xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EZ9AZ5DBZ_KK0ZP0 INGEST_TIME 2014-10-03T21:58:26Z PACKAGE UFE0046453_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
1 TEACHER MATHEMATIC AL IDENTITY AND PARTICIPATION IN AN ONLINE TEACHER PROFESSIONAL DEVELOPMENT (oTPD) PROGRAM By SHERRI KAY PROSSER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FU LFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014
2 2014 Sherri Kay Prosser
3 I thank my parents, Elmer and Mary
4 ACKNOWLEDGMENTS My acknowledgements necessarily begin with my doctoral committee. My chair, Dr. Stephen Pape, gave me opportunities to teach a variety of courses and participate in several research projects. The knowledge that I have gained through these experiences has been invaluable. Dr. Nancy Dan a, my co chair, has been a rock throughout this process. Her calming presence belies the passion that she feels for her work. I would also like to thank my committee members, Dr. Tim Jacobbe, and Dr. Holly Lane, for their time and advice. In addition, I wish to thank Dr. Marty League for her continuous support and guidance throughout my doctoral year s. She held roles as my supervisor, colleague and mentor and I do not know how I would have made it through this process without her I would like to thank my colleagues from Volusia County Schools. I would not have pursued this opportunity without insistence and support of my mentors, Dr. Beattie and Dr. Potter and I would not have persevered in this program without the guidance of Dr. Stephen Bismarck. I am very thankful that I have kept in close contact with each of you over the years and look forward to many years of collaboration. I thank my fellow graduate students that comprised my support network Jonatha n Bostic, Karina Hensberry, and Anu Sharma pr eceded me in the mathematics education program and Yasemin Sert was here with me through the end. Their willingness to listen and their suggestions on how to navigate this process is greatly appreciated Karina was instrumental in helping me survive teac hing my first college level course and i take a break from writing From those who began this journey with me in the special education cohort, I would like to thank Su sie Helvenston and Kate Zim mer in particular.
5 Our close friendship has endured and bee n strengthened by this program and I know will continu e to grow throughout the years. In addition to the love and patience I felt from my family, I wish to acknowledge the friendship of my close f riends Bill Sherrier and Kevin Alligood. Bill is the dearest friend I have ever had and is a fount of wisdom. He helped me to keep things in perspective and encouraged me to find balance in my life through running. He told me that getting a doctorate wa s like a marathon and that my last year was like the final10k: painful, but quitting was not an option! enjoyment. His time and his family made my years here substantially more enjoyable. I would, of course, like to offer my sincerest gratitude to my participants Heide and Brynn, for agreeing to speak with me about their thoughts and experiences
6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ .......... 10 LIST OF FIGURES ................................ ................................ ................................ ........ 11 ABSTRACT ................................ ................................ ................................ ................... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 14 Context of the Study ................................ ................................ ............................... 16 Statement of the Problem ................................ ................................ ....................... 17 Research Questions ................................ ................................ ............................... 18 Methodology ................................ ................................ ................................ ........... 19 Overview of Constructs ................................ ................................ ........................... 20 Teacher Identity ................................ ................................ ................................ 20 Views of Mathematics ................................ ................................ ...................... 21 Narrative Inquiry ................................ ................................ ............................... 22 Participation in Online Teacher Professional Development .............................. 23 Structure of the Dissertation ................................ ................................ ................... 24 2 REVIEW OF THE LITERATURE ................................ ................................ ............ 25 Teacher Identity ................................ ................................ ................................ ...... 25 Historical Constructs ................................ ................................ ........................ 26 Contemporary Constructs ................................ ................................ ................. 27 Identity and narratives ................................ ................................ ...................... 30 Teacher Mathematical Identity ................................ ................................ ................ 31 Mathematics history ................................ ................................ ......................... 32 MKT ................................ ................................ ................................ .................. 35 Mathematic Teacher Identity Development ................................ ...................... 40 Teacher Characteristics and oTPD ................................ ................................ ......... 42 Identity and oTPD participation ................................ ................................ ........ 43 Conclusion ................................ ................................ ................................ .............. 48 3 METHOD ................................ ................................ ................................ ................ 50 Context of the Study ................................ ................................ ............................... 50 Prime Online PD Project ................................ ................................ ................... 50 Prime Online PD Program ................................ ................................ ................ 51 Narrative Inquiry ................................ ................................ ................................ ..... 55
7 P articipants ................................ ................................ ................................ ............. 57 Participant Selection ................................ ................................ ......................... 57 Description of Participants ................................ ................................ ................ 59 Data Collection ................................ ................................ ................................ ....... 62 Archival Data ................................ ................................ ................................ .... 63 Current Study ................................ ................................ ................................ ... 71 Data Analysis ................................ ................................ ................................ .......... 73 Limitations and Delimitations ................................ ................................ .................. 76 Trustworthiness and Credibility ................................ ................................ ............... 77 Credibility ................................ ................................ ................................ .......... 78 Dependability ................................ ................................ ................................ .... 78 Transferability ................................ ................................ ................................ ... 79 Confirmability ................................ ................................ ................................ .... 79 Researcher Subjectivity ................................ ................................ .......................... 79 Structure of the Narratives ................................ ................................ ...................... 82 4 HEIDE: STUDENT CENTERED TEACHER, ENTHUSIASTIC LEARNER ............. 84 Prime Online PD Program Teacher Mathematical Identity ................................ ...... 84 Mathematics History: A Lack of Understanding ................................ ................ 84 Self Efficacy ................................ ................................ ................................ ..... 85 MKT During Prime Online PD Program Activities ................................ ............. 86 Mathematics Teaching and Learning ................................ ............................... 89 Participation in the Prime Online PD Program ................................ ........................ 92 Teamwork and Optimism ................................ ................................ .................. 93 Participation and Satisfaction ................................ ................................ ........... 94 Supports and Hindrances to Participation ................................ ........................ 95 Present Day Teacher Mathematical Identity ................................ ........................... 98 Recent Teaching Experiences ................................ ................................ .......... 98 Ma thematics Teaching and Learning ................................ ............................... 99 Teacher Role and Student Confidence ................................ .......................... 100 Professional Growth and Confidence ................................ ............................. 102 Shifts in Teacher Mathematical Identity ................................ ................................ 104 MCK and MKT ................................ ................................ ................................ 104 Role of Assessment s ................................ ................................ ...................... 105 Teaching with Manipulative Materials ................................ ............................. 106 Summary ................................ ................................ ................................ .............. 106 5 BRY NN: TRADITIONAL INSTUCTION, SPORADIC PARTICIPATION ................ 109 Prime Online PD Program Teacher Mathematical Identity ................................ .... 109 Mathematics His ................................ ................... 109 Self Efficacy ................................ ................................ ................................ ... 110 MKT During Prime Online PD Program Activities ................................ ........... 112 Mathematics Teaching and Learning ................................ ............................. 114 Participation in the Prime Online PD Program ................................ ...................... 117 Applic ability and Openness to Change ................................ ........................... 117
8 Participation and Satisfaction ................................ ................................ ......... 119 Supports and Hindrances to Participation ................................ ...................... 120 Present Day Teacher Mathematical Identity ................................ ......................... 123 Mathematics History: Closing Doors ................................ .............................. 123 Mathematics Teaching and Learning ................................ ............................. 124 Professional Growth and Opportunities ................................ .......................... 128 Shifts in Teacher Mathematical Identity ................................ ................................ 129 Self Efficacy ................................ ................................ ................................ ... 1 30 Manipulative Materials ................................ ................................ .................... 131 Teaching for Co nceptual Understanding ................................ ........................ 132 Summary ................................ ................................ ................................ .............. 133 6 CONCLUSION ................................ ................................ ................................ ...... 136 Research Question 1 ................................ ................................ ............................ 136 Heide ................................ ................................ ................................ .............. 136 Brynn ................................ ................................ ................................ .............. 139 Looking Across the Cases ................................ ................................ .............. 141 Summary ................................ ................................ ................................ ........ 145 Research Question 2 ................................ ................................ ............................ 145 Heide ................................ ................................ ................................ .............. 146 Brynn ................................ ................................ ................................ .............. 147 Looking Across the Cases ................................ ................................ .............. 148 Implications for Practice and Suggestions for Future Research ........................... 150 Implications for Practice ................................ ................................ ................. 156 Suggestions for Future Research ................................ ................................ ... 159 Conclusion ................................ ................................ ................................ ............ 161 APPENDIX A PRIME ONLINE PD PROGRAM WEEKLY ACTIVITIES ................................ ...... 162 B SEGMENT ONE ACTIVITIES ................................ ................................ ............... 163 C PI CONDUCTED INTERVIEWS ................................ ................................ ........... 165 D SEGMENT TWO ACTIVITIES ................................ ................................ .............. 168 E MODULE SURVEYS ................................ ................................ ............................ 170 F SEGMENT SATISFACTION SURVEYS ................................ ............................... 175 G INFORMED CONSENT FORM ................................ ................................ ............. 183 H INTERVIE W ONE ................................ ................................ ................................ 184 I INTERVIEW TWO HEIDE ................................ ................................ .................. 186
9 J INTERVIEW TWO BRYNN ................................ ................................ ................ 189 LI ST OF REFERENCES ................................ ................................ ............................. 193 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 202
10 LIST OF TABLES Table page 2 1 Four Ways to View Identity (Gee, 2000) ................................ ............................ 27 3 1 Data Collection Summary ................................ ................................ ................... 62 4 1 M scores ................................ ................................ ........................ 87 4 2 M scores ................................ ................................ ...................... 113
11 LIST OF FIGURES Figur e page 1 1 Domains of Math ematical Knowledge ................................ ....... 35
12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TEACHER MATHEMATI CAL IDENTITY AND PARTICIPATION IN AN ONLINE TEACHER PROFESSIONAL DEVELOPMENT (oTPD) PROGRAM By Sherri Kay Prosser May 201 4 Chair: Stephen Pape Cochair: Nancy Dana Major: Curriculum and Instruction The purpos e of this study was to gain insight into the r elationship between teacher mathematical identity and participation in an online teacher professional development ( oTPD ) notable variance in learning and implementation after professional de velopment (Lieber et al., 2009). That professional development is increasingly offered online would likely exacerbate these variances. This study extends prior research by examining teacher mathematical identity in an online setting and the relationship between teacher mathematical identity and participation in an oTPD program Two general education elementary teachers were chosen from a cohort that completed an oTPD program Data collection included extant data from the oTPD program (i.e., activities, f or um posts, PI conducted interviews, surveys, measures of mathematical content knowledge and levels of participation ) and interviews two years post oTPD program Data were analyzed using qualitative methods to identify similar and divergent themes within and between cases.
13 Each case study described s mathematics history, self efficacy, mathematical knowledge for teaching (MKT; Chang, 2009), views on mathematics teaching and learning, oTPD program participation, and identity shifts. Cross case analysis revealed that bo th participants had similar negative histories as learners of mathematics, which were related to their views of mathematics teaching and learning. evelopment, and willingness to changes were factors in oTPD program participation. Although their participation levels in the oTPD program varied considerably, both participants underwent shifts in teacher mathematical identity. The participants awa reness and the extent to which their current teacher mathematical identity aligned with the oTPD program content and pedagogy were related to their shifts in teacher mathematical identity. Implications for research and practice as well as limitations will be discussed.
14 CHAPTER 1 INTRODUCTION What teachers bring to learning to teach prior belief and experience affects what they learn. Increasingly, teachers' own histories personal and professional are thought to play an important role in what they learn fr om professional development experiences. (Ball, 1994, p. 20) Recent changes in the national policies of the United States have led to increased accountability for the teaching and learning of K 12 mathematics. Due in part to high stakes assessments, strin gent standards have been set for teachers to be highly qualified in content areas and in instructional practices (National Governors Association Center for Best Practices, Council of Chief State School Officers [CCSSO], 2010; No Child Left Behind Act [NCLB ] of 2001; National Council of Teachers of Mathematics [NCTM], 2000). A rising emphasis on science, technology, engineering, and mathematics (STEM) education puts even greater pressure on inservice mathematics teachers to implement standards based practic es. In addition, teachers of mathematics mathematics (Ball, 1994, p.15), which creates a pressing need for quality professional development (PD). Development of effective PD is an integral part of the effort to reform the teaching and learning of mathematics. Until recently, PD was limited to brief, face to face sessions consisting of whole group lect ure and printed materials. Even when face to face PD is extensive, individualized, and has an onsite support component, there is still
15 2009). Complicating the problem of l ess than effective PD is that, as new and more readily available technology has become available, PD is increasingly being offered online. In fact, online teacher PD (oTPD) is developing so rapidly that best practices for design and implementation have yet to be established (Dede, Ketelhut, Whitehouse, Breit, & McCloskey, 2009). Given this lack of established best practices, the variance in outcomes present in face to face PD would likely be exacerbated in online formats. Teacher characteristics play a majo r part in what is taken away from PD. There have been several early childhood education studies (Brady et al., 2009; Downer, Locasale Crouch, Hamre, & Pianta, 2009; Lieber et al., 2009), reading or language arts studies (Brady et al., 2009; Downer et al., 2009), and special education studies (Dingle, racteristics and corresponding responses to PD, but few exist in mathematics education (Chang, 2009). Substantial research has been done that considers mathematics and their subsequent professional g rowth have been less frequently studied Although there has been a call for additional research in this area, one can 309). growth, PD developers need personal characteristics and how teachers engage in PD. Even though different teachers have different reactions to PD, considering t heir histories as learners and teac hers of mathematics
16 helps explain their approaches in adapting to reform based practices (Drake & Sherin, 2006). Until there are understandings of how individual teachers participate in oTPD programs and why their participation differs, changes cannot be m ade to ensure that the PD content and pedagogy reaches practitioners. As Putnam and Borko (2000) so aptly which a person learns, becomes a fundamental part of what i Context of the Study The context of this study was a follow up to Prime Online: Teacher Pedagogical Content Knowledge and Research B ased Practice in Inclusive Elementary Mathematics Classrooms (Prime Online), a yearlong oTPD program. Th e Prime Online PD program was the result of an Institute of Educational Sciences (IES) funded research project to develop an asynchronous oTPD program for third through fifth grade general and special education teachers. The Prime Online PD program consi sted of 27 modules, within three segments over a period of one year, that focused on developing mathematical knowledge for teaching (MKT), the characteristics and learning needs of students with high incidence disabilities, and practitioner inquiry. Segmen t One examined the NCTM (2000) Principles and Standards for School Mathematics mathematical proficiency, the characteristics of students with learning disabilities (LD) and research based strategies to address their learning needs, and the inquiry cycle ( i.e., develop a wondering, collect data, analyze data, take action, share with others; Dana & Yendol Hoppey, 2009). Segment Two explored number sense, conceptual knowledge of multiplication and division, representation and operations of fractions and deci mal numbers, evidence based practices and Response to Intervention (RTI) for struggling learners, and further involvement in the inquiry cycle. In Segment
17 Three, participants engaged in the inquiry cycle by choosing a Prime Online PD program mathematics c oncept that had been challenging to teach in the past. The participants used progress monitoring to collect and analyze data on their struggling students and their culminating project was shared via synchronous web conferencing software. Each module had f our sections (i.e., Introduction Anticipatory Activity Content and Discussion and Reflection and Assessment) to provide continuity across the weeks. In addition, at the end of each module, one of the facilitators posted a Course Announcement that summa included reflecting on beliefs and experiences, working with manipulative materials, reading practitioner journal articles and PowerPoint presentations, and analyzing videos of lessons. The outcomes of these activities were often forum posts and assignments (e.g., uploaded documents or images). Statement of the Problem In their recent oTPD research agenda Dede et al. (2009) state that program effectiveness is over studied, particularly those using pr e and post intervention surveys to dete rmine participant satisfaction. Not only do most studies ignore data unique to online environments (e.g., discourse analysis, interaction with the content), there is also a lack of studies addressing the need to kno w how and why oTPD works without ignoring the what (Dede et al., 2009). A qualitative study would help fill a gap in the literature regarding how and why teachers grow professionally (Craig, 2011), especially considering the recent emergence of oTPD progr ams. Pre and post measures are not enough; professional growth needs to be analyzed across time to show different stages of teacher learning (Dede et al., 2009).
18 their MKT can help determine if information presented in PD will be accepted or rejected, which has considerable implications for PD designers, implementers, and facilitators. PD is more likely to have an impact (i.e., enhanced knowledge and skills) if it is prolonged, intense, and cont ent specific; provides active engagement with the content; and is relevant to classroom contexts (Garet, Porter, Desimone, Birman, & Yoon, 2001). Data also support the commonly held belief that when teachers from the same grade level work together, greate r improvements in knowledge and practice will result (Garet et al., 2001). The Prime Online PD program had all of these components, but an initial analysis of the participants showed noteworthy variance in teacher satisfaction and MKT acquisition. To bet ter understand underlying factors involved in these, and other, variances, t he purpose of this dissertation was to study the relationship between teacher mathematical identity and participation in an oTPD program. Research Questions Although the Prime Onl ine PD program provided the context of this dissertation, the unit of analysis was the individual teacher. In order to gain insights about how teachers learned from and interacted with an oTPD program, this dissertation utilizes narrative inquiry. Archiv al data were analyzed to determine the stories of the participants and post oTPD program interviews were analyzed to write a narrative of the relationship between teacher mathematical id entity and participation of third fourth and fifth grade general education and special education teachers from Prime Online the following research questions were posed:
19 1. What is the relationship between teacher mathematical identity and participation i n an oTPD program? 2. What do narratives reveal about shifts in teacher mathematical identity two years post oTPD program? Methodology Individuals interact with contexts in different ways based on their histories as learners and teachers of mathematics. Spec ifically al., 2004, p. 177). Comments or feelings that the participants had at any stage of the Prime Online PD program would likel y have changed, particularly as the participants had over two years to integrate activities and implement instructional strategies learned in the oTPD program. It was also possible that some of what participants found important during the Prime Online PD program were not particularly useful in new contexts or with new constraints. This study used narrative inquiry to gain insights into the relationship between teacher mathematical identity and participation within an oTPD program Themes found in a review of the literature (e.g., teacher role, history as a learner of mathematics) provided the basis for the research questions and, ultimately, the questions for the first of the two post Prime Online PD program interviews that were conducted as part of this s tudy. To form an initial understanding of potential part of the Prime Online IES funded research project were analyzed. F ollowing the guidelines put forth by Braun an d Clarke (2006), t heoretical (i.e., deductive or top down) thematic analysis was used on the Content Knowledge for Teaching Mathematics [CKT M] measures, Segment One activities, PI conducted interviews, Segment Two activities,
20 participation chart, module s urveys, and segment satisfaction surveys. In theoretical related literature (Bra un & Clarke, 2006, p. 13). guided the purposive sampling of two participants with whom I would conduct two interviews. The questions in the first interview focused on co nstructs related to teacher mathematical identity and mirrored topics discussed during the Prime Online PD program (e.g., best recent mathematics lesson, perception of teacher role, definition of s Prime Online PD program teacher mathematical identity and participation in the oTPD was written based on the analysis of the Prime Online PD program archival data. Then, based on analysis of the s present day teacher mathematical identity was written. A second set of interview questions was developed, specific to each participant, to clarify or expand upon interesting or conflicting statements that were uncovered during data analysis. The se cond interviews were conducted and analyzed, and then data were integrated into the existing narratives, as appropriate. teacher mathematical identity (i.e., two years post oTPD) was compared to the Prime Online PD prog ram teacher mathematical identity and a third narrative was written to describe shifts in their identity. Overview of Constructs Teacher Identity Identity is an important construct and reshaped in interaction with others in a professional
21 defined in very specific terms (e.g., encompasses knowledge and beliefs as well as sense of self, dispositions, interests, and ideas about work and change; Spillane, 2000) and in more of a broad sense (e.g., integrates personal and professional selves; Day, profession al identity, Beijaard, Meijer, and Verloop (2004) suggested that professional identity is an ongoing process, implies both a person and a context, consists of sub identities, and connotes agency. As identity is not static, a teacher could have a different sub identity identity could be the answer to Mathematics, therefore, provides a different context in which teachers can interpret, learn about, a nd implement reform (Spillane, 2000) and should be investigated as a distinct type of teacher identity. Views of Mathematics The overall purpose of PD is teacher learning that results in increased student pact t heir instruction and must be development as a quality teacher of mathematics (Hannula, Kaasila, Laine, & Pehkonen, 2005) and beginning elementary teachers without a bac kground in mathematics have low teacher efficacy and decreased efficacy development, which can (Chang, 2009, p. 293) carry forward to their adult lives, and these feelings are important factors in the ways teachers interpret
22 mathematics have been studied as: belief of own talent, belie (Hannula et al., 2005, p. 3 95) ; ; nature of math, important e lements in role as a teacher, confidence in teaching math (McDonough & Clark e 2005) ; and view of themselves as learners and teachers of mathematics, view of mathematics and its teaching and learning, and view of the social context of learning and teaching mathematics (Kaasila, 2007; Kaasila et al., 2008) The views of mathematics constru ct used in this dissertation comprise s views of self as learner (e.g., mathematics history) views of self as teacher (e.g., efficacy) views of mathematics teaching (e.g., teacher role) and views of mathematics learning (e.g., how students learn). Narrative Inquiry This dissertation relied heavily on narrative inquiry and theories of identity development. Narrative inquiry studies human experience (Clandinin & Rosiek, 2006 ) and narratives can Narratives have gained favor in mathematics education during the past two decades bec (Chapman, 2008). Through these stories, narrative inquiry allows for a deeper understanding of the process by which teachers learn (Craig, 2011) and how they understand themselves, both personally and professionally (Drake, Spillane, & Hufferd Ackles, 2001; Kaasila, Hannula, Laine & Pehkonen, 2008).
23 Sfard and Prusak (2005) have established a narrative concept of identity that equates identity with stories told by or about a person as identity are further interpreted as either actual or designated identities. Actual identity is based on the current state of matters and designated identity is based on p resent or future expectations. actual and designated identity can then be perceived mathematical identity. Participation in Online Teacher Professional Development Teacher identity is the way in which teachers come to know and understand themselves and is also a mechanism for guiding actions (Drake et al., 200 1). Thus, teacher identity both affects and is affected by participation in PD. Measures of online participation can include the number of forum posts (Hrastinski, 2008; MacAleer & Bangert, 2011; Yang & Lui, 2004), the types of interactions (i.e., studen t student, student teacher; Wise, Chang, Duffy, & del Valle, 2004; Yang & Lui, 2004), quality of writing (Groth, 2007; Hrastinski, 2008), social presence (Picciano, 2002; Richardson & Swan, 2003; Wise et al., 2004), or perceptions of learning (Hrastinski, 2008; Picciano, 2002). In addition, it is argued that perceptions of learning be considered a performance outcome because these perceptions could incite one to seek out more learning opportunities (Picciano, 2002). For the purposes of this study, partici pation is comprised of completion of module surveys and CKT M measures, engagement with Segment One and Two activities, and assignment completion and frequency of posts.
24 Structure of the Dissertation This dissertation consists of six chapters. Chapter 1 i s an introduction and contains an overview of the literature and the context of this study. Chapter 2 reviews three areas of literature: teacher identity and narratives as methodology, teacher mathematical identity, and participation in online teacher pro fessional development. Chapter 3 includes a description of narrative inquiry, a thorough description of the context for this study, participant selection, data collection, and data analysis procedures. Chapters 4 and 5 present the findings for each parti cipant separately as detailed narratives. Chapter 6 discusses the conclusions and implications of this research, as related to the purpose and research questions.
25 CHAPTER 2 REVIEW OF THE LITERATURE The purpose of this dissertation was to examine the rela tionship between teacher mathematical identity and participation within an oTPD program. In this chapter, I present three relevant literature reviews that shaped how I approached this narrative inquiry. I begin by sharing literature that describes the co nstructs of identity and the study of identity using narrative inquiry. Once the importance of identity in teacher mathematical identity construction and development. The final review pertains to the relationships between teacher identity and participation in oTPD program. Teacher Identity Interest in identity is not limited to the field of education; it has also been explored in psychology, anthropology, and philosophy. W ithin the field of education, much disparity exists in the definition of identity. The construct of teacher professional identity is interconnected with the concepts of emotions, agency, self concept, and discourses (Beauchamp & Thomas, 2009). Although it is commonly accepted in educational research that identity is dynamic and changes due to both internal (van Veen & Sleegers, 2006; Zembylas, 2003) and external (Flores & D ay, 2006; Rodgers & Scott, 2008) factors, defining identity is understandably problematic for researchers. Connelly and Clandinin (1990) explain: Identity is a term that tends to carry a burden of hard reality, something like a rock, a forest, an entity. Being true to this identity, being true to oneself, is often thought to be virtue. Yet, from the narrative point of view,
26 identities have histories. They are narrative constructions that take shape as life unfolds and that may, as narrative construction s want to do, solidify into a fixed entity, and unchanging narrative construction, or they may continue to grow and change. They may even be, indeed, almost certainly are, multiple depending on the life situations in which we find ourselves. This is not less true for teachers in their professional knowledge landscape. (p. 95) Historical C onstructs In his seminal piece, Gee (2000) proposed four types of identity: the nature perspective (i.e., N Identities), the institutional perspective (i.e., I identitie s), the discursive perspective (i.e., D Identities), and the affinity perspective (i.e., A Identities; Table 2 instead of identity being solel y an internal state ( p. 100). The four types of identity are presented not as discrete terms ascribed or an achieved identity. The N Identity is a matter of genetics, whereas the I identity is viewed as somewhat imposed by others. For examp le, I Identity could refer Identity can be measured on a continuum; it could be welcomed by an individual or merely passively accepted. Gee (2000) refers to the two extremes of this continuum as bein g a calling or being an imposition While it is unlikely that teachers would consider their roles as an imposition, it is reasonable to expect that there are differences in how individuals view their role as a teacher. The D Identity is constructed and s ustained when other people talk about or interact with an individual in a way that reinforces a trait. D identity is also on a
27 continuum, with one extreme being labels that are pursued (e.g., charismatic) and the other being labels that are attributed (e. g., attention deficit hyperactivity disorder). Although these four perspectives are inextricably linked, the A Identity view most relates to this dissertation. A Identities formed at the behest of a school, for example, A Identities (Gee, 2000, p. 107) and have been used as framework with which to analyze the identity formation of preservice teacher ( PST s) (Jewett, 2012). The affinity perspective based on shared experiences and practices implies social constructionism; h owever, the affinity perspective focuses primarily on bond formed around the common context, not the bond between group members. The common context of this dissertati on is Prime Online an oTPD program. Table 2 1. Four Ways to View Identity (Gee, 2000) Type of Identity Process Power Source of Power Nature identity a state developed from forces in nature Institution identity a position authorized by authorities with in institutions Discursive identity an individual trait recognized in the discourse/dialogue of/with others Affinity identity experiences shared in the practice of affinity groups Contemporary C onstructs otions of teacher identity in recent research studies (Gresalfi & Cobb, 2011; Settlage, Southerland, Smith, & Ceglie, 2009) and dissertations (Jewett, 2012; Krzywacki, 2009). Settlage et al. (2009) used A identity to explain the choices made by PSTs (e.g. when best practices conflicted with a student teaching placement). Gresalfi and Cobb (2011) expanded upon the notions of
28 institutional identity and affinity identity to account for normative identities (i.e., identity established within a context) and t by participation in that context) during PD discussions. Personal identities were examined in terms of whether teachers resisted, complied, or identified with the normative identities, thereby illumi nating the process by which teachers reconceptualized what it meant to teach and learn mathematics. The feelings of closeness with the normative affinity identity (i.e., the PD program group) and the importance the teachers gave to the normative institutional identity (i.e., what it means to be competent ) were the two integral factors in improving instructional practices. A identity can also be used to describe the impact of core identity professional identity (e.g., Sutherland, Howard, & Markauskaite, 2010), but many focus on rsonal practical knowledge (Beijaard et al., 2004). Based on a review of recent research, Beijaard and colleagues (2004) identified four essential features of tea : an ongoing process, interconnectedness between person and conte xt, sub identities, and a sense of agency.
29 Still others take a more global view on identity. According to Rodgers and Scott (2008): Contemporary conceptions of identity share four basic assumptions: (1) that identity is dependent upon and formed within m ultiple contexts which bring social, cultural, political, and historical forces to bear upon that formation; (2) that identity is formed in relationship with others and involves emotions; (3) that identity is shifting, unstable, and multiple; and, (4) tha t identity involves the construction and reconstruction of meaning through stories over time. (p. 733) Because identity integrates emotions and a sense of agency, there are widely recognized and well studied affective constructs related to identity and id entity development. In an effort to understand how PSTs understand their professional identity, Timostsuk and Ugaste (2010) chose participants based on maximum variation sampling (Creswell, 2002). PSTs from large and small program s and from various subje ct area concentrations were individually interviewed to elicit perceptions of their role as a teacher. Textual segments were then coded into four categories based on categorized as environment (e.g., the passing on of content specific knowledge), emotions (e.g., pride in motivating students, fear of own failure), and activities (e.g., successes and setback in classroom instruction). Given the inevitable contradictions between teaching contexts and individual values and beliefs, teacher educators need clarification on which components of professional identity to focus (Timostsuk & Ugaste, 2010).
30 Iden tity and narratives Whereas some scholars (e.g., Connelly & Clandinin, 1999) see narratives as a means of expressing identity, Sfard and Prusak (2005) view identity as stories that can tes that identities are not found in stories; they are stories and allow researchers to answer the question: narratives with identities, stories can be used to stu dy learning and allow the mechanisms of this learning (i.e., identity development) to be told. reifying endorsable and (Sfard & Prusak, 2005, p. 14). To reify is to make something more concrete by using verbs that connote permanency (e.g., be) and adverbs that imply repetition (e.g., usually). Stories about a person are considered endorsable if that person would say the stories are accurate repli cations. most significant stories of all. This is key, given that PD is typica lly social and context bound. Sfard and Prusak (2005) a rgue that it is not experiences but the vision of our or experiences that form identities and that stories that one tells oneself comprise what most people view as identity. Indeed, some view identity as the telling and retelling of stories over time (Rodgers & Scott, 2008). These self told stories are 2005, p. 17). U nfortunately, as researchers and teacher educators, we are not privy to these self told stories and must rely on the
31 re telling of these stories when studying changes in identity. Further complicating matters, teachers have multiple professional identitie s (Rodgers & Scott, 2008) and elementary school teachers likely have a sub identity for each subject they teach. Therefore, when studying the identity of inservice teachers, it is prudent to focus on subject specific identities. Teacher Mathematical Ident ity view of mathematics as a fundamental part of this sub es, and mathematics (Kaasila et al., 2008), views of mathematics teaching and learning (Kaasila et al., 2008), and their content knowledge (CK), pedagogical knowledge (PK), and pedagogical content knowledge (PCK; Hobbs, 2012) are fundamental parts of teacher mathematical identity. Shulman (1986) introduced the domains of CK, PK, and PCK as a way to understand the knowledge needed in order to teach mathematics effectively. These domains have been refined and are now recognized as MKT (Hill et al., 2008). upcoming section on MKT. Constructs related to knowledge and beliefs have framed studies growth. development of beginning elementary school teachers studied by Chang (2009). Based on performance on a mathematics teaching efficacy scale, 64 teachers were placed in low, medium, or high efficacy groups. Six teachers, two from each level, were
32 observations verified that teachers had progressed to the next gradation of an overlapping five the low efficacy teachers had only reached the second gradation and the medium efficacy teachers had reached the third g radation. Both high efficacy teachers went through the first two gradations within the first two months of the school year. One eventually reached the fourth gradation while the other the only one with a mathematics and science background reached the fif th gradation. Teachers in the same efficacy level progressed in a similar manner and differences were attributed to variations in internal and external factors. External factors were administrator and peer beliefs, past experience teaching mathematics (e.g., tutoring), MCK, and educational history. Mathematics history Changes in teacher expectations due to the Principles and Standards for School Mathematics (NCTM, 2000) and the more recent Common Core Stat e Standa rds for Mathematics (CCSS M; CCSSO current identity and other designated identity (i.e., institutionally sanctioned A learners of mathematics and their experiences as teachers of mathematics. Furthermore, in a variety of ways. For example, Ebby (2000) found that a PST with a strong mat hematics background learned that there is a myriad of ways to work through a problem, a teacher with negative experiences in mathematics reconsidered her
33 assumptions, and a teacher with a traditional view of mathematics teaching came to recognize the impor tance of student centered teaching. researcher might seek to determine what events or factors brought about change in the ple, interviewed PSTs at the experiences as learners of mathematics and their views of themselves as teachers of mathematics. At the end of the course, the teachers were again asked about their evaluative language, negatives, repetition, contrastive connectives, or detaile d to teach the subject at the beginning of the course. The positive chang mathematical identity was equally evident by the end of the course, as she felt more satisfied with her lessons and looked forward to teaching mathematics in her next placement. After a secondary analysis of the interviews, Kaasila (2007) pos ited that the how s and why events (e.g., success with a learner centered approach to teaching mathematics) and significant others (e.g., support by peers and supervising teacher). As s een in the previous study, a negative beginning as a learner of mathematics does not preclude one from acquiring a positive view of oneself as a teacher of mathematics. In another study, Kaasila et al. (2008) considered how PSTs viewed
34 themselves as learn ers and teachers of mathematics as well as how they viewed the learning and teaching of mathematics and examined facilitators of change in their beliefs. Data sources included pre and post mathematical experiences a nd views of mathematics. Tasks to measure conceptual included. Of the 269 participants, 21 were interviewed and four were selected for follow up interviews. Two had a po sitive view of mathematics, high self confidence, and mathematics achievement scores in the top 30%. The other two had a negative view of mathematics, low self confidence, and pre test performance in the lowest 30%. Each narrative case study sought to de termine changes in motivations and dispositions by motivated), social dependent orientation (i.e., help seeking), or ego defensive orientation (i.e., low expectatio n). The narratives indicated, either directly or indirectly, whether participants felt their memories of school mathematics, their views of mathematics, and their mathemati cal skills were used to determine initial socio emotional orientations. Then the researchers compared pre and post performance on a mathematics test and analyzed interviews to establish an updated socio emotional categorization of the participants. Of the four case studies presented, three participants had substantial changes in their view of mathematics teaching and one had a moderate change. Three course activities were suggested as the main facilitators of these changes: ( math ematical experiences, ( b) exploring content with manipulative materials, and ( c)
35 learners of mathematics were the least likely to change, the growth made in MCK and the other affective categories were encouraging. It follows, then, that past mathematics participate in a mathematics based oTPD program. MKT Figure 1 1. Domains of Mathemat ical Knowledge for Teaching. From Knowing and using mathematics in teaching (p. 19), by D. L. Ball, H. Bass, H. Hill, L. Sleep, G. Phelps, & M. Thames, 2006, Learning Network Conference: Teacher Quality, Quantity, and Diversity, Washington, DC. The size of the gap between current and designated identities is also important because the designated identity needs to be seen as achievable (Sfard & Prusak, 2005). Thus, the mathematical content of a PD must be accessible to all teachers, given that it would like ly contain both MCK and PCK components. MCK is conceptual knowledge about mathematics, whereas PCK is an understanding of mathematical content, how students learn that content, and how to teach students (Shulman, 1986). A more recent conceptualization, M KT, has four domains (i.e., common content
36 knowledge, specialized content knowledge, knowledge of content and students, encompasses the knowledge required for teaching mathematics ( Figure 1 1; Ball, Sleep, Boerst, & Bass, 2009). Many elementary teachers lack the conceptual knowledge and deep understanding (Ball et al., 2009) that are required to teach mathematics effectively. Content knowledge alone is not enough; teachers with stro ng MCK may not be willing or able to explain concepts to students (Kahan et al., 2003). Indeed, the number of mathematics courses taken by a teacher is not highly correlated with student outcomes (Swars, Hart, Smith, Smith, & Tolar, 2007). Knowledge of h ow students learn (i.e., PCK) is also necessary. Although the majority of teacher identity research is on PSTs, it is important to reiterate that inservice teachers also undergo shifts in identity. PD is a vehicle for teachers to increase knowledge, re cr aft identifies, and question existing practices (Battey & Franke, 2008) and inservice teacher identity development occurs most often during times of educational reform (Day, Elliot, & Kington, 2005; van Veen & Sleegers, 2006 ) when teachers are asked to shi ft their instructional practices and the ensuing view of themselves as teachers of mathematics professional growth but ought to be differentiated to accommodate individua current levels of MKT. PD is a vehicle for teachers to increase knowledge, re craft identities, and question existing practices (Battey & Franke, 2008). Therefore,
37 d evelopment of MKT and subsequent shifts in teacher mathematical identity MKT c an also influence how one sees oneself as a teacher of mathematics. Self as Teacher of Mathematics How one sees oneself as a teacher and learner of mathematics is influenced by MKT, perceptions of teacher role, beliefs about mathematical proficiency, and affective characteristics (e.g., confidence). To study perceptions of what it means to be a good teacher, Grion and Varisco (2007) designed an asynchronous discussion board to analyze the writing and assignments of 47 inservice and PSTs. The focus of the project, which occurred over a 4 month period, was the development of identity and collaborative practices. Each participant wrote two profiles of what constituted being a goo d teacher, one pre intervention and one post intervention. Individuals also shared a challenging situation from their own school years and constructed a case synthesis in groups. Discussion posts were coded as social, cognitive, or teaching speech segmen ts using qualitative methods. The statements were then coded for varying levels of awareness and as teacher focused, child focused or inclusive. As might be expected the word selection of the PSTs showed more growth than did the inservice teachers. The intervention writings about what it means to be a good teacher emphasized words related to feelings and attitudes and love and sweetness and patience and sensitivity were most commonly used intervention writings provided more de tailed descriptions and included more words related to professional skills and knowledge; words such as reflection assessment, and group (i.e., the ability to work in a team) were most commonly used. In contrast, the inservice teachers had well defined p rofessional identities that did not alter much during the brief
38 the finding that inservice teachers were viewed as being rigid in both the forums and the case synthe ses. (2005) designated identity and wa s the basis for a dissertation about PSTs in a masters level mathematics education course (Krzywacki, 2009). Of eighteen possible candidates, two were purposive ly sampled for case study analysis based on background information and perceived evidence of growth during the course. The cases (i.e., John and Mary) differed in their motivation; John was unsure of his motives for wanting to be a teacher and Mary felt s trongly about her decision to become a teacher. In addition, John had completed previous education coursework and Mary had not. Data were collected over one academic year. The main data sources were three semi structured individual interviews and portfol ios, feedback surveys for the course, and essays written about school memories were supporting data. During the initial interviews, the PSTs were asked about their reasons for becoming a teacher, their educational history, their expectations of the course their views on teaching and learning mathematics (e.g., mathematics history), what constitutes being a good teacher, and their perceptions of how to become that good teacher. The second interview asked students to reflect on the responses to the first i nterview, their updated strengths and concerns about being a good teacher, and their updated expectations for the course. The post interview included reflections on a videotaped lesson, what constitutes a good teacher, how they felt about the course, and their strengths and concerns about being a good teacher.
39 Content analysis of the interview transcripts revealed three main categories: conceptions of teaching and learning mathematics, personal process of becoming a teacher, and teacher education programme supporting individual development. Noteworthy differences emerged. John viewed MCK as integral to being a good math motivate students). In regard to closing the gap betwe en actual and designated identities, John included classroom management skills in his revised designated more precise definition by the end of the course. She still st ruggled, however, to relate to her designated identity. The researcher posited that these differences in identity development occurred because John felt his current identity was closely related to his designated identity and he could, therefore, bridge the consistent with her current identity so there was no need (i.e., learning fuelling tension ) for personal goals to develop. For John, the movement toward his designated identity appeared to be associated with awareness and clarification of what constitutes a good teacher (Sfard & Prusak, 2005, p. 20). For Mary, the movement toward her designated identity was associated with affective characteristics and the need for additional MCK. John and Mary both me ntioned concepts related to CK, MCK, and PCK (1986) three domains of teacher knowledge and components embedded in MKT (Ball, oneself as a teacher of mathem atics can influence how one develops as a teacher of mathematics.
40 Mathematic Teacher Identity Development as identity to illuminating facets of professional identity through stories that teachers tell others about themselves. Bjuland Cestari, and Borgersen (2012) used reflective narratives (i.e., discourses and activities) to document an experienced elementary school teach professional identity during a university research project. The PD consisted of 16 workshops over the course of three years and focused on MKT and building communities of inquiry. Data sources were recorded and transcribed and included classroom obs ervations in the first year, an individual interview and a focus group in the second year, and transcripts from workshops in the final year. This range of PD sit uations was selected to illustrate important instances of her identity development processes. Unlike many past researchers, Bjuland and colleagues did not assign identity based on a preconceived model. Instead they sought indicators of identity (i.e., positioning in relation to pupils, reflecting on developing a workshop model in teaching, integr ating and expanding models of teaching, challenging positioning in relation to didactitions [i.e., instructors]) captured in multiple situations over time. positioning herself as the presenter, observer, and coordinator of the mathematics classroom (i.e., first identity indicator). A videotaped lesson was shown to Agnes almost a year after it was taken, and she was asked during the interview to reflect on the process of her studen
41 have been, givi ng insight into her designated identity (i.e., wants to improve her teaching). The researchers explain ed that this data source should not be the sole s professional identity because questions posed in the interview may have caused Ag nes to respond in such a way as to appease the interviewer. The second indicator of identity is that of reflecting. Through comments made in three narratives, Agnes showed an inclination to transpose, implement, and integrate the content and strategies pr esented in the PD workshops. Comments found in the fourth narrative, post intervention, were similar to comments in the first three narratives, lending support to this identity indicator. The integrating and expanding theme was most clearly exposed in Ag s plenary workshop presentation (i.e., the fourth narrative) in which she shared a model of teaching (i.e., Phase Model) that she and her colleagues created. The model stressed the need for students to be challenged and to have agency, giving insight s current identity. During the individual interview (i.e., second narrative), however, Agnes acknowledged that it was her participation in the PD that helped her to see the link between the PD content and the Phase Model. She also explicitly s tated that the PD content helped her to further develop the model. s positioning related to the PD facilitators. She initially viewed the facilitators as trying to tell her how to teach but came to view the facilitat ors more as co participants in the PD. By the end of the second year, Agnes felt free to express her needs and perspectives on improvements for the PD and that the facilitato
42 These identity indicators provide evidence into the process of teacher mathematical identity development. To combat possible misrepresentation by Agnes, a variety of data were collected over the course of three years. In addition, analy sis was The authors argue that this method could contribute to understandin g the factors in s case, Settlage et al. (2009) argue that identity is constructed within a context and is less of an accrual of experience and more of a reshaping as one encounters novel communities, s ettings, and challenges. Mathematics based oTPD could constitute that novel community with novel challenges. Teacher Characteristics and o TPD comprising doing, communicating, th inking, feeling and belonging, which occurs both Valle & Duffy (2009) investigated learner characteristics and learning strategies. Participant characteristics (e.g., age, teaching experience) were compared to eight variables related to asynchronous course navigation (e.g., total time online, proportion of time in messenger mode, explo ration) and four variables about course experience and evaluation (i.e., satisfaction, learning and transfer, previous experience, group learning preference). Cluster analysis revealed three distinct approaches to learning: mastery oriented task focused and minimalist in effort Teachers with a mastery oriented approach
43 showed a high level of effort in course navigation and the effort was learning focused. This group showed their commitment to the course by having high levels of transitions between cour se activities, the number of course resources accessed, and time spent on learning resources. Teachers with a task focused approach showed moderate effort in course navigation, but worked frequently and intensely in order to expeditiously fulfill course r equirements. Teachers with a minimalist in effort approach used the most calendar days to complete the course but spent the least amount of time online. Experienced teachers were more focused on mastery and less experienced teachers were task oriented. The authors speculate that more experienced teachers have less classroom planning, allowing them more time with course materials, and have more background knowledge, allowing them greater comfort in exploring more deeply. A significant number of minimalis t students stated a preference for working in groups; minimalists consistently reported less course satisfaction, lower learning, and ability to transfer. Identity and oTPD participation Teacher characteristics and approaches to learning are clearly associ ated with the quality and quantity of PD participation. Patterns of participation in an oTPD are positively correlated with professional learning (McAleer & Bangert, 2011) which necessitates a discussion of the how and why this correlation exists. Teache r identity (e.g., histories, MKT teacher characteristics) plays a role in oTPD participation. part of who they are as teachers (Drake & Sherin, 2006), which seems inexora bly linked growth and identity development. Surveys, classroom observations, online discussion
44 posts, reflective journals, and focus groups were used to assess the impact of a blended (i.e., online and face to face) mathematics and science/technology PD on 59). The 48 mathematics teachers only felt significantly more prepared to t each only one of the 11 mathematics topics at the end of the two year PD. Conversely, the 33 science/technology teachers felt significantly more prepared to teach three out of the four science/technology topics, even though the science/technology PD was e ight weeks shorter than the mathem atics PD. The authors speculat relatively weak mathematics background knowledge hindered more substantial gains in both content and pedagogical knowledge, as almost half of the teachers had no post hi g h school mathematics coursework designated identities need to be considered so that the mathematics content of a PD program is accessible. Three case studies of paired secondary mathematics teachers offer other suppositions. Ponte and Santos (2005) designed an inquiry based oTPD program around mathematics investigations, reflection, and collaboration. Data analyzed were several semi structured interviews, a survey, message exchanges, and assignments. Al though all six of the teachers had degrees in mathematics or mathematics teaching, potential of the course activities (e.g., discussing readings, doing task oriented activit ies, reflecting; p. 104). For example, the first pair did not understand the term mathematical investigation had minimal participation and reflection in forums related to theory, and even considered quitting. The second pair had reflective comments and
45 frequent posts, which sometimes included references to outside materials. They were much more engaged than the other groups but their participation declined when they felt collaboration with, and forum posts by, their peers were not rich enough to be usef ul. The third group appreciated the potential usefulness of the mathematics investigations, used the course to deepen their knowledge, and collaborated the most. The third group also stated that the written assignments and postings had more value than fa ce to face conversation with school based peers, as it required greater reflection. Such varied experiences of seemingly similar teachers, working in self chosen pairs, serves to highlight the need for further research on the role of mathematical identity posts to determine perceptions of teacher identity and ideologies existing in the discourses about identity. In two consecutive fall semesters, a total of 21 secondary Englis h PSTs participated in a blended methods course. Students were required to and writing activities, the participants had field placements from which to draw on for thei r wiki posts. The reflective and social nature of the tasks allowed students to simulate what it would be like to be an English teacher. The wiki assignments had not been designed to elicit comments about identity construction, so the author examined pos Data sources were threaded discussions, journal entries, and student initiated threads grouped by participant. Content analysis was used to determine codes (e.g., making a d ifference, building student success) and the resulting seven themes appeared at least three times and by roughly half of the participants. The researcher
46 d a surprising lack of discussion regarding content specific pedagogy and curriculum (p. 142). The researcher posited that the students fashioned their instead of wha t the facilitator interpreted as important. This has weighty implications for the development of future methods courses. Another study of PSTs used discussion posts to search for possible changes in the quality of the postings. Sutherland et al. (2010) s lysis and a new construct, teacher voice allowed researchers to compare asynchronous posts from t he beginning and the end of a blended course. categorized into one of three categories, depending on how closely they positioned themselves as future teachers: theoretical if the examples were from pers onal experience as a student linkage if the examples given consider ed possible future practices) or professional application if the examples were discussed from th e viewpoint of a decision maker Engagement was determined by the level of knowledge used i n the post (i.e., explanation elaboration or reflection/application ) and the length of the post coded as each of the levels of knowledge. Development of teacher voice was evidenced by an 11% increase in the level of cognitive engagement and a 22% increa se in paragraphs at the linkage and professional application levels over the 12 week course. The authors
47 learning might be substantial factors in their engagement with the content and resulting professional identity development. Asynchronous discussions of reform oriented pedagogy formed the basis of a graduate level course for ins ervice middle school mathematics teachers. Groth (2007) selected the two participants with the most frequent number of posts, thereby beliefs about standards based instru ction was acquired prior to analysis of the forum views of mathematics instruction. First, posts were coded for levels of participation (i.e., information exchange knowled ge construction development ) based on a model by Salmon (2004). Information exchange was demonstrated when teachers highlighted points from readings, described their professional beliefs and practices, and affirmed the beliefs and practices of others. K nowledge contruction occurred through brainstorming and debating with others about possible resolutions to pedagogical problems. Development happened when teachers recognized the need for their own MKT growth. Second, the language in the posts was used t o infer acquisition of knowledge (i.e., resistance enrichment revision ). Resistance was when new information resulted in an argument or debate. Enrichment was demonstrated when new information was consistent with current beliefs, and revision happened when information was inconsistent with current beliefs.
48 Changes for both teachers were mainly enrichments and small scale revisions. Although both initially showed resistance to alternative algorithms (i.e., invented strategies), one teacher relaxed her r esistance after a debate with other course participants. The author noted that resistance is not an entirely negative construct. Even when some topics cause debate without subsequent revision, those topics still incited participation in the form of addit ional posts. Furthermore, as this oTPD program encouraged reflection, both teachers identified areas for their own future learning. Reflection, collaboration, and communication with peers are important for growth during oTPD (McAleer & Bangert, 2011). Th rough the analysis of course artifacts (e.g., forum posts, assignments) we can begin to infer how participants interact with course content and with each other. Accordingly, Dede et al. (2009) argue for the analysis of data streams that leave a permanent with the course content. Such analysis might provide insight into why some oTPD programs have more impact on instructional practice and learner outcomes than others, which is an understudied topic (Dede et al., 2009). Instead of merely answering questions as to whether a particular oTPD program works, research should also ask Conclusion Although the majority of the research on identity development focuses on PSTs, hroughout their careers (Beauchamp & Thomas, 2009). While there is a call to study the professional identity development of
49 identity development in any context. How identities are constructed has implications not only for the supports essential to participation in oTPD programs but also to quell possible hindrances (Coldron & Smith, 1999). It follows, then, that the association level. relevant aspects of biographical studies (Knowles, 1992) Most studies on teacher professional identity utilize interviews as the primary data sources and are small scale and in depth (Beijaard et al., 2004), which aligns with narrative inquiry as a methodology. escribe, in a cohesive manner, how their identities impacted and have been impacted by participation in an oTPD program.
50 CHAPTER 3 METHOD This dissertation was designed to examine the relationship between teacher mathematical identity and participation wit hin an oTPD program. The goal of this chapter is to describe the context of the study, narrative inquiry, participant selection, data collection, data analysis, trustworthiness and credibility, and researcher subjectivity as they relate to questions that guided my research: 1. What is the relationship between teacher mathematical identity and participation in an oTPD program? 2. What do narratives reveal about shifts in teacher mathematical identity two years post oTPD program? Context of the Study Prime Online PD Project The context of this study was the Prime Online PD program, which was created as a result of the Prime Online design based IES Goal 2 Development and Research project. The purpose of the IES funded research project was to determine the feasibili ty and impact of the Prime Online PD program through an iterative design process. The Prime Online PD program was a yearlong asynchronous oTPD program for third through fifth grade general and special education teachers. During Phase 1, August 2010 Dece mber 2010, the content and measures for the Prime Online PD program were generated Participants were selected from a nearby school district. The goal s of the project were to impact (b) their abi lity to meet the learning needs of students with learning disabilities (LD) in general education classrooms (grades 3 5), and (c) their knowledge and skill in using curriculum based measurement within a model of classroom based research. Phase 2
51 of the IE S funded research project, January 2011 December 2011, was the implementation of the Prime Online program with ten third through fifth grade general education and special education teachers. PI conducted interviews, module surveys segment satisfaction s urveys, weekly activities and forum posts participation data, and measures of MCK and PCK were collected to aid in PD program revision Prime Online PD Program The Prime Online PD program took place from January 2011 December 2011 and, therefore, spanned two school years. Participants were given a $1000 stipend and either nine graduate credit hours or 180 inservice credit hours for completing the PD program. Each participant was provided with necessary materials and supplies including textbooks, audiovis ual equipment, and an NCTM membership to access Teaching Children Mathematics (TCM) articles that were part of the oTPD program content. The oTPD program was presented asynchronously on a virtual le arning environment called Moodle ( Dougiamas 1999 ), with the exception of one activity that required a synchronous group chat and the culminating activity that required web conferencing. The Prime Online PD program consisted of 27 modules within three segments (Appendix A). Segment One, Building the Foundation f or Inclusive Elementary Mathematics Classrooms had eight weeks. Weeks 1 and 2 presented the NCTM Principles and Standards for School Mathematics (NCTM, 2000) and the Strands of Mathematical Proficiency (Kilpatrick, Swafford, & Findell, 2001). In these weeks, participants reflected on themselves as learners and teachers of mathematics, their meaning of mathematical proficiency and instructional practices that support mathematical proficiency. The remaining modules in Segment One focused on
52 supporting s truggling learners. Topics included characteristics of students with LD ; research based practices (i.e., explicit strategy instruction, self regulated learning, self regulated strategy development); RTI; and progress monitoring. Teacher inquiry was also introduced in Segment One. Segment Two Deepening Mathematics Content and Pedagogy consisted of 13 weeks and was heavily focused on mathematics content. Weeks 9 12 concentrated on number sense and conceptual understanding of multiplication and division. Participants completed virtual mathematics activities and used virtual manipulative materials, analyzed invented strategies and common error patterns in multiplication and division, represented the area model and partial products model with manipulative m aterials and then compared those models to the traditional algorithm for multiplication, and solved partitive and quotative division problems with manipulative materials. Weeks 14 18 concentrated on representation of fractions and operations of rational n umbers. During these weeks, participants wrote about their understanding of a fraction, reviewed NCTM (2000) and CCSS M standards ( CCSSO 2010) and related them to the Strands of Mathematical Proficiency (Kilpatrick et al., 2001), read about fraction repr esentation, discussed and worked with partition and iteration models, and wrote a new statement about their understanding of a fraction; g about partition and iteration; worked through modeling fraction addition and su btraction with virtual manipulative materials; read a TCM article and discuss ed need for common denominators; implemented a multi day lesson about the relationship between fractions, d ecimal numbers, and percentages; and explored an NCTM webpage for lesson s appropriate for their students; estimated solutions and explained strategies for multiplication of rational numbers, completed and discussed an online NCTM article, and explained why traditional algorithms for multiplication of fractions and multipl icat ion of decimal numbers work ;
53 created a word problem when given a fraction division problem, reasoned about conceptual meaning of multiplication and division of fractions, and reflected on how their new understandings could impact their future instructional practices; completed online lessons, read TCM articles, related operations with decimal thinking. Three weeks in Segment Two (i.e., 13, 19 20) related these same topics to teaching students with LD by reviewing evidence based practices, concrete semiconcrete abstract instruction, RTI, and co teaching models. Week 21 had the students reflect on mathematics topics that had been c hallenging to teach in the past; develop a wond ering defined as a burning question they had about their mathematics teaching practice (Dana & Yendol H oppey, 2009); and list goals for the coming school year. Segment Three, Studying the Application of Newly Learned Mathematics Content and Pedagogy to St udent Learning contained six weeks of content. As some of the weeks (e. g. Week 24: The Road Map: Developing the Data Collection Plan and Formative Data Analysis) actually covered multiple calendar weeks, Segment Three took place during September through December 2011. After learning more about data collection and progress monitoring, participants began an inquiry cycle related to the Prime Online PD program mathematics content and designing and implementing a research plan to study their instructional p ractices (Dana & Yendol Hoppey, 2009). The final weeks were devoted to writing u p and presenting their findings from the inquiry process to the Prime Online PD program cohort and facilitators via web conferencing. Each module (i.e., Week ) included four se ctions: Introduction, Anticipatory Activity, Content and Discussion, and Reflection and Assessment. At the end of each week, a facilitator posted a Course Announcement that summarized the learning for the
54 week. Course Announcements were automatically emai led to all participants and were used for any important announcement s The Introduction familiarized the participants with the content for the week by outlining the objectives and materials needed (e.g., articles, websites) as well as a list of assignment s and activities contained in the Anticipatory Activity, Content and Discussion, and Reflection and Assessment. The Anticipatory Activity was current knowledge of mathematics concepts, or refle cting on their thinking. In Week 16, Multiplication of Fractions for example, participants estimated operations with rational numbers using mental mathematics and explaining their strategies in a forum post. The Content and Discussion required participan ts to explore a mathematics concept with understanding, read articl es or website lessons, engage with developer created materials (e.g., PowerPoint, video), or work through a virtual mathematics activity. For example, Week 16 contained three Conte nt and D iscussion activities. In t he first activity participants were asked to read an article and watch embedded videos and then post a reflection on what was interesting, why it was interesting, how it could be used in their classroom, and any questions or con cerns on the topic. In the second forum, participants were asked to discuss why the traditional algorithm for multiplying fractions works. In the third forum, participants were asked to represent fractions in a word problem and then hypothesize why they fractions. The Reflection and Assessment typically asked participants to examine their classroom in relation to the content for the week implement an activity from the week, or model and discuss mathematics concepts. In Week 16, participants were asked to provide a conceptual explanation for multiplication of decimal numbers.
55 The oTPD program archival data were used to create narratives of the teacher mathematical identity as it existed throughout thei r participation in the Prime Online PD program Segment One and Two activities, PI conducted interviews, module surveys, segment satisfaction surveys, participation chart, and CKT M scores were analyzed, but five weeks of the PD program were of particular importance. In Segment One, activities from Weeks 1 and 2 were selected because of their best lessons, their perceptions of their role as teachers of mathematics, an d when they felt most and least effec tive as teachers of mathematics In Segment Two, activities from weeks 9, 11, and 15 were selected because they offered a range of topics (i.e., number sense, multiplication, and rational numbers), a variety of interac tive assignments and reflective forums (i.e., a virtual activity, an activity with manipulative materials, and implementation of a multi topic hands on lesson), and overall high levels of participation ( e. g. quality and quantity of posts). Narrative Inqu iry Narrative inquiry refe rs to creating data in the form of stories, the ways of interpreting that data, and the methods of representing data in narrative form (Schwandt, 1997). Narrative inquiry is, more precisely, the study of human experience and nar rative researchers collect and tell stories of those experiences (Connelly & explanations of their actions) are powerful sources for narrative inquiry (Connelly & Clandinin, 1 990) and typical data sources for narrative inquiries include interviews, journals, autobiographies, and documents.
56 The goal of narrative inquiry is to attain narrative truth when composing the research story (Spence, 1984). Narrative truth occurs when th e occurrences have been represented satisfactorily (Kaasila, 2007) and when the explanation conveys conviction (Spence, 1984) and plausibility (Connelly & Clandinin, 1990). Collecting and retelling lived stories is a challenging process. Connelly and Cla ndinin (1990) impart the complexity and multiple levels of narrative inquiry by stating narratives and the jointly shared and constructed narratives that are told in the research writing, but narrative researchers are compelled to move beyond the telling of the lived A narrative (e.g., an interview) is a sequenced series of events that holds One limitation of any narrative is that some people find it difficult to tell a story. This limitation is minimized with episodic interviews, however, because there are many short narratives instead of a single complete narrative (Flick, 2009). Episodic interviews are specific types of narratives that provide in depth focus on situations (i.e., episodes) that are pertinent to the research study. Advantages of episodic interviews are that the interviewer has more flexibility in defining specific events fo r the interviewee to recount. The interviewee can then choose a specific description or story to respond to the interview question. This link between question answer sequences and narratives is a triangulation of different data collection approaches (Fli ck, 2009). The stories obtained in an interview can be considered a way to express identity (Connelly & Clandinin, 1999) or as identity itself (Sfard & Prusak, 2005). When taking the narrative as hors and
57 and contain & Prusak, 2005, p. mathematical identity. This dissertation sought to interpret inservice teachers mathematical identity and experiences in an oTPD program. This study, therefore, form of archival data and interviews, were the primary data sources. Furthermore the interviews conducted for this dissertation were structured so participants could recount events relevant to their teacher mathematical identity. Participants Participant Selection The participants were purposively sampled from the eight inservice tea chers who completed the Prime Online PD program. Purposive sampling is expected in narrative inquiries and increases rigor, trustworthiness, and credibility (Patton, 1999; Patton, 2001). Cases may be purposively sampled as typical deviant or critical ( Patton, 2001), but purposive sampling for maximum variation is most common in narrative inquiries on teacher identity (e.g., Forbes & Davis, 2008) and teacher change (e.g., Smith, 2011). Such studies select a small number of participants with diverse back grounds in order to create rich, detailed narratives and still gain varying perspectives. Due to the breadth of data sources and the expected depth of data analysis in this study, two teachers were selected for this study. Based on my review of the litera ture, views of self as a learner of mathematics (e.g., mathematics history, PD experiences) and views of self as a teacher of mathematics (e.g., self efficacy) are important aspects of teacher mathematical identity. Quantity and quality of participation w ithin the oTPD program (e.g., assignment
58 completion, frequency of posts) were also of interest. Therefore, to obtain thorough and pertinent data to answer the research questions, participants with maximum variation in levels of participation and views of mathematics were invited to participate in this study (Miles & Huberman, 1994) The participant selection process began with an examination of archival data. During this process, I knew participants only by their Prime Online PD program participant number s to limit potential biases. Data sources were chosen that would and their views of mathematics teaching and learning. Segment One activities and PI conducted intervi ews were examined collectively to minimize bias (Newton et al., 2012) and general statements of teacher mathematical identity were noted (e.g., example of best lesson, perception of teacher role). Quantitative data were overall levels of participation sa tisfa ction across the three segments, and measures of MKT. Summary Prime Online PD program teacher mathematical identity and participation in the oTPD. Although several teacher s volunteered to take part in this study Heide and Brynn were selected as participants, as they were expected to provide the maximum variation in participation and views of mathematics sought in this study (all names are pseudonyms). In addition, Heide a nd Brynn had an age gap of over 20 years, dissimilar teaching contexts, and a difference of eight years of teaching experience but they had the same certification and degrees, and both had prior experiences teaching special education. As part of the Prim e Online development project, all eight participants gave
59 consent for analysis of their archival data. Heide and Brynn signed an informed consent form that provided information about this dissertation study (Appendix G). Description of Participants Heide and Brynn taught in the same school district in the Southeastern United States. All teachers in the county were required to follo w the curriculum pacing guide and to use the Gradual Release of Responsibility (GRR) model of instruction (Pearson & Gallagher, 1983). The GRR model integrates four lesson components: focused lesson, guided instruction, collaborative learning, and independent tasks. GRR is sometimes referred to a scaffolded instruction or an I D o (i.e., direct instruction) We D o (i.e., guided i nstruction) and You D o (i.e., independent practice) method of instruction. Neither participant had attended a mathematics PD program since the conclusion of the Prime Online PD program. Heide Heide had the highest scores o n the satisfaction surveys, the highest levels of participation throughout the oTPD program, and completed all surveys and CKT M assessments. The participant selection process revealed that Heide had a negative hist ory as a learner of mathematics but was motivated to learn. She also h ad a very low efficacy as a teacher of mathematics when compared to the others in her cohort. Heide grew up in an outdoor education camp setting and always knew she wanted to teach. She worked in outdoor education for eight years prior to her current teac hing position in the public school system and had certification in elementary education (K 6) and special education (K 12). She also recently acquired cert ification in Earth space science (6 12). Heide was familiar with online learning, as
60 PD experiences were limited to those related to the newly adopted textbook series and one in which an expert was brought in to lecture on how to make mathematics relevant. Heide stated the following reason for joining the Prime Online PD program : [I] was going crazy in the classroom because I integrated all of m y subjects except math . I just never was confident. . I was looking at the kids who were so unmotivated and unengaged so when I saw it [ the Prime Online PD program mean, seriously, I wanted it so bad. [I 2p2] At the time of this study, Heide was in her thirteenth year of teaching, eleven of which were at her current school, a rural K 4 elementary school that served approximately 520 students. She was in her fifth year as a general education teacher, but she taught third th rough fifth grade students with high incidence disabilities in a self contained classroom for the majority of her teaching career. During both school years that spanned the Prime Online PD program, she taught fourth grade general education. She indicated however, that a majority of her students either had disabilities or were in Tier 3 RTI groups (i.e., needing intense instruction). During that time, a part time paraprofessional five days a week and a special education co teacher assisted her two days a week Heide taught third grade general education one year post PD and at the time of this study, two years post PD, taught first through fourth grade gifted science. Brynn participat ion across modules was intermittent. She did not give feedback on any
61 module surveys and did not participate in the final CKT M. The participant selection process revealed that Brynn had a negative history as a learner of mathematics and traditional view s of mathematics instruction. She considered herself to be an average teacher who was concerned about grasping the content in the Prime Online PD program. degree in special education. She had certification in elementary education (K 6) and special education (K 12). The Prime Online PD program was her first mathematics specific PD and her only prior experience with online learning was one synchronous special education course. Brynn c onveyed that she joined the Prime Online PD program because her experiences as a special education co teacher and her current role as a teacher of Grades 4 5 mathematics seemed to align well with the PD ics in an inclusive setting. At the time of this study, Brynn was in her fifth year of teaching at a low socioeconomic status (SES) K 5 elementary school that serves approximately 570 students. The first year of the Prime Online PD program s first year as a general education classroom teacher. She had not taught the year prior to the Prime Online PD program because she stayed home after the birth of her son. For the two years prior to that, however, she had been a special education co teac her and shared the classroom responsibilities with two fifth grade general education teachers. Brynn had a different teaching assignment for both school years that coincided with the Prime Online PD program During the first year, she taught a general ed ucation fourth/fifth grade combination inclusion class. The class was composed of higher achieving fourth
62 graders and lower achieving fifth graders. She recalled that between seven and 10 students had high incidence disabilities (e.g., specific learning disability, emotional/behavioral disability). Due to an increase in enrollment, B became a fifth grade inclusive classroom early in the second year. She has ta ught a combination fourth/fifth grade class since that time. Data Collection Da ta included quantitative and qualitative archival Prime Online PD program artifacts and two post oTPD interviews conducted as part of this study. Archival qualitative data sources were Segment One activities, PI conducted interviews, Segment Two activitie s, and module surveys. Archival quantitative data were the participation chart, CKT M measures, module surveys, and segment satisfaction surveys. The archival data were collected in 2010 2011 as part of the larger study and two post oTPD interviews were collected in July and October of 2013 (Table 3 1) Table 3 1. Data Collection Summary Instrument Month(s) collected Archival Data CKT M pre assessment December 2010 Segment One activities January February 2011 PI conducted interviews February March 2 011 Segment Two activities April May 2011 CKT M proximal assessment August 2011 Participation chart January 2011 December 2011 Module surveys January 2011 December 2011 Segment satisfaction surveys March 2011 December 2011 CKT M post assess ment January 2011 Current Study First Interviews July 2013 Second Interviews October 2013 a Module 13 survey includes questions about hindrances to participati on
63 Archival Data CKT M measures The CKT across a variety of tasks such as error analysis, multiple representations, estimation, and invented strategies ( Hill, Schilling, & Ball, 2004). Three forms of two scales were used: Elementary Number Concepts and Operations Content Knowledge (EL.NCOP KC ; Learning Mathematics for Teaching, 2001a) and Elementary Number Concepts and Operations Knowledge of Content and Students (EL.NCOP KCS; Learning Mathematics for Teaching, 2001b). Hill et al. (2004) report that Forms A, B, and C of both scales have adequ ate reliability (i.e., EL.NCOP KC = .719, .766, and .784, respectively; and EL.NCOP KCS = .622, .657, .698, respectively). The CKT M measures were administered prior to the oTPD program (December 1 16, 2010), after completion of the Segment Two (July 2 9 Aug 5, 2011), and after the conclusion of the oTPD program (Jan 16 22, 2011). Each participant was given a percent correct fo r each of the six CKT M measure administrations Segment One activities Two modules in Segment One were analyzed for this study : Weeks 1 and 2 (Appendix B). Three activities from Week 1 were examined. The Anticipatory Activity (i.e., Classroom Practices that Promote Mathematical Proficiency) asked participants to upload a statement reflecting himself or herself as a teacher of m athematics by describing their best mathematics lesson, their role as a mathematics teacher, and when they feel most effective and least effective as a teacher of mathematics. For the Content and Discussion, participants read A Vision of School Mathematic s (NCTM, 2000). Then the participants were asked to relate their Anticipatory Activity response to
64 the reading and write two posts. The first post was regarding the ways in which their classroom reflected the Vision of School Mathematics (NCTM, 2000) and any perceived barriers in working toward the goals presented in the reading. The second post asked the participants to discuss commo nalities across the first posts such as common barriers to implementation. Finally, the Reflection and Assessment activit y asked participants to consider the content for the week and their statement from the Anticipatory Activity (i.e., who they are as a teacher of mathematics) and upload a reflection on how their views have or have not changed over the course of the week. T hree activities in Week 2 (i.e., NCTM Principles and Standards for School Mathematics ) were also analyzed. In the Anticipatory Activity, participants were asked to reflect on their history as a learner of mathematics in Grades 3 5 and comment on how that has affected their view of what it means to be mathematically proficient. For the Content and Discussion participants read The Strands of Mathematical Proficiency (Kilpatrick, Swafford, & Findell, 2001) and Tying It All Together (Suh, 2007). Then the pa rticipants were asked to find commonalities across their posts, describe their current instructional practices that support mathematical proficiency and express their goal s pro ficiency. In the Reflection and Assessment, participants were asked to consider the content for the week and their response to the Anticipatory Activity (i.e., history as a learner of mathematics and what it means to be mathematically proficient). The pa rticipants then wrote and uploaded a brief statement about how their views had or had not changed about how they view their role in supporting the development of mathematical proficiency. Due to the wording of the six assignments (e.g ., I see my role
65 as a mathematics teacher to be . reflect on your history of learning mathematics ), Prime Online PD program teacher mathematical identity. PI conducted interviews The PIs of the IES funded resea rch project conducted interviews during Segment One (i.e., between February 21 st and March 16 th of 2011) at a place and time convenient for the participants. The PI conducted interviews, which had no time limit and were audiotaped and transcribed, sought the Prime Online PD program content and pedagogy (Appendix C). The PI conducted interviews were semi structured with open ended questions about the following : why they enrolled in the Prime Online PD program, the ir past experiences with mathematics PD and online learning, what has gone well or has been challenging in the Prime Online PD program, how the Prime Online PD program content has been integrated into their current practice, time management with the Prime Online PD program, one forum discussion that has been particularly meaningful or helpful, discussions (i.e., forum posts and responses) with their cohort, the most valuable and least valuable activities, how their thinking about mathematics has changed, an d factors affecting their participation in the Prime Online PD program. Similar to the Segment One activities, the PI conducted interviews contained narratives Prime Online PD program, whi
66 identity ( e.g., efficacy, views of mathematics) and participation (e.g., successes, hindrances). Segment Two activities ed by its responses) during three activities from Segment Two, Deepening Mathematics Content and Pedagogy oTPD program mathematics content (Appendix D). In Week 9, Number Sense, Pro cedural Knowledge, and Conceptual Knowledge participants used mental mathematics to solve a multiplication problem as many ways as possible for the Anticipatory Activity; read about number sense, procedural knowledge, and conceptual knowledge and used vi rtual calculator software for the Content and Discussion; and, for the Reflection and Assessment, di scussed their class. For one forum, entitled Broken Calculator partic ipants were to access an NCTM webpage, read the content, watch the embedded videos, and then work with the virtual software. The virtual calculator required users to compute addition and multiplication After completing the Broken Calculator (NCTM, 2006) activity, participants were asked to discuss the strategies they used, how the applet might be used with their students, and their reflections regarding the online materials. Week 11, Building Conceptua l Knowledge of Multiplication and Division, participants reviewed an RTI video on problem solving as the Anticipatory Activity, completed three Content and Discussion activities, and discuss ed strategies for multiplication for the Reflectio n and Assessment. The Content and
67 Discussion activities included discussing patterns in a synchronous group chat room; using pictures and virtual manipulative materials to examine set, length, and area models for multiplication; and work ing with base ten blocks to model multiplication problems The third Content and Discussion a ctivity, Working with Base Ten Blocks required participants to model several double digit multiplication problems with base ten blocks a nd upload the images of their work. Finally, the participants responded to questions related to using manipulative materials support understanding of the partial products algorithm and the comparison between the partial products algorithm and the traditio nal algorithm for multiplication. In Week 15, Fractions and Decimal Numbers: Addition and Subtraction participants uploaded images of fraction representation as the Anticipatory Activity ; used virtual manipulative materials to add or subtract fractions; r ead about and discussed conceptual understanding of finding common denominators, and working through an activity called A Meter of Candy (NCTM, n.d.) for the Content and Discussion; and search a website for examples of rational numbers activities and resou rces for the Reflection and Assessment. The third Content and Discussion activity, A Meter of Candy asked participants to work through a multi lesson activity with their students, if possible on the connection between fractions, decimal numbers, and perc entages. The foru m asked participants to discuss the features of the activity the activity, how they would typically assess these concepts, which of the suggested assessm ents were most appealing, and what they liked or did not like about the lesson.
68 The three Segment Two activities selected for this study Broken Calculator Working with Base Ten Blocks and A Meter of Candy covered a range of mathematical concepts (e.g., n umber sense, manipulative materials use, the relationship between fractions and decimals) and had particularly high levels of overall participant engagement as measured by the length of initial posts and the number of follow up responses. The content of t he posts and responses gave insights into as learner of mathematics). Additionally, because of the high level of overall engagement, these activities were an indicator of the breadth and dept mathematics activiti es and the ensuing discussions. Participation chart The participation chart was a record of the number of times participants posted or submitted an assignment within each module over the course of the o TPD program. The Prime Online PD program project manager updated the participation chart twice a week. For each task (i.e., post or assignment), the number of submissions made and whether the submissions were completed by the due date were used to calcul ate each participation and requirements met For assignment participation the number was compared to the number of required tasks. A ratio was then determine d to indicate whether the participant met, exceeded, o r did not meet oTPD program expectations. The percentage of late or missing tasks was also calculated. The requirements met ratio for a task represented the number of each he participation percentage may exceed 100% while the requirements met could not
69 exceed 100%. The participation chart provided information regarding consistency of me eting or exceeding oTPD program requirements. These two variables were quantitative indicators of each teacher Module surveys The participants were emailed a link to the untimed module surveys and were asked to completed the anonymous surveys at a time convenient for them. Each module survey had five to seven Before After Likert type items specific to the module content with a 1 indicating not at all true of me and a 4 indicating very true of me (Appendix E). In addition to the Likert type items, each module survey contained several free response items related to the module content For example, participants were asked to explain significant things they had learned what they would have liked to learn more about, and sug ges tions for module improvement Module surveys from Weeks 1 and 2 were chosen because they aligned with the Segment One activities and Weeks 9 12 and 14 16 were used because they aligned with the Segment Two activities being analyzed in this study. For exam ple, the Week 1 mod ule survey asked participants whether they felt that they had the knowledge an d skills to integrate the NCTM P rinciples for School M athematics into their instructional practices. The Week 2 module survey included an item that asked part icipants how much they understood about how their history of learning mathematics had influenced their thinking about what it means to be competent in mathematics. An item in the Weeks 9 12 module survey asked participants if they understood how to use ma operations such as multiplication and division. The Weeks 14 16 module survey
70 contained an item that asked participants if they understood how to use estimation to supp A mean and standard deviation (SD) were calculated for each Before A fter item in the module surveys as a quantitative measure during each week of the oTPD pro ended responses were considered as possible indicators of teacher mathematical identity (e.g., views of teaching and learning mathematics). The Week 13 module survey also included questions about hindrances t o participating in a timely manner such as the 10 Likert type items offering too much work in an individual week and content has become too challenging as possible hindrances. Participants were also asked to provide additional feedback that was not alread y included in the 10 Li kert type items, which added qualitative information regarding levels of participation. Segment satisfaction surveys T he participants were emailed a link to the untim ed segment satisfaction surveys and asked to complete the anonymous surveys at a time convenient for them. The segment satisfaction surveys were administered at the end of Segment One (i.e., Weeks 1 7), Segment Two (Weeks 9 21), and Segment Three (Weeks 25 27) and were constructed using a 4 point Likert scale of strongly agree, agree disagree and strongly disagree (Appendix F). Each segment satisfaction survey consisted of between 46 and 51 questions regarding overall participant satisfaction with module content pedagogy, and technology and support. Survey prompts in cluded the instructional methods presented in the modules are practical in the amount of time they will require in my classroom the assessment and reflection activities helped me understand what I learned through the modules and interacting and sharing i deas with
71 other participants contributed to the overall effectiveness of the modules The response were tallied and a mean and SD were calculated. A mean was also established for surveys as any trends in satisfaction across the segments. Current Study Two interviews were conducted as the main sources of data in this study. Over two years had la psed since the conclusion of the oTPD to give teachers time to incorporate the activities from the Prime Online PD program and to allow shifts in teacher mathematical identity to occur. The interviews were scheduled at a time and place convenient for the participant and were audio recorded with a digital recording device and transcribed verbatim. Throughout both interviews, participants were prompted to tell stories about their experiences by providing detailed recollections about the feelings, thoughts, actions, and contexts of those experiences. A relationship was established with each participant so that both of our voices were heard (Connelly & Clandinin, 1990). I minimized my own comments particularly those that could have been interpreted as judgme ntal so that the In narrative inquiry, it is important that the researcher listen first to the story. This does not mean that the rese archer is silenced in the process of narrative inquiry. It does mean that the practitioner, who has long been silenced in the research relationship, is given the time and space to tell her or his story so that it too gains the authority and validity that the research story has long had. ( Connelly & Clandinin, 1990, p. 4)
72 The first interviews took place in July of 2013, two years post Prime Online PD program upcoming year. Heid interview lasted approximately 40 minutes. The first set of interview questions and current views of mathematics (Appendix H) Most of the topics mirrored those in the archival data that were relevant to this study to encourage responses that would elicit indicators of teacher mathematical identity. For instance, the participants were asked the following : Tell me a story about one of the best recent mathemat ics lessons you taught. Looking back, when are you least effective as a teacher of mathematics? Please tell me a story that would be an example of when you feel least effective. How do you see your role as a teacher of mathematics? Describe how that mi How do you view your role in supporting the development of mathematical proficiency? How might this look in a typical mathematics lesson? How would you feel, think or do? After the first interview s were transcribed and analyzed follo w up questions were determined and the second interviews were held in October of 2013. Both follow up interviews were second interview lasted approximately 40 minutes a approximately 30 minutes. The second int erviews had similar topics for each participant, (e.g., instructional practices, participation, general clarification of statements) but the specific questions were based on the co ngruence or dissimilarities found between teacher mathematical identity from the archival data and the teacher mathematical identity from the first interview (Appendices H & I).
73 For example, Heide was asked if she would have considered her pre Prime Onli ne mathematics into other subjects [I 1p14]. Brynn was asked to compare statements made regarding the use of manipulative materials. While a participant in the Prime Onlin e to have a negative connotation toward manipulative materials. This was contrasted with the second indication, during our first interview, when Brynn called manipulative materials h 1p7]. Other questions were asked for clarification so that I could more accurately interpret their words and meanings. For example, Heide was asked to compare her definition of mathematical proficiency during the Week 2 Anticipatory A ctivity with the one from our first interview and reflect on whether she perceived those as having different meanings. Similarly, Brynn was asked to differentiate the terms quick strategies used in the Week 1 Anticipatory Activity and tricks used in the We ek 2 Anticipatory Activity. Data Analysis such as contact made with participants and reflections including field no tes from each interview session Notes and journaling created an audit trail to docum ent my interactions with participants, subsequent revisions, and decision making processes. The journal was a place to problems that arise during 71). I also used the journal to record the steps taken during this study and keep a log of codes, both of which helped track emerging themes during data analysis.
74 communicate what has been lear ned to others (Hatch, 2002, p. 148). This dissertation expanded upon the notions of teacher identity and professional growth by examining the relationship between teacher mathematical identity and participation within an oTPD program Theore tical thematic analy sis was chosen to examine these data because studies that rely on interviewing as the sole or primary data collection tool are often undertaken with a fairly focused purpose, a fairly narrow set of research questions, and a fairly well structured data set in terms of its organization around a set of fairly consistent guiding questions. When the study was designed, the researcher had as his or her goal to capture the perspectives of a group of individuals around particular topics. If th e study was well designed and implemented, data from the interviews ought to of interest. So the topics that the researcher had in mind when the study was designed will often be logical places to start looking for typologies [i.e., categories, themes] on whi ch to anchor further analysis. (Braun & Clarke, 2006, pp. 152 153) Therefore, a coding scheme based on a review of the literature was developed. Constructs often studied in re lation to tea cher mathematical identity are history as a learner of mathematics (i.e., views of self as learner) efficacy (i.e., views of self as teacher), teacher role (i.e., views of mathematics teaching) and how students learn (i.e., views of mathemat ics learning) which were encompassed in my v iews of
75 mathematics construct. I also searched for statements related to participation and MKT. Each case was analyzed following the guidelines put forth by Braun and Clarke (2006): 1. Becoming familiar with the data. 2. Generating initial codes. 3. Searching for themes. 4. Reviewing themes. 5. Defining and naming themes. 6. Producing the report. First, I became familiar with the data by read ing through its entirety multiple times to search for patterns while keeping my mind op en to possible new meanings During them later (Braun & Clarke, 2006, p. 17). Next, I began the more formal coding process. During the second phase generating in itial codes codes were created to identify parts of the data that seemed interesting and instances of each code were highlighted. Some passages were highlighted for more than one code. After reading each set of highlighted data, the main ideas from each source were recorded on a summary sheet for each participant. The results of the quantitative analysis were then added to the summary sheets and I began to look for meaning or searching for themes At this point I sorted the codes into potential themes, assessed whether the themes were supported by the data, and searched for non examples of my themes (i.e., reviewing themes). This required rereading all of the data, including the parts that were not highlighted. As stated by Braun and Clarke (2006 imply because an analysis starts with a deductive step does not preclude the researchers being aware that other important categories are likely to be in the data or prevent the researcher from 161). I refined the themes by collapsi ng or separating them. I read the extracts within each theme to check for coherence and assessed how
76 accurately the themes reflected the overall data set. During the fourth phase, d efining and naming themes determine how each theme fit into the broader narrative in relation to t he research questions and again sought to identity any sub themes. The final step was to return to the data, look for connections across themes and find compelling excerpts to let th e audience into the Hatch, 2002, p. 159). Limitations and Delimitations There were limitations and delimitations of this study that need to be acknowledged. A limitation was t hat t he population was restricted to the eight teachers who completed the yearlong oTPD program and was further limited by those willing to participate in this dissertation study. However, a small number of possible participants available for final analys identity are small scale and in depth (Beijaard et al., 2004) and participants were sampled for maximum variation as in other studies of teacher identity (e.g., Kaasila et al. 2008; Ponte & Santos 2005). of their instructional practices were not triangulated with classroom observations. This study was de limi ted in data and not on the resul ts of available efficacy scales. Additionally, c ourse usage data (i.e., how often each page and link was accessed a nd over what period of time) were available through the Moodle (Dougiamas, 1999) software but was not ana lyzed Passive en gagement in an on line course sometimes referred to as lurking can still lead to learning due to the (Mazzolini & Maddison, 2003). However, accessing the course without submitting a post
77 or assignment (e.g., read responses) was not considered participation for the purposes of this study. Trustworthiness and Credibility There has been a recent movement to return the construct of qualitative validity to a matter of e thics. Koro research they cannot escape their responsibilities or leave the rigor or trustworthiness of their res measure of trustworthiness and trustworthiness of data is inexorably linked to the trustworthiness of the researcher (Patton, 1999). That which constitutes validity in qualitative r esearch continues to evolve and is influenced by the theoretical perspective taken by the researcher as well as what has become the norm in that particular field of study (Flick, 2009). For example, McMillan and Schumacher (2005) propose 10 strategies to enhance validity and trustworthiness in qualitative educational studies. Multi method strategies, prolonged field work, participant verbatim language, low interference description (e.g., thorough field notes), and negative case searches are considered ess ential strategies while mechanically recorded data, participant review, multipl e researchers, and member checking should thus researchers should strive to make their work transparent (Koro Ljungberg, Yendol Hoppey, Smith, & Hayes, 2009). Transparency and trustworthiness are important components in establishing the credibility of qualitative research (Patton, 1999). I will
78 (1989) four const ructs of trustworthiness credibility, transferability, dep endability, and confirmability and how each related to my dissertation. Credibility Credibility is how well the research describes the phenomena fro m the mutual meanings 2005, p. 324). Triangulation of sources and methods were used to strengthen credibilit y (Denzin, 1989; Lincoln & Guba, 1985). Data were obtained from multiple sources to study the same phenomenon. Interviews were augmented with qualitative and comparis ons among data sources during data analysis (Flick, 2009, p. 27). Studying the same issues over time supports credibility and trustworthiness, and archival data sources were triangulated with each other as well as with the interviews. Dependability Depend ability refers to prolonged engagement, the use of multiple researchers, and the degree to which the researcher has disclosed changes that occurred during the study and how these changes affected the study (Rossman & Rallis, 2003). I worked for three year s on the Prime Online PD program development project as a graduate research assistant This dissertation had two researchers, as the Prime Online PD program methodologist analyzed the quantitative archival data as part of the larger study. Dependability can also indicate the assurance that the inquiry process has been documenting the specifics of observations is helpful for establishing dependability (Brantlinger, Jimenez, Klin gner, Pugach, & Richardson, 2005), which was done through
79 the maintained thr oughout this dissertation and are available for auditing. Transferability Transferability requires the researcher to provide rich details about the study context and assumptions so that others might decide whether similar findings would likely transfer to their own situations (Lincoln & Guba, 1985). Even one teacher can provide information for researc hers due to common experiences and concerns with other teachers (Muchmore, 2002). Initial descriptions of the participants and the Prime Online PD program and content have been provided in previous sections of this chapter in an effort to make the methods transparent. Confirmability Confirmability is the level at which the results can be corroborated by others and can be made transparent by linking assertions to data or through peer reviews, the search for negative cases, and continual data checks. My cha ir, a researcher with educational psychology and mathematics education backgrounds, provided feedback throughout this study. My co chair, a qualitative researcher with professional development and practitioner inquiry backgrounds, also provided feedback d uring data collection and analysis. Direct quotations from oTPD program artifacts and interviews justified my resulting claims but confirmability is inevitably impacted by the subjectivities and role of the resear cher (Miles & Huberman, 1994). Researcher S ubjectivity Because the researcher is the instrument of measure in qualitative analysis, a statement of potential biases is warranted (Hatch, 2002). My professional experiences, education, and perspectives were all factors in both the collection and the a nalysis of
80 the data. By making my subjectivities transparent in this section of the dissertation, I think that this awareness would assist me in applying these subjectivities in order to tch, 2002). Much of my professional life guided my interest in this study. I had quite a bit in common with the teachers in the Prime Online PD program I was an elementary school teacher for five years, three of which were working as a resource teacher for students with high incidence disabilities (e.g., specific learning disabilities). This gave me an understanding of what it is like to teach third through fifth grade students who struggle academically. I then spent seven years as a s ixth grade mathe matics teacher. This gave me a familiarity with the pressure s of teaching mathematics such as county pacing guides, standardized tests, and tive perceptions of mathematics In addition, the fact that I received alternat iv e certification to teach middle grades mathematics gave me commonality and empathy for teachers who did not have a background in mathematics. My experiences over the past three years increased my perspective on and understanding of inservice PD. While working toward my doct oral degree in curriculum and instruction, I took seminars on designing PD and on the research and practice of doctoral seminars. More significant, however was my work a s a graduate research assistant on the Prime Online development project. I attended most of the P meetings, which provided me knowledge of the inner workings of the development, implementation, and revisions of to the oTPD program. I assisted in devel oping the Prime
81 Online PD program participant through her inquiry cycle as my independent project in a doctoral level course on practitioner research. I had occasional online contact with the oTPD program participants as a facilitator during the mathematics content modules, but for the majority of the year my role was that of an eavesdropper Consequently, oTPD program participants might have recalled my role in the Prime Onl ine PD program and had been more willing to engage in this study. Unbeknownst to me at the time, one of my tasks for the Prime Online development project was the antecedent of this dissertation. At the request of the PIs, I completed case studies of the t wo participants identified as most and least satisfied (i.e., highest and lowest score on the Segment Two satisfaction survey). Quantitative data were the participation chart, CKT M scales, and module and segment satisfaction surveys. Qualitative data we re PI conducted interviews, weekly activities, and module surveys. Data were then reduced to illuminate the individual factors (e.g., teaching experience, familiarity with online learning, personal and professional constraints), oTPD program activities, a nd patterns of participation that may have contributed to their level of overall satisfaction with the Prime Online PD program This task piqued my interest because it made me reflect on my own PD experiences in mathematics education. I have often wonder ed why some PD did not have much of an impact and why one particular PD became such a pivotal experience for the development of my own MKT. This pivotal PD experience was school wide, multi year, and interdisciplinary. The mathematics comp onent integrate d the P rinciples for School M athematics (i.e., equity, curriculum, teaching, learning, assessment, technology ; NCTM, 2000 ) and the five strands of mathematical proficiency (i.e., conceptual
82 understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition put forth by Kilpatrick et al. (2001). These same components were explicitly and implicitly embedded in the Prime Online PD program I was curious if it was the content and pedagogy presented or one of the many other fac ets (e.g., teacher agency, inter departmental collaboration) that made such an impact on me and hoped this dissertation would provide me with further insights. In addition, I had a ty formation, as my post graduation goal was to work with inservice teachers of mathematics. Structure of the Narratives A narrative in four sections is used to frame the findings for each participant in this dissertation (Chapters 4 and 5). The first sec Prime Online teacher mathematical identity as re constructed from archival data and the r of mathematics, self efficacy, MKT, and views of mathematics teaching and learning. Next the Partici pation in the Prime Online PD program is presented. This section of the narrative was also generated base d on archival data and includes general feelings expressed about participating in the Prime Online PD program quantitative and qualitative levels of participation, satisfaction with the oTPD program, and supports and hindrances to participation. The third sectio n dentity (i.e., two years post Prime Online PD program) and used the two inter views conducted as part of this study as data sources. The presentation of this section is similar to that of the Prime Online PD program teacher m athematical i dentity (i.e., includes views of mathematics teaching and learning) but the themes are differen t for each participant. Finally, each case co hifts in teacher m athematical i dentity, as revealed
83 by comparing archival data with the in terviews from the current study and a brief of the My interpret ations of the data are supported by pertinent quotations from the archival data and interviews. All quotations were coded with a system to identify the source of the data. For example, Week 1 has an Anticipatory Activity [i.e., Wk1AA], multiple forums in the Content and Discussion [e.g., Wk1F1], and a Reflection and Assessment [Wk1RA]. Week 2 data are denoted in an identical manner. Feedback obtained from the module surveys are signified by the corresponding weeks The module survey for Weeks 9 12 is denoted [MS9 12], Week 13 is denoted [MS13], and Weeks 14 16 is denoted [MS14 16]. Segment Two activity notations are as follows: [Wk9BC] refers to the Broken Calculator activity in Week 9, [Wk11BT] refers to Working with Base Ten Blocks activity in Week A Meter of Candy activity. For the three interviews that were con sidered for this study, representation of the fourth page 1p4], second interview is [I 2p4], and PI conducted in terview is [PIp4].
84 CHAPTER 4 HEID E: STUDENT CENTERED TEACHER, ENTHUSIASTIC LEARNER Prime Online PD P rogram Teacher Mathematical Identity My reflection on my own personal math experiences in school, exposure to the thinking and memories of cohorts, plus th e reading, have made me question everything I do when teaching math, from planning, classroom discussions, testing, to the focus of my reflections [Wk2RA] Mathematics History: A Lack of Understanding Although Heide could not recall specific instances fro m her elementary school mathematics education, she did report two vivid memories from her adolescent years. The first incident occurred when she was kept after school for extra help to learn the metric system. Her teacher told her father, in her presence Her experience in ninth grade Algebra was not any ld never understand it [Wk2AA]. She explained that she was afraid to ask for help because of the favoritism her teacher the football coach showed the boys, leaving her with a constant feeling of failure. Heide recalled her history of mathematics as havin just what teachers long
85 Self Efficacy common theme in multiple forums, assignments, and interviews during the program. She explicitly stated ho w her negative mathematics history had hindered her teaching. teach for greater success is hampered by our own personal lack of overall In Week 2, participants were asked to write about the goals they had for their classroom practice related to improving classroom proficiency. Heide agreed wi th one of her peers that she no longer wanted to learn the mathematical topics along with the students ea ch day. She explained her challenges in adapting to and comprehending the new mathematics curriculum: I find myself reading the math lessons every night . looking ahead to see where this is all going, reflecting on where we've been, and struggling to help kids make the connections. Then I have to analyze the various strategies for modeling the skill, which takes another 1/2 hour. It's the one time out of the whole day when I have to have complete silence because I have to REALLY concentrate in order to understand it. [Wk2F1] she indicated that her negative history drives her to be a better teacher. She credited her past educational experiences as the reason she felt t Heide considered thi k and understanding she gained when her students showed academic growth [Wk1F1].
86 Heide wrote about her best lesson for the Week 1 Anticipatory Activity, which provided details that seemed to contradict her insecurities as a teacher of mathematics. During the lesson, each student had his or her own set of fraction bars (i.e., manipulative materials) and their desks were arranged in pairs. After providing explicit directions for the use of the manipulative materials, Heide g ave her students the opportunity to discover relationships among the pieces (e.g., two fraction bars are equivalent to one fraction bar). She stated that she had students respond to each the lesson. As Heide had not yet read the Principles and Standards for School Mathematics (NCTM, 2000), she did not realize that her instruction was in fact, aligned with the Teaching Principle: o learning mathematics through the decisions they make, the conversations they orchestrate, and the ph ysical M Standards of Mathematical P ractice ( CCSSO, 2010). MKT During Prime Online PD Program Activities According to Heide, much of the Prime Online PD program mathematics content was new to her However, she scored above the mean on the Content Knowledge portion of CKT M Tests 1 and 2; the Test 3 CK score was below the mean (Table 4 1). She considered several Segment Two activities (e.g., examining 12]. She said that she was put in the same position as her current
87 hard I was trying, some of those things were really difficult to 2p9]. Some of the difficulties faced by Heide were based on her lack of familiarity with mathematical models (e.g., set, area, and le ngth) and manipulative material s. Table 4 1 M scores Test 1 Percent Correct Test 2 Percent Correct Test 3 Percent Correct CK KS CK KS CK KS 79.2 65.0 76.9 84.2 56.5 71.4 Mean 71.4 75.0 72.1 72.9 65.8 73.5 Instead of being overwhelmed with all of the new information, Heide wa s open to the topics and activities being presented. She was the only Prime Online PD program participant who indicated that she tried the mathematics a ctivities in Weeks 9 16 with her students. An activity requiring the use of mental mathematics to solv e a 12]. Nevertheless, she recognized the necessity of explanations because students were now expected to defend their mathem atical thinking. She took the Week 9 number sense content (i.e., mental 12]. She also implemented the Broken Calculator (NCTM, 2006) activity from Week 11 and A Meter of Candy activ ity (NCTM, n.d.) from Week 15 in her classroom (Appendix D). Even when she struggled with understanding the mathematics content, Heide my God, I am how old and I have 2p8]. When she noticed that . 2p9]. In addition, she reported that the
88 Broken Calculator (NCT In her forum post, she concisely stated the strategy that she used to work through the applet. She then elaborate d on that strategy (i.e., gave specific steps, numbers, and an explanation of her thinking) when others posted to request help with the assignment. As one of the few participants to try the Broken Calculator (NCTM, 2006) activity with her students, Heide When another participant mentioned that the Broken Calculator (NCTM, 2006) activity would im pede progress on the curriculum pacing guide, Heide responded that the Working with Base Ten Blocks forum in Week 11. When the participant s were discussing the activity (i.e., using base ten blocks to model multi digit multiplication problems), one of her peers stated that base ten blocks would be helpful for struggling students. Heide countered that the manipulative materials might also he lp students who already knew number sense (i.e., conceptual understanding) [Wk11BT]. In her ow n post, Heide reported: Using the base ten blocks made me think COMPLETELY differently about what I was doing. I had to actually think about place values, grouping accordingly, quantities . it is so different than any kind of rote
89 memorization approac h. My mind was on expanding quantities by adding "more" blocks and larger quantities just what multiplication is all about! [Wk11BT] In contrast, when Heide computed those same products using the traditional algorithm, [Wk11BT]. A Meter of Candy forum was effusive. First, she answered all of the questions in the prompt by identifying features that support the interconnectedness of types of rational num bers, the three models presented for rational numbers, and a typical assessment for this topic (Appendix D) The remainder of her lengthy post was a description of the lesson that she implemented over the course of four days. She indicated that her small explained: They continually went back to the materials to prove their thinking and often recognized errors in reasoning and shifted gears wi th the help of their partners . MUCH higher level of thinking than traditionally found within a math lesson on paper! [Wk15MC] Mathematics Teaching and Learning During the first two weeks of the oTPD program, Heide had multiple opportunities to share h er views on mathematics teaching and learning. Heide not only integrated Prime Online PD program activities with her instruction, she also recognized the impact of the oTPD program content on her teaching practices and her views on how students learn. Wh en presenting the Broken Calculator (NCTM, 2006) activity, Heide had to use
90 the I Do, We Do, You Do lesson format. During the You Do part of the lesson, she I had dem Earlier in the oTPD program, Heide reported that although she used to ask her student s how they got their answers, the Prime Online PD program made her see the need to explicitly cooperative learning allowed her time to reinforce skills with students who needed additional support. Activ ities learned in the Prime Online PD program also proved to save time. She reported that the concepts covered in A Meter of Candy (e.g., connections between fractions, decimals, and percentages; (NCTM, n.d.) would take 16]. A Meter of Candy (NCTM, n.d.) provided her with the opportunity to step back and allow students to take more ownership of their learning [Wk15MC] incorporating the use of the strategy notebooks and mathematics journals that were presented in the Prime Online PD program Her students, particularly those wi th intensive learning needs, were overwhelmed when a new strategy was presented every day, which was the ca se with the curriculum pacing guide. Heide wrote that strategy notebooks, complete with a table of contents, would be particularly beneficial for st
91 in the Prime Online PD program life and integrated problem order thinking [Wk2F1]. the Prime Online PD program provide insight into her and her own mathematical profic iency. She spoke of feeling proficient in elementary school because she memorized facts and could remained static in middle school, Heide recalled equating proficiency with students who know the answer, know where to look for the answer, or know how to solve for the mathematics and their willingness to persevere (i.e., productive disposition; Kilpatrick et al., 2001). In a Week 1 forum post, Heide lamented their problem referred to the Broken Calculator (NCTM, 2006) as a think By attempting to follow the county mandated curriculum pacing guide, Heide bolster ed her own confidence about mathematics teaching and learning. When stude of retention was due, in part, to following the curriculum pacing guide, which left little
92 more hands on activities to help students understand the concepts instead of having them rotely follow procedures [PIp16]. In fact, part of her excitement with A Meter of Candy ls to justify their thinking. Heide shared that she gains an immense amount of personal Heide recognized some other benefits of the new district man dated curriculum. sented in the new mathematics series appealed to her visual kinesthetic learning style [Wk1F1]. Heide mentioned several times (e.g., Wk1F1, PIp17) that her own lack of mathematical understanding restricted both her confidence and her ability to integrate mathematics to the extent that she would like. For example, she professed a willingness to use 16]. Similarly, she wrote that she now un derstood how to use manipulative materials in a manner relevant to a particular concept that was Parti cipation in the Prime Online PD Program
93 done so far, this requires a complete paradigm shift in thinkin g towards teaching mathematics. [PIp4] Teamwork and Optimism Throughout the Prime Online PD program Heide made comments about being part of a learning community and her hopefulness about actualizing he r goals for professional growth For example, Heid e made explicit mention of the importance of sharing her newfound knowledge with others. When describing how she routinely the Prime Online PD program when she principal, CRT, intern, ESE [special education] support staff, and my own children this culty with the Broken Calculator (NCTM, 2006) applet, Heide took it upon herself to search the Internet for other versions and posted the information to help her peers that were also having difficulty She found it challenging to change her teaching method 16]. Heide continued to voice apprehension that her mathematics ability would hinder her holdi ng to develop my the time needed to implement her newfound teaching strategies w ith the individual support required by all of her students. Two weeks later she seemed even more strategies effectively [Wk11BT]. As with her initial difficulty in e xplaining her problem
94 solving strategies, Heide surmised that her comfort level would increase once these experiences became a routine part of her teaching practices Although Heide made frequent remarks about the negative impact her mathematics history ha d on her teaching competency, she also expressed high hopes that the Prime Online PD program would facilitate her professional growth. She wrote: I am intrigued, inspired, and challenged with my new knowledge of the five strands necessary for math profici ency. I am highly motivated to explore and implement activities and opportunities that will expand student thinking and ability, and deviate from traditional methods. I am hopeful. [Wk1RA] l build my and reflect on her teaching in order to drive her planning and instruction [Wk2F1]. She was also optimistic that the Prime Online PD program tand up to any Participation and Satisfaction the Prime Online PD program was exemplary. Her overall participation (i.e., the number of submissions divid ed by the number of required submissions) percentage indicated that she submitted 73% more than what was required and only one percent of the submissions were late or not completed. During the mathematics specific weeks, her participation was even higher. She participated overall requirements met (i.e., if she met the minimum requirements) percentage was
95 99%, which was the same as the requirements met percentage for the math ematics specific weeks. Analysis of the segment satisfaction surveys showed that Heide was more satisfied than the average participant for Segment One (3.62, M=3.24) and Segment Two (3.91, M=3.29). She gave the highest rating possible for the content, ped agogy, technology and support, and the overall satisfaction for Segment Three (4.00, M=3.40). During Segment One, Heide was presented with a chart showing that she frequently exceeded the number of required posts. She replied that she did not pay attentio n to how many responses were required but then followed up that statement with lengthy forum responses throughout Segment Two, including a 471 word post for A Meter of Candy ; the next most lengthy post by other participants was 183 words long. Heide participated in all but three of the module surveys and each included responses to the open [emphasis in original] not meaningful and/or possible when so few people participate. I'd rather just get feedback from facilitators, rather than try to post just to fulfill the to her, wheth er it was directed to her or directed to one of her peers [I 2p13]. Heide seemed to view her level of participation as disproportionate. She said that she was s 2p13]. Supports and Hindrances to Participation Unlike the online courses she took as part of her graduate program, Heide found the forum posts to be valuable. In previous courses, Heide just posted to fulfill course
96 requirements During the Prime Online PD program 2p14]. She reported continually seeking out anything written by one of her peers, because she inferred that he held himself to the same high standard as she did. Heide also me ntioned that the positive feedback or encouragement from oTPD 2p13]. Heide had the highest satisfaction level s and the highest levels of participation of the cohort but noted impediments to her complete engagement. In response to a survey question regarding possible hindrances to participation, Heide indicated that the end of the s chool year is a difficult time o f the year to get everything done was a the content has become more challenging and I do not like participating in the online discussions condu cted interview, Heide stated that she did not have much discussion with her colleagues and that she questioned the review the entirety of her posts thus far (i.e., January to June), however, she noticed were posting in a previous week, she did not always go back and read those posts. Other issues specific to online learning also hin interact with the oTPD program content. Even though the Course Announcements a summary posted by one of the oTPD program facilitators at the conclusion of each week, were sent directly to her email, Heide did not read them. She said that she automatically deleted them because she thought they were merely duplicates of her
97 assignment, Heide explained that it was more difficult to work together online and gave hard. If I was maybe younger . 2]. Unlike others in the Prime Online PD program, Heide was not part of a school dyad She said that she missed having a school level PD, where the entire school or a [PIp14]. Heide remarked that she missed interacting in a classroom setting or having Prime Online PD program Heide mentioned that there were times when she had wanted a face to face exper ience so that they could have done mathematics activities as a group. She 2p15]. Heide expressed feeling a lack of connection with her peers and dismayed at the . it would be Prime Online PD program cohort which was problematic when some of her peers completed their forum posts late or did not seem to be as invested in the PD [MS14
98 14 16]. She also had a difficult 2p12]. This tempered the content of her postings because loves 2p12] This is not to say that Heide did not also struggle with the mathematics activities. She logged on in either late evening or early morning because it was the only time that for more ha nds on activities, and noted several times during the first four mathematics specific weeks that jargon was impeding her learning [MS9 2p9]. She spoke of her appreciation of these downtimes and of the oTPD program structure when contrast ing the Prime Online PD prog ram with attending a conference. She would 2p10]. Present Day Teacher Mathematical Identity Recent Teaching Experiences One yea r post Prime Online PD program Heide was moved to third grade, [I1 p2]. She explained that she was required to use certain worksheet based
99 mathematics booklets and t 1p3]. She expressed a feeling that she and her fellow teachers were being the sense of I just feel like 1p3 4]. She was frustrated with the change in her teaching assignment: gains in fourth were fine and I felt like people who know me know how I teach and so it was j where I was and told to teach with worksheets. [I 1p4] her science teaching certification [I 1p4]. However, she was given the opportunity to teach gifted science the following year and decided to remain at her current school. Mathematics Teaching and Learning During our interviews, Heide was asked to reflect on her teaching efficacy. When questioned about what makes her feel least effective as a teacher, she did not respond Prime Online PD program teaching practices. She said that during the first semester of the Prime Online PD program . mathematics the same way that she was taught mathematics [I 1p8]. For example, after presenting a lesson, she would ask students if t hey had any questions before they students about how she had just taught the material and they had not even bothered to read the problem before asking her a question [I 1p8 ]. She would remark to other
100 1p8] In contrast to her pre Prime Online PD program self, Heide often reminded her 1p12]. For instance, she tutored three low achieving students over t he summer and helped them create [I it meant that the students had taken ownership of their learning [I 2p2]. breadth and depth of classroom discussion s. She reported that there was . a lot more talking in math, where before I had this mentality that they is huge. [I 1p7] She explained that concept and, much more quickly, the kids have manipulatives . and discuss, discuss, 1p7]. Then the whole class would share how they solved the problem and considered the reas onableness of their solutions Teacher Role and Student Confidence Post Prime Online PD program, Heide stated that, as a teacher of mathematics,
101 1p11]. She thought her 1p8] Heide asserted that she asked students to justify their statement then became the spark for small group or whole class discussions [I 1p8]. She introduced 1p8] a variety of manipulative materials and explore various strategies inst this 2p4]. secure, timid kids . being afraid to offer anything . enough work with it that they could discuss it and then kids were learning from kids. 1p5]. Heide admitted tha t t here were still students who became frustrated but those students were now quicker to ask their peers than rely on her. In comparison to her earlier statement regarding her pre Prime Online PD program teaching,
102 there ar e no hands that go up. None. Because . if they read it and lmed. It should have been videotaped. Had I only known. [I 1p8 9] Professional Growth and Confidence discussions that impacted her as a teacher of mathematics. She said that when listenin how he got his answer [I me . because, by listening to her students, she said that she learned how other people think about mathematics [I ever happened if kids were not allowed to talk or were encouraged to be thinking about ways 1p13] Heide expected the changes brought about by her Prime Online PD program experience (e.g., increased MCK, awarenes s of how students think and learn) to play an important part in her forthcoming science only teaching position. She plans to 1p16]. She will have two classrooms, one of whi ch is a lab, and her mathematics manipulative
103 materials will be stored in the main building. She intends to bring over what she needs because if a concept (e.g., differences in temperature or weights) comes up that the students do not understand, she was on [I confident in talking and teaching and discussing and setting up mathematical related 2p7]. In fact, when considering that her current teaching assignment was a math only gifted teaching position in prior years, she did not think participated in the oTPD program [I 2p7 ]. Prime Online PD program self was better able to understand her colleagues She questioned why other teachers were not implementing the modeling considered herself among this group of frustrated teachers who thought teaching Prime Online I 1p7]. Heide expressed empathy: I feel r eally bad now when I look around at teachers who and I see it every year who feel so constrained. And math, because I actually love teaching math now, math is an area where traditionally teachers . might . . you open the book and you do what kids are not proficient in math. [I 1p14]
104 Shifts in Teacher Mathematical Identity Before, it was more teacher talk . so the understanding was superficial. drawing pictures. I became much more confident in utilizing the thinking of the kids in the classroom and having them supporting each other appropriately. [I 1p8 9] MCK and MKT o doing whatever she could t o make sure her students learn [I 1p7]. First, she had to get past her own lack of content knowledge in mathematics. Heide admitted on the module surveys that mathematical jargon stunted her progress in Weeks 9 12 (i.e., the m athematics specific weeks). However, her increase in MCK was evident when she compared multiplying multi digit numbers with base ten blocks versus the steps in the traditional algorithm. She said that, because of the knowledge she 1p8]. That was not true of her pre Prime Online PD program self. Mathematics was nderstand the concepts [I 1p7]. Heide frequently mentioned how she was able to integrate all taught mathematics [I 2p3]. Early in the oTPD program Heide stated that the integration of mathematics was one of her goals. Once Heide had a clearer understanding of how to
105 teach mathematics, she incorporated the concepts throughout the day and also int specific curriculum. Role of Assessments Heide expressed more confidence about teaching mathematics before the Prime Online PD program even concluded. During the PI conducted interview, two months into the oTPD program, Heide described her best lesson as being inspired by the Teaching Children Mathematics journal articles from the second week of the program. She said that she had walked into her class on an assessment day and told the students that manipulatives, number lines, graph paper, etc. as necessary and if you're stumped, discuss your thinking with a classmate. Adults will only provide a definition for a word/phrase and/or re ad the problem to you. Otherwise, you're on your own" [PIp3]. In addition to shifting the mathematical thinking away from her and onto the students and creating a more collaborative learning environment, this teaching practice also provided Heide with t watch their problem [Wk2F1]. She the n used this information to help decide how to organize students for future small groups and paired mathematical activities. In Week 9, Heide was she found that she was ab
106 Teaching with Manipulative Materials to more effectively teach m athematics using manipulative materials. During Week 1 she wrote about a lesson in which fraction bars were used to teach fraction equivalency. 2p5]. Until the Prime Online PD program, she had been unable to teach mathematics with the engaging, hands on approach to learning that she favored. Heide explained that she had always liked, and incorporated, manipulative ability to use them as effectively (and often) as 16]. Midway through the oTPD program, Heide learned about teaching addition and subtraction of fractions conceptually prior to procedurally. She s making it impossible for me to teach like I 16]. She now introduces mathematical topics using manipulative materials t understanding. Summary I had a paradigm shift of the highest magnitude [I 1p14]. Heide seemed to truly enjoy talking about her teaching and her experiences with the Prime Online PD program. She was relaxed and laughing durin g both of our
107 interviews. Her comments throughout the oTPD program and interviews, as well as the physical layout of her classroom, support her assertion that she had an integrated, hands on teaching style. She had experience in outdoor education, specia l education, general education and, now, gifted education. She had considered leaving her school of 11 years, however, because of constraints placed on her by school based administration. At the beginning of the Prime Online PD program Heide wrote that s he hoped to learn enough to be able to teach how she knew was best for her students and stand up to any peer pressure regarding the c urriculum pacing guide. During the interviews two years post oTPD program, Heide said that she could tell that some of her Prime Online PD program peers followed the curriculum pacing guide. Heide, however, said that she bout necessarily 1p14]. Heide went from being concerned about being pressured into following the curricul um pacing guide to not caring if she lost her job if sh e did not follow the curriculum pacing guide. Heide professed that i t was only due to the Prime Online PD program that she was able to teach mathematics the way she had always taught the other content areas. Prime Online PD program experience to a combination of her age and her experiences with students with exceptionalities [I 1p15]. She said that . that . something. I knew that there was more to it, that there were other ways that you could show them or teach 2p6]. She explained that, as a special education teacher,
108 1p15]. When asked about personal factors that may have contributed to what she took away from the PD, Heide credited her motivatio 2p6]. She never have a pr 2p10].
109 CHAPTER 5 BRYNN: TRADITIONAL INSTUCTI ON, SPORAD IC PARTICIPATION Prime Online PD P rogram Teacher Mathematical Identity always been a fraid of is that [not being a strong math student] would kind opportunity to do anything that improves my math teaching skills and even my math skills [PIp1] difference between her experiences in elementary school and her experiences in subsequent years [Wk2F1]. In third [Wk2AA ]. Brynn credited her parents for her early success in mathematics. They had her practice counting money at the grocery store and required her to make flash cards to help her memorize basic facts. Brynn recalled only being taught traditional algorithms. She was not taught that there were multiple ways to solve multi digit multiplication or long division problem and surmised that this was because she showed proficiency with the traditional algorithms. Brynn reached he seventh grad e pre algebra [Wk1F1]. After that, mathematics became her most difficult subject and she was also concerned about any science classes that involved mathematics. Her negative experiences in secondary and post secondary education quelled her desire to exte
110 Self Efficacy Brynn said that, regardless of their abilities as learners of mathematics, teachers have an obligation to overcome any challenges for the sake of their students. She reported that she was con fident in her ability to teach mathematics during her first year. have the same negative experiences as learners of mathematics that she had. She mathematical dispositions. Soon after beginning the Prime Online PD program she elping my students move from being put off by math toward Even with her concerns as a learner and teacher of mathematics, Brynn about the Principles for School M athematics (i.e., equity, curriculum, teaching, learning, assessment, and technolo gy) (NCTM, 2000), Brynn remarked on the amount of PD that would be needed to develop her instructional practices to meet those levels. She wrote about the effort that she was going to have to put forth to glean what she wanted from the oTPD program conten t, especially considering its asynchronous format. In the Week 2 Reflection and Assessment Brynn expressed concern that the lack of true Wk2RA].
111 teacher during whole group instruction on fraction equivalence. The lesson was organized in the following mann er: vocabulary review, teacher demonstration, independent practice, remediation or enrichment as needed, and a one question assessment. She purported that this teaching method (i.e., gradual release) created an environment that supported her lingness to learn mathematics. She remembered, however, that she and the general education teacher had doubts even while teaching the lesson. They get the students to a 2p7]. During the PI conducted interview, two months into the Prime Online PD program Brynn was asked if there had been any changes in he r thinking about teaching mathematics. She responded by describing a district mandated calendar lesson that she had done that day. Instead of following the lesson plan she was given, she used 2p2]. She adap ted the calendar lessons to incorporate skills that her students had not yet mastered. She recalled details of the lesson: smaller our quotient gets? As the divisor gets bigger the quot ient gets
112 had them pair up with base ten blocks to kind of If you want to do some number gets But a small part of our day but that wa s very much a teachable moment and before . . Prime Online MKT During Prime Online PD P rogram Activities Brynn scored above the mean in CKT M Test 1 and 2 (Table 4 2). She could not recall why she did not comp lete Test 3 (i.e., posttest). Contrary to her initial concerns When she needed clarification, she asked for it. For example, s he had a revelation d uring the Week 2 Content and Discussion. Brynn did not know if she understood the article on mathematical proficiency because her response was probed by one of the oTPD program facilitators. Even though she thought his question might have been rhetorical What do you think? little mo
113 Table 4 2 M scores Test 1 Percent Correct Test 2 Percent Correct Test 3 Percent Correct CK KS CK KS CK KS 83.3 80.0 73.1 73.7 no response no response Mean 71.4 75. 0 72.1 72.9 65.8 73.5 Brynn also persevered while working through the Broken Calculator (NCTM, 2006) activity in Week 9 (Appendix D) hout fully functioning keys [WK9BC]. She struggled to find a strategy for solving the problems and likened the Broken Calculator (NCTM, 2006) activity not let that discourage her and she even asked others to find out if the m ethod or pattern if there even was one should be taught to the students or if the students should figure it out themselves. She wanted to know which method would be better for a difficult activity, Brynn agreed with a peer who suggested that the Broken Calculator (NCTM, 2006) activity Brynn had less difficulty using base ten blocks for multiplication of larger numbe the manipulative materials to her understanding of place value [Wk11BT]. After completing the activity, Brynn stated that she still preferred the traditional algorithm. She then admitted that she probably favored the traditional algorithm because of her own history as a learner of mathematics. She cautioned that relying on strategies (i.e., the use of manipulative materials) other than the traditional algorithm would not be practical in most situations such as when multiplying two very large numbers. the Prime Online PD program
114 activities was limited because she Broken Calculator (NCTM, 2006) activity, participate in the Week 15 A Meter of Candy (NCTM, n.d.) activity, or provide feedback in Segment Two module surveys (e.g., MS9 12, MS14 16) Mathematics Teaching and Learning During the first two weeks of the oTPD progra m, Brynn had multiple opportunities to share her views on mathematics teaching and learning. Helping students become independent, functioning adults was an important part of how Brynn saw her role as a teacher of mathematics. Brynn wanted her students to make a challenging concept accessible to all of her students [Wk1RA]. Sometimes this meant teaching students tricks or quick strategies. In our second interview, Brynn was asked to clarify the difference between the terms tricks and quick strategies and to give an example of each. She was unable to recollect exactly what she meant by those terms and determined that they were likely synonym ous. She was also unable to give an example of a quick strategy with certainty. Brynn supposed that, given a problem such as 1/2 multiplied by 2/6, a quick strategy would be to simplify 2/6 to 1/3 prior to calculating the product instead of simplifying th e product after multiplying 1/2 and 2/6. During Week 2 Brynn joked that, without being able to reason and come up with a that if students reached the intermediate grades without conceptual understanding and procedural fluency they would need to rely heavily on quick strategies in order to be successful in mathematics.
115 Brynn had very clear ideas about what was important for students to learn to become mathematically proficient. Not only did she emphasize strategies that she thought would ensure success she was also a self mastery. She saw quick strategies as a way for non proficient st udents to have some measure of success in mathematics. When questioned about her stance on using tricks, Brynn wrote: I see the benefit of knowing ways to make math bearable and enjoyable for a child who isn't confident in his computation ability. I find that my students who are stronger students that prefer the traditional methods of computation aren't as interested in the tricks [quick strategies]. They think it holds them up when they could be moving on. The kids who struggle look for an alternative method. [Wk2AA] Additional comments made by Brynn during the Week 2 Anticipatory Activity showed how her thoughts on mathematical proficiency aligned with her thoughts about world applicati on. For Furthermore, she believed : being proficient in math means being able to naturally apply learned operations to various unrelated situations. Instead of only thinking about division when seeing a division symbol, one should recognize situations
116 that requi re division and apply knowledge from another area of math to answer a real life question. [Wk2AA] Part of mathematical literacy is application of concepts to real world contexts, which was a recurring theme for Brynn throughout the oTPD program. She exp lained that she was most effective as a teacher of mathematics when she taught a student to apply a newly developed skill to a real world context. Brynn saw how excited her students became when they came to the realization that ta seeking out new ideas and activities from the other teachers at her school to try out in Because Brynn valued student interactions, she was not averse to relinquishing her role as teacher when the opportunity arose. Because of the state specific standards aders were taught how to classify two dimensional figures when they were in third grade. According to the newly adopted CCSS M ( CCSSO 2010), classifying two dim ensional figures is now a fifth grade standard. Therefore, her fourth graders were, at times, able to teach mathematical concepts to her fifth graders. Brynn reported feeling that student student learning would reinforce the fourth that, most importantly, she enj Brynn explained that her desire for fun and interactive lessons sometimes conflicted with county curriculum mandates. She wanted
117 without fear of falling too far behind according to the pacing guide and being weeks into the oTPD program, however, Brynn began to integrate her learning into her Prime Online Participation in the Prime Online PD Program I also like rea ding a lot of what people are talking about within the forums, like the Content and Discussion, especially when it . makes people kind [PIp3] Applicability and Openness to Chan ge lives and that she expected PD to be relevant to her teaching. While she enjoyed the readings from practitioner journals, she sometimes found the research articles difficult to understand. She preferred articles that provided concrete, practical ways to teach giv ing [you] something you can use Brynn was receptive to trying new ideas an d expanding her views about mathematics teaching. As part of the Week 2 activities, participants were asked to read an article in Teaching Children Mathematics Suh, 2000, p. 163). Brynn was strategies from the article that she planned to impl ement in her classroom: Math Curse
11 8 life mathematics problems to class, Math Happenings involved the teacher posing her own real life mathematics problems to the class (e.g., painting a nursery), and Convinc e Me guided students through the writing process of justifying their answers teaching practices seemed to be a natural consequence of her goal to make learning mathematics a positive experience for her students. Sh e was particularly interested in strategies that wou ld help her struggling learners such as strategy notebooks. These d I tried to do repeated subtraction, but Brynn was similarly open to modifying how students are assessed. She applauded the collaborative assessment implemented by a Prime Online PD program colleague and related it to her experiences as an adult learner: It reminds me so much of MANY of my college courses in which teachers allowed us to collaborate with a classmate during assessment using whatever resources we had collected over the s emester to enhance our understanding of a concept. It's interesting that in elementary school we're encouraged to have students collaborate ALL THE TIME through things like Kagan Structures but when testing day comes, they're on their own. I know that as an adult in college and as a professional, there are very few times when I can't say "I'll check a source and get back to you
119 Participation and Satisfaction the Prime Online PD program could be characteriz ed as inconsistent. Her overall participation (i.e., the number of submissions divided by the number of required submissions) percentage indicated that she submitted 41% more than what was required and 26% of the submissions were late or not completed. S he fared a bit better when the mathematics specific weeks were considered separately (i.e., 146% and 21%). requirements met (i.e., if she met the minimum requirements) percentage was 93%, although 24% were late or not submitted. In this i nstance, the requirements met for the mathematics specific weeks were lower than the overall requirements met (i.e., 87% and 26%). Analysis of the segment satisfaction surveys showed that Brynn was less satisfied than the average participant for Segments One (3.15, M =3.24) and Two (3.20 M =3.29). After Segment Three of the Prime Online PD program Brynn was still less satisfied than the average participant (3.35, M =3.40). respons e were required for each of the three forums considered for this study. Brynn completed the requirements for the Week 9 Broken Calculator (NCTM, 2006) activity, supplied a post but no response for the Week 11 Working with Base Ten Blocks activity, and did not participate at all in the Week 15 A Meter of Candy (NCTM, n.d.) activity (Appendix D). She did participate, however, in the Anticipatory Activity and Forum 1 for Week 15. When asked about this lack of participation, she concurred that it was almost certainly due to Week 15 coinciding with the last week of the school year. Brynn had many opportunities to give feedback to the oTPD program facilitators. When given a
120 an d assignments in a timely manner, Brynn did not respond. Brynn provided open ended feedback on Weeks 1 and 2 module surveys, but did not provide any feedback during her first year as a general education classroom teacher (PIp4). Supports and Hindrances to Participation co nducted interviews, teachers were asked to reflect on their participation levels. When Brynn was shown her statistics (i.e., the number of times posts and assignment requirements were met and completed on time) from the first six modules, she was quick to treated like any other obligation outside of the school day [PIp4 ] Submitting discussion posts was the most challenging part of the oTPD program for Brynn, particularly if she had missed a due date. At the beginning of the oTPD (e.g., Sunday, Wednesday, Friday, and Monday). By Week 4, all assignments were due at the end of each week to give participants more flexibility. Before this change, Brynn would become frust rated, saying, ctivity on Sunday was more difficult in the first few weeks because the due dates kept changing and she
121 interact with others. She explained that if she posted late in the week, that person might not go back and check the discus sion forum. For example, if Brynn was the last person to post and she posed a question to a peer or tried to add to the discussion, that person might not log back on until the beginning of the next week and, even then, might not go back and check the prev then she might be the one who does not go back and check the discussion forum. For example, if Brynn was able to complete her Anticipatory Activity, Content and Discussion, and Reflection an d Assessment by Thursday, she might not log on until when she did log on. A lack of interaction with her Prime Online PD program cohort was mentioned more frequently than any issues regarding scheduling or the timing of discussion posts. connected to the other participants and to have an idea of what went on in their classrooms. During Week 5, participants were asked to provide one example of how the components of self regulated learning could be enacted in their classrooms. Brynn one of my favorites to read, by far, because again this is letting me . [PIp9]. She repeatedly stated how much she enjoyed articles and forum discuss ions that gave her ideas and resources that could be used in her classroom right away
122 (i.e., Theories of Learning and Teaching Practices: Explicit Strategy Instruction) the part icipants were assigned to one of two groups. Each group of four was to work together and respond to three questions after reading a research based article on instructional strategies for students with LD. When faced with this group assignment, Brynn said that she was unsure of how to approach it without being able to meet or call her peers. She said that she did not mind working in groups when in face to face to figure o person doing part of the assignment and then putting it together at the end [I 2p13]. head exchanging in a that she said applied to any type of online learning : says And you can were just talking. [I 2p9] Her concerns about interaction with the facilitators were also specific to the unaware if their posts were being read by the facilitators [I 2p12]. Although she said that she was uncertain if her comments were being read, she clarified tha t this uncertainty applied to any online course, not just the Prime Online PD program As with
123 any asynchronous oTPD, it is unlikely that a participant would receive an immediate response to a posted comment or question. Brynn remembered thinking, as she responded to one of the mathematics prompts, 2p11]. She said that sh e would also never know if a question that she posed in her post was going to be answered, whether by the facilitators or her peers ead, Brynn reported that when a facilitator commented 2p12]. She referred to one instance of facilitator feedback that prompted her to think more deeply about her initial response. She said that this int 2p12]. Although Brynn appreciated individualized attention from the facilitators, she acknowledged that she did not think there was a direct relationship between her own participation and th Present Day Teacher Mathematical Identity Mathematics History: Closing Doors Brynn expanded on her history as a learner of mathematics to include regrets and the long reaching consequences of her negativ e experiences. Once again she pinpointed her downfall as being in her eighth 1p13]. Because of her academic difficulties, Brynn chose to take Algebra I in high school instead of skipping to Algebra I
124 [I 1p13]. Once she got to college she saw many interesting career options, but realized that she did not have the mathematical options by not taking more mathematic s courses [I 1p13]. The resulting doors closed on potenti al career choices fueled her to take my own experiences, and keep in mind my own shortcomings in have to feel like I felt w 1p13] Mathematics Teaching and Learning Dur ing our first interview, Brynn h arkened back to her own mathematics history to explain her feeling of obligation to help her students understand the value of mathematics for th 1p9]. Brynn suggested that mathematical profi 1p10]. She gave examples of people who unwittingly made bogus investments and prof essional athletes whose money was squandered because of mismanagement. Brynn stated a sincere hope that all mathematics teachers can impart on all of their students the importance of knowing enough mathematics to reach their future goals, especially as they are not yet sure of where life may take them. She said that she tried to p rovide a deeper connection and support proficiency by integrating mathematics into other subject areas. Like many other teachers, Brynn realized the value of using real world contexts for mathematics problems. She looked for teachable moments throughout
125 [I 1p11]. While she admitted that it was easiest to discuss the relevance of mathematics when teaching science, Brynn also indicted a desire to incorporate mathematics related literature and set up a c 1p11]. Brynn recalled reading about the Math Happenings (Suh, 2007) strategy : students actually have to search in their own day to day lives, even at home, an think the more aware they are of it the more likely they are to kind of just se I remember doing this at 1p11] During our second interview, Brynn brought up her continued struggle to balance the mandated county curriculum. She credit ed the Prime Online PD program standard at this date, during this time of your 2p2]. She explained that the reality was that students were not progressin g simply because the curriculum pacing guide said that they should Unlike her pre Prime Online PD program self, Brynn responded to this dilemma by teaching fo 2p2]. When asked
126 numbers; introducing it and being able to use base ten 1p3]. Brynn reported that it was difficult to use manipulative materials with the instructional materials (i.e., county adopted textbooks) they were given because each day was a new lesson and a ne [I 1p4] She took out base ten blocks and dry erase boards and had the students pu t their books on the floor. Brynn launched into a discussion with her students about a topic she knew they would understand doing laundry. They talked about dividing the clothes by color and then moved on to the number of loads that can be done if you ha ve x pieces of clothing and the washer holds y pieces of clothing at one time. She modeled problems with base ten blocks on the SMART Board discussion about the various ways to solve each problem. Brynn asserted that the lesson pro 1p4] She strug gled with knowing the value of having a deeper understanding of grade level concepts and the need for time to remediate her st udents. Brynn remarked that she was most ineffective as a teacher of mathematics when many of her students come in each year not knowing their multiplication tables. The first four chapters of the textbook were multiplication and division, but she had to spend weeks practicing basic facts. Brynn indicated that this left her frustrated because she knew where she was supposed to be in the county pacing guide. In addition, as a student who learned her multiplication tables at home, she had difficulty empat hizing with these students. She
127 mathematics history to that of her students [I 1p8]. Being a gifted student, Brynn did not some patience and . classroom [I 1p8]. group instruction. Although she woul d have liked small groups to occur more frequently, it was during this time that Brynn considered herself to be most effective as a teacher of them on a concept all, kind o 1p7]. She used small group instruction to make sure the students not only knew the algorithm, but also had a real 1p7]. This was typically achieved by using manipulative materials. When working in whole group activities, her students knew how to work with t he student sitting next to them and share the set of manipulative materials they were given. There were not enough manipulative materials for each studen t, but Brynn said that putting students in pair s provided more opportunities to interact and participate than their typical groups of four When describing a particularly effective small group activity, Brynn remarked that base ten 1p7]. The mini lesson was modeled after an assignment she did in the Prime Online PD program She wanted to support her 1p7]. Because actual base ten blocks were not on hand, Brynn used drawings of base ten
128 blocks to show a stude nt how to regroup. The student crossed out a ten (i.e., a rod or long), drew out the units and then crossed out the units that were being subtracted. because base ten blocks were used often in class [I 1p7]. In summarizing this story, 1p7]. Professional Growth and Opportunities Participation in the Prime Online PD program growth in several ways. She was more open to oTPD, mathematics PD, and PD in re ubiquitous 1p14]. When Brynn sees an oTPD 1p14]. She made simil ar comments about mathematics PD: reading because I am just constantly trying to figure out how can I be bett er at this so it can better benefit them. [I 1p14] 1p14]. It was this type of thinking that made her take notice of the Prime Online PD program Brynn stated that the most important factors in choosing a particular PD program are if it had good, quality
129 literatur e and if the P D program provided activities that she could 1p15]. Since the conclusion of the Prime Online PD program Brynn became more familiar with the mathematics content specific to her grade level. She was asked by her principal to represent their school and help map out the county pacing guide for the newly adopted mathematics instructional materials. This involved merging the previous mathematics series with the current mathematics series because the new textboo k series only co vers CCSS M ( CCSSO 2010) and the students would still be tested on not aligned with the CCSS M ( CCSSO 2010) be where he or she was expected to be when the pacing 1p16]. Brynn expressed that she often left her room county mandates. She sai d that it saddens her but she had to persevere because 1p15]. She indicated that she believed she could reach all students if she could just keep them a bit longer each day. She wanted to be able to [I 1p15] Shifts in Teacher Mathematical Identity The thing about professional development and I know I would probably get more out of it if I was posting earlier in the week more consistently is that . going to get out of it whatever you put into it, just like everything. [PIp14]
130 Self E fficacy confidence in her teaching abilities, she ignored the textbook and taught the content using manipulative materials and discourse. For example, Brynn mentioned that it took two weeks for her students to get comfortable with the area model. She did not move on to the next lesson to keep up with the curriculum pacing guide because she wanted her students understand the concept She explained the challenges she faced and how the Prime Online PD program impacted her instructional practices. As a novice teacher . . . job and that I teach everything through the year. You feel like that for a lot of years, but I feel like Prime [ Online ] really just kind of brought in my education about what it means to truly teach math and truly help a kid understand math. And I loved Prime [ Online it . . It helped me understand. [I 2p3] Each of her best lessons reported during Week 1 and t hen two years post OTPD program, included remediation and enrichment, the use of manipulative materials, and followed the same gradual release approach (i.e., I do, We do You do ). The more recent lesson, however, included deviation from the prescribed le understanding. Instead of following the textbook lesson on division of whole numbers, she introduced the concept by having the students discuss how their mothers sorted loads of laundry. Brynn had the students mod el the various examples using base ten
131 unit cubes for each piece of clothing. Next, she helped students understand how the words they were using could be translated into an equation and then solved using the manipulative materials She explained that her perception of herself as a teacher of mathematics had changed because, in addition to wanting her students to see the importance of mathematics, she now wanted them to recognize the value of understanding mathematics. Making mathematics relevant to her s tudents was a recurring theme with Brynn. During the Prime Online PD program she stated that she was most effective when she was able to teach students to apply a skill to a real world context. However, she now felt most effective when using manipulativ e materials (e.g., base ten blocks) during small group instruction Manipulative Materials Prime Online PD program attitude toward manipulative materials had changed since her negative statement in Week 2. During our second interview s he ack that 2p4]. Her notion of what constitutes manipulative materials had expanded. She used to think of manipulative materials as She explained . . cents are the same thing as single unit i Prime Online really opened my eyes to how we can represent math . . as a teacher I want to . put it, like I said before, in a 2p4]. Instead of an impractical m eans of computing numbers in real life, Brynn now 2p5]. Her more recent
132 best lesson involved using base ten blocks to do subtraction with regrouping. Since the Prime Online PD program, s he has expanded her instructional repertoire to include ten frames, a manipulative material that she was introduced to while helping a second r students [I 1p7]. Prior to the Prime Online PD program Brynn taught with some manipulative materials, but the tradi tional algorithm was the focus of her teaching. Her post Prime Online PD program self saw manipulative materials as a precurso r to, instead in competition with the traditional algorithm. Teaching for Conceptual Understanding In addition to using manipula tive materials, Brynn supported her students conceptual understanding by using multiple strategies and requiring multiple representations. Her knowledge about and implementation of multiple strategy use began in Segment One. When speaking of her pre Prim e Online PD program self, learners [PIp12]. Two months into the oTPD program, however, Brynn told a PI that [PIp12]. Post Prime Online PD program she and her students have classroom discussions about different strategies they used and which one might be the most practical. Brynn described an example in which her students were learning the area model for multiplication of large numbers. Some students said t hey would still rather use base ten blocks, so she had the students work through several more problems. The class unanimously determined that when it comes to multiplying 42 and 28 with base ten 2p6]
133 She had also learned strategies for teaching multiplication basic fact fluency. In addition to fluency, Brynn said that she expected her students to understand fact families and be able to represent problems with arrays and area mode ls. This was in contrast to her pre Prime Online PD program reliance on teaching with the traditional algorithm. Brynn was asked to reflect on her Week 1 stance on struggling students using tricks in order to have some success in mathematics. She recall ed that during the Prime Online PD program, there were many discussions about being able to work through a traditional algorithm without understanding why the traditional algorithm hm or quick why manipulative materials) she started, the more likely her students were to 2p9]. Summary I value my growth as an educator because I know that, as I grow, my students grow. [I 2p3] Brynn was open to sharing how the Prime Online PD program had changed her views on mathem atics teaching and learning. She took time to ponder her responses during our second interview, sometimes reading her forum posts multiple times so she gestures mirrored the passion in her words when speaking about how important it was for her students to have understanding and appreciation of mathematics Brynn enrolled in the Prime Online PD program to increase her own MCK. Although confident during her first year of teachi ng, she soon realized that her students
134 had difficulty comprehending mathematics concepts. At the beginning of the oTPD program, Brynn wrote that she doubted her ability to grasp the content to the degree that would be needed for classroom implementation. By mid oTPD program these concerns had abated and, during the interviews two years post Prime Online PD program there were no indications that she still doubted her ability to successfully teach mathematics. She conveyed that she appreciated how the oT PD program helped her understand how to use manipulative materials and, as corroborated by comments in the follow up interviews, gave her a deeper understanding of the role of manipulative materials. Brynn post Prime Online PD program self continued to actively seek me about discovering ten frames, she inquired whether I had additional resources that might be applicable stent and when she did participate, her responses were not as lengthy or in depth as most of her peers. She 2p9], which was exacerbated if she posted very early o r very late in the week. Even with her less than optimal participation levels, the oTPD program influenced her instructional practices from the beginning. One early indication was the adaptations she made to the county mandated Calendar Math activities t o better suit the needs of her students. More recentl y, the infusion of conceptually based activities showed her of her goals was for her students to understand and appreci ate the power of
135 detailed, practical ways to teach mathematics for understanding [Wk2RA]. At the end of the second interview, Brynn was asked if there was anything else she wanted to share. Prime [ Online 2p14]
136 CHAPTER 6 CONCLUSION The purpose of this dissertation was to examine the relationship between teacher mathematical identity and participation within an oTPD program. In Chapters 4 and 5, I presented narratives of Heide and Brynn to describe their teacher mathematical identity and participation in an oTPD program. In Chapter 6, I analyze these stories, individually and across the cases, as they related to the two resea rch questions that framed this study. This chapter also summarizes the findings and concludes with implications for practice and suggestions for future research. Research Question 1 What is the relationship between teacher mathematical identity and partic ipation in an oTPD program? The first research question focused on how teacher mathematical identity was identity during the Prime Online PD program was determined by ana lysis of archival data for indicators of views of mathematics (e.g., history as a learner of mathematics, perception of teacher role). Participation was determined by the participation and requirements met ratios from the participation chart, completion o f the surveys and CKT M, and the content of Segment One and Segment Two activities. Heide Since the very beginning of the Prime Online PD program, Heide was open about her negative history as a learner of mathematics. During Week 1, she recalled a pivotal grade mathematics teacher [Wk2AA]. Heide seemed to
137 origina even though she volumes on what it means complex relationship regarding her high levels of participation in the Prime Online PD program. Heide was likewise unguarded when reflecting on how her lack of background knowledge affected her mathematics instruction. During a Week 1 forum post about possible barriers to becoming an ideal teacher of mathematics, Heide stated the she had much to learn and th at mathematics was never her strength. She made many other statements that indicated her low self efficacy as a teacher of mathematics. However, Heide felt a strong sense of duty toward her students, which brought her to the Prime Online PD program. Hei de taught with student centered, integrated lessons in all subjects other than mathematics. She said that she was never confident in mathematics, but also hated teaching out of the textbook because her students were 2p2]. She reported that she very much wanted to make her mathematics instruction as integrated and hands on as the rest of her teaching. Once enrolled in the Prime Online PD program, Heide had difficulty with some of the mathematics activities in Segment Two, which was indicative of her lack of
138 [MF9 12]. This led her to reflect on how her students must have felt. Heide reported that she tried as hard as she could and still h ad a really difficult time understanding the concepts. She also remarked that the mathematical jargon was hindering her learning and that she appreciated being given extra time to be able to truly reflect on the concepts. Considering her difficulty with t he oTPD program mathematics content, it would Prime Online PD program teacher mathematical identity would have had a negative influence on her participation in the oTPD program. ., perfectionism) and values (e.g., sense of responsibility) seemed to fuel her to overcome these challenges. When asked, two years post Prime Online paradigm shift [I motivation, my perfectionism. I mean, I . and I have a responsibility to them [her 2p6]. Heide spoke freely about her lack of mathematics ability and feelings of inadequacy as a teacher of mathematics. Unexpectedly, these facets of her teacher mathematical identity became the impetus for, instead of barriers to, her participation. Heide had the highest participation rate (i.e., 82% more than required in the mathematics specific weeks) and had the lengthiest posts. She likened herself to a 2p13]. Additionally, Heide showed initiative and perseverance when she sought out an alternative resource when the Broken Calculator applet did not work properly. Her
139 perfectionism and sense of duty to her students combined to create an eagerness to learn and subsequent high levels of participation. Brynn hing and learning) conflicted with the Prime Online PD program content, which was likely related influenced her beliefs about mathematics learning. Brynn reported that s he felt like a genius in elementary school and was successful learning and using the traditional algorithms and mastering her multiplication basic facts. Because mathematics came easily to her in elementary school, Brynn reported having difficulty empathi zing with her students when they struggled to achieve mastery. Not surprisingly, given her history as fact fluency. During Week 9, she wrote that using manipulative mat erials to represent multi because that is how she learned multi digit multiplication. She shared that the more proficient students preferred the traditional alg orithm and found alternative strategies a waste of time and the less proficient students favored the alternative strategies. Brynn grappled with Prime Online PD program assignments that presented strategies other than the traditional algorithm. For examp le, in response to a peer in the Week 9 about where I feel as though I NEED paper and pencil to
140 9, she agreed with a peer that the Broken Calculator activity would be an appropriate activity for the brightest s tudents in class. needs of her students. In the Week 11 Reflection and Assessment, she shared that when multiplying two digit numbers by four digit numbers, her students were s uccessful tration when the kids could put their hands on fraction bars and play around with them to answer whiteboard [Wk15AA]. growth were also related to how she valued the content of the Prime Online PD program. She believed [IIp9], instead of as a means of increasing her MKT or encouraging her to think about her instruction differently. She explained that she preferred the readings from the practitioner journals because they provided her with lessons and strategies that could be put to use immediately. Brynn favored teacher centered instruction, felt that her role as a teacher was to provide activities, and the Prime Online PD program presented content and pedagogy she would have been more engaged when the Pr ime Online PD program content
141 provided such activities and less engaged when the activities were focused on her own mathematical thinking or not relevant to her current context (e.g., not appropriate for implementation in a fourth/fifth grade combination c lass). Brynn noted that she felt overwhelmed because this was her first year as a Although she provided open ended feedback on the first two module surveys, she did not complete any for the duration of the oTPD program. She also did not participate in the final CKT M assessment or meet the requirements in a timely manner for many of the assignments and discussion forums. It is unclear if the lack of participation was b ecause Brynn was a late person if she was overwhelmed with her first year as a classroom teacher while also being a student, or if certain activities were not completed merely because they coincided with especially busy times of the school year (e.g., A M eter of Candy during the last week of school), or if the assignments were given low priority because the PD program content did not align with her teacher mathematical identity. The Prime Online PD program also did not match her definition of what a PD pr ogram should be. Instead of being provided with activities for immediate use, she was asked to engage in an intensive and lengthy PD program that included a variety of mathematical tasks. Looking Across the Cases Heide and Brynn had similar histories as l earners of mathematics. Both participants recalled learning elementary level mathematics by traditional (i.e. teacher centered) instruction. More noteworthy, however, was that both participants pinpointed an incident in seventh grade as a turning point i n their mathematics histories, after
142 which mathematics was arduous and a source of stress and poor achievement. That they struggled in, and disliked, mathematics from seventh grade through their college years certainly impacted their understanding of math ematics and how much they were able to participate in the Prime Online PD program mathematics activities. Both participants had concerns about learning in an online PD program and were particularly concerned about learning mathematics in an online format. They stated having difficulty discussing mathematics problems and solutions without the face to face discussions and found group work to be especially problematic. The participants thought that their learning would have been enhanced by more, and more in depth, discussions. Heide missed classroom interactions and wanted to be com munication from the PD program facilitators. Brynn wanted more individual feedback from the facilitators; she thought this would let her know that someone was reading her posts. Heide preferred the feedback she received from the facilitators to the respo classmates [I 2p13]. Although both stated a desire for more communication, each of them also admitt posts and responses. This dislike of online discussions would likely have hindered the outweighed her negative feelings about online learning.
143 Heide and Brynn also had similarities regarding their views of mathematics teaching and learning. Both participants considered empowering their students to be part of their teacher role. Brynn wanted her studen ts to understand how mathematical knowledge could impact their adult lives, such as providing them with more career choices or preventing them from making poor business decisions. Heide wrote that en the opportunity to motivate participants mentioned the importance of relating mathematics instruction to real world contexts. Brynn specifically mentioned the value of the Math Curse, which encourages students to recognize mathematics in their own lives. The desire to inspire their Prime Online PD program activities that have real world application (e. g., mental mathematics, estimation of solutions for multiplication of rational numbers). Many of the Prime Online PD program activities asked Brynn to present and justify her own mathematical thinking, which conflicted with her perception of the purpose o f PD and likely affected her engagement. Heide and Brynn presented a few differences regarding the relationship between teacher mathematical identity and participation in an oTPD program. One difference awareness or openness about t heir need to change how they taught mathematics. Heide was much more critical of herself. For example, in the Week 1 Reflection and Assessment participants were asked to consider their response ng the Principles and Standards for School Mathematics (NCTM, 2000). As part of her post, Brynn noted that
144 the reading, Heide reflected on her teaching practices in the context of one particular lesson. She wrote that she had benefits from the type of instruction that the NCTM promotes were lost participation is the awareness of her inadequate preparation to teach mathematics. She believed that she could be an effective teacher of mathematics whereas Brynn saw herself as an average teacher of mathematics and was somewhat satisfied with her level of MKT. The disparity in the quality and quantity of posts was also related to subsequent teacher mathem atical identity was fueled by her self proclaimed perfectionism where as Prime Online PD program content limited her shift in teacher mathematical identity. Heide had the most frequent and lengthy posts in th e Prime Online PD program, and even explained that she felt bad about posting so much more than her peers. During Segment Two, Brynn posted and responded once during the Broken Calculator activity; Heide posted and responded to or the Working with Base Ten Blocks activity, Brynn posted but did not respond to others; Heide posted and responded twice. Brynn did not participate at all during A Meter of Candy activity; Heide posted 471 words, responded once, and sent a 399 word emai l to the oTPD program facilitators about how much she and her class enjoyed the activity. Even though Heide had considerably more experience with online courses than Brynn, it is unlikely that this familiarity with online
145 learning was the reason for Brynn was more likely due to a combination of her being a late person, it being a busy time of hear, her feeling overwhelmed, and the discrepancy between her priorities at the time and the purpose of the Prime Online PD program. Summary Both participants lacked foundational knowledge because they began struggling with mathematics in seventh grade. A negative history as a learner of mathematics had a relationship with participation levels but was mediate case, a factor of teacher mathematical identity (i.e., efficacy) combined with personal attributes (i.e., perfectionism and sense of responsibility) and a willingness to change seemed to have led to her levels of participati on. Brynn had a strong preference for traditional teaching methods and felt that the purpose of PD was to provide lessons and sporadic levels of participation, as the P rime Online PD program presented reform based (i.e., student centered) instructional strategies as a mechanism for increasing Research Question 2 What do narratives reveal about shifts in teacher mathematical identity two years post oTPD pro gram? The second research question focused on shifts in teacher mathematical identity that occurred since the conclusion of the Prime Online PD program. Interviews were
146 Prime Online PD program teacher mathematical identity with their present day teacher mathematical identity. Heide During our first interview, Heide made a profound statement about the shift in perception of herself as a mathematics teacher. She explained that, post Prime Online PD program, she was able to see that her mathematics instruction had been teacher 1p8]. She stated that, pre Prime Online PD program, she wou ld be frustrated that the students did not understand the lesson and the kids were taught not to think, but I was actually promoting that same thing by doing the talkin g 1p8]. The Prime Online PD program gave Heide the knowledge and confidence in mathematics to extend her MKT beyond the specific content that was presented in the weekly modules. Heide reported that she now used manipulative materials before teaching any lesson formally, even those concepts that were not explicitly taught during the Prime Online PD program. Her classroom was also arranged so her students were able to access a variety of manipulative material s to represent the concepts on which they were working. Her mathematics instruction went from being teacher centered to being student centered. She no longer told them which manipulative materials to use to solve a particular type of problem. Heide had been using student centered activities in all other subjects, but, pre Prime Online PD program, she did not have the knowledge or confidence to use them during mathematics instruction.
147 Brynn e., self efficacy, manipulative materials, and teaching for conceptual understanding) were presented in Chapter 5, but a more in depth analysis is warranted. The most telling shift was in her perceptions of the importance of conceptual knowledge away from quick strategies, which reflected a fundamental misunderstanding about the use of manipulative materials. At the beginning of the Prime Online PD program, Brynn mentioned it would be impractical to rely on manipulative materials as an adult (e.g., shoppi ng with yellow and red counters). Statements such as this exemplify her misunderstanding of the purpose of manipulative materials. She did not see manipulative materials as a nn saw 2p4]. Post Prime Online PD program, her feelings toward manipulative materials had shifted she and her conception of manipulative materials had shifted [I 2p4]. During the second interview, she explained that she now recognizes money as a manipulative dollars and cents could be used interchangeably with base ten blocks (i.e., flats and unit cubes). She said that the Prime Online 2p4]. Working in concert with greater use of manipulative materials is the acceptance Brynn now has for using multiple strategies to solve a problem. In her post about
148 solve each problem [Wk9BC]. Her present day t eacher mathematical identity presents quite a shift. She now teaches her students different strategies and explained how she 2p6]. Brynn still wanted her students to be fluent with the traditi onal algorithm, but they were free to use alternative methods, particularly when learning a concept for the first time. In fact, Brynn found that her students had a greater understanding of the reasonableness of their answers when she began by teaching at the concrete level versus only teaching . quick strategies [I 1p9]. Looking Across the Cases Both participants shifted in their views of mathematics teaching and lear ning to include teaching for conceptual understanding and the importance of manipulative materials. During the Broken Calculator activity, Heide wrote that being taught methods with manipulative materials would have helped her have a conceptual understand ing of as a kid (only if I had teachers who understood how to use the manipulatives hm due to her familiarity with it and its practicality. Two years post oTPD program, both participants stated that they enjoyed using manipulative materials on a regular basis. need to understand how to use manipulative materials effectively. However, neither participant mentioned the need for any facet of MKT to facilitate her professional
149 growth. based instructional practices, including her pre Prime Online seem a ware of the connection between MKT and changes in instructional strategies. [I 1p14]. Bry nn did not use any similar term to describe her growth nor did her retellings support any such intensity in her teacher mathematical identity shift. The difference between the two participants mirrors the identity shifts of John, a PST described by Krzywa cki (2009). John reflected on the gap between his current and designated identity related to his PCK. He was able to visualize himself in this future image and his pre Prime Online PD program self was already using student centered instruction, but she lacked the CK and MKT to integrate these reform based practices into her mathematics instruction. Conversely, John considered his current and designated identities regar ding mathematical competence to be fairly consistent with his designated identity. Thus, there was no learning fuelling tension to motivate him to set goals for professional development. At the beginning of the Prime Online PD program Brynn considered he rself to be an average teacher of mathematics and seemed content with this level of performance, even after comparing herself to the ideals presented in the Principles and Standards for School Mathematics (NCTM, 200). that the ideal image paves the way for 167). Both
150 participants had beliefs about what was needed to become a better teacher of more closely aligned with the content and pedagogy embedded in an oTPD program, more substantial shifts in teacher mathematical identity are likely to occur. In additional, there is a need for learning fuelling tension to act as the impetus for these shif ts. An openness to change is integral to support shifts teacher mathematical identity, but less obvious factors (e.g., self awareness) also play an important part in supporting these shifts. Implications for Practice and Suggestions for Future Research By investigating teacher mathematical identity shifts, this dissertation sought to extend the literature and help those interested in the development and implementation of oTPD programs particularly those with a mathematics content focus. In order to facili tate teacher PD programs, we must first understand how teachers grow professionally and the conditions that encourage that growth (Clarke & Hollingsworth, 2002). The results of this study provide evidence for a description of the complex interactional pat terns that result in identity formation and shifts in teacher mathematical identity. How a teacher views mathematics teaching and learning is related to development of a learning fueling tension, which in turn affects participation and identity shifts. Fo r example, Heide stated that she knew mathematics should be taught in the same integrated, student centered way that she taught other subject areas. She signed up for the Prime Online PD program to bridg e th is gap between her instruction in other areas an d those she implemented when teaching mathematics, which constituted a gap between her current and designated identities. She was eager to learn the PD program
151 content and implement the suggested activities and strategies. Her perseverance and high level s of participation contributed to the paradigm shift that she experienced Brynn joined the PD program without a learning fuelling tension. She signed up for the Prime Online teaching st yle, which was dissimilar to the type of instruction presented in the Prime Online PD program. Without a learning fuelling tension, her participation was sporadic. She had limited interaction with the PD program content and her peers and did not experien ce identity shifts of magnitude. Brynn also did not have a clear vision of a designated identity or who she wanted to become as a teacher of mathematics and, thus, was left without the motivation needed to bridge a ny gap s between her present instructional strategies and those presented in the PD program Both participants made statements about wanting to be a better teacher for their students. The difference was the degree of their learning (2000) construct of identity, Gre salfi and Cobb (2011) found that the keys to teacher institutional identity and the importance teachers place on that normative institutional identity. In other words, how closely one relates to the PD group and how much value one places on the ideals put forth by the PD group is a factor that may impact views of mathematics teaching and learning. Heide had a great need to attain her designated identity whereas Brynn saw her current and designated identities as similar. She saw herself as an average teacher and did not see an urgency to change her own level of MCK.
152 Heide came into the PD program with an openness to learning and a novice state of mind which is c explanation, regard n othing as sacred and know that the possibility of making a mistake driven by a se CK [Wk2RA]. In addition, Heide felt that her weakness in mathematics was inhibiting Prime Online PD program interview, she stated that reflec ting on self as teacher at the end of Week 1 made her realize that her teaching style was actually causing her students lack of critical thinking. Heide needed an increase in MKT for her current identity a proponent of reform based teaching practices to b ecome her designated identity one who could implement reform based teaching practices in mathematics lessons. Because she already used reform based practices in other subject areas, the gap between her current overall teacher identity and her designated t eacher mathematical identity seemed attainable (Sfard & Prusak, 2005). Heide did not see an openness to learning in her peers, however. She 16]. Brynn joined the Prime Online PD program voluntarily but felt overwhelmed by the pressures of being a novice classroom teacher. Her participation was also likely hampered because her traditional teachi ng style did not align with student centered
153 instruction emphasized by the Prime Online PD program. As Brynn did not value alternative strategies, her interest in the PD program content did not seem to be a Prime Online PD program, Brynn wrote that activities such as the Broken Calculator her struggling learners needed rote learning and tricks in order to master the traditional algorithm and have some measure of success in mathematics [Wk1F1]. Because she felt that she was teaching mathematics the way that her students learned best, there was no opportunity for a learning fuelling te nsion to develop. Additionally, as it was her first year as a general education classroom teacher, Brynn seemed to prioritize balancing her responsibilities over a ttaining a designated identity. Her lack of participation, whether due to her busy schedule, Indeed, del Valle and Duffy (2009) found experienced teachers to have the time and comfort level to explore and engage w ith PD experiences more so than their novice counterparts, whose time was taken up with classroom planning. Prime Online She wante d activities that she could implement immediately. She did not expect to, nor see the meaningfulness of, engaging in her own mathematical thinking. She found some of the assignments challenging and she did not persevere in those activities. Other assign ments were not completed at all. She wanted her students to understand the application of mathematics to real world contexts, but did not view mental mathematics and estimation
154 of solutions as a valuable part of a PD program. In her interview with me, Br ynn stated 1p14]. She made no mention about the content of the PD program or if it aligns with her desires and needs. Instead, she said that she looks for PD programs that have pract itioner based articles and activities that are applicable to her classroom context and can be implemented immediately. One factor associated with a learning fuelling tension is teacher characteristics. Teachers with negative histories of mathematics can c hange their beliefs (e.g., Ebby, 2000; Kaasila, 2007), but low efficacy teachers have been found to be the slowest to students mitigated her negative history as a learner of mathematics and her low self efficacy as a teacher of mathematics. Without her perfectionism, her lack of foundational knowledge would likely have substantially hindered her participation in the Segment Two mathematics specific activities. Instead, s he persevered by asking for help from others (e.g., family members, peers) and by seeking other resources (e.g., another version of the Broken Calculator applet). Heide also mentioned her sense of responsibility to her students as a reason for her perseve rance with challenging tasks. Brynn did not speak of any such motivating factors and had lower levels of participation, yet she also underwent shifts in her teacher mathematical identity. The differences in these cases could be explained by how the part icipants viewed their designated identities. Heide seemed to have a clear image of her designated identity. She said that she was frustrated with her current teaching practices rated
155 [I 2p2]. Heide wrote that she had failed in her role as a teacher of mathematics and Prime Online PD program [PIp1]. Brynn saw herself as an average teacher of mathematics and said that when she learned about the Prime Online an opportunity to do anything that improves my math teaching skills and even my mat h indicative of a learning Finally, there were factors associated with the development of a learning fuelling tension that were specif ic to the online context of the Prime Online PD program. For example, both participants mentioned a preference for facilitator, rather than peer, 2p13] when a facilitator asked her to reflect. Brynn explained that facilitator feedback prompted her to think more deeply so that she was made the discus 2p12]. This finding is similar to studies measuring the importance of consultants (Downer, 2009), mentors (Dede, 2006), and coaching (Yang, 2004) in online contexts. Additionally, both participants shared that true discussions were not possible in an Sfard and Prusak (2005) posit that identity development is communication. Without creating narratives o
156 (e.g., participating in discussion forums, reflecting on facilitator feedback), there can be no identity shift Implications for Practice The implications for practice highlighted by this research are applicable to both oTPD program developers and oTPD program facilitators. More so than the content rmal and informal educational of PD programs (Xu & Connelly, 2009, p. 221). An important, and obvious, aspect of MCK efficacy and history as a learner of mathematics and teacher of mathematics may also influence participation in PD programs. Heide suggested that implementation of reform based mathematics instructional practices is about t 1p7] as well as what teachers are 1p14]. Teachers with negative views of self as learner of mathematics have much more to overcome than merely a lack of foundational knowledge; there are also the year s of low self efficacy as teachers of mathematics. Ball (1994) clarifies the basis and impact of this issue: Elementary teachers are themselves the products of the very system they are now trying to reform. An overwhelming proportion are women, and the m ajority did not pursue mathematics beyond what was minimally required. Many report their own feelings of inadequacy and incompetence, and can even recall experiences that became turning points when they decided to stop taking mathematics. Rather than bec oming critical of the way we
157 their own mathematical lacks and to the inherently useless content of mathematics. Those same experiences have equipped them with ideas about the teache what it takes to learn and know mathematics. (p.16) nking and prompt reflection on self as teacher and views of mathematics teaching and learning need to be embedded throughout PD programs Although reading about the Principles for School Mathematics (NCTM, 2000) or CCSS M Standards for Mathematical Practi ce (NGA & CCSSO, 2010) was beneficial, participants stated multiple times that watching videos enhanced their learning more so than articles or PowerPoint presentations. Seeing what these principles look like in action might help teachers increase their a wareness that a gap between their current and designated identity does, indeed, exist and that the gap can be bridged. A more explicit self assessment of current and designated identities may have incited her to challenge her beliefs about manipul ative materials, alternative strategies, and multiple representations thereby f orm ing a learning fuelling tension. Participants should be asked to explicitly acknowledge areas of concern, how they might see their improved self (i.e., designated identity) and the steps that could be taken to achieve that improved self. Specific goals could be set and revisited throughout the PD program. Philipp (2007) agrees with the importance of reflection in teacher change:
158 beliefs by providing practice based evidence if teachers cannot see what they do not already believe? The essential ingredient for solving this conundrum is reflection upon practice. When practicing teachers have opportunities to reflect upon innovative reform oriented curricula they are using, upon their own their beliefs and practices change. (p. 309) In other words, opportunities for reflection and self assessment can support the cultivation of learning fuelling tensions. As facilitators constitute the affective aspect of oTPD programs, they need to support the development of learning fuelling tensions. In addition to opportunities for reflection, participants must have a novice state of mind in order for a PD program to impact teacher change (Turniansky & Friling, 2006). Particularly given that elementary school teachers have varying levels of MKT, participants need trust in others to open himself, his views, values, understandings, knowledge to examination and re control if a participant is a late person or a perfectionist. However, a PD program does have influence over cultivating a novice state of mind by providing a trusting openness to changing their beliefs and pr actices, engagement with PD program content would necessarily be limited.
159 Regarding the implications for practice specific to online contexts it is clear that norms must be established for communication and support. Participants want and appreciate indiv idualized feedback from oTPD program facilitators. In instances where the role of a facilitator is minimized, participants need to be guided as to how to take face to face interactions and discussions of traditional PD programs. While those exchanges cannot be replicated in an online environment, care should be taken to make this a priority of oTPD program design (e.g., structure group assignments for true collaborati on). In addition to being aware of the above mentioned implications, facilitators have an integral role in sustaining engagement (i.e., participation) in oTPD programs. Suggestions for Future Research More effective PD programs can be designed based on an increased understanding of teacher learning (Kazemi & Hubbard, 2008). Research is needed to assessment. Teacher reflection is an indicator of identity development (Bjuland et a l. 2012; Sfard & Prusak, 2005) and is essential to changing beliefs and practices ( Kaasila, 2008; Philipp, 2007). shifts in teacher mathematical identity warrants further study. Heide came into the PD program with a novice state of mind. She was aware that her negative history as a learner of mathematics was affecting her ability to teach her students, which motivated her to engage with the PD program content. She reported that her n on mathematics instruction was student centered, which was corroborated by the physical presence of
160 on going activities and the classroom layout visible during the two interview sessions. Because Heide was already a proponent of reform based instructional practices, she could bridge the gap between her current overall teacher identity and her designated teacher mathematical identity. The PD program deepened her MKT, which allowed her to attain her designated identity of implementing reform based instructi onal practices in her mathematics lessons. It would be helpful to understand if this is a trend amongst teachers who are implementing student centered instructional practices or if Heide was an extreme (i.e., atypical) case. Additionally, it would be help participation was due to a lack of trust in the group, perhaps because one member of the cohort was a teacher at her school. Further analysis could provide insight into G iven her notions of the purpose of PD, Brynn might have participated more during the weeks with activities related to her grade level that she could implement right away. She also stated that she was struggling to balance her responsibilities, as the firs t year of the Prime Online PD program was her first year as a general education classroom teacher. It is not uncommon for novice teachers to participate in PD programs. Further research needs to be undertaken to understand the mechanisms by which otherwi se overwhelmed teachers remain engaged in PD programs. A few suggestions for research are specific to P D programs in online contexts. The remaining Prime Online PD program participants could be interviewed about the relationship between their participati on and the quality and quantity of facilitator feedback. Follow up studies could analyze course usage data to investigate the role of
161 lurking in PD program engagement. In addition, research is still needed about how to best create collaborative group or communities of practice in asynchronous online settings. Conclusion In order for a PD program to support shifts in teacher mathematical identity, participants need a learning fuelling tension and a novice state of mind working in concert with one another. Without a novice state of mind, participants may not feel secure enough to be open to learning the PD program content and their learning fuelling tension may go unresolved. Conversely, participants may voluntarily join a PD program and feel open to learn ing, but without a learning fuelling tension there is no gap between a current and designated identity that needs to be bridged. This was the case with Brynn. In other instances, there is no motivation, or learning fuelling tension, and no openness to le arning, or novice state of mind. Heide suggested such a condition when she suggested that the lack of participation and negative attitudes of her Prime Online PD program peers was due to them being forced to enroll.
162 APPENDIX A PRIME ONLINE PD PROGRAM WEE KLY ACTIVITIES Segment 1 Building the Foundation for Inclusive Elementary Mathematics Classrooms Week 1 NCTM Principles and Standards for School Mathematics Week 2 Classroom Practices that Promote Mathematical Proficiency Week 3 Characteristics of Stud ents with Learning Disabilities Week 4 Tools for Understanding Struggling Learners: Explicit Strategy Instruction Week 5 Theories of Learning and Teaching: Self Regulated Learning Week 6 Research Based Practices: Self Regulated Strategy Development (SRS D) Week 7 Teacher Inquiry, Response To Intervention (RTI), and Progress Monitoring Week 8 Reflecting on Segment One of Prime Online PD program Segment 2 Deepening Mathematics Content and Pedagogy Week 9 Number Sense, Procedural Knowledge, and Conceptua l Knowledge Week 10 Building Conceptual Knowledge of Multiplication Week 11 Building Conceptual Knowledge of Multiplication and Division Week 12 Building Conceptual Knowledge of Multiplication and Division Week 13 Examining Multiplication and Division and Students with LD Week 14 Fractions and Decimal Numbers: Representation Week 15 Fractions and Decimal Numbers: Addition and Subtraction Week 16 Multiplication with Fractions Week 17 Division with Fractions Week 18 Connections to Operations wi th Decimal Numbers Week 19 Examining Fractions and Decimal Numbers and Students with LD Week 20 Revisiting and Expanding Concepts to Support the Learning of Students with LD Week 21 Reflecting on Segment Two of Prime Online PD program Segment 3 Stud ying the Application of Newly Learned Mathematics Content and Pedagogy to Student Learning Week 22 Teacher Inquiry as a Vehicle to Better Understand the Teaching of Mathematics and Struggling Learners Week 23 The Start of Your Journey: Developing Questio ns or "Wonderings" Week 24 The Road Map: Developing the Data Collection Plan and Formative Data Analysis Week 25 Time to Analyze: Summative Data Analysis for the Teacher Inquirer Week 26 Writing Up Your Work Week 27 Assessing the Quality of Teacher Res earch and Sharing Your Work With Others
163 APPENDIX B SEGMENT ONE ACTIVITIES Week 1 Anticipatory Activity: Who Am I as a Mathematics Teacher? In a Word document, reflect upon yourself as a teacher. Through your discussion provide an illustration of who yo u are as a teacher of mathematics. Please use the following stems as the beginning sentences of your statement: One of the best m athematics lessons I taught was . (in your statement, you may include the objectives and the sequence of events; the physi cal layout of your classroom and how that played into your lesson structure; and the interactions between you and your students that reflect the mathematical conversations typical in your classroom) I see my role as a mathematics teacher to be . (prov ide concrete ways in which this role was instantiated within the lesson described above) I am the most effective as a teacher of mathematics when . I am the least effective as a teacher of mathematics when . Upload the Word document (it will not be shared with other participants) and keep a copy for yourself. Week 1 Content and Discussion: A Vision of School Mathematics Read NCTM (2000) Chapter 1: A Vision of School Mathematics. After reading, vision of mathematics teaching and learning depicted in the Standards. Create a post by clicking on "Add a discussion topic" below in which you answer these questions: In of school mathematics? What barriers do you perceive in terms of realizing this goal? After reading through your colleagues' initial posts, discuss the commonalities you see in their statements such as common successes and barriers. You may either create a new discussion thread or respond to a thread one of your colleagues has begun for Part Two. Do not respond to an individual's initial Part One post. Reread the Word document about who you are as a teacher of mathematics from principles for school mathematics. Write and submit a brief reflection statement indicating how the readings and discussion have changed (or not) how you view your mathematics teaching and your role in the classroom. Upload your reflection. Week 2 Anticipatory Activity: Mathematical Proficiency Reflect on your history of learning mathematics. Think about how you learned mathe matics in grades 3 In your post, discuss how you remember learning mathematics in the elementary classroom. Based on this history of learning, comment in your post on what it means to
164 you t o be proficient in mathematics. You will return to these posts in the "Content and Discussion" section later this week. Week 2 Content and Discussion: Strands of Mathematical Proficiency Read through your colleagues' initial posts from the Anticipatory A ctivity discussion forum. Identify common themes among your colleagues' experiences as a 3rd through 5th grade learner of mathematics and their conceptions of mathematical proficiency. Create a post in which you consider the commonalities among your collea history as learners of mathematics and views of mathematical proficiency in relation to the "Strands" readings. In your post, consider the following questions: How is mathematical proficiency as described in the "Strands" readings similar to and/or d ifferent from the common themes you identified among your colleagues' histories as learners of mathematics and statements of what it means to be proficient in mathematics? In what ways do your classroom practices promote the five proficiency strands as dis cussed in the readings? What goals would you set for your classroom practice for improving mathematics proficiency? Respond to at least two colleagues by Sunday. Reread the Word document about who you are as a teacher of mathematics from principles for school mathematics. Write and submit a brief reflection statement indicating how the readings and d iscussion have changed (or not) how you view your mathematics teaching and your role in the classroom. Upload your reflection.
165 APPENDIX C PI CONDUCTED INTERVIEWS General Questions (Asked of Every Participant) 1. What brought you to the Prime Online? What types of experiences have you had in the past with professional development focused on mathematics? What types of experiences have you had in the past with professional development focused on special education? What types of experiences have you had in th e past with on line learning? 2. Tell me a little about your experiences so far with Prime Online. Overall, in your opinion, what aspects of Prime Online are going well so far? What aspects of Prime Online are challenging or frustrating? How have you int egrated Prime Online into your life? Routines? 3. There are a number of components each week to the Prime Online experience, including the following (provide a brief rationale for why we chose to include each component; have the participants talk about e ach component below in terms of whether or not each is meaningful and/or important): an anticipatory activity content and discussion questions for the week assessment and reflection activity 4. Posting plays a big r ole in on line learning. Take me to a post you created during the first 4 weeks of Prime Online (or however many weeks completed at the time of the interview) that you believe exemplifies an important moment in your own learning. (Read the post togethe r). What are some reasons you selected this post? What does it exemplify about your learning? What are your thoughts/feelings about the value of individual posting as a part of your professional learning in Prime Online? Have the teacher identify a post th at was not as supportive of his/her learning. Use the first two probes above. 5. On line discussion with colleagues also plays a big role in Prime Online. Take me to a post created by one of your Prime Online colleagues during the first 4 weeks of Pri me Online (or however many weeks completed at the time of the interview). (Read the post together).
166 What are some reasons you selected this post? What does it exemplify about your learning? What are your thoughts/feelings about the value of on line discu ssion with peers as a component of Prime Online? What were your thoughts when you responded to the post? 6. Cyndy, Steve, and Nancy contribute to the discussions as facilitators of Prime Online both in discussion forums a nnouncements. Take me to a post created by one of the Prime Online facilitators during the first 4 weeks of Prime Online (or however many weeks completed at the time of the interview) that you believe contributed significantly to your learning. (Read the post togeth er). What are some reasons you selected this post? What does it exemplify about your learning? 7. In what ways, if any, has your thinking about teaching mathematics changed as a result of your participation in Prime On line to date? 8. In what ways, if any, has your thinking about teaching struggling learners changed as a result of your participation in Prime On line to date? 9. Have you made any changes to your classroom pr actice so far based on your experiences in Prime Online? (If Yes Describe any changes in practice) Specific Questions (Tailored to the Individual Participation Data on Each Participant) Craft 2 3 specific questions here for the individual based on th e data Marty has provided for us, such as assignment completed on time. Tell me a little bit about week three and what contributed to ed that during weeks 2, 4, and 5, your posts were late. Tell me a little bit about your ex periences during these weeks. What might have been structured Do the due dates facilitate your movement through the modules?
167 end of each module? What contributed to your ability to complete (what Final Questions So far, what has been the most valuable learning that has occurred for you in Prime Online? So far, what do you believe to be the least valuable component of Prime Online? What suggestions do you have for the future development of Prime Online?
168 APPENDIX D SEGMENT TWO ACTIVITIES Week 9 Content a nd Discussion: Broken Calculator Click on the link below and read through pages 4 and 5. Work through Challenges 1 3 and watch the 2 videos on page 4 and the 1 video on page 5. www.nctm.org/eresources/view_article.asp?article_id=7457 Post your thoughts re lated to the following questions: What strategies did you use to solve the addition problems? How did you have to change your approach to work the multiplication problems? How might this activity support students' developing number sense? Write a brief ref lection on the classroom videos. What stood out to you? What Come back to this forum and have a discussion with your colleagues regarding their responses. Week 11 Content and Discussion: Workin g with Base ten Blocks For this activity, you will use base ten blocks to model several multiplication problems. Use base ten blocks placed on a piece of white paper to illustrate each problem. 22 x 13 43 x 21 37 x 14 Outline the partial products on the pa per by using a marker to separate them similar to the model above and take a digital picture of your representation of the n, go to the discussion forum where you will engage with your colleagues in a discussion of two questions. In the discussion forum, respond to the following questions. As always, go back to the forum and engage your colleagues in a discussion regarding t he following two questions: 1. Think about each step you took while modeling each multiplication problem using the base ten blocks. How does the motion with the manipulative materials support an understanding of the partial products algorithm for multipl ication? 2. Compute these products using the traditional algorithm. How is the partial product algorithm for multiplication similar to or different from the traditional algorithm for multiplication? Week 15 Content and Discussion: Building Connections B etween Fractions and Decimals: A Meter of Candy In week 14 we discussed the importance of helping students make connections between the different conceptions of rational numbers (i.e., part whole comparisons, decimal numbers, percents, ratios, etc.). In t he lesson "A Meter of Candy", students are challenged to construct interrelated understandings of fractions, decimals, and percents.
169 Print the lesson (it will come up as a PDF) and associated student materials from the link to the NCTM website below. Link s to student documents are embedded within the webpage. A Meter of Candy You can do this activity in two ways -with or without your students. If you are able to use this lesson with your students, then please do so. The Illuminations website indicates th at this lesson will take 3 class periods, and it will be a richer experience if you are able to use this lesson with your children. If you are not able to use this lesson with your students, then work through each section of the lesson on the website. Eng age your colleagues in a discussion over one or more of the following: Discuss one or more features of the lesson or the ancillary materials that you felt would support student understanding of the interconnections between constructs of rational numbers. T hree models of rational numbers are represented in this activity. Identify each of these models. How are these models of rational numbers related to one another? How might you have typically assessed your students' knowledge of this concept? Which of the suggested Assessment Options appeal to you most and why? Pose a question about something that you did not understand OR that you did not agree with in the lesson.
170 APPENDIX E MODULE SURVEYS Week 1 Before After Items 1. I understand the vision for school math ematics set forth by NCTM. 2. I have reflected on my teaching in terms of the NCTM vision of school mathematics. 3. I understand the principles for school mathematics set forth by NCTM. 4. I have reflected on my teaching in terms of the NCTM principles for school m athematics. 5. I understand the interrelationships between the NCTM vision and principles and my 6. I have the knowledge and skills to incorporate the NCTM vision and principles in my teaching. Free Respo nse Items 7. Please describe 1 3 significant things you learned in this module. 8. Please list 1 3 issues, ideas or topics from this module you would have liked to learn more about. 9. If you were the module designer, how would you improve this module in the aspect s of content delivery, organization, instructional strategies, etc. Week 2 Before After Items 1. I think of mathematical competence as a broad and multifaceted construct. 2. competence. 3. I u nderstand how my history of learning mathematics has influenced my thinking about what it means to be competent in mathematics. 4. I understand classroom practices I can incorporate within my instruction to support cal proficiency. 5. developing conception of mathematical proficiency. Free Response Items 6. Please describe 1 3 significant things you learned in this module.
171 7. Please list 1 3 issues, i deas or topics from this module you would have liked to learn more about. 8. If you were the module designer, how would you improve this module in the aspects of content delivery, organization, instructional strategies, etc. Weeks 9 12 Before After Items 1. I un derstand the definition of number sense. 2. 3. I understand the relationship between procedural and conceptual knowledge. 4. I understand the importance of invented strategies for studen procedural knowledge. 5. conceptual understanding of arithmetic operations such as multiplication and division. 6. I can identify word problems that represent each type of division problem. 7. I can justify to a parent why it is important for students to learn basic operations using manipulative materials. Free Response Items During weeks 9 12, we incorporated several new online components. Please provide feedback on each of the following online components in terms of your learning mathematics content for teaching. Please provide a specific rationale for your 8. Working w ith web based applets such as the Broken Calculator (week 9) 9. Viewing videos of classroom episodes (weeks 10 11) 10. Working in a chat room where you discuss an assignment with your colleagues (week 11) 11. Working with manipulative materials to model arithmetic op erations (weeks 11 & 12) 12. Viewing a video of one of the instructors explaining a concept (i.e., relationship between motion of materials and the division algorithm) (week 12) During weeks 9 12, we focused on a number of topics related to teaching and learni ng multiplication and division. Discuss each of these topics in terms of its usefulness for your practice. Please provide a specific rationale for your statement.
172 13. Number sense (week 9) 14. Relationship between procedural and conceptual knowledge (week 9) 15. Exp laining your strategies for a mathematical procedure (week 9) 16. 17. 18. understanding o f arithmetic operations (e.g., multiplication and division) (weeks 11&12) 19. 11) 20. How would you improve the modules for weeks 9 12 in terms of content delivery? 21. How would you improve the modules for weeks 9 12 in terms of instructional strategies? 22. Is there anything else you might change to improve the learning of mathematics for teaching for subsequent cohorts of participants? Week 13 Before After Items 1. I know that the IDEA ensures that students with disabilities r eceive research based practice instruction by requiring that all school staff have the skills and knowledge to use scientifically based instructional practices. 2. I know the instructional recommendations in mathematics put forth by the authors of the IES Pra ctice Guide. 3. students in my classroom. 4. I can use the Stages of SRSD to evaluate instructional practices. 5. I can use the learning characteristics of students with LD to evalua te instructional practices. Likert Type Items Reasons for having difficulties participating in the online modules in a timely manner. 6. Too much work within an individual week. 7. Too m any consecutive weeks of Prime Online 8. The content has become more challengi ng.
173 9. The end of the school year is a difficult time of the year to get everything done. 10. 11. I am not learning from the module content. 12. Things have happened in my personal life that are interfering with my a bility to complete the assignments. 13. I do not have convenient access to the technology. 14. I do not have facility with the technology. 15. I do not like participating in the online discussions. Free Response Items 17. Provide other feedback that is not included above. Weeks 14 16 Before After Items 1. I understand three models for representing fractions (i.e., set, area, and length). 2. I understand two fraction processes referred to as iterating and partitioning. 3. I understand the connections between language used and the un derlying images students may have of fractions. 4. I understand the controversies related to teaching algorithms for addition and subtraction of fractions procedurally and conceptually. 5. hey compute fractions operations. 6. I understand how to represent the multiplication of fractions in multiple ways. Free Response Items During Weeks 14 16, we focused on a number of topics related to teaching and learning fractions and decimal numbers. Disc uss each of these topics in terms of its usefulness for your practice. Please provide a specific rationale for your statement. 7. Three models for representing fractions (i.e., set, area, and length) (Week 14) 8. Modeling two different fraction processes, refe rred to as iterating and partitioning (Week 14) 9. Connections between language used and the underlying images students may have of fractions (e.g., out of, cut evenly, make copies) (Week 14) 10. of addition and subtraction of fractions (Week 15)
174 11. Teaching algorithms for addition and subtraction of fractions both procedurally and conceptually (Week 15) 12. Working through as activity that supports connections between fractions and decimals (i.e., A Met er of Candy) 13. Using estimation to compute fractions operations (Week 16) 14. Representing the multiplication of fractions in multiple ways (Week 16) 15. How would you improve the modules for Weeks 14 16 in terms of content delivery? 16. How would you improve the module s for Weeks 14 16 in terms of instructional strategies? 17. Is there anything else you might change to improve the learning of mathematics for teaching for subsequent cohorts of participants?
175 APPENDIX F SEGMENT SATISFACTION SURVEYS Segment One Satisfaction Su rvey Content of the Modules 1. The content was clear to me. 2. 3. 4. The content was appropriate for teachers in grade levels 3, 4 and 5. 5. The assignme 6. The instructional methods presented in the modules were appropriate for teaching students in grade levels 3, 4 and 5. 7. The material about supporting students with learning disabilities (LD) w ill help me improve their learning. 8. The instructional methods presented in the modules are practical in the amount of time they will require in my classroom. 9. The out of school planning time required by the instructional methods presented in the modules is reasonable. 10. I will be able to use the instructional methods presented in the modules for helping students with LD and still meet the needs of other students in the classroom. 11. By implementing the instructional methods presented in the modules for helping st udents with LD, I have gained new insights for teaching all students. 12. I will use the instructional methods presented in the modules to meet the needs of students with LD. 13. class and their work to alter my instruction. 14. The modules stimulated my thinking about the content of the modules. 15. The questions I had about the topics in modules were answered in the modules. 16. My expectations for participating in this professional development we re met by the modules. 17. The screening and progress monitoring methods taught in the modules will help me
176 18. Overall, the mathematics content, instructional strategies and teacher research strategies in th e modules would be beneficial for all students as well as students with LD. Pedagogy of the Modules 19. The learning objectives in the modules were clear to me. 20. The activities in the modules helped me to achieve the objectives of the modules. 21. The introduction to the modules provided a clear overview of the modules. 22. The anticipatory activities in the modules connected to my prior knowledge about the material in the modules. 23. The content and discussion in the modules were clear to me. 24. The content and discussion in the modules were at an appropriate level of difficulty for me. 25. The assessment and reflection activities in the modules helped me deepen my understanding about the material in the modules. 26. The assessment and reflection activities in the modules helped me u nderstand what I learned through the modules. 27. The videos in the modules were engaging 28. The assessment and reflection activities in the modules helped me understand what I still need to learn about the materials in the modules. 29. The modules engaged me in an a ctive manner. 30. The modules provided opportunities to reflect on how the information I was learning applied to my own classroom. 31. Interacting and sharing ideas with other participants contributed to the overall effectiveness of the modules. 32. The discussion fo rums stimulated my thoughts about the material in the modules. 33. 34. The videos in the modules enhanced my learning. 35. The activities in the modules helped me to understand the content of the modules. Technology and Support
177 36. The modules were designed in a manner that is appropriate for my computer skills. 37. The modules were easy to navigate. 38. 39. The modules are well organized. 40. The modules are visually consistent. 41. The modules are functionally consistent. 42. I was able to access the videos in the modules. 43. The modules provide a user friendly environment for online discussion. 44. Adequate technical support was provided to enable me to work in dependently. Segment Two Satisfaction Survey Content of the Modules 1. The content was clear to me. 2. 3. 4. The content was appropriate for teacher s in grade levels 3, 4 and 5. 5. 6. The instructional methods presented in the modules were appropriate for teaching students in grade levels 3, 4 and 5. 7. The material about supporting students with learning disabilities (LD) will help me improve their learning. 8. The instructional methods presented in the modules are practical in the amount of time they will require in my classroom. 9. The out of school planning time required by the instruc tional methods presented in the modules is reasonable. 10. I will be able to use the instructional methods presented in the modules for helping students with LD and still meet the needs of other students in the classroom. 11. By learning about the instructional me thods presented in the modules for helping students with LD, I have gained new insights for teaching all students.
178 12. I will use the instructional methods presented in the modules to meet the needs of students with LD. 13. By completing the modules I have learned class and their work to alter my instruction. 14. The modules stimulated my thinking about the content of the modules. 15. The questions I had about the topics in modules were answered in the modules. 16. My expectations for part icipating in this professional development were met by the modules. 17. The screening and progress monitoring methods taught in the modules will help me 18. The mathematics content in the modules is relevant to the topics I teach. 19. As a result of the modules, I have a better understanding of NCTM and common core standards. 20. Overall, the mathematics content, instructional strategies and teacher research strategies in the modules would be beneficial for all studen ts as well as students with LD. Pedagogy of the Modules 21. The learning objectives were clear to me. 22. The activities helped me to achieve the objectives of the modules. 23. The introduction to the modules provided a clear overview of the modules. 24. The anticipatory activities in the modules connected to my prior knowledge about the material. 25. The content and discussion were clear to me. 26. The content and discussion were at an appropriate level of difficulty for me. 27. The assessment and reflection activities in the modules helped me deepen my understanding about the material. 28. The assessment and reflection activities helped me understand what I learned through the modules. 29. The videos were engaging
179 30. The assessment and reflection activities helped me understand what I still nee d to learn about the materials in the modules. 31. The modules engaged me in an active manner. 32. The modules provided opportunities to reflect on how the information I was learning applied to my own classroom. 33. Interacting and sharing ideas with other participant s contributed to the overall effectiveness of the modules. 34. The discussion forums stimulated my thoughts about the material in the modules. 35. 36. The videos in the modules enhanced my learning. 37. The activities in the modules helped me to understand the content of the modules. Technology and Support 38. The modules were designed in a manner that is appropriate for my computer skills. 39. The modules were easy to navigate. 40. nts and communicates information clearly. 41. The modules are well organized. 42. The modules are visually consistent. 43. The modules are functionally consistent. 44. I was able to access the videos in the modules. 45. The modules provide a user friendly environment for on line discussion. 46. Adequate technical support was provided to enable me to work independently. Segment Three Satisfaction Survey Content of the Modules 1. The content was clear to me. 2. 3. The content hel
180 4. The content was appropriate for teachers in grade levels 3, 4 and 5. 5. 6. Engaging in the process of practitioner research presented in the mo dules helped to enhance my understanding of one or more instructional methods presented in segments one and two. 7. Inquiry into my own teaching practices with students with learning disabilities (LD) will help me improve their learning. 8. The process of practi tioner research presented in the modules is practical in the amount of time it required in my classroom. 9. The out of school work time required by the teacher researcher is reasonable. 10. I will be able to use the process of practitioner research presented in t he modules for helping students with LD and meeting the needs of other students in the classroom. 11. By engaging in practitioner research, I have gained new insights into teaching mathematics. 12. By engaging in practitioner research, I have gained new insights i nto helping students with LD or other students with special learning needs. 13. I will utilize the process of practitioner inquiry presented in the modules to meet the needs of students with LD. 14. By completing the modules I have learned how to systematically a nd intentionally study my own mathematics teaching practice. 15. mathematic learning. 16. The modules stimulated my thinking about ways to utilize data to inform my instruction. 17. E ngaging in the teacher inquiry process gave me confidence to make instructional decisions in the best interest of my students. 18. The questions I had about the topics in modules were answered in the modules. 19. My expectations for participating in this professio nal development were met by the modules. 20. The progress monitoring methods taught in the data collection module will help me
181 21. The teacher research strategies taught in the modules will help me to modify my 22. The teacher research strategies taught in the modules will help me to modify my Pedagogy of the Modules 23. The learning objectives were clear to me. 24. The activities help ed me to achieve the objectives of the modules. 25. The introduction to the modules provided a clear overview of the modules. 26. The anticipatory activities connected to my prior knowledge about the material. 27. The content and discussion were clear to me. 28. The conte nt and discussion were at an appropriate level of difficulty for me. 29. The assessment and reflection activities helped me deepen my understanding about the material. 30. The assessment and reflection activities helped me understand what I learned through the mod ules. 31. The video of the interview with a teacher researcher was engaging 32. The assessment and reflection activities helped me complete my own practitioner inquiry. 33. The modules engaged me in an active manner. 34. The modules provided opportunities to reflect on ho w the information I was learning applied to my own classroom. 35. Interacting and sharing ideas with other participants contributed to the overall effectiveness of the modules. 36. The discussion forums stimulated my thoughts about the material. 37. The readings fost 38. The interview with a teacher researcher video enhanced my learning. 39. The activities helped me to understand the content of the modules. Technology and Support
182 40. The modules were designed in a manner that is appro priate for my computer skills. 41. The modules were easy to navigate. 42. 43. The modules are well organized. 44. The modules are visually consistent. 45. The modules are functionally consistent. 46. I was able to access the video in the modules. 47. The modules provide a user friendly environment for online discussion. 48. Adequate technical support was provided to enable me to work independently. Overall 49. I felt my time was well spent in this professional dev elopment program. 50. I will recommend this professional development program to other teachers. 51. I was satisfied with the professional development program.
183 APPENDIX G INFORMED CONSENT FORM Dear Educator, I am a doctoral candidate in the School of Teaching and Learning at the University of Florida conducting research for a dissertation on mathematics based online teacher professional development. I am conducting this research under the supervision of Dr. Stephen J. P ape and Dr. Nancy F. Dana. The purpose of this study is to understand the relationship between participation in an online teacher professional development and how a teacher views mathematics teaching and learning. I am asking you to participate because of your successful completion of the Prime O nline professional development project. With your permission, I would like to interview you on two occasions over the course of three weeks. Each of the two interviews will last no more than one hour and will be audiotaped. The interviews will be schedul ed at your convenience. You will not have to answer any questions you do not wish to answer. Only I will have access to the audiotapes, which will be transcribed by a transcription service and any identifiers will be removed by replacing your name and an y other names mentioned with pseudonyms. The tapes will be kept locked in a cabinet in my home. Your identity will be kept confidential to the extent provided by law and will not be revealed in the final manuscript. There are no anticipated risks, compen sation, or other direct benefits to you as a participant in this study. Your participation, however, will support our understanding of participation in an online teacher professional development. Your participation is voluntary and you may withdraw your consent at any time without penalty. If you have any questions about this research protocol, ple ase contact me or my facu lty supervisors Dr. Stephen J. Pape or Dr. Nancy F. Dana Qu estions about your rights as a research participant may be directed to the IRB0 2 office at the University of Florida If you agree to participate in this study, please sign and return this copy of the letter to me in the enclosed envelope. A second copy i s provided for your records. By signing this letter, you give me permission to report the data I collect in interviews with you. This report will be submitted to my faculty supervisors as part of my dissertation requirements. Also, by signing, you give me permission to use these data in academic presentations and publications. Thank you, Sherri K. Prosser I have read the procedure described above. I voluntarily agree to participate in nline teacher professional development. I voluntarily agree to participate in the interview and I have received a copy of this description. ____________________________________ _______________ Signature of Participant Date
184 APPENDIX H INTERVIEW ON E During this interview I will ask you general questions about your learning and teaching of mathematics, some of which are similar to discussion prompts or writing assignments during the first few weeks of Prime Online In most cases, I will ask you to p rovide a description or tell a story to explain your response. I hope that you will provide rich details in these descriptions or stories. Before we get started, I would like to confirm your background information (i.e., teaching assignments from January 2011 December 2011 [e.g., grade level(s), subject(s), public or lab school], teaching history, degree(s) earned, area(s) of certification, teaching history). Do you have any questions before we begin? 1. I s a participant in Prime Online How have you been? What have you been up to over the last 18 months? Have you engaged in any professional development activities since Prime Online? 2. Tell me a story about one of the best recent mathematics lessons you taug ht (you may include the objectives and the sequence of events; the physical layout of your classroom and how that played into your lesson structure; and the interactions between you and your students that reflect the mathematical conversations typical in y our classroom). 3. If you look back, when are you most effective as a teacher of mathematics? Please tell me a story that would be an example of when you are most effective. 4. If you look back, when are you least effective as a teacher of mathematics? Pleas e tell me a story that would be an example of when you are least effective. 5. How do you see your role as a teacher of mathematics? Describe how that might 6. You may recall reading an article called Tying It All Together by Suh (2007) that discusses mathematical proficiency The author explains that students that are mathematically proficient: understand the concept, learn the procedures with meaning, solve problems using efficient strategies, defend and justify their rea soning, and find mathematical investigations challenging and engaging. 7. What does mathematical proficiency mean to you? 8. Describe a situation in which you would know if a student is mathematically proficient. 9. How do you view your role in supporting the de velopment of mathematical proficiency? 10. How might this look in a typical mathematics lesson? How would you feel, think or do?
185 11. How do you see yourself as a learner of mathematics? Give me an example that explains your answer. 12. Is there anything you would li ke to add that might be relevant to this study?
186 APPENDIX I INTERVIEW TWO HEIDE I would like to begin by asking you a few questions to help me understand your professional background and then continue with some questions about your teaching practices. F inally, I would like you to reflect on some of the statements you made during our first interview as well as your participation in Prime Online 1. Which science certification do you have: Grades 5 9 General Science or Grades 6 12 in Biology, Chemistry, Earth Space Science, or Physics? 2. Have you been at Newberry Elementary since 2003? Where did you teach prior to that? Prior to Prime Online had you only taught 3 rd 5 th grade self contained special education? Is this your fifth year as a general education teache r? 3. Tell me what led you to the teaching profession. 4. What brought you to the Prime Online project? INSTRUCTIONAL PRACTICES 5. You mentioned in your Spring 2011 interview: One thing that just struck me was the strategy notebook that a child could have with a [PIp8] Have you incorporated strategy notebooks in your classroom? 6. use the pacing guide left little time for re teaching [PIp16]. During our first interview, you told me about a two p5]. Has this always been a part of who you are a s a teacher? (i.e., Did you do this before you enrolled in Prime Online? Was this a true shift in identity?)? 7. When speaking about how your students might perceive your role as a teacher of mathematics, you said: definitely facil itation, more the learning in the kids. [I 1p8] What factors brought about this change in your teaching practices?
187 8. During the Prime Online course, you defined mathematical proficiency as: swer, know where During our interview, you compared mathematical proficiency to being able to model a multiplication problem using base ten blocks, similar to one of the Prime Online act ivities. By being able to work through that task, students would show that they have 1p10]. proficiency? Did these definitions of mathematical proficiency impact on your views about the teaching and learning of mathematics during each of those time periods? IDENTITY SHIFT You made many explicit comments about the impact Prime Online had on your teaching of mathematics. I would like to further probe your thoughts on that by asking you several, more specific, questions. 9. grade. [I elaborate on that? Why were you disheartened at this particula r time? What were your concerns with worksheet based instruction? Would this have always been a concern? 10. education background and to your age [I 1p14]. Would you expand on that? Was one of these more important of a factor than the other? Were there other personal factors or attributes that you brought to Prime Online that may have increased what you took away from the PD? 11. ced with Prime Online will have an impact on your teacher identity as a K 4 science teacher next year? Why or why not? Paint me a picture of how that might look. 12. Toward the end of our interview you seemed to empathize with the general education teachers at your school who struggle when teaching math:
188 and I see it every year who feel so constrained. And math, because I actually love ght are not proficient in math. [I 1p14] Would you have considered yourself as one of these teachers, prior to your Prime Online experience? PARTICIPATION At least two times you mentioned the lack of participation by your Prime Online peers. In Week 13 module survey, regarding p timely participation, you wrote: Online "discussions" not meaningful and/or possible when so few people participate. In Week 14 16 module survey, regarding how to improve the learning of subsequent cohorts, you wr ote: Having kept up with assignments, unlike some others, it's been discouraging to witness the somewhat unreceptive and sometimes outright negative attitudes. Did your feelings of discouragement or about the lack of meaning of the discussions affect your own participation? If so, to what degree? Do you think it affected how much you were able to take away (i.e., learn) from the Prime Online experience? 13. In week 14 16 module survey, you wrote: I'd rather just get feedback from instructors, rather than try t o post just to fulfill the requirement of the module. Was there a relationship between instructor input and your participation in the forum posts? (Does the quantity/quality of your posts depend on if the instructor has responded to you or others)
189 APPE NDIX J INTERVIEW TWO BRYNN I would like to begin by asking you a few questions to help me understand your professional background and then continue with some general questions about your teaching practices. Finally, I would like you to reflect on some o f the statements you made during our first interview as well as your participation in Prime Online 1. Tell me what led you to the teaching profession. 2. You said that you had worked as a co teacher for two years. What were your responsibilities when you were co teacher? What did a typical day look like for you? 3. During your first individual interview, in March of 2011, you were asked if you thought your teaching of mathematics had changed since starting Prime Online During your response, you mentioned that yo u no longer use the Daily Depositor the way it was intended. So it starts off where the students have to recognize the pattern of the calendar and I usually make them make a prediction about something that will happen later on in the month. Then we do D aily Depositor, which I month I just wanted them to practice division some more because we did it in a previous chapter. So I had them take the year 2011 and divide it by the date so it was 2011 divided by 11 and then we deposit that much money into the Daily Depositor. [PIp11]) What is the nature of the changes you made? To what do you attribute these changes? 4. You spoke about being involved with the development of the new mathem atics pacing guide for the county. Talk to me about how you became involved in this activity. What factors led to your involvement? 5. How has your perception of yourself as a teacher of mathematics changed over the last two years? 6. How did the Prime Online PD experience influence your teaching practices?
190 STRATEGIES AND MANIPULATIVE MATERIALS: 7. During our interview in July you told me about many activities in which you used n 1p7]. Reflect on your response to a forum prompt you wrote about mathematical pro ficiency during the second week of the Prime Online program: Regardless, if you can analyze a problem, whether in a book or in real life, and perform a series of mathe matical operations to reach the needed answer, regardless of what operations those are (e.g., using multiplication instead of repeated addition or vice versa), you are able to reason and come up with a strategy to problem solve. (I assume that as adults, most if not all of the participants in this class act mathematically in this way, but I don't really know. Some of us might be carrying five bags of those red and yellow counters when we grocery shop.) [Wk2F1]. The end of this statement seems to indicate a negative perspective on the use of manipulative materials. Has your thinking about using these materials changed? To you, what is the role of manipulative materials in the learning process? In the teaching process? 8. After using base ten blocks to model multiplication with the partial products algorithm practical in a majority of situa [Wk11F3]. Reflect on this statement. What do you think you meant by this statement? What is the role of manipulative materials in learning? 9. You also mentioned stra tegy use while describing your best le sson : We showed students how to use a strategy involving multiplication to compare and order students preferred the aforementio ned strategy to finding common denominators to compare and order fractions. When students reach upper elementary grades without number sense or basic facts knowledge, quick strategies are what they rely on heavily to be successful in math [Wk1AA]. What
191 How do you decide when to teach with manipulative materials and when to teach quick strategies? How do you balance the two? 10. You posted this during a discussion about week about mathematical proficiency: I think tricks are so cool. I don't think I missed out as a student by not knowing the tricks, but as a teacher I see the benefit of knowing ways to make math bearable and enjoyable for a child who isn't confident in his computation ability. I find that my students who ar e stronger students that prefer the traditional methods of computation aren't as interested in the tricks. They think it holds them up when they could be moving on. The kids who struggle look for an alternative method. [Wk2 AA] How do quick strategies c 11. You spoke about your mathematics history during our I was never taught any "tricks" for solving multi digit multiplication problems or long division, and I im agine this is because I showed proficiency after being taught the traditional methods of regrouping and DMSB. [Wk2AA] Is your stance on tricks related to your history as a learner of math? PARTICIPATION: 12. As part of my study, I focused on three activities in Segment Two, one of which was A Meter of Candy in Week 15. You responded to the Anticipatory Activity and Forum 1 for that week (Fraction and Decimal Numbers: Addition and Subtraction), but not Forum 3, A Meter of Candy. Do you recall what hindered y our participation during that module? 13. What caused your levels of participation to vary? The prompts (the questions asked in the forums)? You mentioned that posting Can you elaborate? Also the week that we had to do a group Do you have any further recollections about how you felt during tha t
192 5] Were all of those specific to the online format? The instructor involvement? Regarding your response to the post on big kind of open ed up a little more discussion on my end as far as mathematical proficiency goes Wa s there a relationship between instructor input and your participation in the forum posts? (Does t he quantity/quality of your posts depend on if the instructor has responded to you or others?)
193 LIST OF REFERENCES Ball, D L. (1994, November). Developing mathematics reform: What don't we know about teacher learning but would make good working hypothes es? Paper presented at Conference for Teacher Enhancement in Mathematics K 6, Arlington, Virginia. Ball, D. L., Bass, H., Hill, H., Sleep, L., Phelps, G., & Thames, M. (2006, January). Knowing and using mathematics in teaching [PowerPoint slides] Paper p resented at the Learning Network Conference: Teacher Quality, Quantity, and Diversity, Washington, DC. Retrieved from http://www personal.umich.edu/~dball/presentations/ Ball, D. L., Sleep, L., Boerst, T. A., & Bass, H. (2009). Combining the development of practice and the practice of development in teacher education. The Elementary School Journal, 109 458 474. doi:10.1086/596996 Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Educa tion, 59, 389 407. Beauchamp, C., & Thomas, L. (2009). Understanding teacher identity: An overview of issues in the literature and implications for teacher education. Cambridge Journal of Education, 39 175 189. doi:10.1080/0305 7640902902252 Beijaard, D., Meijer, P., & Verloop, N. (2004). professional identity. Teaching and Teacher Education, 20 107 128. doi:10.1016/j.tate.2003.07.001 Bjuland, R., Cestari, M. L., & Borgersen, H. E. (2012). Professional mathematics teach er identity: Analysis of reflective narratives from discourse and activities. Journal of Mathematics Teacher Education, 15 405 424. doi:10.1007/s10857 012 9216 1 Brady, S., Gillis, M., Smith, T., Lavalette, M., Liss Bronstein, L., Lowe, E., . Wilder, T. D. (2009). concepts: Examining gains from an intensive form of professional development and corresponding teacher attitudes. Reading and Writing, 22 425 455. doi:10.1007/s11145 009 9166 x Brantlinger, E., Jimenez, R., Klingner, J., Pugach, M., & Richardson, V. (2005). Qualitative studies in special education. Council for Exceptional Children, 71 195 207. doi:10.1352/0047 6765(2005)43<92:IOEAWA>2.0.CO;2 Braun, V., & Clarke, V. (2006). Us ing thematic analysis in psychology. Qualitative Research in Psychology, 3, 77 101. doi:10.1191/1478088706qp063oa
194 Chapman, O. (2008). Narratives in mathematics teacher education. In D. Tirosh & T. Wood (Eds.), Tools and processes in mathematics teacher edu cation (pp. 15 38). Rotterdam, Netherlands: Sense Publishers. Chang, Y. (2009). development. International Journal of Science and Mathematics Education, 8 271 297. doi:10.1007/s10763 009 9173 z Chang, Y. L., & Wu, S. C. (2007). An exploratory study of elementary beginning mathematics teacher efficacy. In J. Woo, H. Lew, K. Park, & D. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education Vol. 2 (pp. 2 89 2 96). Seoul, Korea: PME 31. Clandinin, D. J., & Rosiek, J. (2006). Mapping a landscape of narrative inquiry: Borderland spaces and tensions. In D. J. Clandinin (Ed.), Handbook of narrative inquiry: Mapping a methodology (pp. 3 5 75). Thousand Oaks, CA: Sage. Coldron, J., & Smith, R. (1999). Active location in teachers' construction of their professional identities. Journal of Curriculum Studies, 31 711 726. doi:10.1080/002202799182954 Connelly, F. M., & Clandinin, D. J. (1990). Stories of experience and narrative inquiry. Educational Researcher, 19, 2 14. doi:10.2307/1176100 Connelly, M., & Clandinin, J. (1999). Shaping a professional identity: Stories of educational practice London, Ontario: The Althouse Press. Craig, C. (2001 ). Curriculum Inquiry, 31, 303 331. doi:10.1111/0362 6784.00199 Craig, C. (2011). Narrative inquiry in teaching and t eacher education. In J. Kitchen, D. Ciuffetelli Parker & D. Pushor (Eds.), Narrative inquiries into curriculum making in teacher education (pp. 19 42). Bingley, United Kingdom: Emerald Publishing. Creswell, J. W. (2002). Educational research: Planning, con ducting, and evaluating quantitative and qualitative research (3rd ed.). Upper Saddle River, NJ: Prentice Hall. Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics ctices. Journal of Mathematics Teacher Education, 12 325 346. doi:10.1007/s10857 009 9120 5 Dana, N. F., & Yendol Hoppey, D. (2009). research: Learning to teach and teaching to learn through practitioner inquir y, 2nd Edition. Thousand Oaks, CA: Corwin Press.
195 Day, C., Elliot, B., & Kington, A. (2005). Reforms standards and teacher identity: Challenges of sustaining commitment. Teaching and Teacher Education, 21 563 577. doi:10.1016/j.tate.2005.03.001 Day, C., Ki ngton, A., Stobart, G. & Sammons, P. (2006). The personal and professional selves of teachers: stable and unstable identities. British Educational Research Journal, 32 601 616. doi:10.1080/01411920600775316 Dede, C. Ketelhut, D. J., Whitehouse, P., Breit, L., & McCloskey, E. M. (2009). A research agenda for online teacher professional development. Journal of Teacher Education, 60 8 19. doi:10.1177/0022487108327554 del Valle, R., & Duffy, T. M. (2009). Online learning: learner characteristics and their app roaches to managing learning. Instructional Science, 37 129 149. doi:10.1007/s11251 007 9039 0 Denzin, N. K. (1989). The Research Act (3 rd ed.) Englewood Cliffs, NJ: Prentice Hall. Dingle, M. P., Brownell, M. T., Leko, M. M., Boardman, A. G., & Haager, D. (2011). Developing effective special education reading teachers: The influence of professional development, context, and individual qualities. Learning Disabilities Quarterly, 34, 87 103. Downer, J. T., Locasale Crouch, J., Hamre, B., & Pianta, R. (2009) Teacher characteristics associated with responsiveness and exposure to consultation and online professional development resources. Early Education & Development, 20 431 455. doi:10.1080/10409280802688626 Dougiamas, M. (1999). Moodle (Version 3) [Softwar e]. Retrieved from http://moodle.com/ Drake, C., & Sherin, M. G. (2006). Practicing change: Curriculum adaptation and teacher narrative in the context of mathematics education reform. Curriculum Inquiry, 36 153 187. doi:10.1111/j.1467 873X.2006.00351.x Dr ake, C., Spillane, J. P., & Hufferd Ackles, K. (2001). Storied identities: Teacher learning and subject matter context. Journal of Curriculum Studies, 33 1 23. doi:10.1080/00220270119765 Ebby, C. B. (2000). Learning to teach mathematics differently: The i nteraction between coursework and fieldwork for preservice teachers. Journal of Mathematics Teacher Education, 3 69 97. Flick, U. (2009). An introduction to qualitative research (4th ed.). London: Sage Publications.
196 Flores, M. A., & Day, C. (2006). Contex identities: A multi perspective study. Teaching and Teacher Education, 22 219 232. doi:10.1016/j.tate.2005.09.002 Forbes, C., & Davis, E. A. (2008). curricular rol e identity for science teaching. Science Teacher Education, 92 909 940. doi:10.1002/sce.20265 Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What makes professional development effective? Results from a national sample of teachers. American Education Research Journal, 38, 915 945. doi:10.3102/00028312038004915 Gee, J. P. (2000). Identity as an analytic lens for research in education. Review of Research in Education, 25 99 125. doi:10.3102/0091732X025001099 Gresalfi, M. S., & Cobb, P. (2011). Negotiating identities for mathematics teaching in the context of professional development. Journal for Research in Mathematics Education 42 270 304. Grion, V., & Varisco, B. M. (2007). On line collaboration for building a teacher pro fessional identity. PsychNology Journal, 5 271 284. Groth, R. E. (2007). group. Contemporary Issues in Technology and Teacher Education, 7 490 520. Hannula, M. S., Kaasila, R., Laine, A., & Pehkonen, E. (2005). Structure and typical In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29 th Conference of the International Group for the Psychology of Mathematics Education: Vol 3. Research Reports (pp. 89 96). Melbourne, Australia: University of Melbourne. Hatch, J. A. (2002). Doing qualitative research in education settings Albany, NY: State University of New York Press. Hill, H. C. (2010). The nature and predictors of elemen knowledge for teaching. Journal for Research in Mathematics Education, 41 513 545. Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content spec ific knowledge of students. Journal for Research in Mathematics Education, 39 372 400. Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers' mathematics knowledge for teaching. Elementary School Journal, 105 11 30. doi:10. 1086/428763
197 Hobbs, L. (2012). Examining the aesthetic dimension of teaching: Relationships between teacher knowledge, identity and passion. Teaching and Teacher Education, 28 718 727. doi:10.1016/j.tate.2012.01.010 Hrastinski, S. (2008). What is online le arner participation? A literature review. Computers & Education, 51 1755 1765. doi:10.1016/j.compedu.2008.05.005 Jewett, M. A. (2012). identity in a methods course wiki (Unpublished docto ral dissertation). University of New Mexico, Albuquerque, NM. Kaasila, R. (2007). Using narrative inquiry for investigating the becoming of a mathematics teacher. ZDM Mathematics Education, 3 9, 205 213. doi:10.1007/s11858 007 0023 6 Kaasila, R., Hannula, M S., Laine, A., & Pehkonen. (2008). Socio emotional orientations and teacher change Educational Studies in Mathematics, 67, 111 123. doi:10.1007/s10649 007 9094 0 cont ent knowledge in their teaching: A framework for research applied to a study of student teachers. Journal of Mathematics Teacher Education, 6, 223 252. Kazemi, E., & Hubbard, A. (2008). New directions for the design and study of professional development: A across contexts. Journal of Teacher Education, 59 428 441. doi:10.1177/0022487108324330 Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics Washington, DC: National Academy Press. Koro Ljungberg, M. (2010). Validity, responsibility, and aporia. Qualit ative Inquiry, 16, 603 610. doi:10.1177/1077800410374034 Koro Ljungberg, M., Yendol Hoppey, D., Smith, J. J., & Hayes, S. B. (2009). (E)pistemological awareness, instantiation of methods, and uninformed methodological ambiguity in qualitative research proj ects. Educational Researcher, 38 687 699. doi:10.3102/0013189X09351980 Krzywacki, H. (2009). Becoming a teacher: Emerging teacher identity in mathematics teacher education (Unpublished doctoral dissertation). University of Helsinki, Helsinki, Finland. Lie ber, J., Butera, G., Hanson, M., Palmer, S., Horn, E., Czaja, C., . Odom, S. (2009). Factors that influence the implementation of a new preschool curriculum: Implications for professional development. Early Education & Development, 20 456 481. doi:10. 1080/10409280802506166
198 Lincoln, Y., & Guba, E. (1985). Naturalistic inquiry Beverly Hills, CA: Sage. Mazzolini, M., & Maddison, S. (2003). Sage, guide or ghost? The effect of instructor intervention on student participation in online discussion forums. Co mputers & Education, 40 237 253. doi: 10.1016/S0360 1315(02)00129 X McAleer, D., & Bangert, A. (2011). Professional growth through online mentoring: A study of mathematics mentor teachers. Journal of Educational Computing Research, 44, 83 115. doi:10.2190/ EC.44.1.e McDonough, A., & Clarke, B. A. (2005). Professional development as a catalyst for changes in beliefs and practice: Perspectives from the Early Numeracy Research Project. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A Roche (Eds.), Building connections: Research, theory and practice. Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia Melbourne (pp. 521 528), Melbourne, Australia: MERGA. McMillan, J., & Schumacher, S. (2005). Research in education: Evidence based inquiry (6th ed.). Boston, MA: Allyn & Bacon. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis (2 nd ed.). Thousand Oaks: Sage Publications. Muchmore, J. A. (2002, December). Methods and ethics in a life history study of teacher thinking. The Qualitative Report, 7 Retrieved from http://www.nova.edu/ssss/QR/QR7 4/muchmore.html National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author National Council of Teachers of Mathematics (2006). Learning number sense from a broken calculator Retrieved from www.nctm.org/eresources/view_article.asp?article_id=7457 National Council of Teachers of Mathematics. (n.d.). A meter of candy Retrieved from http://illuminations.nctm.org/Lesson.aspx?id=3229 National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards for mathematics Washington, DC: Author. No Child Left Behind (NCLB ) Act of 2001, Pub. L. No. 107 110, Â§ 115, Stat. 1425 (2002). Patton, M. Q. (1999). Enhancing the quality and credibility of qualitative analysis. Health Services Research, 34 1189 1209.
199 Patton, M. Q. (2001). Qualitative research and evaluation methods. T housand Oaks, CA: Sage. Pearson, P. D., & Gallagher, M. C. (1983). The instruction of reading comprehension. Contemporary Educational Psychology, 8 317 344. doi:10.1016/0361 476X(83)90019 X Philipp, R. A. (2007). In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (257 315). Reston, VA: National Council of Teachers of Mathematics. Picciano, A. (2002). Beyond student perceptions: Issues of interaction, presence, and performance in an online class. Journal of Asynchronous Learning Networks, 6 21 40. Ponte, J. P., & Santos, L. (2005). A distance in service teacher education setting focused on mathematics investigations: The role of reflection and collaboration. Interactive Educational Multimedia, 11 104 126. Putnam, R. T., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning? Educational Researcher, 29 4 15. doi:10.2307/1176586 Rodgers, C., & Scott, K. (2008). The development o f the personal self and professional identity in learning to teach. In M. Cochran Smith, S. Feiman Nemser, D.J. McIntyre & K.E. Demers (Eds.), Handbook of research on teacher education: Enduring questions and changing contexts (pp. 732 755). New York, NY: Routledge. Rossman, G., & Rallis, S. (2003). Learning in the field: An introduction to qualitative research (2nd ed.). Thousand Oaks, CA: Sage. Salmon, G. (2004). E moderating: The key to teaching and learning online London: Routledge Falmer. Settlage, J. Southerland, S. A., Smith, L. K., & Ceglie, R. (2009). Constructing a doubt free teaching self: Self efficacy, teacher identity, and science instruction within diverse settings. Journal of Research in Science Teaching, 46 102 125. doi:10.1002/tea.20268 S chwandt, T. A. (1997). Qualitative inquiry: A dictionary of terms. Thousand Oaks, CA: Sage Publications. Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity. Education al Researcher, 34 14 22. doi:10.3102/0013189X034004014
200 Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher 15 4 14. doi:10.3102/0013189X015002004 Sinclair, M., & Owston, R. (2006). Teacher professional devel opment in mathematics and science: A blended approach. Canadian Journal of University Continuing Education, 32 43 66. Smith, J. (2011). The meanings of teaching from the perspective of exemplary and experienced teachers (Unpublished doctoral dissertation) The Pennsylvania State University, University Park, PA. Spillane, J. P. (2000). Constructing ambitious pedagogy in the fifth grade: The mathematics and literacy divide. Elementary School Journal 100 307 330. Suh, J. M. (2007). Tying it all up. Teaching Children Mathematics, 14, 163 169 Sutherland, L., Howard, S., & Markauskaite, L. (2010). Professional identity creation: their work as teachers. Teaching and Teacher Education, 2 6, 455 465. doi:10.1016/j.tate.2009.06.006 Swars, S. L. (2005). Examining perceptions of mathematics teaching effectiveness among elementary preservice teachers with differing levels of mathematics teacher efficacy. Journal of Instructional Psychology, 32 139 147. Swars, S., Hart, L. C., Smith, S. Z., Smith, M. E., & Tolar, T. (2007). A longitudinal study of elementary pre School Science and Mathematics, 107 325 335. doi:10.1111/j.1949 8594.200 7.tb17797.x Timostsuk, I., & Ugaste, A. (2010). Teaching and Teacher Education, 26 1563 1570. doi:10.1016/j.tate.2010.06.008 Turniansky, B., & Friling, D. (2006). The Expert Novice. In S. A. Barab K. E. Hay, & D. T. Hickey (Eds.), Proceedings of the 7 th International Conference on Learning Sciences ICLS (pp. 778 784). International So ciety of the Learning Sciences, Bloomington, IN: ISLS. van Veen, K., & Sleegers, P. (2006). a context of change. Journal of Curriculum Studies, 38 85 111. doi:10.1080/00220270500109304 Vosniadou, S. (1994). Capturing and modeling the process of conceptual change. Learning and Instruction, 4 45 69. doi:10.1016/0959 4752(94)90018 3 Wenger, E. (19 98). Communities of practice: Leaning, meaning, and identity New York: Cambridge University Press.
201 Wise, A., Chang, J., Duffy, T., & del Valle, R. (2004). The effects of teacher social presence on student satisfaction, engagement, and learning. Journal of Educational Computing Research, 31 274 271. doi:10.2190/V0LB 1M37 RNR8 Y2U1 Xu, S., & Connelly, F. M. (2009). Narrative inquiry for teacher education and development: Focus on English as a foreign language in China. Teaching and Teacher Education, 25 21 9 227. doi:10.1016/j.tate.2008.10.006 Yang, S. C., & Liu, S. F. (2004). Case study of online workshop for the professional development of teachers. Computers in Human Behavior, 20, 733 761. doi:10.1016/j.chb.2004.02.005 Zembylas, M. (2003). Caring for teac her emotion: Reflections on teacher self development. Studies in Philosophy and Education, 22 103 125.
202 BIOGRAPHICAL SKETCH Sherri Kay Prosser graduated from the University of Florida with a Bachelor of Arts in Education in special education in 1996 and a Master of Education in special education in 1997. After graduation she moved to V olusia County and taught elementary school special education for five years and middle school mathematics for seven years. In 2009, she enrolled at the University of Florida and began work on a Doctor of Philosophy in curriculum and instruction with an emp hasis on mathematics education. Sherri graduated in May 2014 and works as a consultant for Volusia County Schools in the Office of Professi onal Development and Support She creates online teacher professional development relating to best practices in mathe matics and English language arts