Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UFE0051659/00001
## Material Information- Title:
- Culturally Responsive Teaching in an Algebra I Class for Repeating 9th Graders
- Creator:
- Van Buren, Jenny L
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2017
- Language:
- english
- Physical Description:
- 1 online resource (119 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ed.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Curriculum and Instruction
Teaching and Learning - Committee Chair:
- MURATA,AKI
- Committee Co-Chair:
- DANA,NANCY L
- Committee Members:
- JACOBBE,TIMOTHY
GRIFFIN,CYNTHIA CARLSON
## Subjects- Subjects / Keywords:
- algebra -- culturally-responsive -- equity -- mathematics -- remedial-mathematics -- teaching
Teaching and Learning -- Dissertations, Academic -- UF - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Curriculum and Instruction thesis, Ed.D.
## Notes- Abstract:
- Despite efforts to achieve educational equity in the United States, inequalities still exist through policy, practice, and tradition (Bonner, 2014; Gay, 2010; Gorski, 2013; National Council of Teachers of Mathematics, 2014). Many educators fail to consider the opportunity gaps that exist for students across the achievement spectrum (Horn, 2012). To increase equity, many teachers are working to provide culturally responsive instruction that is designed with students' unique learning styles, family values, and cultural and linguistic frames of reference in mind. Research suggests that culturally responsive teaching is critical to creating equitable learning environments in which students who have struggled in the past become successful and engaged in learning (Gay, 2002; Gay, 2010). The purpose of this study was to learn more about what it means to be a culturally responsive teacher for students enrolled in an Algebra I course for repeating ninth graders at a public high school in Anderson, South Carolina. I used practitioner teacher research to examine my practice and reflect on my instruction, focusing on building relationships, communicating high expectations, and increasing student engagement. There were 12 students enrolled in my course at the time of the study. As I worked to create a more equitable learning environment for repeating ninth graders in my Algebra I classroom through culturally responsive classroom practices, I collected data to gain insights into my research question through (1) student interviews, (2) a researcher journal, (3) observation/field notes, (4) lesson videos, and (5) student work samples. I used both formative and summative data analysis. This study illustrates the ways in which I attempted to increase my culturally responsiveness as a teacher and the resulting student responses. The data from this study indicated that relationships between myself and students, as well as among students, impacted how students engaged in learning. In general, students did not engage with mathematics in open and authentic manners. However, I could impact student engagement as the teacher by focusing on the instructional design of the lesson, the appeal of the activity to students' individual learning styles and needs, and my facilitation of the lesson. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ed.D.)--University of Florida, 2017.
- Local:
- Adviser: MURATA,AKI.
- Local:
- Co-adviser: DANA,NANCY L.
- Statement of Responsibility:
- by Jenny L Van Buren.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Applicable rights reserved.
- Classification:
- LD1780 2017 ( lcc )
## UFDC Membership |

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PAGE 1 CULTURALLY RESPONSIVE TEACHING IN AN ALGEBRA I CLASS FOR REPEATING 9TH GRADER S By JENNY LOTT VAN BUREN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION UNIVERSITY OF FLORIDA 2017 PAGE 2 2017 Jenny Lott Van Buren PAGE 3 To our children PAGE 4 4 ACKNOWLEDGMENTS I would like to begin by thanking my doctoral committee. I appreciate your feedback, questions, and the ways in which you challenged me to continually improve my practice as an educator. I had the great fortune of h aving Aki Murata as my committee chair and Nancy Dana as co practitioner research were the perfect combination. I appreciate the ir enthusiastic, celebratory way of challenging me to think deeply about how I, as a classroom teacher, could make an impact and do my part to transform classrooms, schools, and communities for the benefit of all students. I will be forever grateful for t he endless hours spent reading and critiquing my work in order to help me tell my research story. I would like to thank those that have inspired me as an educator. To my students past, present, and future, I hope that you will always view your mistakes a s opportunities to learn rather than as failures. I am proud of your achievements, I celebrate your successes, and I look forward to seeing the great things you will do in the future. To the teachers in my life who inspired me to do more and be more than I ever thought I was capable of, thank you for believing in me and serving as examples of the kind of person I want to be. To my colleagues, thank you for sharing your wisdom and for the encouragement you provide during our daily work. To my UF cohort, words cannot express my gratitude for our friendship and shared experiences. I am forever changed and inspired by learning alongside such diverse, caring, and like minded people. I pray that you keep fighting the good fight and working to change the worl d for the better. PAGE 5 5 Most of all, I would like to thank my family. Thank you for believing in me, supporting me, and encouraging me to work hard to achieve my goals. To my children, thank you for being my reason for everything. I love you more than I ever thought possible. You are the light of my life, my greatest joy, and my hope for the future. I wish you every good thing that the world and life have to offer. PAGE 6 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ 4 LIST OF TABLES ................................ ................................ ................................ ........... 8 LIST OF FIGURE S ................................ ................................ ................................ ........ 9 ABSTRACT ................................ ................................ ................................ .................. 10 CHAPTER 1 INTRODUCTION AND BACKGROUND ................................ ................................ 12 Background and Significance of the Problem ................................ ........................ 13 Purpose of the Study and Research Question ................................ ....................... 16 Significance of the Study ................................ ................................ ....................... 16 Overview of Dissertation ................................ ................................ ........................ 17 2 LITERATURE REVIEW AND PERSPECTIVES ................................ ..................... 19 Challenges ................................ ................................ ................................ ............ 20 Curricular Tracking ................................ ................................ .......................... 20 Student Diversity and Disproportionate Representation ................................ .. 21 Teacher Beliefs, Biases, and Prejudices ................................ ......................... 22 Lack of Interest in Learning Mathematics among Students ............................. 24 Culturally Responsive Classroom Practices and Pedagogy ................................ ... 25 Relationship Building ................................ ................................ ....................... 26 High Expectations ................................ ................................ ........................... 29 Engaging L essons and Learning Activities ................................ ...................... 31 Summary ................................ ................................ ................................ ............... 33 3 RESEARCH METHODS ................................ ................................ ........................ 34 Context ................................ ................................ ................................ .................. 36 Participants ................................ ................................ ................................ ............ 37 Data Collection ................................ ................................ ................................ ...... 38 Student Interviews ................................ ................................ ........................... 38 Researcher Journal ................................ ................................ ......................... 39 Observatio n/Field Notes ................................ ................................ .................. 40 Lesson Videos ................................ ................................ ................................ 40 Student Work Samples ................................ ................................ .................... 41 Data Analysis ................................ ................................ ................................ ........ 42 Researcher Positionality ................................ ................................ ........................ 51 Enha ncing Trustworthiness ................................ ................................ ................... 52 Limitations of the Study ................................ ................................ ......................... 53 PAGE 7 7 Summary ................................ ................................ ................................ ............... 53 4 FINDINGS ................................ ................................ ................................ ............. 54 What are Relations hips between Students and Myself as well as among Students Like? ................................ ................................ ................................ ... 54 Relationships between Students and Myself ................................ ................... 54 styles. ................................ ................................ ................................ .... 55 Trusting relationships ................................ ................................ ................ 57 Relationships among Students ................................ ................................ ........ 59 Answering Question 1: Classroom Relationships ................................ ............ 62 How are High Expectations for Academics and Behavior Communicated and Enacted in my Algebra I Class? ................................ ................................ .......... 64 Academic Expectations ................................ ................................ ................... 64 Behavior Expectations ................................ ................................ ..................... 69 Answering Question 2: Communicating High Expectations ............................. 71 How do Students Engage in Learning Activities in my Algebra I Class, and How Might Student Engagement be Negotiated Depending on their Individual Backgrounds and Prior Experiences? ................................ ................................ 72 Engagement in Collaborative Group Work ................................ ...................... 73 Engagement in Teacher Led Discussions, Notetaking, and Independent Practice ................................ ................................ ................................ ........ 77 Answering Question 3: Student Engagement ................................ .................. 79 Summary ................................ ................................ ................................ ............... 79 5 PRACTITIONER REFLECTIONS ................................ ................................ .......... 81 Continued Self Reflection and Change within my Classroom ................................ 82 Advocating for Changes Outside of my Classroom ................................ ................ 87 Summary ................................ ................................ ................................ ............... 90 6 IMPLICATIONS ................................ ................................ ................................ ..... 91 Implications ................................ ................................ ................................ ........... 91 Implications for Teachers ................................ ................................ ................ 91 Implications for Future Research ................................ ................................ ..... 95 Conclusion ................................ ................................ ................................ ............. 96 APPENDIX: LESSON PLANS FOR UNIT 10: EXPONENTIAL FUNCTIONS ............. 101 LIST OF REFERENCES ................................ ................................ ............................ 115 BIOGRAPHICAL SKETCH ................................ ................................ ......................... 119 PAGE 8 8 LIST OF TABLES Table page 3 1 Student interview themes. ................................ ................................ ..................... 43 3 2 Finalized codes. ................................ ................................ ................................ .... 50 PAGE 9 9 LIST OF FIGURES Figure page 3 1 Printed teacher researcher journal, day 1 ................................ .............................. 40 3 2 Example of student work ................................ ................................ ........................ 41 3 3 Posters for sub question 1 ................................ ................................ ..................... 46 3 4 Posters for sub question 2 ................................ ................................ ..................... 47 3 5 Poster for sub question 3 ................................ ................................ ....................... 47 PAGE 10 10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Education CULTURALLY RESPONSIVE TEACHING IN AN ALGEBRA I CLASS FOR REPEATING 9TH GRADERS By Jenny Lott Van Buren December 2017 Chair: Aki Murata Cochair: Nancy Dana Major: Curriculum and Instruction Despite efforts to achieve educational equity in the United States inequalities still exist through policy, practice, and tradition ( Bonner, 2014 ; Gay, 2010; Gorski, 2013; National Council of Teachers of Mathematics, 2014). M any educators fail to consider the opportunity gaps that exist for students across the achievemen t spectrum (Horn, 2012). To increase equity, many teachers are working to provide culturally responsive cultural and linguistic frames of reference in mind. Research suggests that culturally responsive teaching is critical to creating equitab le learning environments in which students who have struggled in the past become successful and engaged in learning (Gay, 2002; Gay, 2010). The purpose of this study was to learn more about what it means to be a culturally responsive teacher for students enrolled in an Al gebra I course for repeating ninth graders at a pu blic high school in Anderson, South Carolina I used practitioner teacher research to examine my practice and reflect on my instruction focusing on building relationships, communicating high expectations, and increasing student PAGE 11 11 engagement There were 12 students enrolled in my course at the time of the study As I worked to create a more equitable learni ng environment for repeating ninth graders in my Algebra I classroom through cultura lly responsive classroom practices, I collected data to gain ins ights into my research question through (1) student interviews, (2) a researcher journal, (3) observation/field notes, (4) lesson videos, and (5) student work samples. I used both formative a nd summative data analysis. This study illustrates the ways in which I attempted to increase my culturally responsiveness as a teacher and the resulting student responses. The data from this study indicated th at relationships between myself and students as well as among students impacted how students engaged in learning. In general, students did not engage with mathematics in open and authentic manners. However, I could impact student engagement as the teacher by focusing on the instructional design of the needs, and my facilitation of the lesson. PAGE 12 12 CHAPTER 1 INTRODUCTION AND BACKGROUND In educational contexts, equity can be considered as meeting the needs of all students regardless of racial, cultural, or socioeconomic backgrounds. When defining the principles of equity literacy, Gorski (2013) suggested that congruence exists between wh at educators believe due to their own individual biases and prejudices and what educators believe about what it means to be effective in working with diverse ranges of students and families. Equity literate educators recognize the biases and inequities th at impact students and families, and they act immediately to respond. Equity literate educators work to reject stereotypical and deficit views and move toward recognizing communities and families as assets to learning (Dray & Wisneski, 2011; Gorski, 2013; Gay, 2010; Kose & Lim, 2011; Schillwer, 2008; Ullucci & Howard, 2015). Educators that are concerned about equity work to move from aesthetic caring, which academic achievement, to authentic caring, which emphasizes relationship building between school personnel and students. Teachers who authentically care are communities and families and value rela tionships ( Ross & Adams, 2010 ; Gay, 2010; Schillwer, 2008). Overtime, equity literate educators redress biases and inequities, creating and sustaining a bias free and equitable learning environment for every student. Achieving educational equity is an o ngoing challenge for educators, as it requires PAGE 13 13 to provide high quality programs for all s tudents. Low income families may also lack access to the resources and experiences both inside and outside of schools that are afforded by wealthier families. While some educators are highly effective in engaging low income and minority students, the per sistence of the achievement gap suggests that there is still need for improvement (Books, 2007; Bonner, 2014; Darling Hammond, 2010; Gay, 2010; Gorski, 2013; Milner & Howard, 2015; Nieto, 2013; Ross, Bondy, Gallingane, & Hambacher, 2008). Therefore, it is important that educators reflect upon current practices, constantly work to revise their own practices, and advocate for reforms to promote equity (Darling Hammond, 2010; Dray & Wisneski, 2011; Gay, 2010; Milner & Howard, 2015; Nieto, 2013). One way in w hich teachers work to promote equity in the classroom is culturally responsive instruction. The focus of backgrounds, learning styles, family values, and cultural and linguistic frames of reference. The purpose of this study was to learn more about what it means to be a culturally responsive teacher for students enrolled in an Algebra I course for repeating ninth graders at TL Hanna High School. Background and Significance of the Problem Historically, e ducation in the United States has excluded particular cultural groups, either by not permitting particular races, genders, or classes of students to attend school at all or by providing a lower level of education as compared to the education of white middl e class students. For example, in the 19th Century, school reformers such as Horace Mann argued that, while male and female students should attend school together, black students should not be permitted to attend the same schools as white students (Rury, 2 013). Since the 1970s and the rise of the multicultural education movement, Americans are increasingly concerned about PAGE 14 14 inequities that are apparent in learning opportunities and outcomes for students (Gay, 2010; National Council of Teachers of Mathematics 2014). Despite improvements and efforts to achieve equity through various reforms and litigation, including the outlawing of racial segregation in public schools, inequalities still exist through policy, practice, and tradition. An enduring problem in schooling is the failure of American schools in educating poor and minority children (Bonner, 2014; Darling Hammond, 2010; Gay, 2010; Gorski, 2013; National Council of Teachers of Mathematics, 2014; Nieto, 2013). Many examples of American schools disprov e the idea that American schools fact, in contrast to the popular belief that everyone has an equal chance to success, evidence proves that children who are at risk due t o being economically disadvantaged are more likely to attend schools that lack basic conditions necessary to provide a quality education. There are inequities regarding student access to preschool, school funding, school resources, support services, and t echnologies. In addition, students from lower socioeconomic backgrounds are less likely to be educated by well paid, certified, and experienced teachers and are often subjected to lower educational expectations and lower quality curricula (Darling Hammond 2010; Gorski, 2013). poverty, imprisonment, and unemployment rates are rising (Darling Hammond, 2010; Gorski, 2013; Nieto, 2013). Meanwhile, American companies struggle to find suitable candidates for employment, particularly in mathematics and science related fields, threatening the position of the United States as a global leader (Darling Hammond, 2010; National Council of Teachers of Mathematics, 2014; Rice & Alfred, 2014). PAGE 15 15 Women and people of color are a minority in professions related to mathematics and science, suggesting that K 12 experiences along with family and cultural factors impact career choices (Rice & Alfred, 2014). As a white female mathematics instruct or, I am concerned that I may hold biases and prejudices that I may or may not be aware of. Unless I make an effort to change my own thinking and biases, I may unintentionally act on them and negatively impact the most damaging (and unfounded) 12). Societal ideologies are deeply embed ded and often go unnoticed, appearing to be normal, despite negative impacts on students. Therefore, I must work to recognize and change my own biases to be more supportive of all students. It is important for me to constantly reflect upon my own practic es in order to promote educational equity. Prior to this study, I was also concerned as I was new to TL Hanna High School in Anderson, South Carolina, and teaching in a 4x4 block schedule and in an inclusion classroom for the first time. Inclusion classes in South Carolina are general education cl asses in which students with special education services are included with accommodations to instructional practice and/or assessments. Unlike my teaching experiences in previous schools, where classes changed every 50 minutes, students in my Algebra 1 cou rse remained in class for a 90 minute period. All of the students in the Algebra 1 course were repeating ninth graders, and the course met for a full year instead of a semester like other classes at TL Hanna High School. These changes required me to alte r my practice to better support students in my new context, in PAGE 16 16 addition to understanding my students their backgrounds, and the communities they were coming from. Purpose of the Study and Research Question The purpose of this study was to use practitioner research to learn more about what it means to be a culturally responsive teacher for students enrolled in an Algebra I course for repeating ninth graders at TL Hanna High School. Practitioner research is ctice that involves constructing knowledge from experience and other sources (Cochran Smith & Lytle, 2009; Dana & Yendol Hoppey, 2014, York Barr, Sommers, Ghere & Montie, 2006). The research question that guided my study is: What does it mean to be a cu lturally responsive teacher in the context of an Algebra I class for repeating ninth graders? Within the central question regarding what it means to be a culturally responsive teacher in the context of an Algebra I class for repeating ninth graders, I deve loped the following sub questions: 1. What are relationships between students and myself as well as among students like? 2. How are high expectations for academics and behavior communicated and enacted in my Algebra 1 class? 3. How do students engage in learning ac tivities in my Algebra 1 class, and how might student engagement be negotiated depending on their individual backgrounds and prior experiences? Significance of the Study During this study, I implemented instructional strategies, intentionally collected an d analyzed a variety of data, and reflected on my own work as a mathematics educator. I focused on creating an equitable learning environment, implementing PAGE 17 17 culturally responsive practices, and increasing student engagement among students enrolled in an Al gebra I course for repeating ninth graders. Personally, I benefited from this reflective work by improving my own professional practice and increasing my own balance and perspective and to renew clarity of personal and professional purpose (Yor k Barr et al. 2006). In this study context, it is possible that student learning, engagement, and motivation in mathematics increase d as I ma d e changes to my practice based on the data I collect ed and analyze d This study will also potentially impact other teachers at my school as I share my experiences and learning with them. Other teachers may choose to improve their practice in creating equitable learning environments and bei ng more culturally responsive for students as a result of this work. This study provide s insights into how to improve teaching mathematics and regarding social justice for high school mathematics teaching that are applicable to other contexts. It add s t o the existing body of research about social justice, equity, culturally responsive teaching, and mathematics education in that it is a practitioner research study focusing on Algebra I students in an inclusion classroom who we re repeating ninth grade. O verview of Dissertation This dissertation is the result of my realization of the need to increase equity within my own classroom. The purpose of this study was to learn more about what it means to be a culturally responsive teacher for students enrolled i n an Algebra I course for repeating ninth graders at a public high school in Anderson, SC. I hoped that increasing my own cultural responsive ness would lead to more of my students being successful and engaged in learning (Gay, 2002; Gay, 2010). I used practitioner teacher PAGE 18 18 research to examine my practice and reflect on my instruction focusing on building relationships, communicating high expectations, and increasing student engagement In the next chapter, I will discuss relevant literature to identify challenges teachers face when working with diverse groups of students who have previously been unsuccessful in mathematics courses. I will also explore elements of culturally responsive pedagogy as described in literature. Chapter 3 will address my methods of research, and I will discuss the findings in Chapter 4. Chapter 5 will focus on my own reflections as the practitioner researcher for this study, and I will discuss possible implications of this study beyond my own classroom in the final ch apter. In summary, this dissertation will illustrate the ways in which I attempted to increase my culturally responsiveness as a teacher and the resulting student responses PAGE 19 19 CHAPTER 2 LITERATURE REVIEW AND PERSPECTIVES The achievement of educat ional equity, meeting the needs of all students regardless of racial, cultural, or socioeconomic backgrounds is an ongoing challenge for educators. Despite initiatives that attempt to address discriminatory practices in ill exist (Bonner, 2014; Darling Hammond, 2010; Gay, 2010; Gorski, 2013; National Council of Teachers of Mathematics, 2014; Nieto, 2013 ). The purpose of this study was to learn more about what it means to be a culturally responsive teacher for students en rolled in an Algebra I course for repeating ninth graders at TL Hanna High School. The following question guided the study: What does it mean to be a culturally responsive teacher in the context of an Algebra I class for repeating ninth graders? Within t he central question regarding what it means to be a culturally responsive teacher in the context of an Algebra I class for repeating ninth graders, I developed the following sub questions: 1. What are relationships between students and myself as well as among students like? 2. How are high expectations for academics and behavior communicated and enacted in my Algebra 1 class? 3. How do students engage in learning activities in my Algebra 1 class, and how might student engagement be negotiated depending on their individual backgrounds and prior experiences? In this chapter I review literature to identify challenges teachers face when working with diverse groups of students who have previously been unsuccessful in mathematics courses. I also explore elements of c ulturally responsive pedagogy. PAGE 20 20 Challenges Curricular Tracking Curricular tracking in mathematics is a common practice in American high schools and is intended to assign students to courses that align with their perceived level of ability. However, while achievement and ability may be related, many educators overlook the possibility that mathematical competence is not necessarily correlated with student achievement. Some students perform well but lack deep understanding, while other students understand d eeply but perform poorly. The idea that differences in student achievement are a natural consequence of differences in student ability does not take into account the opportunity gaps that exist for students across the achievement spectrum (Horn, 2012). Th e practice of tracking does not solve the problems of low performance and disproportionate numbers of poor students and students of color in low level courses. advanced mathem atics courses, low achievement will continue. For instance, in a one year case study of an American high school mathematics department attempting to decrease failure rates in standard level courses, Buckley (2010) found that, despite redesigning or replac ing existing courses and adding new courses to the curriculum the results did not create more equitable mathematics education settings. The courses lacked critical components of content, were shallow in depth, and did not provide students with opportunit ies for future mathematics course taking. Although the redesigned curriculum permitted a small percentage of students to have access to more mathematics content, the teachers failed to critically analyze the content for individual courses or teaching strat egies. Changing the tracks were not enough to challenge the PAGE 21 21 existing underlying assumptions regarding students who had a history of low performance in mathematics courses. Student D iversity and D isproportionate R epresentation Students in remedial mathema tics classes each have a unique story, with individual backgrounds and experiences, but a single common factor unites them: previous failure in mathematics coursework (Hill, 2010). Students of color have an increased risk of being placed in low level cou rses or being identified as having developmental disabilities. If students have been subjected to racism, classism, or other forms of discrimination, they may have come to believe negative perceptions about their own identities and abilities (Gay, 2010; N ieto, 2013). Unlike their more advantaged peers, low income students have fewer opportunities for second chances when their failure is a result of academic disengagement (Bondy & Ross, 2008). eration when planning learning activities (Hill, 2010). As Blanchett, Klinger, and Harry (2009) pointed out, students of color who are identified as developmentally delayed are also more likely to live in poverty, receive inadequate prenatal care, and h ave limited access to early intervention services that encounter systems and structures that are not prepared to help them navigate services while living life at the in tersection of race, culture, language, and disability, which results in them ultimately receiving culturally unresponsive and inappropriate services and combat deficit thinking and to think deeply about the role of context and other factors PAGE 22 22 external to students that cause the complex phenomenon of disproportionate representation. Teacher B eliefs, B iases, and P rejudices Beliefs including biases and prejudices, about pe ople informs how teachers manage classrooms and relate to people (Bondy, Ross, Gallingence, & Hambacher, 2007; Gay, 2010; Gorski, 2013; National Council of Teachers of Mathematics, 2014; Nieto, 2013). For example, when managing classroom behaviors, some t eachers may mistake culturally defined actions as resistant or disrespectful due to their own cultural assumptions about appropriate classroom behaviors, leading to conflict, the alienation of certain students, or disruption in the classroom environment (B ondy et al., 2007; Gay, 2010; Weinstein, Tomlinson of reference, and personal background will influence how he or she responds to students (Dray & Wisneski, 2011). In addition, biases and preco nceived notions about students and what they are capable of learning may cause a teacher to have higher or lower academic expectations for particular groups of students (Gay, 2010; Gorski, 2013). Whether intended or unintended, research suggests that tea cher actions are concepts, particularly in mathematics. For example, in a statistical study regarding the relationships among teacher differential treatment and relevant math instruction o n African American concept of math ability, math task value, and math achievement, Diemer, Machand, McKellar, and Malanchuk (2006) used structural equation modeling (SEM) to examine data regarding the African American subsample ( n = 618) in the longitudinal Maryland Adolescent Development in Context Study (MADICS), which sampled children PAGE 23 23 from all 23 public middle schools in Prince s County, Maryland. Upon examining the relations among various concepts, the researchers found that relev ant perceptions of their teachers. Prejudices, stereotyping, and racism are proven to impa ct student self esteem, mental health, and stress levels, creating self consciousness and causing disengagement from learning tasks (Gay, 2010). determines how we think, belie ve, and behave, and these, in turn, affect how we teach context in which learning occurs may be due to differences in gender, race, economic status, geographic location language, religion, family structures, abilities, and family and personal history. Teachers that are unaware of how diversity influences both difference as a deficit (Dray & Wisneski, 2011; Weinstein et al., 2004). Teachers are challenged to understand how their own culture and the culture of their students impacts the educational process and act to bring about changes to prevent inequities. Discontinuities between school c ulture and different ethnic groups may interfere with student learning and success in school. Schools are heavily impacted by dominant norms, and educational practices can privilege certain groups of students while marginalizing others. By placing cultur e at the center of analysis of underperforming students, teachers may broaden their educational practices to be less biased and more equitable for all learners ( Bondy & Ross, 2008; Weinstein et al., 2004). PAGE 24 24 Lack of I nterest in L earning M athematics among S tudents Students in poverty generally have less access than their wealthier peers to higher order, engaging, and challenging curricula and are more likely to be subjected to rote memorization or skill and drill instructional practices (Gay, 2010; Gorski, 2 013). This disparity exists not only between low income and higher income schools, but also within schools due to the disproportionate representation of poor and minority students in lower curricular tracks. Since many teachers of lower level mathematics classes focus on simple memory, procedural based problems, and comprehension skills, another challenge faced by remedial mathematics teachers is to create opportunities for students to think critically and at higher levels about problems that they could e ncounter previous teachers have excluded practices designed to engage, empower, and challenge, students may enter new mathematics courses with a preconceived lack of int erest (Bonner, 2014; Fredricks, Brumenfeld, & Paris, 2004). Negative attitudes and lack of interest in school may lead to poor attendance, disruptive behavior, academic failure, and drop out ( Fredricks, Brumenfeld, & Paris, 2004 ). Students who have previ ously struggled to pass coursework and proficiency tests may have decreased interest in learning mathematics due to self esteem or their perceptions of their Horn (2012): Status plays out in classroom interactions. Students with high status have their ideas heard, have their questions answered, and are endowed with the social latitude to dominate a discussion. On the other side, students with low status often have their i deas ignored, have their questions disregarded, and often fall into patterns of nonparticipation or, worse, marginalization. (p. 21) PAGE 25 25 Students who are continually ignored in classroom discussions continually have their questions unanswered and confusions un aired; they rarely have the opportunity to suggest their own ideas. These students, having few opportunities for academic success or engagement, may internalize their lack of success as negative ideas about their value as human beings, deeming themselves failures and making school pointless ability to succeed after repeated failure, can be difficult to overcome and influences student participation and classroom conversa tions that are key to learning mathematics Culturally Responsive Classroom Practices and Pedagogy Strategies for strengthening school engagement and learning must be based on evidence for what works rather than what has been done traditionally in remedial courses (Gorski, 2013; Hill, 2010). It is important for teachers t o be able to provide culturally backgrounds, learning styles, family values, and cultural and linguistic frames of reference i n mind. characteristics, experiences, and perspectives of ethnically diverse students as conduits ssumptions that knowledge and skills are learned more easily and thoroughly when learning is situated within the lived experiences and frames of reference of individual students. In the context of a classroom of students that have been previously unsucces sful, it is and future learning. By making lessons more personally meaningful to students and considering how students will respond to the lesson culturally responsive t eachers engage students in learning by increasing their interest in course content and learning PAGE 26 26 activities. Research suggests that culturally responsive pedagogy is critical to creating equitable learning environments in which students who have struggled in the past become successful and engaged in the learning process (Gay, 2002; Gay, 2010). Relationship B uilding In contrast to educators who practice aesthetic caring, which focuses primarily on adherence to school rules and academic achievement, equity li terate educators work toward authentic caring, which focuses on relationship building between school personnel and students, families, and communities ( Ross & Adams, 2010 ; Gay, 2010; umanity, hold students in high esteem, expect high performance, and take action to not only show that lives to have a positive impact. They persist in the effort t o help students succeed, even when others give up or diminish the possibility for student success (Gay, 2010). Culturally responsive teachers initiate and maintain strong, caring, respectful, trusting and personal relationships with students (Bondy et al ., 2007; Bonner, 2014; Gay, 2002; Ladson Billings, 1995; Nieto, 2013; Ross et al., 2008). For instance, teachers communicate with students to learn about their home lives, beliefs, community, and funds of knowledg e (Bonner, 2014; Morrison, Robbins, & Rose 2008). They share detailed information about themselves to help students get to know them better (Bondy et al., 2007). W and recognizing that communication is deeply rooted in culture assist teachers with instructional content. PAGE 27 27 Culturally responsive teachers strive to make teaching match the cultures of students, fostering classroom interactions that are reminiscent of interactions that may occur in the home of their students (Bonner, 2014; Ladson Billings, 1995; Morrison, Robbin s, & Rose 2008 ). Teachers that effectively implement culturally responsive classroom practices understand that definitions of appropriate classroom behavior are better understand the biases and inequities they may experience. They work to build caring classroom communities in which students feel safe to take risks, respected, and trusting of one another and the teacher (Bondy et al., 2007; Bonner, 2014; Gay, 2010; Gors ki, 2013; Morrison, Robbins, & Rose, 2008; Ross et al., 2008; Schillwer, 2008; Weinstein et al., 2004). Culturally responsive teachers do the hard work of figuring out who they themselves are in relation to others, engaging in critical reflection of thei r own values, assumptions, and biases. They work to realize the ways in which they are privileged and how those privileges may obstruct their understanding of students, families, and communities. They reflect critically to improve their cultural responsi veness for the benefit of all students. They are open to new ideas and are receptive to the opinions of others (Bondy & Ross, 2008; Dray & Wisneski, 2011; Gay, 2010; Ladson Billings, 1995; Nieto, 2013; Weinstein et al., 2004). Culturally responsive teac hers incorporate activities that help students to get to know and connect with other students and establish an atmosphere in which students respect and are kind to one another (Bondy et al., 2007; Gay, 2010; Horn, 2012; Nieto, 2013; Weinstein et al., 2004) The classroom environment is nurturing and cooperative, PAGE 28 28 and activities are designed to assist students in developing a sense of belonging. To assist students in reimagining themselves and their classmates as competent mathematics learners, teachers est ablish norms for students to communicate in small groups. The selection of group members is done randomly, affirming the belief all students can learn from each other (Horn, 2012). Students share resources, encourage each other, and work closely together to achieve academic success (Gay, 2010; Ladson Billings, 1995). When classroom inequities occur, teachers intervene and Morrison, Robbins, & Rose 2008 ). Culturally responsive teachers assist students in recognizing social inequities and why they occur (Ladson Billings, 1995). They honor and responsibility for others in their students (Gay 2002; Gay, 2010; Nieto, 2013). Culturally responsive teachers work to understand the history of racial injustice that has influenced the present lives of people of color and strive to view communities of color as educational assets rather than as defic its that need to be corrected (Gay, 2010; Ladson Billings, 1995; Nieto, 2013; Schillwer, 2008). They reach out to parents that may not be able to attend meetings and other school events, greet parents at every opportunity, and personally extend invitation s for parents to become involved in their the prescribed curriculum by asking students and families to share resources or to serve as teaching resources for curriculum re enhanced academic success while maintaining cultural competence (Gay, 2010; PAGE 29 29 Morrison, Robbins, & Rose 2008 ). By building relationships with families and support academic success, developing a network of adults to assist students in navigating school and other social structures (Schillwer, 2 008). High E xpectations Culturally responsive teachers establish and communicate clear and high expectations for both academic performance and behavior (Bondy et al., 2007; Bondy & Ross, 2008; Bonner, 2014; Ross & Adams, 2010 ; Gay, 2002; Gay, 2010; Gorski 2013; Ladson Billings, 1995; Morrison, Robbins, & Rose 2008 ; Nieto, 2013; Ross et al., 2008). To ensure that expectations are made clear, teachers make sure that students hear, understand, and practice routines, providing both examples and non examples of appropriate behaviors. Students are required to restate the expectations and demonstrate understanding through practice (Ross et al., 2008). Culturally responsive teachers promote academic engagement through insistence, exhibiting a personal power t hat pushes students to meet higher standards personally (Bondy et al., 2007; Bondy & Ross, 2008; Ross & Adams, 2010 ; Gay, 2010; Ross et al., 2008). They model metacognitive activities such as thinking aloud, scaffold instruction, and provide clarification of challenging curriculum to support meeting high academic expectations. Both individual and collective accomplishments are celebrated, as students work together to solve problems, collaborate and model thinking for each other (Gay, 2010; Ladson Billings 1995; Morrison, Robbins, & Rose 2008 ). For example, mathematics teachers may expect students to be able to exhibit mathematical thinking at any time by being randomly called upon to work problems on the board, viewing mistakes as learning opportunities rather than failures (Bonner, 2014). PAGE 30 30 The classroom is a business like setting in which excuses are unacceptable and students persevere through difficult tasks, feeling confident to take risks (Bondy et al., 2007; Bondy & Ross, 2008; Ross & Adams, 2010 ; Gay, 2010; Ladson Billings, 1995; Morrison, Robbins, & Rose 2008 ). Students are not permitted to choose failure, and, rather than focusing on student characteristics and shortcomings, teachers engage in deep reflection and data collection and analysis to identify their own shortcomings and changes needed to their own practice to ensure student success ( Bondy & Ross, 2008; Ladson Billings, 1995). By communicating high expectations and the idea that all students are capable of learning, mathematics teachers can build student confidence in mathematics and personal identi ty. Transitions and movement between tasks become easier as students are able to self direct themselves through varied learning activities while the teacher walks around the room, facilitating activities and generating discussion (Bonner, 2014). The cla ssroom environment is task focused and calm to ensure that everyone has the ability to concentrate and learn. Teachers are kind, but firm, and use both verbal and non verbal communication to make expectations clear without demeaning students (Bondy et al. 2007; Ross et al., 2008). For instance, teachers may repeat requests verbally or remind students of expectations by miming appropriate actions or simply moving closer to misbehaving students to encourage more focus on assignments (Ross et al., 2008). W hen behaviors do not meet expectations, culturally responsive teachers follow through on consequences that are clearly communicated to students in advance (Bondy et al., 2007; Bondy & Ross, 2008; Ross & Adams, 2010 ; Morrison, Robbins, & Rose 2008 ; Ross et al., 2008). PAGE 31 31 When behaviors repeatedly do not meet expectations, culturally responsive teachers collect data to better understand situations, reaching out and respectfully listening to students in order to gain insight regarding how best to intervene and improve behavior (Bondy & Ross, 2008). Culturally responsive teachers engage in deep reflection of their own thoughts and feelings regarding working with particular students, consider alternative explanations for behaviors, monitor their own assumptions, and collaborate with colleagues as well as families of students to plan how to change the classroom environment or take action to improve outcomes. Once a plan is developed, culturally responsive teachers continuously revisit the reflection process and r eassess progress (Dray & Wisneski, 2011). Engaging L essons and L earning A ctivities Teachers that effectively implement culturally responsive classroom practices understand that there is a positive association between student engagement and academic achieve ment. Student engagement is a multidimensional construct consisting of behavioral, emotional, and cognitive factors. Behavioral engagement factors draw on the idea of participation in educational and extracurricular activities in terms of effort, attenti on, and positive versus disruptive behaviors. Emotional engagement factors include positive and negative reactions to teachers, classmates, academics, and school engagement conce willingness to exert the effort necessary to understand complex ideas and master difficult skills ( Fredricks, Brumenfeld, & Paris, 2004 ). Culturally responsive teachers work to incorporate high interest, engaging activities to combat previous experiences that have caused students to become PAGE 32 32 disinterested in mathematics (Hill, 2010). Routes to student engagement may be social or academic and include opportunities for participation, interpers onal relationships, and intellectual endeavors ( Fredricks, Brumenfeld, & Paris, 2004 ). Culturally responsive activities that ensure students have positive first encounters with content before moving to the more challenging parts of lessons (Gay, 2010; Morrison, Robbins, & Rose 2008 ). They attend to student learning styles by allowing opportunities for collaboration, movement, hands on learning, and choice in activities or asse ssment format ( Morrison, Robbins, & Rose 2008 ). Content is presented in a variety of formats and thoroughly explained to ensure that all learners understand prior to moving on to new material (Bondy & Ross, 2008). By providing opportunities for meaningf ul participation in learning, linking curriculum to students, and incorporating experiential learning and group processes throughout the curriculum, teachers create a more equitable learning environment (Bondy et al., 2007). According to the National Cou ncil of Teachers of Mathematics (2014), effective mathematics teaching incorporates activities that are specifically designed to foster student engagement in learning mathematics. Specifically, mathematics teachers should design lessons that enable studen ts to: engage with challenging tasks that involve active meaning making and support meaningful learning; connect new learning with prior knowledge and informal reasoning and, in the process, address preconceptions and misconceptions; acquire conceptual kno wledge as well as procedural knowledge, so that they can meaningfully organize their knowledge, acquire new knowledge, and transfer and apply knowledge to new situations; PAGE 33 33 construct knowledge socially, through discourse, activity, and interaction related to meaningful problems; receive descriptive and timely feedback so that they can reflect on and revise their work, thinking, and understandings; and develop metacognitive awareness of themselves as learners, thinkers, and problem solvers, and learn to monito r their learning and performance. (National Council of Teachers of Mathematics, 2014, p. 9) In contrast to many teachers that believe that students learn mathematics best through memorizing facts, procedurally using formulas, and practicing skills repeated ly, the culturally responsive and effective mathematics teacher centers lessons around students. By carefully selecting activities and facilitating discussions that promote reasoning and problem solving, teachers work toward the goal of helping students m ake sense of mathematical concepts. Summary I n this chapter, I review ed literature to identify challenges teachers face when working with diverse groups of students who have previously been unsuccessful in mathematics courses. Specifically, I discussed ho w curricular tracking, disproportionate representation, and individual teacher beliefs, biases, and prejudices can negatively impact students. In addition, I described how remedial mathematics teachers are challenged to combat student lack of interest in learning mathematics. I defined culturally responsive teaching, and I explored the culturally responsive practices of relationship building, having high expectations, and designing and delivering engaging lessons and learning activities. In the next chap ter, I will describe my research methods and how I used practitioner research to increase my own learning about what it means to be a culturally respo nsive teacher in my context. PAGE 34 34 CHAPTER 3 RESEARCH METHODS To gain insight regarding what it means to be a culturally responsive teacher in the context of my inclusion Algebra 1 class for repeating ninth graders, I used practitioner research to examine my practice and reflect on my instruction. Practitioner resea rch is knowledge from experience and other sources (Cochran Smith & Lytle, 2009; Dana & Yendol Hoppey, 2014, York Barr, Sommers, Ghere & Montie, 2006). As Bonner (2014) sug gested, successful mathematics teachers of traditionally underserved students reflect and revise techniques constantly with an explicit focus on the classroom environment and cultural connectedness. It was my intention to critically examine student behavi ors and perceptions as well as my own practices as a mathematics educator in order to construct knowledge about culturally responsive teaching within my context. Over the course of Unit 10, Exponential Functions, I wrote detailed lesson plans and intentio nally worked to increase the use of culturally responsive pedagogy in my instruction. Instruction for Unit 10 took place towards the end of the spring 2017 semester and lasted approximately 3 weeks. The key concepts for the unit included evaluating and g raphing exponential functions, modeling growth and decay using exponential functions, writing and using recursive and explicit formulas for geometric sequences and arithmetic sequences, and comparing linear, quadratic, and exponential functions graphically tabularly and verbally (Anderson School District 5, 2016). I chose this unit as the focus of this study because these concepts are particularly applicable to real world problem solving. The lesson plans are provided in the A ppendix PAGE 35 35 As the lessons that I planned were implemented, I carefully observed my Algebra I students and their engagement in mathematics and problem solving while collecting data. To measure behavioral engagement, I observed and documented discussions with my students, particularly documenting the presence or absence of positive conduct, the level of involvement in learning tasks, and contributions to class discussions in my field notes. For example, I documented when students were writing, talking about m athematics, asking questions, and assisting their peers. I also documented instances classroom and to my instructional practices ( Fredricks, Brumenfeld, & Paris, 2004 ). For exa mple, I documented students resisting working with particular students or eagerness to get started with activities. I took notes of my observations and wrote reflections in my researcher journal regarding equity issues that arose during class or discussio ns with students. For example, I wrote down my questions or ideas about how I could better communicate with particular students based on my observations and experiences. Each lesson was recorded by video to capture discussions and activities that took pl ace during the lesson. After the lesson, I watched the videos and added notes to my researcher journal in order to remind myself of important details and support my thinking. I conducted interviews to better understand the background and experiences of i ndividual students in order to inform instructional practices. In addition, I reviewed student work samples to reflect upon how well I communicated high levels of expectation regarding student effort and achievement using equity as a critical lens. Prac titioner research was appropriate to answer my research questions because I examined my own practice and reflected upon my own instruction in an attempt to PAGE 36 36 construct new knowledge about my specific practices in my specific context. This study documented m y practices and student responses that occurred as a result of my work as a practitioner researcher. This study emerged from my own sense of urgency to find ways to increase equity for students in mathematics courses through culturally responsive teaching Context This study took place within an Algebra 1 inclusion class for repeating ninth graders at TL Hanna High School in Anderson, South Carolina, during the 2016 2017 school year. In 2014 2015, 52% of students in South Carolina were white, 37.3% were African American, 2.1% were Asian/Pacific Islander, 8% were Hispanic, and 0.6% were American Indian/Alaskan, while 77.6% of South Carolina teachers were white, 15.2% were African American, 1.1% were Asian, 1.1% were Hispanic, 0.2% were American Indian, and 4.8% were not reported. Statewide, the graduation rate for 2014 2015 was 80.3%. 82.7% of White students, 76.7% of African American students, 91.1% of Asian/Pacific Islander students, 77.1% of Hispanic students, and 79.5% of American Indian/Alaskan studen ts graduated (South Carolina Public Schools, 2016). TL Hanna High School is located in Anderson School District 5, and is one of three high schools in the district. In 2015, 1 773 student s attended TL Hanna High School and 90.3% of students at the schoo l scored 70% or higher on the Algebra I/Math for the Technolog ies 2 End of Course test. Statewide, 85.7% of students in South Carolina scored 70% or above on the Algebra I/Math for the Technologies 2 End of Course test in 2015. In 2015, the School Four Year Cohort Graduation Rate at TL Hanna High was 83.3% (South Carolina Department of Education, 2016 ). PAGE 37 37 Participants The participants in this study were students in an Algebra 1 inclusion class for repeating ninth graders at TL Hanna High School in Anderson, South Carolina. Inclusion courses in South Carolina are general education courses in which studen ts who receive special education services are included with modifications to instructional practices and/or assessments. Since TL Hanna High School operates on a 4 x 4 block schedule, each class session was 90 minutes in duration. Unlike other courses at TL Hanna that only last for a single semester, this class met for the entire school year. Since the class was an inclusion class, a special education teacher joined me to assist and co teach for 45 of the 90 minutes daily. As of April 2017, 12 students were enrolled in the course. At the beginning of the school year, the class had 20 students enrolled, but eight students were withdrawn prior to April. Of the eight withdrawn, 2 students were expelled for discipline issues, 2 students moved to other sch ools within the district, 1 student moved to another district, and 3 students were dropped for non attendance. Of the 12 students that remained in the class and were present at the time of this study, five students were female and seven were male. Five s tudents were black and seven were white. Three of the students were 15 years old, six were 16, and three were 17 at the time of the study. Eight of the 12 students had excessive absences from the course (more than 10 days absent). Of the eight students with excessive absences, five had missed more than 20 days of class. Based on their annual household income, 10 of the 12 students qualified for free lunch. Two of the 12 students had IEP accommodations as follows: Student 1: Calculator PAGE 38 38 Student 2: Pre ferential seating (instructional), calculator, small group setting for quizzes and tests, and oral administration One of the 12 students had accommodations of preferential seating near the source of instruction, time and a half for tests and use of calcula tor based on a 504 plan. Data Collection Practitioner research is different from other research in that the work is specific to the context of the researcher, requiring data that documents classroom practices, student learning, and the thinking and refle ctions of the researcher (Cochran Smith & Lytle, 2009). As I worked to create a more equitable learning environment for repeating ninth graders in my Algebra I classroom through culturally responsive classroom practices and pedagogy, I collected data to g ain insights into my research questions through (1) student interviews, (2) a researcher journal, (3) observation/field notes, (4) lesson videos, and (5) student work samples. Student Interviews Individual interviews are a way to capture perceptions throu gh discourse. In addition to regular, unstructured interviews in which I asked students about their thinking and learning during naturally occurring classroom conversations, I also conducted semi structured, more formal interviews with students (Creswell, 2013; Dana & Yendol Hoppey, 2014) Notes about the unstructured interviews were recorded in my field notes on a daily basis. The semi structured interviews were conducted at the beginning of the unit to better understand the background and experiences o f individual students in order to inform instructional practices. The focus of the interviews was elements of culturally responsive classroom practices and pedagogy, specifically relationship building, high expectations, and engaging lessons and learning activities. PAGE 39 39 The prompts below guided the initial interviews, and additional questions were posed Tell me a little about yourself. Tell me about your family. Describe your relationship with your classmates. Tell me about a time when you really liked being in class. Tell me about a lesson that made you feel smart or successful. Tell me about a lesson that you did not like. Tell me about a lesson that you struggled with. Based on what you know about me and this class so far, what things do you think are most important for you to keep in mind in order to be successful in Algebra 1? As you look back on my actions in class so far this school year, what stands out for you? The semi structured interviews were audio recorded a nd transcribed ( Creswell, 2013). Researcher Journal After each teaching session, I reflected on my teaching practices, student actions and conversations, and student work in my researcher journal. By keeping a journal, a person can capture his or her own thinking ( Dana & Yendol Hoppey, 2014) As part of my reflecting on the work that took place in the classroom, I documented successes and areas of need and recorded my thoughts about what steps I should take next to create a more equitable learning enviro nment and improve my practice through culturally responsive classroom practices. Each day I used the same three questions to guide my journal reflection and writing: (1) What did I do today that related to being a culturally responsive teacher?, (2 ) How did students respond?, and (3) What did I learn today PAGE 40 40 about being a culturally responsive teacher? Figure 3 1 illustrates one page from my researcher journal. Figure 3 1 Printed teacher researcher journal, day 1 (Photo courtesy of author) Observation/Field Notes To capture actions as they took place in the classroom, I took detailed field notes to record my observations. Field notes included scripts of dialogue or conversations, classroom diagrams, or notations about what individuals said or did. I attempted to keep my field notes non judgmental and focused on capturing what was occurring (Creswell, 2013; Dana & Yendol Hoppey, 2014). During each class, I recorded notes to capture the behaviors and statements of myself and students. Lesso n Videos Each lesson was recorded by video to capture discussions and activities that took place during the lesson. The video camera was placed in the back of the classroom to record general lesson procedures and interactions. After each lesson, I watche d the video and added additional notes to my field notes and researcher journal PAGE 41 41 in order to remind myself of important details and support my thinking. By observing lev els, nature of interactions, nonverbal behavior, instructional clarity, and the influence Hoppey, 2014, p. 108). When taking notes using the video, I was sure to include both descriptive and reflective notes. My d escriptive notes summarized the flow of activities in the classroom in a chronological way, while my reflective notes focused on the process and conclusions that I made as a result of observing the video (Creswell, 2013). Student Work Samples Student work as artifacts were generated as a normal part of my Algebra I class. Figure 3 2 illustrates an example of student work that was collected during this study. Figure 3 2 Example of student work (Photo courtesy of author) PAGE 42 42 By systematically c ollecting, labeling, and organizing student work documents, I was able to look within and across documents to analyze them and better understand changes in student learning and make instructional decisions. Student work throughout the course of the unit r evealed shifts that I would not have been able to see when looking at a single work sample in isolation (Dana & Yendol Hoppey, 2014). I particularly looked at student work samples to reflect upon how well I communicated high levels of expectation regardin g student effort and achievement. Data Analysis To increase my understanding about what it means to be a culturally responsive teacher for students enrolled in an Algebra I course for repeating ninth graders, I used both formative and summative data anal ysis. In contrast to the research in which the researcher collects all data prior to analyzing it, I analyzed data during the study so that I could adjust my practices as the unit unfolded. When all the data were collected, I then reviewed the data in th eir entirety, paying attention to how each piece of data may be related and reflecting on what I learned. Initially, student interviews were recorded for audio, transcribed, and coded with themes. To code the interviews, I read the transcriptions entirely and made notes about important details that related to my research question and sub questions in the margins (Creswell, 2013). For example, I noted student comments about expectations for behavior in my class and their personal preferences about learning From the notes that I wrote in the margins, I developed initial themes for the interviews (Table 3.1) expectations regarding behavior, perceptions of expectations r egarding academics, positive relationships with me as their teacher, positive relationships with their peers, PAGE 43 43 preferences of classes in which feelings of success occur, and specific learning styles or preferences Table 3 1. Student interview themes. Stud ent Interview Theme Examples of Student Responses Perceptions of Expectations Regarding Behavior . I think you do . not play around so much, and do my work and turn Perceptions of Expectations Regarding Academics Positive Relationships with Me as their Teacher re always happy in the morning . them time to do work and you . When you Positive Relationships with their Peers Preferences of Classes in which Feelings of Success Occur . enjoy graphing . Specific Learning Styles or Preferences . I act ually do better with like actual music going . hands on stuff, I reflected on these themes as I wrote and modified the lesson plans for the unit of study. For example, I strived to incorporate a lot of hands on and collaborative PAGE 44 44 learning activi back to interview data when I reflected on particular student behaviors throughout the unit in order to try to make sense of things that were happening in the classroom. For exam ple, when particular students did not respond well to my instructional decisions, I referred back to their interview responses to try to better understand individual perspectives. Throughout the unit of study, I adjusted my practices as a result of ongoi ng formative data analysis. After each class session throughout the unit, I reflected on my teaching practices, student actions and conversations, and student work in my researcher journal. At the end of each day, I watched the video of the lesson and ad ded notes to my researcher journal. By using my journal, I worked to capture my own thinking, document successes and areas of need, and collect data to inform instructional decisions for future class sessions. For example, I revised some lessons to have group activities and removed some group activities from other lessons as a result of my observations of student behavioral engagement during specific activities. Classroom successes were identified by evidence of relationship building, effective communica tion of high expectations, and student engagement. For example, I noted when students talked about mathematical concepts, answered questions, asked questions, assisted others, took notes, took a leadership role during group processes, and followed the dir ections for lesson activities. Areas of need were identified when relationships could be improved, expectations are unclear, and student engagement was not apparent. For example, I documented when students did not follow directions and inappropriate clas sroom behaviors that I observed. I noted how my thinking on PAGE 45 45 following day. For example, I wrote down my thoughts about how I might improve how students are grouped fo r activities, the physical classroom arrangement, student activity guides, or other aspects of lessons. I made changes as needed, and I reflected on the outcomes after I implemented the changes. Throughout the unit of study, my field notes and student w ork samples were analyzed using equity as a critical lens to reflect upon how well I communicated high levels of expectation regarding student effort and achievement. For example, if I noticed that particular groups of students followed directions more cl osely than others or that particular groups of students performed higher or lower than others, I modified the following lessons accordingly to communicate with all students better, and the changes were noted in my field notes. In addition, student work th roughout the unit was systematically collected, labeled, and organized so that I could look within and across documents for shifts that occurred over time (Dana & Yendol Hoppey, 2014). After all data were collected, I reviewed the data in their entirety fo r summative analysis at the end of the unit. This process involved re reading transcripts, re reading my researcher journal, and reviewing the video data multiple times. As I reviewed the data, I wrote memos in the margins of documents to record ideas an d concepts that occurred to me as I read. For example, my notes at this point included short transcripts from classroom conversations, details about the number of students that were off task at specific times during the lessons and the number of students that were on task as specific times during the lessons. Then, I jotted down short notes about details within my researcher journal and interview transcripts on post it notes. I PAGE 46 46 made decisions about what to be included (and not included) in these short no tes by focusing on my research question and sub questions. I labeled and used a particular pen for each post it note so that I could easily refer back to the full piece of data. For instance, notes about details from my researcher journal were written in green and labeled according to the number of the lesson (day 1, day 2, etc.). I sorted the post it notes by placing them on posters that I created for each sub question for my study. When I determined that a piece of data applied to more than one questi on, I duplicated the post it note so that it would appear on all the posters for which the data applied. Then, I reviewed all the videos again, adding additional notes to my researcher journal and post it notes to the posters. Figure 3 3, Figure 3 4, and Figure 3 5 illustrate the posters I used in the data analysis process. Figure 3 3. Posters for sub question 1 (Photo courtesy of author) PAGE 47 47 Figure 3 4. Posters for sub question 2 (Photo courtesy of author) Figure 3 5. Poster for sub question 3 (Photo courtesy of author) PAGE 48 48 When thinking about coding the data, I expected to find information related to culturally relevant pedagogy, but I also looked for code segments that represented interesting or unusual information that I did not anticipate (Creswell, 2013). For instance, anticipated themes/code segments included themes from literature about culturally responsive teaching such as practices that demonstrated caring, behavior expectations, academic expectations, academic support, and engaging lessons. However, I also underst ood that there may be code segments that I did not anticipate that are related to mathematics or instructional practices. The posters assisted me in grouping the post it notes that were related. I worked on each poster separately, and the initial groupi ngs were more general and less specific. For instance, when grouping the post it notes that I wrote about relationships, I initially divided all the notes into two categories: student teacher relationships and relationships among students. The notes abou t high expectations were divided into behavior expectations and academic expectations. When grouping notes about student engagement, I initially divided the notes based on the activity taking place (group work, notetaking, whole class discussion, independ ent practice). Within these general categories, I created sub groups in an effort to make more specific groupings. As I grouped the post it notes again, I wrote down possible codes and themes along with additional notes regarding my thinking about the d ata. Possible codes and themes at this stage in the analysis regarding relationships included increasing my own understanding of students, the physical arrangement of the classroom, student resistance to working with new and/or different people, students appearing uncomfortable answering questions unless they were confident, students PAGE 49 49 afraid of having the wrong answer, and dependence on teacher support. Possible codes and themes at this stage in the analysis regarding high expectations included the ways in which expectations were communicated (verbal, modeling, physical cue, etc.) and consequences for misbehaviors as well as specific expectations that were communicated multiple times such as respecting c lassmates, following directions and/or completing wor k, listening/focusing/paying attention, explaining reasoning, and thinking deeply about mathematical concepts. Possible codes and themes at this stage in the analysis regarding student engagement included engagement in whole class discussions and notetaki ng, engagement in group work, disengagement in whole class discussions and notetaking, disengagement in group work, and interventions. I also reviewed student work to search for evidence to support the ways in which I was thinking about the data. For in stance, I specifically looked at student work samples to reflect upon how well I communicated high levels of expectation regarding academics since my observations were not enough to determine whether or not students clearly understood the expectation to th ink deeply about mathematical concepts and to explain their reasoning when solving problems. As themes and stories emerged during the data analysis, I referred back to my research questions to determine the relationship (Dana & Yendol Hoppey, 2014). As I reread and reviewed the data multiple times, I was able to refine codes. For example, students appearing uncomfortable answering questions unless they were confident, students afraid of having the wrong answer, and dependence on teacher support were dete rmined to all be related to trusting relationships. I then organized the final codes by sub question, as shown in Table 3 2. PAGE 50 50 Table 3 2. Finalized codes. What are relationships between students and myself as well as among students like? Code/Theme Exampl es of Evidence from Data backgrounds and learning styles Two way communication between myself and students about their home lives, community, and funds of knowledge My own reflection and modification of lessons based on observing students Trust of me as the teacher Students asking for teacher help Student talk about areas of weakness/problems Students taking risks (or not) Student resistance to working with particular students Resistance to randomly assigned groups Trust among students and student risk taking in mathematics Students asking for peer help/support (or not) Talk about areas of weakness/problems among peers Students sharing answers confidently (or not), even when there is a risk of being incorrec t How are high expectations for academics and behavior communicated and enacted in my Algebra 1 class ? Code/Theme Examples of Evidence from Data Communicating expectations to think deeply about mathematical concepts and to explain reasoning Students writing down the formula that they selected to use, showed what numbers were substituted into the formula, and showed steps that were used to simplify the expression Students effectively communicating their reasoning when solving problems Studen ts making connections that they saw to prior learning Communicating behavior expectations Behavior expectations provided in multiple formats Students self directing behaviors and transitioning smoothly between tasks Following through with consequences for inappropriate behaviors Me following through with consequences for misbehavior How do students engage in learning activities in my Algebra 1 class, and how might student engagement be negotiated depending on their individual backgrounds and prior experiences? Code/Theme Examples of Evidence from Data Engagement in Collaborative Group Work Collaboration, movement, hands on learning Students writing, talking about math, using the materials appropriately, and solving problems Disengagement in Collaborative Group Work Students working independently and talking about non math topics during group work Students not working together to solve problems together Engagement in Teacher Led Discussions, Notetaking, and Independent Practice On task behavi ors such as students writing in their notebooks, verbally responding questions, and watching me as I modeled activities Disengagement in Teacher Led Discussions, Notetaking, and Independent Practice Off task behaviors Me reminding students to either get s tarted or to stop disturbing their classmates PAGE 51 51 The finalized codes developed are shown by sub question along with examp les of evidence from the data (T able 3 2) These findings are discussed in more detail in the next chapter. When choosing specific incidents in the data to share in the results, I narrowed the data to include critical incidences that sparked my own learning or changes in my own practice in orde r to become more culturally relevant. Researcher Positionality Creswell (2013) defined qualitative inquiry as including not only the voices of participants and a complex description of the problem, but also the reflexivity of the researcher. As I engage d in the process of practitioner research, it was important to be cognizant of who I am and what I brought to this study. I wanted the reader to understand any biases or assumptions that may have impacted this study. In this section, I describe my past e xperiences and orientations that are likely to shape the interpretation of data for this study. I was raised by a poor single mother in a rural area of Louisiana. From an early age, I believed in the idea of meritocracy. I saw education as a means to esc ape poverty and to better myself for my family, and I believed that my hard work would pay off. I graduated valedictorian of my high school class, and I was the first person in my family to attend college. Looking back on my K 12 experiences now that I a m adult, I see ways in which I was, and ways that I still am, privileged in relation to others. For instance, I see how being white has afforded me advantages over others that I have not earned. I also see ways in which I overcame the disadvantages I exp erienced in my youth. For instance, I realize how my belief in my own ability to succeed was largely influenced by my teachers, with whom I developed strong relationships with. I see the PAGE 52 52 network of people that assisted me in navigating school and getting through college. I brought these experiences to this study, along with a strong desire to help kids succeed. At the time of this study I had been a mathematics educator for 11 years. Prior to August 2016, all of my experience as an educator took place in Florida, six years at a local high school and four years at a virtual school of choice. Although I always had a personal commitment to working to assist students who ha d not previously been successful in mathemat ics courses, this was my first experienc e teaching a class in which all the students did not pass in the previous year. It was also my first experience teaching an inclusion class and on a 4 x 4 block schedule. I realize that my past teaching experiences influenced the decisions I m ade as the teacher and as the researcher in this study but I also believe that, in many ways, I ventured into the unknown. I had a strong desire to learn from this experience, and I was motivated to do everything I could to see students in this class succeed. Enh ancing Trustworthiness In the previous section, I described my positionality as the researcher in this study. To further enhance the trustworthiness and validity of my research, I used the process of triangulation and rich, thick descriptions. Creswell (2013) defined journal, observation/field notes, and student work samples as dat a. By developing rich, thick descriptions of the context, participants, and events, I attempted to assist readers in making decisions regarding the transferability of my work. PAGE 53 53 Limitations of the Study This study is a small glimpse at my experiences as a teacher in an Algebra I class for repeating ninth graders at TL Hanna High School in Anderson, South Carolina. It is not meant to be representative of all Algebra I classes nor all classes of repeating ninth graders. Schools and classrooms vary greatly depending on geographical region, funding, organization, student demographics, and other factors. Individual teacher experiences can also vary widely, and I realize that this study was impacted by my own background and prior experiences as a white female educator with 10 years of experience teaching in public high school mathematics classrooms. However, I hope that teachers in other contexts will be able to transfer important concepts from my experiences during this stu dy to their own settings One limitation of this study is that it was conducted within a single unit at the end of the school year. This study does not capture the events that occurred within my classroom prior to the unit of study or after the unit of s tudy. The limited time frame may have impacted my ability to capture my own professional growth as well as the academic and personal growth of my students throughout the school year. Regardless of this limitation, this study serves as an important contri bution to existing research about culturally responsive teaching and mathematics education and has implications for teachers beyond my context and for future research Summary In this chapter, describe d my research methods and how I used practitioner rese arch to increase my own learning about what it means to be a culturally responsive teacher in my context. Specifically, I described the context, participants, data collected, and data analysis process for this study. In the next chapter, I will present m y findings. PAGE 54 54 CHAPTER 4 FINDINGS This study focused on increasing my understanding of what it means to be a culturally responsive teacher in the context of an Algebra I class for repeating ninth graders. Within the central question, I developed the followi ng sub questions: 1. What are relationships between students and myself as well as among students like? 2. How are high expectations for academics and behavior communicated and enacted in my Algebra 1 class? 3. How do students engage in learning activities in my Algebra 1 class, and how might student engagement be negotiated depending on their individual backgrounds and prior experiences? As described in the previous chapter, in an attempt to answer my research questions, I collected the following forms of data: ( 1) student interviews, (2) a researcher journal, (3) observation/field notes, (4) lesson videos, and (5) student work samples. My findings are organized by the sub questions developed for this study. What are R elationships between S tudents and M yself as w ell as a mong S tudents L ike? In this section, I will use actual classroom interactions from the data to describe relationships between students and myself as well as among students. The section will be organized to focus on two types of relationships separ ately. I will end this section with a summary of the section to answer Research Question 1. Relationships between Students and Myself One of my goals in becoming more culturally responsive is to practice more authentic caring for my students. In contrast to aesthetic caring, which focuses primarily on adherence to school rules and academic achievement, authentic caring focuses on relationship building between school personnel and students, families, and communities PAGE 55 55 ( Ross & Adams, 2010 ; Gay, 2010; Schillwe r, 2008). In my daily interactions with study, I intentionally worked to increase m backgrounds and learning styles so that I could better plan learning activities that would meet individual student needs and interests and increase student engagement in the learning process. Increasing my backgrounds and their daily life experiences. I specifically asked students about their home lives in order to become more aware of their individual backgrounds and daily life experiences. I learned that many of my students came from low income, single parent families. Two of the seven students interviewed did not have relationships at all with either one of their biolo gical parents, and two other students of the seven students interviewed lived with extended family members and not their biological parents. Six of the seven students interviewed had other children living with them as part of the household, and all six de scribed having to assist with the care of younger children in the household. One of the students interviewed also worked part time after school and on weekends, one of the students interviewed enjoyed playing sports in his neighborhood, and another enjoye preferences. For instance, during the interview process, I noted that three out of the seven students interviewed described h ands on learning experiences as lessons that they enjoyed. Two students indicated that they prefer to work alone, two students PAGE 56 56 specifically indicated that they liked working with other students in groups or as a team, and one student described himself as a visual/auditory learner. Five of the seven students described struggling particularly with mathematics. Three of the five students specifically mentioned having difficulty remembering uggle sometimes with knowing what the signs are. Like telling whether you go down or up when you add or I lessons at the beginning of the school year. Another student described disliking learning about transformations of parent functions because he struggled with memory. The other two students that indicated struggles with mathematics specifically mentione d not liking word problems and having difficulty with multi step problems. Five out of the seven students interviewed indicated that they felt successful during lessons or classes in which they found learning easy or caught on quickly. As the Unit 10 le styles and preferences through interactions that took place in the classroom and by observing student behaviors. I intentionally looked for students who were demonstrating behavioral enga gement during lessons, which students were eager to assist others, which students were seeking help from me, and which students were seeking help from their peers. Behavioral engagement factors draw on the idea of participation in educational and extracur ricular activities in terms of effort, attention, and positive versus disruptive behaviors ( Fredricks, Brumenfeld, & Paris, 2004 ). After each lesson, I reflected on my own learning, and I planned future lessons accordingly. As I PAGE 57 57 increased my understandin became even more aware of how establishing and maintaining trusting relationships between students and myself as their teacher impacted student interest and motivation to learn. T rust ing rela tionships I wanted my students to feel safe to take risks, to feel respected, and to trust me as their teacher (Bondy et al., 2007; Bonner, 2014; Gay, 2010; Gorski, 2013; Morrison, Robbins, & Rose, 2008; Ross et al., 2008; Schillwer, 2008). In general, the data indicated the presence of a trusting relationship between my students and me in that the students viewed me as someone who would help them with their lessons. However, some of the data suggested that, at times, particular students either relied t oo much on my support or did not seek support from me at all. Student interview responses as well as particular student behaviors indicated that students trusted me to assist them with problems and answer their questions in class. For instance, during to work with everybody individually . you give them time to do work, and you help students were observed asking me questions, asking me to check their work, or requesting my support. On a few occasions, I noticed stu dents not working at all unless I was physically near them, answering questions and/or assisting. For example, during the first lesson, when exponential growth was initially introduced, I noticed that some students did nothing while I was assisting other groups. I documented my thoughts and questions PAGE 58 58 about how to be more effective in scaffolding support and encouraging more peer interaction as I reflected on my own learning in my researcher journal: Several students did not begin the task until I moved cl oser to the desk. Some still did not begin working until I modeled how to fold the paper with them at their desk and even pointed to where they needed to be writing on the activity guide . When I told students they could work together, they did not ta lk as much as I anticipated. Some students did nothing while I worked with other groups. Perhaps they gave up until they got more support from me. In contrast to incidences in which students appeared to rely too much on my support, there were also inst ances in which I noticed particular students rarely requested support from me or asked questions. When I watched the video of the second lesson, I noticed that I spent less time with these particular students. As a result of my reflection, I was more con sciously aware of my need to look at their work, provide feedback, and solicit their questions. I also wondered if I would spend more time assisting them if they were seated closer to the front of the room, and I decided to also vary the position of stude nts for future lessons. After the fifth lesson, I documented how I attempted to assist students that did not solicit my support in my researcher journal: [They] had not asked me any questions, but they appeared to be working, so I stopped at their table t o provide feedback. They said very little as I provided them with feedback, only nodding their heads to my comments. Later during the lesson, I asked if they had any questions, they shook their heads no. One possible explanation for incidences in which students relied heavily on my support or had limited needs for my support is their relationships with peers. For instance, students may have felt more or less comfortable asking their classmates questions than asking me questions. Next, I will summarize the findings regarding relationships among students. PAGE 59 59 Relationships among Students When I interviewed students, five out of seven students responded either positively or neutrally when asked to describe their relationship with their classmates. Two students specifically indicated that they liked working with other students in groups or as a team, and two students indicated that they prefer to work alone. When planning lessons and setting up the physical classroom arrangement for specific activities, I thought deeply about how students might interact with one another based on what I learned from the interviews and from the previous lessons. I wanted my lessons and classroom environment to promote the development of a caring classroom community in whi ch students interact ed in positive ways and discussed mathematical concepts in order to increase mathematical understanding (National Council of Teachers of Mathematics, 2014). My goal was for students to feel safe and comfortable enough to take risks, re spected by their peers, and trusting of one another (Bondy et al., 2007; Bonner, 2014; Gay, 2010; Gorski, 2013; Morrison, Robbins, & Rose, 2008; Ross et al., 2008; Schillwer, 2008). Resistance to working with particular students. While the interview da ta from specific students suggested that they had already established relationships with their classmates, the data collected during the lessons suggested that some of those students still resisted working with particular students in the class. I was surp rised that four of the students resisted working with randomly assigned partners, while the majority of students, including the two students that indicated they preferred to work alone during the interviews, complied with my directive to pair with a random student. For instance, for the group activity in the seventh lesson, I asked students to draw numbers in order to randomize which students were paired together as well as their PAGE 60 60 physical position in the room. Four students resisted being randomly paired with explained my reasoning for the decision to the students that resisted I also followed through with consequences for students that still did not follow the directive, but the resistance made me question my decision. I wrote the following in my researcher jou rnal after the seventh lesson: I am still unsure about the best w ay to group students. On one hand, I want them to build relationships and step outside of their comfort zones in order to work with other students. I also want to limit the interactions that are not related to the topic of study. On the other hand, the resistance to working with students that are chosen at random rather than by the students themselves is difficult to deal with. The next day, I once again used random assignment to organize students into pairs for a group activity to explore geometric se quences. To assist the students that resisted the random assignment of groups in the previous lesson, I personally spoke to them before class to explain my reasoning for randomly assigning groups. I told them that I wanted them to step outside of their c omfort zones and learn how to work with people that they may not normally work with. I asked for their cooperation, and they agreed. All of the students completed the activity and correctly answered the questions for practice at the end of the activity. During my observation, I noticed students writing, talking about math, using the materials appropriately, and solving problems. I was surprised that, even during the practice time after the activity, when I told students that they could choose to work in dependently or with their partner, many students chose to continue working with their randomly assigned partner. PAGE 61 61 Trust among students and student risk taking in mathematics. Throughout the unit, I noticed that many students seemed to lack trust or feeli ngs of support from their classmates. In general, students were resistant to share or discuss their mathematical thinking when they were unsure about whether or not they were correct. For instance, during the fourth lesson, when students had to identify data and graphs as exponential growth, exponential decay, or neither with a partner, students engaged in meaningful discussion only after coming to their own conclusions. In general, students wrote their own answers, compared answers with the group, and d iscussed reasoning only after I prompted them. When I asked one group of students why they had responded in the ways that they did, they appeared to think that my questioning indicated an incorrect response. One of the students scratched out his response s when I asked him why he had responded that way. Another group of students refused to write their responses at all. When I asked why, they said that they were using pen and wanted to know they were correct before committing the response to paper. I fur ther probed to see if they had discussed the reasoning with their partner, they indicated that they had only compared answers and agreed. I reflected on my need to improve the learning environment to encourage risk taking in my researcher journal: Student s did not seem confident in their responses, as they were unwilling to risk being incorrect when asked why they responded the way that they had. Students generally did not talk about mathematics or work together to solve problems when given a practice act ivity. One student appeared to be waiting for the others in the group to complete the questions so that he could copy down the answers. When using group work, I need to ensure that the task requires the input from all of the students in the group. I als o need to work on improving the learning environment so that students feel safe taking ri sks, even if they are wrong. PAGE 62 62 Answering Question 1: Classroom Relationships Within the context of my Algebra 1 class for repeating ninth graders, building relationsh ips with and among students was particularly challenging. During this study, I and learning styles. During the initial interviews, I learned that many of my students ca me from low income, single parent families, and some lived with extended family members instead of their biological parents. I increased my understanding of my enjoye d or did not enjoy and by observing students in class. Consistent with literature regarding low level and remedial courses, my class had a disproportionate number of students of low SES and of color in comparison to the larger school population. By learn classism, or other forms of discrimination, combined with the history of academic failure could lead my students to have negative perceptions about their own abilities (Gay, 2010; Nieto, 2013). I also saw how these negative perceptions impacted their willingness to work with and trust teachers and/or peers. While the data indicated that students trusted me to help them with their lessons, at times, particular students either reli ed too much on my support or did not seek support from me at all. Although five out of seven students responded either positively or neutrally when asked to describe their relationship with their classmates during the interviews, the data collected during the lessons suggested that some of those students still resisted working with particular students in the class. In general, students were resistant to share or discuss their mathematical thinking when they were unsure about whether or not they were corre ct. T he importance of the right answer appeared to be PAGE 63 63 students acted as if they we re wrong It is possible that these incidences demonstrate a lack of trusting relationships S tudents may have had negative perceptions of their own abilities and/or the abilities of their peers Students could have had p reconceived notions about what their peers or teachers thought about them or remedial students in general due to their prior educationa l experiences Despite my efforts to plan learning activities that would meet individual student needs and interests, I sometimes felt unsuccessful in establishing an environment in which students felt safe to take risks and engaged in meaningful discussi ons about mathematical concepts. As I reflected on my own actions, I realized that I indeed held biases that I was not aware of. For instance, I noticed that I tended to spend more time with students that were vocal and asking me questions during class. When particular students rarely requested support from me or asked questions, I did not realize that I spent less time with them in comparison to the time I spent with other students until I watched the recording of the lesson. As I became more conscious of this bias, I also became more intentional about looking at student work, providing feedback, and soliciting questions. responding to questions, and I thought about possible reasons for their lac k of confidence. I realized the need to alter my questioning strategies and work to improve the classroom environment so that wrong answers were viewed as opportunities to learn rather tha n incidences of failure. I also realized my need to improve how I communicate expectations for both academics and behavior in order to increase positive interactions PAGE 64 64 among students. In the next section, I will discuss how high expectations were communica ted and enacted. How are H igh E xpectations for A cademics and B ehavior C ommunicated and E nacted in my Algebra I C lass? In this section, I will illustrate how high expectations were communicated and enacted in the classroom with illustrations of actual class room interactions and student learning from the data. Th is section will be organized to focus on two different kinds of expectations, academic and behavioral, followed by a summary of the section to answer the Research Question 2. Academic Expectations Co mmunicating expectations to think deeply about mathematical concepts and to explain reasoning. Student work samples collected throughout the unit indicated that students understood the expectation to show work to demonstrate thinking. For instance, when students were asked to solve word problems involving exponential growth, exponential decay, and compound interest, they wrote down the formula that they selected to use, showed what numbers were substituted into the formula, and showed steps that were used to simplify the expression. However, as I previously described, students were resistant to verbally share or discuss their mathematical thinking when they were unsure about whether or not they were correct. As I reflected on the student work samples fro m the first half of the unit, I realized that, while showing work on practice problems and assessments demonstrated student thinking to a certain extent, I also wanted students to effectively communicate their reasoning when solving problems or connections that they saw to prior learning. PAGE 65 65 As a result of my reflections of the first four lessons and student work samples, I intentionally worked to increase the number of students that discussed their reasoning and explained their thinking as they learned. I wanted them to talk through important concepts with their peers and think deeply about mathematical relationships (National Council of Teachers of Mathematics, 2014). I referred back to literature, and I thought revised the handouts and activity guides for the future lessons so that students were required to explain, describe speci fic details, and draw conclusions in writing. I also worked to scaffold my support during activities to allow time for students to think deeply about concepts and discuss with their peers. For example, for the fifth lesson, students were asked to graph multiple transformed functions and draw conclusions about changing h, k and a based on their observations and discussion with their partner. During the lesson, as I circulated the room and provided feedback, I intentionally commented about correct answer s and hints about how to be specific in their responses. When I noticed students struggling, I intentionally did not provide the answer. Rather, I asked more questi ons to assist students in coming to the correct conclusion on their own. For instance, when I noticed that a lot of students were struggling with domain and range, I attempted to intervene by of the graph. I pointed y students understood the expectations, I sat down in the middle of the room and waited. PAGE 66 66 Some of the students asked me to check their work while I waited. In these instances, I provided feedback verbally. When I reviewed the work from the fifth students explained their thinking when writing on the activity guide. All of the groups successfully recorded the correct domain for all of the transformed exponential functions. One group correctly recorded the range on 8 out of 8 functions (100% of the time), one group correctly recorded the range on 7 out of 8 functions (87.5% of the time), and two groups correctl y recorded the range on 5 out of 8 functions (62.5% of the h x 2 negative. Depending on whether the signs are positive or negative, the it along the x k affect k value, and the a affect a PAGE 67 67 es to negative, it reflects over the x 3 makes the line go over the x a is positive, it goes up. If it is negative, it goes down. When a While one of the groups was unable to make a correct generalization regarding the effect of k, all of the groups clearly understood that they needed to state the conclusion they made based on observations of various graphs in writing. Some groups even provided exa mples of specific functions to demonstrate their reasoning. During the seventh lesson, students worked in groups to identify patterns and explore arithmetic sequences. As I circulated the room during the partner activity, I noticed that many groups we re not following the directions to determine the number of circles that will be in the ninth group without making the 7 th and 8 th groups. I asked several groups to explain how they determined the number of circles in the ninth group, and when they indicat ed that they had added the common difference each time, I asked number each time with their partner. Following the arithmetic sequences partner activity, I led the class in a whole group discussion and modeled note taking using the document camera. During the whole class discussion, three students verbally shared their own examples and non examples of arithmetic sequences that they developed and correctly justified why thei r examples met the criteria, or why their non examples did not meet the criteria, for an arithmetic sequence. When looking at the example to determine whether or not a sequence was arithmetic, I intentionally provided a short period of silence to provide adequate time for students to think about the question. I asked students to raise their PAGE 68 68 hands to vote yes or no. Then, I asked them to explain their reasoning. Two students were willing to argue their points in front of the class, and one of the student s specifically used the definition of an arithmetic sequence to justify his reasoning. When I noticed that two students did not vote, so I asked them individually. They said no, and one of them justified the response with the definition. After providing the group with feedback regarding the example, I made sure to point out that students were not allowed to just say yes or no. If a sequence was identified as arithmetic, they had to provide the common difference. If it was not arithmetic, students had t o say why it was notetaking guide, all but one of the students that were present during the lesson provided the common difference for the sequences they identified as arith metic. Four of the students present did not justify their reasoning when determining that a sequence was not arithmetic when practicing, but they did correctly state that it was not arithmetic. In summary, the data regarding academic expectations indica ted that students understood the expectation to show work to demonstrate thinking, but students were resistant to verbally share or discuss their mathematical thinking when they were unsure about whether or not they were correct. To increase the number of students that discussed their reasoning and explained their thinking as they learned, I revised handouts and activity guides so that students were required to explain, describe specific details, and draw conclusions in writing. I also worked to scaffold my support during activities to allow time for students to think deeply about concepts and discuss with their peers. As a result, students improved in explaining their thinking, providing examples, and justifying their responses. However, academic expect ations communicated PAGE 69 69 effectively can only guide student learning when behavioral expectations are also communicated. In the following section, I will illustrate how behavioral expectations were communicated and enacted in the classroom. Behavior Expectatio ns Communicating behavior expectations. During this unit, I analyzed how behavioral expectations were communicated. My goals were to ensure that expectations were made clear through both verbal and non verbal strategies and to shift my focus to changes n eeded in my own practice to ensure student success rather than focusing on student characteristics and shortcomings (Bondy & Ross, 2008; Ladson Billings, 1995). I wanted to create a task focused learning environment in which students we re able to self dir ect through varied learning activities. I wanted my role to be focused on facilitating activities and generating discussion (Bonner, 2014). During the interviews, several students mentioned my behavior expectations. When reviewing the video data, I not ed that I used both verbal and non verbal communication to make expectations clearer. I modeled and explained what I wanted students to do. When I noticed that students were off task, repeated requests verbally or moved closer to students to encourage mo re focus on assignments. As I circulated the room, I frequently pointed with my finger to where students should be writing or to the question that groups should be discussing. If I needed the attention of the entire class and multiple students were off t ask, I would stop instruction, use a verbal cue, and wait for students to indicate that they were on task by looking at me or discontinuing off task behaviors. Following through with consequences for inappropriate behaviors. Some of the students in the class continued inappropriate behaviors, such as talking about PAGE 70 70 non math topics or not working, even when I attempted to intervene. In instances in which verbal and physical cues were not effective, I followed through with consequences by removing students that were not behaving appropriately from the classroom. For example, during the third lesson, I reminded a student several times to get on task and complete the assignment. When he refused by shaking his head no after the final reminder, I asked him to go to the hallway. I called an administrator, and the administrator counseled with the student. When these instances occurred, I reflected on what happened and what I could do to avoid similar situations in the future. One of the biggest challenges I faced during this unit was resistance from four particular students when I randomly assigned partners for group activities. When I pressed to find out the reason for the resistance to working with randomly assigned partners, two students argued that they worked best together. I told them that the pairings would not be permanent. They then complied with my request to work with their randomly assigned partner. The other two students refused to provide a reason for being unable to work with their randomly assigned partners. They resisted even after sitting in the group and became very disruptive. One student insisted that she could not do the work, saying she could not even count to two. Another verbally refused to work multiple times, hit her partner on the arm, and repeatedly raised her voice when talking, disrupting the class. After multiple warnings, I asked both of the disruptive students to leave the room and had an administrator counsel with them. After those two students were removed from the cl ass, the other students finished the activity. The next day, I personally spoke to the students who were removed before class to explain my reasoning for randomly assigning groups. I explained that I wanted them to step PAGE 71 71 outside of their comfort zones and learn how to work with people that they may not normally work with. I asked for their cooperation, and they agreed. As I reflected on explaining my reasoning if needed, and respectfully insisting that s tudents meet expectations. In summary, interview data indicated that students were aware of behavior expectations, and video data indicated that I used both verbal and non verbal communication to make expectations clearer However, in instances in which verbal and physical cues were not effective in stopping inappropriate behaviors among students, I followed through with consequences by removing students that were not behaving appropriately. I also learned that reflectio n is critical in figuring out how to best prevent and change classroom behaviors that are not desirable. For example, after following through with consequences for students that resisted working with randomly assigned partners, it was important that I con sider reasons for the misbehavior, work on building relationships, and communicate my expectations better I needed to consider how to better prevent student resistance to working with randomly assigned partners in the future and possible alternative cons equences when students refuse to work together. Answering Question 2: Communicating High Expectations Communicating high expectations regarding academics and behavior is an ongoing process in which I must be intentional about making sure students understa nd. While there was evidence that students understood the expectation to show work to demonstrate thinking, I was challenged to ensure that students understood the expectation to engage in deep levels of thinking and to communicate their reasoning PAGE 72 72 regardi ng mathematical concepts. In addition to revising written directions so that students were required to explain, describe specific details, and draw conclusions in writing, I learned that I needed to be intentional about talking to students about their thi nking and scaffolding my support during activities. It was important that I allow time for students to think deeply about concepts and discuss with their peers. While I used both verbal and non verbal communication to make expectations clear, and I also needed to model and explain what I wanted students to do without doing the work or thinking for them. When I noticed that students are off task, repeating requests verbally, moving closer to students, and pointing to where students should be writing or to the question that groups should be discussing were useful strategies. Finally, while following through with consequences for inappropriate behaviors may be necessary, it rea sons for the misbehaviors. Thus far, I have discussed the findings regarding classroom relationships and the communication of high expectations. As expected, both classroom relationships and the communication of high expectations impacted the ways in which students engaged in learning. In the next section, I will illustrate how students engaged in learning activities in my Algebra I class and how I attempted to negotiate student engagement. How do S tudents E ngage in L earning A ctivities in my Algebr a I C lass, and H ow M ight S tudent E ngagement be N egotiated D epending on their I ndividual B ackgrounds and P rior E xperiences? This section focuses on student engagement and how I attempted to negotiate student engagement. Using illustrations of actual classroom interactions and student learning from the data, I will illustrate incidences of student engagement, student disengagem ent, and how I intervened or adjusted my practice in order to increase PAGE 73 73 student engagement. The section will be organized to focus on two different kinds of activities: collaborative group work and teacher led activities, notetaking, and independent pract ice. I will close this section with a summary to answer the Research Question 3. Engagement in Collaborative Group Work When I interviewed students prior to the unit, three of the seven students indicated that they preferred hands on learning. Two studen ts specifically indicated that they liked working with other students in groups or as a team during the interviews. Based on the interview data and observations throughout the unit, I anticipated that students would be engaged in hands on activities. For instance, I noted that all students immediately jumped in to folding the paper during the first lesson. During my observation of the second lesson, I noticed students working together to complete the initial hands on part of the activity when they had to record data from the lab on the table and find the percent change and equation by hand. In my researcher journal, I noted that the hands on portion of the activity in the third lesson went quickly and that students appeared to have a prior understan ding of what to do from the previous day. Despite the initial engagement in the hands on learning portions of activities, I frequently recorded off task behaviors and students not working together when they proceeded to analyze what happened during the h ands on parts of activities or practice applying concepts. For instance, students seemed to have more difficulty staying on task and working together when they proceeded to the calculator portion of the activities in the second and third lessons. In gene ral, the data suggested that the frequency of small group discussion regarding mathematics increased when the physical classroom arrangement was set up prior to class, the activity required deep mathematical thought PAGE 74 74 and input from multiple students, and th e expectations for group processes were explicit, presented in multiple formats, and frequently addressed. During the first lesson, when exponential growth was initially introduced, I verbally asked students to work in pairs for the paper folding activit y and in small groups after the whole class discussion. When planning the lesson, I assumed that students sitting side by side would work as partners and that the pairs would join the pair behind them for the small group activity. I did not alter the arr angement of the desks prior to the lesson, and students were seated in their usual seats. As I observed students working, I noticed very little to no discussion. While students appeared to begin working to fold the paper immediately, students primarily c ompleted the activity alone. When it came time to practice after the whole class discussion, many students requested my support instead of seeking support from their peers. Some students did not work at all until I went to their group to assist. I wonde red how changing the desk arrangement would assist students in working together, and I wrote the following in my researcher journal: Perhaps changing the desk arrangement would increase math talk among students and also increase the number of students starting tasks sooner/staying on task. Tomorrow, I will arrange the desks so that students are facing each other in pairs. For the next lesson, I decided to arrange the desks so that group members would be physically closer and facing each other. While s tudent to student discussion increased during the second and third lessons, students did not necessarily work together to solve problems for the entire group activity even after I verbally reminded them that they should be working together. One student ap peared to be doing nothing and watching his partner complete the activity. PAGE 75 75 After the guided notetaking activity during the fourth lesson, I directed students to work in groups to identify data and graphs as exponential growth, exponential decay, or neith er. Then, I directed the groups to work together on practice problems. After the lesson, I reflected on the dialogue that occurred among students in my researcher journal: When students had to identify data and graphs as exponential growth, exponential decay, or neither, students engaged in meaningful discussion only after coming to their own conclusions. In general, students wrote their own answers, compared answers with the group, and then discussed reasoning (after I prompted them) . During the practice, the students appeared to be mostly working independently and talking about non math topics. While some were helping each other with understanding particular problems, they were not working together to solve problems in general . When using g roup work, I need t o ensure that the task requires input from all of the students in the group. After the fourth lesson, I referred to literature for guidance on collaborative learning in secondary mathematics. I realized that some of the activities that I asked students to complete in groups were not really group worthy. According to Horn (2012): Cognitively demanding tasks, of which group worthy tasks are a subset, require students to do more than just apply previously learned procedures. Such tasks re quire high level mathematical thinking, forcing students to make connections to the underlying mathematical ideas and engaging students in disciplinary activities of explaining, justification, and generalization. (p. 61) I realized that students were not a lways engaging in meaningful discussions about mathematics because the activities that I chose did not require them to think deeply or require input from more than one person. I revised some of the activities for future lessons to eliminate the directive for students to work in groups or to be more cognitively demanding by requiring students to make connections and draw conclusions. PAGE 76 76 For the fifth lesson, students were asked to graph multiple transformed functions and draw conclusions about changing h k and a based on their observations and discussion with their partner. Prior to the activity, I was more explicit about the I projected and discussed the following expectations that were written on the activity guide: BOTH partners must work together and contribute to the discussion. Partners must take turns writing responses on the activity guide. Each partner will complet e an assessment of his or her own contributions as well as the contributions of the partner at the end of the activity. I also projected and discussed the rubric that students would use to rate themselves and their partner that I developed after the four th lesson: COMPLETE THIS PORTION INDIVIDUALLY. 1. scale (1 is the lowest, 5 is the highest). These ratings WILL impact the grade. G roup Participation C riteria YOU YOUR PARTNER Shared ideas and answers with the group 1 2 3 4 5 1 2 3 4 5 Stayed on task during the assignment 1 2 3 4 5 1 2 3 4 5 Asked relevant questions when needed to increase understanding 1 2 3 4 5 1 2 3 4 5 I ci rculated the room during the activity, intervening when needed and encouraging peer support. Following the lesson, I reflected on how these changes improved engagement in discussion during the group activity and question ed how I could improve further in my researcher journal: The students were more on task and working together than in the group activities in the previous lessons . It appeared that they were more aware of the behavioral expectations . I wonder if changing up the groups so that students are working with classmates that they do not normally work with would have similar outcomes. For the next group activity, I will use random group assignments. PAGE 77 77 For the seventh and eighth lessons, I use d random assignment to organize students into pairs for group activities. As I described in detail previously, some students resisted at first. However, by the eighth lesson, students were more willing to work with different partners. During the lesson, I noticed students writing, talking about math, using the materials appropriately, and solving problems. In summary, as anticipated, students were generally engaged in hands on activities during collaborative group work. However, the data indicated that the number of off task behaviors and students not working together increased when students proceeded to analyze what happened during the hands on parts of activities or practice applying concepts. While changing the desk arrangement so that group members would be physically closer and facing each other increased student to student discussion, students did not necessarily work together to solve problems for the entire group activity unless the activity required students to think deeply or require input from more than one person. In addition, my role as the facilitator was critical in that I needed to be explicit regarding the expectations for the activity by providing directions in multiple formats, modeling, and circulating the room to intervene when neede d and encourage peer support. I learned that all activities are not group worthy and should be designed so that students are working independently. In the next section, I will describe the findings related to student engagement in teacher led discussions notetaking, and independent practice. Engagement in Teacher Led Discussions, Notetaking, and Independent Practice On task behaviors observed during teacher led discussions, notetaking, and independent practice included students writing in their notebooks verbally responding questions, and watching me as I modeled activities. In general, students responded to PAGE 78 78 questions, but some students only responded to questions when I called on them if at all. Students also generally wrote down notes when I modeled note taking using a copy of the note taking guide and the document camera. Off task behaviors occurred more frequently when students had to formulate their own notes or practice problems independently. In these instances, some students did not write at all. I frequently felt as if I had to remind students to either get started or to stop disturbing their classmates during the lesson. For instance, during the sixth lesson, I documented students talking before beginning t o work on problems and another student singing and walking around the room once he completed a problem in my f ield notes. When watching the recording of the lesson, I wrote the following in my researcher journal: . What I want right students stopped talking and became more focused on me. When we discussed the problem, students wrote on their papers and looked at the board, correcting their work as needed . It seems that explicit statements of behavior expectations are needed every day, along with their work and make sure that they are on task. Throughout the course of the unit, I noted that certain students frequently responded when I did not call on individual students to answer questions, others only responded when I called on them, and a couple of students sometimes did not respond to questions at all. Some students also resisted res ponding to questions in writing. For instance, when I asked students to write their own definitions in the final lesson, only five students wrote a definition in their own words. After I circulated the room, I asked for volunteers to share their definiti ons with the class. Three students shared their definitions, but some students did not write a definition until I wrote a definition on my PAGE 79 79 foldable under the document camera. After this experience, I wrote the following in my researcher journal: Some stu dents are still reluctant to speak in front of the whole class. Others seem more comfortable, confident, and more willing to take risks. Since students are reluctant to ask questions, it is necessary for me to as they are working, and provide feedback and support, even when it is not solicited by the student. Answering Question 3: Student Engagement Based on the data collected for this study, student engagement is heavily dependent on the instructional design of the lesson, the appeal of the activity to lesson. For instance, students were not always engaging in meaningful discussions about mathematics initially because, in some instances, the activities that I chose to be done in groups did not force them to think deeply or require input from more than one person. The physical classroom arrangement, whether or not the activity required deep mathematical thought and input f rom multiple students, and the communication of expectations impacted student engagement and needed to be carefully considered prior to instruction. While students generally appeared to be on task during teacher led discussions, off task behaviors occurre d when students had to formulate their own notes or practice problems independently. In addition, the frequency of verbal responses to questions varied from student to student, suggesting that some students may have felt more comfortable taking risks when responding to questions than others. Summary In this chapter, I discussed my findings related to what it means to be a culturally responsive teacher in the context of an Algebra I class for repeating ninth graders based on data collected during this study, according to the three sub questions I PAGE 80 80 developed related to central research question. In the next chapter, I will discuss reflections of my own learning as a practitioner researcher. PAGE 81 81 CHAPTER 5 PRACTITIONER REFLECTIONS As a white female mathematics instructor teaching in an inclusion classroom and an entire class of repeating ninth graders for the first time, I recognized the need to alter my teaching practices to better support students in my new context. Even before starting this research project, I was aware that in addition to understanding my students and the communities they were coming from, I would need to become more aware of my own biases and prejudices and change them to increase my cultural responsiveness and equity in my classroom. Hence, I designed this research study to learn more about what it means to be a culturally responsive teacher for students enrolled in an Algebra I course for repeating ninth graders at TL Hanna High Sc hool in Anderson, South Carolina. As I reflect on my own learning and the findings of this study described in the previous chapter, I realize that becoming a more culturally responsive teacher in the context of my Algebra 1 class for repeating ninth grad ers meant becoming more aware they engaged in learning. I identify my primary learning as becoming aware of how my students having been previously unsuccessful in math ematics, did not engage with mathematics in open and authentic manners. This manifested in ways that they did not 1) discuss mathematical ideas with their peers, 2) share mathematical ideas in whole group discussions, nor 3) write down their own ideas unt il the correct answer was confirmed. In short, the students were unwilling to make their thinking public. This chapter addresses reflections of my own learning as a practitioner researcher. Specifically, I will describe actions that I plan to take as an educator to alleviate this PAGE 82 82 phenomenon and increase student discourse about their thinking, first in terms of self reflection and changes within my own classrooms, then more broadly about changes outside of the classroom. Continued Self Reflection and Chan ge within my Classroom was due to them having negative perceptions of my beliefs about students who struggle academically. Therefore, I believe that my first initiative shoul d be to identify my own possibl e negative perceptions and work to change them. It is true that, i n many ways, I believe that I changed my own biases and prejudices as a result of this study. For example, I became aware that I unintentionally spent less t ime with students that asked fewer questions, and I intentionally worked to increase equity regarding the time I spent with each student individually. However, although I realized that I needed to improve the learning environment so that students were mor e comfortable and confident in sharing their ideas, there were still incidences in which I felt that I could have done better despite my attempts to use a variety of strategies to improve the situation. One incident was when I removed two students from t he classroom when they became disruptive and resisted working with an assigned partner. While I believe that it is necessary to follow through with consequences for misbehaviors, I also wonder if the misbehavior s could have been avoided or if there was a way to assign a consequence and stop the misbehavior s without removing the students from the classroom. Surely there are other strategies that I could have used that would have be en more effective in improving the particular student behaviors but they di d not come to mind as the incident was occurring PAGE 83 83 Prior to this study, similar incidents had occurred with the same two students. I had previously attempted to move the students to another part of the room and they continued to be defiant refuse to work, and speak loudly across the classroom. On another prior occasion, I moved the students to the hallway to work By moving students to the hallway and leaving the door open, they still had access to the lesson and me, but they could no longer disrupt the rest of the class. On this occasion, the students completed their work quietly, and I felt that I found a quick solution. However, I was told by my administrator that I could not use this strategy anymore due to supervision rules. In short, I reali ze that I do not always know the best way to respond to student behaviors in the moment that the behaviors are occurring. Most importantly, I realize my failure in establishing the kind of relationship that would prevent certain behaviors with these parti cular students. Looking back, I see the cultural barriers between my students and me and I am saddened by how difficult it was to break through these barriers. While I initially thought that my personal background of being a low income student from a single parent family would help me relate to my students, I overlooked the differences in our academic backgrounds. Growing up, I too was part of a curricular tracking program, but, unlike my students, I was placed in an honors track. In contras t to my students, I had n ever in my life experienced failing a course. I was always interested in school I performed well academically, and I was motivated by academic success. Despite my sincere desire and belief that I could establish positive relatio nships with each and every student, it was difficult for me to understand and relate to all of the students in my class. PAGE 84 84 As a high school student, I remember thinking that anyone could make straight e my own personal obstacles of being raised by a single parent with low income. became an adult that I that my academic success, while earned throu gh hard work, was also partly due to my that could lead to academic failure and combat my own deficit thinking Rather than nesses, I need to consider contextual factors and try to imagine how their experiences in my classroom are unique Prior to this study, I thought that I had already worked to improve my own perceptions and biases. While I do see improvements, I regret that there we re instances in which I blame d incidences that occurred while teaching this course on my students. For instance, knowing that my students had previously been unsuccessful, my mind would sometimes blame students for their lack of engagement rather than considering the possibility that there were other invisible factors at work that prevented students from engaging in the lesson When students did not attend school, I would my own relationship with stud ents and their families I realize that these thoughts stem from my own background and upbringing, as well as from the messages sent through cultural patterns and dominant cultural norms in the U.S. society as a whole I remember standing in the hallway after class was PAGE 85 85 and, despite my efforts, I was not always successful in changing student behaviors thus I took the outcome personally instead of objectively At tim es, I would confide in other teachers regarding situations that occurred in my classroom. Frequently, my colleagues would tell me something about how these problems are the nature of the repeater group. At these times, it was difficult to focus on studen ts assets rather than deficits and take responsibility for my own shortcomings as their teacher Putting myself in their shoes, but as a teacher, I realize that I need to be cognizant about how my experiences teaching th is repeater Algebra class may infl uence my future teaching. For example, as my students had a hard time engaging with mathematics content in an authentic manner due to their prior negative experiences and academic failure I see the need to self check my own thoughts and feelings when I f ace a challenging student in the future. I w ill strive to continue to be open and engaging with all of my students and continue to work on my own negative perceptions so that I can better assess what changes are needed within my classroom. I have learne backgrounds, or my perceptions of their backgrounds, as a reason for my own lack of success as a teacher. It is easy to continue current practices with the mindset that all students should conform and adjust to learn in the ways that I teach. However, if I continue the use of traditional teaching practices, I will also continue to marginalize particular students. If I am to do my part to close the achievement gap and increase equity in education, I must accept responsibility for what happens within the classroom when students are not engaged or fail, even when I do not know what I could have done PAGE 86 86 differently, and seek ways in which I can improve. If I do not immediately see a assets as soon as possible to assist my students. In addition to changing my own thoughts to hold stude nts in a more positive regard, as a practical approach, I must shift my tendency to reuse lessons that I found successful with former students without thinking Rather, I must focus on how each student is unique and learns differently. Moving forward I will carefully construct and modify lessons to match the unique learning styles and preferences of the students that I am currently teaching. Looking back on this study, I realize that, while I provided a large variety of activities in order to appeal to multiple learning styles, I did not provide many choices in activities or assessments. Although students frequently had choices regarding their own role within group work, perhaps I could increase student interest in learning by providing choices in the problems to be solved or how students will demonstrate their learning. I also realize that some students may benefit from the option to work alone and that I may have placed too much emphasis on the goal for students to work together. Since I realize now that all activities are not group worthy and some students may not yet feel comfortable sharing their thinking, I will also take more care to consider how students will collaborate during lessons. I need to be more conscientious of how trust and relations hips influence students and their willingness to work together. I now realize that some of my students may have resisted working tog ether because they were trying to protect themselves from the embarrassment of being incorrect This fear PAGE 87 87 of making mistak es publicly could be an aspect of the culture of repeaters that I did not anticipate S ince the resistance to share ideas countered my vison of a collaborative learning environment I tried even harder to e nforce my expectations for students to make their thinking public, but that effort further created a gap in my relationships with particular students Providing options regarding how students will collaborate or the option to work alone m ight have help ed to alleviate this phenomenon In addition, I nee d to create activities that place less emphasis on correct answers and more emphasis on the thinking process. Finally I will continue to take notes regarding my observations when I am teaching and reflect upon how I can improve upon future lessons. My notes will document both successes and areas of need This way, I can record my thoughts about what steps I should take next to improve my instructional practices As part of my decision making process, I will collaborate with other teachers in addition to using literature to get ideas for lessons and how I can improve my teaching. Advocating for Changes Outside of my Classroom While making changes regarding my own biases and classroom practices has potential to benefit my students in the classroom, I am afraid it will not be enough to empower all students throughout their schooling. It is possible that in the current educational climate of accountability and high stakes testing, most teac hers within my context focus on right answers and individual work, potentially leading to students being unwilling to collaborate in other classes. It is also possible that teachers hold more or less negative perceptions about students that inform their i nstructional practices. In order to do my part in helping close the achievement gap among students, I now realize that I need to PAGE 88 88 do a better job of advocating for changes to benefit all students and promoting the idea that all students are capable of lear ning and achieving at high levels. Therefore, it is important for me to work on not only increasing, but improving, my communication skills and influence in order to effectively promote changes within my context that will increase student discourse about their thinking and overall engagement in learning With my colleagues, in both formal and informal conversations, I can bring up problems and encourage supporting students in collaborating and sharing their thinking. For example, as issues arise in our classrooms, my colleagues and I might discuss how to improve situations through informal conversations in the hallway or during our common break times. I might also bring up issues as I collaborate with my colleagues during more formal conversations, such as professional development opportunities or faculty meetings. Initiating professional study groups among teachers to discuss common issues may help us to establish a common set of expectations regarding how students should be sharing their thinking and working together. By sharing about my own experience experiences are like outside of my classroom, to inform my understanding of their behaviors in my classroom, ultimately to better myself as a teacher while also helping other teachers. As a t eacher with the intent to promote changes to increase equity in learning, I also realize that I need to work to build trusting relationships with my colleagues. I need to be a better listener so that my fellow teachers feel comfortable sharing their conce rns with me. I need to remember to express empathy when others complain or seem frustrated and to frame my constructive feedback so that it is perceived as supportive PAGE 89 89 rather than judgmental. When receiving feedback, I need to be open to criticism and the opportunity to learn rather than defensive. I must be more aware of the unique perspectives of others in order to better understand how to work with them. As we share, I will not only seek to learn how to improve upon my own practice, I will also seek w ays I can assist in their improvement efforts (Katzenmeyer & Moller, 2009). I will work with my colleagues to overcome challenges and celebrate small successes towards our larger goals. For example, I have already worked with other teachers to advocate for changes in the way we track students. We opposed the traditional view that students with failing grades should be assigned to a "repeater" course the following year. By working with allies and like minded colleagues, my school administrators have agr eed to make a change to include repeating Algebra 1 students in classes that contain students taking Algebra 1 for the first time next year. I believe that this change is important as it will allow students to not feel excluded because of their academic f ailure while encouraging teachers to have higher expectations for students that have been previously unsuccessful in Algebra 1. Through both formal and informal conversations with my colleagues, I believe we have increased capacity for change, and I think this particular change will have positive outcomes for both students and teachers. I also realize that my role as an advocate will increase as I advance in years of teaching experience and leadership roles within my context. Over time, my relationship wi th my colleagues may change. For instance, if I eventually obtain a more formal leadership role, my relationship with teachers may become more distant. If this occurs, I will need to be even more cognizant of my own position and the positions of others PAGE 90 90 w hen communicating, making sure t hat everyone has a chance to be heard. I realize that I will need to focus further on contextual and cultural changes that promote the development of learning communities in which teachers work together to solve problems. Through shared decision making processes and small steps toward changes changes. Summary In this chapter, I reflected on my own learning as a practitioner researche r as a result of this study. I focused this chapter on the phenomenon of how students in my class as students that had been previously unsuccessful in mathematics, hesitated to engage with mathematics in authentic and open manners, being very much unwill ing to make their thinking public. I described my reflection of th is phenomenon along with a ctions that I plan to take as an educator to alleviate this phenomenon and increase student discourse about their mathematical thinking. Specifically, I discussed my plans for continued self reflection self change and advocating for changes to benefit all students outside of the classroom. In the next chapter, I will address conclusions, along with limitations and possible implications of this study. PAGE 91 91 CHAPTER 6 IMPLICATIONS In the previous chapter, I discussed how through this study, I became aware that my students as students who had previously been unsuccessful in mathematics, did not engage with mathematics in open and authentic manners and reflected on my own learning as a practitioner researcher. I also outlined actions that I plan to take as an educator to increase student discourse about their thinking within my context. This chapter addresses possible implications of this study beyond my own classroom. I will end this chapter with a conclusion to summarize my final thoughts regarding this study and what it means to be a culturally responsive teacher. Implications Although this study was framed in practitioner research as I studied my own practice, there are potential imp lications beyond my context. I will focus on implications specifically for teachers, as this study provided valuab le examples for teachers to cons ider when attempting to increase their cultural responsiveness in teaching mathematics. I will organize implications for teachers according to three critical elements of culturally responsive teaching (relationship building, high expectations, and engagin g lessons) and mathematics teaching. I will end this section with implications for future research. Implications for Teachers Relationship building. backgrounds and experiences are the foundation of cult urally responsive teaching. Strategies to foster relationships with students include talking with students and their families, observing students both in and outside of the classroom, and reflecting on PAGE 92 92 tial biases. It is important for influenced by their backgrounds and experiences. D evelop ing a deep understanding of individual backgrounds, daily life experience s, and learning styles is critical to building relationships and positively impacting student interest and motivation to learn In other words, by building relationships with students, a teacher is better able to align learning experiences to students, an d students feel valued for where they come from when teachers match instruction to students in this way. Having been previously unsuccessful in mathematics, my students resisted discuss ing mathematical ideas with the ir peers in public, sharing i n whole g roup discussions, or even writing down their own ideas until the correct answer was confirmed. It is possible that my students were unwilling to make their thinking public due to negative prior experiences or fear of being judged by myself or their peers. Perhaps in the past, or even in my classroom without my intention or noticing, some of my students might have been ignored or had their questions disregarded due to dominant cultural norms or individual biases and prejudices. Therefore, teachers must be conscientious of students backgrounds and prior experiences when scaffolding support and utilizing strategies to encourage more peer interaction It is important for teachers to be aware that some students may require more assistance than others, and s ome students may solicit support more than others. However, a student not soliciting support does not necessarily mean that support is not needed. For instance, particular students in my class rarely asked questions, and I discovered that I needed to be more intentional about spending time with them PAGE 93 93 individually to review their work and ask them specific questions regarding their thinking. Likewise, a student that frequently solicits teacher support may actually need to be encouraged to work more indepen dently or to seek support from his or her peers. It is difficult, but important, for teachers to maintain the balance of help ing students engage their thinking without doing the thinking for them (Horn, 2012). Getting to know students individually can as sist teachers in maintaining that balance and knowing when to push, when to assist, and when to step back. High expectations. Communicating detailed directions in multiple formats and modeling can increase student on task behaviors and success in explaining their thinking, providing examples, and justifying their responses. Multiple formats, including verbal, written, and physical cues, are necessary to appeal to a variety of learning styles and to assure that students understand expectations. W hen students do not meet the expectations, teachers should consider all possible factors that led to undesirable outcomes. For instance, a teacher might think about his or her own relationship with individual students, relationships between students, and the events that occurred in consequences, teacher r eflection is also critical in figuring out how best to prevent and change classroom behaviors and other outcomes that are n ot desirable. Student engagement. Engaging students in learning means creating opportunities for meaningful participation. Teachers must be aware that r outes to student engagement may be social and/ or academic including opportunities to develop interp ersonal relationships and participate in intellectual endeavors ( Fredricks, Brumenfeld, & Paris, 2004 ). Consistent with literature, the findings of this study PAGE 94 94 indicated that s tudent engagement wa s heavily dependent on the instructional design of the lesso n, the perceived needs, and how well I facilitated the lesson. In teaching, one lesson does not fit all learners. Rather, lessons should be carefully designed and adapted with individual students and learning styles in mind. When planning lessons, teachers should consider whet her or not activities require deep levels of thought an d input from multiple students before determining whether an activity should be completed collaboratively or independently Some students may feel more or less comfortable working with other students as a result of their individual backgrounds. It is important for teachers to consider how students may respond in advance so that alternative options or additional support can be provided if needed. Teachers should also think about how to facilitate the lesson prior to instruction, such as what he or she needs to say and do and how materials should be used. In addition to thinking deeply when planning lessons, it is also important that teachers develop observational insights and skills to determine how students are engaging during lessons, reflect on areas that may need improvement, and consider making changes to impact future lessons. For instance, through my ob servation and reflection I became aware of the need to adjust multiple factors, such as th e physical classroom arrangement and how I communicated expectations. Through reflection and intentionally seeking new strategies and ways to improve, teachers can p ositively impact student engagement in learning. Mathematics teaching. The findings of this study shed light on how some students and particularly students with prior academic failure, have difficulty engaging PAGE 95 95 with mathematics in open and authentic man ners and were in many ways unwilling to make their thinking public, stemming from different reasons. Student discourse about mathematical concepts is essential for learning, as it allows students to reflect on their own think ing It is important that teachers create and sustain a collaborative and supportive learning environment, while also increasing student use of higher order thinking skills Teachers need to be able to recognize and design c ognitively demanding tasks that w ork well for collaborative learning while also providing support for students that may not feel comfortable collaborating with others Specifically, cognitively demanding tasks require students to make connections to th e underlying mathematical ideas, ex plain their thinking, justify their reasoning and make generalization s (Horn, 2012 ; Stein & Smith, 1998 ). However, more emphasis should be placed on thinking and problem solving than on obtaining the correct answer. Through discourse, students can also make mathematical conjectures, refine their work and take ownership o f their mathematical knowledge. Implications for Future Research This study is significant in that it adds to the existing body of research regarding social justice, equity, culturally responsive teaching, and mathematics education in that it is a practitioner research study focusing on Algebra I students in an inclusion classroom who are repeating ninth grade. However, culturally responsive teaching is a broad topic needing to be studied from different viewpoints and contexts. Future research is needed to explore culturally responsive teaching in other mathematics classrooms as well is in classrooms for other subject areas and for grade levels. It w ould be beneficial for other teachers to engage in practitioner research regarding culturally responsive teaching in their specific contexts. In addition, research studies PAGE 96 96 conducted over longer periods of time would be helpful in capturing the growth of s tudents and teachers over time. One aspect of culturally responsive teaching that I believe can be generalized across content areas and contexts is that it requires several shifts in thinking amongst rounds in terms of their assets as opposed to deficits and reject the tendency to use traditional methods when working with remedial students ( Dray & Wisneski, 2011; Gorski, 2013; Gay, 2010; Kose & Lim, 2011; Schillwer, 2008; Ullucci & Howard, 2015). E xpl icit focus on the classroom environment and cultural connectedness are necessary to ensure students feel comfortable and safe to take risks in learning (Bonner, 2014). In supporting teachers to make changes in both their thinking and instructional practic es, research to inform professional development needs and to evaluate existing professional development regarding culturally responsive teaching may be necessary. Conclusion Despite improvements and efforts to achieve equity through various reforms and lit igation, there is a persistent disparity in educational performance among subgroups of U.S. students In my quest to promote equity in my classroom, I designed this practitioner research study to learn more about what it means to be a culturally responsiv e teacher for students enrolled in an Algebra I course for repeating ninth graders. In many ways, this study confirmed what literature suggests as challenges teachers face when working with diverse groups of students who have previously been unsuccessful in mathematics courses The students enrolled in my Algebra 1 course were assigned to a particular section and class period due to c urricular tracking based on their prior academic performance in Algebra 1 Compared to the larger school PAGE 97 97 population, there was a disproportionate number of poor students and students of color in my course, suggesting the need to create more equitable opportunities in mathematics education for my students. There is evidence to suggest that my students may have held negative p erceptions about their own academic identit ies and abilities, However, this study also suggests that teachers have the potential to overcome these challenges by becoming more culturally responsive in their instructional practices particularly by and prior ex periences impacts their engagement in learning Through this study, I have grown both personally and professionally. I became more aware of my own biases, thought deeply about my instructional practices and how I could better impact students, and adjust ed my practices in an effort to improve student outcomes. I realize the power culturally responsive teaching has to positively impact students, as I have experienced stronger relationships with my students and observed how student behaviors and engagement changed as I made adjustments to my practice. While I still may not know all of the answers of how I could have done better in instances in which I felt unsuccessful in reaching every student, I take responsibility for both the successes and areas that n eed improvement within my classroom. South Carolina Algebra 1 End of Course Examination Program (EOCEP). Statewide, the mean score was 69.4%, and 74.7% of the 62,655 students tested passed the exam. The mean score for students in poverty was 64.9%, and 64.7% of the 32,973 students in South Carolina in this group passed the exam. For the 12 students who participated PAGE 98 98 in this study, the mean score on the Algebra 1 EOCEP was 60. 75% and 66.7% (8 of the 12) passed the exam. Of students who participated in this study (10 of the 12) 83% earned credit for Algebra 1. On the morning of May 31, 2017, as I was in my classroom preparing for students to arrive, a voice came over the int ercom calling all teachers to report to the cafeteria immediately. When I arrived to the cafeteria, I saw multiple students on the floor and other students talking to police, administrators, and teachers. When I noticed that two of the students were mine I immediately went to them, concerned that they were either hurt or in trouble. It was two of the girls in my Algebra 1 repeater class, with whom I had in many ways failed to make a connect ion After they had arrived to school that morning, they were i nvolved in a large fight in the cafeteria They were arrested and charged with disturbing schools along with nine of their peers who were also involved in the fight. Before they were taken away, I told them that they had passed the Algebra 1 EOCEP and ea rned credit for Algebra 1. I told them that they were smart, and I asked them to promise me that they would graduate from high school. That was the last time I saw them. Today, I celebrate my students who succeeded, but I feel sad for my students who d id not succeed and for those who were expelled from school for discipline issues I feel responsible for the failure of the students who did not pass the class. I wonder how I could have done a better job in relating to particular students and preventing behaviors that resulted in them no longer being able to be in school. I worry about all of them. I know that those who passed the course are currently in classrooms filled with students who have not experienced failure in the same ways as my students did before PAGE 99 99 and I hope that their teachers are conscientious of how their status as former repeaters impacts their learning. I hope that their new teachers do not assume that they do not want to learn because of their history of failu re and/or somewhat confusing behaviors of disengagement in learning settings For those who did not succeed or were expelled, I hope that someone can reach them sooner than later. For me, culturally responsive teaching is a process of constant improveme nt based on a sincere desire to help students of any background grow personally and academically. In the context of my Algebra 1 class for repeating ninth graders, experiences of a cademic failure impacted their classroom behavior and engagement. It meant that I needed to be aware of and work hard to overcome the barriers that existed between my students and me While content knowledge wa s important, becoming a more culturally resp onsive teacher additionally require d me to shift my mindset more towards teaching students and relationships while maintaining my view towards teaching mathematics. Culturally responsive teaching in the context of this study mean t being intentionally focused and concerned about building caring and trusting relationships with and among students even when it seemed impossible or difficult It was loving students even when their attitudes and behaviors made it difficult It wa s al lowing that constant awareness of relationships to refine my instructional practices, challenge my perceptions, and alter my thinking. It was extremely difficult, and I learned that being culturally responsive does not mean seeking a level of perfection Rather I will always be making progress in becoming PAGE 100 100 more culturally responsive Looking back, I realize my growth, but I also realize that I still have much to learn. My journey continues. PAGE 101 101 APPENDIX L ESSON PLANS FOR UNIT 10: EXPONENTIAL FUNCTIONS Day 1 Lesson Title: Introduction to Exponential Growth Lesson Objectives : Students will (1) evaluate and graph exponential functions (Anderson School District 5, 2016). Materials Needed : activity guides; 2 large rectangular sheets of paper for each student Instructional Procedures : I will distribute a large rectangular piece of paper and activity guide to all students. Using one of the large rectangular pieces of paper, I will demonstrate the procedure, providing oral directions and modeling each s tep. Students will fold the paper in half and record how many sections are formed by the creases in a table. Students refold the paper, fold the paper in half again, and record how many sections are formed by the creases. I will instruct the students to work in pairs and continue folding the paper in half and recording the number of sections until the paper cannot be folded any more. Using the activity guide and working with a partner, students will record answers to the following questions: How many fo lds could you make? How many sections were formed? What function/equation is modeled by the folds and sections created? How did you come up with that equation? What was the pattern you noticed? As students work in pairs, I will circulate the room, redirect ing off task behaviors, assisting pairs and answering questions. Once each pair has successfully completing the first activity, I will provide a second piece of large rectangular paper and instruct the pair to use the piece of paper to model the equation y = 3^ x I will provide the clue that the students will need to fold the paper into thirds. The students will once again record how many sections are formed by the creases on the activity guide and continue folding the paper in thirds and recording the n umber of sections until the paper cannot be folded any more. Students will record answers to the following questions on the activity guide: How many folds could you make this time? How many sections were formed? What was the pattern you noticed this tim e? PAGE 102 102 What function/equation is modeled by the folds and sections created? Upon successful completion of the final activity, students will turn in their work to be checked by the instructor. I will verify that the pair obtained correct answers, answer quest ions, and respond to any concerns the students have. I will then direct student attention to the front of the room, where the general form of the exponential equation will be displayed. I will describe/explain the parts of the general exponential equatio n, and how the paper folding activities were examples and equations of exponential functions. I will tell the students that there are 2 types of exponential functions: exponential growth and exponential decay and ask whether the students would describe the activity and data they collected as exponential growth or decay. I will lead the class in a discussion about how the activity models an equation of the form y = ab x where a is a nonzero constant, b is greater than 0 and not equal to 1, and x is a real n umber. Specifically, I will show how a represents the starting number/initial value, b represents the growth or decay factor, x is the number of periods (or folds of the paper in the activity), and y is the ending number /value (number of sections in the p aper activity). After the whole class discussion, students will work in groups of 3 4 to evaluate and apply exponential functions as practice. For example, students will evaluate the exponential function y = 5^ x for x = 2, 3, 4 and write an exponential function given a word problem scenario ( Anderson School District 5, 2016). Elements of Culturally Responsive Pedagogy: academic success (Gay, 2010; Morrison, Robbins, & Rose 2008 ). I chose the paper folding activity to introduce exponential growth because it allows students opportunities for collaboration, movement, and hands on learning I hope to promote engagement with the paper folding activity, and I believe that the ac tivity will provide students with opportunities for meaningful participation in learning. In this lesson, the paper is used as a representative tool to support student thinking ( National Council of Teachers of Mathematics, 2014). In addition, the lesson incorporates experiential learning and group processes, which are researched based strategies to increase equity (Bondy et al., 2007). To determine whether or not the paper folding activity is effective in implementing elements of culturally responsive pe dagogy to better engage students, I will carefully monitor how each student participates in the activity during the lesson. affective reactions in the classroom and to my instructional practices ( Fredricks, Brumenfeld, & Paris, 2004 ). My note taking will take place both during the lesson and as I write reflections in my researcher journal. In addition, as I watch the recorded videos of lessons, I will make additional notes about the discussions a nd activities that take place during the lesson. For example, I will document which students are following the directions for the activity, which students ask for PAGE 103 103 teacher and/or peer support, who takes the initiatives in group processes while working with a partner, and who provides assistance to peers. Anticipated Response from Students: I anticipate that the students will respond positively to the activity and be engaged in meaningful discussions. However, I also anticipate that students will struggle with writing the exponential functions and need lots of assistance. I anticipate that some of the students will complain that they have too much work, but I hope that they will demonstrate persistence though the challenging parts of the activity, particularly when I ask them to model y =3^ x Days 2 3 Lesson Title: Exploring Exponential Grow th and Decay using Regression Equations Lesson Objectives : Students will (1) evaluate and graph exponential functions (Anderson School District 5, 2016). Materials Needed : activity guides; graphing calculators, cups Instructional Procedures : I will organize the class into pairs and distribute an activity guide, a graphing calculator, a bag of Once all the materials are distributed, I will guide the students through the activity, p roviding oral directions and answering questions while circulating the room. To model exponential growth, students will place two cup/plate, dump out the s, and add another for every with epeated 15 times. For each trial, the new population will be recorded in a table and graphed on a coordinate plane. The students will work together to respond to questions on their activity guides regarding the asymptote, calculate the percent change for each trial, and write an exponential growth function that models the data. They will also use the exponential growth model to make predictions. To model exponential decay, students will count the total number of s in their bag and record this numbe r in trial # 0. They will dump out the s and remove the population will be recorded in a table and graphed on a coordinate plane. The students will repeat this process until they have completed 10 ph ases OR when the population gets below 4, taking care to not record 0 as the population. The students will work together to respond to questions on their activity guides why the number of s cannot be reduced to zero, write the exponential re gression equation and compare values using the exponential decay model they found to their actual data ( lab (Exponential growth and decay, n.d.). PAGE 104 104 Elements of Culturally Responsive Pedagogy: I chose the activity to model exponential growth and decay because it allows students opportunities for collaboration and experiential, hands on learning, while helping them to construct their own knowledge about exponential functions (Bondy et al., 2007; Morrison, Robbins, & Rose 2008 ). The model provides students with an opportunity to think critically and at higher levels. This lesson is an intentional attempt to connect students with the problems encountered in the real world (Buckley, 2010; Ross & Adams, 2010 ; Gorski, 2013; Hill, 2010; National Council of Teachers of Mathematics, 2014 ). By using the candy, I hope to increase student interest and participation in learning. To analyze the effectiveness in implementing elements of culturally responsive pedagogy to better engage stu dents through the activity, I will carefully monitor how each student participates in the activity. I will document discussions with and among my students, particularly noting the presence or absence of positive conduct, the level of involvement in t he learning task, and contributions to the group discussions in my field notes. I will also document activity ( Fredricks, Brumenfeld, & Paris, 2004 ). For instance, I will make notes about how the students are usi ng the behaviors, such as playing with or eating the My note taking will take place both during and after the lesson. I will watch the recorded videos of lessons and document details about which students are following directions, which students ask for teacher and/or peer support, who takes the initiatives in group processes, and who provides assistance to peers in my field notes. Anticipated Response from Students: I anticipate a p ositive response to this lesson. I expect that the students will love this activity because it involves of the students may try to eat the have to be sure to remind them to wait. I also anticipate students needing additional help using the calculators to find a regression equation since this will be their first experience using that particular calculator function. Day 4 Lesson Title: Exponential Growth and Decay (continu ed) Lesson Objectives : Students will (1) evaluate and graph exponential functions (Anderson School District 5, 2016). Materials Needed : interactive notebooks; note taking guides Instructional Procedures : All students will receive a copy of the note tak ing guide, which will include a foldable to review exponential growth and decay models and the general a word prob lem to demonstrate each model. S tudents will individually co mplete the note taking guide while participating in a whole class discussion led by me. PAGE 105 105 For the word problems, I will demonstrate how to how to use highlighting/underlining to help students select important information that they will need to solve problems. Then, students will work together in small groups to practice exponential growth and decay word problems. The questions will be adapted from the practice section of the Interactive Student Guide on lesson 7.6 Growth and Decay (McGraw Hill, 2010). During the discussion, I will circulate the room, redirecting when students become off task. As the students practice, I will continue circulating the room, providing feedback, and answering questions. Elements of Culturally Responsive Pedagogy: In this lesson, I will model metacognitive activities such as thinking aloud, scaffold instruction, and provide clarification of the concepts that were introduced in the previous lesson. I will also model highlighting and underlining text, which is a reading strategy that assists students with comprehension by organizing information. The class will utilize interactive notebooks as a tool to as sist students in meeting academic expectations for concept knowledge/application. I will also take care to promote the use of precise mathematical language that reflects mathematical structures within problems (What Works Clearinghouse, 2015). During the discussion and note taking, I will take care to communicate high standards for both academics and behavior and encourage students to take risks. For instance, during the whole group discussion and note taking I will randomly call on students to respond an d encourage all students to view mistakes as learning opportunities rather than failures (Bonner, 2014). During the practice, students will work together to solve problems, collaborate and model thinking and utilizing the highlighting/underlining text str ategy for each other (Gay, 2010; Ladson Billings, 1995; Morrison, Robbins, & Rose 2008 ). I will circulate the room during the group practice to intervene when needed and encourage peer support (Bonner, 2014; Morrison, Robbins, & Rose 2008 ). Both during and after the lesson, I will document how each student participates in the interactive notebook activity. I will document details about which students are following directions, writing notes, and utilizing the highlighting/underlining text str ategy as well as students who resist completing the notetaking activity. I will also make notes about which students ask for teacher and/or peer support, what is said during the whole group discussion, and how students respond when I randomly call on them in my field notes. During the group work, I will note which students take leadership roles within their groups and which students ask for teacher and/or peer support. Anticipated Response from Students: I anticipate an overall positive response to this l esson, as I think that the underlining/highlighting the text strategy will assist students with word problems. I hope that students will utilize their interactive notebooks to review and study for assessments. PAGE 106 106 Day 5 Lesson Title: Exploration and Inquir y Transforming Exponential Functions using the form y = ab x h + k Lesson Objectives : Students will (1) evaluate and graph exponential functions and (2) sketch the graph of a function from a verbal description showing key features (Anderson School District 5, 2016). Materials Needed : graphing calculators; activity guides Instructional Procedures : I will introduce the activity and organize students into groups of 2. Prior to distributing the materials, I will discuss the following expectations for group work: (1) BOTH partners must work together and contribute to the d iscussion for this activity. (2) Partners must take turns writing resp onses on the activity guide. (3) Each partner will complete an assessment of his or her own contributions as well as the contributions of the partner at the end of the activity. Each pair will receive a copy of the activity guide, and all students will receive a graphing calculator. Students will follow the directions on the activity guide and work together in pairs to respond to the discussion questions. During the activity, I will circulate the room, redirecting when students become off task, providing feedback, and answering que stions. The activity guide questions are below: RECALL: Exponential functions are functions that can be written in the form y = ab x where a b > 0, and b this lesson, you will be exploring the graphs of exponential functions as the equatio ns are transformed by a. Adding or subtracting a constant h to the exponent: y = ab x+h and y = ab x h b. Adding or subtracting a constant k to the function: y = ab x + k and y = ab x k c. Changing the constant a that is being multiplied in the function: y = ab x The Effect of h : Graph the following functions on the same coordinate plane by using y 1, y 2, and y 3 in your graphing calculator. Then, discuss and answer the questions with your group. y = 3 x y = 3 x + 2 y = 3 x 2 1. Describe the shape and position of each graph. Include in your description the domain and range, whether the function is increasing or decreasing, and whether the function is positive or negative. y = 3 x : y = 3 x + 2 : PAGE 107 107 y = 3 x 2 : 2. In general, how d oes changing the value of h affect the graph of the equation? The Effect of k : Graph the following functions on the same coordinate plane by using y 1, y 2, and y 3 in your graphing calculator. Then, discuss and answer the questions with your group. y = 3 x y = 3 x + 2 y = 3 x 2 1. Describe the shape and position of each graph. Include in your description the domain and range, whether the function is increasing or decreasing, and whether the function is positive or negative. y = 3 x : y = 3 x + 2: y = 3 x 2: 2. In general, how does changing the value of k affect the graph of the equation? The Effect of a : Graph the following functions on the same coordinate plane by using y 1, y 2, and y 3 in your graphing calculator. Then, discuss and answer the questions with your group. y = 3 x y = 2(3) x y = (3) x y = (1/2)3 x 1. Describe the shape and position of each graph. Include in your description the domain and range, whether the function is increasing or decreasing, and whether the function is positive or neg ative. y = 3 x : y = 2(3) x : y = 3 x : y = (1/2)3 x : 2. In general, how does changing the value of a affect the graph of the equation? Be sure to include what happens when a is positive vs. negative, when a is greater than 0, and when a is between 0 and 1. Putting it all together : Work with your group to answer the questions below. PAGE 108 108 1. By looking at the equations, work with your group to predict the transformations used to obtain the graph of g from the graph of f Write down your predict ions. f ( x ) = 3 x g ( x ) = 3 x +2 2 2. Graph both equations using y 1 and y 2 in the graphing calculator. Put a check by the predictions you made that were correct. Put an x by the predictions you made that were incorrect. Write down any transformations that were missed by your group an d why you think they occurred. 3. Write an equation that would transform the function f ( x ) = 2 x by shifting the function up 3 units and right 1 unit. Check your answer by graphing both functions in the graphing calculator. 4. Write a n equation that would transform the function f ( x ) = 2 x reflecting the graph across the x axis. Check your answer by graphing both functions in the graphing calculator. Elements of Culturally Responsive Pedagogy: This lesson activity is designed such that students are working in pairs. The purpose of the partner work is to assist students to get to know and connect with other students and establish an atmosphere in which students respect and are kind to one another (Bondy et al., 2007; Gay, 2010; Nieto, 20 13). In this activity, students are expected to work closely together to construct their own knowledge about exponential functions. I expect students to use their prior knowledge and make sense of transformations during this activity (National Council of Teachers of Mathematics, 2014). I will circulate the room during the activity to intervene when needed and encourage peer support (Bonner, 2014; Morrison, Robbins, & Rose 2008 ). To increase student motivation, the activity guide is carefully constructe d so that students build upon prior knowledge about graphing and key features of graphs. The activity is designed to progress from easy to more complex tasks and thinking to ensure that students have positive first encounters with content before moving to the more challenging parts of the lesson (Gay, 2010; Morrison, Robbins, & Rose 2008 ). To analyze the effectiveness in implementing elements of culturally responsive pedagogy to better engage students through the graphing calculator activity, I will car efully monitor how each student participates in the activity. I will reactions to the activity ( Fredricks, Brumenfeld, & Paris, 2004 ). My note taking will take place both during and after the lesson as I watch the recorded video. I will document details about which students are following directions, using the calculator, and writing responses as well as students who resist completing the activity. I will also make notes about which students ask for teacher and/or peer support, who takes the initiatives in group processes, and who provides assistance to peers in my field notes. PAGE 109 109 Anticipated Response from Students: I anticipate an overall positive response to this lesson, but I also a nticipate some resistance to using the calculators. Some students may also want to leave the section about putting it all together blank. If this occurs, I will need to work to communicate high expectations for them to persist through the entire activity Day 6 Lesson Title: Graphing Exponential Functions (continued) Lesson Objectives : Students will (1) evaluate and graph exponential functions and (2) sketch the graph of a function from a verbal description showing key features (Anderson School District 5, 2016). Materials Needed : interactive notebooks; note taking guides Instructional Procedures : All students will receive a copy of the note taking guide, which will include an Exponential Functions foldable to review and practice transformations note taking guide while participating in a whole class discussion led by me. Then, students will work independently to practice evaluating and graphing exponential functions. The questions will be adapted from the practice section of the Interactive Student Guide on lesson 7.5 Transforming Exponential Functions (McGraw Hill, 2010). During the discussion, I will focus on the use of precise mathematical language and recognizin g and generating strategies to solve problems ( What Works Clearinghouse, 2015). I will also circulate the room, redirecting when students become off task. As the students practice, I will continue circulating the room, providing feedback, and answering q uestions. Elements of Culturally Responsive Pedagogy: In this lesson, I will model metacognitive activities such as thinking aloud, scaffold instruction, and provide clarification of the concepts that were introduced in the previous lesson. The class wi ll utilize interactive notebooks as a tool to assist students in meeting academic expectations for vocabulary and concept knowledge/application ( Ross & Adams, 2010 ; What Works Clearinghouse, 2015 ) During the discussion and note taking, I will take care t o communicate high standards for both academics and behavior and encourage students to take risks. For instance, during the whole group discussion and note taking I will randomly call on students to respond and encourage all students to view mistakes as l earning opportunities rather than failures (Bonner, 2014). Both during and after the lesson, I will document how each student participates in the interactive notebook activity. I will document details about which students are following directions and wr iting notes as well as students who resist completing the notetaking activity. I will also make notes about which students ask for teacher and/or peer support, what is said during the whole group discussion, and how students respond when I randomly call o n them in my field notes. PAGE 110 110 Anticipated Response from Students: I anticipate an overall positive response to this lesson, even though I know that this will not be the most enjoyable lesson in the unit. I hope that students will utilize their interactive n otebooks to organize their thinking and study for assessments. Day 7 Lesson Title: Arithmetic Sequences Lesson Objectives : Students will write and use recursive and explicit formulas for arithmetic sequences (Anderson School District 5, 2016). Materials Needed : counting objects, activity guides, interactive notebooks; note taking guides Instructional Procedures : Students will be organized into pairs using random assignment. Each group will be given some objects that are all the same size and shape and an activity guide. I will be sure to verbally describe the expectation for partners to work together and encourage students to work together. The activity guide will instruct students to form a pattern using groups of 2, 5, and 8 objects and to find the number of objects in the next three groups. Once the groups have a solution, they will be asked to write a statement to defend their answer. Students should discover to add 3 to each group. During the activity, I will circulate the room, redir ecting off task students, answering questions, intervening when needed, and encouraging peer support (Bonner, 2014; Morrison, Robbins, & Rose 2008 ). Upon writing the statement to defend their answer, the activity guide directs the students to ask their t eacher to check their work. Once students have correctly completed the first part of the activity, they will proceed to the second part of the group activity, which will include determining the number of items that will be in the ninth group without makin g the 7 th and 8 th groups. Students will create a new pattern that seems to follow the same rules as the previous pattern. If needed, I will recommend starting with a different number of items or adding a different number each time. Each group will justi fy in writing why the new pattern is similar to the original rule. Then, students will find the number of items in the 6th group if the first group has 2, and the number that is added each time if the first group has 3 and the 5 th group has 11. Once agai n, the students will ask me to check their work, and I will provide feedback. Upon successfully completing the second part of the activity and justifying their responses, the students will be instructed to put away their materials and get their notebooks. Once all the students have completed the partner activity, I will lead that class in a discussion using a notetaking guide. The notetaking guide will include a Frayer Model and practice problems. A Frayer Model is a graphic organizer used to assist stud ents in building vocabulary. The Frayer Model in this lesson will require students to define an arithmetic sequence, generate examples and non examples of arithmetic sequences, give characteristics of arithmetic sequences, and/or draw a picture to illustr ate an arithmetic sequence (The Teacher Toolkit, n.d.). Students will individually complete the note taking guide PAGE 111 111 while participating in a whole class discussion led by me. Then, students will work independently to practice. The questions will be adapte d from the practice section of the Interactive Student Guide on lesson 3.5 Arithmetic Sequences as Linear Functions (McGraw Hill, 2010). During the discussion, I will circulate the room, redirecting when students become off task. As the students practice I will continue circulating the room, providing feedback, and answering questions. Elements of Culturally Responsive Pedagogy: In this lesson, the concept of arithmetic sequences will be introduced using a hands on partner activity. The purpose of the partner work with random assignment to groups is to assist students to get to know and connect with other students and establish an atmosphere in which students respect and are kind to one another (Bondy et al., 2007; Gay, 2010; Nieto, 2013). In this act ivity, students are expected to work closely together to construct their own knowledge about arithmetic sequences (National Council of Teachers of Mathematics, 2014). I will circulate the room during the activity to intervene when needed and encourage pee r support (Bonner, 2014; Morrison, Robbins, & Rose 2008 ). To increase student motivation, the activity guide is carefully constructed so that students move from easy to more complex tasks and thinking to ensure that students have positive first encounter s with content before moving to the more challenging parts of the lesson (Gay, 2010; Morrison, Robbins, & Rose 2008 ). Upon completing the hands on activity, I will model metacognitive activities such as thinking aloud, scaffold instruction, and provide clarification of the concepts that were introduced in the hands on activity. The class will utilize interactive notebooks as a tool to assist students in meeting academic expectations for vocabulary and concept knowledge/application ( Ross & Adams, 2010 ) During the discussion and note taking, I will take care to communicate high standards for both academics and behavior and encourage students to take risks. For instance, during the whole group discussion and note taking I will randomly call on students t o respond and encourage all students to view mistakes as learning opportunities rather than failures (Bonner, 2014). I will circulate the room during the independent practice to redirect off task behavior and assist students. Both during and after the les son, I will document how each student participates in the hands on activity as well as the interactive notebook activity. I will document details about which students are following directions, communicating with peers, taking risks in the group process. I will also document details about which students are taking notes as well as students who resist completing the notetaking activity. I will also make notes about which students ask for teacher and/or peer support, what is said during the whole group disc ussion, and how students respond when I randomly call on them in my field notes. PAGE 112 112 Anticipated Response from Students: I anticipate an overall positive response to this lesson, but I expect resistance to working with different students from who students normally work with. I hope that I will be able to effectively deal with the resistance from students to working with others and that students will feel more comfortable working with students that they do not normally work with. Day 8 Lesson Title: Geomet ric Sequences Lesson Objectives : Students will write and use recursive and explicit formulas for geometric sequences (Anderson School District 5, 2016). Materials Needed : balls, activity guides, smart phones, measurement tools Instructional Procedures : Students will be organized into pairs. Each group will be given a ball, a measurement tool, and an activity gui de (adapted from Drop and Catch; Hyde Canzone, & Galasso, 2016 ). I will first model the activity steps, making sure to verbally describe the expectation for partners to work together and that BOTH partners are expected to contribute to the discussion for this activity. The activity guide will instruct students to practice dropping the ball 3 times from the same height and recording the reboun d height using video. Prior to recording the data, students will predict whether or not the data (Bounce #, rebound height) will represent a function, state why or why not, and, if they think it will be a function, predict what type of function and why. Students will approximate how high on average the ball goes on each bounce for 4 bounces and record the data on a table. Students will review the rebound height data and describe the patterns they see using a complete sentence. During the activity, I wil l circulate the room, redirecting off task students, answering questions, intervening when needed, and encouraging peer support (Bonner, 2014; Morrison, Robbins, & Rose 2008 ). Once all students have their data recorded on a table and described the patter n, students will use a graphing calculator to create a scatter plot. Students will be asked to determine whether or linear or exponential model is appropriate to model the data, and defend their answer. The students will be asked to use their theory and ratio to predict the height of the ball on the 7 th bounce. The activity guide will instruct students to ask me to check their work at this point, and I will provide feedback. Once the groups come to a decision, they will follow the steps on the activity g uide to obtain a regression equation. Students will predict the height of the ball on the 7 th bounce using the regression equation and compare the answer to their prediction. Then, continuing to work in pairs, students find the common ratio and write an equation for the n th term of the geometric sequence 2, 6, 18, 54, . given the formula for a geometric sequence. Then, they will use the formula to find the thirteenth term of the sequence. Once all the students have completed the partner activity, I will lead that class in a discussion of how geometric sequences are related to exponential functions. Students will be given a few problems to practice independently at the end of the lesson. PAGE 113 113 Elements of Culturally Responsive Pedagogy: In this lesson, the concept of geometric sequences will be introduced using a hands on partner activity. The purpose of the partner work is to assist students to get to know and connect with other students and establish an atmosphere in which students respect and are kin d to one another (Bondy et al., 2007; Gay, 2010; Nieto, 2013). In this activity, students are expected to work closely together to construct their own knowledge about geometric sequences. I will circulate the room during the activity to intervene when ne eded and encourage peer support (Bonner, 2014; Morrison, Robbins, & Rose 2008 ). To increase student motivation, the activity guide is carefully constructed so that students move from easy to more complex tasks and thinking to ensure that students have po sitive first encounters with content before moving to the more challenging parts of the lesson (Gay, 2010; Morrison, Robbins, & Rose 2008 ). Upon completing the hands on activity, I will provide clarification of the concepts that were introduced in the h ands on activity. During the discussion, I will take care to communicate high standards for both academics and behavior and encourage students to take risks. For instance, during the whole group discussion, I will randomly call on students to respond and encourage all students to view mistakes as learning opportunities rather than failures (Bonner, 2014). I will circulate the room during the independent practice to redirect off task behavior and assist students. Anticipated Response from Students: I anticipate an overall positive response to this lesson, but I expect some continued resistance to working with different students from who students normally work with. Day 9 Lesson Title: Geometric Sequences (continued) Lesson Objectives : Students wi ll write and use recursive and explicit formulas for geometric sequences (Anderson School District 5, 2016). Materials Needed : notetaking guides, interactive notebooks In this lesson, I will lead that class in a discussion using a notetaking guide. Th e notetaking guide will include a Frayer Model and practice problems. A Frayer Model is a graphic organizer used to assist students in building vocabulary. The Frayer Model in this lesson will require students to define a geometric sequence, generate exa mples and non examples of geometric sequences, give characteristics of geometric sequences, and/or draw a picture to illustrate a geometric sequence (The Teacher Toolkit, n.d.). Students will individually complete the note taking guide while participating in a whole class discussion led by me. Then, students will work independently to practice. The questions will be adapted from the practice section of the Interactive Student Guide on lesson 7.7 Geometric Sequences as Exponential Functions (McGraw Hill, 2010). During the discussion, I will circulate the room, redirecting when PAGE 114 114 students become off task. As the students practice, I will continue circulating the room, providing feedback, and answering questions. Elements of Culturally Responsive Pedagogy: During this activity, the class will utilize interactive notebooks as a tool to assist students in meeting academic expectations for vocabulary and concept knowledge/application ( Ross & Adams, 2010 ; What Works Clearinghouse, 2015 ) During the discussion and note taking, I will take care to communicate high standards for both academics and behavior and encourage students to take risks. For instance, I will randomly call on students to respond and encourage all students to view mistakes as learning opportu nities rather than failures (Bonner, 2014). I will circulate the room during the independent practice to redirect off task behavior and assist students. Anticipated Response from Students: I anticipate an overall positive response to this lesson, as it is the last lesson of unit prior to the review and unit test. I hope that students will utilize their interactive notebooks to organize their thinking and study for assessments. 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PAGE 119 119 BIOGRAPHICAL SKETCH Jenny Lott Van Buren graduated Summa Cum Laude from the University of New Orleans in New Orleans, Louisiana in 2006 with her Bachelor of Science in s econdary m athematics e ducation with university honors and honors in her major. She earned her m e ducational l eadership from the University of North Florida in 2010. She earned her Doctorate of Education in c urriculum and instruction from the University of Florida in 2017. Jenny began her teaching career in Orange Park, Florida, where she taught high schoo l mathematics before moving to South Carolina. Her current position is teaching high school mathematics in Greenville, South Carolina. Jenny currently resides in Williamston, South Carolina with her husband and children. Her interests include mathemati cs instruction, culturally responsive teaching, teacher leadership, and teacher inquiry as a form of job embedded professional learning |