THE CONNECTION BETWEEN TEACHER SELF-EFFICACY AND REFLECTIVE PRACTICE

by

Timothy M. Sibbald

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Curriculum, Teaching and Learning Ontario Institute for Studies in Education of the University of Toronto

© Copyright by Timothy M. Sibbald, 2008

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THE CONNECTION BETWEEN TEACHER SELF-EFFICACY AND REFLECTIVE PRACTICE

Doctor of Philosophy, 2008 Timothy M. Sibbald Department of Curriculum, Teaching and Learning University of Toronto

ABSTRACT This study addresses a gap in the literature by developing a theory that bridges reflective practices and self-efficacy. Since self-efficacy has been linked to student achievement, the theory is important as an explanatory mechanism linking reflective practice to student achievement. The theory was developed using primary source datafromfour types of reflective practice groups: Lesson study, peer coaching, mentoring and self-study. Practices within each group were structured but sufficiently flexible to facilitate the development of shared leadership and direction within each group. Data primarilyfromplanning and debriefing meetings was analyzed using grounded theory and constant comparison to develop a substantive theory. The theory demonstrates that reflective practice is linked to self efficacy through a general theory of mam teaching which implies that reflective practice can address teaching issuesfrompedagogical content knowledge to deeper conceptual matters. It also implies that constraints arising in reflective practice are most

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frequently limitations of pedagogical content knowledge which can be mitigated by a community of practice that provides appropriate resources and subject expertise. The findings of this study are of practical significance because they demonstrate that the general theory of math teaching, that is developed, and having a repertoire of pedagogical choices are key factors linking reflective practice to self-efficacy. This implies that reflective practice groups need diverse expertise within the subject area for pedagogical sharing to take place. It also implies that ongoing support for teacher experimentation that avoids staff complacency is important. The community of practice has a crucial role to play when a reflective practice group is unable to resolve specific content issues and needs to supply resources, access to expertise or specific professional development. Leadership (which is assumed not to be mandating the reflective practice) has a supporting role of facilitating release time, scheduling and assistance with administrative details.

m

ACKNOWLEDGEMENTS There are many people to thank for their different contributions to this endeavor. First and foremost, my family has supported me through the entire process. John Ross, who supervised the academic process, always seemed to provide the right advice at just the right time. Jamie Pyper assisted with the inter-rater reliability check and codebook validation. Karen Button assisted by circulating a request for participants quite widely. A variety of colleagues at the school where I teach also provided support in various ways that was much appreciated. I am indebted to the participants who allowed me to follow along in their reflective practice and observe their activities. Many times I drew inspirationfromtheir efforts and more than once I learned new thingsfromthem that I could use in my own classroom. I found the professionalism and efforts of the participants exciting and widely varying which highlighted the importance of their roles as teachers. To each participant I say thank you for letting me come into your school and watch what you do. Lastly, I would like to thank the other members of my thesis committee for taking the time to review and provide feedback on this thesis in its entirety. Their efforts have helped improve the clarity of the overall explanation and the conceptual components of the substantive theory that was developed.

IV

TABLE OF CONTENTS ABSTRACT ii LISTOFTABLES vi LIST OF FIGURES vii LIST OF APPENDICES viii Chapter One: INTRODUCTION 1 Definitions.......................—............... ..—.....—...... ...... . .3 Purpose.......................................................................................—....................—................... 5 Assumptions............................................................................................................................... 6 Limitations................................................................................................................................. 7 Research Questions................................................................................................................... 8 Organization of Chapters......................................................................................................... 8 Background of the Researcher................................................................................................ 9 Chapter Two: LITERATURE REVIEW 11 Classroom Practices................................................................................................................ 11 Influences................................................................................................................................. 14 Pedagogy.........................—...........................................................................—......... 18 Pedagogical Content Knowledge ...... .—.. ... . 19 Curriculum Orientation .... .......................... . ....... .......21 Self-Efficacy 23 Reflective Practices . . 28 Review of Specific Reflective Practices.............. ........ . .. 34 Chapter Three: METHODOLOGY 57 Data Collection and Analysis.. ..........—................ . ............... .— 59 Participants.............................................................................................................................. 65 The Reflective Groups............................................................................................................ 66 Ethical Considerations . .......... . . . .....68 Credibility and Reliability 69 Limitations.............—....... . .—... ...—. . .......... ..70 Generalizability.............. .............. . . ....... .......... 71 Chapter Four: RESULTS 73 Research Question 1............................................................................................................... 73 Research Question 2 . — . . .................. ......—.... ...... 121 Research Question 3.............................................................................................................132 Research Question 4 . . .. . ....143 Chapter Five: DISCUSSION 151 Theory Development............................................................................................................. 151 Research Questions......... .... .......................................................................................... 161 Reconciliation with Research Literature............................................................................ 164 Implications of the Theory................................................................................................... 184 Future Research.. . ......... . ... . . .187 REFERENCES 189

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LIST OF TABLES Table 2.1: Collegial influences

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Table 2.2: Description of observed reflective practices

35

Table 4.1: Equation solving in grades 6,7, and 8. Ontario curriculum expectations (2005)....79 Table 4.2: Summary of lesson study vignettes

88

Table 4.3: Summary of peer coaching vignettes

97

Table 4.4: Summary of pre-service mentoring vignettes

110

Table 4.5: Summary of isolated reflective practitioner vignettes

120

Table A. 1: Event counts

202

Table B.l: Codes, sub-codes and definitions

209

T a b l e d : Frequency of codes and sub-codes

213

VI

LIST OF FIGURES Figure 1.1: Existing partial framework

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Figure 2.1: Osterman and Kottkamp (2004)

framework

Figure 2.2: Lesson study cycle (reproducedfromYoshida, 1999)

31 37

Figure 4.1: Community of practice

154

Figure 4.2: General theory of math teaching

155

Figure 4.3: The substantive theory

156

VII

LIST OF APPENDICES Appendix A: Research Events

202

Appendix B: The Codebook

205

Appendix C: Frequency of Codes and Sub-Codes

213

Appendix D: The Miller-Seller Instrument

215

Appendix E: Interview Guide

222

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CHAPTER ONE INTRODUCTION Reflective practice is a teacher based activity that facilitates adapting one's teaching skills to particular situations. It is a mechanismformodifying classroom methods and testing new methods. Innovation has also been associated with teacher self-efficacy when it is interpreted as the belief in one's ability to produce an intended result in the classroom. It is the interconnection between reflective practice and teacher self-efficacy that this study explores. The particular context of this research is pedagogical changes by mathematics teachers. When a lesson is planned, the consequence of the planning is moderated by teacher selfefficacy and the act of teaching has two consequences. Firstly, it leads to learning by students and therein contributes to student achievement. Secondly, the teacher assesses the results of the lesson and uses the results to gauge the needs for subsequent lessons. For example, if a lesson was deemed to be significantly challenging, as judged by student engagement, then a larger proportion of the next class might be devoted to review and explanation of homework. There are many formalized techniques for reflection, such as action research, peer coaching or study groups, however, their effect on self-efficacy and resulting effects on classroom learning are not fully understood. The goal of this study is to build upon a partialframework,shown in Figure 1.1 and supported in the research literature (see chapter two), to construct a framework that includes the mechanism by which reflective practice, through some process, affects selfefficacy and thereby improves student learning in the classroom. No specific reflective technique was chosen because the literature regarding reflection suggests that many properties extend across many types of reflection. In view of this, having a variety of reflective techniques facilitates some comparison of the different methods. 1

Figure 1.1 Existing partial framework.

Note that this study does not examine reflective practice that is imposed on teachers. In such situations it would be necessary to consider organizational factors that are beyond the scope of this research. It would also require a distinction between self-efficacy and collective-efficacy which would increase the complexity of the analysis considerably. Regardless of the degree of individual reflection, the outcomes are thoughts and hypotheses about teaching in the context of specific lessons. These ideas may be dissected along two of the dimensions identified by Shulman (1986); content knowledge and pedagogical content knowledge. Content knowledge pertains to the complete details of the subject matter being taught which includes a full range of applications and an understanding of how the content fits into the larger body of subject knowledge. Pedagogical content knowledge acknowledges that some content knowledge can be readily adapted to the task of teaching while some cannot. Pedagogical content knowledge is the combination of specific content and the method that makes it particularly suitable to teaching or conducting a lesson. It includes the choice of teaching strategy, the way in which the teacher engages the students and the social development of activities in the classroom.

A teacher's reflections about a lesson's pedagogical content knowledge leads to planning around notions that a teacher wishes to put into practice in subsequent lessons. However, the key element influencing the adoption of the idealized notion is the teacher's belief that they can enact the intentions into specific classroom practices. This raises the need for a theory explaining the link between planning based on reflections and teacher self-efficacy. Definitions This section provides brief definitions of key concepts and contributing factors that were expected to emerge, in one form or another, from the grounded theory methodology of this research. This list was not considered to be exclusive and the possibility of other concepts or factors arising in the course of the study was recognized. These definitions are examined, within the context of the existing research literature, in chapter two. Classroom Practices These are defined as interactions between students and the teacher that pertain to the subject topic being taught (Cobb, Stephan, McClain, & Gravemeijer, 2001). Typically, these take place through the enactment of a lesson plan in a regularly scheduled class. However, the definition is extended to also include events outside of the regular class such as instances of students seeking extra help. Reflective Practices This is a process by which a teacher recalls and thinks about classroom events in the context of planning and pedagogical choices (SchSn, 1982; Stenhouse, 1983). Frequently, it is based on a perceived concern or challenge derived from enacting a lesson plan and, through a multi-step process, results in modifications of the teacher's practices. For example, suppose a teacher has students making measurements as part of a lesson and is surprised by the number of

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students who are confused when they are attempting to interpret their measurements in groups. The teacher consults a colleague who suggests the problem may be due to a mix of metric and imperial rulers which leads to confusion when students compare results. The teacherfindsthis to be the case and then considers two options; only using one kind of ruler or having students determine and discuss the units of measurement that they used. The teacher's choice is then used to plan the next lesson. Influences For this study, an influence is defined as a professional colleague, member of the community, or other source of information that contributes to the reflective process of a teacher (Ross, 1993). This component addresses the fact that teachers do not work in isolation. While influences are likely to be colleagues who may suggest alternative interpretations or methods in response to a teacher articulating a particular concern, they could also be consultations with resources. It is assumed in this study that the teacher is under no obligation to follow any advice that is provided. In this respect, organizational considerations are left for future research. Pedagogy The methods and professional judgments one intends to use in the classroom are considered to be the pedagogy (Tubbs, 1996). That is, the term is used to refer to the intention of a planned lesson. This does not necessarily transpire as planned which makes the distinction with the enacted lesson. As an example, a teacher may plan to have students complete an activity using their textbooks and this may differ from the enacted lesson which has to deal with students who neglect to bring their textbooks.

5 Pedagogical Content Knowledge This is a conceptual representation of teaching options that includes knowledge associated with a specific technique and an understanding of how that technique may fit with other aspects of the subject (Borko & Putnam, 1995). This includes understanding where students might encounter difficulty and particular views regarding good examples for demonstration or a comparative view of different methods of conveying the same content. For example, the line of best fit can be efficiently taught by having students represent points and use a rope as a line to demonstrate that the best fit allows most students to touch the rope simultaneously. This is a tactile approach that differs from chalkboard explanations with exemplars. Curriculum Orientation This refers to the way in which one generally envisions the curriculum being enacted in the classroom. It is interpreted in this work, using Miller and Seller (1990), as being linked to fundamental philosophical views regarding the most effective teaching practices. A teacher who typically emphasizes the articulation of content differs from a teacher who regularly has students in small groups develop ideas through experiments and then convey them to the class through short presentations. Self-Efficacy Self-efficacy is a belief in one's ability to achieve an intended result (Ross, 1998). It is used in this study, within the context of teachers' belief in their abilities to enact new techniques or revamp old techniques to new teaching situations in the classroom. Purpose The purpose of this study was to investigate the way in which reflective practice affected teacher self-efficacy with respect to particular aspects of the classroom. Through the process of

6 reflection, the participants attempted to make or discuss specific changes in practices and those efforts were interpreted through the lens of self-efficacy. While other lenses are plausible, selfefficacy was used because it has been found to have an impact on student achievement. The methodology was multiple case studies (Merriam, 1998) and grounded theory within the theoretical context of Figure 1.1. Through constant comparison, a substantive theory providing a connection between reflective practice and self-efficacy was developed. This was achieved through an examination of four different groups of reflective practitioners who were following different techniques toward different objectives. In terms of current research this study contributes to: •

a better understanding of the reflective practice mechanism as it pertains to self-efficacy,

•

an understanding of what conditions can foster reflective practice within the context of school based staff development,

•

a framework that can potentially be extended to include organizational and collective contributions of teachers, and

•

an improved understanding of the process that teachers experience when they try to substantiate changes in practices with only peer support. Assumptions This study assumed that there was a common substantive theory linking a variety of

reflective practices to self-efficacy. It was assumed that the theory would not be overly dependent on the type of reflective practice. This assumption was supported by Schon (1987) who reported that reflection is supported in a very broad context. Within the reflective practice process, the details of 'reflection in action' (Schon, 1987) were not directly considered. The elements of the classroom that the teacher recalls and

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comparison of expectations leading into a class with outcomes during the class were considered to be the key factors driving the changes in practice that affected the teacher's self-efficacy. The methodology was based on an in depth examination of four case studies and it was assumed that the observation mechanism would not unduly interfere in the reflective practice process. It was also assumed that the teachers who participated were acting in good faith and that any objectives they may have had beyond classroom oriented goals were immaterial to this study. Limitations This research pertains to a mechanism linking reflective practice and self efficacy with minimal dependence on organizational factors or school culture. This was essentially the case; however, some leadership components were involved in a relatively minor way. It was anticipated that the linkage of reflective practice and self efficacy, in this fashion, will lead to future investigations of organizational forms of reflective practice and collective self-efficacy. The implications of teacher self-efficacy and classroom practices for student achievement are based on research evidence reported in the literature. While the evidence supports the idea that improved teacher self-efficacy implies improved student achievement, this study does not directly examine student achievement. Teachers who engage in reflective practice are likely to exhibit a bias in the type of reflective practice they choose to engage in. They are also likely to show a bias in teacher selfefficacy when compared to the overall community of teachers. This suggests that the results of this research are limited to certain causal antecedents that give rise to the willingness to engage in reflective practice in the first place. It is also possible that the researcher's questioning of participants may have caused reflections beyond what would have taken place otherwise.

8 Lastly, this study had teachers observe peers. The students had some familiarity with the observing teachers since they are seen in the school environment. There was concern that having the researcher participate in the observation phase might interfere with the reflective process because it would entail having an unknown person in the classroom. Therefore, the researcher limited his observations to two cases were the impact would be limited and this made it infeasible to examine reflection in action. Research Questions The research questions for this study were: 1) What changes do teachers consider during reflective practice? How do the discussions and efforts toward implementing changes affect each teacher's self-efficacy? 2) How do reflective practices and self-efficacy interact in the process of teacher innovation? 3) To what extent does a teacher's curriculum orientation affect the decisions they make regarding techniques arising from the reflection process? 4) To what extent does the level of formality in the reflective process influence the teacher's efficacy during an innovation? Organization of Chapters The remainder of this thesis is organized as follows: Chapter Two - The research literature is used to ground the overall research purpose. This ties the conceptual definitions to the existing research literature and clarifies the research questions. Chapter Three - The methodology is described and details of the cases that were examined are provided. Using the data collected, emergent themes are explained, a detailed codebook developed, and results of reliability and validity checks are provided.

9 Chapter Four - Results are provided within the context of the codebook using detailed vignettes from the interviews and observations to answer the research questions. This facilitates constant comparison between specific reflective practice groups that were observed. Chapter Five - The substantive theory is developed and results are interpreted within the context of the existing research literature. Practical implications for the use of reflective practice, staff development and future research are discussed. Background of the Researcher This research project had the hidden objective of being the first complete qualitative study that the researcher has done. He has been involved in collaborative qualitative research, particularly qualitative program assessments for the Ministry of Education. In addition, the researcher has been involved in considerable quantitative research which included an investigation regarding the retention of fraction knowledge through the first three years of secondary school. The subject area of the study, mathematics, provided a sound basis for exploring qualitative research as the researcher has been teaching mathematics in the senior division for the past eight years. The researcher is certified, with the Ontario College of Teachers, in the junior, intermediate and senior divisions as a mathematics specialist. During this period of time, the researcher completed a Masters of Arts degree using quantitative analysis to examine fraction performance of students during the first three years of secondary school. Prior to teaching mathematics, the researcher was involved in remote sensing research. This entailed the development of mathematical and statistical algorithms for identifying and modeling various natural and manmade structures that arise in radar imagery.

10 ' The participants in three of the four reflective groups in this project were from the same school board as the researcher but unknown to the researcher prior to the study. Two of the schools were half an hour drive from the researcher's school. One reflective group, in a mentoring situation, was from a private school and the mentor was known informally to the researcher through a course they had both participated in during magisterial studies at OISE. No formal or professional associations between the mentor and researcher exist. The researcher has been informally using reflective practice consistently during his years in teaching. He has been engaged formally in reflective practice as a mentor for pre-service teachers but not in any other variety of formal reflective practice. He has been aware of formal methods through professional development in the school board, where he teaches, and a graduate course regarding the theory of reflective practice. The researcher views reflective practice as a beneficial approach to professional development for groups of teachers who wish to work independently within a school, with minimal support, and in a manner focused on their own needs. He does not view reflective practice as a replacement for other methods of professional development and does not have a preference for any particular kind of reflective practice.

CHAPTER TWO LITERATURE REVIEW This review of the literature is composed of two parts. The first describes previous research on constructs that arefrequentlycited as being essential elements in reflective practice. The second part reviews research on four specific approaches to stimulating reflective practice that were observed. Following the two part review, the research questions are presented. The theory that this research develops builds on the work of Ross (1998) who stated that: Although reflections on the results of one's teaching are the most important contributors to teacher efficacy, other sources of information may be influential. These include observing peers, attempts by colleagues or supervisors to convince teachers they are able to accomplish particular teaching tasks, and physiological responses, (p. 50) While Ross suggests a linkage between reflective practice and efficacy, this study examined whether there are moderating factors, such as pedagogical content knowledge, limiting the scope of options that teachers can use. A reflective practitioner may see something desirable in a peer's classroom, but, without an understanding of the pedagogy that supports the technique, they may be unable to effectively adopt the technique. It is the theoretical linkage between reflective practice and self-efficacy, inclusive of the pedagogical and any other limitations, that this study will establish. A list of key constructs that are salient were defined briefly in chapter one. The definitions are expanded here, tied to existing research and known relationships to the other variables explored. Following the definitions, four specific types of reflective practice that the research participants were engaged in are reviewed. Classroom Practices Classroom practices are the educational interactions and consequential interactions between the teacher and the students in the classroom (Cobb et al., 2001). Typically, this corresponds to planned lessons in a regularly scheduled class. It also extends to instances where events occur 11

12 outside of the classroom but either reference classroom events or are referenced in the classroom at a later time. Of particular interest are events that pertain to changes in teaching practices arising from reflective practice. An example of an event outside regular classes includes a student who seeks assistance outside of regular class time ("I didn't understand the example we did in class today."). Another example is information that arises from third party sources, such as discussions with parents/guardians, guidance counselors or administrators, provided the information pertains to a regularly scheduled class ("John found the homework easy."). This definition of classroom practice distinguishes itfrompedagogy. In this study, pedagogy reflects the lesson plan and intended outcomes arising from a teacher's conception of what will happen in the classroom. Classroom practices, by contrast, are the enacted lesson inclusive of any unforeseen consequences of the planning. Cobb et al. (2001) describe classroom practices as: ".. .students' reasoning as acts of participation in communal practices that they and the teacher establish in the course of their ongoing interactions" (p. 158). This definition of classroom practices is quite broad because students who visit teachers outside of regular classes are frequently a source of reflection or, if they visit a department office, discussion among teachers. Furthermore, they often raise questions that provide direct feedback to the teacher about the strengths and weaknesses of a particular lesson. In these respects, students who have interactions, whether direct or indirect, with the teacher outside of regular classes may be important catalysts to the reflection process. The teaching process is intrinsically tied to the classroom environment where teachers demonstrate practical knowledge that is embedded in classroom methods of practice (CochranSmith & Lytle, 1999). Since the classroom environment is dynamic and fundamentally different

13 than most forms of professional development (Fennema et al., 1996), teachers naturally think about the process. In some cases, this entails using self-reflection to evaluate an individual's experiences (Pajares, 1996). When teaching, students generate cognitive dissonance (Slavin, 1988) within the teacher who has to address it in one manner or another. However, it is this dissonance which leads many teachers into a process of reflection. Those teachers who later reflect on the events may alter strategies or try new methods and, in both cases, they may consult with peers to reconcile the dissonance. The effect of this process may be thought of as a continuum of adaptation (Stodolsky & Grossman, 2000)fromminimal alterations, such as pace or coverage, through to reconceptualizing one's practice. This professional process will impact subsequent classroom practices and is distinctfromtraditional research because the teacher is an integral part of the context (Jacobson, 1998). Overall, the dynamic nature of the classroom environment gives rise to issues which may lead teachers to reflect on classroom experiences. However, teachers vary in the degree of adaptation that they are willing to make in the classroom (Stodolsky & Grossman, 2000) and if they are not inclined toward adaptation or innovation then there is little reason for them to engage in reflection (Warfield, Wood, & Lehman, 2005). At the same time, some teachers engage in reflection permitting different degrees of adaptation, in some cases this process amounts to little more than consulting peers. Perhaps the extent of the reflective process corresponds to the level of development of each teacher's personal theories of teaching. It was the engagement in the adaptation process, specifically through reflective practice, that was the focus of this study because this type of reflection, or practitioner inquiry, potentially facilitates significant adaptations within the classroom (Cochran-Smith & Lytle, 1999; Rosenblatt, 2004).

14 Influences For this study, an influence is defined as an educator, member of the community, or other source of information that contributes to the reflective process of another teacher. This has been described as professional sharing; "the reciprocal exchange of reports of professional experience by practitioners." (Ross, 1993, p. 93) It has to be understood to be inclusive of any cognitive conflict, emotional support, and collaboration that may be present. Colleagues were the primary influence considered in this study because they share the contextual setting of the teacher and often have interactions that are synchronous with day to day events. Other influences, such as professional development, may be out of context, asynchronous with classroom needs or derivedfromother schools that may vary quite markedly. The role of the contextual setting is clearfromobservations that most exchanges (75%) arise between teachers of the same grade level, with proximity of classrooms (15%) and immediate availability (13%) being of lesser importance (Zahorik, 1987). Zahorik also observed that exchanges between teachers have no particular time or location and occur throughout the teacher's day and travels through their school routines. In summary, "The typical collegial encounter involves teachers of the same grade level meeting briefly at a variety of places in the school to discuss materials, discipline, activities, or individualization. [(Defined as meeting needs of a specific student)]" (Zahorik, 1987, p. 392). Bainer and Didham (1994) also found evidence suggesting that".. .support behaviors are distributed across a range of individuals or relationships, as opposed to being centralized in one mentor..." (p. 245). Lee, Bryk, and Smith (1993) state that, even when formal support mechanisms are put in place, informal social networks contribute "professional assistance and personal support" (p. 222).

15 However, when considering colleagues, a distinction is made between collegial and collaborative relationships (Kruse, 1999). A colleague may be collegial in the sense of a person to talk with over coffee or collaborative as someone who cooperates formally in a reflective process (and therefore shares in decision making). Examples of the latter are mentors or peer coaches. Influences derivedfromformal administrative positions are not considered in this study. Therefore, a teacher engaging in reflective practice because of a mandate from the school principal would not be included. This choice respects the suggestion of mandated reflection having limited success (Bainer & Didham, 1994) and circumvents the possibility of significant socio-political influences in the process of enacting changes in the classroom (Lee et al., 1993; Hargreaves & Dawe, 1990). It is believed that this study will facilitate the examination of departmental and systemic influences but these are beyond the scope of this work. A summary of literature findings related to the influence of colleagues is presented in Table 2.1. This list is not intended to be comprehensive but rather to give an indication of the variety and number of considerations that are necessary in considering the influences that teachers receive. De Lima (2001) considered the concept of "closeness" with respect to collegial relationships. He considered a typology of 13 types and concludes that: ..only four were used by at least about one third of the respondents: self-disclosure (referred by 70.9%), provision of help and support (37.2%), sharing of interests and characteristics (30.7%) and relational expression (explicit communication of feelings of closeness)(30.3%). (p. 104) In this study, only the first three seem pertinent to reflective practice. However, it should be noted that the impact of a teacher recounting an experience is likely to have more impact on the listeners than on the speaker and that having dialogue appears to further the impact on the

16 listeners (Ross & Regan, 1993). Therefore, the type of closeness may be an antecedent to the willingness to engage in dialogue rather than a direct cause for professional learning. Willingness to engage may depend on other attributes such as personality, experience and educational philosophy (Wallace, 1998). This is certainly an image consistent with brief interactions between teachers in hallways, breaks and during different duties. Table 2.1 Collegia! Influences Finding

Reference

Interactions with colleagues are positively correlated with teacher

Ashton & Webb, 1986;

attitudes and student performance.

Bainer & Didham, 1994

Collaboration is positively related to efficacy in terms of both the

Tschannen-Moran, Hoy, &

extent of opportunities and extent of observations.

Hoy, 1998; Bainer & Didham, 1994

Help seeking, instructional experiments and positive expectations

Ross, 1998; Smylie 1988

based on improved teaching effectiveness become more acceptable when offered by teachers considered to be colleagues. In the reflective process, collegial interactions provide support

Tschannen-Moran et al.,

and encouragement during setbacks or self-doubt.

1998; Hoy & Spero, 2005

Gains in efficacy were largest for teachers who discussed the

Ross, 1998

implementation of professional development and interpretations of outcomes with their peers. For new methods, the impact on efficacy is greater when they are acknowledged by peers and the principal.

17 Social Ties: Weak social ties facilitate stronger professional associations than strong social ties because they are more conducive to

De Lima, 2001; Zahorik,

independence of thought and actions. However, if the ties are too

1987

weak then trust can be an issue leading to superficial dialogues

Bullough & Stokes, 1994

that avoid risking honesty. Novice teachers are often supported by an informal mentor who

Rowley, 1999

provides "surrogate self-efficacy" by expressing confidence in the ability of the novice teacher to overcome challenges. "Skill flexibility" was better in larger schools, a finding that may

Rosenblatt, 2004

relate to the availability of choice in peers. Different teacher perspectives can be: A) a source of cognitive conflict

A) De Lima, 2001

B) a shared process of dialogue with no clear conflict

B) Hsu, 2004

Availability of time for meaningful reflection with colleagues is a

Manen, 1991; Zahorik,

challenge.

1987; Bainer & Didham, 1994; Wallace, 1998; Hargreaves & Dawe, 1990

Asynchronous network based communication is significantly

Volkman, Scheffler, &

more reflective than face-to-face discourse. Network messages

Dana, 1992; Hawkes, 2001

are frequently posted from home in the evening and these are typically the most reflective.

18 The availability of time is a significant consideration for this study (Manen, 1991; Zahorik, 1987; Bainer & Didham, 1994; Wallace, 1998). The research method itself must not interfere in the reflective process. For many teachers, their level of involvement in reflective practice is negotiated with consideration of time constraints. There are numerous influences that have an impact on the relationships between colleagues and their ability to succeed with reflective practice. The emphasis in this study was to balance the assessment of the role of colleagues with a methodology that did not impact that interaction significantly. Pedagogy Pedagogy, in this study, refers to planned methods of teaching with respect to a particular topic (for example, methods of teaching percent) (Tubbs, 1996). It is important to recognize that a distinction is drawn between pedagogy and the actual events that transpire in the classroom in the same way that a lesson plan and the enacted plan differ. The term pedagogy includes the relative merit of different methods and, in this sense, is teacher dependent. It may also include both theoretical images of teaching and also pragmatic images that are developed through experience. Traditional methods of teaching, for example, pose a pedagogical starting point for most teachers. Such methods are familiar through the experience of being a student and relatively straightforward in their implementation. In mathematics, the use of "telling" provides structure to the classroom and narrows the pedagogical choices to manageable proportions (Smith, 1996). Reflective practices provide a method for teachers to gain insights into teaching practices and, particularly where colleagues or professional resources are involved, provide an opportunity to learn new pedagogical methods. However, the process of addressing new methods or altering

19 one's practice is constrained by one's ability to make the changes. The most effective teaching method, or a method agreed upon by a group of reflective practitioners, is not necessarily one that each individual teacher is able to use properly. Thus each teacher will find self-efficacy modified according to prior experience and level of philosophical agreement with the approach chosen by the group of reflective practitioners. An individual may, for example, be constrained by his professional knowledge such as a teacher who wishes to use cooperative learning (Slavin, 1995) but has not realized that it goes beyond simply using groups, or by his own beliefs about which methods are and are not appropriate (cooperative learning is not consistent with transmissive teaching). The latter can often be recognized among the most senior teachers who may discount professional development on the grounds that it failed to produce the desired results last time it was promoted. Efficacy is sometimes greater when traditional methods are used (Wheatley, 2005). However, emphasizing traditional methods is likely to reduce efficacy on the long term, because teacher effectiveness will be static or in slow decline. Thus teachers who have a strong sense of efficacy likely include those who are willing to try innovations (Tschannen-Moran et al., 1998). Certainly, having more than traditional techniques available facilitates choices that can benefit student learning and teacher efficacy (Ross, 1998). Pedagogical Content Knowledge Given that pedagogy is being defined in a manner that is inclusive of theoretical and practical images of teaching, it is important to recognize that particular methods may or may not be seen as specific instances of some technique with a broader scope (for example, percent as a fraction, percent as a decimal, etc.). The extent to which teachers can make choices is constrained by the extent of their knowledge of the options that could be used. Pedagogical

20

content knowledge (Borko & Putnam, 1995), which represents teaching options conceptually, is the knowledge associated with a specific technique and an understanding of how that technique may fit with other aspects of the subject. For example, when slope is defined as afraction,some teachers recognize the specific technique allowing them to use equivalentfractionsto highlight different but equivalent graphical features. Collaborative reflective practice provides an opportunity for teachers to learn new pedagogical content knowledge with their peers. In math education, the combination of reflection and communication about student thinking relates to teachers learning to teach differently (Warfield et al., 2005). For teachers with high efficacy, the selection of techniques that improve student development is likely to be considered more important than covering the curriculum (Ross, 1998). However, it is necessary that a technique that is developed through reflective practice is used regularly. Not necessarilyfrequently,but it is important for the technique to be exercised if the efficacy increase due to the technique's availability is going to be maintained (Ross, 1998). This implies that the extent of techniques developed through reflective practice and the duration for which they are maintained are important considerations. For mathematics, student achievement is greater when professional development is focused on teaching specific mathematics content and not significantly changed when the focus is general pedagogy (Garet, Porter, Desimone, Briman, & Yoon, 2001). Given the high level of context dependence of reflective methods, the mathematics content focus was met, but the level of math content specificity of reflective practice varied. Kagan (1992) characterizes teachers as "... a unique ecological system of pedagogical beliefs and practices that is inextricably connected to the teacher's personality and prior experiences in life" (p. 159). One approach to addressing this connection was to consider that

21 reflective methods would be successful if the zone of promoted action lay within the individual's ability to extend his zone of proximal development (Blanton, Westbrook, & Carter, 2005). However, exploration of personality zones and prior experiences were not tangible in this study. Rather, it was believed that collegial reconciliation of personality and experience factors would naturally occur during the reflective process. In this way, personality becomes primarily an issue of appropriateness of colleagues and the pedagogical choices that are made. Within this study, the pedagogical focus was mathematics and it was the extent to which teachers believed they had pedagogical options that was pertinent. It was the extent to which pedagogical content knowledge was developed through reflective methods that was relevant. Curriculum Orientation Curriculum orientations refer to beliefs, supported by a philosophical position, regarding the purpose and methods of education (Miller & Seller, 1990). The principles that an orientation are dependent upon are philosophical in the sense that they are not overly dependent on the subject being taught. However, when applied, the curriculum orientation is an important linkage between subject matter knowledge and pedagogical content knowledge that affects student achievement (Kinach, 2002). Within this study, curriculum orientation was separatedfrompedagogy because orientation and efficacy do not have a direct link (Ashton & Webb, 1986). In addition, changes arising from reflective events were expected to induce changes in pedagogy that do not necessarily create changes in one's curriculum orientation (Pajares, 1992). This view is supported in a study of preservice teachers that found subtle changes in personal teaching metaphors that indicated "changes of degree, not kind" (Bullough & Stokes, 1994, p. 210).

22

The interdependence of curriculum orientation and pedagogy suggests that methods that teachers adopt through reflective practice are either chosen because they are consistent with their curriculum orientation or interpreted in a manner that imposes consistency with the curriculum orientation. In either case, the teacher's efficacy should mirror the level of consistency. While several frameworks for curriculum orientation are available (Eisner & Vallance, 1973; Fung & Chow, 2002; Miller & Seller, 1990; Reynolds, Sinatra, & Jetton, 1996) the relative merits of each framework rests with its ability to be melded with the mathematics context and pedagogical changes derivedfromreflective practice. Fung and Chow (2002) included classroom practice categories of 'nurturing', 'social reformist' and 'apprenticeship' which do not match mathematical teaching practices particularly well. Reynolds et al. examine different methods of knowledge acquisition and representation that may reflect different emphases for teaching, such as social perspectives or situated cognition, rather than an encapsulation of the teacher's philosophical approach. Eisner and Vallance (1973) use five conceptions of curriculum which have the same limitation as Reynolds et al.. Miller and Seller (1990) defined transmission, transaction and transformation orientations. The transmission orientation focuses on behavioral approaches and views knowledge as being transmittedfromthe teacher to the student. The transformation orientation focuses on the needs of the whole child and is self-actualizing. The transaction orientation, which is developmental, focuses on cognitive processes and views teaching as a bidirectional transaction between the teacher and student. The transmission and transaction orientations are closely aligned with traditional "chalk and talk" methods and constructivist approaches that the Ontario curriculum endorses and the NCTM has promoted (NCTM, 2000). In view of this alignment, the Miller and Seller framework is used in this study.

23

Self-Efficacy Self-efficacy is a belief in one's ability to achieve an intended result (Ross, 1998). While this is influenced by the level of innovation or change the teacher brings to the classroom, the "belief used in this definition encompasses more than just an attitude. This concept provides a bridge between the intention of a plan and the reality of classroom practices. Furthermore, classroom practices varyfromday to day according to topic and the type of class. In response to the variability, teachers need innovative adjustments to their practices that suit the context which they are teaching in. This study focused on reflective methods of adjustment and innovation within the teacher's normal context. Pajares (1996) articulates Bandura's (1986) conception of self-efficacy as the way in which our ".. .self-referent thought mediates between knowledge and action..." (p. 543). This is a useful starting point because it highlights the bridge that self-efficacy forms between pedagogy, the knowledge of what one wants to do in the classroom, and the action that one takes in the classroom. However, the definition is not suggestive of the types of observations one might make to assess the role self-efficacy may be playing within a given classroom. Ross (1998) defines self-efficacy as a ",.. teachers' belief that their efforts, individually or collectively, will bring about student learning" (p. 50). Goddard, Hoy, and Hoy (2004) and Hoy and Spero (2005) echo this definition with the emphasis that the beliefs do not have to be accurate assessments. The focus of this work only included collective activities to the extent that they occurred within the reflective process, were not imposed and were relevant to the teacher's classroom activities at the time. Ashton and Webb (1986) distinguish two types of teacher efficacy: "sense of teaching efficacy" and "sense of personal teaching efficacy". The latter pertains to a personal view of

24

one's own competence and influences the pedagogical choices that are made rather than the efficacy that directly impacts students. A moderate correlation was found between these two aspects of efficacy. While both are expected to surface in the current research, the former was expected to be somewhat peripheral because it relates to how teachers, collectively, can overcome disadvantages that students bring to school, whereas the latter focuses on the impact of a teacher on classroom practices. It was expected that "personal teaching efficacy" would play a role in the decision to instigate reflective practice (which may well bolster it) and on pedagogical choices making it the more relevant type of teacher efficacy for the current study. A pragmatic explanation of self-efficacy is provided by Tschannen-Moran et al. (1998): ...teacher efficacy as a type of self-efficacy - a cognitive process in which people construct beliefs about their capacity to perform at a given level of attainment. These beliefs influence how much effort people put forth, how long they will persist in the face of obstacles, how resilient they are in dealing with failures, and how much stress or depression they experience in coping with demanding situations, (p. 203) They also found efficacy to be linked to professional commitment. Efficacy has also been linked to innovation including a desire to find better ways of teaching (Allinder, 1994; Guskey, 1988). This leads teachers to construct informal theories that model their classroom context (Kagan, 1992; Smylie, 1988). However, these theories vary in complexityfromthe simplest codification of experience to dynamic reflective practice methods. Subject content is an influential component of efficacy and ultimately the student achievement derived from it (Ashton & Webb, 1986; Ross, 1998). Additionally, Ross (1998) found that efficacy was higher in classes with ".. .high ability and orderly pupils, and when teacher workloads are moderate" (p. 57). These components represented minor variables in the present study.

25 Hoy and Spero (2005) found increases in self-efficacy during student teaching but selfefficacy declined during the first year of teaching. They point to the student teacher phase and induction year as potentially the period when self-efficacy is the most malleable. In view of this, 'years of experience' was considered as a potential explanatory consideration, though the relevance was expected to wane with years of experience. The expected decline in relevance was based on the hypothesis that new teachers are not as likely to maintain a sustained focus on any one aspect of teaching because, of necessity, they need to switchfrequentlyto different foci. This would lead to varying self-efficacy that is indicative of the foci changes rather than reflection on a single aspect of teaching. This view is supported by Korthagen and Wubbels' (1995)findingof no link between reflectivity and innovation in pre-service teachers, even though it has been established for regular teachers (Hoy & Spero, 2005; Stein & Wang, 1988). Wheatley (2005) considered the possibility of directly influencing self-efficacy in teacher education and concludes that "...interpretive methods should be the central focus of research into teachers' efficacy beliefs." This is consistent with the approach taken here where passive observation of self-efficacy was expected to reveal natural efficacy changes but in a manner that facilitates interpretation within the context of the teacher's focus of reflection on a particular element of teaching. In this study, reflection was not expected to arise as a direct result of self-efficacy but rather as a result of classroom processes. In fact, Hawkes (2001) found no significant relationship between self-efficacy and self reported levels of reflectiveness. In the context of the classroom, self-efficacy has implications for the conception of classroom actions and these do not necessarily make one a reflective practitioner. This is highlighted by Smith (1996) who called for research into how teacher efficacy changes as a result of curriculum reforms. However, in the

26 case of the reflective practitioner, it is conceivable that classroom activity is mediated by selfefficacy that arises indirectly from the reflection, a notion supported by Guskey (1988) who concluded that change in belief (i.e. efficacy as a proxy) preceded change in behavior (Pajares, 1992). The purpose of this research is to look at the interconnectedness of reflection and selfefficacy when both are present. Self-efficacy is a "significant predictor of productive teaching practices" (Goddard et al., 2004, p. 4) and is related to classroom innovation (Hoy & Spero, 2005; Stein & Wang, 1988). This does not appear to be consistent with Hawkes' (2001) finding of the lack of a relationship between self-efficacy and reflectiveness. It is possible that innovation arises through mechanisms other than reflective practice, but this does not account for the apparent disconnect from reflective practice. Perhaps the relationship is mediated by classroom events, constrained by pedagogical content knowledge and changes in accordance with a teacher's philosophical perspective. Self-efficacy has an effect on classroom practices that improves student achievement (Goddard et al., 2004; Hoy & Spero, 2005; Pajares, 1996; Ross, 1998). This has also been noted by Ashton & Webb (1986) who state that: Our findings strongly support the hypothesis that teachers' sense of efficacy is related to student achievement. Furthermore, the results support the assumption that teachers' efficacy attitudes are situation-specific. Students' mathematics achievement was significantly related to teachers' beliefs in the efficacy of teaching, (p. 138) The connection to student achievement was not a focus of this study, however, the existence of a link makes the research compelling. The situation-specific nature of a teacher's efficacy is consistent with reflective practice being applied. Reflective practice necessarily starts with a situation in mind that is examined in detail and the self-efficacy of that situation is a plausible motivator of the teacher's reflections.

27

Several mechanisms causing improved student achievement have been suggested. Teachers may: • • •

Expect more of students (Tournaki & Podell, 2005), Focus on more student characteristics as a means to improvement (Tournaki & Podell, 2005), Persist in the face of setbacks (Hoy & Spero, 2005; Pajares, 1996),

•

Respond to classroom events with reflective practice (Goddard et al., 2004).

With respect to the last point, Hoy and Spero (2005, p. 354) ask "Does mentoring provide the kind of support that protects and builds efficacy?" One of the reflective practice groups, followed in this research, was engaged in mentoring. Hoy and Spero (2005) provide some circumstantial support for the connections between self-efficacy, classroom practices and reflection. They point to increased efficacy if classroom success is attributed to controllable causes, a judgment that could arise from reflective practices. They also point out that a student has increased efficacy if they identify with the teaching method and this may also arise when a teacher is using reflective practices. In addition, a "negative classroom climate" is detrimental to learning (Ashton & Webb, 1986) and such a "climate", whether negative or positive, is something of a natural bell-weather for teachers who engage in reflection. Finally, Bandura (1978; see also Goddard et al., 2004) postulated four kinds of experiences that contribute to efficacy beliefs that are useful in articulating elements of reflective practice: Mastery experience, vicarious experience, social persuasion and affective state. Mastery experience resultsfroma successful effort and reinforces the expectation for future successes. Vicarious experience arises when a skill is modeled by a colleague and varies in its impact on efficacy according to the success of the modeling. Social persuasion is feedback and/or encouragement related to a specific effort. Finally, affective state is an alteration of self-efficacy

arisingfromexcitement or anxiety. Each of the four kinds of experiences arose within this study, however, the impact of each of the experiences likely alter efficacy in a temporary fashion because of the need for reinforcement of learned practices. It is necessary to establish if the four experiences link to different aspects of the reflective practice and to explore changes in efficacy within the context of the experiences that generated the current state. Teacher self-efficacy, that is indicative of innovation, teacher effectiveness and improved student achievement, is consistent with many features of reflective practice. In spite of this, a connection between the two has not been explicitly researched. Reflective Practices Reflection has been defined in a variety of ways and the meaning used here follows the Stenhouse/SchOn tradition (Sch6n, 1982; Stenhouse, 1983). Thus reflection is interpreted as a practical means of addressing an issue, whether real or perceived, arising from the context of one's actions. It is a process which spirals through "...stages of appreciation, action, and reappreciation" (Schdn, 1982, p. 132). The process ultimately converges on an appreciation that is sufficient to reconcile the cognitive dissonance (Slavin, 1988) that caused the teacher to reflect on his actions in the first place. The definition of reflection contrasts with definitions used by other authors such as Little (2003) and Killion and Todnem (1991). Little (2003) proposes that teachers "Continuously monitor and adjust instruction based on student results" (p. 281). Realistically, teachers do not have formal results on a daily basis. Little also excludes the teacher appraising attributes of the teaching process that may not be explicit in student results whereas Stenhouse (1983) and Schdn (1982) clearly include self reflection as part of the process. Killion and Todnem (1991) state that reflection must be formal andrigorous,but the approach taken in this study is in keeping with

there being "... a number of more or less sophisticated routines for reflective practice deriving from SchSn's original formulation" (Golby & Appleby, 1995, p. 150). The definition used here also rejects the presumption that reflection is typically not focused on problem solving but rather on understanding pedagogical significance or meaning of events (Manen, 1991). Instead of this exclusive view, the approach used here follows Sch6n who "directed attention to the explicit 'refraining' of experiences by the practitioner in the action context. He suggested that new ways of thinking about puzzles and problems within the practice setting can generate new actions that can be tested in the practice setting" (Russell, 2004, p. 1193). The generation of 'new actions' constitutes problem solving (Korthagen and Wubbels, 1995). Since this study was focused on changes that arisefromthe classroom context, rather than imposed reforms or systemic changes, the notion of'critical reflection' (Farrell, 2004; Valli, 1997) that pertains to social-political perspectives is excluded. Firstly, there is evidence suggesting that it may be difficult to isolate personal philosophiesfromthose adoptedfromthird parties (Roberson, 2000), and secondly, in the context of the mathematics classroom, it is unclear how social-political perspectives could be theoretically linked in a meaningful fashion. In many respects, these features were indirectly included through the influence they have on a teacher's curriculum orientation and pedagogical choices. The focus of this study was in the way that teachers adapt teaching methods at the micro system level (Ashton & Webb, 1986). In this sense, the focus was very much on independent adaptation of teaching without external causes. The influences included in the study are part of the meso system but are utilized as a micro system component because the teacher was not

30

necessarily obliged to follow the advice or suggestions that emergefromany particular influence. The non-explicit socio-political and micro system perspectives constrained the scope of this study to reflective activities that the teacher, or a small group, chooses to pursue. Those activities that may be reflective but are imposed or 'required' of a teacher are not considered. This distinction acknowledges that, engaging in reflection due to directives from school leadership or as part of an organized professional development program, may incur socio-political factors of a collective (i.e. large group) or systemic nature that warrant study beyond the scope of the present study. It is believed that the results of this research will facilitate future studies of critical reflection along with collective (i.e. large group) and systemic programs that use reflection (such as whole school reflection). This study, while constrained in scope, did not dismiss leadership influences. They were acceptable as incidental contributions, such as periodic encouragement, and as contributions that any other colleague might make. The present study was constrained, however, to circumstances where the teacher was not being obligated by the leadership. York-Barr, Sommers, Ghere, and Montie (2001) describe a four step model of reflection. "Description" occurs in class, though it is usually recalledfrommemory and mulled later, "Analysis, interpretation" is the teacher's self assessment, "Overall meaning and application" is a process that also fits within the teacher's self assessment but usually will entail comparison with personal principles and with colleagues through dialogue, lastly "implications for action" is embedded in the pedagogy and self-efficacy components of the currentframework.While this is closely aligned with this research it is not designed to embody limitations that arise because of a

31

teacher's curriculum orientation or experience with particular pedagogical methods, both of which may affect the teacher's self-efficacy. Osterman and Kottkamp (2004) provide another useful idealized model of the reflective process (see Figure 2.1). It is idealized because it does not explicitly consider the practical limitations that may exist for each component. Their model defines reflective practice as an experiential learning cycle with four stages. The model is not entirely appropriate to this study because of the pedagogical focus of the active experimentation in the present research and the possibility that activity is moderated by the teacher's self-efficacy. The former distinction is significant because it separates content oriented changes and pedagogical changes under the assumption that a teacher's approach to experimentation will differ between the two. The moderation by self-efficacy is important because it reflects the teacher's conviction toward working with reconceptualized elements of his typical classroom practices. Problem Identification

Active Experimentation

Problematic Experience

Observation and Analysis

Abstract Reconceptualization Figure 2.1 Osterman and Kottkamp (2004) framework

Using the Stenhouse/Schon approach, it appears that an efficacious teacher would be aware of "exposure to discrepant data" (Timperley & Robinson, 2001, p. 283) in the classroom. If a

32

teacher is a reflective practitioner then this cognitive dissonance would lead to efforts to revise personal theories of the classroom. These efforts would often entail consultation with colleagues (Rowley, 1999; Timperley & Robinson, 2001). The resulting options would be constrained by the teacher's assessment of personal effectiveness in using various pedagogical methods and any constraints the classroom circumstances impose. The teacher's final plan of action would be mediated by self-efficacy as it is brought into the classroom and put into action. At this stage, one reflective practice cycle would be complete and the process might continue as professional judgment dictates. The place of reflection is to facilitate teachers to modify their own teaching practices in the context of their own classrooms (Hsu, 2004; Nolan & Hillkirk, 1991). In some cases, this may be supported through training or resources regarding protocols (Glaude, 2005) that provide guidance for observation and discussions. It allows them to learn about teaching. "Researchers in mathematics education have found that, when teachers focus their reflection and communication on the mathematical thinking of their students, they learn to teach differently" (Warfield et al., 2005, p. 442). For example, teachers have been observed to listen to student opinions about what teaching techniques are working for them and which are not (Montalbano, 2001). Reflection involves the interpretation of classroom actions which are a "powerful source of efficacy information" (Ross, 1998, p. 65) and teachers with higher levels of self-efficacy tend to "set higher goals and persist through obstacles" (p. 57). This supports the idea that reflection and self-efficacy positively reinforce one another. Ross found that peer coaching, as one type of reflective practice, positively affected teacher efficacy through sharing that confirmed beliefs (i.e social persuasion) and indirectly for coaches through participation in other teacher's successes

(i.e. vicarious experience). Montalbano (2001) observed action research giving teachers "...a sense of empowerment, as their own efficacy beliefs seemed to be strengthened" (p. 201). Reflection can be viewed as moderating the implementation of beliefs when they are put into practice (Warfield et al., 2005), however, the use of beliefs as a construct is awkward (Pajares, 1992) and seemingly unnecessary because of the pragmatic context in which this research is being conducted. This view is consistent with reflection being dependent on career stage where novice teachers focus on practical methods, mid-career teachers develop practical theories and leadership skills, and late career teachers preparing for their next life stage (WrightEvans, 1999). Communication contributes to reflection as a means to refraining a problem, a method of generating ideas to address the issue, or a way of identifying aspects of the problem that arise from tacit assumptions. While teachers have empirical knowledge that they readily call on, it often remains intuitive because of the lack of a conceptualization, or schema (Reynolds et al, 1996), that facilitates communication (Skemp, 1978). The lack, or relative simplicity, of teacher conceptualizations allows teachers to routinely change their practices (Richardson, 1994). However, when a conceptualization is developed through reflective dialogue with colleagues, the result is a pragmatic understanding in a broader, yet highly contextualized sense (Hsu, 2004). Further, the embedded position of the teacher within the process is contrary to traditional conceptions of validity (Huberman, 1996) limiting the applicability of the teacher's conceptualization. Huberman acknowledges the potential for reflective practice to generate grounded theoretical results and Russel (2004) suggests that the independent nature of this approach to practitioner inquiry is a significant step for the teaching profession.

While teachers change their practices routinely (Richardson, 1994), it is unclear if this is due to ambiguities of purpose and instabilities in the classroom context (Schdn, 1982). The difficulty is at least partly revealed in the reflection process if tacit knowledge is made explicit and found to be the basis for changes that have been adopted. For example, in the case of "overlearning", there are too many tacit understandings, leading to cognitive dissonance (Slavin, 1988), and explicit appraisal can correct the situation. In view of the tacit-explicit dichotomy of a teacher's knowledge, it is unlikely that reflection can be gauged beyond a categorical classification. For this reason, the research literature pertaining to each of four observed techniques is examined rather than trying to review the multifarious reflective practices that teachers have used. Review of Specific Reflective Practices Four specific groups of reflective practitioners were examined in this study. The details of the groups are discussed in the next chapter; the literature specific to each of the four observed reflective practices is reviewed in this section. The four types of practice were: lesson study, peer coaching, pre-service mentoring, and an isolated reflective practice. The research literature specific to each of these is discussed following a brief comparison of the four reflective practice methods. The four techniques that were observed have distinct features that are highlighted by van Lier's typology of research designs (Freeman, 1998). According to this typology the practices that were observed can be organized according to the degree of intervention (low/high) that is used and the degree of organization (low/high) within the research methodology. As can be seen in Table 2.2, all four quadrants of van Lier's typology arose in the reflective practices mat were observed.

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Table 2.2 Description of Observed Reflective Practices Reflective technique Lesson Study

Degree of

Explanation

Intervention

Organization

Low

High

Teachers as observers but with specific features to watch for.

Peer Coaching

High

High

Teachers co-plan and individually teach to small groups removedfromnormal classrooms.

Pre-service

High

Low

timing but non-specific observations.

Mentoring Isolated

Shared planning and synchronized

Low

Low

Changes based on personal observation

Reflective

and organization based on incremental

Practitioner

needs and assessments.

In the following four sections each of the specific approaches to reflective practice are reviewed in detail. Lesson Study Lesson study is a form of action research (Yoshida, 1999, p. 420; see also Taylor, Anderson, Meyer, Wagner, & West, 2005) that uses the process illustrated in Figure 2.2 (reproduced from Yoshida, 1999, p. 58). The group meeting establishes a focus for the lesson

study that is tied on curriculum objectives (not necessarily a content objective). Subsequently the focal objective is interpreted in concrete terms that can be translated into observable elements of the classroom. In addition, the group meeting collaboratively plans a lesson in a manner that addresses the focal objectives. This is a detailed lesson plan that includes a list of anticipated student responses (Fernandez, Cannon, & Chokshi, 2003; Meyer, 2005; Yoshida, 1999). A teacher implements the lesson while the other teachers observe the classroom. The observing teachers focus on concrete observations pertaining to the focal objective and keep notes for later discussion. The group meeting, which follows a protocol (Chokshi & Fernandez, 2004), provides a forum for discussion of the lesson that was observed. This discussion emphasizes the teaching and considers the concrete observations. Modifications to the lesson are planned and a second teacher teaches the modified lesson while being observed in the same manner as the original lesson. Subsequently, a second group meeting provides a forum to discuss changes that arose from the changes made to the lesson plan. Toward the end of the process, a written report summarizing the lesson plans, the observations and the teachers' reflections about the lesson study is produced. The lesson study process, as outlined, has been described by many authors (Blum, Yocum, & Trent 2005; Fernandez & Chokshi, 2002; Kelly, 2002; Rearden, Taylor, & Hopkins, 2005; Stigler & Hiebert, 1999; Taylor et al., 2005). Some authors split the three rectangular components in Figure 2.2 into two separate items and refer to the illustrated process as having eight steps. There are also differing articulations regarding "filing the records", for example, Blum et al. suggest an overall discussion rather than a written report. Variations of lesson study have been proposed and carry a degree of legitimacy because the method is one technique in a family. The family of techniques share classroom observation,

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collection of data and subsequent analysis (Lewis, Perry, & Murata, 2006). It has been suggested that the technique can be augmented with theoretical grounding in a learning theory and this can provide a lens through which to make classroom observations (Ling, Chik, & Pang, 2006). Another variation, "design study", also seeks a theoretical basis for changes (Pang, 2006; Pang & -A Typical Lesson Study CycleGroup Meetings: (Research A Preparation)

Grade Level Meetings and Alt Teacher Meetings

Group Meetings: (Reflection & Improvement)

Group Meetings: (Reflection & Filing the Records)

Figure 2.2 Lesson Study Cycle (reproduced from Yoshida, 1999) Marton, 2003) which may be more appropriate to deeper reforms that change the teacher's context or require a philosophical change in the teacher's thinking (Oshima et al., 2006). Wagner (2003) succinctly articulates the promise of lesson study: "[lesson study] develops pedagogical content knowledge, it is situated and authentic, it involves discourse communities, and it promotes sustained and generative growth" (p. i). However, the actual mechanism by

38 which lesson study causes change is not understood (Lewis et al., 2006). In particular, claims, such as Wagner's (2003) ignore the small sample sizes and the need for a causal link between specific changes and improvements in teaching. In spite of there being questions regarding the underlying theoretical basis of lesson study many observers (including two of the authors articulating the need for a theory of causation) find tangible changes arisingfromlesson study: .. .seven key pathways to improvement that underlie successful lesson study: increased knowledge of subject matter, increased knowledge of instruction, increased ability to observe students, stronger collegial networks, stronger connection of daily practice to long-term goals, stronger motivation and sense of efficacy, and improved quality of available lesson plans. (Lewis, Perry, & Hurd, 2004, p. 18) With many attributes of teaching potentially benefiting from lesson study, the benefits are considered individually and it is anticipated that groups using lesson study will not observe all the benefits but are likely to see some of the benefits detailed below. Lesson study occurs within the same physical space as the teacher teaches (Yoshida, 1999). The discussion after each lesson can be quite specific to the circumstances of the school and the teachers who are participating in the lesson study. In this respect, teachers are motivated by the way their discussions can immediately be used to adapt changes in classroom practices (Hartman, 2004), a phenomenon Shulman (2003) calls "reflective immediacy". Within the teacher's context, lesson study also provides an opportunity to improve vertical alignment through collaboration (Meyer, 2005). Since there is a sharing of content knowledge between the collaborating teachers, there will likely be a pragmatic appreciation for the content foci of other teachers and some recognition of the relative strengths and weaknesses of students coming from other classes. While lesson study should slowly alleviate issues of discrepant

content knowledge, it may also provide the facility to recognize the content background of one s current students. Yoshida (1999) found that teachers reflected on teaching materials, the process of learning by children, and the method of developing a lesson plan. He provides many examples of discussions between teachers that focus on the different types of solutions that students generate for math questions. The acceptance of different solutions in a positive manner that integrates them into the instruction contributes to the observation that lesson study increases the level of student centered learning (Kinzer, 2005). In addition, Carter (2004) found that early-service teachers' lesson study improved content ownership on the part of students. During the lesson planning process, teachers make a list of anticipated student responses, which, through collaborative discussion, expands the teachers' pedagogical content knowledge (Meyer, 2005). In addition, the observation and subsequent discussion facilitate student-centered changes in pedagogical content knowledge through apprenticeship (Oshima et al., 2006). A key aspect of lesson study is the communication that takes place between teachers. Teachers trying to implement lesson study have experienced difficulties developing the professional dialogue necessary (Hartman, 2004; Puchner & Taylor, 2006; Taylor et al., 2005). Once lesson study is in use a common technical language is used for discussion (Rock & Wilson, 2005; Yoshida, 1999) and it has been suggested that research on the effect of this language development would be useful (Fernandez, 2005). Teachers develop improved collegiality (Carter, 2004; Taylor et al., 2005), confidence (Rock & Wilson, 2005; Wilms, 2003) as well as efficacy (Hurd & Licciardo-Musso, 2005) through their participation in lesson study. These factors are correlated and a predictor of student achievement (Puchner & Taylor, 2006). Some authors have noted a significant increase in

student engagement (Puchner & Taylor, 2006) including student collaboration (Meyer, 2005). A note of caution, however, because teachers who are more collegial tend to be more successful with lesson study (Hartman, 2004) there could be antecedent conditions for lesson study success. For teachers, the planning required for lesson study requires substantial development and has been observed to increase the sophistication of teacher's planning (Stewart & Brednefur, 2005). Planning, reflected explicitly in the lesson plan, should include the choice of problems, manipulatives, wordings, the expected learning process, a method for gauging learning and ties to the curriculum (Takahashi & Yoshida, 2004). Manipulatives include aspects of the classroom, such as the explicit use of the chalkboard (Yoshida, 1999). Leadership should be an active element of lesson study (Friedman, 2005; McLaughlin & Mitra, 2001). Yoshida (1999) attributes part of the success of lesson study to the coherence of the professional development plan at the school level. There may also be facilitations that leadership can contribute through scheduling whether to have common preparation time combined with release time for observation, or if teachers choose to plan after school, a schedule where teachers have different preparation times to facilitate observations (this approach is sustainable by motivated teachers in the absence of available release time). However, Kelly (2002) notes that, where lesson study has been voluntary, it has been viewed as a cornerstone of practice rather than, when mandated, as another tool in the box. In the absence of a causal theoretical explanation, it appears that lesson study owes its success to many features. It may be that theflexibility,with which lesson study addresses so many different elements of teaching, is the reason that it is important. For teachers, the authentic contextual nature of this professional development surpasses the need for theory. The technique addresses personal pedagogical and pedagogical content knowledge needs within the context of

alignment of practice within the lesson study group. In addition, teachers observe changes in their classrooms which they consider beneficial. Collaboration, particularly the development of professional communication, deepens existing collegial relationships and shares expertise. The skill of planning lessons is improved and a collaborative leadership has an opportunity to increase the professionalism of their staff. Lesson study is in its infancy within North America and has many unresolved issues and pitfalls that do not yet have established solutions. The issues fall into three categories: implementation, research and extensions. Implementation issues are based on the details of the technique by which lesson study is performed. Research issues pertain to the details that may affect or be observable in this research project. Finally, issues of extension are based on the larger image of lesson study within the context of the teacher-as-researcher. The most commonly mentioned issue for implementing lesson study is time (Kinzer, 2005; Hartman, 2004; Yoshida, 1999). This includes time for shared planning of a lesson as well as time for classroom observation and post-observation discussion. When the observers of a lesson highlight a student's misconception, it needs to be understood as a point to build on rather than something to avoid (Wagner, 2003). In some instances, observations of student thinking lead to improved teacher pedagogical content knowledge only in the specific activity because of a lack of general pedagogical content knowledge for the topic (Fernandez, 2005; Meyer, 2005). This highlights the concern that lesson study, among a group of teachers who all have weak content knowledge, may lead to the transfer of erroneous understandings (Fernandez, 2002). Concerns about loss of individuality through collective planning were unfounded (Blum et al, 2005; Hartman, 2004). Teachers need to appreciate that the lesson study process will provide

them more options in teaching, not decisions (Watanabe, 2002). A related issue is teacher apprehension about teaching infrontof peers (Puchner & Taylor, 2006). However, all of these concerns constitute an issue of trust that, in a supportive environment, will be resolved with increased collegiality and efficacy. In some instances, teachers have had difficulty appreciating all of the components of lesson study (Pennisi, 2004). Using charts during the implementation may help avoid this difficulty (Wiburg & Brown, 2007). Lesson study is not systematic research (Pang, 2006; Pang & Marton, 2003) and there may be a division of opinions between practitioners and academics regarding the need for improving the analysis component. Yoshida (1999) expresses an interest in having an expert included in lesson study to conduct a proper scientific analysis and this view is supported by Friedman (2005) who does not think that teachers are adequately prepared for the research perspective of lesson study. However, other authors suggest that experts can only make a difference while present (McLaughlin & Mitra, 2001) which implies that sustained lesson study is not practical if it requires an expert. In addition, there are questions about the feasibility of developing scientific tools for expert advising of lesson study because of the sheer number of variables in the classroom that cannot be controlled (Ling et al., 2006). The lesson study environment needs to be stable and unchanging to facilitate changes in teaching techniques. This limits the scope of lesson study to reproducing good practices rather than developing new approaches (Oshima et al., 2006). There is also a concern that teachers' choices for topics in lesson study may show preference to newer ideas rather than revisiting older methods (Lewis et al., 2006).

Perhaps the most basic research result that has been called for is evidence that lesson study improves student achievement (Friedman, 2005; Meyer, 2005). Additional research to make the evidence of improved teacher instruction more extensive is desirable (Meyer, 2005). Both of these issues would be addressed if the mechanism underlying lesson study were understood (Lewis et al., 2006). Some authors have proposed inclusion of theoreticians in lesson studies as an approach to making deeper reforms (Oshima et al., 2006; Pang, 2006; Pang & Marton, 2003). However, having several external advisors increases the potential that the process may not be perceived as teacher led (Fernandez, 2002). There may also be difficulties for the theoreticians to appreciate the practitioners' perspective (Lewis et al., 2006). Lesson study appears to be a specialized form of action research that has potential benefits associated with reflective practice. However, the relative novelty of the method suggests that there are different conceptions of the process and the relative importance of different aspects may vary. Within the context of this research project, several components to be monitored have been identified. What evidence is there of a supportive administrative environment? Are there any contextual changes during the period of lesson study? What interpretation of lesson study are the participating teachers using and how well do they achieve each of the steps in the process? The literature supports the notion that efficacy is being affected by the use of lesson study. There are suggestions for principle variables of this study; pedagogical content knowledge, peer influences, pedagogy and teacher orientation. However, the literature has mixed messages that highlight only a few variables at a time and without looking at how they interact. In short, lesson study makes an excellent case to study.

Peer Coaching Peer coaching is a reflective practice that pairs teachers for collegial support in making instructional decisions (Harwell-Kee, 1999) and changes to teaching practices (Sparks, 1986; Wood & McQuarrie, 1999). The technique contains mentoring as a special case where there is a substantial difference in experience between the pair of teachers (Harwell-Kee, 1999), however, mentoring is treated separately in this review because it was observed as a separate case within the study. The focus of this review is peer coaching where the participants are professionals of equal status (Bruce & Ross, 2008; Conley, Bas-Isaac, & Scull, 1995). The structure of peer coaching includes a pre-conference, teaching observation and a postconference (Wood & McQuarrie, 1999). The process may be repeated with alternating roles (Kohler, Ezell, & Paluselli, 1999). The structure isflexible(Kohler et al., 1999; Nolan & Hillkirk, 1991; Tschannen-Moran et al., 1998) and may be altered to meet contextual needs. For instance, it will have a schedule but one that teachers may change to suit day to day needs. The pre-conference establishes goals and observables (Perkins, 1998) which are then discussed in the post-conference. It is advised that effective conversations follow guidelines for collegial respect, such as excluding direct observations of the teacher, and to improve the quality of questioning (Perkins, 1998). Using guidelines would mitigate concerns that observers emphasize positives and avoid specific pedagogical feedback (Wallace, 1998). Specific goals of peer coaching vary according to the needs of the participants but are generally focused on teaching methods, planning, classroom organization or the assessment of student needs (Bruce & Ross, 2008; Kohler et al., 1999; Wood & McQuarrie, 1999). It may also be used to alter organizational parameters to facilitate improvements (Hargreaves & Dawe, 1990). The goals are achieved through dialogue that highlights different perspectives that

45 ".. .illuminate the boundaries andframesof thought which limit their current perspective..." (Jay & Johnson, 2002, p. 76). Goals that are externally defined have ".. .successfully produced higher implementation of innovative programs and specific teacher behaviors" (Stevens & Slavin, 1995, p. 327). Peer coaching has issues which contribute to the low occurrence rate. Lowden (2005), for example, found that only 12% of a sample of 205 teachers had been observed by peers. Availability and scheduling time can be significant issues (Perkins, 1998; Tamargo, 2005) as well as availability of resources (Conley et al., 1995; Reed, 1996). There may also be issues of peer compatibility and professional apprehension (Perkins, 1998; Wallace, 1998) particularly with the observation component (Kohler et al., 1999). In addition, some teachers resist having a high degree of structure (Kohler et al.) preferring to use implied protocols and assumed expectations (Wallace, 1998). The mechanism by which peer coaching provides benefits is not clear. Generally, support has not been through the articulation of a specific mechanism but by evidencefrompeople who have used it, which makes validity a concern. One such report, where validity is not explicitly addressed, stated: In summary, the data reported here support the peer coaching model as a means of fostering increased professional knowledge, improved teaching, positive changes in participants' attitudes and values, more positive feelings about one's self and one's relationships with others, and, to a more modest extent, improvement of school culture and student learning. (Gordon, Nolan, & Forlenza, 1995, p. 90) The lack of an explicit mechanism is made clear through the claim that coaching ".. .is lacking on the impact of coaching on student achievement or even a relationship between coaching and student learning" (Green, 2004, p. 14). Conley et al. (1995) found peer coaching to produce collegiality which was more often collaborative collegiality (34%) than contrived collegiality

46 (21%). The discussions between peers develop "shared language" and common understandings to make collegial interactions more effective (Showers, 1985; Stevens & Slavin, 1995). However, the discussions tend to be of greater benefit to "...listeners who interrupted professional sharing...than those who did not." (Ross & Regan, 1993, p. 103). Teachers who willingly engage in peer coaching are likely to have higher personal teaching efficacy (Nolan & Hillkirk, 1991; Tschannen-Moran et al., 1998) which is correlated with higher student achievement (Ross, 1992). However, Ross (1992) found no interaction between use of a coach (without observation) and increased efficacy where Bruce and Ross (2008) indicate that "Peer coaching also contributes to increases in teacher self-efficacy." Bruce and Ross observed a bidirectional interaction between efficacy and innovative instruction: Peer input influenced teacher efficacy and innovative instruction as predicted, but equally powerful was the influence of innovative instruction and teacher efficacy on peer input. That is, as teachers implemented standards-based mathematics teaching and increased their efficacy, the quality and importance of peer feedback was also increased. (Bruce & Ross, 2008) This study, however, involved a content oriented professional development component that confounds comparison with teacher initiated peer coaching. The absence of an interaction being observed (Ross, 1992) when there was no observation by the coach is consistent with the premise that self-evaluation rather than peer evaluation may be important; "The self-evaluative feedback and nonevaluative feedback of peer coaching ... are more likely to promote reflection and a desire to change" (Licklider, 1995, p. 55). Self evaluation may be influenced in two ways, firstly, seeing a peer perform a skill may cause one to persevere to implement the skill in spite of personal difficulties and, secondly, a teacher is likely to examine his teaching more carefully prior to peer observation (Licklider, 1995).

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The peer coaching group that was observed in this study was using the PRIME resource and some remarksfromthe research regarding this resource are pertinent. PRIME ".. .can be used to identify and implement specific goals for school improvement" (McDougall, Ross, & Jaafar, 2006, p. 1). However, teachers should focus on one dimension at a time in order to be successful. In addition, leaders need to avoid providing too much direction which could derail teacher's efforts to implement the resource. Pre-Service Mentoring Mentoring is typically directed toward novice or pre-service teachers as was the case in this study. However, the term is used more generally for a peer coaching situation in which there is a hierarchal senior-junior pairing (Conley et al., 1995). Like peer coaching, the structure is somewhat negotiable (Robbins, 1999). Hiffman and Leak (1986) found ".. .mentors were able to provide assistance in addressing their needs by providing encouragement, collegiality, and specific helpful suggestions for the improvement of teaching" (p. 23). The mentor is supportive and provides constructive criticism that is separatefromthe evaluation process (Robbins, 1999) and achieves this through informal conversations (Hiffman & Leak, 1986). Mentors tend to focus on classroom management and other technical skills of teaching rather than theoretically-informed visions of teaching and learning (Elliott, 2002). The mentor may also provide organizational information such as school or board expectations (Hiffman & Leak, 1986). During the mentoring process, the novice teacher makes a transitionfromissue based questioning to making evidence based hypotheses to independently rationalize issues (Mewborn, 1999). In cases where the mentorship involves co-planning and co-teaching, the novice may deepen his analytical views of his teaching ".. .from technical and procedural support to other

philosophical matters." (Cherian, 2007, p. 35). It may be the goal of deepening the novices analytical view that prompted Feiman-Nemser (2001) to suggest that the mentor needs "an explicit vision of good teaching and an understanding of teacher learning" (p. 18) which is a theoretical image of how pre-service teachers develop and how to meet his projected needs. Wang and Paine (2001) suggest that it is sufficient for the mentor to be able to identify the novices' zone of proximal development and work within it to move the novice toward independence. Elliot (2002) suggested "mentoring orientations" which included "collaborative skills", "coaching skills", reflective practice", "learner stance" (learning from all teaching situations) and "trying ideas skills" (trying different teaching techniques). Using the mentoring orientations, Elliot found that".. .orientations were not just a product of the individual but were informed by participants' professional community's normative ways of interacting." (Elliott, 2002, p. 212) and "Mentees' understanding of mentors' guiding orientations and reactions to these orientations mediated productive and less than productive opportunities to learn." (Elliott, 2002, p. 245) The mentors also gainfromthe relationship and, while they have more experience, it is not strictly a unidirectional relationship (Conley et al., 1995; Wollman-Bonilla, 1997). The mentors gainfromreflections that arise because of reflections, on their own part and the novice teacher's part, regarding the mentor's ways of thinking about teaching practice (Elliott, 2002). While mentoring of pre-service teachers must be widespread in order to meet the needs of the overall education system, there appear to be very few quantitative assessments of mentoring experience. In a sample of 205 teachersfromall divisions and levels of experience, 39% were involved in mentoring (compared to 26% for peer study and 38% using reflective practices generally); however, only 5% of the teachers reported having received professional development

pertaining to the mentoring role (Lowden, 2005). Bainer and Didham (1994) reported lower figures in a mid-western state-wide sample where they found that half the school districts had formal mentoring programs but only 15% of their sample was involved in them. The difference in thefiguresmay be due to the distinction between pre-service mentoring and induction programs during the first few years of teaching, however, it may also be due to increased interest in mentoring during the last decade. The matching of the novice teacher with the mentor is problematic because it should be determined based on how well the two people can work together but this does not mean that they need to share personality traits, social background, or type of schooling (Galvez-Hjornevik, 1986). Generally it has been suggested that mentors should not exceed half a generation older than the novice because it would risk the formation of a parent-child relationship. Another issue arising with mentoring is the potential lack of support for changes in practices by mentors; "If the expertise of mentors is embedded in and even, in part, rests on their connection to prevailing teaching contexts, the question of how mentors can support change is complex" (Wang & Paine, 2001, p. 158). Frykholm (1996) found that pre-service teacher training regarding the Standards (NCTM, 2000) was not implemented in placements and that the mentor had more influence than the training received at university. The pre-service training may also be confounded by historical student experiences becoming the basis for teaching even when alternative methods have been presented at the college (Arvold, 1998). However, care is needed in interpreting these claims because the methodology was based on a single participant who had had a successful and satisfying experience in the education system as well as having her preservice teaching experience and subsequent first year of teaching in secondary schools close to her home (i.e. in the same geographical vicinity of the secondary school she attended).

Mossgrove (2006) identified differences between two pre-service teachers and concluded that they resulted from: .. .the curriculum used, the alignment of support, and the mentoring practices each preservice teacher was exposed to impacted their instructional practices, particularly with respect to the cognitive demands of the tasks used in the classroom, the use of representations as a means to build understanding of a concept, and the number of and types of questions asked during the enactment of the lesson. (Mossgrove, 2006, p. 218219) This view is similar to a much older result based on the study of English and History; "The content in pedagogical content knowledge has to be reorganized to take into consideration students, classrooms, and curriculum. This reorganization revolves around teachers' personal values and those embedded in their specialization" (Gudmundsdottir, 1990, p. 47). Arvold (1998) suggests that a teacher may be viewed as a "blending of roles" (p. 220) and that frustration may develop when underdeveloped roles are required. In this sense, mentoring may be a process of developing new "roles" and blending them in a manner that provides utility in the classroom. Self-efficacy may simply reflect confidence in the ability to execute the new role in the classroom when the need arises. Self-regulation of motivation has been associated with efficacy beliefs (Chester & Beaudin, 1996) which, in urban schools, was found to be mediated by "opportunities for collaboration with other professional staff members, supervisor attention to classroom performance, and the availability of instructional resources" (p. 249). This is consistent with mentors conveying a sense that the novice teacher can overcome present hurdles (i.e. efficacy) (Rowley, 1999). This suggests that the researcher's presence in the pre-service teacher's classes may have had an impact on her efficacy because the project contributed to the mentorship collaboration, provided more supervision in the sense that the researcher provided feedback, and the researcher provided

51 some resources (whiteboard markers, extra textbooks, and a demonstration model) (Chester & Beaudin, 1996). Isolated Reflective Practice In recent years, self-study has grown in popularity as is reflected by the start of two dedicated journals, Reflective Practice and Studying Teacher Education (both by Routledge), as well as the formation of a special interest group within the American Educational Research Association. Literature searching was challenging because of the absence of standard terminologies for describing self-improvement forms of reflective practice. Terms that were used in title, abstract, and keyword searches were self-reflection, self-study, self-reflective, selfassessment, self-evaluation, and self-perception. However, relating these terms to pedagogical improvement in schools seemed to be relatively rare and they were frequently used self referentially in other contexts (for example, a recent issue of Studying Teacher Education, 3(1), was dedicated to self-study of teacher educators teaching teachers). The terms "identity" and "contemplative education" revealed a body of literature with some overlap in self-study (with the former term being much more relevant to the mathematics content area than the latter). Other terms, such as "individual", were explored but not sufficiently focused that they required augmentation with other terms already mentioned. While many search terms were explored and references to similar research papers were used, the literature search was not as extensive as the other reflective practices. Self-study arises in other areas of reflective practice where teachers, as part of the practice evaluate individual experiences (Pajares, 1996). For example, Grandau (2005) considers selfstudy to include consultations with a "criticalfriend"in which consultations may contribute fresh perspectives. In another example, Buschman (2001) reports of an action research project that

52 involved self-assessment and reflection journals as components. The study entailed student interviews to explore how well teachers knew the level of understanding of students and it led to an understanding of how to address vocabulary limitations and focus on misconceptions to meet student needs more effectively. While Buschman used self-study within the context of a larger professional development program, it is impossible to separate the two. Practitioner research is sometimes used to refer to self-study however some researchers define practitioner-researcher as requiring systematic inquiry (Louie, Drevdahl, Purdy, & Stackman, 2003). This confusion of terminology highlights a key issue within self-study which is appropriate dissemination of self-study research. Self-study results are considered a form of local knowledge (Anderson & Herr, 1999; Buschman, 2001) that is used to the researcher's own ends and therefore does not need to be systematic, in a formal sense, or address validity beyond the teacher's own personal requirements (Louie et al., 2003; Zeichner, 2007). The authors (of both references) go on to express concerns that reports of self-study generally focus on the individual case and do not make connections to other research that might inform a larger community of researchers. However, this is not a universal view since it has been suggested that self-study, presented within certain guidelines, could be used to "provoke, challenge, and illuminate rather than confirm or settle" (Bullough & Pinnegar, 2001) or be the basis of mentoring other teachers using what was learned (Hoban, Butler, & Lesslie, 2007). Collopy (2003) studied two veteran junior division teachers who were implementing curriculum materials without external support. The curriculum materials downplayed rote memorization and algorithmic problem solving. One teacher was observed to make a superficial implementation, dismissing elements that she did not consider relevant to her vision of

53 mathematics, where the other made genuine changes in her teaching practices. It was concluded that: Ms. Ross adopted a new approach to teaching mathematics, with curriculum materials as the primary source for her professional development. Her instructional practices and, more importantly, her focus during mathematics instruction and her rationales for practices changed in the direction the curriculum materials advocated. (Collopy, 2003, p. 306) Kamina (2005) investigated the implementation of the same curriculum materials by grade five teachers, using lesson study, and found that implementation varied between teachers with many common factors. One account of self-study bore a striking resemblance to a participant in this study and requires special attention (Thomas & Monroe, 2006). In this particular study, one author was engaged in a self-study effort to enact a Standards (NCTM, 2000) based teaching while the other author was a critical friend who provided expert advice later in the process. The classroom is initially described as teacher centered, textbook driven and lacking in differentiated instruction. The teacher had a low opinion of manipulatives. However, after watching a video of math lessons being taught in a manner consistent with the Standards, he concluded; "Seeing this lesson was exciting and made me wonder if I could involve my students in rich discussion and pique their interest in finding solutions to open-ended problems. I did not know how to do this..." (Thomas & Monroe, 2006, p. 172). The result was no change and this was explained as having not understood that problem solving was to be integral to the teaching process rather than the topic to be conveyed. "I had not yet grasped their potential as ways to approach the entire mathematics environment in my classroom" (Thomas & Monroe, 2006, p. 172). Through contact with the second author, the teacher recognized the importance of choosing suitable tasks that

would facilitate questioning and encourage student thinking. The results developed as he developed more meaningful tasks: As I developed tasks that were more meaningful to the students, I discovered that they had a natural interest in mathematics. They seemed to have more enthusiasm and perseverance for mathematics than I had ever observed in my previous years of teaching. (Thomas & Monroe, 2006, p. 179) The teacher described the movement to the Standards based curriculum as a reflective process and, as will be seen in this study, one that has similarities to the self-study participant included in this study. The move to Standards based math teaching that Thomas and Monroe (2006) described is essentially the same as the self-study situation observed in this study. It represents a substantial change in teaching: when teachers decide to change their practices in ways that are consistent with the standards put forth by the NCTM (1989,1991) and other mathematics reform projects, they also decide to transform who they are as mathematics learners and teachers. In other words, they embark on a process of re-forming their mathematics identities. (Drake, Spillane, & Hufferd-Ackles, 2001, p. 2) It should be noted, however, that the change in "identity" could be related to the observation that all teachers in the Drake et al. sample had had negative math experiences and the change could be interpreted as the need to overcome negative experiences in order to teach more positively or a perception that reforms might provide more positive experiences to students: It is clear that they needed to have had at least one, and possibly several positive experiences with mathematics in order to begin to re-form their mathematics practices in ways that were consistent with reformers' visions. It was these positive experiences that allowed them to narrate their mathematics stories with a sense of efficacy and enjoyment. This new perspective on mathematics then led them to believe that reform was possible and motivated them to try it out. (Drake et at, 2001, p. 16)

55 This suggests that willingness to engage in self-study for mathematics may entail antecedent conditions which are consistent with the purposeful sampling used in this study, but is indicative of a significant issue regarding generalizability. It has been suggested that a self-assessment tool may be useful to provide direction to selfstudy but may require support, particularly in mathematics, to implement changes (Ross & Bruce, 2007). However, it appears the interpretation was for relatively fast change in practice which was not observed in the current study. Given that availability of time is a constraint for self-study (Wolpin, 2006), it is likely that support would be necessary for a relatively fast pace of change. Closing remarks on reflective practices The literature regarding reflective practice is extensive and even within specific types of reflective practice there is a substantial body of knowledge. For the most part, the literature gives pragmatic insights into the benefits and limitations of reflective practices. What has been given less emphasis in this review, reflecting a bias in the literature perhaps, is the scope of reflective practice. Gallagher and Gray (2001) demonstrate that reflection has limits based on what one perceives as truth being at odds with multiple perspectives that may be held by others and one's ability to interpret the other perspectives even if they were revealed. Markham (1999) raises issues about the extent to which an individual can separate his selfimagefromthe situation being examined; "The more my own self-image is involved in my interaction with others, the less clearly I can see those others" (Markham, 1999, p. 66). In addition, he suggests that reflective opportunities may be thwarted by authority (such as administrative constraints), instabilities of the teaching context, incompatibilities between personal philosophy and reflection, competition between teachers, and "transference" (p. 68)

56 which amounts to students viewing their teacher through the lens of relationships they have had with previous authority figures. While there are limitations to what reflective practice can achieve, the literature extolling its benefits far outweighs the few papers dedicated to its limitations. The constraints do not pose a dire situation for this research study but do provide a mechanism for comparison with the research literature. There are also, particularly in the case of Markham (1999), concerns of a fundamental psychological nature that are likely to be beyond the scope of this study and will have to be given due consideration when generalizability is considered.

CHAPTER THREE METHODOLOGY The method was multiple case studies (Merriam, 1998) with each of the four cases representing the mostfrequentlyencountered approaches to reflective practice. The sampling was purposeful, specifically theory or concept sampling, following the guidelines set out by Creswell (2005). The analysis used grounded theory with constant comparison between different groups of reflective practitioners. It has been beneficial to examine one particularly rich case and then use pattern matching and progressive refinement to assist in the comparison and contrast of one reflective group with another. Many forms of practice involve two or more teachers working together. Each group of reflective practitioners will be treated as a single "reflective group". The sample will contain independent reflective groups. Within each group, the focus is on individuals, but the group will be treated as a cohesive unit because of the shared understandings it generates and to respect the structure the teachers have put into place. The groups of participants using reflective practices were monitored for a period of time corresponding to the type of reflective practice they were engaged in. In two cases, the period of time was the full duration of the reflective practice groups and the other two cases reached saturation. While lengthier monitoring would have been beneficial in the first two cases, it was not feasible without creating a contrived situation that would cast doubts on the validity of the observations. The technique that developed differed from the original intention. As groups of reflective practitioners were located, it was rapidly recognized that their involvement in reflective practice was a negotiated demand on their time and the researcher was going to have to minimize any additional demands on their time. In view of this, rather than regular isolated interviews of

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participants before and after their meetings, the researcher recorded the reflective practice meetings and all discussions between the participants that they organized. The researcher participated in the discussions to a limited extent but in a manner that allowed probing of the evolution of topics discussed in previous meetings. The researcher's involvement was consistent with "... engaging] in promoting development (teaching) as part of a cycle of interaction and reflection" (Simon & Tzur, 1999, p. 253), however, the researcher's emphasis was inquiring about events that had passed rather than actively engaging in changing the course of the reflective practice. It was also realized that some reflective practice methods are quite short in duration and the time available to observe changes in methods poses a limitation on the depth of observation that is feasible. Three of the four groups that were studied had schedules that were less than two months, for practical reasons, such as the use of release time funding. Additionally, one-on-one semi-structured interviews (and interview guide is provided in Appendix E) were scheduled toward the end of the reflective practice to discuss any issues that were unclearfromthe meetings and to discuss, individually, the interpretation of the Miller-Seller instrument (described below). The methodology evolved to fit with the context of the reflective practice groups. This meant following their schedule and attending their meetings. However, the participation was not intended to extend to the classroom because this would have risked affecting the behavior of students and other observing teachers. The focus was on the teachers' views of their teaching practices and the way in which their discussions and the practices they discussed changed during the course of the reflective practice. In two cases, opportunities to observe classes with minimal impact on the reflective practice arose and the researcher made field notes based on observations.

59 Thefirstopportunity was the mentoring case were a two and a half week block of time provided time for desensitization of the senior students and the other opportunity was peer coaching were two grade level groups met with pairs of familiar teachers in the library and the researcher was able to observefroma distance with little or no impact on the observed lessons. Three key sources of information were collected during the study. Firstly, audio recordings and field notes of the meetings held by the reflective practitioners provided the main source of information. The field notes include various documents that the teachers provided such as seating plans with notes they took during observation, hand outs they used in classes and lesson plans. The second source was a questionnaire that each participant completed providing information about his background, how they came to be involved in the reflective practice group and the Miller-Seller instrument. The Miller-Seller instrument, which is provided in Appendix D, provides seven paragraphs describing different approaches to learning and curriculum. The participant ranks the approaches and subsequently the proximity of pairs of approaches is interpreted as a meta-orientation that has a philosophical foundation (Miller & Seller, 1990). The final source of information was individual interviews that reviewed the interpretation of the Miller-Seller instrument (because it has not been formally validated) and addressed any unclear aspects of the meetings that had been observed. Data Collection and Analysis Data gathered in this study included audio recordings of meetings of the reflective practitioner groups, interviews of participants, collection of artifacts such as notes made by the teachers during classroom observation, and field notes. An overview of the 98 data events collected is provided in Appendix A.

60 The audio recordings were made using a battery powered micro-cassette recorder. The recorder had difficulties with one group where a female participant with a quieter voice was difficult to hear against the backdrop of three males with deeper voices. There was also a recording where the batteries were not at full power and the level of static tended to be higher. In view of these challenges, the audio recordings were used to make computer audiofilesthat could then be played using different filter settings on the graphic equalizer that is available in the Windows Media Player. This allowed reduction of spurious high frequency static and adjustment of thefilterto improve the perception of the quieter participants. The audio recordings andfieldnotes were transcribed. In places where the recordings were unclear, identifiers were put in the transcription that indicated the duration of the unclear portion and allowed for sections that were later deemed potentially important to be revisited for further effort on the transcription. In the few cases where a section of text was deemed sufficiently important to warrant revisiting the audio recording, the interpretation was done by adjusting the filter settings and repeatedly listening to the section with a variety of filter settings until the interpretation was considered to be correct. In all instances where this arose, the researcher believes that an accurate transcription was made. The transcriptions were imported into N6 (QSR International, 2002) for further analysis. In the initial stages, this was done with information from only one reflective group (the lesson study). This was used to develop an initial set of codes that were the basis for an inter-rater reliability check with Mr. Pyper. Mr. Pyper has been a secondary mathematics teacher, department head and, at the time of the reliability check, was teaching in a concurrent mathematics and education pre-service program. The check used text samplesfromone reflective practice group and onlyfromtheir early meetings but was considered necessary to

61 mitigate any oversights in the coding process. The reliability check served to clarity the initial interpretation of codes which was beneficial to the emergence of additional codes when additional reflective groups were examined. It was not feasible to revisit the inter-rater reliability later in the analysis. When a complete set of transcriptions was available for the lesson study group, the information from this group was used to develop a preliminary code book. This was preliminary in the sense that incremental comparisonfromthe addition of information from other reflective practice groups led to constant comparison that facilitated alterations to the preliminary code book. The changes that were required to the preliminary code book were significant primarily because they provided more detail of various pedagogical and pedagogical content knowledge based events that were observed. In this respect, the additional reflective groups that were observed provided information that allowed a refinement of the code book. One fundamentally different code was added that was based on student as observers that did not fit into the pedagogical or pedagogical content knowledge codes. This did not arise in the preliminary codebook because it had not occurred in the discussions by the reflective practice groups to that point. During the construction of the codebook self-efficacy was not coded directly. Rather evidence of influences was gathered and interpreted in terms of its implication for an individual's beliefs in their own ability. The evidence was sought along the way by reviewingfieldnotes and discussed with each participant directly at a later meeting or thefinalinterview to validate the self-efficacy interpretation. The explicit triangulations across meetings was intended to verify that statements regarding influences had endured and were not simply transitory feelings. In this sense, the evidence supporting the self-efficacy construct was derivedfromthe text coded as

62 influences and primarily the sub-codes representing individuals influencing themselves and others within the reflective group. When all transcribed information had been coded and examined, a final codebook was developed (see Appendix B). The codebook was written with examples and examined by Dr. Ross. His feedback focused on improving a few wordings and suggestions about the choice of examples. In some cases, two examples had been included that highlighted different aspects of a particular code and Dr. Ross suggested the example he felt would be the better choice. Following revision of the codebook, it was reviewed by Mr. Pyper who suggested a change of name for one code and questioned the quality of one example. We agreed on the meaning of the latter code and that the issue was the quality of the example. He commented that it generally "made sense" and that the definitions had good clarity. His suggestions, a code name change and improved example, were followed to arrive at the final version of the codebook. With the codebook finalized, the coding process for all transcriptions was reviewed and data from each reflective practice group was extracted according to its relevance to the research questions (a table of coding occurrences is provided in Appendix C that was validated with the researcher's impression of emphases in each case). This was used to prepare individual documents summarizing, for each reflective group, the content that was considered relevant to the research questions. These documents had the content organized according to the research questions. Within the context of each research question, the content was organized into vignettes that corresponded to the chronological progression of events around a single theme. For example, algebra tiles were mentioned multiple times and all such mentions were brought together in chronological order. In addition, a minimal amount of interpretation was added to improve the readability. Each participant was presented a copy of the document pertaining to the

63 reflective group (in all but one case this was in person) and asked to provide a member check that included considering accuracy of the record and whether any important events or observations had been omitted. Member checks were received backfromall but one participant. The one participant was in the peer coaching group where four other participants provided feedback. The feedback acknowledged that the record was "very thorough" and all responding participants indicated that they agreed with the account and that there were no omissions. Three types of changes occurred because of the member check. In one instance, the researcher had interpreted comments regarding SMART Boards where the full group had never actively discussed them. The member checking pointed out a misinterpretation that arosefromthis and the record was corrected. This was the only clear error of content in the member checking. There were four instances of rewordings to clarify the meaning and the researcher changed the wording to reflect the meaning understoodfromthe meetings. Lastly, there were several minor grammatical changes, for example, the lone reflective practitioner indicated a minor grammatical fix up in a quotation transcribed from the taped interviews. Given that it was the same person who was being quoted and had provided the correction, the change was made. Other changes, such as a transcription error of "where" instead of "were", were accepted for correctness. The analysis used axial coding (Kelle, 2005) based on reflective practice as the coding paradigm. Kelle (2005) indicates that coding paradigms are derivedfrommicro-system (or sociological action) theory (Ashton & Webb, 1986) and have a potential of missing macrosystem phenomena. However, in this study, the role of leadership imposed reflective practice has been minimized (it was not feasible to avoid leadership influence altogether) which mitigates this particular area of concern with the use of a coding paradigm.

This axial coding began with the analysis of one particular reflective practice group that appeared, to the researcher, to be well suited to the inductive (Hitchcock & Hughes, 1989) approach. The researcher could recall there being several episodes that highlighted relationships between different categories which might assist in generating the theoretical perspective. The coding was assisted by having organized the content into chronological vignettes for the member checks. The other three cases were then considered with the tentative theory as a starting point in order to contrast any discrepancies that were found. Ultimately, a theory that would explain the four observed reflective groups was sought, however, the process worked incrementally with cases orderedfrommost promising to develop theory to most likely to add support or clarify details of an existing theory. Theoretical codes (Strauss & Corbin, 1994) were developed using an initial coding paradigm,fromKelle (2005), of "causal conditions", "phenomena", "context", "intervening conditions", "action strategies" and "consequences" in the data. This was not to the exclusion of other theoretical codes, but rather a starting point that was open to modification if the process required additional codes. In particular, there was a concern that there were simultaneously two distinct groups of contexts in the sense that teachers were discussing their individual classroom contexts, as highlighted by observations and practices, while engaged in the non-classroom context of a reflective practice meeting. During the theoretical coding, sequences of events chaining constructs together began to emergefromthe vignettes. The emergence of processes linking multiple constructs together sequentially represented underlying patterns and, as more vignettes and the various reflective practice groups were considered, repetitions of patterns providing support of multiple perspectives (Strauss & Corbin, 1994) were recognized.

65 The relative brevity of the codebook that emergedfromthe data was best suited to the development of a substantive theory that was theoretically dense rather than conceptually dense (Shkedi, 2005). In this respect, it was anticipated that there are likely to be many instances where theoretical codes represented a multiplicity of perspectives rather than additional conceptual relationships. In addition, the multiplicity of perspectives represented by the participants, four different kinds of reflective practice in four different divisions, suggests that the emergent theory that is sought will be a mid-level theory (Kelle, 2005; Shkedi, 2005). With these notions in mind, a conceptual chart was drawn and sequences of processes linking multiple events were drawn for different vignettes. This highlighted the different perspectives, the theoretically dense relationship that was established and highlighted instances were certain particular perspectives were only observed within a single group of reflective practitioners. This allowed some limitations of the theory that emerged to be recognized efficiently. Participants The participants were identified in two key ways. Firstly, a staff developer within a public school board circulated information requesting research participants. This was circulated among people who had applied for release time funding under a program aimed at reflective practice for teachers with more than ten years experience. Secondly, the researcher contacted two colleagues he had met two years earlier in OISE graduate courses, one of whom became a participant along with a pre-service teacher he was mentoring, and the other led to a chain of contacts that resulted in the recruitment of another participant. A third approach was pursued wherein a staff developer in a separate school board circulated information requesting research participants to math department heads, however, this solicitation received no responses.

66 Generally the recruitment of participants was challenging and required that my initial objective of focusing exclusively on intermediate and senior divisions was relaxed to include the primary and junior divisions. This was later seen as an asset, particularly the primary division, because it provided a math teaching environment where the researcher was not as readily familiar with the pedagogical content knowledge. The Reflective Groups The participants that were recruited had already engaged in or agreed to engage in some form of reflective practice. They had chosen the format and colleagues for the reflective practice. The four groups represent a purposeful sample because they all contained teachers with considerable experience and/or a strong background in math as a teachable subject. Lesson Study The lesson study group was composed of three intermediate teachers in an elementary (JK8) school with approximately 700 students. The school is located in an urban environment with a mixed socio-economic composition. Grade six EQAO results for the school have been within 2% of the Provincial averages for the last three years. The three teachers involved in the lesson study have 5 to 10 years teaching experience. Two have English backgrounds and one has a physical education background. One teacher was clearly comfortable teaching math and has been involved in math leadership within the school. The group was composed of one teacher from each of grades six, seven and eight. Their focus was vertical alignment within mathematics and they chose to use the lesson study method of reflection.

Peer Coaching The peer coaching group was organized by a vice principal and had four primary teachers. The elementary school is one of several in a moderate sized town and serves JK to grade 8. It has approximately 350 students who are all non-rural. Grade three EQAO results are erratic and vary from less man 5% below to over 20% below the Provincial averages within the last three years. The four teachers, excluding the vice principal, all had over twenty five years experience. They had a variety of backgrounds both academically (English, psychology, arts) and professionally (multi-division, special education, teaching specialists) with them all being equal in the absence of any specific math foci. The vice principal, who perceived her role as being organizational leadership, had a background in psychology and was a math specialist. The focus of the peer coaching was a follow up to earlier professional development that introduced the PRIME math kit (Nelson, 2005). The teachers had not implemented the kits and the peer coaching was to facilitate their efforts to see how and why they might use the kits. Pre-service mentoring The mentoring group paired a secondary math teacher with a pre-service teacher. The school was a private school that serves ages as low as two in a pre-school program to an advanced placement program (a year after grade twelve). The population of the school was 270 and EQAO results were not available. The school was non-denominational but its vision is inspired by the Baha'i Faith. The students at the school were international with high levels of multiculturalism, many are residential students and many werefromwealthy families. The teacher (mentor) has ten years experience in private schools, a background in physics and a Masters of Education. The pre-service candidate was enrolled in a pre-service program at a Western U.S. faculty of education. She has a background in mathematics and physics.

The focus of the mentoring was the reflective elements of the pre-service practician. The situation had two sections of a high level grade twelve math course being taught with one having the mentor as the teacher and the other the pre-service teacher. This facilitated comparisons of techniques and methods that extended beyond the normal pre-service reflective practice. Isolated reflective practitioner The isolated practitioner teaches in the junior division and was identified by a math expert within the school board. An initial telephone interview revealed that she engaged in professional development based on reflection essentially in isolation. The elementary school she taught at served a small town with JK to grade 8 and has approximately 300 students. Many students are from rural areas and an estimated 200 students are bussed. Grade six EQAO results have been consistently more than 8% above the Provincial average for the last five years and over 25% above for two of the last three years. The teacher has over 15 years experience and has taught in both the primary and junior divisions. She has a background in kinesiology and has completed a Masters of Education. Through interviews, this participant recounted her experiences using reflective practice without peer support to come to terms with having started teaching math at the junior level four years ago. This reflects her having to come to terms with curriculum changes and modern views on best practices for teaching math with no collaborative support. Ethical Considerations This study was conducted following the ethical guidelines of the University of Toronto. There were two key components that required special consideration with regard to ethics in this study. One was the inherent risk of shared experience between teachers in a reflective group. This element was increased by the presence of the researcher who was unknown to the

participants. Theriskbetween teachers was understood by the participants and accepted, by them, in the formation of their group. In order to mitigate concern over the unknown researcher, an initial meeting establishing the purpose and protocol of the research was held. In two cases, there was visible relief on the part of participants when they learned that I did not intend to visit their classrooms. Initial concerns were also further addressed through the maintenance of a friendly but professionally respectful relationship (For example, bringing coffee to meetings and posing questions in a constructive rather than critical manner.) In a few instances, participants requested that the researcher stop recording the conversation so that sensitive details could be discussed and this was always done (it only represented a gap of a few minutes in many hours of recordings and did not impact the overall research). The second key ethical risk was the potential revelation of the image of the teachers that would be derived from the data that was collected. To address this concern, member checks were completed that allowed the participants an opportunity to draw attention to any misinterpretations or statements that they might conceive of having an adverse impact on them. No such statements were actually identified but the opportunity was provided, on ethical grounds, through the member check. Credibility and Reliability Several facets of the study were put into place to assure that the results were both credible and reliable. The initial analysis of the interviews used detailed footnotes (over 300) to ground the emerging theory within the context of the interviews. This allowed for cross checking between different meetings of the participants to check that what was said in one meeting was in agreement with statements in other meetings. In many instances, the researcher purposely revisited key discussion topics at different times and in different meetings so that the meaning

was more substantial than a singular discussion. The extensive ties between the analysis and the transcripts also facilitated checking interpretations between different reflective groups. These techniques were aimed at having triangulation within meetings using multiple distinct discussions, between meetings by revisiting topics, on an individual basis by having individual member checks, and across reflective practices and grade divisions by comparing different groups. In addition, the theoretical constructs that emerged were validated during an early interrater reliability check and later with a codebook review (reported in the Results Chapter). In both cases, the explicit meaning of different codes and the interpretation using examples from the interview transcripts were discussed. At the end of the researcher's participation in each group, the researcher met individually with each participant to verify the interpretation of the MillerSeller instrument and to discuss key areas of involvement of the participant in the reflective process. At a later date, member checks were completed by providing a detailed hardcopy document detailing all the information extracted from the interviews in each case. Participants reviewed the document for the reflective practice group individually and provided feedback to the researcher independently of the rest of the group. Limitations The methodology is limited in scope with respect to the variety of types of reflective practice that were observed. The choice was made to examine four reflective groups in depth rather than have a lesser examination spread across a cornucopia of reflective techniques. The teachers who participated were busy and the use of reflective practice added another demand on their time. This led to the methodology having to evolve to suit the circumstances and may constrain thefindingsto those features that can be unobtrusively observed.

71 No statistics were found to indicate the prevalence of reflective practice in mathematics. It is the researcher's perception that it was not overly common which may be indicative of a difficulty in ascertaining the motivations of those teachers who are involved in reflective practice. Mentoring, for example, has been used as an entry point for leadership experience that may be an asset in acquiring a headship. For this reason, there may be a hidden bias within the participants. It is unclear what implications, if any, such a bias would have on the theoretical framework. As Simon and Tzur (1999) point out, "...observations of changes in teachers'.... Growth in knowledge and of and about mathematics, although important, are not necessarily indicators that significant changes in teaching have taken place" (p. 258). To address this concern, considerable cross case evidence was sought, consistency of the emergent theory across a relatively wide variety of situations was sought and a rationale for making changes based on meaningful practical utility was developed. Generalizability In each reflective group, the teacher practices that are discussed are highly contextual and emergefrompractitioner's views of a specific classroom with its established routines (Richardson, 1990). However, while the specific details are not generalizable, the inclusion of a relatively diverse set of reflective groups led to an emergent theory based on constructs that are not contextually specific. The reflective groups rangefromthe primary to the senior division with content varyingfromelementary counting through to topics in permutations and combinations. The schools included both public and private schools and the teachers had levels of experiencefrommonths to over thirty years. The focus of the reflective practice was

significantly different in each case; vertical alignment, facilitating the use of math kits, preservice practice, and adjusting to a modern math teaching practices. The diversity in the reflective groups is in keeping with SchSn's (1982)findingthat reflective practice has quite a diverse basis and can be found in a broad range of contexts. The method supports this by facilitating the development of a substantive theory that has a broad range of contexts contributing to the theory. While it is not assured to cover all eventualities of the reflective practice spectrum, it is anticipated that, like SchQn's research, thefindingswill have utility in a wide variety of settings.

CHAPTER FOUR RESULTS The research questions are the focus of the beginning of this chapter and are followed, in the following chapter, by an explanation of the substantive theory that has been developed. Each research question is addressed using the results of the lesson study, mentoring, lone practitioner and the peer coaching in order. Note that the results for the first research question are considerably longer than the other questions because the first includes descriptions of events that occurred in the group deliberations. The events are subsequently reexamined, but not restated, in the results for the other research questions. At the end of the results for each research question, a summary is given that highlights key relationships that will be used in the substantive theory (presented in chapter 5). Research Question 1 What changes do teachers consider during reflective practice? Haw do the discussions and efforts toward implementing changes affect each teacher's self-efficacy? In all of the reflective practice groups, changes in teaching practices were observed. These, more than any other aspect appeared to be the key learning opportunity for the teachers involved. The support that the group provided combined with classroom implementation within the teachers' context provided a powerful combination for professional learning. In this section the different reflective practice groups are examined individually. For each group, innovations or changes in practice are discussed. The discussion follows a chronological sequence but ignores gaps that may have occurred between meetings. Following the discussions for all the groups, a summary providing comparative information is given and used to assess whether the type of reflective practice had any bearing what was accomplished.

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Lesson Study Other's Mark. Other's mark was explained during a meeting as being similar to a Kagan strategy and amounting to setting up cooperative pairs where each student would receive the mark of the person they were paired with. A strong student would then be motivated to explain how to accomplish the task so that the student they were paired with would earn them a good mark, the weaker student would be relieved of the stress of worrying about the evaluation and could focus on learningfromthe direct peer support they were receiving. The idea of using "other's mark" arosefroma dialogue between the participants regarding the distinction between 'communication' and 'group work'. Initially, the two were treated synonymously by one participant,"... the reality of the education is much more group work, more communication," and the other two participants implied a distinction (for example, "I think there is a sense in which, too much group work impedes communication.") The issue was resolved when the original participant was able to use the distinction to return the discussion to the competitive nature of his classroom by focusing in on the possibility of using group structures; "I am thinking group work based on Kagan strategies." (Note that the topic of communication was pursued in other discussions and is addressed in the section on the 'curriculum achievement chartframework'.)With the teacher of the competitive classroom having now steered the conversation toward the use of Kagan strategies, there was an implied willingness, on his part, to consider options. A colleague then proposed that 'other's mark', while "not an official Kagan strategy", might be a good way to address the competitiveness of students. The other two participants had never heard of the technique and had no knowledge of it.

75 All three of the teachers had received training around Kagan strategies and used associated terminology with a common understanding during their meetings. The description of 'other's mark' as being similar to a Kagan strategy made a connection to familiar concepts for the other two participants. The implication was that this new technique was a feasible extension of what the teachers were already using. Kagan (1995) has expressed concern about grades being allocated inappropriately and, in spite of the name, it was unclear whether the 'other's mark' technique would be used to generate a grade the way it portrays or was really a motivational rouse with a real mark being based on engagement. The participant who was familiar with 'other's mark' explained it to the other two participants. Students who are at different levels of achievement are paired and have to solve a problem. They have to present solutions which will be marked but they will receive the partner's mark for the work. It should be noted that there was some discussion around this marking scheme but it appeared to be a simplistic way of explaining mat engagement in the problem solving process, rather than just the solution, would be marked. The high achieving student is then motivated to make sure that the lower achieving student understands how to solve the problem to the extent that they create a high quality solution. The low achieving student has someone to work with but may be called on, by the teacher, to present the solution in order to insure that they contribute in a significant way to the solution of the problem. Following the initial explanation of the technique, there was a relatively short discussion in which there was agreement that it was an interesting technique that would be interesting to try. A side benefit that was identified was the development of student peer support that extends to other circumstances. The hypothetical difficulty of parents of competitive students phoning the teacher with concerns was also mentioned.

Following an in-class demonstration of the technique, one of the observing teachers asked what the next step in the technique would be. The explanation was that what was observed was the entire technique. Apartfromthis specific question, there was a very positive tone as the discussion shifted to the researcher asking a series of questions regarding the observers' efficacy. Initially, the two participants who were unfamiliar with the technique responded in a positive manner. One stated that it "sounds excellent" while the other was more reserved. Following the classroom observation, in which the teacher familiar with the technique modeled it for the other two teachers, the participants remained positive. Both participants indicated confidence that they could now successfully implement the technique in their own classrooms. That is not to say that they would not have to make adaptations for their own context, but rather that they were confident that they could execute the method and begin developing the adaptations required for their situations. Discussion about adaptations was primarily focused on characteristics of individual students in classes and how to achieve good pairings. One participant indicated that his initial trial for 'other's mark' would be in a language arts class because he perceived that class as the most likely to lead to a successful implementation and that would help him attain a degree of mastery in the technique before trying to implement it in more challenging classrooms (such as his highly competitive math class). He also indicated that he would have no hesitation to discuss any implementation challenges with the teacher who demonstrated the technique. The other observer, while positive about the technique, maintained some reservations about implementing the technique which reflected concern over the potential for this technique to interfere in a successful "culture of collaboration" that she was observed to have in her classroom.

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The 'other's mark' technique appears quite general in its potential application. This technique is not constrained by pedagogical content knowledge and appears to be generally applicable across different subjects. Two constraints that were observed were both specific to the implementation of the technique. Firstly, there were specific contextual details such as the choice of student pairings and, secondly, the pedagogical conflict between this technique and successfully implemented alternative techniques. It is also noteworthy that this particular innovation demonstrated that reflective practice can facilitate a direct transfer of knowledge from one participant to the other two who had been unable to previously use the technique because they did not know of it. Ordering Fractions. The task of putting fractions and decimals in order was mentioned as an example of teachers routinely talking about specific components of teaching. The point was being made by one participant that they, like the other two participants, get together regularly with a grade level colleague, in the absence of formal structure, and discuss areas of concern. Another participant, as an example, recounted having had a conversation (quite separate from the lesson study) the day before about ordering decimals. The discussion had begun during playground supervision and subsequently carried over into part of their preparation time. According to the recounted story, an idea that arose in the discussion amounted to using the students' existing understanding of alphabetical order as a building block for teaching students how to order decimals. The teachers who were engaged in the conversation were familiar to the teacher who had the pedagogical concern. In addition, the meeting was organized around convenience having been initiated during a common supervision time on the playground. The

78 teachers were sufficiently professional to continue the conversation after the supervision was done. This afforded them an opportunity to discuss the matter further. Within the lesson study, this example was being used to demonstrate that teachers routinely consult one another. That a participant could immediately come up with a clear example supports the idea. The events were outside the reflective practice group and there was no follow up to see what classroom implementations arosefromthe discussion. This example appears to have depended on a convenient circumstance arising during the course of the day. It demonstrates that consultation does arise with a minimum of formal reflective practice structure; however, the depth of the ideas arisingfromthe conversation is in contrast to changes, such as 'other's mark' that arose through the reflective process. Curriculum Alignment. The lesson study focused the first meeting on vertical alignment between grades 6, 7, and 8 for equation solving. The discussion around this topic, in the planning stage, highlighted different views and helped create a shared understanding of what solving equations in the intermediate division means. Between the first and second meeting, the participants consulted the curriculum expectations specific to solving equations (provided in Table 4.1). The discussion of equation solving highlighted a variety of distinctions between the grade levels: • •

Grade 8 has integer solutions but grade 7 doesn't. Grade 8 was characterized as ".. .we do ... anywherefromdistributive property and like terms to three different terms by the end of grade eight." • Grade 7 solves equations like x + 4 = 10 but not 2x + 4 = 10 • Grade 6, unlike grade 7, is "through investigation". The discussion continued with the grade 6 teacher suggesting that she could also teach equation solving and this was accepted provided that investigative methods were available to the students.

Table 4.1 Equation solving in grades 6, 7, and 8 Ontario Curriculum Expectations (2005) Grade 6

Determine the solution to a simple equation with one variable, through investigation using a variety of tools and strategies (e.g., modelling with concrete materials, using guess and check with and without the aid of a calculator) (Sample problem: Use the method of your choice to determine the value of the variable in the equation 2 x « + 3 = ll.Is there more than one possible solution? Explain your reasoning.).

Grade 7

Solve linear equations of the form ax = c or c = ax and ax + b = c or variations such as b + ax = c and c = bx + a (where a, b, and c are natural numbers) by modeling with concrete materials, by inspection, or by guess and check, with and without the aid of a calculator (e.g.,"I solved x + 7 = 15 by using guess and check. First I tried 6 for x. Since I knew that 6 plus 7 equals 13 and 13, is less than 15, then I knew that x must be greater than 6.").

Grade 8

Solve and verify linear equations involving a one-variable term and having solutions that are integers, by using inspection, guess and check, and a "balance" model (Sample problem: What is the value of the variable in the equation 30x - 5 = 10?).

The key pedagogical concern raised during the planning phase was that the teachers would be seeing vertical alignment but also updating their understanding of how much the curriculum revision in 2005 had changed the details at the other grade levels. One participant detailed a

number of changes with respect tofractionsin grade seven and eight that surprised the grade six participant. All three teachers were aware of the changes within their own grade level having been in-serviced about this component of the changes. Lesson study is often thought of as focusing on a single lesson which is refined by a group of teachers at the same grade level. In the observed lesson study, a participant noted in a postlesson meeting that adaptation in vertical alignment will be due to both observations of other classes and also differences in the curriculum expectations. Details of curriculum differences were not revisited in the debriefing sessions. When asked about the overall process, the participants indicated that it had provided a broader picture of the curriculum and pedagogical techniques that each teacher could then adapt to his context. What the teachers discussed in the debriefing sessions was consistent with the curriculum expectations that had been described in the planning. The lack of discussion regarding the curriculum in the debriefing sessions suggests that there were no obvious discrepancies in the grade level curricula that were implemented. During the discussion the researcher tried to extend the conversation to a pedagogical content knowledge issue that has arisen for him while teaching equation solving in grades 9 and 10. The issue was that order of operations teaches students to multiply/divide before they add/subtract so that 4 + 2x5 has a value of 14. However, when equation solving is taught and 2A + 4 = 14 is to be solved the rearranging addresses the add/subtract is addressed (i.e. 2A = 10) before multiply/divide (A = 10/2). In this sense, equation solving reverses order of operations (examples such as 2(x+3)2 - 7=11 that arise in grade 10 academic go much further in highlighting the point). The reaction of the intermediate teachers highlighted the breadth of scope of their content knowledge. The grade 8 teacher understood the concern and suggested that "if I were teaching a regular class, now, I can always say 'whatever is furthest away from the variable' do

81 first." The grade 7 teacher felt the process ignored the zero-principle that is required for the manipulation of values in equations and he demonstrated what he meant using algebra tiles. The discussion demonstrated that the focal distinction in grade seven is on understanding how equation solving works, whereas in grades 9 and 10, the process has been much more internalized. While he acknowledged that internalization occurs even as students move from grade 7 to grade 8, he did not readily acknowledge why one would draw a connection between equation solving and order of operations. During this discussion, the grade six teacher appeared uncertain about the discussion, which was substantially more complex than the grade six teacher's content knowledge, and subsequently she asked some questions about how the algebra tiles are used for equation solving. The discussion on curriculum alignment highlighted two key features: the influence of the community of practice on the reflections and, secondly, the contextual dependence of pedagogical content knowledge. The community of practice was evident in the use of Ministry curriculum documents and a subject leadership document to compare the grade level expectations. It also highlighted a connection, through content, between the grade eight teacher and the secondary school teacher. The reflections led to an explicit comparison of perspectives toward equation solving that highlighted specific pedagogical content knowledge approaches to address the changing needs of students as they develop their math skills. Lastly, it is important to note that reconciliation of equation solving and order of operations at the grade ten level highlighted how discussion can be constrained by content knowledge or perceived contextual relevance of the group members, in this particular example the issue was unresolved.

82 Algebra Tiles. Algebra tiles developed as a topicfromthe curriculum alignment discussion. They had been mentioned on a previous occasion as a focal topic, within equation solving, for the lesson study. However, the idea did not take hold because another participant indicated they simply did not know enough about using algebra tiles. The discussion about the relationship between order of operations and equation solving led one participant to demonstrate equation solving by means of algebra tiles on a SMART Board. While the conversation at the time did not pertain to algebra tiles, the grade six participant inquired about their use and this led to a tutorial and subsequent effort at implementation on her part. The tutorial had a detailed demonstration and included several questions and comments that were quite important for the grade six teacher to subsequently use them in her classroom: • • •

What antecedent conditions would she need to set up? Who in the school has class sets of algebra tiles? The comment was made that there are sets that can be used on an overhead for demonstration. • Keeping the numbers in the equations relatively small. While this fits with the pragmatic reason of only having so many tiles in a kit, the actual explanation given was the students perceive equations with larger values as "a more abstract problem." • Concern over whether the integer skills would exceed student ability was addressed as using the tiles as a visual indicator to help them solve the integer aspect without necessarily having a full understanding of integer addition and subtraction. In addition, the conversation pointed out that she would only be having integers in intermediate steps of solving and that the question could be set so that the final answer was positive. While the key tutorial discussion took place at a lesson study meeting, the teacher had subsequently had another shorter review meeting with her colleague when she went to pick up the algebra tiles kits. She later explained this visit as: Well that would be why I went to [participant who gave the algebra tiles tutorial] about the algebra tiles. It is exactly the same thing, I was like; Show me that one more time because I think I have it, but I am not one hundred percent sure.

The grade six teacher took what she had learned about the algebra tiles and implemented them in her classroom quite separatelyfromthe lesson study group observing her class. The algebra tiles were used to introduce equation solving to her class, where the lesson study class involved problem solving using equation solving. The grade six teacher who implemented algebra tiles reflected on how it went: I don't think I would change it, I think I would use it more often, and think of other ways I could use algebra tiles. So I personally, would have done more follow up with it, but I can't necessarily leave that for somebody else [(in reference to having a supply teacher in that day)]. Two weeks after the detailed demonstration of how to use algebra tiles, and before the implementation, I asked the grade six teacher if she thought she would use algebra tiles and she gave a very positive response. After her initial use of them in the classroom, I asked if the conversation at the earlier meeting had put her at ease and she responded "Yes. Oh definitely. I never would have used them before." As for the future, she indicated "I'll go back again." A crucial detail in the use of algebra tiles was the knowledge of where to find the kits in the school. The demonstration that was given was done on a SMART Board and the teacher did not have any kits on hand. This was the single key item that would not have been resolved in more traditional professional development settings. While the focus of the algebra tiles was on the grade six teacher who implemented them, the grade eight teacher indicated that he liked the idea of using manipulatives but the algebra tiles were well understood by his students. He indicated having no doubt about the utility of using algebra tiles because he sees his students using red and blue markers during tests to draw pictures of tiles. However, he indicated many of his students have internalized equation solving and returning to algebra tiles would "confuse them". It was suggested that a novel use for the teacher might be to get his grade eight students, with minimal instruction, to explain how to

solve an equation using the tiles. While this suggestion was made, there was no indication, during the observation period of this study, to suggest that the grade eight teacher would follow up on the suggestion. The lesson study use of algebra tiles highlights many features of reflective practice. It started with an area of concern, 'How to improve equation solving?', and through peer support generated some suggestions. There were several iterations of providing pedagogical content knowledge and addressing pedagogical questions that emerged. However, ultimately all the questions were resolved. The teacher, having learnedfromthe experience (by expanding her pedagogical content knowledge), made the decision to implement the use of the algebra tiles because her self-efficacy had increased sufficiently. However, just prior to actually implementing the tiles, her self-perception caused her to reflect on how well she understood and this caused her to consult her peer for a review of the pedagogical content knowledge. Since this went well, she stuck with her decision to implement the tiles, her self-efficacy was sufficient and she implemented the tiles. Her subsequent post-implementation reflections show that her selfperception has maintained a high level of self-efficacy and that it is quite likely she will use the algebra tiles again. The teacher's experience of learning how to use algebra tiles during a reflective practice to ultimately implementing them and having her self-efficacy increase due to enactive attainment (Bandura, 1986) exemplifies the relationship between reflective practice and self-efficacy for a single classroom technique. Curriculum Achievement Chart Framework ("Kitca"). "Kitca" was a term used in the meetings to refer to a system of assessment and evaluation that is based on the categories listed in the provincial curriculum achievement chart. The acronym is derivedfromthe achievement categories: Knowledge, Inquiry/Thinking,

85 Communication, and Application (KITCA). The achievement chart was not new as it was included in the 1997 curriculum document. However, the alignment of assessment and evaluation with the achievement chart is new for the intermediate division. The assessment and evaluation system has been in place in the secondary schools (giving the researcher familiarity with it) for several years. The system combines the use of weighted achievement categories with using rubrics based on the achievement categories. The key problem this posed at the secondary level was how to use rubric based marks to generate a percentage required for report cards. However, during the secondary division implementation, mere was much discussion of other issues and, not surprisingly, some of these were observed in the preliminary implementation phase of the intermediate division participants. For the intermediate teachers, it appears that new software and a modified report card will be included in the innovation. The KITCA issue represents an innovation that the lesson study teachers were aware of as a focus for the following school year; however, they reported having sketchy information regarding it. As an impending systemic change that is in its preliminary stages, KITCA does not fit the type of classroom changes this research was examining, but, it is included because the lesson study provided a forum in which one teacher's view of the systemic change was observed to change. The importance of this innovation is not the innovation itself but the way in which the reflective practice assisted a teacher in coming to terms with the early stages of the innovation. Two concerns were raised about the KITCA innovation. These do not necessarily reflect the reality of the innovation, however, they do provide an understanding of the level of uncertainty the participants had regarding the innovation. Firstly, using the achievement categories may be in conflict with the present use of Bloom's taxonomy and Kagan structures, and, secondly, concerns regarding theflexibilitythat the school system provides for the weighting factors.

One teacher, who seemed the most concerned about the KITCA innovation, reflected on his lesson that "I need to do a lot more with 'Kitca', a lot more with 'Kitca', starting now." This was reiterated later; "So I reflect, that I know, I know [emphasis], I have to change my style of teaching and more problem based, word problem based, problems." This same teacher said that, through the process of observing his lesson study colleagues, he recognized the difference in their usage of cooperative groups and that the discussions had highlighted how addressing mis pedagogical component would help him to make the changes required for KITCA. This changed the teacher's viewfromwaiting for the change to be delivered to a spirit of tinkering with different facets in preparation for the change. There were two features of the curriculum achievement chartframeworkthat make it relevant to this study. First, is the causality of change where an impending organizational change at the board level caused a teacher to perceive (rightly or wrongly) that his pedagogical methods would not be sufficient for the systemic change. Secondly, that the lesson study provided an opportunity to observe other pedagogical approaches and recognize alternative practices that might facilitate the implementation of the systemic change when it arrives. The second feature led to increased self-efficacy as reflected in the participant's confirmation that after the lesson study he was in a position to explore some options for alternative pedagogical methods that would be useful for implementing the systemic change. SMARTBoard. SMART Boards (SMART Technologies) are an interactive whiteboard that allows the interactive display of a computer screen and also conversion of work done on the interactive whiteboard to a computer file. Thefrontprojection SMART Boards are relatively new in the school board where this research took place but sufficiently abundant that most schools seem to

have some. The novelty of the boards was evident as they arose in conversations at all three public schools the researcher visited. In the early stages of planning, it was established that the three teachers in the lesson study would be teaching grade appropriate lessons around a common content topic. In making the point that the lessons may differ, the only teacher of the three who had a SMART Board made reference to the fact that he might use it where the other two might use other resources available to them. (He later decided not to use the SMART Board.) Throughout the second meeting, the SMART Board provided some utility for the demonstration of algebra tiles and, during scheduling discussions, to display a calendar. In both cases, resources that were not based on the SMART Board were available. When the grade six teacher used what she had learned about algebra tilesfroma tutorial that used the SMART Board, she did so with manipulative kits that would work in her regular classroom. In subsequent meetings, the SMART Board was referred to but simply as a point of reference for the algebra tiles presentation and to discuss locations of students in the classroom. The only usage of the SMART Board during the lesson study classes was for setting up the focal question in the grade seven class. As an educational innovation, the SMART Board represents the confluence of two extremes; its value rests with those who have access to one. The device is a vehicle for presentation and its educational value can only be measured in terms of the available tools and methods for using it. The two teachers and researcher (who also teaches) do not have SMART Boards and showed an interest in being aware of a technology they may be using down the road. There was no sense of immediacy in using the SMART Board, or in the details of what pedagogical content knowledge can be conveyed effectively using it.

88 The SMART Board is included here because it highlights how access to resources confines pedagogical options. At the same time, it is important to recognize that the demonstration of algebra tiles on the SMART Board did not constrain a teacherfromusing a traditional version of the resource. This implies that pedagogical and pedagogical content knowledge have constraints based on access to resources but mat the reflective practice forum provided a means to consider options that might address the constraints. Conclusion For the lesson study group, the focal area of discussion was the classroom and teaching techniques as well as some secondary mentions of influences from the board and school administration. Early planning meetings led to a focal area for peer observation while later planning meetings included dialogue about different approaches to teaching content. Specific methods of teaching were chosen by individual teachers and resulted post-observation discussion. The discussions were particularly insightful where the classroom perspective was discussed and teachers reflected on how implementation in their classrooms would differ. A summary of the different vignettes is provided in Table 4.2 Table 4.2 Summary ofLesson Study Vignettes Vignette

Source of Reflections

Area Affected

Influence on SelfEfficacy

Other's Mark

Peer in group.

General pedagogy

Increase for method. Split for implementation.

Ordering Fractions

Convenience peers

PCK

Unknown

89 outside group. Curriculum

Community of

Subject pedagogy /

Alignment

practice

PCK

Algebra Tiles

Peer in group.

Subject pedagogy /

Small, non-specific

Increase

PCK Curriculum

School administration

Achievement Chart

/ Board

General pedagogy

Small increase

General awareness

No change

Framework (KITCA) SMART Board

School/Board

Peer Coaching The peer coaching group was focused on the use of the Professional Resources and Instruction for Mathematics Educators (PRIME) resourcefromthe very beginning. The kits had arrived at the school the previous school year and the vice-principal had encouraged the peer coaching group to form around the idea of looking at the PRIME resource. In the first meeting, twenty minutes was spent scheduling dates and then the vice principal set the focus on the PRIME resource: So the first thing that I thought we would probably want to launch into would be doing the PRIME math assessment. So if we have time today I thought it would be helpful if we could break them out again and look. Okay, what would I have to do? How would I schedule the kids? How am I going to make it operationalized? The group focused on the PRIME topic of numeracy and operations and several issues arose and were addressed. It is the issues that are the topic of this section rather than the outcomes of using the PRIME resources.

There was concern about the identification of appropriate students using the diagnostic testing. This was addressed, in part, by a teacher who had used the diagnostic tool with a few students the previous year. One teacher provided a synopsis that they needed students who had a good understanding of the basics, were homogeneous with respect to ability and likely to be in attendance on the days of the intervention. For the peer coaching project, the feeling was that the diagnostic test could be used to identify an area of weakness among a small group of students but the time scale of the project would be too short to have a meaningful post-intervention diagnostic. The conclusion was that "it is going to be the perception, the perceptive, or the perceived change in their ability in such a short time" and that".. .the goal is for you [(the teacher)] to become familiar with the kit so that you feel more confident using it in daily practice or as a team." There was also a suggestion that they could revisit the diagnostic tool by talking to each of the students who was in the intervention group about the specific diagnostic questions relating to the intervention to see if there was an improvement, but the general feeling was that there needed to be a delay to be certain of retention. A teacher raised the question of how much time was required for the diagnostic testing and another teacher indicated that it would take twenty minutes or less. The teacher then inquired about testing more than five students in order to select the five that she would use during the intervention and she was reassured that she would have time to do this. Later the teacher asked how easily the diagnostic tool would be to grade and again her peers assured her it was straightforward. There was some debate about whether the peer coaching group would find a common theme among all the diagnostic test resultsfromboth grades one and three. This led to the mantra

91 "Look, see and plan" which implied looking at the diagnostic results to see if there was a theme and then planning the intervention based on what was found. A common theme, counting backwards by twos, was found among students of both grade levels and became the focus. (In the member checking, this was clarified as the focusing on the meaning of counting by twos both forward and back, not just rote learning, but with a focus on understanding.) This group had determined a schedule before determining exactly what they were doing; as a result an issue arose with the timing of the use of the diagnostic tool and having the lesson observation scheduled for the next day. It was felt that having the lessons taught the next day might not give the teachers time to reflect on the meaning of the diagnostic result. In the end, there was so much shared discussion around the lesson planning that the problem was resolved without changing the schedule. There was a discussion of several possible formats for the lesson that was to be observed. In the end, the group decided that the focus was really to be five students who showed difficulties on the diagnostic tool. The decision was made to remove thefivefromthe regular classroom in order to focus a twenty minute lesson on the needs of those five students. The pair of grade level teachers (two taught grade one students and the other two taught grade three students) would then discuss the grade level lesson before switching and having five studentsfromthe other teacher's class. It was also decided that the students would only be taught by their regular teacher and the other teacher would be strictly an observer. A grade one teacher described it this way: [The other grade one teacher] and I kind of have this written out, here is what we have. On May the 4th, I will teach myfivebodies and then we stop and talk, and then she will teach herfivebodies the same day, right, then stop. And when she isfinishedwe stop and talk. Then we will retalk at the end of the two lessons. The "retalk" referred to getting all four teachers and the vice principal together to discuss the progress of the group as a whole.

The PRIME resource was used as a diagnostic tool and some of the ideas specific to the lesson were examined. As the peer coaching progressed, the role of PRIME became less clear, "So mostly we learn about the delivery of the lesson and how it affected the children and what the relationship is to the lesson that we planned." This also arose when applying the diagnostic tool to more than five students was suggested, "Then you will have more of a sense of your class." Here again, the focus was on 'diagnosing' the class rather than focusing on the strategies the resource provided to address the results of the diagnostic for the five specific students. There were mentions of consulting the resource, including a discussion of how to use the resource to get information specific to a pedagogical content knowledge element of the diagnostic tool, but the extent was not entirely clear. As the peer coaching moved into the lesson planning, one participant expressed uncertainty over the methodology when each participant was going to be an observer for a grade level partner and also have a teaching role. "I just don't understand if we are doing kids or we are doing teacher practices, that is what I don't understand." She became more comfortable when the other pair of teachers provided a structure for the peer coaching that clarified the methodology. The process of setting the peer coaching in motion involved organizational leadership providing a focus and a series of reflections pertaining primarily to pedagogical matters and resulted in pedagogical questions that were resolved using reflections of other participants. There were several questions that paralleled research questions such as the validity of using the diagnostic tool in a pre- and post- format to assess changes and the reliability of the diagnostic tool. However, these questions were unresolved except to the extent that teachers rely on the quality of products supplied to them and their individual views after using the resource. It is also

93 noteworthy that leadership was distributed within this group in the sense that the formal administrator was absent for significant portions of time as the teachers directed themselves. Pedagogical Content Knowledge. The peer coaching group was working in the primary division and this was the only one of the four groups where the researcher did not feel he had a comfortable grasp of the pedagogical options for teaching the content. While following the group, the nuances of different techniques and different manipulatives were not fully apparent to the researcher. This limitation was specific to this particular group. During the discussion of the diagnostic tool, one teacher explained how she would verbally associate a value of five with objects such as pens and, as she added a pen into a group, would count up by fives, subsequently when asking how many there were the students would tell her the number of pens rather than the associated count by fives. The researcher suggested the use of plastic bags with collections of five objects in each as a visible way of drawing a distinction between the number of groups (i.e. the bags) and the number of objects in the groups (i.e. number of objects in the bags). This idea was adapted as a stepping stone toward removing the bags and genuinely counting up by a fixed amount (while moving that number of counters). A teacher was surprised by a diagnostic question that asked the students to "Count them any other way other than by ones." Her surprise was in the explicit specification of not counting by ones and she was concerned that her students might not understand what counting by ones was referring to. The concern may also have been one of language where students might have been hindered by the wording of the question as opposed to math ability. One of the other members of the group indicated that they routinely ask this question, which reinforced the ideas

being put forward by the PRIME resource. He further went on to explain the importance of the question as a skill specific to teaching multiplication. During the lessons, the researcher focused primarily on the grade one lesson with periodic attention to the grade three lesson that was occurring simultaneously in a separate part of the room. In other words, the researcher focused on the two grade one teachers, one teaching and the other observing, while five studentsfromthe instructing teacher's class were given a lesson. Subsequently, the teacher and observer changed roles and five different studentsfromthe class of the teacher assuming the instructing role were taught. (The grade three teachers were following the same approach in a different part of the room.) During the grade one lessons, it was noted that, when counting up by twos, the teachers stopped the process at twenty. The researcher also noticed that the grade three students were able to count to twenty by twos but had trouble after twenty. Twenty was stated to be the limit of the expectation for counting by twos. Comparison of curriculum expectations suggests that the limit of twenty is a pragmatic interpretation of the curriculum. The expectations are to "count forward by l's, 2's, 5's, and 10's to 100,..." (Ontario, 2005, p. 33) in grade one, "count forward by l's,2's, 5's, 10's, and 25's to 200,..." (Ontario, 2005, p. 43) in grade two, and "count forward by 1 's, 2's, 5's, 10's, and 100's to 1000fromvarious starting points,..." (Ontario, 2005, p. 55) in grade three. The upper limit appears to pertain to the highest increment (i.e. counting to 200 by 25s in grade two is reasonable) and leaves the limit open to interpretation for the other amounts (i.e. does it mean students in grade two have to count to 200 by ones?). The lessons that were observed used quite a few approaches that included the use of counters, hundred charts, bingo daubers, a walk-on number step (Small, 2005, p. 39), bags with counters, and song. The researcher may well have missed resources because with two

95 simultaneous lessons and unfamiliar pedagogical content knowledge it was difficult to keep track of all the activities. What was clear, however, was the teachers were using a diverse set of approaches in order to observe student responses to the different techniques. The observations regarding pedagogical content knowledge indicated that the teachers were trying a variety of techniques and manipulatives in order to assess what was having an impact and to learnfromwatching each other work with the different techniques and resources. This appeared to have two main effects. Firstly, the teachers were buoyed with excitement by the new ideas they had observed and tried and, secondly, they felt very positive that their efforts had had an impact on student understanding. It should be noted, however, that the group's reflections indicated concern about retention and concluded that the second set of lessons should reinforce understanding of the students. Pedagogy. A concern was raised regarding the assumption that students had the antecedent knowledge necessary for the instruction they would receive. The hypothetical example given was addressing a student's inability to count by five when they can't even count by ones. This concern arose after the use of the diagnostic tool, when counting backwards by an amount became the focus and a grade three teacher indicated that counting forward by an amount was also an issue even though the diagnostic tool apparently had not indicated this. The PRIME resource challenged some fundamental ideas of the teachers: "When I think of Piaget, I think of not being able to change a stage until a child is ready. However, these books say if we did these exercises, with the students, we could change the stages." This may well be a situation that was updating what the teacher had learned in teacher's college over twenty five

years earlier. The researcher recalls being led to believe that the stages identified by Piaget represented hard boundaries when he did his junior-intermediate qualification in 1990. This group benefited from two of the participants having some familiarity with the PRIME resource. Their ability to answer some of the initial questions prompted the process along and all participants gained more hands-on experience than they would have otherwise. Familiarity, however, had not translated into using the resource; "But we did see [PRIME], the people in the math thing, did a workshop a couple of times. Like [name of presenter] did a PRIME thing, so I thought of it a long time ago

I didn't actually use it though."

During thefinalday of the peer coaching, a teacher with a split grade one-two class used the diagnostic tool with her grade two students. The motivation was simply "I wanted to see the difference between the ones and the twos." This implies a degree of satisfaction with what had been accomplished with her five grade one students in the peer coaching. The peer coaching addressed one content component that represents a relatively small portion of both the math curriculum and the PRIME resource. In many respects, the teachers were likely overwhelmed by the sheer volume of information in the resource within the confines of the relatively short amount of release time that was available for the peer coaching. In spite of this, it is interesting to have observed teachers questioning some fundamental knowledge regarding Piaget's stages of development. This suggests that the peer coaching not only provides a forum for teachers to discuss classroom practices but an opportunity, time permitting, to consider much deeper conceptions of teaching. Conclusion The peer coaching group added to the study by extending the scope of inquiry to the primary division in which the pedagogical content knowledge is much more concrete. In

97 addition, the group had contributionsfroman administrator that clearly helped with the organizational details and they used the PRIME resource to identify a focal area and for instructional ideas. A summary of particular vignettes arisingfromthe group is provided in Table 4.3. Table 4.3 Summary ofPeer Coaching Vignettes Vignette

Source of Reflections

Area Affected

Influence on SelfEfficacy

Use of PRIME

Pedagogical content

School/

Subject pedagogy /

Unclear (Small

Administration

PCK

increase?)

Group / PRIME

PCK

Increase

Group / PRIME

Subject pedagogy

Small increase

knowledge Pedagogy

Pre-Service Mentoring Scavenger Hunt Activity. A scavenger hunt was developed by the pre-service teacher and consisted of a set of seven review questions related to permutations and combinations that were placed at different locations around the school. Each question guided the students to the next question. Students were assigned different starting points and, in principle, had to follow the sequence of clues in order to have the answers in the right order. The mentor ran the scavenger hunt during his class and then, in the next class, the pre-service teacher ran it with the modification that a prize was offered for the first group of students to return.

The scavenger hunt was designed by the pre-service teacher in a short period of time and intended to provide some review in an exciting format. The pre-service teacher indicated that she didn't know how to manage the activity, but the mentor used the activity in the first class and modeled a solution by not being overly concerned with managing the senior students. The mentor and pre-service teacher were diplomatic in answering questions about their views on the use of prizes. Their actions indicated a difference of opinion, and a respectful discussion saw them ultimately agreeing that prizes were fine as long as they did not isolate any students: Mentor: For something like that its fine. I used to do that a little. Pre-service teacher: Well I've baked cookies. Mentor: Yeah, fun prizes In the particular circumstance, the pre-service teacher hadn't had time to bring in a prize and was promising one at a later date. The activity, which had been used by the mentor's class the previous period, had not been left fully intact because a question attached to a water fountain had come off during the second period (likely due to condensation on the fountain). In addition, one team returned with a complete set of answers that were in the wrong order. These issues raised questions about student behavior awayfromthe classroom. As a result, the pre-service teacher decided not to issue a prize and, instead, had students consult a moral guide in their student planners. At a later date, the pre-service teacher brought in brownies that were shared among everyone after a game of Jeopardy. The mentor had also run the scavenger hunt activity but had no mention of prizes and none were presented. It was not clear if this may have contributed to the "measured" approach his class took to the task. However, the feeling was that the pre-service teacher's class was much

more excited and got morefromthe activity. In spite of this difference, the mentor felt that the activity was enriching for his class. The pre-service teacher experienced extremes of self-efficacy during the scavenger hunt. Sherightlythought the activity would motivate the students to complete some review questions and had high efficacy as students excitedly engaged in the activity. However, she had been concerned about questions being left in place for the second class and this concern turned out to be valid when a student brought a question back to class to clarify the meaning. The teacher described her feelings as "shock" and "frustration." During the second class, her efficacy rose again when a neighboring teacher inquired about the students in the hall and, upon hearing of the scavenger hunt, said "Your lesson plans are so cool!" The activity demonstrated the use of the school building as a resource but brought about issues regarding management of student behavior awayfromthe class. As the activity progressed, the pre-service teacher had shifts in her self-efficacy as various events, both good and bad, unfolded. It seems likely that her reflections, at the time, influenced her self-perception (both positive and negative) regarding her decision to run the activity and this had both positive and negative self-efficacy consequences. At the same time, the reflections influenced her understanding of the pedagogy of running such an activity and expanded her knowledge of this type of opportunity. The mentor was surprised by the differences between the two classes with one being quite reserved and calm while the other was quite excited and vigorous. These observations were puzzling and led to pedagogical questions that were unresolved. The Fourth Dimension. In secondary school math courses, there are some enrichment opportunities that can be achieved by introducing the idea of having more than three geometric dimensions. This can be

done, for example, through the use of Cartesian vectors with more than three coordinates. It is purely enrichment with no explicit mentions being made in the Ontario curriculum documents, NCTM Standards (2000) or the course textbook (Alexander, Harrison, Lenjosek & Taylor, 2003). The use of the fourth dimension as an extension arose in the mentor's class under a circumstance where he was basing much of his lesson on his memory of the textbook questions rather than a formal lesson plan. Toward the end of a Friday class, the mentor had insufficient time to start a new topic and chose to give the class a textbook question that, along with a four dimensional extension he added, would occupy the students to the end of class. When the question was taken up in the Monday class, the mentor reflected that the students "hadn't really thought about it." One benefit of the extension was some professional content discussion between the mentor and researcher considering the instructional merits of three distinct methods for addressing the four dimensional problem. These methods had been developed earlier in the class for a related three dimensional question. It is interesting to note that this discussion was an antecedent to a later professional discussion while jointly exploring a computer simulation of Rubik's cube in four dimensions. It is fair to say that neither the mentor, nor the researcher, is likely to use the simulators in the classroom. However, the exploration did point out the possibility of finding resources that might make the fourth dimension more meaningful to students through the opportunity to play with simulators. The pre-service teacher did not attempt to introduce the fourth dimension. She viewed it as an enrichment opportunity that fit with the end of a Friday class and something that had potential to give her a pedagogical challenge. She did, however, use the mentor's teaching regarding

101 direction naming (i.e. X, Y, Z and W for the fourth dimension) in her own class within a three dimensional context. In this particular case, the pre-service teacher did not feel comfortable addressing the fourth dimensional enrichment. Her concern was not the content, which she was familiar with through her undergraduate studies, but rather concern with how to represent it to the students visually. The mentor felt substantially more comfortable with the content and it provided an opportunity to share pedagogical content knowledge for making the extension. What made the discussion of the fourth dimension interesting to this study is that it was the only event in the entire study that pushed the pedagogical content knowledge boundary. The content was slightly beyond the ability of the students (which is why they hadn't thought it through) but within the realm of content that the mentor and researcher could see adapting to teaching if effective methods where available. The pre-service teacher understood the content but recognized that it was outside of the pedagogical content knowledge that she believed she could convey to students. While other events show teachers learning and implementing new strategies, this one event highlighted the challenge of two professionals dialoguing about ways to develop the strategies that might allow implementation. Homework Policy. The homework policy for the mentor and pre-service teacher allowed two days for completion rather than one because mere were insufficient textbooks for each student to have one. The researcher arranged a loan of textbooksfromanother school thereby providing an opportunity for the homework policy to be changed if desired. Consistent with the policy, there was evidence of homework being done at both the last minute and on the day between it being assigned and being due.

102 On one occasion, the mentor took up the homework one day after it was assigned in spite of the two day policy. This may have been due to the teacher constructing his lesson as he taught rather than having an explicit lesson plan, a need to emphasize what good explanations look like and an opinion weighing routine homework checking with the number of assessments that were needed. The pre-service teacher based her homework policy, in part, on reflective feedback that the students had provided. She had received feedback three times during the course and the feedback had prompted her to change "the homework to every two days,fromone, and also changing it to completeness rather than accuracy so that it is more formative rather than summative." The preservice teacher, having received enough textbooks for every student to have one, decided not to change her homework policy because she had already altered it once in response to feedback and, had recognized other benefits from the current two day approach. The pre-service teacher had one homework assignment due by email on a Sunday. When asked, the mentor indicated that he was not concerned and would personally have made it Monday. The mentor showed hisflexibilityby letting the pre-service teacher experiment. The events around homework constitute relatively minor adaptive changes. It is natural for a pre-service teacher to experiment and the use of student feedback was an excellent technique. What these events do show, however, is that simply resolving the original issue, namely insufficient textbooks, does not necessarily take into account contextual changes. When the textbook issue was resolved, a change in the homework policy had already been established based on student feedback. It was also late on in the semester. Any change in the homework policy required reflection on the teacher's part, in the current context, to properly consider its impact.

103 The homework issue is useful for isolating some natural personal reflection that the preservice teacher was going through as she experimented with different options in the classroom environment. While the mentor provided a useful contact, the homework issue was not discussed and there was a recognition that the pre-service teacher needed to use her own reflections and self-perception to generate a level of self-efficacy that was sustainable without a mentor's support. The mentor also reflected on the homework policy and chose to override the policy for one of his classes because of a specific pedagogical issue that he felt had to be addressed but otherwise he was satisfied with the policy and did not consider it a point of discussion. Language Teaching. Language was addressed in a variety of ways in the observed math classes. The students were predominantly Chinese with only one student in each class who was not ESL. However, English was the language of instruction and usage was quite good for many students while others regularly consulted pocket electronic translators. What was evident during the reflective practice was a difference in how the mentor and pre-service teacher addressed language and the absence of explicit dialogue during reflective discussions. Language instruction, by the mentor, was incorporated into regular instruction. This was due in large part to having taught ESL for five summers and having taught in Guyana for three years. One example of language being addressed by the mentor was a class where explaining and the meaning of the word 'explain' were focal points. The pre-service teacher, by contrast, was inexperienced in dealing with language. While making permutations of the letters in 'boy', she mentioned 'yob' as the word 'job' spoken with a Spanish accent (her pre-service program was based in the South Western U.S.), in using the standard permutation symbolism nPr she referred to this using the U.S. acronym National Public

104

Radio as a mnemonic and assumed that it would be common knowledge in Canada. It should be noted that the standard symbolism „Pr was also presented in other symbolic forms, including one that is apparently common in China. For the pre-service teacher, language was not yet viewed as a component of instruction. The only observed instance with clear instructional benefit arose while implementing a translation strategy for teaching permutations of letters. The word "wow" was used and the teacher raised each of her hands with three fingers spread to make two letter Ws while opening her mouth to make the letter ' 0 \ This was a spur of the moment attention grabber that the mentor respected; "That was really great, I liked that one." It served to grab the attention of students while addressing the issue of permutations when there is a duplicate letter. The contrasting views of language between the mentor and pre-service teacher were discussed and the researcher pointed out that the mentor had instruction based on language whereas the pre-service teacher gave much less formal attention to it. The discrepancy was immediately recognized and the discrepancy in experience with ESL emerged as the focal explanation. The pre-service teacher stated "I don't think I have really thought about that." This suggests that the pre-service teacher's self guidance was focused on other priorities. The number of tasks that the pre-service candidate was engaged in was a significant constraint on her ability to have addressed ESL needs. She was focused on classroom management, lesson planning and content focused curriculum objectives, as one would expect, and other pedagogical facets, particularly context dependent needs, did not receive as much attention. The discrepancy in language was not addressed through the reflective process. This suggests a constraint on the reflective process imposed by the number of tasks the pre-service

teacher had and a second constraint arisingfromcontextual awareness. Addressing language, as the mentor did, suggests a broadening of one's view of "content" that comes with experience. The issue is not one of self-efficacy so much as what one defines as the scope of the teacher's teaching activities. It is not a criticism of the pre-service teacher, rather an opportunity to see pedagogy and content as growing in scope over the course of several years of experience. Towers ofHanoi. The pre-service teacher had looked at the prices of instructional resources and expressed shock at the expense. She specifically mentioned the 'Towers of Hanoi' and a 'conic section demonstrator' as desirable, but expensive. The researcher offered the loan of a large 'Towers of Hanoi' and the pre-service teacher instantly accepted. The pre-service teacher said she wanted a model of the Towers of Hanoi so that she could provide an in-class demonstration before going to the computer room and having students try it individually using an internet based resource. The Towers of Hanoi that was loaned was not used immediately. On one occasion, before the towers were used* a teacher and student from another classroom visited and started a discussion with the pre-service teacher around the model. This provided indirect support for the pre-service teacher's interest in having a model that would catch student interest. However, the pre-service teacher switched "between the concrete model and offering the website for an electronic version. She appears to have preference to the electronic one but it is likely so that each person can individually try it out." During the process of member checking, the pre-service teacher confirmed the perceived concern regarding student engagement in the activity. This suggests a belief in the content offered by the Towers of Hanoi but some uncertainty in the relative merits of the model compared to an internet simulation.

106 The interest that had been expressed by the teacher and student,fromthe other class, led students to inquire about the model Towers of Hanoi and this prompted the pre-service teacher to begin her lesson regarding it. This began with a handout that was read and an acknowledgement that some of the electronic translators that the students had included the Towers of Hanoi as a game. The pre-service teacher used a handout and demonstration of the Towers of Hanoi followed by a visit to a computer lab. In the computer lab, the students were to determine the fewest moves necessary for towers with different numbers of starting disks. The particular simulator did not work because Java was disabled on the computers in the lab (although it was enabled for the computer in the classroom where she had shown her mentor) and the students were guided to use a search engine to find another simulator. Unfortunately, the students found a simulator with a feature that one could press a button to have the simulator solve the puzzle in the minimum number of moves. This led students to fill out the handout sheet with less engagement than they might otherwise have had. The mentor chose to teach about the Towers of Hanoi using the model that had been loaned. However, during his lesson, the model came apart several times because the mentor was unaware that it had been designed for disassembly and easy storage. In spite of this challenge, he persevered and completed his lesson. He decided not to use the internet simulation because of a shortened schedule. The Towers of Hanoi manipulative broadened the pedagogical options for teaching. They facilitated having an in-class hands-on demonstration and discussion as the mentor did. They could also be combined with the use of the computer lab as the pre-service teacher chose. The pedagogical choice created a situation that required consideration of the relative merits of the

107 options. The mentor and pre-service teacher each addressed the contextual variables presented by their classes. However, they did this separately and, to the researcher's knowledge, did not discuss the approaches because of interruptions in the daily schedule and a significant increase in time demands for the mentor. The key factor was that having the resource available in two forms (model and Internet) caused reflection about the pedagogical components of each leading to individual comparisons of the relative merits of both methods. Teaching permutations using mapping. During a meeting with the mentor and pre-service teacher, the researcher suggested that a method for teaching permutations that might be particularly effective for ESL students would be a 'translation process' that maps letters onto different letters. For example, the letters in 'ski' can be arranged in six distinct ways, if ' s \ 'k\ and T are translated to 'p', ' o \ and 'p' then 'ski' becomes 'pop'. The benefit of this method is that, with two letters being translated into the same letter, the number of permutations can be calculated as six divided by two. This explanation, in the researcher's experience, is quite practical in classroom implementations. At the meeting, both the mentor and pre-service teacher responded positively to this approach. The day after the meeting, the pre-service teacher presented a lesson based on the method. As the reduction of the number of permutations, due to having two letters translate to the same letter, a student was heard to utter "Oh wow." This was not lost on the pre-service teacher who later assessed the class as "They were into it." The mentor judged the method as having value to some students and appears likely to develop the idea in the future; however, he chose not to revisit a class that he had already taught. He was clearly focused on a schedule based on the conclusion of the course and recognized an impending squeeze on the time available. The pre-

service teacher, by contrast, was in herfinalweek and used the opportunity to explore an alternative pedagogical approach. Whiteboard markers. The classroom was equipped with a two panel whiteboard. However, the pens that were being used were close to worn out. The researcher gave a new set of whiteboard pens to the two teachers to see what would happen. There seemed to be a reluctance on the part of the mentor to use the new ones and this led to the pre-service teacher asking "Can I have a new color because these markers are shot." In photos of the whiteboard that were taken after each class, there was more use of color after the donation of the pens. The use of color subsequently declined slowly and within a couple of weeks, the notes were dominated by a single color again. Later, the mentor recounted how a teacher had walked into his class and taken a whiteboard marker. Following this revelation, he acquired two new black pens and a green penfromthe office. Inspection of photosfromthis particular day shows three pens all of which are black. The mentor indicated that it is not uncommon to run short of pens toward the end of a semester but did not attribute this observation to any specific causes. The mentor indicated that his pedagogical use of color is content specific and he gave the example of graphing using different colors for the axes, curves, and points. In general terms, he described calculus as having more opportunities for pedagogical use of color than the discrete mathematics course that was observed. The content dependence idea was supported by the isolated reflective practitioner (gradefive)who stated, "I would change all my blackboards for whiteboards, because I can use color."

109 The use of color for pedagogical purposes was something the pre-service teacher had not considered. While she had shown interest in using color, when asked specifically about pedagogical use, she responded that it was "something to think about." Perhaps in the absence of a pedagogical purpose for changing colors of pens, and the given context of the observed teachers, it is natural for them to rarely change colors. The key contextual factors appeared to have been the convenience of not having to search for pens that move in a shared classroom, absence of choice as a norm that one is used to, and the convenience of not having to provide additional effort to maintain the option of using color (i.e. having to prevent people walking off with markers). The availability of pens did not have much impact. What it did show, like the issue with language, was a larger pedagogical scope arising with the experience the mentor had. Few changes by the mentor. Within the mentoring relationship, there was a disparity of commitments. The pre-service teacher more readily made changes in her classes. However, the mentor has made significant innovations in the past, such as implementing knowledge building software as a professional development forum, and appears to have been highly constrained by workload and time issues. His classes were effective but fairly routine and this facilitated an escalation in his workload due to significant assignments at the end of term, extra-curricular activities, and additional commitments within the school. Conclusion. The pre-service mentoring group expanded the content and pedagogical scope of the lesson study group. The range of experience covered uncertainties of the pre-service teacher through to considerations for teaching about the fourth dimension. The group also provided a seemingly

110 different context because it was a private school and had a vision guided by religious principles. A summary of vignettes is provided in Table 4.4. Table 4.4 Summary ofPre-Service Mentoring Vignettes Vignette

Source of Reflections

Area Affected

Influence on SelfEfficacy

Scavenger hunt

Pre-service teacher

General pedagogy

Increase and decrease

activity The fourth dimension

Mentor and researcher PCK

Unclear

Homework policy

Group

General pedagogy

No change

Language teaching

Researcher

Subject pedagogy

Unclear

Towers of Hanoi

Group

General pedagogy

Increase

Teaching

Researcher

PCK

Increase

Whiteboard markers

Researcher

General pedagogy

No change

Few changes by

Mentor

Not applicable

No change

permutations using mapping

mentor.

Isolated Reflective Practitioner The isolated reflective practitioner recounted a sequence of events that had occurred over a period of years. The details she provided show a migration in pedagogical thinking that has taken place over a period of years and is continuing to evolve.

Ill Philosophical Changes in Math Teaching. The isolated practitioner's account started with a change of principal and math being put in her timetable. She had previously taught two classes language while another teacher taught both classes math. This began a journeyfromviewing math teaching as the transmission of algorithms through to a multiple method, problem centered view. During this process, the curriculum was revised, however, the revision was not a philosophical transition of the curriculum but the change did provide some support that the participant was able to use. When she wasfirstasked to teach math at the junior level, the participant's feeling was: I thought 'Oh, it's a challenge' in uie beginning, I've got a lot of growing to do. I delved into some reading materials and originally had the old MathQuest series over there. I believed mat teaching math, back then, was teaching algorithms. This subsequently came into question through a progression of changes and, most notably, a particular resource, she referred to as 'Van de Walle' (Van de Walle & Folk, 2005), caused the participant to question her philosophy of math teaching. She recounted, for instances, reading a section during a summer and being stunned by the authors stating that they purposely did not supply any answers to add to the authenticity of the problems. The curriculum revision was supported by a Ministry initiative at the junior level which included regular workshops for two teachersfromthe school. At one of the workshops, the participants were given a copy of a resource by Van de Walle and Folk (2005). The participant subsequently adapted specific methodsfromthe resource and it provided ongoing support anytime the participant wants it. The workshops led the participant to understand the curriculum shift in pragmatic terms such as appreciating that a new textbook she was using directly supported the new curriculum.

112 During the interview process, the researcher brought a copy of the NCTM Standards (2000) to show to the participant. The researcher felt that Van de Walle and Folk (2005) was a resource book that developed many ideas from the Standards document into a form that might be more readily usable for teachers. She was unaware of it, but was concerned about it being American. The researcher pointed out that it has considerable similarity with the Ontario curriculum. She then expressed an interest in reading the sections relevant to her division. Through her involvement in the primary-junior task force, the participant has more exposure to the philosophy underlying the new curriculum: So the idea is, problem solving .... the whole idea of this which was really good, the whole concept, I felt, was very very good. It was leaving it more open ended. So that she had a wider range [of possible strategies]. The task force became a source of ideas that had potential for the implementation of best practices for the new curriculum. For example, "the focus this time was this new Japanese method called a Bansho" which is a technique for focusing student attention on the diversity of solution methods used by the class (Kestell, 2006). In the case of the bansho, it is a technique mat could be learned pragmatically with minimal understanding of its connection to the problem centered approach to teaching, however, the participant described the task force presentation including a comparison of closed and open sorting methods and details of the ideas underlying the technique. The participant indicated that her pedagogical techniques did not change as a result of her coming to terms with the change in math philosophy from the algorithmic view to the problem centered view. She indicated mat her pedagogy was closely tied to instructional intelligence methods which did not require the development of new methods to suit the change of philosophy underlying the curriculum. However, she also indicated that"... having taken some training in

113 teaching mathematics, the idea of teachingfroma problem solving base, that did profoundly affect my way I teach math." The participant consistently separated her change in view of math teachingfromher overall philosophy of teaching and view of teaching in other subject areas. While she didn't discount the possibility of her change of view of math teaching having an impact in other subject areas, she indicated that it would be a relatively minor effect compared to other changes that had arisen in the other subject areas. An overall summary of conversation was offered by the researcher, and confirmed by the participant, as: Researcher:.. .that is not changing the overall, who you are as a teacher, it is more instructional, more content specific and instruction specific. It is not changing your view of what teaching is or your philosophy or it's not like you have discovered some new pedagogical that is completely different than what you were doing. Participant: Right, no. One potential improvement was raised and that was altering the daily schedule to improve the implementation of problem centered instruction. Using constructivist methods might be facilitated, the participant suggested, by having longer math classes, more time for deeper exploration and consolidation would be available: ".. .the one thing I keep telling the principal is that to do this kind of math requires huge blocks of time with no interruptions and that is not happening." This issue was unresolved and possibly a reflection of the use, at some schools, of larger blocks of time being scheduled for literacy. The shift in math philosophyfroman algorithmic view to a problem centered approach evolved over a period of time. The participant recounted different sources of change: a professional document, Ministry funded workshops and a board level task force that works to improve math instruction. All three contributed to the transition of pedagogical philosophy. However, the contributions had an underlying philosophical difference from the teacher's conception and this discrepancy created pedagogical questions for her. In turn, while innovating

114 based on her understanding, the reflective mechanism would help resolve the pedagogical issues in preparation for the next cycle of learning. It is important, also, to recognize that there was little content expansion for the teacher, however, her pedagogical scope was broadened by including multiple methods. It is also noteworthy that at least one pedagogical question, what benefits a change in the daily schedule would have, remained unresolved. Accessing the Math Community. The degree of isolation of the participant has changed during the course of her reflections. When she was first told she would teach math at the junior level, it was within the context of her having very little connection with anyone else regarding the subject. There had been little, if any, collaboration between the participant and the other division teachers. In addition, the purchase of math textbooks that corresponded to the new curriculum was dealt with by the principal and a publisher's representative. As she has been involved in the reflective process, her understanding has grown and this has led her to engage with a community of math teachers who are outside of her school but primarily within her own school board. The participant was the junior lead teacher and received information from the new principal regarding workshops funded by the Ministry of education for a pair of teachers at the junior level to help implement the new curriculum. At one of the workshops, the participants received a resource by John Van de Walle and Folk. It also included money for purchasing manipulatives that support instruction which had the added benefit of causing the division teachers to organize math resources, together, in a common storage area that made sharing more viable than previously. In order to address the Van de Walle and Folk (2005) resource, the teacher arranged to allocate a professional development day to read a substantial portion of the Van de Walle resource.

115 It was through the Ministry supported workshops that the participant heard about the Ontario Association of Mathematics Educators annual conference which happened to be held within commuting distance of the participant's school. The participant put in some effort to arrange release time and conference fees so that she could attend. The conference was "A huge eye opener

The bottom line is, I go to this, I'm seeing." She further described the

experience as "... in the process I get to see mathematics, and I am dialoguing with people." This sense of being immersed in a culture of math teaching for the duration of the conference was conveyed in other ways as well. For example, the participant found a guide to problem solving on the back of something she had been given and it "made sense to me." Subsequently, she had made a laminated poster version for the wall of her classroom that she referred students to. She also realized that the ideas of Marilyn Burns were an up-and-coming focus and she also was introduced to the idea of a "balanced math" program. The participant, on a suggestionfromher principal, applied and was accepted to join a board organized task force on primary-junior math which provides leadership to teachers on mathematics. Within the first two meetings, a resource book was given out to members of the task force and they were introduced to an index of children's literature that involves math (for example, it mentions the Sir Cumference book series by Cindy Neuschwander). The group was also briefed on a new Ministry pamphlet "Guide for effective instruction K to 6" that is designed for teachers. The participant volunteered to be involved in the production of regular news brief regarding the teaching of math. An aspect of the math community that is often overlooked is the third-party supply of resources. The participant mentioned using products from a particular store which were primarily worksheets that fit with an older philosophy of math teaching. While she still buys resources

116 from the store, she is now more inclined to modify the format to better match the philosophy of the new curriculum. One aspect of the participant's efficacy was the component of taking on leadership roles in order to create opportunities that she was interested in. For example: .. .1 became the computer contact teacher this year, because I wasn't computer contact person last year, but then I thought we were missing things and our use of computers was going down. And SMART Boards became important to me when I went to OAME and I went to one of our workshops and I saw how to use it. This was combined with using information from professional acquaintances in order to make a successful application. This type of networking demonstrated access to a community but it was general rather than specific to the math subject area. One problem that the participant had was isolation within a small school. With no peers to routinely discuss math with, the only recourse were board specialists who were quite busy dealing with groups of teachers and didn't always have time for individual queries. Her involvement with the primary-junior task force provided a mechanism to access math expertise within an organized group. However, this did not come about through isolated effort; "(interviewer) This looks right up your alley. This looks like it hit you at just the right time. (Participant) Yes, exactly. Exactly, it just kind of fell in my lap. It would not, had the principal not supported me." Her acceptance into the group came in the form of a letter of acknowledgement from the director of the school board. This represented a level of respect and encouragement that was likely relatively rare given her isolation. The respect is in contrast to other professional development which the participant described as: I really like going to math and literacy, I hated [another subject area], I thought they treated adult educators as kids, I didn't like it, I thought [the person in charge] talked down to us And [the math] people treated me as an adult and all of a sudden I am okay with it...

117 The involvement in math specific activities has not been to the exclusion of other professional development. The participant recounted attending a Saturday workshop about brain based learning. She also pointed out that her participation in this research study was also engaging in the community of practice. What made this particular participant so relevant to the process of developing a substantive theory was her growing involvement in a community of practice. Through leadership mechanisms, she has been engaged in active efforts to enact the current vision of the mathematics curriculum. This has provided her with opportunities to discuss pedagogical principles while engaging both pedagogy and pedagogical content knowledge in her classroom. It is important, as well, to note the lack of shared experience recounted in the school. This may be due to differences in priorities or interests within the division staff. Change of Math Textbooks. The participant started teaching math, at the junior level, about four years ago using MathQuest (Kelly et al., 1987) books and continued to use them in spite of having new Quest 2000 (Wortzman, 1996) textbooks. She also used grade four textbooks with some students who were working at that level. However, the participant no longer had a full class set of textbooks when interviewed and used them for differentiated instruction according to need, and as a resource to develop handout sheets where she can copy a full class set. Textbooks matched to the new curriculum were purchased by the school a year after the participant began teaching math in the junior division. However, the participant was only marginally aware of the purchase ahead of time and not familiar with the details of why a new textbook was needed to support the new curriculum. She recounted having difficulties switching to the new textbook because it used a problem centered approach rather than a prescriptive

118 approach. In the first year of implementing the new textbooks, the participant had difficulty getting through all the material in the curriculum. The participant mentioned the use of multi-cultural names in the textbook but explained that it was more important for her students to see a personal connection to the mathematical content. "And another thing I found Nelson had was an awful lot of names they use, like Rajiv and Fadeek, great if you are in Toronto but not great if you are here." One aspect of the change of textbooks mat really helped the participant was the match she found between the textbook and another book (Van de Walle & Folk, 2005) that explained the purpose of the curriculum. This gave her a resource for more information if she found content in the textbook she wasn't certain about. In spite of the benefits of this resource, the participant thought if "I ask the question, if I use the Nelson books and ask [board specialist] and she says yes you are moving in the right direction" which did happen and the participant's direction was confirmed. The participant reflected on the change in her teaching practices that was initiated by the textbook (and supported by other resources she has used): "if you had told me I would teach multiplication this way [(using arrays of different sizes)] two years ago I would have said you're crazy." However, the participant indicated that it wasn't just the textbooks since she had a choice of using older textbooks or newer textbooks and she chose to use the older ones because they aligned with the resources that were available at Scholastic. Since that time, she has found that she still uses both books; "Sure I do, but I use them differently." In some instances she will adapt problemsfromthe older book and redesign them and other times she will use the older book for reinforcement exercises.

119 The isolated reflective practitioner found that the change in textbooks created cognitive dissonance which, on its own, did not cause immediate change. However, it did raise pedagogical questions which stimulated the participant's reflective thinking. This led to other sources of information that ultimately addressed the cognitive dissonance. SMART Board. The participant recounted the history showing the evolution of her math teaching and it was only natural for her to speak of the changes that were on the horizon at the time of the interview. One such change was the acquisition of a SMART Board which would be delivered and installed at a later date. The acquisition process had started after she had gone to a workshop about SMART Boards at an Ontario Association of Mathematics Educators annual conference. While the initial source of information regarding the SMART Board was a math conference, this teacher clearly views it as a general pedagogical tool with uses in many content areas. One benefit is simply having another computer available in the classroom. The participant indicated that she had taken on the role of computer contact teacher in order to help improve the computers in the school and because of her interest in SMART Boards. In addition, she talked to people outside of the school about the application process and consulted with her principal who ultimately wrote the application. The result of this effort was funding for three SMART Boards. She expressed the view that they could have achieved funding for more SMART Boards but "only three [teachers] stepped up to the plate." Willingness on a teacher's part to engage in a change clearly was an important component. The teacher expressed her interest in the SMART Board as; I think, if you have the opportunity for a SMART Board, it's free, it's given, the opportunity is there. Why would you not want to develop your own personal repertoire of skills for how to use one when it is the up and coming technology?

120 The SMART Board was unique to this study because it showed a teacher taking on a leadership role in order to move a process along. The process itself had subject leadership (OAME conference) provide a demonstration which the participant reflected on. This generated ideas for new pedagogical options but did not resolve the issue of how to acquire the resource needed to explore the options. She then took on a leadership role that, along with peer networking, provided the information on how to apply for the resource. At the completion of the observation phase the SMART Board had not been delivered and the teacher had not yet used it in her classroom. Conclusion. The isolated reflective practitioner demonstrated what can be achieved using very little external support. She had made efforts to apply what she learnedfromresources and actively pursued opportunities when she found out about them. Her experience highlighted the role of the community of practice more than any of the other groups. A summary of vignettes is provided in Table 4.5. Table 4.5 Summary ofIsolated Reflective Practitioner Vignettes Vignette

Source of Reflections

Area Affected

Influence on SelfEfficacy

Philosophical changes Resources/

Accessing the math

Administration

PCK

Administration / Self

Subject pedagogy /

community. Change of math

Subject pedagogy /

Increase

Increase

PCK Administration

Subject pedagogy /

No change

121 textbooks. SMART Board

PCK Self / Administration

General pedagogy

No change

Research Question 2 How do reflective practices and self-efficacy interact in the process of teacher innovation? This question pertains to reflections defined as a process by which a teacher recalls and thinks about classroom events in the context of planning and pedagogical choices. The data pertaining to this question highlighted that, while there is a connection in some instances, there is evidence of circumstances where the connection is indirect. There also appeared to be some evidence suggesting the absence of a link, direct or indirect, however, alternate explanations were developed. The answer to this research question is developed for each reflective group. Within the discussion for each group, the connection is explored according to whether it appeared to be direct, indirect, or lacking a connection. Lesson Study In the lesson study group, linkages between reflections and self-efficacy only arose in the post-lesson debriefing sessions. This seemed a natural outcome of the shared experience of classroom observation and the comparison of multiple perspectives derivedfromthat experience. As part of the lesson study, one observer was tracking the questions that the teacher asked according to Bloom's taxonomy. The teacher expressed his self-perception that his questions would be predominantly in the knowledge category. However, discussion revealed that, while there was a bias in the whole-class portion of the lesson, it was followed by diverse questioning astiieteacher circulated and talked to individual and small groups of students. The teacher

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reflected that this was consistent with his trying to meet individual needs. While the teacher's self-efficacy was addressed through his pedagogical technique, it did so while focusing on his need to develop a broader range of questions during the main lesson. In the following quote, the teacher reflects that his self-efficacy is stronger toward traditional teaching approaches than problem centered approaches. However, he reflected on what he saw in a colleague's classroom and showed signs of improved self-efficacy toward problem centered teaching because of the procedural choosing that he had seen explicitly modeled by the other teacher. Teacher: See my eyes are opening because, [as a] math teacher I want a solution, I want the kids to know the answer, you know what I mean, I want a solution to it, whereas the open endedness, develop their own strategies, that's a fairly new concept. Researcher: And you said you were going to adapt (?) based on what you saw in [other participant's] class. Teacher: I liked the way her class knew the strategies. I couldn't name them all, although they had their books but they knew. And I loved the way that all those around me used different strategies to solve the problem. While the time was too brief to say how much change would result from the interaction, this demonstrates a reflection that has increased awareness of pedagogical options that are fundamental to all of math teaching. This suggests that the reflection has the potential to be influencing the teacher's self-efficacy toward using problem centered as an alternate to traditional methods. In one instance, a teacher augmented his lesson using a temperature conversion estimate he learnedfromhis grandparents. He indicated an increase in his self-efficacy because he believes such practical ties add authenticity to his lessons and make a connection between his lesson and daily life. This exemplifies a reflection modifying self-efficacy through the enactment of a lesson that satisfies one's pedagogical principles. In addition, the lesson study discussion afterward explored the possibility of using the estimation accuracy as a topic in future classes and this led

123 to a discussion around the pedagogical content knowledge for comparing accuracy of an estimate with an exact formula. The suggestion generated an interesting discussion but did not alter selfefficacy as could be seen in a subsequent conflicted self-assessment: "But that was way too long. ... Yeah, I probably should have [included a mention of estimation]." A teacher demonstrated a technique called "other's mark." An observing teacher expressed conviction that he can simply implement the technique while another observing teacher indicated that they believe they can implement it but need to think about a suitable pairings of students. In both cases, self-efficacy is sufficiently high, having observed the technique, that classroom implementation is feasible. In this instance, the classroom demonstration led to reflections by the two other teachers and these pertained to the method as a general pedagogical method and comparisons of the method with other techniques. A reflection that a class was much slower than the teacher wished turned into a statement about a desire to be implementing the curriculum achievement chart (KTTC A) framework for evaluation (theframeworkis discussed under its own heading pertaining to research question one). The close proximity of the two disjoint notions suggests a degree of frustration with addressing the achievement chart framework and thereby a reduced self-efficacy. Since this arose during the lesson study, it is questionable whether class being slower was genuinely the issue and the possibility of looking for suggestionsfrompeers regarding the achievement chart framework is credible. The teachers in the lesson study group observed varying degrees of cooperation and competitiveness in three different classes. While this was explicitly discussed, the reflections did not lead directly to a change in self-efficacy: First participant: ...we had conversations about that to, like the competitive side in [another participant's] class was much more apparent than in either of ours.

Second participant: Absolutely, but that makes me more aware. The competitive issue has been mentioned previously but the participant with the competitive classroom showed no sign of change in his self-efficacy. In the initial discussion, it was suggested that the competitive element may be due to the use of groups of four students (as opposed to pairs). However, the teacher later indicated that this likely wasn't the cause of the competitiveness because he did not use groups as often as the other two teachers because he sees himself as preparing the students for secondary school. This is indicative of a situation where the teacher's self-efficacy regarding the "culture of competition" does not change because the multi-faceted nature of the culture is at odds with the simple explanations that are available, and the proposed explanations potentially disregard a pedagogical principle (preparation for secondary school) that is given greater attention in grade eight than grades six and seven. The interpretation of this particular issue was that reflections, based on multiple classrooms, raised the issue as a pedagogical issue that led to pedagogical questions (Do group sizes have an affect on the level of cooperative/competitiveness in a classroom?). However, deciding whether there are two equally valid approaches (cooperative and competitive) and weighing the merits by grade level was beyond the scope of the lesson study. The issue therefore remained unresolved and did not affect self-efficacy. In an early planning meeting, it was suggested that the lesson study could use algebra tiles and another participant revealed that she didn't "know enough" which, after learning about them, was clarified as "I didn't know how to use them." This led to discussion and demonstration of algebra tiles, in a later meeting, that provided sufficient detail to allow the teacher to use them in the classroom. It was the specific detailed discussion of using algebra tiles that included the pedagogical content knowledge of algebra tiles and pedagogical considerations ("keep your

125 numbers really small") that generated sufficient self-efficacy that the teacher had a successful innovation: Researcher: What I was going to ask you was, whether or not, the discussion at the planning meeting about algebra tiles helped put you at ease as far as using them in the classroom. Participant: Yes. Oh definitely. I never would have used them before. She also indicated that she will continue to use them in the future. Lastly, the lesson study group discussed a student who pointed out that the teacher had made a spelling mistake. The teacher did not believe there was a spelling mistake, however, one of the observing teachers pointed out the mistake that the student observed. The teacher responded by explaining how they had made the document in a rush and had difficulties with the computer. If this discussion had any impact on the teacher's self-efficacy, it would have been directed at the response to the student rather than the pragmatic reality of having to periodically create documents in a rush. Only one instance was observed in the lesson study that suggested that reflection on practice and self-efficacy may not be linked. The observation suggests the alternative perspective that it may be the peer observation rather than reflections on practice that cause changes in a teacher's self-efficacy: We will see how it goes. I think there is a real sense I would... If I had gone first, I would [have] very calmly approached this with my own arrogance and all the rest of it. Now it has certainly been colored and, I hate to say tainted, but tainted by what I've seen two other teachers do and I don't know. This is not a reflection about classroom events, nor is it a consideration of the effectiveness of a method. However, while it does not constitute a reflection about classroom practice, it does highlight a response to observing one's peers and the prospect of being observed. It is a teacher's vicarious experience (Bandura, 1986) as they consider the lesson implementation where peers

will be present as observers. This is further evident from the teacher's response to being asked what they thought of the lesson during their post-lesson debriefing: Teacher: Ah, it was pretty good. They seemed engaged. Did they seem engaged to you? Observer: Umhmm, Teacher: See, engaged. The effect of the vicarious experience appeared as preemptive concern over reflections that would emergefrombeing observed. This suggests that reflections, about the reflective practice process itself, can lead teachers to question their self-perceptions and this can have an impact on their self-efficacy. Peer Coaching In the peer study group, there was little discussion that highlighted self-efficacy in an explicit manner. As one teacher said, "We are constantly self reflecting and trying to change." The group had increased enthusiasm as the group got together and was quite positive about the whole experience at the end. They indicated that they had new techniques for the classroom and wanted to continue to develop the PRIME resource. They did not attribute this overall increase in enthusiasm to self-efficacy pertaining to isolated events. However, it appeared that the group had learned and applied new techniques and was motivated to continue their effort. This appeared to be due to increases in self-efficacy which the participants did not recognize as such, or perhaps did not want to admit, having had so many successful years in the classroom. It may be that the level of professionalism of the group of veteran teachers puts an emphasis on the details needed to implement changes and on the pedagogical principles regarding the utility of the changes rather than being explicit about their individual beliefs regarding their ability to implement the changes. It may also point to the need to consider collective self-efficacy in instances where teachers have a well developed professional understanding of their peers.

127 Mentoring The pre-service mentoring group displayed few direct connections between self-efficacy and reflections. One instance on the part of the pre-service teacher was during an activity in which a group of students at thefrontof the class was being used to demonstrate different ways of making permutations. During the activity, the teacher was monitoring both the students at the front but also the students who were supposed to be watching. The teacher used reflection-inaction leading to adaptive classroom management: "Well, I probably would have had them stay up there longer but I felt like I was losing the crowd on the side." Her reference to the "crowd on the side" reflects the declining number of students who were paying attention to the demonstration. Asidefromthis example, the general lack of observations showing a direct connection between reflections and self-efficacy, in this group, appears to illustrate a high degree of focus on pedagogical and pedagogical content details rather than the implications for selfefficacy. There were several instances where self-efficacy changed but it was not directly linked to classroom events. In these instances, reflections about the classroom appear to be moderated by other factors and the moderated reflections were then having an impact on self-efficacy. For example, on one occasion, a reduction in self-efficacy was evidentfromhesitancy that was observed and later acknowledged by the teacher. The cause was an ambiguity in a permutation question she had posed. She had asked how many permutations of the letters in "Korean" would keep the "OR" together but subsequently was unsure whether her answer should include instances with "RO". On another occasion, the pre-service teacher reflected; "I feel like I got started off on the wrong foot and then I got nervous..." This appears to exemplify 'stage fright' that signifies a

128 reduced self-efficacy, specifically physiological efficacy (Bandura, 1986), without any clear causal affect that the pre-service teacher could have addressed. The reflection did not appear to impact later self-efficacy which was probably due to there being no clear causal agent. There was a student in the role of a teaching assistant in the class. This was a challenge that the pre-service teacher was unsure how to handle. It appeared that the pre-service teacher was unable to provide effective leadership for the teaching assistant and the teaching assistant had abdicated much of his responsibilities. The mentor reflected that the circumstance of having a pre-service teacher and student teaching assistant simultaneously was not ideal. While this is a reflection that rationalizes the situation, the impact on the pre-service candidate's self-efficacy pertaining to classroom assistants was negative; "I know that I will be hesitant to choose a student assistant in the future, partly based on this experience." The experience was negative enactive attainment (Bandura, 1986); however, the level of effort and adversity of the initial implementation of the teaching assistant was unclear. There were occasions when reflections did not appear to be linked to self-efficacy. These were observed rarely but one example is: It is interesting because I find that sometimes I'll be like, I knew something was going on because of the giggling, and I didn't know if there was a joke or something. I am so doing what I am doing. In this instance, the pre-service teacher was highly focused on the content that was being conveyed in the lesson that she is not fully aware of other events in the classroom. This demonstrates reflection with an uncertainty about the cause of classroom events. Given the uncertainty, it is unlikely that any significant change in self-efficacy emerges from such an event. However, the interpretation, by the teacher, to consider self-perceptions is consistent with the lesson study participant who expressed self-consciousness about his peers watching him.

There appeared to be little or no effect on the mentor's teaching as a result of the reflective practice (a statement verified in the member check: "agreed!!"). While some pedagogical ideas were shared and he is likely to make use of them, he was constrained by time demands that limited his opportunity to adapt or innovate during the reflective practice. It is conceivable, but was not directly observed, that the researcher's involvement may have influenced the mentor because of the level of observation and feedback provided to both the mentor and pre-service teacher. Both the mentor and pre-service teacher were surprised at the extent of field notes and expressed surprise at some elements of classes that the researcher raised in the interviews. Isolated Reflective Practitioner The retrospective interviews of the isolated reflective teacher did not facilitate finding out how her self-efficacy might have changed as she explored different innovations. Her references to self-efficacy tended to focus on the overall outcomes and issues around the math curriculum rather than her ability to address particular lessons (which she described as "strong"). This focus may also have stemmed from her acquiring resources that allowed her to address any concerns she had about individual lessons and this set aside any significant concerns about her ability. She indicated that when she felt her grasp of the content was insufficient that she knew she had appropriate resources to consult. The few mentions of changes in self-efficacy may also have arisen from her lack of awareness of the bigger picture - she didn't know the extent of the discrepancy between her approach to teaching math and the problem centered approach: Researcher:.. .you believed in yourself because you thought you knew what [math teaching] was? Participant: Yes Researcher: Okay. Participant: I didn't know anything different. Researcher: Happy in one's own ignorance. Participant: Yes. Yes, that is a very succinct way of putting it.

The retrospective interviews raise doubts about effects on self-efficacy as the isolated reflective practitioner put innovations into place. However, an alternate perspective is warranted by some recollections she had of significant changes along the way. She recalled the impact of Van de Walle and Folk (2005) stating that their book didn't contain answers in order to be authentic to real math problems and how this really bothered her. This suggests that the reflective practitioner was cognizant of her self-efficacy. This would permit the explanation that her professional learning, because it was self-paced, did not lead to large changes in her self-efficacy. This is consistent with the view that she was content with her progress because she did not have any external agents challenging her approach with an overall view or expectation for problem centered learning. Whether an external push would have been beneficial or simply exacerbated her isolation was unclear and likely would depend on the level of support that any external push came with. Conclusion to Research Question 2 The evidence is indicative that classroom events can give rise to two main effects. One is self-perceptions and the other is reflections. The first is the less complicated and were readily observed when teachers commented on their classes and then other observing teachers could confirm or correct the teacher's perception. Self perception can lead to reflections but also may have an impact on self-efficacy. Reflections arisingfromthe classroom had two impacts. Firstly, they appear to generate self-perceptions but, more commonly, focused on content issues. These content issues however, were not merely the lesson taught, they extend across a range of content and pedagogical issues that reflect a general theory of math teaching. They pertain to pedagogical content knowledge, as in the case of the algebra tiles tutorial, general pedagogical techniques, such as "others mark", to

131 deeper pedagogical principles, such as whether grade eight is fundamentally different then grades six and seven because it is preparatory for secondary school. While the reflections lead to healthy discussion of options within math teaching, the implications for self-efficacy only emerge after teachers chose what they wanted to do in the classroom. The various options discussed may be followed up at other times but the only impact on self-efficacy arose after teachers expressed an interest in particular techniques. With the various methods of addressing equation solving, the clearest evidence of a self-efficacy change arose after one teacher decided to implement algebra tiles and deemed she would continue to use them in the future. The different reflective groups had different contexts and focused on different aspects of teaching. The lesson study was the most articulate on pedagogical options and details. The peer coaching was the least explicit about self-efficacy and used the widest array of pedagogical techniques ultimately leading to a large assortment of options that were well suited to primary students. The mentoring arrangement had a contrast of time demands with the mentor too busy to be overly innovative and the pre-service teacher articulated the most instances specific to classroom events. Finally, the isolated reflective practitioner appears to have generated her own program of improvement mat focused on movingfromtraditional math teaching to a problem centered approach. This addresses math teaching at a level of pedagogical principles and teaching philosophy that borders on curriculum orientation (Miller & Seller, 1990). The isolated reflective practitioner appears to demonstrate a slow migration that may not have had large changes in self-efficacy because it used self-paced learning combined with a corresponding rate of implementation.

132 In summary, there was a variety of ways that reflective practice and self-efficacy were observed to interact. However, it was not a simple connection and it was necessary to introduce a general theory of math teaching to show the range of considerations. In addition, self-efficacy was unaffected until a teacher chose to implement a technique and that choice required a level of understanding that was sufficiently well developed that they met a minimum level of belief in their ability. Research Question 3 To what extent does a teacher's curriculum orientation affect the decisions they make regarding techniques arisingfromthe reflection process? The survey indicated that the curriculum orientation of all of the participants except one was transformational. The transformation orientation focuses on the needs of the whole child and is self-actualizing. The one person who was not transformational was transactional which is developmental, focuses on cognitive processes and views teaching as a bidirectional transaction between the teacher and student. No participants were observed with the transmission orientation that focuses on behavioral approaches and views knowledge as being transmittedfromthe teacher to the student. Secondary orientations were clear for some teachers, however, they were not clearly relevant within the reflective practice. The majority of statements regarding curriculum orientations arosefromindividual interviews that served to validate the results of the instrument. While the results of the instrument were consistent with the views of the participants, the lack of discrimination may show a commonality of the teachers who choose to use reflection or it may be indicative that the conceptual distinctions between the orientations are not valid. The latter point was highlighted by one teacher indicating that the transaction orientation is certainly promoted by problem centered initiatives.

133 What appeared to be much more relevant than the curriculum orientations survey were discussions around systemic components and principles that the teachers held regarding their methods. Examples of systemic components include the question of whether grade eight differs from grade seven because of the need to prepare students for secondary school or differences between primary and junior divisions because of funding emphases. Principles for teaching are views that appear to be strongly held and were effectively accepted within the reflective practice group. For instance, a teacher of a split grade one and two class stated that "There is no grade one subject they could ever do on their own." Another teacher talked about a rule of thumb his grandmother used for temperature conversions and explained how he feels such grounding of the subject matter is important. The third research question was addressed by examining how teaching principles arose in different discussions and contexts within the reflective practice. The instances are examined according to each group and a further distinction between systemic and instructional principles is made. The objective of this section is to show the diversity of principles mat were discussed to facilitate the emergence of a definition of the concept of a "teaching principle". Lesson Study Systemic Principles. Two issues arose that were systemic in the sense that they were beyond the teachers' scope for taking direct action. While they could pursue change as a result of the reflections, they could not, individually or as a group, enact the change. The two issues were; streaming in the grade eight program and whether grade eight was different than grades seven and six because of the need to prepare students for secondary school. The concern, with respect to streaming, was that variability within the students in the high stream persisted. This led to questions about how the

134 teacher simultaneously runs a pair of tests that are matched to student levels of achievement (levels 3 and 4 on the Provincial achievement chart). The second issue was put forward, by one participant, who suggested that grade 6 is more caring and nurturing and grade 7 is more process oriented than grade 8. The subsequent discussions suggested that the observed grade 6 class was the most cooperative and grade 8 class the most competitive. This seems to link the systemic elements, physical separation of grades eight and nine and the use of streaming in the grade eight program, to instructional elements that the reflective practice brought into question. Instructional Principles. How much competition should there be in a classroom? This was unresolved but the observation and discussion highlighted different levels of cooperation and competition in different classrooms. The teacher with a competitive classroom felt that competition fit with his class being a streamed high end grade eight class and preparation for secondary school. "I think there is a sense in which, too much group work impedes communication." This was indicative of a distinction between working in a socially friendly environment and fulfilling the requirements of communication skills in the provincial curriculum. The statement may also have served to acknowledge that the cooperation-competition dichotomy, which was discussed as a result of observing different classrooms, might not have a simple solution. In one of the lessons, an approximation for converting Celsius to Fahrenheit was used to ground the lesson. The approximation was F=2C+32 whereas the correct formula would use 9/5 as the coefficient of C (i.e. F=l .8C+32). This led to discussion in the debriefing session about the possibility of comparing the two formulae as part of a lesson. The participants agreed, in

135 principle, about the importance of the conceptual idea but differed in opinion about the feasibility of using comparison of two formulae in the classroom. There was a lengthy discussion about the importance of variables being defined by a letter that identifies the quantity being represented. The conversation identified some of the finer details such as T, X, or R being good choices for the number ofridetickets but P being inappropriate because the number does not correspond to the price. A differentiated view was volunteered by one teacher in a manner suggesting that she was looking for confirmation of the idea: You know that, am Irightin this, that it is much of my personal opinion but back it up please, that the lower level students you are not going to engage them too much with the higher level thinking skills questions. The same type of idea arose in the context of some students using fractions as "sheer knowledge" whereas for others their use amounts to problem solving. This did not receive a direct response but it was clearly a fundamental premise that the participant was considering. There was an acknowledgement of the idea at a later meeting but the gap between the suggestion and a direct response implies some doubt, or conditional acceptance, about the idea. "[Another participant] has his kids in groups of four. I don't do problem solving in groups of four, I don't do groups of four work I find it hard to draw accountability. So pairs work I find very easy to." The participants indicated that groups of four are "pushed" to some extent in the teacher evaluation process. This highlights an instructional practice being in potential conflict with a systemic practice. During the lesson study planning, a prototype lesson was copied and distributed to the participants. The prototype lesson involved solving equations; however, it involved mixed fractions in the equation solving. All three teachers agreed that they would not tackle the topic

using fractions and that using fractions would introduce an added complexity to the task. While this is consistent with the curriculum, which introduces equation solving with fractional coefficients in grade nine, the explanation based on complexity suggests that the reason was based on principles. One teacher articulated the principle of introducing students to as many strategies as possible and allowing them to adopt what works best for them. This was an acknowledged strength of the teacher and both of the other participants indicated that they intend to include more of the support they observed for the principle in their classrooms. The lesson study group did not identify principles as such but several came up in their discussions. What was particularly interesting was the range of topics to which the principles apply. It is difficult to think of another forum that would facilitate teachers communicating about such a diverse set of established ideas. In some cases, such as the use of fractions in equation solving, there are pedagogical content knowledge principles. The discussion of student's cognitive responses to fraction questions rangingfromrote algorithmic memory to problem solving is indicative of principles of subject pedagogy. At yet another level, the ideas around effective groupings for problem solving may have wider applicability than just math and could therefore be considered to fit with more general pedagogical principles. Peer Coaching Systemic Principles. When the Miller-Seller instrument was provided to this group of teachers, they asked and had it confirmed that it was to be completed with respect to their math teaching. They subsequently concluded that this was not readily feasible and found it difficult to clearly separate math instructionfromthe other roles they fulfill as primary division teachers. The key area of

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concern, with respect to the instrument, was that social and emotional aspects of the teaching process (such as a caring environment or addressing student's self-confidence) cross content boundaries and cannot be clearly discerned for content specific aspects of teaching. Fetal alcohol syndrome was mentioned as an instance in which a teacher tries a multitude of strategies in response to the student having a learning challenge. It was clear that, in spite of the challenge, the teachers look for strengths that the student can build on. This response to the situation suggests a pedagogical principle of using multiple instructional methods to meet the needs of the students. Instructional Principles. The peer coaching group discussed several principles of teaching that seemed to fit with a shared understanding of student development in the primary years. It seems likely that the level of experience of the teachers in the group either endorsed the principles based on experience or chose not to contest the ideas because of collegial respect. The latter constitutes cultural norms that would clearly hinder reflection. Checking answers was not considered an effective means of evaluation at the primary level because it does not provide sufficient details to be sure of a student's understanding. As one participant put it;".. .you wouldn't know how they were doing it if you didn't watch them." It was considered important to have a demonstration of the skills, by the student, which highlighted the importance of developing new hands-on approaches through the reflective practice. There were several discussions around developmental issues at the primary level. A distinction was made that, in grade one, students can find a partner to provide assistance "... who you think is as smart, or smarter man you...", however, at the grade three level"... they want to be with their friend." This change was acknowledged by all five participants. A second

138 developmental issue arose around whether the PRIME kit was implying that a teacher could overcome Piaget's developmental phases. The discussion focused on one teacher who had taken time to read the preface and introduction of the resource; he indicated that the resource does not claim to overcome Piaget's stages but does facilitate moving forward more effectively when a student is ready. The result of the discussion was an implication that they all needed to delve further into the resource and that there were some considerations of the Piaget stages that they should consider further. A participant explained that he does not use a 100-chart for skip-counting unless the starting value is a multiple of the amount being skipped. For example, to countfromeighteen by three would be okay on a 100-chart because eighteen is a multiple of three, however, counting from twelve by five would not be done on a 100-chart because twelve is not a multiple of five. The participant did not apply this restriction to the number line. The specific requirement for skip counting on the 100-chart was to facilitate the pedagogical content knowledge around teaching multiplication using a 100-chart. The concern was raised that his choice of manipulative might provide a priori information to the students; however, this issue was not pursued. The benefit to the reflective practice appeared to be the depth of consideration for pedagogical practice and the particular discussion highlighted a decision one teacher had made with respect to pedagogical content knowledge. There was a mention of pedagogically moving instruction from a group demonstration, to working with a partner, to individual capability. This draws a parallel to the cooperative to competitive dichotomy brought up by the lesson study participants. As with the lesson study, the range of principles that was discussed is quite large. Some, such as the skip counting principle, are specific to pedagogical content knowledge. The

139 discussions of Piaget's notions of development as well as the pedagogical scheme of moving from group demonstration to individual skill reflect general pedagogy. Pre-Service Mentoring Systemic Principles. The mentoring group was in a private school that was Bah&'i inspired. This is a fundamental systemic difference which was reinforced by professional development based on Baha'i writings and the inclusion of teachers of the Baha'f faith. One participant indicated that, due to periodic time constraints, errors in preparation of examples or model solutions to mathematical problems arise in the classroom. As a consequence, the teacher includes addressing errors that arise as a joint task of the teacher and students. This would not work for a transmissive expert whose expertise would be doubted by students because of the errors. The pre-service teacher was not happy with the desk arrangement that the mentor used. The challenge of the situation was exacerbated by the larger size of her class. However, the preservice teacher did not change, or request changes, to the seating plan because she felt that it "would have caused unnecessary tension and issues, so I just decided to not worry about that aspect." Given that the pre-service teacher did not perceive that she could initiate the change, it is deemed a systemic component even though it was in the classroom. Instructional Principles. At one point, the mentor allowed resubmission of a piece of work and the pre-service teacher disagreed, in principle, to the idea of resubmission. The pre-service teacher had difficulty articulating her concern, having previously said she always allows resubmission, but indicated that it placed too much emphasis on marks and mat there were more beneficial ways the students

could invest their time. The mentor, on the other hand, had spent an entire class emphasizing the meaning and importance of "explaining" a mathematical concept. The pedagogical dilemma of the situation is whether fixing an existing piece of work entails learning through strengthening the weaknesses in the work or would entail minimal learning because of the possibility of consultation with other versions of the marked assignment Some pedagogical content knowledge that the teachers looked for was course specific. For example, a student was asked to use permutations rather than an alternate technique learned in a data management course. In addition, within the course content, there were instances were the mentor demonstrated that he was readily familiar with the numerical basis of the subject matter. He used specific pedagogical content knowledge choices to complete calculations both mentally and rapidly. For example, in one particular question he evaluated

8'

as 2520 very

X * +*» 4far* X>«

rapidly. When mis was discussed it became clear that an 8 factored from the numerator canceled three of the components in the denominator and he had known the value of 7! to be 5040 which he divided by two. During the discussion, the participant pointed out that this is the smallest number divisible by all numbersfromone to nine. The mentor and pre-service teacher had some confusion because of a difference in terminology where the mentor referred to in-class evaluations exclusively as quizzes. The preservice teacher, on the other hand, used the term "test" as well as "quiz" and was uncertain about the nature of an evaluation because the mentor was referring to it as a quiz. There was additional confusion when the mentor made a distinction between 'practice final exam questions', a 'practicefinalexam', and the 'final exam'. Thefirstof these was not marked, the second was marked to allow an improvement on some quiz marks, and thefinalexam was the final summative evaluation of the course.

As the mentor and pre-service teachers' courses progressed, there was a certain status quo in the operation of the classroom. As a result, some issues (such as group formation) that could be addressed through alterations of the classroom environment early in the semester would have become issues that would have a negative impact on the instructional component if they were pursued later in the semester. As in the lesson study and peer coaching, the principles reflect a range that includes pedagogical content knowledge (expectations based on course content), general pedagogy (resubmission of work) and a pedagogical principle (student responsiveness declines through a semester). Isolated Reflective Practitioner Systemic Principles. A problem centered curriculum that does not have distinct, right and wrong, answers was problematic and created a profound change for this participant. It was in direct conflict with her initial view that mathematics was algorithms ".. .you teach the particular skill and there were practices all through it." This change is included as systemic because it was induced by the education system rather than the participant's selection. The participant pointed out that the methodologies of the three teachers in her division were not aligned to one pedagogical approach. She expressed concern that her class was highly differentiated whereas the grade six class, that the students would move into, was not. There is a dilemma between teaching for student need (in terms of the overall curriculum) and teaching the specific grade level curriculum. This also manifests itself with respect to time where the participant indicated weighting the teaching time according to her perception of the importance of the content to student progression within the curriculum.

142 Instructional Principles. The participant expressed concern that the use of names of diverse ethnic origin was not representative of the school's community and she preferred to tailor questions by using student names and their personal interests. This highlights the issue of addressing inclusion within provincially oriented textbooks while the distribution of different ethnicities is both heterogeneous and changing. The lone reflective practitioner focused predominantly on systemic issues and did not articulate instructional issues she would have had to work through to implement the curriculum change she made. The issues she raised speak to systemic issues that can be interpreted as unresolved pedagogical principles. She raised concerns over a lack of pedagogical alignment of her division, appropriate enactment of the curriculum (i.e. placing emphases to improve long term student achievement), and modifying inclusive resources to reflect the lack of ethnic diversity within the school's community. Conclusion to Research Question 3 The original research question was altered because of the level of commonality among the curriculum orientations of the participants. It was unclear if this reflects a characteristic of teachers who choose to participate in reflective practice or is a common understanding that emerges from the community of practice (particularly in the context of the NCTM Standards promoting non-transmissive teaching). The question was altered to examine two perspectives: systemic and instructional principles. Systemic principles were diverse and addressed in a variety of ways. Many issues were left unresolved because they were considered to be beyond the scope of the reflective practice group. Examples include the streaming and cooperative/competitive issues in the lesson study, and the

143 variations in pedagogical approaches within the junior division of the lone reflective practitioner. The peer coaching group mentioned fetal alcohol syndrome as an example of an issue that cannot be resolved by the teacher but is addressed by exploring a multiplicity of teaching strategies. The pre-service teacher indicated a perception that the seating arrangement of the classroom was beyond the scope of changes she might have explored with the mentor. This was in contrast to the isolated reflective practitioner who purposely took on some leadership roles in order to promote changes she felt were needed. Instructional principles were varied and several unresolved issues arose. In the lesson study group, an example issue was the extent to which group work is desirable in different grades. The peer coaching group had a discussion about the implications of always using a specific resource for a specific topic. Instructional principles that were recognized within each reflective group also arose. Practical grounding of concepts such as a rule-of-thumb for temperature conversion and the choice of variable names arose in the lesson study. There was also a shared understanding that what constitutes problem solving may depend on the prior achievement of individual students. The peer coaching highlighted childhood development where there are important differences between students in grades one and three. In the senior division, the mentoring group was unique for having pedagogical content knowledge requirements that were course specific and raising assessment concerns. All of the reflective groups indicated the principle of presenting students with a multiplicity of strategies and allowing the students some choice. Research Question 4 To what extent does the level of formality in the reflective process influence the teacher's efficacy during an innovation?

144 The intention of this question was to examine the implications of how well the reflective practice technique followed protocols (Glaude, 2005) that are often provided in practitioner guides and teacher workshops. However, all of the groups were observed to compose aspects of the reflective practice as they proceeded according to the professional judgment of the participants. None were observed to follow set guidelines or formal protocols for the reflective practice. Where formality did arise was in the role of factors outside the group influencing the reflective practice process. While the original intention was to avoid leadership elements to avoid having to consider collective effects, it was found to be impossible to completely avoid the influence of leadership. This research question was then reconsidered from the point of view of what the influence of leadership (as the formalizing agent) was on the process. The change in this research question is indicative of the 'community of practice' concept emergingfromthe interviews (the term was explicitly used by the lone reflective practitioner). In view of this, looking at different leadership influences is a means to explore the possibility of needing to utilize the concept of a community of practice without predicating the analysis on its relevance. Lesson Study For the lesson study group, leadership was separated between teachers and administration. The group was initiated by the teachers in response to a professional development session that outlined funding opportunities for teachers with over ten years experience. The teachers effectively formed an independent working group that was only connected to the administration by the requirement for release time to facilitate the classroom visits. This group was

145 professionally motivated and did all the planning after school in order to minimize disruption of their classes. Funding for the release time, in this case, included conditions. One participant was deemed ineligible for the funding criterion and having a cameraman present in the classrooms to record the lesson study process so that it could be used as video footage for professional development was also required. These issues were exacerbated by the fact that the group had already invested time in planning and did not apply for funding until they had worked out how much release time they needed and when. Funding for the one participant's release time was provided by OISE/UT and the requirement of a cameraman was waived because of concerns raised by the researcher regarding the impact it could have on the validity of the results. The reflective practice provided an opportunity for participants to discuss components of teaching that were not directly related to the reflection task. On the positive side, participants discussed some professional development content from several months earlier. One negative effect of note was the observation that, with a change in the school's daily timetable to a 'balanced day' (effectively two moderately long breaks rather than one long lunch and two short recesses), a teacher had noted that collegial interaction in the staffroom had seen a significant reduction. It was also noted during the lesson planning that other teachers were being affected because the lesson study was interrupting routine joint planning that was done. This also implies that teachers who keep their classes in step would also have to adapt to the effect lesson study would have on their timing. The influences on the lesson study included effects by the board as the source of funding, school level effects were teachers engaged in some conversation about issues (discussing past

146 PD and effects of a change in the daily schedule on collegiality), and division effects where teachers exchange information within the lesson study but with the possible negative effect of interrupting routine interactions with other division teachers. Peer Coaching A significant aspect of efficacy within the peer coaching group was the selection of the members of the group itself. They indicated that trust was a significant consideration and that there were peers they would not be comfortable working with. They had chosen the group in a manner that insured that trust would not be an issue. This group included an administrator with a partial teaching load. Her administrative component allowed her to offer coverage for teacher's classes provided it did not interfere with her teaching component (although this never actually occurred). She also organized two planning charts that assisted with choosing dates and established that the peer coaching would work in conjunction with the PRIME resource that had arrived at the school at the end of the previous school year. During the process, this administrator was obliged to split her time between the peer coaching group and other administrative duties. As a result, when she did periodically arrive, it became an opportunity for the group to summarize their progress and therein reflect on what they had accomplished and needed to do. Having an administrator affiliated with the group eliminated the challenges associated with teachers taking on the unfamiliar role of writing a request for funding. It also provided an advocate for the group who was asked to speak to the principal about possible ways to continue the peer coaching group in the following school year (one idea required the daily schedule to facilitate a common preparation period each week).

147 While the efforts of the administrator were appreciated, a concern was raised that peer coaching using PRIME is only part of a larger picture that tends to move faster than the teachers can digest it. "We gofromone allegiance to another and leave it all behind." This concern was raised toward the end of the peer coaching sessions and may reflect the feeling that the teachers had only scratched the surface of what the PRIME kits contain. Having recognized the potential of PRIME, the teachers may have been expressing concern that future opportunities to further develop PRIME would be lost to other professional development foci at the board level. The peer coaching group benefited from having an administrator in the group who dealt with the organizational details of the group. At the same time, the teachers expressed concern that professional development is provided at a pace that does not necessarily match their needs. There was clearly much more that they could do with the PRIME kits but an implied obligation to follow the board's lead with other professional development which would impede the effort to use the PRIME kits. Pre-Service Mentoring For the pre-service mentoring group, in school and external leadership were not involved in reflective process. While the pre-service teacher was on a placement from a teacher's college, the reflective process between the host teacher and her was distinctfromthe teacher's college requirements. The level of formality of the reflective practice was not influenced by the teacher's college. An evaluation by a vice principal may have influenced one of the initially observed classes where acclimatization to the researcher's presence could also have been an affect. The vice principal's role was not a routine influence on the reflective process. (Note that the researcher purposely avoided using reflections from the acclimatization period which included the vice principal's evaluation. This was considered a necessary step to ensure validity.)

Staff meetings included teachers presenting examples of best practices, however, the influence of this would not have been a direct influence on the reflective practice process that was observed. It may have contributed to new ideas rather than influence efficacy for preexisting ideas or efforts in progress. The reflective process may have been indirectly influenced by the mentor, pre-service teacher and the school being of the Baha'i Faith. There was no evidence of a direct connection but it likely served to substantiate a common moral understanding and that may have influenced efficacy during changes in practice that involved risk taking, in the reflective process. The school was not part of a board and therefore did not have a larger educational administration to provide opportunities for within-board sharing. In addition, the participants were not involved in any professional teaching organizations. This would constrain the extent to which content leadership would be shared and the size of the professional community they had access to with the same content focus. At the same time, the mentor had experience teaching in South America without having been to teacher's college combined with a Bah&'i influence seems to have provided an experience of being part of a larger community. The pre-service mentoring group was the most sheltered of all the groups from outside influences. The school was private and without a board to promote ideas, the students were primarily from abroad and parents for the classes I observed were unlikely to visit, in addition the mentor was not affiliated with optional teaching organizations. As a result, the changes that arose in this reflective group grew from within and were not influenced from outside the school. Isolated Reflective Practitioner This participant indicated that opportunities for formal reflective practice within her school were not available. While there was a colleague who was "math rich", sharing of materials was

149 based on the participant knowing what to ask for and knowing how to use the materials. One notable exception was her principal covering the participant's classes in order to facilitate her attending a Provincial math conference. The primary-junior task force, at the board level, provided an opportunity to meet other people with expertise. However, this appeared to be primarily a new source of information as opposed to reflective collaboration. For example, the participant sought verification of one of her adopted approaches from an expert in the task force. It was a change in leadership that led to the participant teaching mathematics and beginning her journey of upgradingfromprescriptive methods to a more modern problem centered approach. This also corresponded to a change in the curriculum and the acquisition of new textbooks. The school administration plays an indirect role in supporting the reflective practice of this participant. The principal covered the participant's classes for a day, has provided some professional development resources, recommended the participant to be involved in the primaryjunior task force for math and helped the teacher in the process to acquire a SMART Board. At the same time, the amount of professional resources appears to be substantially less than the participant has personally invested, the principal does not appear to be willing or able to schedule large blocks of time for math which the teacher believes are necessary, and the participant's involvement as the divisional team leader was dropped because of increased expectations from the principal. There appears to be a communicative element but sometimes it appeared to be a case of leveraging within the system, such as when the participant used information from colleagues at other schools in order to cause change within her own school.

The Ministry indirectly supported the reflective process by providing money for development of mathematics in the junior division. This was used to buy manipulatives and also to provide a series of workshops which the participant was able to attend. The lone reflective practitioner began teaching math as an isolated individual and has slowly found a place in a larger math community within the board. The opportunities that have been provided seem haphazard and without coherent alignment, yet they represent a progression of expertise toward the goal of an enactment of a problem centered curriculum. Conclusion to Research Question 4 The role of the community of practice varied greatly between the groups. The peer coaching, for example, included an administrator who dealt with organizational details so that the teachers could focus on the reflective practice. At the other end of the spectrum, the lone reflective practitioner engaged the community of practice in a manner that assisted in meeting the needs of her students. In all of the groups, the community of practice contributed to the reflective process through resources and expertise. However, utilizing the contributions of the community of practice was not automatic and required significant effort on the part of the teachers. The peer coaching group, for example, used the PRIME resource but had to gain an understanding of how the resource was laid out and what it could provide for them.

CHAPTER FIVE DISCUSSION The grounded theory methodology provided insights into the linkage between reflective practice and self-efficacy that the researcher had not anticipated. In hindsight, the method was a good choice because it revealed the researcher's bias toward particular types of interactions within the secondary school community of practice that differ from the other groups. In spite of this, a substantive theory emerged which is presented below. Following the development of the theory, the research questions are revisited, the research literature is examined in terms of the theory, the practical implications and future areas for research are then considered. Theory Development In this section, the lesson study group is used as the focal group to describe the mechanisms that the theory is based upon. The theory is then described directly and subsequently the other reflective practice groups are used to explain the theory in further detail. In the lesson study group, the emphasis was on instructional practices with the participants observing and implementing new methods. However, constraints such as having the knowledge to use a particular technique were critical to implementation. This limitation was directly based on the PCK of the reflective participant. They cannot use a technique if they do not understand the details to the extent necessary for implementation. When a method is within the PCK of a teacher, or satisfactorily explained that the teacher has an image of the requisite PCK (such as was the case for a participant after an algebra tiles tutorial), the teacher has to weigh the options and decide whether the potential of implementing a new technique is better than previously used approaches. This leads the teacher, in isolation, to decide on the lesson they want to enact in the classroom and that decision has a direct bearing on

151

152 his self-efficacy. They may decide not to innovate for contextual reasons or uncertainty over details of a new method, or they may decide to make a change. In either case, self-efficacy will change in response to the choice. The lesson study also highlighted self-perception on two occasions and these led in two different directions. In one instance, an observing participant added details that led the selfperception into reflection because it indicated mat the self-perception was only correct for part of the lesson. The circumstance would naturally lead to questions of whether one's methodology during lessons needed to or could be changed. In another instance, the self-perception was met with reassurance and this likely led to an increase in self-efficacy, much as one feels empowered by having someone else acknowledge their good efforts. During the development of the theory, it was supposed that, in a case of partial reassurance (i.e. you did well on ... but need to improve ...), both a change in self-efficacy and reflection could occur. For instance, a teacher was observed to have good questioning skills on an individual basis when he circulated the classroom, which will have increased his self-efficacy. However, his questioning during the lesson was limited and mis led to reflections about questioning during lessons. There were a few instances during the lesson study where pedagogical matters, whether curriculum interpretations or pedagogical content knowledge, led to questions that where discussed. In some cases, this became a source for further reflections and, in a few cases, they remained unresolved questions. An example leading to further reflection was the algebra tiles where the teacher who ultimately implemented them had to ask quite a few questions before she felt she understood the practicalities of how to use them. In each case, the pedagogical content knowledge led to a question that was answered and the process of reflection was used to amalgamate the new information with the previous understanding. In some cases, for instance

the level of cooperation and competition that is ideal for grades six to eight was unresolved within the lesson study. They recognized what the practices in the school were, but did not have die means to judge the effectiveness of them. Influences on the reflective processfromleadership pertained to a variety of factors. There was the issue of funding which involved a board initiative. Documents were consulted that were derivedfromsubject expertise at the board level and also professional resources that were independent of the board. In addition, there were resources that the school had made available. As the theory developed two issues arose; Firstly, how to include a diverse range of leadership components in the process and, secondly, how to address the range of pedagogical components that were addressed. The former issue was resolved by the isolated reflective practitioner who made reference to her community of practice and this was found to be a good encapsulation of the diverse range of leadership influences. The second issue was resolved using dimensionalization (Charmaz, 2003) by encapsulating different content and pedagogical attributes. A theory of math teaching was developed which portrays the act of teaching as the intersection of pedagogy and content embodied in pedagogical content knowledge. The community of practice was defined, in the sense of Wenger and Snyder (2000), by considering sources of influence from the teacher's (i.e. participant's) perspective. The most frequent influences are classroom based; resources, peers, other teachers of the same grade or division. Lessfrequentlyconsulted influences are the organizational leadership of the school; lead teachers, technology coordinating teacher, vice-principal, principal. When teachers go to board events it is generally to workshops directly related to aspects of the curriculum; however, these typically are developed by board experts who are involved in subject leadership and tied to a larger, subject based, community of practice. Finally, for most teachers, the Ministry of

154 Education, particularly through curriculum documents, is a key influence. This model of the community of practice is illustrated in Figure 4.1.

Figure 4.1; Community of practice

An effort to dimensionalize (Charmaz, 2003) pedagogical and content components in a linear fashion met with some suitable qualities for this study. However, there were aspects mat did not fit agreeably such as the way in which some pedagogical methods could cross subject boundaries (such as teaching how to order decimals using alphabetical orderfromlanguage). Another example would be having a broad base for pedagogical philosophy and techniques but a relatively confined subject base (for example, a teacher with considerable experience teaching in a new subject area). Within the mentoring group, a discussion the researcher had with the mentor regarding the fourth dimension highlighted philosophical views of mathematical content in the absence of discussion of pedagogical techniques. For these reasons, it was felt that pedagogy and content represent two separate, yetfrequentlydependent, aspects of a general theory of math teaching. It should be noted that the specific evidence supporting the general theory that was

155 developed is limited and further study of this particular aspect of the theory is desirable. The general theory of math teaching as derived in this study appears in Figure 4.2. Pedagogical Orientation

Philosophy of Teaching. General Pedagogy

Specific Teaching Method Pedagogical Content Knowledge (PCK) Specific Content Subject Curriculum Philosophy of Subject

Content Orientation

Figure 4.2: General theory of math teaching.

In the general theory of math teaching developed here, pedagogical components correspond to the upper line and content orientation to the lower components. The content orientation illustrates that math content is much larger than the content taught in schools and that there is an interaction between math content in schools and content in the larger math community. The vertical dashed lines highlight the typical affiliation of pedagogy with an equivalent level of content. At the right, the intersection of pedagogy and content is pedagogical content knowledge which combines specific content objectives with a particular instruction technique. The middle pair of circles relates general content, for example graphing, to standard expectations of practice, such as using graphing calculators in math. The subject curriculum can be interpreted as a sequence of content items within the general content. The leftmost pair of circles represents an

156 abstraction of foundational viewsfromwhich the teacher's actions are derived. Philosophy of teaching represents a personalized context of deeply seated views of teaching that corresponds to a personalized and deeply held understanding of pedagogy in a general sense and philosophy of content is a personalized context of deeply seated views specific to the content area. The use of the community of practice and the general theory of math teaching facilitated the construction of the theory providing the linkages between reflective practice and self-efficacy. The theory is presented in Figure 4.3 and then discussed.

Figure 4.3: The substantive theory

To understand the theory, consider a circumstance where a teacher plans a lesson which effectively starts the process at the "general theory of math teaching" because their views shape the lesson. For a given lesson there are likely to be specific content objectives which the teacher pairs with appropriate pedagogical techniques. The teacher has a level of self-efficacy regarding the lesson which may, in part, be due to reflections on past usage (or lack of familiarity if they are innovating) of aspects of the lesson plan. The teacher's self-efficacy supports their effort to enact the lesson plan in the classroom. Regardless of how the lesson actually transpires, the reflective practitioner considers the events of the classroom and reflects on how they fit with the

157 planned lesson (closing the loop to "general theory of math teaching"), what the implications are for their self-efficacy, and information from the community of practice such as how well a curriculum expectation was met. In some cases, events, aspects of planning or interpretation of information from the community of practice are puzzling. Most often these cause the teacher to seek clarification, further information or alter their method; however, in some cases the questions are not satisfactorily resolved. Instances that leave unresolved questions may be perennial questions or simply instances were a teacher is obliged by other constraints to move on without fully resolving an issue. In reflective practice involving a group of people, the reflective practice typically begins with an issue or focus that is derivedfromthe classrooms. This leads to consultation of resources (community of practice) and discussion between the teachers that focus on their individually held theories of math teaching. Note that the theories are personalized in the same sense that the Miller-Seller curriculum orientations may be different for individual teachers within a group. Discussion within the group has a bearing on each teacher's self-efficacy as do classroom observations that occur in the process. The reflective practice continues in this way but the key element of growth is development within the personalized general theory of math teaching facilitating improved self-efficacy. A vignettefromthis study that illustrates the substantive theory particularly well is the lesson study group adoption of algebra tiles in a grade six classroom. Through the vignette conceptsfromthe theory are identified in brackets. In the early planning stages the focal topic of equation solving (specific content) was discussed and the use of algebra tiles (specific method) was mentioned as a possible method that could be explored. When the technique was initially mentioned it was not discussed to any extent and seemed to simply be a passing topic in the

discussion. The participant who ultimately implemented algebra tiles indicated that she had let the topic pass because she had not seen them since teachers' college and did not know enough about them (self efficacy, specific method). At a later meeting the researcher raised an issue regarding order of operations within the context of equation solving (general pedagogy). The conversation that arose reflected different grade level perspectives regarding equation solving (subject curriculum, philosophy of subject) and the grade seven teacher articulated the zero-principle that is fundamental to teaching for understanding at his grade level. In order to make his point, he demonstrated the zero-principle using algebra tiles (resources) on a SMART board (resources). The tutorial evolved to an exchange of questions (pedagogical questions) and answers (specific teaching method, specific content) that provided detailed information regarding the implementation of algebra tiles in the classroom (PCK). Two aspects of the dialogue were particularly noteworthy. Firstly, the issue of what resources were available in the school was raised and addressed (resources). This included a mention of the availability of algebra tiles suitable for use with an overhead projector since a SMART board would not be available to the teacher (resources). Secondly, the concern was raised that algebra tiles implement the zero principle and this implies an ability to add and subtract integers. The concern was that the implied ability exceeded the actual ability of many students (pedagogical questions, philosophy of teaching). The resolution to this issue required that the final answer would be positive (specific content) and that students had a visual indicator to assist them with the integer skills. When the teacher went to pick up the algebra tiles, she went over how to use the tiles one last time. This is indicative of a self-reflection (reflective practice) that reassured the participant

(self efficacy) that they understood how to use algebra tiles (PCK). She then implemented the tiles in the classroom (classroom lesson) and later reflected (reflective practice) that it had been successful (self efficacy). She indicated that she would use them more often and explore other ways of using them in the classroom. The vignette highlights how an area of concern can emergefroma reflective practice discussion. A limitation in pedagogical content knowledge (not general content) prevented the teacherfromusing algebra tiles even though she had seen them in teachers' college. The dialogue in the reflective practice meetings provided detailed information that raised the participant's self-efficacy and facilitated the classroom implementation. It also provided a forum for support and a mechanism for resolving concerns that might have arisen during the implementation. There was evidence that further discussions (within the community of practice) would take place in order to expand the usage of algebra tiles in the grade six class and to share the beneficial experience with another grade six teacher. The peer coaching group provided insight into the influences of the community of practice on the reflective process itself. The administrator who had organized the release time provided charts to the participants to help them plan their time. She also provided leadership by assessing the needs of the group and addressing those needs that lay outside the reach of the group. That said it is also clear how the community of practice could have a negative influence on the reflective process. The other distinct feature that the peer coaching group demonstrated was the role that a resource can have in shaping the pedagogical content knowledge that is used in reflective practice. The PRIME diagnostic tool brought consensus to the group and provided activity ideas that could be tested during the observation phase.

160 While there were two novel aspects to the peer coaching group, there was also considerable support for the rest of the theory. The only area that did not receive direct support was the selfperception component and this was likely due to the amount of experience of the participants and perhaps the size of the group. The mentoring group highlighted the way in which access to resources can affect pedagogical choices. In the instance of acquiring additional textbooks there was no corresponding change in practice. In contrast, the Towers of Hanoi provided a means to change the pedagogical method used for introducing the problem. It also highlighted the interplay of pedagogical comparisons when multiple resources are available. The mentoring group again highlighted many features of the theory. The pre-service teacher showed self-perception leading to both changes in self-efficacy and reflections more than any other person in the study. This supports the supposition that it may not have been observed in the peer coaching group because of their level of experience. The mentor, on the other hand, highlighted the influence of time demands. In both cases, there was considerable dialogue about content and PCK Scope was observed to be different between the mentor, pre-service teacher and the researcher. The lone reflective practitioner highlighted die community of practice and emphasized the role that her principal, board expertise, and subject leadership outside the school played. At the same time, her recounting of adapting to a problem centered curriculum provided insight into the extent to which reflective practice can be beneficial across the content curriculum. Not only did she adapt her understanding of the problem centered philosophy, she enacted it with pedagogical content knowledge that she had not used previously.

161 The theory presented fits with the findings of all four diverse reflective practice groups. There is considerable overlap on key connections within the theory. The key areas where the theory could be improved are the level of evidence supporting the least observed features. These are the connections linking the Ministry (in the community of practice) and the philosophical end of the general theory of math teaching to other components in the model. In addition, the evidence supporting the general theory of math teaching in the form presented is perhaps the weakest component of the theory. That the general theory of math teaching is linked within the substantive theory is not in question, where the evidence is relatively weak is in the support of the structure of the general theory of math teaching. This is an area that requires further study. Research Questions 1) What changes do teachers consider during reflective practice?. How do the discussions and efforts toward implementing changes affect each teacher's self-efficacy? The theory highlights that reflective practice addresses three facets of teaching; Selfperception, the community of practice, and the general theory of math teaching. Self perception was relatively infrequent or the most private. The community of practice placed emphasis on external agency and arose as both a source of information and ideas but also as a discussion point where new ideas were not fully understood. The general theory of math teaching was the most important area for changes and it mediated suggestions arisingfromthe community of practice. The scope of the general theory of math teaching is broad and facilitates a wide range of content specific and pedagogical changes to occur during reflective practice. Discussions within the reflective practice group were professional and generally focused on classroom events and observations. The discussions enabled teachers to transfer ideas among themselves, to problem solve resource content that had little familiarity to any of the participants

162 and to generate ideas that teachers could explore without observation. The opportunities provided by the reflective discussions provided useful approaches for the classroom, increased collaborative interactions and an expansion of components of individual teachers' knowledge of teaching (reflected in the general theory of math teaching) that extended passed the immediate content focus. The response of teachers was an overall positive response to having beneficial opportunities to discuss methods that they perceived would improve their classroom experiences and provide new opportunities to students. The result was an increase in self-efficacy because of increased numbers of options that arose in the discussions. When classroom implementation followed, observation of methods added a level of credibility and immediacy that is typically lacking in other forms of professional development. 2) How do reflective practices and self-efficacy interact in the process of teacher innovation? The theory that emerged shows reflective practice providing information that focuses on the needs of the teacher, whether those needs are content oriented, pedagogical, or resource based. If those needs can be met, or alternative approaches developed where the needs can be met, then the teacher has the opportunity to experiment with a possible solution in the classroom. However, if the needs are not met then resolution of the issue that caused reflection is not feasible. The constraint of meeting needs is multi-faceted and may be organizational or due to limitations in the teacher's PCK or repertoire of pedagogical methods. It was also found that reflections can have an impact on a teacher's self perception because an observing teacher has a different perspective. While this appeared rarely, it modified teachers' self perception and therein affected their self efficacy. 3) To what extent does a teacher's curriculum orientation affect the decisions they make regarding techniques arisingfromthe reflection process?

163 Philosophical orientation represents the components of the general theory of math teaching with the widest scope and this was an area with little evidence in this study. While the orientation end of the general theory of math teaching is an area for further study, the analysis looked at the variety of teaching principles that emerged (and fit in the middle section of the general theory). This provided evidence of an intermediate level of principles that reside between the daily needs of lesson planning and the philosophically based curriculum orientation. The principles that teachers used were broad based and reflected both generalizations of content and pedagogical approaches as well as less coherent elements that had arisen in practice but not to the extent necessary for a generalized understanding (for example, fetal alcohol syndrome was mentioned in terms of a strategy for teaching but not in terms of a generalized understanding of varieties or intricacies of the syndrome). The general theory of math teaching was required in order to address the routine changes within day to day lessons, the principle based stance that guided longer duration planning and the relative unwavering long term curriculum orientation. 4) To what extent does the level of formality in the reflective process influence the teacher's efficacy during an innovation? The level of formality was found to be quite fluid making it impossible to compare the methodological implementations. Details of the operation of the reflective practice procedure were frequently discussed rather than following a specific set of guidelines. In view of this, the role of leadership (as an external formalizing agent) was examined and found to have an impact, generally positive but not without negative effects, through the community of practice. Leaders were typically in a supporting role that was successful when appropriate supports were provided. In one case, the active involvement of a school leader provided direct organizational support and

164 benefits that extended beyond the other groups was observed. However, the involvement of leadership has to be mitigated by the value of the group internally negotiating shared leadership. In addition, two cases had reduced levels of external leadership (the isolated reflective practitioner because she was isolated and the pre-service mentoring in which the school did not have a school board) and this did not prevent their reflective practice being successful. This is an area for future research where the current theory will need to be developed to include active interventions on the part of leadership and the concept of collective self-efficacy is expected to be important. There would also be benefits to examining the characteristics of teachers who successfully use reflective practice with minimal external influences because this would reveal characteristics that are antecedents to reflection and could provide insights into effective methods of assisting other teachers, particularly pre-service teachers, to use reflective techniques effectively. Reconciliation with Research Literature There is a considerable body of research pertaining to reflective practices. In order to facilitate the reconciliation with the theory that has been developed, each of the three parts of the theory will be examined separately. The community of practice (as shown in Figure 4.1) will be examined first followed by the general theory of math teaching (Figure 4.2) and finally the substantive theory (Figure 4.3). Community ofPractice The initial phase of this study did not anticipate the need for the entire community of practice, rather it had been envisioned that a much smaller school based community of practice would be pertinent to the short time spans and contextually specific focus of the reflective practitioners. This was based on a faulty assumption that reflective practice provides

165 considerable autonomy that facilitates individualized responses (Warfield et al., 2005). The community of practice emerged from the analysis because autonomy is distinct from professionalism and it is the latter that makes the community of practice imperative. Thefindingsof this study, with the exception of the isolated reflective practitioner, are based on reflective practitioners who had a degree of formality in so far as they had a formal structure that was recognized by each school's administration. However, in each case, the group had considerable autonomy that fit with a definition of reflective practice that was relatively fluid (Golby & Appleby, 1995) and not an imposed intervention (Killion & Todnem, 1991). While this study focused on the classroom context rather than imposed changes, some aspects of critical reflection (Farrel, 2004; Valli, 1997) were observed when teachers discussed systemic changes that were in progress or changes they had noticed in their school structure. However, these discussions were relatively rare and seemed to arise to explain the larger school context to the researcher rather than as individual explanations. Only in one case (the curriculum achievement chartframework)was critical reflection potentially tied to characteristics of the classroom, however, even in this case the emphasis was on the classroom characteristics that could be addressed within the reflective practice context and the critical reflection appeared to be a personal secondary reflection that was beyond the scope of the group. Consistent with Zahorik (1987), most interchanges observed were between teachers of the same grade or division, often having classrooms in close proximity and having better availability than most other sources of professional support. In some instances, the reflective practitioners had exchanges outside of the reflective practice meetings and these seemed to be coincidental and brief (for example, the lesson study observation plan was drafted while two of the teacher's classes had a shared event in the library).

166 There was considerable discussion around resources and availability of borrowed resources was an issue (Conley et al., 1995; Reed, 1996). This is a fundamental constraint within the theory where the absence of resources can prevent a teacher implementing a technique they learn about through reflective practice. In such cases, there is no connection between the technique learned in the reflective practice and the teacher's self-efficacy because they do not have an opportunity to implement the method. The theory explains observations that teachers using reflective practice learn to teach differently (Warfield et al., 2005) because reflective practice clarifies individual needs that are addressed through both the community of practice and the extension of the teacher's knowledge within the general theory of mam teaching. Since interchanges are frequently between teachers of the same grade or who have classrooms in close proximity the opportunity to facilitate resource borrowing to test innovative practices is possible. There was some evidence of the role of career stage (Wright-Evans, 1999) in the case of the pre-service teacher. The other teachers emphasized practical methods but it wasn't clear to what extent this was utilized to expand practical theories or leadership skills. There did appear to be greater discussion around practical theories for more experienced teachers but, as the theory shows, this appeared to be a reflection of having a larger vision within the general theory of math teaching or, more commonly, the community of practice. The most senior participants were late career and did not show any signs of preparing for their next life stage (Wright-Evans, 1999) which was interpreted, in this theory, as having professional focus within the reflective practice. It was not the purpose of the reflective practice and, therefore, this theory does not discount the possibility, it merely shows that teachers at all career stages can engage in meaningful reflective practice focused on practical methods.

167 The community of practice provides an efficient encapsulation of the different sources of information and influence that can act as supportive factors in the process of implementing techniques learned through reflective practice. There were few instances observed where the community of practice hindered teacher change (disappearing white board markers being a rather minor example). General Theory of Math Teaching The general theory of math teaching (see Figure 4.2) serves two purposes in the theory. Firstly, it supports the idea that developing as a teacher does not mean that one will acquire knowledge of all the instructional practices that are available (an idea analogous to Ross (1986) regarding the scientific method). Secondly, the general theory of math teaching provides an encapsulation of content and corresponding pedagogical methodsfromdifferent scales or durations of implementation. The general theory of math teaching differs from other models of content such as Perkins and Simmons (1988) because it is organized according to the teacher's perspective rather than the students'. While students may be concerned with the level of difficulty such as separation of factual information from problem solving, the teachers in this study demonstrated concerns that were more fundamentally linked to timing. The general theory of math teaching may be thought of as separating content or teaching according to the duration of time that the component will be a focus. Philosophy of teaching will endure through an entire course because it is rooted in personal meaning (Miller & Seller, 1990). The curriculum is likely to be used to organize major units of a course while PCK is a daily consideration. The philosophy of content, general content and pedagogical content knowledge as combinations of subject content and the corresponding

pedagogy, are embedded in lesson planning and it is the stability of each component that determines where in the general theory of math teaching it is positioned. The changes that were discussed by the reflective practitioners reflected different aspects from a continuum of adaptation (Stodolsky & Grossman, 2000) including minor changes in practice, pacing changes and reconceptualization. However, the changes emergedfromthe teaching context rather than a systematic or theoretical derivation. It should be noted, however, that the need to introduce the community of practice and general theory of math teaching both reflect a range of resources that would be suited to addressing needs arisingfroma continuum of adaptation. In addition, the theory includes adaptations within the process of comparing pedagogical and PCK options. The observed reflective practices were conducive to a high degree of math specificity and this was the most actively shared part of the reflective practice. Consistent with Garet et al. (2001), it was more engaging to the teachers than more general discussions. In the case of the peer coaching group, the use of the PRIME resource helped the peer coaching group identify and focus on a specific dimension of math teaching (McDougall et al., 2006). The participants were observed to have difficulties at times with communication (Skemp, 1978), conveying conceptualizations (Reynolds, Sinatra & Jetton, 1996) and conveying some pedagogical thoughts (Hartman, 2004; Puchner & Taylor, 2006; Taylor et al., 2005). These are included in the current theory through the general theory of math teaching where personal limitations or different understandings (such as terminologies) are fundamental and, with the development of appropriate technical language, resolvable communication issues (Yoshida, 1999; Rock & Wilson, 2005). Generally, there was sufficient common ground to facilitate changes in practice (Richardson, 1994) that were highly contextualized (Hsu, 2004) but have

169 immediate utility in the participants' classrooms (Hartman, 2004; Shulman, 2003). The theory reflects this with common ground being overlap between individuals' enactments of the general theory of math teaching and the context arisingfromthe localized format of the reflective practice. Teachers in the lesson study, and the other reflective groups, found that they gained pedagogical options (Watanabe, 2002) through additions to their individualized versions of the general theory of math teaching. There was apprehension, as the theory predicts with the inclusion of self-perception, about teaching infrontof peers (Puchner & Taylor, 2006) and overcoming this helped to develop trust which strengthened the collegial support. Wheatley (2005) suggests that efficacy may be higher when traditional methods of teaching are used while Ross (1998) suggests that having more pedagogical options will benefit teacher efficacy. In this study, it appeared to be a matter of familiarity with techniques rather than whether they were traditional. Higher efficacy was related to how routinely (not necessarily how frequently as Ross (1998) points out) one used a technique and this was what made it teacher and context dependent. More pedagogical options would facilitate better acuity in choosing methods and lead to improved self-efficacy. Curriculum orientation was limited in scope by the narrow range of the participants which may be due to the content focus (Kinach, 2002). Whether the bias awayfromtransmission is due to emphases in the community of practice or due to variety of available instruction tools was unclear. The emphasis in the reflective practices observed was the classroom and, while some deeper questioning of perennial teaching questions arose, they were not viewed as resolvable within the limited scope of the reflective practice. The absence of observations regarding curriculum orientation is consistent with Ashton and Webb (1986) claiming that it is not directly

linked to self-efficacy. However, the theory developed in this study suggests that philosophical orientation is linked, through the general theory of math teaching, to self-efficacy. The reconciliation of these two findings may simply be the time scale on which events happen. Curriculum orientation, as a broad based component of the general theory of math teaching, would derive support from a wide array of daily practices (and therefore not necessarily change in response to a teacher's pedagogical changes as noted by Pajares (1992)). Therefore, any change in curriculum orientation would require broad supportfroman equally wide array of practices and that would not be feasible in a single short term professional development. At the same time, observations over longer periods of time are likely to lack clarity because there would be many other minor changes that distinguishing the two types of changes would be difficult. In this respect, the pre-service teacher was unusual because she appeared to be in the process of establishing her curriculum orientation in a manner that was consistent with it taking considerable time for changes affecting one's curriculum orientation. The theory outlines limitations for reflective practice derived from limitations in the general theory of math teaching regarding the scope of the teacher's knowledge of content and pedagogical options. This has been explained elsewhere as the zone of proximal action having to be within the ability of the individual teacher to extend his zone of proximal development (Blanton et al., 2005). There are also limitations arisingfromthe community of practice through the availability of support or resources. The focus on content and pedagogical methods, in the current theory, facilitates direct connections with the actions of teachers. The theory developed in this study is based on mathematics as a content area. It seems likely that many features of the theory will be equivalent in other content areas. However, as Stodolsky and Grossman (2000) observed, mathematics lends itself to a sequential process of

teaching that may not be applicable in other content areas. In addition, the content area has an impact on self-efficacy and the student achievement derivedfromit (Ashton & Webb, 1986; Ross, 1998). It is unclear whether encapsulation with the general theory of math teaching can be applied in other content areas and whether the inner structure of the theory is independent of the content area. In science, it is not clear how experimentation would fit into the general theory of science teaching and whether classroom use of equipment would require a degree of classroomfocused collective efficacy rather than just the teacher's self-efficacy. Lastly, Arvold (1998) suggests viewing a teacher as a blending of roles which may provide a useful way of viewing the linkages between content components and corresponding pedagogical components within the general theory of math teaching. While this notion has not been examined in this research, it is consistent with observations of discussions focusing on content with specific methods and suggests the possibility of developing the general theory of math teaching further. The general theory of math teaching encapsulates a diverse array of concepts and limitations. Perhaps the main challenge with the concept is the diversity of effects that emerge from teachers having significantly different versions of the general theory of math teaching. While this may be a challenge to comprehend, it may also provide insight into the effectiveness of reflective practice as a method of sharing practices and perspectives. The Substantive Theory The theory (see Figure 4.3) outlines the relationship between reflective practice and selfefficacy which is composed of two pathways. The self-perception pathway is less frequently observed and links reflections directly to self efficacy. The other pathway explains the benefits and constraints that the community of practice and general theory of math teaching generate in

172 making the connection with self-efficacy. In both cases, however, the theory is in agreement with Ross' (1998) suggestion that observing peers, colleagues and other sources of information can influence a teacher's self-efficacy. The theory permits self-evaluation feedback that may promote reflection (Licklider, 1995). It also allows self-reflection to evaluate one's own experiences (Pajares, 1996). Among the observed participants, self-evaluation was most evident for the pre-service teacher; however, this may reflect the explicit, rather than tacit, nature of her knowledge (Lam, 2000) at that stage in her career. The use of self-reflection is explained by the theory by allowing self-perception to interact with reflections (during reflective practice) or to have a direct impact on self-efficacy for future lessons. More typically, reflective practice is a process which spirals through "...stages of appreciation, action, and reappreciation" (Sch6n, 1982, p. 132) which is consistent with the theory developed here. Classroom events provide the opportunities for reflective practice ("appreciation"), this leads to content and pedagogical considerations and comparisons which are then mediated by self-efficacy as the process returns to the classroom where the reflective process can repeat ("reappreciation"). This process was observed, more frequently than self perception, to lead to significant adaptations in the classroom (Cochran-Smith & Lytle, 1999; Rosenblatt, 2004). Bainer and Didham (1994) found support to be distributed within the reflective practice group rather than centralized in single individuals and this was also the case with the groups in this study. There were two instances where individuals had areas of detailed understandings surpassing the other members of the reflective practice group, however, there was a shared sense of professional commitment that served to motivate and many of the participants indicated a

173

desire tofindbetter ways to teach (Allinder, 1994; Guskey, 1988). Even where the pre-service teacher was concerned, the discussions focused on options rather than the mentor conveying his chosen technique. This type of social organization was, in all cases except the isolated reflective practitioner, conducive to persevering in the face of obstacles (Tschannen-Moran et al., 1998). Interactions appeared to correspond to teacher attitudes (Ashton & Webb, 1986; Bainer & Didham, 1994); however, the methodology was not sustained and longer term stability of attitude changes were unclear. Collaboration contributed to efficacy because of opportunities and the observations the teachers made of each other (Bainer & Didham, 1994; Tschannen-Moran et al., 1998). However, the theory shows that this is within constraints imposed by knowledge and the community of practice. In the absence of hindrance, teachers resolve their instructional concerns and create professional bonds that can be used to resolve future issues both of which positively affect efficacy. The willingness of participants to seek help improved (Ross, 1998; Smylie, 1988) within the confines of the short period of observation. There was evidence of contact beyond the confines of the reflective practice group. Support and encouragement in the reflective practice process (Hoy & Spero, 2005; Tschannen-Moran et al., 1998) were evident and interpreted as loops within the theoreticalframework.Two looping pathways start with reflective practice leading to the general theory of math teaching and then to self-efficacy assessment but then branching according to the level of self-efficacy. An efficacious teacher would implement the plan in the classroom, while a less efficacious teacher would consider his self-perception and might choose to reexamine his reflective practice (such as re-consulting his colleagues). In either case, each return to reflective practice facilitates peer support and encouragement in response to the teacher's actions.

Gains in efficacy were observed to be most significant in the middle stages of the reflective practice. In the early stages, the scope of discussion was generally quite broad, because the teachers have to learn about each other's versions of the general theory of math teaching, whereas toward the end there was an air of having discussed what they could. In the middle stages of the process, the discussion was sufficiently focused and novel to create the most significant opportunities for the deepest discussions of aspects of the general theory of math teaching leading to change (Garet et al., 2001; Ross, 1998). Learning among the reflective practitioners about each other's versions of the general theory of math teaching accounts for observations of surrogate self efficacy by the pre-service teacher (Rowley, 1999) when the mentor, but not the pre-service teacher, had a personal version of the general theory that could support efficacy for a particular technique. It also explains the importance of the discussion of perspectives (De Lima, 2001; Hsu, 2004); however, the theory had to address the observation that the discussions, in some case, refined a pedagogical question without resolution. When discussions of perspectives provide innovative teaching practices for an individual teacher, the implementation of the practice is moderated by their self-efficacy (Goddard et al, 2004; Hoy & Spero, 2005; Stein & Wang, 1988). The strength of observed social ties was consistent with the literature (Bullough & Stokes, 1994; De Lima, 2001; Zahorik, 1987). The participants indicated that they had made choices in the formation of their reflective groups to avoid ties where trust would be an issue. The theory explains this in two ways: untrustworthy participants could interfere with one's self-perception and they could also violate theoretical assumptions such as the autonomy of the group by bringing a new component into the theory (such as leadership). There was evidence of "weak social ties" by way of professional relationships that might extend to periodically having lunch

175 together but did not extend outside of the school day. Generally, the groups appeared to have been formed with a purpose in mind rather than for social reasons. Bandura (1978; see also Goddard et al., 2004) postulated four kinds of experiences that are relevant to efficacy. All of these are included in the self-efficacy aspect of the theory. Affective state, because it arisesfromexcitement or anxiety, is the most likely to affect self-perception and, in this respect, is likely to be the weakest form of self-efficacy for successful innovation. Vicarious experiences and social persuasion were observed to be the most common forms of self-efficacy arisingfromreflective practice. Mastery experience is well suited to teacher's individually reinforcing techniques they learnedfromthe reflective practice after the process has been completed. Lesson Study. The lesson study implementation was largely in agreement with the lesson study method outlined in the literature review. It differed in the level of detail in the lesson plans which did not include a list of anticipated student responses (Fenandez et al., 2003; Meyer, 2005; Yoshida, 1999). This appeared to be due to a limited vision of what constitutes lesson study (Pennisi, 2004) but was reconciled by having some discussion of overall anticipated achievement levels of each grade level followed by detailed conversation regarding student responses after the fact. A second minor divergencefromthe literature was the dual focus of having a pedagogical focus within a vertical alignment (i.e. curriculum) objective, this is a minor divergence because of a bias in the literature toward reporting on a single repeated lesson rather than variants such as vertical alignment (Meyer, 2005). There was a written report to satisfy administrative requirements while the participants emphasized the overall discussions (Blum et al., 2005).

Like Yoshida (1999), teachers in the lesson study group focused on teaching materials and the learning process. Unlike Yoshida, the emphasis was not on the development of a lesson plan but, because the focus was vertical alignment, consistency of lessons across the intermediate grades. Lesson study has been observed to increase the extent of student centered learning (Carter, 2004; Kinzer, 2005) and to improve the student focus in pedagogical content knowledge (Oshima et al, 2006), both of which are consistent with observations in this study and the theory which is contextually based and derives meaningfromthe students who the teachers are currently teaching. Lesson study is not systematic research (Pang 2006; Pang & Marton, 2003) and while some suggest including an expert in the process to ensure a proper scientific process (Friedman, 2005; Yoshida, 1999), this would tend to constrain the wide range of discussions that were held. Aside from pragmatic problems of including experts (Fernandez, 2002; Ling et al., 2006; McLaughlin & Mitra, 2001), the attraction of the method is the ability for the discussions to touch on many topics and discuss them to a utilitarian, rather than scientific, degree. Peer Coaching. The observed peer coaching used a common structure (Wood & McQuarrie, 1999) with alternating roles (Kohler et al., 1999). A minor modification was having both pairs, the pair of grade one teachers and the pair of grade three teachers, meet together as a group of four for discussion. Goals were established and later discussed (Perkins, 1998) with the assistance of an administrator. Guidelines for respect and questioning (Perkins, 1998; Wallace, 1998) were not necessary because of a high level of professionalism among the veteran teachers.

The theory is consistent with Ross (1998) who found that peer coaching positively affected teacher efficacy. It also extends Montalbano's (2001) observations, of action research providing a sense of empowerment and strengthening the participant's efficacy, to a larger group of reflective practices. Pre-Service Mentoring. Hoy and Spero (2005) suggest that self-efficacy is more malleable in newer teachers. While this is consistent with the findings in this study, the pre-service teacher also showed the greatest fluctuations of self-efficacy and the most experienced participants showed very little selfefficacy, this appeared to be because there were fewer innovations available to the more experienced teachers. When the veteran teachers discussed new techniques, they were aware of similar techniques and could draw parallels that suggested they could induce surrogate selfefficacy through parallels within their own general theories of math teaching without having to be explicit about it. It also appeared, particularly with the isolated reflective practitioner, that deeper issues in the general theory of math teaching were being considered and these would lead to shifts in self-efficacy over longer periods of time. The observed case of mentoring was a senior-junior pairing (Conley et al., 1995). The mentor provided encouragement, collegiality and constructive advice separate from the evaluation process (Hiffman & Leak, 1986; Robbins, 1999). The mentor's advice was generally informal and oriented toward technical skills rather than theoretical visions of teaching (Elliott, 2002). During the process of mentoring, the pre-service teacher indicated a transition from individual issues toward using evidence and rationalizing issues which was consistent with the literature (Cherian, 2007; Mewborn, 1999). This process fits the theory because the mentoring

178 process facilitates the pre-service teacher working through a phase of overall organizing of the general theory of math teaching and gaining an understanding of the community of practice (This process was suggested by Gudmundsdottir, 1990, but not in mathematics). The initial stages of the process involve addressing individual components of the general theory of math teaching and community of practice that develops into a practical understanding of the interaction of the various components. This is not a theoretical understanding, nor is it a comprehensive image; rather it is a functional image of the individual's making that allows them to operate independently in the own classroom. There are different images of the mentor presented in the literature. They may need a vision of teaching combined with knowledge of how teachers learn (Feinman-Nemser, 2001), an understanding of the pre-service teacher's proximal zone of development (Wang & Paine, 2001) or use a mentoring orientation to mediate the pre-service teacher's learning process (Elliott, 2002). The theory reconciles these images using the notion that pre-service teachers' general theory of math teaching and community of practice (in particular knowledge of resources) will principally be a subset of the mentor's. The mentor had a well developed general theory of math teaching and community of practice which they use to comprehend the pre-service teacher's development. The vision that Feinman-Nemser (2001) refers to is recognition, by the mentor, of the subset of practices the pre-service teacher is using and conveying the mentor's familiarity with interactions between different elements of the general theory of math teaching and community of practice. Knowing how to teach teachers may be overstating a high degree of commonality between the subset of experiences of the pre-service teacher and the mentor. This view is consistent with Wang and Paine (2001) because the zone of proximal development amounts to the mentor recognizing ways to guide the pre-service teacher to expand his subset of

the general theory of math teaching and community of practice to the extent needed to support independence. The use of mentoring orientations (Elliott, 2002) is a way of theorizing the need for a particular kind of relationship between the pre-service teacher and mentor in order for the reflective practice to be collaborative and effective (a concept discussed by Galvez-Hjornevik, 1986). The mentoring relationship is bidirectional (Conley et al., 1995; Elliott, 2002; WollmanBonilla, 1997) and the theory explains this in two ways. Firstly, pre-service teachers often suggest ideas that are outside of the areas typically used by the mentor, or perhaps something the mentor has never tried, and this may provoke reflections by the mentor leading to reconsideration of different approaches (especially where the pre-service teacher adopts a method with success that surpasses the mentor's expectation). Secondly, the mentor, in order to give useful advice, has to consider relationships between practices they see the pre-service teacher using and those proximal elements of their general theory of math teaching and community of practice that will be of benefit to the pre-service teacher. This process of consideration will deepen the mentor's understanding of the general theory of math teaching. Isolated Reflective Practitioner. The lone reflective practitioner highlighted constraints, particularly availability of collaborative peers, imposed by school size (Rosenblatt, 2004). The theory explains this in terms of a constraint in the community of practice that has a direct impact on the reflective practice options. She may have found reflection feasible, in part, because having higher ability students (according to the school's EQAO results) would likely provide a higher level of efficacy (Ross, 1998).

180 Other Reflection Methods. Other models of reflection were found to have some alignment with the theory presented here, however, there were differences arisingfromdifferences in perspective. York-Barr et al. (2001) emphasize a teacher using individualized reflection prior to discussion with the group, however, the formal structures that were used did not facilitate time for such individual reflection and it was the observing teachers who provided the principle focus in post-observation discussions. Osterman and Kottkamp (2004) provided a model that is consistent with the theory developed in this study. Problems were identified in reflective practice discussions which led to observation and analysis within the general theory of math teaching in consultation with information from the community of practice and the results were used for active experimentation. However, there is a discrepancy between what Osterman and Kottkamp (2004) label "abstract reconceptualization" and the process of discussing pedagogical options in the theory that emerged in this study. This theory that emerged reflects the pragmatic approach of the observed teachers and de-emphasizes the abstract theorizing that may occur in a private cognitive process for each individual. The theory, as anticipated in the literature review, does not fit with Little (2003) who indicated mat classroom adjustments were based on formal student results. While these would be assessed by a reflective practitioner, the practitioners that were observed did not gather formal assessments during their process. In terms of the theory, formative assessments might be used as the basis of instructional reflection which would be consistent with the theory (although it was not observed). Summative assessments are not likely to provide reflective information in a timely manner or with details associating them with specific pedagogical methods. Little (2003) also excludes the appraisal of attributes that are not explicit in student results but this was rarely

181 observed in practice. In this study, the teachers were observed to discuss a wide range of contextual features and did not curtail conversation that did not pertain directly to student results. At times, there were tangential conversations that appeared to be more meaningfully interpreted as building collegiality or collaborative capacity rather than in terms of student results. Hawkes (2001) did not observe any relationship between self-efficacy and self-reported levels of reflectiveness. In terms of the theory developed in this study, this could be a result of two factors; there may have been other limiting factors preventing the development of reflections into pedagogical or content changes that would affect self-efficacy during classroom implementation. Secondly, there may have been a discrepancy between self-reported reflectiveness and reflective practice particularly in terms of the level of focus on areas that cause cognitive dissonance within the teacher. In the theory, teachers are aware of differences in classroom methods and consult with colleagues to reconcile those differences (Rowley, 1999; Timperley & Robinson, 2001). The reflective process occurs within the context of teacher's classroom contexts (Hsu, 2004; Nolan & Hillkirk, 1991). The theory accounts for improved student achievement because teacher's expand or fine tune the repertoire of techniques the use, which constitutes teaching differently (Warfield et al., 2005) and responding to student needs (Montalbano, 2001). The theory does not illustrate the impact of improved self-efficacy on improved student achievement (Ashton & Webb, 1986; Goddard et al., 2004; Hoy & Spero, 2005; Pajares, 1996; Ross, 1998). The theory does not directly explain why teachers with higher self-efficacy would expect more of their students (Tournaki & Podell, 2005). However, learning new teaching techniques and expanding a teacher's repertoire may facilitate better matching of pedagogical techniques with content and needs of students. Persistence in the face of setbacks (Hoy & Spero,

182 2005; Pajares, 1996) creates more opportunities for students to learn and improve their achievement. However, persistence likely does not work in isolation and it is likely that student achievement improves because of teachers using reflection to assess classroom events (Goddard et al.). In particular, the developed theory shows that reflective practice provides support that can improve self-efficacy. Time was observed, and noted in the theory, to be a universal factor, both in terms of the amount available and in terms of scheduling, which has been well documented in the literature (Bainer & Didham, 1994; Hargreaves & Dawe, 1990; Hartman, 2004; Kinzer, 2005; Manen, 1991; Perkins, 1998; Tamargo, 2005; Wallace, 1998; Wolpin, 2006; Yoshida, 1999; Zahorik, 1987). Conclusions The theory developed in this study fits with four cases that correspond to the four quadrants of Van Lier's typology of research designs (Freeman, 1998) because it includes both high and low interventions combined with high and low degrees of organization (see Table 2.2). This, along with SchQn's observation that reflection is broadly applicable, suggests that the presented theory, within its limitations, has applicability to other kinds of reflective practice. The theory develops a plausible mechanism that was needed for lesson study (Lewis et al., 2006) including an explanation of improved instruction (Meyer, 2005) and improved student achievement (Friedman, 2005; Meyer, 2005). It also provides an explanation that was needed for improved student achievement in peer coaching (Green, 2004). The theory also explains various problems within peer coaching and mentoring. In order for mentors to support change (Wang & Paine, 2001), they must have a general theory of math teaching, and to a lesser extent a community of practice, that supports the changes. For example,

183 changes toward the NCTM Standards (NCTM, 2000) were observed to be problematic (Frykholm, 1996). The theory accounts for this because the common experiences of the preservice teacher and mentor (Arvold, 1998) would likely represent the initial general theory of math teaching for the pre-service teacher and, while the mentor would be able to work within this subset of practices, they were not in the realm of the Standards. At the same time, the mentor likely had less experience with the Standards than other techniques and therefore would feel lower efficacy to facilitate the growth to independence of a pre-service teacher. These factors would combine with the reality of resources that may not have been fully supportive of the Standards as well. This theoretically based image is consistent with Mossgrove's (2006) comparison of two pre-service teachers. The isolated reflective practitioner fits quite well with an image of self-study in which she generated local knowledge (Anderson & Herr, 1999; Buschman, 2001) to support her own needs (Louie et al., 2003; Zeichner, 2007). While changes in instructional practices can be supported by curriculum materials, reported results appear to be mixed (Collopy, 2003; Kamina, 2005) and do not explain the success of the observed practitioner. One account in the literature (Thomas & Monroe, 2006) bore a striking resemblance to the observed case and this has led to consideration of two possibilities. One is that historical experiences, particularly positive experiences, may play a role in the absence of other factors (Drake et al., 2001). A second explanation, which may build on the first, is that the observed participant and the account given by Thomas and Monroe both included a capacity to draw on theoretical images; in the observed case using knowledge from her masters of education and for Thomas and Monroe supplied by the critical friend. In both cases, this may have provided sufficient capacity for abstraction within the general theory of math teaching for growth with minimal external support.

184 In overall terms, the theory presented is consistent with a considerable portion of the research literature and the few discrepancies have been explained. There are areas that require further consideration, such as clarification of the impact reflective practice may have on the philosophical portion of the general theory of math teaching, but the overall theory provides a framework with which to examine these areas. An additional strength of the theory is reconciliation with the research literature while allowing the maintenance of practical utility that can explain experiences that teachers are familiar with when using reflective practice. Implications of the Theory The implications of this theory are both academic and practical. The academic implications are considered first and the consequential practical implications are developedfromthem. Academic Implications The academic contribution of this theory is the demonstration of a linkage between reflective practice and self-efficacy which, according to the research literature, contributes to improved student achievement. The theory shows the components that need to be in place in order for the link between reflective practice and self-efficacy to be strong. Specifically, the theory demonstrates the central importance of the general theory of math teaching and how limitations in PCK can have a direct impact on a teacher's ability to implement techniques they learn through reflective practices. The theory illustrates that the community of practice has a secondary supporting role that is important but not imperative to the success of reflective practice. The theory shows the scope of changes that reflective practice can impact; in particular, the theory implies that reflective practice can do more than address immediate classroom needs. Reflective practice can impact the principles that teachers use and result in significant conceptual changes (provided there are few significant hindrances to change). Lastly, because the theory has

185 been developed from four different kinds of reflective practice, it demonstrates a theoretical commonality of reflective practice techniques. Practical Implications Practical implications of the theory are: implications for the formation of reflective groups, the role of leadership in support, role of the community of practice in resolving content and pedagogical questions, the need for an ongoingflowof new ideas, time concerns may be lessened by reducing program conflicts and a justification for endorsing reflective practice as a mainstream form of professional development. Reflective practitioners will only be successful if they form a group that has the internal ability to broaden the participant's general theories of math teaching. That is, they must be able to share, explain and resolve issues pertaining to details of pedagogical content knowledge and associated content pedagogy. This suggests that reflective practice is likely not a good choice if the content area is weak or unfamiliar to the teachers in a group. The role of the school, subject and organizational leadership, within the scope of this study (i.e. when not providing a direct mandate for reflective practice), is one of support and removing hindrances. Support in terms of facilitating release time, assisting with scheduling, and assistance with administrative details assists in keeping the reflective practice focused on content. It is equally important that they supply resources where they are needed and assist in the resolution of both content and pedagogical issues when a reflective group is unable to do so. The need for specific guidelines that direct reflective practice was not observed, but having general guidelines facilitated the process. Broad discussions leading to focusing of purpose and negotiation of shared leadership within the reflective practice group appeared to be beneficial.

The central role of the general theory of math teaching highlights the importance for issues around content and associated pedagogy. Ideally, within a reflective practice group methods are developed to improve student achievement, however, the purposeful sampling in this study suggests that less than ideal circumstances could arise and these would be more likely to require the community of practice to assist in providing pedagogical options to the reflective practitioners. This might be in the form of supporting involvement in curriculum workshops, supplying specific content oriented resources (such as PRIME) or calling on appropriate board expertise but it would, ideally, be focused on the particular needs identified by the group. For any particular lesson, having a variety of techniques facilitates pedagogical and PCK comparisons which are beneficial because they allow the teacher to match the choice of method to the specific needs of students. This suggests that the broad range of topics discussed in reflective practice meetings is beneficial. It also suggests that addressing staff complacency is necessary and encouraging ongoing experimentation at a division level combined with sharing of new techniques is beneficial. It also implies the need for an ongoingflowof new ideas, such as one might acquire from subject specific periodicals. Time was found to be a factor in all aspects of this theory. While the issue is omnipresent, several teachers indicated that scheduling of reflective practice was secondary to other commitments that were organizationally required. It appeared that improvements could probably be made by organizational leaders to avoid conflicts between organizational programs and reflective practice. Lastly, the theory demonstrates that the specific advantages of reflective practice are in broadening the content and pedagogical repertoires of teachers. It provides a clear picture of what is required to facilitate the process and indicates what leadership could look for in terms of

187 evidence of change. In view of this, the theory supports the endorsement of reflective practice as an alternative to off-site programs of professional development. While it is not suggested as a replacement for reform programs, it does appear to be a viable approach for teachers to learn practical classroom oriented teaching techniques. This may appear to be clear from the literature on reflective practice; however, monetary support for reflective practice appears to be a scarce reality. Future Research While this theory provides useful insights into some limitations of reflective practice, it is only applicable at the micro-system level (Ashton & Webb, 1986) and needs to be generalized in a manner that would explain how organized reflective practice could be achieved. This would seem to amount to a generalization of the theory to include collective efficacy effects and more direct interventionfromthe community of practice. The theory highlights a mechanism that explains how several types of reflective practice result in improved student achievement. However, it did this by drawing on the connection between self-efficacy and student achievement in the literature. It would be beneficial to use a longitudinal approach to examine the longer term effects of reflective practice and to observe both the reflective practitioners and students in order to examine the link to student achievement through direct observation. This would also allow an investigation into causality and the direction, if one exists, of the linkages between the different components of the theory. The present theory has only been developed in the area of mathematics. While this has been done with a sample covering all divisions, it would be useful to see how the theory needs modification for other subject areas (the need for modifications is implied by Drake et al., 2001). This might provide considerable insights into the organization of knowledge and pedagogy

188 within the general theory of math teaching and this might provide deeper insights into the philosophical orientations that teachers hold. Lastly, the literature has scant information regarding the use of reflective practices. While a few quantitative statistics were found in the course of this work, there would be benefits to knowing how common the different techniques are and what variations are being used. It would also be useful to establish the prevalence of reflection that is being performed informally through day to day activities of teachers.

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APPENDIX A RESEARCH EVENTS In this research study, an event was an interaction with a reflection group or member of a group (with the exception of emails regarding scheduling issues). The main events that took place were meetings organized by the participants in which they discussed reflective practice. During these meetings, the researcher maintained audio recordings,fieldnotes (a count of one representsfieldnotes for one meeting) and collected artifacts (hands outs for example). In addition, each participant completed a survey that included the Miller-Seller instrument and had an individual interview with the researcher (which was also recorded) separatefromthe rest of the reflection group. Table A.1 shows counts of individual events. Table A. 1 Event Counts Reflective group

Recorded

Artifacts

meetings

Surveys and

Field notes

interviews

Total for group

Lesson study

8

11

6

8

33

Peer coaching

5

2

10

5

22

Pre-service

4

7

4

20

35

3

2

1

2

8

mentoring Lone reflective practitioner

202

The recorded meetings are listed below with the duration of the recording, the general purpose of the meeting and the number of transcribed pages of text that were generatedfromthe recording (single spaced, standard 8V2 xl 1 paper). Janl7Audio Jan29Audio Febl4Audio Febl5Audio March7Audio March29Debriefl April3Debrief2 April5Debrief3 Peer Coaching Aprill2 May3ScoresPlanning May4Debrief May7Debrief Mayl8Meetings Pre-Service Mentoring April26Debrief MaylDebrief May3Debrief May4 Lone Reflective Practitioner Aprill9 June2 lMeetingPartl June2 lMeetingPart2

18:58 38:16 34:55 41:24 13:47 1:20:38 1:26:16 1:30:52

Planning. Planning. Planning. Planning. Planning. Debriefing. Debriefing. Debriefing.

7 pages 15 pages 10 pages 8 pages 4 pages 23 pages 23 pages 24 pages

1:35:41 1:29:47 1:01:42 1:01:42 1:47:58

Planning Planning Debriefing Debriefing Meetings

15 pages 6 pages 8 pages 7 pages 17 pages

35:39 5:39 17:43 1:01:40

Debriefing Debriefing Debriefing Debriefing

7 pages lpage 4 pages 15 pages

1:25:03 2:19:07 0:15:13

Interview 28 pages Interview 45 pages Interview included in above.

The artifacts included classroom handouts, lesson plans, observation notes on seating plans, reflective practice scheduling document, and an informational brochure for the private school. These were collected in cases were there was a direct connection to the research study. In some cases, such as a board document, field note references were made and copies were not collected since the researcher had access to the document by other means. Surveys and interviews were completed with one per participant. The only exception in Table A. 1 is the lone reflective practitioner where review of the survey was completed in the

course of a regular interview. In this case, the participant interview was counted as part of the recorded meeting. Field notes were also maintained and these augment the recorded sessions. This was particularly important in the pre-service mentoring instance where fifty pages of hand written notes were collected but recorded meetings were difficult to arrange. There was overlap between the different research events. Most artifacts arose in conversation between participants which was part of a recorded meeting and likely described in the field notes. The level of overlap was maintained purposefully to allow comparison of observations after the fact which provided another method for checking validity of observations.

APPENDIX B THE CODEBOOK The interview text was coded using the lesson studyfirstand subsequently augmented using the other cases. The coding process began with categories derivedfromthe key concepts anticipated in this study. The transcripts of meetings for the lesson study were analyzed chronologically in the initial stage of coding. Subsequently, text with each code was reexamined for themes that highlighted potential sub-codes. In this process, each time a new sub-code was added there was a review of all material that had been assigned that code and associated subcodes to reevaluate the appropriate codification of the text. As all the codes and sub-codes stabilized and new themes were no longer emerging, afinalrevision of all codes was done with particular attention to sharpening the acuity of some sub-codes. For example, remarks pertaining to principles that teachers articulated regarding methodology had been coded as principles relating to PCK, however, on further examination it was concluded that being methodology based meant that the principles were essentially principles of pedagogy which was a separate sub-code. The reconciliation process for the lesson study was completed prior to examination of the textfromany of the other cases. During the construction of the codebook evidence that could potentially be construed as indicative of self-efficacy was coded as "influences". Additional evidence of beliefs was sought through discussion with each participant at a later meeting or thefinalinterview. This process of triangulation across meetings was intended to validate the self-efficacy interpretation and avoid sporadic expressions of influence that did not endure. The code book, as derivedfromthe lesson study texts, was: (l)Pedagogical Content Knowledge (1 l)Resources 205

206

(11 ^Instruction (1 1 2)Planning (1 1 3)Non-teaching (1 3)Observed (1 4)Planning (2)Pedagogy (2 l)Observation Based (2 2)Content Specific (2 3)Resource based. (2 4)Principle Based (3)Influences (3 l)People within the group (3 2)People outside of the group. (3 2 l)Teachers (3 2 2)School leadership (3 2 3)Non-school and PD (3 3)Self assessment (3 4)Time issues (3 5)Student observation (4)Curriculum Orientation (5)Reflective Practice (5 l)Process (5 2)Observations (6)Student specific The analysis continued by coding the text for the peer coaching, pre-service mentoring and lone reflective practitioner in that order. The order was used because it agrees with the use of purposeful sampling. The coding process was initially done without altering the existing code book because a) no text was found that differed sufficiently to warrant the addition of a new code, and b) where the existing sub-codes were marginal the text was added with the intention of reconciling the sub-codes after additions. The reconciliation process had two key elements, one was improving the acuity and definition of the sub-codes and the second was to improve the distinction between sub-codes. As an example of the first, the pre-service participant had mentioned a parent, who happened to be a teacher, as an influence and this did not fit the existing definitions of teachers or out-of-group influences that the coding scheme had. The distinction between sub-codes was evident with pedagogy-observed being a key sticking point and it was

207

recognized that a review of how this is distinguishedfromPCK-observed and reflective practiceobserved was needed. In both instances, some text was coded in both categories for ease of identifying discrepant cases. Within the peer coaching text, it was found that key differences were; •

the way in which the reflective process was established using the guidance of a viceprincipal,

•

the teaching was much more transformative and emotional reactions by the primary students in response to being out of the regular classroom and having more teachers in the vicinity were an active consideration in the learning process,

•

the choice of students because they were using small groups of students rather than whole classes,

•

Two types of observations were made in the coding. One is an observation that had an effect on a teacher ('They did much better than I expected.') and the other was observations that came directlyfromthe reflective process and constituted evidence of the teacher reflecting on the teaching process. In the peer coaching case, the teachers were using the PRIME kit which includes diagnostic testing. The discussion of results has been included as reflections even though the analytic character of the discussion seems somewhat removedfrompersonalized ownership of a genuine reflection and the effect on the teachers involved appears to be uncertain due to their lack of familiarity with the diagnostic,

•

The lesson study group worked as a single unit whereas the peer coaching had a group of four veteran teachers who would reconfigure as two pairs of teachers depending on the grade level specificity of what they were doing.

With the pre-service mentoring key differences were; •

The format of the teaching was observed to be much more transmissive than the other cases, this was particularly evident in the extensive use of the textbook,

•

In the case of the pre-service teacher, influences differed from the other participants and an interpretation of professors and relatives who are teachers needed to be reconciled,

•

The meetings, in this particular case, were difficult to arrange and specific details from thefieldnotes were needed.

•

The observed classes had primarily studentsfromChina and language was an important component as well as considerations of culture

With the lone reflective practitioner differences were; •

The use of resources and professional development opportunities was much more pronounced.

•

The focus was on the transition of the teacher as she reconciled her initial views of math teaching with the problem centered curriculum and resources.

The revised set of codes, along with the revised definitions and examples of the codes are provided in Table B.l.

Planning Resources

Discussions

Table B.l Codes, sub-codes and definitions Sub-Code Code Pedagogical Instructional Content Resources Knowledge (PCK) Classroom Observation

Examples ".. .using literature to teach math. Math and non-fiction. Math in literature. 4 to 6, K to 5. These are the books they are using. These are picture books and then lessons wrapped around mathematics using the literature." PCK around events that have been "Well I don't, we've done fractions, I don't think they could observed in the classroom setting. have, I suppose it was more one half, one third, one quarter, it These may have been observed by was all one over something. But (student name) said, 'They are the teacher or reflective observers. the same because one is half of two and two is half of four.' " Discussions between teachers about (participant 1) "For you, though, the keyword is 'through PCK. This may include discussions investigation'" of interpretations, comparison of (participant 2) "Yup. One variable. So that is using... like... different options or the impact of uh... 2 times N plus 3 equals 11. So we are talking about particular teaching methods on order of operations there, right, we're not just talking through different PCK options. investigation." (participant 1) "But they have already had bedmas at some point with you" (participant 2) "Not previously this year, you would hope that in previous years yes, but..." Resources that are consulted with "There was talk in one of the mathtilingsabout having the respect to planning lessons. These kids cut apart the 100 block into strips and have the kids put it are usually subject specific resources in a linear way." mat assist teachers in the creation of lesson. Includes teacher guides that come with textbooks, board documents, and curriculum documents.

Definition Manipulatives used in the classroom for instruction. Includes algebra tiles, handouts, calculators, etc.

209

Influences

Pedagogical Planning

Pedagogical thinking that is specific "So, anyway, yeah, we were doing, the focus this time was to the content being taught but not this new Japanese method called a Bansho. Have you ever sufficiently specific to particular heard of a Bansho?" details to qualify as PCK. For example, the sequencing of topics or method of organizing students for a particular activity fit with this category. "(Interviewer) Are you using Prime? Methods derived from resources Resources such as teacher resources that come (Participant) We have it, we have access to it. Have I used it? Consulted with textbooks, board documents No. Am I against using it? No. I'm totally for it, I don't have and curriculum documents. any problem with it, it's just that it is a learning curve and I just don't have the time." Principles Pedagogical ideas that have not been "When I give a test, I separate them, I give a level 3 test and a level 4 test." observed and are not specific to a particular content area. This includes ideas that are of a more general nature about what sorts of pedagogical ideas should and should not be used. "And I am like way out there on problem solving, and all that Devices References to teaching devices of a general nature such as SmartBoards, stuff (staff member name) does on the SmartBoard, I am like 'Neat! How did you get the answer?'" overheads, etc. Within the group Influences between the participants. "But you know you set them up really well to, am I right about the read the question three times and pick out the mathematical information, I liked the way that you ... reinforced, you have to look at the question" "[using stories to ground the lesson] is one of my huge Self assessment Judgments about one's own strategies, because with me I need this because I am not as knowledge, ability, etc. technologically based, I don't have as much visualization as I should, and I count on that sort of connection. How the kids can know what it means." "I was amazed; I don't know what it was. They really didn't Student's Student actions are discrepant with

Content Specific

210

Curriculum Orientation

the observations and/or expectations of the teacher or other observers. Teachers outside Colleagues who do not have a the group leadership capacity and are essentially peers on an equal footing. Note that pre-service professors, who are on a consultation bases, are included. Leadership in Influences that arise because of school. leadership actions in the school. Note that the principal is the key leadership role but this also includes vice-principals. For the case study that includes a vice-principal, coding here pertains to the leadership capacity. Leadership These are influences that arise from outside the school sources outside of the school. This includes people in board roles who have an impact, committees that include multiple schools, board initiatives, Ministry initiatives, and conferences. These are leadership in the sense that there is an implied level of obligation for teachers to be receptive and this distinguishes them from resource materials. No sub-codes. Philosophical view of teaching.

influence

"(Participant 1)... there is nothing wrong with competition, kids more engaged, (Participant 2) And in some cases, it is the reality of the world they are going into. (Participant 1) But, at the same time, the reality of the education is much more group work, more communication,"

"And sometimes it is board initiatives. Board initiative influence you, I don't have a choice. That influences you. Curriculum influences you."

"Cause I just felt I had had enough. You know, I just can't do anymore, the principal now wanted weekly, monthly meetings writing stuff up and I don't have time for that. Okay, I have enough things on my plate. So, uh,"

get the one half is the same as two quarters. Nobody knew what the heck that..." ".. .after the staff meeting another teacher poked her head into your room and said to you, your lessons are cool."

211

No sub-codes.

No sub-codes.

Time References

Reflections

Process

Student specific

Reflective Practice

Statements about classroom events directly related to the reflective method or assessing the effectiveness of methods. It also includes comparisons of before and after. It does not include details that define a PCK component or a specific methodology; however, it does include opinions about the effectiveness of either of these. Details directly related to classroom events that focus on students but are not coded as PCK-ClassroomObservations (1,3) or as direct reflections (5,2). Any references to time being a relevant consideration.

Organizational details for the functioning of the reflective practice.

"It caused confusion for you as well as them, and it took a fair bit of time, so then you were in a bit of a time issue as well."

"I got to the point where I assigned this student to these two and I told her, she said 'Which group am I in at this point?' and it was clear she wasn't happy about that. Then I was surprised, she actually went off with the other group."

"what is our plan, talk about our lesson? Your going to give it, I'm going to watch and then give you feedback? And then the next day I give the lesson, we'll talk about it and you give the feedback? Okay, so before I go down .... We know what we are actually going to do." "Would I have taught alternate algorithms prior to everything I have done in the last year? I am teaching differently, not what I would have done a year ago."

212

299 183 257

(2 2)Lesson Specific

(2 3)Resources consulted.

(2 4)Principles

(2)Pedagogy 182 14 89

58 154 141

75

156

54

21

40

16

26

(1 5)Planning Resources

31

70

53

249

(1 4)Discussions

51 64

87

111

(1 3)Classroom Observation

78 375

121

45.3

32.2

52.7

4.6

43.8

19.5

9.7

(x.1%)

Study

service

Isolated

Lesson

Pre-

Coaching Mentoring Reflection

Peer

55

Study

Lesson

(1 2)Instructional Resources

(l)Pedagogical Content Knowledge

Text Codes and Sub-Codes

Frequency of codes and sub-codes

Tabled

service

Isolated

14.8

6.2

213

8.4 26.0 20.4

20.4

5.2 33.0

59.2 54.2

42.4

67.6 22.3

14.7

5.7

17.4 139.2

33.5

13.9

(x.1%)

29.0

(x.1%)

46.5

(x.1%)

Coaching Mentoring Reflection

Peer

Pre-

The text unit counts and proportion within a reflective group (in tenths of percents) for each code and sub-code are given in Table C. 1.

FREQUENCY OF CODES AND SUB-CODES

APPENDIX C

149 37 75 40

(3 3)Self assessment

(3 5)Student's influence

(3 6)Teachers outside the group

(3 7)Leadership in school.

232

(4)Curriculum Orientation

61 2694

43 2600

284 5679

(7)Time References

TOTALS

86

178

43

210

17

37

21

67

63

307

486

(5 2)Reflections

399

117

62

94

3

70

18

49 49

48

73

97

3

(6)Student specific

529

(5 l)Process

(5)Reflective Practice

164

school

(3 8)Leadership outside the

218

12

(3 1)Within the group

(3)Influences

(2 5)Devices

3677

36

4

28

0

83

152

63

35

18

91

0

9

50.0

54.1

85.6

93.2

40.9

28.9

7.0

13.2

6.5

26.2

38.4

2.1

16.5

16.5

80.8

153.5

45.0

23.8

36.2

1.2

18.8

18.8

37.3

1.2

22.6

31.9

66.1

6.3

13.7

7.8

24.9

23.4

26.0

6.7

17.8

27.1

214

9.8

1.1

7.6

0.0

22.6

41.3

17.1

9.5

4.9

24.7

0.0

2.4

APPENDIX D THE MILLER-SELLER INSTRUMENT This instrument is present as it was communicated by Dr. Seller. While it does not appear in Miller and Seller (1990), it has been used for graduate courses in conjunction with the book to legitimize different approaches to teaching and learning. Instructions Below are listed seven approaches to learning and curriculum. You are to rank the seven approaches in the order that most closely represents your thoughts and feelings about an appropriate climate for learning. While you mayfindthat no particular approach completely represents your beliefs, or conversely, that you do not totally disagree with any of them, you should try to provide a ranking which represents those beliefs as closely as possible. You should rank them in terms of your learners, whether they are elementary or secondary school learners, or adults in a college or some other program. Approach A: This approach stresses learner mastery of basic subjects and disciplines in the curriculum. In the elementary school, the stress is placed on basic literacy and computational skills. All other aspects of the curriculum such as physical education, environmental studies, and the arts are seen as less important. In secondary school the learners study various academic disciplines such as math, science, language, and history. The emphasis is on acquiring the conceptual framework and inquiry skills associated with each discipline. Knowledge is viewed as arisingfromthe disciplines and learning not linked with an academic discipline is viewed as less valuable.

215

216 Approach B: This approach is based on a developmental conception and reflects the assumption that all individuals proceed through definite stages of development. The stages are important because if teachers are not aware of the various levels, development can be thwarted. Teachers committed to this orientation listen closely to learner's observations and reasons so that they can attain some sense of the learner's development level. This approach also emphasizes the importance of learner's interaction with a wide variety of materials and experiences, particularly experiences which cause the individual to examine his or her reasons and thoughts in approaching a problem or dilemma. It is through this interaction that development is facilitated. Finally it should be noted that the aim of this approach is not to accelerate cognitive, ego, and moral development but to remove barriers that hinder the learner's natural progression to higher levels of functioning. Approach C: This approach to learning views the learner primarily as a social being. The learner is seen as part of me larger social context and thus must acquire skills to participate in that context. In a democratic society, the learner acquires the values and skills mat are necessary to participate in the democratic process. Thus, the learners learn to critically analyze information, distinguish fact from opinion, and make decisions on public policy issues. In most programs this involves classroom discussion; however, in some programs the learner may be encouraged to participate actively in community activities. Thus the learner might work in a community agency or observer proceedings in a local court. Approach D: This orientation focuses on learner acquisition of specific competencies. Learning tasks are often broken into small, definable units so that learners can master various skills. Programmed

learning and individualized instruction are often employed in this orientation. The computer and other technologies are also used within this approach. Learning is often evaluated with criterionreferenced tests to assess whether the learner has achieved the desired competency. Educators working within this orientation attempt to package the curriculum in an efficient manner so that learning is maximized. Approach E: This approach focuses on a holistic view of the learner. The teacher attempts to work with the whole person - the physical, emotional, intellectual, aesthetic, moral, and spiritual dimensions. In particular, the teacher attempts to integrate analytic and intuitive thinking or the right and left sides of the brain. This is often accomplished through an integrated approach to the arts - drawing, painting, music, dance, and drama. Other aspects of the curriculum such as reading and math are also integrated with artistic activity. The teacher may also use techniques such as movement and imagery so that the learner becomes aware of his or her inner life. Approach F: This approach to curriculum is primarily concerned with the development of intellectual skills. It is process oriented in that it attempts to develop a number of cognitive skills (analysis, synthesis, evaluation). There is also an emphasis on problem-solving skills, and so the learner gains confidence in analyzing problems and evaluating possible solutions. The curriculum contains a wide variety of activities, simulations, and games that stimulate independent thinking on the part of learners. The teacher's role is to help the learner deal with various problems and situations so that cognitive competencies are realized. Thus the teacher often probes learner reasons to stimulate further thinking.

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Approach G: This approach to curriculum emphasizes two basic aims: the development of a positive individual self-concept and skills in communicating with others. The basic emphasis is on personal growth and personal integration. Through a variety of experiences and exercises, this approach attempts to develop a sense of emotional well-being. Techniques such as role playing, classroom meetings, and communications training are employed in the classroom. Learning is seen as a vital and potentially enriching experience in its own right, and the present lives of the learners are a major focus of concern. Interpretation: Ranking the Approaches to Learning and Curriculum The way we rank the orientations provides some insight into our basic beliefs about what we do and how we prefer to go about doing it. Since almost everyone who tries this exercise wants to pick specific ideasfromdifferent approaches, there are some very common groupings of approaches. The groupings can be used to identify the meta-orientations of Transmission, Transaction, and Transformation. By helping others understand and explain their beliefs, it is my experience that people are more open to discussion about the content and approach to a program or course in a school division, department, grade level, etc. The identification of the orientations legitimizes each person's beliefs (it is no longer viewed as a personal attack on "the way I teach") as each orientation is grounded in a variety of theories. First, the letter identification of each approach is replaced with a more common name from the curriculum literature.

219 Approach A: The Disciplines Approach Very characteristic of the "back to the basics" movement. Favors more structure to the content of curriculum. Jerome Bruner is a major curriculum theorist for this approach. Approach B: The Developmental Approach Grounded in the ego development theory of Erik Erikson, cognitive development work of Jean Piaget, and moral development theory of Lawrence Kohlberg. It also includes such things as theories about professional stages of development, and stages of adult development. Approach C: The Social Approach This is the one approach that is present in all three meta-orientations. There are three interpretations of the social role of education. The interpretation chosen determines which meta_ orientation the respondent will likely feel most comfortable with. Interpretation 1: The major social role of education is to transmit the culture of society (Transmission orientation). This may also include professional cultures. Interpretation 2: The major social role of education is ensure that learners are knowledgeable, productive citizens and understand their civic responsibilities or again, responsibilities to a profession (Transaction orientation). Interpretation 3: The major social role of education is to help learners understand what is wrong with society so that they can change it (Transformation orientation). Approach D: The Behavioral Approach B. F. Skinner provides the basic psychology for this approach which favors such things as competency based education, programmed learning, and criterion referenced testing. Technology is also appealing to this approach Skinner, for example, developed teaching machines, and some present day computer application learning programs are an extension of the idea.

Approach E: The Transpersonal Approach The focus here is on the higher self. Educators who practice this approach will talk about and use such techniques as centering, visualization, meditation, and movement education. Approach F: The Cognitive Processes Approach The work of David Ausubel and Carl Bereiter both provide examples of this approach in use. The accent is on problem solving and higher order thinking skills in this approach. Content is considered a vehicle only and is therefore seldom of much concern. Approach G: The Humanistic Approach The focus of this approach is the development of a fully functioning, self-actualizing individual. Hence, the work of people such as Abraham Maslow and Carl Rogers is very important in applying this approach to practice.

The approaches are then used to identify meta-orientation groupings. Transmission Orientation This orientation can usually be identified by the close proximity of Approaches A and D in the ranking exercise. Programs which contain or require specific content or behavioral skills often display this orientation. Transaction Orientation Approaches B and F are usually in close proximity when people identify this orientation. This orientation is most commonly seen as part of programs which encourage a great deal of learner activity and interaction. Activity based learning, inquiry learning, and the use of a lot of group work for discussions and other activities are techniques used a lot. Transformation Orientation

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Approaches E and G exemplify this orientation. This orientation requires a highly personalized curriculum so both content and process are variable according to the individual learners and their needs.

APPENDIX E INTERVIEW GUIDE At the end of each observation period, each participant was interviewed. This interview process has three aims: Verify the interpretation of the Miller-Seller instrument, check the factual details of any events that seemed, to the researcher, to be important but lacking in clarity, and lastly to gain the participant's impressions regarding their personal feelings as sequences of events unfolded. Verification of the Miller-Seller instrument began with a description of the instrument as having three meta-orientations. It was explicitly stated that there is no "correct" orientation and that all three results are equally valid. After an overall description and seeking evidence of understanding on the part of the participant, an explanation was provided of the outcome of the instrument for their particular case. This was explained for them individually with no reference to any other participants. This led to dialogue between the researcher and the participant seeking confirmation of the relative priorities of the three orientations for the participant. Factual details were checked and provided further insights, in some cases, regarding events that occurred between regular meetings. In some cases, the known details fully addressed the events but verification was sought to be sure. In a few instances, the interview was used to review seemingly discrepant statements that had occurred at different times or in different interviews. The interviews were also used to explore the participant's impressions through the course of events. The researcher had not transcribed the interviews prior to the interviews, in most cases, and the events were based primarily on field notes. The interview reviewed events based on the information in the field notes which was later found to be effectively articulated in the

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transcriptions of meetings. However, the interview process assumed a lack of information since duplication provided an additional source of information regarding internal consistency.