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Title:
Examining the attitudes toward mathematics of preservice elementary school teachers enrolled in an introductory mathematics methods course and the experiences that have influenced the development of these attitudes
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English
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Schackow, Joy Bronston
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University of South Florida
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Journals
Reflection
Achievement
Value
Enjoyment
Motivation
Confidence
Dissertations, Academic -- Secondary Education -- Doctoral -- USF   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Summary:
ABSTRACT: The reform movement in mathematics education has recognized affective factors as an important area where change is needed. This study examined the attitudes toward mathematics of preservice elementary teachers entering an introductory mathematics methods course. The methods course utilized constructivist instructional methods, such as the use of manipulatives, cooperative group work, problem solving, and calculators. Qualitative methods were used to explore participants attitudes toward mathematics and the experiences that have led to the development of these attitudes. The study sought to determine the extent to which preservice teachers attitudes toward mathematics changed during the methods course and the correlation between preservice teachers initial attitudes toward mathematics and their achievement in the methods course.Thirty-three university students enrolled in one section of a mathematics methods course for elementary education majors completed the Attitudes Toward Mathematics Inventory at the beginning of the semester and again during week 12 of the 15-week semester. Throughout the semester, participants submitted reflective journal entries in which they reflected on their attitudes toward and experiences with mathematics. The instructor responded to each journal. Participants initial survey scores indicated that they valued mathematics, but their scores for Self-Confidence, Enjoyment, and Motivation were somewhat negative. As a whole, participants showed a statistically significant positive change in attitude on the second survey.In individual interviews, participants who showed significant positive changes in attitude mentioned manipulatives, journals, and the organized format of the course as aspects of the methods course that had positively influenced their attitudes toward mathematics. A statistically significant positive correlation was found between initial attitude survey scores and the methods course departmental final examination, which was used as a measure of achievement. Through their journal entries and interviews, participants offered a clear view of the types of experiences that encouraged the development of positive and negative attitudes toward mathematics. These findings have implications for teacher educators who seek to improve the attitudes toward mathematics of preservice elementary school teachers and for mathematics teachers at all levels.
Thesis:
Thesis (Ph.D.)--University of South Florida, 2005.
Bibliography:
Includes bibliographical references.
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by Joy Bronston Schackow.
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Title from PDF of title page.
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Document formatted into pages; contains 416 pages.

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oclc - 69365255
usfldc doi - E14-SFE0001293
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ABSTRACT: The reform movement in mathematics education has recognized affective factors as an important area where change is needed. This study examined the attitudes toward mathematics of preservice elementary teachers entering an introductory mathematics methods course. The methods course utilized constructivist instructional methods, such as the use of manipulatives, cooperative group work, problem solving, and calculators. Qualitative methods were used to explore participants attitudes toward mathematics and the experiences that have led to the development of these attitudes. The study sought to determine the extent to which preservice teachers attitudes toward mathematics changed during the methods course and the correlation between preservice teachers initial attitudes toward mathematics and their achievement in the methods course.Thirty-three university students enrolled in one section of a mathematics methods course for elementary education majors completed the Attitudes Toward Mathematics Inventory at the beginning of the semester and again during week 12 of the 15-week semester. Throughout the semester, participants submitted reflective journal entries in which they reflected on their attitudes toward and experiences with mathematics. The instructor responded to each journal. Participants initial survey scores indicated that they valued mathematics, but their scores for Self-Confidence, Enjoyment, and Motivation were somewhat negative. As a whole, participants showed a statistically significant positive change in attitude on the second survey.In individual interviews, participants who showed significant positive changes in attitude mentioned manipulatives, journals, and the organized format of the course as aspects of the methods course that had positively influenced their attitudes toward mathematics. A statistically significant positive correlation was found between initial attitude survey scores and the methods course departmental final examination, which was used as a measure of achievement. Through their journal entries and interviews, participants offered a clear view of the types of experiences that encouraged the development of positive and negative attitudes toward mathematics. These findings have implications for teacher educators who seek to improve the attitudes toward mathematics of preservice elementary school teachers and for mathematics teachers at all levels.
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Examining the Attitudes Toward Mathematic s of Preservice Elementary School Teachers Enrolled in an Introductory Ma thematics Methods Course and the Experiences That Have Influenced the Development of These Attitudes by Joy Bronston Schackow A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Secondary Education College of Education University of South Florida Major Professor: Deniss e R. Thompson, Ph.D. Richard Austin, Ph.D. Roger N. Brindley, Ph.D. Lou M. Carey, Ph.D. Fredric Zerla, Ph.D. Date of Approval: October 25, 2005 Keywords: journals, reflection, achievement, value, enjoyment, motivation, confidence Copyright 2005, Joy Bronston Schackow

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DEDICATION To my daughter, Stefanie. Your encour agement sustained me throughout this seven-year endeavor. Thank you for always being there for me, providing everything from supportive hugs to computer assistance to transcribing interviews, and for never complaining about all of Mom’s time spent studying and writing. To my son, Mark. Thank you for always be ing proud of me and for making me so proud of you. To my husband, Sam. You entered my lif e midway through my doctoral program, a decision that would cause many to questi on your sanity. Your love and support made me even more determined to complete this journey. To Grant. I know that you would be proud.

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ACKNOWLEDGEMENTS I would like to express my d eep and heartfelt gratitude to my major professor, Dr. Denisse Thompson. She has been my teacher, my mentor, my colleague, and my friend. Dr. Thompson is the world’s busiest person, but she always made time for me when I needed her. I so appreciated the care and atten tion to detail that she gave in reviewing my manuscripts. I feel very fortunate to have learned from the best. I am also incredibly grateful for th e contributions of my other committee members, Dr. Richard Austin, Dr. Roger Brin dley, Dr. Lou Carey, and Dr. Fredric Zerla. I learned so much about teacher education from observing Dr. Austin’s elementary mathematics methods course, and I thank hi m for his perspective on issues involving mathematics education. Dr. Brindley, qualitative researcher extraordinaire, taught me the value of thick, deep, and rich description a nd how to give voice to my sample, and I thank him for his great sense of fun. I would al so like to thank Dr. Carey for her guidance with the quantitative portions of this study. I felt so fortunate when Dr. Carey agreed to join my committee, and I have come to admire her greatly, both professionally and personally. Dr. Zerla taught me about the history and beauty of mathematics, and I thank him for the mathematician’s perspectiv e that he provided to this study. Thank you to my parents for instilling in me the importance of getting an education. Thank you to all of my family and friends for all of their support and encouragement.

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Lastly, I would like to tha nk the participants in this study. Thank you all so much for openly sharing your memories with me Your candor and trust made this study possible.

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i TABLE OF CONTENTS List of Tables viii Abstract xi Chapter One: Introduction 1 Purpose of Study 6 Research Questions 7 Definitions 8 Chapter Two: Review of Literature 9 The Affective Domain 9 Beliefs About Mathematics 9 Research on Beliefs about Mathematics 11 Teacher Beliefs about Mathematics 12 Beliefs of Preservice and Beginning Teachers 16 Beliefs of Experienced Teachers 20 Attitudes Towards Mathematics 22 Teacher Attitudes Toward Mathematics 23 Attitudes Toward Mathematics of Preservice Elementary Teachers 23 Attitudes Toward Mathematics of Preservice Secondary Teachers 26

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ii Concepts Related to Teacher Attitude 27 Connecting Teacher Beliefs and Attitudes about Mathematics 29 The Effects of Attitudes and Beliefs on Achievement in Mathematics 29 Journal Writing and Mathematics 36 Journal Writing with Elementary School Mathematics Students 39 Journal Writing with Intermed iate and Middle School Mathematics Students 42 Journal Writing with High School Mathematics Students 46 Journal Writing with College Mathematics Students 50 Journal Writing with Mathematics Teachers 52 Journal Writing with Preservice Teachers 53 Phenomenology 54 Conceptual Overview 54 Phenomenological Methodology 56 Reflection 58 Summary and Implications for Teacher Education 59 Significance of Study 63 Chapter Three: Method 66 Pilot Studies 67 Present Study 67

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iii Participants 68 Description of Course 68 Procedure 69 Instruments 74 The Attitudes Toward Mathematics Inventory (ATMI) 74 Journal Prompts 76 Experiences with Mathematics Interviews 77 Changed Attitudes Interviews 79 Final Examination for Mathematics Methods Cour se 80 Data Analysis 82 Surveys 82 Journals 85 Interviews 88 Limitations of Study 89 Chapter Four: Results 90 Question 1: Initial Attitudes Toward Mathematics 90 Question 2: Changed Attitudes Toward Mathematics 98 Change Scores 98 Changed Attitudes Interviews 100 Amelia 101 Jennifer 105 Erin 109

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iv Tessa 112 Stephanie 116 Shelly 119 Yezania 123 Question 3: Relationship Between In itial Attitudes a nd Score on Final Examination 126 Question 4: Journals 127 Journal 1: Feelings at Beginning of Course 128 Positive Feelings about Course 129 Negative Feelings about Course 129 Mixed Feelings about Course 130 Journal 2: Memories of Mathematics in Elementary School 136 Journal 3: Feelings about Mathematics 148 Journal 4: Memorable Experience with Mathematics 160 Journal 8: Use of Reflective Journals in the Methods Course 174 Journals 5, 6, and 7: Relevant Excerpts 181 Journal 5: Boosting Confidence of Students 181 Journal 6: Qualities of Best Mathematics Teac her 182 Journal 7: Qualities of Worst Mathematics Teacher 184 Question 5: Participants with the Most Extreme Attitudes 186 Mary 186 Lisa 190 Hermione 195

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v Torri 199 Reflections from the Re searcher’s Journal 202 Chapter Five: Overview, Conclusions, Im plications, and Recommendations 208 Overview of Study 208 Summary of and Conclusions fr om Research Findings 213 Research Question One: Initial Attitudes Toward Mathematics 213 Research Question Two: Changed Attitudes Toward Mathematics 216 Interviews: Positive Change Scores 219 Interviews: Negative Change Scores 223 Themes Across Interviews 228 Research Question Three: Relationship Between Attitudes and Achievement 230 Research Question Four: Themes from Journals 232 Journal One: Feelings at Beginning of Course 232 Journal Two: Memories of Mathematics in Elementary School 234 Journal Three: Feelings About Mathematics 237 Journal Four: Memorable Experience w ith Mathematics 240 Journal Eight: The Use of Reflec tive Journals in the Methods Course 243 Journals Five, Six, Seven: Rele vant Excerpts 246 Beliefs Expressed in Journals 248

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vi Themes Across Journals 248 Research Question Five: Expe riences of Those with Most Extreme Attitudes 249 Implications and Recommendations for Practice 253 Mathematics Teacher Educators 253 Mathematics Teachers at All Levels 256 Implications and Recommendations for Future Research 261 References 266 Appendices 282 Appendix A: Pilot Study I 283 Appendix B: Pilot Study II 309 Appendix C: Course Syllabus 322 Appendix D: Table of Contents 333 Appendix E: Attitudes Toward Mathematics Inventory 346 Appendix F: Researcher’s Journal Responses 352 Appendix G: Observer Protocol 359 Appendix H: Experiences with Mathema tics Interview Protocol 360 Appendix I: Changed Attit udes Interview Protocol 361 Appendix J: Final Examination 362 Appendix K: Hycner’s Guidelines fo r the Phenomenological Analysis of Data 366 Appendix L: Computation of Inter-Rater Reliability 368 Appendix M: Researcher’s Possible Bias es and Preconceptions 373

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vii Appendix N: Post-Course I ndividual Survey Items 374 Appendix O: Relevant Excerpts from Re searcher’s Journal 379 Appendix P: Samples from Course Note Packet 394 About the Author End Page

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viii LIST OF TABLES Table 1 Results from Evan’s Study with Journal Writing 43 Table 2 Survey Items Grouped by Factors 75 Table 3 Journal Prompts and the Orde r in Which They Were Assigned 78 Table 4 Content of Methods Course Final Examination 82 Table 5 Stratified Random Sampling of Journals for Double Coding 86 Table 6 Initial Attitudes Toward Mathematics: Mean Raw Scores on the Attitudes Toward Mathematics Inventory 92 Table 7 Initial Attitudes Toward Mathematics: Mean Per-Item Scores on the Attitudes Toward Mathematics Inventory 92 Table 8 Means and Standard Deviations on Items from the Attitudes Toward Mathematics Inventory 93 Table 9 Change Scores Per Attit ude Factor: Mean Per Item Scores 99 Table 10 Feelings about Methods Course at Beginning of Course 130 Table 11 Positive Attitudes and Experiences Expressed in Journal One: Feelings About Course 132 Table 12 Negative Attitudes and Experiences Expressed in Journal One: Feelings About Course 133 Table 13 Themes from Journal Prompt: What Do You Hope to Gain From the Course? 134

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ix Table 14 Positive Memories of Mathematics in Elementary School from Journal Two 137 Table 15 Positive Memories of Mathem atics Teachers in Elementary School from Journal Two 139 Table 16 Negative Memories of Math ematics in Elementary School from Journal Two 141 Table 17 General Memories of Mathematics in Elementary School from Journal Two 144 Table 18 Themes from Journal Prompt Two: What Did You Learn as a Future Teacher? 146 Table 19 Positive Feelings About Mathema tics from Journal Three 150 Table 20 Experiences Associated with Positive Feelings About Mathematics from Journal Three 151 Table 21 Positive Experiences with Teachers at Specific Levels of Schooling 152 Table 22 Negative Feelings About Mathematics from Journal Three 154 Table 23 Experiences Associated with Negative Feelings About Mathematics From Journal Three 156 Table 24 Negative Experiences at Specific Le vels of Schooling 158 Table 25 Memorable Experiences that Influenced Attitudes toward Mathematics 161 Table 26 Benefits of Reflective Journal Wr iting in the Methods Course 176 Table A-1 Initial Attitudes Toward Mathematics 287

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x Table A-2 Themes Identified at the Beginning of Elementary Mathematics Methods Course 289 Table A-3 Themes from Journal Prom pt: What Do You Hope to Gain From the Course? 291 Table A-4 Positive Memories of Mathematics in Elementary School 294 Table A-5 Negative Memories of Mathematics in Elementary School 296 Table A-6 Negative Memories of Mathematics Teachers 297 Table A-7 Themes from Journal Pr ompt: What Did You Learn as a Future Teacher? 298 Table B-1 Initial Attitudes Toward Mathematics 312 Table K-1 Inter-rater Reliability: Conditional Misses 370 Table N-1 Means and Standard Deviati ons on Items from the Post-Course Attitudes Toward Mathematics Inventory 374

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xi EXAMINING THE ATTITUDES TOWARD MATHEMATICS OF PRESERVICE ELEMENTARY SCHOOL TEACHERS EN ROLLED IN AN INTRODUCTORY MATHEMATICS METHODS COURSE AND THE EXPERIENCE S THAT HAVE INFLUENCED THE DEVELOPM ENT OF THESE ATTITUDES Joy Bronston Schackow ABSTRACT The reform movement in mathematics educ ation has recognized affective factors as an important area where change is neede d. This study examined the attitudes toward mathematics of preservice elementary t eachers entering an introductory mathematics methods course. The methods course utilized constructivist instructi onal methods, such as the use of manipulatives, cooperative group work, problem solving, and calculators. Qualitative methods were used to explore part icipants’ attitudes toward mathematics and the experiences that have led to the devel opment of these attitudes. The study sought to determine the extent to which preservice teach ers’ attitudes toward mathematics changed during the methods course and the correla tion between preservice teachers’ initial attitudes toward mathematics and their achievement in the methods course. Thirty-three university stude nts enrolled in one secti on of a mathematics methods course for elementary education majors completed the Attitudes Toward Mathematics Inventory at the beginning of the semester and again during week 12 of the 15-week semester. Throughout the semester, participants submitted reflective journal entries in

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xii which they reflected on their attitudes towa rd and experiences with mathematics. The instructor responded to each journal. Participants’ initial survey scores indicat ed that they valued mathematics, but their scores for Self-Confiden ce, Enjoyment, and Motivati on were somewhat negative. As a whole, participants showed a statistically significant pos itive change in attitude on the second survey. In individual interviews, pa rticipants who showed significant positive changes in attitude mentioned manipulatives, journals, and the organized format of the course as aspects of the methods course th at had positively influenced their attitudes toward mathematics. A statistically significant positive correlation was found between initial attitude survey scores and the methods course departmental final examination, which was used as a measure of achievement. Through their journal entries and interviews participants offere d a clear view of the types of experiences that encouraged the development of positive and negative attitudes toward mathematics. These findings have implications for teacher educators who seek to improve the attitudes toward mathematics of preservice elementary school teachers and for mathematics teachers at all levels.

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1 CHAPTER ONE INTRODUCTION The reform movement in mathematics education that began in the 1980s has recognized affective factors as an important area where chan ge is needed. This emphasis on affective issues “is related to the importa nce that the reform m ovement attaches to higher-order thinking. If students are to be active learners of ma thematics who willingly attack nonroutine problems, their affective responses to mathematics are going to be much more intense than if they are merely e xpected to achieve the satisfactory levels of performance in low-level computational skills” (McLeod, 1992, p. 575). Everybody Counts, the National Research Council’s (NRC) report on the future of mathematics education, focused on the need to change the public’s attitude s and beliefs about mathematics (NRC, 1989). The authors pointed out that affective issues involving both children and adults must be considered in order for mathematics education to improve. The National Council of Teach ers of Mathematics (NCTM, 1989, 2000) has also focused on affective issues related to mathematics education. In the Curriculum and Evaluation Standards for School Mathematics (1989), two major goals were established related to affective factors: learning to value mathem atics and developing confidence in one’s own mathematical ability. When discussing mathem atical disposition and engagement in the Principles and Standards for School Mathematics (2000), NCTM stressed the importance of students’ confidence, intere st, perseverance, and curiosity in learning mathematics.

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2 The importance of developing self-confiden t, motivated students who value and enjoy mathematics has been well established, but the means for doing so are not as clear. Although results are not always consistent, studies have s hown that children typically begin school with positive att itudes toward mathematics, but these attitudes tend to become less positive as they get older. By the time students reach high school, their attitudes toward mathematics have fre quently become negative (McLeod, 1992). Vanayan, White, Yuen, and Teper (1997) surveyed 1,320 third graders and 1,320 fifth graders from 60 schools. They found that 79% of the third graders gave a positive response to the statement, “I like math.” The percentage was somewhat lower, 73%, for fifth graders. However, roughl y 60% of third graders and 65% of fifth graders indicated that they liked other subjects more than mathematics. Prawat and Anderson (1994) f ound that fourth and fifth graders reported “twice the amount of negative as compared to posit ive affect while engaged in mathematics seatwork” (p. 219). Mink and Fraser (2002) ad ministered an attit udinal survey to 120 fifth-grade students before im plementing a program integratin g children’s literature. The survey measured attitudes to reading, writing, and mathem atics. Students responded to questions such as, “Do you like mathematics ?” They found that the average item mean for attitudes to mathematics was lower than those for attitudes to reading and writing, indicating that the students liked reading and writing mo re than mathematics. The tendency for attitudes to become more negative continues in middle school. Ruffell, Mason, and Allen (1998) intervie wed 14 middle-school students in the Netherlands. Approximately 50% of the stude nts claimed to like mathematics. They seemed unable to say what they liked about ma thematics, but they were able to say what

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3 they did not like. The author s pointed out that this is an example of how negative memories appear to dominate positive memories when forming attitudes. When describing a bad experience in mathematics, th e words most often us ed were ‘nervous’ or ‘bored.’ The students associated mathematics wi th “repetition, lack of challenge, lack of novelty, routine, sitting at desks, working in silence, working alone, and knowing what to expect” (p. 7). The students generally viewed mathematics as good if “you get it right.” They described the best part of mathematic s as doing investigati ons and working in a group. The general impression given by the studen ts during these interv iews was a dislike of mathematics. By high school, negative attitudes toward mathematics have often become more apparent. Hoyles (1982) found that 14-year -old students tend to correlate their mathematical experiences with feelings of anxiety, shame, and failure. Olson (1998) surveyed high-school geometry students in order to examine their enjoyment of mathematics, previous experiences with mathematics, and perceived usefulness of mathematics in the future. She found that one-third of the students did not enjoy mathematics. When reflecting on their prio r experiences with word problems, 38% described their experiences as frustrating. Several large-scale studies have examined attitudes toward mathematics. In 1995, the Third International Mathematics and Sc ience Study (TIMSS) examined attitudes, beliefs, and opinions related to mathematics. This study collected and analyzed data from students in 45 countries. Student s were asked to state their level of agreement with the following statements: 1) I like mathematics, 2) Mathematics is boring, and 3) I enjoy learning mathematics. Mullis et al. (1997) reported that fourth-grade students had

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4 relatively positive attitudes toward mathema tics. Beaton et al. (1996) found that eighth graders had relatively neutral f eelings about mathematics. In most countries, high school students expressed that they liked mathematics to some degree, but in Austria, the Czech Republic, Hungary, and Lithuania, more than half of the students reported that they disliked mathematics (Mullis et al., 1998). Affect has been a focus of various national studies as well. The National Assessment of Educational Progress (NAEP) has included some mathematics attitude items on its regular assessments. At grade 12, the percentage of st udents who indicated that they liked mathematics has dropped fr om 51% in 1992 to 47% in 2000. However, the percent indicating they are good at mathematic s has increased slightly from 50% in 1992 to 53% in 2000 (Strutchens, Lubienski, McGraw, and Westbrook, 2004). The Seventh Mathematics Assessment of the NAEP in 1996 also included data on student attitudes toward mathematics. Although over half of four th graders and eighth graders agreed with the statement “I like mathematics,” the percen tage of eighth graders who agreed with the statement was lower than the percentage of f ourth graders who agreed. The percentage of twelfth graders who agreed was even lowe r. The frequency of positive responses was greater among twelfth graders who were taki ng mathematics than among those who were currently not taking mathema tics (Mitchell, Hawkins, Stancavage, & Dossey, 1999). Although many students value and enjoy mathematics and are motivated and confident in their abilities to do mathematics, there are still too many who do not feel this way. Negative attitudes toward mathematics ar e “thought to plague lear ners at every level of schooling” (Sherman and Christian, 1999, p.95) These negative feelings become even more significant when considering that some of these students will eventually choose to

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5 become elementary school teachers and will be teaching a subject about which they may have negative attitudes. Several studies have examined the attitudes toward mathematics of preservice elementary school teachers and none of these have indicated a preponderance of positive attitudes. Rech, Hartzell, and Stephens (1993) found that preservice elementary teachers have less favorab le attitudes toward mathematics than the general university population. Cornell (1999) surveyed graduate students who were taking a mathematics instruction seminar for certification as elementary school teachers. He found that about half of the students report ed a dislike of mathem atics. In nearly all cases, positive attitudes were correlated with success and negative attitudes with failure. Philippou and Christou (1998) surveyed 162 first year prospective primary teachers in Greece. They found that “a considerable propor tion of prospective teachers bring to the university negative feelings toward mathematic s, a subject they will soon be supposed to teach” (p. 203). Many believe that in order to teach ma thematics well, one needs to have a positive attitude toward the subject, and that the task of improving the attitudes toward mathematics of future elementary teacher s begins at the university. Sherman and Christian (1999) said that improving the at titudes toward mathematics of preservice elementary teachers is “an important concern for university education courses in order to facilitate positive mathematics attitudes in future elementary pupils” (p. 96). Hungerford (1994) cited the need to improve the mathematic s education of future elementary teachers by altering curriculum and attitudes. He sugge sted that elementary school teachers who do not know much mathematics, who care little about what it means to do mathematics, and who are afraid of mathematics will be unlikely to foster positive attitudes toward

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6 mathematics in their own students. Thus the sy stem continues to move in a vicious cycle that must be broken, and teacher educators mu st carefully consider the impact of their training programs on the attitudes of prosp ective teachers (Philippou & Christou, 1998). Several studies have demonstrated success in improving attitudes toward mathematics of preservice elementary teacher s enrolled in mathematics methods courses. These methods courses utilized constructivist instructional methods such as the use of hands-on manipulatives, cooperative group wo rk, problem solving, and the use of technology. My own experience teaching elementary mathem atics methods courses has reinforced this notion. In addition, my inform al use of reflective journal writing with my students has led me to believe that perhap s this type of reflection can provide an additional tool for teacher educators who seek to understand how attitudes toward mathematics are formed and how to impr ove their students’ attitudes toward mathematics. Purpose of the Study The purpose of this study was to examin e the attitudes toward mathematics of preservice elementary school teachers ente ring an introductory mathematics methods course. The study focused on the following att itudes: value, enjoym ent, motivation, and self-confidence. Qualitative methods were used to explore thes e attitudes and the experiences that have led to the developm ent of these attitudes. The study sought to determine the extent to which preservice teach ers’ attitudes toward mathematics changed during the methods course. The study also examined the correlati on between preservice teachers’ initial attitudes toward mathema tics and their achievement in the methods course.

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7 Research Questions 1. What are the attitudes toward math ematics of preservice elementary school teachers entering an introductory mathematics methods course? In particular, how do preservice teachers sc ore on each of the four attitudinal components being measured: value of mathematics, enjoyment of mathematics, motivation for mathem atics, and self-confidence with mathematics? 2. To what extent do attitudes toward mathematics of preservice elementary school teachers change during the mathematics methods course? To what do preservice teachers whose attitudes toward mathematics were altered attribute this change? 3. What is the relationship between pres ervice elementary teachers’ initial attitudes toward mathematics and thei r grade on the methods course final examination? 4. What do preservice elementary schoo l teachers’ reflective journal entries reveal about their attitudes toward ma thematics and the experiences that have influenced the development of those attitudes? 5. What are the attitudes toward and ex periences with mathematics of those preservice elementary school teachers identified as having the most extreme (either positive or negative) attitudes?

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8 Definitions Attitude toward mathematics. This term will refer to “a general emotional disposition toward the school subject of mathematics” (Haladyna, Shaughnessy, & Shaughnessy, 1983, p. 20). Beliefs about mathematics. This term will refer to the ways in which an individual cognitively understands the natu re of mathematics, as well as “the ways in which a teacher understands classrooms, students, the na ture of learning, the teacher’s role in a classroom, and the goals of education” (Kagan, 1990, p. 423). Reflective journals. In reflective journal writing, students reflect on experiences and organize their thoughts and feelings in order to communicate clearly. Students are often given prompts that di rect their reflection.

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9 CHAPTER TWO REVIEW OF LITERATURE The literature review begins with a de scription of the affective domain and includes sections relating to be liefs about mathematics, attitudes toward mathematics, and the effects of attitudes and beliefs on achievement in mathematics. It also includes sections relating to journa l writing with mathematics, phenomenology, and reflection. The chapter concludes with a summary of these areas as they relate to the present study and a description of the signi ficance of the study. The Affective Domain McLeod (1992) identified three major co mponents of the affective domain in learning mathematics. First, students hold certain beliefs about mathematics. These beliefs can greatly influence a student’s a ffective reactions to learning mathematics. Second, as students repeatedly encounter ma thematical situations they will develop positive and negative attitudes toward mathematics. Third, when students are learning mathematics, interruptions and blockages ar e bound to occur. These can lead to both positive and negative emotional responses in students. Beliefs About Mathematics Although beliefs are mainly cogni tive in nature, they play an important role in the development of attitudes and emotions a bout mathematics. Students’ beliefs about

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10 mathematics can greatly influence the feeli ngs they have about learning mathematics. Schoenfeld (1992) pointed out that some co mmonly held beliefs can undermine students’ problem-solving performance. Some examples of such beliefs are that mathematics primarily involves memorization of rules and procedures, and that if a mathematical problem cannot be solved quickly, it cannot be solved at all. Schoenfeld provided the following compilation of some of the beliefs about mathematics that many students typically hold: Mathematics problems have one and only one right answer. There is only one correct way to solve any mathematical problem—usually the rule the teacher has most recently demonstrated to the class. Ordinary students cannot be expected to understand mathematics; they expect simply to memorize it and apply what they have learned mechanically and without understanding. Mathematics is a solitar y activity, done by indi viduals in isolation. Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less. The mathematics learned in school ha s nothing to do with the real world. Formal proof is irrelevant to proce sses of discovery or invention. (p. 359) Schoenfeld argued that students abstract th ese beliefs in large measure from their own classroom experiences. These beliefs are consistent with the traditional manner in which mathematics has been presented to stud ents. These beliefs can also shape students’ behavior “in ways that have extraordinarily powerful (and often negative) consequences” (Schoenfeld, 1992, p. 359). Such beliefs as thes e conflict with the goa ls of the current

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11 reform efforts (NCTM, 1989, 2000) and may be the cause of the negative affective reactions of many students to efforts to de velop mathematical problem-solving skills (McLeod, 1994). Research on Beliefs about Mathematics Studies have shown that teachers’ beliefs about mathematics teaching and learning are mostly formed dur ing their own schooling and ar e developed as a result of their own experiences as mathematics student s. Their conceptions about mathematics and how it should be taught are deeply rooted and are difficult to change (Thompson, 1992). In order to understand how these beliefs devel op over time, it is important to consider the beliefs of students at vari ous stages of schooling. Students begin to develop beliefs ab out mathematics at an early age. Kloosterman, Raymond, and Emenaker (1996) followed 29 students in grades one through four over a period of three years to determine the stability and developmental trends in their beliefs about mathematics. Th ey found that the students’ beliefs remained relatively stable over th e three years. They also found that students had a narrow view of the usefulness of mathematics, citing such trite responses as “you need math to get to third grade” (Kloosterman et al., p. 49). Thei r perspectives on group versus individual work reflected their own classroom expe riences with each. Those students whose teachers used cooperative learning tended to show approval for it and those whose teachers did not include group wo rk as a part of instruction tended to prefer working alone.

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12 Beliefs of middle school st udents have also been studied. Wilson (1995) surveyed 59 eighth-grade students in order to determin e their conceptions of what it means to do mathematics. She found that the students’ beliefs were predominantly traditional in nature. Even after 9 months of participating in a reform-oriente d class, the majority of the students still felt strongly about including “listening to the teacher explain” in their conception of school mathematics. By high school, students have developed co re beliefs. Fleener (1996) investigated high school students’ beliefs about mathematics during a four-week summer residential mathematics and science program. She pointed out that many of the core beliefs that students have about mathematics are a result of their personal experi ences in mathematics classrooms. The researcher found that the st udents had well-defined core beliefs about the nature and discovery of mathematical truths and the importance of engaging in mathematical inquiry. Their responses suggested that they did not view mathematics as a “dynamic, changing discipline” (Fleener, 1996, p. 316). Teacher Beliefs About Mathematics Teacher beliefs and their influence on t eacher behavior in the classroom have been researched and discussed extensively. Ka gan (1990) defined teacher beliefs as “the highly personal ways in which a teacher unders tands classrooms, stude nts, the nature of learning, the teacher’s role in a classroom, and the goa ls of education” (p. 423). Richardson (1996) listed three categories of experiences that influence the development of beliefs about teaching: (a) personal e xperience, (b) experien ce with schooling and instruction, and (c) experien ce with formal knowledge. In order to understand how a

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13 teacher’s belief system can significantly in fluence how that teacher interprets and implements curricula, it is important to dist inguish between teacher beliefs and teacher knowledge. Two teachers can have similar know ledge, but use very different teaching approaches based on their diffe ring beliefs (Ernest, 1989). Thompson (1992) cited some features of teacher beliefs that distinguish them from knowledge. One of these features is that beliefs, unlike knowledge, can be held with varying degrees of conviction. A nother distinction is that trut h or certainty is associated with knowledge while disputability is associat ed with beliefs. Thompson also pointed out that what we view as knowledge can actually be belief because it can change in light of new theories. Pajares (1992) also discussed some distin ctions between beliefs and knowledge. He pointed out that beliefs are deeply persona l and unaffected by persuasion. He described what he called the “perseve rance phenomenon,” which says that early experiences strongly influence beliefs (p. 317) They become highly resistant to change. The earlier a belief is incorporated into a pers on’s belief structure, the more difficult it is to change. Teachers’ beliefs about teaching and learning can influence their teaching practices (An, 2000; Fang, 1996; Kagan, 1992; Thompson, 1992). Stipek, Givvin, Salmon, and MacGyvers (2001) surveyed 21 f ourththrough sixth-gr ade teachers at the beginning and end of the school year in orde r to assess their beliefs about mathematics and teaching mathematics. The teachers were also videotaped teaching, and the observed behavior was coded to characterize each teacher’s classroom practices. They found substantial coherence between the teachers ’ beliefs and their classroom practices. Teachers who held traditional beliefs about mathematics emphasized performance and

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14 speed in their classrooms rather than emphasizing learning and understanding. These teachers also gave students less autonomy and created a classroom environment where mistakes were viewed as something to be avoided rather than creating an environment where there was no risk of being em barrassed if a mistake was made. The beliefs that teachers hold can also a ffect their students’ beliefs. G. Carter (1997) studied seven teachers and the re lationship between their beliefs about mathematics and the beliefs of their student s. She found that students of teachers with beliefs that were aligned with the NCTM Standards had significantly different beliefs about factors that lead to success than ot her students. These students believed that working hard and striving for understanding were essential for success. Often teachers’ existing belief systems conflict with the pe dagogical techniques and practices that they are being encourag ed by the profession to adopt. Mathematics teachers of today are being as ked to shift their mathematic s instruction away from the traditional teaching that they most likely received as students to a constructivist perspective of mathematics instruction. Di scussing traditional mathematics instruction, Van de Walle (2004) said: Traditional teaching, still the predominant instructional pattern, typically begins with an explanation of whatever idea is on the current page of the text followed by showing children how to do the assigne d exercises. Even with a hands-on activity, the traditional teacher is guiding st udents, telling them exactly how to use the materials in a prescribed manner. The focus of the lesson is primarily on getting answers. Students rely on the teach er to determine if their answers are correct. Children emerge from these experi ences with a view th at mathematics is

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15 a series of arbitrary rules, handed down by the teacher, w ho in turn got them from some very smart source (pp. 12-13). Anderson and Piazza (1996) cited several barriers to reform in mathematics education that inhibit the change process in teachers. They pointed out that many of the beliefs and attitudes of some teachers are “i n direct conflict with those inherent to constructivism” (p. 54). When discussing barriers to the reform movement in mathematics education, Ross, McDougall, and Hogaboam-Gray (2002) reviewed one hundred fifty-four empirical studies on reform in mathematics education that were published between 1993 and 2000. They found that “the most important obstacle [to reform] is that teachers’ beliefs and prior experiences of mathematics and mathematics teaching are not congruent with the assu mptions of the Standards” (p. 132). Ambrose (2001) suggested several avenues for changing belief systems. The first involves the process of reflection and exam ination of personal beliefs. In this way, inconsistencies can be identified. The second involves making connections among beliefs. This allows one to activate new be liefs in situations where they might not previously have been activated. Another way that belief systems can be changed is by developing a new belief that is connected to existing beliefs. Th e last belief change is the reversal of an existing belief. However, this type of paradi gm shift is rare. Wideen, Mayer-Smith, and Moon (1998) reviewed 93 empirical studies on learning to teach. Their review supported findi ngs of others that many traditional teacher education programs have little effect on firm ly held beliefs of preservice teachers. However, they did note some successful programs. These programs typically build upon the existing beliefs of beginning teachers rather than trying to cultivate a reversal of

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16 beliefs (Wideen et al., 1998). The authors expl ained that learning to teach was a “deeply personal activity” (p.161), and the first step involved having beginning teachers examine their existing beliefs. These pr ospective teachers should then be encouraged to consider how their existing belief system correlates with the expectations of the university and the teaching profession. Beliefs of preservice and beginning teachers. Research involving teacher training programs and their attempts to change the belie fs of their participan ts have shown little consensus. Although many of these programs have been successful in showing a significant change in beliefs, some have not Ambrose (2001) descri bed a case study that examined one such program aimed at ch anging preservice teac hers’ beliefs about mathematics and teaching by building on existing beliefs. The program allowed preservice elementary school teachers to work with individual children once a week for eight weeks. The emphasis of the sessions was on problem solving and using a concrete approach to develop conceptual understand ing. Donna, the case-st udy student teacher, believed that children had the ability to so lve problems on their own. She also believed that standard algorithms were the best way to solve problems and were the focal point of mathematics. She viewed effective teaching as involving explaining things to children and being nice to them. Data from field not es, surveys, intervie ws, and written work revealed a change in Donna’s belief about how a child develops self-confidence. Initially she had assumed that a nice, encouraging teac her would lead to a se lf-confident student. She altered her belief when she observed that her student became dependent on her nonspecific praise. She also deve loped the belief that childr en often need a variety of

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17 experiences with a topic in or der to learn it, and that they do not always learn what is taught. Stuart and Thurlow (2000) described another progra m that was designed to change the beliefs of preservice elementary school teachers. A methods course focused on how students’ beliefs about the nature of mathematics, themselves as learners, and teaching-learning processes would influence their decisions about classroom practices. Interviews, journals, mathematics autobiograp hies, exams, and class writings were all used to collect data about th e beliefs of these preservice teachers. The researchers found that students did successfully reevaluate and change their beliefs about teaching mathematics by the end of the semester course The authors reinforced the importance of reflection to allow the students an opportunity to bring their beliefs to a conscious level where they could be examined. Other preservice teaching programs have b een successful in changing the beliefs of their participants as well. Vacc and Bri ght (1999) followed 34 preservice elementary school teachers through their coursework and student teaching. During their mathematics methods course, the participants were introduced to Cognitively Guided Instruction (CGI), an instructional progr am where teachers make instructional decisions based on their knowledge from cognitive science (Car penter & Fennema, 1991, as cited in Vacc & Bright, 1999). Their training program incor porated professional development schools where the student teachers could connect th e theory they were being taught in the methods courses with actual classroom practices. They accomplished this through observation of teachers using CGI in the cl assroom and field experiences where they could put into practice the theo ries they had learned in th e methods course. They found

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18 significant changes in the stude nts’ beliefs and perceptions about teaching and learning mathematics after completing the mathema tics methods course and again after the student-teaching experience. Mewborn (2002) tracked an elementary school teacher over a four-year period, from her first mathematics methods course through her student te aching experience and first two years of teaching. This beginning teacher changed some of her beliefs about teaching and learning mathematics, as well as classroom practices refl ecting these beliefs, during the course of the four years. Mewborn attributed th ese changes to Dewey’s notion of the reflective thinking pr ocess (as cited in Mewborn, 2002). Reflective thought about herself and her teaching experiences th rough journals, a written autobiography, interviews, and informal conve rsations with the researcher allowed her to alter the structure of her belief system. Some teacher-training programs designed to alter the belief systems of its participants have found that prior beliefs were resistant to change (M cDaniel, 1991; Weinstein, 1990). Kagan (1992) reviewed 40 preservice teaching programs and found that preexisting beliefs and prior experiences played a central role in how preservice teachers interpreted the content of educati on courses. Each study that she examined demonstrated that these beliefs were stable and inflexible to change. She identified some essential elements for changing preservi ce teachers’ beliefs. According to Kagan, preservice teachers must have the opportunity to interact with and study students. She also said that university courses must focu s not only on theory, but also on practical strategies and pro cedural knowledge.

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19 Sometimes prospective teachers will appear to have altered their beliefs as a result of the training program, but their classroom practices may not reflect these changes. Frykholm (1996) found that even when preservice teachers’ beliefs appear to be aligned with the teachings of the t eacher-training program, their st udent teaching practices might not reflect these beliefs. His study followe d 44 preservice mathematics teachers over a period of two years. Based on data that we re collected through classroom observations, preand post-lesson conferences, lesson plans, seminar sessions, surveys, and informal conversations, the prospective teachers appeared to be proponents of the goals, content, and recommendations set out in the NCTM Standards documents (NCTM, 1989, 1991). However, the author found that “an overwhelm ing majority of the lessons observed bore little or no resemblance to the values so highly espoused by the student teachers” (p. 665). He suggested that the mathematics co mmunity must broaden, and sometimes even challenge, the belief system s of many present and future mathematics teachers. In similar studies, the same trend was observed. Raymond (1997) followed six firstand second-year elementary school te achers over a 10-month period. Data were gathered through interviews, observations, docu ment analysis, and a beliefs survey. The beginning teachers’ stated beliefs were not always consistent with their teaching practice. Their actual teaching practices were more traditional than their stated beliefs about mathematics teaching and learning. For example, one teacher in the study said that she believed that dividing the class into groups to work with manipulatives was the best way to introduce and deepen the students’ concep tual understanding. However, due to this teacher’s concerns about classroom management and time constraints, her actual teaching practices were more traditional and rarely included this type of activity.

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20 Benken and Wilson (1998) studied a preservice mathematics teacher through her methods courses and student teaching. They fo und that her view of mathematics learning as sequential and built upon a foundation kept he r from allowing her students to explore and investigate problems before teaching th em what she considered the basics. The authors viewed this teacher’s own conceptions about mathematics and mathematics teaching as her main obstacle to implem enting the teaching methods that she had expressed a desire to use w ith her students. These results support the findings of Tabachnick and Zeichner (1984) that preservi ce teachers brought their own perspectives to teaching, and the student teach ing experience tended to solidi fy rather than alter these perspectives. Beliefs of experienced teachers. Studies involving changing beliefs of experienced teachers have also shown conflicting results. Some have demonstrated a change in the teachers’ beliefs as a result of professional development programs (Pligge, Kent, and Spence, 2000; Simon and Schifter, 1993). V acc, Nesbitt, Bright, and Bowman (1998) examined changes in teachers’ beliefs ove r a period of two years of professional development for the program Cognitively Guid ed Instruction (CGI). Teachers attended several professional development workshops an d met on a monthly basis to discuss their progress in implementing CGI. In add ition, each teacher was observed teaching mathematics once a month by experienced CG I teachers. The researchers found that generally the teachers did change their beliefs about teachers’ views of children, the role of teacher and student, and skill acquisition and problem solving. However, the changes

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21 varied by category and grade level, and severa l teachers did not change their beliefs in these areas. Sometimes belief systems are so strong that desired changes in a teacher’s classroom practices are not po ssible. A case study that fo llowed a veteran middle school teacher over a period of two years showed that he was unable to make desired modifications in his teachi ng practices (Wilson and Goldenberg, 1998). The teacher was interviewed and observed regular ly. The researchers reported that his initial teaching style reflected a narrow view of mathematic s and mathematics teaching. Over the course of the study, he was able to increase his focu s on concepts rather than on procedures, but he continued to portray mathematics as a rigid subject to be mastered rather than as a way of thinking or a subject to be explored. The authors beli eved that this view of mathematics was the primary obstacle prev enting him from shifting from a teacherfocused classroom toward a student-centere d class environment where exploration was encouraged. Although the teacher felt that he was making tremendous strides in this direction, the researchers di sagreed. They concluded that his underlying epistemology had changed very little. Consistencies between teachers’ stated be liefs and their classroom practices may be related to their abi lities to reflect on their actions (Gellert, 1999; Lock and Lee, 2001). Thompson (1992) recognized the importance of the process of reflection in connecting beliefs to practice: It is by reflecting on their views and actio ns that teachers gain an awareness of their tacit assumptions, beliefs, and views, and how these relate to their practice.

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22 It is through reflection that teachers develop coherent rationales for their views, assumptions, and actions, and become awar e of viable alternatives. (p. 139) Attitudes Toward Mathematics Attitude toward mathematics is define d as “a general emotional disposition toward the school subject of mathema tics” (Haladyna, Shaughnessy, & Shaughnessy, 1983, p. 20). A positive attitude toward math ematics is generally valued because: 1. A positive attitude is an important school outcome in and of itself. 2. Attitude is often positively, although slightly, related to achievement. 3. A positive attitude toward mathematics may increase one’s tendency to elect mathematics courses in high school and co llege and possibly one’s tendency to elect careers in mathematics or mathematic s-related fields. (H aladyna et al., p.20) McLeod (1992) cited two different ways in which attitudes toward mathematics appear to develop. Attitudes may result from the automatizing of a repeated emotional reaction to mathematics. For example, if a st udent has repeated nega tive experiences with a particular area of mathematics, the intensity of the emotional impact will usually lessen over time. Eventually the reaction will become more automatic and stable and can then be measured by a survey or questionnai re. The second source of attitudes is the assignment of an already existing attitude to a new but related task. A student who has a negative attitude toward one particular area in mathematics may attach the same attitude to a related concept.

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23 Teacher Attitudes Toward Mathematics Although teacher beliefs a nd their role in learning to teach and classroom practices have been studied ex tensively, this has not been th e case with teacher attitudes. The growing interest in cogni tive psychology within the fi eld of education has drawn interest away from attitudes and toward be liefs (Richardson, 1996). However, there have been some studies that have examined t eachers’ attitudes towa rd mathematics. Attitudes toward mathematics of preservice elementary teachers. Research involving teacher attitudes toward mathem atics has focused largely on preservice elementary teachers. Rec h, Hartzell, and Stephens (1993) found that preservice elementary teachers have less favorable atti tudes toward mathematics than the general university population. Cornell (1999) surveyed graduate students who were taking a mathematics instruction seminar for certificat ion as elementary school teachers. He found that about half of the students reported a di slike of mathematics. In nearly all cases, positive attitudes were correlated with su ccess and negative attitudes with failure. Philippou and Christou (1998) studied 162 pros pective primary teachers in Greece and found that “a considerable proportion” of them expressed negative feelings toward mathematics (p. 203). Some studies have looked at teacher training programs designed to improve attitudes toward mathematics of preservice elementary school teachers. In one such study, McGinnis, Kramer, and Watanabe (1998) collected data from 1995 to 1997, as the participants completed a teacher preparat ion program. The Maryland Collaborative for Teacher Preparation (MCTP) is an undergradu ate program for specialist mathematics and

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24 science elementary/middle level teachers. Th e program’s goal is to promote development of confident teachers who can teach mathematics and science using technology, who can make connections between the disciplines, and who can challenge diverse learners by creating an exciting learning environment fo r them. The program focused on developing students’ understanding of key concepts and making connections between mathematics and science. The authors described the program as: compatible with the constructivist perspe ctive (i.e., address conceptual change, promote reflection on changes in thinki ng, and stress logic and fundamental principles as opposed to memoriza tion of unrelated facts). (p. 4) The researchers found that the students’ att itudes about mathematics and science were affected in the desired direction as they progressed through the pr eparation program. The authors pointed out that thei r students were somewhat distinctive because they had expressed an interest in teaching mathem atics and/or science by making connections between the two. Gibson and Van Strat (2001) also conduc ted a longitudinal study that tracked preservice teachers’ attitudes toward teach ing and learning mathematics and science while enrolled in a teacher preparation progr am that utilized constr uctivist instructional methods. The Urban Preservice Degree Arti culation in Teacher Education (UPDATE) program’s goal was to “provide a pathway fo r urban para-educators of color to become certified teachers” (Gibson & Van Strat, 2001, p. 2). Thirteen of the fourteen preservice teachers who participated planned to teach elementary school. The students completed two mathematics content courses and one me thods course. All three were taught using instructional strategies such as collaborat ive group work, problem solving, manipulatives,

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25 and calculators. Their results showed a si gnificant positive change in the preservice teachers’ attitudes. Philippou and Christou (1998) conducted a study in Greece that involved a teacher preparation program whose goal was to help preservice primary teachers acquire mathematical concepts and teaching methods while improving thei r self-confidence in doing mathematics. The program consisted of two mathematics cont ent courses and one methods course. In these courses, students were provided opportunities to experience success with mathematics. There was a focus on mathematics as a “constantly changing creation of human activity” (p. 193). This wa s accomplished by including the historical development of basic concepts and allo wing students to develop conceptual understanding through discussions and hands-on activities. Using a pre-test, post-test design, students were given instruments to measure their attitudes toward mathematics prior to beginning the program, after the firs t course, and after completing the entire program. They found significant differences in attitude at the conclusion of the program, indicating significantly more positive attitudes toward ma thematics. In addition, the prospective teachers participated in 45-minut e interviews where their own evaluations of their feelings prior to the program and of the effectiveness of the program relating to attitudes were given. Quinn (1997) examined the effects of an elementary mathematics methods course that stressed the use of manipulatives, t echnology, and cooperati ve learning in the teaching of mathematics on the attitudes of preservice teachers. He found that the preservice elementary teachers’ attitudes improved significantly after completing the methods course.

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26 In a similar study by Anderson and Piazza (199 6), preservice elementary teachers, as part of their teacher preparation program, were enrolled in a mathematics content course that focused on an inquiry appr oach utilizing group problem solving and manipulatives. Each of the 48 prospective t eachers wrote an essay about their learning experience in the course, and common themes were identified from the essays. Twentyone of the students said that they felt less anxiety about learning/teaching mathematics as a result of the course. Ten of th e students said that they felt a greater sense of confidence. Attitudes toward mathematics of preservice secondary teachers. Much of the research involving teacher attitudes has focused on the elementary school level. However, Camacho, Socas, and Hernandez (1998) surveyed prospective secondary mathematics teachers in Spain about their belief s and attitudes. They found that only 50% of the preservice teachers expressed enj oyment in doing mathematical work. The researchers felt that these results “put into doubt an ability to genera te a positive attitude towards mathematics in the classroom” (p. 323). Wagner, Lee, and Ozgun-Koca (1999) also studied beliefs and attitudes of preservice secondary teachers. They surveyed participants in the United States, Turkey, and Korea about their attitudes toward math ematics, teaching mathematics, and their teaching program. They reported that student t eachers in the United States had more selfconfidence than those from the other countries Teachers from the U.S. also supported the use of manipulatives and tec hnology while those from the other countries did not. The authors pointed out that thei r findings supported the assump tion that student teachers’ beliefs and attitudes were affected by their experiences. Teachers from Turkey and Korea

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27 did not have experience with these materi als as the American teachers did. These researchers recommended that preservice t eachers be encouraged to reflect on their experiences as students and as prospective teachers. Concepts related to teacher attitude. Other areas in the affective domain have been studied that are related to teacher attitude and have implications for mathematics education. Most of the research that has been done in th is area has examined various combinations of these concepts. One such con cept is confidence in learning or teaching mathematics. Self-concept can be thought of as a generalization of confidence in learning mathematics. A variation of self-concept is the notion of self-efficacy, which represents a person’s “beliefs concerning his or her abilit y to successfully perf orm a given task or behavior” (Hackett & Betz, 1989). Mathematics self-efficacy can be distinguished from other attitudes in that mathematics self-efficacy is a problem-specific assessment of one’s confidence in his or her ability to accomplish a particular task. Personal teaching efficacy has been defined as a belief in one’s ability to teach effectively (Huinker and Madison, 1997). Wenner (2001) reported the results of th ree studies concerning teaching efficacy that were conducted over a five-year period. He examined the efficacy beliefs about mathematics and science of prospective and practicing teachers and found that experience had a positive effect on personal teaching effi cacy. Practicing teachers scored higher than preservice teachers on knowing how to support mathematics skill and concept development, effectiveness in monitoring use of manipulatives, ability to teach

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28 mathematics effectively, and understanding of concepts well enough to be effective in teaching. In a similar study, Huinker and Madison (1997) considered the effect of a methods course in mathematics and scie nce on the personal te aching efficacy of preservice elementary school teachers. They based their study on th e premise that the more positive the impact of the methods cour se on the personal teaching efficacy of the preservice teacher, the more likely that pers on is to become an effective teacher. The mathematics methods course was based on a constructivist philosophy and included use of manipulatives, collaborati ve group work, and investig ation and discussion of challenging problems. They found that the methods course enhanced the personal teaching efficacy of the participants. The pres ervice teachers’ confidence in their abilities to teach mathematics effectively had increased. In a study that inves tigated the effect of mathematics attitudes on preservice elementary teachers’ global self-concept, Sherman and Christian (1999) surveyed 88 preservice teachers at both the beginning and the end of a semester-long methods course. Like the previous studies, the methods c ourse focused on the use of manipulatives, problem solving, mathematical discussi ons, and cooperative learning to understand mathematics teaching methods. The student s were given questionnaires for both mathematics attitudes and global self-concep t. There was a significant improvement in the teachers’ mathematics att itudes. However, there was no significant difference in their global self-concepts. The authors conclude d that global self-concept is not easily impacted or altered. They suggested that global self-concept does not change much as

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29 individuals age, and that future studies mi ght investigate global self-confidence of specific age group populations. Connecting Teacher Beliefs and Attitudes about Mathematics Stipek, Givvin, Salmon, and MacGyvers (2001) surveyed 21 fourththrough sixth-grade teachers to assess their self-conf idence and enjoyment of mathematics and mathematics teaching and to see if these attitudes were relate d to the teachers’ beliefs. They found that teachers who held more tr aditional beliefs about mathematics and learning had lower self-confidence and enj oyed mathematics less than teachers who viewed mathematics instruction as inquiry-b ased. These authors speculated that less confident teachers are drawn to beliefs and practices that allow them to teach in a prescribed way, following the proce dures spelled out in the textbook. Philippou and Christou (1998) also discu ssed the correlation between teacher beliefs and attitudes toward mathematic s. They pointed out that because many prospective elementary school teachers have ne gative attitudes toward mathematics, they are likely to view and teach mathematics in a more traditional manner. According to Philippou and Christou, these teach ers’ traditional teaching methods are then likely to promote the development of negative attit udes in their own students. The Effects of Attitudes and Beliefs on Achievement in Mathematics There has been little consensus in th e research literature concerning the relationship between attitudes toward mathem atics and achievement in mathematics. Ma and Kishor (1997) conducted a meta-analy sis of 113 studies that examined the

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30 relationship between attitudes toward mathem atics and achievement in mathematics. In reviewing the literature, they reported that although some researchers have found the correlation between the two to be quite low, ranging from zero to 0.20, others have found statistically significant corre lations ranging from 0.20 to 0.40. Still others have found quite strong correlations above 0.40 (p. 27). In their meta-analysis, Ma and Kishor used 113 studies that examined the relationship between attitude toward mathematics (ATM) and achievement in mathematics (AIM). In total, 82,941 students in 12 grade levels participated in the studies. Most samples were mixed for gender and ethnicity. The smallest sample size was 10 and the largest was 23,132. A study was consid ered to be appropriate for this metaanalysis if it: 1. had a definition of ATM similar to the one used in this study. 2. investigated the relations hip between ATM and AIM. 3. measured ATM and AIM using psychometrically-developed instruments. 4. did not include any experimental interven tions on either attitude or achievement. 5. contained students at the elementa ry and/or secondary school level. 6. reported quantitative data in sufficient de tail for calculation of an effect size. (p. 30) Because each of the 113 studies in the meta-analysis used different instruments to measure achievement, the researchers made sure that “the vast majority” of the AIM instruments were developed psychometrically. The authors noted that this was necessary in order “to ascertain that the reliability and validity of the instruments could be justified” (p. 32).

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31 Researchers found that the effect of ATM on AIM was not strong. The ATMAIM relationship was very similar for both males and females. This relationship between attitudes and achievement was weakest at the elementary school level. Students in grades 1-4 showed a Pearson product-moment r = 0.03, and students in grades 5-6 showed a correlation of r = 0.14. However, the relationship strengthened even more as students reached secondary school. Secondary students showed a correlation of r = 0.26. The authors suggested that future measures of attitudes should involve attitudes toward specific mathematical topics or activities rath er than toward mathematics in general. They also suggested that researchers examine both direct and indirect e ffects when studying the causal ATM-AIM relationship. Large-scale national and inte rnational studies have also examined the relationship between attitudes and achievement. Simich-D udgeon (1996) investigat ed the relationship between mathematics attitudes of Hispanic and Asian students in the 1992 National Assessment of Educational Progress (NAEP) Mathematics Trial State Assessment, by gender and ethnicity, and by their mathematics performance scores. They also examined attitude variables and how well these variab les predicted Hispanic and Asian students’ mathematics achievement. In discussing thes e NAEP results, the author cited various research projects that have studied the re lationship between certain demographic and background variables and mathematics achievement: Confidence in learning mathematics has a positive correlation with mathematics achievement, and gender differences in conf idence levels are usually associated with gender differences in mathematic s achievement (Reyes & Stanic, 1988)

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32 Male students attributed thei r success in mathematics to ability more frequently than female students; female students at tributed their success to effort more often than male students. Female students were more likely than male students to associate failure in mathematics to a lack of ability and to the difficulty of the task (Wolleat, Pedro, Becker, and Fennema, 1980). The perceived usefulness of mathematic s was identified as one of the most important variables in un derstanding sex-related di fferences in mathematics achievement and as an important predic tor of student sel ection of optional mathematics courses (Fennema & Sherman, 1978). Self-efficacy has been a reliable predic tor of whether a student will attempt a task, and the amount of effort and pers everance that he or she will put forth (Radhawa, Beamer, & Lundberg, 1993). Mathematics performance and self-effi cacy measures are significantly and positively correlated with attitudes toward mathematics, and self-efficacy is a stronger predictor of the choice of a mathematics-related major than mathematics achievement variables (Radhawa et al., 1993). Attitudes toward mathematics may be related to (a) majority and minority status within a culture, (b) ethn icity, and (c) a combination of gender and ethnicity (Iben, 1991). For this study, records of 32,009 fourth-and eighth-grade Hispanic and Asian students from the 1992 NAEP Trial State Asse ssment data set were used. Descriptive statistics were used to generate cross tabul ations of students’ b ackground characteristics, including their attitudes toward mathematics and their average mathematics performance.

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33 The students were asked to agree or disagr ee with eight statem ents regarding their attitudes about mathematics. These included such statements as: I like mathematics, I am good at mathematics, and I understand mo st of the mathematics class. Results indicated that most of the attitude variables were signif icant predictors of Hispanic and Asian students’ mathematics achievement. More Asian students than Hispanic students believed that they were “good at math” and “understand math.” These two attitudes were correlated with higher ma thematics achievement at both grade levels. However, results indicated that “like ma th” was not a signif icant predictor of mathematics achievement. The author suggest ed that this finding may mean that a student’s affinity for mathematics does not ne cessarily reflect his or her self-assessment of mathematics ability or judgment of understanding of mathematics instruction. Three attitude variables represented dime nsions of mathematics usefulness: math is mostly memorizing facts, math is used in jobs, and math is for solving problems. Agreement with the statement “math is mo stly memorizing facts” was a significant predictor of low mathematics achievement for both male and female Hispanic and Asian students. Agreement with the statement “math is used in jobs” was a significant predictor of higher mathematics achievement in Hisp anic and Asian students in both gender groups. However, agreement with “math is for solving problems” was a significant predictor of achievement for female and male Hispanic students but not for Asian students of either gender group. Braswell et al. (2001) inves tigated this relationship be tween students’ attitudes toward mathematics and their mathematics achievement on the 2000 NAEP assessment. They reported that students at all three grade levels that we re tested, grades 4, 8, and 12,

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34 demonstrated a positive relationship between performance on the NAEP and attitudes. Those who agreed with the statements, “I li ke mathematics” and “Math is useful for solving problems,” had higher average achieveme nt scores than those who disagreed with these statements. Tocci and Engelhard (1991) investigated the relations hip of attitudes toward mathematics and mathematics achievement us ing data from the Second International Mathematics Study (SIMS) for 3,846 eighth-grad e students from the United States and 3,528 eighth-grade students from Th ailand. One of the attitude dimensions investigated in this study was Mathematics and Myself This was designed to a ssess the extent to which students enjoyed studying mathematics, felt conf ident in their abilitie s to do mathematics, and wanted to achieve in mathematic s. Another attitude dimension was Mathematics and Society. This assessed students’ views about the usefulness and importance of mathematics to society. Researchers also measured Mathematics Anxiety or the extent to which students were anxious about mathematic s. The researchers found that achievement had a statistically signifi cant correlation with both Myself and Society They also found that achievement had an inverse relationship with Anxiety, indicating that a high level of mathematics anxiety was correlated with low mathematics achievement. In 1995, the Third Internati onal Mathematics and Scie nce Study (TIMSS) found a positive relationship between attitudes and achievement in mathematics. Students who performed well in mathematics generally had positive attitudes toward the subject and those with positive attitudes tended to have higher achievement scores (Papanastasiou, 2000a, 2000b). Students in grades 3 and 4 showed a high correlation between mathematics achievement and their perception of the role of luck, talent, and having good

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35 memory, as well as their perception of the ro le of hard work and memorization in being successful with mathematics (Gadalla, 1999). Self-confidence was a st rong predictor of mathematics achievement for girls, but not fo r boys. In grades 7 and 8, attitude explained more of the variation in achievement than it did for the younger students (Gadalla, 1999). The factors that related most to achievement were self-efficacy, educational aspirations, and external attributions of su ccess (Papanastasiou, 2000b). High achievement is not always associated with positive attitudes toward mathematics. Leung (2002) also examined data from the Third Intern ational Mathematics and Science Study (TIMSS). He found that st udents in grades 4 and 8 from the East Asian countries of Hong Kong,1 Japan, Korea, and Singapor e outperformed students from other countries in mathematics achievement However, students from these countries expressed relatively negative attitudes towa rd mathematics. Although over 80% of eighth grade students from Singapore reported that they liked mathematics, students from the other three East Asian countri es disagreed. Japan and Korea were among those countries whose students expressed the greatest dislike of mathematics. Less than 55% of Japanese eighth-grade students and less than 60% of Korean eighth-grade st udents reported liking mathematics. Students from these four countries were among the seven countries (the other three were Latvia, Lithuania, and Portugal) in which over 40% of students reported that they did not think they did well in mathematics. This was unexpected given the high achievement levels of these countries. Leung point ed out that this could be a result of the 1 Note: Hong Kong is currently part of China, but it was considered a separate country at the time the TIMSS study took place.

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36 common culture that these countries share. These cultures stress the importance of humility or modesty. Children are taught not to be boastful, so they may be hesitant to state that they do well in mathematics even if they believe that they do. Leung also said that if students in thes e cultures are consistently taught to rate themselves low, they might internalize the idea and actually have low self-confidence. He also pointed out that the school systems in these countries featur ed competitive examinations and high expectations, which left many students who we re viewed as failures by the system and possibly by themselves. There continues to be a lack of consen sus concerning the link between attitudes and achievement. Many studies have demons trated a correlation between the two. Although some research suggests that attitu de and achievement are not dependent on each other, they do interact with each other in “compl ex and unpredictable ways” (McLeod, 1992, p. 582). McLeod says that as research methodology becomes more flexible and qualitative methods such as in terviews become more widely used, “we can expect research on attitude to make new contributions to the field of mathematics education” (p. 582). Journal Writing and Mathematics The National Council of T eachers of Mathematics (NCTM) has emphasized the importance of communication as an essentia l part of mathematics and mathematics education. “Through communication, ideas beco me objects of reflection, refinement, discussion, and amendment” (NCTM, 2000, p. 60). Writing is an excellent way for students to communicate about mathematics.

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37 The assertion that writing can contribute to learning depends essentially on a Vygotskyan view of the relationship between language and thought, where both language and thought are transformed in the act of representation (Borasi & Rose, 1989). Writing contributes to the learning process because it actively engages st udents in structuring meaning; students can go at their own pace; and it “provides unique feedback, since writers can immediately read th e product of their own thinking on paper” (Emig, 1977, as cited by Borasi & Rose, 1989, p. 348). Writing to learn focuses on learning and writing as processes that involve the learner in actively building connections betw een what is being learned and what is already known. Not all writing activities may qualify as writing to learn. Journal writing is one particular form of writing to learn where students keep a l og or personal notebook. Unlike diaries, journals focus on particul ar academic subjects and go beyond an account of the day’s events. In journa l writing, students reflect on ex periences and organize their thoughts in order to communicate clearly. Students are often invited to reflect on their learning by expressing their t houghts and feelings about what they are learning. The teacher is expected to read the journal en tries and respond to them in a supportive and non-evaluative way. The first six standards in the Professional Standards fo r Teaching Mathematics (NCTM, 1991) involve the process of teachi ng mathematics. Stewart and Chance (1995) described how these standards can be connected to journal writing: Standard 1: Worthwhile Mathematical Tasks: This standard promotes teaching that is based on students’ understanding, interests, experience, and learning styles. Entries from students’ journals give such information.

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38 Standard 2: The Teacher’s Role in Discourse : “The objectives of this standard have connections to journal writing that include listening to students’ ideas, asking students to express their ideas in writing, and deci ding when and how to react and respond to students’ ideas and comments” (p. 94). Standard 3: The Student’s Role in Disc ourse: Journal writing provides students the opportunity to “respond to and question the teacher, to make connections, and to communicate mathematically and otherwise” (p. 94). Standard 4: Tools For Enhancing Discourse : Journal writing can enhance discourse by supplying a setting for using the tool s of discourse. These include pictures, diagrams, analogies, stories, written h ypotheses, explanations, and arguments. Standard 5: The Learning Environment: Jo urnal writing can have a large impact on the classroom environment. Students are of ten asked to write ideas for improving the teaching and learning of mathematics. The implementation of their ideas can make the classroom “more student centered, more collaborativ e, and more supportive of students’ learning” (p. 94). Standard 6: Analysis of Teaching and L earning: This involves interconnected and ongoing assessments for teachers who must ad just and alter their instruction when necessary. One goal of this standard is to observe, listen to, and gather information about students. Journals certainly provid e an opportunity fo r all of these. Countryman (1992) listed some purposes of using journals in mathematics classrooms. These included the following: “to increase confidence, to increase participation, to decentrali ze authority, to encourage inde pendence, to replace quizzes and tests as a means of assessment, to monitor progress, to enhance communication

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39 between teacher and student, to record grow th” (p. 42). She also suggested that when using journals with mathematics students, a teacher should expect language that is “informal, conversational, personal, and contextual” and “questions, observations, doubts, digressions, examples, dr awings, sketches” (p. 43). Journal-writing prompts can be used to help students focus their thoughts in a particular direction. Dougherty (1996) identified three types of journal-writing prompts: (a) mathematical content prompts, which focus on mathematical topics and their relationships; (b) process prompts, which offe r students an opportunity to reflect on their preferred solution strategy and to consid er ways in which they learn; and (c) affective/attitudinal prompts. This type of pr ompt asks the students to write about past experiences and how these experiences have aff ected their attitudes about mathematics. Journal Writing with Elementary School Mathematics Students Journal writing can be adapted to any gr ade level. Even kindergartners can benefit from this type of activity. Fuqua (1998) want ed her kindergarten students to be able to record how they solved a problem and shar e their results with others. Instead of individual journals, she used a big book that served as a class mathematics journal. She decided to call it a ‘Problem-solving Book’ ra ther than a ‘math journal’ because she wanted the focus to be on logico-mathematical thinking. In her article, Fuqua described some of the activities that we re recorded in the class journal. The first entr y occurred when one of the ch ildren brought Halloween rings to school. There were three types of rings: ba ts, spiders, and pumpkins. After the rings

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40 had been distributed, one of the children w ondered if the ring types had been evenly distributed. He asked, “Who has the most—the bats or the sp iders or the pumpkins?” (p. 74). When the teacher asked how this c ould be determined, another child suggested they count and make a graph. Th ey divided the paper into th ree columns. At the top of each column was a picture of one of the ri ngs. Each person wrote his/her name in the column that represented his or her ring. They then counted the total in each column and found that there were five names for each kind of ring. Another day the class was discussing where to go for recess. The teacher told them that they would all need to go to the same place. Two of the children took a piece of paper and wrote at the top, “Do you wot to go on the his or the lidrpagon” (Do you want to go on the hills or the little playground?) (p. 75). Each child signed unde r the heading of his or her choice. Fuqua pointed out that these activities il lustrated the children’s problem-solving abilities. She said that this process made them think about and use various symbols, including letters, words, and drawings, to represent their thoughts. They were also actively involved in reasoning, comparing, counting, and other ma thematical concepts. Scott, Williams, and Hyslip (1992) desc ribed their experiences with journal writing in second grade classrooms. To maxi mize the experience, the students were led through an instructional sequence: motivationa l experiences, verbal interaction, class and individual journal writin g, sharing, and responding. Before each journal entry, teachers presented multisensory mathematics activities that were unusual, unexpected, or used unfamiliar materials. These activities stimulated thinking and lively discussions. The activit ies might involve partners, small groups, furniture rearrangement, special guests, music, color, or a variety of manipulatives. Each

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41 of these activities was followed by an oral language experience. The class might discuss how they would explain their mathematics ac tivity to an absent classmate. Another method involved a soft ball. The teacher woul d begin a sentence such as, “Today we used tangrams and my square was . ..” She would then toss the ball randomly to students. The student who caught the ball had to finish the sentence and begin another one. He or she then tossed the ball again. The class next moved to written comm unication. The teacher wrote students’ comments on the chalkboard or chart paper. She then wrote the ideas in complete sentences in an oversized jour nal to model the writing proces s. This was done as a large group activity until the students became fam iliar with the process of writing about mathematics. Then individual journals re placed the class journal. Some children expanded phrases from the experience chart while others expressed their ideas with pictures and diagrams. Student s could volunteer to share their journal entries with the class, although reading them aloud was ne ver required. The authors reported that “students of all ability levels found their work and progress va lued by peers, teachers, and others” (p. 17). The teachers read the journals and res ponded to them on a weekly basis. In responding to journal entries, teachers modeled correct spelling, punctuation, and grammar although the journal entries were neve r corrected in any way. The desired focus was on thinking and process rather than the finished product. Res ponses always closed with a question encouraging further thought. Th e authors reported that journal entries reflected improved attitudes to ward mathematics as well as growth in mathematical thinking and use of mathematical language. St udents’ self-esteem grew as well. Journal

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42 entries included statements such as, “I thi nk I can be a math teacher,” and “I am smart now” (Scott et al., 1992, p. 18). Journal Writing with Intermediate a nd Middle School Mathematics Students Evans (1984) conducted a study involving tw o fifth-grade classes in the same school. Both classes were taught using th e same procedures and textbook. The only difference in approach was that one of the teachers incorporated the use of journal writing during mathematics class. Three types of pr ompts were used. The first involved asking the students to describe how to do something. Rather than writing to the teacher, students wrote to an uninformed third party. This required them to be more specific in their descriptions. The second type of writing they did was definitions. Describing something new in their own familiar terms gave student s a chance to gain understanding of the new concept. The third type of writing was “troub leshooting.” Students were asked to explain their errors on homework or quizzes. Not onl y did this allow stude nts to analyze their own mistakes, but it also helped the teacher identify students who did not know what they had done wrong. This allowed for immediate reteaching of the concept. The study looked at two areas of the ma thematics curriculum, multiplication and geometry. Evans gave pretests and posttests to both classes. Her results are found in Table 1. In both units, the cont rol group scored higher than the test group on the pretest. However, the test group scored higher than the control group on both posttests. This was especially true on the geomet ry unit. In looking at indivi dual scores, a pattern emerged. Students with the lowest pr etest scores in the test group made the most gains.

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43 Table 1 Results from Evans’ Study with Journal Writing (in percent) Multiplication Geometry Group n Pre Post Pre Post Test 22 41 77 17 70 Control 23 54 76 24 60 Note. From “Writing to Learn in Math,” by C.S. Evans, 1984, Language Arts, 61, p. 834. Gordon and Macinnis (1993) described the use of dialogue jour nal writing with students in grades four, five and six. Dialogue journals ar e distinguished from other forms of journal writing because of the im portance given to the communication between student and teacher. Dialogue journals are wr itten conversations between a student and a teacher. They are intended to provide student s with an opportunity to share privately in writing their reactions, questions, and con cerns about school experiences with the teacher. Two types of writing were used in the journals: prompted writing, where the teacher poses questions and free writing or open-ended writing. Prompts were usually centered on the students’ understa nding of decimals. For example, students were asked to respond to such questions as: “What do you think decimals are?” “Which is larger, 0.5 or 0.42, and why?” Open-ended entries allowed students to write on anything of interest or concern to them in mathematics. For one year, the 180 students from seven classes in fourth, fifth, and sixth grade wrote entries in their mathematics journals. The students were allowed 10-15 minutes to write in the journals during the mathematics class. The teacher read the journals and

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44 returned them by the next week. Approximate ly 25 journals were handed in daily. The authors reported that the students appeared to enjoy the writing. Th ey displayed a “keen interest in the teacher’s response” (p. 39). The authors reported the following observations based on what the student s told them in their journals: 1. Intermediate grade-level students have some conceptual knowledge, but generally speaking, acquire better understand ing of the decimal system by sixth grade. 2. Students were able to assess what th ey viewed as their own strengths and weaknesses. They wanted to be heard, as they related comments about what might help them learn better. 3. Students had sufficient awareness of them selves as learners to indicate what worked or did not work for them. 4. Personal feelings were readily expresse d as trusting relationships were built in the journal communication. 5. Students shared their difficulties and problems with decimal learning because the journal was not seen as an evaluative tool. 6. Students shared their discoveries an d their insights in their journals. The teacher’s responses to the journal entries consisted of comments, questions, and notes of encouragement. No attempt was made to teach mathematical concepts through the journal responses, but misconcep tions and concerns were acknowledged. The students were assured that the problem woul d be addressed. The journals allowed the teachers “the opportunity to reflect on the teaching/learning process and to better meet the instructional needs of each of the students” (p. 42).

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45 DiPillo (1997) designed a project that used journal writing with fifth and sixth grade students. Over an eight-week period, tw enty-six fifth graders and twenty-eight sixth graders wrote in their jour nals for five to eight minutes, three or four times a week, in response to specific prompts. Teachers responded to each journal entry by offering written comments, questions, or encouragem ent. Teachers also wrote in their own journals, describing classroom activities a nd reflecting on how information from the students’ journals had influen ced their instruction. Four cate gories of prompts were used: (a) instructional prompts, such as “What is a fraction?” (b) contextual prompts, such as “What is the hardest thing you learned this w eek?” (c) reflective prom pts, such as “How has writing in your journal helped you in math ?” and (d) miscellaneous prompts, such as “Discuss the school subject that is most like mathematics.” In her article, DiPillo describe d some of the students’ re sponses to the prompts. She found that fifth graders tend ed to make their responses mo re concrete than the sixth graders did by using diagrams. Fifth graders were also more likely to respond by using personal feelings and giving examples. Sixth graders gave more textbook-type definitions, procedures, and explanations. Ov erall, students expressed positive attitudes about mathematics. Many, however, voiced concern with grades. These students frequently indicated negative attitudes toward mathematics. DiPillo reported that the vast majority of the students expressed enj oyment about mathematics journals. Their responses “revealed five differe nt categories of benefits: (a) opportunities to express their feelings, (b) bette r retention of information, (c) incr eased understanding of mathematics, (d) stimulation of thinking about mathematics, and (e) improved writi ng ability” (p. 311).

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46 Jurdak and Zein (1998) conducted a teachi ng experiment to measure the effect of journal writing on achievement in and attitude s toward mathematics. The participants in the study were students between the ages of 11 and 13 who were attending an international school in Beirut, Lebanon. Ma thematics instruction in this school was conducted either in English or French. The journal-writing group received the same instruction as the no-journalwriting group. However, the jour nal-writing group spent 7 to 10 minutes at the end of each class period, th ree times a week for 12 weeks, engaged in prompted journal writing. Results showed that the journal writing had a positive impact on conceptual understanding, procedural knowledge, and mathematical communication, but not on problem solving, school mathematics achieveme nt, and attitudes toward mathematics. The authors suggested that the failure of journal writing to improve school mathematics achievement could be due to the fact that school tests normally measure instructionspecific content rather than general ab ilities such as procedural knowledge and conceptual understanding. They also pointed out that the failure of journal writing to improve attitudes toward mathematics was in sharp contrast with the positive attitudes that the students expressed toward journal wr iting itself. The author s noted that previous studies reporting positive effects of writing on attitudes toward mathematics were conducted at high school or college levels. Journal Writing with High School Mathematics Students Waywood (1994) described work he had done from 1985 to 1990 developing a pedagogical model of journal wr iting in mathematics at a s econdary school for girls. He

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47 explored the theme of student’s questioni ng in order to show a connection between mathematics learning and writing. By 1990 journa l writing had been integrated into the teaching and assessment of the school’s mathematics program for grades 7 through 11. The pedagogical model for journal writing at the school consisted of the following elements: 1. A clear task specification: The journal was to be kept in a separate book. An entry was completed after each mathematics lesson. Each entry was to summarize the day’s lesson, which in cluded discussing the lesson, asking questions, and showing examples. 2. A developmentally structured feedback sheet: This feedback sheet served to connect the task specification with assessment. On it were progress descriptors for each of the required components of the entry. The teacher would circle those criteria that the student had met. 3. A set of assessment descriptors: Th ese were specific to grade levels. Essentially, this was a rubric used fo r assigning letter grades to the journal writing. The journal component counted for 30% of the entire grade for mathematics. The purpose of the assessment descriptors was to maintain consistency with grading in the school. 4. The final step in this process was the inclusion of Journals on the report cards that went home to parents. Waywood examined the questions that st udents in one tenth-grade class had posed in their journals. He wanted to an swer the question: Do questions posed by students in journals contribute to a profile of a student’s learning in mathematics? In

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48 order to answer this question, he presented fi ve short case studies of students’ questioning in journals. Four of the students studied we re in the same tenth-grade class. These students were chosen because th eir journals represented the ra nge of grades given to the journal component of the assessment. The other case examined i nvolved two journals from the same student, one written in grade 9 and the other in grade 11. Waywood (1994) stated that “questioning is clearly an activity to be prized in students of mathematics” (p. 325). He iden tified the following three beliefs about questions that were reflecte d in the progress descriptors used on the feedback sheet: 1. Questions are tools. They can be used to focus and direct thinking as well as focusing and directing learning. 2. The more specific the question the better. 3. There is a distinctly mathematical questioning. Waywood also looked at how grades assigned to journals compared with other assessment tasks in discriminating between and ranking students. In addition to the journals, he included problem solving, proj ects, and skills. Grades for these other components were produced by three teach ers working independently. He found no evidence to suggest that the gr ading of journals was in any way distinguishable from the grading of the other tasks. Waywood’s conclusion was that journal writing did discri minate between students at a particular level. He found that through years of journal use, it was possible to recognize changes in successive journals. Therefore, he found that journals provided meaningful profiles of students’ learning in mathematics.

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49 Stewart and Chance (1995) investigated the use of journals by students and their teachers in four secondary first-year algebr a classes over an entire school year. Two of the classes wrote in journals, two did not. J ournal entries were made three times a week during the last five minutes of class. The pr ompts were in three categories: mathematics concepts and procedures, curriculum issues, and free writing. On M ondays, the instructor gave the students a prompt that focused on the mathematics concepts and procedures being taught. One example was, “Subtrac ting is the same as adding the opposite because…” On Wednesdays, students responded to curriculum prompts such as “One mathematics activity I really enjoy is … b ecause…” Thursdays were reserved for free writing, where the only requirement was that th e entries involve the writers as students of mathematics. Students often wrote about their accomplishments, frustrations, and personal problems that interfered with schoolwork. Students handed in journals at the end of th e class. The instructor read the entries, made occasional written comments, and returned the journals to the students. Writing was not judged on grammatical accuracy, although co mplete sentences and paragraphs were encouraged. No grades were given for journal writing. Students were gi ven pretests at the beginning of the school year and posttests at the end of the year. Achievement was significantly higher for the journal-writing students. Similar tests performed on the changes in anxiety scores suggested for journa l-writing students a decrease in anxiety that approached significance (Stewart & Chance, 1995).

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50 Journal Writing with Colle ge Mathematics Students Using writing in college mathematics courses can improve performance in mathematics (Abdalkhani and Menon, 1998; Austin, 1998). Britton (1992) wanted to incorporate writing into his calculus, precalc ulus, and statistics classes. He had his students write weekly jour nal entries of one to two pages. He read each journal entry and returned them the next class meeting. The primary topic for the journals was a summary explanation of the mathematics studied that week. Frequently the journals included exercises the students had worked and wanted th e instructor to check or to determine why they did not get the correct answer. Other journal material included questions for the teacher, lists of items not understood, and comm ents about the course. The journals were evaluated using three levels: minus, if unsatisf actory; check, if satisfactory; and plus, for an exceptionally good entry. The author stated that his primary re ason for using journals was that students learned better if they had to describe the ma terial they were studying. He discussed three additional advantages for the re quired journals. He found that students made more of an effort to keep up with their work when using journals. The increased communication between students and their instructor was also a definite advantage. One of the indirect advantages that he found was that collecting the journals was also a means of recording attendance without having to take roll during class. Borasi and Rose (1989) did a study invol ving the use of jour nals with college mathematics students. The students were en rolled in a course entitled “Algebra for Professional Programs.” This was a 3 semeste r-hour course that was taken predominantly by business students in their first or second ye ar of college. The writing experience was

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51 structured so as to make journal writing an integral and valued component of the course without modifying the goals and content of th e course, nor the teaching approach usually employed by the instructor. This approach cons isted mainly of lectures by the teacher, followed by the assignment of homewor k practice and follow-up discussion. The students were asked to write thre e entries per week, with the journals collected every other Friday and retu rned on the following Monday, along with comments by the teacher. Credit toward the c ourse grade was assigned for maintaining the frequency and volume of writing, but not on the basis of mechanics or content. Although topics for the entries were intentionally left open and flexible, the instructor produced a list of 36 suggested ideas. These included the following writing topics: Reflect on math ideas or feelings about math. Describe your favorite math class. How should we use class time to best advantage? How do I go about doing word problems? Where do the rules of math come from? In addition to the open entries that they were supposed to write at home, students were occasionally asked to write in their j ournal during class, in response to a topic assigned by the instructor. As a follow-up to the journal writing activity, the students were asked to write an eval uation at the end of the semester by responding to the following open-ended questions: 1. How has writing in your journal aff ected your learning of mathematics? 2. How do you feel about journal writing for this course? 3. What are the benefits of journa l writing for mathematics classes?

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52 4. How could journal writing be change d to be more effective? (p. 351) Twenty-three complete sets of journals a nd evaluations were an alyzed in order to examine both what happened when journal writi ng was used in this specific setting and what meanings the participants attributed to the experience. As a result of a content analysis of these sets of data, a number of potential benefits of journal writing were identified and explored. Researchers found that students demonstrated an increased learning of mathematical content as well as improving their problem-solving skills. They also found that writing in journals provided a therapeutic effect for students when they wrote about feelings and att itudes. The dialogue between st udents and teachers through the journals created a su pportive class atmosphere. Journal Writing with Mathematics Teachers Burk and Littleton (1995) conducted the Pre-Algebra Experience, which was a project developed to improve mathematics in struction in the middle grades. The primary goal of the project was “to stress to mi ddle school mathematics teachers the content necessary for students to succeed in high sc hool algebra” (p. 576). The teachers attended a three-week summer institute. The thirty participants were asked to keep journals during the workshop. The teachers were asked to respond daily to the question “What did you learn today?” The following prompts were among those given to the teachers to encourage reflective thinking: How does what you learned fit with what you already know? Do you think that you can use in your cl assroom what you have learned? How? Why?

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53 Are you confused or surprised by anyt hing you learned today? What? Why? How has what you have learned changed you? The second phase of journal writing began when school started and the participants were actually involved in mathematics inst ruction. Participants were instructed to write in their journal twice a w eek. In these journals, teachers were told to (1) record the date of the entry, which s hould cover only one day and be written on the day of the experience; (2) br iefly describe the context of the sequence of events; (3) select one or two significant in cidents to describe in detail; and (4) analyze this incident. Why is it significant? What is your interpre tation of this incident? What did you learn from it? What questions did it raise for you? The authors found that the use of reflective journals dur ing this project served the following purposes: The journals (a) guide d instruction during the institute; (b) documented the institute’s effectiveness for evaluation purposes; (c) cemented learning and encouraged reflection on that learning; (d) linked understandi ngs gained in the institute to actual classroom practice; and (e) involved teachers in an instructional strategy that they could imple ment with their students. Journal Writing with Preservice Teachers Studies involving journal writing with t eacher education students enrolled in educational psychology courses have shown that students believed that they had benefited from the journals. Some students sa id that they found that journal writing allowed them to reflect on thei r own experiences and made the material more relevant to

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54 them (C. W. Carter, 1997). Othe rs said that the journals re quired them to organize their thinking and encouraged reflection and construction of meaning (Good and Whang, 2002). Garmon (2001) used dialogue journals with 21 students enrolled in a multicultural education course for preservice teachers. St udents were given prom pts for their journal entries. Two of these prompts asked the st udents what they liked/disliked about the journals and whether or not th ey found journals valuable. From this data, the author identified the following benefits of dialogue j ournals and relative frequencies of mention: (1) facilitating learning of cour se material (27%); (2) prom oting self-reflection and selfunderstanding (25%); (3) provi ding procedural convenience s and benefits, such as scheduling and grading policies for journals (21%); (4) providing an opportunity to express ideas (14%); (5) gett ing feedback on ideas and ques tions (8%); and (6) improving the teacher-student rela tionship (5%). The author conclude d that the use of journals in teacher education courses “may offer a number of important benefits for some prospective teachers” (p. 47). Phenomenology Conceptual Overview The word phenomenon is derived from the Greek word phaenesthai, which means to flare up, to show itself, to appear. Thus phenomenology, in a broad sense, looks at that which appears in consciousness, the phenome non, as the “impetus for experience and for generating new knowledge. Phenomena are the bu ilding blocks of human science and the basis for all knowledge” (Moustakas, 1994, p. 26). Phenomenology asks the foundational

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55 question: “What is the meaning, structure, a nd essence of the lived experience of this phenomenon for this person or group of people?” (Patton, 2002, p.104). Phenomenology differs from other sciences in that it gain s insights from the way people describe their experiences “pre-reflectively, without taxonomiz ing, classifying, or ab stracting it” (Van Manen, 1990, p.9). Many have embraced this notion of studying phenomena, and phenomenology has come to mean different things to different people. A German philosopher named Edmund Husserl (1859-1938) was the founder of pure phenomenology. Husserl believ ed that everything in life was directed toward the goal of achieving consistency and harmony (K im, 1989). Husserl’s basic philosophical assumption was that knowledge and unders tanding come from experience. This understanding comes initially from sensory e xperience of phenomena, but later these experiences must be “described, explicate d, and interpreted” (Patton, 2002, p. 106) in order to lead to understanding. Whereas Husserl focused on phenomenol ogical psychology, Alfred Schutz (18991959) approached phenomenology from a soci ological perspective (Wagner, 1970). Sociology is concerned with the subjective meaning and understanding of social action. In this context, action can relate to any human conduct wh ere the acting person attaches meaning to the conduct. This action is cons idered social when it is “directed upon the conduct of others” (Wagner, 1970, p. 8). Schutz made a major contribution to the field of phenomenological sociology by combining his fa miliarity with Husserl’s philosophy with his own extensive knowledge of sociology. His mo st important contributions to this field involved “uncovering, describing, and analyzing the essential features of the world of everyday life” (Psathas, 1989, p. 8).

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56 Phenomenology has also been viewed as an inquiry paradigm, an interpretive theory, a research methods framework, and a majo r qualitative tradition (P atton, 2002). Although the term phenomenology can have several different meanings, they all share a desire to explore how people make sense of their experiences and transform experience into consciousness. Phenomenological Methodology Exploring how human beings make sense of their experiences requires methodology that “focuses on descriptions of what people experience and how they experience what they experience” (P atton, 2002, p. 107). Patton described a phenomenological approach to qualitative research as: capturing and describing how peopl e experience some phenomenon--how they perceive it, describe it, feel about it, judge it, remember it, make sense of it, and talk abou t it with others. (p. 104) Van Manen (1990) explained that consciousne ss is the way people access the world and their own lived experiences. Since one cannot reflect on lived experience while it is actually occurring, “phenomenological reflection is not introspective but retrospective ” (Van Manen, 1990, p. 10). Van Manen (1990) explained that in human sciences such as phenomenology, objectivity and subjectivity might take on meanings that differ from the traditional perspective. Here the ‘object’ refers to the object of the rese archer’s inquiry, so ‘objectivity’ means that the researcher remains true to the object. This involves describing and interpreting the object while re maining faithful to it. ‘Subjectivity’ means

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57 that the researcher can examine the object of the study in a unique and personal way while not being misled by his or her own “unreflected preconceptions” (Van Manen, 1990, p. 20). Moustakas (1994) described steps that a phenomenological researcher can take in investigating and describing how people expe rience some phenomenon. The first step is called epoche This is a Greek word that means to refrain or abstain from judgment. Epoche requires the researcher to set asid e his or her own prior assumptions, judgments, and knowledge. This involves acknowledging personal bias and preconceptions and doing one’s best to eliminate them. The next step is phenomenological reduc tion. This comes from the Latin word reducere, which means ‘to lead back.’ Reduction involves going back to the data and “bracketing out” the researche r’s biases that were identi fied during epoche (Patton, 2002). The process allows the researcher to view the data “on its own terms” (Patton, 2002, p. 485). Reduction entails locating key phras es that are relevant to the phenomenon being studied, interpreting th e meanings of these key phrases, and inspecting these meanings for revelations about the phenom enon being studied. After the bracketing has been complete d, the data are ‘horizontalized.’ This implies that all elements and perspectives of the data are equal in weight. Data are organized into meaningful clusters, and invarian t themes within the data are identified. In doing this, the researcher performs an ‘imaginative variation’ on each theme. This refers to the researcher approaching the phenome non from various perspectives in order to understand that there is not just one single truth to be found. At this point, the researcher is ready to synthesize the “meanings and essences of

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58 the experience” (Moustakas, 1994, p. 144). This involves a structuring of the meaning found in the data. In order to do this, the researcher must be car eful not to allow his or her biases to interfere with the process. Of ten researchers check each other’s work and collaborate to find consensus when discrepanc ies arise. This process adds validity and reliability to the findings. Reflection The Oxford Dictionary (1996) defines the verb refl ect as “meditate on; think about … consider; remind oneself” (p. 1261). In the field of education, much has been written about reflective practice in teachi ng (Canning, 1991; Clift, Houston, & Pugach, 1990; Jay, J.K., 2003; McEntee et al., 2003). Mo st of the ideas concerning reflective practice in teaching are based on the wo rk of John Dewey. Dewey (1933) defined reflection as “active, persistent, and careful c onsideration of any beli ef or supposed form of knowledge in light of the grounds that suppo rt it and the further conclusions to which it tends” (p. 9). Dewey’s view of reflectiv e thought involved identifying a problem and searching for a solution to that problem. Dewey’s notion of reflectiv e thinking lends itself well to the use of reflective practice in teaching. Reflectiv e teachers actively seek solutions to their classroom problems. Alternative soluti ons are considered and re-c onsidered. Because Dewey’s notion of reflection involves “considerat ion of any belief or supposed form of knowledge,” each teacher must consider his or her own personal beliefs and assumptions when making classroom decisions. A reflective teacher considers various solutions to a classroom problem, assesses each potential soluti on with regard to research and personal

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59 beliefs, and then implements the solution. If one solution fails, the reflective teacher will consider other possible solutions in the same manner. Although much has been written about re flective practi ce in teaching and the use of reflection in identifying and considering pe rsonal beliefs, little has been written concerning the use of reflection in examin ing and possibly improving attitudes toward mathematics. This type of reflection invol ves recalling and relivi ng lived experiences. In reflective journal writing, participants reflect on experiences and organize their thoughts and feelings in or der to communicate clearly. Phenomenology explores how people make sense of their experiences (P atton, 2002). Reflectiv e journals are an excellent way for people to relive and make sense of their experiences. Summary and Implications for Teacher Education The reform movement in mathematics educ ation has recognized the importance of affective issues and the connection between these issues and higher-order thinking. Beliefs about mathematics and attitudes towa rd mathematics are two of the three major components of the affective domain in learning mathematics. Teachers’ beliefs have been shown to influence their teaching practices (An, 2000; Fang, 1996; Kagan, 1992; Stipek et al., 2001; Thompson, 1992) and the beliefs of their students (G. Carter, 1997). A teacher w ho regards mathematics as a set of rules needed to produce accurate results will most likely teach the subject differently than a teacher who views mathematics as a way of d ealing with ideas that grow out of problem situations. Because beliefs are based on subj ective evaluations and personal experiences and are so difficult to change, teacher educator s need to be aware of the influence beliefs

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60 have on teacher behavior in the classroom. The ways that prospective teachers perceive and interpret knowledge “may be shaped by belief systems beyond the immediate influence of teacher educators” (Nespor, 1987, p. 326). Thompson (1992) pointed out that “the task of modifying long-held, deeply rooted conceptions of mathematics and its teaching in the short period of a course in methods of teaching remains a major problem in mathematics teacher education” (p. 135). Often teachers’ existing belief systems conflict with the pe dagogical practices they are being encouraged by the profession to adopt. Research involving mathematics belief changes in teacher training programs has shown little consensus. Many studies have demonstrated a change in the beliefs of prospective and new teachers (Ambrose, 2001; Mewborn, 2002; Stuart and Thurlow, 2000; Vace and Bright, 1999), while others have shown an inconsistency between new t eachers’ stated beliefs and their classroom practices (Benken and Wilson, 1998; Fr ykholm, 1996; Raymond, 1997). Studies with practicing teachers have also shown conflicting results. Although some teachers have demonstrated a belief change after a profe ssional development program (Pligge et al., 2000; Simon and Schifter, 1993; Vacc et al ., 1998), others have not (Wilson and Goldenberg, 1998). Several avenues have been proposed for changing teachers’ belief systems. These include reflection and examination of pe rsonal beliefs and building new beliefs upon existing beliefs. The benefits of teacher sel f-reflection have been shown in many of the studies included in this review (Anderson and Piazza, 1996; Gellert, 1999; Lock and Lee, 2001; Mewborn, 2002; Stuart and Thurlow, 2000; Wagner, Lee, and Ozgun-Koca, 1999). This suggests that teacher education program s designed to build on preservice teachers’

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61 existing beliefs by providing them opportunitie s to reflect on thes e beliefs would be effective. Attitudes toward mathematics have not been studied as extensively as beliefs. Studies have shown that stude nts’ attitudes toward mathematics tend to become more negative as they get older and the relations hip between their attitudes and achievement tends to get stronger. Studies have also shown that many preservice elementary school teachers have negative attitudes toward ma thematics (Rech et al., 1993; Cornell, 1999; Philippou and Christou, 1998). This should be a concern for teacher educators because teachers with negative attitudes toward math ematics are unlikely to cultivate positive attitudes in their own students (Hungerfo rd, 1994). “Improving pre-service students’ attitudes toward mathematics is an important concern for university education courses in order to facilitate positive mathematics attitu des in future elementary pupils” (Sherman and Christian, 1999, p. 96). Improving the atti tudes of preservice elementary school teachers is a crucial step in breaking the cycle of teachers with negative attitudes fostering negative attitudes in their own students (Phili ppou and Christou, 1998). Several studies involving teacher training programs that utilized constructivist instructional methods have shown positive resu lts in improving the attitudes and teacher self-efficacy of preservice elementary t eachers (Anderson and Piazza, 1996; Gibson and Van Strat, 2001; Huinker and Madison, 1997; McGinnis et al., 1998; Philippou and Christou, 1998; Quinn, 1997; Sherman and Ch ristian, 1999). Although these results are encouraging, future studies that follow these teachers past their teacher training programs and into their first few years of teaching to se e if the attitude change s remain stable over time would be beneficial.

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62 Preservice teachers bring to teacher edu cation certain attitudes and beliefs that have developed over time and are often resistant to change. However, mathematics methods courses and teacher training programs provide an opportunity to change these attitudes in a positive way, as well as alter belief s so that they are more in line with those set out by the mathematics education profe ssion. The National Counc il of Teachers of Mathematics (2000) has recognized the impor tance of reflection as a component of mathematical communication. Reflection has been effective in changing preservice teachers’ beliefs about mathematics (A mbrose, 2001; Mewborn, 2002; Stuart and Thurlow, 2000). It has also been useful in connecting teachers’ stated beliefs to actual classroom practice (Gellert, 1999; Lo ck and Lee, 2001; Thompson, 1992). Journal writing offers an opportunity for reflective thought that has been shown to encourage learning (Borasi and Rose, 1989; Burk and Littleton, 1995; Carter, 1997; Garmon, 2001; Good and Whang, 2002). This study explored the use of reflection, through reflective journals, as a tool for t eacher educators seeking to both understand how attitudes toward mathematics are formed and to improve these attitudes. More research is needed in this direction, especi ally in the underreprese nted area of teacher attitudes toward mathematics. Teacher attit udes have been linked to beliefs (Philippou and Christou, 1998; Stipek et al., 2001), a nd beliefs have been linked to classroom practices. If we want teachers to adopt ch ild-centered inquiry-ba sed instruction, then teacher educators and researchers need to focus more of their efforts in the area of teacher attitudes toward mathematics.

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63 Significance of the Study The reform movement in mathematics e ducation has recognized the need for change in the area of aff ect. Although the importance of developing self-confident, motivated students who value and enjoy ma thematics has been well established, the means for doing so are not so clear. Childre n’s attitudes toward mathematics tend to become less positive as they get older, and often by the time they reach high school or college, their attitudes have become negative (McLeod, 1992). Researchers have found that many preservi ce elementary school teachers at the university have negative att itudes toward mathematics (C hristian, 1999; Cornell, 1999; Hungerford, 1994; Philippou and Christou 1998; Rech et al., 1993). These researchers have suggested that improving the attitudes to ward mathematics of preservice elementary school teachers at the university should be a main concern of teacher educators. If preservice elementary school teachers can develop more positiv e attitudes toward mathematics, then they will be more likely to foster positive attitudes in their own elementary school students. Perhaps these el ementary school students would then be less likely to develop negative attitudes as they ge t older. If this trend were to continue, eventually we could have fewer preservice elementary school teachers with negative attitudes, and the cycle of teachers passing on negative attitudes toward mathematics to their students would be broken. Several studies have demonstrated success in improving attitudes toward mathematics of preservice elementary teacher s enrolled in mathematics methods courses (Anderson and Piazza, 1996; Gibson and Va n Strat, 2001; Huinker and Madison, 1997; McGinnis et al., 1998; Philippou and Chri stou, 1998; Quinn, 1997; Sherman and

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64 Christian, 1999). These methods courses util ized constructivist instructional methods such as the use of hands-on manipulatives, cooperative group work, problem solving, and the use of technology. However, none of thes e studies used the additional tool of reflective journals where participants refl ected on their own at titudes toward and experiences with mathematics. Journal writing offers an opportunity for reflective thought that has been shown to encourage learning (Borasi and Rose, 1989; Burk and Littleton, 1995; Carter, 1997; Garmon, 2001; Good and Whang, 2 002). However, in these studi es, students’ reflections focused largely on course content and peda gogy. When they did use affective prompts, they focused mainly on feelings and attitudes. In the present study, pa rticipants reflected on their own attitudes toward ma thematics and also the experiences that have led to the development of these attitudes. It has been th e researcher’s experience that this type of reflection seems to promote an awarene ss of and possibly an improvement in participants’ attitudes toward mathematics. The following journal from a participant in Pilot Study II demonstrated this newly acquired insight: I have learned a lot about myself through these reflective journals. I always knew that I didn't like math but I never sat down and tried to figure out why I don't like it. These journals have helped me figure out those reasons and in turn they have helped me like math. After I learned what I didn't like about it and why, I was able to realize that it wasn't math that I didn't like; it was the teachers that di dn't help me overcome this dislike. If I had been taught to do math correctly, I wouldn't have grown up so frustrated with it, and I would have been able to learn it. But because I was

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65 so frustrated with math and my teachers never took time out to help me overcome the frustration, I decided to hate math and not care about it. In addition to the use of a different type of reflection, one that focuses on attitudes and experiences that have led to the devel opment of those attitude s, the present study offers another contribution to the field of mathematics education. Information that is gained from exploring how attitudes toward mathematics are formed can inform not only teacher educators, but also mathematics teac hers at all levels. It is important for mathematics teachers to know which types of experiences, especially classroom experiences, lead to formation of positive at titudes toward mathematics in students and which lead to formation of negative attit udes. Teachers can then try to avoid those situations that negatively infl uence attitudes toward mathematics and promote those that positively affect students’ attitudes. The need for change in the area of at titudes toward mathematics has been well established by the field of math ematics education. This study s ought to affect this change in two important ways. Improvi ng the attitudes toward mathematics of future elementary school teachers and informing practicing math ematics teachers about how to create classrooms where the development of positive attitudes is promoted and the development of negative attitudes is diminished are bot h important contributions to the field of mathematics education.

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66 CHAPTER THREE METHOD In this chapter, the research design that was used to achieve the goals of this study is outlined. The study included both qualitati ve and quantitative data collection and analyses in order to provide a detailed perspe ctive of the attitudes toward mathematics of preservice elementary school teachers and the experiences that have led to the development of these attitudes. The study sought to answer the following research questions: 1. What are the attitudes toward mathem atics of preservice elementary school teachers entering an introductory mathema tics methods course? In particular, how do preservice teachers score on each of the four attitudinal components being measured: value of mathematics, enjoyment of mathematics, motivation for mathematics, and self-confidence with mathematics? 2. To what extent do attitudes toward math ematics of preservice elementary school teachers change during the mathematics me thods course? To what do preservice teachers whose attitudes toward mathematics were altered attribute this change? 3. What is the relationship between preservi ce elementary teacher s’ initial attitudes toward mathematics and their grade on the methods course final examination? 4. What do preservice elementary school te achers’ reflective journal entries reveal about their attitudes toward mathematics a nd the experiences that have influenced the development of those attitudes?

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67 5. What are the attitudes toward and ex periences with mathematics of those preservice elementary school teachers id entified as having the most extreme (either positive or negative) attitudes? Pilot Studies After using reflective journals with pr eservice elementary school teachers for three years, a more formal pilot study wa s completed in spring, 2003 for the purpose of familiarizing the researcher w ith the interview protocol that was used in this study and the use of Hycner’s guidelines for thematic analysis, which are described below. Details of Pilot Study I are f ound in Appendix A. A second pilot study was completed in spring, 2004 for the purpose of determining if any changes in preservice el ementary school teachers’ attitudes toward mathematics occurred during a mathematics me thods course. Details of Pilot Study II are found in Appendix B. Changes that were ma de to the present study based on the pilot studies are noted where applicable in the remainder of this chapter. Present Study This study was a mixed methods study. Quan titative methods were used to answer questions 1, 2, and 3. Qualitative methods were used to provide more detail to answer question 2. Questions 4 and 5 used a phenom enological approach, as described in Chapter Two, to examine more closely the attitudes toward mathematics of the participants and the experiences that have in fluenced the development of those attitudes. This study utilized journals and interviews to capture and describe experiences that have influenced the participants’ attitudes toward mathematics.

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68 Participants The participants in this study were 33 university students enrolled in one section of a mathematics methods course for elementa ry education majors at a major research university in the southeastern United St ates during the fall semester, 2004. Students enrolled in this course typically are juni ors and seniors who are working toward state certification as elementary school teachers of kindergarten through grade 6. Thirty-two of the participants were females and one was male. Twenty-three of the participants were between the ages of 18 and 22, six were betw een the ages of 23 and 27, one was between the ages of 28 and 32, two were between the ages of 33 and 37, and one was over 37 years of age. The researcher taught the course. It was necessa ry to use an intact group due to university scheduling. The class me t once a week for three hours. Description of Course This course is the first of two mathem atics methods courses that elementary education majors must complete. The methods c ourse utilizes construc tivist instructional methods such as the use of hands-on mani pulatives, cooperative group work, problem solving, and the use of calculators. There is also an emphasis on the use of children’s literature in the teaching of elementary school mathematics. The course syllabus (Appendix C) states the following purpose for the course: The purpose of this course is to provide opportunities for preservice teachers to examine their understanding of various ma thematics topics and to construct a vision of mathematics that considers the goals and assumptions of the current reform movement in mathematics educati on. Content, methods, and materials for

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69 teaching elementary school mathematics will be examined with a focus on Problem Solving, Whole Number concep ts, and Rational Number concepts. In this course, preservice teachers were engaged in a variety of teaching/learning activities. These included lectures, discussi ons, cooperative learning activities, question and answer sessions, student demonstrations /explanations, and role-playing. Preservice teachers were expected to present results and problem solutions to their peers. The methods course used the text Elementary School Mathematics: Teaching Developmentally Second Custom Edition, by John A. Van de Walle (2004). This edition was taken from Elementary and Middle School Mathem atics: Teaching D evelopmentally, Fifth Edition by John A. Van de Walle (2004). The cour se covered the firs t 17 chapters of the textbook. The Table of Contents for th e textbook is found in Appendix D. Students were also required to purchase a manipula tives kit which included a book of teaching activities called Hands On Teaching (HOT) Strategi es for Using Math Manipulatives by Carol Thornton & G. Lowe-Parrino, ETA, 1997. Th e kit contained such manipulatives as Pattern Blocks, Base Ten Blocks, Color Tiles, Fraction Tower Cubes, Tangrams, PopCubes, Geoboards, Fraction Circles, TwoColor Counters, Coins, GeoReflectors, Angle Rulers, Spinners, Number Cubes, and Factor Blocks. Procedure Each participant completed the Attitudes Toward Mathematics Inventory (Appendix E) at the beginning of the semester and again during week 12 of the 15-week semester. This allowed the researcher to measure each participan t’s initial attitudes toward mathematics and to assess any change s that may have taken place during the first

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70 11 weeks of the semester. Scores were dete rmined for each of the four attitudinal components measured by the ATMI: value of mathematics, enjoyment of mathematics, motivation for mathematics, and self-confidence with mathematics, as well as a composite attitude score. In addition, demogr aphic information including age, gender, previous mathematics coursework, and previ ous education coursework was collected. The ATMI was also administered during w eek one and week twel ve to students in two other sections of the mathematics methods course for the purpose of making comparisons in results. Two different instru ctors taught these met hods course sections. The first instructor is an A ssistant Professor of Mathema tics Education. After receiving her Ph.D. in 1998, she held a faculty positi on in the Mathematics Department at a university in the Rocky M ountain region. During her te nure there, she taught mathematics content and pedagogical content co urses and was fundamentally involved in the reformation of the undergraduate teacher preparation programs. Although this was her first time teaching the methods course at this university, she had previously taught this methods course in the same state in whic h this study took place. These classes were taught intermittently over a period of six y ears before she moved to the Rocky Mountain region. She was a doctoral student at the time. The second instructor received a B.S. in Mathematics Educ ation in 1993 and an M.B.A. in 2000. She is currently pursuing a Ph.D. in Mathematics Education. She has taught high school mathematics for five year s in both urban and suburban schools. She has also worked with elementary school student s as a private tutor. Although this was her first time teaching the methods course, she observed the researcher’s class each week before teaching her own class. However, she did not try to model he r teaching style after

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71 that of the researcher. When asked about this, she said that she had used some of the researcher’s course content, but as an expe rienced teacher, she “had [her] own [teaching] style, and it would have been difficult for [her] to change that.” The researcher and the ot her two instructors all us ed the same textbook and manipulatives kit. Although each instructor was responsible for the content of her own course, all three classes covered the same content material and took the same final examination. The two other instructors were as ked not to give their classes any written assignments that would involve reflection on their attitudes toward mathematics. Throughout the semester, participants in the researcher’s class submitted reflective journal entries as part of their co urse assignments. The journals were graded only for completeness, with grammar and spel ling errors ignored. Journal entries were submitted by email, and the researcher responded to each entry by email. Rose (1989) cited teacher response as an important benef it of journal writing. “As the teacher writes back to the students, students realize the t eacher hears and cares” (p. 26). Examples of the researcher’s responses are found in Appendix F. The two preservice teachers with the lowe st initial scores on the ATMI and the two with the highest scores were each asked to participate in an individual interview where their attitudes toward and experiences with mathematics were further explored. These Experiences with Mathematics Interviews took place between week six and week eight of the semester. This use of purposef ul sampling allowed “information-rich cases” to be studied in depth (Patton, 2002, p.46). After participants had completed the s econd ATMI, individual interviews were conducted with four preservice teachers w ho showed significant positive changes in

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72 attitude. In addition, three preservice t eachers who experienced negative changes in attitude also participated in individual interviews. The method that was used to determine which participants were interviewed is described in the Data Analysis section that follows. These Changed Attitudes Interviews focused on participants’ ideas about those aspects of the methods course that may have influenced their attitudes toward mathematics. All interviews were conducted in a private conference room. Interviews took place approximately one month after th e completion of the methods course and submission of final grades. The decision to interview these participants after the completion of the course was based on findings from the first pilot study. The Changed Attitudes Interview protocol was used with 2 participants in the pilot study who were inte rviewed six months after the completion of the course. In addition, they were also asked the following question: Would your answers have differed any if you had been interviewed before the end of the course? If your attitudes had ch anged in a negative way, would you have been open about it if interviewed be fore the end of the course? Both of the interviewees said that their answ ers would not have differed if they had been interviewed before the conclusion of the methods course. However, both recommended that any participants whose attitudes had changed in a negative direction be interviewed after the conclusion of the semester. They both acknowledged that securing the participation of those particip ants would be more difficult after the course had ended, but both believed that interviewees with negative attitude change s would be better able to put their experiences in the methods course into perspective after some time had elapsed. As one of the interviewees explained:

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73 [A student] might not like a particular cla ss, so it’s easier to think, ‘Oh, I hate math,’ because they’re in that class and th at’s all they can see. Whereas if they wait until after they’re ou t of that class, they might th ink, ‘Well, that class is over; it wasn’t that bad, looking back.’ Then they might have more of a positive view of math. Both participants focused their responses on this notion of allowing time to pass rather than on the response anticipated by the research er that students might feel uncomfortable discussing the course with the instructor be fore their final grades had been submitted. They both believed that interviews with thos e who experienced posi tive attitude changes could be conducted either during or afte r the conclusion of the methods course. The researcher kept a reflective journal during the methods course. This journal provided a detailed account of what took place during each class, as well as the researcher’s impressions of each class. This perspective was valuable when analyzing interviewees’ references to class activities. In an effort to provide reliability of this account, a doctoral student who has taught th is mathematics methods course several times also took field notes during two of the classes. The observer, without the researcher’s knowledge, randomly chose which classes to attend. One of the observations took place during weeks two to six of the se mester and the other took place during weeks seven to eleven of the semester. In order to establish an observer protoc ol, the observer atte nded one class during the second pilot study and took field notes A comparison was made between the observer’s account and that of the resear cher, and guidelines were established. The observer protocol (Appendix G) asked the observer to record instructional activities and

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74 students’ activities that took place during the class. The observer was also asked to note any observations that might reflect students’ attitudes toward mathematics, especially value of mathematics, enjoyment of mathema tics, motivation for mathematics, and selfconfidence with mathematics. The researcher and the observer agreed that it would be beneficial for the observer to move to a diffe rent location in the classroom approximately every forty-five minutes. This allowed the observer to spend time taking notes in each area of the classroom during a three-hour class. Instruments The Attitudes Toward Mathematics Inventory (ATMI) The ATMI was used to assess preservice elementary teachers’ attitudes toward mathematics. The ATMI contains 40 items (Appendix E). Participants were asked to indicate their degree of agreement with each statement using a Likert-type scale, from strongly disagree to strongly ag ree. The authors of the ATMI tested the instrument for internal consistency and construct validit y. They administered the ATMI to 544 high school students in Mexico City. The st udents represented all grade levels and mathematics levels. The alpha reliability coefficient was .97 (Tapia, 1996). A factor analysis identified students’ self-confidence, motivati on, enjoyment, and value of mathematics as underlying dimensions of student s’ attitudes toward mathematics. Table 2 lists some of the survey items that correspond to each factor.

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75 Table 2 Survey Items Grouped by Factors Self-Confidence: Mathematics makes me feel uncomfortable. I have a lot of self-confidence when it comes to mathematics. Value: Mathematics is a very worthw hile and necessary subject. I believe studying mathematics helps me with problem solving in other areas. Motivation: I am willing to take more than th e required amount of mathematics. The challenge of math appeals to me. Enjoyment: I get a great deal of satisfaction ou t of solving a mathematics problem. I like to solve new problems in mathematics. Note. From “The Attitudes Toward Mathematics Instrument,” by M. Tapia, 1996. Paper presented at the annual meeting of the Mid-South Educational Res earch Association, Tuscaloosa, AL, p.15. The ATMI was later administered to 262 middle school students from a private, bilingual school in Mexico City, with an al pha reliability coeffici ent of .95 (Tapia and Marsh, 2000). Because these samples of mi ddle school and high school students from Mexico were so different from the sample in this study, the ATMI’s reliability with preservice elementary teachers was also de termined. Cronbach’s Coefficient Alpha was calculated using the data obtained in the firs t pilot study (n=31). An alpha coefficient of .98 was found using this sample, indicating a high degree of internal consistency.

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76 Because the ATMI was written for high sc hool students, two of the items were inappropriate for university st udents. Following the suggestion of the ATMI’s author (M. Tapia, personal communicat ion, December 14, 2002) these two items were changed as follows: “High school math courses would be very helpful no matter what I decide to study” was changed to “Math courses would be very helpful no matter what grade level I teach,” and “I would like to avoid using mathematics in college” was changed to “I would like to avoid teaching mathematics.” Journal Prompts Participants were given prompts for each journal entry. Prompts and corresponding due dates were included in th e course syllabus (Appendix C), which the participants received during the first class of the semest er. The researcher had found that providing participants with all of the prompts at once allowed them to think ahead about which of their experiences best applied to each prompt. The prompts were developed by the researcher, with the following exceptions: Item #4: Describe in detail one experience from your pa st that is particularly memorable and influential in your attit udes about mathematics. Where were you? Who was there? What was said? What did you do? How did you feel? (Davidson and Levitov, 2000, p. 18). Items #6: What do you think are the qualitie s of the best mathematics teacher you have ever had? (Stallard and Thompson, 2004) Eight journals were assigned over the cour se of the semester. Fi ve of these related directly to the purpose of this study and were analyzed usi ng methods that are described

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77 in the Data Analysis section that follows. The remaining three journal prompts did not relate directly to the purpos e of this study, but they did relate to the purpose of the methods course. Therefore, participants respond ed to them as well, but these entries were not analyzed unless they cont ained information relevant to the study. Journal prompts and their designated purposes can be found in Table 3. Experiences with Mathematics Interviews Individual interviews were used to explore more deeply the attitudes toward and experiences with mathematics of those pres ervice teachers with the highest or most positive attitude scores and those with the lo west or most negative attitude scores. A standardized open-ended interview format wa s used. The same interview protocol that was used successfully in th e first pilot study was used again, with one change. The following question was omitted: Complete: I enjoy or feel positive about mathematics because … and/or Complete: I do not enjoy or I feel negative about mathematics because … This question was omitted from the interv iew protocol in order to use it as a journal prompt. This change was made in order to allow all participants to reflect on these attitudes rather than just t hose with the most extreme attitudes toward mathematics. The Experiences with Mathematics Interview protocol (Appendi x H) contained nine questions. The first que stion, “Why did you decide to become a teacher?” was included as a way of ‘breaki ng the ice’ before asking par ticipants to focus on their memories and to relive experiences. Four questi ons asked the participan t to reflect on his

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78 Table 3 Journal Prompts and the Order in Which They Were Assigned Prompts related to study Prompts related to methods course 1. Discuss any fee lings (positive or negative) that you have about taking this course. What are you hoping to gain from the course? 5. Many students have low selfconfidence when it comes to mathematics. What will you do as a teacher to boost the selfconfidence of your students regarding mathematics? 2. What are your memories of learning mathematics in elementary school (attitudes, success, etc.)? What can you, as a future teacher, learn from these experiences? 6. What do you think are the qualities of the best mathematics teacher you have ever had? What effect did this teacher have on you as a learner of mathematics? 3. Complete each of the following. Explain your responses. Why do you think you feel th is way? I enjoy or feel positive about mathematics because … and/or … I do not enjoy or I feel negative about mathematics because … 7. What do you think are the qualities of the worst mathematics teacher you have ever had? What effect did this teacher have on you as a learner of mathematics? Table continued on the next page

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79 Table 3 (Continued) Journal Prompts Prompts related to study Prompts related to methods course 4. Describe in detail one experience from your past that is particularly memorable and influential in your attitudes about mathematics. Where were you? Who was there? What was said? What did you do? How did you feel? 8. Discuss the use of reflective journals in this course. What benefits, if any, did they provide? What, if any, were the drawbacks? or her own experiences as a student in a ma thematics classroom at various levels of schooling. Two questions related directly to feelings about the methods course. Changed Attitudes Interviews In addition, individual inte rviews were conducted with four preservice teachers who showed the greatest positive change in at titude and three who experienced a negative change in attitude. These interviews focused on participants’ ideas about those aspects of the methods course that may have influenced their attitudes toward mathematics. The

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80 Changed Attitudes Interview protocol (Appendix I) contai ned 11 questions. Participants were asked how they felt about the use of manipulatives, cooperative learning, problem solving, and journals in the methods course a nd also in teaching math ematics in general. The Changed Attitudes Interview protocol was used successfully with two participants in the first pilot study who were interviewed six months after the completion of the course. It was also used with three participants in the second pilot study who were interviewed immediately following the comple tion of the methods course and submission of final grades. No interview prot ocol changes were made as a resu lt of the pilot studies. Final Examination for Math ematics Methods Course The final examination for the mathematics methods course was a 50-item multiple-choice instrument that included questions about both mathematics content and pedagogy. The test was a departmental exam, a nd all students who were enrolled in any section of the course took the same final ex am. The final exam was used as the measure of course achievement rather than the final course grade in an effort to minimize bias. The test was a multiple-choice instrument, so grading was not subjective. In addition, the test was not written by the researcher, so use of the final exam as a measure of achievement provided validity and reliabilit y. A Mathematics Education faculty member who has taught the methods course for many years oversaw the writing of the departmental exam. All methods-course instru ctors were invited to share opinions about and contribute problems to the test. Thus th e final exam was a collaboration of several professionals with expertise in th e field of mathematics education.

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81 Exam questions were based primaril y on the content of the textbook. The distribution of exam questions from specifi c textbook chapters can be found in Table 4. Descriptions of specific test items are found in Appendix J. As a security measure, two similar forms of the final exam were used in the methods course so that each participant would have a different form than the classm ate sitting on either side. To determine the reliability of the final examination, a Kude r-Richardson formula 20 (KR-20) coefficient was calculated using the software program SA S. A KR-20 coefficien t of 0.71 (n=17) was found for Exam Form A, and a KR-20 coe fficient of 0.73 (n=16) was found for Exam Form B, indicating a relatively high de gree of reliability.

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82 Table 4 Content of Methods Course Final Examination Chapter Topic Frequencya 3 Developing Understanding in Mathematics 1 4 Teaching Through Problem Solving 1 5 Building Assessment Into Instruction 2 9 Developing Early Number C oncepts and Number Sense 1 10 Developing Meanings for the Operations 6 11 Helping Children Master the Basic Facts 1 12 Whole-Number Place-Value Development 4 13 Strategies for Whole-Nu mber Computation 5 14 Computational Estimation with Whole Numbers 4 15 Developing Fraction Concepts 10 16 Computation with Fractions 9 17 Decimal and Percent Concepts and Decimal Computation 6 an = 50 Data Analysis Surveys Scores for each participant on each of the four attitudi nal components, as well as a composite attitude score, were calculated at both the beginning and after the twelfth week of the semester. Possible scores on th e ATMI range from 40 to 200. The author of the ATMI provided scoring guidelines (Tap ia and Marsh, 2000). Most of the items,

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83 such as Mathematics is important in everyday life, use anchors of 1: strongly disagree, 2: disagree, 3: neutral, 4: ag ree, and 5: strongly agree. Ho wever, 11 items were reversed items, so the anchors were also reversed. An example of a reversed item is I am always under a terrible strain in a math class. Reversed items use anchors of 1: strongly agree, 2: agree, 3: neutral, 4: disa gree, 5: strongly disagree. The sc ore is the sum of the ratings. Therefore, a higher score reflects more posi tive attitudes than a negative score. The ATMI contains 10 items dealing with Value, 10 with Enjoyment, 15 with Self-Confidence, and 5 Motivati on. Because there are unequal numbers of items for each attitude factor, the average score per attit ude factor was also found in order to make comparisons more easily. These average per-item scores range from one to five. These scores were used for statistical analyses to answer research questions #1-3 using the software program, SAS. Question #1 asked: What are the attitudes toward mathema tics of preservice elementary school teachers entering an introductory mathematic s methods course? In particular, how do preservice teachers score on each of th e four attitudinal components being measured: value of mathematics, enjoyment of mathematics, motivation for mathematics, and self-confidence with mathematics? For question #1, only the scores from the beginning of the semester were used. Descriptive statistics were computed for pa rticipants’ composite attitude scores, their scores on each of the four attitudinal compone nts being measured, and on each individual survey item. These statistics included mean s, standard deviations, and measures of skewness and kurtosis.

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84 Question #2 looked at the precourse and post-course surv ey scores to determine if preservice elementary teachers’ attit udes toward mathematics changed during the methods course. Question #2 asked: To what extent do attitudes toward mathem atics of preservice elementary school teachers change during the mathematics me thods course? To what do students whose attitudes toward mathematics we re altered attribute this change? Because pre-course and post-course scores are no t independent, a repeated measures t test was conducted using composite survey scores to determine if a significantly significant change in attitude occurred. Each participant’s change score, which was their post-course score minus their pre-course sc ore, was calculated. Change scores could range from –160 to 160, where a negative change score represented a negative change in attitude and a positive change score repres ented a positive attitude change. Those with change scores greater than one standard devi ation above or below the mean change score were considered for individual interviews. These interviews examined to what these preservice teachers attributed this change. Question #3 asked: What is the relationship between preservice elementary teachers’ initial attitudes toward mathematics and their grade on the methods course final examination? A Pearson correlation coefficient was found in order to determine this relationship. The composite attitude score was the independent variable and the methods course final examination grade was the dependent variable. An alpha level of 0.05 was used to indicate whether the obtained correlat ion was statistically significant.

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85 Journals Question #4 asked: What do preservice elementary school teac hers’ reflective journal entries reveal about their attitudes toward mathema tics and the experiences that have influenced the development of those attitudes? Analysis of the qualitative aspects of the study involved l ooking for patterns. Hycner (1985) provided guidelines for the phe nomenological analysis of interview data, and his methods were utilized in this st udy. Although the guidelines were written for analysis of interview data, they are easily ad aptable for the analysis of journal data as well. After reading a journal entry for a sens e of the whole, units of general meaning were delineated. These units were record ed using the computer software program Ethnograph. When a journal expressed multiple themes, these themes were analyzed separately. A more extensive description of Hycner’s guidelines is found in Appendix K Once units of meaning had been identif ied for each journal entry for a given prompt, units of meaning from all journa l entries responding to that prompt were examined. Units of meaning relevant to the research questions were then clustered and common themes identified from the data. Themes were labeled using words that were introduced by the participants themselves wh enever possible, and frequencies of themes were noted. When excerpts from journal entr ies were cited, no names or other means of identification were given. Any names of peopl e, schools, etc. that were included were replaced by pseudonyms. In an effort to provide reliability, a graduate student who was familiar with Hycner’s guidelines repeated the coding process independently on a sample of 18

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86 journals. A stratified random sample of re sponses to this prompt was chosen using subgroups based on the participants’ ages a nd lengths of response. Table 5 shows how these samples were selected. Journal One was randomly selected for double coding. The researcher and the coder independently id entified both units of meaning and common themes from these journals. Prior to collabo ration, a comparison of identified units of meaning for the 18 journals produced an overa ll 71.6% inter-rater re liability. Differences were categorized into five groups. After co llaboration between the researcher and the coder, 100% agreement was reached. A sample of this process is shown in Appendix L. Table 5 Stratified Random Sampling of Journals For Double Coding Age Average Length of Groups Number Randomly Chosen Bracket Journal Responsea From Each Group Traditional Long 1 3 College Age Medium 2 3 (ages 18-22) Short 3 3 Nontraditional Long 4 3 College Age Medium 5 3 (ages over 22) Short 6 3 aShort responses ranged from 70 to 100 words, Medium from 145 to 206 words, Long from 216 to 370 words The graduate student who coded the sample journals also served as the classroom observer described previously. She was extr emely qualified to participate in both capacities. She is an experienced mathema tics teacher who was completing a doctoral

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87 program in Elementary Education with a cogn ate in Mathematics Education at the time of this study; she graduated in August 2005. Sh e had taught the elementary mathematics methods course several times and was therefor e very familiar with the content and nature of the course. Five journal entries from Pilot Study I were used for training and determining inter-rater reliability between the research er and the coder in identifying units of meaning. For this training, the researcher and the code r determined common themes together. Prior to collaboration, a comparison of identified units of meaning for five journals produced an 83% inter-rater re liability. After collaboration between the researcher and the coder, 100% agreement wa s reached. A sample of this process is shown in Appendix L. In investigating and describing how people experience a phenomenon, it is important for a researcher to utilize the pro cess of epoche. This involves first identifying, and then putting aside, any personal biases a nd preconceptions. Therefor e, this researcher attempted to record personal biases and pr econceptions prior to identifying units of meaning in the journals. Sitting alone in a quiet setting, I contem plated my own lifeexperience, focusing on personal experiences th at might influence my interpretations of the data. I then attempted to view the data without the influence of these preconceived notions. This record of the researcher’s pos sible personal biases and preconceptions is found in Appendix M. Due to the confidential nature of reflective journals, there was concern about potential bias in preservice teac hers’ responses if they were aw are that their journals were being used as part of a research study. Ther efore, they were not asked to sign consent

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88 forms for the journals until the end of the se mester. Although they were aware of their participation in the study from the beginning of the semester when they signed consent forms for the survey, they were not aware until the end of the semest er that the journals would also become a part of the study since journal writing was a c ourse assignment. All participants signed consent forms to allo w their journals to be used. Interviews Interviews were used to answ er question #5. Question #5 asked: What are the attitudes toward and expe riences with mathematics of those preservice elementary school teachers id entified as having the most extreme (either positive or negative) attitudes? Interviews were also used to answer the second part of question #2. Question #2 asked: To what do preservice teachers whose att itudes toward mathematics were altered attribute this change? The interviews were audio taped and later transcribed. Hycner’s guidelines were again used to identify common themes. After listeni ng to the tape and reading the transcription of an interview for a sense of the whole, uni ts of general meaning were delineated. Units of meaning relevant to the research questi ons were then delineated and clustered and common themes identified. Once a summary had been written for each individual interview, a second meeting was scheduled with each participan t. A discussion of agreement or disagreement with the research er’s findings allowed each participant to make any necessary corrections or additions Once any needed modifications had been made, themes common to multiple interviews were identified, as were any themes that

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89 were unique to a single interview or a mi nority of interviews. The use of surveys, reflective journals, and interviews al lowed for triangulation of the data. Limitations of the Study 1. The relatively small sample size could limit the quantitative elements of the study. 2. The use of an intact group could lim it generalizability to the entire population. 3. The researcher’s role as instructor of the methods course could limit the validity of the qualitative elements of the study. 4. Possible biases and preconceptions (Appe ndix M) could affect the researcher’s judgment.

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90 CHAPTER FOUR RESULTS The purpose of this study was to examin e the attitudes toward mathematics of preservice elementary school teachers ente ring an introductory mathematics methods course. The study focused on the following att itudes: value, enjoym ent, motivation, and self-confidence. Quantitative methods were used to measure these attitudes. Qualitative methods were used to explore these attitudes and the experi ences that have led to the development of these attitudes. The study sought to determine the extent to which preservice teachers’ attitudes toward math ematics changed during the methods course. The study also examined the correlation betw een preservice teachers’ initial attitudes toward mathematics and their achievement in the methods course. This chapter will present the researcher’s findings in addressing the research questions. Question 1: Initial Attitudes Toward Mathematics Question #1 asked: What are the attitudes toward mathema tics of preservice elementary school teachers entering an introductory mathematic s methods course? In particular, how do preservice teachers score on each of th e four attitudinal components being measured: value of mathematics, enjoyment of mathematics, motivation for mathematics, and self-confidence with mathematics?

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91 Survey scores for each participant on each of the four attitudinal components, as well as a composite attitude score, were ca lculated at the beginning of the semester. Possible scores on the ATMI range from 40 to 200. There are 10 items dealing with Value, 10 with Enjoyment, 15 with Self-C onfidence, and 5 with Motivation. Because there are unequal numbers of items for each at titude factor, the average score per attitude factor was also found in order to make co mparisons more easily. These average per-item scores ranged from one to five, with one i ndicating the most negative attitude and five indicating the most positive att itude. These scores were used for statistical analyses to answer research question 1 usi ng the software program SAS. Descriptive statistics were computed for participants’ composite attitude scores, their scores on each of the four attitudinal components being measured, a nd on each individual survey item. These statistics included means, standard deviations and measures of skewness and kurtosis. Participants’ initial survey scores were highest or most positive for Value of Mathematics, with a mean score of 3.96 on the 5-point scale ranging from strongly disagree to strongly agree. A score of one repr esents the most negative attitude, a score of three represents a neutral pos ition, and a score of five re presents the most positive attitude. The lowest or most negative scores were for Motivation, with a mean score of 2.55. Results from the initial su rvey using raw scores are found in Table 6. Results from the initial survey using average scores pe r attitude factor are found in Table 7. The individual survey item results are found in Table 8. Post-course individual survey items are found in Appendix N.

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92 Table 6 Initial Attitudes Toward Mathematics: Mean Raw Scores on the Attitudes Toward Mathematics Inventory Mean Standard Skewness Kurtosis Deviation Value 39.64 6.40 -0.46 -0.49 Enjoyment 27.94 9.97 0.28 -0.73 SelfConfidence 44.45 16.28 0.09 -1.06 Motivation 12.76 4.53 0.45 -0.54 Composite 124.79 33.83 0.11 -0.70 Note. Value and Enjoyment scores range from 10 to 50, Sel f-Confidence from 15 to 75, Motivation from 5 to 25, Composite from 40 to 200. Table 7 Initial Attitudes Toward Mathematics: Mean Per-Item Scores on the Attitudes Toward Mathematics Inventory Mean Standard Skewness Kurtosis Deviation Value 3.96 0.64 -0.46 -0.49 Enjoyment 2.79 1.00 0.28 -0.73 SelfConfidence 2.96 1.09 0.09 -1.06 Motivation 2.55 0.91 0.45 -0.54 Composite 3.12 0.85 0.11 -0.70 Note. Per item scores range from 1 to 5, with 1 indicating the most negative attitude and 5 indicating the most positive attitude.

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93 Table 8 Means and Standard Deviations on Items from the Attitudes Toward Math ematics Inventory Item Mean Standard Deviation SkewnessKurtosis Value of Mathematics 1. Mathematics is a very worthw hile and necessary subject. 4.39 0.56 -0.13 -0.89 2. I want to develop my mathematical skills. 4.24 0.66 -0.31 -0.66 3. Mathematics helps develop the mind and teaches a person to think. 4.18 0.73 -0.30 -0.99 4. Mathematics is important in everyday life. 4.12 0.82 -0.96 1.02 5. Mathematics is one of the most important subjects fo r people to study. 3.61 0.97 -0.43 -0.71 6. Math courses would be very help ful no matter what gr ade level I teach. 4.24 0.75 -1.38 3.16 7. I can think of many ways that I use math outside of school. 4.00 0.94 -0.98 0.44 8. I think studying advanced mathematics is useful. 3.24 1.09 -0.36 0.01 9. I believe studying math helps me with problem solving in other areas. 3.73 0.98 -0.69 -0.40 Continued on the next page

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94 Table 8 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Value of Mathematics (Continued) 10. A strong math background could he lp me in my professional life. 3.88 0.86 -0.71 0.29 Enjoyment of Mathematics 11. I get a great deal of satisfaction out of solving a mathematics problem. 3.18 1.04 -0.39 -0.56 12. I have usually enjoyed studying mathematics in school. 2.64 1.34 0.47 -0.73 13. I like to solve new problems in mathematics. 2.91 1.10 0.04 -0.17 14. I would prefer to do an assignment in math than to write an essay. 2.42 1.60 0.51 -1.45 15. I really like mathematics. 2.82 1.24 0.47 -0.59 16. I am happier in a math class th an in any other class. 2.06 1.06 0.88 0.40 17. Mathematics is a very interesting subject. 3.03 1.07 -0.22 -0.54 Continued on the next page

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95 Table 8 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Enjoyment of Mathematics (Continued) 18. I am comfortable expressing my own ideas on how to look for solutions to a difficult problem in math. 2.88 1.17 0 -1.02 19. I am comfortable answering questi ons in math class. 2.88 1.24 -0.17 -1.15 20. Mathematics is dull and boring.* 3.12 1.05 0.09 -0.14 Self Confidence with Mathematics 21. Mathematics is one of my mo st dreaded subjects.* 2.76 1.46 0.32 -1.28 22. When I hear the word mathematics, I have a feeling of dislike.* 2.79 1.24 0.43 -0.87 23. My mind goes blank and I am unable to think clearly when working with mathematics.* 3.36 1.19 -0.65 -0.60 24. Studying mathematics makes me feel nervous.* 3.00 1.27 0 -1.19 25. Mathematics makes me feel uncomfortable.* 3.15 1.28 -0.30 -1.14 Continued on the next page

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96 Table 8 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Self Confidence with Mathematics (Continued) 26. I am always under a terrible strain in a math class.* 3.06 1.34 -0.20 -1.09 27. It makes me nervous to even thin k about having to do a mathematics problem.* 3.09 1.31 -0.27 -1.15 28. I am always confused in my mathematics class.* 3.24 1.12 -0.23 -0.68 29. I feel a sense of insecurity when attempting mathematics.* 3.03 1.31 -0.15 -1.21 30. Mathematics does not scare me at all. 2.58 1.23 0.36 -0.92 31. I have a lot of self-confidence when it comes to mathematics. 2.70 1.10 0.21 -1.01 32. I am able to solve mathematics problems without too much difficulty. 2.97 1.07 -0.26 -0.94 33. I expect to do fairly well in a ny math class I take. 3.18 1.07 -0.22 -1.24 Continued on the next page

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97 Table 8 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Self Confidence with Mathematics (Continued) 34. I learn mathematics easily. 2.82 1.13 0.38 -0.58 35. I believe I am good at solving math problems. 2.85 1.00 -0.07 -0.51 Motivation with Mathematics 36. I am confident that I could learn advanced mathematics. 2.85 1.28 -0.18 -1.22 37. I plan to take as much mathematics as I can during my education. 1.94 0.79 0.52 -0.04 38. The challenge of math appeals to me. 2.39 1.06 0.63 -0.21 39. I am willing to take more than the required amount of mathematics. 2.03 1.21 1.06 0.33 40. I would like to avoid teaching mathematics.* 3.55 1.00 -0.33 -0.94 Note. Martha Tapia. ATMI used with permission of author. Scoring for most items uses anchors of 1: strongly disagree, 2: disagree, 3: neutral, 4: agree, and 5: strongly agree. *Scoring for these items is reversed and uses anchors of 1: strongly agree, 2: ag ree, 3: neutral, 4: disagree, 5: strongly disa gree. Therefore, on all items, scores range from 1 to 5, with 1 indicating the most negative attitude and 5 indicating the most positive attitude.

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98 Question 2: Changed Attitudes Toward Mathematics Question #2 asked: To what extent do attitudes toward mathem atics of preservice elementary school teachers change during the mathematics me thods course? To what do students whose attitudes toward mathematics we re altered attribute this change? Change Scores Pre-course and post-course scores are not independent, so a repeated measures t test was conducted using composite survey scores to determine if a significantly significant change in attitude occurred. Each participant’s change score, which was their post-course score minus their pre-course sc ore, was calculated. Change scores could range from –160 to 160, where a negative change score represents a negative change in attitude and a positive change score represen ts a positive attitude change. An alpha level of 0.05 was used to determine whether the results were statisti cally significant. The mean change score for the 33 participants was 17.03 ( SD = 17.59). The median change score was 15, and there were four modes: 7, 11, 15, and 20, each with a count of 2. The change scores were slightly positively skewed (Sk=0.29). The kurtosis was –0.46, indicating that the distribution was platykurtic. The mean per item change score was 0.43 on a five-point sc ale. Scores for all four co mponents increased, with SelfConfidence having the largest per-item change score. Per item change scores for all components are found in Table 9. A repeated measures t test was used to test the null hypothesis that the mean change score in the population was zero. Because t = 5.561 > 2.04 (tcrit), and p < 0.0001, the null h ypothesis was rejected.

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99 Table 9 Change Scores Per Attitude Factor : Mean Per Item Scores Pre-Course Survey Post-Course Survey Change Score Value 3.96 4.19 0.23 Enjoyment 2.79 3.28 0.49 SelfConfidence 2.96 3.48 0.52 Motivation 2.55 2.98 0.43 Composite 3.12 3.55 0.43 Note. Per item scores range from 1 to 5, with 1 indicating the most negative attitude and 5 indicating the most positive attitude. The validity of the repeated measures t-test depends on the assumptions of independence and normality. Although the pre-cour se and post-course survey scores were dependent or repeated measures, the change scores were independent The distribution of change scores was slightly positively skewe d. However, because n (33) > 20, a repeated measures t-test is relatively robust to viol ations of the normality assumption. The effect size, d = Xdiff / S diff, was 0.968, indicating a large e ffect size. In summary, it was possible to reject the null hypothesis of a mean change score of zero (t(33)= 5.561, p < 0.0001). There was a statistically signifi cant positive attitude change. The ATMI was also administered during w eek one and week twel ve to students in two other sections of the mathematics methods course for the purpose of making comparisons in results. Two different instru ctors taught these met hods course sections. The mean composite initial survey score for the first comparison class was 140.2. The mean change score for the 24 participants in the first comparison class who completed

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100 both surveys was 1.0 ( SD = 12.75). The median change scor e was 0, and there were two modes, 0 and 7, each with a count of 3. The change scores were slightly positively skewed (Sk=0.44). The kurtosis was 0.63, indicati ng that the distribut ion was leptokurtic. A repeated measures t test was used to test the null hypothesis that the mean change score in the population was zero. Because t = 0.384 < 2.07 (tcrit), and p = 0.7044, the null hypothesis was not rejected. The mean composite initial survey sc ore for the second comparison class was 140. The mean change score for the 31 partic ipants in the second comparison class who completed both surveys was 1.48 ( SD = 13.08). The median change score was -2, and the mode was -3, with a count of 4. The change scores were moderately positively skewed (Sk=1.16). The kurtosis was 1.82, indicating th at the distribution was leptokurtic. A repeated measures t test was used to test the null hypot hesis that the mean change score in the population was zero. Because t = 0.631 < 2.04 (tcrit), and p = 0.5325, the null hypothesis was not rejected. In summary, it wa s not possible to reject the null hypothesis of a mean change score of zero in either of the comparison classes. Neither comparison class demonstrated a statistically si gnificant attitude change. Changed Attitudes Interviews Statistical analysis revealed that six participants had positive change scores greater that one standa rd deviation above the mean change score. This reflected a change score of at least 35 points. These significan t positive change scores ranged from 36 to 58. The four participants with the greatest posit ive change scores were interviewed in order to explore to what these partic ipants attributed th eir positive attitude change. In addition,

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101 six participants had negative change scores gr eater than one standard deviation below the mean change score. This reflected a change score less than -0.56. These negative change scores ranged from -12 to -1. The four part icipants with the greatest negative change scores were also asked to pa rticipate in interviews. Three of the four agreed to be interviewed. The fourth partic ipant did not respond to the researcher’s requests for an interview. All of these interviews took place four to six weeks afte r the completion of the methods course and submission of final grades. ‘Amelia’ Amelia’s change score was 58, the highest in the class, indicating the greatest positive attitude change. Her score on the pr e-course ATMI was 97 out of a possible 200 points. This represented a mean response of 2.4 per survey item, w ith 1 representing the most negative response and 5 the most posi tive response. Her post-course ATMI score was 155, representing a mean response of 3.9 per item. Her scores increased in all four components: Value increased by 7 points, Enjoyment by 16 points, Self-Confidence by 26 points, and Motivation by 9 points. Amelia began by saying that she believed that her attitudes toward mathematics had improved. She explained that in previous mathematics classes, she had been taught how to do mathematics, but she had not al ways understood the underlying concepts: I’m a lot more comfortable in teachi ng math and math itself…. [Previous teachers] said, “Ok, this is how you do it,” but they never said why. Now that I grasped it in your class so well, I’m very excited to teach it to other kids now. That’s how [my attitude] changed. I [previously thought], ‘I hate math, I don’t want to teach it,’ and now that I’m excited; I’m like, ‘OK, it makes sense to me,’

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102 so I know I can make it make sense to the children. When asked which aspects of the methods c ourse she thought had affected her attitude toward mathematics, Amelia first menti oned the journals and the opportunity for reflection that they provided: The journals, actually, had a big impact because [journal writing] made you sit back and actually think about … the wa y you were taught, and do you want to teach that way, and so on and so forth, and that gave me a chance to [reflect] with the prompting of the questions. Amelia also said that the manipulatives that were used in the course had positively affected her attitude: And also the use of the manipulatives, knowing how to use them, and being able to be comfortable to show someone else how to use them. Because I had people sitting next to me [saying], ‘I don’t unders tand,’ and I was able to explain it to them, which made me comfortable; if I can explain it to an adult, I surely can explain it to a child. I thi nk those [journals and manipul atives] are the two that really stuck out to me that made me sa y, ‘You know what? I think I can do this,’ as opposed to at the beginning of the cla ss where I was [thinking], ‘No, that’s not happening.’ Amelia said that she believed that manipul atives would be useful for mathematics students at all levels: I think all the way through high school you should use [manipulatives] because me, being even a college student, it helps me out a lot to see it in front of me … Like I said, it helped me to learn how to teach [mathematical concepts] to myself

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103 … It gave me a different view on [m athematics] using the manipulatives. When asked about the use of cooperativ e learning, Amelia said that her experience as a substitute teacher had helped her to understand the benefits cooperative learning provides: I’m all about cooperative learning. Substituti ng, I change the seats around because if they were not in groups, I didn’t like it. I believe you can learn from your peers, and even if you might be [more adva nced] than your other peer, that peer can still learn from the lower…. I think it he lps, especially with the lower ones. [The teacher] can’t be everywhere all th e time…. having someone that may be a little bit more advanced helps to show them … and they can explain maybe a little bit different than I can; a child’s pers pective to it as opposed to mine. Amelia also found the use of cooperative learning in the methods course to be beneficial: In the actual methods class, I thought it helped a lot … even being adults [some people] didn’t get it, and it’s a little bit more fun, a little bit more inte ractive than just sitting there and hearing a lecture all the time. It gave us a break and let us use manipulatives with someone else so that we could see how it would be with the child. Amelia said that she “hated problem solving, …especially word problems,” and that she had “always had a hard time” with them. She did appreciate the value of putting a computation problem “into a context that [ch ildren] can relate back to.” She said that, “It helps a whole lot more than putting 30 + 80. Putting it into word problem solving, solving it out … helps the kids a lot.” As for problem solving in the methods course, Amelia said that “those parts were hard where we had to actually do the [problem

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104 solving].” She did benefit from class discus sions regarding how ch ildren might approach problems: I [thought], ‘Oh, a child would get this.’ By you teaching us the methods in the methods class, [I thought], ‘Oh, you know, if you do relate it, it might help this way,’ and you showed us a way to relate it. So I learned a lot, actually, for problem solving. Amelia said that “journals get overlooke d too much in math” and are used more for English. She recognized the benefits of journal writing as a form of assessment: The students being able to reflect on how th ey think they’re doing in math or just put down their thoughts, anything within a journal, gets a better view for the teacher because sometimes you don’t know if a child’s struggling. I mean, you can see it here and there, but I think journals help a lot… You [as a teacher] get a feel for your students, where they’re at, how they feel, and you just get a different insight than you would just sitting there t eaching … You really get an insight into what they’re thinking, what background they come from, as far as math and their attitudes… So I’m for journals all the way around. I love them. At this point in the interview, the researcher showed Amelia her preand postcourse survey scores and the change scores for each attitude co mponent. Viewing these scores confirmed Amelia’s belief that her at titudes toward mathematics, especially her self-confidence, had improved: Seriously, that class was a great class, a nd I really thought I was going to hate it going in, and coming out I’m so much mo re … confident about teaching math. I’m ready to get in there and do it!

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105 ‘Jennifer’ Jennifer’s change score was 45, the second highest in the class, indicating the second most positive attitude change. Her scor e on the pre-course ATMI was 113 out of a possible 200 points. This represented a mean response of 2.8 per survey item, with 1 representing the most negative response and 5 the most positive response. Her postcourse ATMI score was 158, representing a m ean response of 4.0 per item. Her scores increased in all four components: Value in creased by 6 points, Enjoyment by 9 points, Self-Confidence by 24 points, and Mo tivation by 6 points. Jennifer said that she believed her attit ude toward mathematics had “totally changed” [in a positive way]. She explai ned that she had never been a strong mathematics student. When beginning a new ma thematics course, she would tell herself, “Well, I’m not good at math so I’ll settle for a ‘C’.” Then she “had a great teacher” for Social Science Statistics. She “ended up getting an ‘A’ in that course for the first time,” and her self-confidence began to improve. She thought, “OK, maybe I’m good at math.” As a result, Jennifer began the methods course believing that if she “tried really hard and studied, it would be possible to get an ‘A’:” … coming into here [methods course], I still was kind of uneasy, but now after [the methods course], I felt like, ‘All ri ght, I can do math, no question about it.’ I’m even willing to get up and teach kids math when before I would say that was the one subject that I did not want to teach. I [thought], ‘Give me a co-teacher because I don’t want to teach that.’ But now I feel like I could step in there and actually teach math.

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106 When asked which aspects of the course she thought had aff ected her attitudes toward mathematics, Jennifer first mentioned the format of the classes. She appreciated the copies of overhead transparencies used in class that the instructor made available for students to purchase thro ugh a local copying service: In math class normally you are looki ng up, writing, looking up, writing. With the packet … I could sit and real ly pay attention, fill in the blanks, … it just worked much better than saying, ‘Chapter one; you n eed to read this and then we’ll come in and do math problems.’ That’s what I’m used to doing [in math classes], like [a teacher says], ‘This is what you should be reading, we’ll just go over the homework and then learn new examples,’ where this is more hands-on and you feel like you can actually pay attentio n without having to look up, writing, look up, writing, look up, writing. Jennifer also said that the jour nals had influenced her attit ude change. She explained that journal writing provided an excellent means of communi cation between students and teacher: They [journals] made me feel like you r eally care about our math experience and that [students] could be more open about, like ‘I don’t understand this,’ or ‘I’ve had a problem with this in the past.’ It wa s a way outside of class that [a student] could communicate to [the teacher] even if you were shy about certain things or you knew where we were all coming from at the beginning of the course. Jennifer especially appreciated knowing that the instructor was reading and responding to the journals: You actually care about where we’re co ming from; why do we feel this way.

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107 Maybe if I had sent them and you didn’t reply back, I might feel like you were just doing them for, you know, extra points. But the fact th at I knew that you were reading every single one of them and you had something to say back [made me think], ‘Well, OK. The line of communicat ion is open. You’re not just a teacher who taught us math. You are someone w ho cared about where we were coming from and how we’re progressing’…. It made me [think], ‘She’s taking time to read this, so I need to take time to sit down and write something that’s meaningful.’ When asked about the use of manipulativ es, Jennifer said that she thought it was “a great idea” to use them in the elementary school classroom. She had found that using manipulatives was very effective during her internship. Some of her students “weren’t quite getting it,” so she used manipulatives to help them understand. Jennifer also found the use of manipulatives very help ful to her in the methods course: Oh, I really liked [manipulatives]…. I’m a visual learner, so when I can move things around and do hands-on visuals, th en I can actually see what I’m doing. Like with fractions, cause I really dislike fractions, [manipulatives] made it that much easier for me to understand…. Like w ith the fractions, using the pattern blocks with fractions; that made so much sense, like if you can’t fit four of them in there than it can’t be of that main object [unit]…. You can actually see the reasoning behind it. Jennifer said that she thought using c ooperative learning in the classroom was “fine” because you “just turn to the person next to you.” However, she had reservations about the use of cooperative l earning for assignments or fo r studying together outside of

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108 class. She explained that she had benefited fr om this type of cooperative learning in the methods course because she had a classmate with whom she worked well: When I’m in other classes where I don’t know anyone, I really dislike it because they don’t pull their part…. Some times I really like it, bu t I think it’s because I had my one main person [to work with]. I know we work well together. I know we both won’t settle for anything less than an ‘A.’ The fact that I had [classmate’s name], I know that we will work well together… when it’s stuff like [an assignment] or studying together, knowi ng that that person will be there to study with you, just having the extra support. Jennifer said that problem solving was “important because you have to use it throughout your whole life, not ju st in math.” She thought that problem solving should be included “in every single subject.” She believed that her problem-solving skills had improved as a result of the methods course: That was my problem before. If it didn’t work this time, then I would think, ‘Alright, I’ll just wait.’ But now I [think,] ‘If it can’t work this way, let’s turn around and try it this way.’ I pick something else that I know and try to apply it to that. When asked about the use of journal wr iting in teaching mathematics, Jennifer said that she thought it was important: In mathematics you don’t normally think a bout journaling. When you think about journaling, you think English a nd things along that line. Bu t I think it’s important though, like if you don’t understand somethi ng and you’re too afraid to say something about it [in class], then you can write it in your journal. Or you can

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109 write, “I’ve had a really hard time in mat h, but this semester I’m really trying to do good,” and things along that line…. I rea lly enjoyed [the journals]…. It was something I looked forward to doing. At this point in the interview, the re searcher showed Jennifer her preand postcourse survey scores and the change scores for each attitude component. When asked if there was anything else she would like to add about her attitude change or to what she attributed this change, Jennifer said: Just that, from the [methods] course, so ma ny kids may just decide that they can’t do math, and it just takes going about th ings differently, like maybe changing learning styles or setting up study groups and things along those lines, to change an attitude…. I can finally say, ‘Hey, I can do this.’ I may be 23, but now I’m finally realizing that [math] is not that ha rd. If I really apply myself, I have to do the [review] exercises, I have to have a study group before the test. So that’s how my attitude has changed. Before, I thought, ‘Oh, just do the homework and show up.’ I know now that I have to say, ‘OK, it’s Sunday night. Let’s get together to study so we can do good.’... Everyone that we work with all started making ‘A’s on their tests, so it’s just a positive th ing because not only are you doing that for yourself, but also for the person next to you. Li ke, it’s a good part of a friendship. ‘Erin’ Erin’s change score was 44, the third highest in the class, indi cating the third most positive attitude change. Her score on the pr e-course ATMI was 86 out of a possible 200 points. This represented a mean response of 2.2 per survey item, w ith 1 representing the most negative response and 5 the most posi tive response. Her post-course ATMI score

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110 was 130, representing a mean response of 3.3 per item. Her scores increased in all four components: Value increased by 8 points, Enjoyment by 14 points, Self-Confidence by 20 points, and Motivation by 2 points. Erin believed that her attitude toward mathematics had improved since the start of the methods course. She attributed this change primarily to the use of manipulatives and the focus on conceptual unde rstanding in the course: I know [my attitude] changed after the class because a lo t of it was I was not very confident in [math] … I did have a lot more self-confidence at the end of the semester, definitely… I think it was the teachi ng of the concepts that helped a lot, too ‘cause so many times before with math it was like, ‘formula, ok.’ But when you have to understand, it makes it a lot easier …. I found myself ve ry visual with a lot of things and we worked with manipul atives so that helped a lot. And later on in my math career when I started to go downhill with it, it wa s very much like there was no hands-on. It was like, an overhead and teacher ditto…. Also because I had someone [classmate] in there that I knew it also helps with that process so we were able to get together and I coul d even further my understanding of it. Erin said that the use of manipulatives in teaching mathematics was “absolutely necessary” because “it’s such a hands-on thing and by doing that it really helps to cement in what you’ve been learning about.” She reinforced her feeling that the use of manipulatives in the methods course was “v ery helpful … probably the number one thing that helped” her in the course be cause she was a “visual learner.” When asked about the use of cooperative learning in teaching mathematics, Erin said that it is “a really good thing just because some kids are at different levels so when

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111 they get together they are able to teach each other, which is great.” She found the use of cooperative learning in the methods course help ful to her. She explained that sometimes she “didn’t understand a concept or somethi ng and getting together with other people helped a lot.” Erin said that problem solving was one of her “main struggles.” She remembered that there was “a lot” of problem solving in the methods course. She found it “hard at first,” but once she “got the concept down, it was real easy because we would always use manipulatives.” Erin believed that journal writing wa s useful in teaching mathematics: I think it’s good just because it’ll help you [teacher] relate to your students better and, especially if you do math problems w ithin the journaling, you’ll be able to see if they’re [students] st ruggling in a subject and you’ll be able to help out with them. She also shared her views on the use of journal writing in the methods course: In the course, I thought it was good because you [instructor] were able to see how we came to feel about certain subjects in math by the questions that you asked, and it was very good for me because I had forgotten about some things and having to think about it, why my attitudes were a certain way, I was [thinking], ‘Oh, yeah. THAT’S why I don’t enjoy math.’ At this point in the interview, the resear cher showed Erin her preand post-course survey scores and the change scores for each attitude component. Erin seemed to be aware that her attitudes had improved. She responded, “I did have a lot more self-

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112 confidence at the end of the semester, defin itely.” When asked why she thought her selfconfidence had increased, Erin said: I felt a lot more confident in math, like I just understood it. Understanding’s a big part of it because, definitely. With my past experiences with math, not understanding it is very di scouraging so [I would th ink,] ‘Hmm, I don’t really care. I don’t enjoy it.’ I’d just sit th rough the class, wait for it to end. When asked if there was anything else she woul d like to add about her attitude change or to what she attributed this change, Erin sm iled and said, “I like math a lot more!” ‘Tessa’ Tessa’s change score was 41, the fourth high est in the class, i ndicating the fourth most positive attitude change Her score on the pre-course ATMI was 90 out of a possible 200 points. This represented a mean response of 2.3 per survey item, with 1 representing the most negative response and 5 the most positive response. Her post-course ATMI score was 131, representing a mean response of 3.3 per item. Her scores increased in all four components: Value increased by 9 point s, Enjoyment by 10 points, Self-Confidence by 20 points, and Motivation by 2 points. Tessa said that she was sure that her attitude toward mathematics had improved since the start of the methods course: Well, I know for a fact that it did change ‘cause I know in my previous years, math was a dreaded subject, [I was] very intimidated by it. I don’t like it, I don’t want to have anything to do with it… Bu t now after I took th e course, the way that we used the manipulatives; it was just so much easier for the kids to grasp now. If we would’ve had that back then, I know that math would have been one of

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113 my favorite subjects. I know for a fact So, I do know that my attitude has changed and I’m thankful for it ‘cause I know I can take my positive attitude back to the classroom and I know that’s a big help. Tessa believed that the use of manipulativ es in the methods course had positively affected her attitude toward ma thematics. She also felt less pressure in this course than she had in previous mathematics courses because she was able to understand concepts and not fall behind. She thought that this pos itively affected her attitudes as well: I know there were a couple [of things th at affected attitude]. The manipulatives were definitely one … Those really helped out. And there wasn’t a lot of pressure in the class. It was very, not laid-back, but it wasn’t like, ‘you have to learn this in order to learn this.’ ‘C ause I know in my previous math classes, like in high school [teachers said], ‘Well, if you don’t ge t this concept, then you’re definitely not going to get that,’ and that’s how all math is, but I got it the first time [in the methods course] so it was easier to get one concept and go to the next. When asked about the use of manipulatives in mathematics classrooms, Tessa said that she liked “the idea of having an ove rhead set as well as a student class set.” She felt that it had helped her in the methods course and that it would help children to see the teacher using the manipulatives on the overhead projector while they modeled a problem using their own manipulatives. She said that “t he kids could see it and they can physically have it in front of them as well as seeing it on an overhead.” Tessa also felt that she had benefited from using the manipulatives on the tests. She found that when she used the manipulatives, she “could just do the problem, and it was a piece of cake. It was a lot easier.”

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114 Tessa supported the use of cooperative learning in mathematics classrooms, but she also felt that “it’s very important that the kids have independent work as well as group or partner work.” At first Tessa was a bit uncomfortable with the cooperative learning activities in the methods course. She didn’t know many of her classmates and felt “a little shy in the beginning.” However, she soon realized that “the classroom was very relaxed and there was not a lot of pressu re; it was ok to ask my neighbor for help.” She eventually felt very comforta ble asking a classmate for help: Towards the end, if I didn’t get somethi ng, my neighbor did, or the group that we were working with did, then I could [say ], ‘Wait, how did you get that?’ and I wasn’t intimidated, I wasn’t scared to as k them the answer or to show me how they got it and then I could try another one by myself if that was the case, but I know [cooperative learning] definite ly helped out in the course. As she was remembering her discomfort w ith cooperative learni ng activities at the beginning of the methods course, Tessa re called some of the feelings she was experiencing at the time and how th ey changed during the semester: I came in there [methods course] very hesi tant and very nervous and so I was all high strung and everything, but then I was [thinking], ‘Oh, it’s not going to be bad, it’s ok! It’s fine, it’s just math class, it’s OK.’ That’s how I felt towards the end. I was like, ‘Yea, I’m going to math cl ass! What can I learn now with these manipulatives?’ When it came to problem solving in the methods course, Tessa felt very “hesitant” and uncomfortable about not being able to so lve problems by herself. It helped her to know that if she could not solv e the problem herself, she could work with her partner.

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115 She also felt more comfortable about problem solving due to something she had learned in the methods course: From the class, I learned there’s not ju st one way of doing something, and that’s how we used to be taught in our younge r years. There are obviously definite answers for some questions, but there are di fferent ways of going about to get the answer…. From the beginning, I was [thi nking], ‘Oh, I don’t know how to get it,’… but by the end of the class I was very surprised, I was [thinking], ‘Oh, well I could do it this way, or I could do it this wa y,’ and as a teacher I get to be able to express that to the students that there ar e different ways to solving a problem. Tessa said that at the beginni ng of the course and in past courses when she had trouble solving problems, she felt “frustrated.” By th e end of the methods course, she was feeling much more confident about he r problem-solving abilities: I was frustrated ‘cause that was the only way that I knew how to do that problem and if I couldn’t get it, then I felt dumb or I felt stupid or I felt that I wasn’t good in math, but at the end [of the methods course], I was [thinking], ‘Oh, I got it!’ and then ‘Oh, I could do it this way or this way!’ and I was [thinking], ‘Oh, you know what? I can do math!’ I would tell my mom all the time, ‘Mom, my math class is going so good!’ ‘cau se she knows that I’ve had a really big problem with it all through the years so it definitely has changed. Tessa said that she “enjoyed the journals .” She remembered that in her education classes, she had been encouraged to use refl ection as a teacher. She recognized that the journals had allowed her to use re flection relating to mathematics: It [journal writing] allows me to gras p my thoughts, grasp my feelings, and not

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116 only think about them, but put them down, not necessarily on paper, but on the computer physically. It led me to think of other aspects of math that I didn’t think about before…. The journals were great I did like the journals a lot. At this point in the interview, the rese archer showed Tessa her preand postcourse survey scores and the change scores for each attitude component. These scores supported Tessa’s belief that her attitudes had improved, especially self-confidence. She responded, “Yeah, that one probably went up the hi ghest. And I feel it, too, ‘cause I am a lot more confident about math and teaching as well.” ‘Stephanie’ Stephanie’s change score was -12, the lowe st in the class, indicating the most negative attitude change. Her score on the pre-course ATMI was 141 out of a possible 200 points. This represented a mean response of 3.5 per survey item, with 1 representing the most negative response and 5 the most positive response. Her post-course ATMI score was 129, representing a mean response of 3.2 per item. Her scores for the four attitude components changed as follows: Value decreased by 10 points, Enjoyment increased by 1 point, Self-Confidence decrea sed by 1 point, and Motivation decreased by 2 points. It should be noted that changes in Enjoyment, Self-Confidence, and Motivation were too small to be either statistically or practically significant. Stephanie said that she believed her atti tude towards mathematics had improved since the start of the methods course. Her fi rst thoughts were of th e after-school program she runs and how she has already been able to apply what she learned in the methods course:

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117 I definitely think my attitude has change d about math. The biggest thing, I guess, for me is with the after-school thing, is trying to help the people who are helping [the students] with their homework unders tand that [teaching math] is not the way it used to be … One of the biggest things that I use from your class is the base ten rods. Even when I’m helping the kids w ith their homework, I’ll draw [base ten blocks] and I’ll ask them, ‘Do you recognize that? Do you do that in class?’ and they do, and it always helps them get it. It’s really awesome…. I was really fearful about teaching math, but I think that the manipulatives helped, and the book also…. and the reflections were really, really helpful. When asked if there was anything else she would like to a dd about the use of manipulatives in teaching mathematics or in the methods course, Stephanie said that she used “a lot of it to help the kids [in afte r-school program] with their homework.” She has found that the manipulatives “really helped [h er students] make that connection” between multiplication and repeated addition. She r ecalled discussing manipulatives with her sister and “telling her about all the manipula tives and how we have one for fractions and the [fraction] circles, and they make everyt hing so hands-on and it’s really awesome.” Stephanie later mentioned that she “lik ed that [instructo r] had the overhead [manipulatives] so you could show us how to do it as we were doing it.” Stephanie thought that the use of coopera tive learning in mathematics classrooms was “really great. It’s really good for skill building.” She felt that the cooperative learning activities in the methods course we re “good. We were learning how to use the manipulatives.” Stephanie made it clear that she did not like coope rative learning when it involved assignments:

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118 I just don’t like when we have to colla borate and turn stuff in … someone ends up always doing more, namely me, and I re ally don’t like that. I’m taking ESOL online, I’m doing most of the work fo r my group. I absolutely hate it. When asked about the use of problem solving in teaching mathematics and in the methods course, Stephanie just said, “I definitely feel like I’m going to use stuff that I used in the class and the problem solving that we talked about.” Stephanie said that she found the journals “really helpful.” She added, “I’ve even gone back… I go back and read them sometimes and they’re really helpful. They made me think about a lot of stuff that I pr obably wouldn’t have thought about unless you prompted me.” At this point in the interview, the rese archer showed Stephanie her preand postcourse survey scores and the change scores for each attitude co mponent. Stephanie was astonished that her score had decreased, a nd she said once again that she thought her attitude toward mathematics had improved si nce the start of the methods course. She looked at her responses on the post-course survey and observe d that she had not answered ‘strongly agree’ to any of the survey items. She said, “I’m just thinking that, maybe that day, I didn’t feel like I would ‘strongly agre e.’ … because I didn’t write ‘strongly agree’ for anything .” Stephanie then reminded the researcher that Stephanie’s mother had passed away during week 10 of the semester, which was only a few weeks before the participants completed the post-course surv ey. She suggested that perhaps she was not feeling overly positive about anything during that time. When asked if there was anything else that she would like to add about her attitude change or to what she attr ibuted this change Stephanie said:

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119 I just want you to know that I really do use this stuff and I’m trying to train people who work with my after-school program. They’re not elementary ed. majors so they really don’t know everything that I know. I even told my boss, ‘I teach a math training course.’ Because people are helping them [students in after-school program], but they don’t know the right thing to do and they’re just confusing them. Stephanie also added someth ing concerning the journals: When I sit home and I’m writing my reflections and I’m thinking about what I want to say, that’s the time when my emotions aren’t play ing any part in it because I have the computer and I can t ype it and I can write the thing and say, ‘Oh, I don’t want to say that.’ ‘Shelly’ Shelly’s change score was -10, the sec ond lowest in the class, indicating the second most negative attitude change. Her sc ore on the pre-course ATMI was 155 out of a possible 200 points. This represented a m ean response of 3.9 per survey item, with 1 representing the most negative response and 5 the most positive response. Her postcourse ATMI score was 145, representing a mean response of 3.6 per item. Her scores for the four attitude components changed as follows: Value decreased by 6 points, Enjoyment decreased by 4 points, Self-Conf idence decreased by 2 poi nts, and Motivation increased by 2 points. It shoul d be noted that possible change scores ranged from –160 to 160, so these changes were too small to be of statistical or practical significance. Shelly was unsure when asked how she thought her attitude toward mathematics had changed since the start of the methods course:

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120 I don’t know. My attitude about math, I think, is constantly changing, all the time, and sometimes it might just be that day or certain subject s or topics that I have in my head. It’s certain categories or something that to this day still just rub me the wrong way. I can’t get them down. I don’t know; I think a lot of it has to do with the teachers that I’ve come acro ss because everybody has such different teaching styles. Shelly said that she had done well in the me thods course and that she appreciated the organized format of the course: I thought of [the course] as very organized … There we re no surprises, you didn’t throw any surprises at us. You told us thi ngs in particular and it was really clearly stated of how we could go about it and how we could teach it and I just really enjoyed it.… And I remember everything that you did with me and I’ll know how to use that with other kids and then ma ybe they’ll have a better attitude about math, too, because I wasn’t taught the way that you taught or showed me. I had a negative attitude about math for a very long time. Shelly said that she “loved” using the manipulatives in the methods course and that she was already thinking about how she would use manipulatives with her future students: Even when I did that mini-lesson plan that we had to do for your class, I just was thinking of so many ways I could use mani pulatives, and I thi nk that every math problem, like I want to try to use a mani pulative, for the visual and kinesthetic learners. Any math problem that some how provides an example, a hands-on example, I think would probably be the be st way to go. I think that’s great.

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121 Shelly felt that the use of cooperative learning in teaching mathematics was “very important.” She added, “anything where you’re involved w ith other people and figuring things out together, es pecially with peers, you can le arn from their perspective.” When asked about the use of cooperative learning in the methods course, Shelly said: As long as that’s not a constant thing, I think that it wouldn ’t ever bring my attitude down because I know that a lot of people, especially with math, they do like to go back and work on their own a nd do their own work and get their own answer like with problem-solving and skill prac tice. They want to figure it out for themselves and usually if they can’t get it, then cooperative learning is important coming on that end of the deal. But then also you go back and have some independent or individual learning … I don’ t like to start out doing a problem by myself if it’s major or important or just fo r play or fun. I just like to have someone show me and show me and show me agai n. Eventually I do want to do it and I do want to get the right answer and then I al so do want to be able to show someone else so they’ll know how. Shelly viewed the use of problem solv ing in teaching mathematics as “good” and important for children because problem solv ing “is going to help you advance or think deeper, and that’s important for teachers to be able to make their kids do that.” However, she thought that for a child, “it could be a turn -off.” Shelly said that she sometimes found word problems to be “intimidating.” Although she considers herself a “critical thinker,” Shelly doesn’t really like word problems:

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122 I like the basic arithmetic, multiplying, st raight numbers, and I don’t really like word problems. I don’t know why I don’t. I don’t know if it’s because I think there’s a hidden trick or clue that I’m going to forget or skip, and I’m just going to mess the whole thing up. Shelly felt that she had benefited from the reflection that the journal writing in the methods course had provided: They [journals] make you not forget your own little histor y, your little pattern of life, like your personal math diary that you thought you were never going to have, but it’s good. It reminds you of the ups and downs and the positives and negatives, the things that are important…. Everybody needs time to reflect. At this point in the interview, the researcher showed Shelly her preand postcourse survey scores and the change scores for each attitude component. Shelly was surprised to learn that her attitude scores had decreased: I’m surprised…. When I think of that and wh at you’re saying, I really think that it probably would have gone the other way. I do and I think that it has to do with the way that you’ve taught me how to teach children which may have been different than the way that I learned so I just mi ght see it differently now in my head and the whole visual aspect, and then the presen tation of it. I don’t know. I think that you’ve cleared up a few things, even t hough we only did some basic and simple math for pretty young children. I don’t know. I’ve picked up a lot from your class and I think that my attitudes would’ve gotten better just because a lot more makes sense now and that’s just kind of surprising to hear that it went the other direction.

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123 When asked if there was anything else th at she would like to add concerning her attitude change or to what she attr ibuted this change, Shelly responded: I just really wish you could have taught all my math classes, I really do. You actually remind me of one of my other teac hers that I wrote ab out in my journals; she taught Algebra, Geometry all through high school. You kind of resemble her, I don’t know why, in a way. If I could ha ve just had you two, I probably would have gone through Calculus, I think. It’s no t that I’m turned away or intimidated by those upper level subjects. As I said be fore, when I take Math II, I’m taking it with you. I have to. This will probably be my last math class for the rest of my life … I hate the feeling I get when I go into a math class and I ge t, ‘Uhhh, it’s math, so I’m a little nervous.’ And you just don’ t know what’s expected. You just never know how it’s going to be. But at least with you I know how it’s going to be, and I feel confident, and I feel like I’m kind of interested about doing it. ‘Yezania’ Yezania’s change score was -5, indicati ng a negative attitude change. Her score on the pre-course ATMI was 80 out of a po ssible 200 points. This represented a mean response of 2.0 per survey item, with 1 repres enting the most negative response and 5 the most positive response. Her post-course AT MI score was 75, representing a mean response of 1.9 per item. Her scores for the f our attitude components changed as follows: Value decreased by 1 point, Enjoyment decr eased by 4 points, Self-Confidence increased by 1 point, and Motivation decreased by 1 point It should be noted that these changes were too small to be either statistically or practically significant.

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124 Yezania said that she thought her attit udes toward mathematics had improved in some ways and not in others: Yes [I think my attitudes changed] because I liked all the manipulatives we did. See, when I was growing up, and I was in elementary school, we didn’t have any of those [manipulatives]… I’m a visual learner so math was just always [disgruntled noise]. You know, they would write numbers and that was it. And I was thinking, ‘what’s going on?’ So that’s why I’ve even postponed taking College Algebra because I’m so scared. So yes, and no, because I’m still kind of nervous when it comes to math, I’m not s ecure of myself, but it was fun. Let’s put it that way. Yezania thought that the use of manipulatives in teaching mathematics was “awesome.” She is the director of an after-school program where she has used manipulatives with her students: I even used them with my kids in afte r-school programs, and they loved it. They were doing a problem on the worksheet; they were not getting it. Once I took out manipulatives, they were like [noise indicating speed], and they knew how to do it. So I think they’re awesome. Yezania felt that manipulatives had helped he r “a lot” in the methods course as well, “especially with those word problems ‘cause I’m not good with word problems. I would have to draw it out. So I would use my little circles … fraction circles.” Yezania thought that in teaching mathematics, “you need a lot of cooperative groups because there are strong learners a nd then there’s weak learners so maybe together they’ll kind of help each other out.” She felt that maybe she would have done

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125 better in the methods course and benefited mo re from the cooperative learning activities if she had not been sitting with her friends: “I n the math course, it was interesting, I think, because we did too much talking instead of actually doing the work. So, you know, I’m being honest. Sometimes you get kind of sidetracked.” Yezania “really had a rough time” with problem solving in the past. She “learned a lot of new ways to solve” problems in th e methods course, which she said “was good for” her. She explained that she was a “ver y visual” learner, so she benefited from learning problem-solving strategies that used “a lot of writing and drawing:” I learned a lot of new things to make it ea sier on me and a lot of new ways to, like I told you, in my after-school program I have to help kids with their homework. It’s very frustrating when I don’t like math to help them do math. So I’ve learned a lot of new ways to teach them and to teach myself. When asked about the journal writing in the methods course, Yezania said that she “loves writing,” especially her “own opi nions and everything.” She concluded that she “loved ” the journals and was disappoint ed when she took her second mathematics methods course and there were no journals assigned. At this point in the interview, the rese archer showed Yezania her preand postcourse survey scores and the change scores for each attitude com ponent. The researcher explained that although Yezania’s overall sc ore had only decreased five points, her original score had been quite low, with a m ean score of 2.0 per item with 1 representing the most negative response and 5 representing the most positive response. Yezania said that she believed that these results could have been related to the distractions she had mentioned earlier involving sitti ng with her friends in class: “I think it did [affect the

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126 scores]. Especially those problems, rememb er, like the compare stuff [meanings of operations], like sometimes I wouldn’t be gr asping it, but I would be too involved in [chitchat noises] so I woul dn’t grasp it too well.” Yezania stressed that even though her at titude towards mathematics had not improved and her score had decreased, she stil l felt that she had “learned a lot even though.” She added that she had learned a le sson about paying a ttention in class: Yeah, I learned a lot in this [methods c ourse] as now I’m taking College Algebra so I already have this attitude that I ha ve to pay attention, grasp everything that the teacher’s [doing]. I’m always wr iting, whatever she’s doing, I’m writing it down. Question 3: Relationship Between Initial A ttitudes and Score on Final Examination Question #3 asked: What is the relationship between preservice elementary teachers’ initial attitudes toward mathematics and their grade on the methods course final examination? A Pearson correlation coefficient was found using the software program, SAS, in order to determine the relationship between initial attitudes toward mathematics and achievement in the methods course. Achi evement was measured using the methods course final examination. The departmental te st is a 50-item multiple-choice instrument that includes questions about both mathematics content and pedagogy. Information about the authors and content of the final exam is found in Chapter Three (pp. 80-82) and in Appendix J. The use of the final examination as a measure of achievement provided validity and reliability to the results. A reliability coefficient of 0.71 (n=17) was found for

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127 Exam Form A and a coefficient of 0.73 (n= 16) was found for Exam Form B, indicating a relatively high degree of reliability. The composite attitude score was used as the independent variable and the methods course final examination grade was used as the dependent variable. An alpha level of 0.05 was used to i ndicate whether the obtained correlation was statistically significant. A statistically significant Pearson Correlation Coefficient of r = 0.5321 was found, indicating a moderately strong positive correlation (p = 0.0014 < 0.05, n = 33). Question 4: Journals Question #4 asked: What do preservice elementary school teac hers’ reflective journal entries reveal about their attitudes toward mathema tics and the experiences that have influenced the development of those attitudes? Analysis of the qualitative aspects of the study involved l ooking for patterns. Hycner (1985) provided guidelines for the phe nomenological analysis of interview data, and his methods were utilized in this study. After reading a journal entry for a sense of the whole, units of general meaning were deli neated. Hycner defined units of meaning as “those words, phrases, non-verbal or para -linguistic communications which express a unique and coherent meaning” (Hycner, 1985, p. 282). These units were recorded using the computer software program Ethnograph. Wh en a journal expressed multiple units of meaning, these were analyzed separately. Once units of meaning had been identif ied for each journal entry for a given prompt, units of meaning from all journa l entries responding to that prompt were

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128 examined. Units of meaning relevant to the research questions were then clustered and common themes identified from the data. Themes were labeled using words that were introduced by the participants themselves whenever possible, and frequencies of themes were noted. When excerpts from journal entr ies were cited, no names or other means of identification were given. Any names of peopl e, schools, etc. that were included were replaced by pseudonyms. Journal 1: Feelings At Beginning of Course The first journal entry asked participants to discuss any fee lings, positive or negative, that they had about taking the met hods course. Responses were analyzed and positive, negative, and neutral themes were identified. Some of the journal entries expressed multiple themes, and these themes were analyzed separately. Therefore, frequencies may total more than 33. Initially, 19 distinct units of meaning associated with positive feelings about the course and 14 units associated with negative feelings about the course were identified. As themes emerged, those representing similar concepts were combined. For example, the following excerpts were initially categorized, respectively, as ‘Apprehensive about Course’ a nd ‘Nervous about Course.’ “I am very apprehensive about taking this course, considering my math skills aren’t the greatest.” “I am nervous because I wasn ’t always great at math.” They were both later counted as two in stances of ‘Nervous, Worried, Apprehensive about Course.’ The following themes, along w ith their respective percentages of all comments made for this prompt, were identifie d for Journal One: positive feelings about course (17.5%), negative feelings about cour se (11.7%), mixed feelings about course

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129 (3.9%), positive attitudes and experiences not related to course (19.5%), negative attitudes and experiences not related to course (29.9%), and beliefs expressed (17.5%). Table 10 shows the themes that were iden tified related to the methods course. The following journal excerpts are representa tive of data responses for each of these themes: Positive Feelings about Course “My feelings about taking this course ar e positive even though math is not my strongest subject.” “When I was signing up for classes over the summer this was the class that I was most looking forward to.” “I am for the most part very excited about this course.” “I am finding the kit of manipulatives intriguing.” “So I come into this class with some confidence.” “Not only did I think I would do well but I knew that I would be interested.” Negative Feelings about Course “I can honestly say that I had some appr ehension about taking this course, as mathematics has not been a favorite subject of mine.” “I was a tad anxious about this class becau se any class with the word ‘math’ has always made me a little nervous.” “I am a little nervous because math has always been the subject that I have struggled the most with.” “When I got into the school of education and saw that I had to take two more math classes, my heart was broken.”

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130 Table 10 Feelings about Methods Cou rse at Beginning of Course Feelings Frequencya Positive Feelings Feel positive, look forward to 11 Excited about course 6 Like manipulatives kit 5 Confident 3 Interested 2 Negative Feelings Nervous, worried, apprehensive 14 Dislike of mathematics 4 Mixed Feelings Both positive and negative feelings 6 aTotal frequency of 51. Total frequency of 27 positive feelings came from 21 of the 33 participants. This represented 17.5% of all comments made for this prompt. Total frequ ency of 18 negative feelings came from 15 of the 33 participants. This represented 11.7% of all comments made for this prompt. To tal frequency of 6 mixed feelings came from 6 of the 33 participants. This represented 3. 9% of all comments made for this prompt. “At the beginning of summer I didn’t want to think about taking this course at all. Just the idea of math was enough for me to dislike the course.” Mixed Feelings about Course “I would have to say that I currently have very mixed emotions about this course.”

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131 “For me, taking this math course is go ing to be both positive and negative.” Some participants expressed attitudes toward mathematics that were not specific to the methods course. Tables 11 and 12 summarize the positive and negative themes that were identified and the frequencies with wh ich these themes were cited. The journal excerpts that are given for each theme are re presentative of data responses given. Seventeen participants expressed twenty -seven beliefs (17.5% of all comments made for this prompt) about mathematic s, teaching mathematics, and learning mathematics while responding to Journal One. The following are repr esentative of these beliefs: “Everything in the world is made up by some mathematical equation.” “With fine [math] instruction, you can learn anything.” “[Using the textbook everyday] ge ts boring for the children.” By including manipulatives and other hands-on activities, [students] will be able to interact with math instead of just acquiring new information for memorization.” “So many people do not like [math classes].” “Some people have more of a natural intere st and feeling for math than others.” “Once children get a concept, it is such a self esteem booster.” “If a teacher has a negative attitude towards the subject, the students will recognize this and develop a bias at titude towards the subject as well.” Journal One also asked the participants what they were hoping to gain from the course. Table 13 summarizes all of the themes that were identified in addressing this question.

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132 Table 11 Positive Attitudes and Experiences Expressed in Journal One: Feelings About Course Theme Frequencya Representative Journal Excerpt Positive Attitudes Toward Mathematics Likes, enjoys math 10 “Math has always been my favorite subject.” “I enjoy math at the elementary level.” Does well 6 “Math is one of my strongest subjects.” Confidence 1 “I know that I can do math, it just takes time and a good teacher.” Positive Attitudes Toward Teaching Mathematics Wants to teach math 4 “I look forward to teaching math in elementary school.” Good at teaching math 2 “Just last week I taught my nine year-old sister long division. Good times.” Positive Experiences with Mathematics Classes Elementary school 3 “When I was in elementary school, I would get really happy when it was time to work in groups and use manipulatives to learn.” Great teachers 2 “I had great mathematics teachers as a child.” Math came easily 1 “In math [class], the teacher could usually show me once how to do something, and I would just get it.” Loved geometry 1 “I remember algebra, which was OK, and geometry, which, for some reason, I loved.” aTotal frequency of 30 came from 15 of the 33 participants This represented 19.5% of all comments made for this prompt.

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133 Table 12 Negative Attitudes and Experiences Expressed in Journal One: Feelings About Course Theme Frequencya Representative Journal Excerpt Negative Attitudes Toward Mathematics Not good at it 12 “I am horrible at math.” Don’t like/hate 8 “I do not like math.” “I dread the idea of doing anything concerning math.” Not interested 2 “I am not partic ularly interested in [math].” Intimidated 2 “I am intimidated by math and all of its subtopics.” Negative Attitudes Toward Teaching Mathematics Worried about teaching math 6 “I wonder how I am going to teach a subject that I am not very good at.” Methods not same as book 1 “Whenever I try to help ot hers I know what I am doing but my methods are usually not the same as the book describes.” Fractions 1 “I tried to teach a lesson on fractions to 1st graders. The lesson was a flop, the kids were LOST.” Negative Experiences with Mathematics Classes High school and college 6 “I got into high school and sort of forgot the basics and instead memorized formulas.” Struggled 5 “I was never the best math student.” General 3 “I have had good and bad experiences [with math classes], unfortunately more bad than good.” aTotal frequency of 46 came from 24 of the 33 participants This represented 29.9% of all comments made for this prompt.

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134 Table 13 Themes from Journal Prompt: What Do You Hope to Gain From the Course? Theme Frequencya Representative Journal Excerpt Teaching strategies and tools 24 “I [hope to gain] a be tter understanding of the strategies used to teach math well.” Learn to like and appreciate math 13 “My main goal in this class is to try and appreciate math, not only for my sake, but for my future students as well. Help students develop positive math attitudes 11 “I do not want [my students] to have the same fear [of math that] I do.” “I don’t want to make my students dread math.” Gain better understanding of math 10 “I am hoping that through th is course I will learn more about math.” Make math enjoyable, fun 9 “I am excited to learn ho w to make math fun for my students.” Gain confidence with math 9 “I am hoping to learn and feel confident that I can do math after completing this course.” Continued on the next page

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135 Table 13 (Continued) Themes from Journal Prompt: What Do You Hope to Gain From the Course? Theme Frequencya Representative Journal Excerpt Be a good teacher 6 “I am hoping I can come away from this course with the ability to become a good math teacher.” Accommodate diff. learning styles and needs 5 “I want to learn different methods of teaching the same types of problems so that I can use these techniques in classes with students who learn differently and at different speeds.” Help students see math as relevant 4 “I want them to see it as a tool, as something that can positively affect their lives.” Make math interesting 3 “I want to be able to make [math] as interesting to them as possible.” Learn to use manipulatives 1 “I hope to gain [knowledge of] how to use all the manipulatives from our kit.” aTotal frequency of 95 came from all of the 33 participan ts. This represented 100% of all comments made for this prompt.

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136 Journal 2: Memories of Math ematics in Elementary School The second journal entry as ked participants to refl ect on their memories of learning mathematics in elementary school. Initially, 19 distinct units of meaning associated with positive memories and 25 units associated with negative memories were identified. As themes emerged, those represen ting similar concepts were combined. For example, the following excerpts were initia lly categorized, respec tively, as ‘Negative Memory about Word Problems’ and ‘N egative Memory about Fractions.’ “I remember feeling challenged by word problems because I never knew where to start.” “I can remember not enjoying fractions.” They were both later counted as two inst ances of ‘Negative Memory about Learning Specific Topics.’ The following themes, along with their re spective percentages of all comments made for this prompt, were identified for J ournal Two: positive memories of mathematics in elementary school (34.2%), positive memori es of mathematics teachers in elementary school (9.7%), enjoyed mathematics more after elementary sc hool (1.3%), negative memories of mathematics in elementary school (32.3%), negative memories of elementary school teachers (3.2%), began to struggle after elemen tary school (3.2%), negative memories of mathematics teachers after elementary school (2.6%), general memories of elementary school (5.8%), neut ral attitudes (1.3%), and beliefs expressed (6.4%). Those journal entries reflecting positive memories of elementary school mathematics are summarized in Table 14 Those journal entries reflecting positive memories of elementary school mathematic s teachers are summarized in Table 15 .

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137 Table 14 Positive Memories of Mathematics in Elementary School from Journal Two Theme Frequencya Representative Journal Excerpt Specific topics, especially mult. facts 11 “We would have to name all of the times tables under a certain amount of time. We earned stickers on our name card … It was a large boost of confidence after receiving a new sticker.” Good at it; successful 8 “I always remember looking forward to math because I was good at that subject.” Enjoyed it 5 “In elementary school I remember enjoying math so much.” Unit related to real-life application 4 “We did a unit where we learned about money. We created a school store….I love d it and still to this day remember doing well at it and having fun with it.” General positive memories 4 “I have many fond memories of learning math at the elementary level.” Positive attitude toward math 3 “All during elementary school I had a really positive attitude towards math.” Parental involvement 3 “My parents were chaperones for this [math-related] event and I remember being so happy that I could share this experience with them.” Continued on the next page

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138 Table 14 (Continued) Positive Memories of Mathematics in Elementary School from Journal Two Theme Frequencya Representative Journal Excerpt Doing well increased confidence 3 “I was retaining the proce dures quickly and effectively and this gave me a lot of confidence.” Manipulatives 3 “I couldn’t wait to get my hands on the blocks, pennies and geometric shapes. In class, each of us would get our own manipulatives.” Cooperative learning groups 3 “In elementary school I remember enjoying math so much…Often, we would work in groups to solve tough word problems.” Fun 2 “When I was in elementary school, math was fun.” Games 2 “I remember in first grade, we played a game called Around the World … This served as a source of encouragement to me.” Motivated 2 “I always wanted to get good grades and be looked up to.” aTotal frequency of 53 came from 21 of the 33 participants This represented 34.2% of all comments made for this prompt.

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139 Table 15 Positive Memories of Mathematics Teachers in Elementary School from Journal Two Theme Frequencya Representative Journal Excerpt Took time for extra help 4 “My teacher would take the time to explain something until everyone understood what was going on.” Made math interesting 2 “I loved this teacher because she made learning math and science so interesting.” Patient, supportive 2 “I know that I would not have been able to [be successful] without the patie nt supportive help from my teacher, ‘Mrs. W.’ I will never forget her.” Encouraging 2 “I remember my teachers being extremely encouraging towards all of us.” Taught math in many different ways 2 “My teachers would teach us in many different ways. Some days we would use di ttos and have drill and practice and the next we would be using M&Ms to count with.” Made math relevant 1 “One of the reasons I did so well was because my teachers …applied [lessons] to real life situations.” Encouraged group work 1 “The teachers always encouraged lots of group work during math, which I think is wonderful.” Great at explaining 1 “I always had very good teachers who were great at explaining things.” aTotal frequency of 15 came from 7 of the 33 participants This represented 9.7% of all comments made for this prompt.

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140 Although Journal Two asked about memories of elementary school mathematics, two participants said that they began to like mathematics more once they were out of elementary school: Loved algebra: “I enjoyed math more wh en I got to middle school or high school because I love to do algebra.” Started working and succeeding: “I didn’ t start to like [math] until I was in college and really started applying my self. I got pleasure from earning good grades on tests.” Those journal entries reflecting ne gative memories of elementary school mathematics are summarized in Table 16. In addition, the following negative memories of mathematics teachers in elementary school were each mentioned once: Teachers not compassionate: “I had go to Catholic School and some of the nuns were not too compassionate for some of the students.” Teacher mostly used dittos: “[My second grade teacher’s] idea of math class was assigning a bunch of mindle ss dittos that taught nothi ng more than simple addition and subtraction.” Teachers not excited about math: “As I reme mber back to my elementary days of math there are no teachers that taught math that I recall being excited about math.” Teacher bitter: “I had a teacher that w ould mark down our grade if we would ask questions when we did not understand because she was bitter and did not want to be bothered.”

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141 Table 16 Negative Memories of Mathematics in Elementary School from Journal Two Theme Frequencya Representative Journal Excerpt Specific topics, especially multiplication facts Struggled with math General negative memories 11 7 5 “We did many drills for our times tables. If you didn’t pass your two’s and three’s then you didn’t move on. That put pressure because if you didn’t move up then everyone knew, that made me feel dumb.” “I remember excelling in all my other subjects, except math in which I struggled.” “So as you can see all my main memories of learning math are bad.” Did not understand 4 “I remember crying in front of the teacher because it was so hard for me just to get an understanding of math.” Stressful 3 “[Struggling with math ] was a very new and incredibly stressful thing for me to comprehend.” Low test scores 2 “I can recall memories of receiving my Stanford Achievement scores in elementary school and always placing in the average or low percentile in math.” Felt stupid, less smart 2 “I remember feeling less smart when people around me … just ‘got it’ so easily.” Hated math 2 “I remember that I hated doing [math].” Continued on the next page

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142 Table 16 (Continued) Negative Memories of Mathematics in Elementary School from Journal Two Theme Frequencya Representative Journal Excerpt Boring 2 “I can remember as a ch ild, not really having any interest in math. I found it boring.” Games 2 “We played a game called “Around the World” in which we competed to see who could answer the fastest. I dreaded these parts of math.” Not relevant or needed 2 “I remember [thinking] what was the purpose of me doing this when I don’t n eed it later in life.” Teachers did not help 2 “I remember not understanding anything and not having teachers that would go back and help.” Only one way to solve problem 2 “I remember not being taught several ways to work out problems.” Drill work 2 “The drill, drill, drill made me hate math for a little while.” Moved to a new school 1 “Math had always been one of my lesser subjects, but it got even worse in 3rd grade [when] I transferred to a new school.” Homework 1 “I hated the home work…The problems were a lot more difficult than the easy ones we did in class.” aTotal frequency of 50 came from 16 of the 33 participants This represented 32.3% of all comments made for this prompt.

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143 Teacher would yell: “I remember that my teacher used to yell at the class, and when she yelled, a lock of her hair would fall over her forehead and a vein in her neck would stand out.” Five participants said that they liked mathematics in elementary school, but that they began to struggle with it and dislik e it once they reached middle school or high school: “Once I went to middle school I was to tally lost when it came to math.” “I can clearly remember unpleasant ma th experiences in middle school and beyond.” “I did not really start to dislike math until the 7th grade.” “When I got to high school, it got a bit ha rder and I started to shut down from wanting to learn more math.” “Unfortunately, along the way, math becam e one of my most hated subjects.” Although Journal Two asked about memories of elementary school mathematics, four participants mentioned negative memo ries of teachers after elementary school: “In high school I had one teacher that sa id that there was no way I would make it in school because I did not do well in math.” “In high school, I had an extremely rigid, math teacher. She made math class so tense it was hard to feel at ease. Thus, she created a difficult learning environment.” “In my Algebra 2 class [I had] a HORRIBLE teacher.”

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144 “I took College Algebra he re at [university], and my teacher was HORRIBLE!!! He never helped, and every time you would ask for help he would make you feel stupid.” Some of the participants’ memories of elementary school mathematics were neither positive nor negative. These are found in Table 17. Table 17 General Memories of Mathematics in Elementary School from Journal Two Theme Frequencya Representative Journal Excerpt Worksheets 4 “I don’t really remember too much about elementary math; what I do remember is worksheets and ditto sheets and lots of them.” Manipulatives 4 “My memories earl y on are using manipulatives such as bottle tops and pieces of felt.” Teacher would demonstrate or explain 1 “The teacher would show how to do the problem on the board and explain; you w ould read about it in your textbook; then you would do problems in class and also as homework.” aTotal frequency of 9 came from 7 of the 33 participants. This re presented 5.8% of all comments made for this prompt. In addition, two participants expressed neutral attitudes toward mathematics in elementary school: “I believe my basic attitude [was] that math was something I needed to do … it wasn’t particularly painful, but it wa sn’t particularly enjoyable either.”

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145 “I did not dread math -but I did not l ook forward to it either. It was simply something that I had to do.” Journal Two also asked participants wh at they, as future teachers, had learned from these experiences. Table 18 summarizes their responses. While discussing their memories of elemen tary school mathematics, 9 participants expressed 10 beliefs about mathematics, teaching mathematics, and learning mathematics. The following are representative of these beliefs: “[History] is mostly me morization. Math problem so lving is a more complex thought process.” “Parental involvement is important when teaching a child any subject. If the student doesn’t have a good support system then it’s going to be ten times more difficult for the child.” “Kids tend to have their own language a nd can better explain things to each other.” “With the right amount of teacher support most students are capable of not only using the four basic operations of math ematics, but also higher-level math.” “Research says that dancers are strong in math and I believe it to be true.” “I believe that if you make math fun st udents are more apt to pay attention in class and more likely to rememb er what you have taught them.”

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146 Table 18 Themes from Journal Prompt Two: What Did You Learn as a Future Teacher? Theme Frequencya Representative Journal Excerpt Provide extra help when needed 9 “If a student needs extra hel p, I think it is important to stay after school to help them.” Make math fun 8 “As a future educator I want to be the type of teacher that makes math fun.” Make math interesting 7 “As a future teacher, I want to try to make mathematics more interesting for my students than it was for me, if possible.” Foster positive attitudes toward math 7 “I hope as a teacher that I will make an impact on [my students] and that they will love to do math.” Build students’ confidence 6 “I will try to make them feel good about themselves when they have success …It brings out confidence.” Help students feel comfortable 6 “I want my students to feel comfortable in my classroom. I feel once they are then they will be able to meet their full potential.” Make math relevant 5 “As a teacher I must try to help my students understand that math is something they are going to need in their lives at all times.” Continued on the next page

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147 Table 18 (Continued) Themes from Journal Prompt Two: What Did You Learn as a Future Teacher? Theme Frequencya Representative Journal Excerpt Help students feel successful 5 “I need to help my children in the future become successful in everything th at I can because they will then feel successful!!!” Understand math content 4 “I think this will make me a better teacher because I not only know a way to teach multiplication, but I understand the process behind the lesson.” Accommodate different learning styles 3 “As a teacher I will try my hardest to teach math in a way that all children, with different learning styles will learn.” Various types of strategies and assessment 3 “I want [students] to know that there is more than one way to do something.” Use manipulatives 2 “I think using manipulativ es is going to be a great asset and I cannot wait to l earn how to apply them to math.” Encourage group work 2 “I’m definitely going to encourage group work in math.” Continued on the next page

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148 Table 18 (Continued) Themes from Journal Prompt Two: What Did You Learn as a Future Teacher? Theme Frequencya Representative Journal Excerpt Not use too many worksheets 2 “I do not want to have th e students complete as many worksheets as I worked on because I believe it is redundant.” Integrate math with other subjects 2 “As a teacher, I plan on integrating math with literature and all of the other subjects.” Ask colleagues for help 1 “If I don’t remember or can’t do a problem, I will ask my fellow teachers for some help.” aTotal frequency of 72 came from all of the 33 participan ts. This represented 100% of all comments made for this prompt. Journal 3: Feelings about Mathematics The third journal entry asked participants to complete the following statements: I enjoy or feel positive about mathematics because … and/or … I do not enjoy or I feel negative about mathematics because …. Participan ts were also asked to explain why they thought they felt this way. Initia lly, 32 distinct units of mean ing associated with positive feelings and 53 units associated with nega tive feelings were identified. As themes emerged, those representing similar concepts were combined. For example, the following excerpts were initially categorized, resp ectively, as ‘Negative Feelings about Mathematics: Intimidated’ and ‘Negative Feelings about Mathematics: Scared.’

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149 “I am very intimidated by math and all of its procedures.” “I would describe myself as be ing scared [of mathematics].” They were both later counted as two instances of ‘Negative Feelings about Mathematics: Scared, Nervous, Intimidated.’ The following themes, along with their re spective percentages of all comments made for this prompt, were identified fo r Journal Three: positive feelings about mathematics (14.8%), experiences associated with positive feelings about mathematics (13.4%), positive experiences w ith teachers at specific grade levels (3.8%), experiences that improved attitudes toward mathematics (1.9%), negative feelings about mathematics (22.5%), mixed feelings about mathematics ( 1.9%), experiences associated with negative feelings about mathematics (23.0%), negative ex periences at specific grade levels (7.2%), want to improve own attitudes toward ma thematics (4.8%), want to develop positive attitudes toward mathematics in future st udents (1.9%), and belie fs expressed (4.8%). Table 19 shows the identified themes that reflecte d positive feelings about mathematics. Table 20 shows the experiences that participan ts associated with pos itive feelings about mathematics.

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150 Table 19 Positive Feelings About Mathem atics from Journal Three Theme Frequencya Representative Journal Excerpt Finds mathematics enjoyable, fun 10 “I really enjoy learni ng new things about math.” “I do enjoy mathematics most of the time.” Finds mathematics useful, relevant 7 “I know that math is an integral part of our everyday lives.” Feelings related to the methods course 5 “I like the approaches th at we are learning in this class.” Likes constancy of mathematics 4 “I have always liked math because it has a certain way to solve problems, and if you do it right, you will always get the right answer.” Has learned how to approach math 4 “I feel positive about mathematics because I now know how to approach the subject.” Feels confident about mathematics 1 “If I had to complete a [mathematics] problem I think I would feel confident to do it.” aTotal frequency of 31 came from 19 of the 33 participants This represented 14.8% of all comments made for this prompt.

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151 Table 20 Experiences Associated with Positive Fee lings About Mathematics from Journal Three Theme Frequencya Representative Journal Excerpt Positive experiences with math teachers 8 “I enjoy or feel positive about mathematics because I have had good teachers to help me along the way.” Success with mathematics 5 “I enjoy and feel positive about mathematics because math has always been my best subject.” Mathematics came easily 5 “I enjoy or feel positive about mathematics because it has always been so easy to me.” Positive experience in elem. school 4 “At one point in time [elementary school], I truly enjoyed math and it made sense to me.” Feels confident about mathematics 3 “During the CLAST [test] I could not wait for the math sections for a stress relief and a confidence boost…. It all comes back to confidence. Positive experience in college 2 “I think I feel positive about mathematics because the first math class I took when I came to college I actually understood and passed.” Enjoyed parental involvement 1 “ … any time I didn’t unde rstand a lesson, [my dad] was always able to give me the extra help I needed.” aTotal frequency of 28 came from 18 of the 33 participants This represented 13.4% of all comments made for this prompt.

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152 While describing their positive feelings about mathematics, four participants remembered experiences that improved th eir attitudes toward mathematics. The following journal excerpts are representative of the five comments with which this theme was identified: “They [mathematics teachers] have made me enjoy math because they helped me build my confidence. They made math fun and exciting.” “The years that I made good grades in math have boosted my confidence especially in college.” In addition to the experiences associated with positive feelings listed in Table 20, there were some references to positive experi ences with teachers at specific levels of schooling. Table 21 lists the different grade le vels and the frequency with which positive memories were cited. Table 21 Positive Experiences with Teachers at Specific Levels of Schooling Grade Level Frequencya Elementary School 4 Middle School 0 High School 1 College 3 aTotal frequency of 8 came from 6 of the 33 participants. This represented 3.8% of all comments made for this prompt.

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153 The following journal excerpts are examples of data responses at each level of schooling: Positive experiences in elementary school. “I feel positive about mathematics because I have had good teachers. I loved my elementary school teachers because they made learning math fun.” Positive experiences in high school. “ I think the reason why I like math the way I do is from my algebra one teacher in high school. She made it fun and I could understand everything.” Positive experiences in college. “ I had a wonderful teache r that made learning fun. He always made sure we knew what was happening and offered any help if we needed any.” Table 22 shows the identified themes that reflected negative feelings about mathematics. Table 23 shows the experiences th at participants associated with negative feelings about mathematics.

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154 Table 22 Negative Feelings About Math ematics from Journal Three Theme Frequencya Representative Journal Excerpt Negative attitude toward math 8 “I cringe at the word mathematics and feel negative about mathematics.” Feels intimidated, nervous, fearful 8 “I began to fear [math] at an early age, and not much was done to ever change that.” Does not enjoy mathematics 6 “I do not really enjoy math.” Lacks confidence with math 5 “I feel negative about mathematics because I have very little confidence in myself.” Not good at mathematics 4 “I’ve convinced myself that I am not good at math.” Feels frustrated with math 4 “I feel negative about mathematics because when I get stuck on a problem or c oncept I get frustrated.” Does not see need for advanced mathematics 3 “I am not in any job that I need more than the basics, such as adding, subtracting, multiplying, dividing, and fractions.” Views math as unneeded 3 “I always thought why do I have to learn math. I won’t need it later in life.” Continued on the next page

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155 Table 22 (Continued) Negative Feelings About Math ematics from Journal Three Theme Frequencya Representative Journal Excerpt Tries to avoid mathematics 3 “I do not do math unless I absolutely have to.” Does not like nature of math 3 “I do not enjoy or appreciat e that math is a growing or continuing process by adding new concepts to the old ones.” aTotal frequency of 47 came from 14 of the 33 participants This represented 22.5% of all comments made for this prompt. Four participants expressed mixed fee lings about mathematics. The following excerpts are representati ve of these responses: “I have very mixed feelings when it comes to mathematics.” “I feel both positive and negative about math in different situations.” In addition, two participants expressed neutral feelings a bout mathematics. The following excerpt is representative of these responses: “Up to this point, I have felt neither negative nor positive about mathematics.”

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156 Table 23 Experiences Associated with Negative Feelin gs About Mathematics from Journal Three Theme Frequencya Representative Journal Excerpt Struggled with mathematics 13 “I feel negative about mathematics because of the fact that throughout all of my schooling, math was the one and only class I really struggled at.” Disliked ‘traditional’ teaching methods 8 “My past teachers never used manipulatives. It was strictly read the book and do the problems on the board.” Bad teachers 8 “I have had some bad teachers that have made me not like math because I never learned anything from them.” Felt stupid 4 “I struggled a lo t and sometimes I felt like I was dumb or stupid.” Negative experiences with tests 4 “When I studied really hard for the test and still did below average, I immediately began to doubt myself.” Have always disliked math 2 “Growing up I really disliked math and anything to have to deal with it. My mind would shut off when given a math problem.” Felt frustrated, traumatized 2 “When my parents would help me with my math homework I would cry … I would get so frustrated.” Continued on the next page

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157 Table 23 (Continued) Experiences Associated with Negative Feelin gs About Mathematics from Journal Three Theme Frequencya Representative Journal Excerpt Nonspecific negative experiences 2 “I feel negative about mathematics because I have had some negative experiences that have turned me off of math.” Did not enjoy 2 “Naturally because I struggled with [math], I did not enjoy it.” Problem with pacing of courses 2 “As soon as I was caught up with what we were learning, the class had already moved on to a new topic.” Brother’s success with math 1 “The fact that [math] always seemed to come very easy to my brother did not help me much either.” aTotal frequency of 48 came from 22 of the 33 participants This represented 23.0% of all comments made for this prompt. In addition to the experi ences associated with negativ e feelings listed in Table 23, there were some references to negative expe riences at specific levels of schooling. Table 24 lists the different grade levels and the fr equency with which ne gative memories were cited.

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158 Table 24 Negative Experiences at Speci fic Levels of Schooling Grade Level Frequencya Elementary School 2 Middle School 3 High School 7 College 3 aTotal frequency of 15 came from 11 of the 33 participants This represented 7.2% of all comments made for this prompt. The following journal excerpts are examples of data responses at each level of schooling: Negative experiences in elementary school. “I do not enjoy and feel slightly negative about mathematics. I know it stem s partially from my experiences as an elementary student.” Negative experiences in middle school. “As I got into middle high, my math classes used less and less manipulatives and more lecturing became popular. Hands-on activities and working with groups rarely, if ever happened.” Negative experiences in high school. “I feel negative about mathematics because I made bad grades at the high school level.” “The only thing I feel negatively about as far as mathematics is that all through high school I was looked at as a nerd because I was good at mathematics.”

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159 Negative experiences in college. “When I entered [university] I was upset to learn I needed more math. I took the two easies t maths possible, failed them both. I had to retake my math classes for forgiveness.” While describing their feeli ngs about mathematics, seve n participants said that they wanted to improve their negative attitudes toward mathematics. The following journal excerpts are representative of the ten comments with which this theme was identified: “I feel that I am open to new ideas and want to change my opinion about math.” “The new way of teaching sounds like it wi ll be more inviting to the students and to myself… I think that my opinion of math will soon change.” “I am looking forward to making how I f eel about math change for the better.” “If I am going to teach math I need to have a positive attitude about it. I believe now is the time to turn my attitude toward math around.” In addition, four participants said that they wanted to encourage the development of positive attitudes toward mathematics in their future students by making mathematics fun and enjoyable for them. The following jour nal excerpt is repres entative of those responses with which this theme was identified: “I want to find a way to make math mo re fun for students like me whose favorite subject may not be math.”

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160 While discussing their feelings about ma thematics, 8 participants expressed 10 beliefs about mathematics, teaching mathematics, and learning mathematics. The following are representative of these beliefs: “Too much math and/or bad instruction can turn you away from enjoying it.” “Children are able to recognize when [teachers] are uncomfortable with a subject. I feel that when they do recognize this, it in turn makes them uncomfortable and that much harder to teach.” “Schools should focus more on practical mathematics, and then for those children who love math and want to lear n more, they can take special electives to learn it.” Journal 4: Memorable Expe rience with Mathematics The fourth journal entry aske d participants to describe in detail one experience from their past that was particularly memora ble and influential in their attitudes about mathematics. The following themes, along with their respective percentages of all comments made for this prompt, were iden tified for Journal Four: positive memories (56.8%) and negative memories (43.2%). Table 25 shows the frequencies of experiences that reflected positive and negative memories and the grade levels at which these experiences occurred. While describing a memorable experience th at influenced their attitudes toward mathematics, eight participants recalled one special teacher or private tutor whose individual help and encouragement positiv ely affected their attitudes toward mathematics. The following journal excerpts ar e representative of these experiences:

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161 Table 25 Memorable Experiences that Influenced Attitudes toward Mathematics Grade Level Frequencya Positive Memory or Experience Elementary school 8 Middle school 4 High School 7 College 1 No specific grade level 1 Negative Memory or Experience Elementary school 4 Middle school 5 High school 6 College 0 No specific grade level 1 aTotal frequency of 37. Total frequency of 21 positive memori es came from 21 of the 33 participants. This represented 56.8% of all comments made for this prompt. Total fre quency of 16 negative memories came from 16 of the 33 participants. This represented 43.2% of all comments made for this prompt. Together these represented 100% of the comments for this prompt. An experience I had in math was when I was in elementary school. I remember [teacher’s name] was the nicest teacher ever. I remember every time she told us to get out our math books I would feel emba rrassed because I knew we were about to start something that I wasn't very good at. I knew [teacher’s name] could tell that math made me uncomfortable because I was so eager with all the other

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162 subjects and I clammed up when it came to math. One day she took me aside and said, "[student’s name], I can tell you ar e uncomfortable with math time but I know that you are smart and I believe that you can really achieve a lot if you put your mind to it. I want to help you succeed, and I will do whatever it takes to help you get there." After [teacher’s name] said that, I knew that there was no reason for me to feel uncomfortable because I was smart and if she had that much faith in me I must really be something. When I was in the seventh grade … I had a wonderful math teacher named [teacher’s name] She was teaching us … about addition and subtraction using negative numbers. This was a new con cept for me. Before that concept, I understood all of mathematics without a problem. But, those pesky negative numbers kept throwing me off. So, one afternoon when I was particularly frustrated, [teacher’s name] pulled me out of class and handed me a laminated slip of paper. It was a number line that she said I could use when doing my homework as well as tests. I thought that was incred ibly considerate of he r, especially since she made it for me without my having to ask her for it. So, I used that number line for the rest of the year until I felt co mfortable adding and s ubtracting negative numbers without it. It definitely built my trust in Math teachers. My most memorable experience occurred during the summer of my 7th grade year. I had a decent year in prealgebra but my mom insisted that I attend summer school with Mrs. S…. That summer I was introduced to the fun and

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163 exciting side of math. I enjoyed every day in my math class. Everyday we received challenging logic problems to solve and great rewards whether we got the correct answer or just attempted it. Mrs. S. was the most influential math teacher I have ever had. She single handedly opened my eyes to math. One experience that was memorable to me was when I was in high school trying to pass the [county test required for gra duation]…. [The teacher] took time to help students individually so that they woul d understand the concepts. He gave me practice worksheets so I could get the con cept better. He said motivational things to keep our head up like you can do it, just keep trying. He never said anything to keep us down. He knew that math wasn't my skill but he took time and devotion to help me so that I could graduate. Ev en though I hate math with a passion he made me understand how to do it and made me feel confident when I did the test…. I walked in on the test and felt good about every answer I did because of his teaching skills. Four participants recalled experiences that they said changed their attitudes toward mathematics in a positive way. Three of these involved a special teacher or tutor. The following journal excerpts are repr esentative of these experiences: One of my past memories that was memo rable and influential in having turned a negative attitude of math in to a positive one was when I decided to go to a math tutor. I was in high school at the time a nd had a math teacher that would call on me knowing that I did not know the answer I was making a "D" in her class and

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164 she suggested I find a tutor. I did and she was GREAT!!! Not only did she help me with what I was currently doing but she would help ge t me started on the next day’s work so when the teacher asked ques tions I was the first one to put up my hand and was able to know what I was talk ing about. She told me several times that going to the tutor had definitely he lped my grades improve. This made me feel great inside! At the end of tenth grade, I had an extremely influential experience, which changed my attitudes about mathematics. During tenth grade, I had a math tutor that I went to once a week. Even though I had a tutor and worked extremely hard I was barely squeaking by with a C. Needless to say, I thought my tenth grade geometry teacher was terrible. I hones tly think she found pleasure out of torturing her students. As I was preparing for the final, my math tutor could tell my ego had been battered and bruised. Before I le ft her house, she told me, “[Name], you know this material and you will do well on the exam.” She also reminded me to take a deep breath before I start taking the exam and to skip over the problems that I do not know how to solve. Her confid ence in me gave my math self esteem a much needed boost. The following da y I took the exam and her advice. A couple of days later I received my exam results, which showed I made a B-. I was so elated, because I finally made a satisfactory grade in geomet ry. Six participants recalled negative experi ences involving individual teachers. The following journal excerpts are representative of these experiences:

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165 I remember I was in 4th grade, and this really mean lady was my math teacher. She retired the next year, so I think that she was bitter and tir ed of teaching. I had always had a difficult time in math. On e time she called on me during a lesson on fractions, and I had no idea wh at the answer was…so in front of the whole class she said that I was not listening and fo llowing directions, so I could not go to reward that week. I began to cry because I felt that the teacher just did not like me, and I was embarrassed because I was listening, but I just could not understand the concept she was trying to teach. That even affects me today because I think that since that class I ha ve really disliked math. An experience from my past that is particularly memora ble and influential in my negative attitudes abou t mathematics would have to be a statement from my 7th grade teacher. That year I was a part of a class that combined math and science in a two-hour block. I can clearly remember my teacher explaining to us that he hated math and was not going to teach it because he liked science better. Although we touched on math before test s and final grades, we quickly brushed through it and I clearly remember not l earning too much. I also remember the effort that our teacher did put into math was accompanied by yelling and frustration when we weren't getting it correct or we were producing poor scores…. I remember feeling confus ed about the importance of math. If an adult didn't like math and even professed to never use it, then it must not rea lly matter.

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166 The one memory I can think of that left an impression on me was kind of weird. I was in high school in an algebra 2 class. It was the worst math class ever. I wasn't understanding anything…. When [the teach er] went to start writing on the board, with his back towards the class, almost th e entire class started to play cards and do other things. When he did turn around he did notice that the class was doing other things and he didn't even care. He let it continue on. Unfortunately for me, I can’t concentrate when there are other things goi ng on. So for the rest of the year I got nowhere…. I suffered the rest of the year. I almost fa iled. Luckily I got a tutor and that helped me, somewhat!!!! Three participants recalled experiences involving individual teachers that they said changed their attitudes toward math ematics in a negative way. The following journal excerpts are represen tative of these experiences: In 11th grade I had what has to be the worst math teacher ever… I gave up in that class because she would not take time to help me with the problems I was having…. The teacher was an older woman a nd I think that part of the problem was that she could not truly control th e class…. The teacher would write on the board the page numbers she wanted us to l ook over. She would then sit at her desk and grade papers. She would have us read and work on problems. She never went over the class work/homework that we did; she would just hand it back with red marks all over it. If you tried to ask her for help she would just do the sample problem on the board and think that ever yone understood from that point on. She did not try and explain things in different ways, she did not take time to work with

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167 anyone individually. She just was not a good math teacher. I got a “D” in that class, the worst grade I had ever gotten… I felt dumb and that I was not important enough for her to help me. From that moment on I disliked math. In high school I took AP Calculus so I c ould get out of taking one math class in college. I absolutely hated that class b ecause the teacher never did anything. We would always do homework for other classes. So a few months before the final exam, the teacher decided to start teaching so we knew a little bit for the final. I asked him a question about one of the skills and he told me I didn't have to understand why it was done the way it was, ju st memorize the steps. Things like that turn me off to math. While describing a memorable experience th at influenced their attitudes toward mathematics, four participants recalled situa tions in which they experienced feelings of success with mathematics. The following jour nal excerpts are representative of these experiences: One experience that stands out for me wa s middle school geometry and algebra. There is not one particular ev ent, but just the fact that finally there were two parts of math that I "got" and I felt confident about. I really loved solving equations, as they felt like puzzles to me. I am not sure why this particular math came easier to me than others, or if it was just that I loved my teacher [teacher’s name]. ( I had him for both subjects)…. I liked that th e rules and applications connected in my brain and that things finally "click ed" for once in math…. I remember actually

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168 thinking that I might like math, and perhap s could be "good" at it. Alas, then came trigonometry, which had me stumped again. One math experience I had that gave me one of the best feelin gs in the world took place when I was a sophomore in high school taking geometry. When the semester first started, I was struggling a li ttle bit and failed my first test…. My mom asked [geometry teacher from different high school] to tutor me…. I worked really hard and by the second test I made a B and by the next one, an A. From there on out, I made 100's and beyond. At the end of the semester, I had a 104% in the class. To me, it was a great feeli ng because I had worked so hard and it felt so good to know the material as well as I did and to complete the class with an A. One memory that is particularly memorable to me is getting my first A in mathematics last semester in statistics. I studied really hard for this test…. I can clearly remember seeing the 95% at the t op of the paper. I was screaming inside. This was the first time in my whole life th at I received an A on a math test. I kept it in the whole time during class, and as soon as I got out I called my mom as soon as I could to tell her the news. She was so proud of me, and I was so proud of myself. It was a turning point in my math career. For the rest of the semester I continued to make A's on all of my tests. It was amazing.

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169 Three participants recalled memorable experiences involving changing schools. One of these was a positive experience and the other two were negative experiences. The following journal excerpts reflect these experiences: … I was in fourth grade. At the time I lived in New York…. I was upset because I was moving to Florida. This also ha ppened to be my last day of long division, which meant a test…. I remember that I had to force myself to focus on the test and not the move…. It was two months befo re I was placed back into school. I remember how scared I was, not just because I was in a new school, but also because I did not know how behind I would be. I ended up being ahead…. I even tutored one of the girls. I know that you are wondering how this affected my attitude toward mathematics. At first, it made me really dread mathematics. However, I then felt confident after I lear ned that I was ahead. In addition, it was a great way to make friends. I felt pretty confident that day. Probably my most memorable experience involving mathematics occurred when I was in the sixth grade…. I had missed the first term of sixth grade [family was caring for ill relative in another state] …. I had few problems merging back into my other subjects, but math was a differe nt story…. I was not only seven or eight weeks behind, but I was completely in a f og about the new concept [prealgebra]. My teacher was a man, and I had always had women teachers before. I still remember how intimidated I felt when he cal led me up to the front of the class to stand beside his desk so that he could give me individua l instruction while the rest of the class worked on the class assignme nt…. How did this influence my attitude

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170 about math? Oddly enough, I don't think it affected my attitude toward math at all – once I had managed to grasp the concept, I had no trouble working the problems. I think it probably affected my attitude towards male teachers: I tended to see them in my following year s of school as be ing impatient and intimidating. Five participants recalled memorable experiences that involved the use of tricks, drill exercises, and games for learning basic facts. Four of the fi ve recalled positive experiences, and the fifth memorable experience was negative. All of the positive experiences mentioned the positive recognition that the participant received for being successful with the drills and games. The following journal excerpts, beginning with the negative experience and followed by two excerpt s that are representative of the positive experiences, reflect this theme of using tric ks, drill exercises, and games for learning basic facts: I would say my most memorable experien ce involving mathematics would have to be from third grade…. For some reason, I just could not grasp the concept of multiplication, and I failed quiz after quiz on the material. I think it was mainly because there was no logic to it, just memo rization. What I hated most of all was that the tests were timed, which just gave way to more anxiety (i.e. more failure). As if class was not already horrible enough for me, my teacher decided to call my parents and tell them about the difficu lty I was having…. The following afternoon after school, my dad was wa iting for me with a load of blank worksheets he had made for me to practice multiplying. I sat there and practiced over, and over, and

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171 over until it was nighttime. At the time (and for several weeks afterward) I resented and hated my dad for making me do that, but I must say, I never failed another test. One experience that I can recall very well that had an early influence on my math life was in the first grade. When I was very young I really enjoyed math…. [We] usually played a game called "around the world." What would happen was each student would go up against the other in friendly competition and compete to solve a math problem first…. One day I went up against each student and won. I was only the second person in my class to do so. I got a certif icate and lots of encouragement…. I did not always win. Th e teacher made the atmosphere fun and comfortable. I was so pro ud of myself. This early exposure really gave me confidence in math. Later on is when it all changed. When thinking back about my past e xperiences with math, one of my favorite memories was when I learned my times tabl es in the fourth grade…. In my class we had this game that we played called the clock game and we would compete against other classes…. Due to the game we played I was very motivated to learn [times tables] quickly. I made sure to practice them as my homework along with what we did in class during math. This experience with multiplication facts, and the game that we played showed me that math could be fun, and I was capable of knowing my times tables if I just took th e time to practice them and give them

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172 attention. My teacher was very impressed with my knowledge, and I ended up being the best in class at the game. This made me happy and also proud of myself, considering that I did not lik e math very much. The impact of a teacher’s recognition was also noted in another participant’s journal. She was recalling a memorable experience from middle school: … One day Mr. M. [teacher from the pr evious year] came by and said hi to the class as school was letting out. The next day, Mrs. H. [current teacher] told me that Mr. M told her that I was the best student he'd ever had.. That itself is a huge complement, but considering that he had taught for almost forty years, this was the nicest thing that has ever been said about me. I almost cried I was so touched…. Knowing that a teacher recognizes me as such is one of the best feelings in the world. Four participants recalled negative experiences where th ey felt stupid or not very smart. The following journal excerpts are representative of these experiences: I have a younger brother who has always been very good at math while I have always seemed to struggle…. One eveni ng I was at home doing homework at the counter as usual and I asked my dad to ta ke a look at my homework …. [As] he began trying to talk himself through it to understand the problem … my younger brother who was sitting next to me doing his homework as well, looked over and said "Oh that's easy, you just do this." … My dad said to me, "Wow maybe you

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173 should just have your brother help you with your math homework from now on, you won't have to wait for me to get home ." I felt so incredibly stupid that my YOUNGER brother not only knew more than me but could figure out something he wasn't even learning at the time. From then on I think I resigned in my head that I would never be good at math. I remember sitting in my sixth grade math class going over some new math problems. The teacher was instructing in front of the classroom using the overhead. After doing several problems, she started calling on people randomly to give answers to some problems…. I ha d no clue as to what the answer was. I was still trying to understand the problem, and before I knew it, I was being asked to give answers. I just shrugged my shoul ders and said I didn't know. It made me feel so stupid for the rest of the day. I felt like it was my fault and that I should know these things. Now that I look back on it, I really see that the teacher made a big mistake by just assuming that we w ould all grasp the concept after a few example problems. Two participants recalled me morable experiences involvi ng being placed into a more advanced class. One of these was a positive experience and the other was a negative experience. The following journal excerpts reflect these experiences: I remember being tested for “gifted” the summer after 2nd grade. For some reason, I didn't focus well that day, and didn't do too well. After my teachers recommended I be retested the whole ne xt school year, I was re-tested, and the

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174 psychologist that gave the test said I should have already b een in the “gifted program.” I think this, combined with the fact that my mom and her two sisters were math majors, and one of them used to be a math teacher, really made me want to excel in Math. My most memorable math event was when I was in the [name of program] in middle school…. I really think this pr ogram changed my view on math [negatively]. I was always good in math and when I was in 5th grade I had to take a portion of the PSAT to prove that I kne w enough about math to be one of the few to be in a special program offered at only 2 or 3 middle schools in the county. I took it and did very well. I opted to be in this program…. I understood bits and pieces of the class but my mind was in ot her places. We spent two hours everyday working on these items…. I had a personal issu e with my teacher so that didn't help the situation either. Journal 8: Use of Reflective J ournals in the Methods Course The eighth and final journal entry asked participants to discuss the use of reflective journals in the methods course. They were asked about the benefits, if any, as well as any drawbacks related to the journa ls. Initially, 97 distin ct units of meaning associated with benefits of the journals a nd 8 units associated with drawbacks of the journals were identified. As themes emerge d, those representing similar concepts were combined. For example, the following excerpts were initially categorized, respectively, as ‘Provides Means for Informal Communicati on’ and ‘Allows for Personal Dialogue.’

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175 “[The journals] let [the in structor] know what we we re thinking without having to write a formal paper.” “[The journals] are a great way to di alogue more personally with someone, especially a professor.” They were both later counted as tw o instances of ‘Good Means of Personal Communication.’ The following themes, along with their respective percentages of all comments made for this prompt, were identif ied for Journal Eight: benefits of journal writing (89.2%), drawbacks of journal wr iting (3.9%), comments about the methods course (2.5%), and beliefs e xpressed (4.4%). Table 26 shows the identified themes that reflected benefits of re flective journal writing in the methods course. Although the overwhelming majority of id entified units of meaning focused on benefits of the reflective journals, eight participants also mentioned drawbacks in addition to benefits. These drawbacks repres ented 3.9% of the comments made for this prompt. The following are representative of these comments: “On the other hand, there was one drawb ack to doing these journals. It was depressing having to relive my s ophomore year in high school.” “Perhaps the only reason [the journals] ha ve not been helpful is because they’ve prompted me to realize that I have been lacking very much in the area of good math teachers and role models my entir e school career, whic h makes me feel quite uneasy about teaching math myself.” “The only drawback was that a few of the journal topics seemed a little repetitive.”

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176 Table 26 Benefits of Reflective Journal Writing in the Methods Course Theme Frequencya Representative Journal Excerpt Opportunity to reflect on future teaching 26 “The reflection in its en tirety has gotten me to … really think about how I am going to teach my own students.” Opportunity to reflect on past experiences 26 “I think that [the journa ls], really help us, as students, to look back on our past math experience and see what we can learn from them.” Reflection on attitude toward mathematics 20 “The reflective journals in this class have given me a lot of insight about my attitudes toward mathematics.” Reflection on good and bad teaching 12 “The journals helped me to remember what I liked and what I didn’t like about the ways I was taught math.” Reflection on specific plans as future teacher 9 “I learned not to be like my bad teacher, but I learned to be willing to help and reach out to children that need help.” Liked, loved, enjoyed journals 9 “I absolutely loved the us e of reflective journals for this class.” Continued on the next page

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177 Table 26 (Continued) Benefits of Reflective Journal Writing in the Methods Course Theme Frequencya Representative Journal Excerpt Plans as future teacher regarding students’ math attitudes 9 “[The journals] helped me to see that as a teacher I would have to be sensitive to the fears [regarding mathematics] of my students.” Journals were a great idea 8 “I thought the use of refl ective journals was a great idea.” Realized teachers’ impact on students’ attitudes 6 “From these reflections and my own experiences I can see that the teacher plays an extremely important role in shaping a student’s attitude toward math.” Helpful, beneficial 5 “I think doing the reflective journals has been beneficial to me.” Reflection on struggles with mathematics 5 “I was actually provoked to go back into my past and to figure out where it all went wrong with math.” Reflection on future teaching methods 5 “After I reflected on [a pr evious teacher’s methods], I made a point to remember to do those same approaches and math tricks when I become a teacher.” Continued on the next page

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178 Table 26 (Continued) Benefits of Reflective Journal Writing in the Methods Course Theme Frequencya Representative Journal Excerpt Comments about researcher’s responses 4 “Your responses allowed me to know that I am not the only person out there with horror stories about math.” Negative at first, but appreciated later 4 “These [journals] seemed tedious in the beginning. However, after I did the first two I realized the immense benefits of reflective dialogues.” Reflection about mathematics 4 “[The journals] made me think about math in a different way.” Reflection about making math class fun 4 “As I reflected on the best teacher I had, I remembered what he did to make math fun and I want to use those same strategies when I teach.” Plan to use journals with future students 4 “I think that I will use something like this in my classroom so children can e xpress their fears or their joys about math and it will just be between me and the student.” Journals make you think 3 “[Journals] make you think about things that I would not have thought about.” Reflection on methods course 3 “[Journal writing] allowe d me to reflect on my learning in this class.” Continued on the next page

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179 Table 26 (Continued) Benefits of Reflective Journal Writing in the Methods Course Theme Frequencya Representative Journal Excerpt Reflections on specific teachers 3 “Math started to go downhill for me. That was until I got a math teacher who saw my potential and made me believe … I could do it.” Good means of communication 3 “People can be more open through writing than speaking face to face at times.” Reflection on importance of students’ understanding 2 “[The journals] showed me that I want to be a good teacher and make sure th at [my students] have a good understanding of math.” Benefits of reflection 2 “I learned so much more from the journals because they required reflection.” Realized commonalities 2 “I also think that the reflective journals helped me to see that my concerns were not very different from other students in math.” Liked e-mail format 1 “This e-mail form was very convenient for me.” Wish journals were used by more teachers 1 “Overall I really enjoyed the weekly journals, and I wish more teachers would take the time to do them.” Continued on the next page

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180 Table 26 (Continued) Benefits of Reflective Journal Writing in the Methods Course Theme Frequencya Representative Journal Excerpt Provided opportunity for sharing of reflection 1 “It also brought up discussion. For example, when carpooling with [name] to our internship we would talk about some of the stuff that were brought up in our journals.” aTotal frequency of 181 came from all of the 33 participants This represented 89.2% of all comments made for this prompt. “The only drawback for me is that my me mory is not as good as it used to be and I find it hard to recall things that ha ppened when I was in school.” While discussing the benefits and drawback s of reflective journal writing, nine participants expressed beliefs about mathematics, teaching mathematics, and learning mathematics. The following are representative of these beliefs: “Moving children into a harder subject wh en they are not ready is damaging to their self-esteem.” “Many people have some negative views toward the subject [of mathematics].” “So many students don’t like math.” In addition, five participants made commen ts regarding the methods course itself. The following are representative of these comments: “Your class has also taught me the vast amount of resources th at are out there to help me accomplish my goals as a teacher. I never dreamed there were so many

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181 things I could do to make math hands on and fun. I hope my students will benefit from my class as much as I have from yours.” “I also think that you make this class awesome. The manipulatives that you use are easy to understand. I use many strategies that we ha ve learned to train my employees [at an afterschool program] and help them teach the children. I try to steer them away from traditional algorithms and to use base ten blocks, especially with the little ones. I really am taking so much knowledge away from the course. Thank You.” “I have learned so much in this course!!!” Journals 5, 6, and 7: Relevant Excerpts Although eight journals were assigned over the course of the semester, only journals 1, 2, 3, 4, and 8 related directly to the purpose of this study and were analyzed using methods that are described in chapte r 3. The remaining three journal prompts did not relate directly to the purpose of this st udy, but they did relate to the purpose of the methods course. Therefore, participants res ponded to them as well, but these entries were not analyzed unless they contained information relevant to the study. The following journal excerpts containe d data that was pertinent to the research questions of this study: Journal 5: Boosting Confidence of Students The fifth journal asked participants how th ey as future teachers will help boost the confidence of students who ha ve low self-confidence regarding mathematics. Eleven participants responded with comments that re lated to their own experiences and feelings

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182 of low self-confidence with mathematics. The following are representative of these comments: “I can definitely relate to the fact th at many students have a low self-confidence when it comes to mathematics.” “I know what it is like to not have any confidence in your abilities, and not want to raise your hand because you’re afraid. I believe some teachers do not realize how they can so easily negativel y affect their students.” “I think that a lot of children have lo w self-esteem when it comes to math. I was and still sometimes am one of those persons.” “I will also try to show my students that math can be done by everyone....not just boys or the smart kids. This was often relayed to me as a child and I know it can be very discouraging.” “I wish I had someone to build up my confidence when it came to math because I sure did need it during those times.” “I agree with the statement that children grow to have a low self-confidence when it comes to math because I am living proof of those children. I, like many children, grow to either not like mathematics as a whole or become intimidated by it entirely.” Journal 6: Qualities of Best Mathematics Teacher The sixth journal asked part icipants to reflect on the qualities of the best mathematics teacher they ever had and the effect this teacher had on them as learners of mathematics. Twenty-one participants responde d with comments that were pertinent to

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183 the research questions of this study. The following are representative of these comments: “She made me believe in myself.” “[Teacher’s name] had such a positive att itude about mathematics that she made me think learning math was exciting.” “Honestly, I have never really had a ve ry good mathematics teacher. … This is also why I’m a little anxious about teaching math in my elementary classroom one day because I don’t feel like I had a nyone who really modeled for me what a good math teacher is supposed to be.” “Math is still one of my l east favorite subjects (science is my least favorite!) but she was the first math teacher I had that made me feel like I could handle it all…. She did give me a slightly more positive attitude towards math.” “My better mathematics teachers had a good attitude about the subject. They never implied that math was one of the harder subjects and never dreaded teaching it.” “She really kept a positive and upbeat attitude, which wa s visible in all that she did…. She made it possible for me to see math as something exciting and fun.” “She would answer any question we had without making us feel stupid.” “I had a difficult time with math and I hated it. I had a hard time understanding why math rela tes to real life. It to ok me years to realize that we do use math everyday.”

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184 Journal 7: Qualities of Wo rst Mathematics Teacher The seventh journal asked pa rticipants to reflect on the qualities of the worst mathematics teacher they ever had and the effect this teacher had on them as learners of mathematics. Seventeen participants responded with comments or experiences that were pertinent to the research que stions of this study. The follo wing are representative of these comments and experiences: I believe the worst qualities a math teacher can possess are impatience, discouragement, lack of enthusiasm, and inflexibility. Unfortunately, I have had a math teacher who has possessed all these qualities—[teacher’s name]. She is the reason why I have such a strong distaste for math. The worst teacher[s] I ever had [were] throughout my elementary school years. I say this because they did not give me the help I needed. They did not show me that they cared and wanted me to do the work on my own. The effect they had on me is the way I feel about math now, that I dislike it…. They made me feel like I was worthless and that they felt they did not have to teach me the material. The worst math teacher I ever had wa s 10th grade geometry with [teacher’s name]. He was a bad teacher because his overall personality was hard to approach. If I didn't understand someth ing, I was scared to ask him…. After Geometry, I was not too excited about Mat h. I took an easier math course in high school the next year. The qualities of my worst math teacher wa s that she never liste ned to our worries, and she would never explain things more then once, which hurt a lot of people because it made us feel STUPID!!! Ever since this teacher I disliked

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185 mathematics…. Since this teacher I am s cared to ask a question because when we would ask her a question she would emba rrass us in front of the whole class making us feel that we were the stupidest person in the world. My 11th grade math teacher was the worst. She just would not take the time to work with us to help us understand th e problems. She did not use different methods of teaching, it was always the sa me thing…. I think that our class just gave up. I know most of the st udents did just as poorly as I did in her class. We stopped asking her questions because she ne ver explained them to us anyways…. This teacher affected me greatly. I did not want to ever have to take another math course again, I hated math. She would start everyday w ith a lecture about some ne w math strategy and show us examples on the overhead. During the examples, she would call on students to help her out. There was no time for us to process the new information and practice it for ourselves. It was just expected that all of us [would] immediately know what we had just learned. I got called on a few times, [and] when I didn't know the answer, she made me, and others, feel horrible and dumb for not knowing the answer. This obviously had a very negativ e effect on me and ended up making me feel incompetent…. Overall, the effect was damaging to my self-esteem about mathematics. While responding to journals 5, 6, and 7, some participants expressed beliefs about mathematics that were pertinent to the research questions of this study. The following are representative of these beliefs: “[Math] is a difficult subject to get students excited about.”

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186 “I think a lot of the time the students come into [math] with a bias against the whole subject.” “When a person believes they can do some thing they have a much more positive outlook on things and will do better overall.” “Attitude is key when it comes to learni ng. You have to keep an open mind and venture from what you are comfortable with.” “Teachers that make math difficult and uninteresting make kids hate math.” “If you dislike a subject, I really feel that you do not do as well as if you really liked that subject.” Question 5: Participants with the Most Extreme Attitudes Question 5 asked: What are the attitudes toward and experiences with mathematics of those preservice elementary school teachers id entified as having the most extreme (either positive or negative) attitudes? The two preservice teachers with the lowest initial scores on the ATMI and the two with the highest scores were asked to participat e in an individual interview where their attitudes toward and experi ences with mathematics were further explored. These Experiences with Mathematics Interviews took place between week six and week eight of the semester. ‘Mary’ Mary’s score on the ATMI was 63 out of a possible 200 points. She was one of two participants to score 63, the lowest in the class, indicating the most negative attitude

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187 toward mathematics. This represented an average per item score of 1.58, with 1 representing the most negative attitude and 5 the most positiv e. Her average scores on the four attitude components were: Value, 2.5; Enjoyment, 1.1; Self-Confidence, 1.4; and Motivation, 1.2. Mary remembered elementary school ma thematics as “pretty tough.” She had a difficult time understanding concepts, especia lly subtraction. The al gorithm confused her, so she would “end up counting on [her] fingers.” When asked if she remembered anything else from elementary school mathema tics that might have a ffected her attitudes toward mathematics, Mary replied: Just the pressures of trying to unders tand it and not doing so well. Then going home and bringing C’s, which [par ents] didn’t too much like, which brought on more pressure. It was stre ssful. I remember crying because I could not understand what was going on. Middle school brought more probl ems with understanding concep ts such as long division. An after-school tutor helped somewhat, but she still had trouble understanding the concepts: I just could not understand math…. Bringing a ‘C’ home is not good for math, but I was just happy if I could get a ‘C,’ I was happy. I was happy about that. That’s all I got in math was just C’s. If I got a ‘B’ I would throw myself a party! High school was even harder for Mary. She was in her senior year and needed to pass a standardized test in order to graduate She had failed the test the year before and was placed in a test-preparation class. The teacher for this course was very helpful to

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188 Mary. He taught her strategies to use on the te st and made himself available to her when she needed help: When I got to the actual test, I felt comfortable and confident because I remembered the different things that he taught us, and I passed it the first semester. But it was hard because you coul dn’t use a calculator and just grasping all those things, it was hard. But he took time, even after school he would say, “Come to the library and I’ll help yo u with these certain problems.” Mary attended a community college before coming to the university. She passed College Algebra with a ‘C.’ She attended a tutor lab fo r individual help with that course. She took Statistics three times and Liberal Arts Math twice before she passed those courses. When asked if this had affected her attitude toward mathematics, Mary responded: Yes, they were already negative, but th en they got more [negative], especially when I took statistics. The first time, I just did not understand why did I have to take it. Then they told me I could take Calculus, and I said that was a NO!! I thought I’ll take the simpler one, still horri ble, but simpler. The second time with Statistics, the teacher’s tests were all di fferent from what he taught. The third time, the teacher I had was pretty good. He did step-by-step more [which helped] understanding. Then when I got to liberal arts, it was like a mixture of statistics also, but yet it still got co nfusing in the middle. I went to the math lab and everything, but people can only help you so much because they have different people [working], so you just have to get it on your own and just read. And I’ve read the book thousands of times and c ouldn’t understand it. Prior to the start of the methods cour se, Mary was nervous about the course:

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189 At first I thought, “Oh Lord, how am I going to understand the different strategies to use. I was scared. With all these strategies I’m learning, would I be able to use it when I go into actual [classroom]? If I can’t understand, how am I going to be able to teach? Mary said that her feelin gs had changed since the st art of the methods course: Yes, it’s changed because I’m learning diffe rent strategies and ways to use it. I never knew that with the subtractio n problems, there were different [interpretations] and stuff like that. Now I get an understanding of how I could break it down for students so they co uld have a better understanding of it. When asked what a teacher could do to he lp students develop a good attitude toward mathematics, Mary said: I would say more one-on-one time…to see those students who are having the most trouble and … help those students who are not grasping that certain material…. They would have a better unders tanding of it instead of having them lost like me. When asked what Mary would like to add c oncerning her attitudes toward mathematics, she responded: I’m just glad I don’t have to take any mo re math…. Math to me is like, “What’s the use of it?” and “Why do I need it?” Bu t now I have to go to the elementary [level] and teach it to them. That’s one of my biggest fears is the math part, teaching them because I know that I had a very bad understanding [of math] and I don’t want to go in the classroom and have them looking lost like I was.

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190 At the end of the methods course, Mary s howed a survey change score of 14 points, indicating a positive attitude change. Her scores increased in all four components: Value increased by 5 points, Enjoyment by 4 points, Self-Confidence by 3 points, and Motivation by 2 points. ‘Lisa’ Lisa’s score on the ATMI was also 63 out of a possible 200 points. She was one of two participants to scor e 63, the lowest in the class, indicating the most negative attitude toward mathematics. This represen ted an average per item score of 1.58, with 1 representing the most negative attitude and 5 the most positiv e. Her average scores on the four attitude components were: Value, 2.9; Enjoyment, 1.3; Self-Confidence, 1.0; and Motivation, 1.2. Lisa said that she had clear memories of doing well in elementary school reading, and she remembered learning social studies and science. However, she was concerned that she had no memories of elementary school math other than not scoring well on standardized tests: I don’t remember math. That’s what scares me. The only thing I do remember are the [standardized] tests that we got back and those score sheets that say Low, Average, High. My math was always in the Low column and my reading and everything else was in Average and High. Lisa’s memories of middle school mathematics focused on a teacher who taught both science and mathematics in a 2-hour block. He repeatedly told the students that he

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191 disliked mathematics and would spend most of the time on science. When they did do mathematics, the teacher would get angry when the students made mistakes: When we did have [math] tests or we did have assignments, he was so angry when our stuff came out wrong. That’s wh at I really remember about middle school, just “I don’t like math. I don’t like math.” Hearing teachers saying that. “I don’t like math. I hate math.” When asked about her attitudes toward mathematics at that time, Lisa said: I thought it was OK not to like it. If ther e are teachers who don’t like it, why should I like it? … It just always felt like we were pressed for time in middle school, like we were rushing th rough the material. Even in 8th grade, it was a better teacher, a lot better, but we just always seemed rushed. Lisa made good grades in high school ma th, although she did not believe they were deserved. In ninth grade her Algebra I teacher was “a lot of fun and a nice guy.” However, Lisa remembered that he got off th e subject a lot. When they did talk about mathematics, she once again experienced “that rushed feeling:” I got an ‘A’ in that class, and I don’t think I deserved it. I wasn’t getting the material enough to get an ‘A’. But he was one of those who gave an ‘A’ for effort, gave the whole class an ‘A’, but I di dn’t know it well enough to get an ‘A’. In tenth grade Geometry class, Lisa’s teach er was a permanent substitute teacher who was “a joke.” He gave the cla ss assignments and then sat wi th his feet up on the desk: We’d kind of work it out together amongst ourselves. Some students were smarter in it than others, but we didn’t know how to teach each other. So there was a lot of copying. So I got through that class with an ‘A’... didn’t know it at all. It was all

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192 copying. Lisa’s eleventh-grade Algebra II teac her recognized that even though she had good grades from the previous two years, she di d not know any basic concepts. This teacher was very helpful to Lisa: She even said to me, “I don’t understand why you got the high marks that you did if you don’t know these basic concepts.” [She said it] in a very nice way. She spent a lot of time and effort with the class as a whole. She opened her door during lunch periods and mornings and tutoring outside of school. She was amazing. Unfortunately we didn’t have e nough time to get through as much time as we wanted to, but she helped with a lot of the basics that I do have now. So that was a good year. Lisa enrolled in Precalculus her senior year, even though she had already completed the required mathematics credits. However, sh e was struggling and ended up withdrawing from the course: The teacher made sure we knew [that the course was not required] right off the bat. She said, “You don’t have to be here if you don’t want to.” And with that being said, I didn’t take the fourth math a nd I left. I tried. I was there for the first couple of weeks and I was struggling. Sh e was rushing through, saying, “If you don’t know this, you don’t need to be here” type thing. So I left because I didn’t know it. Lisa said that mathematics in college “wasn’t too fun.” She had to retake her courses and was still trying to complete her mathematics requirements: I put it off; I put math off. I started with the Finite Math and Liberal Arts Math. I

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193 had never even heard of that. I didn’t know what that was. But it actually worked out better for me because it’s more wording and more stuff I can memorize. I memorize math. It isn’t the best thing to do, I know that now, but that’s a lot of my strategy…. That’s how I approach mat h. I was never taught another way to do it…. right now I’m struggling with my th ird math to graduate…. I’ve got two more chances. I’ve got the spring and the summer. Again I saved it for the last…. Two times for taking College Algebra, taking it and then dropping it halfway through. I’m not going to try a third time. When asked about her feelings prior to the methods course, Lisa said that she put off taking the mathematics methods courses until he r last few semesters. She said that she avoids mathematics classes until she “absolute ly has to take them.” When asked if her feelings about the methods course had change d since the start of the course, Lisa said: About the class, yes. About my ability with math, probably not….I’m actually learning things that I didn’t know, and that’s exciting. Bu t it’s kind of scary, too, being 22 years old and learni ng basic addition and subtrac tion strategies that I never heard before.… It wasn ’t as scary as I thought it was going to be. We’re learning more strategies vers us like you’re giving us math problems to do. But even with the “brain teasers” that we’ve had, and I couldn’t do them right off the bat, that still makes me nervous. I feel like I’m supposed to know it and I’m supposed to get it quickly, es pecially with my kids looki ng at me for the answer. When asked what teachers can do to help students develop a good attitude toward mathematics, Lisa said: Definitely to say that they like [math], to acknowledge that it is difficult for some

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194 people and people think in diffe rent ways, but it doesn’t have to be as scary. It’s not bad. Math is not bad. I think the best th ing would be to relate it to real life. That way they understand; I’m doing math a ll the time and it doesn’t have to be scary. Lisa expressed concern about her struggles with mathematics and about how those struggles would affect her ab ility to teach mathematics: I just hope it gets better. That ’s what I’m looking forward t o. I really want it to get better. I just am really nervous about not knowing enough for my kids [future students]… I just hope it’s not too late. That’s my bi ggest concern. I’m just concerned. I don’t know how many other pe ople are in my same position. It seems like everyone I talk to says, “Oh, you ’re just not a math person,” and if you hear something enough, you are going to believe it. When asked who had said that to her, Lisa replied: It was in college. My professor that I just got out of the class [said it]. He said, “You’re not a math person.” He just c onfirmed it. It’s been something I’ve [always believed]. I remember thinking [since middle school] that no one around me was a math person. I would bring my math book home, and my parents couldn’t touch it. They didn’t know about it. And if the teac hers didn’t like teaching it, they must not be math people. At the end of the methods course, Lisa show ed a survey change score of 23 points, indicating a positive attitude change. Her scores increased in all four components: Value increased by 3 points, Enjoyment by 4 points, Self-Confidence by 14 points, and Motivation by 2 points.

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195 ‘Hermione’ Hermione’s score on the ATMI was 185 out of a possible 200 points. Her score was the highest in the class, indicating the most positive attitude toward mathematics. This represented an average per item score of 4.63, with 1 representing the most negative attitude and 5 the most positive. Her aver age scores on the four attitude components were: Value, 4.7; Enjoyment, 4.2; Self-C onfidence, 4.9; and Motivation, 4.6. Hermione remembered elementary school mathematics as “just easy.” She made good grades, but “it didn’t really mean anyt hing” to her because she found most of the mathematics to be so easy. She did rememb er learning long division and how it helped her better understand the concept of multiplication: I distinctly remember long division. I re member thinking, ‘Oh times; that’s what times means! Why didn’t they tell me that before? ’ Then on the way home from school that day, she remembered helping others: I remember the day I learned long division. I taught people on my bus and I said, ‘See, this is easy.’ Hermione attended a junior high school ra ther than a middle school. She remembered doing well in 7th grade Prealgebra honors class: Our teacher would give back our tests, and she would call your name and you’d get your test. She’d call my name, and she’ d say, ‘Perfect score.’ Seventh grade was great. The next year Hermione was enrolled in an Algebra I honors class. She really thought her teacher was great even though her grade in th is course was not as good as in previous years:

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196 I had the best teacher. I didn’t get an ‘A,’ I got like a ‘C.’ But I knew a lot… Then the next year I had [teacher’s name] for Geometry. [Algebra I teacher] had come into the class, and I said ‘hi,’ and then he went away. The next day [Geometry teacher] told me that [previous teacher] said that I was the best student he’d ever had. I was like, ‘Oh my gosh!!’ That’s something that’s really important to me. I want to be a good student. When asked how she felt about receiving lo wer grades in mathematics that year, Hermione said that she still felt confident about her mathematics knowledge and abilities: I did get A’s in elementary math. But when I got C’s in middle school math, I knew more and it felt like I KNEW I was good at math. In 9th grade, Hermione took a Geometry course at her junior high school. She did well, and this reinforced her feelings of self-confidence with mathematics: I did really, really good in Geometry…. In Geometry, I got over 100, and I was so happy. I [thought], ‘See, I am good in math,’ and I was all happy. In high school, Hermione took Algebra II, Precalculus, and the AP [Advanced Placement] Calculus course: Then I went to Algebra II, and that was tough. I don’t know why. Now it’s so obvious; the unit circle is so obvious. I don’ t know what my problem was. I had a really great teacher. [Teacher’s name] was awesome, a great teacher. I also had him for Pace Setter Precalculus. It’s just be low AP. You have the test at the end, but you don’t get college credit. I was the only person who passed the test in the Pace Setter class. The Pace Setter class is really, really great. It’s [coursework is done] all in groups and that was another ‘A ’ that I got in math. We had this

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197 project … When everyone brought in their project on the day it was due, the class would all vote [for the best project]. W hoever got the highest [number of votes] would get 100 [points], as though they t ook another test and got 100 on it. Mine won … Then I took AP Calculus. AP Ca lculus was different. I really liked calculus. Calculus made sense … I didn’t get an ‘A;’ I got a ‘C.’ Hermione retook Calculus I in college. Again she received a ‘C,’ but she felt that this time she really understood the mathematics. Sh e did not feel as conf ident with Calculus II: I still got a ‘C,’ but now I knew everything, but I still screwed up on the test. Then I took Calc. II and I don’t know how I passe d it. I must have gotten 100 on the final, because I had an ‘F’ before the fina l. Then I took Calc. III and I dropped it because I wanted to get straight A’s, and I did and then I was happy. The thing about calculus is that I th ink it is really doable. It’s something we could teach [students]. Once they get the concepts, I think everybody is capable of calculus. Hermione then remembered that she had take n another college mathematics class, Finite Mathematics, which she found extremely easy: I took this class that was so easy, so easy it hurt my brain. Finite Math was just like, remember all that stuff that se emed hard in high school, but you knew it wasn’t hard? Well, here you go…. I was so frustrated, and I went to class everyday because [teacher] took roll, and [I would think], “Why am I here?” It was way too easy. It was so weird because people didn’t know what pi was, like they thought it was rational…. So I took the final, and I went over every problem because if I missed one single problem, I w ould be so mad at myself. There were

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198 some bonus problems, and I got th e highest you could get. Prior to the start of the methods course Hermione felt “kind of excited.” When asked if she felt any stress about the course, she said: No, no…. A lot of people are afraid that they don’t know enough, and I’m afraid they don’t know enough. If I don’t know something, I’m willing to learn it. I’m not afraid. I don’t know how a teacher can be a teacher and be af raid of learning something. It’s counter-intuitive. When asked what teachers can do to help their students develop good attitudes toward mathematics, Hermione said: Look at math as a tool, not a chore. So many people hate math. What’s to hate about it? It’s just numbers and operations and things to help you when you need to find the answer to a problem. I don’t know where it starts, but somewhere along the way with people who hate math, they have a bad teacher. You [as a teacher] have to be enthusiastic about what you are doing. You have to want to learn. If you learn in front of the kids and you’re saying, ‘Yeah, we are learning! This is helpful. I can solve things.’ Y ou’re modeling the behavior…. I think math is not the gargantuan task people make it out to be. At the end of the methods course, Hermione showed a survey change score of –6 points, indicating a negative attitude change. Her scores were sti ll extremely high, the fourth highest in the class However, they did decrea se in all four components: Value decreased by 1 point, Enjoyment by 1 point, Self-Conf idence by 3 points, and Motivation by 1 point.

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199 ‘Torri’ Torri’s score on the ATMI was 182 out of a possible 200 points. Her score was the second highest in the class, indicating the second-most positive attitude toward mathematics. This represented an average pe r item score of 4.55, w ith 1 representing the most negative attitude and 5 the most positiv e. Her average scores on the four attitude components were: Value, 4.9; Enjoyment, 4.3; Se lf-Confidence, 4.6; and Motivation, 4.2. Torri’s father was in the military, so she moved and changed schools several times. She was in Germany for grades two th rough four. She noticed a difference in the way mathematics was taught overseas: I did notice that when I was overseas, we us ed more manipulatives in class, more hands-on and different techniques to do mat h. When I came back to the states, it was all ones-place, tens-place, you have to do it that way [focus on algorithm]…. It was a lot more fun until I got [back] over here. …[In Germany, they used] group work, more hands-on stuff, and wh en I got back here, it was more worksheets, out of the book, things like that. In 7th grade, Torri’s mathematics teacher sugge sted to her mother that Torri attend a summer mathematics program that the teacher was conducting for students who needed extra help. Although Torri was doing “pretty well” in pre-algebra, her mother and teacher both thought that she could benefit from the pr ogram. At first Torri was not happy about attending this program, but it turned out to be a gr eat experience: I wasn’t happy at all having to go to su mmer school for math. When we got there, it wasn’t like regular school. Everythi ng was a different kind of activity. She gave us [activities] like riddles…. It wasn’t all like ‘OK, this is a math problem,

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200 this is how you do it. This is the answer.’ It was, ‘now YOU come up with a way to do it. Now YOU come up with a wa y.’ It was always something fun. By high school Torri had moved again. She took Geometry in 9th grade, Algebra II in 10th grade, Trigonometry and Analytic Geometry in 11th grade, and Mathematics Analysis in 12th grade. She had trouble with Geometry: Geometry was not good for me at all… [T he teacher] went real fast. I was [thinking], ‘Uhhhh!! I don’t understand! I’m fa iling!’ I was really failing. And it shocked me because up until 8th grade I was doing so good in middle school. Up until then I was doing really good, and then to get to Geometry and be like, ‘OK, I don’t understand this.’ And then I went to summer school at another school after that and she [teacher] kind of touched on Ge ometry a little bit more then, and then she helped me understand. In college, Torri took Finite Mathematics, Statistics, and Precalculus. She sometimes had trouble understanding the concep ts and would get frustrated when the professor would not adequately explain the mathematics. When that happened, Torri knew that she would have to figure it out herself: She [Precalculus Professor] didn’t really explain how to do the work. If the problem was in the book, she’d rewrite the problem. I’m [thinking], ‘No!! Explain why that does that!’ I kind of had to teach myself. By that point I was used to teaching myself. After 9th grade, I kind of got to a point where I thought, ‘If I don’t understand and my teacher won’t explain it, then I have to figure it out for myself.’ I got a ‘B’ out of [Precalculus]. That was good. Torri felt “kind of excited” prior to the start of the methods c ourse. She was looking

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201 forward to taking methods courses that would focus on how to teach different subjects. When asked if she felt any stress prior to th e beginning of the course, she replied, “No, not at all.” Torri reflected on her elementa ry school experiences in Germany as she shared her ideas about helping students de velop good attitudes toward mathematics: [I would] not scare them. Not just be like, ‘OK, this is math. This is how you do it. There is no other way.’ And if you get a wrong answer, not just [say], ‘OK, that was wrong. You’re wrong. This is the righ t way.’ There are other ways you can get the answer. But that was one reason why I like math because I kind of like being right. So when I got the right answer, I was like, ‘Aha! I got the right answer!’ When I came back here [to the U.S.] there was more emphasis on getting the right answer, like, ‘You did a good job because you got the right answer,’ while in Germany it was more like, ‘Well, you didn’t quite get it, but I see how you tried to do it,’ rather than ‘N o, it’s wrong; You’re wrong.’ When asked if there was anything else she would like to share concerning her attitudes toward mathematics, Torri said: I think I do [have a good attitude toward ma th]. I just hope that when I start teaching, that I can convey that to my students, a good attitude. That’s the only thing I’m really scared about. At the end of the methods course, Torri s howed a survey change score of 18 points, indicating a positive attitude change. Her scores increased in all four components: Value increased by 1 point, Enjoyment by 7 poi nts, Self-Confidence by 6 points, and Motivation by 4 points.

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202 Reflections from the Researcher’s Journal The researcher kept a reflective jour nal (Appendix O: Rele vant Excerpts) during the methods course. This journal provided a detailed account of what took place during each class, as well as the researcher’s impre ssions of each class. The intended purpose of this journal was to provide additional in formation when analyzing interviewees’ references to class activities. In an effort to provide reliability of this account, an additional observer also took field notes during two of the classes. The researcher’s journal and the observer’s notes added some perspective to these results. During her interview, Shelly made some comments that indicate d that she did not like solving “word problems.” With this comm ent in mind, the researcher’s journal was examined for relevant information. This search showed that nearly every methods course class included problem-solving activities. Howe ver, Shelly referred specifically to “word problems.” This type of problem solving was used during four of the classes. Two of these activities involved the researcher pres enting the participants with several word problems and asking them to share their soluti on strategies with ea ch other. The other two lessons involved recognizing and writing word problems demonstrating different interpretations of the basic operations of addition, subtraction, multiplication, and division. Shelly also said that she did not th ink cooperative learning ac tivities should be used all the time because sometimes student s like to figure out problems by themselves. The researcher’s journal reve aled that cooperative learning was always encouraged, but was usually not required. During several clas ses, games and interactive activities were used, so participants were required to work c ooperatively for these. However, most of the

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203 cooperative learning in the clas sroom was optional. The observe r’s notes reinforced this when she wrote, “some working together, bu t most working alone,” and “some working alone, some in groups.” The researcher’s journal and the observe r’s notes reinforced Yezania’s comments about her distraction and lack of focus in class. She sa id that she and her friends sometimes “did too much talking instead of doing the work.” The researcher had noted that the group of four, Yezania and her thr ee friends, were sometimes inattentive. The first mention of this came in the re searcher’s journal from week four: We did number relationships for 10-20 and th en I told them that we were going to play the counting game. I think that most of them enjoyed playing that game. There was a group of four sitting in the back next to [name] that was not really very engaged in the game. It looke d like a couple were sort of playing it, but they were also talking about other th ings. When I came over, the one student sitting closest to the front was not playing at all. When I asked, she said that she was watching the others. I asked if she did not have her blocks with her, and she said that she did, so she took them out. Another entry concerning this group of participants came in week 11: I decided to start with a c ouple of activities while we we re waiting for latecomers. The first was Benny’s Cakes [an activity involving fractions]. They worked on them for about 5-10 minutes while I ci rculated around the room. The group of students who sit [description of where they always sat] was not really engaged--as usual.

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204 The observer made comments in her notes about this group of participants on both of her visits. Because the observer moved to a different part of the room every 45 minutes, she was able to sit near this group for pa rt of each visit. On th e first visit, she sat next to Yezania and her friends and wrote, “student trying to look busy but not doing anything.” It is unclear to which student sh e was referring, but it was either Yezania or one of her friends. On her second visit, the observer again sat next to Yezania and her friends. She noted that students were instruct ed to model decimals by shading a decimal grid. She observed, “[There were] three student s next to me, two did not try it, one did.” It is interesting to note that Yezania was th e only one of the friends to have a negative attitude change score. The researcher’s journal also provid ed insight in an unexpected manner. Throughout the journals and interviews, many participants related positive experiences and attitudes to teachers who showed care a nd concern for them. With this in mind, and in light of the large improvement in attitudes toward mathematics that the journal-writing class demonstrated, the researcher examined her journal for instances where she had noted thoughts or actions that reflected her care and concern for pa rticipants and their affect. The following excerpts are representati ve of these. Please note that the data written in brackets reflect the researcher’s personal reactions a nd thoughts at the time. The first excerpt was written on the first da y of classes. Perhaps the researcher’s memories of students in previ ous classes writing in their jo urnals about how nervous and uncomfortable they felt on the first day prompted these actions and thoughts: When I first got there, the class was very quiet, not really talk ing to each other very much. I came in and started talking to those in front very casually about the

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205 heavy traffic on first week of classes, park ing situation, etc. [trying to establish rapport and put students at ease] The next excerpt was from the researcher’s en try for the second class. The researcher had received a couple of journals where students said that th ey were feeling overwhelmed with the amount of information in the first five chapters: When I brought this up, approximately 10 stude nts agreed and said that they were feeling the same way. This was a good opportu nity for me to point out where in the note packet the review questions we re. We spent a few minutes talking about the idea that I wanted them to read all five chapters to get an overview of the philosophy of the course, but that as far as tests were co ncerned, they would be responsible primarily for things on th e review sheet. [I thought that students seemed relieved by that. I remember thinki ng that this type of communication is another benefit of journals. Something like that might have never come up, but because they were writing that first jour nal about their concerns about the course, it did come up. This allowed me to addre ss this issue with the whole class and probably alleviate the concerns of several students.] The following excerpt was from the day of the methods course’s first test. The researcher was sitting in her office before class, and one of the participants came in. She seemed very stressed, and she said that she was nervous about the test, especi ally since we would be covering new material first. Perhaps as the researcher tried to alleviate the participant’s concerns, the researcher was r ecalling journals from previous students who talked about the anxiety asso ciated with testing: I explained that we would do about an hour of new material and then we would

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206 take a long break. During the break, they would have time to ask questions, get refocused, and she seemed to be relieved a bout that. After she left I decided that I better go to the classroom early because I figured there would probably be several who were stressed. When I got to the cla ss, I could immediatel y tell that I was correct. One of them even said, “Can you f eel the stress in the air?” and I could. I tried to reassure them. I asked them what they were nervous about. Several of them seemed to be saying the same thi ng, which was that they had never taken one of my tests before. They were concer ned about the format. They pointed out that in the review notes, there were many pa ges that had five parts to this and six steps to that, and they were afraid that I would ask them to recall all of them. [I remember thinking that this was a valid concern and that as a student I would have felt the same way.] Because I felt th at so many of them were feeling this way, and I wanted them to not be focu sed on that while we were covering new material, I actually got out a copy of the te st. I told them that I would not tell them the actual questions but I went through the shor t answer section and gave a general idea … name 2 or 3 of 7 steps, etc. They seemed relieved that they wouldn’t have to name all of them. They also seemed concerned that they would have to write long essays and I assured them that they wouldn’t. They did seem very relieved at that point, so I decided to go ahead and start the new material at about 9:15. The researcher’s journal was intended to provide additional information

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207 when analyzing interviewees’ references to cl ass activities. As it turned out, the journal provided a great deal more. It provided the rese archer with valuable insights into her own teaching and interactions with her students. The researcher, as well as the participants, realized the benefits of reflec tion as a result of this study. As one participant said in a journal, “I see the benefits of reflection a nd I am thankful for it. Sometimes you forget how much you can learn from yourself and your experiences, but re flection helps you to see the whole picture.”

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208 CHAPTER FIVE OVERVIEW, CONCLUSIONS, IMPLICATIONS, AND RECOMMENDATIONS This chapter contains four sections. The first section presents an overview of the study. The second section summarizes the resear ch findings of this study and examines the conclusions derived from these findings. This section is organized according to research question. The third section discu sses implications and recommendations for practice. The fourth section focuses on im plications and recommendations for future research. Overview of Study The importance of developing self-confiden t, motivated students who value and enjoy mathematics has been well established (McLeod, 1992; NCTM, 1989, 2000; NRC, 1989), but the means for doing so are not as clear. Although results are not always consistent, studies have shown that children typically begin school with positive attitudes toward mathematics, but these attitudes tend to become less positive as they get older. By the time students reach high school, their attitu des toward mathematics have frequently become negative (McLeod, 1992). Although many students value and enjoy mathematics and are motivated and confident in their abil ities to do mathematics, there are still too many who do not feel this way. Negative at titudes toward mathematics are “thought to plague learners at every level of scho oling” (Sherman and Christian, 1999, p.95).

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209 Studies have also shown that many pres ervice elementary school teachers have negative attitudes toward mathematics (R ech et al., 1993; Cornell, 1999; Philippou and Christou, 1998). This should be a concern for teacher educators because teachers with negative attitudes toward mathematics are unlik ely to cultivate positive attitudes in their own students (Hungerford, 1994; Philippou and Christou, 1998; Sherman and Christian, 1999). This study explored the use of reflecti on, through reflective journals, as a tool for teacher educators seeking to both understand how attitudes toward mathematics are formed and to improve these attitudes. The purpose of this study was to examine the attitudes toward mathematics of preservice elementary school teachers ente ring an introductory mathematics methods course. The study focused on the following att itudes: value, enjoym ent, motivation, and self-confidence. Qualitative methods were used to explore thes e attitudes and the experiences that have led to the developm ent of these attitudes. The study sought to determine the extent to which preservice teach ers’ attitudes toward mathematics changed during the methods course. The study also examined the correlati on between preservice teachers’ initial attitudes toward mathema tics and their achievement in the methods course. The intent of the study was to answer the following research questions: 1. What are the attitudes toward mathem atics of preservice elementary school teachers entering an introductory mathema tics methods course? In particular, how do preservice teachers score on each of the four attitudinal components being measured: value of mathematics, enjoyment of mathematics, motivation for mathematics, and self-confidence with mathematics?

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210 2. To what extent do attitudes toward math ematics of preservice elementary school teachers change during the mathematics me thods course? To what do preservice teachers whose attitudes toward mathematics were altered attribute this change? 3. What is the relationship between preservi ce elementary teacher s’ initial attitudes toward mathematics and their grade on the methods course final examination? 4. What do preservice elementary school te achers’ reflective journal entries reveal about their attitudes toward mathematics a nd the experiences that have influenced the development of those attitudes? 5. What are the attitudes toward and ex periences with mathematics of those preservice elementary school teachers id entified as having the most extreme (either positive or negative) attitudes? The participants in this study were 33 university students enrolled in one section of a mathematics methods course for elementa ry education majors at a major research university in the southeastern United States during the fall semester, 2004. The researcher taught the course; an intact group was used due to university scheduling. The class met once a week for three hours. This course is the first of two mathematics methods courses that elementary education majors must comple te and utilizes constructivist instructional methods, such as the use of hands-on mani pulatives, cooperative group work, problem solving, and the use of calculators. Children’ s literature in the t eaching of elementary school mathematics was also emphasized. Participants’ attitudes toward mathematics were measured using The Attitudes Toward Mathematics Inventory (ATMI), (Tapia, 1996) which contains 40 items (see Appendix E). Participants were asked to i ndicate their degree of agreement with each

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211 statement using a Likert-type scale, from st rongly disagree to st rongly agree. Possible scores on the ATMI range from 40 to 200. Th ere are 10 items dealing with Value of Mathematics, 10 with Enjoyment, 15 with Self-Confidence, and 5 with Motivation. Because there are unequal numbers of items fo r each attitude factor, the average score per attitude factor was also found in order to make comparisons more easily. These average per-item scores ranged from one to five. Descriptive statistics were computed for participants’ composite attitude scores, th eir scores on each of the four attitudinal components being measured, and on each individual survey item. Each participant completed the ATMI at the beginning of the semester and again during week 12 of the 15-week semester. This allowed the researcher to measure each participant’s initial attitudes to ward mathematics and to assess any changes that may have taken place during the first 11 weeks of the se mester. The ATMI was also administered during week one and week twel ve to students in two other sections of the mathematics methods course for the purpose of maki ng comparisons in results. Two different instructors taught these methods course se ctions. The researcher and the other two instructors all used the same textbook (A ppendix D) and manipulatives kit. Although each instructor was responsible for the cont ent of her own course, all three classes covered the same content material and took the same final examination. The two other instructors were asked not to give their classes any writ ten assignments that would involve reflection on their attitudes toward mathematics. Eight reflective jour nals were assigned over the cour se of the semester. Five of the journal prompts (Table 3) related dire ctly to the purpose of this study and were analyzed using Hycner’s guidelines for the phenomenological analysis of data (Appendix

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212 K). This involved reading the journal for a sense of the whole, then identifying small units of meaning. Similar units of meani ng were clustered and common themes were identified. When a journal expressed multiple themes, these themes were analyzed separately. The remaining three journal prompt s did not relate direc tly to the purpose of this study, but they did relate to the pur pose of the methods course. Therefore, participants responded to them as well, but these entries were not analyzed unless they contained information relevant to the study. Journal entries were submitted by email, and the researcher responded to each entry by email (see Appendix F for sample responses by students and the instructor). The two preservice teachers with the lowe st initial scores on the ATMI and the two with the highest scores were each asked to participate in an individual interview where their attitudes toward and experiences with mathematics were further explored. These Experiences with Mathematics Interviews (Appendix H) took place between week six and week eight of the semester. After participants had completed the second ATMI, a repeated measures t test was conducted using composite survey scores to determine if a signi ficantly significant change in attitude had occurred. Those particip ants with change scores greater than one standard deviation above or below the mean ch ange score were considered for individual interviews. These interviews were conducted with four preservice teachers who showed the greatest positive changes in attitude and three preservice teachers who experienced negative changes in attitude. These Changed Attitudes Interviews (Appendix I) focused on participants’ ideas about thos e aspects of the methods course that may have influenced their attitudes toward mathematics. These interviews took place approximately one

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213 month after the completion of the methods c ourse and submission of final grades. All interviews were analyzed using Hy cner’s guidelines (Appendix K). The departmental final examination wa s used to determine the relationship between participants’ initial attitudes toward mathematics and their achievement in the methods course. The final examination was a 50-item multiple-choice instrument that included questions about both mathematics c ontent and pedagogy. The composite attitude score was the independent variable and the methods course final examination grade was the dependent variable. Summary of and Conclusions from Research Findings Research Question One: Initial Attitudes Toward Mathematics Participants had a mean composite survey score of 3.12 on the 5-point scale, with a score of one representing the most negativ e attitude, a score of three representing a neutral position, and a score of five representi ng the most positive atti tude. Therefore, the mean composite score reflected attitudes th at were just above the neutral position. The average initial survey scores were highest or most positive for Value of Mathematics, with a mean score of 3.96. Value was the onl y component with a m ean score above the neutral position of 3.0. The mean scores fo r Self-Confidence and Enjoyment were 2.96 and 2.79, respectively. The lowest or most nega tive scores were for Motivation, with a mean score of 2.55. These preservice teachers, as a whole, valued mathematics and viewed it as important, but they did not enj oy mathematics or feel self-confident or motivated about mathematics.

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214 Table 8 (pages 93-97) lists the mean rating for each individual survey item, grouped by attitudinal component. The hi ghest-scoring item for the Value of Mathematics component was also the highe st-scoring item for the entire survey: Mathematics is a very worthwh ile and necessary subject. The mean rating for this item was 4.39, indicating a positive attitude. The lo west-scoring Value of Mathematics item, I think studying advanced mathematics is useful, had a mean rating of 3.24. Six of the 10 Value item ratings were at least 4.0. All Valu e item ratings reflected at least somewhat positive attitudes. In the Self-Confidence compone nt, the highest-scoring item, My mind goes blank and I am unable to think clearly when working with mathematics, had a mean rating of 3.36. Because this item was stated from a negative perspective, a response of strongly agree was scored as one point, and a response of strongly disagree was rated as five points. Therefore, a mean rating of 3.36 represen ts slightly positive attitudes. The lowestscoring Self-Confidence item, I have a lot of self-confidence when it comes to mathematics, had a mean rating of 2.70. This repres ents attitudes that are somewhat negative, indicating that, as a whole, participants believed th at they did not have a great deal of self-confidence with mathematics. Se ven of the items had mean scores below the neutral position of 3.0, and seven it ems had mean scores above 3.0. The highest-scoring item for Enjoyment, I get a great deal of satisfaction out of solving a mathematics problem, had a mean rating of 3.18. This item reflected an attitude slightly above the neutral position, indicati ng that these preservice teachers did not particularly enjoy mathematics problem solv ing. The lowest-scoring item for Enjoyment, I am happier in a math class than in any other class, had a mean rating of 2.06,

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215 indicating a negative attitude. These future el ementary school teachers did not especially enjoy mathematics classes. Seven of the 10 Enjoyment items had mean scores less than the neutral position of 3.0, reflecting nega tive attitudes concerni ng liking and enjoying mathematics. The statement, I would like to avoid teaching mathematics, was the highestscoring item for Motivation, with a mean sc ore of 3.55. Because this item was stated from a negative perspective and the scoring wa s reversed, a mean score of 3.55 represents moderately positive attitudes. It is notewo rthy that these future elementary school teachers will most likely all be teaching mathematics at some poi nt in their careers, yet as a whole, they had only moderately positive att itude scores about teaching mathematics. The lowest-scoring item for Motivation, I plan to take as much mathematics as I can during my education, had a mean score of 1.94, which repr esents a negative attitude. The mean score for this item was the lowest on th e entire survey, reflecti ng the most negative attitudes. Participants, as a whole, were not interested in taking any more mathematics courses than were required. It should be noted that, at this point in their education, most of the participants had fulf illed their mathematics requirements for graduation, so perhaps this result is not surprising. Researchers have found that many preser vice elementary school teachers at the university have negative att itudes toward mathematics (C hristian, 1999; Cornell, 1999; Hungerford, 1994; Philippou and Christou 1998; Rech et al., 1993). This study confirms these findings. The NCTM (1989, 2000) establis hed goals related to affective factors. One of these goals was that mathematics students learn to value mathematics. The participants of this st udy, as a whole, demonstrated that they valued mathematics. They

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216 viewed mathematics as worthwhile, usef ul, and necessary. This finding should be encouraging to those in the field of mathem atics education. These re sults indicate that perhaps mathematics educators have had some success in achieving this goal. Furthermore, it is promising that these pr eservice teachers valued mathematics, as perhaps they will be more likely to pass on this positive aspect of mathematics to their future students. Unfortunately, results from the other thr ee attitude components in this study offer a more pessimistic view. Another goal of the NCTM (1989, 2000) was that students develop confidence in their own mathematical ability. Results from this study indicate that this goal has not been met with these participants. In fact, mean scores for the components of Self-Confidence, Enjoyment, and Motivation were all below the neutral position of 3.0. The composite survey mean score was 3.12, but if the Value items were removed, the composite mean score for the Self-Confidence, Enjoym ent, and Motivation items would be 2.8, reflecting somewhat nega tive attitudes. These findings should be disappointing to mathematics educators, es pecially when one considers that these participants will soon be teaching mathematics in classrooms of their own. If they do not feel self-confident or motivated about math ematics and do not enj oy doing mathematics, they could pass these nega tive attitudes on to their future students. Research Question Two: Changed Attitudes Toward Mathematics A repeated measures t test was conducted using co mposite survey scores to determine if a statistically significant change in attitude occurred. Each participant’s change score, which was their post-course score minus their pre-course score, was

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217 calculated. Change scores could range from -160 to 160, indicating a negative or positive change in attitude, respectively. The mean ch ange score for the 33 participants was 17.03 ( SD = 17.59), and the median change score was 15. This represented a statistically significant positive change in attitude. The ch ange scores ranged from -12 to 58, with a change score of zero representing no change in attitude. Twenty-seven of the 33 participants, or 82%, had positive change scores. Negative change scores were small in magnitude and therefore lacked practical signi ficance. Mean survey scores for all four attitude components increase d, with Self-Confidence having the largest per-item change score (Table 9). It should be noted that comparisons betw een mean pre-course survey item scores (Table 8) and mean post-course survey item scores (Appendix N) indicate that post-course scores increased on every survey item. The following survey items are among those whose mean scores increased by gr eater than 0.5 on the five-point scale, indicating more positive res ponses to the items: I like to solve new problems in mathematics. I really like mathematics. I am comfortable expressing my own id eas on how to look for solutions to a difficult problem in math. I am comfortable answering questions in math class. Mathematics does not scare me at all. I have a lot of self-confidence when it comes to mathematics. I am able to solve mathematics problems without too much difficulty. The challenge of math appeals to me.

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218 When I hear the word mathematics, I have a feeling of dislike.* I am always under a terrible strain in a math class.* It makes me nervous to even think about having to do a mathematics problem.* I am always confused in my mathematics class.* The ATMI was also administered during w eek one and week twel ve to students in two other sections of the mathematics methods course in order to make comparisons in results and to determine if the methods course alone would lead to attitude changes. Two different instructors taught these methods course sections. These classes also used collaborative group work, problem solving, and manipulatives, but participants were not asked to reflect on their att itudes toward and experiences with mathematics. Neither comparison class demonstrated a statistically significant attitude change. The mean change score for the 24 participants in the first comparison class was 1.0 ( SD = 12.75), and the median change score was zero. Change scores ranged from -25 to 33; 10 of the 24 participants, or 42%, had positive change scores. The mean change score for the 31 participants in the second comparison class was 1.48 ( SD = 13.08), and the median change score was -2. The change scores ranged from -21 to 35; 14 of the 31 participants, or 45%, had positive change scores. It is interesting to note that the pa rticipants in the journal-writing class demonstrated significant positive attitude cha nges, with a mean change score of 17.03, *Scoring for these items is reversed and uses anchors of 1: strongly agree, 2: agree, 3: neutral, 4: disagree, 5: strongly disagree. Therefore, on all items, scores range from 1 to 5, with 1 indicating the most negative attitude and 5 indicating the most positive attitude.

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219 while the comparison groups had mean change scores of only 1.0 and 1.48, which reflect very small attitude changes. Both comparis on classes had some participants with large positive change scores, but in both cases these accounted for less than half of the class. The positive attitude changes that took place in the journal-writing class were greater and accounted for a much larger percentage of the class. Not only did the participants in the journal-writing class demonstr ate a much greater positive ch ange in attitudes toward mathematics than the comparison groups, but also results from the two comparison groups were very similar to each other. Becau se the two comparison classes were taught by two different instructors, this suggests th at perhaps some element or elements that were present in the study class bu t not in the comparison classes aff ected this change. Interviews: Positive Change Scores Question two also focused on those partic ipants whose attitudes were altered and asked to what these participants attributed the change. Six participants had positive change scores greater that one standard de viation above the mean change score, which reflected a change score of at least 35 point s. The four participan ts with the greatest positive change scores were interviewed in order to explore to what these participants attributed their positiv e attitude change. In addition, six participants had negative change scores greater than one standard deviation below the mean change score. Because the mean was so high, this reflected change scor es of less than -0.56. These negative change scores ranged from -12 to -1. Although these change scores were negative, the difference between pre-course and post-cour se attitude scores was very small. Such minor variations could be explained by contextual variable s or measurement error. Even though the

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220 magnitude of their change scores was small compared to those of the positive change score interviewees, three of these participants were also interviewed. All of these interviews took place four to six weeks afte r the completion of the methods course and submission of final grades. All four of the participants with posi tive attitude changes who were interviewed had mean response pre-course survey scor es of between 2.2 and 2.8 per item. These scores reflected attitudes that were slightly to moderately negative as they were below the neutral position of 3.0. All four post-course m ean response survey scores were between 3.3 and 4.0 per item, reflecting somewhat posit ive attitudes to positive attitudes. These four participants not only improved their att itudes, but they went from having somewhat negative attitudes toward mathematics to ha ving positive attitudes toward mathematics. In addition, all four interviewees improved thei r attitudes toward math ematics in all four of the components being measured. Before being informed of the results, all f our of these participan ts said that they believed their attitudes toward mathematics had improved since the start of the methods course. They all had previously struggled with mathematics, felt uncomfortable with mathematics, dreaded mathematics, and lack ed self-confidence with mathematics. Three of them said that they now had a more posit ive attitude toward teaching mathematics as well. When asked which aspects of the course they thought had affected their attitudes toward mathematics, three of these participan ts mentioned the use of manipulatives. They felt that using manipulatives had increased their conceptual unde rstanding. Showing a classmate how to use the manipulatives incr eased their self-confidence about teaching

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221 mathematics. Two of the four also mentione d the journals. They appreciated the prompts relating to attitudes and experiences with mathematics and the instructor’s time in responding to their journals. One mentioned the format of the course and the copies of overhead transparencies that were provided, allowing her to pay attention to the class discussion rather than worrying about taki ng notes (samples found in Appendix P). One of these participants also talked about the benefit of having a classmate who was a close friend. They were able to help each other understand the concepts. The four participants with the greatest improvement in attitude were all very positive about the use of manipulatives in teaching mathematics and said that manipulatives were “useful” and “necessary.” When discussing the use of manipulatives in the methods course, they all found them “v ery helpful” in understanding and making sense of the mathematical concepts. All of these participants expressed po sitive views about the use of cooperative learning in the mathematics classroom. They all found the cooperati ve learning activities in the methods course “helpful” and “beneficia l” because they could ask their classmates for help when needed. All four talked about thei r previous struggles with problem solving. They all said that their problem-solving skills had greatly improved as a result of the methods course. While discussing this, they all made comments that indicated that their self-confidence with problem solving had increased. Tessa expressed this the most clearly: I was frustrated 'cause that was the only way that I knew how to do that problem and if I couldn't get it, then I felt dumb or I felt stupid or I fe lt that I wasn't good in math, but at the end [of the methods c ourse], I was [thinking], 'Oh, I got it!' and

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222 then 'Oh, I could do it this way or this way!' and I was [thinking], 'Oh, you know what? I can do math!' When asked about the use of journals in teaching mathematics, the interviewees said that journals were a “useful” and an “important” form of assessment. Two of them mentioned that journals were usually used with English rather than mathematics. They all said that they “liked,” “loved,” or “enjoye d” writing journals fo r the methods course. They felt that the journals were a great way to communicate with the instructor and also a good way to reflect on their past experien ces with mathematics. As Erin said: In the course, I thought [journal writing] was good because you [instructor] were able to see how we came to feel about certain subjects in math by the questions that you asked, and it was very good for me because I had forgotten about some things and having to think about it, w hy my attitudes were a certain way, I was [thinking], 'Oh, yeah. THAT 'S why I don't enjoy math.' After learning that their at titude scores had increased so dramatically, all four interviewees said that they were not at all surprised. They all expre ssed an increased selfconfidence with mathematics and with teac hing mathematics. As Jennifer said: … coming into here [methods course], I still was kind of uneasy, but now after [completion of methods course], I felt like, 'All right, I can do math, no question about it.' I'm even willing to get up and teach kids math when before I would say that was the one subject that I did not want to teach…. But now I feel like I could step in there and actually teach math.

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223 Interviews: Negative Change Scores The three participants with negative change scores who were interviewed had mean change scores of -12, -10, and -5. Th ese change scores were much smaller in magnitude than those of the interviewees with positive change scores, which ranged from 41 to 58. This is because 82% of the particip ants had positive change scores, and none of the six negative change scores was very ex treme, with -12 being the largest negative change. Nevertheless, these participants were interviewed in order to determine if their decrease in attitude scores was due to anything other than measurement error. It is informative to look at the pre-co urse and post-course mean response survey scores of these participants. The first particip ant, Stephanie, went fr om a pre-course score of 3.5 per item to a post-course score of 3.2 per item. The second participant, Shelly, went from 3.9 per item to 3.6 per item. Both of these participants st arted with moderately positive attitudes and ended with attitudes that were still somewhat positive, as they were above the neutral position of 3.0. Therefore, perhaps it is mo re appropriate to describe these change scores as less strong positive rather than negative It is important to note that both Stephanie and Shelly were surprised when they learned that their attitude scores had decreased. The third interviewee, Yezania, began with a score of 2.0 per item and finished with a score of 1.9 per item. Her attitude toward mathematics began in the negative range, and it ended slightly more negative. Before being informed of the results, St ephanie said that she “definitely” thought her attitude toward mathematics had improved si nce the start of the methods. Shelly said that she was not sure if her attitude had ch anged. She explained that her attitude toward mathematics was “constantly changing,” depending on the mathematics topic being

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224 discussed. Yezania said that she thought her attitude had improved in some ways and not in others. She explained that she thought her attitudes had improved because the manipulatives used in the course had incr eased her conceptual understanding. However, she also said, “I am still kind of nervous when it comes to math. I’m not secure of myself.” It is interesting to note that both Sh elly and Yezania were initially conflicted as to whether or not their attitudes had changed, but Stephanie was initially convinced that her attitudes had improved. However, both Step hanie and Shelly expressed surprise when they learned that their attit ude scores had decreased. When asked which aspects of the methods course they thought had affected their attitudes, none of the three thought of anyt hing that had negatively influenced their attitudes. Two of the three mentioned that th e use of manipulatives was an aspect of the course that had positively affected their att itudes. Both Stephanie and Yezania explained that they work with elementary school stude nts in after-school pr ograms, and they both have had success using manipulativ es with their students as a re sult of the course. One of the interviewees said that the journals were “really, really helpful.” Another said that the organized format of the course had positivel y influenced her attitude. There were “no surprises … [everything] was really clearly st ated.” One also mentioned that the textbook was useful and had positively affected her attitude. The three interviewees described th e use of manipulatives in teaching mathematics as “really awesome” and “great.” They all had positive comments about the use of manipulatives in the methods course as well. One interv iewee said that she benefited from watching the instructor m odel the manipulatives on the overhead projector. Another interviewee said that she “loved” using the manipulatives and was

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225 already “thinking of so many ways [to use] th e manipulatives” with her future students. The third interviewee, Yezania, said that the manipulatives rea lly helped her with problem solving because she is “not very good with word problems.” The interviewees believed that cooperative learning was “really great,” “very important,” and “needed” when teaching math ematics to children. Their comments about the use of cooperative learning in the methods course were more conditional. Stephanie said that the cooperativ e learning activities in the met hods course were “good,” but she made it clear that she does not like coopera tive learning when it involves assignments because “someone ends up always doing more” than the others in the group. It is unlikely that this view was a factor in her small ne gative change score because there was only one assignment in the methods course that coul d have been done as a group assignment, and participants had the choice of working in gr oups of up to four or working alone. Shelly said that the cooperative lear ning activities “wouldn’t ever bring my attitude down,” but she also said that cooperative learning s hould not be “a constant thing.” She explained that sometimes students want to “work on th eir own and get their own answer.” This view could have been a factor in Shelly’s change score as the methods course included many activities where participants were encouraged to work together in class. Yezania’s response to this question was very revea ling. She said that maybe she would have benefited more from the cooperative learning ac tivities in the methods course if she had been more focused: “I think because we di d too much talking instead of actually doing the work. So, you know, I’m being honest. Sometimes you get kind of sidetracked.” Stephanie said that she felt that sh e would use what she had learned about problem solving. Yezania also said that she had “learned a lot of new ways to solve”

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226 problems in the methods course. She explaine d that this was “good” for her because she had struggled with problem solving in the past Shelly said that she had also struggled with problem solving and sometimes found it “i ntimidating.” This view could have been a factor in Shelly’s change score becaus e the methods course included many problemsolving activities. All three of the nega tive-change-score interviewees said that they had benefited from the journals. Stephanie said that she f ound it “very helpful” not only to write them, but also to go back and read what she had previously written. Sh e said, “They made me think about a lot of stuff that I proba bly wouldn’t have thought about unless you prompted me.” Shelly also said that the j ournals were helpful because they “make you not forget your own little history, like your personal math diary…. It reminds you of the ups and downs and the positives and negatives, the things that are important.” After learning that their at titude scores had decrease d, two of the interviewees reacted with surprise. Perhaps this was not unexpected since th eir change scores were so small in magnitude and also because they bot h began and ended the methods course with somewhat positive attitudes toward mathematic s. Stephanie was amazed to learn that her score had decreased. At the beginning of the in terview, she had said that she believed her attitude toward mathematics had improved, a nd she reiterated that at this time. She looked at her second survey and observed that she had not answered ‘strongly agree’ to any of the items. Stephanie then reminded th e researcher that St ephanie’s mother had passed away only a few weeks before sh e had completed the second survey. She suggested that perhaps this ha d influenced her responses on the post-course survey. It is interesting to note that Stephanie had the la rgest negative attitude change of all the

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227 participants in the methods course, although her score still reflected pos itive attitudes. It appears quite conceivable that her small change in attitude could have been influenced by her mother’s recent death rather than by anyt hing that occurred in the methods course. This seems even more plausible because Ste phanie herself felt that her attitude toward mathematics had greatly improved. At the beginning of the interview, Shelly had said that she was not sure if her attitude had changed or not. However, upon learning that her attitude score had decreased, Shelly said that sh e was “surprised.” She added: I really think that it probably would have gone the other way. I do, and I think that it has to do with the way that you've taught me how to teach children which may have been different than the way that I learned … I've picked up a lot from your class and I think that my attitudes would' ve gotten better just because a lot more makes sense now and that's just kind of su rprising to hear that it went the other direction. Perhaps comments that Shelly made duri ng her interview con cerning cooperative learning and problem solving could help explain her slightly less positiv e attitude score. Yezania had said at the beginning of the interview that she thought her attitude toward mathematics had improved in some ways and not in others. When she learned that her attitude score had decrease d, Yezania said that she belie ved that these results could have been related to the dist ractions she had mentioned earl ier involving sitting with her friends in class. As she said, “I think it di d [affect the scores] .… sometimes I wouldn't be grasping [a concept], but I woul d be too involved in [chitcha t noises] so I wouldn't grasp

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228 it too well.” It is interesting to note that both Stephanie and Yezania had theories as to why their attitude scores had decreased, while Shelly did not. Themes Across Interviews Twenty-seven participants in the methods course, or 82%, had positive change scores, indicating improved attitudes toward mathematics. These positive change scores ranged from 1 to 58. Only six participants ha d negative change scores, ranging from -12 to -1. Those interviewees with positive attitude change scor es had change scores that ranged from 41 to 58, which reflected large chan ges in attitude. All four of them began with somewhat negative attitudes and ended with attitudes in the positive range. Those interviewees with negative att itude change scores had change s that were much smaller in magnitude and lacked practical significan ce. The two who began with moderately positive attitudes ended with attitudes that were still somewhat positive, and the one who began with negative attitudes ended slightly more negati ve. Therefore, comparisons between those interviewees with positive cha nge scores and those with negative change scores are difficult to make. We are comparing participants with large positive changes to those with very small negative changes. Perhaps this explains why there are not many differences in the responses of the two groups. In fact, it seems to make more sense to look across all interviewees to determine which aspects of the methods c ourse participants think have positively influenced their attitudes. Of the seven inte rviewees, five mentioned manipulatives when asked which aspect or aspects of the methods course they thought had influenced their attitudes toward mathematics. Three of the seven said that the j ournals had positively

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229 affected their attitudes. Two of the seven said that the organized format of the course had influenced their attitudes in a positive manne r. It is interesting to note that the two comparison classes also used manipulatives, but they did not use j ournals or the same format where participants were provided w ith copies of overhead transparencies. When asked about specific aspects of the methods course, responses across interviewees were similar. A ll seven of those interviewed were very positive about the use of manipulatives in teaching mathematics and also in the methods course. All seven expressed positive views about the use of cooperative learning in teaching mathematics and in the methods course, although three of the interviewees qualif ied their responses. Six of the seven who were interviewed sa id that they had struggled with problem solving. Stephanie was the only one who did not mention this. Six of the seven also said that their problem-solving skills had improved as a result of the methods course. Shelly was the only one who did not express this view. All seven interviewees had very positive comments about journals. They described journal writing as “useful,” and “im portant.” When discussing the journals in the methods course, they said that they “ liked,” “loved,” and “enjoyed” journal writing and found it “helpful” and beneficial to reflect on their past experiences with mathematics. It is interesting to note that there are plausible explanations for all three of the interviewees’ small negative change scores. One was most likely due to the recent death of the participant’s mother. The second could have been a result of the extensive use of cooperative learning activities and problem solving in the methods course. That participant had some negative views about bot h of these. The third participant with a

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230 negative change score believed th at her own lack of focus in the course was a factor in her slightly negative change in attitude score. Several studies have demonstrated success in improving attitudes toward mathematics of preservice elementary teacher s enrolled in mathematics methods courses (Anderson and Piazza, 1996; Gibson and Va n Strat, 2001; Huinker and Madison, 1997; McGinnis et al., 1998; Philippou and Chri stou, 1998; Quinn, 1997; Sherman and Christian, 1999). These methods courses util ized constructivist instructional methods such as the use of hands-on manipulatives, cooperative group work, and problem solving. However, none of these studies used the a dditional tool of reflective journals where participants reflected on their own attitudes toward and experi ences with mathematics. In this study, participants from all three classe s used manipulatives, cooperative group work, and problem solving. However, only the jo urnal-writing class experienced significant positive attitude changes. This suggests that the use of manipulatives, cooperative group work, and problem solving is not the sole explanation for positive changes in attitude toward mathematics. Research Question Three: Relationship Between Attitudes and Achievement A Pearson correlation coefficient was found in order to determine the relationship between initial attitude surv ey scores and scores on the methods course departmental final examination. A statisti cally significant Pearson Co rrelation Coefficient of r = 0.5321 was found, indicating a moderate ly strong positive correlation (p = 0.0014 < 0.05, n = 33).

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231 These findings are strengthened by the use of the Mathematics Education departmental final examination as the measur e of achievement. This examination is a 50-item instrument that includes questi ons about both mathematics content and pedagogy. All students who are enrolled in a ny section of the methods course take the same final examination. This test was used as a measure of course achievement rather than the final course grade in an effort to minimize bias. The test is a multiple-choice instrument, so grading is not subjective. In addition, the test was not written by the researcher. A Mathematics Education faculty member who has taught the methods course for many years oversaw the writing of the departmental exam, but all methods-course instructors were invited to c ontribute problems to the test. Th erefore the final exam was a collaboration of several professionals with expertise in the field of mathematics education. Information about the content of the final exam is found in Table 4 and in Appendix J. A reliability coefficient of 0.71 (n=17) was found for Exam Form A and a coefficient of 0.73 (n=16) was found for Exam Form B. The use of the final exam as a measure of achievement provided validity and re liability of the results. There has been little consensus in the research litera ture concerning the relationship between attitudes toward math ematics and achievement in mathematics. Some researchers have found the correlation between the two to be quite low, while others have found statistical ly significant corre lations ranging from 0.20 to 0.40. Still others have found quite str ong correlations above 0.40 (Ma an d Kishor, 1997). In their meta-analysis of 113 studies, Ma and Kishor (1 997) also found that the correlation tended to become stronger as students reached high sc hool. Participants in grades 1-4 showed a Pearson Correlation Coefficient of r = 0.03, and students in grades 5-6 showed a

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232 correlation of r = 0.14. However, the relationship strengthened even more as students reached secondary school, with seconda ry students showing a correlation of r = 0.26. The results from this study support the notion that attitudes toward mathematics and achievement in mathematics are relate d. The moderate correlation of 0.53 found in this study also appears to support previous fi ndings that the correla tion between attitudes and achievement strengthens as students get older. Participants in this study were university students, so they were older th an the students in Ma and Kishor’s metaanalysis. The previously estab lished pattern of in creasing correlation with age and grade level is supported in th ese findings. Research Question Four: Themes from Journals Analysis of the reflective journals invo lved looking for patterns using Hycner’s guidelines (Appendix K). After reading a journal entry for a sense of the whole, units of general meaning were delineated. When a journal expressed multiple themes, these themes were analyzed separately. Once units of meaning had been identified for each journal entry for a given prompt, units of m eaning from all journa l entries responding to that prompt were examined. Units of meani ng relevant to the research questions were then clustered and common themes identified from the data. Journal One: Feelings at Beginning of Course Table 10 (page 130) shows five themes th at reflected positiv e feelings at the beginning of the course and two themes that reflected negative feeli ngs about the course. The 27 instances of positive feelings came from 21 of the 33 particip ants and represented

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233 17.5% of all comments made for this prompt. These participants said that they were looking forward to the course, excited about it intrigued by the manipulatives, confident, and interested. The 18 negative feelings cam e from 15 participants and represented 11.7% of all comments made for this prompt. These participants said that they were nervous, worried, apprehensive, and disliked mathematics. Some participants expressed attitudes toward mathematics that were not specific to the methods course. There were 30 inst ances of positive attit udes and experiences cited. These comments came from 15 particip ants and represented 19.5% of all comments made for this prompt. Participants said that they liked mathematics, had done well with mathematics, looked forward to teaching ma thematics, and had positive memories of elementary school mathematics. There were 46 instances, from 24 participants and representing 29.9% of all comments made fo r this prompt, of ne gative attitudes and experiences cited, such as being horrible in mathematics, disliking mathematics, not being interested in mathematics, and being intimidated by mathematics. Some were worried about teaching mathematics. Fourteen of these negative experiences were of mathematics classrooms, mostly in high sc hool and college. These involved struggling with mathematics and not doing well, feeling intimidated, memorizing procedures rather than understanding concepts, and horrible teachers. Journal one also asked participants wh at they were hoping to gain from the course. Table 13 (pages 135-136) summarizes the themes that were identified in addressing this question. There were 11 themes identified with a frequency of 95; these came from all of the 33 particip ants and represented 100% of all comments made for this prompt. There were 24 instan ces where participants said that they wanted to learn

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234 strategies and tools for teaching mathematics. Pa rticipants said that they wanted to learn to like and appreciate mathematics, to help their future students develop positive attitudes toward mathematics, to gain a better unde rstanding of mathematics, and to gain confidence with mathematics. They also wanted to learn how to accommodate different learning styles, to help students view mathem atics as relevant, and to make mathematics interesting for their future students. There were more positive f eelings about the methods course at the beginning than negative feelings. However, while explaining their feelings at the beginning of the course, participants expressed more negative attitudes and memories than positive ones. One might find this a bit conflicting. Howeve r, the participants’ responses concerning what they hoped to gain from the course o ffered an explanation for this. Many of them said that they were looking forward to the course and excited about it because they wanted to learn to like and appreciate mathem atics in order to help their future students develop positive attitudes toward mathematics. Several said that they did not want their students to have the same negative attitudes toward mathematics that they had. Many of these participants expressed ne gative attitudes toward and ex periences with mathematics, but they were motivated to improve these at titudes in order to develop more positive attitudes toward mathematics in their future students. Journal Two: Memories of Math ematics in Elementary School Those journal entries reflecting positive memories of elementary school mathematics are summarized in Table 14 (pages 138-139). There were 14 themes, with a frequency of 53, which reflected positive memories of elementary school. These

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235 comments came from 21 participants and re presented 34.2% of all comments made for this prompt. Eleven participants positively recalled rewards associated with learning specific topics, especially multiplication facts. Seven remembered feeling successful with mathematics and enjoying it. Four had fond memo ries of specific proj ects that related to “real life.” Six participants remembered feeling positive, confident, and motivated with elementary school mathematics and the fun of sharing mathematics with their parents. Eight recalled how much they enjoyed the use of manipulatives, cooperative learning groups, and mathematics games. Those journal entries reflecting positive memories of elementary school mathematics teachers are summarized in Table 15 (page 140). There were 8 themes, with a frequency of 15. These comments came from 7 participants and repr esented 9.7% of all comments made for this prompt. Participants fondly remembered elementary school math teachers who took the time to give them extra help when needed, made mathematics class interesting and relevant, and who were pati ent, supportive, and en couraging. They also appreciated teachers who taught mathematics in many different ways and varied their lessons, and those who encouraged group work and were great at e xplaining things. Those journal entries reflecting negativ e memories of elementary school mathematics are summarized in Table 16 (pages 142-143). There were 16 themes, with a total frequency of 50, that reflected ne gative memories of elementary school mathematics. These comments came from 16 participants and repr esented 32.3% of all comments made for this prompt. Four pa rticipants recalled elementary school mathematics as boring, not relevant or needed, and being mostly drill work. Four recalled games that they dreaded and teachers who w ould not provide extra help when needed.

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236 Three remembered only being taught one way to solve problems, difficult homework, and struggles with mathematics that became ev en worse after moving to a new school. In addition, five participants mentioned elementa ry school teachers who were bitter, lacked compassion for students and excitement about mathematics, and were unwilling to help when students asked questions. These res ponses about elementary school teachers represented 3.2% of all commen ts made for this prompt. It is interesting to note that there were more positive memories of elementary school mathematics and elementary school mathematics teachers, representing 43.9% of the comments made for this prompt, than negative memories, representing 35.5% of the comments. Studies have shown that children ty pically begin school with positive attitudes toward mathematics (McLeod, 1992), and these memories of elementary school seem to support those findings. Although journal two asked about memories of elementary school mathematics, some participants discussed their memories of mathematics at other levels as well. Two participants said that they began to lik e mathematics more once they were out of elementary school. However, five participan ts said that they liked mathematics in elementary school but began to struggle with it and dislike it once th ey reached middle or high school. Four participants remembered specific mathematics teachers after elementary school who were discouraging, ex tremely rigid, horrible, and teachers who created tense classrooms and made students feel stupid when they asked for help. Journal two also asked participants what they, as future teachers, had learned from the experiences they had cited. Table 18 (p ages 147-149) summarizes their responses. There were 16 themes identified with a fre quency of 72; these came from all of the 33

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237 participants and represented 100% of all comments made for this prompt. Many of the participants said that they had learned that teachers should provide extra help to struggling students. Teachers should make mathematics fun, interesting, and relevant. Mathematics teachers should foster positive attitudes toward mathematics in their students, build students’ confidence with math ematics, help students feel successful, and create a comfortable classroom environment fo r their students. In a ddition, participants learned that mathematics teachers should unde rstand the mathematics content they are teaching, accommodate different learning styles, let students know that there is more than one way to solve a problem, not use too ma ny worksheets, integrate mathematics with other subjects, and use manipulatives and gr oup work with their students. They should also ask a colleague for help when needed. Journal Three: Feelings About Mathematics Those journal entries reflecting positi ve feelings about mathematics are summarized in Table 19 (page 151). There we re six themes, with a frequency of 31, which reflected positive feelings about ma thematics. These comments came from 19 participants and represented 14.8% of all comments made for this prompt. Participants said that they enjoyed mathematics and found it useful and relevant to their lives. They also felt that they had learned the best techni ques and strategies for them to approach or deal with mathematics, and they liked the constancy of mathematics where the correct steps always lead to the correct answer. It is informative to note that this algorithmic view of mathematics runs counter to the perspec tive that the NCTM is trying to encourage. One participant expressed confidence about mathematics. Five participants expressed

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238 positive feelings related to the methods course The most common of these was that they liked the constructivist teaching methods that they were learning about in the methods course. Those journal entries reflecting the experien ces that participants associated with positive feelings about mathematics are summarized in Table 20 (page 152). There were seven themes, with a frequency of 28, whic h reflected experiences associated with positive feelings. These comments came from 18 participants and represented 13.4% of all comments made for this prompt. Eight of these experiences referred to mathematics teachers who helped their students embrace mathematics, understand mathematics, see the relevance of mathematics, not fear mathematics, and teachers who were patient with their students, made learning fun, and booste d the confidence of their students. Participants also recalled events where th ey experienced success with mathematics and felt that it came easily to them. They reme mbered specific experiences in elementary school and also in college when they were able to understand mathematics and feel successful with it. Three participants recalled specific incidents where success and recognition led to feelings of confid ence about mathematics. Journal entries reflecting negative feelings about mathematics are summarized in Table 22 (pages 155-156). There were 10 themes, with a frequency of 47, reflecting negative feelings. These comments came from 14 participants and represented 22.5% of all comments made for this prompt. Many of th ese participants said that they disliked, hated, and feared mathematics. They said that mathematics made them feel intimidated, nervous, and frustrated. They did not enjoy, were not good at, and lacked confidence with mathematics. Three said that they did not see the need for studying advanced

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239 mathematics. They viewed mathematics as unneeded, and they try to avoid mathematics whenever possible. Three participants said that they did not like the nature of mathematics, where new concepts are built upon previous ones. Table 23 (pages 157-158) shows the experiences that participants associated with negative feelings about math ematics. There were 11 themes, with a frequency of 48, recalling experiences that were associated with negative feelings. These comments came from 22 participants and represented 23% of all comments made for this prompt. The vast majority of these were experiences wh ere participants cited their struggles with mathematics as an explanation for thei r negative feelings. Many remembered bad teachers, feeling stupid when they could not understand, and doubting themselves when they did not perform well on tests. They said that struggles with mathematics made them feel frustrated and traumatized, and they did not like the quick pace of mathematics classes. Others mentioned that they dislik ed ‘traditional’ teaching methods that their previous mathematics teachers had used. While describing their feeli ngs about mathematics, seve n participants said that they wanted to improve their own negativ e attitudes toward mathematics. They recognized that, as future mathematics teache rs, they should have more positive attitudes toward the subject they will soon be teaching. Participants also wrote about their desires to develop positive attitudes toward mathematics in their future students by making mathematics fun and enjoyable for the students. It is interesting to note th at there were more negative feelings about mathematics, representing 45.5% of the comments made for this prompt, than positive feelings, representing 28.2% of the comments. St udies have shown that many preservice

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240 elementary school teachers at the university have negative attitudes toward mathematics (Christian, 1999; Cornell, 1999; Hungerford, 1994; Philippou and Christou 1998; Rech et al., 1993). Participan ts’ responses to Journal Three se em to support those findings. Journal Four: Memorable Experience with Mathematics Experiences were categorized as either positive or negative. Table 25 (page 162) lists the frequencies of these positive and negative memories. Positive memories came from 21 of the 33 participants and represen ted 56.8% of all comments made for this prompt. Negative memories came from 16 of the 33 participants and represented 43.2% of all comments made for this prompt. Eight participants recalled one special teacher or tutor whose individual help and encouragement positively affected their attit udes toward mathematics. In addition, four participants recalled experiences that change d their attitudes toward mathematics in a positive way, and three of these also involved a special teacher or tutor. These excerpts recalled teachers who were sens itive to the student’s struggl es, often initiated help, and who expressed faith and confidence in the student’s ability. The successes that followed and the teacher’s or tutor’s stated confidence in the student led to the student’s increased self-confidence. Four participants recalled positive situati ons where they experienced success with mathematics after struggling with it. They described feelings of exhilaration and confidence. They talked about how good it felt to understand the concepts and feel pride in this accomplishment.

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241 Four participants remembered positive me morable experiences that involved the use of tricks, drill exercises, and games for learning basic f acts. They all mentioned the positive recognition, in the form of certificates, rewards, and verbal encouragement that the participant received for being successful with the drills and games. This success led to feelings of self-confiden ce. All four of these participants found these activities to be fun and enjoyable. However, one participant had a negative experience with this type of activity. She hated the memorization and timed tests involved with learning basic facts. They produced anxiety and feelings of failure for her. Six participants recalled negative experi ences involving individual teachers. In addition, three participants recalled experien ces involving individual teachers that they said negatively changed their attitudes towa rd mathematics. These excerpts recalled teachers who singled out students who were struggling. The teachers embarrassed or punished the students for not knowing the an swer. Students felt dumb or stupid when they could not understand the concept, and the teacher appeared unwilling to take the time to help. This left the students feeling fr ustrated and unimportant. Participants also remembered teachers who admitted disliking mathematics, who stressed memorization rather than understanding, and who did not have control of the class. Four participants recalled negative experiences where th ey felt stupid or not very smart. These all stemmed from incidents wh ere the participant t hought that he or she should have known the answer or answers but didn’t. They involved tests, homework, and class work. These struggles seem ed to diminish self-confidence. Five participants had memories related to changing schools or classes. Two of these were positive. The first involved moving to a new school. The participant was

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242 scared about the possibility that the curricu lum she had been studying in her previous school would be behind that of her new classm ates, but her confidence soared when she realized that she was actually ahead of them and able to help some of them with mathematics. The other positive memory involved being tested for the gifted class, failing the test, and then being retested the next ye ar and passing the test. This was a motivating force for this participant. The three negativ e memories of changing schools or classes involved moving to a new school or class that was more adva nced or further along in the curriculum than the former school or cla ss. This brought on feelings of failure, frustration, and intimidation. Studies have shown that children typica lly begin school with positive attitudes toward mathematics, but these attitudes tend to become less positive as they get older. By the time students reach high school, their attitu des toward mathematics have frequently become negative (McLeod, 1992). When par ticipants responding to Journal Three mentioned experiences that were associated with either positive or negative feelings about mathematics, some referred to specific le vels of schooling. It is interesting to note that four of the eight positive experiences mentioned occurred in elementary school, none in middle school, and one in high school. Ho wever, only two of the fifteen negative experiences cited took place in elementary sc hool. Three occurred in middle school, and seven in high school. These results seem to support McLeod’s findings that attitudes toward mathematics tend to become more negative by the time students reach high school. Results from Journal Four further reinforced this notio n. When describing a memorable event that had influenced their attitudes toward mathematics, the highest

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243 frequency of positive memories occurred in elementary school while the highest frequency of negative memories t ook place in high school. Journal Eight: The Use of Reflectiv e Journals in the Methods Course Those journal entries reflecting benefits of journals are summarized in Table 26 (pages 177-181). There were 27 themes, w ith a frequency of 181, which reflected benefits of journals. These comments came from all 33 participan ts and represented 89.2% of all comments made for this prompt. Tw enty-six of the participants said that they had benefited from reflecting on their pa st experiences with mathematics. Twentysix also said that they had benefited fr om reflecting on their future teaching of mathematics. Twenty of the thirty-three partic ipants said that they had gained insights about their attitudes toward a nd struggles with mathematics. As they remembered their former experiences with mathematics, eightee n of the participants said that they had reflected on the differences between good and bad teaching, and they made specific plans as future teachers based on these reflectio ns. They wanted to be like their favorite teachers and to avoid the qualities and practices of their least-favorite teachers. Nine of the participants said that they had made sp ecific plans as future teachers regarding being sensitive to and trying to impr ove their students’ attitudes toward mathematics. Six said that they had realized thr ough journal writing that teacher s have a large impact on their students’ attitudes toward mathematics. Seventeen of the participants said that they liked, loved, and enjoyed the journals, and that they thought the journals were a good idea. They found them helpful and beneficial. After reflecting on their own mathematics classroom experiences as students,

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244 participants planned to use the teaching met hods and strategies that they found helpful and to avoid those that they did not find help ful. They were determined to make their mathematics classrooms fun and to use jour nal writing with their own students. Four of the participants said that they had negative fee lings about the journals at first, but all four said that after writing one or two, they changed their minds. Four participants said that the researcher’s responses to the journals were a positive aspect of journal writing. Not only did th ey appreciate the researcher’s time and care in responding to their journals, but they also felt that the researcher’s responses added to their insights. Three participants said that the journals ma de them think about things they might not have without journaling. Three felt that the journals were a good means of communication with the in structor because journals offer an opportunity for a student to be more open and bring up questions and issues that might not have been expressed faceto-face. Participants recognized the benefits of reflection and the re alization that their concerns were not so different from those of others in the class. There were five themes from eight pa rticipants, representing 3.9% of the comments made for this prompt, which reflecte d drawbacks of journals. Two participants found it unpleasant to relive bad experiences w ith mathematics. Two said that realizing through reflection that they had not expe rienced many good mathematics teachers to serve as role models made them feel uneas y about teaching mathematics in the future. Two participants mentioned that the jour nal prompts seemed a bit repetitious. One participant said that she had a hard time re membering things that happened when she was in school. The only other drawback mentioned was a technical proble m that interfered with the participant’s ability to send and receive emails.

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245 Despite these few drawbacks, participants were overwhelmingly favorable in their views about writing journals. The journals helped them determine the type of mathematics teachers they wanted to become by allowing them to reflect on their own mathematics teachers and classroom experiences They realized that teachers can have a huge impact on their students’ attitudes toward mathematics, and these future teachers want to encourage the development of positive attitudes in their own students. If their future students have negative attitudes toward mathematics, these participants want to be sensitive to this and to try to improve thes e negative attitudes. Perhaps this type of reflection will help them become better te achers by increasing their awareness about these issues. Rose (1989) cited teacher response as an important benefit of journal writing. “As the teacher writes back to the students, stude nts realize the teacher hears and cares” (p. 26). Responses from participants in th is study reinforced this notion. Several participants mentioned the rese archer’s responses as they reflected on journal writing. They appreciated the research er’s time and care in respond ing and the insi ghts that the responses provided. It is interesting to not e that in recalling memories of former mathematics teachers, participants often re lated positive experiences and attitudes to teachers who took time to help them and who seemed to care about them. Negative experiences and attitudes were often associated with teach ers who were not willing to give struggling students their time and who ga ve their students the impression that they did not care. Many participants said that they had gain ed a great deal of insight about their own

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246 attitudes toward mathematics and the experien ces that had influenced the development of those attitudes. Perhaps the participants’ own wo rds offer the best view of this result: “I thought the use of reflec tive journals was beneficial in this class because I didn't realize how I felt about math until I wrote them.” “I think that the benefit to me of keeping a reflective jour nal in this class is that it has caused me to consider my attitude toward the subject of mathematics and possible causes for the development of this attitude.” “From these journals, I have learned a grea t deal about why I have the feelings I do for math. They allowed me to think back through my life and pinpoint events or people that influenced my feelings and attitudes. I was able to recall events or people that I think shaped why I don't like ma th today. I was also able to recall the events and people that were positive and didn't make me totally turned off from math.” “It made me realize that I don't truly di slike the subject, but it was the teachers who were not helpful and patient that made me feel that way.” “By reflecting I have really come to term s with how comfortable I am with math and how I want my students to feel about math.” Journals Five, Six, Seven: Relevant Excerpts When asked how they, as future educators, would help boost the confidence of students with low self-confidence regarding ma thematics, eleven pa rticipants responded with comments that related to their own e xperiences and feelings of low self-confidence with mathematics. They said that they knew wh at it was like to have low self-confidence

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247 and feel intimidated by mathematics, and they believed that a lot of children and adults lack self-confidence with mathematics. They said that teachers do not realize how great an impact they can have on students’ self-c onfidence with mathematics. Two participants said that they had been encouraged to belie ve that mathematics was just for “boys or smart kids,” and they would try to show their st udents that this was not the case. When asked to reflect on the qualities of the best mathematics teacher they ever had, 21 participants responded with comments that were pertinen t to the research questions of this study. They remembered teachers who had positive attitudes toward mathematics themselves and who modeled this for their students. These positive teachers made learning mathematics fun and exciting. They helped their students believe in themselves and their abilities to do mathematics. They would patiently answer questions without making their students feel stupid. One participant said that she had never had a very good mathematics teacher. When asked about the qualities of the wo rst mathematics teacher they ever had, 17 participants responded with comments that were relevant to the study. They recalled teachers who did not seem to care about their students and their learning. They were not willing to take time to help students who were struggling. They were impatient and difficult to approach, and students were afraid to ask for help. When students did ask a question, these teachers would embarrass them in front of the whole class and make them feel dumb. Some of these teachers would call on students for answers without giving them time to process new material. When th e students could not answer the question, the teachers would embarrass them and make them feel stupid. These teachers lacked enthusiasm for mathematics and would alwa ys use the same boring teaching methods.

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248 Nearly all of these 17 participants said that their worst teachers had a lasting negative effect on them and their attitude s toward mathematics. They sa id that as a result of having this teacher, they disliked or even hated ma thematics and tried to avoid it or take easier courses whenever possible. Participants said that their self-confidence suffered as a result of these teachers and some felt worthless and incompetent. Beliefs Expressed in Journals Although this study focused on attitudes toward mathematics, it is also important to look at the relevant beliefs that participants expresse d in their journals. Studies have shown that teachers who have negative attitudes toward mathematics are more likely to view and teach mathematics in a more traditional manner (Philippou and Christou, 1998; Stipek, Givvin, Salmon, and MacG yvers, 2001). The participants in this study had initial attitudes that were somewh at negative, especially enjoyment, selfconfidence, and motivation. However, the beliefs that they expressed were not particularly indicative of traditional beliefs. In fact, many of them seemed to reflect ideas that were being promoted in the methods cour se. It is important to remember that the initial attitudes were measured at the beginni ng of the semester, but the journals were written throughout the semester. This may indicat e that as participan ts’ attitudes toward mathematics were changing, perhaps their beliefs were as well. Themes Across Journals In reviewing themes from the journals, it is interesting to compare responses of participants with positive memories to those with negative memories and to observe their

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249 contrasting relationship. Those with positive memories talked about feeling successful with mathematics and enjoying it, while those with negative memories recalled struggling with mathematics and finding it boring. Positive memories referred to real-life projects, manipulatives, cooperative learning, and game s, whereas negative memories referred to mathematics content that was not relevant to real life and class work that was mostly drill. Although some said that they liked mathematics and found it useful, many talked about how they disliked and feared mathema tics. They found that it made them feel nervous, intimidated, and frustrated, and they thought it was unneeded, especially advanced mathematics. Although some wrote ab out feelings of confidence, others said that they lacked confidence with mathematics. Those with positive memories recalled ma thematics teachers who were always available for extra help when needed and w ho were patient, supportive, and encouraging. They were very sensitive to their students’ st ruggles and expressed fa ith in their students’ abilities to succeed. These teachers helped their students understand and appreciate mathematics and boosted their self-confidence. Those with negative memories referred to teachers who were not willing to take time to help them when they were struggling and who were bitter and lacked compassion. Th ese teachers would single out struggling students to answer questions in class wit hout giving them time to process the new information. This left their students f eeling embarrassed, frustrated, and stupid. Research Question Five: Experiences of Those with Most Extreme Attitudes The two participants with the most negativ e initial attitude scores, Mary and Lisa, and the two with the most posi tive initial attitude scores, He rmione and Torri, were asked

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250 about their attitudes toward and experiences with mathematics in individual interviews. As might be expected, the two participan ts with extremely positive attitudes had memories and experiences that were much more pleasant than those of the two with extremely negative attitudes. Like the memories and experiences that all the participants shared in their journals, there was a contra sting relationship betw een the two types of experiences. On the one hand, the two participan ts with the most positive initial attitude scores remembered mathematics classes through out their schooling as easy and fun. They remembered feeling successful with and conf ident about mathematics. On the other hand, the two participants with the most negativ e initial attitudes remembered mathematics classes throughout their schooli ng as tough and stressful. Th ey remembered struggling with mathematics, feeling unsuccessful, and la cking self-confidence about mathematics. The two participants with the highest at titude scores took upper-level mathematics courses every year of high school and in colleg e. The two with the lo west attitude scores took only the three required courses in high school and had to reta ke required college courses several times before passing. The high-scoring participants had a good understanding of the concepts, and even when they were confused, they were eventually able to make sense of the mathematics on their own. The low-scoring participants had trouble understanding concepts and had to rely on memorization when the teacher would not explain. The two with the most positive attitude scores were both excited at the beginning of the methods course, while the two with the lowest scor es were nervous and worried about the methods course. They both felt be tter about the course once it had started

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251 because they were learning new strategies that were helping their conceptual understanding. From these interviews, it appears that conceptual understanding was a key difference between the two participants with extremely high attitude scores and the two with extremely low attitude scores. On th e one hand, Hermione and Torri both talked about their abilities to figure out and make sense of mathematics, even when they may have initially been confused. On the other hand, Mary and Lisa referred repeatedly to their struggles with understand ing mathematics concepts. It is interesting to note that those participants with the highest attitude scores did not always make the highest grades. However, even when their grades were somewh at low, they still fe lt confident in their understanding of the concepts. In contrast when Lisa received good grades in high school, she felt that she did not deserve them because she did not have conceptual understanding of the content from those c ourses. Perhaps developing self-confidence with mathematics begins with developing c onceptual understanding of the mathematics. One would expect self-confident students to have experienced prior success with mathematics, and one would usually look at grades as a measure of success. However, it appears that these interviewees relate d their self-confiden ce and success with mathematics to conceptual understand ing rather than to grades. The importance of conceptual understa nding in developing positive attitudes toward mathematics is especially important for preservice teachers because it seems to affect their teaching self-efficacy. Both Mary and Lisa concluded their interviews by saying that they were concerned about teaching mathematics. They said that they feared

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252 they would not be able to teach children mathematical concepts that they had trouble understanding themselves. One of the common experiences that the two participants with extremely negative attitudes shared was especially notewort hy. They both mentioned having one special teacher in high school who helped them. Bo th of these teachers made themselves available to their struggling st udents as much as possible. Students could come for help any time including lunch periods as well as befo re and after school; on e of them was even available for tutoring outside of school. Thes e special teachers helped their students by teaching them strategies and helping them catch up with basics they had missed. Mary said that she felt “comfortable and confident” because of the strategi es this teacher had taught her. This reinforces the notion that students relate positively to teachers who demonstrate that they care a bout their students by taking time to help them. Torri’s experiences in elementary schools here in the U.S. and overseas were also interesting. When she attended a U.S. m ilitary elementary school in Germany, she remembered mathematics being fun. She r ecalled manipulatives, hands-on activities, using different techniques to solve problems, and group work. When she returned to the U.S. in fifth grade, she remembered that the focus was on algorithms, worksheets, and textbooks. Torri also benefited from a summer program in middle school where they used problem-solving activities and focused on st udent strategies. Perhaps these early exposures to constructivist-teaching methods might have influenced Torri’s ability to make sense of mathematical concepts and he r positive attitudes toward mathematics. Studies have suggested that a positive attitude toward mathematics may increase one’s tendency to elect mathematics cour ses in high school a nd college (Haladyna,

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253 Shaughnessy, & Shaughnessy, 1983). Findings from Question Five seem to support this. The two participants with the highest att itude scores took upper-level mathematics courses every year of high school and in colleg e. The two with the lo west attitude scores took only the required mathematics in high sc hool. They both had to retake required college courses several times before passing. Implications and Recommendations for Practice The reform movement in mathematics educ ation has recognized the importance of affective issues and the connection between these issues and highe r-order thinking. The National Council of Teachers of Mathematics has established goals involving students’ dispositions toward mathematics that incl ude value, self-confidence, interest, and perseverance. Results from this study lead to some implications and recommendations for teacher educators who seek to improve the a ttitudes toward mathematics of preservice elementary school teachers. By studying preservi ce elementary teachers’ attitudes toward mathematics and the experiences that have pl ayed a crucial role in the development of these attitudes, teacher educators can use th is information to develop training programs aimed at improving these attit udes. These results also prov ide some implications and recommendations for mathematics teachers at all levels. Mathematics Teacher Educators The literature tells us that many pres ervice elementary school teachers have negative attitudes toward mathematics (R ech et al., 1993; Cornell, 1999; Philippou and Christou, 1998). The results from this study support these previous findings. Teacher

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254 educators need to do more to improve the attitudes toward mathematics of future elementary school teachers, especially thei r self-confidence, enj oyment, and motivation. It is important for teacher educators to r ealize that many of th eir students who have negative attitudes toward mathematics are motivat ed to improve their attitudes in order to be better teachers. The preservice teachers in this study wanted to gain confidence with mathematics and develop positive attitudes towa rd mathematics in their future students. Teacher educators should also know that these preservice teachers wanted to learn how to accommodate different learning styles, how to help their future students view mathematics as relevant, and how to make mathematics interesting for their future students. Teacher educators must do all they can to provide preservice teachers with the opportunity to accomplish all of th ese goals. The contrasting relationship between key elements of positive memories and those of negative memories offered a clear view of the types of experiences that encouraged the development of positive and ne gative attitudes in these participants. In their memories of elementary school, partic ipants associated the use of manipulatives, cooperative learning, games, and real-life proj ects with positive memories. They recalled classrooms where mathematics was interesti ng and enjoyable, a nd the teacher taught concepts in different ways. They remembered feeling successful with mathematics. It is important for teacher educators to stress th e use of these teaching methods with their preservice teachers. However, this is not enough. Participants also associated positive memories with teachers who were patient, supportive, and encour aging. These teachers were always available to he lp struggling students. By ma king themselves available for help, these teachers let their students know that they cared. Preservice teachers should be

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255 encouraged to be patient and supportive w ith their students, providing them with a positive classroom environment where they can feel confident about mathematics. Perhaps a discussion concerning the developmen t of attitudes toward mathematics should become a part of mathematics methods courses. Nearly all of the participants who were interviewed said that they had always struggled with problem solving, and nearly a ll of them said that their problem-solving skills had improved as a result of the methods course. Problem solving is an essential component of the reform movement in ma thematics education, and teacher educators should model teaching with problems in thei r methods courses whenever possible. By encouraging preservice teachers to invent and share their own solution strategies, perhaps future teachers will not only improve their ow n problem-solving skills, but also feel more confident about problem solving and be more likely to use problem solving with their own students. All of the participants were very pos itive about journal writing. They enjoyed writing them, and they found it beneficial to re flect on both their pa st experiences with mathematics and on their future teaching of mathematics. Many articulated that they had gained a great deal of insight about thei r own attitudes toward mathematics and the experiences that had influenced the developm ent of those attitudes. They thought that journals provided an excellent means of communication with the course instructor. Although reading and responding to student journals is qu ite time consuming, it is a worthwhile practice. If teacher educators do not have enough time to read and respond to eight journals as the resear cher did, perhaps assigning fewer would still be useful. Participants thought that the re searcher’s responses to their journals we re another positive

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256 aspect of journal writing. This indicates th at teacher educators s hould not assign journal writing unless they will read a nd respond to each journal. Mathematics Teachers at All Levels These findings provide implications not onl y for teacher educators, but also for mathematics teachers at all levels. Students asso ciated positive memories with feelings of success with mathematics. They recalled th e confidence and pride that accompanied success. It is important for teachers to help their students experience feelings of confidence and success rather th an feeling frustrated, intimid ated, and stupid as many of the participants described. Many participants associated positive fee lings about mathematics with positive teachers who were patient and supportive. Unfortunately many participants remembered mathematics teachers who were much different. These teachers were not willing to provide extra help to students who were struggling. Some teachers even singled out struggling students, causing them to feel embarrassed and humiliated. Some teachers would ridicule, embarrass, and even punish st udents when they asked a question or could not respond correctly to the teacher’s questi on. All mathematics teachers should provide a positive classroom environment where stud ents feel supported and can ask questions without fear of embarrassment. It is also important for mathematics teach ers to note that participants remembered benefiting from the use of manipulatives, coope rative learning, real-l ife projects, games, and teachers who taught concepts in more than one way. Many remembered mathematics classes that used these methods as fun and enjoyable. However, many others had negative

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257 memories of mathematics classes that were bor ing, not relevant to re al life, and mostly drill work. Mathematics teachers at all levels must work harder to create classrooms where students are actively engaged and interested. Conceptual understanding was a key differe nce between the two participants with extremely high attitude scores and the two with extremely low attitude scores. Those with the highest scores talked about their ab ilities to figure out and make sense of mathematics, while the two participants with th e lowest scores referred repeatedly to their struggles with understanding mathematics concepts. This suggests that perhaps developing self-confidence w ith mathematics begins with developing conceptual understanding of the mathematics. Teacher s must provide students with learning experiences in which they experience the ex citement that comes from making sense of mathematics instead of memorizing formulas and rules. This focus on making sense of mathematics is an essential component of the reform movement in mathematics education. Teachers also need to develop students’ motivation toward mathematics. It is important for teachers to help students unders tand the role mathematics plays in fields that they might find interesting or challe nging, such as engine ering, science, and technology, so that they will be more motivat ed to take more upper-level mathematics classes. Helping students realize these type s of connections between the mathematics they are learning in school and its applica tions in the outside wo rld is also strongly encouraged by the reform movement. Although most of the interviewees were ve ry positive about the use of cooperative learning in teaching mathematics and in the me thods course, three of the seven had some

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258 reservations. Two said that they did not like group assignments because someone in the group usually ended up doing more of the work while others did not do their share. Another interviewee said that cooperative learning should not be “a constant thing,” and that students sometimes like to figure things out for themselves. These perspectives should inform all teachers that cooperative learning may not always be right for every student, and that some students may benef it from working alone. When using a group assignment, teachers should devise a means fo r students to assess their own role and level of participation in the group pr oject so that no one will feel that they have done all the work. Results from this study are encouraging. However, improving preservice teachers’ attitudes toward mathematics is not enough. Ma thematics educators should be focused on developing positive attitudes in our youngest stud ents and then doing all we can to help them maintain these positive attitudes as they get older. We mathematics educators must ask ourselves why so many of our students ha ve negative attitudes toward mathematics, and what we can do to avoid this with future students. In journals and interviews from this study, participants repeatedly recalled one special mathematics teacher or tutor who stood out from all of their other teachers. Thes e special teachers were able to help their students feel positive about mathematics and about themselves with mathematics. I would encourage all mathematics educators to strive to be that special teacher to all of their students, as this research i ndicates that one special teacher has the capacity to influence his or her students for many years to come. Throughout their journals and interviews, participants in this study have consistent ly connected positive memories and feelings about mathematics with special teachers who treated them with care and respect and who

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259 helped them understand and feel confident about mathematics. I urge all mathematics educators to adopt the followi ng practices that were associat ed with special teachers who made a positive difference in the lives of their students: Show students that you care about them a nd their learning by being willing to take time to provide extra help for struggling students. Be patient, approachable, and create a comfortable classroom environment where students feel comfortable asking questions. Do not ever embarrass them or make them feel stupid. Encourage students to embrace mathematics rather than fear it. Model a positive attitude toward mathematics for your students. Help students feel successful and confident with mathematics. Help students understand mathematical c oncepts and allow en ough class time for them to process new concepts. Make mathematics fun, inte resting, and relevant. Use teaching methods that accommodate different learning styles, including manipulatives and group work. Let students know that there is more than one way to solve problems and encourage them to share their own solution strategies. These results raise a number of questions about which mathematics educators at all levels should be concerned: 1. Why does the memory of one special t eacher who offered individual help and encouragement, was sensitive to students’ struggles, and expressed confidence in students’ abilities seem to be the exception rather than the rule?

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260 2. Results from this study indicated that participants valued mathematics, but, as a whole, they did not enjoy mathematics or feel self-confident or motivated about mathematics. Do students really value math ematics, or are they merely mimicking what they have been told by teachers re garding the importance of mathematics? 3. Do mathematics teachers themselves genuinely value and enjoy mathematics? How can we encourage students to experi ence the excitement and appreciate the relevance of mathematics if we, as teacher s, do not feel this way ourselves? 4. Journal and interview comments indicated that some students had very positive memories about competitive games and drills, although others remembered such activities as stressful and anxiety pro ducing. How can teachers find the balance needed to help all of their student s develop positive attitudes toward mathematics? 5. Several participants recalled mathematic s classes as “boring” and “dull.” How do we, as teachers, share our enthusiasm a bout and love of mathematics with our students? 6. The vast majority of preservice elementa ry school teachers are females. Only one of the participants in this study was a ma le. How do females’ at titudes toward and experiences with mathematics co mpare with those of males? 7. Previous studies have shown that many preservice elementary school teachers have negative attitudes toward math ematics (Cornell, 1999; Hungerford, 1994; Philippou and Christou, 1998). This study suppo rts these findings. Rech, Hartzell, and Stephens (1993) found that preser vice elementary teachers have less favorable attitudes toward mathematic s than the general university population.

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261 Why are so many students with negative attitudes toward mathematics choosing the field of elementary education? Professionals in the field of mathematics e ducation must find answer s to these questions if we want our students to value and enj oy mathematics and feel motivated and selfconfident about mathematics. Implications and Recommendations for Future Research. Several studies involving teacher-training programs that utilized constructivist instructional methods have shown positive resu lts in improving the attitudes and teacher self-efficacy of preservice elementary t eachers (Anderson and Piazza, 1996; Gibson and Van Strat, 2001; Huinker and Madison, 1997; McGinnis et al., 1998; Philippou and Christou, 1998; Quinn, 1997; Sherman and Chri stian, 1999). All of these studies used constructivist methods incl uding collaborative group work, problem solving, and manipulatives. Results from this study seem to both support and contradict these prior findings. All three classes in this study util ized constructivist-teaching methods, but only those participants in the jour nal-writing class showed a si gnificant positive change in attitudes toward mathematics. Because this study was not experimental in nature or design, it is not possible to determine if journal writing was the reason w hy the attitudes toward mathematics for one class improved dramatically while the ot hers did not. Perhaps the opportunity for preservice teachers to reflect on their attitudes toward and experiences with mathematics was a factor in their positive attitude change. A future st udy that was experimental in nature would help shed light on this issue. If random assignment we re not possible due to

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262 university scheduling, perhaps a random c hoice of which existing classes would be designated as the experimental group and cont rol group would be bene ficial. In order to minimize the effects of irrelevant variables, course activities and procedures could be more standardized than they were in this study. Another possibility would be to have the same instructor teach two classes, one with journal writing and one without journals. This would eliminate the instructor variable. Further studies could also investigate other possible factors that could have influenced these results. University stude nt evaluation forms offered one possible explanation. At the end of each semester, un iversity students are asked to complete an anonymous evaluation of each course and instruct or, and the instructor is able to view them after the course is completed. Twelve participants wrote comments on their evaluation of the methods course, and nine of the comments were relevant to this study. It was interesting that five of the nine anonymously-made comments referred to the organization of the course: I really liked the course because it was so well organized and the new ideas were presented so well. [Instructor] really is clea r about her wants and expect ations from students. She represents materials clearly and in structs clearly, very organized. [Instructor] was very clear on instructions and assignments. [Instructor] makes everything easy to understand. She is well organized. I like how we knew about everything we had to do—helped keep things organized.

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263 These comments were especially noteworthy because Jennifer, the participant with the second highest positive attitude change score, immediat ely referred to the organized format of the methods course when asked which aspects of the course she thought had influenced her attitude change. Perhaps students with negative attitudes toward mathematics feel more comfortable with mathem atics and more in control when they feel organized. Further research is needed in this area. Two of the student evaluation comments suggested another possible factor that could have influenced the large positive attitude change that the participants demonstrated: [Instructor] really provides a comfortable and secure environment and really cares about the students! She was a great instructor—very con cerned and considerate of students. Student journals from this study and from previous classes taught by the researcher clearly spelled out those teach er behaviors that had a ne gative effect on students’ attitudes toward mathematics and those that had a positive effect. Creating a comfortable classroom environment and showing care, co ncern, and consideration for students were all included in the positive category. Perhap s reading students’ re flective journals over the past four years has influenced the resear cher’s teaching style and ways of interacting with students and has increased the researcher’s awareness of and sensitivity to students’ attitudes toward mathematics. Reflections fr om the researcher’s journal (p. 203) offer some insight into this. Furthe r studies are needed to invest igate the effects of teacher behavior on students’ attitude s toward mathematics.

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264 Results from this study supported the view that attitudes toward mathematics and achievement in mathematics are related. Th e strong correlation of 0.53 found in this study also supported previous findings th at the correlation between attitudes and achievement strengthens as students get older. More research is needed in order to investigate further this pattern of in creasing correlation between attitudes and achievement as students get older. Studies have shown that teachers who have negative attitudes toward mathematics are more likely to view and teach mathematics in a more traditional manner (Philippou and Christou, 1998; Stipek, Givvin, Salmon, and MacGyvers, 2001). The participants in this study had initial attitudes that were some what negative, but the beliefs that they expressed were not particularly indicative of traditional beliefs. In fact, many of them seemed to reflect constructivist ideas that we re being promoted in the methods course. It is important to remember that the initial at titudes were measured at the beginning of the semester, but the journals were written throughout the semester. This may indicate that as participants’ attitudes toward mathematics were changing, perhaps their beliefs were as well. Further research is needed to see if this is the case. Because the ATMI measures the attitudes of value of mathematics, en joyment of mathematics, motivation for mathematics, and self-confidence with math ematics and not beliefs about teaching and learning mathematics, a similar study is needed where a pre-course and post-course belief survey is used in addition to a preand post-course attitude survey. Studies have shown that children typica lly begin school with positive attitudes toward mathematics, but these attitudes tend to become less positive as they get older. By the time students reach high school, their attitu des toward mathematics have frequently

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265 become negative (McLeod, 1992). Results fr om this study seem to support McLeod’s findings. Further research is needed to explore why students’ attitudes toward mathematics tend to be positive in elementary school and become more negative as they progress through secondary school. Although these results of improved attitude s toward mathematics are encouraging, future studies that follow these preservice teachers past their t eacher training programs and into their first few years of teaching w ould be useful. Explori ng how their reflection about their attitudes influences their teaching st rategies and investiga ting if their attitude changes remain stable over time would both be beneficial to the field of mathematics education.

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267 Benken, B.M., & Wilson, M. (1998, October). The impact of a secondary preservice teacher’s beliefs about mathema tics on her teaching practice. Paper presented at the annual meeting of the North American Chapter of the Inte rnational Group for the Psychology of Mathematics Education, Raleigh, NC. Borasi, R., & Rose, B. (1989). Journal writing and mathem atics instruction Educational Studies in Mathematics, 20 (4), 347-365. Braswell, J.S., Lutkus, A.D., Grigg, W.S., Sa ntapau, S.L., Tay-Lim, B., & Johnson, M. (2001, August). The nation’s report ca rd: Mathematics 2000 (Report No. NCES 2001-517). Washington, DC: Na tional Center for Education Statistics. (ERIC Document Reproduction Service No. ED453086) Britton, G. L. (1992). Journals and essay examin ations in undergraduate mathematics. In A. Sterrett (Ed.), Using writing to teach mathematics (pp. 104-110). Washington, DC: Mathematical Association of America. Burk, J. & Littleton, P. (1995). Reflective journals: Enhancing mathematics staff development. Mathematics Teaching in the Middle School, 1 (7), 576-583. Canning, C. (1991). What teac hers say about reflection. Educational Leadership, 48 (6), 18-21. Carter, C.W. (1997, March). The use of journals to promote reflection. Paper presented at the annual meeting of the American Educa tional Research Association, Chicago, IL. Carter, G. (1997). The relationship betw een teacher and stud ent beliefs about mathematics. School Science and Mathematics, 97 (2), 62-67.

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280 Wagner, S., Lee, H.J., & Ozgun-Koca, S.A. (1999, January). A comparative study of the United States, Turkey, and Korea: Attitude s and beliefs of preservice mathematics teachers toward mathematics, teachi ng mathematics, and their teacher preparation programs. Paper presented at the annual meeting of the Association of Mathematics Teacher Educators, Chicago, IL. Waywood, A. (1992). Journal writi ng and learning mathematics. For the Learning of Mathematics, 12 (2), 34-43. Waywood, A. (1994). Informal writ ing-to-learn as a dimensi on of a student profile. Educational Studies in Mathematics, 27 (4), 321-340. Weinstein, C.S. (1990). Prospective elemen tary teachers’ beliefs about teaching: Implications for teacher education. Teaching and Teacher Education, 6 279-290. Wenner, G. (2001). Science and mathematics efficacy beliefs held by practicing and prospective teachers: A 5-year perspective. Journal of Science Education and Technology, 10 (2), 181-187. Wideen, M., Mayer-Smith, J., & Moon, B. (1998). A critical analysis of the research on learning to teach: Making the case for an ecological perspective on inquiry. Review of Educational Research, 68 (2), 130-178. Wilson, L.M. (1995). Students’ beliefs about doing mathematics. Paper presented at the annual meeting of the North American Chapter of the Inte rnational Group for the Psychology of Mathematics Education, Madison, WI. Wilson, M. & Goldenberg, M.P. (1998). Some conceptions are difficult to change: One middle school mathematics teacher’s struggle. Journal of Mathematics Teacher Education, 1 269-293.

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281 Wolleat, P., Pedro, J. D., Becker, A. D., & Fe nnema, E. (1980). Sex differences in high school students’ causal attributions of performance in mathematics. Journal for Research in Mathematics Education, 11, 356-366.

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282 APPENDICES

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283 Appendix A: Pilot Study I Purpose of Study The purpose of this study was to examine the attitudes toward mathematics of preservice elementary school teachers enroll ed in an introductory mathematics methods course and to explore these at titudes and the experiences that have led to the development of these attitudes through refl ective journals and interviews The study sought to answer the following questions: 1. What are the attitudes toward mathem atics of preservice elementary school teachers entering an introductory mathematics methods course? 2. What do the reflective journals of pr eservice elementary school teachers enrolled in an introductory mathematic s methods course reveal about their attitudes toward and experi ences with mathematics? 3. What are the attitudes toward and ex periences with mathematics of those preservice elementary school teachers id entified as having the most negative attitudes? 4. What is the relationship between pres ervice elementary teachers’ initial attitudes toward mathematics and thei r grade on the methods course final examination? Methods Participants The participants in this study were 31 university students enrolled in one section of a mathematics methods course at a major research university in the southeastern

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284 Appendix A (Continued) United States during the spring semester, 2003. St udents enrolled in this course typically are juniors and seniors who are working toward state certification as teachers of grades kindergarten through six. Thirty of the stude nts were females and one was male. Twentyseven of the participants were between the ages of 20 and 30, three were between ages 31 and 40, and one was between ages 41 and 50. Th e researcher was the instructor for this course. Procedure Each student completed the Attitudes Toward Mathematics Inventory (ATMI) at the beginning of the semester. The survey scores were used to identify those students with the most negative attitudes. The ATMI (Appendix E) contains 40 items, and students are asked to indicate their degree of agreement with each statement using a Likert-type scale from one to five, from strongly disagr ee to strongly agree. The instrument has been tested for internal cons istency and construct validity and measures the following four components: (1) student’s self-confidence, (2) value of mathematics, (3) motivation, and (4) enjoyment of mathematics (Tapia & Marsh, 1996). Throughout the semester participants subm itted reflective journal entries as part of their course assignments. Journal entries were submitted by e-mail, and the instructor responded to each entry by e-mail. Students were not asked to sign consent forms for the use of journals until the end of the semester. Therefore, at the time they wrote them, students were unaware that thei r journals would be used in a research study. However,

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285 Appendix A (Continued) they had signed consent forms for the survey and knew that the survey results would be part of a research study. The following j ournal prompts were among those given and were the focus of this research project: 1. Discuss any feelings (positive or negative) that you have about taking this course. What are you hoping to gain from the course? 2. What are your memories of learni ng mathematics in elementary school (attitudes, success, etc.)? What can you, as a future teacher, learn from these experiences? The two students with the lowest scores on the ATMI participated in individual interviews where their attitudes toward and experiences with mathematics were further explored. The Experiences with Mathematics Interviews took place during week twelve of a fifteen-week semester. They were audio taped and later transcribed. In addition, two students were interviewe d six months after the completion of the methods course. These Changed Attitudes Interviews focused on participants’ ideas about those aspects of the methods course that may have influenced their attitudes toward mathematics. The Changed Attitudes Interview protocol (Appendix I) asked participants how they felt about the use of manipulatives cooperative learning, problem solving, and journals in the methods course and also in teaching mathematics in general. These interviews were also audio ta ped and then transcribed.

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286 Appendix A (Continued) Data Analysis Surveys. Scores for each participant on each of the four attitudinal components, as well as a composite attitude sc ore, were calculated using the software program SAS. Scores on the ATMI range from 40 to 200. There are 10 items dealing with Value, 10 with Enjoyment, 15 with Self-Confidence, and 5 with Motivation. Because there were unequal numbers of items for each attitude factor, the average score per attitude factor was calculated in or der to make comparisons more easily. These average per-item scores range from one to five. Journals. After reading each student journal en try related to a given prompt or listening to and reading the transc ription of an interview for a sense of the whole, units of general meaning (Hycner, 1985) were delineated Themes were then identified from the data and recorded using the computer software program Ethnograph. Results Surveys: Initial Attitudes Toward Mathematics Participants’ initial survey scores were highest or most positive for Value of Mathematics, with a mean score of 4.17 on the 5-point scale ranging from strongly disagree to strongly agree. A score of five re presents the most positive attitude, a score of three represents a neutral pos ition, and a score of one represents the most negative attitude. The lowest or most negative scores were for Motivation, with a mean score of 2.65. Results from the initial su rvey are found in Table A-1.

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287 Appendix A (Continued) Table A-1 Initial Attitudes Toward Mathematics Mean Standard Skewness Kurtosis Deviation Value 4.17 0.57 -0.31 -0.68 Enjoyment 3.10 1.20 -0.34 -1.48 SelfConfidence 3.23 1.15 -0.30 -1.43 Motivation 2.65 0.96 0.31 -1.45 Composite 3.36 0.91 -0.20 -1.58 Note. Scores range from 1 to 5, with 1 indicating the most negative attitude and 5 indicating the most positive attitude. Surveys: Relationship Between Initial Attitudes and Final Exam Grade A Pearson correlation coefficient was found using the software program, SAS, in order to determine the relationship between initial attitudes toward mathematics and achievement in the methods course. Achi evement was measured using the methods course final examination. This departmental test is a 50-item multiple-choice instrument that includes questions a bout both mathematics content and pedagogy. The composite attitude score was used as the independent variable and the methods course final examination grade was used as the dependent variable. An alpha level of 0.05 was used to indicate whether the obtaine d correlation was statistically significant. A statistically significant Pearson Correlation Coeffi cient of r = 0.39508 was found, indicating a moderately strong positive corr elation (p=.0278 < .05, n = 31).

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288 Appendix A (Continued) Journal One: Feelings About Course The first journal entry asked students to discuss any feelings, positive or negative, that they had about taking the met hods course. Responses were analyzed and positive, negative, and neutral themes were identified. Some of the journal entries expressed multiple themes, and these themes were analyzed separately. Therefore, frequencies may total more than 31. Table A-2 shows the themes that were identified at the beginning of the course. The following journal excerpts are representa tive of data responses for each of these themes: Positive themes about course “I’m excited about taking this course. I always enjoyed math when I was in elementary school, and I am excited to teach it.” “I am very excited about this course. It is my first Elementary Education course, and I am glad to finally be starti ng on classes towards my major.” “The feelings I have about taking this cour se are all positive. I am so excited to learn how to teach my students to become better in mathematics.” Negative themes about course. “Prior to attending class, I was filled with anxiety about being forced to take another math class.” “I was very nervous about taking this cla ss at first. I have never been good at math and I was afraid we would be re learning everything at a quick pace. I definitely was not looking forw ard to starting this class.”

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289 Appendix A (Continued) Table A-2 Themes Identified at Beginning of Elem entary Mathematics Methods Course Feelings Frequencya Positive Feelings Excited about course 10 Feel positive, Look forward to 9 Negative Feelings Nervous, worried, apprehensive 9 Don’t enjoy mathematics 2 Neutral Feelings No feelings yet 3 aTotal frequency of 33. Total frequency of 19 positive feelings came from 17 of the 31 participants. This represented 46.3% of all comments made for this prompt. Total fre quency of 11 negative feelings came from 8 of the 31 participants. This represented 26.8% of all comments made for this prompt. Total frequency of 3 neut ral feelings came from 3 of the 31 participants. This represented 7. 3% of all comments made for this prompt. Negative themes about course (continued). “I have a few negative feelings about this course simply because I don’t enjoy math and never have.” Neutral themes about course. “I truly do not have any feel ings for this class yet.” Some students expressed negative attitudes toward mathematics that were not specific to the methods course. The following are representative of these attitudes: “I have really never been a very good math student. I just don’t get it.”

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290 Appendix A (Continued) “I made an A in my last math class, but deep down in my heart, I felt that it must have been a fluke.” “Mathematics has never been one of my fa vorite subjects, nor ha s it been one of my strongest.” “I am sorry to say I do have a very ne gative attitude about math that goes way, way back to really bad teachers that probably felt like I do about math.” Journal One also asked the students what they were hoping to gain from the course. Table A-3 summarizes the themes that were identified in a ddressing this question and the frequencies with which these themes were cited. The journal excerpts that are given for each theme are representa tive of data responses given. Journal 2: Memories of Mathem atics in Elementary School The second journal entry asked students to reflect on their memories of learning mathematics in elementary school Initially, 13 distinct units of meaning associated with positive memories and 27 units associated with negative memories were identified. As themes emerged, those representing similar concepts were combined. For example, several students had negative memories that focused on learning a specific mathematics concept. Initially these were grouped by topi c, but they were later combined into one category, which was called ‘Negative Memo ry about Learning Specific Topics.’ The following excerpt was initially categ orized as ‘Negative Memory about Multiplication Facts’:

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291 Appendix A (Continued) Table A-3 Themes from Journal Prompt: What Do You Hope to Gain From the Course? Theme Frequencya Representative Journal Excerpt Better understand “I am hoping to gain a better understanding of math math, sharpen fr om your advice and teachings. I find the more math skills 6 people who help me with math, the better.” Gain confidence “I am hoping th at this class will give me some in math 4 confidence so that I am not so scared of math and that I am co nfident in my abilities to be able to teach it well.” Help students to: “I want to be able to reach students like me who have a be confident harder time in math or children who give up too about math 1 easily, never finding the answer.” not give up 1 “I want to learn how to create a safe environment for be comfortable those ‘afraid’ of math. I want students in class to with math 2 not be intimidated by the math process.” not be intimidated by math 1 Continued on the next page

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292 Appendix A (Continued) Table A-3 (Continued) Themes from Journal Prompt: What Do You Hope to Gain From the Course? Theme Frequencya Representative Journal Excerpt Develop teaching “I would like to learn effective ways to teach math. I strategies 14 would also like to know about mistakes that can be made so I can avoid them.” Make math: “I’m hoping to learn how to teach math in a fun and interesting 3 interesting way.” fun 4 Meet needs of “I hope to gain a better understanding about what kids students 2 need, how to use the materials around me, and to become the kind of teacher that all kids learn from!” Change own “I need to change the way that I view math. I have an negative academic lifetime of negative feelings towards attitude 2 math, and I do not want to bring that to my instruction with my students.” aTotal frequency of 40 came from all of the 31 participan ts. This represented 100% of all comments made for this prompt. “I remember that I could not remember my multiplication [basic facts]. I had always been an ‘A’ student in math, but I just could not figure them out.” The following excerpt was initially categ orized as ‘Negative Memory about Multiplication Facts’ and also ‘Negative Memo ry about Fractions.’ It was later counted as two instances of ‘Negative Memory about Learning Specific Topics.’

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293 Appendix A (Continued) “The first thing that jumped into my mind when I thought of my elementary school math days was ‘multiplication tables’ and ‘ugh, fractions’.” Those journal entries reflecting positive memories are summarized in Table A-4. In addition, the following positive memories of mathematics teachers in elementary school were each mentioned once: Teacher as facilitator: “[The teacher] lets you figure it out without telling you the answer. This worked very well, beca use then you figure out where you went wrong on your own, you are more likely to remember the next time.” Teacher provided individual help: “[The teacher] helped me to understand things that I was having problems with by …working with me one on one when necessary.” Teacher was patient: “[The teacher] was very patient, and he never embarrassed anyone. I was not afraid to try in his class.” Teacher provided repetition: “I can reme mber my math teachers as being very repetitive…. For me, this was gr eat. I learn well doing repetition.” Teacher helped students understand: “[The teacher] had a way of making everyone understand the math we had to do in class.”

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294 Appendix A (Continued) Table A-4 Positive Memories of Mathematics in Elementary School Theme Frequencya Representative Journal Excerpt Fun 4 “[Math] was fun. It seemed like I wasn’t doing work, just playing.” Good at it 9 “I was alwa ys good at math when I was younger. I wanted to practice it all the time. The thing I remember the most is how proud my third grade teacher was of me when I memorized my times tables.” Enjoyed it 3 “I think I enj oyed it because I understood it and I made good grades in it.” Use of songs “I did enjoy math in elementary school because I like or the use of manipulatives … It was almost like playing manipulatives 3 with toys.” Work in “We got to do lots of group work which I feel can be groups 2 helpful to students.” Speed tests 1 “I liked the sp eed test because it was a race against your friends.” General “I have great memories of mathematics from elementary Positive 1 school, even though some students may have hated it.” aTotal frequency of 23 came from 13 of the 31 participants This represented 23.5% of all comments made for this prompt.

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295 Appendix A (Continued) Those journal entries reflecting negative memories from elementary school are summarized in Table A-5. Students’ negativ e memories of mathematics teachers are summarized in Table A-6. Some students expressed beliefs about ma thematics, teaching mathematics, and learning mathematics. The following are representative of these beliefs: “Personally I feel that math is one of th e most important subjects for students to learn. Unfortunately, math is usually th e subject that most students hate.” “Math is a very difficult subjec t to teach in my opinion.” “Many kids and even adults hate math and give up.” “Because I struggled with math, my attitude set me up to fail.” Math should not be a subject to be intimidated by, but for so many people (including myself), it is.”

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296 Appendix A (Continued) Table A-5 Negative Memories of Mathematics in Elementary School Theme Frequencya Representative Journal Excerpt Did not do well 13 “On my report ca rds, I excelled in everything but math (and cutting on the lines!)” Did not like 3 “I rememb er math being one of my least favorite subjects.” Frustrating 3 “I remember being very frustrated with math and often giving up.” Boring 2 “I n elementary school, I remember math being very boring and not very fun.” Confusing 3 “To a child, it was confusing. I gue ss I need a concrete strategy to go with my pa rticular learning style.” Too much drill “I rememb er having to write out the “times tables” again and and practice 3 again, dril l and practice. I HATED doing that!! I started zoning out when the teacher said, ‘Open your math books to page …’ and didn’t tune back in until science.” Specific topics, “In elementary school I remember that I could not remember especially my multiplication [facts]. I just could not figure them out.” mult. facts 11 Not challenging 1 “I rememb er it as being fun but not challenging.” Too much “I think I never really liked it because I tend to be slower when it pressure 3 comes to figuri ng things out. I can’t thi nk well under pressure.” aTotal frequency of 42 came from 21 of the 31 participants This represented 42.9% of all comments made for this prompt.

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297 Appendix A (Continued) Table A-6 Negative Memories of Mathematics Teachers Theme Frequencya Representative Journal Excerpt Only taught “The math teachers I had in elementary school only had one way 4 one way of teaching the lessons. This made it difficult for me because I did not always understand the way my teacher was teaching the lessons.” Would not help “She gave us bookwork and told us to read and figure it out. As when a result, I had no clue what was going on and I completely needed 4 lost interest.” Boring, old, lazy, “One teacher severely hindered my ability to excel because she drab 5 was burned out and lazy!” Impatient, moved “In fifth grade I reme mber my teacher was very intimidating on too quickly, and I was always asking friends for help when I didn’t intimidating 3 understand something.” Lack of content “It really was a shame how badly I was taught math, by a knowledge 1 teacher who didn’t know much math herself.” Generally negative “The teacher didn’t seem to know what to do to help either. memory 3 She said that teaching math was her weakness.” aTotal frequency of 20 came from 11 of the 31 participants This represented 20.4% of all comments made for this prompt.

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298 Appendix A (Continued) Journal Two also asked students what the y, as future teachers, had learned from these experiences. Table A-7 summarizes their responses. Table A-7 Themes from Journal Prompt: What Did You Learn as a Future Teacher? Theme Frequencya Representative Journal Excerpt Want to make “What I want my students to learn is that even if math isn’t math fun, their stronge st subject it can still be fun.” interesting 11 Want students to “The most impor tant thing that I learned from my feel confident, experiences is that I do a lot better in a class if my teacher positive 8 and I have confidence in me.” Use manipulatives, “I learned well with manipulatives so I will use those a lot rhymes, songs 4 especially since they have so much more to choose from.” Use cooperative “Using cooperative lear ning is very important in math. learning 3 Students s eem to learn better when they have help from their peers or watching their peers solve problems.” Make math “Practical applications how and why will they need to know relevant 7 this; th is is what I want to emphasize to my students.” Continued on the next page

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299 Appendix A (Continued) Table A-7 (Continued) Themes from Journal Prompt: What Did You Learn as a Future Teacher? Theme Frequencya Representative Journal Excerpt Accommodate “I have learned how to teach using all different kinds of different learning methods so the stude nts may learn in a way that is easiest styles 4 for them.” Make math “Math does not have to be worksheet after worksheet, timed more than tests, and drills. My goal as a teacher is to look both just drill 3 inside and outside the box.” Be patient 4 “As a future teacher, I will strive for patience with my students when I am teaching math.” aTotal frequency of 44 came from all of the 31 participan ts. This represented 100% of all comments made for this prompt. Experiences with Mathematics Interviews ‘Sandra.’ Sandra’s score on the ATMI was the lowest in the class, indicating the most negative attitude toward mathematics. One of Sandra’s most vivid memories of elementary school mathematics was of learni ng multiplication and division basic facts in third grade. She remembered enjoying practici ng at home with her sisters as they played school. Her other memory of elementary schoo l occurred in fifth grade. She was doing well in her mathematics class, so the teacher moved her to a more advanced class. She

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300 Appendix A (Continued) immediately felt lost when she realized that the new class was working on material that she had not yet seen: The first day I knew I was lost and I think I stayed maybe a week in there. And they had already passed what we knew, and to me, it was like, ‘I’m not going to try and catch up,’ you know, I was already lo st, so I went back to the class and enjoyed the fact that I had the highest grade in the class! Sandra’s problems with mathematics began in seventh grade. She began having trouble understanding the concepts: If I didn’t get it after two times of her explaining, I would get so frustrated that I would start crying a nd give up. I didn’t want to know it anymore. In 8th grade I had a better teacher, but still, by that time, there were so many things that I had lost from 7th grade, that it was frustrating for me to try and keep going with it. And even if I caught on in class, by the time I went home to do the homework, I already lost it. Sandra continued to have problems unders tanding mathematics in high school and college. She viewed this as a result of fa lling behind in middle school. When discussing her second attempt to pass a coll ege statistics course, she reca lled, “They went so fast and it was too hard to understand for me.” Sandra was “very scared” at the start of the methods course. She was afraid that she would be “the stupid one.” When asked if her feelings had changed since the start of the course, Sandra seemed somewhat relieved: Sandra: Yes. It’s not like you’re testing us on our abilities; you’re teaching us how to teach and how to make sense of it, and to me, I’ve

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301 Appendix A (Continued) learned a lot of stuff. JS: So, are you finding it stressf ul? Like you anticipated? SE: Not necessarily. The tests are always stressful … and then today when we were doing those questions, as a pair, and [my partner] … came up with an answer and I said, ‘Well, I don’t unde rstand it,’ and it frustrated me, that she could get it a nd I couldn’t. But it turned out … she was wrong, but we worked together to do it and it made sense at the end. She had her way and I had my way. When asked if there was anything else sh e would like to say about her attitudes toward mathematics, Sandra replied: I’m hoping I don’t have to teach a lot of it when I become a teacher …I know I have negative attitudes still about it and I don’t want that to reflect on the students that I’m teaching. I want them to have their own experiences. And I’m not sure that I could overcome that … I’m hoping if I have to that I can forget about th at and move on, but…it’s something I don’t want to do. I’m a little nervous about it. ‘Debbie.’ Debbie’s score on the ATMI was the second lowest in the class. Debbie remembered loving mathematics in elem entary school. She recalled, “I didn’t struggle with it; it came easily, and I enjoyed it.” Middle school mathematics was also a positive experience for her. Her troubles began in high school when she took Algebra: When I got to Algebra, that’s, th at’s when it all happened….I didn’t understand it…. You asked [the teacher] a question and she would tell you

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302 Appendix A (Continued) how to do it, but she didn’t explain… It was kind of like you were a bother to her, a burden, you know? It’s like, ‘Well, you don’t get it so you’re not wasting my time,’ and that’s how I felt, so everyday when I’d go into that class, I …I dreaded it. I dreaded wa lking through the door. I can still see her face now, but I just, I dreaded it. She remembered college mathematics course s in much the same way. She had to take remedial courses before she could take the college level courses. She remembered one course in particular: I actually ended up taking it 3 times before I passed it, 3! And that is very, very hard … it was hard for me like in high school….I was so used to excelling at things and then when I got to high school, I didn’t and so … my self esteem started getting rea lly low as far as academics go. When asked to complete the sentence, ‘I do not enjoy or I fe el negative about mathematics because … ,” Debbie’s response was: “Because it’s scary! …‘Cause I don’t always understand it… I don’t understand why or how it happened.” Debbie described herself at the start of the met hods course as “a nervous wreck.” She added, “Just the word math scares me.” When asked if her feelings had changed any since the start of the class, she said, “No, not a lot. I can’t honestly sa y that they have changed…It just makes me nervous. Math makes me nervous.”

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303 Appendix A (Continued) Changed Attitudes Interviews The Changed Attitudes Interviews took place six months after the completion of the course and focused on participants’ ideas about those aspects of the methods course that may have influenced their attitudes toward mathematics. Interviewees were asked if they thought their attitudes toward mathema tics had changed since the beginning of the methods course. Both said that they believe d their attitudes had improved as a result of the course. They were then asked about thos e aspects of the methods course that may have influenced their attitudes toward mathematics. ‘Lynn.’ Lynn’s score on the ATMI was one of the lowest in the class, with only five of the 31 students scoring lower. When as ked what aspects of th e course she thought had influenced her attitudes, she immedi ately thought of the us e of manipulatives: What I got most out of [the method s course] was the ‘bag of tricks’ [manipulatives kit] as I call it, and how to use them. When I went to school, it was worksheets. There wasn’t even much m odeling going on or anything to help you learn. That’s what I got most out of it, and that’s what I will bring into my teaching. Using the manipulatives defin itely helped my own understanding. When asked about the use of problem solving, Lynn said: I remember a couple of times I got frustr ated because I wasn’t one of the ones who got it. I could see though [how others solved it]. Solving something on your own definitely makes you feel better about math. Reflecting on the use of journal wri ting in the methods course, Lynn said,

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304 Appendix A (Continued) I loved them and I will use them [when teaching] in all subjects…. I think it got some of my negative attit udes out. I remember that I had made a comment in class that you overheard th at I would stay home before I would teach math. I really felt that. But I’m teaching math now [in internship], so obviously my attitudes did change. ‘Brenda.’ Brenda’s score on the ATMI was the third lowest in the class. When asked about the aspects of the course that may have influenced her attitudes toward mathematics, she said: We used manipulatives. Each step was explai ned, why we do this to get this. That is what gives the confidence, and with th e confidence comes liki ng it better. I like understanding why I’m doing something, not ju st mindlessly doing something. Brenda found the cooperative learning in the methods course “helpf ul.” She explained: [The instructor] had time to go around a nd help. I could be talking to someone next to me about the problem instead of waiting for [the instructor] to go to each individual person. Maybe one or two wo rds from someone and I get the whole thing; I understand it. When asked about the use of problem solv ing in the methods course, Brenda said: It was sometimes fun. If it’s challenging, but something [students] could achieve, for me at least, it boosts confidence, like, ‘Wow! I solved the problem. I’m a mathematician.’

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305 Appendix A (Continued) Reflecting on the use of journal writi ng in the methods course, Brenda said: It definitely got us to think about differen t aspects of things, instead of just seeing it one way. If I was remembering a bad teacher, it could make me start hating math again, but I think it’s good to reme mber those things so you know you are progressing or digressing in your attitudes. You know where you stand and if you’ve changed any from where you were. Discussion The majority of the students said that they were excited and positive about the methods course. However, many of the stude nts expressed feelings of nervousness, worry, and apprehension at th e start of the semester. They had encountered negative experiences with mathematics in the pa st that included not understanding the mathematics, not doing well in mathematics, and disliking mathematics. Many of the students, including those with positive and thos e with negative feelings about the course, hoped to develop effective teaching strate gies that would allow them to make mathematics fun and interesting for their stude nts. Several revealed that they hoped to gain confidence in their own mathematical abil ities. Others said that they wanted to change their negative attitudes toward mathem atics in order to avoid passing them on to their future students. When reflecting on memories from their own elementary school mathematics experiences, several students had positive memories. Many recalled mathematics class as fun and enjoyable. These results are in line wi th the literature, which says that children

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306 Appendix A (Continued) typically have positive attit udes toward mathematics when they begin school. However, thirteen students said that they did not do well in elementary school mathematics. Many remembered feeling frustrated, confused, a nd pressured in mathematics class. Others remembered it as boring, with too much dri ll and practice. Eight students specifically recalled negative memories associated with learning the basic facts for multiplication. When considering memories of elementary school mathematics teachers, students referred to teachers who were boring and laz y, who would not offer help when needed, who offered only one approach to concepts who were impatient, intimidating, and who lacked content knowledge. Because the literatu re shows that students’ attitudes typically tend to become more negative as they get olde r, it is noteworthy that so many of these future teachers seem to have developed nega tive attitudes toward mathematics while still in elementary school. Several students made statements that reflected their own beliefs about mathematics and teaching mathematics. Although they seemed to view mathematics as an important subject for students to learn, they also saw it as a subject that is difficult to teach and is disliked by ma ny, if not most, people. As students reflected on their elementary school experiences and considered what they, as future teachers, could learn from these experiences, several participants said that they wanted to make mathematics fun and intere sting for their students, that they wanted their students to feel confident and positive about mathematics, and that they hoped to help their students see the relevance of the ma thematics they were learning. They stressed the importance of helping st udents understand mathematics and providing extra help

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307 Appendix A (Continued) when needed. These future teachers felt that they had benefited from using manipulatives, songs and rhymes, cooperative learning, and ac tivities other than drill and worksheets. They also hoped to be patient with their students, as well as accommodating of their different learning styles. Implications The reform movement in mathematics educ ation has recognized the importance of affective issues and the connection between these issues and highe r-order thinking. The National Council of Teachers of Mathematics has established goals involving students’ dispositions toward mathematics that include value, self-confidence, and interest. By studying preservice elementary teachers’ attitudes toward mathematics and the experiences that have played a crucial role in the development of these attitudes, teacher educators can use this information to deve lop training programs aimed at improving these attitudes. Using manipulatives, songs and rhymes, cooperative le arning, and activities other than drill and worksheets were practices that these preservice teachers associated with positive memories and should be stresse d in an elementary mathematics methods course. The methods course should focus on st rategies that teachers can use to make mathematics relevant to their students’ lives, to help their students develop conceptual understanding of the material, and to accomm odate their individual learning styles. Preservice teachers should be encouraged to be patient with their students, providing them with a positive classroom environmen t where they can feel confident about mathematics.

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308 Appendix A (Continued) By identifying patterns of teacher be haviors, teaching methods, and other memorable incidents that students identify as significant contributors to negative attitudes toward mathematics and by focusing methods courses on alternative teaching methods and teacher behaviors, perhaps students will complete these courses with more positive attitudes. Perhaps they will then be more like ly to pass on to their future students more positive attitudes toward mathematics. In th is way, perhaps the cycle of elementary school teachers with negative attitudes toward mathematics fostering negative attitudes in their own students can be broken. These findings provide implications not onl y for teacher educators, but also for mathematics teachers at all levels. Students asso ciated positive memories with feelings of success and enjoyment of mathematics. It is important for teachers to provide a positive environment for their students, where they can feel comfortable rather than intimidated, engaged and interested rather than bored, conf ident rather than frustrated, and successful rather than defeated. It is up to mathematics teachers at all levels to provide such an environment.

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309 Appendix B: Pilot Study II Purpose of Study The purpose of the second pilot study was to determine if any changes in preservice elementary school teachers’ atti tudes toward mathematics occurred during a mathematics methods course. The pilot study so ught to answer the following questions: What are the attitudes toward mathem atics of preservice elementary school teachers entering an introductory mathema tics methods course? In particular, how do preservice teachers score on each of the four attitudinal components being measured: value of mathematics, enjoyment of mathematics, motivation for mathematics, and self-confidence with mathematics? To what extent do attitudes toward math ematics of preservice elementary school teachers change during the mathematics methods course? To what do preservice teachers whose att itudes toward mathematics were altered attribute this change? What is the relationship between preservi ce elementary teacher s’ initial attitudes toward mathematics and their grade on the methods course final examination? The participants in this study were 38 university students enrolled in one section of a mathematics methods course at a majo r research university in the southeastern United States during the spring semester, 2004. St udents enrolled in this course typically are juniors and seniors who are working toward state certification as teachers of grades kindergarten through six. Twenty-nine of the pa rticipants were between the ages of 18

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310 Appendix B (Continued) and 25, three were between ages 26 and 35, f our were between ages 36 and 45, and two were over the age of 45. The researcher was the instructor for this course. Methods Each student completed the Attitudes Toward Mathematics Inventory (ATMI) at the beginning of the semester and again during week 12 of a 15-week semester. This allowed the researcher to measure each partic ipant’s initial attitudes toward mathematics and to assess any changes that may have taken place during the first 11 weeks of the semester. The ATMI (Appendix E) contains 40 items, and students are asked to indicate their degree of agreement with each statement using a Likert-type scale from one to five, from strongly disagree to strongly agree. Th e instrument has been tested for internal consistency and construct validity and meas ures the following four components: (1) student’s self-confidence, (2) value of mathematics, (3) mo tivation, and (4) enjoyment of mathematics (Tapia & Marsh, 1996). Composite attitude scores were calcul ated at both the beginning and during the twelfth week of the semester. These scores we re used for statistical analyses using the software program SAS. Participants’ change scores, which were their post-course scores minus their pre-course scores, were calculate d. Pre-course and post-co urse scores are not independent, so a t test for repeated measures wa s conducted to determine if a statistically significant cha nge in attitude occurred. Those participants with change scores gr eater than one standard deviation above or below the mean change score were cons idered for individual interviews. These

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311 Appendix B (Continued) Changed Attitudes Interviews focused on participants’ ideas about those aspects of the methods course that may have influenced their attitudes toward mathematics. The Changed Attitudes Interview protocol (Appendix I) asked participants how they thought their attitudes toward mathematics had changed since the start of the course. Participants were also asked how they felt about the use of manipulatives, cooperative learning, problem solving, and journal writing in th e methods course and also in teaching mathematics in general. These interviews were audio taped and then transcribed. Interviews took place during the week following the completion of the methods course and submission of final grades. Results Surveys: Initial Attitudes Toward Mathematics Participants’ survey scores were highest or most positive for Value of Mathematics, with a mean score of 3.64 on the 5-point scale ranging from strongly disagree to strongly agree. A score of five re presents the most positive attitude, a score of three represents a neutral pos ition, and a score of one represents the most negative attitude. The lowest or most negative scores were for Motivation, with a mean score of 2.61. Results from the survey are found in Table B-1.

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312 Appendix B (Continued) Table B-1 Initial Attitudes Toward Mathematics Mean Standard Skewness Kurtosis Deviation Value 3.64 0.73 -1.33 2.25 Enjoyment 2.72 1.10 0.12 -1.22 SelfConfidence 3.08 1.12 -0.28 -1.10 Motivation 2.61 1.01 -0.15 -1.35 Composite 3.07 0.90 -0.08 -1.28 Note. Scores range from 1 to 5, with 1 indicating the most negative attitude and 5 indicating the most positive attitude. Surveys: Relationship Between Initial Attitudes and Final Exam Grade A Pearson correlation coefficient was found using the software program, SAS, in order to determine the relationship between initial attitudes toward mathematics and achievement in the methods course. Achi evement was measured using the methods course final examination. This departmental test is a 50-item multiple-choice instrument that includes questions a bout both mathematics content and pedagogy. The composite attitude score was used as the independent variable and the methods course final examination grade was used as the dependent variable. An alpha level of 0.05 was used to indicate whether the obtaine d correlation was statistically significant. A statistically significant Pearson Correlation Coeffi cient of r = 0.41638 was found, indicating a moderately strong positive corr elation (p=.0093 < .05, n = 38).

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313 Appendix B (Continued) Surveys: Changed Attitudes The mean change score for the 38 particip ants was 9.08, with a standard deviation of 16.44. The median change score was 5.5, and ther e were four modes with a count of 3. These were -2, -1, 1, and 15. The change sc ores were positively skewed (Sk=1.145). The kurtosis was 2.809, indicating that the distribution was leptokurtic. The repeated measures t test was used to test the null hypothesi s that the mean change score in the population was zero. Because t = 3.403 > 2.02 (tcrit), and p = .0016 < .05, the null hypothesis was rejected. The validity of the repeated measures t-test depends on the assumptions of independence and normality. Although the pre-cour se and post-course survey scores were dependent or repeated measures, the change scores were independent The distribution of change scores was positively skewed. Howe ver, because n (38) > 20, the repeated measures t-test is relatively robust to viol ations of the normality assumption. The effect size, d = Xdiff / S diff, was .5521, indicating a medium ef fect size. In summary, it was possible to reject the null hypothesis of a mean change score of zero (t(37)= 3.403, p= .0016). There was a statistically significan t positive attitude change. Changed Attitudes Interviews Statistical analysis revealed that six participants had positive change scores greater that one standa rd deviation above the mean change score. This reflected a change score of at least 26 points. Th e participant with the greatest change score, 66 points, had to leave town immediately after the final exam and was therefore unavailable for an

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314 Appendix B (Continued) interview. The participants w ith the second largest change score, ‘Jasmine,’ and the third largest change score, ‘Linda,’ were interviewed three days after the final examination. ‘ Jasmine.’ Jasmine’s score on the pre-course ATMI was 64. This represented a mean response of 1.6 per survey item where one was the most negative response and five was the most positive response. Her post -course ATMI score was 103, representing a mean response of 2.6 per item. When aske d how she thought her attitudes toward mathematics had changed, she responded: My attitudes changed [positively] by all the activities we did in class, all the hands-on activities, because I’m a hands -on person, and it helped me understand ... like with fractions. I can’t stand fractio ns, but using the manipulatives helped me understand what I was doing. When asked how she felt about the use of cooperative learning, Jasmine said: When I grew up, I didn’t do a lot of stuff with my hands with math. So trying to memorize it, I was like, ‘huh.’ I didn’t unde rstand what I was trying to formulate when it came to math. Working together with other people [in the methods course] helped me, because I didn’t have that when I was little. When asked about problem solving, Jasmine connected problem solving to the use of manipulatives: It was the way that we got to solve the problem [that influenced an attitude change], not the original problem because I always have trouble with problem solving with math problems, but now I know how to look at it in a different way.

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315 Appendix B (Continued) You know, like think of something [to repr esent] the amount you’re looking for, like the [fraction] circles or base ten blocks helped also. Reflecting on the use of reflectiv e journal writing, Jasmine said, The journals made me realiz e that in order to be an effective teacher in a subject that I really don’t like, I have to not just pretend I like th e subject, but at least give an effort to help [my students] understand mathematics, which is hard for any kid. A lot of kids can pick it up, for othe rs it’s a slow process. ‘Linda.’ Linda’s score on the pre-course AT MI was 113. This represented a mean response of 2.8 per survey item where one was the most negative response and five was the most positive response. Her post-course ATMI score was 144, representing a mean response of 3.6 per item. When asked how sh e thought her attitudes toward mathematics had changed, she responded: I think my attitudes got mo re positive. At the beginni ng, before I took the course, I felt kind of negative towards math. I think that I wasn’t taught it well. After the course, I learned different ways to teach kids and help them have a more positive attitude about math. When asked which aspects of the course she thought had affected her attitudes, Linda said: The manipulatives really helped me see th ings better, and I th ink that they would help kids see things better. If I had been taught that way, maybe my attitude [toward math] wouldn’t have been so negativ e. I was just taught to know the rule

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316 Appendix B (Continued) and that’s just how it is, so you neve r really understand why you’re doing it, but the manipulatives help you see it. That was probably the biggest thing for me. When asked how she felt about the use of cooperative learning, Linda said: I think it was really good because someti mes I couldn’t see something, but my group member next to me could. Then she could explain it to me in a different way and show me with the manipulatives and stuff, so I really liked that. When asked about problem solving, Linda responded: I think it’s a really importa nt skill to teach kids. I thi nk it’s harder to teach, but I think that it’s really important that kids get it, because it’s something they’ll use all through life. Reflecting on the use of journals, Linda replied: I think that the journals were really good. It’s probably something I would use in my own classroom. I think that journals are a good way for [students] to reflect, even if it’s not math. They can talk about why they don’t understand something. I think journals are really a good way for [students] to think more about what they’re doing. Maybe they can share them if they want, but they don’t have to if it’s private. When asked specifically about the use of reflec tive journals in the methods course, Linda said: I think they were really good in the cour se too. It helped me think things through more than just, you know, not thinking about it and just doing it. They help you think about it more.

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317 Appendix B (Continued) Three participants had negative change scor es greater than one standard deviation below the mean. These participants had change scores of -23, -18, a nd -9. The two with the most negative change scores were asked to participate in interviews. The participant with the largest change score, -23 points, declined the researcher’s request for an interview. The second participant, ‘Nancy,’ was interviewed four days after the final examination. All interviews took place afte r final grades had been submitted to the Dean’s office. ‘Nancy.’ Nancy’s score on the pre-course ATMI was 144. This represented a mean response of 3.6 per survey item where one was the most negative response and five was the most positive response. Her post -course ATMI score was 126, representing a mean response of 3.2 per item. However, shor tly after the beginning of the interview, it became apparent that a mistake had been made. When asked how she thought her attitudes toward mathematics had changed, Nancy responded: I think [my attitudes] are more positive toward math. When I was in school, [the teachers said] ‘OK, do it. Here’s the pr oblem.’ If you didn’t understand it or you had a problem with the problem, [the teacher would say] ‘Well, watch real closely and I’ll show you again. Three plus three equals six,’ without saying, ‘You can count them’ or ‘Here’s how we do it.’ The researcher then asked Nancy if she t hought her attitudes toward mathematics had changed in a positive way. Her response was, “Yes, definitely.” At this point, the researcher explained that Nancy’s surveys had shown an 18-point negative change in

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318 Appendix B (Continued) attitude score. Nancy was quite surprised to hear this, so the researcher showed her the actual surveys. After briefly examining them, Nancy said that she had used the ranking scale incorrectly on the first survey. She ha d chosen “A” to repr esent “Strongly Agree” when the survey used “A” to represent “S trongly Disagree.” The researcher rescored Nancy’s first survey and found that she actua lly showed a 56-point positive change in attitude score. Nancy confirmed that this sounded accurate to her. At this point, the researcher returned to the interview prot ocol and asked Nancy which aspects of the course she thought might have affect ed her attitudes. She responded: The use of the manipulatives, actually ha ving hands-on and being able to move things… I think a lot of it had to do with the presentation, the scenarios, [problem-solving activities] and just ha ving the students in class share their invented strategies. There were some [p roblems] that I was confused on, and one of the other students said, ‘Oh, well this is how I did it.’ I thought, ‘OK’ because it made a lot of sense to hear that ther e is more than one way to do it. When asked how she felt about the use of cooperative learning, Nancy said: I feel [cooperative learning] is important because [students] can feed off each other with their strategy sharing, and sometimes it’s easier to hear from a classmate versus the teacher. If the te acher’s busy, they can say, ‘How did you come up with this?’ or ‘Do you know how to do this?’ When asked about problem solving, Nancy responded: Personally I was confused w ith a lot of it. Word problem s have always been a big issue with me. I did like the children’s liter ature lesson plans that we had to write,

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319 Appendix B (Continued) and the way you incorporated children’s literatu re into the course. I feel that would be essential with the problem solving to give th em real situations, give them a menu and saying, ‘We’re going to learn about money. This is how much money you have. What can you buy?’ … Ba sically just presenting real life situations for the students. Reflecting on the use of journals, Nancy said: In general, I think it’s a great idea. It incorporates wr iting as well as math, crosscurricular. I think it’s important for stude nts to reflect on what they did and gives them an opportunity to rethink. They can think about it; you di d it, now what did you do? When asked specifically about the use of refl ective journals in the methods course, Nancy replied: I was a little hesitant at first. I thoug ht, ‘I don’t remember. I’ve been out of high school eleven years.’ Having to think back to elementary school, and most of my [memorable] experiences with math were in elementary school, was difficult. But after the first couple weeks of journal entr ies, I thought they were great. I thought, ‘Wow! OK, [my elementary school te acher] never did that, or [another elementary school teacher] always did something one way because the right way was his way or this is how the math books say to do it.’ So, it gave me another perspective on how I want to work with my students. Let me not just say, ‘OK, this is the only way it can be done.’ Nancy was asked if there were anything else she would like to add about her positive

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320 Appendix B (Continued) change in attitude. She said: [At the beginning of the semester] I was te rrified that I had another math class. Although it was learning how to teach it, that scared me even more than me just having to do it. So, then coming in and [t he instructor] was very personable and I really enjoyed the overheads [demonstrati on of using manipulatives]…. Just the explanations and everything made it so mu ch easier to say, ‘OK, that makes sense now. ’ If my teachers had done this in s econd or third grade, maybe I would have enjoyed my math a little more than I did. Discussion Participants showed a statistically sign ificant improvement in attitudes toward mathematics since the start of the methods c ourse. It should be noted that even though Nancy said that she had made a mistake when completing the pre-course survey and that her pre-course ATMI score should have been much lower and her change score much higher, the statistical analysis was not recalculated. The re ported p-value of 0.0016 included Nancy’s incorrect change score of –18 rather than what she said was her true change score of 56. It was the researche r’s view that because the error was found by chance, no changes in data should be made. When asked how they thought their atti tudes toward mathematics had changed since the start of the methods course, all thr ee interviewees said that they believed their attitudes had become more positive. In cons idering which aspects of the course might have affected their attitudes, all three menti oned the use of manipulatives. They felt that

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321 Appendix B (Continued) the manipulatives helped them understand the mathematical concepts rather than just memorizing a rule. Two of them said that they also benefited from watching the instructor’s overhead projector de monstrations with manipulatives. The interviewees found cooperative learning benefi cial. They discussed experiences in class when they were str uggling, but a classmate was able to help by offering another perspective. When discus sing problem solving, interviewees talked about how important it was for teachers to present problems that are relevant to their students’ lives. Two of them said that they benefited from heari ng other students explain their own solution strategies to problems. All three interviewees expressed positive views concerning the reflective journals. Two of them said that they would like to use journal writing with their own students and that they appreciated the valu e of reflection. Two of them said that they thought the journals that they wrote for the methods c ourse provided them with insights that would help them become better teachers. All three in terviewees expressed the notion that if they had been taught mathematics using the met hods prescribed in this course, their own attitudes toward mathematics would have been much more positive.

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322 Appendix C: Course Syllabus MAE 4310: Teaching Eleme ntary School Mathematics I Instructor: Joy Schackow Office: EDU 308-O Office hours: TBA E-Mail: Joys31999@aol.com Prerequisites: Two college level mathematics courses Required: Elementary School Mathematic s: Teaching Developmentally 2nd custom edition, John A. Van de Walle, Allyn & Bacon, Inc., 2004. Hands On Teaching (HOT) Strategies for Using Math Manipulatives book and kit. Carol Thornton & G. Lowe-Parrino, ETA, 1997. Course Packet available at ProCopy, 5219 E. Fowler Ave, www.procopycoursematerial.com Recommended: Principles and Standards for School Mathematics (NCTM, 2000) http://www.enc.org/reform/journals/ENC2280/nf 280dtoc1.htm Sunshine State Standards for Mathematics (Available at Pro-Copy) http://www.firn.edu/doe/curric/prek12/frame2.htm Other Professional Standards for Teaching Mathematics (NCTM, 1991) Resources: Assessment Standards fo r School Mathematics (NCTM, 1995) Florida Comprehensive Asse ssment Standards (FCAT) Elementary school mathematics textbooks (various) Journals (e.g. Teaching Children Mathematic s, Mathematics Teaching in the Middle School, Teaching Exceptional Learners, Computing Teacher, Mathematics Teacher, Inst ructor, School Science and Mathematics, Childhood Education)

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323 Purpose: This course is required in the undergraduate program in Elementary Education. The course provides for the development of knowledge and skills necessary to prepare students to assume roles as teachers of mathematics in elementary classrooms Such a course is recommended by the National Council of Teachers of Mathematics (NCTM) in its Guidelines for the Preparation of Teachers. Goal: Know How and also Know Why That is, you should focus on discovering the reasons behind th e actions in mathematics. The vision of mathematics learning espoused by the National Council of Teachers of Mathematics assumes the following: "Knowing mathematics means being able to use it in purposeful ways. To learn mathematics, students must be engaged in exploring, conjecturing, and thinking rather than only in ro te learning of rules and procedures. Mathematics learning is not a specta tor sport. When students construct personal knowledge derived from mean ingful experiences, they are much more likely to retain and use what th ey have learned. This fact underlies teachers’ new roles in providing experiences that help students make sense of mathematics, to view and use it as a tool for reasoning and problem solving." ( Curriculum and Evaluation Standards fo r School Mathemat ics: Executive Summary, National Council of Teac hers of Mathematics, March 1989, p. 5) Thus, the purpose of this course is to provide opportunities for preservice teachers to examine their understanding of various mathematics topics and to construct a vision of mathema tics that considers the goals and assumptions of the current reform movement in mathematics education. Content, methods, and materials for teaching elementary school mathematics will be examined with a focus on Problem Solving, Whole Number concepts, and Rational Number concepts. “From the perspective of attaining mathematical competence, teaching elementary mathematics does not mean bringing students merely to the end of arithmetic or to the beginning of ‘pre-algebra.’ Rather, it means providing them with a ground work on wh ich to build future mathematics learning" (p. 117). (Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.)

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324 Appendix C (Continued) Objectives: Upon completion of this course, students will have demonstrated the following: 1. Knowledge of the major goals and charac teristics, including scope and sequence, of elementary school mathematics programs, and aspects of theories of learning as applied to the planning of instruction for the teaching of elementary school mathematics. 2. Knowledge of the current developments in education, including research, that may affect the elementary school mathematics curriculum. 3. Knowledge of the properties of a numbe r system and their application in the teaching of elementary school mathematics. 4. Knowledge of pre-number concepts and id eas and their application in the teaching of elementary school mathematics. 5. Knowledge of numeration concepts and prin ciples and their application within the Hindu-Arabic System. 6. Knowledge of whole number concepts a nd principles and computational skills (algorithms) and their application in the teaching of elementary school mathematics. 7. Knowledge of number theory concepts a nd principles and thei r application in the teaching of elementary school mathematics. 8. Knowledge of rational number (fractions and decimals) concepts, principles and computational skills (algorithms) and their application in the teaching of elementary school mathematics. 9. Knowledge of problem solving processes/ strategies and their application in the teaching of elementary school mathematics. Instructional Design A variety of teaching/le arning techniques may be used. The activities include lectures, discussions, cooperative lear ning activities, question and answer sessions, student demonstrations/explana tions, and role-playing. Assigned reading will supplement classroom activities. Be pr epared to present results and solutions to your peers. We will discuss the content of the st ated chapters in your textbook and will do many activities that are a ppropriate to do with child ren. You should not expect, however, that we will be able to cover every item that is mentioned in your textbook. Therefore, you should read the te xtbook chapters carefully and stop by my office if there is an ything that is unclear.

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325 Appendix C (Continued) Course Requirement/Responsibilities 1. Professionalism Because this course is part of an acc redited program that leads to professional certification, students must demonstrate be havior consistent with a professional career. Failure to demonstrate such c onduct will impact a student’s grade, as noted in the course syllabus. In particular, students are expected to a. attend all class meetings. b. prepare carefully for class. Your input into the class discussion is important. Thus, you are expected to be present at the beginning and conclusion of class. c. complete all assignments on time. Students should maintain a file of all graded assignments until after receiving an official grade notification from the registrar. d. collaborate responsibly with colleagues in coursework. e. interact professionally with classmates. Students should demonstrate respectful standards of beha vior during class discussions. Students are expected to conduct th emselves professionally by positively influencing the classroom environment. Students who come la te, leave early, or are absent, rarely contribute ideas, app ear to be participating in discussions extraneous to the class, are observed to be doing work not related to the class, are disruptive, or inattentive, or pa ssive are not behaving professionally. Attendance Unexcused absences and extreme tardin ess almost always adversely influence your grade. I reserve the right to lower th e grade of any student with more than 1 absence. Medical emergencies will be handled on an individual basis. Students who anticipate being absent from cla ss due to the observation of a major religious observance must pr ovide notice of the date(s) to the instructor, in writing, by the second class meeting. I will take attendance each week. Students with Special Needs The College of Education shares the university’s commitment to eliminating barriers to the education of all students accepted and enrolled in our programs and courses. Therefore, I will attempt to follow the policies outlined by the university and articulated by the Office of Disabled Student Academic Services. It is your responsibility to notify me, in writing, by the second class of any disability that may affect your learni ng process. There should be documentation for any services/ accommodations from the Office of Disabled Student Academic Services.

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326 Appendix C (Continued) Tentative Course Outline This schedule is subject to change as we proceed through the semester. August 25 Introduction Chapters 1, 2, 5 September 1 Chapters 3, 4 Developing Understanding in Mathematics Problem Solving Journal #1 Due September 8 Chapter 9 From Kit: Concepts and Number Sense Two-Color Counters Journal #2 Due Unifix Cubes Number Cubes September 15 Chapter 10 (p. 135-142) From Kit: Chapter 11 (p. 156-168) Pattern Blocks Addition and Subtraction Base Ten Blocks Professional Journal Abstracts due Tangrams September 22 Review From Kit: Test 1 All We’ve Used September 29 Chapter 10 (p. 143-154) From Kit: Chapter 11 (p. 168-177) Pattern Blocks Multiplication and Division Base Ten Blocks Journal #3 Due Tangrams October 6 Chapter 12 From Kit: Place Value Base Ten Blocks Journal #4 Due Coin Set October 13 Chapters 13, 14 From Kit: Computation Strategies & Estimation Base Ten Blocks Journal #5 Due

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327 Appendix C (Continued) October 20 Review From Kit: Test 2 All We’ve Used October 27 Chapter 15 From Kit: Developing Fraction Concepts Pattern Blocks Journal #6 Due Fraction Circles Fraction Tower Cubes Two-Color Counters Tangrams November 3 Chapter 16 From Kit: Computation with Fractions Tangrams Journal #7 Due Pattern Blocks Fraction Circles Fraction Tower Cubes Two-Color Counters November 10 Chapter 16, cont. From Kit: Computation with Fractions Pattern Blocks Journal #8 Due Fraction Circles Fraction Tower Cubes Two-Color Counters Tangrams November 17 Chapter 17 From Kit: Decimals and Percents Base Ten Blocks Literature-Based Lesson Plan Due November 24 No Class December 1 Review From Kit: Test 3 All We’ve Used December 6 Final Exam

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328 Appendix C (Continued) Assignments: Assignments are due at the beginning of class on the assigned date, whether or not you are present in class. I reserve the right to refuse to accept late assignments. If accepted, it is likely ther e will be some loss of points. If unforeseen circumstances arise, it is better to talk with me sooner rather than later to attempt a solution acceptable to both of us. Exams should be completed at the scheduled time. I will consider make-ups ONLY in special circumstan ces and ONLY IF you discu ss absences prior to the time of the exam. Grading Criteria: The following represents my current thinki ng about the evaluation for this course. I reserve the right to make changes and/or deletions as needed. Exams: 3 exams of 50 points each 150 Final Exam: Departmental 100 Professional Journal Abstracts (2 @ 20 points each) 40 Literature Based Lesson Plan 40 Reflective Dialogue Journals (8 @ 5 points each) 40 Grading Scale: 94-100% A (4.0) 348-370 points 90-93% A(3.67) 333-347 points 87-89% B+ (3.33) 322-332 points 84-86% B (3.0) 311-321 points 80-83% B(2.67) 296-310 points 77-79% C+ (2.33) 285-295 points 74-76% C (2.0) 274-284 points 70-73% C(1.67) 259-273 points 67-69% D+ (1.33) 248-258 points 64-66% D (1.0) 237-247 points 60-63% D(0.67) 222-236 points 0-59% F (0.0) 0 – 221 points

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329 Appendix C (Continued) A distribution of grades from A to C is typical, with A representing outstanding performance and C representing minimally acceptable performance. The following description outlines general requirements for each grade. A Outstanding Performance The student demonstrates solid conceptu al understanding and in sight as evidenced by participation during in-class discussi ons and activities. The student shows mastery of the course content and is ab le to make extensions or apply that knowledge to new situations. Assignment s/papers/projects are of excellent quality. The student contributes subs tantially to class and shows strong development regarding the teaching of math ematics. The student earns an average score of A as indicated above. B Good Performance The student demonstrates good mastery of th e course content and is able to make some extensions or apply some of the knowledge to new situa tions as evidenced by participation during in-class discussions and activities. Assignments/papers/projects are of good quality but are no t exceptional. The student contributes to class discussi ons. The student shows good development regarding the teaching of mathematics. The student earns an aver age score of B as indicated above. C Adequate Performance The student demonstrates adequate u nderstanding and mastery of the course content but has difficulty extending or a pplying the knowledge to new situations as evidenced by in-class discussion and activities. Assignments/papers/projects are acceptable. The student shows acceptable development toward becoming a teacher of mathematics. The student earns an average grade of C as indicated above. D Below Average Performance The student demonstrates unacceptable understanding and mastery of the course content. Assignments/papers/projects are inadequate. The student shows poor development toward becoming a teacher of mathematics. The student earns a grade of D as indicated above. F Unacceptable Performance The student demonstrates poor performan ce of the course cont ent. Either some assignments are not completed, are often la te, or are of poor quality. The student does not contribute to class discussions. The student lacks development toward becoming a teacher of mathematics.

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330 Appendix C (Continued) Journal Topics: 1. Discuss any feelings (positive or negative) that you have about taking this course. What are you hoping to gain from the course? 2. What are your memories of learning math ematics in elementary school (attitudes, success, etc.)? What can you, as a future teacher, learn from these experiences? 3. Complete each of the following. E xplain your responses. Why do you think you feel this way? I enjoy or feel positiv e about mathematics because … and/or … I do not enjoy or I feel negative about mathematics because … 4. Describe in detail one experience from your past that is particularly memorable and influential in your attitudes about ma thematics. Where were you? Who was there? What was said? What did you do? How did you feel? 5. Many students have low self-confiden ce when it comes to mathematics. What will you do as a teacher to boost the self -confidence of your students regarding mathematics? 6. What do you think are the qualities of the best mathematics teacher you have ever had? What effect did this teacher have on you as a learner of mathematics? 7. What do you think are the qualities of th e worst mathematics teacher you have ever had? What effect did this teacher ha ve on you as a learner of mathematics? 8. Discuss the use of reflective j ournals in this course. What benefits, if any, did they provide? What, if any, were the drawbacks?

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331 Appendix C (Continued) Professional Journal Abstract Summary 1. Select two articles to read and review. The article s should come from either Teaching Children Mathematics or the Arithmetic Teacher. Both journals are available in the University Library (2nd floor). Many articles from these journals are also available online through NCTM’s website. 2. Both articles should deal with a topic from the content of MAE 4310 (problem solving, place value, number sense, whole numbers, fractions, decimals, percents, proportions, estimati on). If you are not sure an article meets this requirement, PLEASE ASK ME FIRST!! 3. Write a 1-2 page paper for each article. a. Summarize the article in your own words. Quotes should be clearly marked as such and page referenc es should be given. Your summaries should contain the essence of each article in a broad sense. b. Provide a critique of the articles This is your personal reaction. Did you like the article? Why or why not? To what extent do you think the article is usable in the elementary classroom? Justify your opinion. c. Provide a bibliographic citation of each article. This should include the name of the article, author’s name, na me of the journal, volume number of the journal, year published, page numbers of the article. d. Your paper should be typed, double-spaced, 12 point font. 4. Grading (for each article) 18 20 points The abstract contains all the essential features. The summaries are clea r and the personal reaction is well justified. Your essay provides a reasoned opinion. The writing flows well, observing proper spelling a nd appropriate grammar. 16 – 17 points At most one essentia l element is missing. The summaries are clear but th e personal reaction is weak or not well justified. The essay lacks a flow of logic. The writing contains a small number of spelling or grammar errors. 14 – 15 points At most two essential features are missing. The summaries are uncle ar or missing important informa tion. The personal reaction is weak or not well justified. The writing contains awkward flow with numerous spelling or grammar errors. If a grade of at least 14 points is not warranted, the assignment will be returned to you for resubmission. A penalty may be assessed should this be necessary.

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332 Appendix C (Continued) Literature Based Lesson Plan 1. Each person or group (no more than four in a group) will write a literature based lesson plan that f eatures a topic from this course. You may work together and write one lesson plan for the group or you may work alone. 2. Please choose from the following topics: Number Sense, Counting, Addition, Subtraction, Multiplication, Division, Pr oblem Solving, Fractions, Estimation NOTE: Please ask me if you are not sure about a topic. Lesson plans that deal with any topics other than those listed above will not be accep ted unless you have checked with me first. 3. The plan should include: a short overview of a children’s book that deals with the chosen topic objectives for the lesson a list of materials for the lesson a description of the mathematics based activities (at least 2) that the elementary school students will be doing Indicate the target grade leve l for the activities you design. Explain how you will evaluate the lesson. ESOL modifications 4. Your lesson plan should be typed and should include a complete bibliographic cita tion. If you use any ideas that are not your own, please cite your sources. Points will be deducted for an y missing components listed above.

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333 Appendix D: Textbook Title Page, Copyri ght Page, and Table of Contents (Used with Permission of Pren tice Hall Publishing Company)

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346 Appendix E: ATTITUDES TOWA RD MATHEMATICS INVENTORY Demographic Information Name ______________________________ Age: (check one) ____ 18-22 Gender: ____ Male ____ 23-27 ____ Female ____ 28-32 ____ 33-37 ____ 38 and over Mathematics courses taken since high school: Course Institute Grade Received Which education courses (if any) have you completed? Course Institute Year Are you part of a cohort? _______________

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347 Appendix E (Continued) ATTITUDES TOWARD MATHEMATICS INVENTORY Directions : This inventory consists of statements about your attitude toward mathematics. There are no correct or incorre ct responses. Read each item carefully. Please think about how you feel about each item. Circle the choi ce that most closely corresponds to how the statements best describe your feelings. Us e the following response scale to respond to each item. Complete your responses for all 40 statements. 1. Mathematics is a very worthwh ile and necessary subject. Strongly Disagree Disagree Neutral Agree Strongly Agree 2. I want to develop my mathematical skills. Strongly Disagree Disagree Neutral Agree Strongly Agree 3. Mathematics helps develop the mind and teaches a person to think. Strongly Disagree Disagree Neutral Agree Strongly Agree 4. Mathematics is important in everyday life. Strongly Disagree Disagree Neutral Agree Strongly Agree 5. Mathematics is one of the most im portant subjects for people to study. Strongly Disagree Disagree Neutral Agree Strongly Agree 6. Math courses would be very helpfu l no matter what grade level I teach. Strongly Disagree Disagree Neutral Agree Strongly Agree 7. I can think of many ways that I use math outside of school. Strongly Disagree Disagree Neutral Agree Strongly Agree 8. I think studying advanced mathematics is useful. Strongly Disagree Disagree Neutral Agree Strongly Agree

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348 Appendix E (Continued) 9. I believe studying math helps me with problem solving in other areas. Strongly Disagree Disagree Neutral Agree Strongly Agree 10. A strong math background could he lp me in my professional life. Strongly Disagree Disagree Neutral Agree Strongly Agree 11. I get a great deal of satisfaction out of solving a mathematics problem. Strongly Disagree Disagree Neutral Agree Strongly Agree 12. I have usually enjoyed studying mathematics in school. Strongly Disagree Disagree Neutral Agree Strongly Agree 13. I like to solve new problems in mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree 14. I would prefer to do an assignment in math than to write an essay. Strongly Disagree Disagree Neutral Agree Strongly Agree 15. I really like mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree 16. I am happier in a math cla ss than in any other class. Strongly Disagree Disagree Neutral Agree Strongly Agree 17. Mathematics is a very interesting subject. Strongly Disagree Disagree Neutral Agree Strongly Agree

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349 Appendix E (Continued) 18. I am comfortable expressing my own ideas on how to look for solutions to a difficult problem in math. Strongly Disagree Disagree Neutral Agree Strongly Agree 19. I am comfortable answering questions in math class. Strongly Disagree Disagree Neutral Agree Strongly Agree 20. Mathematics is dull and boring. Strongly Disagree Disagree Neutral Agree Strongly Agree 21. Mathematics is one of my most dreaded subjects. Strongly Disagree Disagree Neutral Agree Strongly Agree 22. When I hear the word mathematic s, I have a feeling of dislike. Strongly Disagree Disagree Neutral Agree Strongly Agree 23. My mind goes blank and I am unable to think clearly when working with mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree 24. Studying mathematics ma kes me feel nervous. Strongly Disagree Disagree Neutral Agree Strongly Agree 25. Mathematics makes me feel uncomfortable. Strongly Disagree Disagree Neutral Agree Strongly Agree 26. I am always under a terrible strain in a math class. Strongly Disagree Disagree Neutral Agree Strongly Agree

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350 Appendix E (Continued) 27. It makes me nervous to even think about having to do a mathematics problem. Strongly Disagree Disagree Neutral Agree Strongly Agree 28. I am always confused in my mathematics class. Strongly Disagree Disagree Neutral Agree Strongly Agree 29. I feel a sense of insecurity when attempting mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree 30. Mathematics does not scare me at all. Strongly Disagree Disagree Neutral Agree Strongly Agree 31. I have a lot of self-confidence when it comes to mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree 32. I am able to solve mathematics problems without too much difficulty. Strongly Disagree Disagree Neutral Agree Strongly Agree 33. I expect to do fairly well in any math class I take. Strongly Disagree Disagree Neutral Agree Strongly Agree 34. I learn mathematics easily. Strongly Disagree Disagree Neutral Agree Strongly Agree 35. I believe I am good at solving math problems. Strongly Disagree Disagree Neutral Agree Strongly Agree 36. I am confident that I could learn advanced mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree

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351 Appendix E (Continued) 37. I plan to take as much mathematics as I can during my education. Strongly Disagree Disagree Neutral Agree Strongly Agree 38. The challenge of math appeals to me. Strongly Disagree Disagree Neutral Agree Strongly Agree 39. I am willing to take more than the required amount of mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree 40. I would like to avoid teaching mathematics. Strongly Disagree Disagree Neutral Agree Strongly Agree Martha Tapia1 1 ATMI used with permission of author

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352 Appendix F: Researcher’s Journal Responses The following are samples of journals for each prompt and the researcher’s responses to those journals. All of these sample journals come from the four participants who had the largest positiv e changes in their attitude survey scores. Journal 1: Discuss any feelings (positive or nega tive) that you have about taking this course. What are you hopi ng to gain from the course? Participant’s Journal Entry: I am, in a way hesitant to take this cla ss because I know that math has had a big influence in my educational career. I have grow n to be not fond of it, so it is very hard to look at math in a positive manner I think that throughout my schooling years, especially high school and includi ng college, I am intimidated by math and all of its subtopics. Going into the class, I feel that it is ju st hard trying to teach my students something that I am both not particularly interested in and do not like. I do know however, that math is a ve ry important part of our daily lives and simple, common, and basic math procedures do need to be taught and learned. So, by the end of this course, I would like to gain several things. First, I would like to gain a new and positive outlook on math as a subject. I want to be able to look forward to teaching all the subjec ts to my students and this definitely includes math. Also, my outlook on cert ain topics or subjects are picked up on and mocked by students. If this is the ca se, I do not want t hem to also gain a negative experience from math like I did. Secondly, from this course I would like to discover and develop new and improved ways to teach math. I want it to be fun for my students to learn. By in cluding manipulatives or other hands-on activities, they will be able to interact with math instead of just acquiring new information for memorization. Researcher’s Response: Your feelings and experiences with math are not uncommon. I hear these things frequently from students. I th ink it is very important that you and your classmates, as future teachers, focus on ways to make math fun, interesting and relevant for your students so that they will feel that they can do math and can enjoy doing math. We will be talking about these issues throughout the course, and hopefully you will leave with lots of ideas on how to accomplish this. I am glad to hear that you want to improve your attitudes toward math so that you don't pass on negative attitudes to your students. That is certainly a problem that happens sometimes. I hope that this course will help you do that.

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353 Appendix F (Continued) Journal 2: What are your memories of learning mathem atics in elementary school (attitudes, success, etc.)? What can you, as a future teacher, learn from these experiences? Participant’s Journal Entry: In elementary school I loved math. I was ve ry good at it and wasn’t afraid to let everyone know it. My teachers, for the most part, used creativ e ways to facilitate my learning. In elementary, there we re some worksheets, but mostly fun activities. When I got in high school, it got a bit harder and I started to shut down from wanting to learn more math. I hope t hat as a future teac her, I will be able to use creative activities with my students. I th ink it is important to make sure that the children have fun while they learn. I know it made a big difference for me in my younger years. I want to make t he same difference for my students. Researcher’s Response: Many kids do well in elementary school and then begin to have trouble when they get to middle and high school. That is one of the reasons why I believe that it is so important to introduce topics such as geometry and algebra in elementary school. You will learn more about teaching geometry and algebraic thinking in Math II. By exposi ng the kids to these concepts early, maybe they won't have such a hard time when they get to middle and high school. I agree with you that it is im portant to make the math fun for kids. This can go a long way towards building positive attitudes toward math in students. Journal 3: Complete each of the following. Ex plain your responses. Why do you think you feel this way? I enjoy or feel positive about mathematics because … and/or … I do not enjoy or I feel negative a bout mathematics because … Participant’s Journal Entry: I feel positive about mathematics because I have struggled with it in the past. I know this sounds funny, however, if I am going to teach math I need to have a positive attitude about it. I belie ve now is the time to turn my attitude toward math around. Children are able to recognize when you are uncomfortable with a subject. I feel when they do recognize this it in turn, makes them uncomfortable and that much harder to teach. I can't wait to start teaching math because I feel the best way to overcome my attitude is to dive right into it!

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354 Appendix F (Continued) Researcher’s Response: I'm glad to hear that you want to improve your attitude toward math. I agree with you that this is very importan t. I think that teachers need to try and make math class fun and releva nt for their students. They need to encourage students to solve proble ms in ways that make sense to the student, even if it's not the way the teacher solves it. I hope that this course will help you begin to see math in a different light. It is important for teachers to feel positive about the su bjects they teach so they don't pass on negative attitudes to their students. Journal 4: Describe in deta il one experience from your past that is particularly memorable and influential in your attitudes about mathematics. Where were you? Who was there? What was said? What did you do? How did you feel? Participant’s Journal Entry: An experience I had in math was when I wa s in elementary school. I remember [teacher’s name] was the nicest teacher ever I remember every time she told us to get out our math books I would f eel embarrassed because I knew we were about to start something that I wasn't very good at. I k new [teacher’s name] could tell that math made me uncomfort able because I was so eager with all the other subjects and I clammed up when it ca me to math. One day she took me aside and said, [Student’s name], I c an tell you are uncomfortable with math time but I knew that you are smart and I be lieve that you can really achieve a lot if you put your mind to i t. I want to help you succeed, and I will do whatever it takes to help you get there." After [teacher ’s name] said that I knew that there was no reason for me to feel uncomfort able because I was smart and if she had that much faith in me I must really be something. Researcher’s Response: WOW!! What a great story. It brin gs out a very important point that teachers of all grade levels should re member. A student will probably live up to (or down to) the teacher's expectat ions. If you let a kid know that you think they can do the math, they probably will be able to do it because your confidence in them will give them self -confidence. Unfortunately, it often works the other way too. We as teachers need to let our students know that we have confidence in them and thei r ability to do math. You are very

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355 Appendix F (Continued) fortunate to have [teacher’s name] as a role model for being this kind of teacher. You know firsthand what an impact it can have on a student. Journal 5: Many students have low self-confid ence when it comes to mathematics. What will you do as a teacher to boost the self -confidence of your students regarding mathematics? Participant’s Journal Entry: I agree with the statement t hat children grow to have a low self-confidence when it comes to math because I am living pr oof of those childr en. I like many children, grow to either not like mathem atics as a whole or become intimidated by it entirely. This actually is not a good thing and needs to be corrected as soon as possible. I feel that it is up to the teachers now to change their methods of teaching to try to benefit the children and their attitudes about math, rather than hurt them. I also believe that it is the responsibi lity of the up and coming teachers to already have knowledge of this teaching method and use it constantly in their classrooms, especially math lessons. As an up and coming teacher I plan to use several techniques and strategies to make math more enjoyable for my students. Not only do I plan on making my math lessons fun and interesting, I plan on using as many manipulatives as [much as] possible. The more opportunities children get to wo rk with their hands and touch objects to make connections, the easier it will be for them to grasp and understand difficult mathematical concepts. Also in my ma th classroom, I am going to incorporate other subjects into several lessons. Fo r example, art is a great way to allow children to express their knowledge. It is so wonderful because not just one type of art could be used. There is visual, movement/dance, drama, and music. For example, math is all about numbers and ge tting the right beat just like music. These are only a few of my ideas for maki ng math a less horrible subject. It is not horrible and this is what we need to get the children to realize. It can be fun, interesting, and quite exciting. Hopefully by using these forms of instruction, I can change the view points of most of my students in the future. Researcher’s Response: It sounds like you have some great ideas about how to increase your students' self-confidence with math. Ma king math fun and interesting, using manipulatives, and connecting math to other subjects are excellent ways to accomplish this goal. Ultimately we wa nt all students to feel confident in their ability to do math.

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356 Appendix F (Continued) Journal 6: What do you think are the qualities of the best mathematics teacher you have ever had? What effect did this teacher have on you as a learner of mathematics? Participant’s Journal Entry: When I read this subject I got really ex cited because the best math teacher I’ve had accrued last year in college. I took social science stats, for my third math course, by this time i already took finite and liberal arts twice for grade forgiveness. Needless to say the last thi ng I wanted to do was take another math class. A friend recommended [professor’s nam e] so I thought what the heck, here goes another two semesters for the same ma th class. To my surprise he was amazing teacher, and I got an A in the course My first ever A in a math class. The things that made him so amazing was t hat he really took ti me to make sure we understood what was going on (even if he had to explain things 5 times and 5 different ways.) He always found a way for us to understand what he was trying to teach. He always made it a point to rela te topics to real life and show why it is important to learn. He also set up a buddy group system with his past students. These students were TA's who took time out of there [sic] days to meet up with us to review material. Th is helped tremendously because if you didn't get it in class it wasn't too late because someone else would be there to help you. He understood the needs of his students and provided the bes t learning environment for us. Still to this date I’ve turned ar ound my view on math. It even shows this semester in your class. I’ve received 2 A's on my test so far, and I really feel like a lot of that is a direct re sult from [professor’s name]. Researcher’s Response: I definitely agree with you about the impact one teacher can have on the attitudes of his or her students. The qualities that you mentioned for your best math teacher are qualities that I hope you will remember when you begin teaching. Making sure students are understanding the concepts, making the math relevant to students lives, and encouraging students to work together are great ways to be th e kind of teacher that students will recall as one of the best they've had. Journal 7: What do you think are the qualitie s of the worst mathematics teacher you have ever had? What effect did this teacher have on you as a learner of mathematics?

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357 Appendix F (Continued) Participant’s Journal Entry: As sad as this sounds I have had many more bad teachers than good. Or maybe it’s just that the bad time s stick with me more than the good. What all of the teachers have in common is that they a ll seem to rush through curriculum. I always felt like I was not smart becaus e they would move on so fast from concept to concept without properly assessing where I was. I feel that they just wanted to finish the book to say they finished the book. I really hope that I will never be a teacher who rushes my students. I want all of my future students to feel like they can take their time to understand the concept. I feel that if these teachers were to take their time with each student the outcome would be great. Researcher’s Response: Unfortunately, teachers have lots of pressures on them. One of these is covering a certain amount of material in a certain amount of time. However, I agree with you that rushing through it is not a good idea. Students need to make sense of the math before they move on to something new. If the majority of the class does understand a concept, then I believe that the teacher should find a way to work wi th those who don't. This could be before or after school or while others are workin g on another problem or activity. I know that you will be sens itive to students in this situation because of your own experiences. Journal 8: Discuss the use of reflective journals in this cour se. What benefits, if any, did they provide? What, if any, were the drawbacks? Participant’s Journal Entry: I have really enjoyed doing these reflective jo urnals. I think that they really help us, as students, to look back on our pas t math experience and see what we can learn from them. As futu re educators, our students ar e going to be in the same position we were and go through the exact same (or different) experiences we did. Looking back on our good and bad profe ssors, we now know what to do and not do in our own classrooms. Our goal is to allow the students to get a positive outlook and attitude towards math and not a negative one like most of us did growing up. Also through these math j ournals, I have been able to see where I stand on the topic of math and use this for my classroom as well. I have really and truly learned a lot about myself, my future, and my fu ture students and classroom through these reflective journal s. Thank you for this opportunity and I will use my acquired knowledge to the best of my ability.

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358 Appendix F (Continued) Researcher’s Response: I am glad that you have enjoyed th e journals. I think that it is very important for future math teachers to reflect on their own experiences as learners of mathematics, as well as ot her important issues they will face as teachers. As you said, this type of re flection can help you achieve the goal of developing positive attitudes toward math in your st udents, and I think that is wonderful!!

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359 Appendix G: Observer Protocol In observing the class and r ecording your observations of bot h instructional activities and students’ activities, please make special no te of any observation that might reflect students’ attitudes toward mathematics. These attitudes should include value of mathematics, enjoyment of mathematics, motivation for mathematics, and selfconfidence with mathematics. Observed Activity and Approximate Time Spent Personal Insights, Interpretations

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360 Appendix H: Experiences with Ma thematics Interview Protocol Experiences that have Influenced Attitudes 1. Why did you decide to become a teacher? 2. Tell me about your own experiences as a student in mathematics classrooms in elementary school. 3. Describe your feelings about the methods course prior to the start of the course. 4. Have your feelings about the course change d since the start of the methods class? (a) If so, how have they changed? What has influenced the change? (b) If not, what (if anything) has reinforced these feelings? 5. Tell me about your own experiences as a student in mathematics classrooms in middle school. 6. What are some things that a teacher can do to help his/her students develop a good attitude toward mathematics? 7. Tell me about your own experiences as a student in mathematics classrooms in high school. 8. How did you do on the CLAST test? Do you remember your score? How many times did you take it before passing it? 9. Tell me about your own experiences as a student in mathematics classrooms in college. 10. What else would you like me to know concerning your attitudes toward mathematics?

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361 Appendix I: Changed Attit udes Interview Protocol Effect of Course on Attitudes 1. The survey that you recently completed in class showed a cha nge in your attitudes toward mathematics since the start of th is course. How do you think your attitudes toward mathematics have changed? 2. Which aspects of the course do you think affected your attitudes toward mathematics? How did it affect your attitudes? Ask remaining questions only if that aspect of the course has not been mentioned. 3. How do you feel about the use of mani pulatives in teaching mathematics? 4. How do you feel about the use of manipulatives in this course? 5. How do you feel about the use of coopera tive learning in teaching mathematics? 6. How do you feel about the use of cooperative learning in this course? 7. How do you feel about the use of probl em solving in teaching mathematics? 8. How do you feel about the use of problem solving in this course? 9. How do you feel about the use of journals in teaching mathematics? 10. How do you feel about the use of journals in this course? 11. Is there anything else you would like to tell me about your attitude change or to what you attribute this change?

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362 Appendix J: Final Examination The departmental final examination consists of 50 multiple-choice items. Due to security issues, a desc ription of the test items has been included rather than the actual exam questions. 1. Use pattern blocks to represent a fraction. 2. Represent a portion of a rect angular region by a decimal. 3. Identify strategies for le arning basic addition facts. 4. Recognize appropriate invented strategies for multiplication. 5. Using given digits, find the largest whol e number possible, given that one of the digits must hold a specified place value. 6. Use pattern blocks to re present fraction subtraction. 7. Given a number in exponential expanded fo rm, identify the number in standard form. 8. Determine properties of place value. 9. Recognize when a particular esti mation strategy is appropriate. 10. Identify interpretations for subtraction. 11. Identify estimation methods. 12. Recognize under which operations the set of Natural Numbers is closed. 13. Identify an example of the Cardinality Principle. 14. Model a division problem with fraction circles. 15. Identify the product of two fracti ons geometrically on a number line. 16. Identify methods of diagnosing student errors.

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363 Appendix J (Continued) 17. Identify appropriate ways to introduce mathematical ideas to elementary school students. 18. Diagnose a student error involving writ ing a number represented by Base Ten Blocks. 19. Identify a property of multiplication geometrically. 20. Identify properties of equivalent fractions. 21. Determine which rational number best id entifies a point marked on a number line. 22. Identify instructional approaches involving invented strategies. 23. Identify a property about unit fractions. 24. Identify fractions that have a terminating decimal equivalent. 25. Given a sample of a student’s wo rk, diagnose the student’s problem. 26. Identify an anticipated issue for students who are deve loping subtraction algorithms. 27. Given the beginning of a word problem involving subtraction of fractions, identify an appropriate question. 28. Given a word problem for subtraction, identify an appropriate interpretation. 29. Given a particular manipulative, id entify which type of fraction model it represents. 30. Make judgments about appropriate estimation strategies. 31. Use pattern blocks to represent a given fraction. 32. Identify which fraction, decimal, and pe rcent expressions are equivalent to a given fraction.

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364 Appendix J (Continued) 33. Identify appropriate models for representing decimals. 34. Given a pictorial model, identify which operation or operations are being modeled. 35. Given a word problem involving frac tions, choose the appropriate operation needed to solve it. 36. Use pattern blocks to model fractions. 37. Choose which manipulatives would be us eful to model ordering of fractions. 38. Use pattern blocks to model fractions. 39. Choose which of five decimal numbers has the greatest value. 40. Model a fraction subtraction probl em with Fraction Tower pieces. 41. Given a scenario involving estimation, id entify which estimation strategy was used. 42. Given five subtraction word problems, id entify which model or interpretation of subtraction is being used. 43. Given a scenario involving an elementary school class activity, identify which type of learning activity is taking place. 44. Given a word problem involving fractions choose a number sentence that can be used to solve it. 45. Diagnose a student’s error involving fraction concepts. 46. Diagnose a student’s error i nvolving a fraction set model. 47. Identify which fraction operation is illustrated by an area model. 48. Identify an appropriate purpose for using a specific manipulative.

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365 Appendix J (Continued) 49. Diagnose a student’s error with two-digit addition. 50. Identify the problem that is be ing modeled on a decimal grid.

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366 Appendix K: Summary of Hycner’s Guidelines for the Phenomenological Analysis of Data Hycner’s (1985) guidelines for phenomenol ogical analysis of interview data include the following 15 steps: 1. Transcription. This involves not only the literal statements, but also non-verbal and para-linguistic communication. 2. Bracketing and phenomenological reduction. This involves listing and then suspending the researcher’s own presuppositions. 3. Listening to the interview for a sense of the whole. This requires listening to the tape and reading the tr anscript several times. 4. Delineating units of general meaning. The researcher reviews each word, phrase, and sentence in order to iden tify unique units of meaning. 5. Delineating units of meaning rel evant to the research question. The researcher reviews each unit of meaning in order to determine whether it responds to or illuminates the research question. 6. Training independent judges to verif y the units of relevant meaning. This adds reliability. 7. Eliminating redundancies. Previously listed units of meaning can now be eliminated, although their fr equencies should be noted. 8. Clustering units of meaning. This involves grouping together those units of meaning that naturally cluster together.

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367 Appendix K (Continued) 9. Determining themes from clusters of meaning. The researcher examines the clusters of meaning and determines if th ere are one or more central themes that express the essence of the clusters. 10. Write a summary for each individual. The summary should incorporate the identified themes. 11. Return to the participant w ith the summary and themes. The researcher conducts a second interview with the participant so that the participant can determine whether the essence of the first in terview has been accurately captured. 12. Modify themes and summary. If needed, modifications can be made based on any new data that was collected from the second interview. 13. Identify general and unique them es for all the interviews. The researcher looks for themes that are common to most or all of the participants as well as individual variations. 14. Contextualization of themes. The researcher may find it helpful to then place these themes back into their original contexts. 15. Composite summary. A composite summary is then written that accurately captures the essence of the phenomenon being investigated.

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368 Appendix L: Computation of Inter-Rater Reliability Pilot Study Five journal entries from Pilot Study I were used for training and determining inter-rater reliability between the research er and the coder in identifying units of meaning. The researcher identified 40 units of meaning from the five journals, and the coder identified 33 units of meaning. Together they identified six common themes. Prior to collaboration, a comparison of identified units of meaning for the five journals produced an 83% inter-rater reliability. While comparing the researcher’s identif ied units of meaning with those of the coder, a pattern emerged. In nearly every cas e, one coder had interpreted a statement as one unit of meaning, and the other coder ha d viewed it as two units of meaning. The following journal excerpt is an example of this tendency: “ I do remember [math] being one of my least favorite and weakest subjects in school.” The researcher had coded this as two units of meaning: math was one of least favorite subjects, and math was one of weakest subjects. The coder had coded this statement as only one unit of meaning: pa rticipant felt unsuccessful with math. After a brief discussion, both the researcher a nd the coder agreed to consider this excerpt as two units of meaning: math was one of least favorit e subjects, and math was one of weakest subjects. The researcher and th e coder agreed that this stat ement should be categorized as two instances of a negative memory of el ementary school. After collaborati on between the researcher and the coder, 100% agreemen t was reached on both units of meaning and common themes.

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369 Appendix L (Continued) Dissertation Study The researcher and the coder independen tly identified both units of meaning and common themes on a sample of 18 journals from one prompt. Journal One was randomly selected for double coding. A stratified random sample of responses to this prompt was chosen using subgroups based on the particip ants’ ages and lengths of response. The researcher identified 165 units of meaning a nd 11 themes from the 18 journals, and the coder identified 129 units of meaning and 12 themes. The researcher and the coder collaborated to identify 169 units of meaning and 11 themes. Prior to collaboration, a comparison of identified units of meaning for the 18 journals produced an overall inter-rater reliability of 71.6%. Differences fell into five categories. Both the researcher and the coder agreed that the first category was an ‘actual miss,’ and the other four categories were ‘cond itional misses.’ The five categories were: Actual Miss: One coder identified a unit of meaning and the other did not R-2/C-1: Researcher separated the excerpt into two units of meaning and coder kept it together and identified only one R-1/C-2: Researcher kept the excerpt t ogether and identified only one unit of meaning and coder separated the exce rpt into two units of meaning DC: Researcher double-coded an excerpt. This means that the entire excerpt was coded as two different units of meaning. The coder did not realize that she could do this, so she did not do any double coding.

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370 Appendix L (Continued) Sub-theme: Researcher and Coder both categorized unit of meaning as same theme, but differed on sub-theme There were 17 ‘actual misses’ where one coder identified a unit of meaning and the other did not Examples of the four ‘conditional misses’ and their frequencies are found in Table K-1 below. Table K-1 Inter-rater Reliability: Conditional Misses Type of Miss Frequencya Representative Journal Excerpt R-2/C-1: Researcher: 2 units of meaning Coder: 1 unit 15 “I have never had a problem with math and have always liked it.” R: Separated and coded as two instances of Pos. Attitude Toward Math (never had a problem and always liked) C: Coded as one instance of Pos. Attitude Toward Math (liked math) R-1/C-2: Researcher: 1 unit of meaning Coder: 2 units 4 “I am very apprehensive about taking this course, considering my math skills aren't the greatest.” R: Coded as Neg. Attitude about Course (apprehensive) C: Separated and coded as Neg. Attitude about Course (apprehensive) and also Neg. Feelings about Self with Math (poor math skills) Continued on the next page

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371 Appendix L (Continued) Table K-1 (Continued) Inter-rater Reliability: Conditional Misses Type of Miss Frequencya Representative Journal Excerpt DC: Researcher: Doublecoded excerpt Coder: Coded excerpt only once 6 “If I can learn how to teach my students and have them really learn and enjoy the s ubject that idea makes me so happy.” R: Coded twice: Once as Pos itive Attitude about Course (happy about idea of learning how to have students learn and enjoy math) and also as Hope to Gain from Course (help students le arn and enjoy math) C: Coded only once: Hope to Gain from Course (helping children learn and enjoy math) Sub-theme: Researcher and Coder identified same theme, but different sub-theme 6 “I am eager to learn to adapt to different ways of thinking.” Both coded as Hope to Ga in from Course, but coded differently as sub-theme. R: Sub-coded as: Hope to Gain: Accommodate Different Learning Styles and Needs (lea rn to adapt to different ways of thinking) C: Sub-coded as: Hope to Gain: Regaining Personal Understanding of Math (adapt to new thinking) a n = 31

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372 Appendix L (Continued) It should be noted that 31 of the 48 overa ll ‘misses’ (65%) fell into the ‘conditional misses’ category. Counting only th e ‘actual misses’ produced an inter-rater re liability of 90%.

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373 Appendix M: Researcher’s Possible Biases and Preconceptions Researcher’s Assumptions: Preservice elementary school teache rs are capable journal writers. Participants will be willing to share th eir experiences with me honestly and openly through their jour nals and interviews. Participants will take time to reflect on each of the journal prompts, survey items, and interview questions and honestly shar e their attitudes to ward mathematics rather than responding in the manner in wh ich they think I want them to respond. Participants will be able to remember signi ficant events that have influenced their development of attitudes toward mathematics. Researcher’s Expectations: Although many of the preservice teachers wi ll have positive attitudes toward and experiences with mathematics, there will also be many with negative attitudes and experiences. The majority of the participants will have a positive view of journal writing in the methods course. Participants with positive attitudes to ward mathematics will have experienced success with mathematics, and those with negative attitudes will have been largely unsuccessful with mathematics.

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374 Appendix N: Post-Course Individual Survey Items Table N-1 Means and Standard Deviations on Items from the Post-Course Attitudes Toward Mathematics Inventory Item Mean Standard Deviation SkewnessKurtosis Value of Mathematics 1. Mathematics is a very worthw hile and necessary subject. 4.58 0.56 -0.88 -0.20 2. I want to develop my mathematical skills. 4.58 0.50 -0.32 -2.02 3. Mathematics helps develop the mind and teaches a person to think. 4.42 0.61 -0.56 -0.52 4. Mathematics is important in everyday life. 4.33 0.69 -1.16 2.57 5. Mathematics is one of the most important subjects fo r people to study. 3.94 0.83 -0.59 0.17 6. Math courses would be very help ful no matter what gr ade level I teach. 4.39 0.61 -0.45 -0.58 7. I can think of many ways that I use math outside of school. 4.39 0.70 -1.31 2.69 8. I think studying advanced mathematics is useful. 3.30 1.07 -0.34 -0.97 9. I believe studying math helps me with problem solving in other areas. 4.15 0.83 -0.99 0.94 Continued on the next page

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375 Appendix N (Continued) Table N-1 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Value of Mathematics (Continued) 10. A strong math background could he lp me in my professional life. 4.15 0.76 -0.73 0.66 Enjoyment of Mathematics 11. I get a great deal of satisfaction out of solving a mathematics problem. 3.64 1.06 -0.89 0.75 12. I have usually enjoyed studying mathematics in school. 2.97 1.16 0.06 -0.90 13. I like to solve new problems in mathematics. 3.45 0.97 -0.41 0.03 14. I would prefer to do an assignment in math than to write an essay. 2.61 1.58 0.40 -1.48 15. I really like mathematics. 3.45 1.09 -0.26 -0.69 16. I am happier in a math class th an in any other class. 2.55 1.20 0.52 -0.34 17. Mathematics is a very interesting subject. 3.52 0.91 -0.72 0.73 Continued on the next page

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376 Appendix N (Continued) Table N-1 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Enjoyment of Mathematics (Continued) 18. I am comfortable expressing my own ideas on how to look for solutions to a difficult problem in math. 3.48 1.23 -0.39 -0.84 19. I am comfortable answering questi ons in math class. 3.58 1.15 -0.66 -0.19 20. Mathematics is dull and boring.* 3.58 1.00 -1.02 1.14 Self Confidence with Mathematics 21. Mathematics is one of my most dreaded subjects.* 3.24 1.23 -0.17 -1.20 22. When I hear the word mathematics, I have a feeling of dislike.* 3.30 1.10 0.09 -1.38 23. My mind goes blank and I am unable to think clearly when working with mathematics.* 3.82 0.92 -0.91 0.28 24. Studying mathematics makes me feel nervous.* 3.36 1.19 -0.42 -0.92 25. Mathematics makes me feel uncomfortable.* 3.64 1.08 -0.77 -0.23 Continued on the next page

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377 Appendix N (Continued) Table N-1 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Self Confidence with Mathematics (Continued) 26. I am always under a terrible strain in a math class.* 3.85 1.00 -1.45 2.55 27. It makes me nervous to even thin k about having to do a mathematics problem.* 3.76 1.09 -1.18 1.01 28. I am always confused in my mathematics class.* 3.85 0.97 -1.20 1.54 29. I feel a sense of insecurity when attempting mathematics.* 3.48 1.15 -0.42 -0.93 30. Mathematics does not scare me at all. 3.15 1.09 -0.01 -1.14 31. I have a lot of self-confidence when it comes to mathematics. 3.24 1.12 0.20 -1.39 32. I am able to solve mathematics problems without too much difficulty. 3.58 0.94 -0.23 -0.71 33. I expect to do fairly well in a ny math class I take. 3.55 0.90 -0.55 -0.55 Continued on the next page

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378 Appendix N (Continued) Table N-1 (Continued) Individual Survey Items Item Mean Standard Deviation SkewnessKurtosis Self Confidence with Mathematics (Continued) 34. I learn mathematics easily. 3.12 1.17 -0.25 -0.90 35. I believe I am good at solving math problems. 3.45 1.09 -0.72 -0.14 Motivation with Mathematics 36. I am confident that I could learn advanced mathematics. 3.12 1.27 -0.24 -1.11 37. I plan to take as much mathematics as I can during my education. 2.48 1.06 0.37 -0.46 38. The challenge of math appeals to me. 2.97 1.16 0.32 -0.87 39. I am willing to take more than the required amount of mathematics. 2.39 1.12 0.71 0.10 40. I would like to avoid teaching mathematics.* 3.91 0.77 -0.29 -0.15 Note. Martha Tapia. ATMI used with permission of author. Scoring for most items uses anchors of 1: strongly disagree, 2: disagree, 3: neutral, 4: agree, and 5: strongly agree. *Scoring for these items is reversed and uses anchors of 1: strongly agree, 2: ag ree, 3: neutral, 4: disagree, 5: strongly disa gree. Therefore, on all items, scores range from 1 to 5, with 1 indicating the most negative attitude and 5 indicating the most positive attitude.

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379 Appendix O: Relevant Excerpts from Researcher’s Journal The following excerpts from the researche r’s journal are relevant to the study. The portions relating to the re searcher’s observations, actions or thoughts that reflected consideration of participants’ attitudes towa rd mathematics are highlighted in bold font. Please note that the data written in brackets re flect the researcher’s personal reactions and thoughts. Week 1: When I first walked into the room, about twothirds of the class wa s there. The class has (presently enrolled) 34 students, and there were about 22-23 there already. When I first got there, the class was very quiet, not rea lly talking to each other very much. I came in and started talking to those in fr ont very casually abou t the heavy traffic on first week of classes, parking situation, etc. [trying to establish rapport and put students at ease] While they were introducing each other, on e of them said that her partner whom she was introducing hated math. [I was delighted --great way to bring up ATM]. After we finished introductions, I asked how many really liked math. About six or seven raised their hands. Then I asked ho w many really disliked math; I would say that at least 20 raised their hand. [I th ought … GREAT!! Good class for the study. They are already willing to be open about it .] I asked if any of them would be willing to share their reasons why they disliked math, and several immediately raised their hands. The first to raise her hand said that she didn’t like math because she wasn’t good at it. She had always struggled with it. One person said that she disliked it because there’s always only one right answer, and if you don’t get that answer and you don’t do it exactly the right way, then it’s wrong. I explained that in the course we would h ave a different perspective about that and we would learn more about that in the course. Another student said that she had always lik ed math until she got one teacher. I think it was HS Geometry … not sure. This teacher wa s very mean and would look for things to take points off and she turned her agai nst math. She never liked it after that. I pointed out how one teacher or one bad experience could have that effect. Another person said that she had always liked math until she got to [university] and then she had 3 or 4 professors for math classes and all of them were horri ble and turned her off to math.

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380 Appendix O: Continued Another one said that she had always hated math until she had a [university] professor (measurement course?) who was just absolutely wonderful. I asked her what he did that made him so different from the others she’ d had, and she said that they worked in groups, …. I can’t remember what else she sa id, but they were all things that we will be talking about and promoting in the course and I pointed that out to them. [I was thinking that they brought up some good things that I would probably see again in their journals-hoping this will get others thinking and relating.] I then gave them a problem to work, which was the pelicans and turtles problem.1 For the most part, they seemed to get right to it. There was one group of 3-4 who didn’t know how to work it right away, so they had just put it aside and were talking. Another group of 3 and then 2 who were sitting next to ea ch other … all 5 were sitting together, but 2 were working together and three were work ing together (includi ng the one male). He came up with a rather unusual way to solve the problem and I noticed that the females, the 2 who were working with him and the othe r 2, had seemed to just kind of give up what they were trying to do because it wasn’t working and just telling me that he had figured it out for them. [gender issues … females just assuming that the male was better at math than they were?] I also noticed that nobody had worked the pr oblem with a drawing. I realized that we were running out of time (5 min. left), so I asked who wanted to share their solution with the class. Three or four were willing to [very happy about that!!] which is not always the case. The first one actually came up to th e board and demonstrated [this was great to me]. She had used a guess and check method Students seemed surprised to learn that this was OK with me. Another one had made a chart listing each possibility in a very systematic way, which was also a grea t way to do it. A third one just started by dividing 33 in half and then ad justing from there, which was also a great way to do it. I wanted to show them the draw ing, but we ran out of time. [glad I got the opportunity early on to demonstrate the idea of using different solution strategies, sharing them with each other, cooperative group work.] Week 2: I asked about journals – if anyone had a problem emailing journals because sometimes they do. One did, but she had spok en to me before class, so no one raised their hand for that. [that is good … they so metimes get frustrated with the emailing when they have problems.] 1The problem asks students to figure out how ma ny pelicans and turtles there are if we have a total of 33 heads and 102 feet.

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381 Appendix O: Continued By about 9:05 it seemed like just about ev eryone was there, so I started class. As I began to start class, I remembered that I had received a couple of journals where students said that they were feeling overwhelmed with the amount of information in the first 5 chapters. When I brought this up, appr oximately 10 students agreed and said that they were feeling the same way. This was a good opportunity for me to point out where in the note packet the review questio ns were. We spent a few minutes talking about the idea that I wanted them to read a ll five chapters to ge t an overview of the philosophy of the course, but that as fa r as tests were concerned, they would be responsible primarily for things on the rev iew sheet. [I thought that students seemed relieved by that. I remember thinking that this type of communication is another benefit of journals. Something like that mi ght have never come up, but because they were writing that first journal about their concerns about the course, it did come up. This allowed me to address this issue with the whole class and probably alleviate the concerns of several students.] When I got to the transparency on problem-s olving strategies, I reminded them about the problem we did last week with the pelicans and turtles. As we went through the problemsolving strategies, I reminded them about how different people had shared their different solution strategies and how just about all of the strategies listed had been used and presented, with the exception of drawing a picture. I used this as an opportunity to show them a method of so lving this problem with a picture. They seemed to really enjoy that when they realized that a young child could have actually solved that problem that way. I them gave them the problem of using the numbers 1-6 on the triangle so that the sums of each side were equal. They seemed to be engaged … everyone I saw was actively involved in trying to solve the problem. After a few minutes, some still hadn’t come up with any of the four solutions, so I gave them the hint about the corners. I also suggested that those who had found just one solution look at the corners to give them a basis to theorize ab out the other solutions. One person put one of the solutions on the boa rd, and the class discussed it. Then I gave them some time to see if they c ould come up with the other solutions. Then volunteers put the other solutions on the board and we discussed them. [I was very happy about all the participation and involvement.] Then I read them the book Benny’s Pennies. They seemed to stay with me. There wasn’t any talking or anything like some times happens in some classes. So far it seems like a pretty good class as far as that is co ncerned… interest level and involvement. We talked about ideas about how you could plan a lesson around the book, and they had some very good ideas. I was glad that we discussed this, as I thought it might get them started thinking about their lesson plans they will be writing.

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382 Appendix O: Continued When I got to the overhead about Why Teach With Problems, one student raised her hand and she said that she wasn’t sure that she agr eed with the idea of i nvented strategies. She was questioning the idea of not teaching the traditional algorithm and wouldn’t it hurt the children if they never were taught it. I told th em that some people believe that they don’t ever have to teach it … that it’s just one of many ways to do it. I said that I could understand her perspective though because the kid could have a teacher in the future who could require him/her to do it th at way. I stressed though that th is should come later, after they’ve had a chance to invent their own al gorithms and it’s presented as just another way to do it, rather than THE way to do it. She seemed fairly satisfied with that response, but I wasn’t totally sure that she accepted what I had said. [I was glad she brought it up though because I want them to feel comfort able sharing their views even if they’re different from mine or the textbook’s.] Week 3: Classes were cancelled due to Hurricane Frances Week 4: When I got to class, I told the class that b ecause we had missed the previous class due to Hurricane Frances, the first test would be put off a week. I told them that the journal abstracts that were due today could be handed in next week since many had been without power or Internet access. The majority of them did have theirs ready though. I asked them to go ahead and do journal 3 for next week so the journals would not get behind. [I thought that they might have some anxieties about getting behind, and I tried to address that right away and put them at ease by telling them that if they did not have their paper done, they could turn it in next week.] I told them that we needed to figure out a way to make up the material from the missed class. We discussed the options, which were to do an hour of new material before the first two tests or an hour after the tests or to co me the week of Thanksgiving. I distributed the ballots I had made for them so they could vote privately rather than raising hands. We discussed each of the options. I told them my own view of the pros and cons of each. They shared some comments and questions abou t them as well. Then they voted to have an extra hour of class before each of the first two tests. [I overheard a student say to another student that she was glad I let them decide rather than ju st telling them … I was hoping they would noti ce and appreciate that.] We began class by talking about number sense and different ways that a teacher can use everyday classroom activities to develop numb er sense. They had some good ideas and then we went over the ones on the over head. We talked about the three levels1 at which children are able to represen t knowledge and the prenumber concepts. Then we did some activities involving prenumber concepts. We began with the Geopieces2 and sorting, then 1 These are the Concrete, Pictor ial, and Symbolic Levels 2These are pieces of construction paper that differ in size, shape, and color.

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383 Appendix O: Continued we played the two-difference game. [We did 4 tr ansparencies and then went to activities, which I thought was good. I have noticed in the past that with any mo re than that, they get restless. They seemed very happy when they saw that we were going to be doing activities and using manipulatives. Several had mentioned in journals that they were looking forward to using the manipulatives.] After break, we got to the overhead on differe nt types of counting and then we used the calculators. I showed them how to do the calculator races [one student mentioned that she had read an article for the assignment about using calculators this way. She seemed pleased that we were doing something that she had read about … it was validated] and also we did the number re lationships 1-10. One student mentioned that her 3-yr-old child was counting using rote counting. I mentioned that it would be interesting for her to watch him go th rough this development of number sense. [relevance!!] I showed them how to use the calculators to do one and two more than and one or two less. We talked about 5 and 10 frames and using unifix cubes or groovy boards for partpart-whole relationships. They seemed to get that pretty well. I had them each pick a number between 1 and 10 and show how to get it using each of those different methods. There were a few who seemed very confused about what I wanted them to do, but after I helped them a little, they seemed to get it. We did number relationships for 10-20 and then I told them that we were going to play a counting game.1 I think that most of them enjoyed playing that game. There was a group of four sitting in the back next to (n ame) who was not really very engaged in the game. It looked like a couple were sort of pl aying it, but they were also talking about other things. When I came over, the one stude nt sitting closest to the front was not playing at all. When I asked, she said that she was watching the othe rs. I asked if she did not have her blocks with her, and she said that she did, so she took them out. [I am wondering if this is the one w ho wrote the journal about not lik ing to play games, etc. in these methods classes … I will find out next time when I figure out her name … she could already have a neg. attitude that is influencing her reaction to the course.] The rest of the class seemed to be engaged in the activity. After that, we did the M&M Counting Book. They did seem to enjoy that, especially eating the M&M’s. That was where class ended. 1This is a board game designed for children w ho are learning to count It involves rolling a number cube, counting that many spaces on a game board, and selecting that number of blocks.

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384 Appendix O: Continued Week 5: I reminded them about the test next week and that we would have class for an hour first. I told them how many questions were on it and reassured them that they should have plenty of time and that we would move to another location if they weren’t finished. I started by talking about the properties for a ddition. Then I gave them 5 examples on the board for them to try and determine which properties were being used. Before going over them, we talked about the difference between commutative and associative. They shared some good strategies for determining one from the other. By the time we went over them, they seemed like they did know them. I remember mentioning or hinting that these would make a good test question, and I no ticed that when I said that, they seemed a lot more interested in making sure they had it straight. We did the zero facts and talked about some word problems that they could relate these to. They gave some good examples. Then we talked about relating addition and subtraction facts. Then we started on the different interpretations for addition and subtraction word problems. We started with the join (Van de Walle, 2004, p. 136). I had them work each of those types using base-ten blocks. I explained that we want the kids to make sense of these prob lems and this would be a good way for them to do it rather than focusing on whether they are adding or subtracting. One student (name) asked if you could have the kids act out the problem using actual pennies. I reminded them that acting out the problem was one of the problem-solving strategies we had listed, and that it was an excellent suggestion. After the join, we did the same thing with separate (Van de Walle, p.137). It took quite a while, and I started realizing that time was going to be an issue today. Then we did the part-part-whole (Van de Walle p. 137) and then the compare (Van de Walle, p. 137). I got a lot of good participation as far as sh aring their methods, which I was glad of since it’s not always the case, but I was sens ing that they had had enough of this. So, that’s when we took our break. After the break, I had told them that I was going to give them some word problems to write, which I generally do next because I think it’s a good way for them to understand the different types and also helps them to identify each type when they share them. However, I decided that I better put that o ff until the end because I knew the test was next time and I didn’t want to get behind. They had seemed to be getting a bit restless with the problem types, so I decided to start th e second half with a couple of activities. First we did the pattern blocks. I had them determine how the different pieces were related. Then I gave them the problem where the small green triangle is a cake that sells for $1. We talked about the different ways one could spend $3 on cakes. Then I asked them to see if they could find the 12 ways that one could spend $8 on cakes. I stressed that they should write equations or number sentences for this. Some of them thought

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385 Appendix O: Continued they had found 13 ways, but when we went ov er it, they had one tw ice. I think it was a good challenge for them to know how many there w ere and to try and get them all. I didn’t have time to do tangrams there. I really wanted to, but I didn’t have time. I think I will do that next week. After that I did the number line slide rule activity1 with them. They seemed to really like that. I heard several of them say, “Oh that’s cool!!” and things like that, so I was glad that I took the time for that. We did addition and subt raction and also talked about how it works. I knew that we had to get to basic facts si nce that’s on the test next week. We did a couple of overheads about learning basic facts, different strategies and order of learning them. I tried to really reinforce what is and is not a basic fact. They came up with some great ideas when I asked them how they would do 6 + 8 mentally. They actually ended up bringing up many, if not most, of the different strategies that we were going to be discussing. That was great --a big time saver when we got there. Once we got to the chart, we had already talked abou t most of the strategies, so I was able to go through it rather quickly. I told them th at they didn’t need to get it all down since it was in the book. Then we talked about the “hard” subtract ion facts, and they shared some great strategies about how they did those [such a good class!!]. Then I did have them write word problems, but we only had ten minutes left at that point. I told them to just write one or two and we would stop at 11:45 and share them. We shared maybe 3 or 4. I would have rather gotten to more, but I told them that we would take time next time before the test to go over more if they want ed or they could email me if they had questions. That was the end of class. Week 6: At about 8:40 a.m., I was sitting in the offi ce, and one of the students (name) came in. She seemed very stressed and she said that she was nervous about the test and about having class before the test (as the class ha d voted to do in order to make up the missed day). She said that she wouldn’t hear a word I said because she would be tuning it out and trying to stay focused on the test material. I told her that I understood and that for that reason, I had decided not to cover th e next new material, multiplication and division, because that would be too much like what was on the test. Instead we would jump ahead to place value, which wo uld be something very different. That didn’t really seem to alleviate her stress, so I added that we would do about an hour 1This activity involves making tw o number lines and lining them up so that they model addition and subtracti on of whole numbers.

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386 Appendix O: Continued of new material and then we would take a long break. During the break, they would have time to ask questions, get refocused, a nd she seemed to be relieved about that. [This is why I didn’t want to have class on the same day as the test, but unfortunately with the hurricane, I had no choice.] After she left, I decided that I better go to the classroom early because I figured there would probably be several who were stressed. When I got to the class, I could immediately tell that I was correct. One of them even said, “Can you feel the stress in the air?” and I could. I tried to reassure them. I asked them what they were nervous about. Several of them seemed to be saying the same thing, which was that they had never taken one of my tests before. They we re concerned about the format. They pointed out that in the review notes, there were many pages that had five parts to this and six steps to that, and they were afraid that I would ask them to recall all of them. [I remember thinking that this was a valid co ncern and that as a student I would have felt the same way.] Because I felt that so many of them were feeling this way and I wanted them to not be focused on that while we were covering new material, I actually got out a copy of the test. I told them that I would not tell them the actual questions, but I went through the short answer section and gave a ge neral idea … name 2 or 3 of 7 steps, etc. They seemed relieved that they wouldn ’t have to name all of them. They also seemed concerned that they would have to write long essays and I assured them that they wouldn’t. They did seem very relieved at that point, so I decided to go ahead and start the new material at about 9:15. We started with tangrams since we didn’t have time to do them last week. I asked them to explore the different relationships between the pieces like we had done with the pattern blocks. We went over them and they seemed to be Ok with that, for the most part. Then we did the first 2 overheads about numeration systems, base ten system, and then the worksheet about the Roman Numerals, etc. They seemed to be interested in that and asked a lot of good questions [made me feel confident that they were getting something out of the lesson--that they weren ’t so focused on the test that it was a waste of time. I did notice that a few of th em were studying their notes instead, but I didn’t say anything.] Then I introduced base ten blocks. I wanted to do that so that when we do get to the rest of the lesson on place value, they would ha ve a basic understanding of how the blocks work. I had them model 2 or 3 numbers and then it was 10:00, so I stopped. I decided to start the test at 10:20, so everybody took a 5 min. break. Then those who wanted to review could come back fo r review and those who didn’t want to review could take a longer break. When I got back at 10:05, most everyone was back.

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387 Appendix O: Continued We did review. They seemed to be worried about identifying different types of add and subtract word problems, so I made up some and they were to tell me which type it was. I was pleased that the group as a whole seemed to be getting them, which made me feel that they were prepared for the test. At 10:20 we stopped and I passed out the test. At that point, we had about an hour and a half. Everyone finished within the time although two students were there until 11:45. However, one of them had arrived and started the test late. During the test, it seemed like the majority of questions asked were about the pattern block equation problem. Most of those were people who had been absent last week. [I remember thinking that this was good --hopefully they would get the message that just studying the book and notes isn’t enough--they do need to come to class as well.] Week 7: I began passing back tests. It seemed like there were more abse nt than there. I told them to help each other figure out what they had mi ssed if possible. If they were stuck, I would answer questions. I walked around and answered questions individually because I have found that they are sometimes more likel y to ask that way than in front of the whole class. Several more had arrived by this ti me. A few questions were asked and answered. There was a question about basic fact strategies and another about a multiplechoice question. There was one question regarding the next test. I deci ded to have it on the day it was scheduled, although it may not cover everything it would have. There was a question about whether new material cov ered that day would be on the test. I reassured them that it wouldn’t be. We talked about the properties of multiplic ation and how they could reduce the number of basic facts a student has to learn. I said that they were similar to the addition properties, so the only really new one was the distributive property. I said that we would be discussing how elementary school students could use the distribut ive property later in the course. Then we talked about connecting multiplication and division and also division with zero. I showed them why it is sometime s defined and sometimes undefined. I think that most of them were with me, although I’m not sure how many. It may have been over the heads of some. Then we began talking about the four type s of word problems for mult/div. We went through each type, and I asked them to model the problem. Then we discussed why it was that type. Just like w ith the add/subt ones, they had good ideas and were willing to share their strategies. There are probably about 10 who are good about sharing with the class [so glad!!—not always that many in other classes] Then we talked about the basic fact strategi es. When we got to “nifty nines,” several shared their own strategies about nines. Ma ny had not heard of some of these, and

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388 Appendix O: Continued they seemed intrigued. We talked about remediation a nd how failure to master basic facts should not be a ba rrier to doing real math I asked what they thought that meant. One said that it referred to math about th e real world. Another said that students should be able to use manipulatives, etc. to help. I said that they were both right, and that when the purpose and objective of a lesson was problem solving and not computation, students should be able to use calc ulators or manipulatives I also told them that they might have students who were great at problem solving a nd logical thinking and reasoning who had trouble remembering basi c facts due to learning disabilities. Week 8: We began by going back to place value. I re freshed their memories about what we had done before --expanded notati on and base ten blocks. I asked them to model a number with blocks and also to wr ite it in expanded notation. Then we talked about the number 323, and how the leftmost 3 was 100 times as large as the other three. We did a few more examples including one with a decimal point. Then I had them cut out a Place Value Viewer and write the number 3647. We talked about the different ways some one could withdraw that much money from the bank. They tried to come up with all the different ways using denominations that reflect base 10 place value concepts. They seemed to “get it” for the mo st part. I told them that they could use a blank one on the test. I also said that we woul d skip Base-ten Riddles for now and come back. I wanted to make su re I got through the stuff for the test. We did 3 overheads about invented vs. tradit ional algorithms. Then I gave them an addition problem to model with base ten bloc ks. After we went over that, I gave them another one to draw out on paper. I told them that they would have to do this on the test. I went around and looked at everyone’ s problem and told them whether or not it would be acceptable on the test. I told them that I was looking for them to demonstrate modeling each of the numbers and also the regrouping. Then I showed them the low-stress algorithm for addition. One student said that she wished she had been shown that as a child b ecause it was so much easier. We talked about how it enforced place value concepts rather th an just being a mechanical process w/o understanding. I gave them a subtract ion problem to model with blocks. After we went over that, I gave them one to draw out, and I repeated the process of checking each one. Then we took a break. After the break, we began mult. I gave them a problem to model with blocks. We talked about how you could do it two different ways: 13 groups of 4 or 4 groups of 13. All but one did 4 groups of 13. Then they drew it, a nd I checked them. We went over the low

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389 Appendix O: Continued stress algorithm for mult. They seemed more concerned wi th how I might ask this on the test than how it worked. I showed that it could be done left to right or the other way, which was an advantage. Then we did the ar ea or rectangular model. They did seem to grasp this faster than some classes in the past. I showed it to them on graph paper and encouraged them to try it on th e graph paper in their notes. Next we talked about the dist ributive property and how kids c ould use it. I asked them to use the distributive property to simplify 15 x 12. They had several different ways they had done it. I tried to reinforce that this was a good thing. Then we got to the lattice method. I told them that this was my favorite. [I think many of them were “wowed” as I had hoped.] I gave them one to try. I skipped the rest of the stuff from this chapter and to ld them that we would cover it before the test. It would not be on the test, but it could be on the final. Then I skipped ahead to estimation strategies. We went th rough them and talked about how to make them relevant to kids. They should always be presented in a contex t of a situation where estimation is warranted. One student (name) gave some good examples. I had covered everything for the test, so we went back and did Base-Ten Riddles. Most of them seemed to get them pretty well. I enco uraged them to use the blocks, but it seemed like many, if not most, weren’t. [I think they were ready to go.] Week 9: Today was scheduled as a test day, but we had to have class before the test in order to get caught up from the missed hurricane day. I got to class at about 8:45. I wanted to get there early due to the test. I walked around the room for about 15 minute s, answering individual questions. Most of them concerned review questions from the review sheets. At 9:00, I told the class that we would spend about 45 minutes on new material. Th is material would not be on this test or the last test, but it coul d be on the final exam. Someone asked if the la st test would be harder since it covered more material. Actu ally I don’t think it does cover more material. I didn’t address that though. I ju st said that I had never received any feedback reflecting that point of view. I did remind them that usually the first test covers much more though. Someone said that they (the class) were giving me feedback that the way we had done it with two tests instead of one was much more desirable. No one disagreed, and several nodded their heads in agreement. I said that I also agreed with that. I began with the overhead about thinking strategies. As I was going through that, I noticed that many were not paying any attention and were studying for the test. I stopped and said that I knew that they were studying instead of listening. I told them that I understood how di fficult it was to focus on new material with a test

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390 Appendix O: Continued coming, and that I would never have planned it this way. I reminded them that it was Mother Nature who was responsible for the change in plans. I assured them that they would have plenty of time to study before the test. We would stop at 9:45 and break until 10:00. Then we would rev iew until they were ready … probably about 15 min. They seemed a bit relieved and put their notes aside and were more attentive. We modeled a division problem with Base Te n blocks and talked about how the model connected to the traditional algorithm. Several of them seemed quite interested in this. I have noticed that the long division al gorithm is not very widely understood, and that students are often interested in better understanding it. Then we did the Arrow Math with the hundreds chart. Many seemed interested and challenged by this. They also seemed, as a group, to pick it up quickly. The last thing we covered was Diagnosing Student Errors. I told them that this could be on the final … hint, hint … I wanted to make sure they didn’t overlook it since it wouldn’t be on any of their tests. They seemed to pick this up pretty quickly too. [I hope they aren’t just acting like they understand by not asking questions because they are distracted by the upcoming test … this reaffirms my practice to not teach class on test days … this is what happens if the test is after class, and if the test is first, they either leave or are unhappy becaus e they either have to wait around for others to finish or feel rushed with the test because others are waiting.] Week 10: I began class by passing back test 2. Overall they did very well. This is the first time since switching to the new book th at I have broken the first te st into two tests. This has definitely proven to be a good idea and one that I will continue to use. I walked around the room and answered questions individually. There weren’t very many--just a few. As I circulated among the class, one student said that she had a que stion about the lesson plan assignment. I said that I would address this with the whole class. I brought it up now. I referred them to the assignment sheet. I told them to be su re that they checked with me if they had any doubt if their b ook met the required topic and also to be sure and include each required component. I explained that they must include a summary of book and that points would be deducted if anything were missing. We began by talking about rati onal numbers. They seemed to already know or pick up quickly the concept of rational numbers. We talked about the meaning of a fraction. Someone said that she wished she had learne d this as a child. I used the opportunity to emphasize that kids need to have co nceptual understanding of what a fraction means before getting to algorithms for comp utation, etc. They seemed to agree.

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391 Appendix O: Continued I asked them to figure out what each pattern block piece would equal if the yellow were the unit. We repeated this with two yellows as the unit, then with one red as the unit. Most of them seemed to r ealize that the blue would be 2/3, which usually causes confusion. Then we moved to tangrams and they figured out what each piece equaled if the big triangle were the unit. Someone asked why they were called tangrams. I told them that there was a story about that and they would learn that in math II. I once again had to admit to being ‘tangram challenged.’ They always enjoy that. [Maybe it makes me seem more “normal?” Unfortunate ly it is the truth!!] We then went through the three types of frac tion models, including fr action circles, graph paper, paper folding, Cuisenaire rods, fracti on tower cubes, and tw o-color counters. By then they were ready for a break. After the break, I passed out the Cuisenaire r ods and told them to work together on the Parts and Whole worksheet. There were questio ns as I circulated about the Cuisenaire rods mostly, with a few on the pattern blocks and counters. When everyone was finished or nearly finished, I put the answers on the overhead. Two or three wanted to challenge an answer, which I welcomed. I tried to le t them know that I certainly could have made a mistake, and that I welcomed their challenges. I was able to help them see where they were wrong, but I also tried to po int out that they want their students to do this (challenging the teacher) and that it was a good thing. There wasn’t too much time left, so I read them a story, Two Ways to Count to Ten. We talked about learning activities that could be used with that story to teach the concept of counting by 2’s or 5’s. Someone suggested using manipulatives to model ten and separating them into equal groups. I also suggested a follow-up activity of having the animals count to a bigger number, like 20 or 50, and the different ways they could do that. That was the end of class. Week 11: I decided to start with a coupl e of activities while we were waiting for latecomers. The first was Benny’s Cakes. They worked on them for about 5-10 minutes while I circulated around the room. I gave the class a couple of hi nts to get them thinking about some other ways to do this. After a few minutes, I put up the solution on the overhead. We discussed how you would know that the pieces were equal on some of them. I made the point that as a teacher, they wouldn’t necessarily have to think of all of thes e, but they should be able to see which are right if a student came up with these. [trying to alleviate their fears about not always knowing or th inking of all the possible answers] Then we cut out the fraction circ les and made the manipulative. They seemed to like that, which is the norm.

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392 Appendix O: Continued Next we went over a few overheads dealing with fraction number sense. I tried to stress the importance of developing number sense and es timation with fractions and related it to similar concepts we have discussed for w hole numbers. The first dealt with ordering fractions that have numerator of 1, using a conceptual pe rspective rather than an algorithm. They shared their strategies and how they could help kids make sense of this. The next one used benchmarks of 0, , and 1 to determine which fractions were larger. The third used estimation for addition of mixed numbers. Again, they shared their strategies for each aft er doing the first together. After the break, I read the story Remainder of One They seemed to enjoy it [sometimes hard to tell if they like to hear stories or think it’s a waste of time. I try to discuss how they could plan a lesson around it to make it relevan t to the course.] After the story, we did the equivalent fraction rule, which I tried to relate to algebra, and then we began the operations w/fractions problems. The first was addition. They felt that the problem wa sn’t worded quite right, so I changed it to reflect their suggestions. They were able to model the problem and share their strategies well. Many used fraction tower pi eces and many used drawings. The subtraction one caused some confusion. So me of them thought it should be of the rather than of a whole. This is very common and happens in every class I teach. When we got to the multiplication problem they seemed to better understand the difference. One of the girls in the group who talk in class an d come in late from break every week seemed annoyed with me when I tried to explain to her that it was subtraction. She had the right answer, but she didn’t view it as subtraction. [It took a lot of patience on my part to ke ep patiently explaining it to her and not just say that maybe if she listened in class occasionally and showed up on time, she might better understand!!] Week 12 (second survey given during this class): I began by reminding them that lesson plans were due next week and I answered a few questions about that. I also reminded them th at all 8 journals should be completed now and that if they were not all sent, th ey should take care of that this week. [I always like to begin class by making sure everyone knows where they should be and also answer any questions about upcoming as signments, tests, etc.] At this point, we began ne w material. Today’s lesson was on computation with fractions. I once again pointed out that modeling and understanding should come before algorithms. We began with addition and subtraction. I gave them a problem to model w/pattern blocks and write a word problem. I stress ed that w/addition a nd subtraction word problems, the unit must be the same. I gave them a few problems to try modeling as I circulated around the room. Most of them se emed to be getting it, although some had problems. I heard someone say that she wished they had done fractions this way when she was a kid … or something to that effect [YES!!!!].

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393 Appendix O: Continued When we got to multiplication, there were so me problems. We began with the area or rectangular model. They seemed to unders tand the problems where the fractions were less than one. I have learned to do this on the board rather than the overhead so that they can see how I’m dividing each side of the unit rectangle. One student (name) mentioned that she really liked this and th at I didn’t need to help her as usual because she was getting it. I told them ho w I could usually tell when people don’t understand this by the way they divide the si des on a test. This alerted them to the fact that this would most likely be on the test, so they were very attentive. After break, I read the book Betcha!! We talked a little about le ssons a teacher could plan around the book. One student (name) said that she was planning to use that book for her lesson plan. After the story, we went back to multiplication of fractions. This time we were working with mixed numbers. I showed them the leng th x width idea and also the graph paper. I had never shown the graph paper before a nd had decided to add it the night before. At first they seemed confused and I regretted it but then I realized that before, they had just been using the area formula to figure it out and now with the graph paper they were understanding it better. [Stude nt’s name] made the comment that the other way was just using an algorithm and the graph paper actually showed you the area. I guess I will continue to use that. Even though it seemed to cause quite a bit of confusion, they asked for several more problem s to try, and by the end, I think most of them had gotten it. I felt like I needed to move on in order to get through division, so I told one student who was struggling that I would show her more after class. She ended up telling me that she had gotten it by the time class ended. Next we worked on division of fractions with pattern blocks. One stude nt (name) asked if it was OK to use a different manipulative ra ther than pattern blocks. She was using fraction circles. I said that they were both area models, so it was OK. I reminded them that as teachers they would want to use all three types of models though. I also said that I could ask specifically about pattern blocks on the test [wanted to make sure they would study that so they wouldn’t be caught off guard]. They had a little trouble with the division, but eventually most of them seem ed to be getting it. I noticed that (name) and also (name) were having problems with this By the end of class, I think that they were understanding though. We ended class at about 11:50.

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394 Appendix P: Samples from Course Note Packet HOW CHILDREN LEARN MATHEMATICS Three levels at which children are able to represent knowledge: Level 1: ___________ Level of Representation Level 2: __________ Level of Representation Level 3: ___________ Level of Representation

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395 Appendix P: Continued COUNTING 1. ________Counting: number sequences are used to verbally recite numbers by rote, without referring to objects. 2. _________ or ___________ Counting: counting involves matching objects or events with a number name. 3. _________ Numbers: indicate the relative position of an object in an ordered set (ex. 1st, 2nd, 3rd ). 4. ___________ Principle: The understanding that the last count word names the quantity of the set. Ex: After counting six objects, a child is asked “how many are there?” If the child answers that there are six without recounting, he or she is said to have the ___________ principle.

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396 Appendix P: Continued Strategies for Learning Basic Facts + 0 1 2 34567 8 9 0 1 2 3 4 5 6 7 8 9 Zero Facts One-More-Than Facts Two-More-Than Facts Doubles Near-Doubles Make-Ten Facts Ten-Facts Can Use Make-Ten Extended, Counting On, Doubles Plus Two, or just memorize remaining

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397 Appendix P: Continued CONNECTING MULTIPLICATION AND DIVISION If you know 6 x 4 is 24, then you also know: DIVISION BY ZERO 0 DIVIDED BY N = 0 (0/n) Why? N DIVIDED BY 0 IS NOT POSSIBLE (n/0) Why? 0 DIVIDED BY 0 IS NOT POSSIBLE (0/0) Why? VOCABULARY FACTOR X FACTOR = PRODUCT DIVIDEND/DIVISOR = QUOTIENT

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398 Appendix P: Continued Review: Chapters 10, 11: Multiplication/Division 1. Identify the different types of word problems for multiplication and division and create a word problem for each interpreta tion or type. If given a word problem, be able to identify the appropriate type. Some samples: a. Sarah has 5 bags of candy. Each ba g has 6 pieces of candy in it. How many pieces of candy does Sarah have? b. Tom earned $6 this week. He saved 3 times as much money as he saved last week. How much did he save last week? c. An ice cream store has 3 types of cones and 4 flavors of ice cream. How many different combinations of one-scoop ice cream cones can you get? d. A rectangle has length of 3 cm and width of 4 cm. How many square centimeters are there in the rectangle? e. Amy has 30 pieces of candy to share with 5 friends. How many pieces will each friend get? f. Amy has 30 pieces of candy to share. She wants to give each of her friends 6 pieces. How many friends will get candy? g. Tom earned $2 last week. This week he earned 3 times as much. How much did he earn this week? h. Tom earned $6 this week. Last week he earned $2. How many times as much money did he earn this week as last week? i. An ice cream shop can make 12 diffe rent combinations of one-scoop ice cream cones. If they have four di fferent flavors of ice cream, how many different types of cones do they have? 2. What two division sentences are related to the multiplication sentence 6 x 2 = 12? 3. a. Explain why 5 0 and 0 0 are both undefined. b. Why is 0 5 defined? 4. Provide examples that illustrate that: a. division is not associative or commutative b. subtraction is not as sociative or commutative 5. Be able to identify and distinguish basic multiplication and division facts from those that are not considered basic facts. 6. For each multiplication fact strategy, list some facts for which the strategy can be used, and explain the thi nking process involved in using that strategy. 7. How can you help children who have been drilling their basic facts for years and still have not mastered them? 8. Illustrate and explain how the distri butive property can be useful in doing mental computations. 9. What does the author of your textbook sa y about speed drills (timed tests) for basic facts?

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399 Appendix P: Continued DECIMAL GRID How would you represent: a. 0.05 b. 0.2 c. 0.18

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ABOUT THE AUTHOR Joy B. Schackow received a Bachelor’s De gree in Elementary Education, with a minor in Mathematics, from the University of Iowa. She taught grad es 4 and 5 for four years before receiving her Master’s De gree in Mathematics Education from the University of South Florida. She then taught high school mathematics for three years and community college mathematics for 15 years. During this time, she also worked as a mathematics tutor for a private college prep aratory school where she tutored middle and high school students enrolled in va rious mathematics courses. She entered the Ph.D. program at the Univ ersity of South Florida in 1999 in order to pursue her interests in mathematics teacher education and attitudes toward mathematics. While in the Ph.D. program, she worked as a Graduate Teaching Assistant, as well as holding a two-year position as a Visiting Instru ctor. These positions involved teaching mathematics methods courses and supervising interns.