1 E XPLORING TEACHER IMPLEMENTATION OF REFORM ORIENTED MATHEMATICS INSTRUCTION USING THE EARLY CHILDHOOD LONGITUDINAL STUDY KINDERGARTEN CLASS OF 1998 99 By JUNGAH BAE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVE RSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014
2 Â© 2014 Jungah Bae
3 To Jaeyoung Kwon
4 ACKNOWLEDGMENTS Pursuing a PhD is diff icult but was even more difficult for me as an international student. It was a lonely path, but I was not alone. First, my committee chair, Professor Cynthia Griffin, always guided me during this arduous journey. I am particularly indebted to her: She p rovided me with opportunities to work with her and to participate in an incredible research project regarding improving mathematics education for all students. She continually and convincingly conveyed a spirit of adventure in research and excitement in t eaching. She also spent endless hours listening to my ideas, proofreading my writing , and offering excellent suggestions. I would also like to acknowledge and sincerely thank my committee member, Professor James Algina. Under his mentoring, I learned n ot only statistical and methodological skills and knowledge, but also how to be a researcher and educator. Without his guidance and continual help, this dissertation would not have been possible. His influence has been especially valuable because he set a great example for me as a researcher and educator with his willingness to nurture students in every phase of their learning, his careful attention in their conversations, and his patience in watching over their progress. In addition, many thanks go to P rofessor Mary Brownell and Professor Paul Sindelar for their excellent guidance, caring, and patience with me. Special thanks also go to all staff members in the department and to Carrie for her support in my writing as well as her friendship. Above al l, I want to thank my husband Jaeyoung for his love, support, and great patience at all times. My family in Korea and all my friends have given me their unequivocal support throughout, as always, for which my mere expression of thanks does not seem suffic ient. I also thank all my special students whom I have met, known, or taught. And finally, I praise and thank God. His voice always raises me up:
5 Be vigilant. Stay firm in the faith. Be brave and strong. Let everything you do be done in love. ------1 Corinthians 16: 13 14
6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 9 LIST OF FIGURES ................................ ................................ ................................ ....................... 10 LIST OF ABBREVIATIONS ................................ ................................ ................................ ........ 11 ABSTRACT ................................ ................................ ................................ ................................ ... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 14 Statement of Problem ................................ ................................ ................................ ............. 14 A Brief History of Reform oriented Mathematics ................................ .......................... 17 Teacher Implementation of Reform oriented Mathematics Instruction .......................... 20 Rationale for the Study ................................ ................................ ................................ ........... 26 Early Mathematics Learning: The Selection of the 1 st to the 3 rd Grade Data .................. 29 The Desirable Outcomes: The Selection of Variables of Prior Achievement, Motivation, and Reform oriented Teaching ................................ ................................ 30 Interactive Relationshi ps between Student and Teacher Variables ................................ . 34 Purpose of the Study ................................ ................................ ................................ ............... 36 Conceptual Framework ................................ ................................ ................................ ........... 38 Research Questions ................................ ................................ ................................ ................. 40 Significance of the Study ................................ ................................ ................................ ........ 41 Definitions of Terms ................................ ................................ ................................ ........ 42 Delimitations ................................ ................................ ................................ ................... 43 Limitations ................................ ................................ ................................ ....................... 44 2 REVIEW OF THE LITERATURE ................................ ................................ ........................ 46 Characteristics of Reform oriented Mathematic Education ................................ ................... 47 Reform oriented Mathematics Education and Teacher Outcomes ................................ ......... 54 Reform oriented Mathematics Education and Student Outcomes ................................ .......... 61 Motivational dimensions of reform oriented mathematics education ................................ .... 65 Various student populations in reform oriented mathematics education ............................... 70 3 METHOD ................................ ................................ ................................ ............................... 76 Description of Data and Sam ple Selection ................................ ................................ ............. 76 Primary Sampling Unit, Stratification, and Sampling Weights ................................ ...... 77 Construction of Analysis Samples ................................ ................................ ................... 79 Instruments ................................ ................................ ................................ ............................. 80
7 Teacher questionnaires ................................ ................................ ................................ .... 80 Self description questionnaire mathemati cs (SDQ Math) ................................ ............... 81 Direct cognitive assessment ................................ ................................ ............................ 82 Analytic Approaches ................................ ................................ ................................ .............. 84 RQ1. Do interpretable teacher profiles of teacher practice emerge using the mathematics instructional items in the ECLS K 3 rd grade data set? ............................ 84 RQ2. Do the mathematics self concept items f rom the self description questionnaire (SDQ) that are included in the ECLS K 3 rd grade data set define a single mathematics self concept SDQ factor? ................................ ............................. 87 RQ3. Is the relationship between 1 st grad e mathematics achievement and 3 rd grade mathematics achievement mediated by mathematics self concept? ............................ 87 relationship by mathematics self concept between 1 st grade mathematics achievement and 3 rd grade mathematics achievement? ................................ ............... 90 RQ5. Do differences exist in the parameters defined with the teacher latent classes as the moderator for two mathematics performance groups: struggling learners and typical learners? ................................ ................................ ................................ .... 93 4 RESULTS ................................ ................................ ................................ ............................... 9 6 Research Quest ion 1: Teaching Profiles of the 3 rd Grade Teachers ................................ ....... 96 Research Question 2: Factor Structure of Mathematics Self concept ................................ .. 105 Resear ch Question 3: Mediation Modeling of Prior and Subsequent Achievement via Mathematics Self concept. ................................ ................................ ................................ 107 Research Question 4: Moderation Effect of Teaching Profile on the Mediation Relationship ................................ ................................ ................................ ...................... 108 Research Question 5: Difference in moderated mediation effects by student groups .......... 112 5 DISCUSSION ................................ ................................ ................................ ....................... 117 Interpretation of Findings ................................ ................................ ................................ ..... 118 Three Subgroups of 3 rd Grade Teachers Who Share Similar Teaching Profiles Identified by the Latent Class Analysis ................................ ................................ ..... 119 A Learning Mechanism Prior and Subsequent Achievement Mediated by Mathematics Self concept Explored by the Mediation Modeling ............................. 125 The R elationship between the Implementation of Reform oriented Mathematics and the Learning Mechanism Established by the Mediation Modeling Different Influences for Three Teaching Profiles Explored by the Moderated Mediation Modeling ................................ ................................ ................................ .................... 127 Lack of Evidence of Reform oriented Mathematics as Best Practices for Elementary School Students, Including Low achievers Findings from Two Moderated Mediation Models ................................ ................................ .................... 131 Implications of the Present Study ................................ ................................ ......................... 136 Practical Implications of the Findings ................................ ................................ ........... 137 Policy and Research Implication s of the Findings ................................ ........................ 141 APPENDIX
8 A VARIABLES ................................ ................................ ................................ ........................ 147 B M PLUS PROGRAMS ................................ ................................ ................................ .......... 149 LIST OF REFERENCES ................................ ................................ ................................ ............. 154 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ....... 172
9 LIST OF TABLES Table page 3 1 Sample S ize and Case Size ................................ ................................ ................................ 77 3 2 Weights, PSUs, and Strata ................................ ................................ ............................... 79 3 3 The Highest Proficiency Level of 1 st and 3 rd Grade Samples ................................ .......... 83 3 4 Mathematics IRT Scale Score ................................ ................................ ........................... 84 4 1 Model fit statistics. ................................ ................................ ................................ ............ 97 4 2 AN COVA Results ................................ ................................ ................................ ........... 110 4 3 Model Tests of Multiple Group Approach to Moderated Mediation Modeling 1 .......... 112 4 4 Moderated Mediation Associ ation by Teaching and Learning Groups .......................... 115
10 LIST OF FIGURES Figure page 3 1 Mediation modeling ................................ ................................ ................................ ........... 89 3 2 Variations on moderated mediation modeling in Preacher et al. (2007) ........................... 91 3 3 Moderated Mediation Model ................................ ................................ ............................. 92 4 1 Pr ofiles of Response Probabilities Across Response Categories: Traditional Instruction Variables of Textbook; Worksheet; Concepts/Facts; Procedural; and Math Tests ................................ ................................ ................................ ........................ 100 4 2 Profiles of Response Probabilities Across Response Categories: Components of Explicit Reform ori ented Instruction Variables of Calculators; Measuring Instruments; Manipulatives; and Computers. ................................ ................................ .. 101 4 3 P rofiles of Response Probabilities Across Response Categories: Reform oriented Instruction Variables of Group Work; Real Life; Talking; W riting; Discussion; and Project ................................ ................................ ................................ .............................. 102 4 4 Profiles of Response Probabilities Across Response Categories: Topic Variables (Number/Operation; Measurement; Geometry; Statistics; and Algebra) ........................ 103 4 5 Profiles of Response Probabilities Across Response Categories: Specific Mathematics Concept and Process Variables (Place Value; Fraction; Estimation; Analytic Reasoning ; Communication; and Shape) ................................ .......................... 104 4 6 Standardized loadings for 1 Factor Confirmatory Mo del of Mathematics Sel f concept ................................ ................................ ................................ ............................. 106 4 7 Mediation: Effe cts of Prior Achievement on Subsequent Achievement mediated by Mathematics Self concept ................................ ................................ ................................ 108 4 8 Unconditional mediation model test with combined data set with teacher and student data ................................ ................................ ................................ ................................ ... 109 4 9 Multiple Group Approach for Moderated Mediation Modeling: There Different Mediation Relationship by Teaching Profile of Anti reform, Active Impleme nter, and Proactive Implementer ................................ ................................ .............................. 110 4 10 Student Group Moderated Mediation Modeling: There Different Mediation Relati onship by Teaching Profile of Anti reform, Active Implementer, and Proactive Implementer ................................ ................................ ................................ ..................... 113
11 LIST OF ABBREVIATIONS ECLS K The Early Childhood Longitudinal Study, Kindergarten Class of 1998 99 . The ECLS K data provide information on childr en's early school experiences beginning with kindergarten and following children through middle school. NCTM National Cou ncil of Teachers of Mathematics was found in 1920. As the its publi shed standards have been highly influential in the direction of mathematics education in countries around the world, in particular for the 1990s reform efforts in the United States. STANDARDS The phrase "NCTM Standards," or just "Standards"(capitalized), will be used for the National Council of Teachers of Mathematics recommendations for K 12 curriculum, teaching, and assessment contained in the four volume set (Curriculum and Evaluation Standards for School Mathematics [1989; 2001], Professional Standards for Teaching Mathematics , and Assessment Standards for School Mathematics ), all published in Reston, VA, by the NCTM. SDQ M Self Description Ques tionnaire Mathematics is one sub test of t he Self Description Questionnaire (SDQ) , which is a mu lti dimensional instrument designed to measure seven facets of self concept hypothesized in Shavelson's hierarchical model. It asks student to complete several questions on mathematics self concept. Currently SDQ III version was developed and used.
12 A bstract 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 EXPLORING TEACHER IMPLEMENTATION OF REFORM ORIENTED MATHEMATICS INSTRUC TION USING THE EARLY CHILDHOOD LONGITUDINAL STUDY KINDERGARTEN CLASS OF 1998 99 By Jungah Bae August 2014 Chair: Cynthia Carlson Griffin Major: Special Education A secondary analysis of the Early Childhood Longitudinal Study Kindergarten (ECLS K) data set was conducted to investigate early mathematics learning with in the context o f reform oriented mathematics teaching. The national level, process driven reform movement has been driven by three underlying assumptions concerning the impact of this effort. First, the effort would guide teachers to implement reform oriented instruction with support from policy documents (e.g., NCTM Standards ) and instruments (e.g., NSF funded curricula). Changes in classroom teaching would positively influence student outcom es, including achievement and motivational dimensions. And, by improving mathematics teaching in the early childhood years, . To explore these assumptions, t he present study adopted con ceptually and methodologically alternate frameworks: (a) a person oriented analysis was used to capture instruction from the perspective of reform oriented mathematics teaching ; (b) mediation modeling was used to empirically test a unidirectional associati on between mathematics achievement and mathematics self concept; (c) the established learning mechanism was introduced in the subsequent models as the outcome variable; and, (d) moderated mediation modeling was employed to examine differences in the
13 learni ng mechanism by teacher subgroups and by student groups in accordance with reform policy expectations . Results showed that mathematics self concept significantly mediated the relationship between prior and subsequent achievement during the period from 1 st grade to 3 rd grade. However, none of the three teacher subgroups successfully re established this learning mechanism for early elementary school students as a whole, or among groups of typical and struggling learners . I mplications for research and policy a re discussed.
14 CHAPTER 1 INT R O DUCTION Statement of Problem Since the early 1990s, a national level, process driven reform movement has produced influential documents outlining instructional content and pedagogical approaches in mathematics education (e.g., National Council of Teachers of Mathematics [NCTM] Standards, National Science Foundation [NSF]). The NCTM Standards based curricular materials (e.g., NSF funded textbooks) and professional development programs were expected to link policy, instruction, and student outcomes (Cohen & Hill, 2000; Swanson & Stevenson, 2002). These reform movements were initiated with three underlying assumptions. First, reformers believed that the reforms could aching and learning, leading to changes in c lassroom practices. Second, the s e changes in mathematics classroom s can positively influence student outcomes, such as motivation to learn mathematics and, ultimately, achievement. Finally, these desirable chan ges were targeted to all students. Success entailed improving achievement as well as closing the achievement gap. Accordingly, how these three assumptions were manifested has been a key question for more than two decades. However, several obstacles exis t that preclude successful effects on implementation at the classroom level, such as ambitious and ambiguous goals, insufficient resources, and an environment with a broad array of circumstances (Cohen, Moffitt, & Goldin, 2007). These obstacles are unfavo rable for mathematics education, in general, but at the elementary school level, in particular. First, ambitious goals for changes in mathematics teaching and learning require a sharp departure from extant practices, so called traditional instruction, suc h as switching from teaching mathematics procedures (e.g., computation) to securing a conceptual understanding in students (Ball, 2001). NCTM and NSF supported, reform oriented
15 mathematics also presents ambiguities for policy developers and implementers because this type of teaching, learning, and improvement in mathematics was not carefully articulated with specific approaches and detailed examples. The school and classroom environments were also unfavorable for reform. US elementary school teachers ha ve been historically and chronically subjected to a lack of professional knowledge to teach mathematics (Ball, 1990; Borko et al., 1992; Ma, 1999). In addition, the situation can be exacerbated by negative experiences they may have had as mathematics lear ners, their opposing belief systems about mathematics teaching and learning, and a general lack of confidence in teaching mathematics (Bursal & Paznokas, 2006; Harper & Daane, 1998; Gresham, 2008). Furthermore, the emphasis on reform oriented mathematics education in the process of mathematics learning (e.g., problem solving, reasoning, communication) rather than solely on the content areas to be learned is challenging for teachers to implement, especially novice teachers (Stein, Engle, Smith, & Hughes, 20 08). That is, teachers are expected to capitalize on their expertise in making sense of the process components (i.e., NCTM Standards) and implement ambitious pedagogy across multiple mathematics topics. This is a challenging task for most teachers but ev en more so for those who lack solid knowledge of mathematics and teaching mathematics and show negative dispositions toward mathematics. Although this serious dilemma existed between the goals of challenging mathematics instruction and the realities of mathematics classrooms, teachers were viewed as capable of bridging the gap by virtue of the information included in the policy documents, curricular materials, and professional development (PD) programs created and aligned with the NCTM Standards. Furthe rmore, teacher implementation of reform oriented mathematics instruction was expected to positively influence student outcomes (Swanson & Stevenson, 2002; NCTM, 1991, 2000) , including achievement and motivation (Senk & Thompson, 2003). These beneficial
16 ef fects were expected to apply to the early mathematics learning of all students, both typically developing and struggling learners. That is, reformers believed that teachers would be empowered to achieve the strategic goals of the process driven mathematic s reform policies using the documents and resources provided to successfully implement the tenets for allowing all students to mindfully engage in challenging mathematics learning (Cohen & Hill, 2000; Swanson & Stevenson, 2002). However, the literature on the early elementary school level includes mixed or limited results regarding the implementation and its impact on reform oriented mathematics instruction, thereby necessitating further research (Ross, McDougall, Hogaboan Gray, & LeSage 2003). The main q uestion arising from this problem is how researchers conceptualized components of the process for successful implementation of reform oriented mathematics. In addressing this question, the argument of the current study is that three areas need exploration with regard to both conceptual and methodological research domains: Distinctions must be made in the literature between what teachers are expected to implement and how teachers implement reform oriented mathematics instruction. A need exists for altern desirable outcomes beyond achievement alone. Few studies have focused on comparisons of students at different levels of mathematics learning and the use of reform oriented mathematics instructio n, yielding mixed results. The present study thus aims to explore the relationships between teacher and student factors in early mathematics achievement during a period of mathematics education reform that was influenced by a series of mathematics standar ds publications put forth by the NCTM (1989, 1991, 2000). The Early Childhood Longitudinal Study Kindergarten Class of 1998 99 (ECLS K) offers a means for examining the influence of reform efforts on mathematics teaching and student learning using this ex tant, large scale data set. In addition, the ECLS K data set allows
17 for examination of important conceptual and methodological domains in educational research. To substantiate the significance of this study, background information regarding reform orient ed mathematics and its implementation by teachers is provided in this chapter. A Brief History of Reform oriented Mathematics After a plethora of historical events over the last several decades, from William hing in schools, to the progressive life adjustment movements of the 1930s, to the New Math period of the early 1950s through the 1960s, to the Back to the Basics movement of the 1970s, and A Nation at Risk (1983) of the 1980s, a general consensus has been reached on the importance of quality mathematics education, including that it be taught beginning in the early school years and by quality mathematics teachers, and be marked by accountability for high standards (Klein, 2003; Schoenfeld, 2004). Unfortuna tely, it has been harder to arrive at an agreement about how to achieve those goals. The history of the NCTM Standards (1989, 2000) development illustrates this struggle. The 1989 NCTM Standards supp ort idea l s of a progressive and utilitarian viewpoint on mathematics education which have played a pivotal role in developing state level frameworks. On the other hand, the Standards and its implementation have faced a number of criticisms, primarily but not exclusively, from mathematicians and parents, incl uding, (1) the de emphasis on mathematical content to be learned, (2) shortcomings inherent in the constructivist pedagogy, (3) reservations about the effectiveness of the Standards on students from low income families, and (4) the drive by precarious poli tical slogans or market forces, not empirical evidence (Raimi & Braden, 1998; Sykes, 1998; www.Mathematicallycorrect.com). For now, through the unprecedented process of accepting input from relevant constituencies, the NCTM Principles and Standards for S chool Mathematics (2000) has attempted to find stability between the two extremes. This has altered the equilibrium by shifting
18 the focus away from the radical disposition and placing it on typical algorithms and computational fluency. Now that the mathe matics instruction inspired by the NCTM 2000 Standards remains based on the progressive and utilitarian perspectives of mathematics teaching and learning, contemporary state level standards have continued to evolve so as to strike a balance between reforma tive and traditional principles. For example, in a series of standards analyses by the Thomas B. Fordham Foundation, The Sunshine State Standards and the relevant state level guides of Florida were evaluated in the 1990s as unclear in terms of what studen ts for problem solving (Klein, 2005; Raimi & Braiden, 1998). More recently, the panel of ed the current 2008 state level standards and guides as excellent in terms of clarity, specificity, and content rigor. The report explains that Florida could achieve superior standards, even to the Common Core Mathematics Standards (CCMS), by emphasizing the mathematics content that is most important at each grade level (Carmichael, Martin, Proter Magee, & Wilson, 2010). Of note, this is an important advancement given that the CCMS is regarded as the new national standards, adopted by all but a few states (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). The extreme shifts from the New Math to the Back to the Basics or the California Math Wars , depict the struggle to find a balance between different epistemological perspectives on mathematics teaching and learning; some including traditionalists who consider school mathematics to be a mental discipline, while others consider it as authentic preparation for placed on the academic subject (i.e., the core mathematical topic s ), or the pedagogical principles
19 (e.g., the alignment with the learner characteristics). History, however, teaches that it is important to b e vigilant against considering this issue as an oversimplified dichotomous question. Not all mathematics topics are included in the standards because the standards should inform how to teach selected mathematics topics and how to prepare teachers to profi ciently teach them. Delivering rich mathematical topics without pedagogical considerations could Schoenfeld, 2004). Also, hands on activities, mathematical discussions, and real life problem solving do not necessarily assure conceptual learning, mathematical proficiency and, by extension, mathematical literacy to the extent that is required to be successful in society (Cohen, 1990). It should be noted that any one of the five mathematical proficiency stran ds (i.e., conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition) should not and cannot be weighted against the others, as they are all interdependent and interwoven (NRC, 2001). After all, the c areful examination of actual teaching and learning of what has been determined as school mathematics is key, not the unilateral arguments based on epistemological perspectives on mathematics teaching and learning (Schoenfeld, 2004). In sum, reformative eff orts inspired by a series of mathematics standards publications (NCTM, 1989; 1991; 2000) have influenced the process oriented instruction that emphasizes mathematical problem solving, thinking, and communication in the classroom since the 1990s (Senk & Tho mpson, 2003). Also, the balanced viewpoints and elaborated notions of mathematical proficiency (NRC, 2001) and instructional practices (RAND Mathematics Study Panel, 2002) offer valuable guidance to educators, researchers, policy makers, and parents on ma thematics teaching and learning. Currently, however, it is difficult to conclude that mathematics education research has a considerable body of evidence indicating that those efforts
20 have been successful during this intensive period of mathematics educati on reform. Without such convincing proof, it is difficult to know whether reform efforts have been or will be linked to changes in the teaching practices and/or student outcomes reformers intended. Teacher Implementation of Reform oriented Mathematics In struction Regarding implementation of reform oriented mathematics, one can assume two reactions, active adoption and implementation shortfall. However, policy implementation research does not link active adoption to mindless compliance or implementation s hortfall to policy resistance by implementers, which emphasizes the importance of sense making by implementers su ch as teachers (McLaughlin, 2005 ). Depending on how teachers make sense of the new practices, which requires them to understand policy message s and their environment, an unintended failure of implementation can occur. Thus, the implementation of reform oriented and learning before f undamental changes o ccurred . Several qualitative studies have explained the potential facilitators and barriers to successful enactment of reform oriented mathematics instruction, focusing on (a) teacher knowledge and cognition (e.g., Cohen, 1990; Spillane, 2000 a ; Spillane & Zeulis, 1999; Spillane, their changes in belief systems (Brown, 2003; Gellert, 2000; Skott, 2001), (c) teacher interactions with mathematics curriculum (Collopy, 2003; Remillard, 1999, 2000; Remillard & Bryans, 2004), and making process of mathematics content and reform oriented mathematics teaching (Drake, 2006; Drake, Spillane, & Huffered Ackles, 2001). Collopy (2003), for example, explored the learning proc ess of two veteran elementary teachers while they were piloting a reform oriented curriculum. These teachers shared such similarities as no support from additional professional development and concerns about conflicts
21 between reform oriented mathematics ( e.g., problem solving and conceptual understanding) and mandated high stakes tests (e.g., standard algorithms and procedural knowledge). The teachers also displayed a high level of belief in their self efficacy as mathematics oriented curricular materials and subsequent teaching practices differed substantially. One le the other reported considerable changes in philosophy and demonstrated changes in mathematics instructional practices. The difference in implementation may occur because the actual classroom practice is shaped by the reform messages in a complicated w ay. In response to the new suggestions, teachers should understand what reform oriented mathematics means through various cognitive activities, including reading, interpreting, learning, and ultimately sense making. In this process, prior knowledge and e xperiences, as well as beliefs about the subject, teaching, and student learning, are involved in changing established schema (Spillane et al., 2006). The characteristics of schema that are newly constructed can explain why some teachers use manipulative materials (Cohen, 1990; Spillane & Zeuli, 1999). This unintended failure of implementation occurs because implementers might attend to superficial features or familiar ideas, hindering them from making more fundamental transformations of their schema. Meanwhile, teachers who are able to fundamentally transform their existing schema of mathematics teaching become successful implementers who integrate the underl ying ideas of reform texts into their teaching practices. They do so by rejecting their previous beliefs and knowledge base (Spillane et al., 2006).
22 However, this epistemological transformation is neither easy nor simple in any subject matter, but the c ase of mathematics teaching can be even more difficult (Spillane et al., 2006; Burch & Spillane, 2003; Drake, 2006; Drake et al., 2001). Many elementary teachers have reported that their prior experiences in mathematics classrooms were negative and devoid of reform oriented mathematics education. Their prior experiences negatively influenced their confidence and self efficacy for teaching mathematics, in particular , reformative approaches (Bursal & Paznokas, 2006; Harper & Daane, 1998; Gresham, 2008). Ho wever, the reform policy instruments (e.g., curriculum materials and PD programs) were expected to contribute to changes in belief systems, knowledge bases, and thereby teaching practices despite complexities associated with those changes. For example, Dr ake and colleagues (2006; Drake et al., 2001) oriented mathematics, finding different responses to reform messages in different teachers. According to Drake and colleague s ( 2006; Drake et al., 2001 ), learning identity refers to the perceptions we hold of ourselves as learners and teachers. Elementary teachers' identities in mathematics often differed from those of literacy or reading in that they saw learning opportunitie s for literacy in all aspects of life, while they tended to limit learning opportunities for mathematics to a formal classroom setting. Teachers shared a common story about literacy; if they encountered difficulties with literacy teaching and learning, th ey were similarly committed to active problem solving. The story about mathematics teaching and learning is more divergent, including three different identities: (a) turning point, (b) failing, and (c) roller coaster. Arguably, recent positive experience s from PD opportunities focused on reform oriented instruction and curriculum can empower teachers to change their perceptions of themselves as mathematics learners and teachers. In particular, "turning point" teachers set forth
23 their intentions to contin uously change their identities and teaching practices. However, these teachers narrowed their attention to affective dimensions of mathematics instruction rather than various components including mathematics content. Meanwhile, "roller coaster" teachers who recently experienced mixed success and failure revealed a desire to improve their students' experiences of learning mathematics by improving their own mathematics content knowledge (Drake et al., 2001). Among teachers who had negative mathematics life stories but current positive experiences, two groups were identified: The" turning point" was related (a) to teaching and learning and (b) only to teaching. Only teachers in the former group sought ways to incorporate what they were learning about reform oriented mathematics education into their teaching practices. Among these "turning point" teachers, two groups were further identified: (a) content based and (b) tool based. "Content based" teachers noticed and interpreted reform oriented messages in a broader way so that fundamental transformations took place in their epistemological views on mathematics content and the teaching and learning of mathematics. In contrast, "tool based" teachers focused on the value of manipulative materials and developed a narrow definition of reform oriented mathematics as, for example, the frequent use of manipulative materials to make mathematics learning fun. Consequentially, observations of "content based" teachers' classrooms revealed that much more had been achieve d in the implementation of reformative teaching, such as mathematical discourse between students, justifications for answers given, and the use of multiple solutions and representations (Drake, 2006). In the same vein, other studies ( Remillard , 1999, 2000 ; Remillard & Brians, 2004) also
24 agenda, and beliefs about teaching and learning mathematics led to differences in teacher learning via teacher interactions with curr iculum materials. reform texts, but also their environments. Persistent tensions that school district leaders face in instruction reform efforts (Knapp, 2008) should also a pply to classroom teachers. However, maintaining a singular reform agenda with calls for satisfying multiple agendas which often compete with each other and confronting unanswered questions are difficult. A notable example is the nature of teaching math ematics to struggling students. In the case of two veteran teachers in their challenging mathematics classrooms. To accommodate the academic needs of low ac hieving students, the teacher who adhered to traditional instruction replaced the new reform oriented curriculum with a traditional curriculum. However, the teacher who was open to new main goals in her classroom but a ddressed the missing content in the reform oriented material based on her understanding of student performance as well as her newly changed understanding of mathematics education as derived from the new curriculum. Therefore, student characteristics, in p articular the level of mathematics oriented mathematics. This necessitates including various student populations in evaluating the implementation of reform mathematics. Collecti knowledge bases, and thereby teaching practices is complex, such policy instruments as curricular materials and PD programs affected this transformation process, according to several
25 link to student outcomes should be investigated. Therefore, the c urrent study extended the body of policy implementation literature by first using large scale da ta and then linking instruction to student outcome variables. To achieve this end, not only student but also teacher data from the 1st grade to the 3rd grade i n ECLS K were selected for this study. This data set allows for the examination of both cognitive and non cognitive variables in mathematics learning as well as instructional variables in mathematics teaching at the early elementary level. Of note, the current study does not directly evaluate policy implementation. It is not possible for a secondary analysis of large scale data to generate information regarding school or district contexts, specific policy instruments or interventions, or outcomes for an evaluation of the link from input to output. Instead, a key argument in this study is that it is necessary to depict what mathematics classrooms looked like during a time of mathematics reform efforts at the national level rather than at the level of the specific locations receiving special policy interventions. This argument is plausible given that in the 1990s m athematics reform movement was sweeping the nation, and US mathematics classrooms made changes, to some degree, due to unprecedented reform eff orts and lively debates regarding those classrooms. According to the TIMSS 1995 and 1999 video studies ( Jacobs, Hiebert, Givvin, Hollingsworth, Garnier, & Wearne, 2006), many teachers have become aware of the NCTM Process Standards (Problem Solving, Reaso ning and Proof, Communication, Connections, and Representation) and believed themselves to be implementers of teaching practices inspired by those standards even before official publication of the new version (NCTM, 2000). Progress continued in the implem entation of these reform oriented practices, according to the TIMSS 2007 International Report (Mullis, Martin, & Foy, 2008).
26 Research gaps exist in conceptually and methodologically different approaches to portray a picture of US mathematics classrooms f ocused on early mathematics learning during a time of reform efforts. Thus, the current study aims to synthesize the body of literature regarding reform oriented instruction and student mathematics learning and to present alternate approaches to capture t he features of teaching (i.e., teaching profiles from the latent class analysis) and to evaluate its effect on the desirable outcome (i.e., effects on a learning mechanism from the moderated mediation modeling). The nature of this study, however, highligh ts the importance of justifying the rationale for the selection of sub set(s) of data and variables from the large scale data set. In the following section, the rationales for selecting the grade levels and variables of teaching and students are presented . Rationale for the Study A recent content analysis of textbooks spanning over 100 years has revealed that elementary school mathematics materials have undergone significant changes between 1991 2000 (Baker et al., 2010). During that time period, more a dvanced topics were included in classroom materials (e.g., mathematical reasoning, abstraction, conceptual understanding, multiple representations, and strategies) for even very young students, increasing the level of cognitive demand. As such, the many y ears of reform appear to have paid off. For example, Chicago Public Schools (CPS) have achieved modest growth in elementary and middle school mathematics achievement despite challenges, such as the widening gap in scores and growth rates between African A merican students and other racial/ethnic groups, and the inconsistent growth patterns in low performing schools (Luppeschu et al., 2011). Despite the modest results in CPS, the evaluation of reform policies and the implications of these assessments vary d epending on how researchers conceptualize the desired output and the process from input to output. As an evaluation of policy efforts, the current study is designed to delineate not only the
27 conceptual, but also methodological research domains, for bolste ring our understanding of the influence of reform efforts in mathematics education. First, the present study highlights the dynamic environment and process of education (Ball & Forzani, 2007). A key to policy success is bridging the gap between policy m akers and practitioners in terms of knowledge and implementation because policy and practice are mutually dependent (Cohen, Moffitt, & Goldin, 2007). By nature, policy sets up ambitious aims that can create professional incompetence by calling for new kno wledge and changes in practice. Depending on the knowledge and capabilities of policy makers, various policy instruments can be developed in order to clearly inform and support the acquisition and utilization of new knowledge by practitioners. Therefore, the central question on educational policy effectiveness is how efforts enable teachers as practitioners, who are the ultimate policy makers (or breakers), to effectively implement what is intended (Cohen et al., 2007). Hence, this study addresses the im plementation of reform mathematics at the classroom level as one of the important contributors to policy success. Second, examining the educational process implies attention to the interactions among the content, teacher, and students, as well. To enhance understanding of the exchanges among important stakeholders, education research must situate the instructional dynamic of constituents (e.g., beliefs, knowledge, or/and capability of teachers and students) into its conceptual framework of study (Ball & Fo rzani, 2007). Studies conceptualized in this way can offer more specific answers as to why certain approaches may succeed or fail by providing evidence of an intervening mechanism for achievement causality (Chen, 1990; Chen, Donaldson, & Mark, 2011), whic h distinguishes answers suggested by education research from other fields addressing similar educational questions (Ball & Forzani, 2007).
28 Finally, one of the most important agendas for education researchers relates to how those interactions among conten t, teachers, and students are methodologically formulated. Structural equation modeling (SEM) has the advantage of being methodologically rigorous, as well as providing a precise evaluation of the intervening mechanism(s) (Chen, 1990; Chen, et al., 2011). Various SEM approaches (e.g., latent class analysis, confirmatory factor analysis, and mediation and moderated mediation modeling) can be employed to investigate complex educational processes. With regard to conceptual and methodological frameworks, the reform policy in mathematics should be studied in the same vein by closely examining the interactive relationships of content, teachers, and students. Mathematics reform efforts have pursued changes in the entire system of mathematics education. The poli cy efforts situated in process driven strategies that capitalize on the standards, curricular resources, and professional development opportunities serve to link policy and practice (Cohen & Hill, 2000; Swanson & Stevenson, 2002). The emphasis on problem solving, reasoning, and communicating in the standards and curriculum was expected to lead to changes in mathematics classrooms so that teachers and students could engage actively in mathematical inquiries and discourse with various strategies and meta str ategic plans (H iebert et al., 1996). Teachers have been subjected to unprecedented demands on their content and pedagogical knowledge when charged with providing students with meaningful problems, tasks, and interactions (Spillane, 1999; 2000b; Spillane & Zeuli, 1999). Meanwhile research evidence has shown that States have responded to reform mandates by actively adopting content and professional standards and accepting performance standards and aligned assessments to some degree. Contrary to the presumed lack of alignment between
29 state and classroom level reforms, classrooms in one state that actively adopted reform standards reported more reform based teaching practices in classrooms (Swanson & Stevenson, 2002). The final analysis in the current study will include examinations of classroom level factors to which stronger and more complex influences have been attributed. When one considers reports and/or direct observations suggesting that reform principles have replaced existing practices as evidence of policy success, there is a caveat. The case does not necessarily equate with policy success when local leaders make sense of disjointed aspects of reform policy (e.g., de mathematize such principles as problem solving and hands on activities) based on their form focused rather than function focused understandings (Spillane, 2000b ). This has also been true for the case when teachers have used the reform rhetoric to rename traditional practices or selectively challenge their behavioral regularities (e.g. , adopting discussion practices) of mathematics instruction without the fundamental transformation of the epistemological regularities (e.g., discussion grounded in mathematical concepts and discourse norms) of mathematics instruction (Spillane & Zeuli, 19 99). Given these concerns, ongoing evaluations with various frameworks are necessary and assessments must be focused on the interactive relationships among reform mathematics, teachers, and students. As such, the present study attempts to link teaching p ractices to student factors including mathematics motivational constructs and achievement in an effort to evaluate the implementation of reformative mathematics education. Early Mathematics Learning: The Selection of the 1 st to the 3 rd Grade Data Early m athematics performance is most important in predictive modeling of the consequent achievement outcome within the period from kindergarten to the 5 th grade (Classen, Duncan, & Engel, 2006). Even after accounting for the predictive functions of social and
30 e conomic status (SES), gender, reading difficulty, special education placement, and learning behaviors, the mathematics difficulties of kindergarten children in this study remained unchanged during subsequent years in elementary school. Furthermore, kinder garten students who showed mathematics difficulties, which can be presumably attributed to the lack of informal knowledge of mathematics in preschool, showed consistent and substantially lower achievement and growth rates (Morgan, Farkas, & Wu, 2009). Giv en that more formal and intensive mathematics instruction typically begins in 1 st grade, it is necessary to investigate the potential mechanism of the trajectory as well as risk factors in early elementary mathematics education. On this account, data from the 1 st and 3 rd grade levels in the ECLS K dataset were chosen for the modeling in the current study. The Desirable Outcomes: The Selection of Variables of Prior Achievement, Motivation, and Reform oriented Teaching Such cognitive variables as numeracy , literacy, and previous achievement have strong and pervasive effects on current achievement in mathematics (Bodovski & Farkas, 2007; Hemmings, Grootenboer, & Kay, 2011; Reynold, 1991; Spinath, Spinath, Harlaar, & Plomin, 2006). At the early elementary s chool level, initial mathematics performance is highly correlated with subsequent performance not only in mathematics but also in reading (Cla e ssen s , et al., 2006). The finding that effort and attitudes associated with mathematics do not exert added influ ence on current achievement in secondary mathematics learning (Hemmings et al., 2011) seems to reaffirm the belief that the rich get richer (i.e., Matthew effects , Stanovich  ). These cognitive factors and the impact of them are often perceived as st able and uncontrollable more frequently in mathematics learning due to its hierarchical nature and dependency on prerequisite knowledge and skill so that prior achievement is evidently a valid variable in the examination of a student achievement mechanism. Accordingly, 1 st grade test
31 score s are linked to the 3 rd grade score s as an initial point in models to explicate the mechanism of early mathematics achievement for the present study. However, the nature of mathematics learning is complex and dynamic, no t being explained simply by prior achievement. Quality learning experiences and affective/behavioral variables are influential throughout the school years (Bodovski & Farkas, 2007; Newton, 2010). For secondary school level students, early tracking and ma king progress in courses taken were associated with increased academic attainment at the 12 th grade and overall growth rates. This continuous exposure to advanced mathematics mitigated the senior slump in mathematics learning , and also exerted a stronger influence on mathematics achievement and growth for students who were initially low performing than for students who had a higher initial level. Such non cognitive variables as educational expectations, self esteem, and behavioral problems also had a sign ificant effect on achievement as well as on growth patterns (Newton, 2010). When it comes to elementary school students, academic engagement plays a pivotal role in high achievement (Bodovski & Farkas, 2007). Academic engagement was associated with achie vement not only positively but also more strongly than the instructional time. When students were grouped into four clusters by their kindergarten attainment, larger gains from K to 3rd grade for initially advanced students can be interpreted as a functio n of more active engagement in the learning activities given that those students had higher engagement levels. Even when a greater amount of instructional time was provided, the first quartile students who were in the lowest achievement level at kindergar ten showed less learning engagement and the smallest growth in achievement throughout their first four years of schooling. More importantly, there was a negative interaction between the fourth quartile group and their learning engagement level in predicti ng growth, indicating that the strength in the relationship between engagement
32 and growth was greater for the group of students who were initially low achievers than for those in other performance groups (Bodovski & Farkas, 2007). These results also bring to light that research needs to attend to the interplay among prior achievement, student engagement in learning activities, and instruction on gains in mathematics performance, in particular for students who perform poorly from the beginning. Yet, it wa s hard to identify measures that could be construed as a mathematics specific engagement variable from the ECLS K data set. As such, this study incorporated academic engagement logically rather than explicitly modeling it. Incorporating a motivation comp onent in the model can serve as an example. Research (Fredricks, Blumenfeld, & Paris, 2004) recognizes three types of academic engagement: behavioral, emotional, and cognitive. Behavioral engagement is pertinent to student conduct and academic behaviors in schools and classrooms (Finn, 1993; Finn, Pannozzo, & Voelkl, 1995; Finn & Rock, 1997). Emotional engagement emerges as affective reactions, such as boredom and excitement at learning activities (Skinner & Belmont, 1993). Cognitive engagement is defin ed as psychological investment in learning or mastering contents presented (Fredricks, et al., 2004). All notions of the three types are similar and interwoven, and also can be interconnected to motivation (Blumenfeld, Kempler, & Krajcik, 2006). Addition ally, including a reform oriented teaching variable is another example of amalgamating academic engagement into the study models. Students were able to engage behaviorally, emotionally, and cognitively (Fredricks, et al., 2004) in the academic activities when the instruction was challenging and supportive. In order to make the learning environments challenging and supportive, higher order thinking and deeper levels of understanding are reinforced and mathematical conversations make frequent connections to the outside world of mathematics (Marks, 2000; Stodolsky, 1988). These characteristics of
33 mathematics classrooms certainly reflect what the reform policy has called for. Therefore, in theory, modeling components of motivational beliefs and reform orient ed instruction makes it possible for findings from this study to imply the student learning mechanism involving academic engagement, to some degree. Taken together, this study supports the notion that prior achievement levels may contribute to subsequent performance in the ECLS K data. The student learning mechanism, however, is not a simple one such that teaching practices and other student level variables, such as mathematics self concept, can be used to better understand the complex process of achievem ent and the difficulties students experience learning mathematics. Reform driven instruction can be more thoroughly examined if instructional practices exert an influence on the presumed academic engagement of students and in turn, on the causal relations hip of prior achievement to other student level variables in the era of mathematics reform. The primary purpose of including teaching variables, of course, is to investigate the interactive relationships between reform oriented mathematics teaching, teach er instruction, and student learning. Limited research has been identified that incorporates teaching relevant factors in the examination of the early mathematics achievement and difficulties of students. Rather, researchers have focused on SES, gender, learning behaviors, and difficulties in other subjects such as reading (e.g., Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Jordan, Kaplan, Locuniak, & Ramineni, 2007; Jordan, Kaplan, Olan, & Locuniak, 2006; Morgan et al., 2009). All of the factors mentione d above are student level and were not linked to teaching level components in these studies. In the current study, by contrast, one of the most proximal variables to mathematics teaching (i.e., instructional practice) will be examined as it relates to tea chers and students.
34 Interactive Relationships between Student and Teacher Variables Teachers possibly exert a positive influence on student achievement by creating classroom learning environments inspired by reform principles. Often, this influence inter acts with student background characteristics, the curriculum focus, and achievement tests used. However, there is a lack of research devoted to the examination of the interactive influences of teacher, content, and students on student outcomes in the cont ext of reform oriented mathematics education. A line of research attempt ing to establish a link between student and teacher components is the opportunity propensity model (O P model) of achievement by Byrnes and his colleagues (Byrnes, 2003; Byrnes & Mill er, 2007; Byrnes & Wasik, 2009; Jones & Byrnes, 2006). The intent of using this integrative model is to fit together the various pieces of the mathematics achievement puzzle. From a structural equation model incorporating as many factors as possible that motivational beliefs (one of the propensity components in the O P model), teaching practices (one of the opportunity components), and achievement were found (Byrnes & Mille r, 2007). The research team defined effective teaching as a balance between meaning , and practice based approaches. As such, higher scores on the balanced instruction rating indicated an emphasis on such reformative principles as conceptual understandi ng and problem solving as well as seemingly traditional principles including sufficient practice opportunities. It was found concept was related to the balanced instruction rating and also to the level of teacher responsiv eness perceived by students. As one of the opportunity factors, the balanced instruction rating predicted 10th grade as well as 12th grade achievement, though the concept, one of the pr opensity factors, also predicted the outcomes for both grade levels as did the strongest
35 predictor, prior year achievement (9th grade). Links between teacher instruction and achievement and between student motivation and achievement revealed significant c oefficients in the multiple regression models as well as significant path coefficients in the SEM models. Since researchers did not model the paths between propensity and opportunity factors, there is no indication what the causal direction of the relatio nship between motivation and instruction would be (Byrnes & Miller, 2007). Thus, it is necessary to examine a possibility that student motivational constructs and teacher instructional practices have a n association with each other, which in turn influence s student mathematics achievement. Another study using the O P framework of achievement highlights the importance of models with data of forty two students fro m three different Algebra II classes taught by a highly skilled teacher. Results showed differences in achievement on the end of course exams depended on cognitive, meta cognitive, and affective components of learning. Propensity variables, such as prior knowledge, general aptitude, self regulation of performance and time use, and the frustration level about learning mathematics, accounted for an additional 34% of variance to the initial model into which the opportunity variables were only entered. All b ut the general aptitude significantly predicted student achievement in the full model. Accordingly the researchers concluded that providing an opportunity is necessary but not sufficient so that fostering cognitive, meta cognitive, and affective component s must also be emphasized in planning and implementing instruction. These two studies focused on the mathematics learning of secondary students. The current study will include early elementary school students, thereby creating an opportunity to
36 investig ate the influence of a motivational construct on the mathematics learning of younger students and its relationship with other student and teaching level variables. Purpose of the Study The primary purpose of this research was to examine the major contrib utions of both teacher and student factors on the mathematics achievement of students during the period of mathematics education reform influenced by a series of NCTM mathematics standards publications (NCTM, 1989; 1991; 2000). First, important variables involved in mathematics teaching and learning were explored. To achieve this, latent classes or latent variables of teacher and student factors were identified in order to create alternative methodological approaches for characterizing mathematics instruc tion and for confirming the dimension of mathematics self concept of students. Then, following the identification of viable instructional profiles of teachers and motivational dimensions of students, analyses examining various relationships among signific ant factors of teachers and students were conducted. Next, secondary analyses were accomplished using the ECLS K 3rd grade data set along with subsets of the sample from the ECLS K 1 st grade Fall and Spring data. The ECLS K data set was chosen because it is a nationally representative sample of approximately 22,000 children. Also, it is comprehensive, as well as longitudinal, so that researchers are able to examine various aspects of and changes in children's cognitive and behavioral development, includi ng their family and schooling environments. It is notable that the ECLS K 1 st grade Fall round is not the regular cycle of data collection and was limited to a sub sample. For this study, three sets of achievement data including the 1 st grade Fall data w ere used to longitudinally link achievement data from 1 st to 3 rd grade, and therefore the models relevant to achievement and corresponding analyses were restricted to subsets of the ECLS K sample.
37 Advantages of conducting a secondary analysis of pre col lected data include (a) its potential for resource efficiency in time and cost and (b) it allows for inferences to be made to the general population from nationally representative samples. Above all, researchers are able to employ a variety of frameworks, designs, and analysis techniques when using large, preexisting datasets offering potential insights for formulating new research foci. As such, this type of research may contribute to defining and clarifying the aspects of a research problem for future i nvestigations (Kiecolt & Nathan, 1985). There are, however, a number of limitations in the secondary analysis of existing datasets. One limitation refers to the difficulty in finding the most salient data and variables within the data set to achieve rese arch objectives and/or answer research questions. Some types of research topics lend themselves more readily to secondary analysis, but others do not. It is not easy to find a dataset for which the formulation and collection processes merge cohesively wi th the secondary research objectives. Even when an appropriate dataset is selected, the variables contained therein might be poorly operationalized in the light of the secondary research objectives (Kiecolt & Nathan, 1985). Given these limitations, the s econdary research of pre collected data will not yield the aforementioned contributions to scholarship unless it is anchored in a theoretical and/or conceptual framework. For this study, all of the variable selections, analysis designs, and interpretation of results were based on the conceptual framework and knowledge base from relevant research areas. For example, person oriented analysis (Campbell, Shaw, & Gilliom, 2000) was employed to generate subgroups of teachers with shared profiles ( Nylund, Asparo uhov, MuthÃ©n, 2007) in terms of instructional practices. The number and profile descriptions of the classes were not determined a priori, but rather derived from the data. As such, consideration was given to model fit indices as well as conceptual models in which possible instructional profiles can be hypothesized so as to
38 determine whether uncovered classes represent conceptually reasonable subgroups of teachers math ematics self concept was knowledge based because the intent of this analysis is to evaluate whether items from the self description questionnaire (SDQ) in the ECLS K 3rd grade data set supports the motivational structure of mathematics learning indicated b y the field, such as the collective work of Marsh et al. (1985; 1990; 1991). Furthermore, the final models intended to test the mediation and moderated mediation relationships must be based on the conceptual framework that highlights the interactive relat ionships among the mathematics content, teachers, and students in evaluating educational effectiveness (Cohen & Ball, 1999; Cohen, Raudenbush, & Ball, 2003). Accordingly, the purpose of this study is to examine models with which to conceptualize and test hypothesized relationships between important determinants of mathematics achievement within the context of reformative mathematics education. Based on the conceptual framework, this research emphasizes the interactive relationships among reformative mathe matics, teacher instruction, and student characteristics focused on academic motivation at the center of the inquiry, in order to better understand the effectiveness of reform efforts in mathematics teaching and learning. Conceptual Framework Cohen and hi s colleagues (Cohen & Ball, 1999; Cohen, et al., 2003) suggest a framework by which the researcher investigates the interactive relationships between the content taught, the ove schools (including resource investments and instructional approaches) have yielded very few significant results and that there have been conflicting opinions about the effects of those inputs among researchers. These can be attributable to a simple pe rspective on inputs and outputs in education
39 research and policy namely, a simple linear link between resources as input and student achievement as output. However, it is necessary to take into account the interaction among the three elements of content, students, and teachers because instruction consists of interactions among them. Accordingly, Cohen and Ball and Cohen et al. suggest that research should consider the instructional system as the cause, not the resources per se (e.g., providing curriculum and professional development materials for reformative mathematics). In the instructional system, students and teachers interact around the content to form knowledge, coordinate instruction, provide incentives, and manage the environment as interdependent agents (Cohen & Ball, 1999; Cohen et al., 2003). By examining interactions between these three components of the instructional system (i.e., content, teacher, and students), the effectiveness of reform oriented investments is determined by evaluating var ious reactions of teachers and students around the content. Based on this perspective, studies could be designed to offer more specific answers for why programs might succeed or fail by explicating an intervening mechanism for achievement causality (Chen, 1990; Chen, et al., 2011). This framework emphasizes the importance of knowledge use of and incentives for interdependent agents, and research on mathematics reform also highlights those components. Since m athematics reform challenges teachers and stud ents to attain authentic and higher level mathematics, the success of it certainly reflects their knowledge use and willingness to exert effort. The teacher is expected to use professional knowledge to organize instruction in a new way and students must u se their prior knowledge and intellectual potential to engage in learning. Also, the success can be determined by how both of these interdependent agents are willing to use their knowledge and skills in the reformative classroom.
40 After all, probing more or less complex interactions of what teachers and students bring to reformative mathematics classrooms may reveal the effects of investments since the 1990s in nurturing students to become proficient at a high level of mathematics. With this conceptual f ramework, what is unique about the present study is that it attempts to examine the triangular relationship among both teacher and student factors within reformative mathematics contexts, whereas previous research efforts have focused on examinations of di screte subsets of relationships, such as the relationship between student motivation and achievement, or between teaching and student motivation (e.g., Stipek, Givvin, Salmon, & MacGyver, 1998; Stipek, Salmon, Givvin, Kazemi, Saxe, & MacGyver, 1998). For this type of research, it is important to note that structural equation modeling (SEM) has the advantage of being methodologically rigorous as well as providing a precise evaluation of the intervening mechanism. Research Questions Based on the conceptual framework that student learning takes place through interactions among the content, student characteristics, and teacher instruction (Cohen & Ball, 1999; Cohen et al., 2003), this study is designed to examine these interactions, including (a) the provisio n of relevant teaching, (b) student motivational beliefs, and (c) student achievement, in order to better understand the effectiveness of mathematics education reform efforts that began in the 1990s. the following research questions: 1 Using a person oriented analysis (Campbell, et al., 2000), do interpretable teacher profi les of teacher practice emerge using the mathematics instruction items in the ECLS K 3rd grade data set? 2 Do the mathematics self concept items from the SDQ that are included in the ECLS K 3rd grade data set define a single mathematics self concept factor as indicated by the work of Marsh et al. (1984; 1990; 1991)?
41 3 Is the relationship between the 1st grade mathematics achievement and the 3rd grade mathematics achievement mediated by mathematics self concept? 4 derate the mediated relationship of ematics self concept between 1st grade mathematics achievement and 3rd grade mathematics achievement? 5 Do differences exist in the parameters defined with the teacher latent classes as the moderator for two mathematics performance groups: struggling learners and typical learners? Significance of the Study The present study has theoretical and practical significance, particularly in the field of mathematics education and special education for students with mathematics disabilities or difficulties. Theoretically, despite a growing body of research addressing teaching practices in the context of reform oriented mathematics education, little attention has been paid to (1) capturing teaching profiles which view instruction as a totality, (2) including the learning mechanism as a desirable outcome, and (3) incorporating all students at the elementary school level in an evaluation of school mathematics. This study enables researchers to conceptualize teaching pra ctices and student mathematics learning as a complex or dynamic relationship rather than a linear relationship from teaching to achievement. This study also unfolds how an alternate conceptualization can be methodologically explored with the use of struct ural equation modeling. Practically, the findings of this study have implications for understanding and interpreting mathematics reform efforts in the US. As policy moves from makers to implementers, the policy message is not presented simply by rejecti on or adoption. Instead, teachers construct the policy intent and then determine what it entails for their current practices. The composition of their class, such as the prior achievement level of their students, is a key factor in this process. Hence, understanding how teachers implement ambitious instructional principles and how the patterns of
42 has potential for making significant contributions to policy design and implementation strate gy. As the focus of this study is on the national level of teacher implementation, a secondary analysis of large scale data was selected, using various and alternate modeling techniques. This also makes a contribution to educational program evaluation, p roviding alternate models that capture teaching practices and evaluate the effectiveness of programs on student outcomes. Definitions of Terms NCTM Process Standards ( http://www.nctm.org/sta ndards/content.aspx?id=322 ) focus on how people learn mathematics and develop mathematical reasoning. The five processes are integral components that cut across all mathematics content areas and involve all grade levels from prekindergarten through grade 12. Thus, instructional practices are expected to incorporate those components for every mathematics topic for every learner from the early level of school mathematics. Problem Solving refers to build ing new mathematical knowledge through problem solvi ng. Learners are expected to apply and adapt a variety of appropriate strategies to solve problems and monitor and reflect on their problem solving process. Reasoning and Proof refers to recogniz ing reasoning and proof and investigat ing mathematical conje ctures. Learners are expected to develop and evaluate mathematical arguments and proofs. Communication involves organiz ing and consolidat ing mathematical thinking through communication. Learners are expected to use the language of mathematics to express mathematical ideas precisely in their communication with peers, teachers, and others and to evaluate the mathematical thinking and strategies of others Connections refers to recogniz ing and us ing connections among mathematical ideas. Learners are expected to understand how mathematical ideas interconnect and apply mathematics in contexts outside mathematics. Representation involves creat ing and us ing representations to organize, record, and communicate mathematical ideas. Learners are expected to select , apply, and translate among mathematical representations to solve problem s or model mathematical phenomena.
43 Rather than focusing on mastery of mathematical rules and procedures, reform oriented mathematics emphasizes a much broader outcome, which constit utes mathematical proficiency (National Research Council, 2001). These five instructional goals are intertwined: Conceptual understanding comprehension of mathematical concepts, operations, and relationships Procedural fluency skill in carrying out proced ures flexibly, accurately, efficiently, and appropriately Strategic competence ability to formulate, represent, and solve mathematical problems Adaptive reasoning capacity for logical thought, reflection, explanation, and justification Productive dispositi on habitual inclination to see mathematics as sensible, useful, and Delimitations The present study was performed with secondary analysis of an extant data set; the study did not invol ve an original research design with primary data collection. The focus of the secondary analyses in this study was the relationships between teacher and student level factors within mathematics education. The study also relied on Cohen and his colleague by which the researcher investigates the interactive relationships among the content taught, the Advantages of conducting a secondary analysis of pre collected data in clude (a) its potential for resource efficiency in time and cost and (b) its allowance for inferences regarding the general population from nationally representative samples. Above all, researchers can employ a variety of frameworks, designs, and analysis techniques when using large, preexisting data sets, offering potential insights for formulating new research foci. Thus, this type of research contributes to defining and clarifying various aspects of a research problem for future investigation (Kiecolt & Nathan, 1985). However, a number of limitations exists in the secondary analysis of existing data sets. One limitation is the difficulty of locating the most
44 salient data and variables within the data set to achieve the research objectives and/or answe r research questions. Some types of research topics readily lend themselves to secondary analysis, but others do not. It is not easy to locate a data set for which the formulation and collection processes merge cohesively with secondary research objectiv es. Even when an appropriate data set is selected, the variables contained therein might be poorly operationalized in the light of the secondary research objectives (Kiecolt & Nathan, 1985). Given these limitations, the secondary research of pre collecte d data will not yield the aforementioned contributions to scholarship unless it is anchored in a theoretical and/or conceptual framework. Limitations By the nature of secondary analysis, only variables included in the ECLS K data set could be modeled for t he present study. Thus, several limitations should be noted. First, variables used to create subgroups with similar teaching profiles were included in the ECLS reports. Other measures regarding instructions, such as c lassroom observation or teacher logs, were not available in the ECLS K data set and, ctices ( e.g., Remillard & Bryans, 2004), the characteristics of the curricular materials that teachers are using are important contextual factors in investigating teaching practices. Data on this information are also unavailable in the ECLS K data set so that the study did not examine reform oriented instructions within the context of the curriculum materials used. Although the nature of data collected by the complex survey was addressed by using the primary sampling unit (PSU), strata, and corresponding weights, it was not possible for the latent class analysis in which only teacher level variables were used. The ECLS K data set does not provide appropriate PSU, strata, and weight for the analysis employing only teacher level
45 variables. Also, note that results should not be interpreted as information of a nationally representative teacher population, but rather as information of teachers of a nationally representative student population.
46 CHAPTER 2 REVIEW OF THE LITERATURE The literature on reform or iented mathematics has argued that reform policy instruments, such as curricular materials and professional development programs, have guided teachers to transform their teaching practices. This transformation occurred via knowledge acquisition and episte mological turns with regard to mathematics, school mathematics, and mathematics teaching and learning. In other words, the actual teaching practices varied albeit with similar supports from policy instruments. The variations can be attributable to differ ent characteristics of knowledge acquisition and epistemological transformations. Accordingly, studies have focused on knowledge and cognition (Cohen, 1990; Spillane, m athematics and changes in belief systems (Brown, 2003; Gellert, 2000; Skott, 2001), teacher interactions with mathematics curriculum (Collopy, 2003; Remillard, 1999, 2000; Remillard & making process of mathematics and ref orm oriented mathematics (Drake, 2006; Drake, et al., 2001). Since the aforementioned literature falls under qualitative methodology focusing on teacher implementation of reform oriented mathematics instruction, a larger sample of teachers, links between teacher implementation and student outcomes, and various learner populations have not been explained. In the present chapter, a review of the literature is presented in a way that addresses this gap. This literature review focuses on empirical studies th at examine various outcomes for covers five emerging themes from the empirical research on reform oriented mathematics: ( a ) characteristics of mathematics instruc tion, ( b ) teacher outcomes, ( c ) student outcomes, ( d ) motivation dimensions, and ( e ) demographics of various student populations. Before turning to
47 the evidence, it is necessary to acknowledge the search and review procedures used for the present review. First, peer reviewed articles were identified by such electronic databases as Education Full Text and EBSCO Host Platform for Academic Search Premier and PsycINFO. Searches were conducted using combinations of keywords of NCTM standards , reform , and mat hematics. By the nature of this study (i.e., mathematics teaching and learning during a reform era), emphasis was given to work published since 1990. A hand search of relevant journals was also conducted including Journal for Research in Mathematics Educ ation , Journal of Mathematics Teacher Education , Educational Studies in Mathematics , Exceptional Children , Journal of Special Education , and Journal of Learning Disabilities . An ancestral search was also conducted using relevant articles, reports, and boo k chapters cited in chapter 1 of this study. Sources were selected based on the following inclusion criteria: ( a ) the study was conducted with teacher or/and students in the mathematics classrooms at the elementary or middle school levels; and ( b ) outcome s relevant to reform oriented mathematics were defined, collected, and analyzed including teacher beliefs, teaching practices, affective domains of students, or student academic performance. A similar search process was done using key words of mathematics self concept and reform ; and special education, mathematics disabilities, difficulties, achievement gap , and reform . Unlike the reform oriented mathematics literature, studies on high school or college level mathematics were not excluded, d ue to a lack o f research on motivational aspects and various student populations in the co ntext of mathematics reform. Characteristics of Reform oriented Mathematic Education No single set of attributes characterizes reform oriented mathematics teaching and learning in literature. However, NCTM policy statements (1989; 1991; 2000) including the Principles and the Process Standards have guided the ways i n which studies distinguish reform oriented from traditional approaches to instruction. For example, McKinney and col
48 studies (McKinney, Chappell, Berry, & Hickman, 2009; McKinney & Frazier, 2008) used six fundamental principles: equity, teaching, learning, assessment, technology, and curriculum (NCTM, 2000) to categorize the pedagogical practices envisioned by reformers. Based on detailed descriptions of these six fundamental principles, various items have been developed and surveyed to identify mathematics education in studies. Items such as high expectations for all students and differentiated instruction we re considered pertinent to the equity principle, while various practices including hands on activities, using manipulatives, and problem solving were considered pertinent to the teaching and learning principle. Furthermore, alternate assessment forms of r eflection, interview, and portfolio were used as indicators to address the assessment principle. The use of software or calculators was recommended for the three principles of technology, teaching, and learning. Clarifiers considered as pertinent to three principles of curriculum, teaching, and learning. Since how students learn mathematics has acquired a greater importance in reform oriented mathematics, the Process Standards , which refers to how students develop and use mathematical knowledge and skills, could be used as a base for reform oriented instruction indi cators. As such, this has functioned as a framework for developing observation or survey instruments designed to capture reform oriented mathem atics classrooms in studies (e.g., Jitendra, Griffin, & Xin, 2010). Studies have defined such classrooms as reform oriented when curricular materials, mathematics tasks, teacher actions, and learning activities were more oriented toward the five NCTM (200 0) processes of ( a ) problem solving, ( b ) reasoning and proof, ( c ) communication, ( d ) connections, and ( e ) representation. The five processes are integral components that cut across all mathematics content areas. Accordingly, those components have been us ed directly or revised to capture the degree to which
49 students experience reformative learning environments unrestrictedly throughout mathematics topics taught. Instruments used to assess teaching practices evaluate teacher and student use of real life pr oblem solving, conjectures, multiple representations, and mathematics communication in the reform oriented curriculum (e.g., Saxe, Gearhart, & Seltzer, 1999; Tarr, et al., 2008) and task development (e.g., Henningsen & Stein, 1997). Often, the component o f connections can be combined into other components of problem solving and communication , as in the case of problem solving connected to real life situations or communication connected to prior knowledge. The underlying premise for research is that reform oriented mathematics teaching can be defined as skill sets that are observable and learnable. For example, teachers who were supported by reform oriented curriculum have cultivated a more stimulating learning environment characterized by conceptual under standing, multiple approaches to doing mathematics, and mathematical discussion despite the variation in curriculum implementation (Saxe et al., 1999; Tarr, et al., 2008). The emphasis shift s from factual knowledge toward making sense of knowledge, from a representative strategy toward varied approaches for problem solving, and from transmitting of knowledge toward collaboratively constructing of knowledge. Furthermore, as a consequence of teacher learning, pre service elementary school teachers showed a tendency toward reform oriented teaching when they were observed and rated using the Reformed Teaching Observation Protocol (Sawada & Piburn, 2000, [RTOP]) after having completed reform oriented methods courses (Jong, Pedulla, Salomon Fernandez, & Cochran Smith, 2010). Additionally, in service elementary teachers exhibited a similar tendency after participating in a mathematics content focused professional development (PD) program (Hamilton et al., 2003; Cohen & Hill, 2000). Moreover, teachers showed prof essional growth in
50 teaching mathematics to students in high poverty middle schools as a result of comprehensive reform efforts including reform oriented materials and relevant PD supports (Balfanz, Mac Iver, & Byrnes, 2006). Research using large scale dat a has also successfully distinguished between different mathematics classrooms. Smith et al. (2005) utilized the data from the 2000 NAEP 8th grade teacher surveys. Factor analysis revealed that 11 questions captured the three dimensions: be considered reform oriented instruction because both address higher order thinking and conceptual understanding, which is the foundation of mathematics reform teaching, and because the former is often achieved by the successful use of the latter. Quest ions clustered within each dimension are as follows: Conceptual emphasis: (1) developing reasoning and analytical ability in problem solving, (2) developing mathematical communication, and (3) developing an appreciation for the importance of mathematics; Conceptual strategies: (1) mathematical writing, (2) mathematics projects and report writing, (3) discussion with other students, (4) problems that reflect real life situations, (5) working in small groups or with a partner, and (6) mathematical discuss ion; Procedural teaching: (1) mathematics facts and concepts, and (2) skills and procedures. In another study using the 2000 NA E P 4th grade database, 9 dimensions of reform oriented instruction emerged from teacher survey questions (Lubienski, 2006). Three survey questions on the frequency of tests taken, the use of textbooks, and the use of computers were excluded because the author believed that possible confounding factors behind those questions potentially distorted the relationships between reform oriented practices and other outcome
51 variables such as student achievement. In other words, the question of how closely a teacher adheres to the textbook may capture a different nuance according to the curriculum used in each school. Excluding those que stions, 24 survey questions remained for the factor analysis. Results identified 9 variables, including the use of (1) a calculator, (2) manipulatives, (3) multiple choice tests, (4) an emphasis on facts and skills, (5) collaborative problem solving, (6) historically less taught topics (i.e., geometry), (7) mathematical writing tasks, (8) reasoning and proof, and (9) Standards , that determined whether teaching could be considered reform oriented or not. The dimension of collabo rative problem solving attained the highest about problem solutions attained the highest level of loading (.84) among 5 questions pertinent to this dimensio n (Lubienski, 2006). In regard to observation studies, Gearhart and her colleagues (Gearhart, Saxe, Fall, Seltzer, Schlackman, Ching, et al., 1999; Saxe, Gearhart, & Seltzer, 1999) have developed a measure by which the level of opportunity to learn aligne d with tenets of reformative mathematics was captured. Validation procedures of the measure included rater agreement examination and a principal component analysis, suggesting two important dimensions of opportunity to learn that teachers and students joi ntly created in reformative classrooms: (1) opportunities to work on conceptual issues in whole class discussions of problem solving by emphasizing mathematical thinking; and, (2) opportunities to use (oral and written) numeric representations (the graphic representation was omitted due to low technical quality). TIMSS video studies (Jacobs et al., 2006) developed more specific codes to define the features of classrooms where the 5 NCTM Process Standards were employed. To indicate the standard of Problem Solving , for example, various dimensions were examined from lessons
52 including: (1) given time to work individually and collaboratively in the small group; (2) work assignment during those private work; (3) number and length of mathematics problems; (4) pr ocedural complexity captured by number of steps and sub questions before the final solutions; (5) real life connections and use of real world objects (e.g., boxes, maps, or newspapers); (5) given opportunities for students to explore alternative solution m ethods and discuss them; (6) number of problems with multiple answers; (7) use of technology. Research in the special education context has defined reform oriented instruction in a similar way. A study by Jitendra et al. (2010) articulated the Process S tandards and formulated operational definitions of five processes to conduct a textbook contents analysis and classroom observation. Authors identified the incidence of each code within the five processes in order to quantify teaching. The category of Pr oblem Solving , for example, included: (1) opportunities to let students use a learned skill to solve math word problems; (2) opportunities to solve problems in different contexts; and (3) opportunities to apply and adapt multiple strategies. Results depi cted that teachers tended to implement the Standards incorporated in the textbooks at relatively similar levels. Otherwise, teachers exerted their own efforts to adhere to the Standards in their instruction, but at varying degrees amongst observed teacher s (Jitendra et al., 2010). Another study conducted by Jackson and Neel (2006) attempted to quantify classroom content and activity using codes of: Content area: (a) algorithm instruction (traditional), (b) concept instruction (reform oriented), and (c) unrelated to mathematics; and Classroom activity: (a) independent activity, (b) teacher directed/facilitation of activities, (c) small group activity (reform oriented), and (d) other. According to observations across three different settings for student s with emotional and behavior al disorders (EBD), instruction in the general education classroom included more reform oriented components such as concept instruction and small group activities. Both the
53 resource and self contained classrooms devoted more t ime to algorithm instruction by focusing on factual knowledge and mathematical procedures. Also, these settings limited mathematical discussions to facts and procedures, rather than discussing mathematical concepts and reasoning. The conclusions from the research relevant to special education, however, should be interpreted with caution. Studies on reform oriented mathematics from this field often have been motivated by the agenda of how students with disabilities need to access the general education cur riculum and instruction so as to meet the legislative mandates by the Individuals with Disabilities Education Improvement Act (IDEIA, 2004) and the No Child Left Behind Act (NCLB, 2002). Meanwhile, it remains unclear if the shift from direct/explicit to c onstructivist approaches of mathematics teaching corresponds to the evidence based practices for this student population (Baxter, Woodward, & Olson, 2001; Baxter, Woodward, Voorhies, & Wong, 2002; Jones & Thomas, 2003; Mayrowtz, 2009; Woodward & Baxter, 19 97; Woodward & Brown, 2006). Hence, with regard to the agenda of how to facilitate the success of students with disabilities, it may be necessary for research to define reform oriented mathematics classrooms from multiple perspectives, for example a balan ced approach (Jones & Southern, 2003). Collectively, this body of research has attempted to distinguish the reform orientation from various instructional practices. The NCTM Process Standards often have provided the conceptual framework to develop instru ments. Classroom observation tools have been designed to document a range of classroom experiences embracing instructional components aligned with these Standards . With regard to teacher reported practices, the language and content of survey questions se emed to be appropriate for making inferences to operationalize the survey using principles inspired by the NCTM documents. Factor analysis was most often used and its results corresponded with conceptualizations of reform oriented teaching, to some extent . It is,
54 however, noteworthy that there has not been an attempt to use Latent Class Analysis (LCA) to generate profiles of teaching practices in light of mathematics reform, as has been done in the current study. Notably, most of studies employed the dich otomous framework, whether certain reform principles were more likely to be witnessed or not regardless of data type from the teacher reported surveys or classroom observations. This dichotomous perspective offers a hypothesized model of teaching practice distinction between conceptual goals and conceptual strategies Hamilton, et al. (2003) also noted that two dimensions of reformative and traditional practices, wh ich emerged from their factor analysis, should not be construed as mutually exclusive notions. A teacher might satisfy both principles by craftily conflating them. For example, in traditional reform oriented teaching as well. Therefore, there is a need for a more detailed and nuanced model to characterize instructional pra ctices in the context of mathematics reform. This underscores the significance of using person oriented analysis (i.e., LCA) for the present study and offers another hypothesized model of teaching profiles and their descriptions, which will be defined by the data driven subgroups. Reform oriented Mathematics Education and Teacher Outcomes Reformative practices have been identified by a body of research using classroom observations and teacher surveys. Overall reform oriented teaching was found to be mor e prevalent than before, but still limited in mathematics classrooms since the 1990s (Jacobs et al., 2006; Lubienski, 2006; Wenglinsky, 2004). More recent descriptive results on teacher reported practices illustrate a steady increase in such practices in the classroom (Mullis, Martin, & Foy,
55 professional knowledge and skills. According to secondary analyses using 2000 NAEP data (Lubienski, 2006; Wenglinsky, 2004), f ourth grade teachers tended to emphasize mathematics facts and routine problems, but used relatively less mathematical reasoning and communicating in their mathematics instruction. They also tended not to use such reformative components as small group wor k, manipulatives, and mathematical writing and projects through real world problems in the instruction. At least, the factor of calculator use was popularly introduced as a reformative component, but this factor failed to be positively associated with stu dent outcomes at the student level as well as at the school level (Lubienski, 2006). This result may be considered as form focused rather than function focused understandings by practitioners (Spillane, 2000b). In other words, it must have been easier fo r schools and classrooms to begin reforming instruction with calculator use rather on activities rather than mathematical reasoning activities as their approach to reform implementation. Also it is notable that TIMSS video studies (Jacobs et al., 2006) revealed that reform oriented instructional practices still scarce in the U.S. middle grade classroom. Some increases in reform oriented teaching practices were achieved ( e.g., significant increase in lessons that contained at least one problem with real from 1995 to 1999 but that those improvements were very limited (e.g., little structure and few opportunity for explori ng reasoning or connection standards). A changing trend in reform implementation suggests the importance of ongoing evaluation by linking various factors to student outcomes in the context of mathematics reform. Yet, s teady progress continues in the
56 empl oyment of reform oriented practices. According to the TIMSS 2007 International Report (Mullis, et al., 2008), more than 50 percent of US fourth grade students had teachers who reported devoting more than half of their lessons to doing word problem solving activities (51%), student explanations (63%), and connections between mathematics and daily life (65%). Here we do not arguably construe increased reports of reform components as function focused understandings (Spillane, 2000b) or the fundamental transf ormation of epistemology in terms of mathematics teaching and learning. T here are commonly perceived factors that are attributable to the implementation of reform practices in the United States: structural or organizational characteristics such as class s ize and greater teacher autonomy. In this vein, Desimone, Smith, Baker, and Ueno (2005) has evaluated the five potential barriers to such practices and compared the United States to other high achieving countries, suggesting a counter example for the asso ciation between structural or organization characteristics and reform implementation in the U.S. classroom. Using TIMSS 1999 data and HLM for the nested nature by countries, Desimone et al. (2005) revealed that the barrier factor of greater teacher auto nomy and thereby less consistent and more varied teaching was not related to reform implementation. The U.S. did not show greater variance in the reform oriented teaching: the coefficients of conceptual teaching (i.e., reform oriented) were similar across countries including the U.S. The assumption that larger class size prevents greater emphasis on the conceptual teaching also failed to be supported contrary to common belief, a significant and positive coefficient for class size and conceptual teaching i n the U.S. were found. Importantly, differences in relationships of reform oriented teaching with other factors, such as teacher beliefs and knowledge, were found between the U.S. and high achieving
57 countries including Japan and Singapore (Desimone, et al ., 2005). For example, Japanese teachers did not seem to choose computational or conceptual instruction based on the achievement level of student (i.e., no significant coefficients of computational, conceptual, and the ratio of the two approaches on the c lass average achievement). Singapore teachers were selective about conceptual teaching in that they were more likely oriented to conceptual teaching for the high achieving class (i.e., a positive coefficient of conceptual teaching on the class average ach ievement). On the contrary, two coefficients were significant and positive for conceptual teaching and the ratio of conceptual to computational teaching for U.S. teachers (negative but insignificant on computational teaching at the conventional p value, . 05), indicating that they were more likely oriented to less computation and more conceptual teaching for the high achieving class. More importantly, years of teaching experience, used as a proxy of professional knowledge or skills, remained significant: t eaching experience was negatively related to computational teaching but positively to the ratio of conceptual teaching to computational teaching in the U.S. More experienced teachers were less likely to use computation teaching, but they were more selecti ve about conceptual teaching based on student achievement levels. As such, m ore widespread implementation of reform practices seems knowledgeable and skillful teachers more likely transform their fundamental understanding of mathematics teaching and learning and challenge their existing practices when reform policy calls for a change. Teacher expertise has not sufficiently measured and examined in establishing a relationship with the implementat ion of reform oriented instruction by studies; nonetheless, a linkage between professional knowledge or skills and the reform oriented teaching may be relatively uncontentious. Knowledgeable and skillful teachers in mathematics content and
58 mathematics ped agogy are more likely to include new and challenging principles in their instruction. Studies on teacher learning in the context of mathematics reform suggest this relationship (Cohen & Hill, 2000; Hamilton, et al., 2003; Hill & Ball, 2004; Jong, et al., 2010). As a result of learning from the reform oriented method courses, beginning elementary teachers were found to use reformative practices (e.g., 2.18 is greater than 2, which is indicative of reformative teaching on the Reformed Teaching Observation Pr otocol, RTOP) in their first two years of teaching and this reform oriented instruction was positively linked to student achievement on district developed tests (Jong, et al., 2010). Meanwhile in service teachers could also obtain new understanding of mat hematics and mathematics teaching from the opportunity to learn, which in turn influenced their instructional practices. Cohen and Hill (2000) examined focused, PD. Findings suggest that teachers were mor e likely to use reform oriented teaching (e.g., discussing different ways to solve mathematics problems) and less likely to use traditional teaching methods (e.g., practicing for or administering tests on computational skills) as a consequence of mathemati cs content focused biased characteristics (e.g., familiarity with Standards ). It is notable that the same positive result was not found in PD addressing generic topics such as family involvement, cooperative lear ning, and equality issues in mathematics teaching, implying that concrete and content specific learning opportunities facilitated teacher learning, which in turn influenced student learning in the context of mathematics education reform (Cohen & Hill, 2000 ). Teacher knowledge for teaching mathematics was directly measured in an attempt to investigate the relationship with teacher learning from content focused PD (Hill & Ball, 2004). Mathematical knowledge for teaching was measured by the items on number sense and
59 operations and pre and post assessments were analyzed after submitted to validation procedures. Overall, PD participants showed gains in mathematics knowledge assessments, indicating the benefits of extended and content focused PD in mathematic s. Additionally, sufficient opportunities for the sake of learning reform principles (mathematical analysis, reasoning, communication, and representation) related to improvements in mathematics teaching and learning; because it has been shown that program s studied influenced teacher knowledge gains when they actively engaged in activities exploring mathematics content based on the reform oriented principles. This intensive PD support empowers teachers to change their practices by promoting their awareness and expertise about various educational principles. Stipek and her colleagues (Stipek, Givvin, Salmon, & MacGyver, 1998; Stipek, Salmon, Givvin, Kazemi, Saxe, & MacGyver, 1998) attempted to conceptualize the reformative as well as motivating mathematics c lassrooms by amalgamating common principles from two different literature bases. They different levels of supports. Supports were provided in the form of PD and curricular resources focused on both reformative and motivational aspects of teaching (Stipek, Givvin, et al., 1998). The differing levels of support included: (1) the use of reformative curriculum with reform , and motivation focused PD; (2) the use of reformative curriculum with a minimum of collegial support for teachers; and (3) the use of traditional curriculum. Results indicate that teachers in both groups using reformative curricula tended to emphasize mastery learning, de emphasize performance , and encourage student autonomy during their teaching. Given that teachers from the group with curriculum and collegial support discussed motivational issues in teaching mathematics with research team, a minimum amount of collegial support using curricul ar
60 resources probably increased their awareness of important teaching principles and influenced actual instruction. However, results also highlight that teachers who were provided with more intensive support (i.e., PD) showed more positive results in term s of their teaching and student motivation. Only teachers from the curriculum and PD group showed a significant difference in the degree to which teachers emphasized effort in mathematics learning over those in the traditional curriculum group. Also, stu dents whose teachers were more intensively supported via PD were less likely to have a performance orientation than those of teachers in both the curriculum collegial support and traditional curriculum teaching groups. The PD group of teachers seemed to b significant correlations between student and teacher ratings on student motivational dimensions. the ir students appeared to be most concerned about their performance and the perceived ability was relatively low in the curriculum support group only. Taken together, Stipek, Givvin, et al., (1998) suggest that curriculum materials could exert some positive influences on practices that are consistent with principles of reformative mathematics instruction as well as motivation enhancing but that implementation by insufficiently prepared teachers may negatively influence student motivation. However, it remain s unknown whether this modest negative effect on student motivation could lead to a negative effect on student learning, as the student outcome was not included in this study. In conclusion, studies reviewed in this section suggest that reform oriented mat hematics teaching can be challenging in that, teachers must learn and exhibit new professional knowledge, perspectives, and skills about mathematics teaching and learning (Spillane, 1999; Spillane & Zeuli, 1999). As noted earlier, the mathematics reform m ovement since the 1990s
61 can be characterized as process driven rather than input or output driven: reform has been guided by influential documents outlining instructional content and pedagogical approaches, and the corresponding curricular materials and P D programs have functioned as instruments linking policy, instruction, and student outcomes (Cohen & Hill, 2000; Swanson & Stevenson, 2002). Thus patterns in teaching practices, if any, could be construed as the teaching level outcomes of our reform inves tments. This teaching level outcome should be examined by relating them to student relevant factors, as an attempt to evaluate the policy efforts. Reform oriented Mathematics Education and Student Outcomes Early evaluation attempts have often been based on curriculum use and its effectiveness. Studies of reform oriented curricula have shown a positive relationship between curricular Math Trailblazer (Teaching Integrated Math ematics and Science Curriculum, 2004) provides students with opportunities for learning mathematical concepts through investigations of scientific problems. Carter, et al. (2003) found that this curriculum is effective for improving student achievement on traditional standardized tests. It is notable that schools with high poverty levels and demographic diversity showed increases in the percentage of students who exceeded the our year average prior to the curriculum implementation). Students in schools located in diverse communities using Everyday Mathematics (EM; Bell et al., 2004) also outperformed their counterparts in schools using traditional curricula in suburban areas ( Carroll & Isaacs, 2003) as well as samples taken from the National Assessment of Educational Progress (Fuson, Carroll, & Drueck, 2000), dispelling concerns about reform oriented mathematics curricula and the performance of chronically low performing studen ts. In particular, these students showed substantial improvements in learning mathematical concepts; an aspect of the curriculum that has
62 historically received less emphasis in favor of mental computation and geometric knowledge (Carroll & Isaacs, 2003). It has also been found that implementing reformative curricular materials exerted positive differential impacts on motivational aspects in mathematics learning (Battistich, Alldredge, & Tsuchida, 2003). The aforementioned studies illustrate positive influ ences of reform oriented mathematics curricula on several student level outcomes. These outcomes include standardized achievement tests, content specific measures of mathematics learning, various historical and overall performance comparisons to district and state goals, as well as comparisons to existing data from other studies. Yet, findings from these studies do not shed much light on the influence of the teacher or on the interactions between students and teachers in the context of reform oriented mat hematics instruction. As such, more studies exploring the desired changes in teacher knowledge and skills and the influence of these changes on student outcomes are needed. In that vein, Hamilton, et al. (2003) tested the regression model by which teache reported use of reform oriented instruction were associated with student performance on open ended as well as multiple choice mathematics assessments. Results indicated a positive but weak relationship between reform oriented teaching and student outc omes on these open ended and multiple choice tests, while controlling for prior achievement level and differences in their backgrounds. Other research using teacher reported data also extends empirical evidence for connecting reform oriented mathematics, corresponding instructions, and student outcomes. performances and enhanced teaching practices as the consequence of reformative PD experiences. A series of seconda ry analyses using 1996 and 2000 NAEP data sets established
63 positive relationships between reform oriented practices and test scores (Lubienski, 2006; Wenglinsky, 2002; 2004). Students in the 1996 NAEP data set whose teachers reported an emphasis on highe r order thinking or active learning performed better than students with teachers who did not make this report (Wenglinsky, 2002). The analysis of the 2000 NAEP data produced a list of teaching practices beneficial or detrimental to student achievement. W englinsky found that taking tests, emphasizing facts, and doing projects were negatively related to achievement scores whereas emphasizing routine problems and geometry were positively related to achievement at the school level. Hands on learning using bl ocks and objects and collaborative small group learning, which are representative signals to reform base teaching, turned out to be non significant to achievement scores in the 2000 NAEP test (Wenglinsky, 2004). Using the same data set and HLM modeling as to positively predict student achievement at the school level. These mixed results may disappoint one who holds a dichotomous view between reformative and traditional. However, we need to attend to little variance in those reform relevant variables in the 2000 NAEP data. It would be difficult to examine the relationship between instructional components and student outcomes wit hout any reasonable extent of enacting those components. Ongoing evaluations with various methods, contexts, and data may help draw stronger, more consistent, conclusions about the effectiveness of certain instructional components, including reform orient ed. For example, a randomized experimental study, in which the effects of two reformative components on several student outcomes were evaluated, was able to provide evidence of not only the effectiveness of reform oriented teaching but also the complexi ty of the teaching and learning mechanism (Ginsburg Block & Fantuzzon, 1998). Of instructional components aligned
64 with the reform principles, authors focused on problem solving and collaborative learning with peers. The problem solving condition (PS) enc ouraged students to develop multiple strategies and share them in a group, placed a high value on the problem solving process, and explored mathematics using manipulative materials. The peer collaboration condition (PC) was composed of a reciprocal peer t utoring program and group reward system. The results revealed that the PS and PC positively impacted not only the achievement in computation and word problems, but also affective outcomes including academic motivation, academic self concept, and social se lf concept. Interestingly, combining the two components simultaneously was not able to exert a stronger effect on student outcomes than providing each intervention with components in isolation. Although the authors provided an explanation that this findi ng may be attributable to the short duration of intervention, more sophisticated research may explain the reason for this unexpected effect of the combination group. Classroom observation research also obtained evidence for associations between reform ori ented teaching and student achievement outcomes. Gearheart and her colleagues ( Gearhart et al., 1999; Saxe, et al., 1999) found that the performance on factions was predicted by teaching practices in which teachers emphasized conceptual understanding of s tudents and demonstrated conceptual assessment of student thinking. The predictability of this reformative practice, however, was limited to problem solving tests and students with knowledge of basic concepts as opposed to computation tests and students w ithout the rudimentary understandings of fractions. Tarr and his colleagues ( Tarr, Reys, Reys, Chave, Shih, & Asterlind, 2008 ) also found that reform oriented curricula could have an impact on the performance of middle school students whose teachers imple mented these practices to a moderate or high degree. A combination of
65 reform oriented curriculum and practices contributed more to achievement on the criterion referenced and constructed response tests as opposed to norm referenced multiple choice tests. Collectively, mathematics education reform efforts have made positive but modest contributions to teacher learning, teaching practices, and therefore student learning. From the review of empirical studies, emerging themes include the degree of implementa tion, the student population, and the type of achievement assessments used. The present study addresses the issue of student population by exploring the commonalities and differences in relationships at different student achievement levels (Research Quest ion 5). In addition, the current study is designed to demonstrate the degree to which or patterns of reformative components implemented in classrooms in the ECLS 2001 wave. Unfortunately, this study is not able to address the third theme (i.e., the type of assessments used), because IRT scaled total scores are available as the continuous outcome variable in the data set. It is notable that other data sets such as the TIMSS assessments contain various forms of achievement scales (e.g., Knowing, Applying, and Reasoning) so that researchers could observe, if any, different trends in relationships by the type reform oriented classroom. In the next section of this revi ew, motivational aspects and studies on various student populations in the context of reform oriented mathematics education will be discussed. Motivational dimensions of reform oriented mathematics education Mathematics self concept includes beliefs in st This construct is an important pred ictor of mathematics achievement across school levels (Dermitzaki, Leondari, & Goudas, 2009; Marsh & Hau, 2004; Marsh et al., 1991; Pajares &
66 Miller, 1994). When self perceptions about abilities (i.e., mathematics self concept) for mathematics performance were measured at the subject matter level, this domain specific measure of self concept could exert influence not only directly on mathematics performance (Marsh et al., 1991 ; Marsh & Hau, 2004; Marsh & Yeung, 1997) but also indirectly through the influen ce on motivational and volitional strategic behaviors (Demitzaki et al., 2009). In addition, the self perception about capability at the task level, which can be captured the Miller, 1994; Pajares & Graham, 1999). Based on social cognitive theory, Pajares and Miller (1994) found that task specific self efficacy had a stronger effect on performance than mathematics self concept did and that self efficacy had an indirect effect on performance, through the strong influence on self concept. However, it is beyond the scope of the present review to examine differences between mathematics se lf concept and self efficacy because findings of differential effects of these two constructs on mathematics performance were mixed depending on the theoretical framework in each study. For example, Norwich (1987) found a stronger effect on self concept w ithin hierarchical regression models in which self concept was entered first based on his own conceptual framework. The prime focus of the present study is to explore the triangular relationship of student, teaching, and learning in reformative mathematic s classrooms, by using that a model in which one of two motivational constructs is introduced will enable examination of those relationships without access to dat a for the other construct.
67 potential mediator of relationships between mathematics achievement and other important determinants, such as gender and mathematics cour ses taken. In a model including motivational beliefs variables, there was no direct gender/prior experience effect on achievement but a significant gender/prior experience effect on self efficacy, which in turn affected achievement (Pajares & Miller, 1994 ). Since the authors did not statistically test the indirect effect, their study results raise the possibility that dimensions of motivation may present as mediation or moderation effects on the achievement mechanism. Examining such indirect (i.e., media ted or moderated) effects by the motivational belief system may render a considerable implication to important than, or at least equally important to, the p revious experience per se. With this type of research evidence, educators can capitalize on the motivational aspects of teaching in changing predetermined relationships between prior and subsequent performance. rning and performance are important predictors and mediators of achievement, we can expect that teaching practices designed to enhance these beliefs may realize such desirable outcomes as active engagement, self regulated learning, greater allocation of ef fort and persistence, and ultimately better achievement (Bong & Skaalvik, 2003). Also, given that the belief system of ability has a reciprocal relationship with previous learning experiences (Marsh & Yeung, 1997) but an independent contribution from prev ious reasonable and positive knowledge, perceptions, and convictions about their mathematics learning must be emphasized in planning and implementing instruction.
68 There is a paucity of research concerning reform oriented teaching relevant to the construct of student motivation. Reform oriented teaching appears to be positively associated with attitudes toward mathematics and motivational dimensions of mathemat ics learning. For example, an analysis of student letters (Bay, Beem, Reys, Papick, & Barnes, 1999) was conducted concerning their perceptions and reflections after a reform oriented curriculum was implemented. Middle school students tended to positively perceive their experiences in favor of hands on activities, group work, and mathematics content connected to real world problems. Student reports also included positive attitudes toward mathematics learning and mathematics related careers. Although some students acknowledged difficulties in problem solving and doing projects, the authors suggested these reports of difficulties possibly link to how teachers challenge and guide students in the problem solving, because student struggles were found in certai n sets of teachers whereas barely found in another sets. One study identified in which motivation dimensions of both teachers and students were examined in the context of reform oriented mathematics was conducted by Stipek, Salmon, et al. (1998). This r esearch team presumed that effective instruction suggested by the motivation literature were parallel to those aligned with reform oriented principles. Reformative mathematics classrooms require a learning orientation rather than performance orientation w ith the goal of improving conceptual understanding rather than simply getting correct answers. Also, a reasonably positive self concept and a willingness to take risks allow students to actively engage in problem solving and mathematics activities emphasi zed in the reform oriented mathematics instruction. Intrinsic motivation and positive emotions of students and teachers can also lead to their greater creativity and flexibility, which in turn enables them to focus on learning, the problem solving process , and the use of multiple strategies.
69 In their study (Stipek, Salmon, et al., 1998), data on reform oriented teaching, motivation were collected through the use of videotapes, field notes, and questionnaires. Collected data were submitted to validation processes (e.g., factor analysis). In particular, the Learning Orientation subscale captured the emphasis on effort and learning, de emphasized performance, and the encouragemen on conceptually oriented test items, but not on procedurally oriented ones. Addit ionally, students were more likely to show help seeking behaviors and positive emotions in classrooms wherein teachers more often showed a positive affect and enthusiasm and created a safe learning environment. However, there was no significant correlatio n between teaching practices relevant to teacher positive emotions and student performance on either conceptual understanding or procedural skills. Therefore, teaching practices relevant to learning orientation facilitated ment whereas those relevant to teacher affect influenced only the motivation outcome, but not the achievement outcome. Interestingly, there was a counterintuitive finding associated with the relationship between teaching and learning. Since motivation en hancing teaching is considered analogous to reformative teaching in this study, the primary goal is on conceptual understanding, which is an important aspect of reform oriented teaching. The motivation dimensions (perceived ability, mastery orientation, h elp seeking, positive or negative emotions, and enjoyment) were significantly associated with procedural learning only, but not with the conceptual learning, on the posttest of fractions (Stipek, Salmon, et al., 1998). The authors reasoned that enhanced m otivation of students engendered more active engagement in learning mathematics including
70 practicing skills, which could result in the improvement in procedural performance. Another possible explanation for no significant association between motivation an d conceptual learning is that it may be difficult to expect an immediate result from enhanced motivation to improved conceptual understanding. When it comes to conceptual understanding, it may be necessary to place an equal emphasis on the quality of the instructional practices and mathematics content delivered during instruction, which evidently have a direct influence on student conceptual understanding and also an indirect influence on it through the impact on student motivation. Various student popula tions in reform oriented mathematics education The last section in this review is devoted to the literature concerning reform oriented teaching across educational settings including various student populations. Although research has succeeded in identifyi ng some effective approaches to teaching mathematic s to groups of students, less is known about the beneficial effects of reform oriented mathematics instruction on closing the achievement gap, or responding to the learning needs of students from the certa in populations. Race/ethnicity and special education are representative of those research domains in which less is known. Some reform oriented practices have been identified as promising practices primarily from correlational studies. Due to the nature of correlational studies, we cannot however overlook the possibility that these instructional practices may be a proxy for classrooms that are composed of high or at most typical achievers, rather than the practices themselves producing the increase in ach ievement. Thus without the empirical evidence focused on populations of learners with specific characteristics, we cannot draw conclusions about the benefits of reform oriented mathematics instruction for all types of students. In this vein, little resea rch is available that specifically addresses the learning needs of special education students, students of minority races and ethnicities, and those from low SES backgrounds to address the expectations for as well as challenges of these populations in the
71 context of reform mathematics education. Survey results reveal that teachers in high poverty urban schools tend to use fewer reform oriented practices (McKinney et al., 2009; McKinney & Frazier, 2008). Researchers ( Jackson & Neel, 2006; Jones & Thomas, 2 003; Maccini & Gagnon, under preparedness for reform oriented teaching that is focused on students with special needs. Specifically, Maccini and Gagnon (2002, 2 006) found that special education teachers were not familiar with the NCTM Standards and they often lacked the resources necessary to implement reform oriented teaching practices. Moreover, observations in different types of classrooms for EBD students fo und more reform oriented teaching in general education classrooms whereas both the resource room and self contained classrooms tended to limit teaching to algorithm instruction for factual knowledge and mathematical procedures (Jackson & Neel, 2006). Whe n it comes to classroom discourse, which is critical practice within reform oriented teaching, discourse patterns in classrooms with students with disabilities lacked the rich verbal opportunities supported by the reformative perspective (Berry & Kim, 2008 ; Griffin, Jitendra, & League, 2009; Griffin, League, Griffin, & Bae, 2013). The characteristics consistent with the NCTM communication Standard tended to be absent within classrooms for students with disabilities, including student to student exchanges a nd verbal interactions initiated by students. The lack of suggested as an important pedagogical approach for this population of students in the context of reform ( Berry & Kim, 2008; Griffin, et al., 2009). Even when definitions and explanations of mathematical concepts were observed, some of them were mathematically incorrect, raising an issue of teacher content knowledge (Griffin et al., 2013).
72 Few studies have focuse d on the associations between reform oriented teaching and students struggling to learn mathematics, yielding mixed results. Wenglingsky (2004) claimed that teaching practices had a potential to close the achievement gaps among minority students based on the result that the within school variance turned to be non significant after introducing factors of reform oriented teaching practices in the HLM models. However, this study should not be construed as the evidence in support of reform oriented teaching b ecause the teaching factors included all of the items in the 2000 NAEP teacher survey without any procedures to establish the validity of instruments. Thus, such factors as devoting instructional time to mathematics and homework, using the textbook, takin g tests, and emphasizing facts/routine problems made some contributions to the change in the within school variance in conjunction with other items more closely aligned with reformative principles. Yet, it was found that emphasizing measurement as a topic area was disproportionately beneficial to black students while the topic of data was to Latino students. Using project work, reasoning, and communication in the classroom, which are considered reform oriented practices, remained non significant to closin g achievement gaps by race/ethnicity. Meanwhile, Lubienki (2006) only HLM models so as to investigate the relationships between this type of teaching and a chievement gaps by race/ethnicity, failing to find a strong relationship. This result raises questions about the claim that equal access to reform oriented teaching in mathematics addresses the achievement gaps between different groups of students. Ongoi ng evaluations using more sensitive measures and various methodological approaches are indicated. The special education community seems to embrace this notion that more research is needed to determine the impact of reform oriented teaching on the mathema tics learning of
73 students with disabilities. Researchers ( Baxter et al., 2001; Baxter, et al., 2002; Mayrowtz, 2009; Woodward & Baxter, 1997; Woodward & Brown, 2006) have questioned whether special education students could be successful in these mathemati cs classrooms, suggesting that they may become minimally engaged, assume passive roles, and engage in non mathematical activities. In these classrooms, students are expected to participate actively in mathematical reasoning and communication while connect ing with diverse mathematical and non mathematical concepts. A disparity emerged between these characteristics of reform oriented teaching and the characteristics of special education students (Geary, 2004) and evidence based practices designed for them ( Baker, Gerstein, & Lee, 2002; Gersten, Chard, et al., 2009). As empirical study examples, observations on special education students in the general classroom (Jitendra et al., 2010 Standards was n ot able to yield differential improvement on word problem solving. oriented mathematics experiences. Mathematics writing was suggested as a promising tool to improv e the situation (Baxter, Woodward, & Olson, 2005) where special education students are more likely to be bystanders in the discussion oriented classroom (Baxter et al., 2001). This conclusion is however tentative because the study did not test the relatio nship between this possibly effective tool and student learning. Griffin et al. (2013) also failed to find a substantial pattern between classroom discourse practices and student achievement. Students with disabilities have shown improvements in some out come assessments even though teachers who built less productive discourse communities in their mathematics classrooms taught them. Taken together, it is necessary to remember that reform efforts to improve mathematics achievement should target all studen ts. Furthermore, legislative mandates reinforce this
74 research agenda must address the achievement gap in mathematics that exists between subgroups of low performi ng students by race/ethnicity and SES as well as the extant contradictions that endure between evidence based mathematics instruction from which special education students can benefit and reform oriented mathematics instruction. These dilemmas underscore the importance of including these populations in evaluations of reform oriented teaching. For the current study, the highest proficiency level in the ECLS K mathematics achievement data was used to define the student group of typical and struggling learne rs. It is notable that the ECLS K Public use Data Files include the variable of Individualized Education Plan (IEP) on school records. Since it was not possible to abstract the relevance with mathematics learning difficulty or disability from the variabl e of IEP status (e.g., mathematics IEP goals), a different achievement variable (1 st grade spring wave the highest proficiency level) was selected to define student mathematics performance levels. This strategy allows for additional examinations of main m odels depending on student groups of typical and struggling learners in mathematics. In conclusion, the literature reviewed by this chapter suggest important issues to justify the research questions posed and analyses used in the current study. The exist ing evidence base highlights the need for ongoing evaluations of mathemat ics education reform efforts with careful attention to conceptual and methodological modeling. Further examinations need to focus on the interactive relationships among student, teac her, and mathematics that enable us to observe how reform efforts influenced teacher, and in turn students. To date, a person (teacher) oriented approach was neither used to characterize reform oriented teaching practices nor to simultaneously evaluate th e interactive relationships in the context of mathematics education reform via medi ational modeling . Both approaches were used for the current study. This type of
75 research however requ ires a large scale data set, in which a reasonably large sample size a nd repeated (or longitudinally) observations are involved, and thereby the ECLS K Public use data set was selected for the current study. The ECLS K data set provided an opportunity for comprehensive analytic approaches in the examination of (1) the chara cteristics of mathematics instruction for the early elementary school year; (2) the mathe matics learning mechanism (medi ational relationships) related to mathematics self concept; and (3) commonality or difference in the learning mechanism according to tea ching profiles and student learning performance groups.
76 CHAPTER 3 METHOD The present study was conducted by performing a secondary analysis of the Early Childhood Longitudinal Study Kindergarten Class of 1998 99 (ECLS K) in order to investigate the stu dent learning mechanism in the context of reform oriented mathematics education promoted for the last two decades. The mechanism for the current study was conceptualized as the interactive relationships of teaching, student motivation, and student achieve ment. The current chapter describes the methodology and analytic procedures used to address five research questions. The chapter includes descriptions of the: (a) ECLS data set and samples selected for the study, (b) assessment results included in the a nalysis, (d) variables selected, and (e) analytic approaches employed for the individual research questions. Description of Data and Sample Selection The ECLS K data used in the current study is part of The Early Childhood Longitudinal Program carried ou t by the National Center for Education Statistics (NCES) within the Institute of Education Sciences. The majority of data was obtained from the public use files on the ECLS K CD ROM, however, the item level data from the Self Desc ription Questionnaire Mat hematics (SDQ Math) were obtained from the NCES website ( http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2010070 ). A nationally representative sample of 21,260 kindergartners has been followed from the base year (1998 1999) through the 8 th grade (2006 2007), with repeated assessments of their cognitive and behavioral development and family and schooling environments in the ECLS K study (Tourangeau, Nord, LÃª, Sorongon, & Najarian, 2009) . For the current study, the data (a) as reported by teachers were obtained from the data set categories of cognitive assessment,
77 behavioral development, and sch ooling environment, respectively. The ECLS K includes five rounds (i.e., points in time) of data collection gathered in the spring of kindergarten, 1 st grade, 3 rd grade, 5 th grade, and 8 th grade. Two additional rounds were collected in the fall of kinder garten and 1 st grade. This data collection process allows the present study to obtain and use three mathematics achievement sets administered during the early elementary school years (from 1 st to 3 rd grades). It is notable that the 1 st grade fall collect ion reflects a 30 percent subsample of schools. As such, in the current study, the models in which student achievement scores at differing rounds are linked involve this subsample. The sample sizes and case sizes are presented in the Table 3 1 below. Tab le 3 1 Sample Size and Case Size Data collection Round Unweighted Sample Size 1 (Respondents) Cases exported 2 1 st grade fall 1999 Approximately 5,300 4,389 1 st grade spring 2000 21,357 (17,324) 21,409 3 rd grade spring 2002 21,357 (15,305) 21,409 (4643 ca ses at the teacher level) Note. The sample size and respondents are based on the ECLS (Tourangeau, et al., 2009). The number of cases is from the exported data set focused on the target variables in the study (e.g., mathematics s cores and SDQ Math scores). Primary Sampling Unit, Stratification, and Sampling Weights The ECLS K data set is based on a multistage sampling design. The primary sampling units (PSUs) were geographic areas consisting of counties or groups of counties. T he 24 largest PSUs as measured by number of 5 year olds in the area were selected with c ertainty. These 24 PSUSs are called self representing PSUs. The remaining PSUs were placed in 38 strata of approximately equal size and two PSUs were selected from eac h stratum with a probability proportional to the size of the PSU. These 76 PSUs are called non self representing PSUs. From within each PSU schools (the second stage sampling units) and students (the third stage sampling units) within schools were then sel ected. Stratification divides the population into
78 different groups that are considered independent of those from another strata (Cochran, 1977). Accordingly, the probability weights for samples will likely be different depending on certain strata. The u se of PSUs in the analysis is to account for clustering, so as to address the issue of attenuated standard errors that lead to false positives in the significance tests. By accounting for the stratification, the standard error of the estimates is reduced. That is, the variance is smaller within the strata than in the sample as a whole. The ECLS K provides a range of cross sectional and longitudinal weights for child and parent level data. The PSU identifiers and strata are used as the nesting variable in the implementation of the Taylor Series linearization method and the sequential numbering of strata and PSUs was constructed separately for each weight. For the present study, in which variables from the various data collection rounds are modeled, the appropriate cross section and longitudinal weights were selected according to the research questions. It is notable that there are no separate weights for the Teacher Questionnaires, School Administrator Questionnaire, Student Records Abstract Form, Spec ial Education Teacher Questionnaires, Adaptive Behavior Scales, or Facilities Checklists . As such, the analysis by which teacher level data is explored without any connection to student data was conducted without accounting for clustering as in the presen t study (i.e., Research Question 1). Weights and PSU and strata identi fiers are presented in Table 3 2 .
79 Table 3 2 . Weights, PSUs, and Strata RQ Variable: Description Number of cases Weighted number RQ2 C5CW0: Spring 3 rd grade cross sectional child w eight C5TCWPSU/ C5TCWSTR: Sampling PSU and stratum spring third grade C weights 14,470 3,938,513 RQ3 RQ4 RQ5 C245CW0: Longitudinal weights for child data from spring kindergarten, spring first grade, and spring third grade, alone or in conjunction wit h any combination of a limited set of child characteristics. C245CPSU / C245CSTR: Longitudinal sampling PSU and stratum from spring kindergarten to spring third grade. 13,694 3,843,642 Construction of Analysis Samples For Research Question 1 (RQ1) rega rding the 3 rd grade instruction from the teacher reports, the data, which was originally exported at the child level, was reorganized to the teacher level. From the sampled 3 rd grade students, 16,383 cases had teacher data as well as mathematics assessmen ts from students. Teacher level data were created using the following data cleaning process: Child level cases missing a Round 5 teacher ID (T5 ID) and scores on all of the mathematics instruction questions (i.e., no teaching level data were collected) we re deleted. When two or more child level cases had the same T5 ID as well as the same scores on the mathematics instruction questions, cases subsequent to the first were deleted. Missing data were cleaned up by selecting only one case among cases which h ad the same T5 ID and had missing data on all teaching variables. These cases were treated as missing data (coded using 999 in this study) in the analysis.
80 Cases were deleted when two or more child level cases had the same T5 ID and did not have scores on any mathematics instruction questions. When a child level case did not have a T5ID, but did have scores on the mathematics instruction questions, the case was included in the data set. Although the majority of the child level cases with the same T5 ID had the same tea ching data, this did not happen for all child level cases with the same T5 ID. For these cases, it was presumed that the teacher attempted to respond differentially based on target students and all such cases were included in the data. T he number of cases included in the analysis was 4,643 at the teaching level. A total of 1 ,299 cases had missing data on all items in the mathematics teaching practices survey. For Research Question 2 (RQ2) regarding the 3 rd grade mathematics self concep t assessment, the exported data included 14,381 cases. For the Research Questions 3, 4, and 5 (RQ3, RQ4, RQ5) for which three different achievement data sets were incorporated, the analytical sample included the participants in the 1 st grade fall round. The 1 st grade fall data collection was limited to a 30 percent subsample of schools and 27 percent of the base year children. The fall 1 st grade sample of schools was a 30 percent equal probability subsample of schools from all 24 self representing PSUs a nd a 60 percent subsample of schools from 38 non self representing PSUs, with one PSU selected at random from the pair of PSUs selected from each of the 38 strata. Approximately 5,300 students were included, but the number of child level cases containing mathematics scores for the spring, 1 st grade data was 4,389 at the student level. Instruments Teacher questionnaires Of the ECLS practices. Among questions on Form A, questions in gr oups Q53 and Q54 are specific to questions in Q53, asking about the types of mathematical activities in which the teacher engages the class; (b) 8 questions in Q54 Skill, asking about the types of skills the teacher emphasizes;
81 and (c) 5 questions in Q54 Topic, asking about the mathematical topic areas the teacher addresses. The 5 mathematics topics consist of Numbers and Operation, Measurement, Statistics, Geometr y, and Algebra. addition, teachers could select 9, do you address each of the fo an 1 to 4 rating scales were treated as missing in the current study. All items for Q53 and Q54 are presented in Appendix A. Refused Do not know ECLS K research team. Again, responses in these categories were treated as missing data in the present study. As such there are four response categories for non missing scores in the teacher level data set. Missing data was coded 999. Since weights were not used for the teaching level analys es, the results are not weighted. Self description questio nnaire mathematics (SDQ Math) perceived interest and competency in mathematics learning: C5 Q6. Work in math is easy fo r me C5 Q12. I cannot wait to do math each day C5 Q16. I get good grades in math C5 Q22. I am interested in math
82 C5 Q26. I can do very difficult problems in math C5 Q30. I like math C5 Q36. I enjoy doing work in math C5 Q41. I am good at math A four point rating scale is used for this instrument and includes the following Among the 6 SDQ scales, the four scales of reading, mathematics, all subjects, a nd peer relationships were adapted from the Self Description Questionnaire I (Marsh, 1990). The reliabilities (alpha coefficient), weighted means, and standard deviations are presented in Table 3 2 below. ECLS K 3 rd grade students reported greater than 3 point with 4 representing the highest rating. SDQ Math for the 3 rd grade level ( C5SDQMTC Math ) revealed to be reliable with C a lpha coefficient of .8 9 (Mean, 3.16 and Standard deviation, 0.79) for the 14,379 cases. Direct cognitive assessment The mathematics cognitive assessment was designed to measure conceptual and procedural knowledge, as well as problem solving across mathematical topic areas including: (a) number sense and number properties and operations; (b) measurement; (c) geometry a nd spatial sense; (d) data analysis, statistics, and probability; and (e) patterns, algebra, and functions. The following nine proficiency levels were identified in the mathematics assessments from kindergarten through eighth grade: (a) Number and Shape; (b) Relative Size; (c) Ordinality and Sequence; (d) Addition and Subtraction; (e) Multiplication and Division; (f) Place Value; (g) Rate and Measurement; (h) Fractions; and (i) Area and Volume. The levels can be used as the ordinal variable as data are c ollected from kindergarten through 8 th grade mathematics assessments. The developers of the proficiency levels assumed that proficiency at a higher level implies proficiency at lower levels as well and report that the data collected in ECLs K were
83 general ly consistent with this assumption. The percentages of 1 st and 3 rd graders in the ECLS K sample who mastered the proficiency levels are presented below in Table 3 3 as percentages. At the 1 st grade fall (subsample), 28% of ECLS K students performed at or below the lowest three levels, but at the spring data collection, only 6 percent of them remained at the low performance level. For the 3 rd grade students, 4 percent of them performed at the lowest proficiency level. Table 3 3. The Highest Proficiency Level of 1 st and 3 rd Grade Samples Variable: Description Below Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Level 8 Level 9 C3R4MPF 1 9 18 42 24 5 1 0 0 0 C4R4MPF 0 1 5 20 48 22 5 0 0 0 C5R4MPF 0 0 0 4 17 30 31 16 2 0 Note. C3R4MPF is 1 st g rade fall Math Highest Prof Level Mastered. C4R4MPF is 1 st grade spring Math Highest Prof Level Mastered. C5R4MPF is 3 rd grade spring Math Highest Prof Level Mastered. NCES used item response theory (IRT) methods to generate adaptive tests in which the stages in which a series of routing questions determined the appropriate level of difficulty at the first stage prior to the matched questions based on score s on the routing test (17 items for the routing test and 123 items for the adaptive test). Test takers were not asked to write or verbally explain their answers or solving processes, and were provided with paper, pencil, and other manipulative materials. The use of IRT scale scores is considered the most appropriate metric for analyzing achievement data across different test form administrations and across different grade levels. The descriptive statistics for the IRT mathematics scale scores are present ed below in Table 3 4. All three mathematics tests showed a high level of reliability (0.94~0.95) indicating good scale precision, and the weighted mean of the IRT scale scores continued to increase across grade levels (from 43.57 to 98.77).
84 Table 3 4. Mathematics IRT Scale Score Variable: Description (Reliability of IRT scores) Range Weighted Mean (Standard deviation) C3R4MSCL: 1 st grade fall Mathematics IRT Scale Score (0.94) 0 174 43.57 (14.22) C4R4MSCL: 1 st grade spring Mathematics IRT Scale Score (0.94) 0 174 61.50 (17.66) C5R4MSCL: 3 rd grade spring Mathematics IRT Scale Score (0.95) 0 174 98.77 (24.96) Note . Reliability is based on the reliability of theta in IRT score estimation. Analytic Approaches RQ 1 . Do interpretable teacher pr ofiles of teacher practice emerge using the mathematics instructional items in the ECLS K 3 rd grade data set? The person oriented approach using latent class analysis (Rost & Langehein e , 1997 [LCA]) was used to identify groups of teachers engaging in diffe rent patterns of teaching practice. Unlike the extant body of research on teaching practices (e.g., Smith et al., 2005; Lubienski, 2006) that has often focused on variable oriented analysis (e.g., factor analysis), this study employs a person oriented ana lysis to define teaching practices . The person oriented analysis was selected in that dimensional identity might be uncertain in studies of instructional practice that rely upon teacher reported data. Dimensional identity refers to how a set of variables defines and confirms a trait of behavior or development (Bergm an, 2006; von Eye & Bergman, 2003). A variable from relatively more frequent responses to a set of teaching practices such behaviors (i.e., variables in the analysis) from the different dimensional arenas when teacher behaviors function as their instruction in the classroom. Developmental and personality psychologists have attended to this matter and suggested a person oriented approach to consider
85 all components simultaneousl y (Campbell, Shaw, & Gilliom, 2000; von Eye & Bogat , 2006; von Eye & Spiel, 2010). Accordingly, by using person oriented analysis, the present study aims to identify teaching practices from a perspective that presumes each individual teacher or the functi oning of teaching behaviors in the classroom acts as a holistic totality, but not as an aggregation of fragmented units. LCA in M plus Version 7 ( MuthÃ©n & MuthÃ©n, 2012) was used to generate profiles of teaching practices in light of mathematics reform. L CA uses observed variables to create subgroups with similar patterns of scores in individual variables. These subgroups are referred to as latent classes (Rost & Langehein, 1997). In the present study, the 26 mathematics teaching variables from the Teach er Questionnaires were introduced to identify the latent classes of teachers with similar teaching practice profiles. The 2 and 3 class models were tested and the ultimate number of classes was determined based on the model fit indices. Model fit indice s include: log likelihood (LL) value and replication of LL, Bayesian information criterion (BIC), and entropy (Nylund, Asparouhov, & Muthen, 2007). The LL serves as the basis for such model fit indices as AIC and BIC. AIC is a measure of the goodness of fit that considers the number of model parameters whereas BIC considers the number of parameters as well as the number of observations. Lower AIC and BIC numbers are desirable (Nylund, et al., 2007). Entropy refers to the uncertainty in classification (V ermunt, & Magidson, 2002). M plus reports a rescaled version of relative entropy, which has been defined by [0, 1]. Values near one indicate high certainty in classification. Estimation of LC models requires starting values for all parameter estimates. Starting values are initial estimates of the parameters. These estimates are improved by iterating the estimation procedure. Final estimation of LC parameters can be dependent on the starting values
86 and produce estimates that are not optimal. To addres s this problem, it is common practice to estimate models by using multiple sets of starting value and/or optimization number and confirm that the best estimates occur for these multiple sets. The best estimates are those with the largest LL. Estimation w as conducted with 1000 sets of initial stage (IS) starting values. Ten iterations to improve the estimates were conducted for each set. The 250 estimations with the largest LL values were identified and estimates from the 10th iteration were used as new starting values. For each of the 250 final stage (FS) starting values sets the estimation procedure was carried out until no further improvement in the estimates was possible. The estimations with the largest LL values were then identified. Overall, t h e same LL obtained for differing starting values and optimization numbers (replication of LL) provides evidence for a global maximum in the parameter estimation , to some degree : the best log likelihood of 77311.797 was replicated (a) 40 times with 500 IS and 40 FS starting values (b) 97 times with 400 IS and 100 FS starting values , and (c) 247 times with 1000 IS and 250 FS starting values , for the 3 class model. The fact that few of LL values reached the different final values, however, suggests a possibi lity that the algorithm might converge to local maxima for some random starts. Appendix B shows the M plus syntax. The statistical model is , where denotes the i th of Q observed teaching practices indicators, denotes the probability of belonging to la tent class k denotes the distribution of the indicators conditional on the parameters of the distributions, and denotes the distribution of the indicators conditional on the parameters of the distributions in the k th latent class.
87 RQ2 . Do the mathemati cs self concept items from the self description questionnaire (SDQ) that are included in the ECLS K 3 rd grade data set define a single mathematics self concept SDQ factor? The purpose of RQ2 is to assess the factor structure of the SDQ Math with the ECLS K 3 rd grade sample by utilizing confirmatory factor analysis (CFA). A model consisting of one dimension (i.e., mathematics self concept) seemed plausible given that there is support for mathematics self concept as a higher order factor of several observed variables regarding self concept (Marsh, 1990), and that there is support for the academic specificity of mathematical self concept (Marsh et al., 1991). Consequently, it was hypothesized that this latent variable model should establish a good model fit, and therefore, the eight items of SDQ Math should predict mathematics self concept of 3 rd grade students in the ECLS K sample. The statistical model is if , where denotes the of P observed mathematics self concept indicators ( , denotes the of P item specific latent variables, and denote the threshold parameter for response cat egory c , For and for . The symbol denotes the latent mathematics self concept variable and is a residual for the item sp ecific latent variable i in the factor analysis model. M plus Version 7 ( MuthÃ©n & MuthÃ©n, 201 2) was used for this analysis. RQ3 . Is the relationship between 1 st grade mathematics achievement and 3 rd grade mathematics achievement mediated by mathematics s elf concept? The purpose of RQ3 is to assess concept and achievement are related. There is reasonable support for the predictability of achievement by mathematics self concept (Dermitzaki, et al., 2009; Marsh & Hau, 2004; Ma rsh et al., 1991; Pajares & Miller, 1994). That is, domain specific self concept measured at the subject matter or
88 task level is associated with mathematics performance. Previous research has been limited to studies on discrete links between mathematics self concept and achievement; consequently, a triangular relationship from prior to subsequent achievement through mathematics self concept cannot be confirmed. To address this limitation in the body of extant literature, the current study has employed me diation modeling. Mediation is related to a chain of questions that an initial cause variable (X) affects the mediating variable (M), which in turn affects the outcome variable (Y). There are various strategies that can be utilized in order to establish a mediation effect. According to the causal steps approach introduced by the classic work of Baron and his colleagues (Baron & Kenny, 1986; Judd & Kenny, 1981), four conditions should be met: (1) significant direct effect, path (X to Y); (2) significant path (X to M); (3) significant path (M to Y); and (4) significant additional indirect effect estimated. A significant indirect effect is supported when the direct effect is changed by introducing M. Meanwhile, the product of coefficients strategies ar e based on the idea that the product term of paths of and is a proper quantification of the indirect effect. Most strategies regarding the product coefficient test the point estimate of the mediation effect for significance by dividing it by its stand ard error and comparing the result to the standard normal distribution (the standard Z method). Sobel (1982) devised the most commonly used standard error for the product estimate. Finally, re sampling or bootstrapping strategies are used to construct co nfidence intervals within which a range of possible estimates of the mediation effect is provided. The significant mediation effect is supported when zero lies outside the confidence interval (Fairchild & MacKinnon, 2009; MacKinnon, 2008; MacKinnon, Fairc hild, & Fritz, 2007).
89 For the current study, a basic mediation model was constructed as depicted in Figure 3 1. It was hypothesized that the effect of prior achievement on subsequent achievement would be co ncept. That is, the model first implies that achievement in 1 st grade should have a direct effect on achievement in 3 rd grade. In addition, it is expected that an indirect relationship in which changes in the level of prior achievement would be related t o the level of mathematics self concept in 3 rd grade, which, in turn, would be related to achievement in 3 rd grade. The 1 st grade and 3 rd grade IRT scale scores were used as the prior (X) and subsequent (Y) achievement, respectively. For mathem atics self concept, the average score on the eight SDQ Math items were introduced as the mediator (M) in the model. Figure 3 1. Mediation modeling The statistical model is , where Y denotes 3 rd grade mathematics achievement, X denotes 1 st grad e mathematics achievement, and M denotes the latent mathematics self concept variable, defined by the equations in Question 3. The are intercepts and the are residuals. The Latin letters denote regression parameters. M plus Version 7 ( MuthÃ©n & MuthÃ©n, 2012) was used for the analysis. X: 1 st grade Achievement M: 3 rd grade Math Self c oncept Y: 3 rd grade Achievement
90 RQ4 . by mathematics self concept between 1 st grade mathematics achievement and 3 rd grade mathematics achievement? Researchers have advocated e valuations of moderator and mediator effects in the same study because the mediated relationships among variables may depend on others (Baron & Kenny, 1986; Kraemer, Wilson, Fiburn, & Agras, 2002). Moderation occurs when the strength of an established rel ationship is changed depending on a third variable, the moderator (W). This variable can affect the relationship by enhancing or reducing its magnitude, as well as changing the direction (Baron & Kenny, 1986; Fairchild & MacKinnon, 2009). Historically, r esearchers have combined mediation and moderation by analyzing them separately and then interpreting together, which is referred to as the piecemeal approach (Edwards & Lambert, 2007; Preacher, Rucker, & Hayes, 2007 ). Another typical approach is the subgr oup approach . In this approach, the sample is divided into subgroups based on the moderator variable (e.g., subgroups by ages) and a mediation model is examined separately in each subgroup. If the significance of the estimated indirect effects differs be tween subgroups, then inferences are drawn for the moderated mediation. Finally, in the moderated causal steps approach , the mediation analysis is conducted following the causal steps procedure with the product ter m with W, the moderator, added within the framework of regression analysis. Besides its shortcomings, such as statistical power and issues inherent in high dependence on the direct effect, none of the approaches mentioned above provide a clear mechanism for modeling (i.e., which paths are affect ed by the moderator) or the magnitude of conditional indirect effect (Edwards & Lambert, 2007). To avoid these problems, the current study framed the moderation in the context of structural equation modeling to express direct, indirect, or total effects a cross different levels of W , testing the conditional indirect effects.
91 Depending on which path W influences, there are five variations on moderated mediation modeling when the focus is on indirect effects (Preacher, et al., 2007). For example, in the sec ond model, represented in Figure 3 2 (i.e., Model 2), W moderates the path (X to M), thereby considering the moderated indirect effect as the sum of two products of coefficients: (Fairchild & MacKinnon, 2009). For the c ase of Model 5, W moderates both paths of (X to M) and (M to Y); thus, the moderated indirect effect is calculated by the product term of path and path relevant coefficients: . If zero is not contai ned in the interval, the moderated effect is supported (Preacher et al., 2007). Edwards and Lambert (2007) have presented additional variations in moderated mediation modeling including the direct effect to be examined. [Model 2] [Model 5] Figure 3 2. Variations on moderated mediation modeling in Preacher et al. (2007) Results of moderated mediation modeling indicate direct or indirect effects as a function of the W . The significant coefficient of Y on XW indicates that the relationship between X and Y (i.e., direct effect) differs depending on W. The significant coefficient of M on XW or Y on MW indicates that the paths consisting of overall indirect effects diffe r as a function of W. Finally, the overall indirect effect conditioned on the level of W is determined by the
92 significance of the product term(s) of two paths supplemented with coefficients relevant to W and its product terms with X or M. For the curre nt study, it was hypothesized that an indirect effect of prior achievement (X) on the subsequent achievement (Y) via the student level of mathematics self concept (M) would be different across different levels of instructional practices (W), referred to as a moderated mediation effect (MacKinnon, 2008). More specifically, student mathematics self concept (M) influences the achievement mechanism from the 1st grade level (X) to the 3 rd grade level (Y), however estimates of indirect effects can be different a cross students whose teachers have different teaching profiles identified by the LCA. Specifically, the class of teachers considered supportive of reform to moderate the overall i ndirect effects of prior achievement, via mathematics self concept, on the subsequent achievement, such that the relationships would be stronger for students whose n model is shown in Figure 3 3. Figure 3 3. Moderated Mediation M odel With regard to the statistical model, if there were two observed classes of teacher practices, the model would be X: 1 st grade Achievement M: 3 rd grade Math Self concept Y: 3 rd grade Achievement W: Teaching Profile Membership
93 , where all terms previously have been defined. With K latent classes of teacher practices the model is , where the k subscript indicates that the parameter varies across the latent classes. The variation captures the moderation of the mediated relationship between 1 st grade mathematics achievement and 3 rd grade mathematics achievement (see MuthÃ©n, 1988 2004). M plus Version 7 ( MuthÃ©n & MuthÃ©n, 2012) yielded the point estimates and confidence intervals. Additional model testing was conducted to directly examine if the indirect effect established by RQ3 was the same across classes of teacher practices. The multiple groups modeling approach was employed. The teacher profile class variable was used as the grouping M plus . Bas ed on the missing data theory, the overall model for each group is estimated by MODEL descriptions from the whole data set. Also, the model of group designated by the label is estimated based on the different specifications of MODEL followed by a label. The M plus program was formed to test the omnibus hypothesis, H 0 : a 1 b 1 = a 2 b 2 = a 3 b 3 , where the subscripted number denotes the three different classes of teacher practices. Pairwise comparison tests were also specified. The M plus programs are presented in Appendix B. RQ5 . D o differences exist in the parameters defined with the teacher latent classes as the moderator for two mathematics performance groups: struggling learners and typical learners? RQ5 examined the difference, if any, in the established m oderated mediation effects involving prior achievement, mathematics self concept, subsequent achievement, and teaching profile between struggling and typical learners.
94 In addition to prior achievement, another student level variable relevant with achieve ment was introduced to the model to define student group of typical and struggling learners. Accordingly achievement data form the 1 st grade spring data was introduced as prior achievement variable because the moderator should be exogenous in the mediatio nal loop (i.e., 1 st spring achievement to 3 rd grade mathematics self concept to 3 rd grade achievement). The student groups were identified by the variable, The Highest Proficiency Level, from the ECLS K 1 st grade fall data. There are 10 categories, Level s 1 to 9 and below Level 1, that are used to classify student performance. Twenty six percent of ECLS K 1st graders could be categorized as struggling learners in mathematics, with 1% at the lowest level, 5% at the second lowest level, and 20% at the thir d lowest level given that these figures align closely with the empirical criteria to identify students with mathematics difficulties (25 th percentile; Mazzocco & Myers, 2003). It is notable that the report s of ECLS K stude nts at the second lowest level (6 %) and at the third lowest level ( 26%) also correspond to the prevalence of mathem atics learning disabilities of 5 8 % ( Geary, 2004) and the con ventional cutoff score to identify children as having mathematical difficulties in the bottom 25% to 30% ( e.g., F uchs, Fuchs, & Prentice, 2004; Geary, Hoard, & Hamson, 1999; Hanich, Jordan, Kaplan, & Dick, 2001; Jordan, Hanich, & Kaplan, 2003 ), respectively. Within the subset of the sample based on the 1 st grade fall assessment, individuals in the three lowest perfo rmance levels on the 1 st grade spring Highest Proficiency Level were coded as struggling learners (1); other children in the sample w ere coded as typical learners (0). Following a multiple groups modeling approach, differences in mediation or moderated me di ational relationships by student group were examined. Two grouping variables are involved in RQ5, including the teaching profile group and the student performance group. In addition to the different classes of teacher practices, the student performance variable (1 for struggling
95 learners and 2 for typical learners) was introduced as the grouping variable. Results across models with different grouping variables allowed for comparisons of differences, if any, in parameter estimates. The statistical mod el expression is presented below. With two classes of teacher practices, the model would be , where all terms previously have been defined, and the m subscript indicates that the parameters of the moderated mediation model vary across struggling and ty pical learners. With K latent classes of teacher practices included in the model, the model is , where the k and m subscripts indicate that the parameter varies across the latent classes and struggling and typical learner groups. The M plus programs a re presented in Appendix B.
96 CHAPTER 4 RESULTS The present study explored a mathematics learning mechanism of young students at the early elementary school level through secondary analyses of the Early Childhood Longitudinal Study Kindergarten Profile o f 1998 99 (ECLS K) dataset. The purposes of the study were to (a) determine whether a set of mathematics instruction questions included in the ECLS K teacher questionnaire would be useful for interpreting latent profiles that represent subgroups of teache rs using teaching practices aligned with empirical evidence on reform based mathematics education; (b) confirm the dimension of mathematics self concept of young students at the early elementary school level; and (c) explore whether selected variables are related wi th each other to construct medi ational a nd conditional (moderated) medi ational relationships. In this chapter, analytic decisions and findings associated with each of the research questions are presented. Research Question 1: Teaching Profiles of the 3 rd Grade Teachers The purpose of Research Question 1 is to determine whether distinct and interpretable latent profiles could be identified and introduced into subsequent models for exploring mathematics learning mechanisms. To select the optimal latent profile model, data driven results must be interpretable and defensible by the model fit indices as well as the literature. In the present study, latent profiles represent the unobserved categorical variable concerning mathematics instruction, whic h were identified using the 26 observed variables reported by teachers in the ECLS Having 4 ,643 teaching level cases and 26 teaching variables, the latent profile analysis for 2 and 3 class models were conducted. For each model, 2 and 3 class solutions were chosen
97 based on the literature that regards mathematics instruction as dichotomous (reformative vs. traditional), or as more complex (e.g., Smith et al., 2005) To evaluate the model fit, log likelihood value (LL), Akaike (AIC), Bayesian information criterion (BI C), and entropy are presented in Table 4 1. Smaller AIC and Sample Size Adjusted BIC numbers are desirable (Nylund, Asparouhov, & Muthen, 2007). Entropy refers to classification uncertainty (Vermunt, & Magidson, 2002). M plus reports a rescaled version o f relative entropy, which is defined on [0, 1]. Values near one indicate high certainty in class identification. AIC and BIC favor the 3 class model , leading to the 3 class model as the optimal latent class model. Table 4 1. Model fit statistics. 2 c lass 3 class # of Parameters LL AIC BIC Sample Size Adjusted BIC Entropy 155 78955.658 158221.316 159169.129 158676.623 0.830 233 77311.797 155089.595 156514.372 155774.025 0.834 N for each profile 1603 (47.9%) 1741 (52.0%) 1022 (30 .6%) 750 (22.4%) 1572 (47.0%) Average Latent Profile Probabilities for Most Likely Latent Profile Membership 3 profile / 26 items 1 2 3 1 0.931 0.067 0.002 2 0.051 0.914 0.035 3 0.005 0.064 0.931 Note . LL = log likelihood; AIC = Akaike; BIC = Bayesian Information Criterion; Replications are based on sets of 1000 start values with the display set showing the top 250 values.
98 To avoid the possibility of local maximum solutions that may threat en the validit y of the results, more models with multiple sets of IS and FS (400/100, 500/40, and 1000/250) were examined. Overall, t he same LL was obtained for differing sets , showing evidence of a global maximum in the parameter estimation. Finally, the selected 3 c lass model represented 3 subgroups of teachers with similar teaching profiles that were distinct from other su bgroups on the 26 variables. In short, teachers assigned to a latent class have, in general, similar teaching profiles in terms of the frequency level of implementation across the 26 ECLS K teaching variables. Prior to presenting the results of the LCA, I present my interpretation of the three latent classes. This should assist the reader to interpret the results. Latent class 1 was labeled as the Anti reform Profile because the teachers assigned to Class 1 showed resistance to implementing reform oriented instructional practices. Latent Class 2 was labeled as the Active Implementer Profile because teachers in the class had the highest level of im plementation of reform oriented components. Their frequency of implementation of traditional practices was similar to the frequencies for the other two groups. Latent Class 3 was labeled Proactive Implementer Profile because teachers in this class , repor ted moderately frequent implementation of reform oriented instruction, but with a reduced emphasis on traditional instructional components. Results of the LCA conducted in M plus provided probability scales representing the proportion of cases associated with each response category for each variable within a latent class . These results can be used to interpret and attach meaning to each class . The odds ratio results can also be used and highlight where differences are most pronounced. Since a smaller n umber (i.e., 1) in the response category indicates more frequent implementation of a teaching indicator (e.g., everyday), an odds ratio above 1 indicates a larger proportion of teachers in the first latent class pared to one of the other latent classes. An
99 odds ratio less than 1 can be interpreted in reverse because the odds smaller than mean the class relative to one of the other classes . For ex LC2: >1 : 8.506. This result means that the odds ratio for a score larger than 1 is 8.506 for Class 1 versus Class 2 and indicates that the odds that teachers classified in Class 1 respond with less than frequent use of group work are 8.5 times larger than the odds for teachers in Class 2. For >1 , 0.840 for Class 1 versus Class 3 indicates odds for that teachers in Class 1 report less th an frequent use of textbooks is 84% of the odds for teachers in Class 3. Latent class odds ratio results ar e presented in the Appendix C. To assist in understanding the interpretations, figures for each class across variables are presented. Before creat ing figures, the 26 variables were categorized into 5 conceptual clusters: oriented instruction; (c) 6 for reformative instructional practices; (d) 5 for mathematical topic areas ; and (e) 6 for specific mathematical skills. Within each cluster, variables that shared a similar visual pattern regarding probability differences by 3 classes might have indicated that these variables differentiated teacher subgroups in a similar way. Variables presented in Figure 4 1 often have been used as indices to traditional teaching practices within the extant literature. odds ratios for Class of Anti reform (in blue) versus Class of Proactive Implementer (in green) at the first response category ( >1 ) were found, indicating that teachers in Proactive Implementer Class ractices.
100 Figure 4 1. P rofiles of Response Probabilities Across Response Categories : Traditional Instruction Variables of Textbook; Worksheet; Concepts/Facts; Procedural; and Math Tests. ( Note . Horizontal axis scale is 1 (more fr equent implementation) to 4 (no implementation) for each variable. As such, these variables effectively differentiated Class of Anti reform and Class of Proactive Implementer revealing that teachers assigned to Anti reform Class had a greater possibility of implementing these types of instructional practices wh en compared with teachers in Proactive Implementer Class . would be used as an index of less reformative instruction, these two variables do not seem to item, teachers in Proactive Implementer Class show an increased probability of implementing Anti reform Class and Active Implemente r Class at the second response category ( >2 ). It is noteworthy that Active Implementer Class (in red) showed the highest probability of more frequent implementation (first or first two response categories) across all 5 variables. This finding suggests th at teachers in Active Implementer Class may be engaged actively in various forms of teaching practices regardless of the different forms of mathematics teaching debated within the research community. According to the overall probability patterns of these variables, a conclusion could
101 be drawn that teachers in Proactive Implementer Class have mathematics teaching profiles characterized by significantly less implementation of traditional practices. Figure 4 2. P rofiles of Response Probabilities Across Response Categories : Components of Explicit Reform oriented Instruction Variables of Calculators; Measuring Instruments; Manipulatives; and Computers. considered indicative of reform oriented mathematics instruction (e.g., Cohen & Hill, 2000). Since these practices are resource relevant, they were plotted together in Figure 4 2. These 4 variables effectively differentiate Class 1 from Class 2 teachers , and somewhat effectively differentiate Class 1 from Class 3. Teachers assigned to Class 1 showed significantly less implementation of teaching relevant to these variables in comparison with Class 2. However, not all of these variables were effective en with more frequent impleme ntation (i.e., the first or first two response categories). For example, 1 2 a month . This ost
102 reform oriented instruction, it might be harder to utilize computers almost everyday for their instruction when compared to their use of manipulative materials . Figure 4 3. P rofiles of Response Probabilities Across Response Categories : Reform oriented Instruction Variables of Group Work; Real Life; Talking; Writing; Discussion; and Project. Regarding these variables, teachers in Anti reform Class reported les s implementation of reform oriented practices in comparison with those in Active Implementer and Proactive Implementer Class characteristic of traditional mathematics teaching. The odds ratio calculations, by which teachers in Active Implementer and Proactive Implementer Class were compared, yielded significant values less than 1 at the response level of 1 or greater, indicating that teachers in Active Implementer Class reporte d more frequent implementation of these practices than teachers in Proactive Implementer Class . Given that the ECLS K 3 rd grade data collection was performed during 2001 2002, and the revised NCTM Standards were disseminated during this time period, Class of Active Implementer revealed rapid and active adoption of these innovations. It is showed a slight different pattern in responses across three classes of teachers. That is, the response category of non implementation (4), det ermine the nature of three
103 Classes, indicating Class of Anti reform reported the highest level of response on non implementation. Figure 4 4. P rofiles of Response Probabilities Across Response Categories : Topic Variables (Number/Operation; Measurement; Geometry; Statistics; and Algebra) Reform policy has emphasized mathematical topic areas as Measurement, Geometry, Statistics, and Algebra. These topic areas share a similar pattern in the plots in Figure 4 4, suggesting that teachers in Active Implement er Class reported more active adoption of these topics (at the response level of 1 or greater), followed by those in Proactive Implementer Class (at the response level of 2 or greater). Again, teachers assigned to Active Implementer Class indicated a prof ile of frequent implementation across all 5 variables, whereas teachers in Proactive Implementer Class reported less frequent teaching of the Number/Operation area, but significant active implementation of topic areas highlighted in reform policy documents , when compared to Anti Reform Class teachers, or those characterized by a more traditional mathematics instruction profile.
104 Figure 4 5. P rofiles of Response Probabilities Across Response Categories : Specific Mathematics Concept and Process Variabl es (Place Value; Fraction; Estimation; Analytic Reasoning; Communication; and Shape) Finally, Figure 4 5 features results for variables regarding specific mathematics concepts (Place Value, Fractions, Estimation, and Shape) and mathematical thinking proces s (Reasoning and Communication) that indicates an emphasis on skills envisioned by the NCTM Process Standards . Active Implementer Class teachers reported the most frequent implementation across all six variables. Conversely, plots for Class teachers yiel ded a similar visual pattern but only for Estimation , Reasoning, and Communication. Teachers in Proactive Implementer Class showed a profile in which they took a moderate approach to implementing reform oriented skills, but also revealed reservations abou t using traditional instruction. According to the LCA, the 3 rd grade teacher sample was not distributed evenl y across the three subgroups. Of the 4,643 3 rd grade teachers who were identified in the spring 3 rd grade round of data, 1299 did not have any sc ores on the mathematics instruction questions. Of the remaining 3,344 teachers, 30.6% were assigned to the Anti reform profile, 22.4% were assigned
105 to the Active I mplementer profile, and 47% were assigned to the Proactive I mplementer profile. It is notab le that the largest class was the Proactive I mplementer profile. Research Question 2 : Factor Structure of Mathematics Self concept The purpose of Research Question 2 was to confirm the dimension of mathematics self concept of young students at the early el ementary school level using confirmatory factor analysis (CFA) to assess the factor structure of the Self Description Questionnaire Mathematics (SDQ Math) from the ECLS K data with a sample from the 3 rd grade spring wave. The tested model consists of one competency in mathematics learning, including: (1) Work in math is easy for me; (2) I cannot wait to do math each day; (3) I get good grades in math; (4) I am interested in math; (5) I can do very difficult problems in math; (6) I like math; (7) I enjoy doing work in math; (8) I am good at math. Previous research by Marsh and colleagues (1990, 1991) suggested a factor structure of mathematics self concept with similar questi ons. As such, it was hypothesized that a single factor model of mathematics self concepts could be confirmed using the SDQ Mathematics items for the ECLS K 3rd grader. The complex survey design was addressed by using the PSUs, strata, and weights in the anlysis. There were 14,381 student level cases and eight variables that were introduced into the model. The data set contained 7,006 cases with missing data on all variables, and these cases were excluded from the model estimation. Cases with missing d ata for some of variables were not excluded because M Plus uses all available data to estimate the model by using the Full Information diagonally weighted least squares estimation when cases with missing data are present. To evaluate the overall model fit , several fit indices were examined. The chi square goodness of fit statistic was statistically significant ( ), suggesting t hat
106 the model did not fit the data. However, this is rarely used as a sole index due to its sensitivity to large sample size. Such indices as the Comparative Fit Index (CFI) and the Tucker Lewis index (TLI) also were examined. By employing the standard suggested by Hu and Bentler (1999), the model demonstrated acceptable fit (CFI = 0.978; TLI = 0.969) becaus e values close to or above .95 are indicative of a good fit. Although the root mean square error of approximation (RMSEA, 0.097 [p=.000]) does not satisfy the standard by MacCallum et al. (1996), which considers 0.01, 0.05, and 0.08 as indicative of excel lent, good, and mediocre fit respectively, the value is not greater than 0.1, which is the cut off not to employ the model (Browne & Cudeck, 1993). Taken together, the model with one dimension of mathematics self concept was accepted without considering p ost hoc modifications. Figure 4 6. S tandardized loadings for 1 Factor Confirmatory Model of Mathematics Self concept. ( Note . N= 14381 and 7006 cases with missing data on all variables exclu ded from the model estimation) .854 (.004 ) Mathematics Self concept Everyday e .372 .778(.006) Grades e .759 .729 (.006 ) Interests e . 22 6 .88 (.004 ) Difficult e . 491 (.01) Like e .133 .931(.003) Good at e .270 Easy e .394 Enjoy e .179 .906 (.004 ) .792(.005 ) . 468
107 The CFA results are described graphically in Figure 4 6. The figure shows that the eight observed variables in rectangles could capture the dimension of mathematics self concept for the ECLS K 3 rd graders with the standardized fact the latent variable of mathematics self concept. These two variables are more closely related with the self appraisal of ability than responses directed at interests and enjoyment in mathematics learning. R esearch Question 3 : Mediation Modeling of Prior and Subsequent Achievement via Mathematics Self concept. Research question 3 tested the hypothesized causal relationship that achievement at 1 st grade would exert a significant effect on achievement at 3 rd grade, and that these direct effects of concept. The mathematics IRT scale scores from the 1 st grade fall wave were introduce d as the initial variable into the model, while those from the 3 rd grade spring wave were introduced as the outcome variable. Accordingly, the longitudinal PSU, strata, and weight variables for the analysis using data from the kindergarten spring wave thr ough 3 rd grade spring wave were used to address the nature of the complex survey data. It is notable that this analysis is based on a sub sample of ECLS K data because the analytical sample included the participants for which data were collected from the special data collection wave, 1 st grade fall wave (N = 4,389). The number of cases missing on all variables was eight. The mediation modeling results with unstandardized estimates are graphically described in Figure 4 7. The direct effect ( ) was signifi cant. Both paths formed a medi ational mechanism from the prior achievement and mathematics self concept to subsequent achievement, and also were significant ( ). The estimate indicates that
108 students ha ving higher scores on the prior achievement had higher scores on the mathematics self concept, as well as on the subsequent achievement. Finally, the indirect effect was significant (indirect effect: 0.008 (0.003), p <.01); therefore, mathematics self con cept significantly mediates the relationship between the first grade and third grade achievement. Figure 4 7. Mediation: Effects of Prior Achievement on Subsequent Achievement mediated by Mathematics Self concept Research Question 4 : Moderation Eff ect of Teaching Profile on the Mediation Relationship For Research Question 4, a combined data set was creating to link the 3 rd grade teacher level variable (i.e., teaching profile) with the student level variables. Although Research Question 3 already ad dressed the medi ational relationship among 1 st grade achievement, 3 rd grade mathematics self concept, and corresponding achievement, a meditation model again was tested as unconditional mediation modeling with the combined data set. Student level cases wi th missing data on teaching profile data were deleted (N=4,286). Aside from slightly changed parameter estimates, a significant unconditional mediation was confirmed (path a: = .005 (.001), p=0; path b: = 2.057 (.474), p=0; indirect effect: 0.011(0.003), p <0.01). 1 st grade Achievement (X) 3 rd grade Math Self concept (M) 3 rd grade Achievement (Y) = 1.35 (.031), p =0
109 Figure 4 8. Unconditional mediation model test with combined data set with teacher and student data A preliminary analysis to determine the relationship between student achievement and teaching profile was performed . An ANCOVA was conducted in which 1 st grade achievement, the teaching profile categorical variable (i.e., n = 927 for Class 1; n = 505 for Class 2; n = 1118 for Class 3), and 3 rd grade achievement served as the covariate, independent variable, and depend ent variable, respectively. The results of this analysis are described below in multiple forms, including: (a) descriptive results; (b) model implied results, evaluated at the values of 44.35 (the grand mean) on the covariate; and (c) ANCOVA results. A prior achievement by teaching profile interaction was significant so that teaching achievement. Students with teachers in the Proactive Implementer group (i.e., Class 3) showed the Anti reform (i.e., Class 1) group. Interestingly, students with teachers in the Active Implementer group demonstrated the lowest sampl e mean, but the highest adjusted mean on 1 st grade Achievement (X) 3 rd grade Math Self concept (M) 3 rd grade Achievement (Y) = 1.351 (.03), p =0
110 Table 4 2 . ANCOVA Results 3 rd grade Mathematics Achievement Teaching Profile (n) Mean SD Adjusted mean (SE) Anti reform (927) 100.14 23.60 99.06 (0.531) Active Implemente r (505) 98.67 25.84 100.88 (0.72) Proactive Implementer (1118) 100.43 24.49 100.45 (0.48) Total (2550) 99.98 24.45 Source MS F (df), p Partial Eta Squared Intercept 366177.90 1404.797 (1), p =0 .356 Teaching Profile (TP) 685.20 2.629 (2), p =0. 072 .002 Prior Achievement (PA) 769443.45 2951.877 (1), p =0 .537 TP by PA Interaction 1308.67 5.021 (2), p =0.007 .004 Finally, moderated mediation modeling was examined. In testing the moderated mediation effects, employing a multiple group approach m akes it easier to interpret the results because it provides separate mediation estimates for each group. Figure 4 9 illustrates the different mediation models by the three teaching profile groups. For the Anti reform group, the path from mathematics self concept to subsequent achievement was not significant. For the Active Implementer group, the path from prior achievement to mathematics self concept was not significant. The Proactive Implementer group showed significance for all paths at Alpha level .0 7, but the indirect effect was not significant (0.008 [0.005]). Anti reform group. Figure 4 9. Multiple Group Approach for Moderated Mediation Modeling: There Different Mediati on Relationship by Teaching Profile of Anti reform, Active Implementer, and Proactive Implementer. ( Note . Dashed arrow represents non significant paths ) 1 st grade Achievement (X) 3 rd grad e Math Self concept (M) 3 rd grade Achievement (Y) = 1.256 (.059), p =0
111 Active Implementer group Proactive Implementer group Figure 4 9 . Continued Model tests were performed to ask directly if the indirect effect was significantly different across th e three teaching groups. Omnibus hypothesis tests indicated that indirect effects for all teaching profile groups were non significant. In other words, there was no difference in indirect effects across three different teaching profiles. That is, none o f the teaching profiles were strong enough to exert differentiated effects on the student learning mechanism involving mathematics self concept. 1 st grade Achievement (X) 3 rd grade Math Self concept (M) 3 rd grade Achievement (Y) = 1.31 (.107), p =0 1 st grade Achievement (X) 3 rd grade Math Self concept (M) 3 rd grade Achievement (Y) = 1.372 (.052 ), p =0
112 Table 4 3 . Model Tests of Multiple Group Approach to Moderated Mediation Modeling 1 Indirect effect estimate (S E) Omnibus hypothesis test Pair wise test Anti reform 0.003 (.005) Active Implementer 0.001 (.013) Proactive Implementer 0.008 (.005) Wald Test of Parameter Constraints 1.360 (df=2), p =0.507 Anti reform vs. Active 0.002(.014) Anti reform vs. Pro active 0.005 (.007) Active vs. Proactive 0.007 (0.015) Research Question 5 : Difference in moderated mediation effects by student groups Research Question 5 examined the difference in moderated mediation effects involving prior achievement, mathem atics self concept, subsequent achievement, and teaching profile between struggling and ty pical learners. And thus, this model takes accounts of difference, if any, between student ability with respect to the learning mechanism moderated by teaching profi les. Following a multiple groups modeling approach , differences in moderated medi ational relationships by student group were examined. Dummy code W defines two student groups: Struggling Leaners (1) and Typical Leaners (0). Results across models with t he different teaching profile grouping variable allowed for comparisons of differences, if any, in parameter estimates. Unlike models for RQ3 or RQ4, different variable of prior achievement (1 st spring) was used. Hence, m ediation modeling was conducted again to confirm the mediational learning mechanism when 1 st spring achievement was used as X variable . This unconditional mediation model results revealed significant paths and indirect effect (path : 0.004 [0.001] , p =0; path : 1.940 [0.245] , p =0; pa th : 1.077 [0.013] , p =0; and indirect effect: 0.007 [0.001] , p=0 ). In doing so, it is notable that corresponding variables of strata, weight, and PSU were all replaced
113 (i.e., mediation modeling with C45CSTR, C45CPSU, and C45CW0 but moderated mediati on modeling with C245CSTR, C245CPSU, and C245CW0). Anti reform Implementer Active Implementer Figure 4 10 . Student Group Moderated Mediation Modeling: There Different Mediation Relationship by Teaching Profile of Anti reform, Active Implementer, and Proactive Implementer. ( Note . Dashed arrow represents non significant pa ths ) 1 st grade Achievement (X) 3 rd grade Math Self concept (M) 3 rd grade Achievement (Y) = 0.912 (.05), p =0 Struggle (W) = 0.036(.134) 1 st grade Achievement (X) 3 rd grade Math Self concept (M) 3 rd grade Achievement ( Y) = 0.954 (.091), p =0 Struggle (W) = 0.232 (.253)
114 Pr oactive Implementer Figure 4 10 . Continued . Figure 4 10 depicts three mediation models by the teaching profile groups , of which each the student group variable was entered as a moderator . For the Anti reform group, non significant moderation effe ct on each of paths incorporated into the medi ational association (i.e., paths of , , and ) was found . Correspondingly , an incomplete mediational association for this teacher sub group stay s the same after introducing a moderator of student group. Similarly, results for the Active I mplementer group revealed that the moderator exerted no influence on paths . Path holds a negative coefficient albeit non significant, wh ich is counterintuitive. Also the indirect effects were non significant in both student groups (se e Table 4 4), failing to establish the triangular learning mechanism for early elementary school students as a whole, or among groups of typical and struggli ng learners who have been taught by the Active Implement teacher. Results for the Proactive Implementer group revealed that the moderator exerted an influence on some of paths and on the whole mediational association. Meanwhile the path from 1 st grade Ach ievement (X) 3 rd grade Math Self concept (M) 3 rd grade Achievement (Y) = 0.89 (.056), p =0 Struggle (W) = 0.212 (.134)
115 1 st grade achievement to self concept was significantly moderated; the path from self concept to 3 rd grade achievement was not significantly moderated. The coefficient of moderation related was negative for path , indicating that the associat ion from mathematics self concept to achievement was less strong for the struggling learner group than for the typical learner group. Additionally, this association turned out to be weaker (path coefficients: 0.004 from Figure 4 9 a nd 0.001 from Figure 4 10) after taking accounts of student group difference. It is notable that results of model for RQ4 (i.e., teaching profile moderated mediation) with the same X variable as that for RQ5 (i.e., 1 st spring achievement) revealed the sam e path coefficient of 0.004. Finally, all three paths were significant and positive and the indirect effect was significant at the alpha level of 0.07 (see the third diagram in Figure 4 10). Therefore, it suggests that a significant mediational associati on of three student level variables for typical learners who have been taught by the Proactiv e Implement teacher . T able 4 4 . Moderated Mediation Association by T eaching and Learning Group s Note. Significant mediational association is presented in bold. Significance w as deter mined at the alpha level of 0.05 for paths and at the alpha level of 0.07 for indirect effect. Teaching Group Learning Group Path Path Indirect (SE) Anti reform Struggling 0.002 (.008) 1.161 (2.07) 0.002 (.01) Typical 0.008 (.002)* 1.723 (.882) 0.013 (0.007)* =0.066 Active Implementer Struggling 0.003 (.007) 2.007 (3.01) 0.005 (.019) Typical 0.004 (.003 ) 2.995* ( 1.446 ) 0.011 (.01) Proactive Implementer Struggling 0.009* (.004 ) 3.672* (1. 219 ) 0.0 33 (.0 15 ) )* =0.032 Typical 0.01 * (.002 ) 2.159* (.9 72) 0.0 22 (.011 ) * =0.052
116 The path from 1 st grade achievement to 3 rd grade mathematics self concept was significantly moderated only in the Proactive implementer groups. The path was significant for both group of typical and struggling learners but the moderation enhanced the path for typical learners but changed the nature of association for struggling leaners relative to the path ( path coefficient of 0.004) in the unconditional mediation model test. The results indicate that struggling learners exhibit a counterintuitive association between 1 st achievement and 3 rd grade mathematics self concept in classes taught by Proactiv e I mplemen ter teachers . Accordingly this significant result for the struggling learners was not accepted as confirming a hypothesis that these students could establish the mediational learning mechanism in classes taught by Proactive Implementer teachers. Furtherm ore the result worth noting is significant paths of and for the typical leaners in the Proactive Implementer teaching sub group despite the marginally significant indirect effect .
117 CHAPTER 5 DISCUSSION The purpo se of the present study was to explore relationships between instructional practices and a student learning mechanism in the context of 1990s mathematics reform efforts . Research questions were addressed by conducting secondary analyses o f the Early Child hood Longitudinal Study Kindergarten Profile of 1998 99 (ECLS K) data set. The ECLS K data set allows cross sectional and longitudinal information about students as well as teachers. All analyses in conjunction with student level data were performed usin g appropriate weights so reported findings represent a national sample of 3rd grade students and their mathematics learning in the 2001 2002 school year. Latent class analyses were conducted to explore whether mathematics instruction al variables included in the ECLS K data set would generate empirically and conceptually interpretable subgroups of teachers who show different features of teaching. Confirmatory factor analysis and mediation modeling were performed to determine whether items from the self de scription questionnaire (SDQ) in the ECLS K 3rd grade data set could support the motivational structure of mathematics learning indicated by the field, such as the collective work of Marsh et al. (Marsh, 1990; Marsh & Shavelson, 1985; Marsh & Yeung, 1997); and whether a triangular relationship involving prior and subsequent achievement mediated via mathematics self concept could be suggestive of a learning mechanism at the early elementary school level. Following selection of three subgroups of teaching p rofile s and a medi ational mechanism of mathematics achievement via mathematics self concept, the moderation mediation modeling was used to explore what relationships could be found between teaching profiles and a student learning mechanism from the data co llected during an era characterized by persistent mathematics reform efforts. Finally, the current study addressed the need for reform oriented
11 8 mathematics efforts to be focused on all students. Accordingly, the study included the final research question asking whether the findings for a certain learner group (i.e., struggling learners) would differ in comparison t o the whole population. The purpose of this chapter is to interpret findings, discuss implications of them, and provide suggestions for practi ce and future research. F ive major findings found in this study are summarized as follow s : 1. Three teaching profiles o f 3 rd grade teacher s were identified. The Anti reform group is characterized b y less frequent implementation of reform oriented approaches . Teachers in the Active Implementer subgroup most frequently implemented both reform oriented and traditional instruction al practice s. And, the Proactive Implementer subgroup featured the second highe st level of reformative implementation but also with reduced use of traditional practices. 2. A learning mechanism in which mathematics self concept mediate d the impact of prior achievement on subsequent achievement was suggested. 3. An interaction between teaching profile s and prior student achievement was foun d. For the Active Implementer subgroup, the gap between mean scores and adjusted mean scores were largest, raising a possibility that students with a lower level of prior achievement are more likely to attain improvement in subsequent achievement in mathe matics class rooms taught by teachers characterized by the Active Implementer profile. 4. The moderation effect of the teachin g profile subgroups on the medi ational mechanism of student learning was not found. Results however raised a possibility of an assoc iation between the Proactive Implementer subgroup and the learning mechanism. 5. No difference was found between struggling and typical learners. Results however raised a possibility of an association between the Proactive Implementer subgroup and the learn ing mechanism for the typical learner group. Interpretation of Findings M athematics reform efforts that began in the 1990s produced guiding documents, curricular materials, and professional development programs to encourage and guide policy implementation instruction that rested upon their deep understanding of both mathematics content and skills for implementing concept ually b ased
119 pedagogy. Th e s e efforts also address the learning needs of all students across various student populations and also diverse student outcomes including mathematics achievement and enjoyment in mathematics learning. Th e s e assumption s w ere supported by theory (e.g., constructivist theory , Bruner  ) and previous research (e.g., Senk & Thompson, 2003) . However , most of the qualitative studies had examined teachers of reform implementation, without attending to student learning (e.g., C ollopy, 2003) while quantitative studies were based on the dichotomous perspective (reform oriented vs. traditional) or focused only on a si ngle outcome , such as student achievement (e.g., S mith et al., 2005) or motivation (e.g . , Stipek, Salmon, et al., 19 98). Accordingly a fundamental purpose for conducting the present study is to determine whether reformers assumptions are confirmed through a nalyses of a large scale data set using alternate modeling techniques involving various variables. The sequence of research questions was explored not only to support existing empirical evidence on reform oriented mathematics, but also to help unpack more nuanced relationships between mathematics reform efforts and teacher implementation and between teaching and lea rning captured by a nationally representative sample of U.S. students and their teachers in early mathematics classrooms. Three Subgroups of 3 rd Grade Teachers Who Share Similar Teaching Profiles Identified by the Latent Class Analysis A 3 class model was selected because it was supported by the model fit indices and the distinction between subgroups was conceptually interpretable by the research. Each c lass represents a group of teachers who share a similar teaching profile and e ach profile is distinct f rom each other. Using 26 variables that were collected from teacher reports o f their mathematics instruction, the three subgroups were constructed . To assist in the interpretation,
120 each group was given distinct label s, including: Anti reform , Active Impl ementer , and Proactive Implementer , and each was used as a categorical variable for the subsequent analys e s. The resulting profiles glean ed from this analysis can be viewed a mathematics instruction delivered during an era of intense reform gi ven that the current analysis was performed using data f rom teachers who taught 3 rd graders during in the 2001 2002 school year. However, s ome of these teachers might have attempted to incorporate suggestions provided by the National Council of Teachers o f Mathematics (NCTM) S tandards (1989; 1991; 1995; 2000) into their teaching in response to their districts and a and/or their own participation in professional development programs on reform oriented mathematics instruct ion. Others might show certain profiles as a consequence of such indirect influences as vigorous national level discourse on reform oriented mathematics throughout 1990s. Still others may not have had much, if any, opportunities for learning during this time period. Hence, the three subgroups offer a sampling of the range of possible teaching profiles national ly of 3 rd grade mathematics education in the early years of the more recent era of mathematics reform. F urther, f indings f o r these three types of teaching profile s seem plausible given that research views reform policy implementation as involving complex transformations of message s they receive , such as adop tion or resistance (Coburn & Stein, 2006; Spillane et al., 2006). The scrutiny o f the probability scales across all 26 variables characterized the Anti reform group as a group resistan t to the reform oriented instructional principles. The Active Implement er group featured a high level of adoption but no decrease in their traditional teaching practices , while the Proactive Implementer group appeared to change their teaching by reporting use of a high level of reformative principles as well as a reduced leve l of traditional principles.
121 Notably, the current secondary analysis using ECLS K data failed to find a profile by which teachers report ed the most frequent implementation of reform oriented instruction but substantial reduced implementation of the tradit ional components, as the reform policy ultimately seeks. Teacher change. The l iterature suggest s several types of teacher change as a consequence of mathematics reform efforts (Coburn, 2004; Spillane, 2000; Spillane & Zeuli, 1999). Some could learn eno ugh to fundamentally transform their beliefs and knowledge, but others might regress to traditional forms of teaching after a short attempt at implementing reformative practices . Some could focus on such surface level components as using manipulative mate rials and small group work, and others might integrate new approaches onto existing practices without fundamental ly transform ing their classroom norms or routines. Accordingly, teachers in the Active Implementer subgroup who reportedly implement ed reforma tive components most frequent ly can hard ly be interpreted as influenced by successf ul reform efforts, for no clear epistemological transformation from traditional to reformative was observed. Polikoff (2013) argued that the alignment between instruction a nd various standards envisioned by policy efforts i s associated with teacher professional knowledge and learning. Although teachers ma k e great efforts to change their instruction to align with policy message s (Hamilton & Berends, 2006), the alignment be tween what teachers do and what they are expected to do is low to moderate ( Kurz, Elliott, Wehby, & Smithson, 2010; Polikoff, 2012; Porter, McMaken, Hwang, & Yang, 2011; Porter, Smithson, Blank, & Zeidner, 2007) and the level of alignment is positively rel ated to factors that are relevant to teachers professional knowledge of the content and curriculum, including their educational and career experiences attributabl e to different teacher learning and professional growth as a consequence of the
122 curricular materials and professional development programs provided t o teachers . T hat is, t eachers with the Proactive Implementer profile may understand the conception of refo rm oriented instruction as a fundamental change in the epistemology of mathematics teaching and learning. They might be able to replace mathematical fact s or procedure focused teaching with problem solving and mathematical communication activities. Alter natively, s urface level knowledge of reform ideas is problematic for impacting implement at i o n and positively impacting student learning. Even secondary student teachers who had solid mathematics subject matter knowledge and substantial experiences with re form oriented mathematics teaching from strong preparation programs viewed the reformative principles as a body of mathematics content, rather than a theoretical transformation in mathematics teaching and learning, leading to difficulties enacting reform o riented instruction (Frykholm, 1999). As such, t he current study targeted elementary school teachers in order t o explore the probability scales for each teaching class across variables f or characteristics of reform implementation. Two different types of reform oriented components the use of manipulative materials or the use of group work have been identified as indicative of implementation without deep understandings of reform oriented ideas given that teachers could employ those instructional approaches based on their form focused rather than function focused understanding s (Spillane, 2000 b ). Also, i t may be easier f o r local leaders to attend to these component s to assess implementation, or the lack thereof, of reform oriented instruction . However, tho se reformative components did not appear to differentiate the Active or Proactive Implementer classes of teachers in the current study. Teachers in the Active Implementer profile showed the highest level of implementation of all reformative components inc luding surface level manifestations
123 (the probability of using manipulative materials variable: Active Implementer 0.403 vs. Proactive Implementer 0.191 with the significant odds ratio at the first response category) as well as deeper level practices (the p robability of discussion variable: Active Implementer 0.572 vs. Proactive Implementer 0.429 with the significant odds ratio at the first response category). Thus more frequent observations on certain reformative indices failed to show differen ces between teaching profiles. Rather, the reduced implementation of traditional components was more likely to differentiate different teaching profiles in the current study. Another possible source of variation in implementation may be related to student character istics as individual students decision making before, during, and after instruction (Bernstein Colton & Sparks Langer, 1993 ; Shavelson & Stern, 1981). As shown by the ANCOVA result for the preliminary analysis of Research Ques tion 4, p rior student achievement was found to significantly interact with the teaching profile s . Given th e persistent achievement gap by demographic subgroup s of students ( NCES, 2013) and differential association s between student demographic characteristics and i nstructi onal practices (Wellingky, 2004), it may be more challenging for teachers whose class room composition i ncludes many low achiev ing students to implement ambitious mathematics instruction. Unlike the interpretation aforementioned, teachers holding t he Active Implementer profile may be able to understand the reform message, make the epistemological change, and thereby attain a fundamental transformation in the use of instructional practices, but their classroom composition may hinder them from making a sharp departure from extant practices, such as switching from teaching mathematics procedures only (e.g., computation) to securing conceptual understanding. Without a clear consensus on the positive association s between students with special n eeds and
124 reform oriented mathematics, it is difficult to conclude this variation in teacher response s to policy message s as an implementation failure. Overall, the current study failed to find evidence on reform policy implementation as reformers ha d ex pected a fundamental transformation with the increase in reform oriented practices and decrease in existing practices. T eachers holding the Active Implementer profile showed an increase in reform oriented practices but a lso an adherence to existing practi ces . Teachers who could be considered proactive in terms of reformative mathematics showed a possibility of making a sharp departure from extant practices but failed to implement the reform oriented instruction at the highest level. Th is result correspon ds to that of a study using the large scale data collected in the similar time period (Jacobs et al., 2006). V ideo recording s collected by the TIMSS 1995 and TIMSS 1999 studies were analyzed from the lens of reform oriented mathematics education. Although U.S. 8 th grade teachers reported familiarity with reform oriented instruction as a consequence of attending the NCTM conference or reading reform policy publications, their teaching depicted in the videos reflected traditional instruction. For example, c lassroom organization of group work for problem solving was more often observed in the TIMSS 1999 videos but not a statistical ly significant increase from the TIMSS 1995 recordings reformers exp ected. Practices in terms of Reasoning and Proof, Communication, Connections, and Representations were worse: such mathematical reasoning practices as deductions or generalization s , problems involv ing connections across various concepts, and student prese ntation and discussion were rarely observed (Jacobs et al., 2006). Therefore, we could conclu de that a national level picture of both 3 rd and 8 th grade mathematics teaching painted by
125 two large scale data sets collected in the 1999 or 2000 school year s di splay a failure to full y implement reform oriented instruction. In conclusion, the latent class analysis was able to capture mathematics instruction that is conceptually meaningful. Even more nuanced information regarding mathematics teaching or the imp lementation of reform oriented mathematics was offered by the use of this alternate approach. Therefore, in addition to a factor analysis approach, th is person oriented approach should be considered as a suitable analytic strateg y for evaluating teacher i nstruction. A Learning Mechanism Prior and Subsequent Achievement Mediated by Mathematics Self concept Explored by the Mediation Modeling Mathematics reform has been derived from a knowledge base pertaining to what mathematics content is worth knowing, h ow mathematics is taught effectively, and how students learn mathematics. Research and evaluation on mathematics reform need to address all those aspects. The third aspect was incorporated into the present study by investigating the student mathematics l earning mechanism and placing it into the center of this study as an outcome variable. In investigating the learning mechanism, the present study is based on a social cognitive perspective (Bandura, 1986; 1997). That is, the study attended to the interp lay between cognitive (i.e., academic achievement) and non cognitive (i.e., motivation) facets of learning. Mathematics achievement encompasses complex relationships of various constructs. Perception s of ability and confidence in ability are important fa cet s of a learning mechanism. Furthermore, research on motivation science has suggested a theory positing meditation or moderation relationships among various cognitive, motivational, and affect ive constructs leading to the final outcome variable (e.g., P intrich 2003). Yet, no known research has attempted this hypothesized mediation/moderation relationship tested directly from the statistical modeling.
126 One of the constructs related to self appraisal of ability and self confidence in mathematics learning is mathematics self concept (Marsh, 1990; Marsh & Shavelson, 1985; Marsh & Yeung, 1997). As such, a model including mathematics self concept is worth considering for exploring the mathematics learning mechanism. Since it has been found that students at the elementary or early middle school grade levels have more malleable self concepts by which achievement precedes mathematics self concept in the causal ordering (Byrne, 1998; Helmke & van Aken, 1995; Muijs, 1997; Skaalvik & Hagtvet, 1990 ; Skaalvik & Vala s, 1999), the current study conceptualized and tested a model in which prior and subsequent achievement we re mediated by the mathematics self concept of th ird grade students. The significant triangular relationship was found. Therefore, students who expe rienced success in learning at first grade perceive d themselves competent in and interested in learning mathematics and th e s e positive beliefs associated with mathematics self concept in turn led to higher achievement in third grade. Using the same data s et as in the present study, Byrnes and Wasik (2009) also contended (p. 168) factors for early elementary students by showing strong (p. 169) such as SES and parent expectati on s with the inclusion of factors consisted of cognitive propensity captured by prior achievement and motivation/self regulatory propensity measured by an approach to learning scale . Over the early grade levels from kindergarten to th ird grade, these factors were greatly associated with subsequent achievement and were significantly predicted by family SES and parent expectation s that were also critical determinant s of mathematics achievement. The B study (2009)
127 suggests that early elementary students learn mathematics or benefit from mathematics learning opportunities depending on how able and willing they are to do so . Furthermore, the current study directly tested the mediation model involving those cognitive and motivational factors for early mathematics learning. Based on this finding, one logical inference is that one cannot meaningfully evaluate any educational support, such as reform oriented mathematics for the current stu dy, without cons idering how students are able and willing to take advantage of the supports or opportunities provided. If the mathematic reform efforts worked successfully, changes in the final outcome would occur from innovatively working on student ab ility as well as motivation. In this vein , the current study used a learning mechanism established by the meditation modeling as the outcome variable in an attempt to evaluate teaching within the context of reform oriented instruction. The Relationship between the Implementation of Reform oriented Mathematics and the Learning Mechanism Established by the Mediation Modeling Different Influence s f or Three Teaching Profiles Explored by the Moderated Mediation Modeling Modeling the outcome variable involvin g cognitive as well as motivational factors is important for the examination of reform based instruction (Blumenfeld, 1992; Pintrich, 2003). Research has suggested that student motivational constructs, provision of reformative instruction, and achievement were interrelated in a positive way and to some degree (Stipek, Salmon, et al., 1998) . These findings, however, came from examinations of discrete subsets of relationships, such as the relationships between student motivation and achievement or between t eaching and student motivation. By contrast, th e current study attempted to examine the relationship between provisions of relevant teaching and the learning mechanism involving motivational beliefs of students to better understand the effectiveness of ma thematics education reform efforts that began in the 1990s.
128 The mathematics reform movements intended education to move toward inquiry oriented constructivist learning in classrooms. These classrooms we re intended to be different from existing classrooms in the nature of the academic activities, classroom organization, and the interactions between teachers and students. Given that different classrooms can be regarded as different cultures or contexts in the understanding of teaching and learning, an hypot hesis could be formulated that differences in instructional practices lead to different meanings in terms of motivational constructs and more importantly, different functional relationship s with outcomes (Pintrich, 2003). Blumenfeld (1992) pointed out tha t the constructivist classroom environment i s one important contextual factor, in particular serving as a moderator, in the examination of the role of a It is notable that the teaching profile s in the present study w ere estimated as a categorical variable based on the most probable latent class membership. Consequently, the study was able to model change, if any, in t he significant medi ational relationships of student variables across the three di stinct teacher groups. Preliminary analysis using ANCOVA suggested a possibility of a relationship between the teaching profile subgroups and student achievement. The significant interaction between prior achievement and the teaching profile s indicates t hat it is misleading to interpret the descriptive scores on subsequent achievement across the three teaching profile subgroups. Students of teachers in the Proactive Implementer class were shown to have the highest achievement level, followed by students of teachers in the Anti reform class, and thus students of teachers in the Active Implementer class had the poorest achievement level. The adjusted average to prior achievement was highest for the subgroup of Active Implementer , followed by Proactive Impl ementer. Also the higher level of achievement in the Anti reform
129 class than in the Proactive Implementer class was no longer supported when the previous achievement was accounted for. Students who were low achieving at the first grade scored higher at t he third grade assessment when taught by Active Implementer teachers than by teachers in the other teaching profiles. It is notable that this teaching profile was built up based on the frequent use of both reform oriented (e.g., manipulative materials or real life mathematics) as well as traditional (e.g., o py, 2003; Remillard, 2000) and difficulty with successful ly implement ing reform oriented practices for low per forming students (Ba x ter, Woodward, & Olson, 1999; Saxe et al., 1999) might be helpful for explaining this finding. Teachers whose class composition is more prone to be low to average achievement might be unable to make a sharp departure from traditional to reformative practices, leading to a failure to transform their teaching practices and showing only an active implementation of the reform message. When accounting for the low level of prior achievement, students achieved academic success in the classro om where learning opportunities from not only reformative but also traditional practices were abundant. However, it was not known from the present study whether this academic success would be achieved by reform oriented teaching or by traditional teaching for struggling students. The multiple group approach depicts how each learning mechanism is established differently for the three teaching profile subgroups. The triangular learning mechanism was not true for the teachers holding the Anti reform or Act ive Implementer profile s due to the non significant path from mathematics self concept to subsequent achievement for the former as well as due to the non significant path from prior achievement to mathematics self concept for the latter. For the Proactive Implementer profile subgroup, thr ee paths that consist of a medi ational
130 relationship were significant at the alpha level of 0.07 but the indirect effect was not, failing to establish a statistically significant learning mechanism involving mathematics sel f concept for this type of teaching. Collectively any of teacher subgroup s w ere not able to establish a significant medi ational mechanism of mathematics learning but the Proactive Implementer profile subgroup alone showed a possibility to do so. The refo rmers suggested a theory of change and convincing research evidence, in that reform efforts would change teacher knowledge and beliefs and this could lead to change s in teaching practices and in turn improve ment in student learning (Swanson & Stevenson, 20 02; Senk & Thompson, 2003). Differences in conceptualization and analytic strategies might explain the contradictory or disappointing results in the present study. The medi a tional learning mechanism was regarded as the student outcome rather than one ind ividual outcome variable such as achievement or mathematics self concept. It is difficult to find a significant medi ational relationship moderated by some other variables, for example the teaching profile variable in the present study. Substantively, it is not easy for a teacher to positively impact the student motivational dimension as well as achievement simultaneously by successfully implementing certain instructional principles, for example reform oriented instruction in the present study. For elemen tary school teachers to implement the reform oriented approach, they need to understand how each approach should be used for each mathematics content area. This goal is not easily attainable by most teachers, thereby requiring sufficient professional deve lopment opportunities for them. In this sense it is notable that the data on teaching practices used by the present study were collected during the 2001 2002 school year , which is the initial stage of revised the NCTM Standards (2000) era. It would be pr emature to expect that all the efforts envisioned by the revised NCTM Standards (2000) were finally paying off at the time. This
131 argument relates to features of effective professional development programs, such as content focus, active learning, coherence , duration, and collective participation ( Desimone, 2009). For teacher professional development in mathematics or science, the programs could take place over multiple years so as to build a consistent knowledge base for changes in teacher knowledge, teach We need to be vigilant about the temptation for a quick fix (Schoenfeld, 2002). Reforming mathematics education entails complex problems: U.S. students have not shown the satisf actory achievement o n national and international mathematics measures and closing the achievement gap between typical and certain populations of students such as t ho s e of color, with low SES, or with special needs h as been hard to accomplish . Furthermore, reform oriented mathematics instruction began with contradictory assumptions about mathematical cognition that students can develop mastery of skills from the activities focused on applications and problem solving rather than that the mastery of skill com es first and then application and problem solving later. Not only all stakeholders in the educational system but also the public have to understand and assent to this assumption of new mathematics teaching and learning. To do so, we were required to perc eive mathematics teaching and learning in a broader spectrum by which problem solving, reasoning, and communication are the ultimate goals of the school mathematics. Thus, steady improvements and incremental changes should be sought in the evaluation of r eform oriented mathematics (Schoenfeld, 2002). Lack of Evidence of Reform oriented Mathematics as Best Practices for Elementary School Students , Including Low achievers Findings from T wo M oderated M ediation M odels Overall, this study lacked the evidence t o support the effectiveness of reform oriented instruction when the learning mechanism was introduced as the outcome variable. To discuss this finding, it is helpful to observe the findings from studies on evidence based practices for
132 elementary school st udents (Slav in & Lake, 2008) and low achieving students (Baker, Gersten, & Lee, 2002), including students with learning disabilities (Gersten et al, 2009). According to the Slav in 8 ), providing reform based curricular materials al ong with appropriate professional development (PD) revealed that the evidence for the effectiveness was limited overall. Additionally, the results were mixed across studies, for instance, Everyday Mathematics , one of the representative reform oriented cur ricular programs, has produced p ositive 0.15 effect size for 2 to 3 years of (2001) study but negative 0.25 effect size for the one year implementation in Woodward and ply employed the provision of materials and corresponding PD as independent variables so that the links between the quality or the extent of reformative instruction implementation and the outcome were not explored directly. It seems nonetheless premature to draw conclusions on the effectiveness of reform oriented instruction to improve achievement or close achievement gaps, given that none of the studies explicitly claiming to support reformative principles showed evidence for the ir effectiveness in the li terature synthesis by Slav in and Lake ( 2008). Rather, programs that incorporate components such as classroom and time management, motivati on strategies, and peer tutoring strategies showed strong evidence of effectiveness (Slav in & Lake, 2008). Those in structional components do not necessarily parallel reformative principles. Presumably, commonalities exist between key components of class wide peer tutoring (positive 0.33 effect size) and such reformative principles as small group work or mathematical d iscussion; however, how closely those two are aligned when enacted is unknown. Therefore, the evidence on the degree of implementing reformative principles and its relationship with outcomes are barely found in the literature on best practices for element ary mathematics
133 education. A similar argument could be made about the association between evidence based practices for low achieving students, including learning disabilities and reform oriented instructions. Baker and colleagues (2002), in a synthesis of empirical research for low achieving students, contended that explicit instruction (EI, 0.58 weighted effect size) is more effective than teacher facilitated/contextualized instruction (0.01 weighted effect size). In their synthesis, studies were coded principles, and problem within strategy instruction. These practices are in st ark contrast with principles suggested by the reform movement , such as student centered inquiry activities . Fur 65) produced the overall effect size of 0.01 in their meta analysis. Based on these findings, the authors did not draw a conclusion about the debate between reformative versus traditional instruction . Instead, they suggested critical issues in future re search on the reform oriented mathematics instruction for low achieving students. They brought up the topic of the quality of the control group and the characteristic s of dependent measures of learning used in evaluating the effectiveness of reform orient ed mathematics for this particular learner group. A m ore recent synthesis ( i.e., Gersten et al., 2009) focused on students with learning disabilities found an identical pattern of results in terms of the strong evidence in support of EI (mean effect size, 1.22). Furthermore, it has been reported that low performing students became minimally engaged, as sumed passive roles, and engage in non mathematical activities in reformative mathematics classrooms focusing on
134 problem solving, communication, reasoning, and connecting the concepts (Baxter, Woodward, & Olson, 2001). Again, so far, we lack evidence that low achieving students, including those with learning disabilities, benefit from the reform oriented classroom. As such, it is not surprising that teache rs in the reform oriented Active or Proactive implementer group failed to establish the significant medi ational learning mechanism involving mathematics self concept and achievement for any learner group of students or for struggling students in the presen t study. The richness of the mathematics content for low achieving students. Concerning the provision of appropriate instruction, it is necessary, and even desirable, for teachers to adopt such approaches as explicit modeling, cumulative reviews, and many opportunities to practice is that a greater number of the aforementioned opportunities may decrease opportunities for roblem solving and mathematical communication with teachers and peers given the limited time and resources. Accordingly, mathematics education for low achieving students becomes insufficient in terms of the richness of the mathematics content to which stu dents are exposed. Accordingly, th e ongoing endeavor to infuse reformative principles into practices for low achievers is crucial for issues concerning (a) access of students with disabilities to the general education curriculum and their progress and (b ) higher expectations within the ambitious curriculum and test standards for all students. The former indicates that struggling students, including those with mathematics learning disabilities, are also expected to be successful in the general education c lassroom that is evolving to ward reform oriented instruction . The latter suggests that these learners should also be literate in advanced competency areas, such as
135 applying concepts to problem solving and actively engaging in mathematical reasoning and co mmunication. The National Assessment of Education Progress (NAEP) also specifies the fourth grade students performing at the Basic level should show some evidence of understanding the mathematical concepts and procedure grade students performing at the Proficient level should consistently apply integrated procedural (http://nces.ed.gov/nationsreportcard/mathematics/achieveall.asp) . Thus, one import ant agenda is to design and construct instructional strategies and programs through which the two approaches o f EI and reform oriented mathematics merge. In this vein, the program of Enhanced Anchored Instruction is worth noticing. The Bottge and colleag ues (2007) designed and provided a problem solving mathematics program for middle or high school students with learning disabilities as well as emotional and behavioral disabilities (Bottge, Rueda, & Skvington, 2006). The program consisted of multimedia b ased problems and related hands on activities encompassing a range of mathematical concepts. For immerse students in the topic of applying conceptual and procedural k nowledge of fractions to real life projects. Video based problems and applied projects were expected to motivate students. Moreover, students were exposed to opportunities to identify the main questions and relevant information and to apply learned conce pts in the course of problem solving. Classroom structures were interwoven with all five strands of mathematics proficiency, including conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive d isposition (NRC, 2001), which are the area s that reform oriented mathematics has pursued.
136 Yet, the EI components cannot be ignored in the program. The foundation al skills were taught directly, and such components as constant progress monitoring, immediate feedback, and more challenging mathematics may be beneficial when academic and personal supports are provided to students to help them succeed in these kinds of mathematics learnin g activities (Middleton, 2004). Better engagement, motivation, and learning in reformative classrooms may depend on particular pedagogical approaches that purposefully provide scaffolds, ad hoc instruction, and opportunities for students to succeed. Thes e academic and personal supports learning. Programs like this one need to be developed for younger students to build mathematical reasoning and to practice procedura l fluency simultaneously. Implications of the Present Study The present study extends previous research by considering the interactive relationship s among reform oriented mathematics (content), the teacher, and students by conducting secondary analyses o f a large scale, nationally representative data set. From the present study, three subgroups of teachers with distinct teaching profiles emerged from the data within the context of mathematics reform policy. Furthermore, the mediation relationships betwe en prior and subsequent achievement via the mathematics self concept was tested empirically. The two learning mechanism, were linked to evaluate reform efforts usi ng the alternative conceptual and methodological framework. In doing so, the learning mechanism was introduced as the desirable outcome variable. The findings from the final two research questions support existing research and theory that suggest the nee d for the research to examine reform oriented mathematics for all students.
137 Practical Implications of the Findings How students develop motivational beliefs is one important agenda of the learning science s because motivation plays a critical role in learni ng mathematics, at least as a mediator or moderator (Pintrich, 2003). The current study attempted to test this argument empirically and found a significant mediation effect of mathematics self concept on the relation between prior and subsequent achieveme nt. First, findings suggest that mathematics teaching needs to address how students self appraise their own capacity for mathematic s learning from the ir prior experience s affec tive and motivational response s to mathematics learning, considering that the construct of mathematics self concept includes enjoyment of and interest in mathematic learning. Lastly, mathematics self concept should be included as one of the outcomes in th e evaluation of mathematics teaching. Pintrich (2003) suggested specific instructional principles that fulfill the motivational dimension of mathematics learning. Tasks, activities, and materials through which mathematics content are delivered should be s timulating, interesting, personally meaningful, and useful to students. Students would more often engage in as well as be challenged by personally meaningful activit ies . For instructional activities or tasks to be meaningful to students, they should be i ntroduced to students as problem solving tasks using real life setting s or fictitious scenarios. Meanwhile feedback, reward, and evaluation structure s should be clear, accurate, and focused on knowledge development and the learning process rather than the learning product. By stressing effort, strategies, progress, and self improved standards, teachers make it possible to build evaluation structures that are corrective as well as constructive. Emphasizing mastery rather than performance goal s allow teach and processes used during learning opportunities. Classroom discourse is also a key aspect .
138 Teachers should focus on utility, mastery, and understanding of the content th rough interaction s with stude nts. By doing so, the mathematics classroom becomes the community of learners that is supportive, cooperative, and collaborative in its pursuit of social and academic goals. The target population was early elementary school students in the current study. The significant mediation model emphasizes the importance of early experience s with success by showing that early mathematics performance significantly influences mathematics self concept as well as subsequent achievement. First grade teachers need to m onitor the performance level on a number of competency skills, including (a) counting, (b) number recognition, number comparisons, (c) nonverbal calculation s , (d) simple story problems, and (e) number combination s (Jordan, Kaplan, Locuniak, & Ramineni, 200 9). A solid understanding of these rudimentary number concepts is closely tied to their success in the first grade mathematics curriculum, which would lead to sound mathematics self concept s as well as later achievement. Specific mathematics topics in th e first grade include: (a) counting by 2s, 5s, and 10s, (b) writing all numbers between 1 and 100, (c) identifying relative quantity, (d) adding and subtracting single digit numbers, and (e) reading two digit numbers. Furthermore, for the relatively advan ced topics, first grade teachers could guide students to be exposed to content on (a) recognizing the value of coins and currency, (b) using measuring instruments, (c) telling time, (d) estimating quantities, (e) writing math sentences to solve problems, a nd (f) recognizing fractions. Regarding these topics, too easy or too difficult content w as found to significantly but negatively predict the achievement at the end of first grade (Byrnes & Waskik, 2009) so the alignment between levels of content and stud ent development is critical. Consistent with other studies using large scale data from teacher reports (Byrnes & Wasik, 2009) and videotaped observations (Jacobs et al., 2006), the present study also
139 demonstrated that only a minority of U.S. teachers rep orted a active implementation of reform oriented principles or sharply transformed their instruction from the traditional to the reformed in 2001 school year the successful implem entation of few, and the long 2005, p . 59) . Although the reform movement that have been sweeping the nation placed hope in make it possible to instill new ideas into teachers, yielding changes in teaching and learning. Yet, we can not blame only teachers for the failure to transform their knowledge, skills, and dispositions, as the problem of implementation needs to be regar ded from a broader perspective by including issues on teacher learning. A question should be answered whether teachers have been afforded helpful professional learning opportunities to implement more ambitious mathematics instruct s (2004 ) observations of typical professional development programs for elementary mathematics teaching, teachers lack access to genuine learning opportunit ies that would allow them to explore important mathematical ideas s about them. This was also true for the programs that were rated highly and unclear about underlying ideas. Teachers were exposed to one of the reform oriented instructional principles, which involved asking students to explain their solutions. However, mited to reciting procedures for the final answers rather than mathematical properties or reasoning. Teachers engaged in hands (p. 160) way (Spillane, 2000b ).
140 Certainly, these activities do not reflect the expecta tions of the NCTM Process Standards of Reasoning and Proof , Communication , or Representation . Features of effective professional development programs, such as content focus, active learning, coherence, duration, and collective participation (Desimone, 20 09) could inform what type of learning would be helpful for teachers. There is, however, a need to further elaborate the se features as they relate to reform based mathematics. For example, for the Content focus that emphasizes both subject matter content and learning process es , the amount and depth of mathematics explored is central . Further, by Active learning and Collective participation , it may be necessary to determine whether participants are active and collectively participating about the mathemati cs itself. Activities and discourse should be geared toward professional learning of mathematics concepts or reasoning and the ways in which they are linked with the mathematical cognition of students. These kinds of learning opportunities c ould enable t eachers to explore mathematical ideas and student learning of mathematics and extend to their teaching contexts or other areas within/outside mathematics as well as engender changed epistemological beliefs and conceptions regarding mathematics teaching and learning. Again, the desirable outcome may complicate the concept of fruitful learning opportunities for teachers and students even more. In order to show great promise for the positive effect on the learning mechanism (i.e., motivation and performance simultaneously), i t is required not only to implement reformative practices effectively, but also to amalgamate them with motivation enhancing strategies successfully. Taken together, teacher education programs should ensure a substantial amount of conten t and pedagogical content knowledge of (a) mathematical concepts, methods, and language to be taught; (b) instructional approaches integrating the five Process Standards into motivation
141 cognitive, linguistic, physica l, social emotional development as well as their prior experience s and informal knowledge. In addition to knowledge acquisition, teachers should make substantial changes in their beliefs about mathematics learning and teaching by situating themselves in t he learning context where mathematical concepts and reasoning are actively and collaboratively explored. This type of exposure to reform oriented mathematics learning is important, given that most teachers have limited experience going beyond procedure s , making sense of mathematical ideas within meaningful context s , and constructing learning by testing, debating, and revising ideas in the classroom that allows teachers and students to engage in those activities. Policy and Research Implications of the Fi ndings The current study attempted to fill a gap in the research evidence. By using the large scale data, a larger sample of teachers were explored for their mathematics teaching in the context of reform oriented instruction. By modeling the mathematics learning mechanism as an outcome, the study also attempted to demonstrate whether the reform oriented instruction was able to exert an impact on the learning mechanism, not simply on the discrete variable of achievem ent or affect . Moreover, it was examine d whether the effects of reform oriented mathematics instruction was consistent across student groups of typical and struggling learners. However, it is notable that the results, such as a non significant moderated mediation effects by instructional profi les for the whole student group and each group of typical and struggling learners do not suggest that the pendulum needs to swing to traditional teaching methods. Rather, the present study has implications for teacher educati on research focusing on genuin e learning opportunities for teachers. Additionally, potential methodological contributions of the present study can give us new insight s into what directions may be taken in further research. First, the significance of study using large scale surveys ne eds consideration . Although the use of randomized field trial s has been most encouraged, large scale survey studies can be
142 used to examine many critical research questions. In particular, studies using large scale data allow for the evaluation of the ref orm effects, and thus provide information on large scale patterns to policy developers. The large scale data and the use of statistical techniques offer the opportunity to examine the dynamic relationships among various teacher and student level factors, including meditation and moderated mediation modeling in the current study. This type of picture depict s the level of implementation of certain pedagogical principles across the nation to inform randomized field trial stud ies of critical conditions and c omponents to be incorporated into the experiment (Berends & Garet, 2002). A methodologically important aspect of the present study was the use of latent class analysis (Muthen & Muthen, 2007) by taking a person oriented analysis to identify the character istics of instruction in the context of mathematics reform. Previous investigations have often used variable oriented analyses to capture the nature and degree of implementation of reform oriented mathematics. As such, studies are not often free from the controversy over the dimensional identity involved in the distinction between reformative and traditional practices (e.g., Lubienski, 2006 and Wenglinsky, 2004; 2006). Also, it is notable that a possible interaction exists between various behaviors (i.e. , variables in the analysis) from the different dimensional arenas when teacher behaviors function as their instruction in the classroom. Accordingly, by using person oriented analysis, the present study aims to identify teaching practices from a perspect ive that presumes each individual teacher or teaching behaviors acting as a holistic unit , but not as an aggregation of fragmented units (Campbell, Shaw, & Gilliom, 2000; von Eye & Ann, 2006; von Eye & Spiel, 2010). To my knowledge, the present study is the first to report teaching profiles driven from a large, nationally representative data set . This approach might be used in policy evaluation s and
143 future research to characterize teaching and examine further the relationships among relevant outcomes. I t is notable that the study used a third grade sample collected in the spring of 2002 when it was early in the implementation period of reform oriented mathematics, as the experimental curricula were published in the mid 1990s and the revised NCTM Standard s was released in 2001. Thus, it may have be en too early to expect the wide implementation of reformative ideas from the front lines, and the consequential impacts on the learning mechanism. tion needs , r esearchers might examine the extent to which the teaching profiles obtained from the present study are replicated using (a) other samples of fifth and eighth graders in ECLS K data set for the examination of other mathematics content and the m other samples of young students in a new ECLS study which is currently being conducted, and (c) other populations across countries from the international evaluation data sets such as TIMSS 2007 or 2013. These t ypes of comparative analyses might enhance understandings of the laten t cluster modeling) to latent class regression modeling to classify teachers into subgroups and develop regression models simultaneously. The analysis of this kind should allow refining of the characteristics of each subgroup by the use of demographic and other covariates such as professional development experience s . However, the findings from this study are subject to the limitation s of survey data in the investigation of instructional practices. Although survey data ha ve been reported as mean ingfully capturing mathematics instruction to some degree, future stud ies should explore the reliability and validity of using survey data for mathematics education (Mayer, 1999) by
144 linking survey data with different types of measures such as classroom obs ervation s (Stipek & Byler, 2003), instructional logs (Rowan, Jacob, & Correnti, 2009; Rowan, Harrison, & Hayes, 2004), or cognitive interviews with teachers (Desimone & Floch, 2004). Furthermore, given that the survey measure allows for a nationally repre sentative data set to be collected with cost effectiveness, which is useful for informing policy decision s and evaluation s , developing survey measure s should be continued. Notably, the current study indicated that several teacher survey the nature of subgroups. In this sense, another possible area of future research would be to better conceptualize the construct of procedural knowledge in mathematics learning and the nature of using tests for mathematics teaching. In particular, it is note worthy that one often equate s procedural knowledge with superficial or r o te knowledge while simply equating conceptual knowledge with deep or meaningful knowledge. So, the former was often indexed to traditional instruction and the latter to reform oriented mathematics. But scholars ( Baroody, Feil, & Johnson , 2007; de Jong & Ferguson Hessler, 1996 ; Star , 2005 ) h ave suggested an alternative reconceptualization of procedural knowledge by disentangling the knowledge type and quality: the procedural al consistency and local generalization tied only to specific contexts (Star, 2005). Two types of knowledge even interrelate with each other in learning certain mathematics concepts (Baroody et al., 2007). Arguments aforementioned however call for empiri cal study. Empirical findings that answer questions regarding the nature of mathematics would be of great help in developing measures of mathematics instruction.
145 Above all, the current study will serve as a base for future studies to include various stu dent subgroups in the examination of policy implementation. If priorities, values, and goals proposed by reform oriented mathematics Standards are committed to nurturing students to become p also should seek to explore what teachers need to know and change in the teach ing of ambitious mathematics to students with special needs. The work of this kind is, however, limited gaps in the mathematics education literature and in the special educati on literature. Furthermore, the recent legislative trend including No Child Left Behind (NCLB, 2001) or the Individuals with Disabilities Education Improvement Act (IDEA, 2004) continued to emphasize the access to rigorous curriculum and accountability fo r students with special needs. As such, a shared responsibility between general and special educators is emphasized and therefore one of the most significant issues is the alignment among instructional foci consistent with the NCTM Standards and al so empi rically validated by special education research (Greer & Meyen, 2009). There is obviously a great need for research explicating what challenges could be experienced by both general and special education teachers in teaching mathematics to students with sp ecial needs and how those challenges should be addressed. Taken together, the reform based instruction is ambitious not only for students but also for teachers and most classroom teachers in the United States have not experienced mathematics learning of t his type themselves epistemological beliefs and pedagogical approaches in terms of mathematics learning and teaching. There is, therefore, a definite need for research and policy efforts concerning how to help teachers learn to teach in these ways. Additionally, the instructional principles suggested by research and policy are stated in a general form and there is no single right way to implement
146 them under various situations. Thus , local leaders a nd teachers are required to adapt them to their own contexts. In fitting principles to their own goals, they may take multiple pathways to create learning environment s for their students because the affordance and constraints that teachers in real classro oms may encounter vary. The key issue for future research and policy seems to be to try to understand effective and efficient ways to implement these principles and to empirically examine them in terms of (a) how they are implemented in certain classroom contexts; (b) under what conditions; (c) with what type of students and teachers; (d) how different their influences are across various outcome variables; and (e) how different principles may support or conflict with others principles.
147 APPENDIX A VARIABLE S A categorized list of the variables that was used in the study. Variable Names Purpose for Proposed Study Variable ID in Data Set Student SDQ Variable 6. Work in math is easy for me 12. I cannot wait to do math each day 16. I get good grades in math 22. I am interested in math 26. I can do very difficult problems in math 30. I like math 36. I enjoy doing work in math 41. I am good at math To capture the dimension of mathematical self concept. To be introduced to the final model as the mediator. C 5SDQMTR Items Level Data Set Teacher Instruction Variable Q 53. How often do children in your class engage in the following? a. Solve mathematics problems from their textbooks b. Solve mathematics problems on worksheets c. Solve mathematics problems in s mall groups or with a partner d. Work with measuring instruments, e.g., rulers e. Work with manipulatives, e.g., geometric shapes f. Use a calculator g. Take mathematics tests h. Write a few sentences about how to solve a mathematics problem i. Talk to the class about their mathematics work j. Write reports or do mathematics projects k. Discuss solutions to mathematics problems with other children l. Work and discuss mathematics problems that reflect real life situations m. Use a computer for math Q 54. In this mathematics class how often do you address each of the following? To capture the dimension of mathematical teaching practices. To be introduced to the final model as the modera tor. A5SOLTXT A5SOLSHT A5GRPPTM A5MSINST A5MANIPU A5USECAL A5MATEST A5MWRITE A5TLKMAT A5MPROJ A5MDISC A5PRBLIF A5MCOMP
148 TOPIC a. Numbers and Operation, b. Measurement, c. Statistics, d. Geometry, e. Algebra SKILL f. Learning mathematics facts and concepts g. Learning skills and pro cedures needed to solve routine problems h. Developing reasoning and analytical ability to solve unique problems i. Learning how to communicate ideas in mathematics effectively j. Recognizing the properties of shapes and relationships among shapes k. Understanding place values with whole numbers l. Reasoning, writing, and comparing fractions m. Making reasonable estimates of quantities A5NUMOP A5MEASUR A5STAT A5GEOM A5ALGEBR A5MTHCON A5PRSOLV A5ANALYT A5COMIDA A5SHAPE A5PALCEV A5FRACT A5ESTIM Student Achievement Variable Fall_1 st grade IRT scale score Third grade IRT scale score Spring_1 st grade IRT scale score Spring_1 st grade Math Highest Proficiency Level Mastered To be introduced to the mediation or moderated mediation model as the prior and subsequent achie vement. To define the student performance group of typical and struggling learners. C3MSCALE C5MSCALE C4R4MSCL C4R2MPF
149 APPENDIX B M PLUS PROGRAM S 1. Latent Class Analysis Title: ECLS K 3rd teaching class; Data: file is c: \ users \ ufzzangah73 \ desktop \ teaching no psu 999 noname.csv; Variable: names are T5ID ID SOLTXT SOLSHT GRPPTN MSINST MANIPU USECAL MATEST MWRITE TLKMAT MPROJ MDISC PRBLIF MCOMP NUMOP MEASUR GEOM STAT ALGEBR MTHCON PRSOLV ANALYT COMIDA SHAPE PLA CEV FRACT ESTIM; Usevariables are SOLTXT SOLSHT GRPPTN MSINST MANIPU USECAL MATEST MWRITE TLKMAT MPROJ MDISC PRBLIF MCOMP NUMOP MEASUR GEOM STAT ALGEBR MTHCON PRSOLV ANALYT COMIDA SHAPE PLACEV FRACT ESTIM; CLASSES = c (3); missing are all (999); categorical are SOLTXT SOLSHT GRPPTN MSINST MANIPU USECAL MATEST MWRITE TLKMAT MPROJ MDISC PRBLIF MCOMP NUMOP MEASUR GEOM STAT ALGEBR MTHCON PRSOLV ANALYT COMIDA SHAPE PLACEV FRACT ESTIM; define: ANALYSIS: Type = MIXTURE; STARTS = 500 40; OUTPUT: TECH1; !sampstat; SAVEDATA: file is threeclassout.dat; save is cprob;
150 2. Mediation Modeling title: child mediati on; data: file is C: \ Users \ ufzzangah73 \ Desktop \ ecls mediation data noname.csv; variable: names are CHILDID C3R4MSCL C5SDQMTR C5R4MSCL C245CSTR C245CPSU C245CW0 C4R4MSCL C4R4MPF; usevariables are C3R4MSCL C5SDQMTR C5R4MSCL C245CSTR C245CPSU C245CW0; missing are all (999); WEIGHT is C245CW0; STRATIFICATION is C245CSTR; CLUSTER is C245CPSU; ANALYSIS: TYPE = COMPLEX; MODEL: C5SDQMTR ON C3R4MSCL (A); C5R4MSCL ON C5SDQMTR (b ); C5R4MSCL ON C3R4MSCL; MODEL CONSTRAINT: NEW (ind); ind=a*b; OUTPUT: TECH1; TECH3;
151 3. Moderated Mediation Modeling RQ4. 1 st Fall 3 rd MSC 3 rd Spring Achievement moderated by Teaching Profiles title: child moderated mediation fina l dataset; data: file is ECLS K combined noname.csv; variable: names are x m y C245CSTR C245CPSU C245CW0 C4R4MSCL C4R4MPF CHILDID R3SAMPLE T5ID Prob1 Prob2 Prob3 Class; usevariables are x m y C245 CSTR C245CPSU C245CW0 Class ; missing are all (999); grouping is class (1 = anti 2=active 3=pro); WEIGHT is C245CW0; STRATIFICATION is C245CSTR; CLUSTER i s C245CPSU; ANALYSIS: TYPE = COMPLEX; iterations=50000; MODEL: y on m (b1) x; m on x (a1); model active: y on m (b2) x; m on x (a2); model pro: y on m (b3) x; m on x (a3); MODEL TEST: 0=a1*b1 a2*b2; 0=a2*b3 a3*b3; MODEL CONSTRAINT: new(anti_act anti_pro act_pro anti acti pro ); anti_act=a1*b1 a2*b2; anti_pro=a1*b1 a3*b3; act_pro=a2*b2 a3*b3; anti=a1*b1; acti=a2*b2; pro=a3*b3;
152 RQ 5 . 1 st Spring 3 rd MSC 3 rd Sp ring Achievement moderated by Teaching Profiles and Student groups (defined by 1 st fall) title: RQ5 final dataset; data: file is C: \ Users \ ufzzangah73 \ Desktop \ ECLS K RQ5 noname.csv; variable: names are ID C3R4MPF m y C245CSTR C245CPSU C245CW0 x TID P1 P2 P3 Class; usevariables are x m y C245CSTR C245CPSU C245CW0 Class struggle xst mst; missing are all (999); grouping is class (1 = anti 2=balance 3=pro); WEIGHT is C245CW0; STRATIFICATION is C245CSTR; CLUSTER is C245CPSU; de fine: if C3R4MPF le 2 then struggle=1; if C3R4MPF ge 3 then struggle=0; xst = x*struggle; mst = m*struggle; ANALYSIS: TYPE = COMPLEX; iterations=50000; MODEL: m on x (ans_anti) struggle xst (da_anti); y on m (bns _anti) x (cns_anti) struggle mst (db_anti) xst (dc_anti); model balance: m on x (ans_bal) struggle xst (da_bal); y on m (bns_bal) x (cns_bal) struggle mst (db_bal) xst (dc_bal); model pro: m on x (ans_pr o) struggle xst (da_pro); y on m (bns_pro) x (cns_pro) struggle mst (db_pro) xst (dc_pro); MODEL CONSTRAINT: new(indns_ant indns_bal indns_pro inds_ant inds_bal inds_pro as_anti bs_anti cs_anti as_bal bs_bal cs_bal as_p ro bs_pro
153 cs_pro); indns_ant=ans_anti*bns_anti; indns_bal=ans_bal*bns_bal; indns_pro=ans_pro*bns_pro; inds_ant=(ans_anti+da_anti)*(bns_anti+db_anti); inds_bal=(ans_bal+da_bal)*(bns_bal+db_bal); inds_pro=(ans_pro+da_pro)*(bns_pro+db_pro); as_anti= ans_anti+da_anti; bs_anti=bns_anti+db_anti; cs_anti= cns_anti+dc_anti; as_bal= ans_bal+da_bal; bs_bal=bns_bal+db_bal; cs_bal= cns_bal+dc_bal; as_pro= ans_pro+da_pro; bs_pro=bns_pro+db_pro; cs_pro= cns_pro+dc_pro; save data: results are check.dat;
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172 BIOGRAPHICAL SKETCH Jungah Bae was born in Masan, Korea. She received the Bachelor of Art and Master of in her 5th year of study in the Special Education Ph.D . Program at the University of Florida. In August 2014, she will graduate with a focus in mathematics learning disabilities and a minor in quantitative research methodology. Prior to her career in academia, she spent 10 years as a Special Education teac her for the Seoul Metropolitan Department of Education in Korea. She taught in inclusive elementary schools serving students 7 to 12 years of age with special needs. Her students had various disabilities ranging from specific learning disabilities to aut ism spectrum disorders, as well as moderate intellectual disabilities . During her tenure as an educator, Jungah gained professional experience in developing instructional and curricular materials related to the content areas of reading, mathematics, and science for students with special needs. Additionally, Jungah was an author of Government Designated Textbooks for Special Education. During the past five years, Jungah participated in various research projects at UF. In particular, with Prime Online (Teacher Pedagogical Content Knowledge and Research based Practice in In clusive Elementary Mathematics Classrooms funded by U.S. Department of Education, Institute for Education Sciences), she has had the opportunity to engage in research concerning teacher PD for mathematics education in the inclusive elementary school, gaini ng valuable insight into topics on mathematics teaching and learning, and ultimately teacher education. Outside of academics, Jungah is very involved in community service. She spent 4 years as a teacher for the Korean Catholic Community in Gainesville, a nd developed education programs of Sunday school class, First Communion, and Altar Training , tailored to the cultural
173 and linguistic characteristics of Korean American children and youths. Also she is now serving as a representative to College of Educatio n for UF Korean Student Association. Jungah enjoys music including playing the guitar and flute and also outside activities including swimming and playing tennis.