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Evaluating the efficacy of the devleoping algebraic literacy model

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Title:
Evaluating the efficacy of the devleoping algebraic literacy model preparing special educators to implement effective mathematics practices
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English
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Ray, Sharon N. E.
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University of South Florida
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Pre-service teacher preparation
Content area instruction
Mixed methods design
At-risk students
Instructional framework
Dissertations, Academic -- Special Education -- Doctoral -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Summary:
ABSTRACT: For students with learning disabilities, positive academic achievement outcomes are a chief area of concern for educators across the country. This achievement emphasis has become particularly important over the last several years because of the No Child Left Behind legislation. The content area of mathematics, especially in the higher order thinking arena of algebra, has been of particular concern for student progress. While most educational research in algebra has been targeted towards remedial efforts at the high school level, early intervention in the foundational skills of algebraic thinking at the elementary level needs consideration for students who would benefit from early exposure to algebraic ideas. A key aspect of students' instruction with algebraic concepts at any level is the degree and type of preparation their teachers have received with this content.Using a mixed methods design, the current researcher investigated the usage of the Developing Algebraic Literacy (DAL) framework with preservice special education teacher candidates in an integrated practicum and coursework experience. Multiple survey measures were given at pre-, mid-, and post- junctures to assess teacher candidates' attitudes about mathematics, feelings of efficacy when teaching mathematics, and content knowledge surrounding mathematics. An instructional knowledge exam and fidelity checks were completed to evaluate teacher candidates' acquisition and application of algebraic instructional skills. Focus groups, case studies, and final project analyses were used to discern descriptive information about teacher candidates' experience while engaging in work with the DAL framework.Results indicated an increase in preservice teachers' attitudes towards mathematics instruction, feelings of efficacy in teaching mathematics, and in the content knowledge surrounding mathematics instruction. Instructional knowledge also increased across preservice teacher candidates, but abilities to apply this knowledge varied across teacher candidates', based on their number of sessions working with students within their practicum site. Further findings indicate the desire of preservice teachers to increase the length and number of student sessions within the DAL experience, as well as the need for increased levels of instructional support to enhance their own experience. This study provides preliminary support for utilizing the DAL instructional framework within preservice teacher preparation experiences for future special educators.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
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by Sharon N. E. Ray.
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Title from PDF of title page.
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Document formatted into pages; contains 408 pages.
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Includes vita.

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aleph - 002007320
oclc - 402487748
usfldc doi - E14-SFE0002797
usfldc handle - e14.2797
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Evaluating the Efficacy of the Developing Alge braic Literacy Model: Preparing Special Educators to Implement Eff ective Mathematics Practices by Sharon N. E. Ray A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Special Education College of Education University of South Florida Major Professor: David Allsopp, Ph.D. Albert Duchnowski, Ph.D. Helen Gerretson, Ph.D. Nancy Williams, Ph.D. Date of Approval: August 18, 2008 Keywords: pre-service teacher preparati on, content area instruction, mixed methods design, at-risk students, in structional framework Copyright 2008, Sharon N. E. Ray

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Dedication I would like to dedicate this dissertati on to my family members, who have each helped me in their own way to make completing this dissertation a real ity. To my brother Ryan, I would never have written this dissert ation before our growing up together – you sparked my interest in special education. To my sister, you lived with me during the final months of writing this dissert ation, then and always you kept me sane by reminding me to eat right, exercise (which you helped me do re gularly), vent my frustrations, and stay fashionable in spite of all th e work I had to complete. To my mom, you have spent my whole life encouraging me to get the most out of life and making it what I want. You convinced me from an early age that women could do anything, not just in theory but in actuality. To my dad, you have always said “ life is hard work” – now I finally realize why you have worked so hard yourself – because when you work hard, you make something of your life and that is its own reward. To my husband, each and every day I have known you, you have inspired me by your ex ample to do my best and be a better person. To my son, by having you it fundame ntally changed who I was and who I am, and for that I am forever grateful. Without these special people, and the unending help of God the Father, I would never have finished this dissertation. For that reason, this finished product is dedicated to all of you because each one of you helped me more than you may ever know.

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Acknowledgements This dissertation would never have be en completed without the help and encouragement of many people. First, I want to thank my doctoral committee: Dr. David Allsopp, Dr. Albert Duchnowski, Dr. Helen Ge rretson, and Dr. Nancy Williams. I would also like to especially thank my major pr ofessor Dr. David Allsopp, who never gave up on me, even when I sometimes stopped believing in myself. Additionally, I would like to thank the doctoral cohort who “adopted” me when I returned to USF to finish my doctoral studies: Dr. Debbie Hellman, Anne Townsend, Sandy May, and Teri Crace. It was our many gatherings of support and fr iendship that helped me get through these many doctoral years. Furthermore, I would like to thank two docto ral students, Jennie Farmer and LaTonya Gillis-Williams, whose help and support in the last few months of this dissertation were immeasurable. At the same time, I would also like to thank the USF Department of Special Education for supporting my endeavors over th e last six years, especially Dr. Daphne Thomas, Dr. Anne Cranston, Dr. James Pa ul, Dr. Kleinhammer-Trammill, Ms. Jaye Berkowitz, and Ms. Amelia Ayers. I would al so like to show my appreciation for my many professors at USF during my doctoral st udies, I learned and gained so much as a person and an educator through my experiences with them. I would like to especially thank Dr. Robert Dedrick for his inva luable help in understanding SPSS. Finally, I want to thank my good friends, who gave me support from near and far as I worked on finishing this dissertation, especially Jennifer Lane, Heather Nghiem, Mylinh Shattan, Helen Toscano, Lisa Richardson, and the Pearls. Thanks to all of you!

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i Table of Contents List of Tables x List of Figures xv Abstract xvii Chapter 1 – Introduction 1 Statement of the Problem 1 Theoretical/Conceptual Framework 7 Purpose 15 Research Questions 16 Overarching Question 16 Major Inquiry Areas within the Research Question 16 Significance of the Study 17 Definition of Terms 18 Delimitations 22 Limitations 23 Chapter 2 – Literature Review 26 Overview 26 Literature Search 27 Federal and State Impetus 28 Content Area Learning 30

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ii Diverse Populations 32 Professional Development 35 Self Efficacy 38 Attitude 39 Content Knowledge 41 Pedagogical Knowledge 44 The DAL Framework 53 Algebra Background 53 Algebraic Literacy 56 Role of Literacy in the DAL Framework 57 Mathematics Practices within the DAL Framework 62 The Framework’s Development 68 Chapter 3 – Methodology 59 Introduction 70 Overarching Research Question 73 Major Inquiry Areas with the Research 73 Participants 74 Selection of Participants 79 Ethical Considerations 81 Quantitative Instruments 81 Qualitative Instruments 89 Analysis of Final Papers on the DAL Experience 89 Pre and Post Focus Groups 90

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iii Case Studies 91 Procedures 92 Quantitative Procedures 93 Qualitative Procedures 95 Research Design 97 Mixed Methods Design 97 Quantitative Design 98 Qualitative Design 99 Chapter 4 – Results 106 Overview 106 Demographics of Participants 107 Description of Case St udy Participant Selection 109 Format of Results Information 109 Quantitative Findings 110 Mathematics Teaching Efficacy Beliefs Instrument 110 Descriptive Statistics for the MTEBI 111 Inferential Statistics for the MTEBI 117 Mathematics Beliefs Questionnaire 123 Descriptive Statistics for the Mathematical Beliefs Questionnaire 125 Inferential Statistics for the Mathematical Beliefs Questionnaire 135 Mathematical Content for Elementary Teachers 146

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iv Descriptive Statistics for the Mathematical Content Knowledge for Elementary Teachers 151 Descriptive Statistics fo r the Mathematical Content for Elementary Teachers 146 Inferential Statistics for the Mathematical Content Knowledge for Elementary Teachers 153 Instructional Knowledge Exam 160 Descriptive Statistics fo r the Instructional Knowledge Exam 161 Inferential Statistics for th e Instructional K nowledge Exam 166 Fidelity Checks 170 Summary of Quantitative Findings 176 Qualitative Findings 177 Final Project Analyses 177 Efficacy Theme 182 Attitude Theme 184 Content Knowledge Theme 186 Instructional Knowledge and Application Theme 187 Summary of Final Project Analyses 189 Focus Groups 189 Efficacy – Pre Focus Groups 191 Attitude – Pre Focus Groups 194 Content Knowledge – Pre Focus Groups 197 Instructional Knowledge – Pre Focus Groups 200 Instructional Application – Pre Focus Groups 203

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v Efficacy – Post Focus Groups 206 Attitude – Post Focus Groups 209 Content Knowledge – Post Focus Groups 212 Instructional Knowledge – Post Focus Groups 214 Instructional Application – Post Focus Groups 217 Focus Groups Summary 219 Summary of Qualitative Findings 220 Case Studies 221 Olivia: Mathematics Teaching Efficacy Beliefs Instrument – Overall Efficacy 222 Olivia: Mathematics Teaching Efficacy Beliefs Instrument – Self-Efficacy 223 Olivia: Mathematics Teaching Efficacy Beliefs Instrument – Outcome Expectancy 224 Summary 225 Kari: Mathematics Teaching Efficacy Beliefs Instrument – Overall Efficacy 225 Kari: Mathematics Teaching Efficacy Beliefs Instrument – Self-Efficacy 226 Kari: Mathematics Teaching Efficacy Beliefs Instrument – Outcome Expectancy 227 Summary 228 Taylor: Mathematics Teaching Efficacy Beliefs Instrument – Overall Efficacy 228 Taylor: Mathematics Teaching Efficacy Beliefs Instrument – Self-Efficacy 229

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vi Taylor: Mathematics Teaching Efficacy Beliefs Instrument – Outcome Expectancy 230 Summary 231 Comparison of Case Study Efficacy Instrument Results 231 Olivia: Mathematical Beliefs Questionnaire – Overall Beliefs: Constructively Worded Items 232 Olivia: Overall Beliefs – Tr aditionally Worded Items 233 Olivia: MBS – Constructively Worded Items 234 Olivia: MBS –Traditionally Worded Items 235 Olivia: TMBS – Constructively Worded Items 236 Olivia: TMBS –Traditionally Worded Items 237 Summary 238 Kari: Mathematical Beliefs Questionnaire – Overall Beliefs: Constructively Worded Items 239 Kari: Overall Beliefs – Tr aditionally Worded Items 240 Kari: MBS – Constructively Worded Items 242 Kari: MBS –Traditionally Worded Items 243 Kari: TMBS – Constructively Worded Items 245 Kari: TMBS –Traditionally Worded Items 246 Summary 247 Taylor: Mathematical Beliefs Questionnaire – Overall Beliefs: Constructively Worded Items 248 Taylor: Overall Beliefs – Traditionally Worded Items 249 Taylor: MBS – Constructively Worded Items 250 Taylor: MBS –Traditionally Worded Items 251

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vii Taylor: TMBS – Constructively Worded Items 252 Taylor: TMBS –Traditionally Worded Items 253 Summary 255 Comparison of Case Study Beliefs Instrument Results 256 Olivia: Mathematical Content for Elementary Teachers 257 Kari: Mathematical Content for Elementary Teachers 258 Taylor: Mathematical Content for Elementary Teachers 259 Comparison of Case Study Mathem atical Content for Elementary Teachers Results 260 Olivia: Instructional Knowledge Exam 261 Kari: Instructional Knowledge Exam 261 Taylor: Instructi onal Knowledge Exam 262 Summary 263 Olivia: Review of Entire DAL Project 264 Kari: Review of Entire DAL Project 269 Taylor: Review of Entire DAL Project 273 Comparison of Case Study Entire DAL Final Projects 276 Olivia – Final Paper Analysis 277 Kari – Final Paper Analysis 279 Taylor – Final Paper Analysis 281 Comparison of Case Study Final Analysis Papers 282 Olivia – Exit Interview 283 Kari – Exit Interview 288

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viii Taylor – Exit Interview 293 Comparison of Case Study Exit Interviews 298 Overall Case Studies Summary 299 Chapter 5 – Discussion 301 Conclusions 303 Mathematics Teaching Efficacy Beliefs Instrument 305 Mathematics Beliefs Questionnaire 309 Mathematics Content Knowledge for Elementary Teachers 312 Instructional Knowledge Exam 316 Fidelity Checks 318 Final Project Analyses 321 Case Studies 325 Focus Groups 330 Limitations of the Study 334 Threats to Internal Validity 334 Threats to External Validity 335 Threats to Legitimation 335 Implications of Research Findings 335 Developmental-Constructivism 335 Theory to Practice Gap 338 Recommendations for Future Research 340 References 343 Appendix A: Literacy Instructional Practices Within the DAL Framework 350

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ix Appendix B: Algebraic Literacy Library With Sample Book Guide 353 Appendix C: Mathematics Instructional Prac tices Within the DAL Framework 358 Appendix D: DAL Model Visual Conceptualization 360 Appendix E: DAL Initial Session Probe 362 Appendix F: DAL Full Session Notes 364 Appendix G: Mathematics Teaching E fficacy Beliefs Instrument (MTEBI) 366 Appendix H: Preservice Teachers’ Mathematical Beliefs Survey 369 Appendix I: Mathematical Content Know ledge for Elementary Teachers Survey 373 Appendix J: Instruct ional Knowledge Exam and Scoring Rubric 378 Appendix K: Fidelity Checklist for DAL Initial Session Probe 399 Appendix L: Fidelity Checklist for Full DAL Session 401 Appendix M: Focus Group Questions 405 About the Author End Page

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x List of Tables Table 1 Reliability Information for the Mathematics Efficacy Beliefs Instrument 83 Table 2 Reliability Information for the Mathematical Beliefs Instrument 84 Table 3 Reliability Information for the Content Knowledge Instrument 86 Table 4 Alignment of Research Key Questions and Instruments 101 Table 5 Demographic Characteristics of Teacher Candidate Participants 108 Table 6 Descriptive Statistics for the MTEBI 112 Table 7 Repeated Measures Analysis of the Mathematics Teaching Efficacy Beliefs Instrument 118 Table 8 Correlation Matrix for Full Effi cacy Instrument Across Pretest, Midpoint, and Posttest 119 Table 9 Correlation Matrix for Self-Effi cacy Subtest Across Pretest, Midpoint, and Posttest 120 Table 10 Correlation Matrix for Outcome Expectancy Subtest Across Pretest, Midpoint, and Posttest 121 Table 11 Correlation Matrix for Self-Efficacy and Outcome Expectancy Subtests at Pretest 122 Table 12 Correlation Matrix for Self-Efficacy and Outcome Expectancy Subtests at Midpoint 122 Table 13 Correlation Matrix for Self-Efficacy and Outcome Expectancy Subtests at Posttest 123 Table 14 Descriptive Statistics for th e Mathematical Beliefs Questionnaire 126 Table 15 Repeated Measures Analysis of the Mathematical Beliefs Questionnaire 136 Table 16 Correlation Matrix for Full Belie fs Instrument Acro ss Pretest, Midpoint, and Posttest 138

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xi Table 17 Correlation Matrix for MBS – Constructively Worded Items Across Pretest, Midpoint, and Posttest 139 Table 18 Correlation Matrix for the MB S – Traditionally Worded Items Across Pretest, Midpoint, and Posttest 140 Table 19 Correlation Matrix for TMBS – Constructively Worded Subtest Across Pretest, Midpoint, and Posttest 141 Table 20 Correlation Matrix for TMBS – Traditionally Worded Item Subtest Across Pretest, Midpoint, and Posttest 142 Table 21 Correlation Matrix for TMBS and MBS Subtests at Pretest 143 Table 22 Correlation Matrix for TMBS and MBS Subtests at Midpoint 144 Table 23 Correlation Matrix for TMBS and MBS Subtests at Posttest 145 Table 24 Descriptive Statistics for Mathem atical Content for Elementary Teachers 147 Table 25 Repeated Measures Analysis for Mathematics Content for Elementary Teachers 154 Table 26 Correlation Matrix for Full Cont ent Knowledge Survey Across Pretest, Midpoint, and Posttest 155 Table 27 Correlation Matrix for Basic Arithmetic Subtest Across Pretest, Midpoint, and Posttest 156 Table 28 Correlation Matrix for Algebr aic Thinking Subtest Across Pretest, Midpoint, and Posttest 157 Table 29 Correlation Matrix for Basic Arithmetic and Algebraic Thinking Subtests at Pretest 158 Table 30 Correlation Matrix for Basic Arithmetic and Algebraic Thinking Subtests at Midpoint 159 Table 31 Correlation Matrix for the Basi c Arithmetic and Algebraic Thinking Subtests at Posttest 159 Table 32 Descriptive Statistics for the Instructional Knowledge Exam 165 Table 33 Correlation Matrix for the Inst ructional Knowledge Exam and Efficacy, Attitude, and Content Knowledge Posttests 167

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xii Table 34 Correlation Matrix for the Inst ructional Knowledge Exam Subsections and Question Types 168 Table 35 Fidelity Checklist Results on Initial Instru ctional Sessions 173 Table 36 Fidelity Checklist Results on 1st Full Instru ctional Sessions 175 Table 37 Fidelity Checklist Results on 2nd Full Instructional Sessions 176 Table 38 Final Analysis Paper Themes and Codes 179 Table 39 Olivia: Mathematics Teaching E fficacy Beliefs Instrument – Overall 223 Table 40 Olivia: Mathematics Teaching Efficacy Beliefs Instrument – SelfEfficacy 224 Table 41 Olivia: Mathematics Teaching Efficacy Beliefs Instrument – Outcome Expectancy 225 Table 42 Kari: Mathematics Teaching Effi cacy Beliefs Instrument – Overall 226 Table 43 Kari: Mathematics Teaching Efficacy Beliefs Instrument – SelfEfficacy 227 Table 44 Kari: Mathematics Teaching Efficacy Beliefs Instrument – Outcome Expectancy 228 Table 45 Taylor: Mathematics Teaching E fficacy Beliefs Instrument – Overall 229 Table 46 Taylor: Mathematics Teaching Efficacy Beliefs Instrument – SelfEfficacy 230 Table 47 Taylor: Mathematics Teaching Efficacy Beliefs Instrument – Outcome Expectancy 231 Table 48 Olivia: Mathematical Beliefs Questionnaire – Overall Beliefs: Constructively Worded Items 233 Table 49 Olivia: Mathematical Beliefs Questionnaire – Overall Beliefs: Traditionally Worded Items 234 Table 50 Olivia: MBS – Constructively Worded Items 235 Table 51 Olivia: MBS – Traditionally Worded Items 236

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xiii Table 52 Olivia: TMBS – Constructively Worded Items 237 Table 53 Olivia: TMBS – Traditionally Worded Items 238 Table 54 Kari: Mathematical Beliefs Questionnaire – Overall Beliefs: Constructively Worded Items 240 Table 55 Kari: Mathematical Beliefs Questionnaire – Overall Beliefs: Traditionally Worded Items 242 Table 56 Kari: MBS – Constructively Worded Items 243 Table 57 Kari: MBS – Traditionally Worded Items 245 Table 58 Kari: TMBS – Constructively Worded Items 246 Table 59 Kari: TMBS – Traditionally Worded Items 247 Table 60 Taylor: Mathematical Belief s Questionnaire – Overall Beliefs: Constructively Worded Items. 249 Table 61 Taylor: Mathematical Belief s Questionnaire – Overall Beliefs: Traditionally Worded Items. 250 Table 62 Taylor MBS: Cons tructively Worded Items 251 Table 63 Taylor: MBS – Tradit ionally Worded Items 252 Table 64 Taylor: TMBS – Constructively Worded Items 253 Table 65 Taylor: TMBS – Traditionally Worded Items 255 Table 66 Olivia: Content Knowledge Results 258 Table 67 Kari: Conten t Knowledge Results 259 Table 68 Taylor: Content Knowledge Results 260 Table 69 Olivia: Instructional Knowledge Exam 261 Table 70 Kari: Instructi onal Knowledge Exam 262 Table 71 Taylor: Instruc tional Knowledge Exam 263

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xiv Table 72 Olivia: Final Analysis Paper Themes 278 Table 73 Kari: Final Analysis Paper Themes 280 Table 74 Taylor: Final Analysis Paper Themes 282

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xv List of Figures Figure 1 Major Inquiry Areas 72 Figure 2 Efficacy Full Survey Box Plots 115 Figure 3 Self-efficacy Subtest Box Plots 116 Figure 4 Outcome Expectancy Subtest Box Plots 117 Figure 5 Mathematics Beliefs Questionnaire Full Survey Box Plots 131 Figure 6 MBS – Constructively Worded Items Box Plots 132 Figure 7 MBS – Traditionally Worded Items Box Plots 133 Figure 8 TMBS – Constructively Worded Items Box Plots 134 Figure 9 TMBS – Traditionally Worded Items Box Plots 135 Figure 10 Mathematics Content for Elementary Teachers Full Survey 152 Figure 11 Basic Arithmetic Subtest Box Plots 153 Figure 12 Algebraic Thinking Subtest Box Plots 154 Figure 13 Efficacy – Pre Focus Group 1 194 Figure 14 Efficacy – Pre Focus Group 2 195 Figure 15 Attitude – Pre Focus Group 1 197 Figure 16 Attitude – Pre Focus Group 2 198 Figure 17 Content Knowledge – Pre Focus Group 1 200 Figure 18 Content Knowledge – Pre Focus Group 2 201 Figure 19 Instructional Know ledge – Pre Focus Group 1 203 Figure 20 Instructional Know ledge – Pre Focus Group 2 204

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xvi Figure 21 Instructi onal Application – Pre Focus Group 1 206 Figure 22 Instructi onal Application – Pre Focus Group 2 207 Figure 23 Efficacy – Post Focus Group 1 209 Figure 24 Efficacy – Post Focus Group 2 210 Figure 25 Attitude – Post Focus Group 1 212 Figure 26 Attitude – Post Focus Group 2 213 Figure 27 Content Knowledge – Post Focus Group 1 215 Figure 28 Content Knowledge – Post Focus Group 2 215 Figure 29 Instructional K nowledge – Post Group 1 217 Figure 30 Instructi onal Knowledge – Post Focus Group 2 218 Figure 31 Instructiona l Application – Post Focus Group 1 220 Figure 32 Instructiona l Application – Post Focus Group 2 221

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xvii Evaluating the Efficacy of the Developing Alge braic Literacy Model: Preparing Special Educators to Implement Eff ective Mathematics Practices Sharon Ray ABSTRACT For students with learning disabilities, positive academic achievement outcomes are a chief area of concern for educators acr oss the country. This achievement emphasis has become particularly important over the last several years because of the No Child Left Behind legislation. The content area of mathema tics, especially in the higher order thinking arena of algebra, has been of par ticular concern for student progress. While most educational research in algebra has been targeted towards remedial efforts at the high school level, early intervention in the f oundational skills of alge braic thinking at the elementary level needs consideration for st udents who would benefit from early exposure to algebraic ideas. A key aspect of students’ instruction with alge braic concepts at any level is the degree and ty pe of preparation their teachers have received with this content. Using a mixed methods design, the current researcher investigated the usage of the Developing Algebraic Literacy (DAL) fram ework with pre-servi ce special education teacher candidates in an integrated practic um and coursework experience. Multiple survey measures were given at pre-, mid, and postjunctures to assess teacher candidates’ attitudes about math ematics, feelings of efficacy when teaching mathematics, and content knowledge surrounding mathematics. An instructional knowledge exam and fidelity checks were completed to eval uate teacher candidates’ acquisition and

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xviii application of algebraic instru ctional skills. Focus groups, ca se studies, and final project analyses were used to discern descrip tive information about teacher candidates’ experience while engaging in work with the DAL framework. Results indicated an increase in pr eservice teachers’ attitudes towards mathematics instruction, feelings of efficacy in teaching mathematics, and in the content knowledge surrounding mathematics instruction. Instructional knowle dge also increased across preservice teacher candidates, but abi lities to apply this knowledge varied across teacher candidates’, based on their number of sessions working with students within their practicum site. Further findings indicate the de sire of pre-service teachers to increase the length and number of student sessions within the DAL experience, as well as the need for increased levels of instructi onal support to enhance their own experience. This study provides preliminary support for utilizing th e DAL instructional framework within preservice teacher preparation experien ces for future special educators.

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1 Chapter 1 Introduction Statement of the Problem One of the largest difficulties facing the educational system in the United States is providing enough “highly qualified” special educators in the curriculum area of mathematics, especially in the higher level th inking skills of algebr a (Bottge, Heinrichs, Chan, & Serlin, 2001; Gagnon & Maccini, 2001; Mizer, Howe, & Blosser, 1990; Witzel & Miller, 2003). According to the NCLB Interim Report (2007), the current number of special educators who consider themselves “highly qualified” was reported at 52% across grade levels. Overall, special education teacher s were almost four times as likely to selfreport they were not “highly qua lified” compared to their gene ral education teacher peers. In the 2002-2003 school year, 57% of districts na tionally reported that they expected to have difficulty recruiting “highly qualified” special education teachers in the upcoming school year, and 60% said the same for mathematics teachers ( NCLB Interim Report 2007). At the same time, only 12% of student s with mild disabili ties achieve successful outcomes by the time they reach secondary mathematics courses, which includes taking algebra classes and other hi gher level mathematics course s (Kortering, deBettencourt, & Braziel, 2005). Because Algebra I is a gra duation requirement in most states, it is considered a secondary gate-keeping course (Chambers, 1994; Maccini, McNaughton, & Ruhl, 1999). Successful completion of Al gebra I can open educational and career

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2 options, so it is imperative that learners who struggle with mathematics be provided wellprepared instructors to meet their mathem atics content and disability needs. With the recent mandate of No Child Left Behind (NCLB 2001) and the latest reauthorization of the Individuals with Disabilities Education Act ( IDEA 2004), special educators must be “highly qualified” in spec ial education and in the content areas they teach. Students of all ability levels are expected to achieve at least adequate yearly progress (AYP) in all subject areas, includi ng mathematics, and schools and districts are being held accountable fo r students’ achievement ( NCLB 2001). The need to prepare “highly qualified” special educators in mathem atics is obvious, but how this preparation can be accomplished is not. Preparing e nough “highly qualified” sp ecial educators in mathematics is a multi-faceted problem. Factors that contribute to the problem include: 1) few teachers seeking entry into the special education field; 2) limited amounts of time and program structures to integrate pedagogical knowle dge and application with mathematics content during special educators’ pre-service preparati on; and 3) increasing student diversity in public education cl assrooms around the country which requires specialized and differe ntiated instruction. The first factor, recruitment, has been an ongoing issue for many years. In 2001, 26,000 new special education teachers were n eeded to fill open positions throughout the country. In that year, approximately 20,000 students graduated from special education teacher preparation programs nationally. Ho wever, out of these 20,000 special education graduates, about half were already employed “ out-of field” as teache rs. Therefore, the shortfall of needed special educators wa s actually 16,000 (Boe, 2006). The difficulty in recruiting special educators is in itself a mu lti-part dilemma. First, educators, across

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3 instructional areas, are one of the most poorly paid groups of professionals that require a college education (Hammer, Hughes, McClur e, Reeves, & Salgado, 2005). On top of salary, individuals who enter the special edu cation profession meet with classroom and student situations that require specialized training a nd skills (Rice & Gossling, 2005). Third, research by Marso and Pigge (1986) sugge sts that many special educators enter the field because of a life experience that pivotally influences them. While these experiences motivate many individuals to become educator s, they are not situations that can be replicated by institutions of higher educati on to increase recruitment, since these events take place by chance in everyday life. Four th, Singh and Stollof (2007) have indicated that personal dispositions play a key ro le in predicting teacher commitment and preventing burnout. They have found that pe rsonal dispositions that embrace social justice, equity, and cultural sensitivity allow special educators’ abilities to better cope with the amount of work responsibilities a nd professional challeng es that result from working with learners with multiple learning and behavioral needs. These particular dispositions are not always innate in future educators, but may require time and be difficult to cultivate (Singh & Stoloff, 2007). In short, creating a fertile environment for increased special education recruitment is a definitive challenge where some aspects can be negotiated by teacher preparation program s (i.e., dispositional change) and some cannot (i.e., motivational life experiences). Special education graduates have t ypically been knowledgeable in the pedagogical strategies and beha vioral management technique s important for instruction with diverse learners. As the second fact or in developing “highly qualified” special educators alludes, few of these graduates ha ve been well-versed in the application of

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4 these various strategies and techniques within content areas such as mathematics. Beyond just experience in the content area s, even fewer newly-graduated special education teachers have considered them selves content area experts (Laarhoven, Munk, Lynch, Bosma, & Rouse, 2007). Pre-service special education programs have struggled with juggling foundational educational courses with instructional cl asses and directed field experiences (Darling-Hammond & Ball, 1998). Adding additional content area preparation to program requirements extends the time and coursework necessary for teacher preparation, potentially acting as a de terrent to individuals selecting special education teacher preparation programs. NCLB is slowly changing the culture and dynamics of the public education classroom; as a result, despite the aforem entioned challenges, the National Council for Accreditation of Teacher Education (NCATE) (2006) is advocating for the implementation of innovative university pr ograms that incorporate coursework, practicum, and trans-disciplinary knowledge fo r the future special education teachers of such classrooms. Multiple areas are now vying for undergraduate special education teacher candidates’ academic time, and it is difficult to work in coursework that meets both the pedagogical and content area need s of individuals w ho attend university programs aimed at preparing them for K-12 certification. While graduates in special education are now being expected to be qualif ied across content areas, as well as grade levels, mathematics content is one of these critical areas. Many special education teacher prepar ation programs already implement preservice professional development that include s coursework across specific curricular areas in addition to special education. An ongoing challenge is to incorporate this

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5 content knowledge gained through coursework into fieldwork and practicum experiences (Edwards, Carr, & Siegel, 2006). To address this issue, many unive rsities have special education programs that incorporate multiple levels of practicum experiences, with increasing levels of involvement in public school settings (McInerney & Hamilton, 2007). Multiple studies have indicated the di fficulty of teacher candidates in successfully transferring knowledge gained through college class work to application-based K-12 classrooms. Some of these issues stem fr om the university support necessary to support and scaffold undergraduates’ learning in th ese situations (Alls opp, Alvarez-McHatton, DeMarie, & Doone, 2006). Another problem is having the type of practicum opportunities where teacher candida tes have the freedom to expl ore ideas gained in their university setting (Sands, Duffield, & Parsons, 2007). Finally, the research-to-practice gap is a concern, because many of the strate gies and practices be ing advocated at the university are not being taught, supported, or utilized by th e schools and districts where undergraduates are placed for practicum experiences (Biesta, 2007; Bryant, Fayne, Gavish, & Gettinger, 1980; Carnine, 1997). The diversity of students in the United St ates’ public education classrooms is the third undeniable challenge in preparing e nough “highly qualified” special educators. Throughout the country, public schools are filled with children having different cultural and ethnic backgrounds, different le vels of severity and type of disabilities, and different levels of English language proficiency. In the fall of 2002, there were nearly six million students with disabilities being served in th e United States. Out of these students 48.3% were diagnosed with learning disabilities, 18.7% with speech or language impairments, 9.9% with mental retardation, and 8.1% with emotional and behavi oral disabilities ( 26th

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6 Annual Report to Congress 2004). The ethnic make-up of schools is increasingly diverse as well: the curr ent total public school popul ation is 57.9% White, 19.2% Hispanic, 17.3% African-American, 4.5% Asia n, and 1.2% American Indian students (NCES, 2004). The increasing numbers of st udents with limited English proficiency is another concern. Currently, there are 3 million students or 7% of the school-aged population considered to be English Language Learners (ELLs) (NCES, 2001). Special education teacher preparation programs have difficulty keeping pace with these everincreasing numbers of students with multiple learning needs. Confronting these pre-service special edu cation teacher preparation issues is a complex endeavor that will require multiple poin ts of focus. One area of focus must be on ways to effectively prepare teachers to possess the pedagogical skills and content knowledge to address the varying needs of a diverse student populat ion in content area learning. A promising approach for accomplishing this task at the preservice level is to embed content area instruction within app lication-based instruct ional frameworks in these special education teacher preparation pr ograms. Yet, researchers must be aware that any such frameworks within a pre-se rvice special education teacher preparation program are not targeting one area of teacher professional development but multiple ones. Teacher candidates enter preparation progr ams with different levels of self-efficacy in content area instructional abilities, different attitude s about content area learning, different grasps on content-related peda gogical knowledge and its application, and different amounts and forms of content area kno wledge. To this end, the current research study evaluated the instructional implications of using one su ch content area instructional framework, the Developing Algebraic Literacy (DAL) model, within a special education

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7 teacher preparation program, exploring its possibilities for affec ting change across the aforementioned areas within teach er candidates. Theoretical/Conceptual Framework The current study has its foundation in a c onstructivist framework. Within this type of study, the research er implements numerous mean s of collecting data that incorporate multiple facets of a given problem. Using the data collected, the investigator works to interpret the data and gain an understanding of the research question by employing the many types of information gath ered to facilitate the understanding and meaning-making process by comparing new information to what is already known (Cronje, 2006). According to Bruning (1995), constructivism involves “selecting information and fitting it with previous ly known knowledge structures”. DarlingHammond (2000) presents this constructivist fr amework in an educational context. She describes the ideal modern teacher as “one w ho learns from teaching rather than one who has finished learning how to teach” (170). This developmental social constructivist approach to pre-service teach er preparation envisions t eacher candidates constructing their instructional ab ilities, not through simple cour sework and knowledge memorization, but directly through application-based teach ing experiences, where beginning classroom situations help teacher ca ndidates understand learning needs and grow in their instructional capabilities in fu ture situations. DarlingHammond (2000) indicates that teacher candidates’ construction of new in structional understandings and competencies are facilitated by an inquiry based appro ach to learning teaching skills. Teacher candidates’ employ this systematic inquiry through observations of their instructional impact and reflection on this impact to develop future teaching practice (Darling-

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8 Hammond, 2000). The current study employed th is developmental social constructivist approach to pre-service teach er preparation using an appl ied instructional framework within a special education teacher prepara tion program, with the goal of observing and evaluating the influence of the instructi onal framework on factor s that have been identified in the literature as pertinent to successful pre-service professional development in content area instruction. One factor related to teacher success in content area instruction, which has received attention in the literature, has b een self-efficacy, the belief teachers have about their ability to bring about possible student outcomes (Enochs, Smith, & Huinker, 2000). In general academic studies of self-efficacy at the college level, it has been found that “the stronger the students’ beliefs in their efficacy, the more occupational options they consider possible, the greater the interest th ey show in them, the better they prepare themselves educationally for different career pursuits, and the grea ter their persistence and success in their academic coursework” (Ba ndura, Barbaranelli, Caprara, & Pastorelli, 1996, p. 1206-1207; Betz & Hackett, 1986; Lent, Brown, & Hackett, 1994). At the inservice education level, high perceptions of self-efficacy correlate with positive teaching behaviors “such as persiste nce on a task, risk taking, a nd use of innovations” (Enochs, Smith, & Huinker, 2000, p. 195). Czerniak (1990) found that teachers with high levels of self-efficacy correlated with teachers’ use of student-centered and inquiry based learning, while teachers with low levels of self-efficacy were more likely to employ teacher lecture and passive student learning act ivities. Although self-efficacy among teachers appears to be important, the current cultu re of accountability and high stakes testing may have a negative impact on teachers’ se nse of self-efficacy because th e stressors associated with

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9 this culture negatively impact teachers’ abilit ies to function at their instructional best (Puchner & Taylor, 2004). Indeed, the develo pment of flexibility and resiliency to sustain teaching self-efficacy appears to be an important area of emphasis for pre-service teacher preparation programs at the moment. In the current study, self-efficacy towards teaching mathematics was an area of inquiry when evaluating the experience of preservice special education teacher candida tes during the implementation of the DAL mathematics instructional framework. Another factor that has recei ved attention in the literatu re is teacher candidates’ attitude towards and beliefs about the s ubject area of instruction (Charalambous, Phillipou, & Kyriakides, 2002; Dwyer, 1993). An individual’s feelings about a specific body of knowledge can significantly impact the person’s approach to dealing with that set of information. Negative teacher perceptions of a content area can result in it having reduced instructional time, student engageme nt in learning activities, and instructional decision-making, which can result in lowered student achievement. Positive teacher perceptions of the same content area can resu lt in enhanced student achievement because of increased time spent on the same variable s (Ernest, 1991). As Hersh (1998) asserts, knowing teachers’ attitudes and beliefs toward s mathematics instruction is essential because it impacts the way they present mathem atical concepts since “the issue is not what is the best way to teach mathematics, but what mathematics really is all about” (13). The role that teacher preparation programs can have in cultivating positive teacher attitudes towards mathematics is defined by th e University of Maryland System (1993) as the “development of professi onal teachers who are confid ent teaching mathematics and science using technology, who can make conne ctions within and among disciplines, and

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10 who can provide an exciting and challenging l earning environment for students of diverse backgrounds” (p. 3-4). Dwyer (1993) suggests two primary ways for teacher preparation programs to collect these attitudinal data, “through observing subjec ts and/or by asking subjects what they believe” (p. 4). Theref ore, it was important to consider the possible changes in teacher candidates’ attitude to wards mathematics instruction during the implementation of the DAL instructional framework. A third relevant factor is the degree to which pre-service teac hers are exposed to and provided opportunities to apply effective pedagogica l practices. For special educators the development of instructional know ledge is indeed a multi-faceted endeavor. Integrating special education instructional knowledge with specific content strategies, such as those practices from mathematics education, is challenging. As professionals prepared to work with students who have be havioral and learning difficulties, special education teachers must learn research-based instructional strategies for enhancing educational experiences for learne rs with disabilities in general, but at the same time have intimate knowledge of the strategies that partic ularly facilitate content specific learning in areas such as mathematics (Maccini & Ga gnon, 2006). Among these accepted practices are slower-paced and more structured pres entations of learning concepts; multiple modalities for instructional presentations in cluding visual, auditory, and kinesthetic; memory aids including word books, acronym s, and classroom routines; explicit instruction with modeling; stra tegy instruction; graphic orga nizers for visual information display and organization; cont inuous student progress monito ring; and moving from more concrete to more abstract concepts in sequencing instructiona l progression (Allsopp, Kyger, & Lovin, 2006; Baker, Gersten, & Lee, 2002; Gagnon & Maccini, 2001;

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11 Kortering, deBettencourt, & Braziel, 2005; M accini & Hughes, 2000; Witzel, Mercer, & Miller, 2003; Witzel, Smith, & Brownell, 2001). When students with learning disabilities study mathematics, these strategies assi st them in comprehending and retaining mathematics information. As Cawley, Parmer, Yan, and Miller (1996) found, student s with learning difficulties do not typically learn concepts in a sequential path of increasing difficulty, but in an erratic, gap-ridden course, wher e retention difficulties are problematic. Specifically in the area of algebra, students w ith mild disabilities of ten struggle with the abstract nature of th e symbols and notation associated w ith this higher level mathematics where Witzel, Smith, and Brownell (2001) reco mmend the use of manipulatives to tie the abstract concepts of algebra to more concre te materials. Interventions that teacher candidates’ learn in their teach er preparation programs can increase their knowledge level of instructional strategies, which can positively impact their future students’ mathematics outcomes (Ashton & Crocker, 1986; Darling-Ha mmond, 2000). Relative to this research study, changes in teacher candidates’ inst ructional knowledge of mathematics were evaluated during exposure to the DAL framework. Ideally, teachers who gain instructional knowledge will transfer this information to the application stage, where they transl ate their instructional knowledge into actual practice. The nature of prof essional development practices that effectively support this important transformation has received some a ttention in the litera ture. The amount of time available to prepare teachers is one variab le that seems to be important. As stated by Nougaret, Scruggs, and Mastropieri (2005), on e of the greatest challenges in special education teacher preparation is having enough time within programs for teacher

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12 candidates to transfer that instructional knowledge in th e academic sense to knowledge that can be implemented flexibly in actual instructional situations in the classroom. A review of the literature by Darling-Ha mmond, Chung, and Frelow (2002) found that the more time and intensity spent in coursewor k, practicum, and student teaching, the better prepared individual teacher candidates believed they were to take on the challenges of their own classrooms. Golder, Norwich, and Bayliss (2005) found a nother variable was experience with individualized instruction. Teacher candidates who were pl aced in school settings to teach special education students one-on-one within a larger classroom situation demonstrated improved understanding of indi vidual student learning needs, enhanced levels of content knowledge, and increased abil ities in differentiati ng instruction through this individualized experience. However, key areas that were reported as needing enhancement were the univ ersity-school partnership, university supervision and communication, and university guidance on sch ool-based assignments. One goal of the current investigation was to evaluate the experience of teacher candidates with the instructional application aspect of th e DAL framework. As teacher candidates implemented this mathematics framework in the practicum site, it was observed if and when teacher candidates were able to tr ansfer mathematics strategy instruction knowledge to actual application. The rese archer provided ongoing support to teacher candidates to facilitate linkages between in structional understandings and application on site, as well as maintained an open dialogue with school site administration on teacher candidate and student performance.

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13 One of the issues in transferring pedagogical knowledge to pedagogical application is the ability of teacher candidates’ to understand the underlying components of the instructional strategies well enough to implement stra tegy steps consistently with fidelity. According to Smith, Daunic, and Tayl or (2007) fidelity is “a critical factor in determining the efficacy, effectiveness, and successful dissemination of an educational practice…ensuring that the pr ofessionals who are responsible for its implementation deliver an intervention under study with accuracy and conformity” (122). Fidelity to an intervention or framework’s model is the pr imary way to ensure that students are consistently being exposed to the same instru ctional elements when a new intervention is being evaluated for its effect, applicability, and outcomes. The idea of fidelity is integral to understanding if an intervention under investigation is responsible for increased student knowledge. Interventions that are implemented continuously and consistently with fidelity have more justification for positive student outcomes, than those outcomes be ing due to outside sources (Bellg et al., 2004). A key part of designing and implemen ting any instructional framework that will be utilized in pre-service professional prepar ation is developing a viable means of monitoring teacher candidates’ ability to implement that in structional framework with fidelity (Duchnowski, Kutash, Sheffield, & Va ughn, 2006). For this particular study, a fidelity measure was used to monitor teacher candidate application of the DAL framework. Fidelity checks using this instrume nt served a dual purpose. The first was to evaluate the teacher candidates’ abilities to transfer information learned about the framework to actual application. The second wa s to facilitate res earcher understanding

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14 of the possible relationship between student outcomes and teacher candidate implementation, if any such relationship exists. Finally, for a framework to enable sp ecial education teacher candidates’ successful instruction in the c ontent area of mathematics, specifically elementary level algebraic thinking, there must be a mechan ism for assisting teach er candidates’ in obtaining proficiency in the con cepts and skills for instruc tion. With many instructional frameworks, future teachers are expected to pick up on desired core concepts through implicit instruction and learning activities. However, in research done by the United States Department of Educa tion (2003), a dual emphasis is advocated for teaching preservice teachers pedagogical knowledge and subject area conten t explicitly. According to Boe, Shin, and Cook (2007), special edu cation teacher education programs that concentrate on these areas of teaching intensively, have resulted in enhanced teacher preparation outcomes for future special educat ion teachers in dealing with the formidable instructional and subject area challenges they will meet. Future teachers need not only pedagogy, but the nuts and bolts of the curric ulum they must teach (NCLB, 2001). Being prepared to educate within the standards-base d learning environments of the twenty-first century is imperative for all teacher candidate s. The current investigation employed an initial intensive training work shop that split instruction be tween the content knowledge of algebraic thinking and pedagogical technique s for struggling learners. As the study progressed, ongoing training was incorporated on content matters, with the teacher candidates provided with informational Powe rPoints and handouts on the content area being taught.

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15 By grounding this exploration of the DAL framework for mathematics instruction within the context of change in teacher candidate self-effi cacy, attitude, instructional strategy knowledge and application, and cont ent knowledge, the viabil ity of the specific instructional framework was explored. Observing DAL’s application across multiple domains was considered essential to cult ivating a successful framework for use in preparing future teachers, who in turn promote positive student learning outcomes. Understanding teacher candidates’ experiences with the DAL model along multi-faceted lines was the core to cons tructing the researcher’s understanding of a complex educational issue through its component pa rts (Cronje, 2006; Darling-Hammond, 2000). Purpose The purpose of the current study was to investigate teacher candidates’ exposure and responses to the Developing Algebraic Literacy (DAL) instructional framework within a second semester professional deve lopment experience for undergraduate special education teachers. The scope of the inve stigation encompassed several elements of teacher preparation involving: 1) feelings of self-efficacy in mathematics instruction; 2) attitude towards mathematics instruction; 3) instructional knowledge and application of mathematics-based pedagogy; and 4) content knowledge for mathematics instruction. The study probed the viability of using a syst ematic mathematics framework, specifically in the content area of elementary level algebraic thinking, with pre-service special education teacher candidates. Through its implementation, the study aimed to inform the limited amount of knowledge currently availabl e on preparing special educators to teach mathematics to struggling learners.

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16 Research Questions Overarching Question The following research question was addressed through the current study: What changes related to effective ma thematics instruction for struggling elementary learners, if any, o ccur in teacher candidates during implementation of the DAL instructional framework in an early clinical field experience practicum for pre-se rvice special education professional preparation? Major Inquiry Areas within the Research Question The following inquiry areas broke the resear ch question down in to investigational components: 1.) What changes, if any, occur in special education teacher candidates’ feelings of self-efficacy about teaching mathematic s from the beginning to the end of a pre-service instructi onal experience using the DAL framework? 2.) What changes, if any, occur in specia l education teacher candidates' attitudes towards mathematics instruction from the beginning to the end of a preservice instructional experience using the DAL framework? 3.) What changes, if any, occur in special education teacher candidates' understanding of instructiona l strategies for struggling learners in mathematics from the beginning to the end of a preservice instructiona l experience using the DAL framework? 4.) What changes, if any, occur in sp ecial education teacher candidates’ application of instructiona l strategies for struggling learners in mathematics

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17 from the beginning to the end of a preservice instructiona l experience using the DAL framework? 5.) What changes, if any, occur in specia l education teacher candidates’ content knowledge of elementary mathematics, including algebraic thinking, from the beginning to the end of a pre-service instructional experience using the DAL framework? Significance of the Study The current study provides information th at informs special education teacher preparation at the undergraduate level in se veral capacities. Firs t, teacher candidates were provided with an initial intensiv e workshop, as well as continued training throughout the semester through ongoing seminars that touched on issues related to DAL implementation and content area knowledge in mathematics. At the same time, researcher support was given to teacher candidates in the context of ongoing DAL implementation and collaboration with school site personnel. Through these measures within the study, the capability and viabi lity of providing preservice professional development in an ongoing and developmental ma nner received targeted investigation. Furthermore, since the DAL framework wa s taught and applied within the context of special education coursework and pract icum experiences, integration of special education and content specific instructiona l practice was evaluated. Traditionally, mathematics content is taught within math ematics education courses, while general instructional techniques for st udents with disabilities are taug ht within special education specific courses. By meshing the two areas in the current study, future opportunities may

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18 be widened for integrating both areas with in special education teacher preparation programs. Lastly, since most courses are taught in the university setting, separate from the applied setting (i.e., school s), the DAL model research provided insight into the possibilities of constructing teach er preparation experiences that link course instruction to applied school experiences e xplicitly. Along this investig ational line, information was gathered on a special educati on teacher preparation experien ce that employed coursework application imbedded in fieldw ork experiences. To this end, this final element of investigation showed promise in indicating whether structured opportunities for learning and practice can scaffold increased usage of instructional strategies, through establishing a direct linkage between knowledge acquis ition and implementation. Definition of Terms “Highly Qualified” Teachers According to NCLB (2001), “To be deemed highly qualified, teachers must have: 1) a bach elor's degree, 2) full state certification or licensure, and 3) prove that they know each subject they teach” (sec. 1119). Existing teachers can achieve “highly qualified” status by going through a state-approved alternative method called, High, Objective, Uniform State Standard of Evaluation (HOUSSE). Attitude This term is defined as the emo tions, feelings, and beliefs held by a teacher in regards to a particular subject ar ea or instructional tas k, with a corresponding set of particular behaviors that the teacher enacts based on a specific emotional, feeling, or belief trigger.

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19 Self-Efficacy This dispositional concern involves the level of teachers’ beliefs in their own instructional abilities and actions as adequate vehicles to effectively convey content knowledge to students. Content Knowledge These specific skills are the abilities and guidelines associated with a particular academic subject. It is these concepts, through instruction, that educators aim to teach students to increase their academic achievement. Algebra This set of skills is advocated by the National Council of Teachers of Mathematics (NCTM) as pertaining to the Alge bra Standard of mathematics curriculum. Skills included within the sta ndard are: “understanding patt erns, relations, and functions; representing and analyzing ma thematical situations and structures using algebraic symbols; using mathematical models to represent and understand quantitative relationships; and analyzing change in various contexts” (NCTM, 2000). AYP This acronym stands for Adequate Y early Progress, which is the amount of academic progress that students are expected to make within one y ear with appropriate instruction. Schools must show that student s with disabilities ar e meeting established goals for academic progress during one academic year through alternative measures when these learners do not meet criteria for stat e-mandated standardized assessments (NCLB, 2001). ELLs These students are known as English Language Learners (ELLs) because they speak a language other than English as th eir first language. A student is considered an ELL when he or she is in one of the acqui sition stages of Eng lish language speaking and writing skills that is not yet consider ed comparable to typical English-speaking

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20 classroom peers and requires s upplemental educational servi ces beyond what is offered in the regular education classroom. Students At-Risk. Learners with this designation are ones not necessarily labeled with a disability categorization, but they coul d have one. Students given this label are ones that because of environmental, econom ic, language, or learning difficulties are considered vulnerable for having difficulty in achieving academically at the same level and at the same rate as their learning peers. Targeted instruction may result in students’ not needing identification for or being re moved from special education services. Learning Disabilities These disabilities are ones that impair the normal cognitive functioning required for basic academic tasks. This group of disabilities results from deficits in sensory, processing, or memory difficulties within a student of normal intelligence. Determination of learning disa bilities is typicall y diagnosed through the completion of intelligence a nd achievement testing, indica ting a significant discrepancy between a student’s actual intellectual ability level and the level at which that student is currently able to achieve. Sensory Deficits. These deficits include impairme nts in one or more of the senses, affecting auditory, visual, or tact ile detection of information, which impedes learners from integrating sensory information from their environment within cognitive processing functions. Processing Deficits. These difficulties impair lear ners’ abilities to break down information into understandable pieces once th at information has been activated through one of the sensory channels. The information still enters the brain from sensory functions but learners have difficult y in making meaning out of this information and then

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21 formulating responses to it. Students with this type of disability benefit from strategies targeted at helping them break down and ma ke sense of the information they acquire from their multiple senses. Memory Deficits These problems are associated with the long-term and shortterm retention of information that has been obtained through sensory functions and processed through cognitive mechanisms. Anot her prevalent memory difficulty is with “working memory”, which is the ability of students to readily retrieve learned information for usage during application situ ations. Students with memory deficits benefit from the usage of memory aids for re tention and retrieval of information. Fidelity This term is typically used wh en discussing the degree to which an intervention or framework is implemen ted along the guidelines of its designed instructional steps. Fidelity is d eemed important for successfully employing interventions or instructional frameworks so they will result in the most positive student outcomes possible. Application-Based This idea involves any piec e of knowledge that is not only retained by a teacher candidate, but applied by that pre-service teacher in a specific learning situation that invol ves taking the knowledge gained through instruction and employing it with actual learners in the classroom. Developmental Social Constructivist Approach This approach to teacher preparation is advocated by the work of Darling-Hammond (2000), and focuses on teacher candidate learning experiences that construct new instructional knowledge and abilities by building on previously learned educational ideas and practices. In the course of growth and development through structured coursework and fiel d experiences, teacher

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22 candidates’ professional pract ice is established through th e social interactions of instruction, collabora tion, and active lear ning activities. Sunshine State Standards. These standards are the State of Florida’s curriculum guidelines that structure public school inst ruction in grades K-12. These standards provide teachers a framework for instructi on with students in th e general education classroom, with suggested modifications for diverse learners. Th ese standards are the interpretation of federal mandates for curriculum advocated by NCLB (FDOE, 2008). Title I Schools These schools receive additiona l funding from their particular school districts because their st udent economic levels are below that of the district mean. When a school has 40% of its students below it s district’s socioeconomic mean, a school will be designated as Title I and will be given additional funding by its district to organize, fund, and facilitate programs that will benefit all students in attendance at the school (DOE, 2007). Delimitations The current study contained certain deliber ate limitations. The participants for the study were selected based on their Level II Cohort status within the Special Education Department at a research university, which limited the individuals eligible for participation in the study. At the same time, the study also situated all teacher candidate participants inside on e Title I school setting within a large urban sc hool district in the southeastern United States. This placement was made for the manageability of the many teacher candidate participants with the time resources of the researcher, as well as the prior established partnership between the pa rticular Title I school and the university’s Special Education Department.

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23 Limitations Results of the study have been interpreted cautiously in view of several potential limitations. First, instrumentation posed a threat to validity. To alleviate these threats, the chief quantitative instruments employed in this study were three surveys that had previously established normative reliability and validity information. Moreover, multiple instruments were employed to collect inform ation pertaining to the research questions, rather than relying on one measure. Add itionally, for qualitative data collection, focus group probes developed by the researcher were focused on key ideas targeted in the quantitative research instruments, attempti ng to secure additional perspectives on an underlying core set of ideas related to teacher preparation. Triangulation was used with both quantitative and qualitativ e data as a strategy to addr ess potential limitations of individual measures. Another possible th reat was maturation, because during the 10 week period of the study, it would be expected that teach er candidates would experience growth in all areas of the study: sel f-efficacy, attitude, instructional knowledge and application, and content knowledge. However, since all teacher candidate participants were progressing through the program with the same coursework and during the same time frame, maturation would be expected to occur concurre ntly across teacher candidates, evenly distributing this effect across all participants. A third possible source of validity concerns pertain to testing effects. Since the quantitative survey instruments were employe d multiple times in the study, at pre-test, midpoint, and post-test, it is t hought possible that teacher can didates’ responses may have been impacted by the number of times the su rveys were administered and the short period of time between these administrations. To co mbat this threat to validity, quantitative

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24 surveys, while employed, were supplemented by qualitative information gathered through focus group responses, case study analyses, a nd evaluation of teacher candidate produced artifacts from the experience of this app lication-based interven tion. Fourth, student absences may have impacted individual te acher candidates’ abili ties to connect their training in the DAL framework with its ac tual implementation with students. To minimize the impact of student absences each teacher candidate was assigned two students for remediation using the DAL framew ork to provide for multiple applications of the framework each week, or allow for at least one application per teacher candidate each week in the case of one student’s abse nce. Fifth, observati on and evaluation bias were considered additional possible threats to validity. To address the observational bias, multiple observers were trained in the DAL framework with the teacher candidates, and these observers made observations of teacher candidates using a structured fidelity checklist. Additionally observers spent time together observing teacher candidates applying the framework, until 90% agreement was reached between observers. With evaluation of teacher candidates’ test questi on responses, three inde pendent raters also judged all teacher candidate test responses. Then, the raters regrouped and reviewed individual student response ratings for disc ussion and agreement purposes with the same 90% agreement level used. With focus group responses, the researcher also completed frequent member checks to ensure that teacher candidates’ re sponses adequately portrayed their feelings and ideas. Organization of Remaining Chapters The remaining chapters explain the current research in more detail. Chapter 2 provides an overview of the federal and st ate impetus for improving special education

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25 teacher preparation, as well as provides further depth in the exploration of the elements of self-efficacy, attitude, instructional know ledge and application, and content area knowledge as essential components of a pre-se rvice special educati on teacher preparation program. The development of the DAL and its accompanying contextualized application library, the Algebraic Literacy Library (ALL), are also presented. Chapter 3 provides information on the study’s methodological construction. Chapter 4 reports both quantitative and qualitative data collection results. Chapter 5 provides a discussion of the study’s results, and research implications and recommendations for future research.

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26 Chapter 2 Review of the Literature Overview Investigating the usage of a framewor k for teaching mathematics within an undergraduate special education teacher prepar ation program involves multiple facets of exploration. The professional development of these pr-eservice teachers involves the complex interaction of several variables. Prep aring teachers to instruct learners who are at risk for difficulties in mathematics invol ves not only the instructional strategies necessitated by these students’ learning needs, but also specific pedagogy targeted at acquiring mathematics content knowledge. Teacher candidates themselves must be trained in the skills a nd abilities associated w ith their specific subjec t area for instruction. To help students achieve successful outcom es in content area learning, such as in mathematics, teacher candidates must have an understanding of the underlying concepts associated with the subjects to be taught. Future teachers also bring with them dispositional characteristics, such as f eelings and beliefs about the content of mathematics, pedagogy surrounding mathematics, and learners’ aptitude in regards to mathematics that can impact their instructional effectiveness (Beswick, 2006; Dwyer, 1993; Seaman, Szydlick, Szydlick, & Beam, 2005). Thus, dispositional concerns surrounding teacher candidates’ approach to in struction are also a viable dimension for study along with content area a nd instructional knowledge.

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27 In addition to discussing professiona l development elements surrounding teacher candidates, this review also analyzes the re search related to the proposed framework for mathematics instruction and its corresponding ap plication library, the Algebraic Literacy Library (ALL). The current framework unde r investigation for pre-service special education teacher development, the Deve loping Algebraic Literacy (DAL) model, incorporates research-based elements relevant to educating diverse learners. It also includes distinctive linkages be tween what is known about systematic reading instruction and algebra instruction, which is the framework’s targeted mathematics content area. Embedded within the DAL model is the use of a “context” for learning algebra, another concept taken directly from the resear ch on reading and learning engagement (Blachowicz & Fisher, 2006; Chamberlai n & Leal, 1999; Gipe, 2006; Hill, White, & Brodie, 2001; Richardson & Miller, 1997). The purpose of the DAL framework is to facilitate the acquisition of basi c algebraic skills for struggling learners in mathematics at the elementary grade levels, using the integr ation of mathematics, reading, and special education pedagogy. Through the employment of the DAL in a specia l education teacher preparation experience using a structured, deve lopmental social cons tructivist approach, the goal of this study was to evaluate teacher candidates’ experiences with and responses to: feelings of efficacy about mathem atics, attitudes towards mathematics, comprehension and usage of mathematicsbased pedagogy for str uggling learners, and mathematics content knowledge surrounding algebraic thinking. Literature Search A review of the literature was comple ted through the usage of multiple search terms and databases found through the research er’s university library. The researcher

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28 used the following search terms for literature location purposes in the Educational Resources Information Center Online Com puter Library Center (ERIC OCLC), PSYC INFO, and Wilson Omnifile databases: “pre -service teacher preparation”, “professional development + special education”, “pre-servi ce teacher preparation + special education”, “algebraic thinking”, “algebraic thinking + disabilities”, “alg ebra instruction + learning disabilities”, “mathematics instruction + disa bilities”, “teacher attitudes”, “teacher + mathematics attitudes”, “teacher self-efficacy ”, “teacher + mathematics self-efficacy”, “reading + algebraic thinking”, “algebraic thinking + assessments”, “algebra + statistics”, “algebra + achievement”, “elementary mathematics + assessments”, “elementary mathematics + algebraic thinking”, “lear ning engagement + reading”, “learning engagement + mathematics”, “Caldecott Awa rd Winners”, “Caldecott + reading”, “teacher recruitment”, “teacher recruitment + special education”, and “student population + public schools.” Federal and State Impetus Recent federal improvement efforts in American public education have their roots as far back as A Nation at Risk (National Commission on Excellence in Education, 1983). This report brought American schoo ls’ student failure statistics to the forefront of public consciousness. Amongst the reports’ findings, teacher preparation qua lity was cited as one of the pivotal areas influencing impr oved student outcomes (National Commission on Excellence in Education, 1983). Goals 2000: Educate America Act (1994) took the findings of A Nation at Risk (1983) to a new level, prov iding funding to “develop clear and rigorous standards for what every child should know and be able to do.” The legislation specifically allotted funding to improve teacher preparation by increasing

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29 training and development opportunities th rough attending workshops, networking, observing, and collaborating ( Goals 2000 1994). The No Child Left Behind ( NCLB ) and Individuals with Disa bilities Education ( IDEA ) Acts have now taken teacher preparation one step further and focused it towards special education teacher professional deve lopment in the content areas (IDEA, 2004; NCLB, 2002). In its core provisions, NCLB has measures targeting improved student outcomes for all learners, while IDEA specifically focuses on learners with disabilities. Both pieces of legislation contain teacher pr eparation and quality standards that target teachers’ proficiency in pedagogical and conten t area knowledge in an effort to increase student performance (IDEA, 2004; NCLB, 2002). Even though educators are afforded increased preparation opport unities under current mandates, accountability for student content learning primarily still falls on thei r shoulders. While educational law is formulated by the federal government, in terpreted by the individual states, and operationalized by the districts, educators mu st still find their own workable methods and tools for meeting all learners’ needs within this demanding and rigid standards-based framework. In 2004, Harvard University administered the No Child Left Behind: The Teachers’ Voice survey, gathering a representative sample of teachers’ views on NCLB (NEA, 2008). In this study, teachers from Ca lifornia and Virginia, were asked to share their thoughts on the key tenants of the NCLB legislation. The findings from this research indicated that teachers believe more funding is needed for increased resources, including curriculum and instru ctional materials. Other comments suggested a need for increased administration quality for leadership in instructi onal matters, as well as more

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30 time for collaborating with experienced teach ers. Enhanced teacher commitment and increased professional support for teachers with in low-performing schools were also seen as high priorities (NEA, 2008). Rebell and Wolff (2008) from The Campaign for Educational Equity at Teachers College, Columbia University, assert along the same lines that NCLB still needs to actualize the number of “highly qualified” teachers. They indicate that urban and minority students, mo re so than other learners, tend to have teachers that are not “highly qualified.” These researchers advocate an increase in instructional resources and support for te achers at schools de signated as “needing improvement.” Content Area Learning At the forefront of the standards movement and increased teacher preparation is a focus on student achievement gains in readi ng and mathematics. Public education is driving students of all levels and abilities to learn more, at faster rates, and with fuller depth than in past decades ( IDEA 2004; NCLB 2002; NRP 2000). This reality requires researchers and educators alike to more comprehensively examine those aspects of content specific learning, like elements of mathematics and r eading, which are critical to success for struggling learners. To this end, researchers and educat ors should not be reluctant to look across content areas to lear n from relevant successes and failures. For example, much might be learned about how to more effectively teach mathematics by examining recent advances in reading instruction (Jamar & Morrow, 1990; Sherwood, 1991; Von Drasek, 2006). In reading, the emph asis is on phonemic awareness, which is considered the building blocks of more adva nced arenas of fluency and comprehension, necessary for written content understanding throughout the lifetim e (Mercer & Mercer,

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31 2005; NRP, 2000). In mathematics, a similarly relevant area might be algebraic thinking. In contrast to the common thinking that al gebra is the manipulation of numbers and symbols that is emphasized in high school Al gebra courses, algebr aic thinking actually spans the K-12 mathematics curriculum targetin g higher order and crit ical thinking skills (Cai, 1998; Carpenter & Levi, 2000; Ka put & Blanton, 2000; NCTM, 2000). Algebraic thinking integrates number a nd number sense (i.e., one’s understanding of what number represents and how numbers re late to one another) with the processes of analysis, reasoning, prediction, and problem solv ing (Gersten & Chard, 1999). Much like the interrelation between phonemic awareness, the ability to unde rstand and manipulate the sounds of spoken language, and phonologi cal awareness, the ability to apply phonemic awareness to print, these two impor tant components of algebraic thinking are building blocks to greater mathematical understanding and awareness. The development of number sense is similar to the deve lopment of phonemic awareness because number sense, like the spoken word, is key to the language of mathematics. Understanding how the elements of mathematics “language” can be manipulated to represent different ideas is critical to mathematical comprehension. Using these understandings to analyze, reason, predict, and problem solve are akin to maki ng meaning out of print in reading for the development of literacy skills. One can begin to communicate and therefore think mathematically when these skills are u tilized with algebrai c thinking, developing mathematics literacy (Gersten & Chard, 1999; Kaput & Blanton, 2000; NCTM, 2000). To help their students develop such compet encies in mathematics, teachers must integrate these important components of al gebraic thinking within a framework that incorporates effective mathematics instructi onal practices. This study incorporated such a

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32 framework for teaching algebraic thinking for struggling learners: the Developing Algebraic Literacy (DAL) model. The DAL framework takes the idea of algebraic thinking one step further, to be called “algebraic literacy ”, defining competency in algebraic skills as an actual form of literacy just as comprehension is in reading. An important component of the DAL mode l is the use of narratives that situate algebraic thinking concepts/skills within m eaningful contexts. The Algebraic Literacy Library (ALL), a library of award-winning children’s books was used for this purpose. The integration of the DAL framework and the ALL represents a trans-disciplinary approach to presenting, redefi ning, and reinforcing the basic skills necessary in algebraic literacy, providing a context for student e ngagement in meaningful problem solving across cultural backgrounds and disability designations. The focus of this study was on the pre-service teachers’ impl ementation of this instructi onal process during a 10 week early clinical field experience that integr ated features of a developmental social constructivist approach to teacher educati on, and the instructional framework’s possible influences on pre-service teachers’ developmen t in the areas of self-efficacy and attitude towards mathematics instruction; pedagogical knowledge and application in mathematics instruction; and content ar ea knowledge surrounding algebraic thinking concepts when teaching students at-risk for difficulties in mathematics. Diverse Populations A pedagogical framework such as the DAL has potential for improving mathematics outcomes for struggling learners, particularly given the current classroom climates in our schools. The classrooms of today look much different than the ones of yesterday. Twenty years ago, the student popul ation in most areas was demographically

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33 over 70% White students (Fry, 2006). Toda y, the situation has undergone phenomenal evolution. According to Richard Fry from the Pew Hispanic Research Center, in the 2005-06 school year, the population of Hispanic individuals co mposed 19.8% of the total student population in schools nationwide. This figure is a 7% jump in just 10 years, considering that Hispanic individuals ma de up just 12.7% of the school-aged population in the 1993-94 school year. In terms of Afri can American students, the number has not jumped but maintained and grown slightl y, rising from 16.5% in the 1993-1994 school year to 17.2% in 2005-06. In the same 10 y ear period, the populati on of White students in public schools has fallen from 66.1% to 57.1% (Fry, 2006). With these population changes, and current overall population distri butions across the count ry, administrators and teachers have to constantly be rethinki ng curriculum with innovative materials and educational strategies to m eet the needs of this new national student body, which is distinctly heterogeneous compared to the re latively homogeneous learners of 20 years earlier. This call for diversified pedagogy is not just a suggestion, it is an urgent cry. In 1994, the Goals 2000: Educate America Act was first enacted with $105 million assigned to improve educational outcomes on eight national goals, one of which was to increase the high school gra duation rate to 90% across th e country’s population. While overall growth has been seen through gr aduation rates in th e total public school population rising to 73.9% in the 2002-03 school year ( CCD, 01-02, 02-03), students from minority backgrounds have not had the same positive story. In 2001, the average graduation rate for African American stude nts was only 56% and for Hispanic students merely 52% (the Manhattan Institute, 2007). Graduation rates acro ss the national student

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34 population continue to be dismal, and the st ory may currently even be worse than the numbers show because of the lack of sta ndardization across districts and states in collecting high school graduati on and drop-out rates. At the same time, schools are being held accountable for the reading and mathematics outcomes of students with disabil ities more so than in previous years. Considering the wide variety of disabilities and the multiple ways these disabilities can impact learning for students, this current pr actice may seem unfair to both educators and students with disabilities alike (Gagnon & Maccini, 2001; NCTM, 2000; NRP, 2000). However, many of these learners are in the general education classroom for the majority of their instructional time, learning the same ideas and concepts as their general education peers. In fact, according to data from the 2002-03 school year, there are approximately six million students with disabilities being served in public schools, with over three million of these same students integrated in to the regular education classroom for over 60% of their academic instructional time ( 26th Annual Report to Congress 2004). Many disability advocates welcome an emphasis on accountability for students with disabilities. For years, students with disabilities were warehoused within special classrooms in schools, where educational em phases were placed on behavioral targets while students atrophied academically. St udents were not exposed to content rich environments that led to positive adult out comes from their educational experiences (Thompson, Johnstone, Thurlow, & Altman, 2005; Wagner, Newman, & Cameto, 2004). Consistent exposure to the general education curriculum is changing this situation for learners with disabilities, and opening up opportunities for instru ction that could be targeted to meet all students’ learning n eeds within the general education classroom

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35 (Herman, 2007; Kozik, 2007). To this end, the DAL framework has been constructed, not only incorporating instruc tional practices that span both reading and mathematics education, but also enveloping best practices for teaching students with diverse learning needs, including mild disabilities. While the primary focus of inquiry in this study is the DAL’s possible effects when used within a de velopmental social cons tructivist approach to pre-service preparation for special educators, the DAL’s influence on these students’ achievement is also of high interest. Professional Development Increasing student outcomes is one of the primary reasons why professional development for teachers at the pre-service a nd in-service levels is receiving increased attention nationally. This development ha s come under increased scrutiny since an influential report by the United States De partment of Education (USDOE) (2002) indicated that college programs targeting the pr eparation of future teachers have little to no impact on future educators’ readiness a nd performance in their first classrooms. While this information caused pre-service teacher development to withstand closer scrutiny, it opened the door for further re search, since the USDOE based its findings primarily only on the data provided from Title II reports of higher education colleges and universities. In the same year, Dar ling-Hammond and Young (2002) reviewed the empirical research base more extensively su rrounding teacher prepar ation and discovered indications contrary to that of the USDOE study. These rese archers found evidence that the impact of teacher preparation depends on several factors including duration, target skills, and professional e xperiences of the prepara tion (Darling-Hammond & Young, 2002). Programs that were structured to cultivate well-prepared educators “to teach

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36 subject matter, develop curriculum, a nd handle classroom management” involved university experiences that included integrat ed programs that focused on instructional techniques, as well as subject matter to be taught (Boe, 2007, p. 159). Teacher candidates with programs that involved “extensive prep aration in pedagogy and practice teaching obtained a much higher level of full certification” than others entering the teaching field (Boe, 2007, p. 168). Darling-Hammond, Holtzma n, Gatlin, and Heilig (2005) also found that teachers who were certified produced st udent achievement results that were higher than their uncertified edu cation peers. In a study c onducted by Sands, Duffield, and Parson (2007), findings showed that when t eacher candidates’ progress was not closely monitored by staff and targeted feedback wa s not given on progress, teacher candidates’ learning outcomes varied by individual student. In better understanding the role of preservice professional development on future special education teachers, the work of Darling-Hammond (2000) sheds light on the fundamental elements of constructing prepar ation programs that re sult in well-prepared teacher graduates. Darling-Hammond review ed the professional development literature over the last 30 years and determined that even with teacher preparation, as imperfect as it may be, it had resulted in “fully prepared and certified teachers” that were “generally better rated and more successful with students than teachers without this preparation” (2000, p. 167). She also indicated that conten t area instruction had been influenced by teacher preparation “in fields ranging from mathematics and science to vocational education, reading, elementary education, and early childhood education”, and she asserted that “teachers who have greater knowledge of teaching and learning are more

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37 highly rated and are more effective with stude nts, especially at tasks requiring higher order thinking and problem solvi ng” (Darling-Hammond, 2000, p. 167). When investigating the weaknesses in pr e-service teacher preparation programs, Darling Hammond (2000) cited several key issues, including th e problematic time limitations on four-year preparation; th e fracture between content knowledge and instructional coursework; th e separation between college coursework knowledge and school-based application of this knowledge; th e deficiency of systematic instructional methods taught to and employed with teacher candi dates in a clinical setting; and the lack of overall resources provided to teacher pr eparation programs through their colleges of education. Instead of just indicating problems, Darling-Hammond (2000) provided solutions that are viable to colleges and uni versities. These ideas focus primarily on internal teacher candidate ch ange through the developmental social construction of new knowledge based in the establishment of understandings and practice in education. Darling-Hammond (2000) pulls on the wo rk from Dewey (1929), which calls on institutions of higher educa tion to “empower teachers with greater understanding of complex situations rather than to control them with simplis tic formulas or cookie-cutter routines for teaching” (p. 170). Dewey’s work calls for inquiry-base d learning in teacher preparation, where teachers cu ltivate their instructional de cision-making based on their own knowledge and application of teaching practices (1929). Darling-Hammond takes the work of Dewey a few steps forward, advoc ating teacher inquiry that targets student learning outcomes. Through this reflection on practice, she assert s that teachers will better understand individual learning differenc es, develop instructi on that reaches all learners, and view knowledge from the multiple perspectives that learners bring to

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38 today’s classrooms. As Darling-Hammond (20 00) indicated, teacher change through preservice professional preparation is a developm ental constructivist endeavor that must be targeted at the areas of teacher self-efficac y, attitude, and content knowledge, as well as pedagogical knowledge and practice. Self-Efficacy Within the context of pre-service teac her preparation programs, professional development experiences can occur within coursework, fieldwork, and supervised teaching experiences. Therefore, these areas can be targeted for the development of relevant competencies. Some aspects of t eacher preparation are w ithin the control of teacher educators, while others are not. For example, future teachers bring with them attitudes, experiences, and beha viors that stem from their life experiences and that might negatively impact their development as eff ective teachers. Such aspects cannot be controlled but some can be reasonable targ ets for change (Enochs, Smith, & Huinker, 2000). One of these domains is teacher candida te feelings of self-efficacy about teaching (i.e., instructional self-efficacy). Instructional self-efficacy includes the fee lings and beliefs that teachers have about their abilities to teach and provide info rmation to students that enhance student learning of core skills and abilities surrounding a particul ar subject (Allinder, 1995; Enochs, Smith, & Huinker, 2000). Heightened levels of teacher self-efficacy have been linked to increased student learning outcomes In a study conducted by Czerniak (1990), teachers with higher self-efficacy were more likely to employ instructional techniques that were varied and met the learning needs of their students to a greater extent compared

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39 to teachers with lower feelings of self -efficacy, who tended to struggle more with presenting content area material and ha ving students retain that material. According to Bandura, Caprara, Barbarane lli, Gerbino, and Pastorelli (2003), selfefficacy plays a crucial role in many lif e functions, not just instructionally: Perceived self-efficacy plays a pivotal role in this process of self-management because it affects actions not only dir ectly but also through its impact on cognitive, motivational, decisional, and affective determinants. Beliefs of personal efficacy influence what self-regul ative standards people adopt, whether they think in an enabling or debilitating manner, how mu ch effort they invest in selected endeavors, how they persevere in the face of difficulties, how resilient they are to adversity, how vulnerable they are to stress and depression, and what types of choices they make at important decisional points that set the course of life paths. (p. 769). Hagerty (1997) maintains that self-efficacy is so integral to teaching and learning that it is “the key element in student achie vement of individual classroom tasks and mastery of subject matter in all disciplines ” (p. 1). Bruning, Shraw, and Ronning (1999) characterized teachers with hi gh levels of self-efficacy as better serving the diverse learning needs in classrooms. They describe d these teachers as better at structuring learning time that was adequate for all lear ners, more consistently employing effective behavior management strategies, and more of ten using praise for student efforts. Attitude Another construct that relates to teachers ’ instructional effectiveness is attitude towards instruction. Attitudinal consideratio ns are especially important for prospective special education teachers. Many times teacher s who instruct learners with disabilities face additional challenges compared to regula r education teachers, because these students require increased depth of instruction and gr eater teacher understanding about the impact instructional practices have on the stud ents’ learning needs (Maccini & Gagnon, 2006;

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40 Witzel, Smith, & Brownell, 2001). These ch ildren also typically need repeated exposures, a structured l earning environment, and more time for understanding and processing key ideas (Allsopp, Kyger, & L ovin, 2006). Therefore, individuals who choose to work with these types of lear ners need to possess a positive and committed attitude towards these students with excepti onalities in light of the many instructional challenges involved. Many times these students ha ve also been made to feel unwanted or inferior in the general education classroom and special education teachers have the increased task of dealing with significant be havior and self-esteem issues in addition to instructional/learning needs (Montague, 1997). A positive teacher attitude towards the content area of instruction can result in equally positive student feelings about themselves in connection with that specific content area, as well as stave off teacher burnout (Mercer & Mercer 2005; Singh & Stoloff, 2007) According to White, Way, Perry, and Sout hwell (2005/2006), researchers typically agree that: students enter teacher education pr ograms with pre-existing beliefs based on their experience of school; these beliefs are robust a nd resistant to change; these beliefs act as filters to new knowledge, accepting what is compatible with current beliefs; and beliefs exist in a tacit or implicit form and are difficult to articulate (p. 36). As a result, these researchers assert “that negative beliefs may contribute to negative classroom teaching strategies, which may in turn contribute to negative pupil beliefs,

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41 attitudes and performance outcomes. If thes e pupils then go on to become teachers, a cycle of negativity may be created” (White et al., 2005/2006, p. 36). Pre-service preparation programs aim to target and change attitudes that might interfere with instructional success, before these non-productive attit udes have the chance to impact students’ classroom achievement Beswick (2006) has identified several features of professional development programs in mathematics education that have been found to affect change on attitudes about both instruction and th e content area of mathematics. These features include: “havi ng pre-service teachers actually engage in doing mathematics, increasing awareness of a nd encouraging reflection on the students’ own beliefs, encouraging reflection on th eir own practice teaching, the use of collaborative group work, and providing altern ative models for mathematics teaching” (Beswick, 2006, p. 37). For pre-service sp ecial education professional development programs to successfully target and change negative attitudes towards mathematics instruction, an awarenes s of prevalently held attitudes towards mathematics instruction must be cultivated, as well as knowledge of research-based effort s to enhance positive attitudes towards this subject ’s instruction accumulated. Content Knowledge The development of teacher subject area content knowledge is a third important component of teacher preparation. The ma thematics content knowledge level of many special education teachers is pa rticularly lacking, especially at the elementary level. According to Matthews and Seam an (2007) teachers at the elementary level often have multiple gaps in content knowledge, including the mathematics subject area. These gaps can have considerable impact on student performance when these teachers cannot

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42 accurately convey important concepts to students in their classrooms (Matthews & Seaman, 2007). In several studies completed by researchers over the last two decades, it has been shown that most mathematics teac hers at the elementary level possess only procedural knowledge (i.e., computation) for such concepts as fractions, decimals, and integers, rather than the conceptual k nowledge that underlie s these algorithmic procedures (Adams, 1998; Fuller, 1996; St acey, Helme et al., 2001; Zazkis & Campbell, 1996). Interestingly, this real ity appears to be more pronounced for teachers in the United States. For example, Ma (1999) found that when American teachers were compared with Chinese educators responsible for teaching the same content with story problems, American educators were 60-80% less likely to have the necessary understanding of these concepts than the Chinese teachers. Elementary level special education teachers often possess less mathematics content knowledge than the general elementary level teacher, because of their programs’ emphases on learning strategies and behavi or management techniques over content knowledge (Mercer & Mercer, 2005). A key probl em in effectively preparing teachers in mathematics content knowledge is the lack of time devoted to coursework in mathematics and mathematics education (McGowen & Da vis, 2002). Indeed, McGowen and Davis (2002) suggest that educators must keep in mind “what is theoretically desirable for a content area course for pre-service teachers versus what might be practically obtainable in one or two semesters” (p. 1). Teacher pr eparation in mathematics must simply require more than the typical one or two courses in mathematics education because content knowledge in mathematics takes a great deal of time to experience and cultivate (McGowen & Davis, 2002).

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43 To give a broader view of the breadth of this content area problem, DarlingHammond (1997) presents disparities in levels of teacher preparation when looking at teacher’s content knowledge from a state to stat e perspective. Some states, typically in the Southern and Western United States, ha d more than 50% of their mathematics teachers without even a minor in mathematic s, while other states, typically in the Northeast, had a mere 15% of their mathem atics educators without this credential. Strutchens, Lubienski, McGraw, and Westbr ook (2004) discovered that this statistic varied across student ethnicit y as well. They found that amongst eighth grade students, 80% of White students had teachers whos e certification included secondary level mathematics, while only 72% of Black and Hisp anic students had teachers with the same level of preparation. On the positive side, teachers who do currently complete preparation in mathematics are being exposed to mathematic s content knowledge at levels not seen in earlier times. Hill and Lubienski (2007) discuss how teachers are now being provided with two types of mathematics content know ledge: common and specialized. Common knowledge refers to the basic mathematics con cepts and ways to compute answers, while specialized content knowledge incorporates multiple ways to represent and solve mathematics problems using both manipulatives and other representations (Hill & Lubienski, 2007). With respect to the algebraic thinking area specifically, teachers typically enter professional development programs holding und erstandings of algebra skills taken from high school algebra courses. However, teach ers are better prepared to convey algebraic learning to their students as ab stract representations used in problem solving if their

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44 professional development facilitates the more global skills involve d in algebra than simply the variables and equations taught in Algebra I (Stump & Bishop, 2002). When teachers understand that algebra encompasse s patterns, relations, and functions, along with representing and solving mathematical e quations, as well as analyzing change in different situations, they are equipped w ith the content knowledge needed by their students for more complete algebr aic understanding (NCTM, 2000). Pedagogical Knowledge Pre-service special education profe ssional development in mathematics instruction certainly should emphasize more than content knowledge. It must also emphasize the development of teachers’ knowledge and application of best instructional practices. This pedagogical knowledge is multifaceted in the case of algebraic learning for struggling students. These learners us ually respond best to instructional methods similar to ones used with students diagnosed with learning disabi lities (Garcia, 2002; McKenna & Robinson, 2005; Mercer & Mercer, 2005). While not all students who are at-risk will eventually be identified with a disability, many will. Those students, who are at-risk for other reasons, incl uding English language learni ng and socioeconomic risk factors, can also benefit from instruction ta rgeting individuals with learning disabilities because this pedagogical approach differentia tes instruction for individualized learning needs (Garcia, 2002). Thus, instructing struggling students in algebraic thinking necessitates knowledge about general methods for teaching students with disabilities, knowledge about general effective mathema tics instructional methods, and knowledge about effective algebraic instructional methods (Jamar & Morrow, 1990; Maccini, McNaughton, & Ruhl, 2000; Mercer & Mercer, 2005; Swanson, 2001).

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45 When addressing the learning needs of stude nts with disabilities, there are a few core pedagogical principles to keep in mind. First, students with learning disabilities, typically have difficulties in one or more academic areas. These areas differ by student, and there is no one template for disabili ty manifestation. Difficulties can include problems with cognitive processing, metacogniti on, attention difficulties, and perceptual problems (Mercer & Mercer, 2005). As a resu lt, instruction for these learners must incorporate many different ways of a ssisting these students in accessing and understanding curriculum. Am ongst the strategies advocated for usage with struggling learners are visual organizer s, hands-on and varied mate rials, explicit instruction, modeling, scaffolding, mnemonic devices, multi ple exposures to concepts, and strategy instruction (Fuchs & Fuchs, 1998; Merc er & Mercer, 2005; Schumm & Vaughn, 1991; Swanson, 2001). Visual organizers present information using multiple modalities. They not only incorporate oral information, but present conc epts in a visual format. These organizers do not just provide the informa tion for students to see, but al so use forms of diagramming and illustration to make conn ections between target concep ts and previous learning, as well as target concepts and applications in every day life (Meichenbaum, 1977; Swanson, 2001). Using varied hands-on and manipula tive materials is another instructional strategy that is imperative with learners who have difficulty with academic tasks. Using materials that are high-interest and tangibl e helps enhance student learning by engaging the students in the learning task, and making new ideas less abstract and more concrete for student comprehension (M ercer & Mercer, 2005).

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46 Explicit instruction, modeling, and scaffold ing are all instructi onal techniques that deal with how teachers construct their pres entation of material for struggling students (Kameenui, Jitendra, & Darch, 1995; Palinesar 1986). Many students in the general education classroom are asked to pick up on instruction implicitly through classroom interactions and activities. For students w ith additional learning needs, this form of instruction is not always successful because many students need specific concepts taught directly to them through exp licit instruction. As a result employing explicit instruction using a direct teacher explanation of learni ng concepts, does not leav e students who have academic difficulties guessing at target lear ning ideas (Mercer & Mercer, 2005; Swanson, 2001). Modeling results in even more powerfu l student outcomes when connected with explicit instruction, be cause teachers not only explain to students exactly how to break down and access learning targets, but they physically show students how to complete academic tasks. Many students benefit from this visual demonstration, where teachers can use talk-alouds, showing not only the skill but explaining their thought process when working on that particular skill. After expl icit instruction and modeling is completed by the teacher, many struggling st udents still need scaffold ing, where independence in learning tasks is facilitated by gradually lessening levels of teacher support (Ellis & Lenz, 1996; Lenz, Ellis, & Scanlon, 1996). At-risk students are often not successful if they are allowed to simply take on academic task s on their own after just teacher-directed instruction. These situations of tentimes result in student fail ure, which can cause learners to doubt their abilities as stude nts and lead to a cycle of le arned helplessness, where these learners give up on academic tasks before ha rdly trying them (Mercer & Mercer, 2005). Using the three-part instructional pro cess of explicit inst ruction, modeling, and

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47 scaffolding can increase students’ knowledge acq uisition and their abi lities to retain and implement their new learning skills. A great number of students with learning disabilities have difficulty processing and retaining concepts. Their teachers must us e a variety of instruc tional strategies to facilitate this processing and retention. One such method is the mnemonic device. According to Nagel, Schumaker, and Deshle r (2003), this instruc tional strategy helps students access and remember learned inform ation by tying the concept to a mental image, keyword, or first-letter mnemonic device, which reduces strain on memory faculties. Another method that assists learners’ cont ent retention is multiple exposures to learning material. Many times struggling stud ents cannot process and retain information that is only presented to them once or twice. These learners need a variety of activities with a particular learning c oncept, so that they become comfortable with the academic content presented and cannot only recite learned information but be able to flexibly apply and use that new knowledge (Scruggs & Mast ropieri, 1994). A last method that can assist struggling students with learning new material is strategy instruction (Borkowski, Weyhing, & Carr, 1988; Graham & Harris, 1996). Learners in the general education classroom are expected to possess many internal metacognitive strategies that assist them in monitoring their own abilities on specifi c learning tasks. However, many struggling learners have not developed these mechan isms and do not possess these abilities. Teaching these learners stra tegies for monitoring their own thinking during learning tasks, as well as teaching them strategies for best acquiring information, has a significant positive impact on these students’ abilities to comprehend new information (Borkowski, Estrada, Milstead, & Hale, 1989; Levin, 1996; Miller & Seier, 1994; Swanson, 2001).

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48 There is a large array of instructional strategies that can be employed with struggling students in a general sense; but at the same time, these strategies can also be employed in conjunction with content specific techniques fo r these learners. According to Miller and Mercer (1997) learners w ho struggle with mathematics often have difficulties attending to details of algorithms; visual-spatial concerns that impact numerical operations; auditory processing defici ts that negate abilities to follow multiple part mathematics directions or sequences; and memory concerns that can impede the retention of mathematics ba sic facts and more complicat ed algorithms. Mathematics instruction for struggling learne rs, like instruction in genera l for these students, must be multi-faceted in nature. While the aforementi oned instructional practices can facilitate the enhancement of learning outcomes in general by targeting students’ learning characteristics, there are al so instructional practices sp ecifically advocated for the mathematics content area for these student s (Allsopp, Kyger, & L ovin, 2006; Mercer & Mercer, 2005). Strategies for mathematics in struction that have a strong research base include teaching big math ideas, using peer-a ssisted instruction, implementing a concreterepresentational-abstract (CRA) sequence of instruction, employing authentic contexts, facilitating structured lan guage experiences, and conducting continuous monitoring of student progress (Baker, Gersten, & Lee, 2002; Kronsbergen & Van Luit, 2002; Mercer & Mercer, 2005; Miller Butler, & Lee, 1998). First, teaching big math ideas is important for learners at-risk for failure because they oftentimes need to see the greater mathematics picture to understand how new mathematics learning targets fit into a larger scheme of ideas (Mer cer & Mercer, 2005). This presentation format helps students ma ke sense of overarching concepts, thereby

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49 assisting students to see the bigger picture, how concepts ar e connected, before having to worry about grasping the smaller nuances of instruction (Allsopp, Kyger, & Ingram, n.d.). Peer-assisted instruction has also shown it self to be helpful to students at-risk for mathematics difficulties (Allsopp, 1997; Fant uzzo, Davis, & Ginsburg, 1995; Fuchs, Fuchs, Phillips, Hamlett, & Karns, 1995; Kr oesbergen & Van Luit, 2002). This strategy has struggling students work with other learners in their clas srooms to facilitate learning mathematics-related ideas. This instruc tional practice has individuals who understand key mathematics instruction work together wi th others who may struggle with the same material. Both students gain from this gr oup work because the student who has mastered the concepts is able to in ternally reinforce that know ledge through explanation and example, while at the same time the student w ho struggles with the content is able to gain clarification, practice, and a comfortable wo rk environment for gaining new mathematics skills (Baker, Gersten, & Lee, 2002; Kroesb ergen & Van Luit, 2002; Mercer, Miller, & Jordan, 1996). The CRA sequence of instruction builds o ff the general strategy of using hands-on materials, but it takes this instruction to a ne w level, offering a gr aduated progression of skills from concrete to representational to abstract (Allsopp, Kyger, & Lovin, 2007; Baker, Gersten, & Lee, 2002; Mercer & Me rcer, 2005; Miller & Mercer, 1997; Witzel, Mercer, & Miller, 2003). In the CRA sequence, concepts are first pr esented with tangible concrete materials. Once problem-solving is ma stered at this level, the student is exposed to the same mathematics ideas at the re presentational level through pictures and drawings. After students have gained th e ability to solve problems with these representations, the targeted learning concept is finally taught using abstract numbers and

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50 symbols. Employing this method, students are moved towards understanding a given concept using an incremental systematic process (Kroesberg en & Van Luit, 2002; Mercer & Mercer, 2005). Using authentic contexts is another wa y to facilitate cr eative problem-solving abilities in struggling students (Allsopp, Kyge r, & Lovin, 2006; Bottge, Heinrichs, Chan, & Serlin, 2001; Kroesbergen & Van Luit, 2002). With this type of mathematics learning, students are presented with learning situati ons which have meaningful problems for student solutions. This strategy facilitates l earners’ engagement in mathematics tasks, as well as assists students in understanding the pa rticular situations that call for certain forms of application-based problem-solving (Mercer & Mercer, 2005). Another strategy, structured language experiences, opens up an element of mathematics understanding that is currently advocated by NCTM (2000), but is difficult for struggli ng learners to access (Montague, 1997). Mathematics standards are cu rrently structured so that students are asked to not only understand how to solve sp ecific mathematics problems and compute answers accurately, but how to explain and justify their prob lem-solving process. Some students naturally pick up these skills fr om mathematics dialogues that happen in the classroom; however, students at-risk for math ematics failure rarely do (Allsopp, Kyger, & Ingram, n.d.; Allsopp, Kyger, & Lovin, 2007). Providing structured language experiences allows these stude nts opportunities to practice wr iting and talking about new concepts in specific ways with teacher guida nce. Students are ther efore given support in developing these mathematics’ communi cation abilities (M ontague, 1997). A final strategy for increasing mathema tics learning outcomes is continuous student progress monitoring (Calhoon & Fuch s, 2003; Collins, Carnine, & Gersten, 1987;

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51 Kline, Schumaker, & Deshler, 1991; Port er & Brophy, 1988; Stecker, Fuchs, & Fuchs, 2005). In mathematics, this progress mon itoring is exceptionally important, because skills are oftentimes cumulative in nature, with early learning building towards later, more complex concepts. Teachers must track struggling students’ progress through concepts carefully, so that gaps or partic ular areas of struggle are pinpointed early on with learning new concepts. Through c ontinuous progress-mon itoring, teachers can observe where individuals are succeeding and where they need additional help, and as a result they can target instruction to the sp ecific needs and areas of each learner (Allsopp, Kyger, & Lovin, 2006). While many of the above general math ematics strategies can be used for instruction in algebra, Gagnon and Maccini (2001) have identified several areas of difficulty for students with learning and behavi oral problems in mathematics specific to algebraic learning, which can be targeted fo r enhanced student outcomes in algebraic thinking: Difficulty in processing information whic h results in problems learning to read and problem-solve Difficulty with distinguishing the rele vant information in story problems Low motivation, self-esteem, or self-efficacy to learn due to repeated academic failure Problems with higher level mathematics that require reas oning and problemsolving skills Passive learners – reluctant to try new academic tasks or sustain attention to task Difficulty with self-monitoring or se lf-regulation during problem-solving

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52 Difficulty with arithmetic, computational deficits (p. 8) Witzel, Smith and Brownell (2001) also advocat e three principles for teaching algebra to students with disabilities, including: 1. Teach through stories that connect ma th instruction to students’ lives. 2. Prepare students for more difficult ma th concepts by making sure students have the necessary prerequisite knowl edge for learning a new math strategy. 3. Explicitly instruct stud ents in specific skills using think aloud techniques when modeling (p. 102). At the same time Gersten and Chard (1999) advocate a progressive approach to teaching number and number sense, which are the building blocks of algebraic instruction. They emphasize a constructiv ist approach that helps students with disabilities “(a) learn the conventions, la nguage, and logic of a discipline such as mathematics from adults with expertise; and (b) actively construct meaning out of mathematical problems (i.e., try a variety of strategies to solve a problem).” Earlier research also can guide edu cators regarding effective alge bra instruction for struggling learners. Case and Harris (1988) worked with students with lear ning disabilities whose problem-solving abilities benefited from “s elf instruction”, where students helped themselves in problem-solving by drawing pi ctures as part of their problem-solving methodology. At the same time, work by Bennett (1982) showed the benefit of using graphic organizers when inst ructing students on basic alge braic thinking information. Montague and Bos (1986) illustrated the benefits of strategy instruction for learners with disabilities in algebra when they used strategy instruction for multi-step algebraic problems. Students who had been taught specif ic strategies to solve problems in this

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53 situation fared better than those individuals who had not been taught such skills. As can be seen, the instructional base of strategies fo r struggling learners in algebra skills is still developing, pulling from general pedagogy targ eted to learners with disabilities, mathematics pedagogy for struggling learners and algebraic specific instruction for struggling students. Important factors surroundi ng professional development, including self-efficacy, attitude, content knowledge, and pedagogical knowledge and application specific to mathematics and learners who are at risk ha ve been explored. An examination of the content and instructional pract ices involved in the DAL inst ructional framework, and its corresponding contextual library, the ALL, will be described in the next section. In this way, a better understanding of th e possible utility of this fr amework within a pre-service special education teacher preparat ion program may be gained. The DAL Framework Algebra Background Some educators may view arithmetic skills as the keys to mathematics success, but in the 21st century, students must possess much mo re than basic skills. Students must be able to think and reason ma thematically. A core curricul um strand for developing this mathematical thinking is algebra. Algebra is a critical area that sp ans all domains of the NCTM (2000) standards and includes an inte rrelated maze of “algebras” which include algebra, algebraic thinking, algebraic reasoning, and algebr aic insight. Having a firm grasp of this algebra-related terminology helps not only individuals using the DAL framework for instruction, but also any teacher who wants to help her students grasp algebraic concepts. As Kaput and Blan ton mention, educators who have a strong

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54 foundation in the algebra curriculum st rand can actively work on “algebrafying” curriculum for enhanced mathematics learni ng for all their students (2000, p. 2). To provide clarity to alge braic vocabulary, the terms al gebra, algebraic reasoning, algebraic thinking, and algebraic insight are all centered on the same core ideals, but each encompasses definitively different aspect s of developing students’ mathematical reasoning. To start, when most people speak about “algebra”, they are talking about the high school coursework at the Algebra I and II levels, which are usually taken in eighth or ninth grade and tenth or el eventh grade, respectively (Gom ez, 2000). In this case, the word “algebra” refers to the curriculum taught in these two secondary classes, encompassing increasingly complex manipul ations of unknowns and variables using symbols and equality signs across contexts (Gagnon & Maccini, 2001). However, there are times when people generically use the term “algebra” to refer to a circumstance when someone solves a problem using an unknown or variable quantity (Bass, 1999). This second situation leads to a muddyi ng of the waters with defini tions. In this second sense, it would be more reasonable to say the pers on is utilizing “algebrai c thinking” to solve the problem. This situation is more aptly described as “algebraic thinking” because it uses students’ higher order thinking abilities to make models and represent problems with unknown amounts, rather than simply focusing on solving equations for specific variables (Austin & Thompson, 1997; NCTM, 2000). In ma ny cases, “algebraic thinking” will be done by students much younger than eighth or ni nth grade, who have not fully developed an understanding of the concept of “variable. ” Many times with “algebraic thinking”, the foundational ideas of equation construction a nd solution identification are initiated and practiced for later exposures w ith Algebra I content.

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55 This skill set incorporated under “algeb raic thinking” is typically thought to develop from a base of competencies in arith metic processes that are cultivated in the early elementary school levels and involve num erical computations where the entities in the problem-solving process are known (A ustin & Thompson, 1997; Gersten & Chard, 1999). “Algebraic thinking” can evolve from arithmetic abilities because it is also a method of problem-solving, except with a more complex approach than with arithmetic skills alone. As “algebraic th inking” is learned, a student’s critical thinking and problem approach skills change from selecting com putational processes for achieving answers to understanding and analyzing currently known data to determine missing outcome information (Ortiz, 2003; Radford, 2000; Ur quhart, 2000; Zazkis, 2002). After some time and exposure to “algebraic thinking” based problems, “algebraic reasoning” may subsequently develop. While “algebraic thin king” is a way of approaching a problem, “algebraic reasoning” is the ab ility of students to take this learned approach and generalize it to new and sometimes more co mplex situations and problems (Lubinski & Otto, 2002). When teachers see students a pproaching novel mathematics problems, and finding methods and strategies to answer these unknown questions without prompting, they can ascertain these learners have intern alized the concepts of “algebraic thinking” for application as their own problem-solving tool through “algebraic reasoning” (Morris & Sloutsky, 1995; National Center for Improving Learning, 2003). This ability to think and use the tool s of “algebraic thinking” readily for “algebraic reasoning” is vital not only for the success of students in Algebra I and II courses, but for many types of everyday problems that involve unknown entities and require critical thinking to me diate and plan solution paths. In fact, Pierce and Stacey

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56 (2007) take algebraic ideas one step further with their vision of what they call “algebraic insight”, which they depict as having two centr al components. They assert that “first, it [algebraic insight] involves thinking carefully about the properties of the symbols being used and the structure and key features of each algebraic expression…secondly, algebraic insight involves thinking about the possi ble links between algebraic symbols and alternative representations” (Pierce & Stacey, 2007, p. 3). This idea of “algebraic insight” evolves as students progress from si mply reasoning and thi nking algebraically to the point of comprehending and utilizing the ab stract symbol system involved in formal secondary algebra and beyond. Algebraic Literacy Now that the terminology surrounding algebr a, as well as its importance has been clarified, a new term “algebraic literacy,” a key component of the DAL intervention, will be introduced and operationalized. For the purpo se of this study, “algebraic literacy” is defined as a student’s accurate and consistent ability to use language to describe algebraic concepts; employ materials to illustrate conc epts; utilize graphic organizers to show connections between target con cepts and other learni ng; provide rationales to solve issues surrounding concepts; and use problem-solving a nd computation to answer questions on concepts. With the addition of “algebraic li teracy” to the algebraic knowledge base, the goal is to give the algebra curriculum area a developmental context, which was heretofore not included. “Algebraic literacy” seeks to combine the underlying core skills that are desired for competency by th e high school level, with an understanding that these algebraic skills will progress in degrees of abstraction and complexity as students’ progress in their mathematical education. As a result, like literacy in reading, “algebraic

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57 literacy” should be cultivated from the earliest years in school so that by the secondary level that literacy is at an advanced level. Role of Literacy in the DAL Framework As mentioned earlier, parallels can be observed between the content areas of reading and mathematics, specifically algebra. One such parallel disc ussed previously is the connection between the building blocks of reading (i.e., phonemic and phonological awareness) and the building blocks of al gebra (i.e., number and number sense). An instructional emphasis on these “building blocks” can help young learners develop understandings about reading and about al gebra respectively. The DAL framework places an emphasis on developi ng number and number sense. The DAL framework also incorporates several effective instruc tional practices that are advocated in reading/literacy. Interestingly, most if not a ll of these instructional practices are also advocated in the mathematics education literat ure as well, albeit ap plied for the purpose of learning mathematics. The purpose is not to explicitly teach reading strategies per say within the DAL framework but to use literature and certain reading in structional practices to engage learners in problem-solving, making connections, and facilitating student retention of ideas and informa tion. Literacy instructional pr actices used within the DAL framework are included in Appendix A. The first literacy instructional practice incorporated in the DAL framework is promoting learner engagement using text. For learners, to be more interested, and thus more receptive towards instruction, research has found that educational attention needs to be focused on initial instructi onal activities that promote ideas that are relevant and meaningful to young children (Jamar & Mo rrow, 1990; Von Drasek, 2006; White, 1997).

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58 With the teaching of reading, teachers never hesitate to pull out a colorful and exciting children’s book to incite this engagement for reading ta sks (Gipe, 2006; Richards & Gipe, 2006). In mathematics, reading one equation after another in a mathematics textbook or looking at groups of sticks, blocks and shapes simply does not qualify as a high interest activity for grabbing most lear ners’ attention, neithe r do these activities promote concentration on learning tasks relate d to algebra. Thus, the DAL instructional framework employs children’s literature to incorporate what Von Drasek (2006) calls the “wow factor”, where students’ attention is captivated for algebraic learning through the usage of colorful children’s books. In order to integrate learner e ngagement using text into the DAL framework, Caldecott Award winning books were selected. Specific Caldecott texts were chosen based on severa l criteria, and 20 selected books make up the initial Algebraic Literacy Library (ALL). Th e DAL framework incorporates the texts of the ALL to stimulate learner interest in probl em-solving situations based on the stories’ contexts. The ALL consists of 20 books selected from Caldecott Award and Honor books from the 2000-2007 timeframe. These Caldecott books were specifically chosen for the ALL for two reasons. First, Caldecott books di ffer from other stories in terms of their connectedness between visuals and storyline. Each book’s illustrations have to be integral in depicting and de veloping the storyline of the book at hand. A criterion for Caldecott Award winners is that they are dis tinguished from other books with pictures in that the illustrations essentially provide the child with a visual e xperience of the story (ALSC, n.d.). Second, because of the wide ly recognized importance of the Caldecott

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59 Award, using these books in the library helps st udents become literate in the stories and tales that embody American culture. From the original 33 Caldecott books fr om the 2000-2007 time period, literature was eliminated from the final library if the book had an absence of print; a revised version of a time-old fairy tale that was believ ed too familiar to be engaging; or a set of non-continuous poetry that did no t lend itself to a complete storyline. Since many of the target students for the ALL are from mu ltiple cultural backgrounds and disability categorizations, particular attent ion was paid to selecting books from this time period that did express ideas and information that were culturally relevant or representative of cultural differences and disabi lities. Student engagement has a two-fold purpose in the final 20 Caldecott books: 1) gaining students’ interest through r eading, and 2) accessing contexts that are ripe for algebraic probl em-solving. A complete listing of the ALL books is included in Appendix B, with a sa mple book guide that was provided for each ALL book for teacher candidates’ instructional usage. A second literacy instructional practi ce employed within the DAL framework, which also has shown results in mathematic s instruction, is making connections between previous learning and new concepts curre ntly being taught (G ersten & Chard, 1999; Gipe, 2006; NRP, 2003). Reading, as an academic area, is typically seen as cumulative in nature, with one core component buildi ng off of the next, with phonological awareness growing from phonemic awareness and implic it comprehension developing from explicit comprehension abilities, as just a couple of examples (NRP, 2003). For each reading ability listed in a scope and sequence chart of skills, a learner grasps concepts more clearly if he or she can relate that particular skill to its place in the spectrum of total

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60 reading skills he or she has already lear ned (Mercer & Mercer, 2005; NRP, 2003). The same is true of mathematics, topics of earlier instruction are springboards for more complex mathematical learning (Allsopp, Kyge r, & Ingram, n.d.; Lee, et al., 2004). By combining instruction in the DAL that employs literature with algebraic skills, teachers can spread a wider net to not only catch those students who can connect algebraic ideas to previous mathematics learning, but also t hose individuals who can be engaged in mathematics through their love and u nderstanding of reading concepts. A third literacy instructional practice infused within the DAL through the ALL is the idea of grasping the figurative “big pi cture” (Richards & Gipe 1996). Many learners, who struggle with both reading and mathematics, benefit from an instructional situation where the main goal is to see the larger con cepts within the scope of the lesson (Maccini, McNaughton, & Ruhl, 1999). While this strategy lends itself well to illustrated children’s literature, reading with any t ype of engaging children’s litera ture can stimulate children’s thinking about the larger issues or themes pr esented in the tale, rather than reflecting on the basic component parts of reading, such as word recogniti on, vocabulary, and story construction (Ouzts, 1996). At the same ti me, learners who struggle with mathematics often need the same format for beginning c ontent presentation, to be introduced to new concepts more holistically or as larger mathematical chunks (Allsopp, Kyger, & Ingram, n.d.). Building off the larger ideas and them es gained through reading the award winning children’s literature, the DAL introduces the “big ideas” of algebraic literacy connected to the NCTM (2000) Algebra strands. A fourth literacy instructional practice employed is active questioning while reading, which can be used in pre-readi ng, during reading, and pos t-reading activities

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61 while using the ALL. If a learner simply picks up a book and begins reading it without preamble, vital reservoirs of a reader’s potential interaction with the text are not accessed (Raphael & Pearson, 1985). Befo re a student begins reading, it is important the stage be set for the particular book by stimulating a learner’s knowledge on the topic at hand. While reading, an individual also needs to have specific questions that he or she wants to answer by reading the text. Af ter reading, it is essential th e student ponders which of his or her questions was actually answered. If a student approaches a re ading task in this active way, he or she will understand and ga in much more content from the book read (Blackowicz & Fisher, 2006; Me rcer & Mercer, 2005). With the DAL’s incorporation of the active questioning strategy, th e end goal is to create soli d comprehension of the ALL story contexts for problem-solving. Student in terest is gained th rough questions in the area for problem-solving; and as a result, students have incr eased clarity on the particulars of solving the specific problems tie d to the learning context. In this way, student problem-solving is enriched, b ecause a key barrier to problem-solving, understanding the problem situation, has been broken down. A final literacy instructional practice us ed in the DAL is providing structured language experiences. McKenna and Robi nson (1990) advocate su ch experiences by asserting that “to be literate in, say, mathematics is not to kn ow mathematics per se but to be able to read and write a bout the subject as effective m eans of knowing still more about it” (McKenna & Robinson, 1990, p. 168). In this way, the basic languag e arts skills of reading and writing are presented as the chie f instruments of developing literacy in specific content areas such as mathematics, not just as tools simply linked to the learning of English course materials. Moreover, in the words of Vogt and Shearer “clarity in

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62 stating problems, use of concre te examples, analysis of abst ract concepts, and application of concepts to next contexts”, illust rates a clear connec tion between reading communication capabilities and their possibl e application to the complexities of mathematics (2007, p. 137). Vogt and Shearer expound on not the skills of reading itself, but the desired outcomes of the reading task for the application purposes of understanding and then communication. This id ea is particularly relevant because it is nearly identical language skills that are valued in the algebra area specifically (Allsopp, Kyger, & Lovin, 2006; Steele, 1999). Before their successful completion of secondary level Algebra coursework, students are required to state algebraic problems in their own words; utilize and understand materials and problems on a continuum of levels from concrete to abstract; and fi nally generalize learned skills to every day situations for utilization (Steele 2005; W itzel, Mercer, & Miller, 2003) Structured language experiences cultivated within the DAL framework provide opportunities for increasing deftness at communicating mathematically re levant ideas for algebraic understanding, affording much needed practice on these sk ills before the secondary level (Allsopp, Kyger, & Ingram, n.d.). Mathematics Practices w ithin the DAL Framework The DAL framework’s target student popul ation is struggling learners, who are having difficulties in mathematics. Therefore, the DAL employs mathematics instructional techniques target ed to learners with individual and complex learning needs, which are similar to the instructional met hods used with students who have been diagnosed with a mild learning disability. These mathematics instru ctional practices are

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63 integrated with the literacy instructiona l practices already described. Mathematics instructional practices used within the DAL framework are included in Appendix C. The first mathematics instructional practice employed in the DAL, ConcreteRepresentational-Abstract (CRA), is at the center of the DAL’s instructional activities. It has been found that learners who struggle with mathematics benefit from exposure to and work on new concepts along a continuum of incremental levels which progresses from actual tactile manipulatives, to pictures or representations, to abstract symbols (Gagnon & Maccini, 2001; Witzel, Mercer, & Miller, 200 3). With the use of the CRA continuum, it is important to note that le arners may progress at varying rates through the levels of materials depending on their rate of understa nding, requiring more time with concrete objects with one particular concept for mast ery while sailing through all three levels to abstraction for another concept’s full gras p (Cai, 1998; Maccini, McNaughton, & Ruhl, 1999; Mercer & Mercer, 2005). Within the DAL model, the CRA sequence of instruction is used in all of the framework’s steps to facilita te in depth comprehension of algebraic concepts. A second core mathematics instructional pr actice involves authentic contexts for problem-solving. When students learn mathema tics, or any academic subject for that matter, this learning is facilitated when cen tered on a situation w ith which students can draw connections (Jamar & Morrow, 1990). Wh en children are presen ted problems, it is much easier for them to grasp the reason for the issue or difficulty at hand when the problem has circumstances that develop purpos eful associations between the child and the problem. Not only does this context ease students’ understa nding of novel math problems, but it also stimulates students’ motivation in solving the actual problems

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64 because they are interested in the answer s and outcomes of the problems (Kortering, deBettencourt, & Braziel, 2005). If students s ee reasons for solving the problems and are interested in them, their involvement with the problems will heighten their responsiveness to learning problem approach es and methodologies (Allsopp, Kyger, & Lovin, 2006). For the purpose of the DAL fr amework and this study, texts from the ALL were used to provide authentic contexts for algebraic problem-solving in all three steps of the DAL model. A third mathematics instructional practice used in the DAL is explicit instruction, along with teacher modeling for problem-solv ing. Oftentimes, students who struggle with mathematics require very detailed expl anations of how to solve novel types of problems, and it is difficult for them to attempt new problems based on a few written guidelines (Witzel, Mercer, & Miller, 2003). While these students benefit from written descriptions and visual demons trations of problems, they also gain tremendously when teachers “walk through” sample problems of the type to be solved shortly by students. This modeling is particularly effective when the teacher utilizes “talk-alouds” to explain his or her thinking, as he or she systemati cally shows the execution of problem-solving steps (Maccini, McNaughton, & Ruhl 1999). In truth, some mathematics teachers themselves may be against the use of solely explicit instruction fo r algebraic learning because of their belief that this instructi onal format does not allow students to attempt strategies experimentally on their own for problem-solving (Witzell, Smith, & Brownell, 2001). This firmly held belief is the reason that while modeling and explicit instruction are instructional practices that can be utili zed in the DAL model’s third step for teaching new skills, their usage is recommended in conj unction with other instructional practices

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65 that promote risk-taking and e xperimentation. In combination with these other strategies, students are provided a supporte d learning environment that promotes access to multiple mathematics concepts and processes. A fourth effective mathematics instructional practice integrated in the DAL framework is that of scaffolding instru ction, which is a structured pedagogical methodology that moves students to greater independence with problem-solving in incremental steps (Mercer & Mercer, 2005). This graduated progres sion of comfort and competency in mathematics skills helps lear ners with mathematics difficulties progress mathematically from A to B to C rather than be expected to zoom from A to Z without support. Furthermore, within the framew ork of scaffolding, students’ toolboxes of mathematics abilities can be enhanced with work on metacognitive strategies (Maccini, McNaughton, & Ruhl 1999). These strategies involve each student thinking about the information in a problem and understanding how his or her own thought processes work and can be employed in solving this problem. By cultivating this ability, the child is increasing his or her ability to answer novel problems correctly, because he or she is better equipped to monitor cognition about mathematics problems and how to find solutions to them (Gagnon & Maccini, 2001). Many students with di sabilities and other diverse learning needs benefit from having me tacognitive strategies modeled and their use scaffolded for them before they are ab le to incorporate them independently in problem-solving (Witzel, Smith, & Brownell, 2001). The DAL model uses scaffolded instruction explicitly in its third step to he lp build student abilities and independence in problem-solving.

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66 A fifth effective mathematics instructiona l practice incorporated within the DAL framework is the usage of visual organi zers (Baker, Gersten, & Lee, 2002; Swanson, 2001). Visual organizers include Venn Diagrams, flow charts, outlines, webs, classification trees, and sketches among ot hers. Through the use of such tools, connections between previous mathematics l earning and current learning targets can be drawn (Mercer & Mercer, 2005). Following inst ruction, such organizers can be used again to draw ties between what students have learned algebraically and applications in their everyday life. The success of these orga nizers is directly tied to the instructional ideals associated with strugg ling learners. First, these learners benefit from being exposed to instruction that uses multiple moda lities, visual being one of these modalities. Second, these students also benefit from inst ruction where information and connections are explicit, and are not left for student s to just discern through problem-solving (Kroesbergen & Van Luit, 2003). Visual organi zers arrange information in a systematic way that specifically helps learners pro cess concepts and see pathways through this clarity of presentati on (Allsopp, Kyger, & Lovin, 2006). W ithin the DAL model, visual organizers are employed in the third step to illustrate connections between new learning objectives and previously learned ones, as well as connections be tween new learning and future learning areas. The sixth effective mathematics instructional practice that the DAL framework incorporates is providing multiple opportunities for practice of algebraic and other mathematical concepts. Many times students appear to grasp mathematical concepts when these learning points are teacher-directed in class. Additionally, students can also seem to grasp concepts directly after they have been taught the new ideas and have

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67 practiced one or two problems (Lee, Ng, Ng, & Zee-Ying, 2004). However, it is imperative that students are given many opportunities to apply newly developing mathematical understandings so that they beco me proficient with th em and are able to use them efficiently, as well as retain th em for the future (Allsopp, Kyger, & Ingram, n.d.; Mercer & Mercer, 2005; Witzel, Smith, & Brownell, 2001). Multiple opportunities for practice are incorporated throughout all th ree steps of the DAL framework. The final effective mathematics instructional practice implemented within the DAL framework is continuous student progr ess monitoring, which is employed within the DAL framework as the basis of inst ructional decision-making for each session (Allsopp, Kyger, & Lovin, 2007; Allsopp, Kyger, & Lovin, 2006). During each student session using the DAL framework, student perf ormance data are collected on the fluency and accuracy of problem-solving through the first step, Building Automaticity. During the second step, Measuring Progr ess, student information is also collected in terms of learners’ abilities to read, solve, answer, and justify problem solutions to algebraic problems. Using these two forms of data from a session, teacher candidates make instructional decisions for their next inst ructional session (Kroes bergen & Van Luit, 2002). Student information that shows l earner comprehension of concepts and independence of skill application will indicat e to teacher candidates to move students ahead in the algebraic concepts to be taught. Student information that indicates learner inability to grasp concepts and/or difficulty applying these skills will be used as the basis for slowing down instructional presentation of material and revisi ting currently taught concepts.

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68 The Framework’s Development While the DAL framework is a relati vely new model, it had been under development by a group of researchers from special education, ma thematics education, and measurement for two years prior to th e current study. In its first year of development, the DAL framework’s three core steps were solidified: Building Automaticity, Measuring Progress and Making D ecisions, and Problem Solving the New. Building Automaticity was established as the first step in the framework as a mechanism for students to revisit key concepts and sk ills that had been taught, and work towards proficiency in those areas. With the second step of Measuring Progress and Making Decisions, teachers were afforded a mean s of presenting multiple opportunities to evaluate students’ use of the problem-solving pr ocess: read, represen t, solve, and justify, and as a result discern learners’ levels of algebraic concept unde rstanding via CRA. Finally, Problem Solving the New allowed teach ers the time and structures within each instructional session to present new alge braic ideas to students, focusing in on connection-making, communication, and inte gration of differe nt problem-solving strategies. A visual conceptu alization of the DAL model is included in Appendix D. After over a year of development, the DAL framework was piloted with a group of students in a Title I school’s summer progr am. With this group of learners, the DAL was first implemented with students of mi xed elementary school levels, ranging from second through fifth grades. From this appl ication, changes were made in several key components of the DAL. One such element was the DAL’s skills assessment and scoring rubric. This evaluation was formulated on th e basis of the four skill areas surrounding algebraic thinking advocated by NCTM (2000). After field-testing, additional items and

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69 question types were added to this initial asse ssment to ensure the qua lity and quantity of questions employed to pinpoint target skills for the DAL’s application with students. Upon field-testing, other changes were made in the DAL to facilitate ease of instructor usage, as well as implement st ructures better refined to meet student learning needs. Based on these changes, a finalized version of the DAL Initial Session Probe, included in Appendix F, and the DAL full session fr amework, included in Appendix G were developed. As the result of this previous research, the cu rrent study, while exploratory in nature because of its involvement of teacher candidates for the first time, has already incorporated a firm situation on instructional strategies groun ded in current literature and practice, as well as refinement and revision as a result of its application with elementary level learners. Through this review, the rich literature base for the current study, involving the DAL model’s application with pre-service special education teacher candidates, has been highlighted. The professional developmen t literature advocates application-based undergraduate teacher preparati on programs that integrate co ursework with structured and supported field work experiences that ta rget teacher efficacy, attitude, and content knowledge, in conjunction with instructi onal knowledge and application. The mathematics and reading strategies unfold as in tegral tools in assis ting struggling learners to better access the higher order concepts in mathematics, specifically targeting a new form of literacy, in this case algebraic lite racy. In Chapter 3, the methodology of how the current study explores the DAL’s implementa tion with undergraduate special educators is presented.

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70 Chapter 3 Methodology Introduction This study, which used a mixed methods design, had the purpose of evaluating experiences of pre-service sp ecial education teachers when implementing a mathematics instructional framework for struggling learne rs (DAL) during an early clinical field experience, and determining how that fr amework and the support provided through a developmental social constructivist approach to teacher preparation may influence future teacher’s professional development in several important areas. The setting of this study was a multi-campus, research university in the Southeastern United States and a Title I school site within a large, nei ghboring urban school district. At this particular university, the College of Education had undergone r ecertification by the National Council for Accreditation of Teacher Edu cation (NCATE) in 2005. At the same time, the College of Education was ranked within the top 50 universi ties in the country for teacher preparation in 2007 (US New and World Report, 2007). As a result, the conceptual framework of the university’s College of Education has been centered on the improvement of teacher preparation. To this end, this study was firm ly aligned with the uni versity’s College of Education’s role in developing exemplary pe dagogical practice in higher education for the professional training of future classroom teachers. Additionally, in the curre nt political climate of accountability set by No Child Left Behind ( NCLB 2001) and the latest reauthorization of IDEA (2004), more than ever

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71 colleges and universities are working towards the construction of education programs that are grounded in research-based pedagogy situated within specific content areas, such as reading and mathematics. In light of this emphasis and the “highly qualified” teacher mandate set forth by the above-mentioned legi slation, this study was timely in that it addressed the important integration of research -based instruction within a critical content area, mathematics, for the purpose of improvi ng the preparation of special education preservice teachers. As illustra ted in Figure 1, this research project’s primary goal was to investigate the experiences of pre-service special education teach ers when implementing the Developing Algebraic Literacy (DAL) inst ructional framework fo r struggling learners within a highly structured early clinical field experience incorporating elements of a developmental social cons tructivist approach (Darli ng-Hammond, 2000) to teacher education. Outcomes measured included self -efficacy in teaching mathematics, attitudes toward teaching mathematics, knowledge of mathematics content, and understanding and application of research-based mathematics in structional practices for struggling learners, as shown in Figure 1.

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72 Figure 1. Major inquiry areas. ________________________________________________________________________ Special Education Teacher Candidate Professional Development in Mathematics Instruction Affective Changes in Teacher Candidates’ Ideas about Mathematics Instruction Mathematics Content Knowledge Changes in Teacher Candidates Instructional Changes in Teacher Candidates’ Ideas about Mathematics Instruction Attitude Knowledge Application SelfEfficac y

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73 Overarching Research Question The following research question was addressed through the current study: What changes related to effective ma thematics instruction for struggling elementary learners, if any, o ccur in teacher candidates during implementation of the DAL instructional framework in an early clinical field experience practicum for pre-se rvice special education professional preparation? Major Inquiry Areas within the Research Question The following inquiry areas broke the resear ch question down in to investigational components that were explored using both qua ntitative and qualitat ive research tools: 1.) What changes, if any, occur in spec ial education teach er candidates’ feelings of self-efficacy about teachi ng mathematics from the beginning to the end of a pre-service instructional experience using the DAL framework? 2.) What changes, if any, occur in spec ial education teacher candidates' attitudes towards mathematics instruction from the beginning to the end of a pre-service instructi onal experience using the DAL framework? 3.) What changes, if any, occur in special education teacher candidates' understanding of instructiona l strategies for str uggling learners in mathematics from the beginnin g to the end of a preservice instructional experience using the DAL framework? 4.) What changes, if any, occur in special education t eacher candidates’ application of instructiona l strategies for struggling learners in mathematics from the beginning to the end of a preservice instructiona l experience using

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74 the DAL framework? 5.) What changes, if any, occur in sp ecial education teacher candidates’ content knowledge of elementary mathematics, including algebrai c thinking, from the beginning to the end of a pre-service instructiona l experience using the DAL framework? Participants This mixed methods study employed a convenient sampling technique by seeking participation from undergraduate teacher candi dates enrolled in th e Level II practicum within the researcher’s Depart ment of Special Education. Participants came from the Level II cohort, who began their enrollment in the Department of Special Education in the fall of 2007 and are expected to complete their profession al preparation in the spring of 2009. Before participating in the study, all cohort members completed their Level I coursework and practicum, which included a fo undational course in special education, a foundational course in mental retardation, a pe rspectives course on learning and behavior disorders, as well as a two-day weekly pr acticum connected with the two foundational courses. During the current study, Level II undergraduate teacher candidates participated in the following coursework linked to the Level II practicum: Clinical Teaching in Special Education ( 3 credits ) Within this course, the focus involved “e ffective teaching principles, instructional management procedures, and specialized teaching techniques for exceptional students” (Department of Special Education, 2007). Behavior Management for Special Needs and at Risk Students ( 3 credits )

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75 The core competencies within this class were “techniques to prevent, analyze, and manage challenging and disruptive classroom behavior as well as teaching social skills” (Department of Special Education, 2007). Both courses were linked to the Leve l II practicum through Key Assessments, which are departmental gate-keeping measur es. These assessments evaluate teacher candidates’ progress in developing inst ructional/behavior management skills, professional dispositions, and field cont ent knowledge through interactions and experiences with elementary level students. Through these key assessments students are required to demonstrate learned instructiona l skills, to synthesize information from various sources for the purpose of making inst ructional decisions, a nd to reflect on their professional practices. In actuality, passing the two Key As sessments in the Level II practicum is integral to teacher candidates pr oceeding to the final tw o semesters of their special education program. I ndividual students who do not achieve the pre-determined competency criteria are required to repeat the Level II coursework and practicum before they can continue with thei r program of study. As a result, the Level II student population was targeted because the Level II se mester is considered a critical one in the development of pedagogical and content ar ea knowledge for these future special educators. The overall focus of the Level II Practicum is to provide teacher candidates with a variety of field experiences that assist them in understanding how to implement individualized instruct ional practices related to academic and behavior outcomes. During the semester of this study, teacher candidates participated in a clin ical practicum at one school site on Mondays where they engaged in one-to-one academic instruction with

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76 students struggling in reading and mathematic s. Teacher candidates also completed a service-learning project as pa rt of the Monday field experi ence. On Tuesdays, teacher candidates were assigned to individual elemen tary classroom sites at schools in the local school district. For this part of the Level II practicum, teacher candidates completed a behavior change project with a selected student in their particular classroom placement and assisted their supervising teacher thr oughout the day with inst ructional activities, classroom management, materials development, and other classroom and student needs. Teacher candidates participated in pract icum throughout the full teacher work day on Mondays and Tuesdays (7:30am 3:30pm). Th erefore, teacher candidates worked with a variety of elementary level st udents in public school setti ngs in one-on-one, small group, and whole class situations. This study wa s carried out during the Monday portion of the Level II practicum. The Monday public school setting was a large, urban school district in the Southeastern United States with a diverse student body in terms of cultural, economic, and disability characteristics. This particular semester the anchor site for the Monday “clinical instruction” portion of the Level II practicum was a Title I school where 97% of students were of minority background, 95% of students were on “free and reduced lunch”, almost 10% of students were Englis h language learners (ELLs), and 24% were students with disabilities (Hillsborough C ounty Public Schools, 2007). Each teacher candidate engaged in individua lized reading and mathematic s instruction on Mondays at this school site. Teacher candidates were initially assigned two reading students and two mathematics students for individualized inst ruction, and they con tinued to work with

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77 these students throughout the semester unless their students withdrew from the school site. Engagement in reading preparation a nd instruction began at the beginning of the semester for teacher candidates using the Univ ersity of Florida Lite racy Initiative (UFLI) instructional framework, while mathematics preparation and instruction using the DAL framework began several weeks later. For the teacher candidates’ preparation for DAL instruction, the initial training workshop and ongoing support mechanisms were structured using a similar format to that of the UFLI. This parallel form of preparation and support was followed because of UFLI’s usage along developmental social constructivist lines within the Department of Special Education’s Level II coursework and fieldwork experiences fo r at least three years. The DAL’s usage within the practicum included a similar training and support sequence to the UFLI, employing the same developmental constructivist principles of meaning making through scaffolded and supported learning experiences. The initial DAL intensive training works hop included an entire teacher-length day of presentations, discussions, and handson activities for learning the essential components of the algebra standard advocat ed by NCTM; understanding research-based instructional strategies fo r struggling learners; and co mprehending the key steps and features of the DAL framework. For several weeks before teacher candidates began their own implementation of DAL instruction, ongoi ng follow-up seminars were provided for the last hour and a half of their Monday practicum day on DAL related training. Additionally, the researcher wa s available to teacher candi dates for discussion, support, and questions all day every Monday during th e training with and implementation of the DAL framework. These elements of in tensive training workshop, active teacher

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78 candidate involvement in the learning proce ss, application of inst ructional framework, and university support during implementation we re identical to that employed with the UFLI reading framework. Elementary level students who worked with teacher candidates for 35-45 minute sessions weekly using the DAL framewor k were identified by their school’s administration and teaching staff based on the criteria of being at-risk for failure in mathematics. “At-risk for failure in math ematics” was defined as having consistently received poor grades in mathematics course s or having scored a failing, or passing score of the lowest level, on the most recent state-mandated standardized mathematics assessment. Due to the particular anchor school site’s 90% y early transition and relocation rate for students, at least two elementary students were selected to receive instruction from each teacher candidate to ensu re that throughout the entirety of the DAL model’s application each teacher candidate w ould most likely have at least one student instructional session per week. After teacher candidates began DAL in struction with these students, the researcher, as well as two university profe ssors and three doctoral students who had attended DAL training, provided ongoing support to teacher candidates through observations with feedback, debriefing sessi ons, discussions, and probing questions. The researcher used a developmental social c onstructivist approach in structuring the supported DAL instructional experience, al lowing teacher candidates to implement instruction; reflect, evaluate and plan future instructi onal sessions based on learning experiences; collaborate with school personnel, other teacher candidates, and university support staff to make sense of instructional knowledge and applicatio n; and question their

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79 understandings and experiences within the DAL instructional experien ce. The researcher was available to students w ithin their Monday practicum experience, as well as through visiting the teacher candidates’ Clinical Teaching course for additional support and questions. Selection of Participants During the study, there were originally 28 teacher candidates enrolled in the Level II practicum and coursework experience. Fr om these 28 individuals, teacher candidates’ participation was requested by the researcher within their Level II practicum and connected coursework. Out of these 28 teacher candidates, 27 agreed to participate and signed Institutional Review Board (IRB) appr oved informed consent forms. From the study’s original 27 participants, five teacher candidates withdrew from or discontinued participation in the special e ducation teacher preparation pr ogram during the semester, so were not included in the study’ s final participant group. Besi des these five individuals, three other teacher candidates were excluded from the final participant group. One of these students exhibited extensive absen ces, and the other two teacher candidates experienced significant health issues over th e course of the semester, being unable to complete course and practicum work along the same timeline as other participants. These three participants were all excluded from the study’s final participant group because it was thought that their experien ce in the Level II cohort cour sework and practicum did not represent that of the typical pr e-service special edu cation teacher. As a result, the study’s final participant group containe d a total of 19 individuals. In an effort to not unduly burden teacher candidates’ workload, the resear cher did not require the participants to

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80 complete any projects or surveys that were not already considered a part of their requirements for Level II coursework. From this base group of teacher candidate participants, three were chosen to have their DAL comprehensive experience and pr oject performance evaluated as individual case studies. Selection criteria for case studi es were determined by several factors. At the conclusion of the practicum, the two prof essors who were involved with teaching the teacher candidates’ two courses and practicum were asked to indi vidually rank teacher candidate participants as fa lling into one of three categor ies: top performing third, middle performing third, and bottom performi ng third. These rankings were based on the teacher candidates’ achievement on course-relat ed tests, assignments, and projects, as well as practicum feedback from their supe rvising teachers and observations made by their university supervis ors. These professors then cam e together with their individual rankings to reach agreement on which stude nts should be included in each grouping. Two possible case study participan ts were then chosen randoml y from each of these three groupings, with one targeted for case study part icipation and the other as a backup in case of difficulties with the first person’s particip ation. Case study participants were chosen based on this three-tiered ranking of perf ormance so as to evaluate the possible differences in teacher candida te experiences within the structured and supported DAL instructional framework in rela tion to their achievement within the full scope of their preservice program. Case study analysis by abil ity level was deemed especially important for informing future pre-service special education teacher preparation programs’ development to meet the learning needs of a greater variety of future teachers, by providing information and understanding of teac her candidate experiences from a variety

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81 of ability levels within an applicationbased, developmental social constructivist instructional framework. Ethical Considerations Before beginning the study, the current investigation was examined by the Institutional Review Board (IRB) of the resear cher’s university to en sure that adequate preparation for the safety and confidentia lity of all teacher candidates had been completed. After the study was approved by the IRB, the researcher requested participation of all the eligible Level II te acher candidates, and obtained consent from all individuals willing to participate in the st udy. All Level II teacher candidates had the ability to choose not to participate in the study without penalty, academically or professionally. Teacher candidate s who agreed to participate in the study did not receive any academic or personal benefits for their agreement to participate. At the same time, all Level II teacher candidate s, study participants and non-participants, completed the same assessments and assignments. Quantitative Instruments The study utilized a mixed methods design, implementing both quantitative and qualitative assessment measures to ascertain triangulation of data for reliability and validity purposes. For the quantitative porti on of this researc h, several types of instruments were used. First, multiple survey s gathered information pertinent to efficacy, attitude, and content knowledge from teacher candidates. The first of these surveys was a self-efficacy mathematics instruction measure. For this purpose, Enochs, Smith, and Huinker’s 21-question, Likert scale, Mathem atics Teaching Efficacy Beliefs Instrument (MTEBI) (2000) was employed to collect pr e-, midpoint, and pos t-test efficacy

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82 information from the teacher candidates, in cluded in Appendix H. The survey did not contain sample items, but before its admini stration the researcher clearly explained the questionnaire’s purpose and di rections for completion. The MTEBI was chosen as the instrument to assess efficacy in this research because it is a comprehensive assessment tool for pre-service teacher self evaluation of efficacy in mathematics instruction. It is cons tructed with Likert scale items that gather information on two types of teaching effi cacy, Personal Mathematics Teaching Efficacy (PMTE), measured by 13 survey items, and Mathematics Teaching Outcome Expectancy (MTOE), measured by 8 survey items. Pe rsonal Mathematics Teaching Efficacy (PMTE) relates to the teacher candida tes’ perceptions of their own self-efficacy in teaching mathematics, and Mathematics Teaching Outcome Expectancy (MTOE) relates to teacher candidates’ expected student outcomes ba sed on their instruction (Enochs, Smith, & Huinker, 2000, p. 194). Moreover, this measur e was also selected because of its high reliability, with an alpha coefficient of .88 for the PMTE subsection and .75 for the MTOE subsection. These alpha coefficients in dicate high internal consistency reliability for survey questions in measuring the effi cacy constructs they aim to evaluate. Additionally, the researcher gene rated alpha coefficients for this instrument based on the study population’s responses. This information is included in Table 1. This instrument was presented to teacher candidates at three poi nts in this investigation to evaluate the changes in their perceived self-efficacy in mathematics instruction abilities over the course of the study.

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83 Table 1 Reliability Information for the Mathematics Efficacy Beli efs Instrument (Cronbach’s alpha) To gain insight into how teacher candidates’ attitudes towards mathematics instruction changed through an experience with the DAL framework, the Preservice Teachers’ Mathematical Beliefs Survey by Seaman, Szydlik, Szydlik, and Beam (2005) was implemented and is included in Appendix I. This second instrument uses items that assess if individuals view mathem atics as “creative and original” or if they perceive it as having a “rule bound and law governed natur e” (Seaman et al., 2005, p. 199). The items probe the teacher candidates’ views about the mathematics content area in general and mathematics instruction specifically. The overall Preservice Teachers’ Mathematical Beliefs Survey is constructed from 20 Li kert scale items, which have the goal of obtaining attitudinal information towards teach ing mathematics to students of varying ability levels. A Rasch analysis was used by th is survey’s authors to determine that this instrument has a person separation reliability between .70 to .84 across items, and an item separation reliability of .98 across the four major attitudinal domains accessed through the study (Seaman et al., 2005, p. 201), indicating th e survey has relatively consistent and reliable student responses across survey items and items themselves are extremely Pre Mid Post Mathematics Teaching Efficacy Beliefs Whole Instrument .80 .82 .80 Self-Efficacy Subtest .83 .86 .84 Outcome Expectancy Subtest .71 .84 .83

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84 consistent as a whole in assessing teache r candidates’ attitudes towards teaching mathematics. As with the efficacy survey, th e researcher generated alpha coefficients for this instrument based on the study population’ s responses. This information is included in Table 2. As with the MTEBI, the Preser vice Teachers’ Mathematical Beliefs survey was administered at three poi nts during the research to ga ther pre-, midpoint, and posttest information from teacher candidates. Table 2 Reliability Information for the Mathematical Beliefs Instrument (Cronbach’s alpha) ________________________________________________________________________ Pre Mid Post ________________________________________________________________________ ________________________________________________________________________ According to studies completed by Adam s (1998) and Stacey, Helme, Steinle, Baturo, Irwin, and Bana (2001), an overwh elming percentage of elementary school teachers are deficient in their basic mathematics skills. However, one essential characteristic mandated by federal legisla tion for “highly qualified” teachers across subject areas is that educator s possess proficiency in the co ntent knowledge of the subject Mathematical Beliefs Questionnaire .83 .90 .90 Constructivist Mathematics Beliefs Questions .69 .85 .90 Traditional Mathematics Beliefs Questions .72 .62 .74 Constructivist Teaching Mathematics Beliefs Questions .67 .89 .69 Traditional Teaching Mathematics Beliefs Questions .56 .80 .68

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85 area in which they plan on teaching. In this same vein, special educators in elementary schools are now expected to possess the sa me amount and degree of content knowledge as their general education teaching peers. As a result, a 20-item instrument by Matthews and Seaman (2007) called the Mathemati cal Content Knowledge for Elementary Teachers was administered to al l special education teacher candidate participants as this study’s third survey, included in Appendix J. Th is particular survey was selected because while the DAL framework focuses on algebraic thinking at the elementary level, it was deemed important that teacher candidates’ ov erall content knowledge be evaluated for the elementary level, as abilities in basic numb er and number sense from the arithmetic skill strand are the foundational competencies for learning algebraic thinking. The Mathematical Content Knowledge a ssessment uses a combination of openended response and multiple choice items to determine the current elementary level mathematical content proficiency of the i ndividuals taking the assessment. While the Mathematical Content Knowledge survey was originally tested by its authors using a population of elementary school teachers, it was also deemed appropriate for special education teachers at the same level, becau se like general education elementary level teachers, special education teachers are t ypically prepared as generalists, who are expected to teach a broad array of content areas. The survey developers’ Cronbach’s alpha for this instrument was calculated to be .80, indicating that the test has a high internal consistency reliability in co llecting content knowledge in elementary mathematics across items. The researcher al so generated alpha coefficients for this instrument based on the study population’s re sponses, providing additional reliability on researcher-devised subtests of basic arithme tic and algebraic thinking. This information

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86 is included in Table 3. As with the other two aforementioned surv eys administered to teacher candidates, this content knowledge instrument was administered at pre-, midpoint, and post-test points. In total, teacher candidates were administered three survey instruments in regards to mathematics instruction: self-efficacy, attitude, and content knowledge respectively. Table 3 Reliability Information for the Conten t Knowledge Instrument (Cronbach’s alpha) ________________________________________________________________________ Pre Mid Post ________________________________________________________________________ Content Knowledge Instrument .74 .79 .84 Basic Arithmetic Questions .54 .67 .71 Algebraic Thinking Questions .58 .62 .69 ________________________________________________________________________ Besides the three survey instruments in the current study, another important aspect of the investigation involved measures that evaluated the amount of mathematics instructional knowledge retained and applied by teacher candidates. This facet of DAL model training and implementation by participan ts was assessed in two ways. First, an exam administered within the Clinical Teachi ng course by the course instructor was used to measure the amount of information retained about effective math ematics instructional practices for struggling learners. Since DAL instruction was imbedded within the Clinical Teaching course via the model’ s workshops and on-going trainings through practicum and course activitie s, several Clinical Teaching test questions focused on the mathematics instruction conten t taught in connection with the DAL intervention, with the test included in Appendix K. During the par ticular semester under research, the Clinical

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87 Teaching course had two foci for instruction, the teaching of reading during the first part of the semester and the teach ing of mathematics during the latter, of which the DAL model was an essential aspect. As a result, the teacher candidates were evaluated on their retention of information provided on the in struction of mathematics and algebraic thinking skills as part of the course exam in the second half of their Clinical Teaching class. Since DAL instruction included training in best instructional practices for struggling learners in mathematics gene rally, and algebraic thinking instruction specifically, student answers on all Clinic al Teaching test questions relating to mathematics instruction for struggling learne rs were used as measures of teacher candidates’ retention of peda gogical knowledge for mathematic s instructional practice. To ensure content validity on the clinical t eaching exam, the professor of the course, in conjunction with the researcher, designed the final exam questions based on the teacher candidates’ experiences with mathematics in struction via the DAL framework in both the Clinical Teaching class and adjoining practicum To this end, the course professor had written the mathematics instruction textbook us ed in the Clinical Teaching course, and had previously worked with the current resear cher as part of the research development team in designing the DAL framework. Thus, the content of the final exam was based on both the course text and DAL framework, wh ich overlapped in their description and usage of many instructional practices for learners at-risk for mathematics difficulties. The second way data collection occurred in the instructional knowledge area was through observation of teacher ca ndidates’ abilities to apply their knowledge of effective mathematics instruction for struggling learners, following their training guidelines for

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88 DAL implementation. This application was measured through DAL model observation fidelity checklists. These checklists were completed on teacher candidates during three different instructional sessions. Two types of fidelity checklists were developed and are included in Appendix L and Appendix M, resp ectively. The first checklist was for the DAL framework’s initial session probe, wh ich included fewer steps for implementation than a regular DAL session, because initial sess ions only included one section of steps: Measuring Progress and Making Decisions. The second checklist was for a typical DAL session, which included all sections and steps: Building Automaticity, Measuring Progress and Making Decisions, and Problem Solv ing the New. To evaluate the abilities of the teacher candidates to apply the DAL consistently along DAL training framework guidelines, three independent raters obs erved the teacher candidates’ one-on-one instruction with students. Ratings were us ed to assess both the accuracy of specific teacher candidates’ implementation of effec tive instructional pract ices and the teacher candidates’ implementation as a whole group. These ratings were also employed to measure how consistently teacher candida tes implemented effective instructional practices across observations. Each rater used the same fidelity checklist, and all three raters observed instructional sessions together until 90% agreement was reached between raters on steps within specific observations. After this percentage was reached, raters divided observations into three groups, with each group being relatively equal. Each group consisted of approximately four teacher candidates and was a manageable sample for observation by each of the three raters. Each teacher candidate in the obs ervation group was observed at pre, mid, and post points in the DAL framework’s implemen tation, unless teacher candidate or student

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89 absence prevented the observation from occurring. In this way, the researcher triangulated quantitative data be tween teacher candidate survey s, test question responses, and observation fidelity checklists to mo re fully probe the teacher candidates’ experiences implementing the DAL model within a pre-service special education teacher preparation program. Qualitative Instruments As part of this mixed methods study, quali tative research elements were used in tandem with quantitative means, allowing for data collection that provided rich description. The overlap in data collection between the quantitative and qualitative methods was purposeful and had the aim of providing in depth information on the multiple aspects involved in special educat ion teacher candidates’ preparation as professional educators. Analysis of Final Pape rs on the DAL Experience As part of their experience with the DAL model in their Level II practicum, all teacher candidates completed a final paper on their instructional involvement and learning through the application of the DAL model. For all study participants, this single document underwent an independent documen t hand review by the researcher as the study’s first means of qualitativ e data collection. This fi nal paper required students to reflect on their learning thr oughout the 10 week duration of the framework’s usage, as well as reflect on any personal and professiona l changes that had occurred throughout the DAL training and application. The researcher evaluated all participants’ final papers looking for themes, ideas, and changes that had developed through the course of the teacher candidates’ progression with the DAL mo del, as well as the commonality of these

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90 items across teacher candidates’ papers. This review probed the large ideas and themes that emerged from the full group of participants versus specific individual experiences. Pre and Post Focus Groups The second qualitative tool was the empl oyment of focus groups with the teacher candidate participants in th e Level II practicum cohort. The purpose of the focus groups was to obtain a shared or group perspec tive on teacher candida tes’ ideas about mathematics instruction within a format that did not have predetermi ned response items. In this way, the open-ended nature of th e focus group conversation allowed for the collection of clarifications on teacher candidates’ ideas about mathematics instruction that were not necessarily accessible through survey responses. There were two focus groups of approximately 9-10 people, each conducted by the researcher, who is trained in focus group methodology. The specific size of the focus groups was chosen for two reasons. First, guidelines for focus group composition recommend between 6-12 participants for these gro ups (Morgan, 1988). Second, the current participant group consisted of 19 teacher candidates, and in terms of time constraints within teacher candidates’ practicum day, it was thought mo st reasonable to conduct two focus groups of approximately 35-45 minutes each at bot h pre and post points in the study. The researcher used the same 15 foundation questi ons in the focus groups at both pre and post, given in Appendix N, as the basis for accessing teacher candidate self-efficacy, attitude, content knowledge, and instructional knowledge and application information in regards to mathematics instruction. All fo cus groups were audiotaped for accuracy of information, as well as tracked through notes taken by an assisting doctoral student in

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91 special education. Frequent member checks we re also completed to ensure that teacher candidate responses accurately reflec ted candidate thoughts and ideas. Case Studies The third qualitative technique involved the implementation of case studies. To this end, three teacher candidates, from the group who volunteered to participat e in the study, were chosen to have their DAL mode l experience analyzed in a specific and comprehensive manner by the researcher. As mentioned previously, the two professors who taught the Level II cohort their courses and supervised their practicum ranked all study participants as in the highest performi ng third, the middle performing third, or the lowest performing third of the cohort for the current semester. Based on these rankings, the researcher randomly selected two case study participants from each grouping, with one being approached for participation and one being used as an alternate if the first person was not willing or available to be a case study participant. For the purpose of the case study analysis, three specific DAL fr amework elements were evaluated. First, for the duration of the DAL mo del instructional experience, teacher candidates made and kept “session notes” which served as their planning and instructional logs of information for thei r instructional periods with students. Additionally, teacher candidates reflected week ly on their instructional experience using the DAL model, focusing their response s around prompts involving how they were implementing the model, what they were lear ning from their experiences, and how they might use this learning in the future. In the end, teacher candidates produced a final paper that synthesized their experience, includi ng personal and professional growth areas. While all teacher candidates pr oduced these three forms of doc uments as part of their

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92 participation in the Level II practicum and c oursework, for the three teacher candidates involved in the case study component of th e research, the researcher used these documents as one piece of obtaining a more co mplete picture of three individual teacher candidate learning situations w ithin the entirety of the total sample of participants. Second, to obtain more specific reflections and experiences of the three case study participants, an individual exit intervie w was conducted with each case study participant at the end of the study. Third, the individual results for each case study participant on the three administered surveys at all points, as well as on the course examination, were extracted from the total participant group. Th ese individual results were then evaluated in isolation with comparison made to the la rger group. While information gained through the case studies was not genera lizable to other members of the study, it facilitated the exploration and understanding of the learning process that teacher candidates’ undergo when experiencing professional developmen t that integrates a research-based instructional framework within a particular content area, such as the implementation of the DAL model. Procedures Because the study employed both qual itative and quantitative research methodologies, multiple procedures were used to ensure proper collection of data utilizing both approaches. All data colle cted via surveys, exam questions, fidelity checklists, final project examination, focus group transcripts, and case study analysis were kept confidential by the researcher maintaining all da ta collected through the study in a locked filing cabinet. Additional procedures speci fic to the quantitative and qualitative methodologies were employed to en able information collection that was both

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93 reliable and valid. By using a mixed methods approach, the researcher sought to explore, understand, and delineate the experiences of and responses to using the DAL instructional framework within a pre-serv ice special education teacher education program. Quantitative Procedures The quantitative procedures of the study encompassed administering multiple surveys at pre-, midpoint, and post-test junctu res, as well as collec ting responses to onetime Clinical Teaching test questions, and ma intaining pre, midpoint, and post fidelity observation checklists on DAL framework a pplication. In terms of the content knowledge survey, it was administered at the beginning of the first week of training with the DAL framework, before any training or experiences had begun, because it was thought that any interaction during the DAL experience might impact the pretimeperiod results for this particular survey. The other two instruments, the efficacy and attitude ones, were administered to teacher candidates during the teach er candidates’ first week of training with the DAL model. In the case of teach er candidate absence, teacher candidates were assessed within one week of this initial time frame for consistency. The researcher also attempted to access absent indi viduals even before the next instructional period, so that exposure to practicum and c ourse content would be equitable with the other teacher candidates for survey purposes. In this way, the data were consistently collected from the same beginni ng time frame for all three su rveys. All midpoint survey information was gathered during the fifth w eek of the DAL’s implementation. Finally, the surveys were administered one last time at the conclusion of the DAL framework’s implementation, which was week ten of the t eacher candidates’ expe rience with the DAL

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94 model. Using these three sp ecific time frames allowed for consistent survey data collection across all t eacher candidates during the durati on of the study. Additionally, the researcher was the person administering all three of these surveys at pre, midpoint, and post junctures, allowing for sta ndardization of administration across types of surveys, as well as across time periods for each survey. The Clinical Teaching exam, which was used to evaluate participants’ knowledge of mathematics instruction, was administered during the week immediately following the DAL framework’s last application. The teach er candidates responded to exam questions within the regular spectrum of their Clinic al Teaching course exam. Questions on the exam for mathematics instruction involved a combination of multiple choice and short answer questions. Three independent exam question evaluators were involved in assessing the accuracy of teacher candidate responses for reliability and validity purposes in determining the accuracy of knowledge ga ined by teacher candidates. Independent raters used a researcher-dev eloped scoring rubric for evaluating all exam short answer questions. This rubric employed a 5-point sc oring system for each question that defined exam question answers from 5, “a full complete answer”, to 1, “an incorrect answer.” All evaluators assessed an identical sampling of three teacher candidate test questions independently, and then regrouped to compare ratings. This process was completed until 90% agreement was reached with scoring these questions acr oss raters. Following that agreement, the three evaluators each then independently scored the remaining teacher candidates’ test questions on mathematics in struction and came back together to reach consensus on all teacher candidates’ test eval uations. While three in dependent evaluators determined the accuracy of teach er candidate test responses fo r the purpose of this study,

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95 the course’s teaching faculty independently evaluated exam responses for the purpose of determining grades for this course assessment. Finally, in terms of collecting quantitative da ta via fidelity observation checklists, teacher candidates were observed by all three raters at one time, until 90% inter-rater reliability was obtained between raters for each observation. Then, a subgroup of approximately twelve teacher candidate partic ipants was divided into three subsections between the three raters, and each of these participants was observed at regularly scheduled intervals at the be ginning, middle, and end of the framework’s implementation by one of the three raters. All teacher candidates were observed for each fidelity check point within the same one-week period to ensu re consistency across time-periods in data collection. Teacher candidates were also observed by one of the three raters for a standard time period, one instructional sess ion, which ranged from 30-40 minutes, to allow for regularity across raters in the time frames allotted for observations. Qualitative Procedures The qualitative procedures of the study were set within a constructivist frame, utilizing focus groups, case studies, and final project analyses as t ools in facilitating the researcher’s knowledge construction a nd meaning making processes for the understanding of the DAL model’s facility as an instructiona l framework within a special education teacher preparation program. Fo r the focus groups, the researcher ensured reliability and validity of the data by completing both pre-point focus groups in the first two weeks of the DAL model framework’s in itial usage and then both post intervention focus groups during the final week of the fr amework’s usage. Within the focus groups, the same 15 researcher-developed questions focusing on teacher candi dates’ attitudes,

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96 self-efficacy, content knowledge, and pedagogical knowledge and application, were employed during pre and post points. These questions were developed based on survey items, test questions, and checklist item s on quantitative measures. Focus group questions sought greater detail and specific information on teacher candidates’ shared group attitudes, self-efficacy, content know ledge, and pedagogical knowledge and application that could not be obtained through quantitative means, but could inform the researcher’s understanding of the larger idea of using the DAL within a teacher preparation program. For accuracy, the res earcher employed the assistance of another doctoral student experienced in focus gr oup methodology to take notes that were compared with the tape recorded comments of focus group participan ts. Additionally, the researcher used frequent member checks wh ile conducting the groups to ensure that the oral responses accurately conveyed the feelings and ideas of the teacher candidates. In regards to the case stu dy process, participants were divided into three groups ranked on their Level II coursework and pr acticum achievement and performance by the Level II cohort’s professors. In this way, the researcher aimed to evaluate and discern a clear picture of the DAL model experience fo r a participant with high level achievement, average achievement, and then low achievem ent within their Level II practicum and coursework. Through this process, the rese archer obtained an understanding of how the DAL framework was experienced by participants across ability levels. Artifacts that were gathered from case study partic ipants included weekly “se ssion notes”, weekly personal reflections, final cumulative projects, and ex it interview transcripts and notes. Using these pieces of information, the researcher ha d multiple, specific written data pieces to

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97 analyze for feelings, ideas, and changes that teacher candidates had during the course of their experience with the DAL framework. The Atlas.ti software program was us ed to help analyze qualitative data collected from written transcripts of focus groups, case study teacher candidate interviews, and final projects of all participants from the Level II practicum. Responses were transcribed and typed using a word processing program. The Atlas.ti software program was used to facilitate the codi ng and categorizing of teacher candidates’ thoughts and ideas. The design of the software enabled the researcher to easily code written comments and then connect these code s, so categories and trends in the data could be seen by the researcher. The open codes generated by the researcher for the focus groups and final projects were categorized into larger themes and ideas across the full group of participants. Coding employed with the case study artifacts enabled the researcher to analyze the individual experien ce of each case study participant. For the case study document artifacts, the research er employed a hand review of teacher candidate session notes, weekly reflections, and final projects, looking for ideas and themes across teacher candidates’ work. Research Design Mixed Methods Design The study was organized as a mixed methods investigation with information obtained through quantitative surveys used in conjunction with the da ta collected through qualitative means. Both types of research methodologies were util ized to provide the researcher with multiple forms of data and information to best understand teacher candidates’ experiences within a struct ured, social-developmental constructivist

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98 preservice teacher preparation experience. Th e goal of the researcher was to utilize qualitative coding, categoriz ation, and analysis, in c onjunction with quantitative statistical information regarding central te ndency, repeated measures over time, and effect sizes to develop an understanding of teacher candidate change in attitude, selfefficacy, content knowledge, and pedagogical k nowledge and application when using the application-based DAL inst ructional framework. Quantitative Design Statistical measures used with the quanti tative data included descriptive statistic calculations, as well as inferen tial statistics in th e form of a repeated measures ANOVA. To this end, on the three survey instruments involving self-efficacy, attitudes, and content knowledge, calculations of mean, median, mode skewness, and kurtosis were generated to provide descriptive informa tion on the teacher candidates’ responses at three points: pre, midpoint, and post-test. The repeated measures ANOVA was employed to detect significant changes in survey scores for the participant group over time. Cohen’s D was used to generate effect sizes based on the statistical calcula tions of the repeated measures ANOVA for each survey. The researcher looked for changes in sta tistical data as the teacher candidates’ progressed through their DAL experience. For all statistical survey data, a comparison of information was made across pre, midpoint, and post-test administrations, as well as across participants. For the Clinical Teaching test questions, de scriptive statistics were generated for teacher candidates’ responses on individual que stions. Comparisons of data were made for each participant between types of test questions, multiple choice versus essay questions and descriptive ve rsus application-based essa y questions, as well as

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99 comparisons done for test questions across the teacher candidate sample. For the fidelity checklists several forms of analysis were us ed. Percentages were calculated for each teacher candidate’s fidelity in implementing th e steps of the DAL framework. Since each teacher candidate was observed three times, fidelity percentages were then compared across time periods for each teacher candidate, as well as across the group of participants at each time period. For each set of observati ons, the fidelity percentages were totaled for the participant group as a whole, and the mean calculated for each observation set (i.e., first set of observations second set of observations, th ird set of observations). Qualitative Design With multiple qualitative measures employed in the current study, it was necessary to use several tools for data coll ection and analysis purposes. For the case study portion, session notes, weekly reflections, final projects, and exit interviews were analyzed using a combination of researcher hand review and electr onic review using the Atlas.ti software. For session notes, th e researcher copied, hand-reviewed, and highlighted teacher candidate planning and strategy implementation, since these session notes were written on pre-de signed DAL lessoning planning forms. The researcher evaluated these session notes in regards to ideas and themes that emerged from teacher candidate writing on instructional knowledge and implementation, as well as attitude, efficacy, and content knowledge. Weekly reflections were also copied and handreviewed like the session notes, using a highl ighting system to code similar ideas and themes. Final projects, focus groups, and cas e study exit interviews were scanned into the researcher’s computer, so they could be uploaded to the Atlas.ti qualitative analysis software. A similar process was employed w ith these data pieces, as with the hand-

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100 reviewed ones, but with the re searcher using the electronic so ftware to assist in coding, categorizing, and theme analysis. Specific teacher candidate expressions related to attitude, self-efficacy, content knowledge, and instructional knowle dge and application were identified and analyzed. A grounded theory approach was used to develop theoretical understandings and conclusions, where collected data were used as the basis of theory development for the investigated research ques tion (Glaser & Strauss, 1965). The researcher used the qualitative themes that emerged to constr uct a greater understanding of the Level II cohort’s experiences and responses to the DAL framework in regard s to attitude, selfefficacy, instructional knowledge and a pplication, and content knowledge in mathematics. A complete listing of major i nquiry areas, quantitative and qualitative data collection measures, and data analysis methods are provided in Table 4. In Chapter 4, results collected by the different quantitative and qualitative data collection methods are presented, along with accompanying analysis.

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101 Table 4 Alignment of Research Key Questions and Instruments Specific Questions Data Collection Instruments Analysis 1.) What changes, if any, occur in special education teacher candidates’ feelings of self-efficacy about teaching mathematics from the beginning to the end of a preservice instructional experience using the DAL framework? Quantitative ~Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) by Enochs, Smith, and Huinker (2000) at Pre, Midpoint, and PostTest Points Qualitative ~Pre and Post-test focus groups with teacher candidates on feelings of self-efficacy related to mathematics instruction ~Weekly reflections on feelings of self-efficacy from 3 case studies ~Analysis of feelings of selfefficacy about mathematics instruction from fi nal papers of all students on the DAL model experience Quantitative ~Descriptive statistics involving mean, mode, median, skewness, and kurtosis, Repeated Measures ANOVA Qualitative ~Document Hand Review ~Transcription of Teacher Candidate Comments ~Open Coding of Ideas ~Usage of Inductive Reasoning in Identifying Categories and Themes

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102 Table 4 (cont.’d) Specific Questions Data Collection Instruments Analysis 2.) What changes, if any, occur in special education teacher candidates' attitudes towards mathematics instruction from the beginning to the end of a preservice instructional experience using the DAL framework? Quantitative ~Preservice Teachers’ Mathematical Beliefs Survey by Seaman, Szydlik, Szydlik, and Beam (2005) at Pre, Midpoint, and Post-Test Points Qualitative ~Pre and Post-Test focus groups ~Weekly reflections on attitude towards mathematics instruction from 3 case studies ~Analysis of attitude towards mathematics instruction from final papers of all teacher candidates on the DAL model experience Quantitative ~Descriptive statistics involving mean, mode, median, skewness, and kurtosis, Repeated Measures ANOVA Qualitative ~Document Hand Review ~Transcription of Teacher Candidate Comments ~Open Coding of Ideas ~Usage of Inductive Reasoning in Identifying Categories and Themes

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103 Table 4 (cont.’d) Specific Questions Data Collection Instruments Analysis 3.) What changes, if any, occur in special education teacher candidates' understanding of instructional strategies for struggling learners in mathematics from the beginning to the end of a preservice instructional experience using the DAL framework? Quantitative ~Clinical teaching short answer test questions on mathematics pedagogical strategi es (evaluated for correctness by 3 parties for reliability purposes) Qualitative ~Pre and Post-Test focus groups ~Weekly reflections on instructional knowledge from 3 case studies ~Analysis of instructional knowledge from final papers of all teacher candidates on the DAL model experience Quantitative ~Percentage of accuracy between and across test questions ~Descriptive statistics involving mean, mode, median, skewness, and kurtosis Qualitative ~Document Hand Review ~Transcription of Teacher Candidate Comments ~Open Coding of Ideas ~Usage of Inductive Reasoning in Identifying Categories and Themes

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104 Table 4 (cont.’d) Specific Questions Data Collection Instruments Analysis 4.) What changes, if any, occur in special education teacher candidates’ application of instructional strategies for struggling learners in mathematics from the beginning to the end of a preservice instructional experience using the DAL framework? Quantitative ~Fidelity measures utilized for mathematics strategies within the DAL model, as well as fidelity measures for the DAL implementation process (baseline: 3-5 teacher candidates evaluated by all 3 raters with 90% agreement) Qualitative ~Pre and Post-Test focus groups ~Weekly reflections on instructional application from 3 case studies ~Analysis of instructional application from final papers of all teacher candidates on the DAL model experience Quantitative ~Fidelity percentages between DAL steps and across DAL participants ~Descriptive statistics involving mean, mode, median, skewness, and kurtosis, Repeated Measures ANOVA Qualitative ~Document Hand Review ~Transcription of Teacher Candidate Comments ~Open Coding of Ideas ~Usage of Inductive Reasoning in Identifying Categories and Themes

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105 Table 4 (cont.’d) Specific Questions Data Collection Instruments Analysis 5.) What changes, if any, occur in special education teacher candidates’ content knowledge of elementary mathematics, including algebraic thinking, from the beginning to the end of a preservice instructional experience using the DAL framework? Quantitative ~Mathematical Content Knowledge for Elementary Teachers by Matthews & Seaman (2007) at Pre, Midpoint, and PostTest Points Qualitative ~Pre and Post-Test focus groups ~Weekly reflections on attitude towards mathematics instruction from 3 case studies ~Analysis of attitude towards mathematics instruction from final papers of all teacher candidates on the DAL model experience Quantitative ~Descriptive statistics involving mean, mode, median, skewness, and kurtosis, Repeated Measures ANOVA Qualitative ~Document Hand Review ~Transcription of Teacher Candidate Comments ~Open Coding of Ideas ~Usage of Inductive Reasoning in Identifying Categories and

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106 Chapter 4 Results Overview In the current study, the Developing Algebraic Literacy (DAL) model, a structured instructional framew ork for teaching algebraic thi nking to at-risk learners, was implemented with a group of undergraduate special education teach er candidates during an early clinical field expe rience. The purpose of the study was to explore teacher candidates’ experiences as th ey received training in the DAL model, as they provided one-to-one instruction using the DAL model, a nd as they received structured support and feedback from practicum faculty in their Level II clinical practicum. Five key elements of teacher preparation were investigated: 1) self-efficacy for teaching mathematics, 2) attitudes toward teaching mathematics, 3) knowledge of mathematics content, 4) knowledge and understanding of re search-based mathematics in structional pr actices for at-risk learners, and 5) applic ation of research-based mathematics instructional practices for at-risk learners. During the course of this study, particip ants engaged in Clinical Teaching and Behavior Management coursework, as well as participated in a twoday a week practicum experience. One day each week of this prac ticum was at a Title I school site, within a large urban school district in the Southeastern United States where the teacher candidates received training and support while imple menting the DAL framework. Data were collected from 19 teacher candidates using bot h quantitative and qualitative research

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107 methods. Moreover, three participants were selected for the purpose of conducting case study analyses. In order to select case study participants, all partic ipants were divided into three ranked subgroups based on thei r overall Level II achievement based on their performance on course-related tests, assignm ents, and projects, as well as practicum feedback from their supervising teachers and observations made by their university supervisors. One student from each of th ese ranked groups was chosen as a case study participant to gather more specific and de tailed information on teacher candidates’ experiences while using the DAL framework. Demographics of Participants In this study, the 19 teacher candidate pa rticipants varied across age, university status, years in college, and ethnicity as s hown in Table 5. All teacher candidates were female students enrolled in th e Level II special ed ucation undergraduate coursework and practicum. In terms of age, a majority, 63.2%, were between the ages of 20 and 24, which is the typical age of undergraduate upperclassmen within most universities. There were also clusters of participants in th eir later twenties, w ith approximately 15.7% between 25 and 29, and between 35 and 44, respec tively. One particip ant was an outlier on the age variable, and she fell between 55 a nd 59. The teacher candidates were split between holding Junior and Se nior status within the uni versity. Slightly more participants indicated they were Seniors at 52.6%, and one student did not indicate her status at all. The teacher candidates varied in the number of years they had attended college or university, but most of the ove rall participant group, 88.6%, had been in college for three years or more. One person reported herself as in college for only one year, and another indicated she had been in college for just two years. The number of

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108 participants in college for three and four years was equal at 26.3% for each year. The largest group of teacher candida tes, 36.8%, reported that they had been attending college for five years. The ethnic background of pa rticipants was primarily white (63.2%), with minority participants including Hispanic/L atino (15.7%), Black/African American (10.5%), Native American/Alaskan Native (5.3%), and Other (5.3%). Table 5 Demographic Characteristics of Teache r Candidate Participants (N=19) F % Gender Female 19 100 Male 0 0 Age 20-24 12 63.2 25-29 3 15.7 30-34 0 0 35-39 2 10.5 40-44 1 5.3 45-49 0 0 50-54 0 0 55-59 1 5.3 Cohort Status Level 2 19 100 University Status Junior 8 42.1 Senior 10 52.6 Not Indicated 1 5.3 Number of Years Spent in College One Year 1 5.3 Two Years 1 5.3 Three Years 5 26.3 Four Years 5 26.3 Five Years 7 36.8 Ethnicity Hispanic/Latino 3 15.7 American Indian/Alaskan Native 1 5.3 Black/African American 2 10.5 White 12 63.2 Other 1 5.3

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109 Description of Case Stud y Participant Selection Within the overall participant sample, th ree students were chosen as case studies. Each of these individuals was selected ra ndomly from one of the three ranked groupings of teacher candidates: upper performi ng third, middle performing third, and lower performing third. This selection of individua ls for case study was done so the researcher could gather specificity of information on i ndividual experiences w ith the DAL model for participants with different academic perfor mance levels. Case study participants were considered representative of the typi cal individual, and her experiences and achievements, for a partic ular ranked grouping. Format of Results Information The current study involved da ta collection using both qua ntitative and qualitative methodologies. Quantitative information was co llected via three surv ey instruments, a course exam, and fidelity checklists. Qua litative information was gathered using pre and post focus groups, final project reviews, and case study analysis. For ease of understanding, resulting data from the curr ent study is presented by data collection methodology, with case study analysis being pres ented in its own section because of the length of data and analysis provided. Each of these methods gathered information on one of the five aforementioned key elements for teacher preparation identified by the researcher. These five elements were believ ed to be critical investigation areas when exploring the study’s overarch ing research question. The main research question of this study was: What changes related to effective ma thematics instruction for struggling

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110 elementary learners, if any, o ccur in teacher candidates during implementation of the DAL instructional framework in an early clinical field experience practicum for preser vice special education professional preparation? Quantitative Findings In this section, data collected through quantitative measures will be presented and analyzed. This information includes findi ngs from pretest, midpoint, and posttest administrations of survey instruments involvi ng self-efficacy for teaching mathematics, attitudes toward teaching mathematics, and knowledge of mathematics content. In statistical calculations involvi ng these survey instruments, participant numbers may vary slightly between administrations. There are tw o reasons for these differences: 1) at times teacher candidates were absent for a given su rvey administration and they could not be accessed within a similar time period as other pa rticipants for that administration, or 2) survey results were only included for particip ants when they completed over 75% of a particular survey’s questions Additionally, results from an instructional knowledge course exam and fidelity checklist find ings are included and interpreted. Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) The first survey instrument explored teacher candidate perceived efficacy when teaching mathematics to elementary leve l students. The MTEBI (Enochs, Smith, & Huinker, 2000) was employed to collect this efficacy info rmation using a total of 21Likert scale items, divided between two subt ests. On this efficacy measure teacher candidates were asked to respond to “I” statem ents about their feelings of efficacy in

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111 mathematics instruction usi ng a 5-point scale. The re sponse options included: (1) Strongly Disagree, (2) Disagree, (3) Uncertain, (4) Agree, an d (5) Strongly Agree. The instrument’s first subtest, Self E fficacy, included questions involving teacher candidates’ perceptions of their abilities to currently teach, as well as develop their teaching abilities (ie., I will co ntinually find better ways to teach mathematics.). The instrument’s second subtest, Outcome Exp ectancy, included questions about teacher candidates’ perceptions of antic ipated student responses to their mathematics instruction (ie., The teacher is generally responsib le for the achievement of students in mathematics.). Enochs, Smith, & Huinker (200 0) assert that “behavior is enacted when people not only expect specific behavior to result in desirable outcomes (outcome expectancy), but they also believe in th eir own ability to perform behaviors (selfefficacy)” (p. 195-196). These ideas assist teacher educators in understanding the importance of efficacy development in any teacher preparation program. While the survey’s items were worded both positively and negatively to access teacher candidate perceptions, all items were recoded so that a rating of “5” indicated high perceptions of efficacy in teaching and affecting student re sponses through instruction, and a “1” rating indicated low perceptions of the same ideas. Descriptive Statistics for the MTEBI For analysis of teacher candidates’ responses on the efficacy instrument, SPSS was employed by the researcher to generate statistical data. When completing this analysis, information on mean, median, range standard deviation, skewness, kurtosis, and standard error of mean were generated. Descriptive statistics are given in Table 6.

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112 For the most part, these statistics suppor ted a normal distribution of the efficacy instrument’s results. Table 6 Descriptive Statistics for the MTEBI ________________________________________________________________________ MTEBI Mean Median *Gain Range SD Skewness Kurtosis Standard Score Error of Mean ________________________________________________________________________ *Gain scores are reported as percentage differences from pretest scores. Mean scores from teacher candidate res ponses indicated that overall perceptions of efficacy increased slightly from pretest to posttest on the full survey, moving from a Full Survey Pre (N=15) 3.37 3.48 1.38 0.42 -0.35 -0.56 0.11 Mid (N=18) 3.64 3.69 16.56%1.62 0.38 -1.06 2.06 0.09 Post (N=19) 3.72 3.67 21.47%1.95 0.46 -0.27 0.96 0.11 Self Efficacy Pre (N=15) 3.35 3.31 2.00 0.53 -0.20 0.42 0.14 Mid (N=18) 3.60 3.69 15.15%2.31 0.51 -0.97 2.24 0.12 Post (N=19) 3.49 3.62 8.48% 1.62 0.48 -0.20 -1.09 0.11 Outcome Expectancy Pre (N=15) 3.39 3.63 1.75 0.56 -1.05 -0.03 0.14 Mid (N=18) 3.70 3.81 19.25%2.25 0.56 0.01 0.50 0.13 Post (N=19) 3.58 3.63 11.80%2.50 0.52 0.52 2.19 0.12

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113 starting mean of 3.37 to an ending mean of 3.72. On the full survey, gain scores also show a rise from pretest to midpoint with a 16.56% increase and from pretest to posttest with a 21.47% increase. The m eans of both subtests showed increases at midpoint, but saw decreases from midpoint to posttest on these subtests. Even with this downward movement from midpoint to posttest on these su btests, an overall in crease was still seen between pretest and posttest. On the self -efficacy subtest, the gain score was 15.15% between pretest and midpoint and 8.48% betw een pretest and posttest. Mean scores on the outcome expectancy subtests were highe r than on the self-efficacy subtests, showing that teacher candidates held more positive perceptions about effective instructional practices being linked to positive learning outcomes than about their own actual instructional abilities to affect this change Gain scores supported these findings on the outcome expectancy subtest with a 19.25% in crease from pretest to midpoint, and a 11.20 rise from pretest to posttest. Box plots of the mean scores for the full efficacy instrument in Figure 2 give a visual picture of the score di stributions and the data movement from pretest to midpoint to posttest for participants. Box plots of pretest and postte st scores are similar normal distributions. Posttest scores show a decrea se from midpoint scores but posttest scores have a higher median as well as range of scor es, than scores at pretest. The midpoint box plot illustrates a distributi on that has an outlier in th e lower range, but also shows participants’ scores increased considerably from pretest, with the interquartile range of all scores nearly all at or above the median point of pretest scores.

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114 Box plots for the self-efficacy subtest in Figure 3 show pretest scores with an outlier in the lower range, as well as midpoint scores with two outlier s in the lower range. At midpoint, except for the two outliers, the sc ores have a much more compact range and higher median than at pretest. While the plots show posttest sc ores decreasing from midpoint, these final scores evidenced no outliers and the median remained above the pretest median level.

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115 Box plots of the outcome expectancy s ubtests in Figure 4 show high variability between participant scores at each administ ration. Of the three sets of scores, the midpoint ones have the most norma l distribution. The scores at posttest show the greatest variability with outliers in bot h the upper and lower ranges. While these posttest scores must be interpreted carefully in light of these outliers, median scores can be seen to move only slightly from pretest to pos ttest, with a rise at midpoint and then a dip back to pretest level at posttest.

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116 When looking at individual questions’ desc riptive statistics, it was found that item 2, “I will continually find better ways to teach students mathema tics” had the highest mean score (4.35) from teacher candidates at pretest. This statistic indicated teacher candidates’ answered between “Agree” and “Strongly Agree” levels that they will actively seek out resources to improve their mathematics instruction. Item 17, “I wonder if I will have the necessary skills to teach mathematics” received the lowest mean response (2.00), indicating that many teacher ca ndidates’ did not question that they would have the abilities to teach mathematics effec tively. At posttest, the highest mean score

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117 was for item 15, “I will find it difficult to us e manipulatives to e xplain to students why mathematics works”, showing that teacher ca ndidates thought that te aching learners using manipulatives would be a hard task for them The lowest mean score at posttest was shared between Item 17 and Item 18, “Given a choice, I will not invi te the principal to evaluate my mathematics teaching”. These results indicate that teacher candidates’ continued to have faith in their ability to le arn how to teach mathematics, and would even invite their future principals into thei r future classrooms while engaging in this instruction. Inferential Statistics for the MTEBI Since the efficacy survey was administ ered to teacher candidates on three occasions during the semester, a repeated measures analysis was completed to see whether there were any statistically signi ficant differences between results of the different administrations for the full efficacy survey and its subtests. Results from the repeated measures analysis are presented in Table 7. For the full survey, self-efficacy, and outcome expectancy results, no statis tically significant di fferences were found between response scores at pr etest, midpoint, or posttest be cause significance for all measures was indicated at the p>.05 level.

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118 Table 7 Repeated Measures Analysis of the Mathem atics Teaching Efficacy Beliefs Instrument ____________________________________________________________________ Measure Source df SS MS F p ____________________________________________________________________ Efficacy Whole Time 1 0.215 0.215 1.839 0.198 Within Group Error(Time)13 1.517 0.117 Self-Efficacy Time 1 0.101 0.101 0.464 0.508 Within Group Error(Time)13 2.832 0.218 Outcome Expectancy Time 1 0.492 0.492 2.014 0.179 Within Group Error(Time)13 3.175 0.244 The final statistical analyses on the efficacy instrument involved evaluating correlations for relevant within test and between test correlations across the three administration time points for the full efficacy instrument and its subtests. Within test correlations were completed to see if ther e was any relationship between the multiple administrations of the full instrument, as we ll as any associations between the multiple administrations of each subtest. Between s ubtest correlations were performed to assess possible connections between teacher candida te self-efficacy and outcome expectancy responses at each administration. Results for the full efficacy instrument indicated a moderate correlation (r=. 759, p<.001) between the midpoint and posttest admi nistrations of the full efficacy instrument as seen in Table 8. This finding depicts a possible connection between how teacher

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119 candidates responded to efficacy items at mi dpoint and how they responded to these items at posttest. No other statistically significant correlations were found between administrations of the full efficacy instrument. Table 8 Correlation Matrix for Full Efficac y Instrument Across Pretest, Midpoint, and Posttest ______________________________________________________ Efficacy Efficacy Efficacy 1 2 3 ______________________________________________________ When correlation analyses were run on the self-efficacy subtests, a moderate correlation was also found between midpoint a nd posttest administrations of the selfefficacy subtest (r=.754, p<.001), while a st rong correlation was also found between midpoint and posttest on the outcome exp ectancy subtest (r=.818, p<.001) as shown in Tables 9 and 10. These results indicate possible connections between how teacher candidates answered self-efficacy questions at midpoint and posttest, with an even Efficacy 1 Pearson Correlation 1 0.535 0.575 Sig. (2-tailed) 0.049 0.025 N 15 14 15 Efficacy 2 Pearson Correlation 0.535 1 0.759 Sig. (2-tailed) 0.049 0 N 14 18 18 Efficacy 3 Pearson Correlation 0.575 0.759 1 Sig. (2-tailed) 0.025 0 N 15 18 19

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120 stronger possible connection seen betw een midpoint and posttest for outcome expectancy. The other within test correl ation analyses did not yield statistically significant results. Table 9 Correlation Matrix for Self-Efficacy Subtes t Across Pretest, Mi dpoint, and Posttest ___________________________________________________________________ Self 1 Self 2 Self 3 ___________________________________________________________________ Self 1 Pearson Correlation 1 0.447 0.560 Sig. (2-tailed) 0.109 0.030 N 15 14 15 Self 2 Pearson Correlation 0.447 1 0.754 Sig. (2-tailed) 0.109 0 N 14 18 18 Self 3 Pearson Correlation 0.560 0.754 1 Sig. (2-tailed) 0.030 0 N 15 18 19

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121 Table 10 Correlation Matrix for Outcome Expectan cy Subtest Across Pretest, Midpoint, and Posttest ________________________________________________________________________ Outcome 1 Outcome 2 Outcome 3 ________________________________________________________________________ Between test correlation analyses in dicated no statistically significant relationships between the self-efficacy and out come expectancy subtests at pretest, midpoint, or posttest as seen in Tables 11-13. Outcome 1 Pearson Correlation 1 0.583 0.498 Sig. (2-tailed) 0.029 0.059 N 15 14 15 Outcome 2 Pearson Correlation 0.583 1 0.818 Sig. (2-tailed) 0.029 0 N 14 18 18 Outcome 3 Pearson Correlation 0.498 0.818 1 Sig. (2-tailed) 0.059 0 N 15 18 19

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122 Table 11 Correlation Matrix for Self-Efficacy and Outcome Expectancy Subtests at Pretest ______________________________________________________ Self 1 Outcome 1 ______________________________________________________ Self 1 Pearson Correlation 1 0.182 Sig. (2-tailed) 0.515 N 15 15 Outcome 1 Pearson Correlation 0.182 1 Sig. (2-tailed) 0.515 N 15 15 Table 12 Correlation Matrix for Self-Efficacy and Outcome Expectancy Subtests at Midpoint ______________________________________________________ Self 2 Outcome 2 ______________________________________________________ Self 2 Pearson Correlation 1 -0.021 Sig. (2-tailed) 0.935 N 18 18 Outcome 2 Pearson Correlation -0.210 1 Sig. (2-tailed) 0.935 N 18 18

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123 Table 13 Correlation Matrix for Self-Eff icacy and Outcome Expectancy Subtests at Posttest ______________________________________________________ Self 3 Outcome 3 ______________________________________________________ Self 3 Pearson Correlation 1 0.013 Sig. (2-tailed) 0.957 Outcome 3 N 19 19 Pearson Correlation 0.013 1 Sig. (2-tailed) 0.957 N 19 19 Overall, correlation results on the full e fficacy instrument indicated that teacher candidate responses throughout th e entire efficacy instrument at midpoint were associated with their responses on the instrument at posttest. A similar a ssociation was seen between midpoint and posttest results for the self-efficacy and outcome expectancy subtests. These associations indicate that how teacher candi dates felt about their efficacy of mathematics instruction at midpoint was c onnected to how they fe lt about this efficacy at posttest. However, teacher candidates’ perceptions of self-efficacy and student outcome expectancy did not evidence a c onnection at any of the administrations. Mathematical Beliefs Questionnaire The second survey investigat ed teacher candidate attit ude towards mathematics in general, as well as mathematics instructi on. The importance of collecting attitudinal information towards mathematics instruction is summarized by “teachers’ beliefs about subject matter and about the nature of teach ing indicate something about the culture of the educational system that produced them ” (Seaman, Szydlik, Szydlik, & Beam, 2005).

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124 Since in most teacher preparation programs uni versity faculty are atte mpting to change or “undue” many of these attitudes, it is important that these stakeholders have an idea of what these attitudes entail. The Mathematical Belie fs Questionnaire (Seaman, et al., 2005), which consists of 40-Likert scale item s on a 6-point scale, was used to collect attitudinal information. Since th is survey instrument uses the term “beliefs” for what the researcher has operationalized as “attitudes” in this study, these tw o terms will be used interchangeably in this analysis and be c onsidered to have the same meaning. The questionnaire’s response options include: (1) Strongly Disagree, (2) Moderately Disagree, (3) Slightly Disagr ee, (4) Slightly Agree, (5) Moderately Agree, and (6) Strongly Agree. Questionnaire items are organized accordi ng to two subtests: the Mathematics Beliefs Scale (MBS) and the Teaching Mathema tics Beliefs Scale (TMBS). Each subtest incorporates items along two themes within it s response statements, including: ones that address constructivist attitudes about mathema tics (ie., The field of math contains many of the finest and most elegant creations of the human mind [MBS], Children should be encouraged to invent their own mathemati cal symbolism [TMBS]) and ones that present traditionalist views about mathematics (ie ., Solving a mathematics problem usually involves finding a rule or formula that appl ies [MBS], Teachers should spend most of each class period explaining how to work sample specific problems [TMBS]) The purpose for the inclusion of constructivist a nd traditionalist items was to discern an overall theoretical perspec tive on teacher candidate att itudes about mathematics in general and mathematics instruct ion. Results for the attitude instrument are reported by subtest (MBS or TMBS) and response item persp ective (constructivist or traditional) for a

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125 total of four areas of information for each Mathematical Beliefs Questionnaire administration. Descriptive Statistics for the Ma thematical Beliefs Questionnaire Statistical analysis of teacher candidates’ responses on the Mathematical Beliefs Questionnaire was completed using the SPSS. Fo r the purpose of descriptive statistical analysis, data on mean, median, range, sta ndard deviation, skewness, kurtosis, and standard error of mean were generated. De scriptive statistics are given in Table 14. These data indicated a fairly norm al distribution of results.

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126 Table 14 Descriptive Statistics for the Ma thematical Beliefs Questionnaire ________________________________________________________________________ Math. Mean Median *Gain Range SD Skewness Kurtosis Standard Beliefs Score Error of Questionnaire Mean ________________________________________________________________________ Full Survey Pre (N=18) 3.57 3.58 1.38 0.37 0.18 -0.45 0.09 Mid (N=18) 3.85 3.85 11.11%1.92 0.49 0.67 0.51 0.12 Post (N=19) 3.72 3.67 7.00% 1.95 0.46 -0.27 0.96 0.11 MBS – Construct. Worded Pre (N=18) 3.81 3.90 2.15 0.55 -0.15 -0.21 0.13 Mid (N=18) 4.03 3.95 10.05%3.00 0.74 1.22 1.86 0.18 Post (N=19) 3.94 4.20 5.94% 3.20 0.76 -0.94 1.58 0.17 MBS – Tradition. Worded Pre (N=18) 3.23 3.20 1.80 0.55 0.28 -0.97 0.13 Mid (N=18) 3.41 3.45 6.50% 2.00 0.53 0.34 -0.18 0.12 Post (N=19) 3.41 3.60 6.50% 2.10 0.61 -1.04 0.18 0.14 TMBS – Construct. Worded Pre (N=18) 4.11 4.16 2.05 0.52 -0.63 0.35 0.12 Mid (N=17) 4.31 4.20 10.58%2.50 0.79 -0.10 -1.13 0.19 Post (N=19) 4.16 4.10 2.64% 1.60 0.49 -0.14 -1.10 0.11

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127 Table 14 (cont.’d) ________________________________________________________________________ Math. Mean Median *Gain Range SD Skewness Kurtosis Standard Beliefs Score Error of Questionnaire Mean ________________________________________________________________________ *Gain scores are reported as percenta ge differences from pretest scores. Mean scores of teacher candidates’ res ponses to the overall attitude instrument revealed an increase from pretest (3.57) to midpoint (3.85) with a slight decrease at posttest (3.72). These results indicate that teacher candidates’ overall responses on the items fell between the “Slightly Agree” and “Slightly Disagree” ratings. During the course of this study, this agreement rose slightly. The gain score on the overall instrument from pretest to midpoint was 11.11% and from pretest to posttest was 7.00%. Within the different subtest areas, constr uctively worded items on both the MBS and TMBS had higher means of agreement then tr aditionally worded items. These scores show that teacher candidates had a stronger iden tification with a constr uctive approach to mathematics learning and teaching. The gain score for constructively worded items on the MBS were 10.05% between pretest and midpoint and 5.94% between pretest and posttest. For the TMBS, constructively word ed items had a gain score of 10.58% and 2.64% between pretest and midpoi nt and pretest and posttest respectively. However, while means for the constructively worded items fell between 3.80 and 4.30, traditionally TMBS – Tradition. Worded Pre (N=16) 3.31 3.39 1.70 0.48 -0.29 -0.67 0.12 Mid (N=17) 3.62 3.50 11.52%2.60 0.66 1.44 2.85 0.16 Post (N=19) 3.38 3.30 2.60% 1.70 0.51 0.42 -0.71 0.12

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128 worded items’ ratings were not far below th at level with scores between 3.20 and 3.60. The gain scores for traditionally worded items on the MBS were consistently 6.50% between pretest and midpoint and between pretest and posttest. On the TMBS, traditionally worded items had gain scores of 11.52% between pretest and midpoint and 2.60% between pretest and posttest. These m ean ranges show that although traditionally worded items had lower “agreement” levels than constructively worded items, but the difference between the two mean ranges was not large. On constructively worded items of both the MBS and TMBS subtests, teach er candidates’ agreement increased at midpoint but then decreased at posttest. While posttest agreement levels were lower than at midpoint, they were still hi gher than at pretest. On traditionally worded items on both the MBS and TMBS, teacher candidate response patterns over the three administrations differed between the two subtests. With the traditionally worded items on the MBS, teacher candidates’ responses increased in agreement from pretest (3.23) to midpoint ( 3.41), and then maintained the same level from midpoint (3.41) to posttest (3.41). The response cons istency between midpoint and posttest indicate that traditionalist views of mathematics in general did not diminish over the latter part of the study. Teacher candi date responses on the TMBS traditionally worded items reveal a pattern similar to the constructively worded items, beginning at 3.31 at pretest, increasing to 3.62 at midpoint, and decreasing to 3.38 at posttest. These results showed a minimal increase in agreem ent with the traditionalist approach to mathematics instruction over the course of the study. Mean score results from the full beliefs in strument and its subtests indicate that while students’ means rose or fell slightly throughout the study, they maintained fairly

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129 similar ratings from pretest to posttes t on items worded both traditionally and constructively. On all item types, except MB S traditionally worded items, increases were seen in agreement levels between pretest a nd midpoint, with sli ght decreases between midpoint and posttest. With the MBS, an incr ease in agreement was seen between pretest and midpoint, which was then maintained at po sttest. This information illustrates the possible resistance to change of the long-held traditionalist beliefs that teacher candidates have about mathematics in general. Box plots of the full beliefs instrume nt scores in Figure 5 show normal distributions at both pretest and midpoint ad ministrations. Posttest scores evidence outliers in both the upper and lower score ranges, which indicate high variability in participant responses at this administrati on. The box plots also show slight score movement from pretest to midpoint to posttest with midpoint scores depicting a small increase before falling to pretest level at posttest.

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130 In Figure 6, box plots of constructively worded items on the MBS section show a normal distribution at pretest, while midpoi nt shows an upper level score outlier and posttest depicts a lower level score outlier. While all three box plots have similar compact interquartile ranges, the median sc ore at posttest shows an increase from both pretest and midpoint levels.

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131 Box plots in Figure 7, show a ll three administrations of traditionally worded items on the MBS section having normal distributions bu t with larger interquartile ranges at all three administrations than constructively wo rded items on the same section. While the plots illustrate median scores that increas e at each subsequent administration, posttest scores show the highest median level with th e greatest variability of score distribution. Posttest scores show the largest amount of va riability particularly in the lowest 25% of scores. This highest median score level coupl ed with the largest ra nge of scores of all three administrations illustrates that while the median scores rose at posttest, there was a large difference in the response levels amongs t participants at th is administration.

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132 On the TMBS constructively worded item box plots in Figure 8, midpoint and posttest scores evidence normal score distribu tions. Pretest scores have a lower range score as an outlier in their di stribution. Another important vi sual seen in the box plots is the larger interquartile range of scores at midpoint than at either pretest or posttest. Median score levels are stable across all thr ee administrations with a slight dip at both midpoint and posttest.

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133 In Figure 9, box plots show that the pret est and midpoint administrations of the traditionally worded items on the TMBS have normal distributions. While the median score level shows an increase at midpoint and then a decrease at posttest, it is at the posttest administration that an upper level score is seen as an outlier. This information depicts that while participants’ overall agreem ent with traditionally worded items on the TMBS decreased at posttest, this lower le vel of agreement was not seen across all participants.

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134 When evaluating teacher candidate means fo r individual questions at pretest, item 7, “There are several different but appropria te ways to organize the basic ideas in mathematics” and item 36, “Teachers must frequently give students assignments which

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135 require creative or investigativ e work” had the same highest mean scores (both at 4.92). This mean score indicated that teacher candi dates rated these constructively worded items at approximately the “Moderately Agree” leve l. Item 1, “Solving a mathematics problem usually involves a rule or formula that applies” had the lowest mean score (1.92), showing that teacher candidates on the w hole chose “Moderately Disagree” on this traditionally worded item. At posttest, the highest mean score (4.61) was on item 26, “Teachers should provide class tim e to experiment with their own mathematical ideas. Item 21, “The teacher should always work sa mple problems for students before making an assignment” received the lowest mean scor e (2.06). As at pretest, on the posttest administration the item with the highest mean was constructively worded, and the item with the lowest mean was traditionally worded. Inferential Statistics for the Mathematical Beliefs Questionnaire Due to the multiple administrations of th e beliefs instrument throughout the study, a repeated measures analysis was completed to see whether there were any statistically significant differences between administrati ons of the full beliefs instrument, and between administrations of its four subar eas. Results from the repeated measures analysis are presented in Table 15. For the fu ll attitude survey and all four subtests, no statistically significant diffe rences were found between pretest, midpoint, or posttest teacher candidate responses because significan ce for all measures was indicated at the p>.05 level.

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136 Table 15 Repeated Measures Analysis of th e Mathematical Be liefs Questionnaire ____________________________________________________________________ Measure Source df SS MS F p ____________________________________________________________________ Attitude Whole Time 1 0.163 1 2.039 0.175 Within Group Error(Time)14 1.122 0.08 MBS Constructivist Time 1 0.343 0.343 0.793 0.386 Within Group Error(Time)16 6.925 0.433 MBS Traditional Time 1 0.199 0.199 0.841 0.373 Within Group Error(Time)16 3.785 0.237 TMBS Constructivist Time 1 0.136 0.136 0.421 0.526 Within Group Error(Time)15 4.85 0.323 TMBS Traditional Time 1 0.439 0.439 2.136 .168 Within Group Error(Time)13 2.671 0.205 The last statistical analyses performed on the beliefs instrument used correlational analyses for determining possible relationshi ps from within test and between test correlations across the three administrations of the full beliefs instruments and its four subareas (MBS – traditionally worde d, MBS – constructively worded, TMBS – traditionally worded, MBS – constructively worded). As with the efficacy instrument, within test correlations were completed to see if there was any re lationship between the

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137 multiple administrations of the full instrument, as well as any associations between the multiple administrations of each subarea. Between subarea correlations were performed to assess possible connections between teacher candidate at titudes about mathematics in general and teaching mathematics. For the full beliefs instrument, a str ong correlation was found between teacher candidate responses at pretest and midpoint (r=0.88, p<.001) and a moderate correlation was found between pretest and posttest (r=.751, p=001) as shown in Table 16. These results indicate that how teacher candidates responded on the pretest survey were closely associated with how they responded on th e midpoint and posttest surveys. This association may indicate a possible resistance to change for teacher candidates’ attitudes about mathematics. Other correlations on the full beliefs survey showed no statistically significant relationships.

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138 Table 16 Correlation Matrix for Full Beliefs Instrume nt Across Pretest, Midpoint, and Posttest ______________________________________________________ Beliefs 1 Beliefs 2 Beliefs 3 ______________________________________________________ Beliefs 1 Pearson Correlation 1 0.88 0.751 Sig. (2-tailed) 0 0.001 N 17 15 17 Beliefs 2 Pearson Correlation 0.88 1 0.546 Sig. (2-tailed) 0 0.023 N 15 17 17 Beliefs 3 Pearson Correlation 0.751 0.546 1 Sig. (2-tailed) 0.001 0.023 N 17 17 19 The next correlation analysis was perfo rmed on the MBS constructively worded items as seen in Tables 17 and 18. For th ese response items, a strong correlation was found between pretest and midpoint (r=.805, p<.001). However, no other significant correlations were found for this subtest’s admi nistrations. This information shows that teacher candidates’ constructive beliefs a bout mathematics in general midway through the study have a possible associ ation between their beliefs at the outset of the study. For the traditionally worded items of the MBS, a moderately strong correlation was found between pretest and midpoint (r=.722, p= .001) and pretest and posttest (r=0.654, p=.003). Other correlations between administrati ons of the TMBS were not found to be statistically significant. In regard to responses to traditio nally worded items on the MBS,

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139 there appears to be a consistent relations hip between responses at pretest and other administrations, indicating that traditionally held attitudes towards mathematics in general may be resist ant to change. Table 17 Correlation Matrix for MBS – Constructively Worded Items Across Pretest, Midpoint, and Posttest _____________________________________________________________________ MBS-Con1 MBS-Con2 MBS-Con3 _____________________________________________________________________ MBS-Con1 Pearson Correlation 1 0.805 0.521 Sig. (2-tailed) 0 0.027 N 18 17 18 MBS-Con2 Pearson Correlation 0.805 1 0.474 Sig. (2-tailed) 0 0.047 N 17 18 18 MBS-Con3 Pearson Correlation 0.521 0.474 1 Sig. (2-tailed) 0.027 0.047 N 18 18 19

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140 Table 18 Correlation Matrix for the MBS – Traditionally Worded Items Across Pretest, Midpoint, and Posttest _____________________________________________________________________ MBS-Trad1 MBS-Trad2 MBS-Trad3 _____________________________________________________________________ MBS-Trad1 Pearson Correlation 1 0.722 0.654 Sig. (2-tailed) 0.001 0.003 N 18 17 18 MBS-Trad2 Pearson Correlation 0.722 1 0.552 Sig. (2-tailed) 0.001 0.018 N 17 18 18 MBS-Trad3 Pearson Correlation 0.654 0.552 1 Sig. (2-tailed) 0.003 0.018 N 18 18 19 Correlations for the TMBS were also generated and analyzed for both constructively and traditionally worded item areas shown in 19 and 20. For constructively worded items, a moderate co rrelation occurred between pretest and midpoint (r=.695, p=.003). This result indicat es an association between constructive attitudes towards mathematics instruction at the beginning and midpoint of the study. Other correlations between administrations of the TMBS constructively worded items did not yield any statistically signi ficant relationships. For trad itionally worded items on the TMBS, a moderate correlation was found betwee n pretest and posttest responses (r=.669, p=.005). Analysis of the remaining data for the traditionally worded items of the TMBS

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141 did not indicate any other statistically signi ficant correlations. The information gathered from the TMBS correlation analysis indicate that teacher candidates’ traditional beliefs about teaching mathematics at the beginning of the study may have some relationship with their traditional beliefs at the conclusion of the study. Table 19 Correlation Matrix for TMBS – Constructively Worded Subtest Across Pretest, Midpoint, and Posttest _____________________________________________________________________ TMBS-Con1 TMBS-Con2 TMBS-Con3 _____________________________________________________________________ TMBS-Con1 Pearson Correlation 1 0.695 0.589 Sig. (2-tailed) 0.003 0.01 N 18 16 18 TMBS-Con2 Pearson Correlation 0.695 1 0.197 Sig. (2-tailed) 0.003 0.448 N 16 17 17 TMBS-Con3 Pearson Correlation 0.589 0.197 1 Sig. (2-tailed) 0.01 0.448 N 18 17 19

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142 Table 20 Correlation Matrix for TMBS – Traditionally Worded Item Subtest Across Pretest, Midpoint, and Posttest ________________________________________________________________________ TMBS-Trad1 TMBS-Trad2 TMBS-Trad3 ________________________________________________________________________ TMBS-Trad1 Pearson Correlation 1 0.494 0.669 Sig. (2-tailed) 0.072 0.005 N 16 14 16 TMBS-Trad2 Pearson Correlation 0.494 1 0.516 Sig. (2-tailed) 0.072 0.034 N 14 17 17 TMBS-Trad3 Pearson Correlation 0.669 0.516 1 Sig. (2-tailed) 0.005 0.034 N 16 17 19 Within test correlation analyses were completed between the MBS and TMBS subtests, for both constructively and traditiona lly worded items, at pretest, midpoint, and posttest in Tables 21-23. At pretest a nd midpoint, no relevant correlations between subtests were evident. At posttest, a moderate correlation was found between the traditionally worded items on the MBS and TMBS (r=.649, p=.003). On the whole, this information shows a lack of association betw een traditional and cons tructivist attitudes about either mathematics in general or math ematics instruction. The one exception is the association between traditional attitudes about mathematics between the two subtests at posttest. This association may be partly due to the fact that statistics showed that

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143 traditional beliefs about mathematics in general rose between pretest and midpoint, and then maintained constant through posttest, indi cating that teacher candidates beliefs about mathematics in general may be deeply rooted. Table 21 Correlation Matrix for TMBS and MBS Subtests at Pretest ________________________________________________________________________ MBS-Con1 MBS-Trad1 TMBS-Con1 TMBS-Trad1 ________________________________________________________________________ MBS-Con1 Pearson Correlation1 0.429 0.417 0.339 Sig. (2tailed) 0.076 0.085 0.199 N 18 18 18 16 MBS-Trad1 Pearson Correlation0.429 1 0.369 0.35 Sig. (2tailed) 0.076 0.131 0.184 N 18 18 18 16 TMBS-Con1 Pearson Correlation0.417 0.369 1 0.444 Sig. (2tailed) 0.085 0.131 0.085 N 18 18 18 16 TMBS-Trad1 Pearson Correlation0.339 0.35 0.444 1 Sig. (2tailed) 0.199 0.184 0.085 N 0.16 16 16 16

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144 Table 22 Correlation Matrix for TMBS and MBS Subtests at Midpoint ________________________________________________________________________ MBS-Con2 MBS-Trad2 TMBS-Con2 TMBS-Trad2 ________________________________________________________________________ MBS-Con2 Pearson Correlation1 0.429 0.417 0.339 Sig. (2tailed) 0.076 0.085 0.199 N 18 18 18 16 MBS-Trad2 Pearson Correlation0.429 1 0.369 0.35 Sig. (2tailed) 0.076 0.131 0.184 N 18 18 18 16 TMBS-Con2 Pearson Correlation0.417 0.369 1 0.444 Sig. (2tailed) 0.085 0.131 0.085 N 18 18 18 16 TMBS-Trad2 Pearson Correlation0.339 0.35 0.444 1 Sig. (2tailed) 0.199 0.184 0.085 N 16 16 16 16

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145 Table 23 Correlation Matrix for TMBS and MBS Subtests at Posttest ________________________________________________________________________ MBS-Con3 MBS-Trad3 TMBS-Con3 TMBS-Trad3 ________________________________________________________________________ MBS-Con3 Pearson Correlation1 0.492 0.563 0.5 Sig. (2tailed) 0.032 0.012 0.029 N 19 19 19 19 MBS-Trad3 Pearson Correlation0.492 1 0.215 0.649 Sig. (2tailed) 0.032 0.376 0.003 N 19 19 19 19 TMBS-Con3 Pearson Correlation0.563 0.215 1 0.378 Sig. (2tailed) 0.012 0.376 0.111 N 19 19 19 19 TMBS-Trad3 Pearson Correlation0.5 0.649 0.378 1 Sig. (2tailed) 0.029 0.003 0.111 N 19 19 19 19 Overall, the results of these correlat ional analyses show some relationship between how teacher candidates responded at pretest to attitude items and how they responded on other administrations. Howeve r, the association between traditionally

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146 worded items seemed to be more consistent throughout the length of the entire study than constructively worded items. This information indicates that traditional beliefs about mathematics may be more firmly held and resistant to change when held by teacher candidates than constructive beliefs, wh ich appeared more open to change. Mathematical Content Knowledge for Elementary Teachers The third instrument evaluated teach er candidates’ content knowledge of elementary level mathematics. Teacher can didates’ accuracy of mathematics knowledge at their target grade level fo r instruction was believed important in the light of current “highly qualified” teacher mandates, which re quire special education teacher candidates to be prepared in the subject area of instruction, as well as in the pedagogical techniques for at-risk learners ( IDEA 2004). The Mathematical Content Knowledge for Elementary Teachers (Matthews & Seaman, 2007) survey was used to assess teacher candidates’ content knowledge proficiency. This measur e utilizes a total of 20 questions involving basic arithmetic and algebraic thinking skills at the elementa ry school level. Questions are a mixture of open-ended calculation and multiple choice items. For scoring purposes, items were marked as either correct or incorrect with no partial credit given for responses. While teacher candidates’ responses were scored for the entire test originally, the researcher then divided questions into two groupings, ba sic arithmetic and algebraic thinking, and scored these questi ons as two different subtests with 11 questions relevant to basic arithmetic and 9 questions pertaini ng to algebraic thinki ng. Descriptive Statistics for the Mathematical Content Knowledge for Elementary Teachers For teacher candidates’ responses on th e Mathematical Content Knowledge for Elementary Teachers survey, SPSS was used to generate descriptive and inferential

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147 statistics. In terms of desc riptive statistics, mean, media n, range, standard deviation, skewness, kurtosis, and standard error of mean were generated. Descriptive statistics are given in Table 24. These statistics in dicate a normal distribution of results. Table 24 Descriptive Statistics for Mathematic al Content for Elementary Teachers ________________________________________________________________________ Math. Mean Median *Gain Range SD Skewness Kurtosis Standard Content Score Error of for Elem. Mean Teachers ________________________________________________________________________ *Gain scores are reported as percenta ge differences from pretest scores. When evaluating the content knowledge measure, the teacher candidates’ overall mean scores for the entire survey and its subtests were calculated for the pretest, Full Survey Pre (N=19) 0.36 0.35 0.50 0.16 0.33 -0.87 0.04 Mid (N=18) 0.42 0.40 9.38% 0.60 0.18 -0.02 -1.16 0.04 Post (N=18) 0.38 0.30 3.12% 0.65 0.20 0.30 -1.03 0.05 Basic Arithmetic Pre (N=18) 0.41 0.41 0.73 0.21 0.40 -0.61 0.05 Mid (N=18) 0.45 0.46 8.47% 0.64 0.18 0.09 -0.34 0.04 Post (N=18) 0.41 0.41 0.00% 0.73 0.21 0.40 -0.61 0.05 Algebraic Thinking Pre (N=18) 0.34 0.28 0.67 0.21 0.16 -1.22 0.05 Mid (N=18) 0.38 0.39 9.09% 0.67 0.22 -0.08 -1.37 0.05 Post (N=18) 0.34 0.28 0.00% 0.67 0.21 0.16 -1.22 0.05

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148 midpoint, and posttest administrations. For this purpose, all correct answers were coded as 1s and all incorrect answers were coded as 0s. On the full content knowledge survey, the pretest mean score was 7.11, which was just slightly over 35% of problems correct, with individual scores ranging from 3 to 13. The midpoint mean score was 7.95, which was just slightly under 40% of problems correct, with individu al scores ranging from 4 to 14. The posttest mean score was 6.31, which is just over 30% of problems correct, with individual scores ranging from 0 to 13. L ooking at the overall means for items by the full survey, the basic arithmetic subt est, and the algebraic thinking subtest, the fluctuation of these means follows a similar manner at each administration point. From pretest to midpoint, the mean on the full content surv ey increased from 0.36 to .42, and from midpoint to posttest the mean decreased from .42 to .38. These scores indicate that teacher candidates were more likely to achieve an item score of 1, a correct score, at midpoint than at any other administration. Th e gain score from pr etest to midpoint was 9.38% and from pretest to posttest only 3.12%. On the basic arithmetic subtest, mean scores followed the same pattern from pretes t to midpoint to posttest moving from .41 to .45 back to .41. Algebraic thinking subtest scor es also had this incr ease/decrease pattern as well, going from .34 to .38 back to .34. Gain scores on these tw o surveys showed an 8.47% increase on the general arithmetic subt est from pretest to midpoint and a 9.09% increase on the algebraic thi nking subtest form pretest to midpoint. Both subtests evidenced no gain between pretest and posttest Between the two subtests and the full survey, the basic arithmetic subtest ha d the highest mean scores during each administration, indicating that teacher candi dates marked correct answers for basic arithmetic questions somewhere between 41% and 45% of the time versus between 36-

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149 42% of the time for the full survey and 3442% of the time for the algebraic thinking subtest. In terms of the overall results fo r elementary level mathematics skills, teacher candidates scored in the deficient range in overall accuracy in solving elementary level mathematics problems, having the most trouble with algebraic thi nking questions across administrations. Box plots in Figure 10 show the content know ledge full survey scores at all three administrations as having normal distributions While median score levels show little movement between administrations, it is seen through the interquart ile ranges that the differences between scores that make up the inner 50% increased at each administration. This increase in variability illustrates that while the median level of scores remained similar, the level of difference among individu al scores of participants rose.

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150 In Figure 11, box plots show pretest and pos ttest scores with normal distributions on the basic arithmetic subtest. Midpoint scores contained one upper level outlier. Median scores show little movement across a ll three administrations. The interquartile range of participant scores is more compact at midpoint than at either pretest or posttest. At pretest, the lower 25% of scores shows a larger range, while at posttest the upper 25% of scores shows a greater span. The vari ability at preand posttest shows that

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151 participants’ performance was less consistent across the group at both the beginning and end of the study then at its middle. The box plots in Figure 12 depict extremel y similar median scores across all three administrations of the algebraic thinking subtes t. All administrations also show a normal distribution of participant scores. A large difference in interquart ile range scores was seen at both midpoint and posttest, with thes e scores being more closely clustered at pretest. However, the upper and lower 25% of scores showed the greatest variability at pretest.

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152 When evaluating teacher candidate means for individual questions at pretest, it was found on item 3, all teacher candidates scor ed a 1, or correct answer. This question was a basic arithmetic multiple choice pr oblem, which involved selecting the correct number sentence that represented 43 x 38 to the nearest 10. While every teacher candidate achieved a correct answer on item 3, the question with the lowest mean was item 19, also a basic arithmetic problem, for which none of the teacher candidates obtained a correct answer. This problem invol ved selecting the corr ect conceptualization

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153 for explaining the process behind a two-digit multiplication problem. On the posttest administration, item 3 remained the item with th e highest mean, while the lowest mean of 0, where no teacher candidates answered corr ectly, was shared between items 7 and 20, both algebraic thinking problems. Item 7 wa s a multiple choice question, where students had to figure out a range of number valu es for two unknown numbers in an averaging problem. Item 20 was also a multiple choi ce item where students had to determine the theoretical conceptua lization of subtraction with regrouping. Inferential Statistics for the Mathematical Content Knowledge for Elementary Teachers Since the content knowledge survey, like th e attitude and efficacy measures, was administered at pretest, midpoi nt, and posttest, a repeated me asures analysis was run to determine if there were any statistically signi ficant differences between the results of the full survey and two subtests for the three administration points. Results from the repeated measures analysis are presented in Table 25. For the full content knowledge survey and its two subtests, no statistica lly significant difference was found between pretest, midpoint, or posttest teacher candi date responses at the p>.05 level.

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154 Table 25 Repeated Measures Analysis for Mathem atics Content for Elementary Teachers ____________________________________________________________________ Measure Source df SS MS F p ____________________________________________________________________ Content Whole Time 1 0.022 0.022 0.837 0.374 Within Group Error(Time)16 0.418 0.026 Basic Arithmetic Time 1 0.025 0.025 0.79 0.387 Within Group Error(Time)16 0.499 0.031 Algebraic Thinking Time 1 0.019 0.019 0.449 0.512 Within Group Error(Time)16 0.672 0.042 As a final part of the da ta interpretation for the co ntent knowledge instrument, correlational analyses were performed on the full content knowledge survey and its two subtests to evaluate within test and between correlations. As with both the efficacy and beliefs instruments, within test correlations were completed to see if there was any relationship between the multiple administratio ns of the full instrument, as well as any associations between the multiple administra tions of each subtest. Between subtest correlations were completed to determine po ssible connections between teacher candidate levels of basic arithmetic skills and algebraic thinking abilities. Initially, a correlational analysis was comp leted between the three administrations of the full content knowledge survey as seen in Table 26. These resu lts showed there was a moderately strong correlation (r=.745, p<.001) between the pretest and midpoint administrations of the content knowledge surv ey. This information indicates a possible

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155 relationship between the teacher candidates’ accuracy of content knowledge at pretest and midpoint. However, other correlations fo r the full content knowle dge instrument did not yield statistically sign ificant correlations. Table 26 Correlation Matrix for Full C ontent Knowledge Survey Across Pretest, Midpoint, and Posttest _____________________________________________________ Content 1 Content 2 Content 3 _____________________________________________________ Content 1 Pearson Correlation 1 0.745 0.488 Sig. (2-tailed) 0 0.04 N 19 18 18 Content 2 Pearson Correlation 0.745 1 0.591 Sig. (2-tailed) 0 0.013 N 18 18 17 Content 3 Pearson Correlation 0.488 0.591 1 Sig. (2-tailed) 0.04 0.013 N 18 17 18 When analyzing correlations across ad ministrations of the two subtests, a moderate correlation (r=.652, p=.003) was s een between the pretest and midpoint administrations of the basic arithmetic subt est and the algebraic th inking subtest (r=.641, p=.004) in Tables 27 and 28. These results ar e indicative of a proba ble association across pretest and midpoint administrations for bot h basic arithmetic and algebraic thinking

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156 items. Other correlations performed on the tw o subtests across admi nistrations were not statistically significant. Table 27 Correlation Matrix for Basic Arithmetic Subtest Across Pret est, Midpoint, and Posttest __________________________________________________________________ Basic Basic Basic Arithmetic1 Arithmetic 2 Arithmetic 3 __________________________________________________________________ Basic Arithmetic 1 Pearson Correlation 1 0.652 0.432 Sig. (2-tailed) 0.003 0.073 N 19 18 18 Basic Arithmetic 2 Pearson Correlation 0.652 1 0.56 Sig. (2-tailed) 0.003 0.019 N 18 18 17 Basic Arithmetic 3 Pearson Correlation 0.432 0.56 1 Sig. (2-tailed) 0.073 0.019 N 18 17 18

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157 Table 28 Correlation Matrix for Algebraic Thinki ng Subtest Across Pretest, Midpoint, and Posttest __________________________________________________________________ Algebraic Alge braic Algebraic Thinking 1 Thinking 2 Thinking 3 __________________________________________________________________ Algebraic Thinking 1 Pearson Correlation 1 0.641 0.419 Sig. (2-tailed) 0.004 0.084 N 19 18 18 Algebraic Thinking 2 Pearson Correlation 0.641 1 0.511 Sig. (2-tailed) 0.004 0.036 N 18 18 17 Algebraic Thinking 3 Pearson Correlation 0.419 0.511 1 Sig. (2-tailed) 0.084 0.036 N 18 17 18 Correlation analyses were then comp leted between the subtests at each administration of the content knowledge measur e as shown in Tables 29-31. Unlike other instruments in this study, the content know ledge instruments’ subtests, the basic arithmetic and algebraic thinki ng skills had correlations at all three administrations. At pretest and midpoint, the two subtests had moderate correlations with (r=.662, p=.002) and (r=.687, p=.002) respectively. At posttest, the correlation was strong between the two subtests (r=.819, p<.001). These result s show a probable relationship between

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158 teacher candidates’ abilities to accurately an swer basic arithmetic and accurately answer algebraic thinking items. Table 29 Correlation Matrix for Basic Arit hmetic and Algebraic Thinking Subtests at Pretest ______________________________________________________ Basic Algebraic Arithmetic 1 Thinking 1 ______________________________________________________ Basic Arithmetic 1 Pearson Correlation 1 0.662 Sig. (2-tailed) 0.002 N 19 19 Algebraic Thinking 1 Pearson Correlation 0.662 1 Sig. (2-tailed) 0.002 N 19 19

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159 Table 30 Correlation Matrix for Basic Arit hmetic and Algebraic Thinking Subtests at Midpoint ______________________________________________________ Basic Algebraic Arithmetic 2 Thinking 2 ______________________________________________________ Basic Arithmetic 2 Pearson Correlation 1 0.687 Sig. (2-tailed) 0.002 N 18 18 Algebraic Thinking 2 Pearson Correlation 0.687 1 Sig. (2-tailed) 0.002 N 18 18 Table 31 Correlation Matrix for the Basi c Arithmetic and Algebraic Thinking Subtests at Posttest ______________________________________________________ Basic Algebraic Arithmetic 3 Thinking 3 ______________________________________________________ Basic Arithmetic 3 Pearson Correlation 1 0.819 Sig. (2-tailed) 0 N 18 18 Algebraic Thinking 3 Pearson Correlation 0.819 1 Sig. (2-tailed) 0 N 18 18

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160 Overall correlation results indicated that there was a relationship between teacher candidates’ ability to correctly answer questions between pretest and midpoint administrations on the full survey, basic ar ithmetic subtest, and algebraic thinking subtest. This association illustrates a possi ble connection between the teacher candidates’ content knowledge abilities at the start of the study and how they performed at the midway point. Linkages were also seen at each administration between the content knowledge and algebraic thinking subtests, providing support to the literature (Baker, Gersten, & Lee, 2002) that suggests connect ions between learne rs’ abilities in fundamental mathematics skills and higher order algebraic thi nking skills. Instructional Knowledge Exam The fourth area of investigation was t eacher candidates’ knowledge of researchbased mathematics instructional practices fo r struggling learners. The instructional knowledge that was assessed consisted of inform ation relevant to mathematics instruction for at-risk learners in conjunction with th e DAL framework that was presented in the practicum by the researcher, and reinforced in the Clinical Teaching course by the professor. The researcher conducted the trainings and support for the DAL framework within the practicum, while the course professor utilized his self-written course textbook and his knowledge of the DAL framework (he was one of the designers of the DAL along with the researcher) in class. The professor designed the course exam to assess teacher candidate instructi onal knowledge on several levels. The exam consisted of two sections: mu ltiple choice and short answer essay. The multiple choice section contained 25 questions 15 on instructional strategy knowledge and 10 on learning characteristics. The short answer essay section contained 8 questions,

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161 with a total of 21 sec tions, on elements of effective inst ructional practices and application of these effective instructiona l practices within the DAL fr amework. 10 of the sections of the essay questions pertained to effec tive practices alone, and 11 of the sections involved their application w ithin the DAL framework. For scoring purposes, each multiple-choice question was given 1 point if corr ect, and 0 if incorrect. For short answer essay questions, each subsection of each essay question was scored on a 5 point scale, with (5) indicating a complete answer to the question and a (1 ) indicating an answer that was not directed at the question asked or was incorrect. The instructional knowledge exam was scored as a whole, as two subsections of multiple choice and essay, and as four subsections: instructional practices (multiple choice), learning characteristics (multiple choice), instructional practices (essay), and instructional practices application (essay). The scoring process was implemented by the rese archer and two outside raters trained in the DAL framework for the purpose of this study, but grading of the exam was done separately by the professor and was not incl uded in the research. Raters each scored a random sampling of 5 tests, and regrouped to compare results. Since 90% agreement between scoring was seen from these 5 tests, this level of agreement was considered sufficient and raters then scored the rest of the tests independently. After completion of all scoring, the raters came back together a nd discussed their indivi dual evaluations of each question for each participant to reach agreement on any scoring differences among raters. Descriptive Statistics for the Instructional Knowledge Exam For statistical analysis of teacher candidates’ responses on the instructional knowledge exam, the researcher used SPSS to generate descriptive and inferential

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162 statistics, as with the other measures em ployed. As with othe r quantitative measures employed in the study, data on mean, media n, range, standard deviation, skewness, kurtosis, and standard error of mean were ge nerated. Descriptive statistics are given in Table 32. These statistics indicated a normal di stribution of results, with the exception of the multiple choice test. On this section of the instructional exam, most students answered correctly on most items. When evaluating the instruction exam, the first item analyzed was the full exam’s score for each teacher candidate. The mean score total for teacher candidates on the whole exam was 80.82 out of a total possible 130 points, or 62% of questions answered correctly. Within the test, multiple choice and essay questions were scored using two different scales. Multiple choice questions we re either scored as 1 for correct, or 0 for incorrect. Short answer essay questions were scored on a 0 to 5 point scale, with 5 being a fully correct answer and 0 being an incorrect answer. As a result, the different types of items must be interpreted separately. For multiple choice questions, the mean score was .91. This result is close to 1, indicating th at many teacher candidates performed well in this section with most of them scoring a 90% or above on items. Dividing the multiple choice questions into two categor ies, instructional practices and learning characteristics, students achieved a mean of 13.74 out of 15, or 92%, on instructional practice questions and 9.00 out of 10, or 90% on learning characte ristic questions. Fo r all essay questions, the mean score was 58.11 out of 105, or approximately a 55%. Breaking the essay questions into effective instru ctional practices and applicatio n of these strategies, teacher candidates had a mean score of 31.53 out of 50, or 64%, on effective instructional practices, and 26.58 out of 55, or 49%, on application of these strategies. For questions

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163 scored under the effective instructional prac tices essay category, the mean score was 3.15, indicating that teacher candidates ofte n received a score of “a few main parts included.” On application essay questions the mean score was 2.43, indicating that teacher candidates achieved “a small part” correctly on items but missed most major points. This exam was only administered at the end of the semester, so there were not multiple administrations with which to compare teacher candidates’ results. However, overall results from the content exam were indicative that instructional strategy and learning characteristics multiple choice questions were answered at proficiency levels. Essay questions as a whole were answered ju st under beginning competency at 55%, but when broken down into instructional practices and application of these practices, it was found that questions on the instructional prac tices themselves were answered with a beginning competency level while applicati on questions were below this level of beginning competency. When appraising individual answer re sponses on the multiple choice questions, several items received a m ean of 1, both on instructi onal strategies and learning characteristics. The multiple choice item w ith the lowest mean score for instructional strategies was item 3, which as ked teacher candidates to corr ectly identify a mathematics instructional practice not emphasized for teaching problem solving strategies. The multiple choice item with the lowest mean score for learning characteristics was item 24, which involved teacher candidates’ correctly identifying learning char acteristics using an individual in a golfing context. In the essay section on instru ctional practices, the question with the highest mean (3.63) invol ved stating “the overall purpose of an instructional strategy”, from a choice of: the CRA sequence of instruction, structured

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164 language experiences, monitoring and charti ng student performance/progress monitoring, and explicit teacher modeling. On this quest ion, teacher candidates were most likely to choose CRA for the overall purpose descrip tion. The instructional practice essay question with the lowest mean (2.68) include d describing “how the language experience instructional practice for str uggling learners is applied wi thin the Developing Algebraic Literacy (DAL) instructional process”. For application essay questi ons, the question with the highest mean (3.26) was on describing “what effective mathematics instruction practice for struggling learners is exemplified by a strategy that is implemented during the third step of the DAL process and invol ves the use of the LIP strategy.” The application essay question with the lowest mean (2.43) included describing “what effective mathematics instruction practice for struggling learners is exemplified by a strategy that is implemented during the second step of the DAL process and is used to evaluate student abilities to read, represent, solve, and ju stify given a narrative context that depicts an algebraic thinking concept.” In summary, teacher candidates achieved proficiently on multiple choice questions, with the most frequently inco rrect questions involving determining which instructional practice had not been taught as an effective practice for problem solving and determining learning characteristics within a golf-based context. With essay questions, teacher candidates achieved just below beginni ng competency rate, indicating more work needed in both understanding instructional practices and their application. On the effective instructional practi ce essay questions, the question that was scored highest was one where students were asked to describe the purpose of one inst ructional strategy, out of a choice of four possible ones. The lowest mean score involved describing the

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165 structured language experience strategy fo r use with struggling learners. With application essay questions, the item with the highest mean score involved identifying the effective instructional practice used in the instructional strategy within the LIP section of the DAL, while the question with the lowe st mean surrounded doing the same for a strategy that involved using narrative text. Table 32 Descriptive Statistics for the Instructional Knowledge Exam _______________________________________________________________________ Instructional Total Item Median Range SD Skewness Kurtosis Standard Knowledge Mean Mean Error of Exam Mean _______________________________________________________________________ **These items have median and range calculated on the total mean versus the item mean. All other medians and ranges are calculated based on item mean. Pre (N=19) Full Survey** 80.84 1.76 85.00 79.00 0.440.50 0.05 4.68 *Multiple Choice** 22.74 0.91 23.00 5.00 0.07-1.21 0.71 0.10 -Instruct. Practices 13.74 0.92 0.93 0.27 0.07-1.32 2.36 0.15 -Learn. Barriers 9.00 0.90 1.00 0.40 0.07-1.0 2.36 0.03 *Essay** 58.11 2.77 61.00 76.00 0.940.43 -0.09 0.22 -Effective Practices 31.53 3.15 3.10 3.00 0.910.25 -0.79 0.21 -Applic. 26.58 2.43 2.45 4.18 1.100.37 0.05 0.25

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166 Inferential Statistics for the Instructional Knowledge Exam After descriptive statistics were analyze d, correlational analyses were completed between the instructional exam and the thre e other instruments also administered at approximately the same time, which were th e posttest surveys for efficacy, attitude, and content knowledge. These correlations were completed to evaluate possible relationships between instructional knowledge and the othe r teacher preparation factors of efficacy, attitude, and content knowledge. These fi ndings are presented in Table 33. No significant correlation was found between these ot her surveys and the in structional exam. Correlational analyses were also completed between the full multiple choice, learning characteristic (MC), instructional practice (MC), full essay, instructional strategies (Essay), and application (Essay) sections of the instructional exam. A moderate correlation was found between th e full battery of multiple choice questions and the ones on instructional practi ce (r=.671, p=.002), and a strong correlation was seen between the full battery of multiple choi ce questions and the learning characteristic ones (r=.822, p<0.001). These results indicate there is an association be tween how teacher candidates performed on the full group of multiple choice questions and how they performed on the two specific types of questions within it. For the essay portion of the exam, very strong correlations were found between the full batt ery of essay questions, and ones on both effective instructional pract ice (r=.907, p<.001) and app lication (r=.948, p<.001). These data indicate an extremely strong associat ion between how teacher candidates performed on the full group of essay questions and the two different types of questions. The data generated between the types of questions and the two di fferent subtests depicts a

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167 relationship between how teacher candidates answered on the subtest as a whole and how they answered on specific quest ion types in the subtest. Table 33 Correlation Matrix for the Instructional K nowledge Exam and Efficacy, Attitude, and Content Knowledge Posttests ______________________________________________________________________ Efficacy 3 Beliefs 3 Content 3 Instruction ______________________________________________________________________ Efficacy Pearson Correlation 1 0.003 0.465 0.152 Sig. (2-tailed) 0.989 0.052 0.533 N 19 19 18 19 Beliefs Pearson Correlation 0.003 1 0.21 0.047 Sig. (2-tailed) 0.989 0.402 0.848 N 19 19 18 19 Content Pearson Correlation 0.465 0.21 1 0.511 Sig. (2-tailed) 0.052 0.402 0.03 N 18 18 18 18 Instruction Pearson Correlation 0.152 0.047 0.511 1 Sig. (2-tailed) 0.533 0.848 0.03 N 19 19 18 19

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168 Table 34 Correlation Matrix for the Inst ructional Knowledge Exam Subs ections and Question Types ______________________________________________________________________________________________________ Multiple Choice MC-Inst. Prac. MC-Learn. Char. Essay Essa y-Inst. EssayPrac. App. ______________________________________________________________________________________________________ Multiple Choice Pearson Correlation 1 0.671 0.822 0.362 0.52 0.198 Sig. (2tailed) 0 0.002 0 0.127 0.022 0.417 N 19 19 19 19 19 19 MCInst. Prac. Pearson Correlation 0.52 0.305 0.462 0.907 1 0.727 Sig. (2tailed) 0.022 0.204 0.046 0 0 N 19 19 19 19 19 19 MCLearn. Char. Pearson Correlation 0.198 0.207 0.105 0.948 0.727 1 Sig. (2tailed) 0.417 0.395 0.667 0 0 N 19 19 19 19 19 19

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169 Table 34 (cont.’d) ______________________________________________________________________________________________________ Multiple Choice MC-Inst. Prac. MC-Learn. Char. Essay EssayInst. EssayPrac. App. ______________________________________________________________________________________________________ Essay Pearson Correlation 0.362 0.268 0.279 1 0.907 0.948 Sig. (2tailed) 0.127 0.267 0.247 0 0 N 19 19 19 19 19 19 EssayInst. Prac. Pearson Correlation 0.671 1 0.13 0.268 0.305 0.268 Sig. (2tailed) 0.002 0.595 0.267 0.204 0.267 N 19 19 19 19 19 19 EssayApp. Pearson Correlation 0.822 0.13 1 0.279 0.462 0.13 Sig. (2tailed) 0 0.595 0.247 0.046 0.595 N 19 19 19 19 19 19

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170 Fidelity Checks The fifth area of investigation was teach er candidates’ abilities to implement instructional practice knowledge for teaching mathematics to struggling learners within the DAL framework at their practicum site. To evaluate teacher candidates’ abilities to convert their knowledge about effective mathem atics instructional practices into actual practice, observations were conducted using fi delity checklists. These checklists were employed during observations of a subgroup of teacher candidates within the full participant group. Two different types of observation checklists were developed. The first checklist was for the DAL initial session probe, which is a shortened version of the full DAL session. This initial probe uses only 7 sections of the full DAL process, which fall under Step 2: Measuring Progress & Ma king Decisions in a full DAL session. The second checklist was for the DAL full instruc tional session, which includes a total of 34 implementation sections. In both types of DAL fidelity checklists, most sections of DAL implementation are required to use the model in accordance with framework guidelines. However, there are a few steps that may be considered “Not Applicable” because of student learning needs. For example, students may not require “problem-solving assistance” in a particular step, so that se ction would be marked “NA” and not included in the total number of sections required for fidelity calculations. Within the study, three evaluators observ ed teacher candidates’ implementation of DAL instruction until at least 90% agreemen t was reached on section ratings between evaluators. Three observation sessions were required for agreement purposes. Then, raters independently observed teacher candi dates performing instruction. The original

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171 goal of the study was to have raters observe approximately one-third of teacher candidate participants through thre e sessions: one observation at the start of DAL instruction, one at midpoint, and one at the end of DAL instructional implemen tation. Several intervening variables prevented the researcher from attainin g this goal. The reasons for difficulties in collecting observational fidelity checklist data were manifold. Many teacher candidates were not able to hold three sessions that included an initial session probe and two full sessions, which would have been ideal for data collection purposes. With the study being only ten weeks, unexpected challenges were met with school issues and programs, student illness and withdrawal from school and teacher candidates’ absences from practicum. During the course of the study, two instructional days were lost because of a “lock-down” for safety reasons on one day, and scheduling issues over picture day on another. Additionally, elementary school student absence was high including several students withdrawing from school. At the same time, on at least tw o instructional days, 4-5 teacher candidates were absent from pr acticum due to illness or personal reasons, which is typically an unusual occurrence. Finally, the initial DAL assessment for instruction took many teacher candidates 3-4 in structional sessions to complete, reducing the overall number of instructional sessions they completed. All of these reasons decreased the number of teacher candidates w ho were able to conduct three instructional sessions above and beyond the in itial DAL assessment. As a result, the possibilities for observing instructional sessions for fide lity checks were greatly diminished. Fidelity data on initial in structional sessions is contai ned in Table 35. It includes observations of 9 teacher candidates. Fideli ty of implementation of the DAL framework was high in these initial sessions, with a mean of 95% fidelity on all sections in the DAL

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172 initial session probe, with all but two teach er candidates showing 100% fidelity to the instructional model. Additionally, teacher candi dates were able to complete all sections contained within the initial probe during thei r instructional sessions. Only one teacher candidate did not have the same number of “total initial probe sections completed” because her student had one section, which in cluded teacher candidate assistance with problem solving, that was not needed in the instructional process be cause the student had no difficulty with any problem presented. As a result, that particular section was omitted by the teacher candidate, and was marked “Not Applicable” by the observer, and was not counted in fidelity calculations.

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173 Table 35 Fidelity Checklist Results on Initial Instructional Sessions ________________________________________________________________________ Participant Initial Probe Total Initial Total Sections Initial Probe Number Sections Probe in Initial Probe Fidelity Percentages Accurately Sections (Accurately Completed Completed Completed Sections/Sections Completed) ________________________________________________________________________ 2 7 7 7 100.0% 3 5 7 7 71.4% 4 7 7 7 100.0% 5 7 7 7 100.0% 6 7 7 7 100.0% 9 7 7 7 100.0% 11 7 7 7 100.0% 17 5 6 7 83.3% 19 7 7 7 100.0% Total: 59 62 63 95.0% From the initial observation group of 9 participants, midpoint observations then involved a reduction in approximately half the participants, as seen in Table 36. Participants who were observed showed a noticeable decrease in their ability to implement DAL instruction along framewor k guidelines. This difficulty with implementation may have been due to the fact that the framework contains a total of 34 sections of implementation at the full sessi on level. Teacher candidates may have had difficulty in remembering the order and compone nt parts for sections for implementation.

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174 At the same time, the number of “session sections completed” can be seen to vary across participants for the first full session because for fidelity calculations the total of 34 DAL sections was not included, but simply the numbe r of sections that we re covered in that DAL instructional session by that particular teacher candidate. None of the teacher candidates completed all 34 sect ions in their first session. Since the DAL framework is cyclical in nature, teacher candidates were not expected to complete all 34 sections in one session, especially while just learning the m odel. Teacher candidates were taught to move through sections in orde r until the end of an instructi onal session and then pick up where they had left off in th e next instructional session. When observing during this second round of observations, raters noted that teacher candidates had difficulties implementing the model. The main reason for the decrease in fidelity was that many teacher can didates did not cover the sections in order or left out key parts of sections for a vari ety of reasons. Some teacher candidates told raters they could not remember how to accu rately implement the key parts of certain sections. Others mentioned they thought th ey could eliminate “unimportant” parts of sections for time purposes. A few teacher ca ndidates deleted whole steps (i.e., all the sections under Step I: Building Fluency) because they felt they had spent too much time on a particular earlier section and should m ove forward towards the end of the process, which involved introducing a new skill. When teacher candidates actually attempted the key parts of a particular instructiona l section, they typically employed pedagogy accurately and in accordance with DAL guidelin es. As a result, the chief issue with fidelity was teacher candidates omitting key parts of sections, entire sections, and even whole steps during implementation. For a ll participants observed, a mean of 60.3%

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175 fidelity to the model’s guidelines was seen across implementation of the full DAL session. Table 36 Fidelity Checklist Results on 1st Full Instructional Sessions ________________________________________________________________________ Participant 1st Full Session Total 1st Full Total Sections 1st Full Session Number Sections Session in Full Session Fidelity Percentages Accurately Sections (Accurately Completed Completed Completed Sections/Sections Completed) ________________________________________________________________________ 2 6 10 34 60.0% 3 12 20 34 60.0% 4 13 15 34 86.7% 9 2 4 34 50.0% 11 10 20 34 50.0% 17 4 9 34 44.4% Total: 47 78 204 60.3% In Table 37, the final table of fidelity implementation information is presented, with only two teacher candidates being observed. In this particular session, one teacher candidate spent a considerable amount of time reading the context for problem solving with her student. Due to instructional session time limits, there was only enough time for the teacher candidate to implement two secti ons of the DAL process in her session, which she did with fidelity. The other teacher candida te was able to implement most of the full DAL session, but said she became confused during the sections of Step 3: Problem Solving the New, while trying to employ the making connections inst ructional strategy.

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176 As a result, she ended up skipping several sec tions. As a result, the mean fidelity to the DAL instructional framework for these two participants was 90.3%, which may not be totally accurate in reflecti ng the average fidelity, since it involved only two teacher candidates, one of which only made it through only two DAL sections. Table 37 Fidelity Checklist Results on 2nd Full Instructional Sessions ________________________________________________________________________ Participant 2nd Full Session Total 2nd Full Total Sections 2nd Full Session Number Sections Session in Full Session Fidelity Percentages Accurately Sections (Accurately Completed Completed Completed Sections/Sections Completed) ________________________________________________________________________ 2 2 2 34 100.0% 17 19 26 34 73.1% Total: 31 28 34 90.3% Summary of Quantitative Findings Quantitative results revealed an increase relationship between all survey instruments on efficacy, attitude, and conten t knowledge between pretest and midpoint, with a decrease seen on all of these instrume nts between midpoint and posttest. Subtests on these instruments also exhi bited a similar pattern. This information indicates that while teacher candidates increased agreement with items on these surveys, or accuracy in the case of the content knowledge, at the midway point of the study, these increases were not sustainable for the full length of the study. Instructional knowledge exam results indicated proficiency in id entification of instructi onal practices and learning

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177 characteristics, with continued work needed on the articulation of both instructional practice components and their application within the DAL framework. Fidelity checks showed that teacher candidates clearly could implement initial probe sessions of the DAL framework with fidelity, but needed continue d practice in this fidelity for full length DAL sessions. Qualitative Findings In this section, data collected through qualitative measures will be presented and analyzed. This information includes findings from final DAL project paper analysis for all teacher candidate participants, and two sets of pre and post focus groups. Within final DAL projects, the researcher coded teacher candidates’ ideas along the key elements identified within the study for special e ducation teacher preparation in mathematics instruction, involving attitude efficacy, content knowledge, and instructional knowledge and application. For focus groups, transcribed discussions were used to identify teacher candidates’ thoughts and ideas along the same key elements. Final Project Analyses To achieve greater clarity on teacher candidates’ experiences with the DAL framework in all five areas of investigation, teacher candidates’ final DAL projects were evaluated. These final analysis projects resulted from a cumulative DAL assignment where teacher candidates were asked to comp lete a summative paper on their learning. The writing assignment’s completion was guided by four prompts: a) what you have learned through your expe riences receiving training in K-5 algebraic thinking, training in the DAL instruction process and assessing and teaching your students using the DAL instruction process;

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178 b) how you will use what have learned for the future as a teacher; c) how (if at all) it has impacted how you feel about teaching mathematics; d) what areas of mathematics instruction (teaching mathematics to struggling learners) you believe you need to target for further professional development and why. Based on these guidelines, 17 of the 19 part icipants successfully completed this analysis paper. For the two participants w ho did not complete the paper, they chose not to turn this final DAL document in to th e Clinical Teaching professor for grading purposes and so the researcher did not have access to final documents for these two participants. The products of th e 17 papers that were turned in varied in length from 1 to 17 pages, as well as in th e content presented, even t hough the above written content guidelines were provided. All teacher candidate s’ papers were scanne d into the Atlas.ti analytical software to assist the researcher in coding teacher candidate writing. During this analysis, candidates’ written statements were coded along four general themes: efficacy in teaching mathematics, attitude towards mathematics instruction, content knowledge, and instructional knowledge and a pplication of instru ctional practices. Instructional knowledge and application were included as one theme because of the selfdisclosing nature of the assignment. As a result, it was unknown whether many of the discussed instructional practi ces were implemented, implemented effectively, or just conceptualized by participants. In Table 42, the number of descriptor codes for each theme is given, along with the types of codes under each theme, and frequency of occurrence of themes, as well as intensity of effect size for each type of quote.

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179 Table 38 Final Analysis Paper Themes and Codes ________________________________________________________________________ Theme Number of Frequency Types of Desc riptor Intensity Descriptor of Codes in Theme Effect Sizes Codes in Occurre nce (Percentage Theme of Total) ________________________________________________________________________ Efficacy 4 97 *Positive SelfEfficacy *Negative SelfEfficacy *Positive Student Outcomes *Negative Student Outcomes 17.1% Attitude 4 69 *Constructivist Mathematics Instruction (CMI) *Traditional Mathematics Instruction (TMI) *Constructivist Mathematics Learning (CML) *Traditional Mathematics Learning (TML) 12.2%

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180 Table 38 (cont.’d) Content Knowledge 14 76 *Manipulatives *Patterning *Student Performance *Equations *CRA *Explicit *Making Connections *Mathematics General Knowledge *Teacher Candidate Knowledge *Resources *Reasons *Progress *Standards *Structured Language Experience 13.4% Instructional Knowledge and Application 8 325 *Resources *Strategies *Learner Characteristics *Learning Environment *Individualized Instruction *Collaboration *Pacing *Development 57.3% Totals: 30 567 100% During the analysis and coding process, 567 different participant statements were coded using a total of 30 specific code s along the 4 major themes believed to be crucial in undergraduate special education teacher preparation in the content area of mathematics. Statements regarding efficacy in mathematics instruction were coded under

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181 4 categories: positive self-efficacy, negative self-efficacy, positive student outcomes, and negative student outcomes to parallel the type of information gathered through the Mathematics Teaching Efficacy Beliefs Instru ment. Under the 4 efficacy categories, 97 specific comments, 17.1% of all coded statemen ts, were analyzed and coded as involving efficacy concerns. Statements regarding attitudes and beliefs towards teaching mathematics were also coded under 4 categorie s: constructivist ma thematics instruction (CMI), traditional mathematics instruction (TMI), constructivist mathematics learning (CML), and traditional mathematics learni ng (TML). These coding categories were selected to parallel the attitudinal data collected via the Mathematical Beliefs Questionnaire. Using the four attitudina l coding categories, 69 teacher candidate statements, 12.2% of all coded comments, were analyzed and coded as involving teacher candidates’ attitudes towards mathematics instruction. Statements regarding content knowle dge were coded under 14 categories including patterning, student performance, equations, CRA, explicit, making connections, mathematics general knowledge, teacher can didate knowledge, res ources, standards, structured language, manipulatives, reasons, and progress. Coding was not limited to match the Mathematical Content Knowledge for Elementary Teachers survey, because it was believed that there were many relevant te acher candidate comments that were made that did not specifically touch on just basic arithmetic or alge braic skills, which were the categories on the content survey. Many content knowledge code s involved types of content knowledge, influences on what c ontent knowledge is taught, and how students demonstrate particular forms of content knowle dge, which were all considered valuable points to be considered for analysis on this critical preparation element. Items were

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182 coded under one of the 14 categories if they pertained to content knowledge described by teacher candidates within their instructional se ssions or in their own learning process. Using the 14 content knowledge coding categories, 76 teacher candidate statements, 13.4% of all coded comments, were analy zed and coded as pe rtaining to content knowledge involving the teacher candidates’ DAL model experience. Statements regarding instructional k nowledge and application we re coded under 8 categories including resources, strategies, learner characteristics, learning environment, individualized instruction, collaboration, pacing, and development. Statements were coded as involving instructional knowledge if they discussed specifi cs of instructional strategy implementation, mentione d external factors relevant to instruction, or depicted relevant student learning characteristics for instruction. Using the 8 instructional knowledge coding categories, 325 statements 57.3% of all coded comments, were analyzed and coded as related to some form of understanding or usag e of instruction for struggling learners in mathematics. Efficacy Theme Specific comments made about efficacy in mathematics instruction made up the second most significant coding category overa ll. Statements coded under this theme included 57 comments which indicated positive perceptions of self-efficacy or student outcomes. The 40 remaining comments were co ded as negative views on the same two variables, showing teacher candidates expressing negative views less often than positive ones. Some teacher candidate statements evidenced student perceptions that the DAL framework had made an immediate impact on their efficacy, such as “With the practice utilizing this framework and studying the stra tegies used during the DAL sessions, I feel I

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183 have learned an effective process to teach sk ills and concepts rela ted to mathematics.” Another similar comment remarked, “I be lieve it has given me some good ideas on strategies to use when teaching math. I found the CRA, justification, and making connections to outside ma terial very important.” A number of efficacy comments were focused in on teacher candidates looking positively forward to ways in which they could further enhance their mathematics instructional efficacy. In this vein, one st udent stated, “Going th rough the DAL training and working one-on-one with math student s has made me more comfortable with teaching this subject. I feel I still have much to learn about understanding and teaching mathematics. I have never been very suffici ent in this subject and I have a hard time being enthusiastic about teaching the material This also makes it difficult for me to relate its importance to experiences outside the classroom.” Negative comments about efficacy tended to focus on teacher candida tes’ lack of comprehension of the DAL framework, deficiencies in tr aining and preparation with the DAL, and outside factors that detracted from teacher candidates being able to facilitate in struction. One such articulate comment along these lines include d, “At the beginning of the DAL process I was apprehensive about its effectiveness in he lping struggling learners in mathematics. Although we were given training on how to im plement the process I was not confident with it. I could not grasp the concept of how we were going to teach algebraic concepts by using a book. I understand the point that was made numerous times about how math and reading is inter-related; I just cannot figure out how.” Comments involving student outcomes as a result of instructional efficacy focused on the reasons why teacher candidate s felt instructional strategies affected

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184 positive change in student learning. Negative st udent outcomes were often attributed to a lack of mathematics instructional efficacy re sulting from factors outside the control of teacher candidates, such as a “lack of the right tools”, “supervisor modeling”, “instructional time constraints” and “student attendance”. Attitude Theme Statements about attitudes regarding math ematics instruction were framed around two different approaches. The first appro ach was a more traditional, rigid, and memorization-based view, which many teacher candidates felt they had experienced at some time during their k-12 school experi ences. The second approach was a more creative, flexible, and developmental and constructivist view of mathematics. For the attitude theme area, statements that invol ved teacher candidate attitude about the mathematics subject area in general, as well as teaching mathematics, were coded as either constructivist or trad itional. Attitudinal comment s involving constructivist DAL framework experiences with mathematics instruction far outweighed the formal comments made about teaching mathematics. This constructivist emphasis in teacher candidates’ statements may have been due to the fact that the DAL framework was designed based on current developmental NCTM process and curriculum standards, as well as the DAL experience being structured using a social-developmental constructivist approach to teacher preparation. Along these lines, one student said, “I was able to see the benefits of breaking things down and repres enting them first on a concrete level, then the representational level, and finally the abst ract level. I could see how this benefited both the students I was working with. It s eemed that they suddenly had “aha” moments when they suddenly understood a concept once it was represen ted on a different level.”

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185 Another student explained her recent change s in her formerly traditional views of teaching mathematics with, “Before we started this program, I felt that only people with a math degree should be teaching math. Howeve r, I know now that this is not true. Teaching math requires a teacher who can s caffold and provide information that is meaningful to students.” Ideas that exhibited more traditional views of teaching mathematics included, “I did not have a chance to do a “get to know you” activity because I was too rushed to make up for lost time”, indicating the teacher candidate’s rigid belief that a certain number of mathematics target concepts ha d to be covered during a certain amount of instructional time. Another teacher candidate indicated that one of her students “needed a thorough review each session of the previous se ssion” presenting this review as wasted instructional time and material that the t eacher candidate had to direct the student through, rather than present as further mathem atics exploration and discovery material for the student. Teacher candidates’ expression of attitudes about the general subject area of mathematics tended to concentrate on either their enjoyment of mathematics learning with statements such as, “I love mathematics” or their learning characteristics that thrived from building their own mathematical understand ings, with “I am one that feels at times that I am not learning anything, until I sit dow n and try to complete a paper or project showing or telling what I l earned.” More formal ideas about general mathematics learning seemed to stem from a general disl ike of mathematics that had developed from early mathematics learning experiences, a be lief that mathematics content should be “delivered” to them as well as students, and a lack of seeing connections between

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186 mathematics and everyday life. As one teacher candidate mentioned, “I have never been very sufficient in this subject and I have a ha rd time being enthusiast ic about teaching the material. This also makes it difficult for me to relate its importance to experiences outside the classroom.” Content Knowledge Theme Teacher candidate statements about cont ent knowledge in their final analysis papers focused primarily on their students’ performances on algebr aic thinking related content during DAL sessions. Most of these statements were made in reporter-like fashion about students’ grappl ing with and mastering concepts, which were presented during instruction. For instance, one teach er candidate commented on her student’s content knowledge with, “In our initial session together, Demarcus demonstrated the ability to read, represent, solve, and justify growing patterns at the representational level. Based on this information, we started our next session at the concrete level of patterning to help build automaticity.” The overw helming content area of discussion was patterning, specifically growing patterns. The reasons for this focus may be due to the DAL’s initial skills assessment, which all teacher candidates administered to their students, and the fact that pa tterning was the first area a ddressed by this assessment. Teacher candidates were trained to target their initial DAL instru ctional sessions on the first area on the initial assessment where students produc ed incorrect answers. For a majority of the students involved in the pr acticum, this area consisted of growing patterns. Following growing pa tterns, the second most discussed content area was setting up mathematical representations and finding their solutions, which are the subsequent skills assessed after patterni ng in the initial DAL evaluation measure. One teacher

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187 candidate’s comments about her student setti ng up multiplication problems illustrates this point, with “Student B learned that multiplicat ion tables represent groups of numbers and she learned how to group them.” While many comments identified a partic ular mathematical area by name (i.e., growing patterns), other comments focused on students’ means of expressing current mathematical understandings: “using a level of CRA”, “explicitly demonstrating”, “by connections between previous learning”, “employing resources”, and “providing their answers and justification orally.” The teach er candidates’ recognition of these different forms of expression for content knowledge were deemed important, because they showed that teacher candidates saw direct connec tions between the content knowledge students were actually learning and their abilities to ar ticulate their understand ing of this content using the instructional met hods the teacher candidates ha d employed with them when teaching. Instructional Knowledge and Application Theme Comments made by teacher candidates in regards to instructional knowledge incorporated the majority of coded statements made throughout the final analysis papers. The teacher candidates’ papers were f illed with examples of their usage and understanding of practices ta ught within the DAL framework. As part of their preparation in using the DAL model, strategies for reading and mathematics instruction presented in Appendix A and C respectively, we re explicitly taught to teacher candidates to facilitate instruc tion in algebraic learni ng. To this end, within their final analysis papers teacher candidates discussed the st rategies of “modeling, explicit instruction, active learner engagement, authentic contexts explicit inst ruction, progress monitoring

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188 and instructional decision making, metacognitiv e strategy instruction, structured language experiences, connection making across c ontent areas, connection making between concepts in the same content area, and scaffolding”. Additionally, teacher candidates included many statements regarding instruction that were not discussed explicitly within the framework, but may have been more im plicitly presented. These surrounding ideas included, “differentiated instru ction, collaboration, pacing of instruction, safe learning environments, external learning barr iers, flexibility, and planning”. The most discussed area of instruction included the usage of CRA, which is the one instructional strategy incorporated in ev ery step of the DAL process. Most of the comments surrounding the usage of CRA were positive, including statements linking understanding of instructional practices to their implement ation within the practicum, such as, “Through my teacher, and especially with Sunflower (student pseudonym), I was able to understand what concrete, representa tional, and abstract le vels of understanding are, and how to deliver instruction at each level.” Another example included, “CRA is a great concept that a teacher s hould use when teaching mathematics to at-risk learners. I never understood the importance of breaking down into these 3 components until I actually started to do it with my students.” The second inst ructional strategy that drew the most student comments was the use of oral language abilities to build and convey mathematical understandings. Interestingly, te acher candidates were ta ught explicitly to use structured language expe riences within the DAL fram ework, in the written form within the student notebook. However, ma ny teacher candidates showed through their final projects that they cons idered the oral language ab ilities exerci sed during the problem solving process (ie., read, represent, justify, solve) as valuable structured oral

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189 language experiences that de veloped communication abilities in mathematics. One teacher candidate described this experien ce through, “Another thing I loved about the DAL (and I plan to implement in my classroom) is for people to justify their answers. It did seem silly to ask ‘why is that pattern considered a growin g pattern’ and wait for ‘because you are adding more each time’ but it was helpful to see their thought process. Once we got into more complex problems, I saw it was harder for them to explain and that is when I found it imperative that they provide an expl anation.” Summary of Final Project Analyses Throughout the entirety of their final anal ysis papers, and the statements within these papers, teacher candidates described their ideas and development through their instructional experience. Wh ile ideas involving instructi onal strategies were expounded upon at length, many pertinent teacher candida te ideas about mathematics instructional efficacy, attitude towards mathematics, and c ontent knowledge gave an indication of the teacher candidates’ thought processes while teaching students mathematics within their practicum experience with the DAL. This in formation indicated that teacher candidates on the whole had more positive feelings of efficacy than negative ones, and constructivist views about mathematics and mathematics teach ing outnumbered traditional attitudes. A focus on patterning skills and student mean s of expressing conten t knowledge were the main ideas presented in the area of mathematics content knowledge. Focus Groups Focus groups were completed with all te acher candidate participants at two different points within the study. This data collection method was used to complement information gathered through the survey instru ments, course exam, final paper analysis,

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190 fidelity checks, and case studies employed in the study. The purpose of the focus groups was to obtain a more holistic perspective of the full group of teacher candidates on the five elements of teacher preparation under investigation: efficacy about mathematics instruction, attitudes towa rds mathematics instruction, content knowledge for mathematics instruction, inst ructional knowledge about teach ing mathematics to at-risk learners; and applica tion of instructional knowledge for teaching mathematics to at-risk learners. The first groups, the pre focus gr oups, were conducted after the initial week of training with the DAL framework. The sec ond set of groups, the post focus groups, took place on the very last day of the study, afte r the teacher candidates had completed their final instructional sessions with their student s. The total group of 19 teacher candidate participants was split randomly between th e two focus groups, with one group having 9 people and the other 10. The members of Focu s Group 1 were the same at pretest as at posttest, and the case was the same for Pre-Focus Group2. For each round of focus groups, the teacher candidates were pulled at th e end of their instructional day at their practicum site during their usual seminar time. For each of the five elements identified as relevant to spec ial education teacher preparation in mathematics, “big ideas” e xpressed in each focus group are listed by focus group administration and focus group number (either 1 or 2). These “big ideas”, listed in figures, were determined from analyzing transcribed focus group sessions, as well as notes taken in each session. The ideas presented are ones that received multiple mention within each group or multiple agreement by pa rticipants in each group. Analysis of these ideas is presented by key element at each administration.

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191 Efficacy – Pre Focus Groups Figure 13. Efficacy – Pre Focus Group 1. Encouraged about teaching algebra, sin ce did not know before training that it started with patterns Need to learn more strategies to facilitate problem solving Learning to teach mathematics will be a continuous process Comfortable teaching mathematics, because like mathematics Feel ready to teach concepts of algebr aic thinking have learned in training Comfortable teaching mathematics, because have middle school children at home who have learned the ty pe of mathematics we’v e been talking about When you have to teach something, you do what you have to do to learn it Collaboration between peers (e specially through this cohort experience) helpful in developing instruction Think will be challenging to teach, but excited to try it Learned helplessness can develop from poor math teaching

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192 Figure 14. Efficacy – Pre Focus Group 2. Feel uncomfortable talking to students about algebraic concepts, because don’t really understand them Apprehensive about working with 4th and 5th grade students Hard when been out of elementary school for a long time, and don’t use those math skills everyday We were not taught mathematics in an application based way in school, so we will have a hard time teaching it that way Okay teaching mathematics if have curri culum or written material to go from Comfort level depends on type of students we are teaching If had to teach regular algebra, couldn’t do it Comfortable teaching the highest ski lls on the DAL assessment, but not comfortable beyond the assessment Have confidence from taking mathema tics education course and learning mathematics strategies (student who had taken the mathematics education course) Feel don’t know any strategies for teaching mathematics effectively Word problems a challenging area to teach Most comfortable teaching concepts learned most recently Feel could teach patterns and basic equations Have to feel comfortable with specif ics of content to teach it to people Feel like don’t have a good grasp on ma th, because always have struggled with math

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193 Figure 14. (cont.’d) Feel at a disadvantage teaching mathem atics right now, because haven’t had mathematics education course yet Comfortable with regular mathematics During the pre-focus groups, teacher candida tes voiced a mixture of feelings in terms of efficacy and their instructional abili ties at the outset of the study. Many of the teacher candidates made comments about f eeling uncomfortable with mathematics instruction because they had never taught mathematics or used the DAL model previously. For instance, one specific comment included, “I get some of it, but not all or it and that’s not good enough for my kids.” Other common comments were ones of excitement, such as, “I think it will be ch allenging, but I’m excited.” While teacher candidates seemed aware of their own lack of experience, they also appeared to be looking forward to the instructional challenge at hand. While the majority of teacher candidates presented feelings of apprehensi on about performing math ematics instruction, especially with the use of mathematics st rategies, the second group evidenced specific concerns about mathematics instruction being different than when they were in school with, “I was taught differently so I don’t understand how they are being taught now.” They viewed this possibility as a drawback to easily learning to teach mathematics skills.

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194 Attitude – Pre Focus Groups Figure 15. Attitude – Pre Focus Group 1. Allow multiple ways of problem-solving Openmindedness – be willing and able to learn from your students To be successful in mathematics, students need the right tools Requires some rote memorization of multiplication facts Mathematics develops logi cal, problem-solving skills Promotes higher level (cri tical) thinking skills Promotes abstract concept development For best mathematics learning, applica tion to real life si tuations needed Early school experiences influenced now poor views of mathematics Students must develop problem-solving sk ills incrementally, teachers should not just give students answers Having mathematics knowledge is very important Sometimes how you get to a math answer not valued, but that’s actually the most important part More important to know the math process than the outcome Math almost like another language Flexibility is important

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195 Figure 16. Attitude – Pre Focus Group 2. You don’t use mathematics every day like you use reading Need to think outside the box when teaching mathematics Need multiple methods of problem-solving to find answers Own experience math was not positive, because did not learn multiple ways to solve problems Math is important because it is in every part of life Do not have a positive view of ma th because not confident in it Really like math, because had really good instructors Practical application of math when lear ning it, helped student really like it Foundation of how they were taught math influences how they now feel about math At-risk students need effective math instruction to beat cycle of failure Algebra is something you ha ve to learn to get through school, never going to use it Algebra encourages higher level thinking skills Can’t stand math – if don’t see connections will give up There’s always one right answer with math Don’t see creativity in math Attitude towards the mathematics subject area, and mathematics in general, seemed to vary across teacher candidates. So me teacher candidates spoke about their fear or dislike of mathematics that was tau ght to them in k-12 classrooms, “My own

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196 experience learning math was not positive, because I did not learn multiple ways to solve problems.” These feelings seemed to st em from their own experiences learning mathematics when they were younger, some of which were traditionally-based. Others, who mentioned they liked mathematics, stated the reason as being either because they found math easy or they had experienced te achers who had positively influenced their mathematics learning experiences through constr uctivist learning activit ies, “I had a great mathematics teacher, who was always open-minded and willing to learn from her students.” Either positive or negative, t eacher candidates’ own childhood school learning experiences had a large impact on how they currently viewed teaching mathematics. Most of the teacher candidates also saw the va lue in effective mathematics instruction for all learners, including student s at-risk for mathematics difficulties, but emphasized that connections must be made between mathematic s and real life situa tions. For instance, one teacher candidate said, “Math is important because it is in every part of life.” As a whole, teacher candidates’ current attitudes voiced about mathematics instruction seemed more constructivist with comments made a bout how “students should develop skills incrementally” and needing to “think outside the box” when teaching mathematics. However, at the same time, a few more trad itional views of instruction were presented, including, “there’s always one right answer in math” and “math is very rule-based.”

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197 Content Knowledge – Pre Focus Groups Figure 17. Content Knowledge – Pre Focus Group 1. General Mathematics Students need to be able to justify their answers Math is a computational process Need to teach students basic skills for everyday life, but doesn’t have to be through rote – counting up is one such strategy Answers in mathematics ar e either right or wrong Word problems need to be taught more Algebra Algebra is balancing equations Algebra is symbols & numbers Patterning Some algebra skills applicable to real life Other algebra skills just seem like learn for test Patterns are everywhere is everyday life Used to think algebra was a bunch of form ulas and gibberish, not involving other things like patterning Builds upon basic arithmetic skills Formula based Algebraic skills in the DAL assessment are understandable

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198 Figure 18. Content Knowledge – Pre Focus Group 2. General Mathematics When you’re just talking about numbers, that’s basic math Fractions are a trouble ar ea for a lot of people Don’t understand 5th grade math of children at home Basic skills like balancing checkbook a nd consumer math important for students Math should focus on what you’re going to use most often, not what never going to use Math has basic rules There are steps to be learned for problem solving Algebra Algebra is when you are ta lking about variables Algebra is when there’s letters in math A big part of algebra is the FOIL method Don’t understand concepts of algebraic thinking Think algebra content should be exposure only at the elementary level, not counted as part of grading Repeating and growing patterns Problems involving Xs and Ys Equations Comments about content knowledge were split between algebra and basic mathematics. Teacher candidate remarks a bout general mathematics learning tended to

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199 focus on mathematics “without variables” that involved computation and skills that could be applied in every day life. One teach er candidate actually expressed general mathematics skills as “when you’re just talk ing about numbers.” Most views on what algebra entailed focused on very traditional algebraic ideas based on balancing equations and usage of variables, such as “algebra is when you have variables” and “problems involving Xs and Ys”. A few statements we re made about algebraic thinking involving the new skills the teacher candidates had learned in the DAL training with specific mention of “repeating and growing patterns”. Of these DAL-related algebraic thinking skills, the one specific concep t that teacher candidates hone d in on was patterning, even though at this point they had already been exposed to deve loping patterns, functions, and relations; representing and solv ing equations, and analyzing ch ange in various contexts. The reason for the focus on this skill may be because it was the first aspect of algebraic thinking at the elementary level taught to t eacher candidates. Another reason may be that from their reactions and comments during the training workshops, teacher candidates seemed most comfortable with learning patte rning from among all of the skills taught during the training sessions.

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200 Instructional Knowledge – Pre Focus Groups Figure 19. Instructional Knowledge – Pre Focus Group 1. Must teach students at their leve l with ways they can understand Present multiple ways of problem-solving Making connections is important, but can be difficult based on the setting of your school Writing problems on the board – singling out students to answer them – doesn’t work Teachers need to understand math concepts, to be able to explain them to others Math should be taught as a process Key to understanding math as language is developing the vocabulary to go with it No reinforcement for getting partial answers in most of mathematics Important to relate mathematics to outside interests of students Use of manipulatives helpful Don’t rush instruction – no point in doing Intimidation doesn’t work with students to help them learn Make it okay to make mistakes Can tie in own experiences of learning math to help break down skills for students – because can relate to their struggle Planning for individual needs importa nt to teach at student’s level Using marker boards can be help ful for individual students Drawing your way out of a problem works

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201 Figure 20. Instructional Knowledge – Pre Focus Group 2. Explicit instruction is helpful for mathematics learning Need to know multiple ways of teaching concepts Base 10 blocks and other manipulatives helpful Have learned how to break down skills and use CRA from the DAL assessment Learner engagement – make algebraic thinking fun Need calculators for so me types of math Make problems applicable to real world using money and shopping Should use assessment to see if students can handle algebra Multiplication charts – strategy that won’t work Sometimes math songs are helpful Group work and cooperative learning helpfu l in math – because can see things from different perspectives Students need to have explanati ons behind methods they are using With instructional knowledge, teacher ca ndidates volunteered a large variety of strategies they thought important for usage w ith at-risk learners, including: CRA, making connections, explicit instructi on, using a variety of material s, learning the “mathematics” language, and learner engagement. Even after just the initial training, which came before these two pre-focus groups, and no direct a pplication with students, teacher candidates spoke about many of the strategies presented as key instructional practices within the DAL model. The reason may have been that these practices were still fresh in teacher candidates’ minds. Additionally, besides spec ific strategies, an emphasis of the first

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202 group was that teacher candidate s felt that support for student efforts in problem-solving, whether right or wrong, should be encourag ed. One teacher candidate mentioned, “There’s usually no reinforcem ent for partially correct answ ers, but there should be.” Another suggested supported problem solving through “the usage of cooperative learning groups to see different problem solving perspe ctives.” There was a definitive emphasis on using praise with the building blocks of student mathematical efforts, just as teachers often use with student attempts at sounding out long, multisyllabic words in reading, whether the final attempts at these words are correct or not. As one participant mentioned, “There’s always praise for soundi ng out parts of words correctly, but rarely any for getting partway to a math answer.” Teacher candidates felt that this incremental praise, for students mastering small parts of problem-solving, should merit more positive attention from teachers.

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203 Instructional Applica tion – Pre Focus Groups Figure 21. Instructional App lication – Pre Focus Group 1. Right now, so much to learn at once – trying to apply coursework to student sessions, but need more time with learning and more time with student Practice in practicum setting helps not with content application, but more how to work with children and their behaviors Really learn mathematics concepts by teaching them Our planning and reflection helps us to better meet the needs of our students Cohort setting valuable for support in teaching mathematics Small group environment fo r instruction is helpful In future, will seek out additional course work to help in content knowledge for instruction For instruction, will seek out In ternet resources and textbooks Need to learn the standards Question whether what we are learni ng here will apply at other schools

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204 Figure 22. Instructional App lication – Pre Focus Group 2. Need more time learning to teach mathematics, because harder than teaching reading Think this practicum experience teaching reading and mathematics will give them more to go on their first year of teaching Difficult because what doing in practicum and final internsh ip cannot prepare you for every type of mathematics learning situation between K-12 Important to think about what you’re doing – plan and reflect on how you’re teaching is going with your students Previous experiences teaching algebra have made teaching mathematics in the current practicum easier Need the learning of mathematics instruc tion spread out over a semester before engage in teaching it DAL training helping them to learn how to plan and organize instruction, and track student progress Relating mathematics to real world and pur pose for it – current strength of teacher candidates Feel hands-on experience will help them to teach mathematics Will seek out mathematics strategies text for teaching mathematics In the area of instructional application, many of the teacher candidates’ comments were more theoretical than anything else, si nce they had not begun working with students in algebraic thinking. However, many teacher candidates voiced that they thought the

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205 application aspect of the mathematics instru ctional strategies within the DAL framework would be helpful in them knowing better how to work with students in their first classrooms. Along these lines, one teacher ca ndidate said “DAL is helping me better learn how to plan and organize instruction” while another indicated, “I feel hands-on experience will help me in teaching mathem atics.” Additionally, many of the teacher candidates seemed eager to explore resources as future aids in instruction, such as curriculum texts and peer or mentor support relationships. These comments expressed a willingness by teacher candidates to reach out for assistance in the area of mathematics when actually applying instructional knowledge with ideas such as, “I will seek out mathematics strategies texts for teaching math ematics” and “The cohort aspect of this practicum is very helpful in terms of l earning how to teach mathematics. Teacher candidates voiced a desire to expand their knowledge gained from their coursework and related experiences, as they had more opportuni ties to apply skills in the classroom. In fact, one candidate mentioned that she planned to seek out “more mathematics courses for learning content kno wledge.”

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206 Efficacy – Post Focus Groups Figure 23. Efficacy – Post Focus Group 1. Feel still need to develop the mathema tics language skills to explain anything higher than patterning Wouldn’t use the DAL model for instruc tion again, because it’s too cumbersome trying to make the connections between al gebraic thinking con cepts and the texts that we’re using within the process Feel the steps in the DAL process are good for learning mathematics, for instance, making connections between different ideas Have good feelings about teaching math Still feel like I don’t kn ow anything about teaching math, except for patterns Feel like preparation in the special e ducation program in teaching the content areas, like mathematics, has a big impact on children because we can speak their (children’s) language now Don’t feel like really have any strategies for teaching mathematics, because just introduced to them and not really sure of them Feel defeated using the DAL framework, b ecause feel like never going to make it through all the steps and students will now be stuck on patterning Feel not enough time during DAL for student learning I know as much about mathematics and teaching mathematics now as when I started with the DAL

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207 Figure 23. (Cont.’d) I have the theory that if I know it, I can make someone else know it by using my way or inventing a new way Feel still need to know better how to teach math to other people Figure 24. Efficacy – Post Focus Group 2. Just don’t know if I can teach math to at-risk learners Still confused and not comfortable with the DAL, but think it could be valuable for at-risk students, if we knew how to use it better DAL model was difficult to understand – fe lt the problem was in the design of the program Hope that by taking the teaching mathematics course this summer, will better understand how to teach mathem atics to at-risk learners DAL model good for activating what kids already know about concepts and how can extend that usage and learning Do not feel prepared to teach students at risk for difficulties with mathematics at this point Still difficult to explain ma th concepts to other people I’m not good at the mathematics strategies, I’m not getting them Do feel more comfortable working with students one-on-one from DAL and UFLI experiences Didn’t feel like was teachi ng students a skill through DAL

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208 During the post focus groups, different types of comments were heard about efficacy in mathematics instruction then in pre focus group meetings. At this point, the teacher candidates had been involved in preparation with and implementation of the DAL framework for a ten-week period. Compared to the comments on efficacy from the prefocus groups, which were filled with appr ehension and excitement about the unknown, this later round of comments was spoken from the frame of experience. Many teacher candidates voiced concern that having atte mpted instruction using the DAL, they were now more aware of all the aspects of mathem atics instruction that they still did not understand. More than one teacher candidate sa id that she, “did not feel prepared to teach at-risk learners at this point.” For several teacher candidate s, this feeling was converted into the desire to seek out further learni ng with, “I hope by taking the mathematics methods course this summer th at I will better understand how to teach mathematics to at-risk learners”, while others had internalized difficulties with instruction by doubting their own overall abilities as educators by saying, “I’m not good at mathematics strategies, I’m not getting them.” Still others voiced that they thought the difficulty with instruction was due to the de sign of the DAL framework itself, “The DAL model was difficult to understand, I felt it was due to the design of the program.”

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209 Attitude – Post Focus Groups Figure 25. Attitude – Post Focus Group 1. Math is very important for a child at-risk for mathematics failure Basic concepts like patterning are importa nt to higher level mathematics further down the road It’s important to cultivate students’ understanding of basic concepts in general mathematics Sometimes children have memorized formulas, which is not good, because when they really need to understand what’s going on behind those things they don’t I used to have a great fear of math, but now that I’ve worked with it, I’ve lost some of that fear To be a good math teacher, you have to know it, and be able to understand and explain it Feel like you have to be a math teacher to be able to explain math to people ( some focus group members) Feel like you have to know the language of the people you’re talking to, and be able to explain ideas to these people ( other focus group members) A lot of bad experiences in math were because teachers knew math, but couldn’t explain it Helping fill in students gaps with mathematics learning can be like figuring out a puzzle I like math, so math really comes easily to me

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210 Figure 25. (Cont.’d Math is strict and has a lot of rules Math is like a puzzle that works out I’m not going to teach math, so I’m not goi ng to plan on it or worry about it – I chose the population I want to work with based on the fact that I won’t have to teach math Want to be a math teacher who’s always trying to learn from and be open to students Figure 26. Attitude – Post Focus Group 2. Think mathematics important to at-risk learners Mathematics learning has to be active for at-risk learners With at-risk students, not sure about the value of learning alge braic properties in their overall mathematics learning I’m not strong in math, so feel like I needed to be prepared to teach mathematics like an at-risk learner because my weakness is in math Wish had been taught math instruction mo re prescriptively than trial and error method for students’ needs Afraid of negatively impacting student le arning and perspectives on mathematics Just because you know math, doesn’t mean you can teach math – just may have some skill with higher level concepts

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211 In terms of teacher candidate attitude towards mathematics and mathematics instruction, some comments from the pre fo cus groups resurfaced, while new issues also appeared. First, the importance of mathema tics instruction for at-risk learners was still valued, as well as the significance of cultivating a positive student outlook on mathematics seen through, “Mathematics is very important to at-risk learners” and “Basic patterning skills are important to higher level mathematics learning further down the road.” Many of the teacher candidates agai n reflected on how it was their own traditional experiences at the elementary level with ma thematics that had turned them off from mathematics learning with, “A lot of my bad experiences in math were because knew teachers math but could not expl ain it.” They were determin ed as a group not to “do” the same to the students they now teach. An additional concern included that many teacher candidates were concerned about teaching mathematics because they realized how students are now taught math is much differe nt, more constructive, from the way they were taught mathematics themselves. They had found it difficult to teach in this “new” conceptual way because they needed to know “the language of mathematics” and “the language of their students” to teach mathem atics effectively. A second new topic was that many teacher candidates thought the languag e element of mathematics, being able to explain concepts to students and then havi ng students do the same, was integral to students’ mathematics learning with “To be a good math teacher, you have to be able to explain mathematics.” Many felt they had ga ined this new concern about “explaining mathematical ideas” as they had attempted to demonstrate seemingl y simple mathematics concepts to students within the DAL, and f ound it was not an easy task. Both a mixture of constructivist and traditional attitudes about teaching mathematics were still presented

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212 with a few teacher candidates expressing that “math is like a puzzle that you have to figure out”, while others thought of math as “strict with a lot of rules.” Content Knowledge – Post Focus Groups Figure 27. Content Knowledge – Post Focus Group 1. I can teach patterning, but still have a hard time justifying answers Graphing – graphing is an area of knowledge strength Everything else but gr aphing is a weakness Statistics – it’s a different type of thi nking, you can relate basic mathematics skills more easily with it I’m going to get a tutor to help me understand concepts I don’t get that I’m still trying to understand – to refresh my memory Figure 28. Content Knowledge – Post Focus Group 2. Taught patterns because co mfortable with patterns In my future teaching, I’m going to follow the curriculum and what I should do – and then I will be okay Think scope and sequence of skills is important for mathematics instruction Understand algebraic thinking skil ls – confident to teach them One of the interesting developments from the pre to post focus groups was the narrowing of the discussion on content knowledge. In the first focu s groups, extensive comments were made about the different elemen ts of basic arithmetic and algebraic type problems. However, in the second round of fo cus groups, very little time was spent by

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213 teacher candidates discussing the nature of these areas, but primarily time was spent on one of the DAL’s content focal areas, pattern ing. This change in teacher candidate comments may be due to the way the question or phrase was asked in post focus groups. It could also be due to the fact that many of the teacher candidates almost exclusively focused on patterning skills while worki ng with their students within the DAL framework, because patterning is one of the most basic algebraic skills assessed for proficiency in the DAL initial assessment and is where many students had exhibited difficulty. At the same time, teacher candidates di d make comments about the connectedness of mathematics curriculum, stated with “I think the scope and sequence of skills is important for mathematics instruction.” Teacher candidates also mentioned other mathematics areas which they felt they were prof icient in such as st atistics and graphing. These comments seemed to stem from their fr ustration with the curr ent algebraic thinking they were teaching, as well as difficulties with other areas of content. One such comment included, “I like statistics – it’s a different type of thinking. You can relate basic mathematics skills more easily with it.”

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214 Instructional Knowledge – Post Focus Groups Figure 29. Instructional Knowledge – Post Group 1. In math, it’s important to teach kids ways to remember things, so they can do it again CRA is useful Use of manipulatives is a good idea Planning and reflection are important b ecause they help you make connections between ideas and concepts Think it’s necessary to relate mathematic s in a way that students will understand If you reflect on your instruction, easier to see where studen ts struggling with content and where you too may be st ruggling with content or going wrong Language you use to explain ideas to students is very important Instruction can really impact the way a child understands concepts Copying from the board is not a good mathematics strategy Kill and Drill – doesn’t work for teaching mathematics Connecting learning to past experiences is helpful Building on strategies students already know is a good instru ctional technique Sometimes schools and administrative staff will have mathematics resources that will help and guide you in your learning about mathematics curriculum – these resources would be great to have at your school when you are a new teacher Being well-prepared and getting extra re sources is important to good mathematics instruction

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215 Figure 30. Instructional Knowledge – Post Focus Group 2. Planning and reflection in teaching math ematics helps you know what you should be doing and keeps you from making so many mistakes Explicit instruction important for in dividuals who are at-risk learners We needed more modeling to better understand the DAL process, more reiterations too Needed to be more explicitly ta ught DAL and have it broken down into steps/parts Important to be flexible and base instruc tion off of students’ individual interests CRA is a great tool Making connections between learning topics /areas in mathematics is important Manipulatives are valuable to use in mathematics Relating mathematics to kids’ own lives is essential Individualized instruction is an important tool for students at -risk for difficulties in mathematics One strategy is not going to work for all ki ds, so need to have a bag of tricks full of instructional strategies With the instructional know ledge piece, teacher candidates also approached this topic from a different angle than in pre focus groups. Teacher candidates again spoke extensively about strategies that were taught through the DAL, as they did in the pre focus groups as well, including “CRA is us eful” and “Use of manipulatives is a good idea.” However, a few other ideas that corr esponded with these strategies were also

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216 discussed, including instructiona l flexibility, using students’ individual interests, and differentiating instruction while teaching. As one teacher candidate sa id, “One strategy is not going to work for all kids, so you need to have a bag of tricks full of instructional strategies.” The other shift in focus for instructional know ledge was teacher candidates’ statements about what strategies best he lped them learn and retain mathematics instructional strategies. It was interesting th at many of the same strategies used within the DAL framework itself, were ones that teacher candidates felt would enhance their actual learning process of teaching mathematics. For instance, one participant said, “I needed more modeling to understand the DAL process” and another mentioned, “I needed to be taught the DAL more explic itly.” The last shift in emphasis was on planning and reflection. Teacher candidates voiced ideas that these two concepts were integral to facilitating instructional sessions and improving the quality of these sessions. One candidate stated this idea succinctly with, “Planning and reflection in teaching mathematics helps you know what you shoul d be doing and keeps you from making so many mistakes.”

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217 Instructional Applicatio n – Post Focus Groups Figure 31. Instructional App lication – Post Focus Group 1. I’ve learned teaching math can be fun Think needed more time to work with stude nts on math to be able to explain it to the students better Needed more time with the DAL to be able to teach with it effectively Through this practicum experience, feel have learned to relate to kids Needed more time spent on explicit strategy learning for us to be able to apply these strategies with students Found out by teaching math in this practicum that the “having to explain” piece is very helpful, because found out where I am having trouble and need help with mathematics when trying to explain it Going to use textbooks to try and access c ontent when have to teach mathematics in the future Very helpful to be taking teaching mathematics and practicum at the same time – helpful for thinking of ideas for practicum

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218 Figure 32. Instructional App lication – Post Focus Group 2. Hard to teach mathematics through reading, if child is a struggli ng reader as well Scope and sequence chart would be a good tool to use with the DAL Didn’t have peer support while teaching DAL as with UFLI, because peers didn’t understand the DAL DAL would be easier to apply in a classr oom or resource setting, than with oneon-one instruction After this experience, st ill have more things th at I want to know about mathematics instruction, so I can better teach stud ents in their classrooms During the practicum, did a lot of research on the Internet on algebra, so would be comfortable in telling students how to do things with algebra The more you do something like teaching math, the easier it is to do it In the future, I will find a mentor to he lp me with my mathematics instruction I will continue to learn more about teaching math Think would have been helpful with this practicum to have had teaching mathematics class beforehand so w ould have had background knowledge for instruction In the future, workshops will be helpfu l in gaining help with instruction Through this process, I think I better unde rstand the process of students’ thinking and why they think that way – know better where kids are coming from Through this experience, feel comforta ble with K-5 instruction, but not 6th grade and up

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219 Lastly, teacher candidate responses had a mu ch different vein a nd tone to them for instructional application. Ma ny of the issues brought up ab out instructional application concerned items that teacher candidates str uggled with during thei r application of the DAL model, such as the amount of time for mathematics instruction and how much they felt they still needed to learn about teach ing mathematics in such a structured and systematic manner. One such comment included, “After this experience I still have more things that I want to know about mathematics in struction, so I can bett er teach students in their classrooms.” There were also sugges tions made about the DAL model itself, and concerns with the DAL framework’s implementation, with “DAL would be easier to implement in a classroom or resource setti ng than with one-on-one instruction.” These ideas included more time and support for unders tanding the model, as well as a different setting, such as classroom or small group, for application. One teacher candidate stated that she “needed more time with DAL to be able to teach it effectively”, while another mentioned she “didn’t have peer support wh en teaching with the DAL” because her peers did not understand the process. Focus Groups Summary Overall, comments in the pre focus groups seemed to be positive yet anxious about efficacy in terms of learning a new form of instruction, DAL, and how teacher candidates were going to apply this knowledge in practicum. After the DAL experience, the bulk of teacher candidates’ comments we re filled with frustration and a new realism about the problems associated with actu ally working with and applying the DAL framework. Attitudes about mathematics instruction tended to be more constructivist at both pre and post, but traditional views were al so presented. A large impact on attitude

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220 also seemed to be early mathematics learni ng experiences in k-12 e nvironments. Content knowledge was focused on traditional ideas of algebra and basic mathematics skills during pre focus groups, and changed to pr imarily encompass patterning and related algebraic learning ideas during post focus groups. The post focus groups had more specifics about the DAL framework and inst ruction versus the pre focus groups, where teacher candidates had not really begun to implement instruction. The second set of focus group comments seemed to have less idealism, and more of the voice of the “experienced” teacher after he or she has undergone a real-life teaching experience and realized the issues connected to instruction for at-risk learners. Summary of Qualitative Findings Final project analysis and focus groups provided valuable insights into the efficacy, attitude, content knowledge, and in structional knowledge and application elements identified as pertinent to teache r preparation in mathematics instruction. Throughout teacher candidates’ statements and comments, it was seen that feelings of efficacy were higher before actual instruction was begun with students. It appeared that working with students, and perhaps being f aced with different challenges as a result, negatively impacted teacher candidates’ feeli ngs about their instru ctional abilities in mathematics. While both constructivist and traditional attitudes were presented by candidates, their comments were predominantly constructivist and these views were maintained through the end of the study. Ma ny teacher candidates i ndicated their ideas about attitude towards mathematics and math ematics instruction had their foundation in k-12 learning experiences. Content knowle dge was viewed traditionally, as numbers being involved in basic mathematics and sym bols and letters being th e root of algebra,

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221 until after experiences working with the al gebraic thinking concepts within the DAL model. This experience expa nded candidates’ ideas most in algebraic thinking pertaining to patterning and representing mathematical situations. For in structional knowledge, teacher candidates expressed familiarity with the pedagogical practi ces taught within the DAL framework; but in terms of instruc tional application, some teacher candidates voiced some difficulty applying them with st udents. However, other teacher candidates explained that their students made large gains in content under standing through learning facilitated through thes e instructional practices. Case Studies Case study data were collected on the thr ee individuals selected from within the ranked groups of overall particip ants as part of th e qualitative data collection and analysis process. The aim of this data collection wa s to clarify individual learning experiences within the total group of part icipants, as well as capture more specific information not available through quantitative methods on teach er candidate self-efficacy for teaching mathematics, attitudes toward teaching math ematics, knowledge of mathematics content, knowledge and understanding of re search-based mathematics in structional pr actices for at-risk learners, and applicati on of research-based mathematics instructional practices for at-risk learners. For the pur poses of this study, the participant from the high achieving group will be called Olivia, the participant from the mid achieving group will be called Kari, and the participant from the low achieving group will be called Taylor. For each case study participant, several types of da ta were accumulated including individual quantitative data on the three surveys and c ourse exam. Additiona lly, qualitative data were amassed in the form of complete DAL project artifacts, final analysis papers, and

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222 research exit interviews. The data for each case study participant are presented below by participant and data collection method. Olivia Mathematics Teaching Efficacy Beli efs Instrument Overall Efficacy For the Mathematics Teaching Efficacy Beliefs Instrument pretest, Olivia was absent because of illness and was not able to make up the test in the available time period. However, she was present for both midpoint and posttest administrations. At midpoint, on the overall efficacy measure, Olivia did not mark any ite ms as “Strongly Disagree” or “Disagree”, indicating that she did not view any of her mathematics rela ted teaching abilities negatively in terms of efficacy. She did ma rk “Uncertain” and “Agree” an equal number of times, with both having 28.6%. Olivia al so noted that she “Strongly Agreed” with statements about efficacy in mathematics just under half of the time (42.8%). At posttest, there was some change in Olivia’s responses She indicated a decrease in her overall feelings of efficacy, with 4.8% of her re sponses “Disagreeing” or showing negative feelings of efficacy compared to none of th ese responses on the midpoint survey. While at the same time her “Agree” re sponse level went up slightly to 33.3%, but her “Strongly Agree” responses evidenced a considerable decrease to 23.8%. Olivia’s results on the full Mathematics Teaching Efficacy Beliefs Instrument show a slight decrease in her feelings of efficacy in teaching mathematics from midpoint to posttest. Table 39 shows data on Olivia’s overall efficacy survey.

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223 Table 39 Olivia: Mathematics Teaching Effica cy Beliefs Instrument Overall __________________________________________________________ Pretest Midpoint Posttest __________________________________________________________ Strongly Disagree 0.0% 0.0% Disagree 0.0% 4.8% Uncertain 28.6% 38.1% Agree 28.6% 33.3% Strongly Agree 42.8% 23.8% Self-Efficacy. At the midpoint administration, on th e survey items directly related to self-efficacy in mathematics instruction, Oliv ia indicated only positive feelings of selfefficacy. She did not respond to any items as “Strongly Disagree”, “Disagree”, or even “Uncertain”. The majority of her midpoi nt self-efficacy responses were “Strongly Agree”, showing this highest efficacy rating 61.5% of the time. At posttest, results were much more spread out across ratings, with a small amount of negativ e feelings of selfefficacy (7.7%) and “Uncertain” indications (15. 3%). While Olivia maintained her level of “Agree” statements, her “Strongly Agree” statements fell sharply to 38.5%. The results of the self-efficacy que stions show a decrease in Oliv ia’s feelings from midpoint to posttest. Table 40 presents the da ta on Olivia’s self -efficacy subtest.

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224 Table 40 Olivia: Mathematics Teaching Efficacy Beliefs Instrument – Self-Efficacy _________________________________________________________ Pretest Midpoint Posttest _________________________________________________________ Strongly Disagree 0.0% 0.0% Disagree 0.0% 7.7% Uncertain 0.0% 15.3% Agree 38.5% 38.5% Strongly Agree 61.5% 38.5% Outcome Expectancy. Olivia did not indicate any nega tive feelings of efficacy in affecting student outcomes in mathematics at midpoint or posttest. However, she did respond with a high level of uncertainty about her feelings, selecting 75% of her answers on both administrations as “Uncertain”. During the midpoint administration, Olivia’s positive feelings for outcome expectancy were equally split between “Agree” and “Strongly Agree”, while at posttest, all of he r positive responses had fallen slightly to “Agree” with no “Strongly Agree” responses indicated. The results for the outcome expectancy subtest indicate a slight decrease in the strength of Olivia’s positive feelings from midpoint to posttest. Olivia’s outcom e expectancy subtest results are shown in Table 41.

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225 Table 41 Olivia: Mathematics Teaching Ef ficacy Beliefs Instrument – Outcome Expectancy __________________________________________________________ Pretest Midpoint Posttest __________________________________________________________ Strongly Disagree 0.0% 0.0% Disagree 0.0% 0.0% Uncertain 75.0% 75.0% Agree 12.5% 25.0% Strongly Agree 12.5% 0.0% Summary. An evaluation of Olivia’s efficacy resu lts indicate that her feelings of overall efficacy declined slightly on the full survey. Her agreement with self-efficacy and outcome expectancy subtest items also d ecreased from midpoint to posttest. On the overall survey and the self-efficacy subtest, at least some of Oliv ia’s responses moved into the negative efficacy range. Howeve r, on the outcome question portion, while Olivia’s strength of efficacy decreased to a small degree, none of here responses shifted to indicate negative feelings. Kari Mathematics Teaching Efficacy Belief s Instrument Overall Efficacy Kari’s pretest scores on the overall Mathematics T eaching Efficacy Beliefs Instrument indicated a slight negative sense of efficacy with 9.5% of responses marked “Disagree” as seen in Table 42. Approximately a third of her answer s indicated she had fee lings of uncertainty

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226 in regards to her efficacy, while slightly mo re than half (57.2%) were responses that noted positive perceptions about her efficac y. At midpoint, there was a considerable increase in Kari’s feelings of efficacy, where she indicated 90.5% positive “Agree” statements for efficacy. At pos ttest, Kari’s scores had fallen considerably, with an equal 4.8% rate for both “Strongly Disagree” and “Disagree”. Additionally, she marked more items as “Uncertain” than at midpoint, and he r positive feelings dropped considerably to only 61.9%. On the overall instrument, Kari’s re sults indicated that while her feelings of efficacy rose at midpoint, they fell back to below pretest levels at posttest. Table 42 Kari: Mathematics Teaching Effica cy Beliefs Instrument – Overall _________________________________________________________ Pretest Midpoint Posttest _________________________________________________________ Strongly Disagree 0.0% 0.0% 4.8% Disagree 9.5% 0.0% 4.8% Uncertain 33.3% 9.5% 28.5% Agree 57.2% 90.5% 61.9% Strongly Agree 0.0% 0.0% 0.0% Self-Efficacy. Kari’s pretest results on the self -efficacy questions indicated that she held some negative perceptions of he r self-efficacy with 15.4% of her responses, shown in Table 43. At the same time, she also had almost as many responses of “Uncertain” as she did positiv e feelings of efficacy. At midpoint, Kari’s responses changed considerably with 84.6% of her res ponses indicating positive perceptions of self

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227 efficacy, and no responses that were negative. At posttest, Kari maintained a high level of “Agree” statements at 53.8%, but her numbe r of “Uncertain” responses increased. Kari also responded 7.7% of the time as “Disagree”, indicating negative efficacy perceptions at posttest. These self-efficacy resu lts indicated that while Kari’s feelings of self efficacy rose considerably from pretest to midpoint, they then declined slightly again at post-test. Table 43 Kari: Mathematics Teaching Efficacy Be liefs Instrument – Self-Efficacy __________________________________________________________ Pretest Midpoint Posttest __________________________________________________________ Strongly Disagree 0.0% 0.0% 7.7% Disagree 15.4% 0.0% 0.0% Uncertain 38.5% 15.4% 38.5% Agree 46.1% 84.6% 53.8% Strongly Agree 0.0% 0.0% 0.0% Outcome Expectancy. At pretest, Kari indicated predominantly positive feelings of efficacy towards student outcomes with 75% of her responses. The remaining responses were “Uncertain” and did not s how negative feelings of efficacy towards outcome expectations. At midpoint, 100% of Kari’s responses were “Agree”, showing a high level of efficacy in expected student re sponses to her mathematics instruction. At posttest, Kari’s results had changed slightly, with not only a decrease in positive feelings of efficacy, but also 12.5% of her responses marked as “U ncertain” or “Disagree”.

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228 Outcome expectancy agreement results show a considerable rise at midpoint, with a slight decrease at posttest, included in Table 44. Table 44 Kari: Mathematics Teaching Ef ficacy Beliefs Instrument – Outcome Expectancy ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Disagree 0.0% 0.0% 12.5% Uncertain 25.0% 0.0% 12.5% Agree 75.0% 100.0% 75.0% Strongly Agree 0.0% 0.0% 0.0% Summary. On the entire efficacy instrument, as well as the two subtests, Kari showed positive growth in her perceptions of efficacy for teaching mathematics from pretest to midpoint. At posttest, Kari’s scor es experienced a declin e on both subtests, as well as the whole instrument. On the self e fficacy subtest, Kari’s score decreased, but not below pretest levels. Howe ver, on the overall instrument and outcome expectancy portion, Kari’s scores decreased belo w pretest levels at posttest. Taylor Mathematics Teaching Efficacy Belief s Instrument Overall Efficacy On the full Mathematics Teaching Efficacy Beliefs Instrume nt, Taylor responded in an almost even split between total negative f eelings of efficacy (47.6%) and total positive feelings of efficacy (52.4%), with no indications of “U ncertain” feelings. At midpoint, her scores

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229 had risen considerably with a majority of responses (90.5%), indicating positive feelings of efficacy. However, there was still a sm all percentage, 9.5% of responses, showing negative feelings of efficacy. At posttest, Tayl or’s scores fell to a marked degree, with a drop to 52.4% in positive feeli ngs and a rise in overall ne gative feelings to 38.1%. “Uncertain” responses also appeared at 9.5%, after not being indicated on either previous survey administration. Overall efficacy resu lts showed a gain at midpoint and then a sharp decrease at posttest, as seen in Table 45. Table 45 Taylor: Mathematics Teaching Effi cacy Beliefs Instrument – Overall _________________________________________________________ Pretest Midpoint Posttest _________________________________________________________ Strongly Disagree 4.8% 0.0% 4.8% Disagree 42.8% 9.5% 33.3% Uncertain 0.0% 0.0% 9.5% Agree 47.6% 90.5% 52.4% Strongly Agree 4.8% 0.0% 0.0% Self-Efficacy. At pretest, Taylor indicated 38.5 % negative perceptions of selfefficacy when teaching mathematics compared to the majority of her responses which were positive perceptions (61.5 %), shown in Table 46. At midpoint, a large change in responses occurred with 100% of her responses indicating agreement, meaning that all of her responses were marked positively for her perceptions of her self-efficacy in teaching mathematics. Posttest results showed a slig ht decrease from midpoint results with 15.4%

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230 of responses indicated as “Un certain” and a small percentage of responses (7.7%) marked as negative feelings of self-efficacy. Taylor’s results indicated positive growth in perceptions of self-efficacy towards math in struction, which were stronger at midpoint than at posttest. Table 46 Taylor: Mathematics Teaching Efficacy Beliefs Instrument – Self-Efficacy _________________________________________________________ Pretest Midpoint Posttest _________________________________________________________ Strongly Disagree 7.7% 0.0% 7.7% Disagree 30.8% 0.0% 0.0% Uncertain 0.0% 0.0% 15.4% Agree 61.5% 100.0% 76.9% Strongly Agree 0.0% 0.0% 0.0% Outcome Expectancy. In terms of her perceived instructional efficacy on student learning at pretest, Taylor indicated a predominantly nega tive view with 62.5% of her responses marked as “Disagree” and only 37.5% marked positively. At midpoint, Taylor, evidenced a large change in her perceptions, with 75% of her responses being “Agree” or positively related to her instructional efficacy. A shift in the opposite direction occurred for Taylor’s results at posttest, with 87.5% of her responses indicating negative feelings about her efficacy in affecting student res ponses through her instruction. The results show a shift from predominately negative outcome expectancy views at pretest to

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231 predominantly positive views at midpoint. At posttest, results ma de a major shift, indicating even more negative views than at pretest, included in Table 47. Table 47 Taylor: Mathematics Teaching Ef ficacy Beliefs Instrument – Outcome Expectancy _________________________________________________________ Pretest Midpoint Posttest _________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Disagree 62.5% 25.0% 87.5% Uncertain 0.0% 0.0% 0.0% Agree 25.0% 75.0% 12.5% Strongly Agree 12.5% 0.0% 0.0% Summary. Results of the Mathematics Teaching Beliefs Instrument indicate an increase in perceptions of efficacy across the total instrument, as well as the subtests from pretest to midpoint. However, from midpoint to posttest, all results decreased. Outcome expectancy showed the most marked decrease followed by the total instrument, and then the self efficacy subtest, which experienced a minor decrease. Comparison of Case Study Efficacy Instrument Results. During the course of the study, the three case study participants’ individual results on the efficacy survey instrument paralleled the quantitative data collected on the total participant group as a whole, showing an increase on the full efficacy instrument and its subtests between pretest and midpoint, but then a decrease be tween midpoint and posttest. Looking at the

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232 individual results between case study participants, Olivia’s decreases at posttest were minimal on the full instrument, as well as self-efficacy and outcome expectancy subtests. Kari and Taylor’s results were different, showing considerable decreases, especially in the area of outcome expectancy which droppe d below pretest levels. These results indicate that Kari and Taylor’s feelings that they could pos itively affect student learning outcomes in mathematics diminished during the latter ha lf of the study. Olivia Mathematical Beliefs Questionnaire – Over all Beliefs: Constructively Worded Items. Both the Mathematics Teaching Efficacy Be liefs Instrument and the Mathematical Beliefs Questionnaire were administered on the same day. As a result, Olivia was also absent for the Mathematical Beliefs Questi onnaire pretest and was not able to make up the test in the availa ble time period. Along with the prev ious instrument, she was present for both midpoint and posttest administra tions. At midpoint, for all items worded constructively and flexibly about mathematics, Olivia marked primarily responses that indicated her agreement with these beliefs. However, this agreement was somewhat tentative because 70% of her responses noted only “Slight” or “Moderate Agreement”. Posttest results showed a considerable change in Olivia’s overall constructivist beliefs towards more traditional attitudes with 60% of her responses indi cating some form of disagreement with more informal and flexible ideas about mathematics. While Olivia’s views shifted towards more formal ideas a bout overall mathematics teaching at posttest, she still evidenced no instances of “Strongly Ag reeing” with these more traditional ideas, seen in Table 48.

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233 Table 48 Olivia: Mathematical Beliefs Qu estionnaire – Overall Beliefs: Constructively Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% Moderately Disagree 0.0% 15.0% Slightly Disagree 5.0% 45.0% Slightly Agree 25.0% 25.0% Moderately Agree 45.0% 10.0% Strongly Agree 25.0% 5.0% Overall Beliefs – Traditi onally Worded Items. With all items on the Mathematical Beliefs Questionnaire which were worded more traditionally towards mathematics, Olivia noted fairly strong di sagreement (90%) at midpoint, shown in Table 49. This disagreement is consistent with he r responses when compared to the overall constructively worded statements, with whic h she predominately agreed (95%). At posttest, Olivia showed more agreement with more formal ideas about mathematics, with 70% of her responses indicating some form of agreement with these views. These results are again consistent with her responses to constructively worded items, where 60% of her responses were in disagreement with these more developmental beliefs.

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234 Table 49 Olivia: Mathematical Beliefs Qu estionnaire – Overall Beliefs: Traditionally Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 50.0% 0.0% Moderately Disagree 20.0% 30.0% Slightly Disagree 20.0% 40.0% Slightly Agree 5.0% 25.0% Moderately Agree 5.0% 5.0% Strongly Agree 0.0% 0.0% MBS – Constructively Worded Items. On the MBS subtest at midpoint, Olivia indicated 90% agreement with ideas supporting more flexib le and creative ways of approaching the learning of mathematics, incl uded in Table 50. At posttest, her beliefs had shifted to an equal split between agreement and disagreement with this constructivist approach. However, within her agreement re sponses, Olivia had 10% of items where she “Strongly Agreed” compared to no items wher e she “Strongly Disa greed”. While her results, indicated that Olivia’s ideas became mo re traditional, her ideas were still slightly more constructivist towards the mathematics subject area.

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235 Table 50 Olivia: MBS – Constr uctively Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% Moderately Disagree 0.0% 30.0% Slightly Disagree 10.0% 20.0% Slightly Agree 50.0% 20.0% Moderately Agree 40.0% 20.0% Strongly Agree 0.0% 10.0% MBS –Traditionally Worded Items With items on the MBS worded in a more traditional and rigid approach towards the acad emic area of mathematics, Olivia indicated disagreement with 90% of items at midpoint, seen in Table 51. These results are exactly opposite and consistent with items worded positively towards constructivist views at midpoint on the same subtest. At posttest, Olivia’s views did not shift towards more agreement with traditional vi ews. However, her disagreement became less strong with 60% of her responses “Slightly Disagreeing” with formal ideas about mathematics. While Olivia’s responses to traditionally worded items is not in opposition to her responses on constructivist items, her answ ers indicate less inclin ation towards formal mathematical ideas than is shown through her disagreement with constructivist ideas.

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236 Table 51 Olivia: MBS – Traditionally Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 20.0% 0.0% Moderately Disagree 30.0% 30.0% Slightly Disagree 40.0% 60.0% Slightly Agree 10.0% 10.0% Moderately Agree 0.0% 0.0% Strongly Agree 0.0% 0.0% TMBS – Constructively Worded Items. On the TMBS at midpoint, Olivia indicated beliefs that consistently agreed with instructing math in a constructivist manner, with 100% of her responses as “Moderately Agree” or “Strongly Agree”. At posttest, Olivia’s responses on this s ubtest notably decreased showing only 30% agreement with constructivist ideas about teaching mathema tics and 70% disagreement. These results indicated that Olivia’s views about mathem atics instruction became more traditional during the latter part of th e study, included in Table 52.

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237 Table 52 Olivia: TMBS – Constructively Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% Moderately Disagree 0.0% 0.0% Slightly Disagree 0.0% 70.0% Slightly Agree 0.0% 30.0% Moderately Agree 50.0% 0.0% Strongly Agree 50.0% 0.0% TMBS –Traditionally Worded Items. With items involving more formal approaches towards mathematics instru ction, Olivia answered with “Strong Disagreement” for 80% of the items at midpoint. These responses match Olivia’s responses to constructively worded items on the same subtest that indicated 100% agreement with more flexible and creativ e views about teaching mathematics. At posttest, Olivia’s disagreement with form al instruction decrea sed to 50% of her responses, with also a decrease in the degr ee of this disagreement to “Moderate” and “Slight” rather than “Strong”. While these responses are consistent with Olivia’s results towards constructivist items, the strength of agreement with formal instruction ideas (50%) is less than that i ndicated by her disagreement with the positively worded

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238 constructivist items (70%). These results s how a slight shift towards more traditional mathematics teaching attitudes in the la ter half of the study, seen in Table 53. Table 53 Olivia: TMBS – Traditionally Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 80.0% 0.0% Moderately Disagree 10.0% 30.0% Slightly Disagree 0.0% 20.0% Slightly Agree 0.0% 40.0% Moderately Agree 10.0% 10.0% Strongly Agree 0.0% 0.0% Summary. At midpoint, Olivia’s responses acros s survey items indicated a strong agreement with informal, constructivist view s on the overall attitude survey, and on the general mathematics and teaching mathematics s ubtests. At posttest, Olivia’s responses on all positive statements about flexibly a nd creatively approaching mathematics on both subtests and the full survey indicated a d ecrease in these views. While Olivia’s agreement with constructively worded items considerably decreased, her agreement with positively worded statements about formal mathematics instruction did not increase to the

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239 same extent, indicating that while her agreem ent with constructivist ideas did wane she could not then agree with positively framed traditional views instead. Kari Mathematical Beliefs Questionnaire – O verall Beliefs: Constructively Worded Items. On the overall Mathematical Beliefs Ques tionnaire, Kari initially showed 70% agreement with constructivist ideas about ma thematics. At midpoint, this agreement increased to 90%, with 25% of her responses indicating “Strong Ag reement” with these ideas. At posttest, agreement with this informal approach towards mathematics decreased slightly with only 60% agreement. Responses at posttest that disagreed with constructivist views rose to 40%. Kari’s re sults indicated that dur ing the middle of the study her attitudes towards mathematics took a decidedly more constructivist turn, but by posttest these feelings had decr eased to approximately match pretest responses, included in Table 54.

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240 Table 54 Kari: Mathematical Belief s Questionnaire – Overall Belie fs: Constructively Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 0.0% 0.0% 0.0% Slightly Disagree 30.0% 10.0% 40.0% Slightly Agree 55.0% 30.0% 45.0% Moderately Agree 15.0% 35.0% 15.0% Strongly Agree 0.0% 25.0% 0.0% Overall Beliefs – Tradi tionally Worded Items. With items worded towards more traditional views of mathematics on the full Mathematical Beliefs Questionnaire, Kari marked 85% of her responses in agreement w ith more formal views of mathematics at pretest. This indication c ontradicts her responses to pos itively worded constructivist statements, to which she responded positive agreement 70% of the time, seen in Table 55. At midpoint, her views grew more strongl y positive for traditional views, with disagreement at 15% and only in the “Slightly Disagree” cate gory. Again, this agreement with traditional views is in opposition to her responses to items worded more constructively where she indicated 90% agreem ent with those statements. At posttest,

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241 Kari’s agreement with traditional view s of mathematics decreased showing 55% disagreement with these ideas and only 45% ag reement. This decrease in agreement with traditional views moved her results to be para llel with her responses to positively worded constructivist items that showed 60% agreement with this more flexible approach and 40% disagreement. Kari’s results on the overall Mathematical Beliefs Questionnaire indicated the appeal of both c onstructivist and traditional items for Kari at pretest and midpoint. At posttest, her views seemed to be equally split between approaches, and for the first time her responses consistent between the two types of items. The dual positive emphasis on the two types of items may be due to Kari’s attitudes towards mathematics not being stabilized at pretest and midpoint, a nd developing to a more solidified state at posttest to be slightly more constructivist.

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242 Table 55 Kari: Mathematical Beli efs Questionnaire – Overall Beliefs: Traditionally Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 5.0% 0.0% 20.0% Slightly Disagree 10.0% 15.0% 35.0% Slightly Agree 40.0% 55.0% 25.0% Moderately Agree 45.0% 25.0% 20.0% Strongly Agree 0.0% 5.0% 0.0% MBS –Constructively Worded Items On the MBS subtest, Kari indicated strong agreement with positively wo rded creative statements a bout approaching mathematics with 70% agreement. At midpoint, this agreement jumped up to 100%. At posttest, Kari’s agreement fell noticeably. Her responses shifted to 60% disagreement with constructivist ideas, and remained only at 40% agreement with th em. Additionally, the strength of this agreement decreased with all agreement only at the “Slightly Agree” level. These results indicated a considerab le increase in constr uctivist ideas about mathematics content at midpoint, but a decrease to below pretest levels at posttest, shown in Table 56.

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243 Table 56 Kari: MBS – Construc tively Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 0.0% 0.0% 0.0% Slightly Disagree 30.0% 0.0% 60.0% Slightly Agree 60.0% 10.0% 40.0% Moderately Agree 10.0% 40.0% 0.0% Strongly Agree 0.0% 50.0% 0.0% MBS –Traditionally Worded Items For items worded positively towards a more formal approach towards mathematics, Kari showed 90% agreement at pretest, shown in Table 57. These results are in opposition with her 70% agreement with constructivist ideas, also at pretest. At midpoint, Kari showed an increase in the strength of her agreement with traditionally worded items, s howing that 10% of he r responses rose to “Strongly Agreeing” with these ideas. Agai n, these results are contradictory to her responses on positively worded constructivist items about mathematics, which actually rose to 100%. By posttest, Kari’s agreemen t with traditional views decreased to only 30%. This shift, while not in complete ag reement with her respons es to constructivist statements, is noticeably more balanced between the constructivist and traditional

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244 approaches at posttest. The results indicated that both constructivist and traditional views of the mathematics subject area appealed to Kari at pretest and mi dpoint. Her responses indicated that she had perhaps not definitively established her own beliefs and thoughts on the mathematics content area at those admini stration points. At posttest, her results from the two types of items on this subtest we re in greater agreement, and seemed to be fairly equally balanced between the two approaches.

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245 Table 57 Kari: MBS – Traditionally Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 0.0% 0.0% 30.0% Slightly Disagree 10.0% 10.0% 40.0% Slightly Agree 50.0% 50.0% 20.0% Moderately Agree 40.0% 30.0% 10.0% Strongly Agree 0.0% 10.0% 0.0% TMBS – Constructively Worded Items. Between pretest and midpoint, Kari’s responses to items indicating agreement with cons tructivist approaches to mathematics instruction rose slightly from 70% to 80% agreement, included in Table 58. At posttest, her percentages of agreement remained the same from midpoint (80%). These results indicated a decided and stable agreement with constructivist ideas about teaching mathematics over the course of the study.

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246 Table 58 Kari: TMBS – Constructively Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 0.0% 0.0% 0.0% Slightly Disagree 30.0% 20.0% 20.0% Slightly Agree 50.0% 50.0% 50.0% Moderately Agree 20.0% 30.0% 30.0% Strongly Agree 0.0% 0.0% 0.0% TMBS – Traditionally Worded Items. On the more traditionally worded statements on mathematics instruction, Kari showed 80% agree at both pretest and midpoint, shown in Table 59. These resu lts do not align w ith her answers on constructivist items that s howed 70% and 80% agreement with constructivist items on pretest and then midpoint. At posttest, agre ement with more formal views of teaching mathematics had decreased to 60%. These posttest results indicated a more even distribution between her agreement with form al and constructivist ideas. While not in total agreement with he r responses on constructively wo rded items at posttest, they are not contradictory. Kari’s results indicated that the constructivist approach towards teaching mathematics appeals to Kari, but at the same time, so does the more traditional

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247 approach. Only at posttest did Kari’s stre ngth of agreement with traditional views of mathematics instruction begin to lessen. Table 59 Kari: TMBS – Traditionally Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 10.0% 0.0% 10.0% Slightly Disagree 10.0% 20.0% 30.0% Slightly Agree 30.0% 60.0% 30.0% Moderately Agree 50.0% 20.0% 30.0% Strongly Agree 0.0% 0.0% 0.0% Summary. Results of Kari’s responses to the overall Mathematics Beliefs Questionnaire and the MBS subtest, indicate so me inconsistency in Kari’s views about mathematics. This inconsistency may be due to her ideas in the ar ea being still in the developmental stage. By posttest, her ideas seemed to be more stabilized with equal agreement between the two sets of ideas However, on the TMBS subtest, the constructivist approach to mathematics instruc tion had consistent app eal to Kari across administrations, while traditional approaches held strong across all three administrations as well. She experienced only a slight decr ease in agreement with traditional items at

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248 posttest. These results possibly indicated that Kari’s ideas about engaging in mathematics instruction did not fully devel op towards one approach or the other during the course of the study, but remained where he r beliefs started in between the traditional and constructivist approaches. Taylor Mathematical Beliefs Questionnaire – Over all Beliefs: Constructively Worded Items. On the full Mathematical Beliefs Questionnaire, Taylor showed 80% agreement with constructivist mathematics ideas at pret est, seen in Table 60. At midpoint, this agreement increased slightly to 85%. At postt est, this agreement increased to 100%, with 85% of her responses being “Moderately Agree” or “Strongly Agree”. Across administrations, Taylor indicated no res ponses of “Strongly Disagree”, and during posttest she exhibited no form of disagreement at all. These results indicated Taylor’s inclination towards constructivist views about mathematics.

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249 Table 60 Taylor: Mathematical Beliefs Qu estionnaire – Overall Beliefs: Constructively Worded Items. ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 15.0% 10.0% 0.0% Slightly Disagree 5.0% 5.0% 0.0% Slightly Agree 10.0% 0.0% 15.0% Moderately Agree 55.0% 70.0% 70.0% Strongly Agree 15.0% 15.0% 15.0% Overall Beliefs – Traditi onally Worded Items. On the overall survey, items that were worded more traditionally in their ap proach received only 40% agreement from Taylor’s responses at pretest, included in Table 61. This agreement decreased to 30% at midpoint, showing consistency between her re sponses on constructiv ist items where she indicated 85% agreement. At posttest, he r agreement with more formal views of mathematics had decreased even further to 25%. While these results show very little agreement with traditional views of mathematic s, they were not in complete accordance with Taylor’s 100% agreement w ith constructivist items. The results indicated that while

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250 agreement with formal approaches to math ematics decreased incrementally throughout the study, agreement with c onstructivist views incr eased considerably. Table 61 Taylor: Mathematical Beliefs Qu estionnaire – Overall Beliefs: Traditionally Worded Items. ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 30.0% 0.0% 5.0% Moderately Disagree 30.0% 40.0% 55.0% Slightly Disagree 0.0% 30.0% 15.0% Slightly Agree 0.0% 5.0% 5.0% Moderately Agree 40.0% 25.0% 15.0% Strongly Agree 0.0% 0.0% 5.0% MBS – Constructively Worded Items On the MBS subtest, Taylor indicated a majority (70%) of her pretest responses in agreement with creative and flexible attitudes about the mathematics subject area, included in Table 62. At midpoint, this agreement rose by 10% to 80% agreement. At posttest this agreement incr eased to 100%, with no responses indicating disagreement with constr uctivist beliefs about mathematics. The results show an incremental increase over th e course of the study in Taylor’s alignment with constructivist views about mathematics in general.

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251 Table 62 Taylor MBS: Constructively Worded Items ____________________________________________________________ Pretest Midpoint Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 20.0% 10.0% 0.0% Slightly Disagree 10.0% 10.0% 0.0% Slightly Agree 0.0% 0.0% 10.0% Moderately Agree 50.0% 70.0% 70.0% Strongly Agree 20.0% 10.0% 20.0% MBS –Traditionally Worded Items With items worded more traditionally towards mathematics learning, Taylor’s in itial responses at pretest showed 60% disagreement at the “Strongly Disagree” or “Moderately Disagree” le vel. At midpoint, this disagreement decreased to only 50%, w ith the degree of disagreement moving to only “Moderately Disagree” or “Slightly”. These responses are in opposition to both Taylor’s pretest and midpoint responses on constructivist items, to which she responded with 70% and 80% agreement respectively. Re sponses at posttest to traditional items indicated a decrease in agreement with thes e views, with 70% of responses marked as “Moderately Disagree” or “Slightly Disagree”. These posttest results are more in line with Taylor’s responses on posttest construc tivist items, which were in 100% agreement at posttest with these ideas. The results of this subtest indicated that while the

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252 constructivist approach held strong appeal to Taylor ac ross the study, at pretest and midpoint so did formal ideas about mathematics. These results could be due to Taylor’s lack of development in her own thinking about mathematics at these two time points. Taylor’s development of mathematics view s showed some stabilization at posttest. While traditional views never held the same level of appeal as constructivist ones for Taylor, it was only at postte st where she showed 100% agreement with informal and developmental constructivist ideas, and a d ecrease in her agreement with traditional views at the same time to 30%, seen in Table 63. Table 63 Taylor: MBS – Traditionally Worded Items ____________________________________________________________ Pretest Midpoi nt Posttest ____________________________________________________________ Strongly Disagree 30.0% 0.0% 0.0% Moderately Disagree 30.0% 40.0% 60.0% Slightly Disagree 0.0% 10.0% 10.0% Slightly Agree 0.0% 10.0% 0.0% Moderately Agree 40.0% 40.0% 30.0% Strongly Agree 0.0% 0.0% 0.0% TMBS –Constructively Worded Items On the TMBS at pretest, Taylor indicated 90% agreement with constructiv ist statements about teaching mathematics, included in

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253 Table 64. At midpoint, the overall percentage of her agreement remained the same but strengthened in the amount of agreement, with “Strongly Agree” statements increasing from 10% to 20%. At posttest, Taylor’s agreement with informal and developmental statements about mathematics instruction rose to 100% agreement. These results indicated an incremental increase of Taylo r’s constructivist views throughout the course of the study in regards to teach ing mathematics. Table 64 Taylor: TMBS – Constructively Worded Items ____________________________________________________________ Pretest Midpoi nt Posttest ____________________________________________________________ Strongly Disagree 0.0% 0.0% 0.0% Moderately Disagree 10.0% 10.0% 0.0% Slightly Disagree 0.0% 0.0% 0.0% Slightly Agree 20.0% 0.0% 20.0% Moderately Agree 60.0% 70.0% 70.0% Strongly Agree 10.0% 20.0% 10.0% TMBS –Traditionally Worded Items Taylor’s agreement with formal mathematics instruction statements began at 40 % at pretest, seen in Table 65. However, a noticeable decrease in agreement was s een at midpoint (10%). While Taylor’s responses to constructivist items and traditional items are not contradictory at pretest, the

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254 formal items did receive more agreement than would have been expected from Taylor’s level of agreement with cons tructivist items. At midpoint, Taylor’s responses to traditional items with 10% agreement were in accordance with her responses (90%) of agreement on informal and developmental items. At posttest, levels of agreement with formal items increased slightly to 20%, but for the most part showed an overall maintenance of 80% disagreement, with 10% being “Strongly Disagree ”. These results indicated some agreement with formal math ematics instruction at pretest, with a considerable decrease in agr eement with traditional mathem atics instruction ideas from pretest to midpoint, with this de crease maintained at posttest.

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255 Table 65 Taylor: TMBS – Traditionally Worded Items ____________________________________________________________ Pretest Midpoi nt Posttest ____________________________________________________________ Strongly Disagree 30.0% 0.0% 10.0% Moderately Disagree 30.0% 40.0% 50.0% Slightly Disagree 0.0% 50.0% 20.0% Slightly Agree 0.0% 0.0% 10.0% Moderately Agree 40.0% 10.0% 0.0% Strongly Agree 0.0% 0.0% 10.0% Summary. The overall Mathematical Beliefs Qu estionnaire showed that Taylor’s agreement with constructivist views increa sed considerably over the study, while her formal views showed an incremental decrease At the same time, Taylor’s views on the MBS subtest showed agreement with both cons tructivist and traditional views at pretest and midpoint, but posttest evidenced a noticeab le decrease in her responses towards formal beliefs. This dual attraction of formal and constructivist statements at pretest and midpoint may have been due to Taylor’s s till developing views on mathematics, which seemed more definitive at posttest with both sets of statements indicating a more constructivist belief about mathematics learni ng. On the TMBS, Taylor showed stronger agreement with constructivist items across all administrations, but did not evidence

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256 considerable disagreement with formal inst ruction until midpoint, which was retained at posttest as well. Again, this initial agreem ent with both formal and developmental views of teaching mathematics may be due to Tayl or’s own learning and construction of her ideas regarding mathematics teaching. Comparison of Case Study Beliefs Instrument Results. Throughout the study, Olivia’s responses on the full beli efs instrument and its subtests was similar to that of the total group of participants. Her results illust rated an increase in agreement with items between pretest and midpoint, and a decrease between midpoint and posttest. However, Kari and Taylor’s response patt erns were different than Olivia ’s and the larger participant group. Kari’s attitudes about mathematics and mathematics instruction did not appear to have been firmly established in her mind as constructivist or traditional at pretest or midpoint, because her responses on the two different types of items were often contradictory to one another. However, at pos ttest, Kari’s ideas, wh ile still not decidedly constructivist or traditional, a ppeared to have stabilized to an equal combination of both approaches. At the same time, Taylor’s at titude responses showed a similar lack of establishment to Kari’s, with her responses to constructivist and traditionally worded questions often being in opposition to one anothe r. However, Taylor’s responses differed from Kari’s. Even though Taylor had this same contradiction between her responses to constructivist and tradi tional statements, throughout the st udy she maintained consistently high agreement with constructivist statements even when she showed agreement with traditional statements. At posttest, Taylor’s views also seemed to have stabilized, similar to this occurrence with Kari. A marked di fference with Taylor was that her attitudes

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257 toward general mathematics and mathematics instruction became decidedly constructivist. Olivia Mathematical Content Knowledge for Elementary Teachers. On the Mathematical Content Knowledge for Elemen tary Teachers survey, Olivia exhibited difficulty with the overall content of the meas ure (60%), as well as both the arithmetic (54.5%) and algebraic thinking ( 66.7%) subsections. At pretes t, her algebraic thinking accuracy level was slightly higher than her ba sic arithmetic skills. Olivia’s accuracy on the overall survey fell with each administ ration, being 35% at midpoint and 25% at posttest. Her basic arithmetic results fell from 54.5% at pretest to 36.4% at both midpoint and posttest. Olivia’s algebr aic thinking score fell considerably from pretest (66.7%) to midpoint (22.2%), with another decline seen at posttest with an 11% score. These results indicated that Olivia started the study at beginning competency level in content knowledge, and her abilities actually decreased to the deficient level over the course of the research, included in Table 66.

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258 Table 66 Olivia: Content Knowledge Results ____________________________________________________ Overall Basic Algebraic Arithmetic Thinking ____________________________________________________ Pre 60.0% 54.5% 66.7% Mid 35.0% 36.4% 22.2% Post 25.0% 36.4% 11.1% Kari Mathematical Content Knowledge for Elementary Teachers. Kari experienced difficulty on the Mathematical Content Knowledge for Elemen tary Teachers survey at pretest, achieving only 15% accuracy on the full survey, seen in Table 67. Her responses were slightly more accurate on basic arithme tic questions (18.2%) than algebraic thinking ones (11.1%) for her subtest results. At midpoi nt, Kari scored slightly higher on the overall measure (20%), while increasing he r basic arithmetic le vel to 27.3%. Her algebraic thinking skills remained steady at 11.1%. Kari’s overall content knowledge accuracy increased again at posttest to 25%. However, her score on basic arithmetic questions fell to her pretest level (18.2 %), while her response s on algebraic thinking items increased to 33.3%. These results indicated that Kari began the study with deficient overall levels of content knowledge. Her basic arithmetic skills did not show consistent gains throughout the study, but algebraic thinking sk ills did show some improvement. While her overall achievement on the content knowledge measure increased during the study, as well as her algebraic thinking performance, her accuracy levels remained deficient across all areas.

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259 Table 67 Kari: Content Knowledge Results ____________________________________________________ Overall Basic Algebraic Arithmetic Thinking ____________________________________________________ Pre 15.0% 18.2% 11.1% Mid 20.0% 27.3% 11.1% Post 25.0% 18.2% 33.3% Taylor Mathematical Content Knowledge for Elementary Teachers. Taylor began the study with 40% accuracy at pretest on the to tal Mathematical Content for Elementary Teachers survey, included in Table 68. At pretest, her highest score was on basic arithmetic skills (54.5%) with a lower score on algebraic thinking items (22.2%). During the midpoint administration, Taylor’s overall accuracy increased to 50%, while her basic arithmetic and algebraic thinki ng levels also increased to 63.6% and 33.3% respectively. At posttest, Taylor’s overall accuracy remained consistent with her midpoint accuracy, but basic arithmetic accuracy decreased (45.5%) and algebraic thinking accuracy increased to its highest level of 55.6%. These results indicate d that Taylor had a minimal level of overall content knowle dge at the start of the study, which rose slightly over the course of the research. He r basic arithmetic skills did not show any noticeable improvement over the study, while her algebraic thinking skills showed a steady increase from deficient to minimal levels.

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260 Table 68 Taylor: Content Knowledge Results ____________________________________________________ Overall Basic Algebraic Arithmetic Thinking ____________________________________________________ Pre 40.0% 54.5% 22.2% Mid 50.0% 63.6% 33.3% Post 50.0% 45.5% 55.6% Comparison of Case Study Mathematic al Content for Elementary Teachers Results. The overall results for all three case stu dy participants indicat ed they began the study without competency in the content ar ea of elementary level mathematics and completed the study with the same skill level. Case study participant results were consistent with the total participant group’s re sults in that deficient levels of content knowledge were seen in both cas e study participants and in th e larger partic ipant group. However, while the total part icipant group experienced an increase in content knowledge from pretest to midpoint, and a decrease from midpoint to posttest, the case study participants did not experience the same pattern of movement in their content knowledge. Olivia actually decreased in all areas of th e content, including her overall score and the subtest areas. Kari increased her scores in both the overall conten t and algebraic thinking area over the course of the rese arch, but she started at such deficient levels of content knowledge that even with her improvements sh e remained in the deficient range for all content areas. Taylor also increased in bot h the overall content a nd the algebraic thinking area, with her level of algebrai c thinking showing cons iderable growth. Yet, her levels of

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261 content knowledge remained just below begi nning competency level at the conclusion of the study. Olivia Instructional Knowledge Exam. Olivia’s overall performance on the instructional knowledge exam resulted in a score of 58%, in cluded in Table 69. On the subtests, her results showed high variability. Olivia ev idenced proficiency in understanding multiple choice items with a 92% score, with scores on the instructional practice and learning characteristic questions be ing nearly equal However, knowledge levels on both effective practice and application essay area s indicated that Olivia had only minimal abilities to explain he r ideas on these points accurately. These results indicate that while Olivia can recognize correct ideas on learning characteristics and instructional practices, she has difficulty with explicitly articulating th e specifics of these effective practices and their application within the DAL framework. Table 69 Olivia: Instructi onal Knowledge Exam ________________________________________________________________________ Total Exam MC Total Eff. Prac. Learn. Essay Eff. Prac. App. (MC) Char. Total (Essay) (Essay) (MC) ________________________________________________________________________ Kari Instructional Knowledge Exam. Kari’s overall achievement on the instructional knowledge exam indicated a 48% accuracy le vel, shown in Table 70. On the multiple Raw Score 76/130 23/25 14/15 9/10 53/105 25/50 28/55 Percentage 58% 92% 93% 90% 50% 50% 51%

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262 choice questions, Kari showed a high level of competency with 92% accuracy, with her scores almost equivalent between learning ch aracteristics and instru ctional practices. However, on the essay portion of the exam, Kari had difficulty effectively explaining the particulars of these practices and their appl ication within the DAL framework. With the effective practice essay questions, Kari s howed beginning levels of competency in articulating her ideas (60%), while on applic ation questions Kari was deficient in her conveyance of understanding. These results in dicate that Kari can recognize correct ideas about learning characteristics and instru ctional practices, when directly presented with these ideas, but still st ruggles with mastering and de scribing these instructional practices on her own. In terms of instruct ional strategy application within the DAL framework, Kari is unable to perform this task with any level of accu racy. Kari’s results indicate a firm ability in identifying releva nt learner characteristics and instructional practices, with further work needed on being able to describe those practices herself. She evidenced little knowledge of the ability to apply he r knowledge within the DAL instructional framework. Table 70 Kari: Instructional Knowledge Exam ________________________________________________________________________ Total Exam MC Total Eff. Prac. Learn. Essay Eff. Prac. App. (MC) Char. Total (Essay) (Essay) (MC) ________________________________________________________________________ Raw Score 62/130 23/25 14/15 9/10 39/105 30/50 9/55 Percentage 48% 92% 93% 90% 37% 60% 16%

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263 Taylor Instructional Knowledge Exam. Taylor’s performance on the instructional knowledge exam indicated a 65% accuracy level, seen in Table 71. On multiple choice items, she evidenced a high level of master y with a 96% score, with her scores on learning characteristic and instructional practi ce questions being nearly equivalent to one another. On the essay portion, Taylor demonstrated beginning competency with understanding instru ctional practices with a 70% score. However, with the application questions in the essay porti on, further work was needed in describing the implementation of effective practices within the DAL framework, as indicated by Taylor’s score of 52%. While Taylor’s results showed that she need s continued work on articulating information about effective instructional practices and their applica tion, she does evidence some understanding of instructional practices and a beginning grasp of their application within the DAL framework. Table 71 Taylor: Instructi onal Knowledge Exam ________________________________________________________________________ Total Exam MC Total Eff. Prac. Learn. Essay Eff. Prac. App. (MC) Char. Total (Essay) (Essay) (MC) ________________________________________________________________________ Summary. All three teacher candidates achieved at different levels on the content knowledge exam as a whole. Kari’s overa ll performance was the lowest at 48%, followed by Olivia at 58%, and then Taylor at 65%. These scores evidence a deficient Raw Score 85/130 24/25 14/15 10/10 61/105 35/50 9/55 Percentage 65% 96% 93% 100% 58% 70% 16%

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264 grasp on overall instructional concepts by Ka ri, with beginning understandings presented through Olivia’s achievement. Taylor shows the most familiarity with the instruction content, but her results still indicate a need for increased overall competency. In the subareas, there were much different results. In the multiple choice area, all teacher candidates scored above 90%, showing master y at the identification level for learning characteristics and instructional practices. On the effective instruc tional practices essay section, Taylor responded with some degr ee of competency, while Kari showed beginning levels of competency and Olivia i ndicated additional assistance in grasping these ideas. On the last sec tion of applying thes e strategies within the DAL framework, Taylor and Olivia scored fa irly equally, at 51% and 52% respectively, indicating needing continued help to fully understand the applica tion of instructional st rategies within the DAL. At the same time, Kari scored a 16%, noting a need for re-teaching these concepts for her grasp of these ideas. Olivia Review of Entire DAL Project – Efficacy. During the DAL experience, Olivia voiced positive comments in her work about her ability to teach mathematics in a way that was engaging for her students and where sh e felt that she had helped them each gain a greater understandi ng of targeted concepts. One example of such a comment was during a reflection on a weekly instructional session where she menti oned, “I pointed out the oranges to my student and she had an aha moment. I learned the significance of connecting to the text and she was learning what times table[s] are representing.” Olivia’s positive feelings of efficacy may have been impacted by the fact that she had two students for instruction throughout the enti re study. Not all teach er candidates within

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265 the study had two students during the entire se mester, because of st udent attendance and school withdrawal. Each of O livia’s students was present at all instructional sessions, except for one absence for each. Another outsid e factor that may ha ve supported Olivia’s efficacious perceptions was she was pres ent for all DAL framework training and implementation, except for one time early on in the training process before implementation had begun. According to Olivia’s DAL framework arti facts, there were other factors unique to Olivia’s instruction that may also have contributed to her positive feelings of instructional efficacy. When reviewing Olivia ’s initial DAL assessment of her students, her summaries of the assessment results cl early indicated that one student missed assessment items beginning with creating a nd extending patterns, and the other first missed items involving representing mathema tical models involved with multiplication based problems. Olivia followed the guideline s of the DAL training explicitly, and began instruction for both students at the concep t which they had both first missed on the assessment. As a result, she was able to im plement mathematics teaching at the student’s instructional level, and both of her student s evidenced gains in proficiency on target skills. One of her students showed 100% accuracy with patterning skills at the representational level, and the other student demonstrated 87.5% accuracy with multiplication problem set up and solution at the re presentational level. A final aspect of her instruction that may have caused Olivia positive feelings of efficacy was that she assisted her students in progres sing incrementally in their un derstandings of concepts by moving them up through the levels of abstrac tion in CRA accurately, rather than jumping between concepts without a leveled progression.

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266 Attitude. Through her experience with the DAL framework, Olivia approached mathematics using a constructivist/developmental attitude towards instruction. This approach was evidenced through several of O livia’s documented actions. While Olivia could have followed some parts of the DAL process and skipped other parts based on time constraints or the need to move her students through the continuum of algebraic thinking skills, she did not and followed the DAL’s developmental and structured approach throughout her instruct ion. Instructional decisions to do otherwise would have indicated her belief in possibly more trad itional views of math ematics instruction. Additional constructivist belief indicators in cluded Olivia’s usag e of student progress monitoring during each step of the process. She did not move her students forward in terms of skill level (ie., CRA) or type of skill unless this information was indicated through the required mastery per centages in Step I: Building Automaticity and Step II: Measuring Student Progress. It was onl y when students had gained successful proficiency levels with targeted skills in Step I or could succes sfully complete the problem solving steps (i.e., read, represent, justify, solve) on a c oncept that she moved forward in the skills she targeted th rough her instruction in Step III. Content Knowledge. Olivia’s description of her grasp and usage of algebraic thinking content knowledge within the DAL fr amework, evidenced her understanding of the scope and sequence of alge braic skills, as well as spec ific comprehension of the intricacies surrounding patterni ng and representing multiplication equations. A reflection statement that indicated her content know ledge understandings was “I decided the objective for the session was growing patterns at the representational level because in the Initial Probe the student coul d clearly use manipulatives a nd representations to extend

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267 patterns.” Olivia also showed she had a handle on decision-making with content knowledge through completing the initial DAL a ssessment with each of her students, and then successfully used those results to ascer tain where instruction should begin. This content knowledge decision-making was illustrated through one of her assessment result summaries where she noted, “Student B understands patterns, including sorting, identifying and describing patterns, and extendi ng and creating patterns. The first skills that needs improvement in the hierarchy of the given assessment are representational multiplication and division therefore I will begi n instruction at this point.” Olivia provided succinct descriptions of student performance on the initial assessment, and then used those results for her initial session pr obe content. Additionally, from the initial assessment, Olivia decided to only target one algebraic thinking skill at one level within each DAL framework step, so broke down each targ et skill into its individual levels of conceptualization for its gr eater understanding. An ex ample is shown through her comment about her goal for one student’s in struction, “I decided the objective for the session was growing patterns at the representational level.” Olivia’s successful utilization of algebraic thinking content know ledge within DAL sessions is contrary to the results of her own content knowledge su rvey. Her ability to understand content knowledge within her instructi onal experience may have been due to the limited nature of the content she taught, which included just patterning and representing multiplication problems. It could be also due to her disclo sure that she sought out ways of learning and understanding the concepts on her own, such as through Internet research and discussions with university support staff, before instructional sessions.

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268 Instructional Knowle dge and Application. Olivia’s DAL project documents show that her understanding of instructional knowle dge and its applications within DAL are tied intimately to the way she taught her student s their target skills. For both students, Olivia’s project included examples of mu ltiple practice opportunities at both the representational and abstract levels of CR A. Her weekly reflections also provided descriptions of using concrete manipulatives involving plastic shapes to illustrate problems. Additionally, Olivia’s reflections also illustrated how she used learner engagement with concepts and student prac tice within a self-deco rated student notebook to motivate students’ learning. These sa me ideas are ones that she spoke most descriptively about in her exit interv iew and scored most highly on during the instructional knowledge exam. Olivia evidenced furthe r instructional applica tion through her usage of The Man Who Walked Between the Towers book as an authentic contex t for instruction. Olivia used this text with both of her students, but devised different types of instructional activities for each student. For one stude nt, she implemented the book as a source of different types of patterns found in the main character’s experience in New York. With her other student, Olivia used the same text for sources of multiplication problems to be devised and solved. This dual context usage illu strated Olivia’s ability to think about the context to be implemented, and how it coul d be incorporated with multiple learning targets to individualize instru ction. In another instance of instructional connection making, Olivia attached a “Mathematics Strate gy” sheet within her DAL project that she designed herself, modeled after the reading st rategies sheet that was handed out by the researcher for assisting students with UF LI’s beginning reading strategies. This

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269 “Mathematics Strategy” sheet incorporated 9 mathematics strategies involving levels of representation, language experi ences, and mathematics resour ce utilization, as well as metacognitive strategies. This chart was one that Olivia pres ented to both her students in their last instructional session to have and use in future mathematics situations. This mathematics strategy chart showed Olivia ’s connection making between reading and mathematics strategy instruction. Kari Review of Entire DAL Project – Efficacy In several places within her reflections during the DAL framework, Kari made stat ements about being unsure of how to implement the DAL model and voiced negative feelings about her in structional efficacy. One such comment in one of her reflections included, “Today we did the initial session probe during our session. It was really inte resting because I did not really understand what I was suppose to be doing so I had to go with what I thought I was suppose to do and make a lesson that.” Through her commen ts, it seemed that her greatest difficulty was in understanding the steps in implementing th e process, as she mentioned in her final analysis paper with, “I feel like I need a lo t more work in the pr oject to understand the concept completely. I have a very genera l knowledge of what I thought I was suppose to be doing and even though I went and asked numerous people about how to do this DAL process it never really came to me completely.” This difficulty may have been due to the fact that Kari had a limited number of sessi ons with both of her students because of her own absence due to illness one time, and then one of her students being out on another occasion. While Kari indicated difficulty in understanding and implementing the DAL, her notes within her project di d not show efforts to seek clarification from university

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270 support staff within the practicum as Olivia ’s did. Additionally, her implementation difficulties may have also been related to th e fact that her reflections on her weekly experiences were very concrete, focusing on what happened in sessions and what could be done in the future, rather than probing her own understanding of student responses to instruction, her own comprehe nsion of the DAL process, and honing her problem-solving abilities for student learning di fficulties. For instance, one of her reflections focused on the following information for what she lear ned in her session, “A fter completing the initial session probe I decided based on what I did with my student that she was at an instructional level and at the representationa l level of growing pa tterns. She really understood the concept of concrete but was still not at the independent level on the representational level of growing patterns so that’s why I think I should start there next week for our session. I think that with a littl e more help and hands on lessons she will be able to really understand and get the concept of growing patterns and what they really are doing.” Attitude. Kari’s overall attit ude towards instructional implementation through the DAL framework was formal in nature, fo cusing on step completion and navigating through the sequence of instru ctional skills. Through her session notes and weekly reflections, Kari indicated that her goals for her students were to “move” them through the instructional content to be learned. A specific instance of this attitude was seen through a reflection about one of her student s’ progress through th e initial assessment with, “I will continue with th e assessment hopefully we will finish it because he is moving rather slow thro ugh the test.” In several of her reflections, Kari noted the length of time it was taking her students to complete their problems. At the same time, she

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271 commented on the fact that when her stude nts questioned their own problem solving processes, it further slowed the flow of problem completion. For example, Kari noted about one of her students, “I learned that she takes a long time to finish a problem when it comes to something that she not understand. She questions everything she does which in the end takes her longer to complete the pr oblems.” This particular statement was indicative of a very traditional view of mathem atics instruction where Kari saw herself as the director of curriculum poi nts for student learning. At the same time, she seemed frustrated by her student’s c onstructivist efforts to make sense of her methods of finding solutions. While the attitudinal survey that Kari completed indicated her valuing constructivist statements about mathematic s and mathematics instruction, feelings evidenced through her project artifacts showed her instru ctional practices as being primarily teacher-directed and traditionally structured. Content Knowledge. In terms of content knowledge, Kari’s project documents indicated difficulty in accurate instruc tional decision-making based on student content knowledge performance, as well as troubl e understanding the scope and sequence of skills to be taught in algebraic thinking. Bo th of Kari’s students showed their first difficulties on the DAL initial assessment with concrete growing patterns at the creating level. While her students appeared to still need further instruction on that skill at that same level after the initial probe was also completed, Kari noted that she moved one student on to the representati onal level. Kari’s instruc tional decision-making at this juncture leaves a question to whether Kari understood the patterning content or required DAL proficiency percentages enough to make accurate data-based decisions on when and why to move students up to the next representation level or skill to be taught. At the

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272 same time, while Kari’s reflections and not es indicated that she planned on moving one of her students up from the concrete to re presentational level, in actuality Kari’s examples and materials showed that she persis ted in having both of her students work on the same skill at the same level, concrete growing patterns, through both Steps 1 and 2 in her next instructional session. This lack of instructional follow though, as well as failing to employ the CRA sequence accurately from Step s 1 to 2, may indicate that Kari did not clearly understand the connectio ns and differences between the levels of understanding (ie., CRA) and the component pa rts of patterning concepts. Instructional Knowle dge and Application. Throughout her sessions, Kari utilized both learner engagement and CRA to facilitate her instruction. Becaus e of her failure to use specific incremental increases in repres entational levels with patterning at the growing pattern level, little to no student pr ogression in learning skills was seen. Kari taught her students both identifying growing patterns at the concrete level during all sessions. Her greatest difficulty seemed to surround the use of CRA, which is used explicitly within each step of the DAL framework. This difficulty was evident when she continued to teach both of he r students within each step of the DAL framework at the identifying growing pattern level using concrete manipulatives. Additionally, based on Kari’s notes, her goal for the second step of her last instructional session with students was to employ high interest materials invol ving candy for student engagement. In her effort to use learner engagement, Kari faile d to follow the graduated levels of CRA, which her documents indicate should have been picture or drawing representations for Step 2. Kari’s instructional efforts also faile d to show individualized instruction, with her

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273 implementation of the same book, at the same skill level, and with the same type of manipulatives with both of her students. Taylor Review of Entire DAL Project – Efficacy. Throughout her inst ruction using the DAL framework, Taylor’s notes on student pe rformance, as well as her reflections, indicated her inability to eff ectively teach and help her students progress in understanding algebraic thinking concepts. Her students’ l ack of success in algebraic learning may be due in some part to outside factors. In Taylor’s situ ation, one of her students had excessive absences that allowed Taylor to only complete the beginning assessment and the initial session probe with this student. At the same time, Taylor’s own absence during the intensive fu ll day of DAL model training, as well as again during one of her instructional days, may have furt her affected her level of inst ructional effectiveness. It also reduced the number of inst ructional sessions she comple ted with both of her students and her number of opportunities for affecting learner outcomes in algebraic thinking. Even in the face of these challenges, Taylor did write that she felt she had positively affected her students’ learning by solidifying the differences between growing and repeating patterns with them. However, this feeling of efficacy was not supported by any data, because Taylor’s pr oject indicated no specific notes on her first student’s skill performance during his initial se ssion. After this in itial session, Taylor was unable to see the student again because of student absen ce. With Taylor’s second student, she collected data during the initia l session probe that indicated further work was needed with growing patterns. When she taught her firs t full session with the student, she conducted Step 1: Building Automaticity on creating repe ating patterns. While her data collected

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274 during this step indicated positive student performance with 5/5 items completed successfully at the abstract le vel, the number of required ac curate items for proficiency indicated continued work needed on this level to raise the accuracy and fluency rate to at least 9/10 in one minute. Unfortunately, Tayl or’s session ended early because of student needs, and Taylor was unable to continue her instruction. Again, as with the first student, Taylor mentioned that she saw limited student progress with this second student, this time in the development of language abilities to describe the formation of growing and repeating patterns. However, her limited collected data and observations simply indicated her student was working towards proficiency level on creating repeating patterns. Attitude. While Taylor had limited opportunities to engage in instruction with her students, her project artifacts indicated that she employed a constructivist approach to instruction to facilitate stude nt learning within the session s she did have. Her project notes depicted her usage of CRA to help students develop their ideas on concepts involving growing patterns. She also stressed the use of language with oral discussion and student justification during the multiple opportunities for practic e that she provided her students. Taylor’s one main instance of more traditional instruction was seen within one of her weekly reflections’ emphasis on di rect instruction when beginning teaching on growing patterns with, “I explained that a repeating pattern was the same set over and over but a growing pattern grew each time it repeated. Once I explained this to Rodniqua, I asked to complete some growing patterns.” Content Knowledge. The bulk of Taylor’s refl ection comments focused on her work with her students in the patterning skill area. Both of Taylor’s students evidenced

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275 difficulties on growing patterns at the creating level. Taylor was able to successfully use the DAL initial assessment re sults to accurately pinpoint these difficulties and begin instruction on this skill with both students. During instruction using the DAL’s initial session probe, Taylor targeted instruction on creating growing patterns for each of her students. When Taylor’s second student di d progress to his first full session, Taylor chose creating repeating patterns to begin Step 1: Building Automaticity. This skill level is several levels under where the student evidenced his inst ructional leve l of creating growing patterns. A more appropriate inst ructional choice would have been extending growing patterns or describing growing pattern s, which are one and two levels below the student’s instructional level, respectively. This jump back wards in skills for Step 1, indicated Taylor’s possible difficulty in unde rstanding the scope and sequence of skills in the patterning area of algebraic thinking Instructional Knowle dge and Application. While Taylor tried to employ CRA, explicit instruction, and oral structured langua ge experiences in her teaching, her limited sessions and number of steps completed in each session impeded her from having more opportunities to use many of the po ssible instructional strategies that can be used within the DAL framework. Through Taylor’s inst ructional session not es, she indicated introduction of target learning concepts with growing patterns thr ough explicit in struction with modeling, which is appropriate for at-ris k learners. At the same time, she tied concrete manipulatives to the context of problem-solvi ng, which involved patterning using beans and bread. Additionally, Taylor indicated specific instances where she afforded students opportunities to develop oral language abilitie s to explain their problem-solving during pattern formation. Un fortunately, Taylor was never able to

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276 implement the student language notebook for written structured language experiences because she did not make it to Step 3: Pr oblem Solving the New with either of her students. Comparison of Case Study Entire DAL Final Projects. Each of the case study participants presented a uniquely different experience through their DAL project artifacts. Olivia’s illustrated one of growth in e fficacy and employment of a constructivist approach towards instruction, using diverse me thods of instruction and multiple ways to understand content knowledge for her instructi on. Kari’s project showed her confusion with the DAL framework’s steps, instructi onal practices, and content, resulting in poor perceptions of self-efficacy, l ack of student progress, and employment of few different forms of instruction. Taylor’s project highli ghted a lack of efficacy and student progress due to the outside factor of absence. Howeve r, Taylor maintained a mostly constructivist approach to instruction, attempting to empl oy multiple forms of instructional practice within her limited sessions. Taylor’s lack of understanding of the scope and sequence of algebraic skills may have also influenced he r students’ lack of pr ogression in algebraic skills. Results from these analyses indicat ed that the top-achieving participant grasped the key pieces of the DAL experience a nd was able to develop her abilities along identified critical elements for teacher preparation in mathematics. Yet, the midachieving teacher candidate, struggled in grasping the DAL framework, as well as content knowledge and instructional pract ices, while the low-achieving participant struggled primarily with her lack of sessi on experiences and in depth understanding of algebraic thinking content.

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277 Olivia Final Analysis Paper – Efficacy Within her final analysis paper, Olivia had 6 specific instances of speaking directly about her feelings of efficacy when using the DAL framework and its related instructional pr actices, shown in Table 72. Her comments were equally balanced between positive and negative comments about her efficacy in facilitating instruction. One specific negativ e efficacy comment included “outside factors affecting the number of sessions we were able to conduct hi ndered her (referring to her student) learning and mine”. On the other hand, one of her positive statements included that the “DAL was an organized process of t eaching” which she felt helpful in facilitating her instructional abilities. Attitude. Olivia also made statements regard ing her attitude towards mathematics instruction on a total of 6 occasions. With in these comments, she had 5 instances of a constructivist nature a nd 1 instance of a more traditional approach to mathematics. One of her comments along constructivist lines me ntioned “goal setting invites students to actively engage in their e ducation”, showing her attit ude of encouraging student involvement in and enjoyment of mathematic s learning. The only formal statement that she made regarding mathematics instruction was that she viewed herself as having to “teach strategies to (her) students” rath er than viewing strategy knowledge and application as a guided discovery process explored by students. Content Knowledge. Within her paper, Olivia discussed 2 specific items involving content knowledge, each on a different topic. One of these comments was regarding her first student’s grasp of patte rning, and the second comment was about her other student’s conceptualization of multiplication model problems. In one of her

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278 statements, Olivia mentioned that her stude nt working on multiplication eventually began to comprehend multiplication as a way of “for ming groups”, showing Olivia’s realization that while multiplication understanding had firs t escaped her student, he then developed a way of comprehending the ideas behi nd that specific skill. Instructional Knowledge and Application. The majority of comments that Olivia made in her final analysis of the DAL expe rience referred to instructional knowledge, with 7 codes and a total of 10 statements. Within her statements, she included the ideas of “progress monitoring”, “systematic stru ctured instruction”, “planning”, “making connections across content areas”, “CRA”, “multiple practice opportunities”, and “building student confidence through instruction”. Many of these quotes identified instructional practices explicitly covered in the DAL process, in cluding CRA, making connections, multiple practice opportun ities, and progress monitoring. Table 72 Olivia: Final Analysis Paper Themes ________________________________________________________________________ Element Number of Frequency of Intensity Effect Descriptor Codes Occurrence Sizes (Percentage in Theme of Total) ________________________________________________________________________ Efficacy 2 6 25.0 % Attitude 2 6 25.0% Content Knowledge 2 2 8.3% Instructional Knowledge 7 10 41.7%

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279 Kari Final Analysis Paper – Efficacy. During her final analysis paper, Kari made 5 specific comments about her efficacy in teac hing mathematics, seen in Table 73. Her comments had only 1 code because they were only negative in regard to her abilities to teach mathematics. However, the reasons for these negative feelings of efficacy were not focused in on her own abilities, but on outside factors related to her preparation, such as “not being given the tools” to facilitate math ematics instruction successfully, and external environmental factors such as “not ha ving nearly enough instructional time”. Attitude. Kari’s attitudes towards mathematics and mathematics instruction were presented 6 times during her paper. The ma jority of her attitudinal comments were constructivist in nature, cons isting of 4 statements, while traditional statements about teaching mathematics were only indicated 2 times For instance, Kari viewed instruction within the DAL as a shared or constructed learning experience betw een her and students when describing instructional aids as “the tool s I needed to complete the process with my student.” However, at another point Kari mentioned “having to teach concepts to her students”, noting a more formal and directiv e approach to instructing mathematics. Content Knowledge. During her writing, Kari made no mention of ideas directly regarding content knowledge in conjunction with her own understandings or her students. This finding is consistent with her scores on the mathematics content area survey, where she exhibited low levels of content knowledge across all areas of elementary level mathematics. It is not surprising that she would not discuss mathematics content knowledge, with which she had evidenced difficulty in grasping.

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280 Instructional Knowle dge and Application. Kari’s statements about instructional strategies and knowledge covered 3 coding ar eas: “modeling”, “planning”, and “multiple practice opportunities”. “Modeling” and “mu ltiple practice opportunities” are specific instructional strategies taught directly within the DAL framework. Planning, while not specifically taught, is emphasized within the DAL as integral in having successful student sessions. An interesting spin on these t echniques was that Kari thought that more opportunities for practice and more modeling demonstrations for the DAL framework should be utilized by the faculty in prepari ng the teacher candidates to implement the DAL framework. As a result, the strategies taught within the model were ones that she felt needed to be employed for her own lear ning rather than her a dvocating their direct usage with students. Table 73 Kari: Final Analysis Paper Themes ________________________________________________________________________ Element Number of Frequency of Intensity Effect Descriptor Codes Occurrence Sizes (Percentage in Theme of Total) ________________________________________________________________________ Efficacy 1 5 33.3% Attitude 2 6 40.0% Content Knowledge 0 0 0.0% Instructional Knowledge 3 4 26.7%

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281 Taylor Final Analysis Paper – Efficacy, Attitude, and Content Knowledge. Within the final analysis paper of the DAL experience, Taylor wrote an extremely short one page evaluation of her experience with the fram ework, included in Table 74. There was no specific mention as to her efficacy in mathem atics instruction within her writing. Also, no statements about her attitude towards mathematics instruction were evident. Additionally, Taylor did not state any inform ation in regards to the content knowledge element of mathema tics instruction. Instructional Knowle dge and Application. The only comments that Taylor made in her final analysis paper were in regards to instructional practices. Within her statements, she spoke about 3 di stinctive instructional strategi es: “explicit instruction”, “structured language experiences”, and “learner engagement”. Each of these strategies was taught to teacher candidates within th e DAL framework’s initial instruction and ongoing support. One specific feature that Taylor focused on was ensuring her instructional efforts make “the most basic of ideas” detailed a nd engaging, so that students do not lose interest in lear ning more fundamental concepts.

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282 Table 74 Taylor: Final Analysis Paper Themes ________________________________________________________________________ Element Number of Frequency of Intensity Effect Descriptor Codes Occurrence Sizes (Percentage in Theme of Total) ________________________________________________________________________ Efficacy 0 0 0.0% Attitude 0 0 0.0% Content Knowledge 0 0 0.0% Instructional Knowledge 3 4 100.0% Comparison of Case Study Final Analysis Papers. The final analysis paper of each case study student was different. With O livia, the student with the highest overall Level II course and practicum achievement her comments covered the breadth of professional development elements covered w ithin this study. On the other hand, Kari, the mid-achieving participant, focused in on ex ternal factors affecting her abilities to efficaciously execute instruction, while never mentioning the content she taught. Her depth of comments on content knowledge also se emed in line with here deficient scores on the content knowledge survey. At the sa me time, Taylor, who was from the lowachieving group of participants, turned in the sh ortest of the three fi nal analysis papers, which seemed in accordance with her overall pe rformance evaluation from her professors in Level II coursework and pr acticum. Interestingly, she on ly wrote about instructional practices in her paper. These instructional concepts were ones dire ctly taught to the

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283 teacher candidates within the scope of the DAL experience and training. Taylor did not mention any concepts, ideas, and experiences that were implicit w ithin her exposure to the DAL framework. These results indicate that the person most completely affected across identified critical elements for mathem atics instruction for at-risk learners was Olivia, the high performing teacher candida te. Kari, the mid-performing teacher candidate, appeared to have gained mostly surface level underst andings across these critical elements, except for content knowledge that appeared to have not been affected by the DAL experience. Taylor, the low performing teacher candidate, articulated learning in explicitly taught instructional practices in bot h the instructional knowledge and application realms, but other areas were not recognized as experiencing gains. Olivia Exit Interview – Efficacy. Within the exit interview process, some key ideas impacting Olivia’s feelings of efficacy in teaching mathematics were illustrated by her comments. For the most part, Olivia mainta ined a high perception of her efficacy in her instructional abilities and her instructional effects on her students. She mentioned that she had entered the DAL experience comfortabl e with elementary mathematics, having two of her own children in middle school. As we began the DAL training, this feeling of comfort increased, because as she said, “I had no idea that patterns we re part of algebraic thinking, and I was thinking ‘p atterns, whoo-hoo’! You know, I just didn’t think it was that important.” As her experience with DAL continued, Oliv ia noted that during the middle of the study, she had doubted her abilities to teach he r students more than before starting with the DAL framework, for two reasons. First, she realized she did not know or could not

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284 remember how to represent multiplication problems. As she stated, “I had gaps definitely, as far as representing multiplicati on. I had to learn that myself…. Now, how I felt about implementing it, I felt a little uneas y because it was new and I had to learn how to do it.” While this first issue momentarily raised concerns in her head about her abilities to teach the multiplication concep t to her one student, she resolved these concerns by accessing resources at her dispos al, including university support staff and peers. Second, Olivia discovered that she had misinterpreted her othe r student’s abilities with patterning, which she described as, “I thought she breezed th rough the patterns, but then I misjudged that and I reassessed her. She had already told me that she had problems with math… And when I reassessed her, I realized she didn’t ha ve patterns.” In terms of helping Olivia devel op her instructional efficacy, she said that at the same time she was participating in the Level II pr acticum, she was also taking a mathematics education course that focused on pedagogical ideas surrounding mathem atics instruction. She stated she used the mathematics educati on course’s text as a resource with, “the teaching math book had some great suggestions for books (indicating sources for authentic contexts)”. Olivia also menti oned conferencing with the practicum support staff, including myself the researcher, as m eans that developed her instructional efficacy with multiplication. Time constraint was the only real detract ing outside factors th at Olivia mentioned as influencing her abilities to teach her stude nts algebraic thinking. Both of her students were absent on one occasion dur ing the process; and anothe r time, she had a shortened session with one student because she felt bad about pulling the student from a “preferred” computer activity. Olivia mentioned she had made her greatest reali zations in developing

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285 her instruction in the last five minutes w ith each student, which again left her wanting more time in the overall DAL experience. Sh e described this experience with, “It all came together at that “aha” moment, like in the last five minutes that I was with the student, I was like what strategy can I teach the student, and I thought partitioning, it just like came to me… Just like came to me in the la st five minutes that I had left with her. And, I wish I could have like really taught her that, that was like the “aha” moment.” Attitude. In regards to her attitude to wards mathematics and teaching mathematics, comments made in Olivia’s exit interview were decidedly constructivist in her approach to learning mathematics instruct ion and facilitating her students’ abilities to gain new mathematical ideas. Olivia’s remarks focused in on multiple ways to help students learn mathematics strategies. She also emphasized that these strategies had to be ones that motivated and engaged students in their own learning, such as, “In the UFLI, we were never up to the goal setting because we had the lower level books. So, when I did it for the math and I got to see how exc ited the student was to set a goal… and, that helped motivate them.” Throughout her interview, O livia stressed the idea that she was still forming her own understandings about mathematics instru ction and this process was an ongoing one, not thoughts that had been traditionally taught and memorized by her. In terms of her current mathematics learning, she stated, “I think what I got most out of it (the DAL training) was concrete, representational, a nd abstract, and actually showing this is concrete, this is representative this is abstract.” Within our conversation, she mentioned specific strategies that helped her learn mathematics included visual and kinesthetic learning activities, modeled demonstrations of ideas, as well as application and

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286 discovery-based experiences with new instruc tional practices to gauge her ability to use them. She specifically said, “I am not an audio person, I have to see it and do it. So, that’s why I’ll write the whole time the t eacher’s talking, because otherwise I won’t process.” Olivia also went on to say he r DAL training would have been enhanced by tapping into technology to meet her multiple modality learning needs through, “…even like a visual podcast to see the interaction with the teacher and the student, the professor and the student.” Olivia’s description of both her mathematics instructional activities with students and her own mathematics lear ning show a developm ental-constructivist perspective. Content Knowledge. During the interview, there were two main focal points of discussion about the content area of the DAL experience: patterning and representing mathematics multiplication-based problems. The reason for this emphasis was most likely because these areas were ones she work ed on with her students in DAL sessions. With the patterning concept, originally O livia had thought the content was not that complex or important. However, as she becam e involved in instruction, she realized the complexities of this skill area and the helpfuln ess of using manipulatives, especially with one student on patterning. She mentioned that with, “…one of them (of her students), the manipulatives, the concrete manipulatives, they were definitely helpful.” In terms of multiplication, one of Olivia’s chief realizati ons was that she herself was still grappling with fully understanding ways to conceptualize the ideas behind the automatic process involved in answer finding. However, sh e described her time learning a conceptual understanding of multiplication as “a very help ful experience”. She also spoke about her

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287 growth with mathematics over the semester with, “Just as we were just wrapping up, I was just getting it. Like I was just getting on my game.” Instructional Knowle dge and Application. When asked about instructional knowledge gained through the DAL experience, Olivia stated that CRA was the most significant of these ideas. She also me ntioned that she understood this method of instruction, and she felt that during the preparation with the DAL, it had been clearly explained by, “I think what I got most out of it (the DAL model training) was concrete, representational, and abstract. Those were like the major co mponents that I got out of it and that was obviously represented well if that ’s what I got out of it.” Additionally, she indicated that she valued the DAL framework as an instructional tool because, “I think it’s a good experience, I think it’s a good pro cess only because I don’t know of any other process. So, it’s nice to have a process.” O livia voiced a desire to learn “processes” for teaching mathematics, and this framework wa s her first towards that goal. While in general she advocated the use of having a structured approach to teaching at-risk learners mathematics in a systematic and incremental way, she noted that there were aspects of the DAL framework that she thought could be streamlined for instructional purposes. She said, “Session notes. The session notes sheet was a bit busy for me. I think if it was simplified a little bit, I think I could have followed it a little bit more. And, I know it wasn’t complicated… the way it was set up, I guess it just wouldn’t be the way I would set it up. I would want even simpler.” Summary. Overall Olivia expressed the lear ning of instructional practices and structured mathematics teaching methods as positive experiences through the DAL framework. She also described her jour ney towards mathematics as continually

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288 developing, along with her abilities to effec tively teach and understand mathematics. While she mentioned needing more time to develop her mathematics instructional abilities further, she felt she had grow n in her current abi lities through the DAL experience in conjunction with her mathematics education class. Kari Exit Interview – Efficacy. When talking with Kari about her feelings of efficacy in teaching algebraic thinking within th e DAL framework, she focused on two key elements that she felt had negatively impacted her ability to instruct her students more fully in their learning: time and preparation. In terms of time, in general she felt there was not enough of it for either her preparation with the model or her implementation of it. She explained that after the initial traini ngs with the DAL, which included several 1-2 hour seminars and a whole day workshop, she still felt “confused.” Consequently, she thought that lengthened trai ning time would have improved her understanding. She believed this change would have significantly helped her, because she knew that in general she really liked math, as she menti oned, “I’m strong in math personally.” As a result, she felt the reason she was “confused” with the process was not the content but the use of the framework. Overall, she voiced th at she did not feel she knew what she was doing with, “I really didn’t do that many sessions, and I rea lly didn’t get what I was supposed to be doing…so I was kind of just winging it.” In other issues involving her time concern, Kari felt that besides greater preparation time, more instructional session ti me would have also been beneficial. She thought the period of time for implementati on was too short, since it was begun with students nearly halfway through the semester. Additionally, she had fewer sessions than

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289 other individuals because she was out sick wi th the flu and then one of her students had been sick. Kari felt that mo re instructional sessions woul d truly have been helpful, because she said that with UFLI, the reading program also learned and implemented by the teacher candidates, she had felt confused with its application and usage in the beginning of the semester. However, her feelings about UFLI had changed over the many weeks of the semester, when she had time to implement the process and learn from her mistakes, and also to make connections with her students. She believed that her mathematics instruction would have been mo re effective with both students if she had increased opportunities to work with them as with her UFLI students. As she stated, “I really don’t think I knew what I was supposed to do or I wasn’t confident in it, so I didn’t really know what to do, and then it kind of ended. So I didn’t get to like grow or anything like with the UFLI, where ‘oh I ki nd of made mistakes, oh I shouldn’t have done that, oh I should have done this’, and gone from there.” It is pertinent to mention that even though Kari felt that time was one of the major barriers in her efficacy of implementation with the DAL, she was the ca se study participant who had struggled on the content knowledge assessment, scoring deficient on the whole content knowledge measure, as well as the subtests of both ba sic arithmetic and alge braic thinking skills. These results do not match her perceptions of strength in the content area of instruction, and may have had some effect on her efficacy in instruction. Attitude. Kari’s ideas of how she could have been better prepared with the DAL model shed light on her attitudes about teac hing and learning mathematics. One of the main deficiencies that Kari felt was elemen tary to her difficulties with using the DAL was the lack of explicit instruction during DAL training. Her comments reflected quite a

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290 traditional view of how she and the other teacher candidates s hould have been prepared to use the DAL framework, by “being told exac tly” how it should be implemented with students. Kari commented on this idea by mentioning that faculty should have approached DAL preparation with teacher ca ndidates as, “like this is what you should do.” She voiced that if this type of traini ng would have been provided, then she could have turned around and done the same for he r students, “explicitly taught the learning targets to them.” Kari’s ideas along these lines included, “I think the math should be more directed like how the UFLI was. I thi nk the UFLI was explicit ly taught to us, I kind of think the DAL was not expl icitly taught to us.” At the same time that Kari expressed th is desire for a more formal type of preparation, she did also mention a few key constructivist ideas a bout her approach to learning and teaching mathematics. One of these thoughts included that she felt it was the normal learning process for her and ot her teacher candidates to be somewhat confused about the DAL framework when they were taught about it in training sessions. Kari seemed to value the use of applying a pr ocess to help internalize learning its parts and intricacies more completely. Her second thought along these line s was that increased time for the framework’s application, allo wing a developmental time period for learning the process, was critical for both her abso rption and understanding of the DAL and her students benefiting from it. Third, Kari ha d felt it was valuable to have the DAL framework in a setting wher e all teacher candidates and university supervisors were together during the practicum day in a resour ce room type setting, where clarifications could be made and understandings develope d on an ongoing basis. She mentioned, “It was like a class and I liked havi ng you and the other professors there to be like, well this

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291 is what everybody is saying, well let’s just go ask them and let’s see what the correct answer is and how to do it, instead of waiting until class or three days later when it’s not that important any more.” Content Knowledge. In regards to the area of content knowledge, Kari’s comments centered on the fact that she had felt at first that both of her students struggled with basic concepts in alge braic thinking, but she had been mistaken in one of these cases. She mentioned that her female studen t had come to their sessions saying she knew she needed additional assistance in mathema tics. However, the second student simply mentioned he liked coming with her because he did not like his teacher. In the situation with the first student, Kari had been happy that at one point in the semester the student had come back to her saying that she had us ed a concept in the classroom that week, which she had learned with Kari in their previous session. Kari had thought this comment very positive, and she realized she had actually taught th e student a key idea with which she had been having trouble. W ith Kari’s second student, she had believed his difficulties had been with not understanding some key id eas with growing patterns. Yet, she said when she began working with hi m she realized that th e student actually did understand these concepts much better than she initially had thought. When asked how she dealt with this situation, she said she pr oceeded with her lesson with the student on the concepts, but had presented the instruction to him as more of a review than anything else. She said she handled it with the stude nt as, “We went off on ‘like you know this’. This is a review then.” She felt that the reason the student performed better on patterning in the session than on the assessment was due to the wording of some of the assessment items. While the assessment did not have a formal script, she said the guidelines for

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292 introducing the items were what had guided her instructions, and she felt that this language and vocabulary were difficult for th e student. In the fi rst few sessions, she realized that with further probing and disc ussion with the student, he really did understand patterning ideas but perhaps just needed these ideas activated in his own language. Instructional Knowle dge and Application. Taking Kari’s content knowledge findings with this last student into account, it is important to remember from the Entire DAL Project review that the researcher had discovered that although Kari found her initial assessment’s results were not complete ly accurate appraisa ls of her student’s abilities, she had difficulty in actually impl ementing instructional changes. When she conducted her initial probe in the target area of the assessment, she had noted she would elevate the student one level, fr om the concrete to the representational in the next session, based on her findings that he actually underst ood the material at the concrete level. However, in her next session, Kari continued to target creating grow ing patterns still at the concrete level, during both steps 1 and 2. Kari’s actions are a reflection of two key parts of her instructional difficulties with the DAL, implementing data-based decisions focused on student performance and correctly us ing the levels of CRA for instruction. As Kari realized her student had a better grasp of patterning material than she had thought, her instructional decision was to move the st udent up to the next le vel of representation for the skill. However, because of possibly faulty understanding of how to apply and use CRA, Kari did not actually implement the in structional change sh e had intended. While Kari mentioned that she thought explicit instruction and CRA were valuable instructional

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293 strategies with her students, it appears that she needed continue d work on understanding and implementing CRA. Summary. Kari’s remarks throughout her in terview voiced frustration with her experience with the instructi onal framework because of time constraints and the design of its initial assessment’s instructions. She also emphasized the need for greater preparation with the DAL process. Additi onally, Kari felt that more formal means of instruction with her own preparation would have aided in her usage of the framework. Taylor Exit Interview – Efficacy. When completing the exit interview with Taylor, she honed in on some key aspects that affected her self-efficacy in instruction, and the ability of that instruction to impact student learning outcomes. Be fore beginning training with the DAL framework, Taylor mentioned that sh e felt confident in her abilities to teach algebraic thinking at the elementary level, because of what she perceived as the “low level content” of the instruct ion. She stated, “As far as anything in elementary school, I felt like I had a pretty good handle on it.” Ho wever, from the start of the DAL training and implementation, Taylor indicated that bei ng absent for health reasons during the one full-day of DAL training at the start of the e xperience, left her feeling uncertain about her abilities to teach mathematics. While she had spent some individual time playing “catch up” with the researcher, she did not feel she grasped the process from the start, she felt this situation negatively impacted her ability to implement algebraic thinking instruction. She stated these ideas with, “I don’t think I ev er really got a concrete handle on what the process was exactly. I don’t think it ever really became clear, lik e I think I’m one of those people that needs to understand why I’m doing what I’m doing. I need to know

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294 what the purpose is for something and if I didn’t understand how some thing benefited the whole process or if I didn’t understand w hy something was being done, then, it just wouldn’t stick in my brain, it ju st wouldn’t retain that. Ther e’s a lot of things, I think I just don’t understand the applic ation of them, and why we do that.” Additionally, when she mentioned she had asked other teacher candidates about implementing the process, she said they could not help her because they were also “lost”. Other variables that she gave as affecting her instructional e fficacy were a low number of sessions for implementation because of student absences, as well as what she called a “mini-crisis” at the school every time she came to the school site for the practic um experience. Attitude. In terms of her attitude toward s mathematics instruction, Taylor displayed a combination of constructivist and formal approach ideas. She felt discouraged when she saw her upper-level el ementary students evidence gaps in their understanding of patterns. Upon working with her students, w ho were in grade levels far beyond patterning, she felt that these gaps we re due to their teachers not developing conceptual understandings of skills, but simply having students memorize abstract concepts. On this topic she mentioned, “It ki nd of reinforced to me the thought that math teachers are teaching, okay this is A + B = C, and this is what you do to get your answer, but they don’t ever explain why that is. Or what the significance is.” Taylor voiced dissatisfaction with this traditional method of teaching mathematics that had been occurring for a long time, and she mentioned th at she had experienced it 15 years before in her own schooling. She also discussed the inability of her stude nts to explain their own understandings of concepts, and their desire to just give her the answer rather than explaining how they arrived at the answer or completed the problem-solving process.

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295 One this idea, she said, “They could do it, a nd they could extend or repeat a pattern or whatever but they didn’t, when they went to explain what was going on…they didn’t know.” Taylor’s ideas are consistent with constructivist ideas of building mathematical understandings by constructing knowledge through the comprehension of one’s own thought processes and means of finding solutions. At the same time, Taylor seemed to s till describe herself engaging in thought processes and practices that were more in accordance with formal instruction methods too. She found little merit in the amount of in struction she completed with her students, because she felt she was simply “reviewi ng concepts” rather than facilitating understanding and retention through this work. She also descri bed student deficiencies in skills as “gaps” in th eir learning. Yet, when she talk ed about designing instruction to meet student needs in these gaps, her descrip tion of how to accomplish this feat was akin to someone shoveling informati on into these holes rather than students bridging these “gaps” through connection building and guided discovery experiences. This duality of perspectives on teaching mathematics was simila r to her views collected on the attitude questionnaire, which showed her inclinat ion towards both types of instructional approaches. Content Knowledge. With the content knowledge area, Taylor spoke about how she had spent all of her time with both students on the “m ost basic” algebraic learning area of patterning, which she felt they really should not have had as a target area for instruction since they had only missed a couple of questions on that skill area. Taylor felt that expecting 100% accuracy on certain area s of the DAL assessment were too high of expectations for any learner, and she felt resulted in students receiving instruction in

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296 basic areas where small clarifications, such as with vocabulary, were all that was needed. She lamented the fact that her students c ould actually do more complicated, as well as abstract levels of algebr aic problem-solving, but str uggled with understanding the concrete and representational levels of skills. Taylor said that before beginning the DAL instructional process, she was unaware that these types of gaps and difficulties could happen in learning what she considered “founda tional skills”. Anot her key problem that she saw specifically with the DAL framework and algebraic thinking instruction was that time spent on integrating reading instruction and related target skills detracted from students’ abilities to focus on mathematics c ontent. Taylor viewed the combination of reading and mathematics instructional strategi es, as not facilitating further mathematics content comprehension, but placing a dual emphasis on unrelated reading content. Additionally, Taylor felt sh e had spent too much time on gathering materials and planning rather than focusing on the actual al gebraic content ideas for instruction. She mentioned, “I think I would have like[d] to ha ve spent less time on making sure that I had the stuff, [and] more time on making sure th at my lesson made sense and was kind of you know logical and applicable the student’s li fe, because I spent so much time making documents with like pictures that I could cut out, and all that.” Instructional Knowledge and Application. According to Taylor, her usage of instructional practices include d her implementation of explic it instruction with modeling, CRA, the problem-solving process, and structur ed oral language experiences, which she felt were all helpful in student learning. However, she remarked that her own understanding of the DAL framework was ne gatively impacted by the instructional format of being “told about the process” rather than having her other own modalities for

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297 learning accessed. Taylor vocalized these ideas through, “I mean you can only say this so many more times before you say this isn’ t going to do any good saying it. So, I don’t want to say add you know another day of inst ruction in, because I don’t know if that’s gonna do anything. I guess just making sure that everyone has the opportunity to do the entire process as the tutor st andpoint, and then again as th e student standpoint.” The other ideas that Taylor mentioned th at had impacted her abilities to use the DAL framework for instruc tion included the model’s difficulty in implementation because of its open-endedness and demands fo r teacher candidates to engage in large amounts of outside planning. When she comp ared the DAL model to the UFLI, which was the framework taught in the same prac ticum for reading instruction, she also remarked that the DAL was “less intuitive” in its application and experiences because the model did not facilitate great er understanding of instruction through multiple exposures to it. She evidenced concerns about th e detailed nature of the DAL process for implementation with, “I think there were thi ngs I just forgot. Like some steps, and maybe I’m wrong, that just didn’t have someth ing on the form, I would forget, like on the first part, you time the activity that you do, but I don’t know, but when it came time to do it, I couldn’t remember what to do. What do I write in? Do I write in the time? Do I write in what they got wrong?” Lastly, in te rms of instructional overlap between reading and mathematics strategies, Taylor did not thi nk they were readily apparent and that she found herself trying to make arbitrary connecti ons between the literature books used with the DAL and the concepts for instruction. Summary. Within her full DAL experience, Taylor felt that she struggled considerably with executing efficacious instru ction with students due to a lack of DAL

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298 training and her inability to remember all th e pieces in the DAL process. However, Taylor did mention that she thought her stude nts made qualitative gains in understanding and were able to explain specific patterni ng concepts through he r instruction. While Taylor mentioned specific instructional st rategies that she learned through the DAL experience that helped her students make mean ing of mathematics concepts, she felt that many improvements could be made within the framework itself. Comparison of Case Study Exit Interviews. Olivia, Kara, and Taylor’s comments each depicted unique experiences with the DAL framework that affected the development of their mathematical instructional abilities in different ways. One common theme across all participant remarks was the need for gr eater amounts of time fo r both instruction and training, as well as the employ ment of more diversified pe dagogy with teacher candidates for their preparation to use the DAL framewor k for instruction. While Olivia enjoyed the social-developmental construc tivist approach to the DAL instructional experience, and seemed to grow across the identified critic al elements in mathematics instructional abilities, Kara and Taylor felt differently a nd at the same time exhibited key areas of difficulty in growth. Kara felt that she need ed more direct instruction and experiences with the model, and Taylor believed that she simply needed mo re understanding of the model, which would be facil itated by hands-on instructiona l activities. Kara’s remarks focused on outside factors influencing her ab ilities to learn and use the DAL framework, while Taylor looked at both her own learning style and pers onality in conjunction with other factors for difficulty in learning th e framework. Kara’s greatest barrier in increasing mathematics instructi onal abilities seemed to be a lack of understanding of her own deficits and needs in learning to teach mathematics. Taylor’s challenges with

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299 mathematics teaching appeared to stem from a lack of mathematics content knowledge and understanding of the instructi onal practices and structure of the DAL model. Overall Case Studies Summary Through the DAL framework, Olivia, the top-achieving teacher candidate, seemed to experience the most well-balanced growth across the five identified elements relevant to mathematics instruction preparati on for at-risk learners. The reasons for this growth seemed to stem from her construc tivist approach to in struction and learning, rather than solely academic ability. Kara, the mid-achieving teacher candidate, appeared to experience limited growth in instruc tional knowledge unders tanding and content knowledge with the DAL framework. It is probable that Kara’s abilities did not experience even growth across the five iden tified elements relevant to mathematics instruction preparation for at-risk learners be cause of her view that her challenges with mathematics instruction were primarily related to external forces outside of herself. As with Olivia, academic abilities did not seem to be the sole factor affecting Kara’s mathematics instruction abilities. Taylor, the low-achieving participant, appeared to have experienced greatest growth in the criti cal elements of content knowledge and instructional strategy knowledge. Taylor seemed focused in on explicitly taught elements of the DAL framework, and seemed to experien ce gains in all areas taught specifically within the context of the training experien ce. Taylor’s greate st challenges with mathematics instruction progress appeared to be at least partly academic, with her inability to grasp the reasoning behind many of the pieces that facilitate effective mathematics instruction. However, upon receiving instructio n and application

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300 experiences, Taylor’s primarily constructivist approach to le arning seemed to facilitate gains in her content and in structional knowledge. In Chapter 5, conclusions based on the resu lts and analysis from this chapter, are discussed. Possible limitations of the current study are also addressed. The chapter ends with implications for research and practice with suggested directions given for future study.

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301 Chapter 5 Discussion This research investigated an appli cation-based instruc tional framework for elementary level algebraic thinking instruc tion within a preservice special education program. Unique to this study, the goal wa s the integration of content-based knowledge and instruction within the coursework a nd practicum of an undergraduate special education preparation experi ence employing a social-devel opmental constructivist approach. The purpose of the study was to in form the usage of the Developing Algebraic Literacy (DAL) framework as an instrument for facilitating preservice special education teachers’ development in mathematics content area instruction. Specifically, the current investigation expl ored the teacher candidate experience with the DAL framework as part of thei r Level II practicum and coursework, where students took Clinical Teaching and Behavior Management courses in connection with a two-day a week practicum experience. The teacher candidates were exposed to instruction and preparation with the DAL framework thr ough an initial intensive workshop and ongoing support seminars within the school site where they implemented one-to-one mathematics instru ction one day per week. In conjunction with this preparation, further instruction and suppor t were provided on site through informal observations and individual feedback thr oughout the practicum day, and by researcher visits and guest lectures dur ing the Clinical Teaching cour se. At the same time, the Clinical Teaching professor collaborated with the researcher to provide additional support

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302 to teacher candidate participants. The inve stigation involved a tota l of 19 participants from which the researcher collected data to inform the exploration of the key research question: What changes related to effective ma thematics instruction for struggling elementary learners, if a ny, occur in teacher candidates during implementation of the DAL instructional framework in an ear ly clinical field experience practicum for preservice special educati on professional preparation? To best evaluate the changes that oc curred in teacher candidates through their DAL experience, pivotal elements identified by the researcher through the literature base of mathematics, language arts, and special education were used as key factors in monitoring teacher candidate change. Thes e elements included teacher candidates’ attitudes towards mathematics instructi on, feelings of efficacy about teaching mathematics, pedagogical understanding and application for at-risk learners in mathematics, and actual mathematics content knowledge for instruction. These elements of the research question were explored under the major inquiry areas: 1.) What changes, if any, occur in sp ecial education teacher candidates' attitudes towards mathematics instructi on from the beginning to the end of a preservice instructional experi ence using the DAL framework? 2.) What changes, if any, occur in spec ial education teach er candidates’ feelings of self-efficacy about teachi ng mathematics from the beginning to the end of a preservice instructional experience using the DAL framework? 3.) What changes, if any, occur in special education teacher candidates'

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303 understanding of instruc tional strategies for st ruggling learners in mathematics from the beginning to th e end of a preser vice instructional experience using the DAL framework? 4.) What changes, if any, occur in special education teacher candidates’ application of instructional strategi es for struggling learners in mathematics from the beginning to the end of a preservice instructional experience using the DAL framework? 6.) What changes, if any, occur in special education teacher candidates’ content know ledge of elementary mathema tics, including algebraic thinking, from th e beginning to the end of a pr eservice instructional experience using the DAL framework? The remainder of this chapter is organized by: 1) conclusions that were reached through the data collected; 2) possible limitations to the current study; and 3) significance and implications of the research. The findings of the study are presented in the conclusions section of this chapte r by data collection method. Conclusions The current study was devised to further th e research base for preservice special education professional development experi ences, which have the goal of preparing “highly qualified” special educa tion teachers, prepared to no t only teach learners at-risk for academic difficulties but the specific content of the mathematics curriculum area (NCLB, 2001; NCTM, 2002; Maccini & Ga gnon, 2002). The ongoing need for greater understanding of instructiona l interventions, frameworks, and methods employed within preservice teacher preparation programs is imperative for enhancing the preparation

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304 experiences of future special education teachers so that they are better positioned to help their future students achieve positive academic outcomes (Gagnon & Maccini, 2001; Baker, Gersten, & Lee, 2002). The literature demonstrates that lear ners who are at-risk for academic failure because of disability, ec onomic, or social causation are more likely to engage in positive learning experiences and school success when they are taught by teachers that are prepared to both meet thei r diverse educational characteristics and who possess the content area and pe dagogical knowledge to teach specific subject areas, such as mathematics, effectively (Bottge, et al., 2001; DarlingHammond, 2000; DarlingHammond, 1999). In the current climate of NCLB (2001) and IDEA (2004), combined with the increasing diversity of the K-12 st udent population (Fry, 2006) the need for such teachers is even more pressing. Because of the wide emphasis on reading in struction research in recent years, the current study incorporated findings of this re search base with the existent mathematics and special education literature to inform its development. The DAL instructional framework also integrates practices supporte d by these research bases. While employing the DAL framework as an applied instructi onal experience within a social-developmental constructivist special education practicum, the investig ation found that overall teacher candidate agreement with constructivist at titudinal statements about mathematics and mathematics instruction increased during the co urse of the study, as well as candidates’ levels of content knowledge in algebraic thinki ng. Identification of learner characteristics and effective mathematics instructional practi ces for at-risk learners was mastered, while the articulation of instructional practice specifics showed beginning competency. However, deficiencies in understanding how to apply these instructi onal practices within

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305 the context of the DAL framework were evidenced. While teacher candidates’ perceptions of their instructional efficacy in mathematics was low at the end of the study, teacher candidates’ were able to implement th e steps of the DAL framework with fidelity over 50% of the time. At the same time, teacher artifacts indicated that teacher candidates had beginning understandings of differentiated instru ction and effective mathematics instruction, but needed conti nued work on understanding specific elements of targeting effective mathematics instruction spec ifically to individual st udent’s needs. Mathematics Teaching Efficacy Beliefs Instrument The Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) was used to collect quantitative information on teacher candida tes’ sense of self efficacy in regards to their mathematics teaching abilities, as well as their beliefs that effective mathematics instruction can impact positive student mathematics learning outcomes. Results from this full instrument showed that teacher candida tes had a mean score ranging from between 3.37 and 3.72 out of a possible 5 from pretes t to posttest, indicating that teacher candidates’ agreement with statements re garding efficacy fell between “Uncertain”, which was a score of 3, and “Agree”, which involved scoring an item as 4. Overall results on this instrument were consistent with the norming groups of the MTEBI and the Science Teaching Efficacy Belief Instrument (STEBI) (Enochs, Smith, & Huinker, 2000), on which the mathematics survey was based. Wh ile these results indicate that the current study’s teacher candidates did not have negative views about their efficacy, which would have involved scores between 1 and 2, thes e numbers did not indicate a significant change in levels of efficacy between pretest and posttest. However, the level of positive

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306 agreement with efficacy statements did show some minimal increase from pretest to posttest. Mean scores for the MTEBI’s two subtests self efficacy and outcome expectancy, also increased from pretes t to posttest, means were between 3.35 and 3.49 for self efficacy and 3.39 and 3.58 for outcome expectancy. Both of these subtests showed upward movement between pretest and posttest for levels of agreement with statements involving personal effectiveness in instruction and student responsiveness to effective instruction. However, neither set of results showed statistically significant growth in feelings of efficacy in mathematics instruction. At the same time that growth was seen between the pretest and posttest for the self-efficacy and outcome expectancy subtests the greatest gains on both subtests were actually seen between pretest and midpoint. While posttest results were higher than pretest results, a noticeable dip in mean efficacy scores was seen between midpoint and posttest. This decrease could be due to a couple of reas ons: 1) teacher candidates initially felt greater levels of efficacy wh en beginning newly learned instruction, but these feelings began to decrease over the latter course of the semester as teacher candidates saw the difficulty in affecting st udent change in mathematics through their instructional efforts and/or 2) teacher candidates’ stress leve l may have been elevated at the time of the posttest administration of the survey because it was the week before final examination week at their university. In terms of specific response items, it was encouraging to see that the highest agreement in terms of efficacy at pretest was on item 2, “I will continually find better ways to teach students mathematics”, which s hows an inherent dedication to seeking out

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307 more effective instructional methods for stude nts who struggle in ma thematics. On the other end of the spectrum, at pretest teacher candidates had the lowest agreement with item 17, “I wonder if I will have the necessa ry skills to teach mathematics”, which indicates that teacher candidat es thought they would in fact be able to develop these skills. Item 17’s response mean shows that teacher candi dates entered the study with some level of confidence in their ability to learn how to teach mathematics. At posttest, teacher candidates had the highest agreement with item 15, “I will find it difficult to use manipulatives to explain to students why math ematics works”. This result was surprising considering that one of the emphases of th e DAL instructional fr amework was using the CRA sequence of instruction, where concrete materials ar e essential for breaking down the complexities of new algebraic concepts. The lowest mean responses at posttest were shared with item 17 and item 18, “Given a c hoice, I will not invite the principal to evaluate my mathematics teaching.” Res ponses to items 17 and 18 give evidence to some sustained feelings of efficacy th roughout the study, since teacher candidates maintained a positive outlook a bout their abilities to learn mathematics instruction, and could even see themselves inviting school princi pals to observe their instruction. their In comparison to the norming group, the current st udy produced consistent results for scores on items 2 and 8, but not item 15 involving the use of manipulatives. This difference may be due to the special education bac kground of the teacher candidates, which was elementary education for the norming group, or the teacher candidates’ difficulty with their diverse student population, which al so differed for the norming group (Enochs, Smith, & Huinker, 2000).

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308 Preservice special education programs are typically generalist preparation experiences where future teachers are prepared with instructional practices that can meet student learning needs across subject areas (Boe, Shin, & Cook, 2007; DarlingHammond, 2000). To this end, this prepara tion typically involves only one or two courses specifically in reading instruction and mathematics instruction while most elementary education programs require severa l courses in both r eading and mathematics (Boe, 2006). With the study participants, two reading courses and one mathematics education course are required as part of th eir special education pr eparation program, but the teacher candidates were scheduled to ta ke their mathematics education course the semester following this study. As a resu lt, teacher candidates may not have felt comfortable teaching students with manipulativ es as fully as those individuals in the instrument’s norming group simply because th e majority of them had not taken their mathematics education course at this poi nt and had limited knowledge and experiences with mathematics instruction. This could e xplain the low rating on the survey item about manipulatives. Previous learning with ma nipulatives through their special education program may not have specifically covered the targeted use of manipulatives for mathematics learning, while the current study only had limited time to do so with them. Additionally, the stud ent population of the current st udy participants included only students at-risk for mathematics failure, wh ile the norming group had participants whose target students were typical learners. While the at-risk student population requires usage of diversified pedagogy, each student’s learni ng needs are different and specific teacher candidates may not have employed manipula tives with their students, depending on individualized instruct ional needs. All of these fact ors may have resulted in the low

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309 rating of manipulative usage by teacher candi dates in this study (Gagnon & Maccini, 2001). Mathematical Beliefs Questionnaire The Mathematics Belief Questionnaire was employed to collect quantitative data on teacher candidates’ attitudes towards math ematics in general and the teaching of mathematics (Seaman, et al., 2005) The ques tions on the survey were broken down into four categories: trad itional beliefs about mathematics, traditional beliefs about teaching mathematics, constructivist beliefs about mathematics, and constructivist beliefs about teaching mathematics. Results of the instru ment indicated that teacher candidates had greatest attitudinal agreement with items i nvolving constructivist ideas about teaching students mathematics, followed by agreement with items involving constructivist ideas about mathematics in general. Through the course of the study, this agreement with constructivist mathematics principles increased from pretest to posttest, with the mean of constructivist teaching mathematics ideas moving from 4.112 to 4.157 on a scale of 6 and the mean of general constr uctivist mathematics beliefs moving from 3.811 to 3.942. The rating of 3 on the measure indicated “slight ly disagree”, and the rating of 4 on the measure indicated “slightly agree”. While both constructivist mean score tendencies indicated that teacher candidates tended to “s lightly agree” with constructivist attitudes, these ratings were not far from tending to “sli ghtly disagree.” It also bears mentioning that agreement with traditionalist views on mathematics in general and mathematics instruction was not far from the same level of agreement, with the mean of traditionalist teaching beliefs moving from 3.311 to 3.382 a nd general traditi onalist mathematics beliefs moving from 3.233 to 3.409 during the c ourse of the study. These overall results

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310 of the Mathematics Beliefs Questionnaire are consistent with the normative data of the instrument (Seaman et al., 2005), although actua l agreement levels with constructivist views of general mathematics and teaching math ematics are slightly less than that of the norming population. This lower agreement leve l of study participants may be due to the norming population consisting of general edu cation classroom teachers, who typically experience more courses in mathematics educ ation than do special educators. At the same time, special education preparation programs as a whole tend to focus more on instructional pedagogy involving knowledge acquisition, repetition, retention, and application to meet the learni ng challenges that at-risk and st udents with disabilities face, versus the exploration, discove ry, and formulation advocated emphasized in elementary and mathematics education programs (Golde r, Norwich, & Bayli ss, 2005; Mercer & Mercer, 2005). As with the MTEBI, gains in attitudi nal agreement were seen across all four domains of beliefs between pretest and midpoi nt, while agreement levels experienced a drop across all four domains from midpoint to posttest. The reasons for this decrease are thought to be due to the same reasons as noted previously for efficacy score decreases in the latter part of the study including teacher candidate challenges in affecting student learning outcomes in mathematics through th eir instruction and the stress level experienced by teacher candidates at th e end of their academic semester. On the specific response items of highest agreement, teacher candidates indicated the highest attitudinal agreement at both pretest and postte st with item 21, “The teacher should always work sample problems for st udents before making an assignment”, which showed a more traditionalist viewpoint for teaching mathematics. While the lowest

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311 agreement at both pretest and posttest was seen for item 37, “Students should be expected to use only those methods that their text or teacher uses”, indicating a more constructivist viewpoint for student learning of mathematics. This dichotomy of thought is indicative of the mixture of both traditionalist and cons tructivist ideals that were held by the special education teacher candidates at this point in their professional development in regards to mathematics instruction. The norming population from the Mathematics Belief Questionnaire had higher agreement with the latter statement, and less agreement with the former. Indeed, the highest rating of the norming group dealt with items 24-26 on the survey, which involved students building th eir own mathematical ideas and problemsolving abilities (Seaman et al., 2005). The differences between the current study’s viewpoints and the norming group may be due to the norming group consisting of teachers involved in elementary education, while the current group included special education teachers’ whose student populati on needs are different and require more individualized consideration. While teacher candidates in elementary education are often taught to employ inquiry-based instruction with their st udents, preservice special education teachers are often taught that the us age of explicit inst ruction with modeling assists retention of new concepts for student s with processing and memory deficits (Boe, Shin, & Cook, 2007). Along these lines, the t eacher candidates in the study may have rated item 21 higher than other responses ba sed on their professional preparation as a whole emphasizing explicit instru ction with modeling, or as a re sult of this instructional method being advocated within the scope of the DAL instruct ional experience itself. Out of all the elements evaluated for teacher change using the DAL framework, attitudinal beliefs of the teacher candidates a ppeared the most consistent and resistant to

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312 change. This quality was indicated by th e highest and lowest agreement items for mathematical beliefs remaining the same from pretest to posttest. At the same time, correlations were seen between administrations on each subtest area of the instrument, where earlier scores on specific subtests correlated with posttest scores on the same instrument. This information is crucial for teacher preparation programs’ development of subject area preparation for speci al educators, because it is indicative of the difficulty in affecting change in teacher candidate attitudi nal beliefs about mathematics in general and mathematics instruction. In response to this knowledge, special education teacher preparation programs can ask perspective t eacher candidates targeted questions about mathematics attitudes to gauge whether these individuals possess attitudes that are more reflective of constructivist ideals before acc epting them into preparation programs. At the same time, programs can also focus mo re specific course objectives on teacher candidates’ abilities to reflect, understand, a nd develop constructivi st attitude towards mathematics learning through an emphasis on reflective writing, di scussion, exploratory activities, and coopera tive learning. Mathematics Content Knowledge for Elementary Teachers The Mathematics Content Knowledge for Elementary Teachers survey was the instrument used to collect quantitative data on the teacher candidates’ understanding of elementary level mathematics knowledge in general mathematics and algebraic thinking (Matthews & Seaman, 2007). Results on this m easure indicated that this group of special education teacher candidates had deficien cies in overall mathematical knowledge, including the areas of general mathematics a nd algebraic thinking. Mean results included a 35% accuracy rate on the overall measure, and 40% and 34% on the two subtests

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313 respectively. While small gains were seen with each of these scores from pretest to midpoint, none of the scores reached near 60%, which could be considered beginning competency with these mathematics content skills. Additionally, all scores fell to approximately pretest levels at the posttest administration. While the current study explicitly taught el ementary level algebraic thinking skills to teacher candidates through its initial tr aining workshop, and supported these skills through seminars throughout the study, it is evid ent that future res earch endeavors should dedicate a greater amount of seminar time a nd activities to the area of content knowledge enhancement in teacher candidates. In this investigation, initial and ongoing training and support were split between the five domain s deemed essential to special education teacher candidate development in mathematics in struction. Yet, current results indicate a specific need for more attention in conten t knowledge preparation. While the teacher candidates’ levels of content knowledge wh en entering this training experience is discouraging, it does provide valuable information to teacher preparation programs by indicating a great need for in tensive time spent on content area knowledge within special education teacher preparation programs. W ith little to no movement seen in posttest scores from pretest, it is also indicative that a different and possibly more extensive approach must be taken for teacher candi dates to absorb and apply this content knowledge. Current teacher candidate results are consistent with the normative control group for the instrument, and slightly below th e scores of the normative treatment group. So, while the teacher candidates in the curren t study had low scores in terms of content knowledge, similar ability levels were also seen in the norming group which consisted of elementary level educators (Matthews & S eaman, 2007). Facilitati ng enhanced content

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314 knowledge acquisition appears to be a time in tensive process, and it is recommended through special education and mathematics education literature alike that content knowledge be targeted through increased cour sework requirements in mathematics, as well as extended learning periods for this coursework for developmental learning experiences over time (Carnine, 1997; Char alambous, Phillipou, & Kyriakides, 2002). From this study’s results, benefits including connection-making have been seen through integrating pedagogical and content knowledge preparation in mathematics. Further exploration may indicate increased positive re sults if this integration is employed throughout entire teacher preparation programs versus just a single ten-week period. As with the MTEBI and the Mathematic s Belief Questionnaire, gains in content knowledge were seen in both basic mathematic s and algebraic thinking from pretest to midpoint, but then scores dropped across both area s from midpoint to posttest. In fact, unlike the two previous instruments, drops in scores at posttest br ought teacher candidate scores back to actual pretest levels rather than evidencing any overall gains. While the main reasons for the decrease are thought to be due to similar factors to the decrease in efficacy and attitude scores experienced by the teacher candidates in th e latter part of the study, a few additional variables may be responsible. First, because of the complex nature of the content knowledge element, it is thought that instruction in this area may have needed more time during ongoing prepar ation experiences fo r understanding and retention of information. Another factor may ha ve been the type of instruction used with teacher candidates in learning content knowledge skills. For instance, the researcher provided lecture and PowerPoint materials in conjunction wi th hands-on application and practice activities. From comments in fo cus groups about what helps the teacher

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315 candidates best learn new information, teach er candidates self-disclosed that using multimodal and hands-on methods best meets their learning needs. Future training efforts may want to focus primarily on ha nds-on activities, targeting more kinesthetic methods than used in the current study, while providing teacher-directed lecture and Powerpoints more as supplementary aids. A dditionally, the focus of explicit instruction for the teacher candidates in this study was th e particular set of algebraic thinking skills that were needed for student instruction within the DAL framew ork. Content knowledge results indicated a need for more in dept h instruction and experiences with general arithmetic skills as well, since content knowledg e skills were low in this area as well. A last factor that may have affected teache r candidate learning of content knowledge was external variables such as teacher candidate absence, scheduling issues at the school site, and number of instructional sess ions. All of these variables were external issues during the current ten-week study that may have infl uenced teacher candidates’ ability to retain content knowledge because they impacted teacher candidates’ ability to learn and practice new content information. Further exploration of content knowledge learning through the DAL experience in another study, could examine these important variables more closely. In terms of specific response items, teach er candidates’ area of strength involved question 2, which consisted of converting a num erical model into a word representation of the same idea. Questions that evidenced specific difficulty were ones that involved a more conceptual and abstract understanding of mathematical concepts. While these results are not identical to the highest and lowe st scored items in the normative data, they

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316 are consistent with items that were generally answered correctly or incorrectly (Matthews & Seaman, 2007). Instructional Knowledge Exam An instructor-made instructional knowledge exam was utilized to obtain data on teacher candidate understanding of pedagogica l knowledge taught within the context of the DAL instructional framework. Questions of two varieties were presented: multiple choice and short answer essay. The items on the test covered three types of information: identification of learning characteristics a nd instructional strategies; articulation of component parts of and instru ctional strategy usage; and ap plication of instructional strategies within the context of the DAL instructional framework. The mean overall scores of teacher candidates on the content knowledge exam was approximately 62%, indicating beginning competency in understanding of teaching at-risk learners using effective and research-b ased practices. Strengt h was seen in teacher candidates’ abilities to identify learning ch aracteristics and instructional strategies through multiple choice questions, with an accu racy rate of 91%, indicating mastery in this particular area. The ab ility to explain component part s of instructi onal practices, which was assessed through exam essay questio ns, was at the beginning competency rate across participants with a m ean score of 60% accuracy. Application essay questions, involving the strategies’ usag e within the DAL model, we re answered correctly by teacher candidate participants less than 60% of the time, which is below competency level for applying these research-based instru ctional strategies for at-risk learners in mathematics in the context of this instructional framework.

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317 These results indicate that teacher ca ndidates have mastered the recognition of student learning character istics and instructional strategies at the identification level. However, it shows that teacher candidates ma y have difficulty when asked to personally articulate elements of instructional strategi es, when using their ow n words to explicitly explain the components of strategies. Teacher candidates’ difficulties with applicationbased questions, which involve relating the strategies to the DAL framework itself, illustrate that while teacher candidates can iden tify and explain instructional strategies to some extent, they continue to need furthe r instruction and support on the usage of this knowledge in applied situations. While teacher candidates’ performance on the instructional knowledge exam mi ght at first seem discourag ing, the results may actually demonstrate promise given that the current research study was conducted in the teacher candidates’ second semester (out of five semesters) in the program. Teacher candidates, being early on in their program, may not ye t have fully developed the study habits necessary for retention of instructional know ledge. At the same time, these future teachers recently entered their special educa tion teacher preparation program at different professional levels of development, and so me may need additiona l time in making sense of instructional practices for application pur poses because they may be in the beginning acquisition stage of these skills (Boe, 2006; Darling-Hammond, 2000). Additionally, this semester represented teacher candidates’ first direct instructional experience with students where they were responsible for assessment, planning, and the delivery of instruction in a specific content area. Fi nally, the relationship of higher scores on multiple choice identification items and lower scores on essay application items is expected due to the nature of the difficulty of essay versus multiple choice questions, as

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318 well as the more in depth and specific natu re of application-based questions (DarlingHammond, 1999). Fidelity Checks During the course of the study, a subgr oup of teacher candidates was monitored for their ability to implem ent the DAL framework and its imbedded instructional strategies with fidelity. For the purpose of the fidelity checks, the researcher developed an observational fidelity checklist in c onjunction with the DAL framework’s primary development expert for independent raters to monitor teacher candi dates’ abilities to implement the DAL instructional framework us ing the steps they we re taught during their preparation and training with the DAL model. Results from this fidelity monitoring indicated teacher candi dates’ abilities to implement shorter initial DAL sessions, calle d the Initial DAL Sessi on Probe, with a high rate of fidelity, approximately 95%. Du ring these observations, teacher candidates appeared to have mastered the majority of th e session steps. This mastery may have been evident for several reasons. First, the ini tial session includes only a total of 7 possible steps, which limits the number of elements th at need to be remembered and used within the session. Second, the goal of the initial session probe is to further explore students’ mathematical understandings to ensure that initial assessment results accurately reflect students’ algebraic thinking abilities and n eeds. Teacher candidates, who spent 2-3 sessions conducting initial DAL skill asse ssments, may have found themselves more comfortable with the informal assessment na ture of this initial session then with the subsequent longer and more instructional full length DAL sessions. Third, teacher candidates may simply have had a high rate of fidelity in these first sessions because the

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319 initial session probe was taught earlier on in the preparati on process than the full DAL session, which was taught further on in the preparation sequence be cause of its later implementation in the DAL process, as well as the multiple training sessions needed to fully explain and teach the 34 steps in the full session. After initial session probe fidelity ob servations, midpoint and post fidelity observations yielded results w ith sizable decreases in fide lity. Midpoint sessions, which included all teacher candidates implementing th e full DAL session, rather than the Initial Session Probe, saw fidelity levels decrease to 60%. This decrease indicated only beginning levels of competency in implementi ng the entire DAL framework with fidelity. This decrease must be interpreted cautiously for several reasons. Initially, it must be noted that the number of teacher candidates th at were actually observed decreased by one third from the initial to the midpoint obser vation. This decrease was caused by multiple factors, including student absences, teacher candidate absences, school site scheduling, and the length of time needed to get through the initial DAL assessment. It is also important to note that the full session employed at the midpoint observation included many more steps, approximately five times as many components as the Initial Session Probe. Lastly, the full session probe did not merely include informal assessment of student skills, with which teacher candida tes had practice through implementing the DAL initial assessment, but also instructional pr actices with which the teacher candidates had limited practice. Final fidelity observations experienced an increased mean accuracy rate from midpoint levels, with a percentage of 90%, but these data must be in terpreted guardedly. For the previously noted reasons included under the midpoint fidelity observations, the

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320 final sessions again showed a decrease in the number of teacher candidates that were observed. In the case of final fidelity checks, the availability of teacher candidates for observation was reduced to two individuals from the initial and midpoint observation groups. At the same time, because of the 10-week time period of the study, many teacher candidates’ pre, midpoint, and post observati ons were conducted one week after another, instead of having several week gaps for pr actice and developmental growth of teacher candidates. As a result, large generalizations about the abilities of teacher candidates to implement the steps of the DAL framework with fidelity could not be made for the group of participants, except that there was minima l evidence that teacher candidates’ abilities to implement the framework rose once they were more familiar with the implementation process. Other important findings from fidelit y observations and teacher candidate debriefings on those observations were the influence of outside factors on teacher candidates’ implementation of DAL session elements. One of these factors was that teacher candidates felt they should skip certain steps to “catch up” and get to a certain point in the DAL framework during each sess ion. Another variable included teacher candidates’ belief that some st eps were more crucial than ot hers, and they thought it was up to their discretion to omit st eps they thought were unimportant or not as relevant to the particular skill being taught. Teacher candi dates also mentioned difficulty remembering key DAL elements because of the amount of a ssignments and expectations made of them in the context of their course work and practicum experiences. Finally, teacher candidates indicated that they were more inclined to skip steps in the process entirely rather than implement those steps incorrectly and possibl y providing misinformation or instruction to

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321 students. Since teacher candidates’ experien ce in implementing instruction of any kind is new to this semester, the development of be ginning decision-making abilities can be seen through their choices. Further guidance a nd ongoing dialogue with university educators may help to guide these individual teacher ca ndidates in making more informed decisions about the process of effectively implem enting mathematics instruction (DarlingHammond, 2000). While faculty support with in the current DAL experience was available for this purpose, it appears that te acher candidates may need further training and assistance in seeking out suppor t and collaboration on specifi c issues that arise during instruction ( Betz & Hackett, 1986; Czerniak, 1990) Final Project Analyses As part of the completion of their instructional experience with the DAL framework, teacher candidates were asked to complete a final written project on what they learned, how they felt, and how they would apply their abili ties gained through the DAL instructional experience. When qualitative coding in the form of thematic analysis was completed on all teacher candidates’ fi nal papers, teacher candidates’ comments were coded along the major elements of pr ofessional development involved in the study, including attitude towards mathematics in struction, self-efficacy about mathematics instruction abilities, content knowledge in elementary math ematics, and instructional knowledge and application fo r teaching mathematics to at-risk learners. Along these lines, the majority of teacher candidate project statements referred to instructional practices and their applica tion for teaching at-ris k students algebraic thinking. This result was not surprising, since the core empha sis of the teacher candidates’ DAL experience involved tr aining and support on how to implement

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322 mathematics-based instructional strategies and the actual DAL framework itself. An interesting connection between teacher candidate instructio nal knowledge and the content knowledge possessed by students was that the focus of teacher candidate content knowledge statements encompassed students’ expressions of their developmental understandings of content knowledge using the same methods and modalities employed by the teacher candidates as instructional stra tegies to assist stude nts in learning that content. For instance, one teacher candida te expressed how a student explained his understanding of patterning con cepts using the different levels of the CRA sequence. Another indicated that her student was able to explicitly explain and then model the difference between growing and repeating pa tterns. Through their understanding of mathematics instructional strategies, teacher candidates were able to specifically articulate aspects of student curriculum abilities, that withou t knowledge of this mathematics-based vocabulary, the teacher candidates may not have been able to identify. Another important aspect of teacher candidate content knowledge statements was their lack of reasoning or analytic explanat ion behind the students’ mathematics abilities or lack thereof. Additionally, the focus of many content knowledge comments was the area of patterning, which may be due to the f act that teacher candida te preparation using the DAL framework first targeted student defici encies in patterning kno wledge. It is also indicative that teacher candi dates possibly focused on patterning when teaching their students, because it was the algebraic con cept with which they were most familiar because of its elementary nature to algebraic thinking or the fact that is was the first skill

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323 taught in the scope of the DAL framework content knowledge and possibly most easily remembered. Teacher candidate comments about efficacy in regards to content knowledge were more positive than negative in regards to mathematics teachi ng abilities with struggling learners. Negative comments may be attributed to the fact that this particular practicum experience was teacher candidates’ first in rega rds to teaching mathematics to their target student population. In fact, the actual mathem atics methods course that will be taken by special education teacher candidates will not occur until the semester following the current study. This being the case, teacher candidates may have had stronger feelings of efficacy at midpoint, after they had in itially begun and gotten accustomed to implementing the DAL framework. However, these feelings of efficacy may have dwindled by the end of the study when final analysis projects were completed due to teacher candidates’ experiencing frustration w ith their own instructi onal abilities or their students’ progress. This idea of decreased f eelings of efficacy at the time of the final analysis papers is also su pported by the fact that posttest scores on the quantitative efficacy measure decreased from midpoint to po sttest. According to the literature base on instructional efficacy, sustainability of high self-efficacy is difficult within the current school climate, as well with the challenges of today’s students and classrooms (Bandura, Barbaranelli, Caprara, & Pa storelli, 1996; Dwyer, 1993; Enochs, Smith, & Huinker, 2000). Pinpointing specific e xperiences and learning activities that positively impact efficacy during applied instructi onal situations may shed furt her light on the difficulty of maintaining self-efficacy in instructional practice.

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324 Teacher candidates’ comments on their att itude towards mathematics in general and mathematics instruction followed along th e lines of the constr uctivist mathematics culture that has been cul tivated during the teacher ca ndidates’ own k-12 learning experiences with mathematics. The major ity of teacher candidate participants fell between the ages of 20 and 30 years old, which indicates that most of these individuals attended schools and learned mathematics during the time of instructional emphasis on developmental and meaning-making experiences for stimulating growth in mathematics knowledge. Thus, it seems that since most of the teacher candidates learned mathematics initially through constructivist methods, they ma y have been more apt to entertain these attitudes now as a special education t eacher (Seaman et al, 2005; Darling-Hammond, 2000). As evidenced by the quantitative survey, t eacher candidates’ attitudes and beliefs about instruction seemed especially resistan t to change. This idea was reinforced by many teacher candidates’ final projects mentioning ideas and feelings about teaching mathematics that stemmed from their own el ementary mathematics experiences. This information is helpful to teacher preparat ion programs in two regards. First, it emphasizes the need to create positive, act ive, and meaningful mathematics learning experiences for students that have the po ssibility of affecti ng students’ lifelong relationship with mathematics learning outcome s. While teacher pr eparation programs cannot actualize these lo ng term types of experiences for current teacher candidates, they can work to facilitate these ideas for l earners currently in k-12 schools through the development of teacher candidate instructiona l practices. In tur n, these instructional practices may positively affect future teacher s that are currently attending our public

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325 schools. Second, it emphasizes the need for ongoing cultivation of specific constructivist instructional beliefs throughout entire teacher preparation programs for these ideas to truly be impacted and changed for the l ongterm through program experiences (Marso & Pigge, 1986). Teacher preparation programs n eed to evaluate their courses and field work to best determine if these programs have incorporated experiences for teacher candidates that involve meani ngful learning activities that assist them in making sense and constructing new knowledge through their pr ofessional preparation. Case Studies For the case study portion of the researc h, three teacher candidates were selected to have their DAL experiences explored on a mo re individual and specifi c level. Each of these individuals was selected based on thei r achievement in academic coursework and fieldwork experiences during th eir Level II semester. One person was representative of the top-achieving third of the participants one for the mid-achieving third of the participants, and one for the lowest-achieving th ird of the participants For all three of the case study individuals, quantitative survey results and the instructional exam were analyzed in conjunction with final analysis projects, overall DAL project artifacts, and exit interviews. Using these data, some general informa tion about teacher candidate experiences with the DAL framework were gleaned. Fi rst, a common comment by all three teacher candidates was that more time was needed w ith the preparation and training aspects of the DAL framework, as well as the amount of time teacher candidates had with students. This comment bears consideration because of its mention across all three case study participants, as well as other evidence of increased time needs found when fidelity checks

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326 were difficult to complete, which was cau sed by unexpected time barriers during the study’s duration. Second, the mid and low achie ving participants, Kara and Taylor, both voiced issues involving understanding and im plementing the DAL framework because of the pedagogical techniques used to prepare the teacher candidates for DAL instructional usage. Both of these teacher candidates affirmed that they needed more hands-on practice with elements of the DAL model be fore actual implementation with students. This adaptation of training activities should be considered in light of all three teacher candidates individually scor ing below competency level on the application essay questions on the instructional exam, as well as the mean score of a ll participants being below competency level. These results indicate a possible need for a different pedagogical emphasis being used with teach er candidates’ training with the DAL framework. While multiple modalities and hands-on learning were incorporated in conjunction with lecture presen tations during the DAL traini ng, it appears that perhaps these instructional strategies need increased usage while teacher-directed presentations may need to be employed more as supplements. For the top-achieving participant, Olivia findings from her complete DAL project review, final analysis paper, case study interview, and final instructional exam indicated that she was successfully able to understand the DAL framework and related instructional practices. Her qualitative results also showed that her feelings of efficacy increased because of her ability to understand the inst ructional project and see change in her students’ performances. While her quantitative results indicated that Olivia experienced a decrease in efficacy from midpoint to posttes t, her pretest information was not available because of her absence at the pretest admini stration, so it is not known whether an overall

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327 increase in efficacy would have been seen from pretest to posttest quantitatively. In other quantitative survey results, Olivia hailed mo re to the constructivist framework in her beliefs, indicating she is more likely to facil itate student-centered learning activities and support student exploration of mathematics id eas. Content knowledge results indicated a weakness in the subject area of basic elemen tary mathematics and algebraic thinking, but from Olivia’s comments about seeking out help within the DAL experience through collaboration, as well as indi vidual research, it is believe d that as a future special educator in mathematics, Olivia would us e multiple methods to access specific content knowledge to overcome these content deficits Since seeking out side resources and assistance from faculty and staff was unique to Olivia, it may be a variable warranting further exploration, considering her growth in all critical elements of mathematics instructional abilities. Although university a nd faculty staff were available to teacher candidates during every practicum day us ing the DAL framework, it appears that developing self-advocacy skills in seeking out this help may be a necessary component in furthering teacher candidates’ instructional abilities in mathematics. For the mid-achieving participant, Ka ra, findings from her DAL experience indicated that she will need c ontinued targeted experiences in developing her abilities to teach at-risk learners mathematics. The reas ons for this need are several. First, her feelings of efficacy rose and then fell from the beginning to the end of the study according to her quantitative efficacy survey, indicating a need for her continued development of effective mathematics pedagogy. At the same time, attitudes and beliefs about mathematics had not stabilized to be ei ther decidedly traditi onal or constructivist, flip-flopping back and fort h between survey administra tions. These mixed results

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328 indicate a need for more in depth explor ation and reflection on her feelings about mathematics and mathematics instruction. Re sults from Kara’s content knowledge exam showed extremely deficient understandings of al l areas of elementary level mathematics, which indicated a need for furthe r instruction in light of Kara ’s concurrent low levels of instructional practice understa nding, collaboration and ot her resource finding skills, feelings of efficacy, and attitudinal foundation towards mathematics instruction of at-risk learners. She has not developed many of the other identified critical elements necessary to support her development of content knowledg e skills. Kara’s final paper analysis provided supporting evidence for this lack of content knowledge. While other critical elements of special education teacher devel opment in mathematics instruction received attention within her final analysis paper, no references were made to content knowledge, indicating a lack of comprehe nsion of the importance of und erstanding these concepts for instruction. Statements made in Kari’s exit interview and thr oughout her entire DAL project also included comment s involving her lack of unde rstanding of instructional practices and the scope and sequence of skills within the DAL context. One dichotomy that was evidenced th rough data collection methods was the difference between Kara’s perceived competen ce with general mathematics and algebraic thinking content, and her performance with content on the content knowledge exam. Kara felt that she had a good grasp on ma thematics, considering that she “liked” mathematics, felt it came easily to her, and had tutored students in mathematics previously outside of her professional traini ng. Kara’s success as a special educator in mathematics would be improved through furt her exposure to fieldwork experiences specifically geared towards teaching mathema tics. A special emphasis should be placed

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329 on content knowledge enhancement since that appears to be a key area of need, especially in developing Kara’s awareness of what types of c oncepts she still needs to master. For the low-achieving participant, Taylor, results showed a teacher candidate that is open-minded about learning to teach math ematics, but currently needs extensive preparation to teach mathematics successfully. Her fear of mathematics, as well as her limited view of what elements construct eff ective practices for teaching mathematics to students with disabilities, seem to be curre nt barriers in her mathematics instruction abilities. During the study, qualitative data collected from Taylor indicated that she viewed mathematics with some anxiety. A specific comment, “I chose my target population for future teaching (students with severe or profound me ntal retardation) based on the fact that I won’t have to teach them math”, explained her strong feelings about mathematics. At the same time, her comments indicated that she felt comfortable with basic elementary level mathematics. Throughout the course of the study, it was found through Taylor’s performance on the content knowledge survey administrations that though she star ted participation in the study with deficient levels of both basi c arithmetic knowledge and algebraic thinking skills, her ability level gradually increased through the course of the DAL experience. This progress shows that part of her trepid ation about teaching mathematics may be due to her lack of understanding and exposure to mathematics skills, which appears amenable to change through remediation. Additionally, Taylor’s resu lts on the efficacy survey instrument indicated that while the begi nning of the DAL experience increased her feelings of efficacy, these fee lings changed in the latter pa rt of the study. These results illustrated that Taylor would need further e xposure to teaching mathematics to encourage

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330 sustainable change in her feelings of efficacy when teaching mathematics. However, like the content knowledge component, Taylor’s atti tude towards mathematics in general and mathematics instruction also seemed amenable to change. Within the research, her initial views of teaching mathematics were more traditional, but by posttest had shifted to consistently more constructivist. Taylor’s final DAL project most extens ively showed her very limited ideas of what constructs effective mathematics instruction for at-risk learners. All of her comments in this project involved instructi onal knowledge and app lication versus any comments on attitude, efficacy, or content know ledge. This information, coupled with the fact that Taylor had limited interacti on with her DAL students because of students’ absences and sessions cut shor t because of student scheduling, indicate that increased experiences with the ideas surrounding math ematics teaching preparation would highly benefit Taylor’s future abilities to teach mathematics successfully Focus Groups Focus groups were conducted at pre a nd post points of the study, splitting all participants randomly into two focus groups at each point. During the course of the focus groups, several key ideas about te acher candidates’ experience came to light. One of the prevalent comments included that teacher candidates’ own k-12 experiences with mathematics, whether positive or negative, ha d a large impact on their current views of mathematics. This information is important for special education teacher preparation programs as they recruit for and structure their undergraduate preservice programs. Teacher candidates’ comments about mathematic s attitudes also appeared resistant to change through the study’s quantita tive attitudinal survey admi nistrations, as well as case

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331 study, final project, and focus group comment s. As a result, teacher preparation programs must ask key questi ons about perspective teacher candidates’ views of and experiences with mathematics learning to best select individuals for their teacher preparation programs (Boe, 2006: Boe, Sh in, & Cook, 2007; Dar ling-Hammond, 2000). Additionally, time must then be invested in these programs to further develop positive and constructivist attitudes towa rds mathematics. It appears that semester long efforts of facilitating mathematics instruction ma y be too short for this purpose. In terms of efficacy, the majority of t eacher candidates spoke about how they felt they were entering the curre nt study with little to no coursework and practical experiences in teaching mathematics. When ending their participa tion in the research, many participants commented that they needed more work and study in teaching mathematics after this 10 -week study. Many teacher candidates emphasized their understandings and benefits from hands-on learning and application used within the DAL’s preparation, but felt they needed more of these training experiences to best understand the DAL framework and mathematics instruction. These comments are important for teacher preparation programs as they set up courses and fieldwork experiences. Teacher candidates evidenced a need for sequential field experiences that require increased understanding and applicati on of concepts as they progress through their programs. Also, it seemed that since teacher candidates’ had little prior knowledge and strong fears about teaching mathematics, they would benefit from having mathematics methods courses at the beginning of their professional development, rather than in the last year, as the participants in this study (Boe, 2006). Additionally, it would be helpful to have more than one of these c ourses and practical expe riences, since teacher

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332 candidates evidenced a need and desire to have more direct mathematics teaching experiences on an ongoing basis. In terms of mathematics instructiona l knowledge, teacher candidates’ comments evidenced that they were able to retain information about the mathematics instructional strategies for at-risk learners in mathem atic taught within the scope of the DAL framework. As also shown through their instru ctional exams, the participants as a whole were able to master the identification of learning characteristics and instructional strategies for their target population. For t eacher candidates, this instructional knowledge gain was important, considering that most of the participants entered the program with little or no knowledge of these strategies at the outset of the st udy. Total understanding, usage, and comfort with these strategies w ould need more time, further courses, and additional field experiences to develop base d on data collected through teacher candidate instructional exam scores, final projects, and focus groups. At the same time, many teacher candidates voiced that they felt what th ey had learned as instructional strategies were not really pedagogical practices, because they viewed instruc tional strategies as involving multiple structured steps. Many in structional strategies employed within the DAL were more holistic and/or complex and were not necessarily step-oriented (Allsopp, Kyger, & Lovin, 2006). Further time would al so need to be spent on emphasizing the utility of these pedagogical strategies for mathematics learning with diverse populations, as well as on direct application of these ski lls with students. A dditionally, since teacher candidates employed instructiona l strategies within the DAL framework’s contexts of the Algebraic Literacy Library’s (A LL) literature, it was thought that instructional strategy application may have seemed more complex to teacher candidates imbedded within such

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333 complicated storylines as those included in the library’s Caldecott Award winning literature. From teacher candidate feedback, it appeared that great er clarity might have occurred with instructional strategy application within different or more limited contexts. Another more effective route may have been to spend more time with the strategies in isolation before having teacher candidates use them imbedded within this literature-based context. Most teacher candidates indicated that they entered the study perceiving that algebraic thinking at the elementary level involved the numbers and symbols of the secondary classroom. Throughout the study, it appeared that many teacher candidates continued to perceive algebraic thinking in this way. Teacher candidates who did internalize the ideas su rrounding basic algebraic th inking including patterning, representing mathematical models, sett ing up and solving basic equations, and monitoring change across different situations viewed these skills as very elementary pieces of algebraic thinking and seemed to doubt the need and usability of them in the total scope of algebra. Mean content know ledge survey scores supported these teacher candidate comments, since these scores were below competency level for the participants as a whole, with most partic ipants expressing difficulty w ith conceptual understandings of both basic mathematics and algebraic thin king skills. Individual difficulties with content were also observed through the case study participants w ith all three having individual content knowledge scores below comp etency levels. These results indicate a strong need for more intensive content knowledge exposure for preservice teacher candidates in special education who will be teaching mathematics (Baker, Gersten, & Lee, 2002). This need could be satisfi ed through innovative teacher preparation

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334 programs involving content knowledge instru ction imbedded with in instructional knowledge and application experi ences. If future special ed ucators are better able to understand mathematics content, as well as inst ructional perspectives, they will be better equipped to teach these skills to students who are struggling in mathematics. Limitations of the Study Threats to Internal Validity Instrumentation, maturation, testing e ffects, observational bias, and student absences were all thought to be possible threats to the internal validity of the study at the outset of the investigation. All of these po ssible threats were assessed by the researcher before the study was begun and a minimization of these threats through study constructs was attempted. During the course of the st udy, two more possible threats to validity came to the surface. The first was that many teacher candidates were unexpectedly absent from coursework and practicum. Th ese absences were controlled for by ensuring that teacher candidates had to “make up” mi ssed practicum days, but because of the set up of the practicum experiences and student schedules, teacher candidates were unable to make up individual missed DAL instructional se ssions with students. The second threat was different unexpected events at the coope rating public school site, such as lockdowns, picture days, and lack of instructi onal space, which occu rred and could have caused large numbers of students to be unable to participate in instructional sessions on particular days. To overcome this thr eat, the researcher worked with school administration to secure flexible and viable school space and instructional time, versus just allowing sessions to be missed by students.

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335 Threats to External Validity The chief threat to external validi ty during the study was the researcherdetermined elements that would be monitore d to evaluate change in teacher candidate professional development in teaching algebr aic thinking to at-risk learners. These variables were determined to be efficacy a bout teaching mathematics, attitude towards mathematics and mathematics instruction, content knowledge of mathematics, and instructional knowledge and application with at-r isk learners. If an evaluation of teacher candidate change in a preservice applicatio n-based teacher preparation was completed again using the DAL framework, these elements may be conceptualized differently by other investigators. Thus, the current el ements considered essential to teacher professional development in mathematics instruction are unique to this study. Threats to Legitimation While the study incorporated 19 participan ts and a mixed methods approach at one university and one school site, these resu lts would not be generalizable to other settings because of the limited size of th e population. The current study’s results are indicative of possibilities for further lines of inquiry at other sites and with more participants. For generalizability, a larger undergraduate teacher candidate population, and more research studies in more locations would need to be completed. Implications of Research Findings Developmental-Constructivism The current study was conducted within a developmental social constructivist frame (Darling-Hammond, 2000). Along these lines, the research provides valuable insight into undergraduate teacher candidate knowledge and skill construction by being

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336 involved in an application-base d training experience. Key el ements that were brought to light included individu al differences in developmental level and progression; length of time needed for the development of teaching abilities for mathematics instruction for atrisk learners in mathematics; and types of in struction and activities needed to facilitate teacher candidate change in abilities to teach mathematics to at-risk learners. In terms of individual differences in teacher candidate development and progression of teaching skills, the case studies particularly illustrated this idea. Each teacher candidate entered the study at di fferent developmental levels, based on differences among their feelings, abilities, a nd knowledge about teac hing mathematics. Along these same lines, each of the participants while involved in the same instructional experience through the DAL framework, cha nged along the five different aspects (attitude, efficacy, content knowledge, and in structional knowledge and application) identified as important to developing instru ctional abilities in mathematics. The topachieving participant was able to juggle the ta sk of instructional pr actice and application with her students, by truly adapting a constr uctivist approach to her own learning through using instructional sessions as situations to test instructional know ledge and application; seeking feedback and assistance from univers ity staff for problem solving concerns and issues within her individual instructiona l sessions; and working to establish new mathematics instructional unde rstandings and abilities by making sense of learning in coursework and practicum sessions by inco rporating them in her own instructional meetings with students. The mid-achieving part icipant struggled more with instructional practice and application, being unable to see he r own deficiencies in mathematics ability, and lacking comprehension of key instructi onal principles within the DAL framework.

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337 The low-achieving participant battled with her fear of mathematics, but progressed in her constructivist attitude towards mathematics instruction and her content knowledge for instruction. While this teacher candidate viewed herself as needing much more assistance with teaching mathematics, her ski lls greatly increased over the course of the study. Keeping these observati ons in mind, designers of futu re special education teacher preparation programs can design programs to f acilitate a wider array of teacher candidate abilities, and work to individualize the experi ence of teacher candidates within a larger teacher preparation program, which appear s to be necessary for increased teacher candidate progress in teaching ab ilities. Indeed, many of the same instructional practices we implement as effective differentiated and individualized l earning with our k-12 struggling learners, could be e ffective in meeting the needs of the teacher candidates who will be working with these students (Boe, Shin, & Cook, 2006). Throughout the different data collection methods employed within the study, a consistent comment by teacher candidates was th at the teacher preparation experience for teaching mathematics needed to be longer in both the training and application pieces. These comments are worthwhile in inform ing the development of future special education programs, where experiences that span an entire semester or longer appear to be needed. Because of the sheer nature of any teacher preparation program being a developmental process over the course of years, not a semester, it would seem conceivable that teaching mathematics be inco rporated throughout an entire preparation program involving multiple semesters rather than as an isolated experience. In this way, greater connectivity would be seen between teaching mathematics and teaching in the other content areas like reading and writing. Additionally, teacher candidates would be

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338 able to build on their cumula tive experiences for both knowle dge attainment and practice in teaching mathematics. On the last idea of the types of prep aration experiences, teacher preparation programs could vary pedagogical practices to bett er meet the needs of teacher candidates. Throughout the study, participants described multi-sensory and hands-on formats that assisted their learning process or would do so in the future (Darling-Hammond, 2000; Darling-Hammond, 1999). Teach er preparation programs of ten advocate these exact, student-centered experiences in the classroom s of their future teachers. However, it appears that further time and development n eeds to be spent on cultivating this same pedagogical practice for the teacher candidates’ own learning. In this way, teacher candidates would have a living and breathing model of how this form of instruction can effectively meet learning needs. However, changes such as these in teacher preparation programs would require teacher preparati on to rethink the traditional university classroom experience and its dynami c with connected practica. Theory to Practice Gap A greater understanding of undergraduate teacher preparation can inform teacher preparation program design and implementation. However, it can only facilitate change in these programs’ design when in depth and reflective efforts are made to redesign and rework such programs by university special education administ ration (Boe, Cook, & Shin, 2007; Darling-Hammond, 200 0). Faculty and staff have to have an openmindedness of approach and flexibility of design with these programs. Instead of viewing teacher preparation programs as sta tic entities, a constantly expanding and exploratory view must be taken by programs in developing future teachers’ abilities.

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339 As mentioned earlier in this study’s re view of the literature, NCATE (2005) requires undergraduate special education t eacher preparation programs to incorporate field work components in their teacher prep aration programs. However, many of these practica lack the linkages to program cour sework, the faculty support to facilitate connections between academic learning in cour sework and application in field work, and the efforts of universities and school distri cts to work towards the common goal of improving teacher preparation through suppor ted and integrated experiences between public school classrooms and university coursewo rk. As evidenced through this study, to teach content area learning such as mathematics, teacher candidates need extended, as well as progressively increasing levels of inst ructional responsibiliti es and expectations. These types of experiences can only be stru ctured for future teachers by establishing structured partnerships between public schools and colleges of education built on flexibility, mutual support, and communication. In the particular universi ty-school partnership in this study, all of the above elements of effective partnership building we re valued, but at times were difficult to successfully incorporate. For instance, there were specific difficulties in navigating an effective and productive relationship between the teachers working within the school site and the teacher candidates working with students using the DAL framework. These difficulties could have been stimulated by the teachers at the Title I school site being faced with a large number of academic perf ormance criteria laid out by the school’s district because of the school’s poor academ ic performance measured by the state’s academic monitoring system. This fact ma y have caused teachers to see teacher candidates not only as providers of mathema tics instruction via the DAL, but also as

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340 assistants in improving stude nt test scores through pr oviding help with student mathematics test-taking. While this task was not a goal of the teacher candidates’ courses, practicum, or this research, it was difficult for teacher candidates to successfully communicate their purpose and academic goals for students to teachers within the school, even with supervisor support. Additionall y, the problem of shared mathematics goals by teachers and teacher candidates may also have been aggravated by a lack of communication between teachers and teacher candidates. As evidenced by several comments in teacher candidate final projects because of the “pull-out” nature of the teacher candidate instructional experience w ith the school’s students, several teacher candidates did not realize the importance of collaborating with the classroom teachers until the end of the instruc tional experience. Through r ecognizing these difficulties, insight into the challenges f acing strong university-school pa rtnerships can be better understood. Recommendations for Future Research While this study is a beginning investigat ion into using the Developing Algebraic Literacy (DAL) framework in a beginning fi eldwork experience, further research is advocated based on the current findings. The DAL has currently been explored along the five dimensions identified by the researcher as pertinent to special education teacher preparation in mathematics: efficacy about mathematics instruction, attitude towards mathematics instruction, content knowle dge for mathematics instruction, and instructional knowledge and application for teaching at-risk students mathematics. Further research would be necessary in seve ral areas to expand current findings. One of these areas would be to implement findings found in the current st udy in regards to the

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341 time for preparation and application of the mode l. At the same time, each of the variables identified by the research as important to special education te acher preparation in mathematics would need to be evaluated in is olation to better iden tify the impact of that particular element on overall preparati on to teach mathematics. Additionally, investigations involving a grea ter number of participants, in a variety of college and university settings throughout th e country, would facilitate a more comprehensive idea of the utility of the DAL within preservice spec ial education teacher pr eparation programs. Lastly, collecting student outcome data resulting from teacher candidates using the DAL framework would give more evidence of the util ity of the actual framework with learners. In summary, future research endeavors along the lines of the current investigation would expand the ideas surrounding mathematics content area instru ction abilities for future special educators. Mathematics conti nues to be a key problem atic for learners atrisk for school failure (Allsopp, Kyger, & Lovin, 2006; Baker, Gersten & Lee, 2002), while at the same time, the number of speci al educators continues to have difficulty keeping pace with the growth of students need ing specialized and targeted instruction in mathematics (USDOE, 2003). Of key concer n is the stimulati on of not only basic arithmetic skills with these students, but ones such as algebraic thinking that activate higher order thought processes th at enable students to not only compute answers, but comprehend, represent, and problem solve. The development of these types of skills must be developed early on in students’ learning careers, esp ecially in learners requiring extra learning assistance because of learni ng disabilities or other environmental and learning factors. Changing the way we appro ach special education teacher preparation in the content area of mathematics has the poten tial to change the educational and job

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342 possibilities for a valuable section of the st udent population which has yet to be fully reached mathematically.

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343 References Allsopp, D., Kyger, M., & Lovin, L. (2006). Te aching algebraic thinking. In P. Hudson & S. Miller (Eds.), Designing and implementing math ematics instruction for students with diverse learning needs (pp. 432-488). Bo ston: Pearson. Allsopp, D.H ., McHatton-Alvarez, P., DeMarie, D., & Doone, E. (2006). Bridging the gap between theory and practice: Conn ecting course and field experiences. Teacher Education Quarterly, 33(1), 19-35. Ashton, P., & Crocker, L. (1986). Does te acher certification make a difference? Florida Journal of Teacher Education 3 73-83. Baker, S., Gersten, R., & Lee, D.S. (2002) A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103 (1), 51-73. Bandura, A., Barbaranelli, C., Caprara, G., & Pa storelli, C. (1996). Mu ltifaced impact of self-efficacy beliefs on academic functioning. Child Development, 67 1206-1222. Betz, N. E., & Hackett, G. (1986). Applica tions of self-efficacy theory to career development. Journal of Social and Clinical Psychology 4 279-289. Biesta, G. (2007). Why “what works” won’ t work: Evidence-based practice and the democratic in educational research. Educational Theory, 57 (1), 1-22. Boe, E. (2006). Long-term trends in th e national demand, supply, and shortage of special education teachers. The Journal of Special Education, 40 (3), 138-150. Boe, E., Shin, S., & Cook, L. (2007). Does teacher preparation matter for beginning

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344 teachers in either special or general education? The Journal of Special Education, 41 (3), 158-170. Bottge, B., Heinrichs, M., Chan, S., & Serl in, R. (2001). Anchoring adolescents’ understanding of math concepts in rich problem-solving environments. Remedial and Special Education, 22 (5), 299-314. Bryant, N. D., Fayne, H., Gavish, B., & Gettinger, M. (1980). Narrowing the gap between research and practice. Technical Report #4 Washington, DC: Office of Special Education. Carnine, D. (1997). Bridging the research-t o-practice gap. Exceptional Children, 63(4), 513-521. Cawley, J. F., Parmar, R. S., Yan, W., & Mi ller, J. H. (1996). Ar ithmetic computation abilities of students with learning disabi lities: Implications for instruction. Learning Disabilities Research and Practice 11, 230-237. Czerniak, C. (1990). A study of self-efficacy, anxie ty, and science knowledge in preservice elementary teachers. Paper presented at the annual meeting of the National Association for Research in Science Teaching, Atlanta, GA. Chambers, D. (1994). The right algebra for all. Educational Leadership, 51 85-86. Charalambous, C., Phillipou, G., & Kyriakides, L. (2002). Towards understanding teachers’ philosophical belie fs about mathematics. Presented at the Annual Meeting of the Internatio nal Group for the Psychology of Mathematics Education, Norwich, England. Cronje, J. (2006). Paradigms regained: Toward integrating objectivism and

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345 constructivism in educational de sign and the learning sciences. Educational Technology Research & Development, 54 (4), 387-416. Darling-Hammond, L., & Ball, D. (1998). Teaching for high standards: What policymakers need to know and be able to do. CPRE Joint Report Series Philadelphia, PA: Consortium for Policy Research in Education. Darling-Hammond, L. (2000). How teacher education matters. Journal of Teacher Education, 51 (3), 166-173. Darling-Hammond, L., Chung, R., & Frelow, F. (2002). Variation in teacher preparation: How well do different pathways prepare teachers to teach? Journal of Teacher Education, 53 (4): 286-302. Duchnowski, A. J., Kutash, K., Sheffield, S., & Vaughn, B. (2006). Increasing the use of evidence-based strategies by special education teachers: A collaborative approach. The Journal of Teaching and Teacher Education, 22 (7), 838-847. Dwyer, E. (1993). Measuring teacher attitude towa rd low achievers in mathematics. Paper presented at the annual meeting of the Mid-South Educational Research Association. New Orleans, LA. Edwards, C., Carr, S., & Siegel, W. (2006). Influences of experi ences and training on effective teaching practices to meet the n eeds of diverse learners in schools. Education 126(3), 580-592. Enochs, L., Smith, P., & Huinker, D. (2000) Establishing factor ial validity of the mathematics teaching efficacy beliefs instrument. School Science and Mathematics, 100(4), 104-202. Florida Department of E ducation. (2008). Next Generation Sunshine State Standards.

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346 Retrieved June 30, 2008, from http://www.floridastandards .org/FLStandardSearch.aspx Gagnon, J., & Maccini, P. (2001). Preparing students with disabilities for algebra. Teaching Exceptional Children, 33 (2), 8-15. Gagnon, J., & Maccini, P. (2001). Preparing stud ents with disabilities for algebra. Teaching Exceptional Children, 43 (1), 8-15. Golder, G., Norwich, B., and Bayliss, P. (2005) Preparing teachers to teach pupils with special educational needs in more in clusive schools: Evaluating a PGCE development. British Journal of Special Education, 32 (2), 2005. Hammer, P., Hughes, G., McClure, C., Reeves, C., & Salgado, D. (2005). Rural teacher recruitment and retention practices: A review of the rese arch literature, national survey of rural superintendent s, and case studies of programs in Virginia. Charleston, West Virginia: A ppalachia Educational Laboratory. Kortering, L., deBettencourt, L., & Braziel P. (2005). Improving performance in high school algebra: What students with learning disabilities are saying. Learning Disability Quarterly, 28 191-203. Lent, R. W., Brown, S. D., & Hackett, G. ( 1994). Toward a unifying social cognitive theory of career and academic inte rest, choice, and performance. Journal of Vocational Behavior, 45, 79-122. Maccini, P., & Gagnon, J. (2006). Mathematic s instructional practices and assessment accommodations by special and general educators. Exceptional Children, 72 (2), 217-234. Maccini, P., & Hughes, C. (2000). Eff ects of problem-solving strategy on the

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347 introductory algebra performance of seconda ry students with learning disabilities. Learning Disabilities Re search & Practice, 15 (1), 10-21. Maccini, P., McNaughton, D., Ruhl, K. (1999). Algebra instruction for students with learning disabilities: Implica tions from a research review. Learning Disability Quarterly, 22 (2), 113-126. Marso, R. N., & Pigge, F. L. (1986). Relationships between student characteristics and changes in attitudes, concerns, anxie ties, and confidence about teaching during teacher preparation. Paper presented at the annua l meeting of the American Educational Research Associ ation, San Francisco, CA. Mizer, R., Howe, R., & Blosser, P. (1990). Mathematics: Promising and exemplary programs and materials in elemen tary and secondary schools. Mathematics Education Information Report. Washington, DC: Office of Educational Research and Improvement. McInerney, M., & Hamilton, J. (2007). El ementary and middle schools technical assistance center: An approach to s upport the effective implementation of scientifically based practices in special education. Exceptional Children, 73 (2), 242-255. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: NCTM Nougaret, A., Scruggs, T., & Mastropieri, M. (2005). Does teacher education produce better special educ ation teachers? Exceptional Children, 71 217-229. Puchner, L., & Taylor, A. (2004). Less on study, collaboration and teacher efficacy:

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348 Stories from two school-based math lesson study groups. Teaching and Teacher Education, 22 922-934. Rice, C., & Gossling, D. (2005). Recruiti ng and retaining male special education teachers. Remedial and Special Education, 26 (6), 347-356. Sands, D., Duffield, J., & Parsons, B. (2007). Evaluating infused content in a merged special education and general educat ion teacher preparation program. Action in Teacher Action, 28 (4), 92-103. Singh, D., & Stollof, D. (2007). Measuring teacher dispositions. Paper presented at the National Fifth Annual Symposium on Educat or Dispositions, Er langer, Kentucky. Smith, S., Daunic, A., & Taylor, G. (2007). Treatment fidelity in applied educational research: Expanding the adoption and applic ation of measures to ensure evidence-based practice. Education and Treatment of Children, 30(4), 121-134. University of Maryland System. (1993). Special teachers for el ementary and middle school science and mathematics: A propos al submitted to the National Science Foundation Teacher Preparation and Enhancement Program. Unpublished manuscript. U.S. Department of Education. (2006) 26th Annual report to Congress on the implementation of the In dividuals with Disabilities Education Act, 2004 Washington, DC: U.S. Department of E ducation, Office of Special Education and Rehabilitative Services. 26th Annual Report to Congress 2004) U.S. Department of Education. (2007). Improving basic programs operated by local education agencies (Title I, Part A). Retrieved December 12, 2007, from http://www.ed.gov/programs /titleiparta/index.html

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349 U.S. Department of Education. (2003). Meeting the highly qualif ied teachers challenge: The secretary’s second annual report on teacher quality. Washington, DC: Author. U.S. Department of Education, National Center for Education St atistics (U.S. ED/NCES). (2001). The Nation’s Report Card: Mathematics 2000. By J. Braswell, A. Lutkus, W. Grigg, S. Santapau, B. Ta y-Lim, & M. Johnson. NCES 2002-517. Washington, DC. U.S. Department of Educa tion. (2007). State and lo cal implementation of the No Child Left Behind Act Volume II – Teacher Quality Under NCLB: Interim Report. Washington, DC. Van Laarhoven, T., Munk, D., Lynch, K., Bosma, J., & Rouse, J. (2007). A model for preparing special and general educati on peservice teachers for inclusive education. Journal of Teacher Education, 58 (5), 440-455. Witzel, B., Mercer, C.D., & Mi ller, D. (2003). Teaching algebr a to students with learning difficulties: An investigation of an explicit instruction model Learning Disabilities Research & Practice, 18 121–131. Witzel, B., Smith, S., & Brownell, M. (2001). How can I help students with learning disabilities in algebra? Intervention in School and Clinic, 37 (2), 101-104.

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350 Appendix A: Literacy Instructional Practices within the DAL Framework

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351 Strategies Description Points for Usage Example Applications Engagement The establishment of student attention and interest in instructional tasks through the usage of stimulating materials that present ideas that have meaningfulness and relevancy for student learning *Utilize applicable and relevant materials *Allow opportunities to manipulate high interest stimuli *Provide time for meaningful student responses to experiences *Use of Caldecott Award & Honor Winning Books *Student experiences with concrete manipulatives and eye-catching representations *Investment in Student Solution Ideas Big Picture (Holistic) Introduction of larger reading concepts such as theme, problem, or thoughts and feelings evoked by the story as a whole, rather than the component parts of reading including phonemic awareness, phonological development, and vocabulary progression *View stories as whole entities to understand and explore *Discuss student thoughts on story theme, plot, and resolution to develop shared understandings *Cultivate competence with elements of setting, character, and plot to stimulate students' thinking on larger story issues *Implement teacher-guided discussions *Use shared book experiences for multiple genres exposure *Employ games that compare and contrast themes expressed in different literature pieces *Integrate multiple modalities when getting at key concepts (ie. visual, dramatic, and written) Active Questioning Involves reading with purpose by focusing on what is currently known on a topic, what information gain is desired, and what information is actually presented in text when it is read and discussed *Tap into previous knowledge before actually reading *Figure out ideas or curiosities for upcoming reading that can guide students' thinking while reading *Follow up after reading with discussions or activities that explore what has been gained during reading *Question-Answer Relationship (QAR) strategies to explore answer finding in text *Directed Reading Thinking Activities (DR-TAs) to develop students' thought processes *K-W-L to stimulate prior knowledge, questioning, and learning

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352 Making Connections To further instruction in a given content area, instruction on new concepts is tied to previous learning, as well as having the relevancy explained between new concepts and the total scope of learning *Begin instruction by reviewing previous learning on a related subject *Explain how the new skill is related to previous learning *Preview how the new learning bridges content to future learning *Utilize students' personal experiences and relate them to what is currently being learned *Establish connections between reading skills learned through explicit instruction and their application in children's literature *Employ connections that can be made within and across the content areas, incorporating both reading and mathematics learning Structured Language Experiences Guided oral and written opportunities that focus and develop students' abilities to communicate important learned concepts and their applications *Make specific goals for students' verbal and written communications on a specific topic *Provide guidelines for outlining pertinent points for discussion or writing *Pair or group students in ways that develop individual strengths and abilities through interactions *Provide prompts for specific written response information on a topic *Allow students to explain their own constructed understandings of concepts by providing them oral or written opportunities to explain a new concept as a "teacher" *Provide compare/contrast opportunities for students to share understanding and construct new group understandings

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353 Appendix B: Algebraic Literacy Library with Sample Book Guide

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354 Caldecott Algebraic Literacy Library Algebraic Literacy Strand Chodos-Irvine, Margaret. (2003). Ella Sarah Gets Dressed San Diego: Harcourt, Inc. Cronin, Doreen & Lewin, Betsy (ill.). (2000). Click, Clack, Moo: Cows that Type New York: Simon & Schuster. Understand patterns, relations, and functions Falconer, Ian. (2000). Olivia New York: Atheneum Books for Young Readers. Rappaport, Doreen & Collier, Bryan (ill.). (2001). Martin's Big Word: The Life of Dr. Martin Luther King, Jr New York: Jump at the Sun: Hyperion Books for Children. Weatherford, Carole & Nels on, Kadir (ill.). (2006). Moses: When Harriet Tubman Led Her People to Freedom New York: Jump at the Sun: Hyperion Books for Children. Gerstein, Mordicai. The Man Who Walked Between the Towers New York: Square Fish. McCarty, Peter. (2002). Hondo & Fabian New York: Henry Holt and Company. Rohmann, Eric. (2002). My Friend Rabbit Brookfield: Roaring Book Press. Represent and analyze mathematical situations and structures using algebraic symbols Thayer, Ernest & Polacco, Patricia. (1997). Casey at the Bat New York: The Putnam and Grosset Group. Williems, Mo. (2004). Knuffle Bunny: A Cautionary Tale New York: Scholastic.

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355 Kerley, Barbara & Selznick, Brian (ill.). (2001). The Dinosaurs of Waterhouse Hawkins New York: Scholastic. Muth, Jon. (2005). Zen Shorts New York: Scholastic. Use mathematical models to represent and understand quantitative relationships Savant, Marc. (2001). The Stray Dog New York: Scholastic. Taback, Simms. (1999). Joseph Had a Little Overcoat New York: Viking. Woodson, Jacqueline & Lewis, E. B. (2004). Coming on Home Soon New York: G. P. Putnam's Sons. Giovani, Nikki, & Collier, Brian (ill.). (2005). Rosa New York: Henry Holt and Company. Henkes, Kevin. (2004). Kitten’s First Full Moon Greenwillow Books. Jenkins, Steven, & Page, Robin (ill.). (2003). What do you do with a Tail Like This? Boston: Houghton Mifflin. Analyze Change in Various Contexts Juster, Norton & Raschka, Chris (ill.). (2005). The Hello, Goodbye Window. New York: Michael Di Capua Books: Hyperion Books for Children. Williems, Mo. (2003). Don't Let the Pigeon Drive the Bus! New York: Hyperion Books for Children.

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356 Source Chodos-Irvine, M. (2003). Ella Sarah Gets Dressed San Diego: Harcourt, Inc. Target Area *Understand patterns, re lations, and functions -Recognize, describe, and extend patterns su ch as sequences of sounds and shapes or simple numeric patterns and translat e from one representation to another (NCTM, 2000) Target Grade Levels Early Elementary Story Synopsis Written and illustrated by Margaret Chodos-Irvin e, the main character Ella Sarah has a mind of her own, especially about what s he wants to wear. No one in her family seems to understand her se nse of fashion. Throughout the st ory, her mother, father, and sis ter attempt to convince her that more practical and less colorful outf its would be more suitable. However, Ella Sarah is unconvince d. Exasperated, she finally decides to dress herself in these co lorful clothes, since no one else will help her do it. The outfit ends up being the perfect outfit for her get together with friends, who seem to be th e only ones who understand her fashion sense. Reading Instruction *Active Questioning Strategy -Utilize “I Wonder” to stimul ate ideas and questions that students have before reading the book, which are answered when reading the book, and discussed as a class after reading the bo ok (Richards & Gipe, 1996). *Big Ideas -Develop the story’s theme through dr amatic reenactments with class members, and as a group determine the main theme of what has taken place in the book *Structured Language Experience -Within Cooperative Learning Groups (lis ted under Mathematics Instruction), students spend time discussing how their individual patterns are the “same” and “different” and “why”

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357 Mathematics Instruction *Explicit Instruction -After reading the story, the teacher w ill spend time explaining the core concept of “pattern” and describe different ways patterns can be constructed *Teacher Modeling -The teacher will have an enlarged mode l of Ella Sarah from the book, and show how each piece of Ella Sarah’s clothing can have different pa tterns based on a choice of different sized wrappi ng paper or wallpaper pieces *Cooperative Learning Groups -Students will be given a chance to c onstruct their own patterns by all being given their own eighteen inch model pe rson, and being asked to dress these people with their own pattern s of wrapping or wallpaper pieces. When finished decorating their figures, the teacher should set group guideli nes for structured oral discussions on the “sameness” and “diffe rences” of the patterns that group members have made (more information listed under Readi ng Instruction) *Concrete [in Concrete-Represe ntational-Abstract (CRA)] -Concrete materials will be utilized thr oughout this activity for the demonstration of patterns on Ella Sarah’s clothing by the teacher, as well as the students’ pattern construction on their figures Extension *If students grasp the concepts of recognizing, describing, and ex tending patterns through the usage of concrete materials, the n visual representations can be provided that ask st udents to identify and describe patterns. *If students grasp the concepts of recognizing, describing and extending patterns th rough the usage of representational materia ls, then abstract symbols (ie. numbers) can be ut ilized with students to have them recogni ze, describe, and extend presented patterns.

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358 Appendix C: Mathematics Instructiona l Practices within the DAL Framework

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359 Strategies Description Points for Usage Example Applications CRA An instructional style that utilizes a leveled presentation of mathematics concepts that progresses from concrete materials, to pictorial representations, to abstract symbols. *Rate of progression between the levels will be individual for students *Initial presentations of materials should begin with concrete or tactile materials *As concepts are grasped at the concrete level, presentations will progress to visual representations of concepts *Use concrete materials that involve aspects of story content *Involve materials at the concrete and representational level that are presented in the children's story and/or of high interest to children Authentic Contexts Situations in which learning can take place through problems that are meaningful and involve real life situations for application *Establish situations that are meaningful and relevant to children *Provide contexts that extend children's typical presentation of material *Ensure that contexts extend easily to real life application for problem solving *Children's literature is employed for the context based situations for learning *Stories and situations presented in literature provide rich situations for actual problemsolving Explicit Instruction with Modeling Teacher guided explanations of new concepts that specifically expound on the nature of the new material and how it is used Modeling is often used in conjunction with explic it instruction to provide working examples of the new material in action. *Used when it is unlikely that students will pick up on subtle clues within exploratory learning *Employed with initial instruction on novel concepts *Best implemented in conjunction with other learning strategies *Teacher demonstrations on white boards at the front of the classroom *Teacher explanations using technology at the front of the classroom *Teacher modeling using high interest materials Scaffolding Facilitating student understa nding and application of new concepts through graduated steps towards independence rather than through instruction and independent appli cation immediately *Should begin with material with which a student is already familiar *Steps should be incremental, and may differ from student to student in terms of how large each increment is *Used for more difficult concepts with supports gradually decreasing *Can occur within ma thematics and reading content alone, as well as between reading and mathematics content Metacognitive Strategies When learners have the ab ility to think about their thought processes and how they apply these effectively for problem-solving *Students may need to develop awareness of these abilities first, before efforts at using these skills are applied *Many times children need modeling and scaffolding to successfully implement these strategies on their ow n for problem-solving *Developing self-monitoring skills for answers that making sense *Checking that answers provide the requested information in questions Student-Centered Learning Learning that focuses on students' experiences, grasps, and outcomes with activities and learning experiences, rather than t eacher directed instruction that focuses on giving the information to learners *Students should be made to feel involved in and masters of their own learning *Overall learning goals and objectives should be clearly defined *Cooperative learning groups *Paired learning teams *Student explorati on with concrete manipulatives and visual representations to make meaning Multiple Opportunities for Practice Providing learners many diffe rent ways of practicing and reinforcing skills, which typically should involve a variety of modalities and s ituations for retention of skills *Varying practice methods should be incorporated *Teachers should closely monitor students progress during th ese opportunities *Practice opportunities should have gradually decreasing levels of support based on student need *Activities that facilitate learners’ constructing their own understandings *Communication of ideas and problem application *Practice involving manipulatives and instructional games *Written worksheets or journal entries

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360 APPENDIX D: DAL Model Visual Conceptualization

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362 Appendix E: DAL Initial Session Probe

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364 Appendix F: DAL Full Session Notes

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366 Appendix G: Mathematics Teaching E fficacy Beliefs Instrument (MTEBI)

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369 Appendix H: Preservice Teachers’ Mathematical Beliefs Survey

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373 Appendix I: Mathematical Content Know ledge for Elementary Teachers Survey

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378 Appendix J: Instructional Know ledge Exam and Scoring Rubric

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379 EEX 4846 Final Exam – Effectiv e Mathematics Instruction Spring 2008 150 points Multiple Choice (2 pts each; 50 pts. Total) Directions: Write the letter of the best answer next to each question. Effective Instructional Practices 1. Which of the following instructional strate gies/techniques is NOT emphasized within the effective mathematic s instructional practice explicit teacher modeling ? a. cuing important features of the target mathematics concept/skill b. telling students what to do and when to do it c. using examples and non-examples d. using think alouds 2. Which of the following instructional strate gies/techniques is NOT emphasized within the effective mathematic s instructional practice scaffolding instruction ? a. providing specific corrective feedback b. providing specific positive reinforcement c. fading teacher direction from high, the medium, to low d. providing general feedback on student performance 3. Which of the following instructional strate gies/techniques is NOT emphasized within the effective mathematic s instructional practice teaching problem solving strategies ? a. teaching general problem solving strategies b. asking students to discover strategies on their own c. teaching specific learning strategies for particular mathematical concepts/skills d. modeling strategies 4. Which of the following instructional strate gies/techniques is NOT emphasized within the effective mathematic s instructional practice structured cooperative learning ? a. playing games for fun for the purpose of motivating students b. assigning students roles and ensuring that all students have the opportunity to engage in different group roles/responsibilities c. teaching behavioral expectations d. ensuring that all students have multiple opportunities to respond

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380 5. Which of the following instructional strate gies/techniques is NOT emphasized within the effective mathematic s instructional practice monitoring/charting student performance/progress monitoring ? a. assigning students grades of A, B, C, D, or F every day for their work b. frequently assessing students’ performance c. providing a visual displa y of students’ performance d. engaging students in goal setting 6. Which of the following instructional strate gies/techniques is most reflective of the effective mathematics instructional practice C-R-A sequence of instruction ? a. teaching students at the abstract le vel first and then moving down to representational or concrete levels if necessary b. using only commercial manipulativ es at the concrete level c. discouraging students from draw ing pictures because they will not be allowed to do this on state assessments d. grounding abstract mathematical concepts and skills in concrete experiences, first using discrete materials and th en teaching drawing strategies. 7. Which of the following instructional strate gies/techniques is most reflective of the effective mathematics instructional practice instructional games ? a. they should be motivational, provid e multiple opportunities to respond, and include a tangible way to monitor students’ performance b. they should primarily be fun for students c. they should only include commercial games (store bought) since this lets students know that they are important d. they should provide multiple opportunities to respond regardless of whether they are motivational to students or not 8. Which of the following instructional strate gies/techniques is most reflective of the effective mathematics instructional practice building meaningful student connections ? a. linking what students know to what they are going to learn b. identifying what students will learn and linking what students know to what they are going to learn c. linking what students know to what th ey are going to le arn and providing a rationale for why what students will learn is important in their lives d. linking what students know to what they are going to lear n, identifying what students will learn, providing a rationale for why what students will learn is important in their lives

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381 9. Which of the following instructional strate gies/techniques is most reflective of the effective mathematics instructional practice structured language experiences ? a. telling students what they should know through “teacher talk” b. encouraging students to use different ways to communicate what they understand about the mathematics they are learning c. using a foreign language as a novel m echanism for reaching students who are having difficulty with mathematical concepts d. making students write down in words what they did to solve a problem in stepwise fashion 10. Which of the following instru ctional strategies/techniques is most reflective of the effective mathematics instructional practice explicit teacher modeling ? a. telling students what they need to know and what they need to do b. using multiple techniques to make mathematical concepts/skills accessible including techniques such as multi-sensory methods, examples and non-examples, cueing, and think alouds c. allowing students to discover the mean ing of mathematical concepts without teacher direction d. providing students with multiple oppor tunities to respond in order to build proficiency 11. The primary purpose of the effective mathematics instructional practice C-R-A sequence of instruction is a. to help students build conceptual unde rstandings of abstract mathematical concepts b. to make mathematics fun for students c. to build students’ sensory motor abiliti es through handling objects and refining fine motor abilities thro ugh drawing pictures d. to “dumb-down” mathematics for struggling students 12. The primary purpose of the effective mathematics instructional practice explicit teacher modeling is a. to make teaching efficient so that teachers can cover as much material as possible in the mathematics curriculum b. to provide students with a “bridge” that allows them to access the meaning of mathematical concepts c. to make sure that students do it the “right way” d. to ensure that the classroom operates in an orderly fashion without behavioral disruptions

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382 13. The primary purpose of the effective mathematics instructional practice scaffolding instruction is a. to incorporate cooperative learni ng into your instructional plan b. to incorporate peer tutoring into your instru ctional plan c. to provide students with a ppropriate levels of teache r support for the purpose of helping students demonstrate increasing levels of understanding of a target mathematics concept/skill d. to provide a way to manage student be havior during mathem atics instruction 14. The primary purpose of the effective mathematics instructional practice monitoring/charting student pe rformance/progress monitoring is a. to continuously measure student performance in order to make efficient instructional decisions based on data b. to test students for the purpose of assigning grades c. to teach students how to make graphs and charts d. to place students into di fferentiated learning groups 15. The primary purpose of the effective mathematics instructional practices such as instructional games, struct ured cooperative learning, and self-correcting materials is a. to provide students with f un activities to do so that they do not get bored with mathematics b. to develop social skills in students c. to provide students with multiple oppor tunities to respond to a mathematics learning task in order to deve lop proficiency and maintenance d. to have several different activ ities planned for “Fun Fridays” Learning Characteristic Barriers 16. The learning characteristic metacognitive deficits is a barrier to learning mathematics for struggling learners because a. it inhibits students from thinking about wh at they are learning mathematically, making connections, employing strategies and monitoring their own learning b. it makes students think about too many thi ngs at one time thereby confusing them c. it inhibits short term memory d. it inhibits long term memory

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383 17. The learning characteristic learned helplessness is a barrier to learning mathematics for struggling learners because a. it results in students refusing to help ot hers thereby lessening their chances of learning through working with others b. it makes teachers tired of always having to answer students’ qu estions resulting in teachers telling students answers rather th an them figuring them out on their own c. it causes attention deficits d. it results in students failing to take risks in problem solving due to past experiences of failure 18. When students have difficulty being aware of their own learning, difficulty employing strategies, and difficulty monitori ng their own learning in mathematics they are exhibiting which of the following learning characteristic barriers? a. memory deficits b. learning helplessness c. cognitive processing deficits d. metacognitive deficits 19. When students who do not have sensory impairments have difficulty accurately perceiving mathematics accurately when it is presented exhibit which of the following learning characteristic barriers? a. memory deficits b. learning helplessness c. cognitive processing deficits d. metacognitive deficits 20. In class, you were briefly presented a pi cture and then were asked to write an appropriate title for a story ba sed on the picture. Many student s wrote titles that did not accurately represent the picture. This expe rience was an illustra tion of which learning characteristic barrier? a. visual processing deficit b. auditory processing deficit c. attention deficit/distractibility d. memory deficit 21. Which of the following statements be st portrays true attention deficits? a. students are unable to attend

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384 b. students also have hyperactivity/impulsivity c. students “hyper-attend” meaning they actually attend to so many things that they have difficulty attending to what is most important d. students engage in behaviors that are distractible to others Foundations 22. Four instructional anchors for ensuring mathematics learning success of struggling learners include all of the following except a. teaching the big ideas in mathematics a nd the big ideas in doing mathematics b. understanding learning characteristics a nd barriers for students with learning problems c. using standardized high stak es testing to grade schools on their effectiveness in teaching mathematics d. making mathematics accessible through the use of responsive teaching practices 23. Which instructional anchor for mathematics learning suc cess of struggling learners has as its purpose to use data for the purpose of instructi onal decision-making? a. teaching the big ideas in mathematics a nd the big ideas in doing mathematics b. understanding learning characteristics a nd barriers for students with learning problems c. using standardized high stak es testing to grade schools on their effectiveness in teaching mathematics d. using continuous assessment/progress monitoring 24. In class, you were asked, “what is: 4+3+ 4+5+5+3+5+3+4?”, with the answer being “even par for nine holes of golf.” This was an example of the importance that __________ has/have for meaning related to mathematics. a. context b. disability c. conceptual understanding d. numbers and mathematical symbols 25. When students are taught only the procedur es/algorithms of mathematics (e.g., 2 x 4 = 8; x = 1/8), they often never acquire a. procedural understanding b. conceptual understanding c. contextual understanding d. the ability to do math facts efficiently

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385 Short Answer/Essay (100 points total) Directions: Respond in writing to each question. Make sure that you address all parts of each question. You can use the ba ck of the page if you need more room be sure you clearly mark the question number that each response addresses. Effective Instructional Practices 26. (20 pts) Select one of the effective math ematics instructional pr actices for struggling learners listed below (CIRCLE THE IN STRUCTIONAL PRACTICE YOU CHOOSE TO WRITE ABOUT). For the in structional practice you select describe the following points: 1) its overall purpose; 2) a general summary of how it can be implemented; 3) the important elements/components of the practice; 4) at least two l earning characteristic barriers for struggling learners and how th e practice addresses each characteristic. C-R-A Sequence of Instruction Structured Language Experiences Monitoring and Charting Student Pe rformance/Progress Monitoring Explicit Teacher Modeling

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386 27. (30 pts) Describe how each of the follo wing effective mathematics instructional practices for struggling learne rs is applied within the De veloping Algebraic Literacy (DAL) instructional process. Be specific in terms of where in the DAL process each practice can be implemented and how it is implemented. Building Meaningful Student Connections Language Experiences C-R-A Sequence of Instruction

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387 28. (10 pts) A strategy that is implemented during the thir d step of the DAL process involves the use of graphic or ganizers. Describe what eff ective mathematics instruction practice for struggling learners this strategy exemplifies a nd its primary purpose in terms of student learning. 29. (10 pts) A strategy that is implemented during the thir d step of the DAL process involves the use of the LIP strategy. Describe what effective mathematics instruction practice for struggling learners this strategy exemplifies a nd its primary purpose in terms of student learning.

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388 30. (10 pts) A strategy that is implemented during the thir d step of the DAL process involves encouraging students to communicate about the algebr aic thinking concept they are learning. Describe what effective math ematics instruction practice for struggling learners this strategy exemplifies and its pr imary purpose in terms of student learning. 31. (10 pts) A strategy that is implemented during the second step of the DAL process is to evaluate their abilities to read, represent, solve, and ju stify given a na rrative context that depicts an algebraic thinking concep t. Describe what effective mathematics instruction practice for strugg ling learners this strategy exemplifies and its primary purpose in terms of student learning.

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389 32. (10 pts) A strategy that is implemented during each step of the DAL process is to situate target mathematics concepts/skills within a narrative text. Describe what effective mathematics instruction practi ce for struggling learners this strategy exemplifies and its primary purpose in terms of student learning. BONUS (up to 5 points) What is the primary purpose of the first step of the DAL process? What stage of learning are students developi ng during this step?

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390 Instructional Exam Scoring Rubric Student Name:________________________ 26. (20 pts) Select one of the effective math ematics instructional pr actices for struggling learners listed below (CIRCLE THE IN STRUCTIONAL PRACTICE YOU CHOOSE TO WRITE ABOUT). For the in structional practice you select describe the following points: 1) its overall purpose; 2) a general summary of how it can be implemented; 3) the important elements/components of the practice; 4) at least two l earning characteristic barriers for struggling learners and how th e practice addresses each characteristic. C-R-A Sequence of Instruction Structured Language Experiences Monitoring and Charting Student Pe rformance/Progress Monitoring Explicit Teacher Modeling Rubric 1.) Its Overall Purpose a. 5 points – Thorough and complete explanation b. 4 points – Main point covered, but minor details may be missing c. 3 points – Some of the main point c overed, one or two larger details may be left out d. 2 points – A small piece of the main point is covered, but a majority of the explanation is missing e. 1 points – Vague idea of the overall poi nt, but little evidence of specific understandings f. 0 points – Answer is not rele vant to the question asked 2.) A General Summary of How it can be Implemented a. 5 points – Thorough and complete explanation b. 4 points – Main points covered, but minor details may be missing c. 3 points – Some main points covered, one or two main points may be left out d. 2 points – One or two main point s covered, but many are left out e. 1 points – Vague idea of the overall con cept, but little evidence of specific understandings f. 0 points – Answer is not rele vant to the question asked 3.) The Important Elements/Components of the Practice a. 5 points – Thorough and complete desc ription of all elements/components b. 4 points – All elements/components c overed, but descriptions may be lacking depth c. 3 points – Most elements/components covered, and descriptions may be lacking depth and one or two descriptions may be missing

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391 d. 2 points – Some elements/components covered, and descriptions may be lacking depth and some desc riptions may be missing e. 1 points – One or two elements/components covered, and descriptions may be lacking depth or missi ng for all elements/components f. 0 points – Answer is not rele vant to the question asked 4.) At Least Two Learning Characteristic Ba rriers for Struggling Learners and How the Practice Addresses each Characteristic a. 5 points – Thorough and complete expl anation of learning characteristic barriers, and comprehensive explanat ion of how the practice addresses each characteristic b. 4 points – Mostly complete explanation of learning characteristic barriers with a general explanation, that l acks some key specifics, of how the practice addresses each characteristic c. 3 points – Learning characteristic barri ers are given but explanation of them may be lacking, with a general explanation, that lacks some key specifics, of how the practice addresses each characteristic d. 2 points – One of the learning characte ristic barriers and its explanation may be left out, with an explanation of how the practice addresses just that one characteristics e. 1 points – Some indication of learni ng characteristic barriers and explanation of how the practice addr esses one or both, but identification and explanation may be vague and unclear f. 0 points – Answer is not rele vant to the question asked

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392 27. (30 pts) Describe how each of the follo wing effective mathematics instructional practices for struggling learne rs is applied within the De veloping Algebraic Literacy (DAL) instructional process. Be specific in terms of where in the DAL process each practice can be implemented and how it is implemented. Building Meaningful Student Connections Language Experiences C-R-A Sequence of Instruction Rubric 1. Where in the DAL Process Each Practice can be Implemented a. 10 points – Thorough and complete explanation of where the practice should be implemented b. 8 points – Main points covered for where the practice should be implemented, but minor details may be missing c. 6 points – Some main points covered for where the practice should be implemented, one or two major details may be left out d. 4 points – A general idea of where th e practice should be implemented is given, but more specific information is left out e. 2 points – Vague idea of where the practice should be implemented, but little evidence of specific unde rstandings of the location f. 0 points – Answer is not rele vant to the question asked how it is implemented a. 20 points – Thorough and complete explanation of implementation b. 16 points – Main points covered, but minor details may be missing from implementation explanation c. 12 points – Some main points covered, one or two main points may be left out from implementation explanation d. 8 points – One or two main points co vered, but many points are left out from implementation explanation e. 4 points – Vague idea of the overall imple mentation, but little evidence of specific understandings f. 0 points – Answer is not rele vant to the question asked

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393 28. (10 pts) A strategy that is implemented during the thir d step of the DAL process involves the use of graphic or ganizers. Describe what eff ective mathematics instruction practice for struggling learners this strategy exemplifies a nd its primary purpose in terms of student learning. Rubric 1. Description of the Effective Mathematics Instruction Practice that the Strategy Exemplifies a. 5 points – Thorough and complete desc ription of the practice that the specific strategy exemplifies b. 4 points – A description that include s most key points about the practice that the specific strategy exemplifie s, but minor details may be missing c. 3 points – A description th at includes some main poi nts about the practice that the specific strategy exemplifies, one or two main points may be left out d. 2 points – A description that include s one or two main points about the practice that the specific strategy exemplifies, but many points are left out e. 1 points – Vague description of the practice that the specific strategy exemplifies, but little evid ence of specific understandings f. 0 points – Answer is not rele vant to the question asked 2. Primary Purpose of the Strategy in terms of Student Learning a. 5 points – Thorough and complete explanation b. 4 points – Main point covered, but minor details may be missing c. 3 points – Some of the main point c overed, one or two larger details may be left out d. 2 points – A small piece of the main point is covered, but a majority of the explanation is missing e. 1 points – Vague idea of the overall point but little evidence of specific understandings

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394 29. (10 pts) A strategy that is implemented during the thir d step of the DAL process involves the use of the LIP strategy. Describe what effective mathematics instruction practice for struggling learners this strategy exemplifies a nd its primary purpose in terms of student learning. Rubric 1. Description of the Effective Mathematics In struction Practice that the Strategy Exemplifies a. 5 points – Thorough and complete desc ription of the practice that the specific strategy exemplifies b. 4 points – A description that include s most key points about the practice that the specific strategy exemplifie s, but minor details may be missing c. 3 points – A description th at includes some main poi nts about the practice that the specific strategy exemplifies, one or two main points may be left out d. 2 points – A description that include s one or two main points about the practice that the specific strategy exemplifies, but many points are left out e. 1 points – Vague description of the practice that the specific strategy exemplifies, but little evid ence of specific understandings f. 0 points – Answer is not rele vant to the question asked 2. Primary Purpose of the Strategy in terms of Student Learning a. 5 points – Thorough and complete explanation b. 4 points – Main point covered, but minor details may be missing c. 3 points – Some of the main point cove red, one or two larger details may be left out d. 2 points – A small piece of the main point is covered, but a majority of the explanation is missing e. 1 points – Vague idea of the overall poi nt, but little evidence of specific understandings f. 0 points – Answer is not relevant to the question asked

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395 30. (10 pts) A strategy that is implemented during the thir d step of the DAL process involves encouraging students to communicate about the algebr aic thinking concept they are learning. Describe what effective math ematics instruction practice for struggling learners this strategy exemplifies and its pr imary purpose in terms of student learning. Rubric 1. Description of the Effective Mathematics Instruction Practice that the Strategy Exemplifies a. 5 points – Thorough and complete de scription of the practice that the specific strategy exemplifies b. 4 points – A description that in cludes most key points about the practice that the specif ic strategy exemplifies, but minor details may be missing c. 3 points – A description that incl udes some main points about the practice that the specif ic strategy exemplifies, one or two main points may be left out d. 2 points – A description that incl udes one or two main points about the practice that the specific stra tegy exemplifies, but many points are left out e. 1 points – Vague description of the practice that the specific strategy exemplifies, but little evid ence of specific understandings f. 0 points – Answer is not rele vant to the question asked 2. Primary Purpose of the Strategy in terms of Student Learning a. 5 points – Thorough and complete explanation b. 4 points – Main point covered, but minor details may be missing c. 3 points – Some of the main point c overed, one or two larger details may be left out d. 2 points – A small piece of the main point is covered, but a majority of the explanation is missing e. 1 points – Vague idea of the overall poi nt, but little evidence of specific understandings f. 0 points – Answer is not rele vant to the question asked

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396 31. (10 pts) A strategy that is implemented during the second step of the DAL process is to evaluate their abilities to read, represent, solve, and ju stify given a na rrative context that depicts an algebraic thinking concep t. Describe what effective mathematics instruction practice for strugg ling learners this strategy exemplifies and its primary purpose in terms of student learning. Rubric 1. Description of the Effectiv e Mathematics Instruction Prac tice that the Strategy Exemplifies a. 5 points – Thorough and complete desc ription of the practice that the specific strategy exemplifies b. 4 points – A description that in cludes most key points about the practice that the specif ic strategy exemplifies, but minor details may be missing c. 3 points – A description that incl udes some main points about the practice that the specific strategy exemplifies, one or two main points may be left out d. 2 points – A description that include s one or two main points about the practice that the specific strategy exemplifies, but many points are left out e. 1 points – Vague description of the practice that the specific strategy exemplifies, but little evid ence of specific understandings f. 0 points – Answer is not rele vant to the question asked 2. Primary Purpose of the Strategy in terms of Student Learning a. 5 points – Thorough and complete explanation b. 4 points – Main point covere d, but minor details may be missing c. 3 points – Some of the main point c overed, one or two larger details may be left out d. 2 points – A small piece of the main point is covered, but a majority of the explanation is missing e. 1 points – Vague idea of the overall poi nt, but little evidence of specific understandings f. 0 points – Answer is not rele vant to the question asked

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397 32. (10 pts) A strategy that is implemented during each step of the DAL process is to situate target mathematics concepts/skills within a narrative text. Describe what effective mathematics instruction practi ce for struggling learners this strategy exemplifies and its primary purpose in terms of student learning. Rubric 1. Description of the Effec tive Mathematics Instruction Practice that the Strategy Exemplifies a. 5 points – Thorough and complete de scription of the practice that the specific strategy exemplifies b. 4 points – A description that in cludes most key points about the practice that the specif ic strategy exemplifie s, but minor details may be missing c. 3 points – A description that incl udes some main points about the practice that the specif ic strategy exemplifies, one or two main points may be left out d. 2 points – A description that in cludes one or two main points about the practice that the specifi c strategy exemplifies, but many points are left out e. 1 points – Vague description of the practice that the specific strategy exemplifies, but little evidence of specific understandings f. 0 points – Answer is not rele vant to the question asked 2. Primary Purpose of the Strategy in terms of Student Learning a. 5 points – Thorough and complete explanation b. 4 points – Main point covered, but minor details may be missing c. 3 points – Some of the main point c overed, one or two larger details may be left out d. 2 points – A small piece of the main point is covered, but a majority of the explanation is missing e. 1 points – Vague idea of the overall poi nt, but little evidence of specific understandings f. 0 points – Answer is not rele vant to the question asked

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398 BONUS (up to 5 points) What is the primary purpose of the first step of the DAL process? What stage of learning are students developi ng during this step? Rubric a. 5 points – Thorough and complete explanation of the purpose and correct identification of stage of learning b. 4 points – Main points covered on the purpose, but minor details may be missing, and correct identifica tion of stage of learning c. 3 points – Some main points covere d on the purpose, one or two main points may be left out, and correct id entification of st age of learning d. 2 points – One or two main points covered on the purpose, and identification of stage of l earning may be off-target e. 1 points – Vague idea of the overall purpos e, and identification of stage of learning may be off-ta rget or left out f. 0 points – Answer is not rele vant to the question asked

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399 Appendix K: Fidelity Checkli st for DAL Initial Session Probe

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400 Initial Session Probe Yes No NA Teacher notes students’ skill level in each of the four problem-solving areas. *Teacher candi date gives student a chance to read the context and problem for problem solving. *Teacher candi date gives student a chance to repr esent the problem for solving. *Teacher candi date gives student a chance to solve the problem. *Teac her candidate has student ju stify his or her proble m-solving. *Teacher c andidate provides concrete, r epresentational, or abstract materials for student's pr oblem solving. *Teacher candidat e provides student assistance in problem-sol ving when needed. Teacher determines directi on (skill and level), based on data gathered from pr obe, for first full session.

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401 Appendix L: Fidelity Checklist for Full DAL Session

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402 Observation #___________ DAL Fidelity Checklist Tutor:_________________________ Date:__________________________ Observer:______________________ School:________________________ Step 1: Building Automaticity Yes No NA Students practice problem-solving with familiar target learning objecti ves and narratives. *Teac her candidate points out st rategies student uses for problem-solving. *Teacher candidate recommends strat egies to use. *Teacher c andidate reinforces student's successes. *Teacher c andidate provides concrete, representational or abstract materials for student's pr oblem solving. Students respond to a timed probe consisting of specific response tasks on this same learning objective. *Teacher candidat e provides all probe tasks at the same re sponse level. Teacher and students record data from the timed probe for goal-setting and decision-making purposes. *Teacher c andidate and student discuss student performance. *Teacher c andidate and student make goals for future sessi ons for timed probe. *Teacher c andidate and student record student performance on data tracking sheet. Step 2: Measuring Progress & Making Decisions Yes No NA Teacher notes students’ skill level in each of the four

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403 problem-solving areas. *Teacher candi date gives student a chance to read the context and problem for problem solving. *Teacher candi date gives student a chance to repr esent the problem for solving. *Teacher candi date gives student a chance to solve the problem. *Teac her candidate has student ju stify his or her proble m-solving. *Teacher c andidate provides concrete, r epresentational, or abstract materials for student's pr oblem solving. *Teacher candidat e provides student assistance in problem-sol ving when needed. Teacher determines Step 3, Problem Solving the New’s target learning object ive and appropriate level for student instruction. Step 3: Problem Solving the New Yes No NA Making Connections to Existing Mathematical Knowledge is where the t eacher first provides an advance organizer that addresses three important items. *Teacher candidate gi ves student a graphic organi zer for making connections. *Teacher candidate links new target learning objective to pr evious mathematics instruction. *Teacher candidate identifies the new target learni ng objective. *Teacher candidate provides a rationale for the new ta rget learning objective. Problem Solving is where the new problem narrative is introduced; it is at th is point that the student reads the story aloud, represents the problem situation, solves the problem, and provides justification for their response and approach. *Teacher candi date gives student a chance to read the context and problem for problem solving. *Teacher candi date gives student a chance to repr esent the problem for solving. *Teacher candi date gives student a chance to solve the problem. *Teacher candidat e has student justify his or her

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404 problem-solving. *Teacher c andidate provides concrete, r epresentational, or abstract materials for student 's problem solving. *Teacher candidat e points out strategies student uses for problem-solving. *Teacher candidat e recommends strategies to use. *Teacher candidat e provides student assistance in probl em-solving when needed. Communicate Mathematical Ideas is where the teacher elicits, from th e students, something she found interesting about the problem and spends a few minutes of focused time engaging students in using language to describe the mathematical idea. *Teacher candidate discusses an interesting mathematical idea from the le sson with the student. *Teacher candidate has student draw a picture representati on of the mathematical idea. *Teacher candidate has student label the picture representati on of the mathematical idea. *Teacher candidate has student write a brief description of the mathematical idea. Make Connections to Student s’ Interests is where graphic organizers are utilized. *Teacher candi date gives student a graphic organizer for maki ng connections from the new mathematical idea to student interests. *Teacher c andidate discusses how the mathematical id ea relates to student interests. *Teacher candida te and student use the graphic organizer to show connections between the mathematical idea and student in terests.

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405 Appendix M: Focus Group Questions

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406 Focus Group Questions Attitude 1. How important do you think algebraic th inking is in a child’s mathematic curriculum? Mathematics in the total scope of the academic curriculum? 2. How would you describe the nature of al gebraic thinking in general? Rulegoverned? Haphazard? Etc.? What about mathematics in general? 3. How do you feel about teaching algebrai c thinking to students at-risk for mathematics failure? What makes you f eel this way? What about teaching mathematics in general to students at-risk for mathematics failure? Self-Efficacy 4. How prepared do you feel to teach algebr a to elementary students at-risk for mathematics failure? What makes you feel this way? How prepared do you feel to teach mathematics in general to students at-risk for mathematics failure? 5. How much impact do you think you as a pr ofessionally traine d teacher can/will have on students with low-algebra achi evement? What about low mathematics achievement in general? 6. How much impact do you think your pla nning and reflection on your mathematics instruction will impact how your stude nts progress through algebraic thinking material? What about how they progr ess through mathema tics material in general?

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407 Instructional Knowledge Information 7. How well do you feel you understand the inst ructional strategi es presented for teaching algebra? Teaching mathematics in general? 8. What do you think some sound pedagogical st rategies are for teaching algebra? Mathematics in general? 9. What strategies, if any, do you think woul d not work for teaching algebra to atrisk learners? Mathematics in general? Instructional Knowledge Application 10. Describe your comfort level in utiliz ing mathematics strategies for teaching algebra. For mathematics in general? 11. Describe how ready you feel to use mathem atics strategies for teaching algebra. For mathematics in general? 12. Describe how likely it would be for you to review instructional strategies for teacher algebra that we have discu ssed and then apply them once you feel prepared. For mathematics in general? Content Knowledge 13. How would you describe your level of unde rstanding of elementary algebra content? How would you describe your level of understa nding of general mathematics at the elementary level?

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408 14. What do you think your greatest strength in terms of content knowledge is for algebra? For mathematics in general? 15. What do you think your greatest weakness in terms of content knowledge is for algebra? For mathematics in general? 16. Are there any strategies you will use to ma ke yourself more comfortable with the content knowledge of algebra? Of mathematics in general?

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About the Author Sharon Nichole Estock Ray earned a Bach elor’s of Science degree in Psychology with a Minor in Communication Sciences and Di sorders, and a Master’s of Education in Special Education with a Concentration in Learning Disabilities from James Madison University. She has also earned a Doctorate degree in Special Education with a Cognate in Literacy from the University of South Fl orida. Sharon has made several presentations on the national and local levels involving math ematics and reading instructional practices for learners with special needs. Sharon has her own remediation service for learners with special needs. She also works part-time teaching university level educa tion courses. Her inte rests include teacher preparation, content area learning, at-risk learners, and transition services.


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Evaluating the efficacy of the devleoping algebraic literacy model :
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ABSTRACT: For students with learning disabilities, positive academic achievement outcomes are a chief area of concern for educators across the country. This achievement emphasis has become particularly important over the last several years because of the No Child Left Behind legislation. The content area of mathematics, especially in the higher order thinking arena of algebra, has been of particular concern for student progress. While most educational research in algebra has been targeted towards remedial efforts at the high school level, early intervention in the foundational skills of algebraic thinking at the elementary level needs consideration for students who would benefit from early exposure to algebraic ideas. A key aspect of students' instruction with algebraic concepts at any level is the degree and type of preparation their teachers have received with this content.Using a mixed methods design, the current researcher investigated the usage of the Developing Algebraic Literacy (DAL) framework with preservice special education teacher candidates in an integrated practicum and coursework experience. Multiple survey measures were given at pre-, mid-, and post- junctures to assess teacher candidates' attitudes about mathematics, feelings of efficacy when teaching mathematics, and content knowledge surrounding mathematics. An instructional knowledge exam and fidelity checks were completed to evaluate teacher candidates' acquisition and application of algebraic instructional skills. Focus groups, case studies, and final project analyses were used to discern descriptive information about teacher candidates' experience while engaging in work with the DAL framework.Results indicated an increase in preservice teachers' attitudes towards mathematics instruction, feelings of efficacy in teaching mathematics, and in the content knowledge surrounding mathematics instruction. Instructional knowledge also increased across preservice teacher candidates, but abilities to apply this knowledge varied across teacher candidates', based on their number of sessions working with students within their practicum site. Further findings indicate the desire of preservice teachers to increase the length and number of student sessions within the DAL experience, as well as the need for increased levels of instructional support to enhance their own experience. This study provides preliminary support for utilizing the DAL instructional framework within preservice teacher preparation experiences for future special educators.
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