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discriminatingly

    Pi-Jen Lin, Wen-Huan Tsai

    Using Research-Based Cases to Enhance Prospective Teachers Understanding of Teaching Mathematics and

    Their Reflections

    Abstract

    This study was designed to improve prospective teachers understanding of

    teaching mathematics and their reflection by using research-based cases. Forty-three participants were enrolled in the Mathematics Method Course. The participants were provided with five videotaped cases to watch and followed by the case discussion immediately. Teachers reflective journal was a major means of

    data collections. The results show that the prospective teachers are not familiar with the topics including semantics structures of addition word problems, the meaning of a fraction involved in either continuous or discrete quantity, the framework of underpinning measurement. The prospective teachers were concerned with the pedagogical content knowledge including ways of presenting a concept to students, teachers role, grouping, instructional strategy, physical arrangement, and assessment. The results indicate that the prospective teachers have greater understanding of the complexity of the classroom in which there are various effective factors to the classroom events, including students levels of development,

    instructors skills of questioning, reinforcement. The research-based case improved the prospective teachers’ reflections on their journals, in terms of their abilities to

    observe discriminatingly, to view a situation from multiple perspectives, to make logically defensible judgments, and to produce alternative suggestions for action. Key words: cases, teacher education, reflection, mathematical journals

F. L. Lin (Ed.) Common Sense in Mathematics Education,231-272

    Proceedings of 2001 The Netherlands and Taiwan Conference on Mathematics Education, Taipei, Taiwan, 19 23 November 2001.

     232

    Problems Related to Teacher Education

    Reacting to social pressures and drastic changes in the educational conditions in Taiwan, the unified textbook used for the recent two decades was replaced by the newly approved textbook examined and approved by the review committees from

    the Ministry of Education since the new Curriculum Standard was issued in 1993 (Ministry of Education of Taiwan, 1993). The philosophy underpinning the 1993 version of the curriculum reform reflects a constructivists perspective. There are

    four components emphasizing on new curriculum: problem solving,

    communication, reasoning, and mathematical connections. The four elements are the same as the focuses of the Standards of United States (NCTM, 1989). The standard-oriented approach of teaching has shifted its focus from teacher-centered to learner-centered. Teachers are encouraged to cultivate the atmosphere of students discussion in classrooms in which teachers roles shift from a problem

    solver to a problem poser. On the other hand, students roles shift from a replicater

    of teachers solutions to a problem solver on their own. Nevertheless, the success of innovative curriculum relies on whether teacher education programs provide the opportunities for teachers to learn to transform the emphases of curriculum into classroom practices.

    The research on teacher education shows that even though teachers are exposed to theories of learning and teaching in some teacher education programs, they are often not able to apply this knowledge to classroom practice (Cooney, 1999; Richardson, 1993; Schifter, 1996a; b). Prospective teachers who may be competent in executing effective teaching skills in simulated situations are not necessarily competent in applying those same skills in particular classroom contexts. To bridge the chasm between theory and practice, some teacher educators began to promote reflection as a means of helping teachers become better solvers of classroom problems (Krainer, 1999, Wood et al., 2001). Many teacher educators who help prospective teachers acquire more types of the knowledge worked with prospective teachers in the ways of structuring prospective teachers‟ thinking, and

    of training them to be more analytic about what they see in classrooms: teaching

     233Using Research-Based Cases to Enhance Prospective Teachers Understanding of Teaching Mathematics and Their Reflections

    them to raise questions about what is being taught and what is being learned and to generate hypotheses about better approaches. Some teacher educators suggest that learning others‟ experiences and helping others learn from our own experience is a way to help teachers translate their theories into classroom practice (Carter, 1993; Richardson, 1993).

    One of the ways we learn from others‟ e experiences and help others learn

    from our own experience is through stories, narratives or cases that reflect aspects

    of normal classroom experience and raise issues. A conviction that using cases may

    be helpful for those who need to learn to think in new ways about dealing with complex teaching problems has been gaining momentum (Carter, 1993; Harrington, 1995; Richardson, 1993). Cases, as defined by Merseth (1996), have three characteristics: (a) cases are realistic, (b) they rely on careful research, and (c) they foster users to develop multiple perspectives.

    The use of cases involved in the research on teacher education includes both case discussion and case writing. Case discussion can play a critical role in expanding and deepening pedagogical content knowledge (Barnett, 1991; 1998). Discussing cases fosters personal reflection through an external process (Shulman & Colbert, 1988). Some researches have examined the influence of discussing cases on what and how teachers learn to teach (Richardson, 1993; Merseth, 1996).

    In Lins study (2000a) examined the effect of cases on teachers knowledge when

    cases are constructed by a collaborative team with university professors and the teachers participating in a school-based teachers professional development project.

    In recent years, the authors investigated a series of research on examining the effect of using cases on both in-service and prospective teachers understanding of

    teaching mathematics and reflection. The uses of cases include constructing cases, writing cases, and case discussing in the way of presenting a case with a video or in a written form. The paper reported here is the result of one of the series of the researches.

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    Research-Based Cases in the Reported Study

    The research-based cases presented in this study refer to the cases to be used in this study are adopted from the authors‟ investigation of the previous three-year

    research project granted by National Science of Council (NSC88-2511-S134-002, 89-2511-S134-001, 89-2511-S134-012). All of the research-based cases developed collaboratively by the authors and classroom teachers that were inspired by observing the teachers involved in the previous three-year research. Then, by discussing the issues addressed in the participant teachers lessons and recording

    the discussion in the form of a complete case. In the study I, the procedures of developing a complete case in a written form includes four stages: observing classroom, initiating the first version of the case, discussing for revising the first version and producing the second version of the case, discussing for revising the second version and producing the third version, and a discussion for revising the third version and then finalizing the ultimate version (Lin, 2000c). In the previous three-year research on cases constructed collaboratively by university professors and classroom teachers, it revealed that the construction of cases could enhance teachers‟ pedagogical content knowledge, improve their awareness of and their

    competence in dealing with the difficulties students encountered when learning mathematics, and promote their abilities to reflect on classroom practices. However, the effect of cases on teachers professional knowledge of mathematical teaching

    was examined particularly for those who participated in the research context in which the cases were constructed. It is still considerable that whether using cases in a written form would influence the teachers professional knowledge of

    mathematical teaching of those who are in-service and did not participate in the research context in which the cases were constructed. To answer the research question, the authors conducted another research project funded by National Science of Council (NSC89-2511-S-134-018-X3) coming after the three-year research project. The following project intended to investigate the extent of influence of using cases on in-service teachers mathematical teaching. Moreover,

    the written form of each case conducted in the previous three-year research would

     235Using Research-Based Cases to Enhance Prospective Teachers Understanding of Teaching Mathematics and Their Reflections

    be re-examined and re-modified in study II if the required components of each case were included completely and described clearly from the users perspective.

    The cases can be presented not only with a written form but also by videos. The video conducted by the authors working with a team of classroom teachers who are potential to recognize salient aspects of the case. The users of these videos can observe the cases presented them to see what and how teachers interacted with students in the classroom. Using videotaped cases appeared potentially efficacious because the videotaped format allowed prospective teachers to experience the classroom in a way unparalleled by a written form, while removing the complications of a live observation (White, 1993). Thus, the authors initiated two further studies, Study III and Study IV, following the previous research on using the cases presented in a written form to investigate the influence of the videotaped cases on teachers knowledge of pedagogical mathematics content. Study IV was designed to estimate the effect of the cases in videotaped from on in-service teachers knowing of mathematical teaching. The work of data collection has finished, but the data analyses are still on processing. The result will soon be reported elsewhere. Study III was designed to examine these videos on prospective teachers understanding of teaching and their reflection. This paper will present the result of Study III. The research-based cases were investigated in different studies for various uses of cases and the users. Figure 1 depicts a series of research on cases.

    The users The use of cases

    Develop & Study? write In-service teachers Study? Research-based Written form cases Study?

    pre-service Study? Video form teachers

     Figure 1 A series of Research on Cases

     236

    Reflection as Problem Solving

    Reflection, as defined by Dewey (1933), occurs when a person encounters a situation that is puzzling, troubling, or uncertain and for which no precedent exists. Deweys idea of process of reflection is not linear. That is, the aspects of reflection do not follow in an orderly progression one after the other. However, Schons (1983)

    idea about reflection focuses on the process of framing, selecting an action, and evaluating consequent results. The process of framing problem, reframing problem, and generating alternatives are three essential components of implementing problem solving in classroom contexts. The quality of reflection can be improved by developing the abilities to observe accurately and discriminately, to view a situation from multiple perspectives (White, 1993). Thus, the strategy capitalizing the groups power to analyze the case rather than relying on individual perceptions is more likely to strengthen ones ability to establish different view of multiple

    perspectives. By doing so, the quality of reflection will be improved.

    To understand the prospective teachers ability of identifying a problem in

    their own practices, Pugach (1990) asked them to point out the problems they are concerned with after observing a video-case. The prospective teachers selected the following problems to address: (a) improving the quality of teachers verbalization,

    (b) improving classroom management, (c) increasing teachers positive responses,

    (d) improving the clarity and content of teachers way of speaking, and (e)

    increasing the time teachers wait for students responses. These findings are

    supported by results of a study conducted by Gore and Zeichner (1991). Many prospective teachers often began with a focus on discipline and classroom management, which are unconnected to curriculum and instruction. The study reported here concentrates on the problems presented in someone elses teaching

    rather than those of ones own teaching. Information is supplied according to what prospective teachers considered problematic when watching a video taped from a real classroom, what perspectives they took, and what types of information they would use to react in a similar problematic situation.

     237Using Research-Based Cases to Enhance Prospective Teachers Understanding of Teaching Mathematics and Their Reflections

    The Use of Case-Method Instruction

    Cases can be used in many ways. Case methods are employed as stimulants to reflection in order to frame the conversation between mentors and novices, as techniques used to enrich field experiences, and as tools applied to orient individuals to particular ways of thinking (Merseth, 1996). Merseth (1996) categories cases of teacher education into three purposes: (a) Cases can be exemplars used to honor the best practice or to make the effective teaching more public and more available for others to analysis and review (Sykes & Bird, 1992). (b) Cases can be used to practice decision-making and problem solving in which case portray problematic situations that require problem identification and analysis, decision making, and action definition. (c) The purpose of cases is to stimulate personal reflection. Teacher educators suggest that using cases is a powerful means to develop habits and techniques for reflection (Kleinfeld, 1992).

    The format of Kleinfelds (1991) study is used for conducting the discussions,

    asking prospective teachers to spot the critical issues in a case, to view the issue from different actors‟ perspectives, and to generate possible alternatives for action that the teacher might consider to take. Kleinfeld reported that prospective teachers participating in the case method instruction showed significantly greater abilities to analyze an educational problem. The result of Kleinfelds study proves that using

    case method instruction to help prospective teachers acquire and improve problem-solving capabilities indeed works. Cognitive dissonance provides additional support in the use of case discussion. Social interaction between peers who hold different perspectives to bear on a problem seems to be more likely to lead to cognitive dissonance and is a highly effective means of inducing cognitive development. Therefore, video-case instruction mentioned in this study includes small discussions with watching video taped from a real classroom. To structure case discussion, the first author of the paper, who is the leader of the discussion, plays different roles to guide, probe, direct, give feedback, and observe the exchanges and contributions among the prospective teachers.

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    Teacher educators support using the case-method instruction to improve prospective teachers reflection as problem solvers (Merseth, 1996; Shulman, 1989). During the discussion, prospective teachers have to offer various viewpoints for consideration, considering the varieties of ways of viewing the same phenomena. Debating the merits of the various viewpoints, prospective teachers would have to articulate justifications for their own viewpoints, clarify the meaning for themselves and suggest alternatives that are easier for their classmates to view the situation. Through the process of discussion, prospective teachers will have greater understanding than that they would have by only viewing the phenomena from a single point.

    This study was designed to examine that the extent of the use of research-based cases instruction can influence the ways in which prospective teachers analyze the teaching events. The research-based cases with videotaped format appeared potentially efficacious because (1) the videotaped cases allowed prospective teachers to experience the realistic classroom situations; (2) the cases were selected from a cohort of videos that were developed by the researchers and a group of in-service teachers who participated in a three-year research on teachers‟

    professional development to help prospective teachers recognize salient aspects of the case; (3) the strategy capitalizing the groups power to analyze the case rather

    than relying on individual perceptions.

    Research Questions

    The purpose of this study was to determine whether research-based cases used to provide case-like experiences in a mathematics method course as a part of the professional development program for prospective teachers. The case-method instruction could be used to develop the knowledge of teaching mathematics and to influence their reflection. It focused on how prospective teachers perceived events in the case, what phenomena they considered problematic, the pedagogical actions they might take in similar circumstance along with their explanations of their choices, and their thought about the possible consequences of their actions. The

     239Using Research-Based Cases to Enhance Prospective Teachers Understanding of Teaching Mathematics and Their Reflections

    research questions include:

    1. What do prospective teachers describe an event after watching the

    research-based videotaped cases?

    2. What problems do prospective teachers choose to address after

    watching the research-based videotaped cases?

    3. How does case-method instruction improve prospective teachers

    capabilities of identifying a problem?

    4. How does case-method instruction improve prospective teachers

    reflection?

    Research Method

    This study was designed to improve prospective teachers understanding of

    teaching mathematics and their reflection by using research-based cases. The use of research-based cases presented with video form has been integrated into a course called the Methods for Teaching Elementary School Mathematics (Mathematics Methods Course). Using research-based cases was not the only instructional strategy employed but was seen as a central part to accomplish the goals of the mathematics method course. The two-hour Mathematics Methods Course once a week with continuing thirty-two hours accomplished in a school semester consists of two kinds of activities in which each one was conducted in two stages. The first stage was designed to help prospective teachers comprehend how to teach. The second stage was designed to understand explain and illustrate how prospective teachers put their understanding of teaching into practice. During the first stage, prospective teachers were supplied with five cases of teaching presented with videos which were conducted by the authors in a three-year research project. Each case was discussed with two hours in the course in every week. In addition, the prospective teachers journals were the resources of documenting their reflection. In the second stage, prospective teachers were asked to observe a classroom teaching, and the instructor was the same one in the previous video. Also,

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    prospective teachers were asked to demonstrate a teaching.

    Participants

     The participants of the study were 43 prospective teachers attending in a one-year Elementary Teacher Education Post Baccalaureate Program at Hisn-Chu Teachers College. The range of participants ages is from the 23s to the 35s,

    including those who have recently graduated from a four-year university and those who have teaching experiences of several years or working at home. All participants were enrolled in the Mathematics Methods Course.

    The Case Discussion Session

     The case discussion session was designed to provide the opportunity of analyzing a video case with four or five people in a group, and then to discuss its implication and importance as they perceived them. The case discussion was followed right after they watching a video case. The 43 participants were divided into eight groups of 4 or 5 persons. One of the intentions of this experience encouraged prospective teachers to analyze teaching from various perspectives, and to make the meanings they construct from the situations of teaching and learning they observed more explicit. The other intention of the case discussion was to help the researchers to identify the prospective teachers capability of framing a problem

    and solving the problem identified. The researchers were concerned with: (1) How do prospective teachers describe an event? (2) What perspective do they take? (3) What problems do they choose to address? The participants were not provided extra supporting information such as guiding questions or considerable points. The Video-cases

     Five videos selected from a cohort of videos conducted from a three-year research project were observed in the Mathematics Methods Course. Each video consisting of the narratives taped from a real teaching that occurred in the classroom from the first to the fourth grade was observed in each class of the course. The narrative of each video represented a number of issues related to the teacher-learner interactions of teacher-led activities in a typical elementary

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