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Preservice Secondary Teacher ‘Moves’
ABSTRACT
This paper discusses the experience of one preservice secondary mathematics teacher as she and her group prepared a pre-calculus lesson in the context of lesson study. The study looks at how this preservice teacher used different compensation ‘moves’ to direct the conversation away from her mathematical knowledge in order to protect her identity as a knower of mathematics. Although the preservice teacher was a strong undergraduate mathematics student, she feared being labeled as ‘dumb’ and sought to deflect attention away from her weak conceptual understanding of secondary mathematics. This paper seeks to investigate the culture in which preservice secondary mathematics teachers develop their beliefs and how those beliefs influence prospective teachers’ behavior during mathematical conversations. The study acknowledges that belief systems are formed from years of experience within the mathematics school culture and provide clues as to the motivation for using ‘moves’ to keep from revealing a weakness in mathematics.
A PRESERVICE SECONDARY TEACHER’S MOVES TO PROTECT HER VIEW OF HERSELF AS A MATHEMATICS EXPERT
Julie Stafford-Plummer
Blake E. Peterson
INTRODUCTION
In 1990, Ball reported on a study in which she investigated three beliefs that were commonly held among mathematics education majors. The beliefs were that high school mathematics was not difficult, the mathematical knowledge needed to teach secondary mathematics was primarily gained prior to college, and any additional mathematical knowledge that was needed was gained as part of a bachelor’s degree in mathematics. She found, however, that many preservice teachers who held these beliefs lacked a conceptual knowledge of high school mathematics and tended to explain concepts in terms of procedures. The purpose of this study was to investigate a form of Japanese lesson study as a possible vehicle to dislodge these commonly held beliefs described by Ball (1990).
LITERATURE REVIEW
To lay the groundwork for this study, we will discuss beliefs in general as well more specific beliefs related to intelligence, mathematics, and mathematics teaching.
Beliefs
One focus of the reports of the Third International Mathematics and Science Study was the culture of mathematics teaching. Stigler and Hiebert (1999) defined teaching as a cultural activity which possesses cultural scripts that are “learned implicitly, through observation and participation and not by deliberate study” (p. 86). They also stated that the culture of mathematics teaching was unique to a country and greatly influenced the way in which mathematics teachers viewed their profession (Stigler and Hiebert, 1999). This meant that the culture largely impacted what the teachers believed about their subject matter, their philosophies about student learning and their role as mathematics instructors (Stigler and Hiebert, 1999). These beliefs then contributed to cultural norms and actually “generat[ed] and maintain[ed] cultural scripts for teaching” (Stigler and Hiebert, 1999, p. 89). This set up a cycle where beliefs about teaching were influenced by the culture, and the culture in turn preserved these commonly held beliefs. Members of the culture took on the cultural beliefs, and then over time, as one acted on those beliefs, he or she helped to perpetuate the cultural norms.
Cultural beliefs about intelligence and aptitude for learning
Cross-cultural research has highlighted the different views that societies have about an individual’s aptitude for learning and capacity for retaining information. One of the most striking differences between the U.S. and Asian countries, like Japan, is the relationship between a student’s capacity for learning and the student’s innate ability. Stevenson et al. (1990) found that when Japanese students and their mothers heard the quote, “The best students in the class always work harder than the other students,” the majority agreed with the statement. They disagreed, however, with “The tests you take can show how much or little natural ability you have.” These responses from the Japanese highlight the different beliefs about schooling held by U.S. students, who offered the opposite response to both statements by disagreeing with the first quote and agreeing with the second. Stevenson et al. (1990) summarized this idea by saying that in the United States “achievement is attributed to innate ability” (p. 23). Thus mathematics students in the U.S., who do well, are thought to have natural mathematical ability.
Williams and Montgomery (1995) looked at how gifted high school students’ “academic self-concepts are determined in relation to both internal and external comparisons. An external comparison refers to the belief that one’s mathematics performance is better than other students in the mathematics class: subsequently, this student reports high mathematics self-concept” (p. 401). They found that there is a high correlation between how students perform in mathematics and how they perceive themselves in the community (Williams and Montgomery, 1995). Students from the United States learn, subconsciously, how their ability to perform in school mathematics is closely associated with how the community views them. This U.S. focus places an interesting pressure on prospective mathematics teachers. Because they are majoring in mathematics, many look at them as having a “natural ability” for mathematics. Since their mathematical success is not measured by effort as much as it is by ability, any stumble in mathematical achievement strikes at how these prospective teachers see themselves.
Cultural beliefs about the mathematics teacher’s role
The role of the teacher in a traditional U.S. classroom could be described as a supplier of knowledge. The teacher’s responsibility then, is to oversee students’ access to mathematical information. This leads to teachers who impart knowledge in “pieces that are manageable for most students” (Stigler and Hiebert, 1999, p. 92) through direct modeling (Hiebert et al., 1997). This style of instruction is common in the U.S. traditional mathematics classroom and strengthens the image that the teacher knows more than students. It also leads to the belief that the teacher is the all-knowing mathematical authority in the classroom (Stigler et al., 1996). Students then learn that what the teacher says is correct, and rarely develop enough confidence to depend on their own knowledge.
Cultural view of mathematics majors and their views of themselves
Through years of being a participant in the mathematics classroom, students acquire beliefs about what it means to teach and be a teacher (Clark, 1988; Cooney et al., 1998). As a result, prospective teachers bring preconceived notions about the role of the mathematics teacher with them to teacher education, such as the teacher being the mathematical authority in the classroom (Stigler et al., 1996). Over time teacher candidates assimilate into their belief systems that the two, teacher and all-knowing authority, are synonymous. Teacher candidates grow accustomed to the direct modeling method of learning mathematics. Researchers (Cooney and Shealy, 1997; Cooney et al., 1998) have elaborated on this idea of learning but in relation to learning how to teach. Many times “preservice teachers press [teacher educators] to tell them the right way to teach” (Cooney et al., 1998, p. 311). Not only do preservice teachers expect to learn mathematics in a delivery and receiving mode, but they also expect to learn how to teach in this manner as well.
This particular cultural norm seems to have the most significant impact on the way society and teacher candidates view mathematics education majors. Cooney (1994) commented that this “orientation of authority can provide a way of conceptualizing how [preservice] teachers view mathematics and their roles as teachers of mathematics and thereby provide a basis for impacting their beliefs” (p. 628). When teacher candidates feel that a teacher is the mathematical authority, it seems natural that during teacher education they will slowly take on this identity. The culture and even the preservice teachers, for that matter, expect that once teacher candidates have completed a teacher education program, they will become the mathematical authority in the classroom (Stigler and Hiebert, 1999).
Ball (1990) studied three commonly held cultural beliefs surrounding teacher education. Ball first reported that often the culture of mathematics teaching views high school mathematics as lacking in difficulty. There seems to be an assumption that high school mathematics consists of procedures and prescribed operations. Therefore, when the culture comments that high school mathematics lacks difficulty, this is in reference to secondary procedures and not the mathematical principles underlying the procedures. The second cultural belief Ball reported was that mathematics courses prior to college had prepared prospective teachers sufficiently to teach high school mathematics. The last belief was that subject matter knowledge could be secured through the process of pursuing a mathematics degree.
Other researchers have come to similar conclusions about the culture’s view of mathematics majors. Stigler and Hiebert (1999) pointed out how “in the United States, teachers are assumed to be competent once they have completed their teacher-training programs" (p. 110). This includes the belief that preservice mathematics teachers know secondary mathematics as well as how to teach it. Once the preservice teacher graduates from a mathematics teacher program, the culture assumes that a beginning teacher is prepared to take on the sole responsibility of teaching students mathematics. Peterson and Williams (2001) studied conversations between student and cooperating teachers. They found that the topic of conversation rarely focused on mathematical content but rather on administrative or managerial issues (Peterson and Williams, 2001). Studying the same group of subjects, Durrant (2001) found that the cooperating teachers felt “intimidated by the mathematical knowledge student teachers supposedly obtain[ed] during their preparation for teaching” (p. 952). As a result, both student and cooperating teacher assumed discussing mathematics was not important (Durrant, 2001). What these student and cooperating teachers did not realize was that the teacher candidates’ experiences with secondary mathematics were usually limited to their initial exposure in the secondary grades (Cooney, 1994). The student teachers actually needed discussions of mathematical content to become better teachers.
Comparing this cultural practice to a Japanese norm, one finds the opposite to be true. In Japan “beginning teachers…are considered to be novices who need the support of their experienced colleagues” (Shimizu, 1999, p. 111). This particular belief is just a part of the overall beliefs that the Japanese culture holds about teachers. In Japan teaching is viewed “more as a craft, as a skill that can be perfected through practice and that can benefit from shared lore or tricks of the trade” (Stigler et al., 1996, p. 216). Thus it makes sense that novice teachers would not be left alone. However, in the U.S. there is a prevalent belief that “teaching is an innate skill, something you are born with” (Stigler and Hiebert, 1999, p. 86). Therefore, beginning teachers are expected to be capable of teaching mathematics from day one. This belief influences not only how beginning teachers are viewed, but also the cultural attitude toward preservice teachers. Not only are they viewed as possessing a natural ability to teach, but also after four years of college they should know how to teach secondary mathematics. Therefore, this cultural belief traps prospective teachers in a system that prevents them from being exceptionally knowledgeable about secondary mathematics (Stigler and Hiebert, 1999).
Summary
It has been established that the teaching culture places certain expectations on preservice teachers that they are or should be the mathematical authority. These expectations often force these new teachers to accept the cultural belief that they should have no weaknesses in mathematics even though their understanding of the underlying meanings of high school mathematics is fragmented and fragile. To further investigate this conflict between personal and cultural beliefs about new teachers being mathematical authorities and the reality that their mathematical understandings are actually weak, we chose the environment of Japanese lesson study to study these cultural beliefs.
RESEARCH DESIGN AND METHODOLOGY
Setting
Stigler and Hiebert (1999) describe the practice of lesson study as a commonly used method of professional development in Japan. They state “In lesson study, groups of teachers meet regularly over long periods of time (ranging from several months to a year) to work on the design, implementation, testing, and improvement of one or several ‘research lessons’ (kenkyuu jugyou)” (p. 110). A research lesson is the actual lesson that results from the collaborative efforts of a lesson study (Lewis, 2000). Although collaborative work between teachers can quickly become a case of “show and tell,” where each teacher describes what they do when they teach a certain topic, Stigler and Hiebert (1999) emphasize the critical component of focusing on student thinking in the lesson study process when they say, “The lesson-study process has an unrelenting focus on student learning. All efforts to improve lessons are evaluated with respect to clearly specified learning goals, and revisions are always justified with respect to student thinking and learning” (p. 121).
The basic format of lesson study below is drawn from Stigler and Hiebert (1999), Lewis (2000, 2002), and Fernandez et al. (2001).
- Goal Setting: Select a learning goal for the research lesson (the lesson that results from the lesson study process). Identify goals for student learning and long-term development.
- Planning: Plan, as a group of teachers, the learning activities and the sequence of these activities that will aid students in achieving the learning goal. This planning often begins by referring to a variety of resources including textbooks and individual teacher experience.
- Research Lesson: One member of the lesson study group teaches the research lesson to a group of students while the other members of the lesson study group carefully observe and take note of how the students are responding to the learning activities and making sense of the mathematics.
- Revise: Based on the data gathered regarding student responses and how those responses fit with the learning goals, revise and refine the research lesson to better meet the desired learning goals.
- Teach Research Lesson a 2nd time: After the research lesson has been adequately revised, have another member of the lesson study group teach the lesson again to a different class. Again the other members of the lesson study group as well as other teachers in the school or community observe the student reactions and responses to the learning activities.
- Report: Write a report describing the lesson study group’s reasoning and rationale behind the purpose and sequencing of the components of the research lesson and summarize the student responses.
Although most Japanese lesson study is done by practicing elementary teachers, this study adapted the above principles to preservice secondary mathematics teachers in the United States. This lesson study format was chosen to look at preservice mathematics teachers’ beliefs because Lewis (2000) found that “competing views of teaching bump against each other” (p. 17). Lewis (2000) stated “it would be interesting to look at research lessons as a potential influence on teachers’ content knowledge development” (p. 17), which would indicate that lesson study could also be fertile ground for investing teachers’ mathematical understandings.