriate times to teach traditional and appropriate times to teach in a reform-minded way.

Cassey was the least affected by her cooperating teacher's beliefs and practices. Mrs. Masters gave Cassey complete freedom to design the classroom and teach lessons in any way she deemed appropriate. Mrs. Masters' only mandate was a daily review that took approximately five minutes to complete. Cassey did not regularly adhere even to this request.

The socio-political events that transpired before and during the study created several layers of complexity. These events served to indirectly and directly affect the preservice teachers practices and beliefs. Initially, the political nature of the state mandated test has helped shape the cooperating teachers' beliefs, and more importantly, their practices. This has manifested itself in the form of daily reviews and weekly fact tests. With this added burden the preservice teachers were severely limited in their ability to implement a reform-minded pedagogy.

A second unforeseen factor that affected two of the preservice teachers was lack of content knowledge. In both cases the preservice teachers felt more comfortable teaching a traditional lesson directly from the book when they did not have a strong understanding of the content of the lesson. Jackie stated that she would continue to teach traditionally when she did not understand fully the concepts she was teaching. Cassey changed her room around to help compensate when the content became increasingly difficult. She went back to teaching directly from the text.

A third factor was the establishment of classroom social norms. Although two of the three cooperating teachers influenced their preservice teachers' actions, ultimately it was the preservice teachers themselves who had to deal with classroom social norms. In different classroom situations, each of the preservice teachers had moments where they decided to teach traditionally as a means of alleviating what they deemed to be inappropriate student behavior. William stopped his cooperative learning method in problem-solving situations and assigned worksheets. Jackie taught directly from the book and did not allow any conversation. Cassey rearranged the desks back into rows and eliminated group work altogether. In each instance, the preservice teacher reacted to classroom management issues by turning to a traditional method of teaching.

These preservice teachers' experiences were immeasurably altered by the state mandated test. Kamii & Lewis (1991) indicate that "in spite of the consensus among mathematics educators that standardized achievement testing encourages lower-order thinking, these continue to be used throughout the country as if they revealed The Truth about students' knowledge of mathematics (p.4)." With the heavy emphasis on standardized testing scores teachers and administrators who are transforming teaching, learning, and school culture are faced with a dilemma between teaching to the test or teaching for understanding. Tsuruda (1994) states:

It is an unfortunate reality that assessment policies often drive curriculum practices. Although teachers may want to change how and what they teach in their classrooms, they are bound by their districts to administer tests that measure skill. Since constructivist teaching, which emphasizes thinking and communicating, is not the quickest way to help students acquire specific skills, and since all teachers feel a certain degree of pressure to have their students do well, teachers are caught in a dilemma. (p. 7)

Research supports a recursive relationship among planning, instruction, assessment, and reflection (Grundy, 1987). According to NCTM (1993) "assessment should reflect the mathematics that is most important for students to learn" (p. 3). And, "assessment should enhance mathematics learning" (p. 4). The assessment should allow every student the opportunity to show mathematical power. Also, assessment should display each student's mathematics learning. As is evident with the state mandated test, and specifically at this school, none of these assertions are met. Hence, the student-teachers are forced to teach in a traditional manner that is consistent with their cooperating teachers' teaching methods in order to quickly provide concepts in a rote way so that students pass a specific assessment. The student-teachers were never able to fully integrate an alternating pedagogy.

The strongest recommendation coming from this research deals with the cooperating teachers. Potential cooperating teachers need to be identified prior to any student placements. These teachers' philosophies should be compatible with that which is advocated in the methods courses. If cooperating teachers are teaching in a reform-minded manner, it indicates that the district and school administrations accept different teaching philosophies and value reform-minded approaches as viable alternatives to be traditional methods of teaching. This alone will facilitate the preservice teachers' comfort with and understanding of constructivist learning theory.

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Title Annotation: | teacher preparation for mathematics teaching |
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Author: | Pourdavood, Roland G. |

Publication: | Focus on Learning Problems in Mathematics |

Geographic Code: | 1USA |

Date: | Jan 1, 2002 |

Words: | 8513 |

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